THE
PRACTICAL  STAIR-BUILDER,
A COMPLETE TREATISE ON THE ART OF BUILDING
STAIRS AND HANDRAILS,
WITH  A MANUAL.  OF
ELEMENTARY   DESCRIPTIVE   GEOMETRY   AND   PRACTICAL   GEOMETRICAL   CONSTRUCTIONS,
DESIGNED FOR
CARPENTERS   AND   BUILDERS,
ILLUSTRATED  WITH  THIRTY  ORIGINAL  PLATES,
C.   EDWARD   LOTH.
PROFESSIONAL STAIR-BUILDER AT TROY, N. Y.
LITHOGRAPHED  AND  PRINTED BY JULIUS BIEN,  NEW  YORK.
A. J. BICKNELL & CO.,  ARCHITECTURAL BOOK PUBLISHERS,
TROY,   N.Y.,   and   SPRINGFIELD,    ILLS. 1869.
BY
694.8
L91
Entered according to Act of Congress, in the year 1867, by
C.   EDWARD   LOTH,
i n the Clerk's Office of the District Court of the United States for the Southern District of New York.
INTRODUCTION.
Several good books have been published on "Stair-Building" in the English language. They treat principally on Hand Railing, while the planning and construction of the Stair Partis neglected. The theory and principles of the scientific part seem to be but imperfectly understood by the authors, which causes errors and difficulties in the practical execution.
The books published on the subject in the German and French languages are edited by scientific men who have no practical experience. They give ground plans and elevations of the different parts, but do not mention the subject of the practical execution of the Handrail, and the workmen have to get out the crooks as best they can, on some system similar to the renowned Peter Nicholson's, which causes a great waste in labor and material.
A thorough understanding of the planning and construction of both the Stairs and the Rails, is obviously necessary to a successful Stair-Builder, and it has been my aim in publishing this book, to cover the whole ground.
In this country, where skilled labor is valuable, especially at the present time of our progressive age, any mechanic, particularly any one connected with Building, is expected to do a neat and substantial piece of work in the shortest time and with the use of the least material possible.
To accomplish this, he, particularly the Stair-Builder, must be a good linear draughtsman.
Linear Drawing is mostly a practical application of the principles and facts arrived at by the demonstrations in Elementary, Constructive and Descriptive Geometry; the necessary Definitions, Theorems and Problems of that science are introduced in the forward part of the book in a practical, comprehensive manner, and illustrated on Plates I. to IX.
Plates X. to XXIX. illustrate every conceivable case and form of Stairs and Rails, which may come under the stair-builder's notice,,
The frontispiece is a perspective view of the stair-case Fig. 1, PI. XXIX., finished in Gothic style, with wainscoting along the wallstring.
By the system followed in this book, the wreath-pieces are all sawed square through the plank, all joints made at once, and the Rail, as also the Stairs, may each be put together and finished in the shop for an entire story, before being taken to the building.
In no case need the thickness of the plank be more than the width of the rail; the instructions given to ascertain the length of odd balusters will be found of service.
The explanation of the Plates is explicit in every particular. That of the practical part of Plates X. to XXIX. is written in such a manner, that a very careful study of the theoretical part is not absolutely necessary.
However, a careful perusal of the chapter on "Theory of Handrailing" is recommended, as it will instruct the reader understandingly on the subject, and the young carpenter, who wishes to advance higher than the common average of what is called a good workman, cannot employ his leisure hours to any better advantage, than to study the whole of this work thoroughly.
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Although fitted by education, which I was fortunate enough, to secure in Prussia, to employments considered by some as more genteel than that of building stairs, it so suits my taste, combining a healthy bodily and mental exercise, that I have made stair-building my exclusive business during the last fifteen years in this, my adopted country.
The extensive experience gained during that time, aided by the application of science and a natural mechanical skill, enabled me to examine the subject thoroughly, and bring it to that perfection, which allows of no " guess-work" whatever, but is mathematically certain, subject only to such variations, as are caused by the mechanical manipulations of the materials used.
During that time I have had many different men in employ; some as finished workmen, more under instructions. This gave me an opportunity to learn the wants and the best modes of teaching those carpenters who wish to become conversant with this particular branch of their business, and it has been my endeavor to present in this book the whole process systematically, as it is carried on under my personal supervision.
I do not claim anything contained in it, as having never been known or practiced by any body else, for such would be an absurd presumption in these days; but I do know, that it contains more solid and specific information than any one book ever published on the subject in any language, and I feel confident, it will supply the want of a complete and reliable Self-Instructor in the art of stair-building in every detail, at once comprehensible to the novice, as well as instructive to the more accomplished workman.
THE   AUTHOR
Troy, N. Y., November, 1867,
P1.I.
PLATE   I.
ELEMENTS    OF    GEOMETRY.
DEFINITIONS.
All bodies or solids are contained in space, and are limited portions of the same, supposed to be occupied by matter.    They may be considered in three relations.
1st. Matter, substance or material.
2d. Figure, shape or form.
3d. Extension, size or volume.
Geometry has nothing to do with the first, but considers solids only in regard to their extensions and forms in an abstract sense.
A solid^ being a limited portion of space, that boundary, which encloses and divides it from the rest of the same, is called its surface.
The boundary of a surface is called a line, and the limit of a line, a point.
Solids are extensions of three dimensions: length, breadth and thickness.
Surfaces have length and breadth, but no thickness.
Lines have length only. A point has no dimensions, but indicates a certain place or position only.
Lines are either straight or curved ; a straight line extends its whole length in one and the same direction ; it is also called a right line.
A curve line changes its direction at every point of its extension; such as the circle, ellipse, &c.
Broken lines are made up of straight lines, not lying in the same direction; and lines made up of straight and curve lines in any direction, are called compound lines.
Surfaces are either straight and plane, or curved.
If a surface is so constituted that a straight line between any of its points wholly coincides with such surface, then it is called a plane ; all surfaces which are not planes, are curved. The surface of a board, for instance, planed true, so that a straight edge applied to it in any direction will be in contact throughout its whole extent, is a plane.
When two lines meet at a common point, they intersect ; that portion of the surface included between them is called an angle ; this term also applies to that portion of space included between two or more intersecting surfaces.
Lines, surfaces, angles and solids, being extensions of one or more dimensions, which may be measured and thus represent quantities, are in a general term, called Magnitudes.
A figure represents or expresses the form or shape of one or more of the above magnitudes, and indicates by lines their relative extension and position.
Geometry, as a general term, is the science of measurement of extension, and the consideration of the properties and relations of the different geometrical magnitudes, by means of such measurement.
Magnitudes are equal, when one being applied to the other, they coincide throughout their whole extent.
That part �of Geometry which treats of lines, angles and surfaces lying in one plane, is called Plane Geometry.
An axiom is a self-evident truth.
A demonstration is a train of logical arguments brought to a conclusion.
A theorem is a truth, which becomes evident by reasons of a demonstration.
A problem is a question proposed which requires a solution.
The shortest distance between any two points is that indicated by a straight line between them; only one straight line can be drawn between two points, but the same may be produced or extended infinitely in either direction-
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A line is designated by two letters. In speaking of a line, a straight line is meant, nnless otherwise mentioned.
Two lines are parallel, when situated in one plane, they will never intersect, however far they may be produced in either direction; the shortest distance between them will be the same on any point of either line.    Thus A B parallel to C D and E F.    Fig. 1.
When two straight lines meet at one point with their respective ends, they form a plane angle. The two lines are called the sides, and the point of intersection, the vertex of that angle.
Angles are mostly designated by a figure < and three letters; the one at the vertex always in the middle.
Thus, in Fig. 2, A B and B D are the sides, and B the vertex of < A B D. If the line A B is produced in the direction of C, then two adjacent angles ABD and CBD are formed, which may either be unequal (one smaller than the other) Fig. 2, or equal, as in Fig. 3; when both angles are equal, they are called right angles (or square angles), and the line B D is said to be perpendicular to the other line A C.
Any angle smaller than a right angle is called acute, as < D B C, Fig. 2, and any angle larger than a right one is called obtuse, as < ABD; in that case, line BD is said to be inclined to the line AG.
When two lines intersect, as A D and E F, Fig. 4, they form four angles; those situated as < ABE and < DBF are called opposite or vertical angles.
When a straight line A D, Fig. 4, intersects two parallel lines, as E F and Q- H, such angles, situated as < A B E and < B C Gr on the same side of the intersecting line, are called alternate angles ; angles situated as E B C and B C G- on the same side of the intersecting line, are termed interior angles.
A plane figure is any portion of a plane, inclosed on all sides by lines.
A plane figure inclosed on all sides by straight lines is called a Polygon or rectilineal Figure.    Polygons have different names according to the number of their sides.
A triangle is one inclosed by three sides ; a quadrilateral by four sides, a pentagon by five, hexagon by six, octagon by eight.
Triangles are classified in regard to the length of their sides and the size of their angles.
An equilateral triangle has its three sides equal, Fig. 5.
An isosceles triangle has only two of its sides equal, Fig. 6.
A scalene or irregular triangle has all its three sides unequal, Fig. 7.
An equiangular triangle has all its angles equal, Fig, 5.
An acute angled triangle is one with all three angles acute, Fig. 5 and 6^
An obtuse angled triangle has one angle obtuse, Fig. 7.
A right angled or rectangular triangle is one in which one angle is a right one ; the side opposite the right angle is termed the hypothenuse, and the other two sides, respectively, the base and perpendicular. Fig. 8 is a right angled triangle ; < A B C is a right angle, A C the hypothenuse, A B the base, and B C the perpendicular.
Quadrilaterals are classified into parallelograms, trapeziods, and trapeziums.
Parallelograms have their opposite sides parallel; there are four kinds�the square, rectangle, rhombus and rhomboid.
A square, Fig. 9, has its four sides equal, and all angles are right angles.
A rectangle, Fig. 10, has four right angles, but only its opposite sides equal
A rhombus or lozenge, Fig. 11, has all four sides equal, but only the opposite angles equah
A rhomboid, Fig. 12, has both its opposite sides and opposite angles equal.
A trapezoid, Fig. 13, has only two of its sides parallel, and all sides and angles may be unequal.
The trapezium, Fig. 14, has no two of its sides parallel.
A polygon is termed equilateral when all its sides are equal; it is equiangular when all its angles are equal.
A regular polygon is one which has all its sides and all its angles equal.    The equilateral
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triangle, Fig. 5, and the square, Fig. 9, are regular polygons; also see Fig. 17, PL II., a regular hexagon, and Fig. 18, a regular octagon.
Two polygons are mutually equilateral, when their sides, placed in the same order, are equal, each to each; two polygons are mutually equiangular, when their angles, placed in the same order, are equal, each to each.
A line which joins the vertices of two angles not adjacent in any figure, is called a diagonal, and that side of a figure on which it is supposed to stand, is called the base. A C and B D, Fig. 9 and 11, also A C, Fig. 14, are diagonals. The perometer of a polygon is the sum of all its sides.
theorems.
Theorem 1.   The sum of two adjacent angles is always equal to the sum of two right angles.
The sum of < A B D and < CBD, Fig. 3, is equal to the sum of two right angles. Theorem 2.  Opposite angles are equal.
<  A B F, Fig. 4, equal to < E B D.
Theorem 3.  Alternate angles are equal, and the sum of two interior angles is equal to two right angles.
<  A B E, Fig. 4, equal to < B C Gk
The sum of < E B 0 and < B C Gr is equal to two right angles. Theorem 4.  An equilateral triangle is also equiangular.
The triangle, Fig. 5, is equilateral as well as equiangular. Theorem 5.  In an isosceles triangle the angles opposite the equal sides are equal.
<  B A C, Fig. 6, equal to < A B C.
Theorem 6.  All angles of a scalene triangle are unequal.
Theorem 7.   The sum of all angles in any triangle is equal to the sum of two right angles. Theorem 8.  In any right angled triangle, the sum of the two angles formed by the hypothenuse with the base and perpendicular, is equal to one right angle.
The sum of < B A C and < A 0 B, Fig. 8, is equal to one right angle. Theorem 9.   The diagonals of all parallelograms bisect each other. Theorem 10.   The diagonals of the square are equal, also those of the rectangle. Theorem 11.   The diagonals of the square are perpendicular to each other. Theorem 12.   The diagonals of the square and the rhombus bisect their respective angles.
Fig. 9, < A B D equal to < C B D, and < A D B equal to < C D B.
THE    CIRCLE.
DEFINITIONS.
The circumference of a cirole, Fig. 15, is a continuous curve line, described by a point A moving in a plane around a fixed point O, at an equal distance, A O. That portion of the plane, terminated by the circumference, is called the circle ; any straight line as A O, the radius, and the fixed point O, the centre of the circle.
Any straight line, as A B, passing through the centre O, and terminated each way by the circumference, is called a diameter ; it is evidently double the length of the radius.   Fig. 16.
By the definition of the circle it follows: that all radii of the same circle are equal, also that all diameters are equal.
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Any part of the circumference, as C D or P G, is called an arc.
Right lines, as P Gr, connecting the extremeties of arcs and not passing through the centre, are called chords ; when extended beyond the circumference they become secants.
Any portion C O D of the surface of a circle, contained "between two radii and the arc connecting their extremities, is called a sector.
A segment is that part of a circle included between a chord and its arc, as F Gr H.
A straight line, E D, which touches the circumference of a circle in one single point, is called a tangent.
Any angle formed by two radii, the vertex being in the centre, is termed a central angle.
Any angle, B C L, Fig. 21, formed by two chords, or a diameter and a chord, the vertex being in the circumference, is called an inscribed angle.
All regular polygons may be inscribed in a circle, or circumscribed to a circle.
The sides of an inscribed polygon are equal chords, as AB, CD, EF, Fig. 17, PL II., which are sides of a regular hexagon inscribed in a circle of the radius 0 A.
The sides of a circumscribed regular polygon are tangents to the circle, as AB, CD, EF, which are sides of a regular hexagon circumscribed around the circle of the radius O Gr.
If the circumference of a circle is divided into a certain number of equal parts (or small arcs), and the two radii running to the ends of one of them are considered as the sides, and the centre of the circle as the vertex of an angle, then the arc is a unit of measure for that angle in reference to the division of the whole circle. It is the custom to divide the circumference for this purpose into 360 units or equal parts, called degrees ; each degree is divided into 60 minutes, and each minute into 60 seconds. To express an angle of forty-five degrees, fifteen minutes, and thirty seconds, it is written thus:   45� 15' 30".
To farther illustrate the meaning of this division into degrees, suppose the arc A D of the angle A 0 D, Fig 21, PL I, to contain 60 parts or degrees of the circumference described by the radius O A. If the sides of this angle be produced until they intersect the circumference described from the same centre, O, by the radius O B, then the arc B C will contain again 60 parts or degrees of the larger circle, but the angle itself, or the inclination between the two lines D O and A O, remains unchanged.
Two diameters perpendicular to each other, will evidently divide the whole circle into four right angles of 90� each.
The sum of two right angles, as B O F and J O F, is 180�; if < B O C contains 60�, then < J O C must contain the remainder of 180�, namely, 120� and < F O C 30�, or the remainder of 90�.
The diagonal of a square divides the right angle into two equal parts of 45� each, as < J O H and < O H J.
The sum of all angles in a triangle, being equal to two right angles, or 180�, each angle in an equilateral triangle, which is also equiangular, must contain 60�.
THEOREMS.
Theorem 13.  Any diameter divides the circle into two equal sectors, called semi-circles. Theorem 14.  Any two diameters at right angles to each other divide tlie circle into four equal sectors, called quadrants. Thus, J O F and B O F, Fig. 21, are quadrants. Theorem 15.   Tangents are always at right angles to the radius at the point of contact.
Thus, E D is perpendicular to J) O, and E B perpendicular to B O.    Fig. 16. Theorem 16,   Any radius,  which is perpendicular to a chord,   loill bisect  the chord   and its arc. Let the radius O H be perpendicular to chord F Gr, then F J and J Gr are equal, and arc F H equal to arc H G.
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Theorem 17.  Any perpendicular bisecting a chord, will pass through the centre.
Let O J be perpendicular to and bisect F G ; it will pass through O, the centre of the circle. Theorem 18.   A  circle   may   be   described   through   any   three   points   not   in   the   same
straight line. Theorem 19.  Any inscribed angle is measured by one half of its arc. Theorem 20.   The diameter is to the circumference as 1 to 3. 14 or approximate 1 to 3|
PROPORTIONS.
DEFINITIONS.
Proportion of magnitudes of the same kind, is the relation they bear to each other in regard to one being more or less than the other ; the measure of such proportion is called their ratio ; it is expressed by the quotient arising from dividing one by the other.
Thus, a line A, which is four inches, and another B, six inches long, are in the ratio of �, or four to six ; the first is two-thirds the length of the second.
This relation between the lines A and B is written thus:  A :  B ::  4:  6 ::  2 :  3.
When another pair of lines, C and D, bear the same ratio to each other as A does to B, then the four are said to be in proportion. For instance : let C be 12 inches and D 18 inches long, then such proportion would be designated thus�
C : D :: 12 : 18 :: 2 : 3 and consequently
A : B :: C : D or substituting their numerical measures
4 : 6 :: 12 : 18
When A is one inch, B three inches and C one foot long, then T> must be three feet long, to be in proportion with the others. On this principle depend the constructions of drawings on a small scale.
The first and last terms are called the two extremes, and the second and third the two me ans.
The product of the two extremes is equal to the product of the two means.
Three magnitudes of the same kind are said to be in proportion, when the first has the same ratio to the second, as the second has to the third ; the middle term is then said to be a mean proportional between the other two.
Thus, when
A : B :: B : C then B is the mean proportional between A and C.
theorems.
Theorem 21.  If in any triangle a line is drawn parallel to either side, it will divide the other two sides proportionally. Let D E, Pig. 19, be parallel to A B5 then
CD:DA::0E:EB or                                   CD : G A:: CE: C B.
If C D is three times the length of D A, then C E will be three times the length of E B. Theorem 22.  If in any triangle a line is drawn parallel to one side, then any lines drawn from the opposite vertex, will divide that side and the parallel proportion ally.
2
10
Let D E, Fig. 19, be parallel to A B, then
AG:GB::DF:PE.
If A G is double the length ot G B, then D F will be double the length of F E.
Theorem 23.  In any right angled triangle, a line drawn perpendicular from the vertex of the
right angle to the hypothenuse, is a mean proportional between the segments of the
hypothenuse.
Let A C B, Fig. 18, be a right angle, and C D perpendicular to A B, then
AD : DC :: DC : DB.
If D B is nine parts long and A D four of the same parts, then D C will be six of such parts
in length;
9 : 6 :: 6 : 4.
D C is the mean proportional between B D and A D. Theorem 24.   The square of the hypothenuse of a right angled triangle is equal to the sum of the squares of the other two sides. Let CAB, Fig. 20, be a right angled triangle.
If A C is three parts long and A B four of such parts, then B C will be five of those parts in
length.    The square of A C will contain nine of the small squares of each part, and
the square of A B sixteen of such small squares.    The sum of both is equal to 25, or
the number of small squares inscribed in the square of the hypothenuse B C.
Remark.�This explains the practice of laying three rods, respectively three, four and five
feet, or six, eight and ten feet long, together to form a right angled triangle, while framing timbers
and other work.
EQUALITY   AND   SIMILARITY   OF  FIGURES.
DEFINITIONS.
Two figures are alike or equal, when, one being applied to the other, they coincide in all their parts. All the sides and angles of equal figures in their respective situation, each to each, are equal; or, in other words, such figures are mutually equilateral and equiangular.
Two figures are similar, when they are mutually equiangular, and have their sides about the equal angles respectively situated, proportional.
In figures which are mutually equiangular, those angles which are equal, each to each, are called homologous angles, and those sides which are like situated, respectively to the equal angles, are called homologous sides.
In every plane triangle there are six parts : three sides and three angles, which are so related to each other, that, when one side and any two of its other parts are given, the remaining ones may be found by geometrical construction.
THEOREMS.
Theorem 25.  Triangles having one side and any two of the other homologous parts alike, are equal. Theorem 26.  Mutually equilateral triangles are equal.
Theorem 27.  Mutually  equiangular   triangles   are   similar,   and their sides  are   mutually proportional. Triangles HOC and J F C, Fig. 19, are similar.
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Theorem 28.   The diagonal of a parallelogram divides the same into two equal triangles.
Theorem 29.   Two squares are equal, when their sides are equal.
Theorem 30.   Two rectangles are equal, when their long and short sides are equal.
Theorem 31.  Lozenges are equal, when one of their sides and one of their homologous angles
are equal. Theorem 32.  Rhomboids   are   equal,   when   their   short and  long  sides   and   one   of  their
homologous angles are equal. Theorem 33.   Two irregular quadrilateral figures are equal, when three of their homologous sides
and two of the homologous angles formed by them are equal. Theorem 34.  Two irregular quadrilateral figures are also equal, when two of the homologous sides
and three of the homologous angles are equal.
PLATE    II.
GEOMETRICAL   CONSTRUCTIONS.
The instruments used in solving the following problems are, a straight edged ruler to draw the straight lines, and a pair of dividers with lead pencil attached to one leg, to describe circular lines.
Problem 1. At a given point in a given line, to construct an angle equal to a given angle.
Let A, Fig. 1, be the given point in the given line A B, and < c a b the given angle. Prom the vertex a as a centre, describe with any radius the arc b c, terminating in the sides of the angle ; then, with the same radius, from A as a centre, describe the indefinite arc B C, lay off on B C the arc as b c, and draw a line through A and C ; then angle CAB will be equal to the given angle cab.
Problem 2.  Through a given point to draw a line parallel to a given straight line.
First solution: Let D, Fig. 2, be the given point, and A B the given line. Describe from any convenient point of the line A B, a semicircle passing through the given point D, then lay off from B to C the arc A D, and a line D C, will be parallel to A B.
Second solution: Let C, Fig. 3, be the given point, and A B the given line. From C draw any straight line C A, forming any angle with A B; then construct angle D C A equal to angle CAB, and the side C D of the second angle, will be parallel to A B.
Problem 3.  To bisect a given arc, or a given angle.
Let A D B, Fig. 4, be the arc to be divided into two equal parts. Describe with any equal radii from the points A and B, two arcs cutting each other at C ; then a line drawn through C and the centre O, will bisect the arc and also the angle A O B. By the same process the angle DOB may again be bisected, and angle ExO B will be then equal to one-fourth of the given angle A O B.
Problem 4.  To bisect a given straight line.
Let A B, Fig. 5, be the line to be bisected. With any equal radii more than one-half of A B, describe arcs cutting each other above and below the line, as in D and E; then a line, D E, will bisect A B in C, their point of intersection.    D E will also be at right angles to A B.
Problem 5. At a given point in a given straight line, to erect a perpendicular to that line.
Let A, Fig. 6, be the given point in the given line B C. Lay off from A each way, equal distances as B and C ; from these points describe arcs with equal radii, cutting in D ; then a line A D, will be the perpendicular.
Problem 6. At the end of a given line to erect a perpendicular to that line.
Let A, Fig. 7, be the end of the given line B A. Describe arcs with equal radii from A and any point B, cutting in D ; then with the same radius, from D as a centre, describe the semicircle B A C, intersecting the line B D produced at C ; a line A C will be the perpendicular.
Problem 7. From a given point A, outside of a given line B C, to let fall a perpendicular to that line.
Let A, Fig. 8, be the given point, and B C the given line. From A as a centre, describe an arc with any radius, cutting the line as in B and C ; from these points describe intersecting arcs either above or below the line as in D, then the line A D will be the perpendicular.
Problem 8. From a given point outside of, and nearly opposite the end of a given line, to let fall a perpendicular to that line.
Let A, Fig. 9, be the given point, and B C the given line. From the point A draw any line A C, intersecting B C as at C ; bisect A C as in D, and describe the semicircle over A C, intersecting C B as in B ; then the line A B will be the perpendicular
Problem 9. To construct a triangle, the three sides being given.
Let the lines marked A, B and C, Fig. 10, be the given sides. Draw D E and make it equal to the side A; then describe from D an arc with the side C as radius, and from E an arc with the side B as a radius, cutting the former in F ; draw the lines D F and E F, and D E F will be the triangle required.
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By reference to the figure, it will be noticed, that triangle D E Gr is equal to triangle D E F, both containing the sides given; therefore, when the figure is to be constructed in reference to some other figure, then the situation of the three sides must be taken into consideration, and the point F or G found as required.
Problem 10. Two sides of a triangle and the angle formed by them being given, to construct the triangle.
Let the lines marked A and B, Fig. 11, be the given sides, and c d e the given angle. Construct the angle given as in Problem 1, with the sides produced indefinite ; make one side D 0 equal to A, and the other D E equal to B, then CDE will be the triangle.
Problem 11. One side and two angles of a triangle being given, to construct the triangle.
Let A, Fig. 12, be the given line, < db c and < d cb the given angles. The two. angles will either be adjacent to the given side, or one will be adjacent and the other opposite. In the first case, draw a line B C and make it equal to A ; construct at B an angle equal to db c, and at C an angle equal to d cb ; the sides of these angles will intersect in D, and BCD will be the triangle required. In the other case, make B C equal to A, and construct at B an angle equal to db c as before; then construct at any point E of the line B E, angle D E Gr equal to deb, and draw through C a line C F parallel to E Gr; then B C F will be the triangle required.
Problem 12.  The side being given, to construct the square.
First solution : Draw two lines, A B and A D, Fig. 13, of indefinite length, at right angles ; make them equal to the side given, and describe arcs from B and D as centres with the given side as radii, cutting at C ; then A B C D will be the square required.
Second solution : Draw a line A B, Fig. 14, equal in length to the side of the square to be constructed ; describe arcs AEC and BED, with AB as radius, and A and B as centres, cutting at E ; bisect the arc E A by the line B Gr at F, describe an arc D F C from E as centre and E F as radius, intersecting the first arcs at C and D, and A B C D will be the square.
Problem 13.  To construct a lozenge, one angle and the side being given.
The solution is similar to the first of the preceding problem. Construct an angle, DAB, Fig. 15, equal to the given angle, and make its sides, A D and A B, equal to the given side of the lozenge ; from the points B and D describe arcs with that same side as radii, cutting in C, and A B C D will be the figure required.
Problem 14. To construct an irregular quadrilateral, three sides and both angles formed by them being given, in relative position.
Let Fig. 16, D A, A B and B C, be the given sides, DAB and ABC the given angles. Draw a line A B equal to the given one, then construct the given angles DAB and AB C as situated, and make their sides A D and B C of the length given, completing the figure by joining C D. Any irregular figure may be most accurately and conveniently copied from another already given, by dividing the latter into triangles, and constructing these according to Problem 9 in their respective situation.    Thus, Fig. 15 is a copy of Fig. 11, PL I, and Fig. 16 that of Fig. 14, PL I.
Problem 15.   To construct a figure similar to another given figure.
Any regular or irregular polygon may be constructed similar (either enlarged or reduced in proportion) to a given one, by dividing the same into triangles, as when making a copy equal to it, and then drawing homologous sides in the proportion required, either by the method applied Fig. 19, PL I., or using the scale, Figs. 14 and 15, PL IV. Thus, ab cd, Fig. 16, PL IL, is a figure similar to A B C D, the sides being reduced one half. It will be noticed, when any of the homologous sides of similar polygons are parallel, all of them will be parallel.
Problem 16.  To construct a, regular hexagon.
It is one of the peculiar properties of the circle, that the radius will space off six equal chords on the whole circumference. The construction of a regular hexagon is therefore very simple, when one side of the same is given. Describe a circle with the given side O A, Fig. 17, as radius, space off the circumference, draw the chords, and a regular hexagon, as A B C D E F, will be the result. In case the distance G H, between two of its parallel sides is given, describe a circle with half of that distance, as O G, for a radius, space off the six arcs as before, and draw the perpendiculars to the radii at the points of division;  as A B tangent to O G at G, and those tangents will form the
14
hexagon required.    Connecting three alternate points of those divisions of the circumference, will produce an equilateral triangle, &$ ace.
Problem 17.  To construct a regular octagon.
When the distance AB, Fig. 18, between two parallel sides is given, construct a square CDEF, its sides equal to that distance ; draw the diagonals CE, DF, and set off from their intersection O, the distance D F, equal to O A and O B. Draw a line H G, perpendicular to the diagonal through F, and where this line intersects the sides of the square, will be the angles, and H Gr, one side of the octagon. To complete the figure, set off the distance EH or EG from the remaining corners on all sides of the square, and connect the points thus found.
In that case, when the side of the octagon to he constructed is given, draw a line C B, Fig. 19, indefinite, and set off A B equal to the side given ; bisect A B by a perpendicular O D, Problem 4, and draw AF parallel to DO. Then bisect the angle OAF, Problems, and make E A equal to A B ; bisect again E A by a perpendicular H O, intersecting D O at O, which will then be the centre of the octagon. The completion of the octagon is easily seen by the figure without further description.
Problem 18. Any square being given, to find the points to form an octagon.
Let A B C D, Fig. 20, be the given square ; draw the diagonals A C, B D. From any corner B, as the centre, and half of the diagonal O B, as the radius, describe the arc O E, intersecting the side in E. Then C E will be the distance to be laid off each way from the corners of the square, to form the same into an octagon.
Second method : Draw the diagonals as before ; from any corner C, as the centre and the side of the square as the radius, describe the arc B F, intersecting the diagonal C A at F ; from F draw F G parallel to the side A B, and A Gr will be the distance to be laid off each way from the corners of the square, to form it into an octagon.
HemarJc.�This problem finds frequent practical application, when making the square base of a newelpost octagon ; set the guage by the distance found, as C E or A Gr, and run a guage-mark each way from the corners.
Problem 19.  To divide a given line into any number of equal parts.
Let A B, Fig. 21, be the given line. From A draw an indefinite line A C on any convenient angle, and set off as many equal parts of some convenient length toward C, marked 1, 2, 3, 4, 5, as the given line is to be divided into ; connect the last point of division (5) with the other point B of the given line, and draw parallels to 5 B through all the points of division ; these parallels, 11, 2 2. 33, &c, will intersect and divide the line AB into the required number of equal parts.
Problem 20. To find the side of a regular polygon, for any number of sides, to be inscribed in a given circle.
All regular polygons may be inscribed in a circle. Several methods for constructing a polygon of an even number of sides have already been given, as that of the square, hexagon, octagon, &c. The method now to be described, is mostly applied for polygons with an unequal number of sides, as the pentagon, of five sides, heptagon, of seven sides, nonagon, of nine sides, &c. Let A B, Fig. 22, be the diameter of the given circle. Make B D equal to A B, and divide A D (this being twice the length of the diameter) into as many equal parts, as the polygon is to have sides ; in this diagram, A D is divided into five equal parts, and let D 4 be one of these divisions; make B C equal to D 4, and describe arcs from A and B as centres, and the diameter as radius, cutting as in E; connect E and C, which extended, will intersect the circumference as in F; the chord B F will be the side of the required pentagon.
Problem 21. To find the radius of the circle, in which a regular polygon of any number of sides may be inscribed, the side of the polygon being given.
Let AB, Fig. 23, be the given side. From A and B as centres, and AB as radius, describe arcs, cutting as in 0 ; bisect A B by a perpendicular indefinitely extended, which of course passes through C. It is evident, that a circle, described from C as a centre and A B as a radius, will contain a regular hexagon, of which A B is a side. Divide C12, which is equal to A C or A B, into six equal parts, marked 7, 8, 9, 10, 11, 12. These divisions are the centres of the circles required, passing through A and B.    For instance:   a polygon of nine sides, of the given side A B may be
15
inscribed in a circle, described from 9 as a centre, and 9 A or 9 B as radius. If a polygon of more than twelve sides is required, lay off as many more of these parts on the perpendicular toward D, as are required ; thus, the mark 15, will be the centre for the circle of an inscribed polygon of 15 sides, and so on. The larger the number of sides, the more the centres thus found will deviate from the true centre of the required circle.
PLATE   III.
GEOMETRICAL   CONSTRUCTIONS.
(CONTINUED.)
Problem 22.  To find the centre of a given circle or arc.
Take three points anywhere in the circumference or arc, draw the chords A B and B C, Fig. 1, bisect them by the perpendiculars E O and D O, and O their point of intersection, will be the centre sought.
Problem 23.  To draw a tangent through a given point, in a given circle.
Let A, Fig. 2, be the given point, and O the centre of the given circle. Draw the radius O A, and erect the perpendicular B C, which will be the tangent required.
Problem 24., Through a given point outside of a, given circle, to draw a tangent to the circle-
Let A, Fig. 3, be the given point, and O the centre of the given circle. Connect A with O, and describe a circle, of which A O is the diameter; where this circle intersects the given circle, as in B and C, will be points of contact, and A B and A C will be the tangents required.
It will be noticed, that from any point outside of a circle, two tangents may be drawn, and that both will be of equal length.    (Compare Figs. 16 and 21, Plate I.)
Problem 25. To draw a tangent through a given point in a given arc, the centre not being Jcnown.
Let A, Fig. 4, be the giveii point and B A C the given arc. From A set off equal portions of the arc as B A and A C ; conceive a chord, drawn from B to C, and bisect the same by a perpendicular F Gr, which evidently must pass through A, and extended, also through the unknown centre of the circle. Through the point A of this line, erect the perpendicular *D E, which will be the tangent sought.
Problem 26. Through a given point, outside of a given arc, to draw a tangent to the same, the centre not being Jcnown.
Let A, Fig. 5, be the given point and C B F the given arc. Draw any convenient secant A C, and describe a semicircle ADC, AC being the diameter ; to this diameter erect a perpendicular B D at B, intersecting the semicircle at D ; then from A as a centre and A D as a radius, describe the arc D E F; cutting the given arc at E and F, and the lines A E and A F will be the required tangents.
Problem 27.  To draw a tangent to a given circle, parallel to a given line.
Let A B, Fig. 6, be the given line, and O the centre of the given circle. From O draw a line O C perpendicular to the given line A B, intersecting the circle at C ; through C, perpendicular to O C, construct the line D E, which will be the tangent sought.
Problem 28. To describe a circle, to which a given line at a given point in the same, will be a tangent, the periphery of the circle to pass also through another given point, not contained in the given line.
Let A, Fig. 7, be the given point in the given line AB, which is to be tangent to the circle, and C the given point, through which the circumference is to pass. At A draw a line A O perpendicular to A B ; then draw the chord A C and bisect by a perpendicular D O ; both perpendiculars will evidently pass through the centre and 0, their intersection, will be the centre of the circle sought.
Problem 29.  To describe a circle in a given triangle.
Let ABC, Fig. 8, be the given triangle. Bisect any two of the angles of the triangle, and O the intersection of the bisecting lines, will be the centre for the required circle.
All three of the bisecting lines A O, B O and C O will meet in one common point, and it is evident, the sides.of the triangle will be tangents to the circle.
Problem 30.  To find a fourth proportional to three gfoen lines.
Let A, B, C, Fig. 9, be the given lines. Draw two indefinite lines D Gr and D H, making any convenient angle ;  from D lay off D E equal to A, D F equal to B, and E Gr equal to C ;  draw the
PI. III.
Fig. I.
Fh/.U.          \z
Ki>
Fig YE.
\F
Fig. IK.
17
line E F, and through the point G, a line GH parallel to EF;   then FH will be the fourth proportional to A, B and C.    (Fig. 19, PL I.)
A: B:: C : FH.
Problem 31. Two lines are given; one of these to he mean proportional between the other one and a third line: to find that third line.
Draw two lines of indefinite length, Fig. 10, forming any convenient angle, as in the preceding problem. Make A C equal to that one which is to be a mean proportional, and A B equal to the other of the given lines ; make B D also equal to A C, and draw D E parallel to B C; then C E will be the third, or the line required.
AB: AC :: AC: CE.
Problem 32.  To find a mean proportional between two given lines.
Draw an indefinite line ; make A B, Fig. 11, equal to one of the given lines, and B C equal to the other. Over A C as a diameter, describe the semicircle ADC, and from B erect the perpendicular B D, cutting the semicircle in D; B D will be the mean proportional sought. (Fig. 18, PL I.)
Problem 33. To find the lengtli, or stretchout of the are of a given semicircle or quadrant, by construction.
The length of an arc is what is commonly understood by the stretchout and may be developed approximate by the following construction :
To find the stretch out of the semicircle ABC, Fig. 12, draw a tangent EF of indefinite length, and describe arcs from A and C as centres, with the diameter as radius, cutting in D ; draw lines A D and C D through the point D and the extremities of the diameter, produce the same until they intersect the tangent, and the line E F will be the length sought. E B and B F each, are equal to the stretchout of the quadrants A B and B C. In the same manner any portion of the arc, as B H, may be developed by drawing a line from the point D through H produced, intersecting the tangent in J ; then J B is the stretchout of H B.
To find the stretchout of the quadrant B C only, describe arcs from O and C as centres, with O C as radius, cutting as in G; produce Gr C, intersecting the tangent in F, and B F is the stretchout.
Hemar~k.�Although this construction developes the stretchout of the semicircle and quadrant very nearly in true length, that of the segment varies more therefrom; for instance: E J falls short of A H, and J B is longer than H B.
THE   ELLIPSE.
It has been mentioned before, that a circular line is the most simple, as well as the most practically applied of all the single curves.
The Ellipse is the next most important of that class, and every workman ought to be well acquainted with its construction and proportions.
DEFINITIONS.
An Ellipse, Fig. 13, is a continuous curve line, described by a point A, while moving around two fixed points B and C, called the foci, which are contained in a straight line D E, termed the major axis, in such a manner, that in any one of the diiferent positions of the point A, the sum of its distances from the two foci is equal to the given major axis D E.
Thus AC+AB=DE, and when A assumes the position at F of the curve
FC+FB^DE, and also DB+DC=DE.
The line F G-, being perpendicular to, and bisecting the* major axis D E at O, is called the minor axis.    The major axis is also termed the long or transverse, and the minor axis, the short
3
18
or conjugate one.    The extremity of an axis, as D or F, is called a vertex, and the intersection 0 of the two axes, the centre of the ellipse.
Lines as AC, AB, FC, &c, drawn from any point of the curve of an ellipse to the foci, are termed vectoes ; and a line as A L, bisecting the angle formed by two vectors, is a normal to the curve, at the point A. Any Jine as MN, perpendicular to a normal AL, touching the curve at one single point A, is a tangent to the curve, at the point of contact. Either axis divides the ellipse into two equal parts, called semi-ellipses, and both axes divide the same into four equal parts, termed elliptical quadrants.
THEOREMS.
Theorem 35. By the definitions of the ellipse given above, it follows:  that the distances C E and
DB, Fig. 13, are equal;  also, that F Q and F Beach, or G G and G B each, are
equal to one half of the major axis D E. Theorem 36. If from any point K, in the curve,   an arc is described  with one half of  the
major axis as radius, cutting the minor axis as in H, and a line H K drawn through
these points, then:   that part as JK, between the major axis and the curve, will be
equal to one half of the minor axis. If from K an arc is described with one half of the minor axis as a radius, cutting the major
axis in J, and K J produced until it intersects the minor axis in H, then :  UK will be
equal to one half of the major axis. Theorem 37. If one of the vectors as B A is extended, forming an angle CAP with the other vector,
then tangent A JSF wiU bisect that angle. It is evident that perpendiculars to either axis through the vertex, are tangents to the ellipse.
Thus four such tangents will form a rectangle PQRS,   Fig. 14, whose sides are
respectively equal to the major and minor axis of the ellipse. Theorem 38. Let F D, Fig. 15, be a tangent to the elliptic curve at A, A B and A C the vectors,
and F Gr parallel to one of the vectors, say A C ; further, let the semicircle F D J Gr be
described from O as a centre, and one half of the major axis as radius ; then, First�A line 0 D, running from the centre of the ellipse to the point where the tangent
intersects the semicircle, will be parallel to the other vector A B. Second�The line BG, running from that point of intersection to the other extremity of
the diameter of the semicircle, will always pass through O, or that focus of the ellipse,
from which the first vector was drawn, and to which the diameter F G is parallel. Third�Triangles COG and B OF being equal, such lines as OG and B F, connecting the
foci with the extremities of the diameter, are equal and parallel. Fourth�Tangent F D, and the line D G passing through the focus C, are at right angles.
They are inscribed in a semicircle.    (See Fig. 17, PL I.) Fifth�If vector OEis parallel to vector BA, then a line D E, drawn from D the intersection
of tangent F D and the semicircle, to E the intersection of the vector CE with the
elliptic curve, is also a tangent to that curve at the point E. Theorem 39. Let A B, Fig. 16, be tangent to the ellipse at B ;  S and P the foci, and the segment
B"Gr of vector B P, equal to one half of the major axis H J ;  further, let the line Q C
pass through the pdint G, and O the centre of the ellipse,  and where this line
intersects the curve as at C, let C A be a tangent to the ellipse ;  then,
ly
Tangents A 3 and A O will be so situated, as to form a parallelogram with those lines B 0 and CO, which connect the points of contact with the centre of the ellipse.
Theorem 40. If from O, Fig. 16, as a centre, with one half of the axes as radii, arcs, as J T and F L, are described, also a line drawn from the centre, intersecting the arcs as in F and T, then, First�Lines, as FE and TE, drawn perpendicular to their respective axis, will intersect
the curve of the ellipse in one and the same point E. Second�Let E D be a tangent to the curve at the point E, intersecting the minor axis K L extended as in D; then, a line F D, drawn perpendicular to 0 Ffrom the point F, will intersect said minor axis in the same point D.
Theorem 41. Let Q R, Fig. 13, be tangent to the curve at Q, and C R perpendicular to this tangent from the nearest focus C ; then, a vector B Q, passing through the further focus B and the point of contact Q, will produced, intersect the perpendicular OB extended at a point 8, which will be the same distance from the tangent, as the tangent from the nearest focus ; that is, SB will be equal to BO: and farther, the distance B 8, will be equal to the major axis D E.
Theorem 42. Let XV and X Z, Fig. 13, be tangents to the ellipse, and W Y an arc, described from
the further focus 0 as a centre, the length of the major axis D E as a radius ; also the
arc WBT described from X as a centre, and the distance X B to the nearest focus, as
a radius, intersecting the first described arc at W and Y ; then,
First�Lines as OV and 0Z, drawn from the further focus and passing through the
points of contact will produced, inter sect,the arcs in the same points, as Wand Y. Second�Tangent X Vwill bisect angle WXB, and tangent XZ angle Y XB.
Theorem 43. The circumference of an ellipse is equal to that of a circle, whose diameter is equal
to a mean proportional between the major and minor axis.
Bemark.�It is well to bear in mind, not to confound a mean proportional with the average of two lines.
PROBLEMS,
Problem 34. The major and minor axis being given, to find the foci.
Let D E, Fig. 13, be the major, and F Gr the minor axis ; take one half of the major axis, as O E, as radius, and describe arcs from either vertex of the minor axis, as 0- or F, cutting the major axis in B and C, which are the foci.
Problem 35. To construct an ellipse, the two axes being given
There are several methods to construct an ellipse.
First method.�This is based on the definition of the ellipse as given. Find the foci B and C, Fig. 13, as in preceding problem ; the four vertexes D, F, E, Gt are, of course, four points in the ellipse.! Open the dividers to any part of the major axis, as E b, and describe arcs from both foci; then take the remainder b D of that axis as radius, and describe arcs also from both foci, cutting the first arcs as in V and 5" which will be two points of the elliptic curve ; any number of points as c\ a\ A, c" may be similarly obtained on both sides of the axis, and a curve line traced through these various points will form the ellipse. The other half of the ellipse is constructed in the same manner.
BemarJc.�To trace elliptic curves and other sweeps, on a small scale, a curved ruler is used; but on a large scale, a tapering strip of hickory or some other hard wood, about one-fourth thick on one, and one-sixteenth on the other end, by about one-half to three-fourths wide, is employed to good advantage; it will bend naturally to most any curve.
Second method.�This is based on Theorem 40. Let D E and G F, Fig. 14, be the given axes, the vertices of which are four points in the ellipse.   To find any number of intermediate points, as
20
a, b, c, d, in the quadrant between D and F, describe circles from O as centre and one half of each axis as radii, draw lines as O a\ 06" from the centre O, intersecting both circles in points a\ a", &', 6" ; from the points a\ V draw lines parallel to the major, or perpendicular to the minor axis, and from the points a", 5", . . . lines parallel to the minor, or perpendicular to the major axis ; the intersection of these parallels or perpendiculars as a, 5, c, are points of the elliptic curve, which is traced through them as in the preceding method.
One quarter of the ellipse, as D F, being constructed in this manner, the other three quarters, being equal to it, may be found as follows: produce the ordinates c c", dd" ; draw also ordinates at corresponding distances from the centre on the other half of the major axis, equal to the first, as O It" equal to O d) ", O f? equal to 0 c"' ; then set off each way from the maj or axis the length of the original ordinates respectively, as d?" dP'",#'" #"" and yfc'" 7c each, equal to <?" d, &c.
Third method.�Let DE and GF, Fig. 14, be the major and minor axis;a from centre O describe a circle with O G or one half of the minor axis, as radius, and at the point of its intersection with the major axis, erect a perpendicular e? e" ; then take one half of the major axis in the compasses and describe from the same centre O an arc Ee", intersecting the perpendicular in 6" ; draw the inclined line Oe" and erect a convenient number of perpendiculars, intersecting the* smaller circle in points, as g\ 7i\ i\ 7c\ and the inclined line in points g*\K\ $'\7c"' ; from these letter points describe arcs from centre O, intersecting the major axis in points g"'\ 7i"'', �"', #'" ; erect perpendiculars to the major axis from these latter points, and transfer upon these the distances from the major axis of the corresponding points in the smaller circle. For instance, make A'" h equal to the perpendicular K upon the major axis, &c. ; the curve is then traced through the points E, <7, 7i, i9 #, so found. One quarter of the tcurve being thus found, the balance may be constructed as in the preceding method. This construction is based upon the fact, that the intersection of a cylinder by a plane inclined to its axis, forms an ellipse, which will be more clearly demonstrated in descriptive Geometry. In this case, the angle GO^'? would be the inclination of such an imaginary plane, with the axis of, and intersecting a cylinder, ^hose base is equal to the circle of which the minor axis is the diameter.
Fourth method.�Let D E and G F, Fig. 13, be the major and minor axis, the vertices of which will be four points of the ellipse ; to obtain any number of intermediate points of the curve, mark one half of the lengths of the major and minor axis from one end on a straight ruler, or a strip of paper, as represented by the line HJK ; JK being equal to O G, and H K equal to O E ; then lay the straight edge in such a manner, that H coincides with the short and J with the long axis, 3-nd the end K will indicate one point of the curve, which is traced through the several points so obtained.
This last method is practically applied in the use of the trammel, to trace continuous elliptical curves.
The teammel, of which Fig. 16 is a top view, consists of two grooved bars, halved flush ovei each other at right angles ; V is a straight bar passing through two moveable blocks M and N"; P is also a moveable block on the bar V, with a pencil holder- These parts are similar to the large compasses represented in a side view in Fig. 16, PL IV., M and "N have each a block a, fitted in the groove and fastened to the underside by means of a screw, around which it turns easily.
To draw an ellipse with this instrument, (which any intelligent wood-workman may construct wi^h little trouble,) place the centres of the grooves on a line with the major and minor axis ; fasten the pencil-holder P, and blocks M and N", by means of the set-screws to the bar V, so that the distance from centre of the pencil to the centre of block "N is equal to O K, or one half of the minor axis, and the distance between pencil and centre of M equal to O J, or one half of the major axis ; press the grooved cross-bar firmly to the paper or board, and move the pencil-holder around ; if the parts work nicely together, a very true ellipse will be described. It is evident, that the grooved bar must be a trifle longer than the difference between one half of the short and long axis ; also, that to describe one half only, or less of the ellipse, three arms of the grooved bar are sufficient. (See Fig. 2, PL XV.)
Another very simple and convenient mode of drawing an ellipse, is by means of a string and two bradawls or pins, as represented in Fig. 15.    When the axes, as HJ and K L, are given, find
21
the foci B and C, and stick the pins in these points ; take a piece of twine (which should Tbe pliable, but not stretch too much) with a sling at one end to receive a pencil; place the latter at either K or L, and pass the twine around both pins and the pencil, as represented in the figure. Carry the pencil around with an equal steady pressure outward, and an ellipse will be the result.
This method is correct in theory, because the pins are stationery, and the sum of the distances at which the pencil M iskept from them, is equal in all positions to the major axis. The unsteady pressure against the string and its extension, causes the deviation from a true elliptical curve ; but any one, with a little practice may describe ellipses which are true enough for all practical purposes.
Fig. 18, PI, IV., represents a small tool to facilitate the drawing of ellipses by means of the string. Around a hollow tube a, in which a pen or pencil b is fitted, two rings c and d turn easily; c has an eye to fasten one end of the string, and the other also one through which to pass the other end ; the latter may be fastened readily at any length by a set screw. Both rings thus turning on the tube, the pen may be held at all points normal to the curve, and a very true ellipse described.
Problem 36. One only of the axes, and one point of the curve other than a vertex being given, to construct the ellipse: also to set the trammel.
Let G F, Fig. 13, be the given axis, and K one of the points of the curve. Find the centre O x)f the ellipse by bisecting the given axis, and a line D E drawn at right angles through O, must indicate the position of the other axis ; from K as a centre and Gf- O as radius, describe an arc, (jutting fke line of position of the long axis in J ; draw a line through K and J, and proceed to intersect the given axis as in H ; then K H will be one half of the long axis. (See Theorem 36.) Both axes being then known, construct the ellipse by any of the preceding methods.
This problem also shows haw a trammel can be set and the ellipse described at once, when either one of the axes and one point in the curve are given. Place the trammel in proper position over the lines indicating the axes ; let K 0, Fig. 16, be one half of the given axis, and P the point of the curve ; set the sliding block N" and the pencil P the distance K O apart, and, placing the slide N in the groove, fasten M by the set-screw and describe the ellipse ; the distance between the pencil and the block M will, of course, be one half of the unknown axis.
It is well to call attention to the fact, that the nearer the given point is situated toward the vertex of the unknown axis, the more accurate the trammel can be set; and the reverse, if the given point of the curve happens to be near the vertex of the given axis, the trammel cannot be set very accurate in regard to the unknown axis.
Problem 37. To construct a tangent at a given point of an ellipse, the foci being known.
First method.�Let A, Fig. 13, be the given point, and B and C the foci; draw the vectors A C and A B ; bisect the angle B A C formed hy the vectors, and erect a perpendicular M N" to the bisecting line A L, through the given point A; the perpendicular M 1ST is the tangent required. (See definition.)
Second method.�Draw the vectors as before, produce one of them, as AP and bisect the angle CAP; the bisecting line HT M will be tangent.    (See Theorem 37.)
Third method,�Let A, Fig. 15, be the given point. Draw the vector to the nearest focus, as A B, and from O a line O D parallel to it; then describe from O as a centre, with one half of the major axis as a radius, an arc, cutting the parallel OD in D ; AD will be the tangent required.
Problem 38. To construct a tangent at a given point of an ellipse without the use of the foci, the axes bHnq given.
Let E, "Fig. 16, be the given point and K L and H J the axes ; from O as a centre, and O L as a radium, describe an arc L F ; from E draw a line E F perpendicular to the short axis, intersecting the arc in F; also draw OF and erect at F, a line perpendicular to it, intersecting the short axis produced in D ; then E D will be the required tangent.    (See Theorem 40.)
Problem 39. To construct a tangent to a given ellipse from a point outside of the curve, the foci being Jcnown.
Let X, Fig. 13, be the given point, B and C the foci; describe an arc Y W from the furthest focus C as a centre, with the length of the major axis D E as radius ; describe also an arc YBW from X as centre with the distance to the nearest focus B as a radius, cutting the other arc in Y and
22
W ; draw lines C W and C Y from the furthest focus C to W and Y, intersecting the elliptic curve in V and Z ; then the lines X V and X Z will both be the required tangents.    (See Theorem 42.)
Problem 40. To construct a tangent to an ellipse from a given point outside of the same, both axes being given, without the curve.
Let X, Fig. 13, be the given point, and D E and GF the axes. Find the points W and Y and draw the lines C W and C Y as in the preceding problem ; then instead of drawing the tangents through the points of intersection V and Z at once, bisect the angle W X B by the line X V, which will be tangent to the ellipse ; V, or the point of intersection of the tangent, with the line C W, will be one point of the curve, and the point of contact.    (See Theorem 42.)
Problem 41. An ellipse and a straight line outside of the curve being given, to construct a tangent to the curve parallel to the given line.
Let D GE F, Fig. 13, be the given ellipse, and TU the given line ; D E and GF being the axes, find the foci B and C ; from the focus C, nearest to the given line let fall a perpendicular C S ; from the further focus B as a centre, and the length of the major axis as a radius, describe an arc, cutting the perpendicular in S ; connect B and S by a right line, whose intersection Q with the curve will be the point of contact; a line Q R through the point Q and parallel to the given line, will be the required tangent.
Problem 42. An elliptical curve, its centre and the foci, also a point in the curve, are given: to construct a tangent to the ellipse through the given point, also another tangent in such a manner, that both tangents and the two lines running from the points of contact to the centre of the ellipse, will form a parallelogram.
Let B, Fig. 16, be the given point, H J and K L the axes, and S and R the foci; draw a line B R from the given point to the further focus ; on this line set off from B to G one half of the length of the major axis ; draw a line from G to the centre O of the ellipse, produce each way intersecting the curve in the two points C and Q ; draw also the line B 0 ; through B draw a line B A parallel to O C, and through C a line C A parallel to O B intersecting each other at A ; then B A and C A will be the required tangents, and A B 0 C a parallelogram. A similar parallelogram would be formed by AB produced, (it being parallel to OQ,) and a line through Q parallel to OB. (See Theorem 39.)
Problem 43. A parallelogram being given, to construct an ellipse to which the four sides of the parallelogram shall be tangents.
Let mb en, Fig. 5, PL XVIII, be one half of the given parallelogram; b m and c n being one half of two of the parallel sides, it is evident, the major axis of the ellipse must be contained in the centre line m n, and o, the centre of that line, must be the centre of the ellipse ; also a perpendicular o e from the centre o to the side b c of the parallelogram, must be the short axis. Describe a circle from o as a centre, with o e as radius, and produce o e and m b until they intersect at/; from f construct a tangent/^ to that circle, and from the point of contact g draw a g parallel to m n, intersecting the side mb of the parallelogram at a, which will be a point of contact, and mb a tangent to the ellipse. (See Theorem 40.) From a draw a line ah7c parallel to go, and a7c is the length of half of the major axis. (See Theorem 36, and fourth method of constructing an ellipse.) Both axes being then known, construct the ellipse as convenient.
P1IV.
Fig. I.
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PLATE    IV.
Problem 44. To construct a curve resembling an ellipse by means of compasses.
Remark.�When compasses are applied to any true ellipse, it will be found that the curve, in parts, very near coincides with circular arcs.
The axes of an ellipse divide the curve into four equal parts; therefore, the circular arcs resembling nearest the curve, must pass through the vertices of the ellipse, and the centres from which they are described, must lay in the lines of the axes. To construct an ellipse by means of compasses, the transverse and conjugate axis must be given. Fig. 1, represents a semi-ellipse described from five, and Fig. 2, one described from three centres.
Comparing the two figures, the respective axes of which are alike, it will be noticed, that Fig. 1 very nearly resembles in shape the true ellipses drawn on Plate III., while Fig. 2 shows a decided abruptness in the curve, where the large and small arcs meet, as at D and G, which in Fig. 1 is remedied by introducing another arc between the two.
The construction of Fig. 1 is as follows :
Let B O and O C be one half of the axes; construct the rectangle O C E B, draw the diagonal C B, and let fall from E a perpendicular to the same ; produce both this perpendicular and the conjugate axis until they intersect as in H, which point will be the centre of the larger curve passing through the vertex C ; next find a mean proportional to O B and O C, or one half of the axes; make O F equal to O 0 and describe a semicircle over F B as a diameter, and produce O C, intersecting the semicircle in G; O G will be such mean proportional. (See Problem 32.) Make CM equal to OG, and describe the arc LML' from H as a centre and HM as radius ; also make B K equal to the mean proportional 0 G and describe the arc K L from J (the point where the perpendicular E H intersects the transverse axis) as a centre and J K as radius, cutting the other arc at L. Draw a line through L and J produced; make M L' equal to M L5 and A J' equal to J B, and draw H U and U J produced; then L and J will be the centres of the arcs on one side, and L' and J' on the other. Describe the central arc V C b and the small arcs a B and cS A; then, if the construction is accurately made, ha, L&, U B\ and 1/ a5 will be equal, and the arcs a b and a?V described from centres L and L' will form with the others the continuous curve A a] V C b a B.
Fig. 2 shows two constructions of the semi-ellipse from three centres ; both produce about the same result.
The quadrant AD G of the curve is found in the following manner:
Let A O and O C be the semi-axes ; construct the rectangle A E C O, and draw the diagonal A C; bisect the angles E A C and E C A, the bisecting lines intersecting in D ; let fall a perpendicular D J from D to A C, which produced, will intersect the conjugate axis or its extension in H. If the construction is accurately made, J A will be equal to J D and H D equal to H 0. J and H are the centres from which the arcs A D and D C are described.
To construct the quadrant C G B in the manner as shown in the figure, draw the diagonal C B, C10 and 0 B being the semi-axes; describe an arc C L from O as centre and the spmi-minor axis O C as radius; then take LB, or the difference between the semi-axes, and transfer from 0 to F on the diagonal; bisect F B, the balance of the same, by the perpendicular G K H, intersecting the minor axis produced in H. From H as a centre and H 0 as a radius, describe the arc C G, cutting the perpendicular in G. If the construction is accurately made, K G will be equal to K B and the arc GB, described from K as centre, will form with the first arc, the curve C G B.
The other quadrants, if a semi-ellipse is required, are constructed by either method in the same manner.
24
THE   PAEABOLA.
DEFINITIONS.
A Parabola, A Gr/, Fig. 3, is an open curve, described by a point A moving in such a manner, that in all positions its perpendicular distance from a fixed straight line E C, called the directrix, and that from a fixed point B, termed the focus, are equal.
A line D F, perpendicular to the directrix C E and passing through the focus B, is termed the axis, and divides the curve into two symetrical parts.
The point Gr, in which the curve passes across the axis, is termed the apex, and is always situated midway "between the directrix and focus.
Any line A H, bisecting that angle, formed by a line A B connecting any point A of the curve with the focus B, and an ordinate A E perpendicular to the directrix from the same point, is a tangent, and A the point of contact.
A line A L, perpendicular to the tangent at the point A, is a normal.
If from any point A of the curve, an ordinate A M is drawn perpendicular to the axis, and also a normal A L from the same point, then the distance M L between the points of their intersections with the axis, is always equal to a similar ordinate of the focus, as B V or B V.
Problem 45. To construct a parabola, the focus and directrix being given.
Let B, Fig. 3, be the focus and C E the directrix.
The following construction is based on the definition of the parabola given above.
Draw lines as Bb\ cd, dd\ parallel to the directrix, take their distances from the same, as D B, D c, D d, as radii and from B as centre, describe arcs, cutting the corresponding parallels in points V c? W, which will be points in the curve. The apex Gr being always midway between D and B, the curve is traced through the points so found.
Problem 46. To construct a parabola, the axis, apex and one point in the curve being given.
Let Gr F, Fig. 3, be the axis, Gr the apex and 4 the given point of the curve. Let fall from 4 a perpendicular 4 F to the axis, produce it and make Ff equal to 4 F; divide 4 F into any number of equal parts, say four, and draw parallels to the axis through the points of division. Divide also F Gr into the same number of equal parts, and draw lines from the point f through these points of division on the axis, producing the same until they intersect their corresponding parallels in points as 1' 2' A, which will then be points of the curve, v
Problem 47, To construct a tangent at a given point of a parabola, the directrix and focus being given.
First�Let A, Fig. 3, be the given point, D E the directrix and B the focus. Connect A and B, draw a line A E from the given point perpendicular to the directrix, bisect the angle B A E formed by them, and the bisecting line H A will be the required tangent.
Seeond-�The axis and focus being given. Let Gr F be the axis and B the focus ; draw the ordinate A M from A, also either ordinate B V from the focus, and lay off the latter from M, the foot of the former,on the axis toward the opening part of the curve ; as ML equal to B V ; then AL will be a normal, and a line AH perpendicular to the normal at the point A. will be the required tangent.
VOLUTES   AND   SCROLLS.
DEFINITIONS.
Volutes or scrolls, as Figs. 4 and 5, are a combination of quadrants of circles, their radii diminishing or increasing in a certain ratio ; the centre part of the scroll terminates in a full circle, called the eye. They find application in constructing the lower parts of staircases, caps of Ionic columns, &c.
Problem 48.  To construct a volute or scroll.
Several methods are employed; they differ only in the mode of ascertaining the centres of the circular arcs, but produce about the same result.
25
Let AB, Fig. 4, "be the diameter of the scroll; divide the same into an unequal number of equal parts, say nine. At division 5 and the point B draw perpendiculars to AB indefinitely, and set off Bf and 5 E equal to one of the divisions; draw the line Af intersecting the perpendicular from 5 at D, and from this point D sett off D O on A/, toward f, equal to one half of one of the divisions. Through O draw the diagonals O 5 and O E produced ; also draw E/ parallel to A B, being intersected by 5 O at F ; from F draw F g perpendicular to the diameter, intersecting E O at Gr; from Gt draw G h parallel to the diameter, intersected again by the diagonal 5 0 at H; fr<M H draw H i at right artgles to the diameter, intersecting the diagonal E O at J, &c.; the points 5, E, F, Gr, H, J will be the centres to describe the scroll from. Commence with 5 as centre, and A 5 &s radius, connecting at A with a straight line at right angles to the diameter A B, and describe the quadrant A e; next take E for a centre and describe the quadrant ef with E e as radius, &c, taking successively F Gr H and J for centres, describing the quadrants fg. gh,h i, and .closing by a full circle, described with J i as radius.
This diminution of radii may be carried further or closed before, as the purpose of the scroll requires.
Fig. 5 is described similar to Fig. 4; divide again the diameter A B into an uneven number of equal parts. Over divisions 5, 6, draw a square with one of the divisions for a side ; divide the square into sixteen equal small squares, use 5 as first centre with A 5 as radius, and describe the quadrant Ac; next use 0 for a centre and Cc as radius, describing the quadrant cd; the third centre is in D, using D d as radius, describing the quadrant de; and so on, using E, F and Gr as centres successively, diminishing the radius and closing the scroll by a full circle described from Gr as a centre and Gtg as radius.
Probletn 49. To connect two straight lines forming any angle, by a curve or easing, to which the lines, at certain points, shall be tangents.
The following construction is generally employed:
Let AB and B C, Fig. 6, be the given lines, and A and C the points at which the^curve is to connect; divide A B and B C each into the same number of equal parts, indicated in the figure by 1, 2, 3; connect the respective points, as represented in the figure by the dotted lines, and the points of their intersection are points of the curve which is traced through the same
THE   SOLIDS.
In regard to form they may be divided into two classes.
First�Solids bounded by plane surfaces, or polyhedrons.
Second�Solids bounded by both curved and plane, or curved surfaces only; to this class belong the solids of revolution.
The Polyhedrons coming here under consideration are two : the prism and the pyramid.
A prism is a polyhedron, in which two of the faces are equal polygons, with their planes and homologous sides parallel, and all the other faces parallelograms.
The equal polygons are termed the bases, the parallelograms the lateral faces, and comprise the convex surface; a line perpendicular between the two bases, is the altitude of the prism.
A prism is termed right, when the lateral faces or sides are at right angles to the bases ; in that case, all the sides are rectangles, and their length and the altitude are equal.
Prisms are called triangular, quadrangular, hexangular, &c, as the bases are triangles, quadrilaterals, hexagons, &c.
A prism is called regular, when its bases are regular polygons; and a prism, whose bases are parallelograms, is termed a parallelopipedon.
A rectangular PARALLELOPiPEDON is one whose bases and lateral faces are rectangles, represented by Fig. 13, PI. V*                                                                                                     ,   , .
4
26
A cube, Fig. 12, PI. V., is a rectangular parallelopipedon bounded by six equal square faces.
A pyramid is a solid, bounded by a polygon, called the base, and as many triangles uniting in one point, called the vertex, as there are sides to the base. The perpendicular from the vertex to the base, is the altitude, and the sum of the lateral triangles, the convex surface of the pyramid.
A right pyramid is one whose base is a regular polygon, and in which the perpendicular from the centre of the base passes through the vertex.
The slant height of a right pyramid is the perpendicular let fall from the vertex to either side of the polygon, which forms its base.
Pyramids are termed, as the prims, according to the form of their bases, triangular quadrangular, &c.
If a pyramid is cut by a plane parallel or inclined to its base, a second base is formed ; that part lying between the two bases, is called the frustum of a pyramid, or a truncated pyramid ; if both bases are parallel, the perpendicular between them is its altitude.
Fig. 14, PL V., represents a right quadrangular pyramid. ABCD is the base, SS the altitude, and the part enclosed between the bases ABCD and E F Gr H, the frustum.
A rectangular prismoid is a solid resembling the frustum of a quadrangular pyramid; the upper and lower bases are similar rectangles, having their homologous sides parallel.
The convex surface is made up of four trapezoids, and the perpendicular distance between its bases is the altitude.
; The solids of revolution coming here under consideration, are:  the cylinder, cone and sphere.    They are the three round bodies treated of in elementary geometry.
A cylinder is a solid, generated by the revolution of a rectangle about one of its sides as an axis.
The convex surface of a cylinder is described by the revolving side parallel to the axis, and the circles described by the sides perpendicular to the same, are the bases.
These bases being in that case perpendicular to the axis, the cylinder is termed a right
CYLINDER.
If the cylinder is cut by two parallel planes inclined to its axis, in such a manner, that the entire convex surface is intersected by each plane, then that part between them is termed an oblique cylinder, and the bases assume the form of ellipses.
The altitude of a cylinder is the perpendicular distance between its bases.
Fig. 15, PL V., represents a right cylinder; S S is the axis and altitude.
RemarTt.�Although, when speaking of a " Cylinder," a right cylinder of revolution, that is, one with a circular base is referred to, a right cylinder may have any other closed curve, the ellipse, oval, &c, as the form of its bases, and is then called an elliptical cylinder, &c.
A right cone is a solid, generated by the revolution of a right angled triangle about the perpendicular side as an axis.
The circle described by the revolving side of the triangle, forms the base, while the hypothenuse describes the convex surface of the cone. The perpendicular of the triangle is the axis, the hypothenuse the slant height, and the terminating point the vertex of the cone.
A cone is called oblique, when its base makes an angle other than a right one, with its axis. |      The altitude of a cone is the perpendicular distance from its vertex to the plane of the base.
If a cone is cut by a plane parallel or inclined to the base, another base is formed, and that part between the two, is called a truncated cone or frustum.                                                   [
The frustum is a right one, when the planes of both bases are parallel and perpendicular to the axis.
Fig. 16; PI. V., shows a right cone ; S is the centre of the base, S S' the axis, also in this case the altitude, S' the vertex, A S' or B Sf the slant height, A B E F a frustum.
A sphere is a solid generated by the revolution of a semicircle about its diameter as an axis. The sphere is terminated by a curved surface, every point of which is equally distant from a point within, called the centre. This distance is termed the radius, and any line passing through the centre and terminated on both sides by the surface, the diameter or axis of the sphere.
27
Every section of a sphere by a plane in any position will be a circle, and that by a plane passing through the centre, is called a geeat circle of the same; its diameter is equal to that of the semicircle, by the revolution of which the sphere is generated.
MENSURATION   OP   SURFACES.
Area denotes the superficial contents of a figure.
Equivalent figures are those, which have equal areas. The area of a surface is found by ascertaining, how many times such surface contains some other given one, which is considered as a unit of measure.
Lines are measured by considering an inch, a foot, a yard, &c, as unit of measure. Surfaces are measured by considering a square inch, square foot, square yard, <fec, as a unit of measure.
A square iisrcH is a plane surface, enclosed by a square, whose sides are one inch in length. A square foot is a plane surface,  enclosed by a square,  whose sides are one foot in length, &c.
It is most convenient to express the superficial contents of a figure in the same square unit, as that unit of length, by which its sides are measured.
Let the sides of the square, Fig. 9, PL I., be one inch in length, then it contains one square inch. Let the base AB of the rectangle Fig. 10, be two inches in length, but the width AD the same as the sides of the square Fig. 9, then Fig. 10 contains Fig. 9 twice in its surface, or two square inches. Let in Fig. 20, PL I., those small squares indicated by dotted lines, represent one square foot each, and the side A C be three feet in length, then the square A C D E contains
3+3=9 square feet. Thus the following table of surface measure will be evident:
1 square yard=9 square feet. 1 square foot=144 square inches. The area of the various plane figures is calculated by certain rules:�That of the simpler figures,   as  triangles,   parallelograms and trapezoids,   depend on their bases and altitudes, and it is therefore necessary, that the meaning of these two terms, in regard to those figures, is distinctly stated.
The altitude of a triangle is the perpendicular let fall from the vertex of one angle to the opposite side, or that side produced ; such side is called a base.
The altitude of a parallelogram is the perpendicular distance between two opposite sides; such sides are then called bases.
The altitude of a trapezoid is the perpendicular distance between its two parallel sides. Rule 1.    To find the area of a square, a rectangle, or parallelogram.
Multiply the base by the altitude. Rule 2.    To find the area of a triangle.
Multiply the base by one half of the altitude. I        It will be noticed, that the area of a parallelogram is twice that of a triangle of the same bkse and altitude. Rule 3.    To find the area of a trapezoid.
Add the two parallel sides, and multiply one half of their sum by the altitude. Rule 4.    To find the area of any quadrilateral figure.           *
Divide the quadrilateral by a diagonal into two triangles, and the sum of their surfaces will be the area required. Rule 5.    To find the area of a circle.
First�Multiply the circumference by one half of the radius.
In this solution the area of a circle is treated as that of a triangle whose base is equal to the circumference, and whose altitude is equal to the radius of the circle.
4�
;              Or, second�Multiply the square of the radius by 3.1416, or approximate, by the
' common fraction %\.
Or, third�Multiply the square of the diameter by 0.785, or approximate, by the common fraction f. Rule 6.    To find the area of a sector of a circle.
First�Multiply the length of the arc by one half of the radius. Or, second�Find the area of the whole circle, to which the area of the sector will be in the same ratio, as the number of degrees contained in the arc, to 360.
Example.�What is the area of a sector, the circle being 14 feet diameter, and the ;  J   arc of the sector containing 60 degrees ? ''; '?                                        Circumference of circle=14 x 3^=44.
Area of circle=44 x J14=308 square feet. Area of sector   ^-=51i square feet. Rule 7.    To find the area of a segment of a circle.
Find the area of the sector of the same arc as the segment;  also that of the triangle formed by the chord and the two radii ; then
First�If the segment is less than the semicircle, subtract the area of the triangle from that of the sector, and the remainder will be the area of the segment.
Second�If the segment is larger than the semicircle, add both the area of the sector and that of the triangle, and their sum will be the area required. RuM 8.    To find the area of any irregular figure.
Divide the figure into parallelograms, trapezoids, triangles, segments, &c, as most convenient, and the sum of their various areas will be the area of the given figure. Rule 9.    To find the area of an ellipse.
Multiply the product of the two axes by 0.78,  or approximate, by the common fraction f.
MENSURATION   OF   SOLIDS.
The mensuration of solids is divided into two parts : Ffrst.�Meiisuratipn of their surfaces. Second.�Mensuration of their solidity. The unit of measure for plane surfaces is a square.
The unit of measure for solidity is a cube, whose face is equal to the square used as a TOit in the ipe^sipexnent of the surfaces.
Let the faces of the cube, Fig. 12, PL V., be one square inch, then it is said to contain one cubic inch.
Let the base of the rectangular parallelopipedon, Fig. 13, be also one square inch, but its altitude or height two inches, it is evident that it will take two cubes as Fig. 12, to form a solid as Fig. 13 ; it is then said, that the latter solid contains two cubic inches. The following is a table of solid measure:
1 cubic foot=1728 cubic inches. 1 cubic yard=27 cubic feet. 1 cubic rod=4492|- cubic feet. Rtyle 10.    To find tjie surface of a right prism.
Find the sum of the sides of the base, and multiply by the altitude ; the product will give the area of the convex surface.   If the whole surface is required, add the areas of both bases. Rule 11.    To find the solidity of a prism.
Multiply the areq, of the ba$e by the cbltitude.
w
Mule 12.    To find the surface of a right pyramid.
Multiply the sum of the sides of the base by one half of the slant height, and the product will be the convex surface.    In case the entire surface is required, add the area of the base. Mule 13.    To find the solidity of the pyramid.
Multiply the area of the base by one-third of the altitude. Rule 14.    To find the surface of the frustum of a right pyramid.
Add the sum of the sides of both bases, and multiply by one half of the slant height ; the product wilt be the area of the convex surface. The sides of the frustum are trapezoids. Rule 15.    To find the solidity of the frustum of a pyramid.
Find the area of both bases, also a mean proportional between the two ; add the three together, and multiply their sum by one-third of the altitude. Rule 16.    To find the solidity of a rectangular prismoid.
Add the areas of both bases: also four times the area of a section parallel to and midway between them, and multiply the sum by one-sixth of the altitude. Rule 17.    To find the convex surface of a cylinder.
Multiply the circumference of the base by the altitude. Rule 18.    To find the solidity of a cylinder.
Multiply the area of the base by the altitude. Rule 19.    To find the convex surface of a cone.
Multiply the circumference of the base by one half of the slant height; when the entire surface is required, add the area of the base. Rule 20.    To find the solidity of a cone.
Multiply the area of the base by one-third of the altitude. Rule 21.    To find tlie conyex surface of the frustum of a right cone.
Take one half of the sum of the circumferences of the bases, and multiply by the slant height of the frustum. Rule 22.    To find the solidity of the frustum of a right cone.
Find the area of both bases, and a mean proportional between them: add the three, and multiply their sum by one-third of the altitude. Rule 23.    To find the surface of a sphere.
First.�Multiply the circumference of a great circle by the diameter. Or, secoiid.�Multiply the square of the diameter by the quotient 8.14 Rule 24.    To find the solidity of a sphere.
First.�Multiply the surface by one-third of the radius. Or, secoii&.-^-Multiply the cube of the diameter by 0.5236 This latter rule shows the solidity of a sphere to be about ��$ more than one half of the cube of the diameter.
MEASUREMENT   OF   LUMBER   AND   TIMBER.
i
The foregoing rules for the mensuration of surfaces and solids find practical application in
the measurement of lumber and timber.
The contents of boards and planks are generally calculated by board measure.    One foot
of boaed measure means a piece of board, one foot square on the face by one inch in thickness.
Timber is measured by the cubic foot, which means a block of one foot square on the face by
twelve inches thick.    It takes therefore, twelve feet of board measure to make one cubic foot, or
one solid foot.
30
The superficial contents of boards one inch in thickness, are their contents in board measure. The contents of planks in board measure, are found by multiplying their superficial measurement by the number of inches in their thickness.
The surfaces of boards of parallel width and square ends are rectangles, and their contents are found by multiplying their length by their width. The length of boards is generally measured in fee^ and their width in inches. Multiplying their length in feet by their width in inches, and dividing the product by twelve, gives at once their contents of hoard measure in feet.
Example 1. How many feet of board measure are contained in a board thirteen feet long by fifteen inches wide?
_ij^=jjhl=l6i feet.
When the length of hoards is twelve feet, the contents are as many feet hoard measure, as the number of inches in width.
Example 2. How many feet of board measure are contained in a board twelve feet long by ten inches in width \
^y0 =10 feet.
Also boards twelve inches in width, contain as many feet board measure, as the number of feet in their length.
Boards whose length and width are given in feet and inches, and the contents are required exactly, hoth the length and width must he reduced to inches, and their product divided by 144.
Example 3. How many feet of board measure are contained in a rectangular board seven feet five inches long by one foot eleven inches wide ?
7's^ii"., 8i94x43s==j^l=14 feet 31 inches.
or approximate 14J feet.
When the number of inches in the length and width, are convenient fractions of the foot, the calculation may be somewhat shortened by expressing the dimensions in those fractions.
Example 4. A board is 9' 8" long by V 4" wide ; what are its contents in board measure %
9' 8"=9f feet.       V 4"=li feet. Contents,  9|xl-J=yxf=ilA=12| feet. This calculation may also be carried out in the following manner:
The board is 1-J- feet wide; if it were 1 foot wide, it would contain 9f feet; now add one-third of 9f to it, and their sum are the contents.
9f+^=9f+3f=12f feet. The surfaces of tapering hoards are trapezoids, and their contents calculated by Rule No. 3. The ends of the board are the parallel sides, and its length the altitude.    One half of the sum of the width at the ends, is equal to the width midway between them.   Therefore, to find the superficial measure of a tapering straight-edged hoard, multiply its length hy its width at the middle.
Example 5. How many feet of board measure are contained in a board 16' long, its width being 14" at one end and 11" on the other ?
-lA+ii xi|=y xf = 2_^3.=i6f feet. Whenever the length of a board is, as in this instance, a convenient fraction more or less than twelve feet, the calculation may be carried out thus:
If the board were 12 feet long it would contain  14:^lt=12^ feet. Now sixteen feet is equal to twelve feet and one-third of twelve feet; consequently, add one-third of twelve and one-half, to twelve and one-half, and their sum are the contents.
12i+Aja=12�+4fr=16t feet. If the length of the board were nine feet, take the contents of a twelve foot board and subtract one quarter of the amount.
13J-^=iaj-3J=9f feet. Example 6. What are the superficial contents of a piece of panel work of a triangular shape, the base being fourteen feet long and the height nine feet six inches %
Apply the Rule No. 2.    Superficial contents,  14 x -^=14 x 4f=66J feet. Example 7. What are the contents of a plank twelve feet long, fourteen inches wide and four inches thick, in board measure %
31
Contents of face, 14 feet.   Contents of plank, 14 x 4=56 feet.
A plank is thirteen feet long, eleven inches wide, and one and one-fourth inches thick; how many feet of board measure are contained in it %
Face,  ^ai^iy^iiii feet. Contents,   11-^ x \\z-X�g- x ^�i^-�14J| feet board measure.
Such planks are surveyed as containing fifteen feet.
Example 8. What are the contents in board measure, of a solid gateway two and one-half inches thick, with a half circular top ; the width is six feet and the height eleven feet to the top of the arch?
The gate being half circular at the top and six feet wide, the radius is three feet. The face will therefore, contain a rectangle of 11�3=8 teet high, by six feet wide, and also a semicircle of six feet' diameter.
Surface of rectangle, 8 x 6=48 feet.
Employing Rule No. 5�first, to find the area of the semicircle, we have Surface of semicircle, 6 x 3}.x ij_e x 22 x 3_ 396 _i4i, feet
_                    �                                              �      ,                        2               2x7x238
Total surface, 48+141=62^ feet.
Multiply by the thickness, 2^ x 62^=155^ teet, board measure.
Timbers of parallel width and thickness, with square ends, are rectangular parallelo-pipedons ; their contents in cubic feet are found by Rule No. 11, relating to prisms.
Example 9. How many cubic feet are contained in a stick of timber eighteen feet long, fourteen inches wide, and eight inches thick %
The dimensions of the section being given in inches, the surface must be reduced to square feet by dividing the product of width and thickness by 144.
The surface of a section or the base of the timber is, 3f^t~Tk\ square feet. Solid contents, Uf x 18=J^-=14 cubic feet.
Tapering straight-edged timbers, with square sections on each end, are frustums of right pyramids, and their solidity is calculated by Rule No. 15.
Example 10. What is the solidity of a piece of timber twenty-four feet long, one end of which is fifteen inches, and the other six inches square?
The larger end,........15x15=225
The smaller end,.......6 x 6= 36
The mean proportional, ...            =90
Sum of the three,......            =351 square inches.
Multiply by one-third of the length, that is, eight feet, and divide the product by 144.
-H^-=19i cubic feet. Remark.�The mean proportional between two numbers is the square root of their product. The square root of a number is that number, which, when multiplied by itself, will produce the given number as product.
As it is not always convenient to find such a mean proportional, the solidity of this kind oi timbers may be calculated by the next rule.
Square-edged timbers, which are tapering, but whose sections are not squares, are rectangular: prismoids, and their contents are found by Rule No. 16.
Ejxample 11. What are the solid contents of a piece of timber twenty eight feet in length, sixteen by twelve inches on one end, and fourteen by ten inches on the other % The timber will evidently be fifteen by eleven inches midway.    Therefore,
Contents,  i*�u�.i�^+mi12llii x 28 = i92 + i4o + 66o x28=-^|^=-^p.=3224T cubic feet.
In order to demonstrate that the result is the same by both rules, find the solidity of the piece of timber of the same dimensions as given in Example 10, by Rule No. 16.
Length 24 feet; section at one end 15 inches square, and on the other 6 inches square.
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Add 15 and 6, and one half of their sum will be the side midway, that is 10J inches. Area of large end, 15x15 =225 square inches. "       small   "      6x6 = 36 "       midway,   10ixlOJ=110t
Solidity 225 + ^i4xlloM)xy=
�JHx4=W=1&i cubic'feet. In practice, when the ends do not vary much and an accurate calculation is not needed, one half of the sum of the surface of both ends multiplied by the length, gives the solidity approximate. Take for an example the timber 28 feet long, 16 by 12 inches on one end and 14" by 10'' on the other.   Half the sum of the surface of both ends,   192 + 140�166 square inches.
Solidity 4H-x28�32A cubic feet, which is but very little more than the true measurement. (See Example 11.)
When the two ends vary considerably, then the error in this mode of calculation becomes more apparent.
Take for an example the piece of timber of the dimensions in Example 10 ; length 24 feet, Large end 15 inches square, 225 square inches. Small end  6          "               36            "
Half their sum,            AJi=130i
Solidity ^l>� x 24=^^=21f cubic feet, which is 2J cubic feet more than the true solidity found in Example 10.
Hound logs are generally tapering, and their solidity is calculated by Rule No. 22, relating to frustums of cones.
To avoid the inconvenience of finding a mean proportional, the solidity may be found similai to that of a rectangular prismoid. Find the sum of the area of the two ends, add four times that of a circle midway, and multiply their sum by one-sixth of the length.
Example 12. How many cubic feet are contained in a log twenty feet long : the butt end being twenty-eight and the top end fourteen inches diameter ?
"Surface of butt end, 14x14x3^=616 square inches, top end,   7 k 7x3^=154 The radius of the circle midway must be one half the sum of the radii of the ends, or 10% inches. Surface of circle midway 10| x lOf x 3|=346| square inches.
Solidity of log,   eie ii54^46^*4 xy^
=Wx�=40ff oubio feet.
The solidity of this log, found by Eule No. 22, is 49f� cubic feet. Domes on churches and public buildings have the form of a half sphere. Example 13. How many square feet are contained in the surface of a dome twenty-one feet diameter ?   Apply Rule No. 23.
Circumference, 21x3^=66 Surface, 1^x^=1^=3693 square feet.
TOOLS,
In regard to position in nature, lines and surfaces may either be horizontal, vertical or inclined.
When a sheet of water is at rest, its surface is said to be horizontal or level.
When a line is freely suspended in still air with a sufficient weight attached for steadiness, it will assume a certain position in the direction to the centre of gravity in the earth, and is said to
be   VERTICAL Or PLUMB.
A line or surface which is neither horizontal nor vertical, is said to be inclined.
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A horizontal line or surface, and a vertical one, are always at right angles to each other. Although vertical lines on different places on the surface of the earth make an angle with each other (because all show toward the centre), that angle is imperceptible within the extent of any one building; and therefore, all vertical lines are considered parallel.
A large sheet of water at rest represents the true shape of a horizontal surface. Although its roundness or curvature is perceptible to the naked eye in the distance of a few miles, all horizontal lines as far as they concern any one building, are considered straight.
Fig. 7 represents a very simple tool called a plumb rule, with which every carpenter is familiar. It consists of a true piece of board with parallel straight edges and a guage-mark through the centre. At the upper end A, over the guage-mark, a string is fastened with a plumb-bob B attached. If either side of this tool is applied to any surface or straight edge, and the line of the plumb-bob coincides with the centre line guaged on the board, it must be plumb or vertical.
Fig. 8 represents an instrument of simple construction to ascertain whether a surface or straight line is level or horizontal. The lower edge C D being straight, a line A B is marked at right angles to it on the upright standard ; over this line is a string A and plumb-bob B arranged, in the manner as the plumb rule. As a horizontal and a vertical line are always at right angles, the lower edge of the tool must be level, whenever the square line coincides with the freely suspended plumb-line.
Fig. 9 represents what is termed a spirit level. Its action depends upon that property of water or any other liquid, to assume a horizontal surface and exert laterally an equal pressure in all directions, whenever at rest. A slightly rounded glass tube B, so far filled with spirits that only a small bubble of air A remains, is fitted to a true rectangular piece of wood or metal in such a manner that, whenever the lower edge C D of the instrument is placed upon a level surface, the air bubble A will assume a position in the centre of the tube. As soon as the instrument is declined either way, the equal pressure of the liquid on both sides of the bubble will be disturbed, and force it toward the raised end of the instrument.
The plumb rule, Fig. 10, is arranged upon the same principle. Whenever the edge C D is in a vertical or plumb position, the bubble A is at rest in the centre of the tube B.
Usually both instruments are combined in one; they are more convenient than those represented in Figs, 7 and 8.
Whenever any number of parallel lines are to be drawn in a position, as the parallels and ordinates in the construction of the ellipse Fig. 14, PI. III., or the parabola Fig. 3, PI. IV., the T-square AB and set-square or angle CDE represented in Fig. 11, are used. The T-square consists of a thin, parallel straight-edged plate B, made of straight and close-grained hard wood, and attached at right angles to an arm A. This arm is slid along a straight edge of the drawing board, and all lines drawn along the edge of the plate B must be parallel. The arm A is usually applied to the left hand edge of the drawing board.
The angle C D E is a rectangular triangle, made of the same material. Two sides are usually of equal length, and therefore, the angles at D and E are 45�.
Whenever one side, as C E, is slid along the edge of the plate B, all lines drawn along the side C I) must be parallel to each other and to that edge of the drawing board, along which the arm A slides, and also at right angles to all lines drawn along the edge of the plate B. For instance: Let C E, Fig. 3, represent the edge of tho drawing board ; then all parallels, as DF, 11', 22', are drawn blong the edge of the plate of the T-square, and all ordinates or perpendiculars as cc', dd', &c., are drawn along the edge of the set square.
4l11 lines drawn along the hypothenuse E D, Fig. 11, would form an angle of 45� to all lines drawn along the plate B, which is often very convenient in making drawings. For instance, the construction of the regular octagon, Fig. 18, PL II., is facilitated by its use.
There are also set squares, as B C D and B' C D', Fig. 12, with one angle of 60� as at B and B', and one of 30� at D and D'. They are used to advantage in constructions such as the regular hexagon, Fig. 17, PL II.
Whenever parallel lines on any angle to the face of the drawing, as B5, 44, 3 3, Fig. 21, PL II., are to be drawn, a simple straight ruler A, Fig. 12, PL IV., which may be placed in any
34
position on the drawing board, is used in conjunction with a set square. For instance : to draw lines parallel to & d\ place one edge as C D of the set square to coincide with & d? ; while pressing it tight to the paper in that position with one hand, apply the ruler A to the hypothenuse B D, with the other ; then press down the ruler, and release the set square, sliding it along the edge of the ruler, and lines, as c d, drawn along the edge C D, will be parallel to & d\
In this same manner lines are drawn perpendicular to each other. For instance : let a line be drawn perpendicular to c d ; place the set square and ruler in position, so that side C D coincides with the line c d; then, while holding the ruler tight to the paper, slide the set-square, and a line as V c\ drawn along the edge B C, will be perpendicular to c d.
Some practice is required to carry out these operations accurately and expeditiously, and to find the many uses to which they may be applied in drawing.
Fig. 13, represents an instrument called a peoteactoe, found in most sets of drawing tools. It is used to read the size of angles in degrees, and to transfer the same. It consists of a semicircle ABC, AC being the diameter and O the centre, divided into 180 parts or degrees. To read the angle A O B for instance, place the centre O at the vertex of the angle, and let the edge A O coincide with one of the sides ; the other side passing through the division marked 60, shows that the angle is one of 60 degrees.
THE    SCALE.
The real size of an object cannot always be drawn on paper, on account of the room it would occupy, but can be represented in its true proportions, when drawn, as it is termed, on a small scale.    (See page 9.)
A scale is a straight line, divided in the same proportion as the unit of measurement, by which the object represented is already, or is to be constructed.
In building, the unit of measurement employed is the foot, divided into twelve inches; inches are again divided into halves, quarters, eighths, &c.
When one inch in length on the drawing, represents one foot in length in reality, then the scale is said to be one inch to the foot; that is, in other words, the drawing is one-twelfth of the real size of the object.
A drawing on a scale of one quarter of an inch to the foot, will occupy ^ part of the real surface of the object represented.
Drawings for buildings are usually made on a scale of either one inch, one half inch, one quarter inch, or one-eighth inch to the foot, in order to facilitate the measurement by a common foot rule, which every mechanic uses.
To construct a scale, draw a straight line and divide whatever unit is to represent one foot, into twelve equal parts, which represent one inch each ; from there space off any number of units required, and number them from left to right 1, 2, 3, &c, representing as many feet.
Fig. 14 shows four different scales marked on the left hand, J, \, f, IN. They are one-quarter inch, one-half inch, &c, to the foot respectively.
All the drawings of stairs and rails contained in this book are made by these scales.
To take off dimensions, as the length AB on the \ scale for instance, place one foot of the compasses in 11 and the other in A; the measurement of that opening will be eleven feet and three inches; the same distance on the f scale would be only one-third of that, or three feet nine inches. Likewise, the opening of the compasses from C to D on the inch scale is one foot eight inches, while the same opening on the \ scale is double o� that, or three feet four inches.
Fig. 15 shows what is termed a diagonal scale, one inch to one foot. Although fractions of an inch may be measured on the other scales by the eye, this scale is used whenever great accuracy is required.
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To construct this scale, proceed to space off on a straight line a number of inches representing feet, marked from left to right, 0, 1, 2, 3; draw eight right lines parallel to it, equally distant apart, marked 2, 4, 6, 8, and erect the perpendiculars through the points of division representing feet. Divide the one at the left on the upper line 8 B into twelve equal parts, to represent inches, and through these points of division draw the parallel diagonals as o A, &c, A B representing one inch, and the eight parallels being equally distant apart, the diagonal o A will cut off on those parallels, fractions in eighths ot an inch between itself and the perpendicular o B. The parallel diagonal lines will also divide the first foot on the lower line into twelve equal parts marked from right to left, 0, 2, 4, 6, 8, 10.
To take off measures on this scale, read the feet and inches on the bottom line as usual, and the eighths of an inch on the parallels from the bottom line up ; for instance, from C to F measures one foot nine inches and three-eighths of an inch.
To take three feet six and five-eighths inches between the compasses, place one foot in the intersection of the third perpendicular with the fifth parallel at E, and the other on the intersection of the same parallel with the sixth diagonal at D.
Fig. 16 is the side view of compasses, used to draw circles of large diameter. Two sliding blocks B and C, with set screws, are fitted to a straight rod A, about one inch wide by half an inch thick, and made of some hard wood.    B has a steel point and C a pencil holder D.
Fig. 17 represents an adjustable triangle, used to take the actual size of angles between the walls of buildings. It consists of three straight rods with half rounded ends and fastened together by screws at B and C, while the ends at A slide over each other. Whenever an angle \t to be measured, press A B and B C against the walls and mark where the end of C A comes on A B; thus, the angle ABC can be transferred to paper.
By inspecting the figure it will be seen, that the tool can be folded up and conveniently carried.   (See Figs. 10, 11, 12, PL II.)
Fig. 18 is the pencil holder used to draw ellipses by means of a string and two pins. It is mentioned in detail in the explanation of Fig. 15, PL III*
PLATE    ~V\
STUDY   OP   PROJECTIONS.
INTRODUCTION.
The representation of any object contained in space, by means of a geometrical drawing on a plane surface, as a sheet of paper for instance, is a Projection.
The art or science of the correct and practical execution of these projections, according to the known principles of geometry, is called Descriptive Geometry, and may4 be divided into two systems:
First�Scenographic Projections or Natural Perspective, which represents an object as it really appears in a certain position to the eye of the observer, viewed from a limited distance.
Second�Orthographic or Geometric Projections, also, Drawings in Ground Plan and Elevation, which represents an object as it would appear in a certain position to the eye of the observer, could it be viewed from such an infinite distance, that the sight lines may be assumed as parallel. The latter system is generally referred to, when speaking of drawings in projection or descriptive geometry; it is the most useful in making what are called construction or working drawings, as they show the object in real proportions and form. It is evident, that, in order to give a correct idea of an object, it must be viewed in different positions, as the top, front, or sides. It is customary, because most convenient, to represent the object on two planes, assumed to be at right angles to each other; one horizontal, called the horizontal plane of projection, and the other vertical, termed the vertical plane of projection.
Working drawings consist of horizontal projections or plans, vertical projections or elevations, and sections,
A horizontal projection, or ground plan, is that view of an object, as it would appear on the surface of the horizontal plane of projection, could the eye of the observer be placed vertically over every point represented.
A vertical projection, or elevation, is that view of an object, as it would appear on the surface of the vertical plane of projection, could the eye of the observer look at every point represented horizontally, and perpendicular to that vertical plane of projection.
These views show the exterior contours or outline of the object, but are not always sufficient; whenever interior parts, which are hidden from view by the outside surfaces, are needed to be shown, then sectional views are drawn.
A section is that view, as the object would appear, were it cut by an imaginary plane, and all those parts between that plane and the eye of the observer, removed.
Sections may be horizontal, vertical, inclined, or oblique, as circumstances may require, and are thus named from the position which the cutting or sectional plane assumes.
i A horizontal section is a view in which the cutting plane assumes a horizontal position ; as the eye of the observer looks at every point represented perpendicular to the horizontal plane of projection, it is therefore, a horizontal projection, as the plan of a building.
A vertical section is a view, when the sectional plane is vertical; the eye of the observer looks at every point represented horizontally and perpendicular to the vertical plane of projection ; it is therefore, a vertical projection, such as an elevation. The position of the cutting plane is mostly assumed to be parallel to the vertical plane of projection.
An inclined section is a view in which the cutting plane assumes an inclined position to either one of the planes of projection.
An oblique section is a view in which the cutting plane assumes an oblique position, that
87
is, inclined to both planes of projection.
In a set of drawings for a building, what is termed the plak of any one story, is a horizontal section through that story ; the horizontal cutting plane is assumed at some convenient height above the floor, to show all the window and door-openings, interior arrangements, &c.
The elevation is a view of the front. There are also side and rear elevations, giving a view of the side or rear of the building, whenever the particulars of either cannot be ascertained by the rest of the drawings.
In all these elevations, the sight lines are assumed to be perpendicular to the main vertical surface, that the view represents.
The section is generally a vertical section, to show the thickness of walls, height of stories, depth of timbers, height of window and door-openings, cornice, arrangement of roof, &c. The cutting plane is assumed to be parallel to some one of the main vertical surfaces of the building.
Sections are Gross sections, when the cutting plane is assumed to pass crosswise ; or longitudinal, when the same run lengthwise through the building.
The outlines of buildings form mostly rectangles; the surfaces are either vertical or horizontal, also at right angles to each other; or, when they are inclined, as roofs for instance, they form an angle with one of the main surfaces only, but are perpendicular to the others.
The main part of the materials used in the erection of a building are in the form of rectangular prisms, such as cut stones, brick, timber, &c, and are joined together parallel and at right angles ; when they are inclined or arched, some of their surfaces are either in a horizontal or vertical position.
The lines in both projections run, therefore, mostly parallel or at right angles to each other; or, when they are inclined or curved in one projection, they are parallel or perpendicular in the other.
Plans of such rectangular work are readily made, easily understood, and carried out in practice without much difficulty. There are cases, however, when certain parts of a building, such as arches, roofs, stairs, and all so-called splayed worlc, oifer some difficulty in their planning, comprehension, and practical execution, because the materials used are joined together in oblique positions, and frequently form when finished, warped or twisted surfaces.
It is the aim and purpose of the study of projections, as demonstrated in descriptive geometry, to assist the workman who, by applying the problems practically, overcomes the difficulty.
No mechanic is more benefitted by its study than the stairbuilder ; it will enable him to lay out every line, and cut every joint with absolute certainty and precision For this reason a comprehensive elementary course in this science is here introduced.
ELEMENTARY   DESCRIPTIVE   GEOMETRY.
DEFINITIONS.
The two planes of projection on which objects are represented, are considered to be of infinite extension, intersecting each other at right angles. They will evidently divide space into four right angular parts, termed diedral angles, whose vertices meet in the line of intersection of both planes; this line is called the ground or base line. To make this idea more comprehensible, let the planes be assumed to be of limited extent, as represented in Pig. 1, PL V. in a perspective view. JNOK represents the horizontal, and TLMS the vertical plane of projection ; Q R is their line of intersection or the ground line, which divides each plane into two parts, called respectively, the forward part as J Q R K, and the hack part as Q N O R of the horizontal plane of projection; also the upper part as QLME, and the lower part as T Q R S of the vertical plane of projection.
38
The eye of the observer, and also the object to be projected, are assumed to be in that diedral angle, formed by the forward part of the horizontal and the upper part of the vertical plane of projection. This angle is therefore termed the first diedral angle, or simply the first angle. The othei three angles are named the second, third and fourth angles, as indicated in the figure.
A ball or sphere appears in circular form in all positions. Let Gt represent one as being suspended in space in the first angle.
Conceive the eye as looking down perpendicular Irom above, and the sphere will appear as the shaded circle H on the horizontal plane of projection ; then conceive the eye as looking from the front against it, perpendicular to the vertical plane, and the sphere will appear as the shaded circle V on the vertical plane of projection.
H represents the horizontal projection, and V the vertical projection oi the globe GK If both planes of projection are considered fixed, the exact position of the globe in relation to them, may be conceived and located by means of its projections H and V ; it would be known : that the sphere is as far off the vertical plane in a perpendicular direction from V, as the horizontal projection H is distant from the ground line ; also, that it is as far above the horizontal plane perpendicular over the projection H, as the vertical projection V is above the ground line.
Conceive the vertical plane thrown back on the horizontal plane, the ground line Q R being the axis to swing on, then the upper part of the vertical plane would coincide with the hack part of the horizontal one, and the lower part of the vertical with the front part of the horizontal plane. The two planes would then lie in one plane, but the projections H and V would not change their relative position in regard to the ground line.
On this principle,diagrams in descriptive geometry are drawn, representing both projections on one and the same sheet of paper. The ground line is drawn parallel to the assumed front of the draughtsman ; the surface between himself and the ground line represents the forward part of the horizontal and the lower part of the vertical plane of projection, and the surface beyond oi above the ground line, represents the upper part of the vertical and the back part of the horizontal plane of projection.
Projections of solids, are given by the projections of their boundary surface; those oi surfaces, by the projections of their boundary lines, and those of lines by the projections of two or more points contained in them.
All objects are, whenever convenient to do so, projected as being contained in the first angle ; their projections will therefore fall on the front part of the horizontal plane, and the upper part of the vertical plane of projection. When speaking of either plane of projection, that forming the first angle is referred to, unless otherwise mentioned.
The figures in PL V"., represent the projections of the point, lines, surfaces and solids, as being contained in the first angle*
PROJECTION OP A POINT.
If we conceive any point as being contained in space, and perpendicular lines drawn from that point to both planes of projection, these perpendiculars are termed the projecting lines of that point; those points, where the perpendiculars intersect the planes of projection, are called its projections.
The projections of a point are indicated by small letters; the horizontal one by the unaccented, and the vertical one by the accented letter.
Fig. 2 is a perspective view of the projection of a point A.
A a is its projecting line to the horizontal plane, and a its horizontal projection. A a' represents the projecting line to the vertical plane, and ar the vertical projection of the point A.
If from the points of projection a and a', lines are drawn perpendicular to the ground line Q R, it is evident that they will meet that ground line in one and the same point a�. The lines thus drawn, are called the ordinates of the point A, and are termed respectively; the horizontal and the vertical ordinate ; a a� is the horizontal, and a' a� the vertical ordinate.
39
A little reflection must show the following facts :
First.�The projecting lines A a, A a', and the ordinates a a0, a' a�, are all contained in one plane perpendicular to both planes of projection, and form a rectangle.
Second.�The projecting line A a is equal to the vertical ordinate a' a0, and the projecting line A a' equal to the horizontal ordinate a a0.
Third.�If one projection of a point is given, it is known, that the point is contained in a projecting line from that point of projection.
Fourth.�If both projections of a point are given, the point is definitely located at the intersection of its projecting lines.
Fig. 2 ^ shows how this projection is carried out graphically on paper. Like letters refer to the same points in the perspective view.
J Q R K is the horizontal plane, Q R the ground line, and Q L M R the vertical plane ; a is the horizontal, and af the vertical projection ; a a� is the horizontal, and af a0 the vertical ordinate.
Ordinates are indicated by fine dotted lines.
In speaking of a point, its projections are referred to ; for instance, the point a-a' means the point A in space.
It has been remarked before, that by the revolution of the vertical plane back on the horizontal one, the relative position of the projections of a point, in regard to the ground line, is not altered; therefore, on the diagram, the ordinates, a a� and a' a� of the point a-a', form one line, perpendicular to the ground line Q R. If only oneprojection of a point is given, it is known, that its other projection must be contained somewhere in the line drawn through the given projection, perpendicular to the ground line.
PEOJECTIOlsr OF A LIKE.
Lines are projected by projecting a number of points' contained in them. Their projections are indicated by small letters, as mentioned in the projections of a point. In speaking of a line in space, its projections are referred to ; for instance, a b�a" b', means the line A B in space.    (Fig. 5.)
The projections of a line are drawn in continuous fine lines, while the ordinates are fine dotted lines.
A straight line is projected, by projecting any two points contained in it and connecting their respective projections by straight lines.
A limited straight line is projected, by projecting its end points and connecting their respective projections by straight lines.
In regard to the position of straight lines and their projections, the following facts become apparent, by examining the figures referred to in the several cases:
First.�When a line is perpendicular to either plane of projection, its projection in that plane will be a point, and in the other, a line perpendicular to the ground line and equal to its real length.
Fig. 3 represents the perspective and Fig. 3 ^ the graphical projection of a line A B, which is perpendicular to the vertical plane of projection. Its vertical projection is a point ar; its horizontal projection a h is perpendicular to the ground line, and equal to A B in length.
Fig. 4 is a perspective view, and Fig. 4 A the graphical construction of the projection of the line A B, which is perpendicular to the horizontal plane; its horizontal projection is a point a, and its vertical one a line a' V, perpendicular to the ground line and equal to the real length of A B. Second.�When a line is parallel to both planes of projection, its projections will be parallel to the ground line and equal to the length of the line, in both planes of projection.
Fig. 5 shows in perspective, and Fig. 5 ^ the graphical construction of the projection of a line A B, parallel to both planes of projection; its projections a b and a' bl are parallel to the ground line Q R and equal in length to A B
40
Third.�When a line is parallel to one plane of projection but inclined to the other, then its projection in that plane to which it is inclined, will be parallel to the ground line and shorter than the real length of the line ; but the projection in that plane to which the line is parallel, will appear inclined to the ground line, and equal in length to the line itself. Fig. 6 is a perspective view and Fig 6 ^ the graphical diagram of the line A B, parallel to the horizontal, but inclined to the vertical plane of projection.
Its vertical projection, a! &', is parallel to the ground line, and shorter than A B ; but its horizontal projection, a b, appears inclined to the ground line and equal to A B.
Fourth.�When a line is inclined to both planes of projection, its projection in both planes will be inclined to the ground line (with one exception, to be explained hereafter,) and shorter than the given line in space.
Fig. 7 is a perspective view of the projection of a line A B inclined to both planes of projection. Fig 7 A is the graphical diagram. Both projections a b and af V are inclined to the ground line and shorter than A B.
These diagrams make it plain that, when one projection of a line is given, it is known that the line in space must be contained in a plane, which is perpendicular to and intersecting in the given projection that plane of projection, in which the line is projected.                                        ,
Such a plane is termed a projecting plane of that line. For instance : let a 5, Fig. 7, be the horizontal projection of a line; then it is known that said line must be contained in a vertical projecting plane, which intersects the horizontal plane of projection in the line a b.
It is also plain that, when both projections of a line are given, the line must be common to two projecting planes which are perpendicular, one to each plane of projection, and necessarily intersect each other. The line itself must therefore be the intersection of its projecting planes.
When the projection of a line is a point, it is known, that the line must be contained in a projecting line of that point of projection. The exception referred to in the fourth case is, when a line is inclined to both planes of projection, and contained in a pro-' jecting plane which is perpendicular to both planes of projection. In such a case, its projection in both planes of projection, will be perpendicular to the ground line, but shorter than the real length of the line.
Fifth.�If two lines intersect in space, their projections will intersect in one or both planes of projection, with that exception, when both lines are contained in & plane perpendicular to both planes of projection, as mentioned in the preceding paragraph.
PROJECTION OF SURFACES.
These are determined by the projection of their boundary lines, and are therefore governed by the rules given in the preceding chapter for the projection of the point and line. |       Figs. 8 to 11 represent the projections of plane surfaces in various positions.
Whenever a surface is perpendicular to either plane of projection, but parallel to the other, its projection in the former plane will appear as a line, and in the other in its real form and size.
Fig. 8 represents the projection of a rectangular surface A B C D, in a vertical position, and parallel to the vertical plane of projection.
Fig. 8 ^ is the graphical construction of its projection, which appears as a line a b in the horizontal plane, and as the rectangle a' V c( d' in the vertical plane ; all the boundary lines appear in real length.
Fig. 9 shows the projection of a rectangle, which is perpendicular to the vertical but inclined to the horizontal plane of projection.
41
In the graphical construction of Pig. 9 A its horizontal projection appears not as large as the given surface, and its vertical projection a' &', as a line equal to the real length of the side A B.
Fig. 10 represents the projection of a regular octagon in a horizontal position ; its vertical projection a' bf d dJ appears as a straight line, while its horizontal projecticn assumes the real shape and size of the given surface.
Fig. 11 represents the projection of a triangular surface A B C in an oblique position.
Both projections will assume different formed triangles than that of A B C in space.
Lines can never appear longer in their projections than they are in reality, although they do appear shorter, as the case may be.
Angles may appear both larger and smaller in their projectionss as they may be situated ; for instance: angle BAG appears in its horizontal projection h a c, Fig. 11 ^, as an acute, while its vertical projection is an obtuse angle.
Remark.�This latter subject will be treated in detail in problems relating thereto.
In regard to the indication of surfaces by letters, it may be mentioned that the visible lines only are indicated by their letters ; for instance : Fig. 9 the side C D is hidden from view in the vertical projection, by the side A B ; this projection is therefore indicated a! b\ although the projection of C D is the same as that of A B in the vertical plane.
Remark.�The perspective views, Fig. 2 to Fig. 11, are drawn in so-called parallel perspective, which is somewhat different from real perspective. The latter method, as has been mentioned in the introduction, shows the apparent size of an object, but its real dimensions could not very conveniently be ascertained by it.
The former, or parallel perspective, allows to show three dimensions in real size in perspective, and for small objects, in which the principal outlines run parallel and at right angles, may combine the advantages of a geometrical plan, elevation and perspective view in one diagram.
Thus, all the ordinates and the distances along the ground line Q R, as also all lines which are perpendicular to either plane of projection or parallel to the ground line, show in real size in the above-mentioned perspective figures, and may be compared by a pair of compasses with the respective graphical constructions in projection.
As many men conceive the relative situation of lines more readily in a perspective view than in a geometrical plan and elevation, and this book being designed to be a self-instructor: it was deemed advisable by the author to introduce these diagrams, in order to impress upon the student's mind theN first principles of descriptive geometry more distinctly.
PEOJECTIOH   OF   SOLIDS.
Solids are projected by projecting their surfaces, edges and outlines, according to the rules already given for the projection of the point and line.
Whenever a solid is projected for the purpose ot showing its size and form only, without regard to the position it many occupy in connection with other objects, it is represented as resting with its base on the horizontal plane of projection.
If such base is a plane surface, the vertical projection of it will be a horizontal line, and coincide wiljh the base line of the drawing.
Tjhe horizontal projection is drawn at some convenient distance from the ground line, bearing in mind that both projections of one and the same point are always contained in one and the same perpendicular to the ground line.
It is customary, to notate the projections of solids by the large letters of the alphabet, using the unaccented letter for the horizontal, and the accented one for the vertical projection ; only the points and lines visible are indicatedby letters.
Figs. 12 lh to 17 & show the projections of six elementary solids, treated of in the elements of geometry.    Their corresponding perspective views are the Figs. 12 to 17.
6
42
A cube Fig. 12 being enclosed by six equal squares, shows as a square in both projections. The upper side, ABCD, being the one seen as horizontal projection, is indicated by these letters, Fig. 12 ^. The front face E F A B in the perspective being the one visible in the vertical projection, is indicated by the accented letters E7 F7 A' B7. A quadrangular parallelo-pipedon, is projected in Fig. 13 ^. Its horizontal projection shows a square, ABCD, and its vertical one, a rectangle E' F7 A' B7.
A right quadrangular pyramid is projected in Fig. 14 fa. The horizontal projection is a square equal to its base ; the vertex being shown in the centre S, which is the intersection of the diagonals A D and B C ; these latter represent the slanting edges of the pyramid in horizontal projection. The vertical projection A7 B7 S7 is an isosceles triangle whose base is equal to one side of the base, and whose altitude is equal to the axis S S7 (prospective view) of the pyramid ; the equal sides A7 S7 and B7 S7 represent the slant height.
To project a right cylinder, as shown in Fig. 15 fa, construct a circle, A B, of the diameter of the base, which will be the horizontal projection. On the ground line construct a rectangle, A7 B7 07 D7, its base equal to the diameter A B, and its altitude equal to the axis of the cylinder; this rectangle will be its vertical projection.
The vertical projection of a right cone is similar to that of the pyramid. The horizontal projection is a circle A B, Fig. 16 fa9 of the same diameter as the base o* the cone ; the centre S indicates the vertex of the same.
The projection of a sphere, Fig. 17 fa, is a circle equal to a great circle of the same, in both planes of projection.
Fig. 18 is a perspective view of a right rectangular parallelopipedon &HMJABOD, connected with a quadrant of a hollow cylinder, JKLMCDEFof equal thickness and height. The hollow cylinder is the space included between two cylindrical surfaces of different radii, generated from the same axis.
Fig. 18 fa shows their projection. The horizontal projection of the rectangular parallelopipedon is a rectangle, ABCD, equal to its base; to construct that of the hollow cylinder, draw a line D O perpendicular to the face D B, and from the centre O describe the two circular quadrants C E and D F ; connect E and F to complete the figure.
The vertical projection shows the face A B as a rectangle GP H7 A7 B7 with A B as base, and of the same height as that of the prism ; the rectangle H7 K7 B7 F7 is the vertical projection of the hollow surface of the cylinder as seen in the position indicated in the horizontal projection.
Fig. 19 is the perspective view of an octagonal newel-post5 and is here introduced to illustrate the practical application of the rules for the projection of solids.
It consists of a right octagonal prism E F Gr H, called the Base ; the shaft ABCD has the form of the frustum of a right octagonal pyramid.
The part C D E F between the base and the shaft, as also the parts above the shaft are called the turned members, and are solids of revolution ; the uppermost member is termed particularly the cap.
To construct the projections of this newel-post, as represented in Fig. 19 fa, draw an octagon E J K F of the size of its base, which is the horizontal projection of the outlines ; then draw the dotted lines from the corners to the centre O, representing the lateral edges of the shaft, and construct the smaller octagon equal to the required size of the upper base A B of the same.
These lines are indicated by dotted lines, on account of their being hidden from view in the horizontal projection by the upper-turned members.
To construct the vertical projection, square over the ordinates from the visible edges E J K F and draw a horizontal line E7 F7 at the height of the base from the ground line Gr7 H7.
The perpendiculars through the points E' J7 K7 F7 are the visible edges of the base in their vertical projection.
To facilitate the construction of the rest of the figure, draw an ordinate of the centre O through the whole height of the newel-post; space off the different heights of the shaft and turned members, and draw horizontal lines A7 B7 and C D7 through these points, from the centre line 0 O7 set
48
off each way on the horizontals the distances for the respective edges of the shaft, taken from the horizontal projection, and connect the points so found, which will give the vertical projection of the slanting edges of the shaft.
It will "be noticed, that the central faces of the "base and shaft appear in real size, while those on each side show foreshortened. The edges formed by the different mouldings of the turned members and cap, appear in vertical projection as horizontal lines; having drawn the shape of the mouldings on one side, their respective points on these horizontals are transferred over to the other side, equally distant from the centre line, and thus the whole drawing is accurately completed.
PLATE    VI.
DESOKIPTIVE     GEOMETRY-Continued.
Heretofore the projections of Points, Lines and Planes contained in the first angle only, were considered, and, although objects are projected as being contained in said first angle, it will occur frequently in the solution of problems, or the development of working drawings, that auxiliary lines and planes, which intersect or pierce the planes of projection, pass into or through the other angles.
The rules which govern their projections in such cases will now be considered,
POSITIONS  OF THE  POINT.
A point may assume nine different positions in regard to the four planes of projection. It may be situated in either one of the four diedral angles, or in either one of the four planes of projection, or in the ground line. These nine different positions are represented in a comprehensive view in Pigs. 1 and 2. Let the thick drawn lines in Pig. 1 represent a perpendicular section through the four planes of projection ; J is the ground line, which appears as a point; the four diedral angles are indicated by the arcs, as 1st, 2d, 3d and 4th, and the different situations of the point in space by the large letters from A to J, while their projections are marked by the respective small letters ; the unaccented for the horizontal, and the accented for the vertical projection.
In this view, the projecting lines appear in their real length and correspond to the respective ordinates in Pig. 2, which represent the projections of the point in the 3ame situations, with the notation used in graphical constructions.    Q J is the ground line ;
A, is situated in the first angle; its projections a-af, Fig. 2, have been considered already in the previous plate.
B5 is situated in the 2d angle ; its projections are b-bf, Pig. 2. In this case both projections fall above the ground line.
When it is taken into consideration, that in a diagram as Pig. 2, the space on the paper above the ground line, represents both the upper vertical and the back part of the horizontal plane, and further, that the unaccented letter stands for the horizontal projection of a point, it is evident that:
Whenever an unaccented letter indicates the projection of a point above the ground line, such projection must be the one on the back part of the horizontal plane of projection.
TMs may take place in three situations: 1st, when the point is situated as B in the 2nd angle; or 2nd, as C in the third angle ; or 3rd, as G- in the back part of the horizontal plane. The projections of C are the reverse to those of A ; the horizontal one, cf, is above, and the vertical one, c, below the ground line-As the lower vertical coincides with the front horizontal plane of projection, both being represented on the paper below the ground line, and the accented letter stands for the vertical projection, it becomes evident that:
Whenever an accented letter indicates the projection of a point below the ground line, such projection must be the one on the lower vertical plane of projection.
This may take place in three cases: 1st, when the point is situated as C in the 3rd angle; or 2nd, as D in the 4th, or 3rd, as H in the lower vertical plane of projection.
D is situated in the 4th angle ; both projections fall below the ground line ; it will Tt>e noticed they are reverse to the projections of the point B.
Whenever a point is contained in one of the planes of projection, the point itself will coincide with its projection in that plane, and its other projection will fall into the ground line.
E is situated in the front part of the horizontal plane; its projection in that plane is indicated by the unaccented letter e, and the vertical one, which falls in the ground line, by the accented el'.
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45
F being situated in the upper vertical plane, its projection in that plane is indicated by the accented/', and its horizontal one is /, in the ground line.
Gr is situated in the back part of the horizontal plane ; its horizontal projection indicated by the unaccented letter g, falls therefore above the ground line, and g] in the ground line, indicates its vertical projection.
H is situated in the lower vertical plane; the projection in that plane falls below the ground line, and is indicated by the accented hf; the horizontal projection h is contained in the ground line.
A point, J, contained in the ground line, coincides with both of its projections, j-f.
This notation of the point in the different positions ought to be well understood, in order to conceive readily its situation in space by its noted projections, as also to notate properly any point found in the course of a graphical construction.
POSITIONS   OF THE LINE.
When a4 line pierces or passes through a plane of projection, the point of intersection is called a trace of the line.
For instance: the line A B in Fig. 3 pierces the horizontal plane at the point A, which is called its horizontal trace, and the vertical plane at B, called the vertical trace of the line A B.
A line which pierces either one of the planes of projection but runs parallel to the other, can have but that one trace.
A line running parallel to both, cannot pierce either one, and consequently has no trace.
Any infinite line oblique to both planes of projection, pierces both the horizontal and vertical plane, and has consequently two traces.
This latter class deserves a more particular notice.
There are four different positions such a line may assume ; they are represented in Figs. 3 to 6, in perspective views.
That portion of the line, which passes through the first angle, is indicated by a full, thick line, while those portions, which are hidden from view by the front part of the horizontal and the upper part of the vertical plane of projection, are drawn in thick interrupted lines.
Fig. 3 shows a line A B, passing through the first angle into the second and fourth one, piercing the horizontal plane at A, and the vertical one at B.
Fig. 4 represents a line A B, as passing from the first angle through the second into the third, piercing the upper vertical plane at B, and the back part of the horizontal one at c.
Fig. 5 shows the line A B, as coming from the second angle, passing through the third into the fourth, piercing the back part of the horizontal plane at B, and the lower vertical one at c\
As no part of this line passes through the first angle, the whole of it is hidden, and indicated by an interrupted line.
Fig. 6 represents the line A B, as coming from the first angle and passing through the fourth into the third, piercing the front horizontal plane at B and the lower vertical one at c\
positions of planes.
*The positions of planes in space in relation to the planes of projection, are indicated by their intersections with the latter.
The lines of intersection of any plane with the planes of projection, are called its traces ; they will evidently meet at one and the same point in the ground line, except those parallel to it, and are notated by two large letters of the latter part of the alphabet.
The trace in the horizontal plane of projection is termed the horizontal trace, and that in the vertical one, the vertical trace. For instance: P Q, Fig. 7, is the horizontal and Q P' the vertical trace of the plane P Q P'; it will be noticed that in this notation, the middle letter is used for the point common to both traces in the ground line; the other in connection with it unaccented for the horizontal, and accented for the vertical trace.
46
In speaking of a plane, its traces are referred to.
The different positions a plane may assume, are represented in Figs. 7 to 9 in perspective, and their corresponding graphical notation in Fig. 7, A? to Fig. 9, /&.
The visible traces in the front part of the horizontal and the upper part of the vertical plane of projection, are indicated by thick continuous lines, while those in the back part of the horizontal and the lower vertical plane, being invisible, are indicated by thick interrupted lines.
An inspection of these diagrams, will convey the truth of the following rules in regard to the traces of planes in the different positions.
The traces of a plane perpendicular to both planes of projection, are perpendicular to the ground line.
Plane P Q P7, Fig. 7 ^, has both traces, P Q and Q P7, perpendicular to the ground line.
A plane perpendicular to one of the planes of projection only, and inclined to the other, has its trace perpendicular to the ground line in that plane of projection, to which it is inclined; the other trace is inclined to the ground line.
tPlane R S R7, Fig. 7 /&, is perpendicular to the vertical plane of projection and inclined to the horizontal; its horizontal trace, R S, is perpendicular, and its vertical one, S R7, inclined to the ground line.
TUT7represents a plane perpendicular to the horizontal plane of projection; its vertical trace, U T7, is therefore perpendicular, and the horizontal one, T U, inclined to the ground line.
Both traces of an oblique plane are inclined to the ground line.
A plane is said to be oblique, when it is inclined to both planes of projection.
Fig. 8 ^ represents two such oblique planes. *
The traces of P Q P7 form, one an acute, and the other an obtuse angle with the ground line on the same side ; while the angles formed by the traces of plane R S R7 with the ground line, are both acute on one side and obtuse on the other.
A plane parallel to the ground line has its traces parallel to the same.
There are four different positions of plants of this class; they are represented in Figs. 9 and 9 &.
In order to convey a clear idea of these positions, an auxiliary plane perpendicular to both planes of projection is assumed in the perspective view, in which their traces are indicated by dotted lines.
A plane parallel to either one of the planes of projection must be perpendicular to the other, and can intersect only that one plane to which it is perpendicular.
R Q Q" is a plane parallel to the vertical plane of projection, and consequently perpendicular to the horizontal one ; its trace R Q is parallel to the ground line.
R' Q' Q77 represents a plane parallel to the horizontal plane of projection ; its trace is R7 Q7.
A plane parallel to the ground line, and inclined to both planes of projection, has two traces, both running parallel to the ground line.
Each trace is notated separately by either one or two letters. R Q-R7 Q7 is such a plane; R Q is its horizontal and R7 Q7 its vertical trace.
The fourth position is that of a plane passing through the ground line and either of the dietral angles. T U U77 is such a plane; it has only one trace, T U, which coincides with the ground line, and its position is only then definite, when another trace in some auxiliary plane is given, as the trace U U" in the figure.
PROBLEMS  IN  ORTHOGRAPHIC  PROJECTION.
After these general definitions and statements in regard to the projection of a point, line and plane, the necessary problems are introduced, which find practical application in the development of working drawings.
In every problem in descriptive geometry, certain parts given and required, are distinctly stated, the same as in the problems in geometrical constructions.
47
Auxiliary parts are such, which, must be introduced, to find or develop the parts required In order to distinguish the different lines readily from each other, the following notations are
observed, with some of which the reader is already familiar:
All visible given or required traces of planes are drawn in thick continuous lines, �� All  visible  projections   of given   or   required  lines   are  drawn  in  continuous fine
lines,--------------------------
Traces of all auxiliary planes, as also those of given or required planes which are hidden
from view by the planes of projection or other objects,  are drawn in interrupted thick lines,
Projections of all auxiliary lines, as also those of invisible given or required ones, are drawn in interrupted fine lines,--------------------------------------------�------
All ordinates, whether given, required, or auxiliary, are drawn in fine dotted lines,...................�
The same letter is used for the same point in all its different positions, which are distinguished by the number of accents.
Developed lines or surfaces are drawn in rather thick continuous lines, and the various points of them are indicated by the same Capital Letters, which are used small in either projection.
For instance: in Fig. 9, Plate IX, h is the horizontal and hf the vertical projection of a point; h" is the position of the same point in one and h,n in another auxiliary plane, while H indicates: that same point in the developed figure.
The graphical constructions of the problems on Plate VI are indicated by the letter ^, and are accompanied by corresponding views in parallel perspective, in which all the ordinates and all distances along the ground line are the same as in the graphical constructions ; this will enable the reader to comprehend their meaning more readily. In the problems on Plates VII, VIII and IX those perspective views are dispensed with.
PROBLEM   lo
The traces of a line being given^ to find its projections.
Each of the traces of the line is also a point of its projection in that plane of projection to which the trace belongs ; the other projection of the trace is contained in the ground line, where its ordinate intersects. Lines drawn through either one of the traces and the respective projection of the other thus found, are the projections of the line.
It is evident, that in case the traces are contained in the same perpendicular to the ground line, the projections of the line will coincide with the ordinates of its traces.
The different positions of lines passing obliquely through the diedral angles, are shown in diagrams, Figs. 3 ^ to 6 ^.
Let a and b', Fig. 3 ^, be the given traces ; a being the horizontal trace, is also the horizontal projection of that end of the line ; the vertical projection of the same point is contained in af, the intersection of its horizontal ordinate with the ground line; b' is the vertical trace, and its horizontal projection is b ; thus a b and af V are the projections of the line whose traces are a and bf.
Let c and br, Fig. 4 ^, be the given traces; c is evidently a trace in the back part of the horizontal plane; its vertical projection is c'; V is the vertical trace, and b its horizontal projection, and c! ~br therefore the vertical projection of the given line and c b the horizontal one.    As this line evidently passes through the second angle, the extension of these projections, as & a and V ar are the projections of that line in the first angle.
!Let b and cf, Fig. 5 /&, be the given traces, b is contained in the back part of the horizontal and c' in the lower part of the vertical plane ; this line passes evidently through the third angle, and no part of it will be visible in the first one ; its projections, b c and V d, are indicated by dotted lines.
Let b and d, Fig. 6 ^ he the traces of a line, b is the horizontal trace, and & evidently that in the lower part of the vertical plane ; c and V are the other projections of these traces ; consequently b c is the horizontal and bf cf the vertical projection of the line.
These projections extended, give in a b and a' br the projections of the line in the first angle.
48
PROBLEM   II.
The traces of two intersecting planes being given, to find the projections of their intersection.
If the definition of a plane is considered, it is evident, that the intersection of two planes must be a straight line, which passes through the intersection of their traces; the solution of this problem therefore is identical with that of problem I.
To illustrate the different positions of intersecting planes, four cases are represented in Figs. 10 ^ to 13 ^, all contained in the first angle.
Fig. 10 /& shows the intersection of two oblique planes, PQP; and P S P'.
P being the intersection of the horizontal traces, and P' that of the vertical ones, these points are also the respective traces of that intersecting line whose projections are required ; P p and P' p' are the projections.
Fig. 11 ^ represents the intersection of an oblique plane P S P', with PQP' which is perpendicular to the horizontal, but inclined to the vertical plane of projection.
A little reflection must convince, that in such a case, where one of the intersecting planes is perpendicular to either plane of projection, the required projection of the intersecting line must coincide with the trace of the perpendicular plane in that plane of projection, to which it is perpendicular.
In this case therefore, the horizontal projection of the intersecting line coincides with the horizontal trace P Q, while P' p' is its vertical one.
For the same reason, in Fig. 12 /^ which represents the intersection of an oblique plane P S P;9 with a plane P Q P' perpendicular to the vertical plane of projection, the vertical projection of the interseeting line coincides with the vertical trace Q P', while P'jp' is the horizontal projection required.
Fig. 13 ^ shows the intersection of two planes, PQP7 and R S E/, both of which are perpendicular to the horizontal plane of projection.
It is evident, that the intersecting line must also be perpendicular to the horizontal plane of projection, or, in other words, a vertical line ; its horizontal projection n coincides with the' intersection of the horizontal traces, and its vertical projection is a vertical line nf mr.
The case is similar when both intersecting planes are perpendicular to the vertical plane of projection.
PROBLEM   III.
The projections of a line being given, to find the traces.
*
It was shown in Problem I, that the trace of a line is also a point of its projection, and the intersection of its ordinate with the ground line is a point of projection of the same line in the other plane of projection ; therefore, when the projections of a line are given, find the intersection of one of the projections with the ground line, and where the ordinate from that point intersects the other projection of the line, must be its trace.
Let a b and af &', Fig. 14 ^\, be the given projections of a line ; to find its horizontal trace, extend a! bf intersecting the ground line at d; from d draw the horizontal ordinate, intersecting a b extended in c, which will be the horizontal trace required ; the vertical trace df is found in the salme manner-Figs. 3 ^ to 6 ^ represent lines projected in different positions, passing through the diedral angles ; in Problem I, the traces were supposed as being given and the projections required ; now let the reader suppose, the projections are given and the traces required, and work out the problem.
PROBLEM  IV.
To construct the traces of a plane, which passes through three given points.
It has been shown in elementary geometry, that any two intersecting lines, also any three points, are always contained in one plane.
49
It is evident, that the trace of any line contained in a plane, must also be a point in the trace of that plane.
To solve the above problem, find the traces of two lines passing through the three given points, which will be two points of the required traces in each plane of projection.
Let a-a\ b-b\ c-c\ Fig. 15 A be the three given points ; employ the solution of Problem III, and/ will be the horizontal and e' the vertical trace of a line passing through the points a-a! and b-bf; d is the horizontal trace of the line passing through b-b' and c-c', and a line drawn through /and d is evidently the horizontal trace P Q of the plane required.
This trace P Q intersects the ground line at Q, and as both traces of a plane intersect the ground line in one and the same point, a line drawn through Q and e' must be the vertical trace of the plane required.
This saves the time to find the vertical trace of the line passing through b-b' and c-c'.
PLATE   VII.
PROBLEMS   IN   ORTHOGRAPHIC   PROJECTION � Continued.
PROBLEM  IV.
To pass a plane through three given points.
To pass a plane through three given points means to construct the traces of the plane containing the three points given. This problem is identical with Problem IV, represented in Pig. 15 t&, PL VI.
Pig. 1 represents another position of the three given points, a-a\ b-b\ c-c\ P is the horizontal trace of the line a b�a' b\ and / that of b e�b[ c'\ P/is therefore the horizontral trace of the plane required, intersecting the ground line at Q.
P' is the vertical trace of a b�a' &', and gf that of a c�af cr; the line through Py is therefore the required vertical trace ot the plane. If the drawing is made accurate, both traces will meet the ground line at the same point Q.
PROBLEM  V.
A line and a point outside of it, being given, to pass a plane through the saws.
This problem is only a modification of the preceding one ; it may be solved in the same manner, by projecting a second line through the given point and intersecting the given line, and find their traces.
Another solution is presented in Fig. 2, based on the fact, that two parallel lines are contained in a plane ; the projections of lines parallel in space being parallel to each other, construct such a line through the given point, and their traces will be points in the traces of the required plane.
Let a b and a' b' be the projections of the given line, and n-n' those of the given point; draw n d parallel to a b, and nf df parallel to a' b\ which will be the projections of the auxiliary line parallel to a b�af V through the point n-n'; b being the horizontal trace of the given line, and e that of the auxiliary, the line P Q drawn through these points will be the horizontal trace of the plane; the vertical trace Q P' is found in the same manner. If the construction is accurate, both traces will meet at the same point Q, in the ground line.
Fig, 3 is a solution of the same problem, with the given line and point in another position,
PROBLEM VI.
*
Two parallel lines being given, to construct the plane passing through them.
The solution of this problem is simply to find the traces of the parallel lines, which are points in the traces of the required plane.
The given lines may assume different positions ; they may be oblique to both planes of projection, or parallel to the ground line, or parallel to one of the planes of projection only and inclined to the other.
Figs. 2 and 3 illustrate the cases in which the given lines are oblique to both planes of projection.. If we assume the line a b�a' bf and the auxiliary one drawn parallel to it through the point n-n' as the parallel lines given, the explanation of the figures in the last problem will be sufficient.
In Fig. 4 a b�a! V and c d�d d' are two lines parallel to the ground line ; in this case it is necessary to construct an auxiliary plane R S R', and find the traces of the given lines in this plane.
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51
For convenience, this auxiliary plane is assumed to be perpendicular to the horizontal plane of projection ; b and d are the horizontal projections of the traces, and, as their vertical projections are found in the intersection of the ordinates with the vertical projections of the given lines, b* df are those vertical projections.
A little reflection will convince that the horizontal trace R S of the auxiliary plane, is also the horizontal projection of the intersecting line of that plane with the plane passing through the given lines, and that a line V dr is the vertical projection of the same intersecting line, because the points b' df are contained in the given lines and the two planes. Then it follows, that R/, which is the intersection of the intersecting line with the vertical trace of the auxiliary plane, is also a point in the vertical trace of the required plane, and likewise, R must be a point in the horizontal trace of the required plane ; the traces must run parallel to the ground line; hence P R and P7 R7 are the traces of the plane passing through the given parallel lines.
Fig. 5 illustrates the case of two parallel lines a b�aJ V and c d�& dr parallel to the vertical plane of projection, piercing the horizontal one only, as in e and// the trace P Q passing through these two points, is evidently the horizontal trace of the required plane, intersecting the ground line at Q. The vertical trace must run parallel to the vertical projection of the given lines, hence a line Q P7 drawn through Q parallel to a' V or & df, is the vertical trace required.
PROBLEM VII.
One trace of a plane, and one point of the other trace being given; to construct the other trace, when both do not meet within the limits of the diagram.
If two parallel planes are intersected by a third one in any position, their lines of intersection must be parallel; then, if an auxiliary plane is constructed through the given point of the other required trace intersecting the given plane, and also another auxiliary plane parallel to the first, the trace of the intersecting line of the latter plane must be a second point in the required trace.
Let P7 R7, Fig. 6 be the vertical trace, and R the given point in the horizontal trace required. For convenience, construct an auxiliary plane, R S R7, perpendicular to both planes of projection through the given point R ; also let T U T7 be a second auxiliary plane, also perpendicular to both planes of projection. Conceive the plane R S R7 revolved around its vertical trace S R7 as an axis, then R7 r" will be its line of intersection with the given plane, as it appears when thrown back into the vertical plane of projection. Construct from T7 a line T71" parallel to R7 r", which will evidently be the line of intersection of the second auxiliary plane with the given plane, also in a revolved position similar to the first; then draw the arc t" T intersecting the trace at T, which point will be the second point in the required trace T R.
PROBLEM   VIII.
Two lines being given, which are neither parallel nor intersect each other; to pass a plane through
one of the lines parallel to the other.
If an auxiliary line is constructed intersecting the given line, through which the plane is to pass, and parallel to the other given line to which the plane is to be parallel, then the plane passing through the so intersecting lines, must evidently fulfill the conditions of the problem.
Let a b�a' V, Fig. 7 be the given line to which the plane is to be parallel, and c d�c' d' the given line through which the required plane is to pass. Then construct a line e f�e' ff parallel to a b�aJ V and intersecting c d�d d' at the point n-nJ, then P Q P7 is the plane required.
PROBLEM   IX.
To pass a plane through a given point, parallel to another given plane.
The trace of an auxiliary line through the given point and parallel to the given plane, must evidently be a point in the trace of the required plane.
52
Let P Q P', Fig. 8, be the given plane, and a-a! the given point; for convenience, construct the auxiliary line a b-a/ 5', through the point parallel to the vertical plane of projection ; its horizontal projection will then Tbe parallel to the ground line, and its vertical projection parallel to the vertical trace of the given plane ; it will pierce the horizontal plane of projection at 5, and as the traces of parallel planes are parallel, RSE' are the traces of the required plane.
problem x.
To find the projections of the point, where a line pierces an oblique plane.
The required point is contained in the intersection of the projecting plane of either projection of the given line with the given plane ; the intersections of the projections of that intersecting line and those of the line itself, are evidently the projections of the required point.
Let P Q P', Pig. 9, be the given plane, and a b-a' bf the given line ; for convenience, construct the projecting plane a d df, perpendicular to the horizontal plane of projection ; then & d' is the vertical projection of its intersecting line with the given plane; this vertical projection intersects that of the given line in nr, which is then the vertical projection of the piercing point; n is the horizontal projection of the same point.
PROBLEM  XI.
To construct a line through a given point, perpendicular to a given plane / also to find the point,
where the line pierces the same.
Whenever a line is perpendicular to a plane, its projections are perpendicular to the respective traces.
To construct a line according to the given problem, draw perpendiculars to the traces through the projections of the given point.   To find the piercing point, proceed as in last problem.
In Pig. 10 a position is chosen, to illustrate the last problem-more fully.
Let P Q P' be the given plane, and a-a' the given point; then the line b d perpendicular to P Q through a, is the horizontal, and bf d perpendicular to Q P' through a', the vertical projection of the line required. To find the point of intersection of said line with the given plane, construct the projecting plane b e e', perpendicular to the horizontal plane of projection ; its vertical trace e e' in this position, will fall into the lower part of the vertical plane of projection, to intersect the vertical trace of the given plane in e'; d is the intersection of their horizontal traces, and d' in the ground line, the vertical projection of the same point; then e' and d' are evidently two points of the vertical projection of the line of intersection of the auxiliary projecting plane with the given plane. Where this vertical projection intersects that of the given line, as in cr, is the vertical projection of the point, where the given line pierces the given plane, and c is its horizontal projection.
PROBLEM  XII.
To construct a plane perpendicular to a given line, through a given point contained in that line.
The traces of the required plane are perpendicular to the projections of the given line, and if, therefore, one point of either trace can be found, both traces are easily constructed; to find that p0int, assume an auxiliary plane perpendicular to the horizontal plane of projection and also perpendicular to the horizontal projection of the given line, through the given point; the horizontal traces of this auxiliary and the required plane, are evidently parallel; their line of intersection must be a horizontal line passing through the given point, and where this horizontal line pierces the vertical plane of projection, must be one point of the vertical trace of the required plane.
Let a b-af bf Fig. 11 be the given line, and n-n' the given point in it; n mm'is a, vertical auxiliary plane, perpendicular to the horizontal projection a b of the given line, and n' m' is the vertical projection of the horizontal line of intersection of the two planes, which pierces the vertical plane of projection at m'. A line Q P' perpendicular to a' bf through m\ is the vertical trace, and a line Q P from Q parallel to m n or perpendicular to a b, the horizontal trace of the required plane.
53
PEOBLEM XIII.
To construct a plane perpendicular to a given line, and passing through a given point not
contained in the given line-
The solution of this problem is identical with the preceding one.
Let a b�a' b', Fig. 12, be the given line and n-n' the given point; then mr will be a point in the vertical trace of the required plane P Q P'.
PROBLEM XIV.
To construct a plane perpendicular to another given plane, through a given line not contained
in the given plane.
By the preceding problems, the fact is-made evident that, when the projections of a line are perpendicular to the traces of a plane, the line itself is perpendicular to that plane ; and farther, when a line is perpendicular to a plane, any other plane passing through that line, must also be perpendicular to that same plane; therefore a plane passing through two intersecting lines, one of which is perpendicular to a given plane, must itself be perpendicular to that given plane.
Let P Q P', Fig. 13, be the given plane, and a b�af bf the given line ; construct an auxiliary line c d�cf d' perpendicular to the given plane and intersecting the given line at n-n\ then the plane RSR' passing through the two intersecting lines, is the plane required.
DEVELOPMENT    OF    LINES     AND     ANGLES.
DEFINITIONS.
All straight lines and plane surfaces are developable, when their projections are given. The Plane containing them may be revolved into either plane of projection, and thus show the figure in its real form and size.   This process is termed Development by Transposition.
All curve lines, also most of the surfaces of single curvature, are developable.
Surfaces of single curvature are those, through any point of which a straight line may be drawn in one direction only ; such as the surface of a cylinder, a cone, a screw, &c. Of these, the surface of a cylinder or a cone is developable, that is, it may be rolled out on a plane, or cut out of some pliable material, as paper for instance, in such a shape, that when bent in the proper curve, it will cover the surface of the solid accurately without ruffling the material.
The process of developing curve lines and curved surfaces, is termed Development by Transformation, and is further explained under PI. IX.
The inclined surface of the screw thread, although of single curvature, belongs to the class of so-called warped or twisted surfaces, which are non-developable.
I Surfaces of double curvature, as that of a globe for instance, are also non-developable.
The next following problems have reference to the development of straight lines, the angles formed by them, and plane surfaces, by transposition.
problem xv.
To find the real length of a line in any position, its projections being given.
This problem finds a solution in the definition of the projection of a line; the development of either of the projecting planes of the line will give its real length.
Let a b�a' V, Fig. 14, be the given projections of a line ; if the projecting plane of the horizontal projection a b, is conceived to revolve back into the horizontal plane of projection,
64
retaining that projection as the fixed axis to swing on, it will evidently assume the shape of the trapezoid a b A B ; a A and b B are perpendicular to a b and equal to the respective vertical ordinates of a' and V ; A B is the real length of the line.
The same result would be obtained, if perpendiculars to the vertical projection a' br should be erected from a' and &', equal in length to the respective horizontal ordinates of a and 5.
PROBLEM XVI.
To find the real size of the angle formed by two lines intersecting in space, their projections
being given.
As the lines intersect, they are contained in a plane ; if either trace of this plane is found, and the plane with the lines revolved back into that plane of projection to which that trace belongs, retaining the same as the fixed axis to revolve on, it must show the real length of the lines and the angle formed by them.
As the traces of both lines are also points in the trace of the plane containing them, the solution may be expressed in the following rule : Find the traces of the lines in one of the planes of projection, also their real length between those traces and the point of their intersection ; describe arcs with their real lengths as radii from their respective traces as centers, and connect the traces with the intersection of the arcs by straight lines ; the angle formed by them will be the required angle.
Let a b�af V and c b�cr b\ Fig. 15, be the given lines ; their horizontal traces are d and e respectively; from b erect perpendiculars b bf/ and b b'n to the horizontal projections, and make both equal to the vertical ordinate of b\ then d bn and e b'" will be the real lengths of the respective lines ; describe the arcs as indicated from d and e as centers, cutting in B, and d B e will be the angle required.
Fig. 16 shows another solution of this problem upon the same principle as the preceding one, with that difference, that the plane containing the lines is really revolved into the horizontal plane of projection..
Let a b�a' bf and c b�& V be the projections of the given lines ; d and e are their traces found by the usual method, consequently line d e is the horizontal trace of the plane in which the given lines are contained. If this plane is revolved back into the horizontal plane of projection, the point b-bf must move in a vertical plane, whose horizontal trace b f is perpendicular to the trace d e (which is the fixed axis of the revolving plane,) and the perpendicular distance between said point b-b' in space and f in the axis, remains the same after being revolved as before. If this distance in its real length is laid off from,/ onfb extended, it will evidently give the point b-bf in position after being revolved. Asfb is the horizontal projection of this distance, and the vertical ordinate of V also known, it may be easily found by the usual method. (Problem XV.) From b erect a perpendicular to b f lay off the vertical ordinate to 5" and f b" will be the real length of said distance. The arc b" B described from /asa center, will intersect/ 5 extended in B, which is then the point b-V in its revolved position. Connect B with the horizontal traces d and e of the given lines, and angle d B e is the one required.
It is evident, d B and e B will be the real lengths of the given lines from their intersection to their respective horizontal traces.
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PLATE   VIII.
PROBLEMS   IN   ORTHOGRAPHIC   PRO JECTION�Continu bd. DEVELOPMENT  OF  ANGLES  FORMED   BY  INTERSECTING  PLANES.
PROBLEM XVII.
To find the real size of the angle formed by both traces of an oblique plane, the traces being given*
What is required in this problem, is that part of any oblique plane in real size, which is contained between its two traces, or in other words, between both planes of projection. This is found by revolving the oblique plane into one of the planes of projection, around one of the traces as an axis ; in doing this, any point of the other trace will move in a plane perpendicular to the trace used as the fixed axis and will maintain the same distance from the intersection of both traces in the groundline, after the revolution, as before.
Let PQT' Fig 1 be the given traces, and the plane be revolved around its horizontal trace as an axis ; through any point P' of the vertical trace, construct the auxiliary plane P" S P' perpendicular to P Q. The point I3/, after the revolution into the horizontal plane of projection, must be contained in the trace P" S extended ; as its distance from Q remains the same, describe the arc p/ p// from Q as a center, and Q P' as a radius, and its intersection P" with the trace, is the position of P' after the revolution. A line drawn through Q P" gives the trace after being revolved, and P Q P" is the angle and real size of the plane required.
A little reflection will convince, that the angles formed by the traces of all planes perpendicular to both or either one of the planes of projection, are right angles.
DEFINITION.
If two planes intersect in space, and a line is drawn in each plane from any one point of the intersecting line perpendicular to it, the angle formed by the two lines, is called the diedral angle of the two planes.
A third plane, containing the two perpendicular lines thus drawn, would evidently intersect both of the intersecting planes, and be perpendicular to each of them. Therefore, the idea of the diedral angle between two intersecting planes, may be defined as being that part of the Surface of a third plane intersecting both perpendicular, which is inclosed between them.
The diedral angle is always referred to, when speaking of the angle formed by two intersecting planes
PROBLEM XVIII.
To find the real size of the angle formed by any oblique plane with either plane of projection.
As the trace of a plane is the line of its intersection with one of the planes of projection, any auxiliary plane perpendicular to both, will intersect them in lines perpendicular to such trace, and indicate the diedral angle, according to the definition given.
Hence this problem is solved, by constructing an auxiliary plane perpendicular to that plane of projection and its trace of the oblique plane, for which the angle is required, and then revolving the intersected part of the auxiliary plane back into the other plane of projection.
Let P Q P', Fig 2, be the given traces of an oblique plane. To find the angle formed with the horizontal plane of projection, construct the auxiliary plane PSP' perpendicular to both ; the horizontal trace P S of the auxiliary plane will be perpendicular to the horizontal trace P Q of the oblique plane, and the vertical trace S P' perpendicular to the ground line.    If this plane is
56
revolved about its vertical trace as the fixed axis, back into the vertical plane of projection, its intersected part will appear as P77 S P7; P77 S being equal to P S. In this revolved position of the auxiliary plane P77 S represents its line of intersection with the horizontal plane of projection, and P77 P7 that with the oblique plane, both being evidently perpendicular to the trace P Q in space, and the angle S P77 P7 formed by them, is the angle required.
To find the angle formed by the oblique plane with the vertical plane of projection, construct the auxiliary plane TEf perpendicular to the vertical trace Q P7; RTT/; is then its intersected part, revolved back into the horizontal plane of rejection, and > R T" T the angle formed by the oblique plane with the vertical plane of projection.
Fig. 3 represents an oblique plane P Q P7 in a different position.
To find the angle formed with the horizontal plane of projection, construct the auxiliary plane P S perpendicular to the horizontal trace. As this plane does not intersect the given vertical trace in the upper part of the vertical plane of projection, a second auxiliary plane S R P7, perpendicular to the horizontal plane of projection and parallel to the horizontal trace of the given plane, is constructed. If the intersected part of the first auxiliary plane is revolved about its horizontal trace P S, into the horizontal plane of projection, it will appear as S P S;; S S7 being equal to R P7, and > S P S' is evidently the required angle.
To find the angle formed with the vertical plane of projection, the horizontal trace is extended into the back part of the horizontal plane of projection, as Q P77. The auxiliary plane P77 R R7 is drawn perpendicular to the given vertical trace Q P7, being thus located in the second diedral angle. R R77 P" is this plane, after being revolved back into the horizontal plane of projection. > R R77 P" is evidently the angle formed by the oblique plane with the vertical plane of projection in the second diedral angle, and>RR"if is the angle formed by the same planes in the first diedraL angle.
Fig. 4 represents the same oblique plane ; its traces P Q P' form the same angles with the ground line as in Fig. 3
To find the angle formed by the oblique plane with the vertical plane of projection, the same construction is employed, as that in Fig. 3 to find the angle of inclination with the horizontal plane of projection. P7 S7 S77 is the auxiliary plane after being revolved into the vertical plane of projection, and S77 P71 the required angle, which is equal to R R" t found by the construction employed in Fig. 3.
PEOBLEM XIX.
To find the real size of the angle formed by an inclined or oblique plane, with any vertical plane not coinciding with the vertical plane of projection.
The solution of this problem is similar |o that of the preceding one, withtthis difference, that the line of intersection of the two given planes must first be found, and then an auxiliary plane constructed perpendicular to it, to find the diedral angle ; while in the previous problem, it will be observed, this line of intersection (it being the trace of the oblique plane) was given.
Fig. 5 represents a plane Q P�Q7 P7 inclined to both planes of projection and parallel to the ground line, which is intersected by a vertical plane P R P7, making an angle with the ground line.
To find the diedral angle formed by the two planes, revolve the intersected portion of the vertical plane about its horizontal trace P R back into the horizontal plane of projection, as P R P77; then P P77 is evidently the line of intersection of both planes in its real length. Construct the auxiliary plane perpendicular to this line of intersection ; pass its horizontal trace R Q, which of course is perpendicnlar to P R, for convenience through R or the point where the given vertical plane PEP' intersects the ground line. The line R S, drawn from R perpendicular to P P77, is evidently the line in which the auxiliary plane would intersect the vertical plane P R P7 in space ; make R S7 equal to R S (as indicated by the arc) and Q R S7 will be the intersected portion of the auxiliary plane, after being revolved about its horizontal trace Q R back into the horizontal plane of projection, and R S7 Q is the required angle.
57
Pig. 6 shows an oblique plane P Q P7, intersected by a vertical plane PEP; perpendicular to the ground line; revolve the intersected portion of the latter about its horizontal trace f R, by carrying the intersection P' of the traces round to the ground line in P" ; PEF' will be this plane after its revolution, and P P77 their line of intersection. As the vertical plane is also perpendicular to the ground line, the horizontal trace of an auxiliary plane perpendicular to it, must fall into the ground line or run parallel to it; for convenience, pass again this trace through the point of intersection R of the vertical plane with the ground line, as R Q or the ground line. R S being again perpendicular to the intersection P P77 and carried over to S7 as indicated by the arc S S', Q R S7 is the intersected portion of the auxiliary plane after its revolution about its horizontal trace R Q back into the horizontal plane of projection, and > R S7 Q is the angle required.
Pig. 7 represents the intersection of an oblique plane P Q P7, with a vertical plane P R P7 making an angle with the ground line. After reading the description of the two preceding figures, this will be easily understood ; P R P77 is the vertical plane developed, R S perpendicular to the intersection P P7/ and R T perpendicular to the trace P R ; T R S7 is the auxiliary plane developed, and > R S7 T the required angle.
Remark.�The angles developed in this, as also those in the last problem with the vertical plane of projection, are technically called the spring-bevels, and are applied to the wreath pieces of stair-rails, &c.    Their theory ought to be well understood by the reader.
PROBLEM  XX.
To find the real size of the diedral angle formed by two oblique planes in any position.
The solution of this problem is similar to' the preceding ones. Construct the line of intersection of both planes, also from any one point in it, perpendiculars to the same, one in each plane, and the angle formed by the two perpendiculars will be the diedral angle wanted.
Let PQP' and PRP' Pig. 8 be the given planes; then P p�Yr p! will be the line of intersection (see Problem II.), and from any point n-nr construct the perpendiculars. It is not necessary to find the projection of these perpendiculars, if their traces and real lengths are found. In order to do this, revolve the oblique planes with the perpendiculars contained in them, about those of their traces, which the perpendiculars will intersect. In this case the planes are revolved about their horizontal traces. While revolving the plane P Q P7, the point n-nr describes an arc in a vertical plane, whose horizontal trace n r is perpendicular to the given horizontal trace P Q ; the real length of the line perpendicular from n-nf to that trace, may be found ; n r is its horizontal projection, and the vertical ordinate of n' to the ground line is also known ; n r being perpendicular to P Q, make c r equal to the vertical ordinate of n\ and n e is the real length. Lay off r n,,r equal to n c, and nnr is evidently the point n-nf after being revolved into the horizontal plane of projection; likewise, the line nr" P is that part of the line of intersection of both given planes, shown as P n�p' n! in its projections, and a line drawn perpendicular to the same in the given plane PQP', would also be perpendicular to P n'n in the revolved position ; therefore, draw n,n a perpendicular to P n"' intersecting P Q in a, which point is evidently the horizontal trace of the perpendicular to be found, and n'n a its real length.
The trace b and real length n/; b of the perpendicular in the other given plane P R P', is found in the same manner, viz: draw n t n" from n perpendicular to P R, make t d equal to the vertical ordinate of n\ and tn" equal to dn. If the drawing is accurately made, n" P and ^77/P will be equal, as both lines represent the same line n T?�p' n! in its revolved position. Then draw n" b perpendicular to n;/ P ; b will be the trace, and nff b the real length of the other perpendicular. The traces and real lengths of the perpendiculars being found, describe arcs from the respective traces a and b as centres and the real lengths as radii, cutting in N, and a N b will be the required diedral angle.
8
58
PROBLEM   XXI.
The angles formed by any oblique plane with both planes of projection being given, to find
its traces.
The diedral planes indicating the angles formed by any oblique plane with either plane of projection, are right-angled triangles (see Problems XVII. and XVIII.)? and a perpendicular from the right angle to the hypothenuse, is evidently the perpendicular distance of the oblique plane from that point of the ground line, in which the diedral plane intersects the same. Therefore, when both diedral planes are constructed through one and the same point of the ground line> they will have this perpendicular distance in common ; consequently* the perpendiculars from the right angle to the hypothenuse will be alike in both, and the traces of the hypothenuses will be points in the required traces of the oblique plane.
Let > a and > b, Fig. 9, be the given angles. Construct both diedral planes through any point R of the ground line; describe a semicircle v u from R as centre, with any convenient radius ; draw a line as S' Pr, making the angle a with the ground line and tangent to the semicircle at u ; then S' R P' will be the diedral plane revolved about its vertical trace R P'. Also construct in like manner the diedral plane P R T in its revolved position, containing the given angle b. The vertical trace P' of the hypothenuse S' P' is evidently a point in the required vertical trace, and the trace P of the hypothenuse P T, a point in the required horizontal trace. If the diedral plane S' R P' revolves back into its real position, the point S' will describe the arc S' S contained in the horizontal plane of projection, and the required horizontal trace must obviously be a tangent to this arc. Therefore, draw a line P S from P tangent to this arc, which will be the required horizontal trace.    S R is the horizontal trace of the diedral plane P' R S' in real position.
Likewise the point T will describe the arc T T/ in its revolution, and a line P' T' drawn through P' tangent to this arc, will be the required vertical trace. When the construction is accurate, both traces will meet in the same point Q of the ground line. A little reflection will convince, that whenever the sum of both angles given is equal to 90�, the plane will be parallel to the ground line, and whenever that sum is equal to 180�, the plane will be perpendicular to it. In all cases, when the sum of both angles given is more than 90� and less than 180�, the position of the plane will be an oblique one.
The radius R v with which the semicircle is described, is evidently the perpendicular distance of the plane PQP' from the point R in the ground line ; if both are given in conjunction with the angles, the traces will be definitely located. Whenever it is required to pass either trace through a givei?. point, both angles being given, find the direction of the traces first without regard to the giyen point, and then draw the required traces parallel to the directions found, and passing through the given point.
PROBLEM   XXII.
The traces of a plane in any position being given, also the horizontal projection of a figure contained in it; to find the angle which the plane makes with the horizontal plane of projection, and to develop the plane with the figure in it, in real size.
The first part required in this problem, is found as in Problem XVIII and the development of the plane and figure in it, may be done as in Problem XVII. Another method, preferable for the purposes of this work, is the following : Construct the auxiliary diedral plane, to find the angle of inclination of the given plane with the horizontal plane of projection, and, instead of revolving, the auxiliary plane about its vertical trace, as was done in Problem XVIII, revolve the same about its horizontal trace into the horizontal plane of projection; then revolve the given plane with the figure in it about its line of intersection with the auxiliary plane, also into the horizontal plane of projection.
Let P Q P', Pig. 10, be the traces of the given plane, and a b c the horizontal projection of a triangle contained in the same.    Through any convenient point P, construct the auxiliary diedral
59
plane, P R P', perpendicular to the horizontal trace P Q. At its intersection R with the ground line, erect a line R P" perpendicular to its horizontal trace P R, and carry the distance of the intersection P' of the vertical traces from the ground line, round to P" as indicated by the arc' P' P" ; then the right angled triangle P R P" will be the auxiliary plane revolved into the horizontal plane of projection, while <RPP is the angle required, and P P" the line of intersection of both planes in real length. It is evident, that this line of intersection and the horizontal trace of the given plane, are perpendicular to each other in space ; therefore, to revolve the given plane about that line of intersection, draw from P a line P Q' perpendicular to P P", and make P Q' equal to P Q ; join Q' and P", which will then be equal to the vertical trace Q P'. The right angled triangle q/ p p// jQ ^a^ par|. 0;j> ule gjven piane jn revolved position which is intersected by both planes of projection and the auxiliary plane PEP'.
Any line contained in a plane parallel to either trace in space, will also appear in its projection parallel to that trace in that plane of projection, to which said trace belongs; so, if lines are conceived from the vertices of the triangle in space parallel to the horizontal, trace P Q, tlieir horizontal projections a a", b b", and c c" will be parallel to it. If these parallels are extended until they intersect the line P P", as in a'", V" and d", then the distances P a"\ P V" and P d" are evidently the real perpendicular distances of the points in space from the horizontal trace, and a" arn, b" bf" and c" d" are the vertical distances of the same points above the horizontal plane of projection. Further, as P Q' is the horizontal trace P Q in its revolved position, the lines a"' A, V" B and d" C parallel to P Q and equal in length respectively to a" a, b" b and c" c> must be the lines conceived through the vertices of the triangle also in their revolved position. A B C is the triangle in real size, and in its true position in regard to the given traces of the plane containing it.
In case the real form and size of the figure alone is required, find the angle of inclination R P P", and draw perpendiculars to P R from the points a, 5, c until they intersect P P" in a"\ b'" and d" ; from these points of intersection draw perpendiculars to P P", and lay off on them the respective distances of the points a, 6, c from P R, as in A B C, which will be the required figure.
Remark.�It is important for the reader to understand this problem thoroughly, in order to conceive readily the problems relating to the intersection of solids by planes, as represented on Plate IX.
EXAMPLES  IN PROJECTION   OF LINES AND SURFACES.
These Examples are here introduced to accustom the reader to the mental conception of the position of lines stated in language, and also to produce the same on paper.
Figs. 11 to 17 represent the projection of a given line, A B, under various conditions.
EXAMPLE   I.
Project the given line, A B, on an angle of 30� to the horizontal plane of projection, and parallel to the vertical one ; the end point A to be contained in the horizontal projection at a.
As the given line A B Fig. 11 is to be parallel to the vertical plane of projection, it will appear in real length in its vertical projection and parallel to the ground line in the horizontal one ; the point A being contained in the horizontal plane of projection at a, its vertical projection is in the ground line at a'. *
Draw an infinite line, a1 b', from a1 on an angle of 30� to the ground line, and make it equal to A B, then a' b' will be the vertical projection of the given line; from the given horizontal projection a of the point A, draw an infinite line parallel to the groimd line, also the ordinate from V, and a b is the horizontal projection of A B.
60
EXAMPE   II.
Project the given line A B, as being contained in a vertical plane making an angle of 30� with the ground line, and the line itself an angle of 45� with the horizontal plane of projection ; the point A of the line being contained in the horizontal plane of projection at a.
As the line is to be contained in a vertical plane making an angle of 30� with the vertical plane of projection, the horizontal projection of the line will make the same angle with the ground line ; therefore, draw from a, Fig. 12, a line a c parallel to the ground line, and also an indefinite line, a b, on an angle of 30� to a c; in this line a b must be contained the horizontal projection of the given line. To determine the length of this projection, draw from a an indefinite line a b" making an angle of 45� with a �, and make a b" equal to A B ; from b" let fall a perpendicular to a b intersecting in 5, and a b is the required horizontal projection. The vertical projection of A is given in a'; to find that of B, draw the indefinite ordinate from b and make its vertical one 6� V equal to b b" ; then a' V is the vertical projection of A B under the conditions given. It is evident a b b" is the projecting plane of the given line revolved into the horizontal plane of projection.
EXAMPLE   III.
Project the line A B as follows :
Let the projecting plane of its vertical projection make an angle of 30� with the horizontal plane of projection and the line itself an angle of 15� with the vertical one ; the projections of the point A being given.
Let a-a\ Fig. 13, be the given projections of A. The graphical construction of this diagram is similar to the last one. The vertical projection of the given line will make an angle of 30� with the ground line ; draw a line af cf through a' parallel to the ground line, and af V on an angle of 30� with the same ; draw also an indefinite line ar b", making the angle of 15� with a' b\ and set off af b" equal to the given line A B ; from b" let fall a perpendicular to af b\ and br will be the vertical projection of B. To find the horizontal projection of B, draw a c parallel to the ground line, also the ordinate from V and set off c b equal to V b" ; then 5 is the horizontal projection of B,
EXAMPLE  IV.
Project the line A B under an angle of 45� to the horizontal, and 25� to the vertical plane of projection ; the projection of A being given, and to be the nearest one of the line to the ground line.
In the two previous examples, the direction of one projection was immediately obtainable from the conditions given.    The construction of this diagram is somewhat more complicated.
It is evident that, one point of a given line being fixed, if it revolves around said point, keeping at a certain angle with the vertical plane of projection in all positions of its revolution, it will describe the curved surface of a right cone ; the fixed point being the vertex, and the circle described by the other end of the line, the base of the same ; the circle will be contained in a plane pairallel to the vertical plane of projection. Again, if the line revolves around the fixed point, keeping at a certain angle with the horizontal plane of projection in all positions of its revolution, it will describe the curved surface of another right cone, the fixed point being its vertex, and the circle described by the other point, its base ; said circle of the base will be contained in a plane parallel to the horizontal plane of projection. A little reflection will convince, that both circles must intersect each other in two points, whenever the sum of both angles given is less than 90�, but when the sum is equal to a right angle, both circles will come in contact at one point only. In the first case these points of intersection, and in the second the points of contact, will be the points in space which the revolving end-point of the given line must occupy, in order to have the line make the given angles with the planes of projection.
61
Four such cones may be described by the given line under these conditions : one above and one below the given point, the bases of both being horizontal; also one in front and one in rear, the bases of both being parallel to the vertical plane of projection. Thus it becomes evident, that in case the sum of the given angles is less than 90�, there are eight different positions, and in the other case when the sum of the angles is equal to 90�, there are four different positions of the line complying with the required condition In the former case, both projections of the line will be inclined to the ground line, while in the latter, they will be at right angles to the same.
The graphical construction of the example given above, is carried out for one of the eight positions in Fig. 14.
Let a-oJ be the given point A of the line A B ; from a draw a line parallel to the ground line, and also another making an angle of 25� with the same, on which set off a b" equal toAB; an indefinite line b b" through b" parallel to the ground line, will represent the horizontal projection of the vertical circle in which the point B would revolve ; then draw from b" a perpendicular to the ground line intersecting at bY a horizontal drawn through a\ and describe an indefinite arc from a' as center, and a' bY as radius, representing the vertical projection of the same circle of which b" b is the horizontal one. From af draw an indefinite line af b,fr on an angle of 45� with a' b\ and set off a' b"f also equal to A B ; a horizontal line through b"r is the vertical projection of the horizontal circle, which the point B would describe, were the line A B revolving on an angle of 45� to the horizontal plane of projection. Then, from b,tf draw also a perpendicular to the ground line, intersecting the horizontal through a in &"", and describe an arc from a as centre with a b"" as radius, representing the horizontal projection of the same circle, of which b' V" is the vertical one. The respective intersections b V of these arcs with the horizontals, are the horizontal and vertical projections of the point B, and a b and a! V the projections of the line in a position to comply with the given conditions. If the diagram is accurately made, the ordinates of b and b! will meet the ground line in one and the same point.
Fig. 15 represents the line A B in a position in which the sum of its angles of inclination to both planes of projection is equal to 90� ; its projections a b and a' V are perpendicular to the ground line, a b A B evidently represents the projecting plane of a b revolved back into the horizontal plane of projection ; c is the horizontal and d! the vertical trace of the line.
When the line A B and the angles of inclination are given in the revolved position, the diagram shows plainly, how the projections are obtained; also in the reversed case, when the projections a b and a' V are given, how to find the line in real length, its angles of inclination and traces, without further explanation.
Fig. 16 represents the line A B in the same position as found in the construction of Fig. 14. Suppose the projections a b and a' V to be given; develop the line in its real length as demonstrated in Prob. XV; a b A B is the projecting plane of a b revolved into the horizontal plane of projection ; c is the horizontal trace of the given line, and b cB the angle made with the horizontal plane of projection ; it will be found to be equal to 45� and A B equal to the given line. Likewise a' b' A! B' is the projecting plane of a' V revolved into the vertical plane of projection ; d! is the vertical trace and A' B' equal to the real length of the line; > V &' B' equal to 25�, is the angle made by the line with the vertical plane of projection.
example v.
Fig. 17. A triangle A B C is given in real size, also a point a in the horizontal plane of projection.
Project the triangle on an angle of 60� to the horizontal plane of projection, the vertex A resting in a; the horizontal trace of the plane containing the triangle, to make an angle of 30� with the ground line, and the side A B an angle of 45� with the horizontal trace.
As the vertex A is to rest in a, the horizontal trace of the plane containing it, must pass through the same point; therefore, draw the trace P Q through a, making the given angle of 30� with the ground line Q R.    The plane of the triangle is to make an angle of 60� with the horizontal
.62
plane of projection : in order to find the other trace, construct an auxiliary vertical plane a R P' (see Problem XVIII, Fig. 2) through any point, say a, perpendicular to the trace P Q, and revolve the same into the horizontal plane of projection ; to do this, draw a line a P" on an angle of 60� to a R, also a perpendicular from R, both intersecting in P', then a R P" is the auxiliary plane revolved ; carry R P" round to P' and Q P' is the vertical trace of the plane containing the triangle.
To construct the required projections of the triangle, conceive the plane P Q P7 containing it, revolved about its horizontal trace P Q, and draw the triangle in this position, according to the conditions given. The vertex A is to rest in a, which remains unchanged, and the side A B to make an angle of 45� with the horizontal trace ; then a B C will be the triangle in its revolved position.
To construct its projections, draw from <x, B and C perpendiculars, and from B and C parallels to P Q, intersecting the perpendicular from a in b" and c" ; carry the distances of bn and c" from a, round to the auxiliary plane in b"J and d/;, as indicated by the arcs; extend the perpendiculars from B and C, and draw from 6//7and d" parallels to the trace P Q, intersecting the former in b and c, which will be the horizontal projections of B and C, and a b c that of the triangle. The perpendicular distances of b'n and dn from the horizontal trace a R of the auxiliary plane, are evidently the heights of these points above the ground line in their vertical projection, and are transferred as indicated by the dotted lines and arcs, intersecting the ordinates from 6 and c in V and d ; al being contained in the ground line, the triangle al V d is the required, vertical pro* jection of A B C.
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PLATE   IX.
INTERSECTION   OF   SOLIDS   WITH   PLANES.
The figure produced by the intersection of any solid with a plane, is called a Section of such solid.
It is evident, that the sections of one and the same solid lby planes in different positions, will he different figures ; those by planes parallel to the base, will be either equal, as in the prism and cylinder, or similar to it, as in the pyramid and cone ; they have been mentioned under Plate V.
The sections produced by planes inclined to the base or axis will now be considered, as also the development of the intersected parts of the surface.
INTERSECTION OP PLANE�SIDED BODIES WITH PLANES.
The section of any plane sided solid by any plane in any position, will be a polygon.
If the cutting plane is so situated, as to cut all the lateral edges of a prism or a pyramid, the figure of the section will be a polygon of as many sides, as the solid has lateral faces respectively.
The construction of the projections of such sections consists simply in the projection of those points, where the edges pierce or intersect the cutting plane (see Problem X.), and the connection of the respective points so found by right lines.
The projections of a section being thus found, and the figure of the section itself being a plane figure contained in the cutting plane whose traces are given, the development of the section is simply an application of Problem XXII.
This general statement will suffice to explain the principles of the construction of Pigs. 1, 3 and 5.
The development of the intersected parts of the surface will be explained in each case separately.
PROBLEM  XXIII.
To construct the section of a right rectangular prism, intersected by a plane perpendicular to the vertical plane of projection, hut inclined to the horizontal one ; also to developethe section and the intersected part of the surface in real size.
Let P Q P' Fig. 1 be the intersecting plane, and abed the horizontal projection of the prism. For simplicity's sake, the prism is assumed to have a quadrangular base, one side being parallel to the ground line. It is evident, that in this case the horizontal projection of the base is also that of the section, and the vertical projection of such section is a' V, or a line coinciding with the vertical trace of the cutting plane, limited by the vertical projection of the edges perpendicular to the groundi line. To develope the section in real size, revolve the cutting plane containing the figure about its vertical trace : it is plain, that the intersecting line of the sides a b and c d with the cutting plane, jnust be parallel to the trace Q P', and those of the sides a d and b c, perpendicular to the same. ' On these perpendiculars lay off from Q P' the respective horizontal ordinates of a, b, c and d, and A B C D is the developed section.
Fig. 2 is the development of the intersected part of the surface between the horizontal plane of projection and the cutting plane.
Oh a straight line dd, lay off the sides of the prism as d a, a b, b c and c d, equal to the corresponding horizontal projections Fig. 1, erect perpendiculars representing the lateral edges of the prism, and make them equal to the corresponding vertical projections; <#DABCD^is then the developed surface of the intersected prism below the cutting plane.
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PROBLEM XXIV.
To construct the section of a right rectangular prism, intersected by an oblique plane; also to develope the section and the intersected surface in real size.
Let P Q P', Fig. 3, be the traces of the intersecting or cutting plane, and ab cd the horizontal projection of the prism, which is also that of the section; the sides stand on an angle to the ground line.
The development of the figure is the same as in Problem XXII. Through any point, say 6, construct the auxiliary plane PEP' perpendicular to P Q, and <EPP;/ is the angle made by the oblique plane with the horizontal plane of projection; draw perpendiculars to P R through a,b,c, d, until they intersect P P" in a", d'\ B and c" ; from these points erect perpendiculars to P P7/ on which lay off the corresponding horizontal ordinates to P R, and A B C I) is the developed section. As the auxiliary plane PHP7 passes through 5, the corresponding point B of the developed figure is also contained in P P". The vertical ordinates, such as b B, are evidently the heights of the points where the vertical edges of the prism intersect the oblique plane above the horizontal plane of projection ; therefore, to find the vertical projection of the section, layoff these heights, on the corresponding vertical projections of the edges from the ground line, and af V & d; will be the required projection.
Pig. 4 represents the development of the intersected surface, the construction of which is the same as that of Fig. 2.
Remark.�It will be noticed that, while in Fig. 2 the lines D A and B C are parallel to the ground line, all the lines in Fig. 4 indicating the intersection, as D A, A B, &c, make an angle with it, because the heights of the edges are all of different length.
It will also be noticed that, in order to develope the section only, the horizontal projection of the same alone is needed, and that projection is at the same time, the horizontal projection of the prism itself.
Thus, the triangle A B C in Fig. 10, Plate VIII., developed from its horizontal projection a b c, is also the developed section of a right triangular prism, of which a b c is the base, intersected by the plane P O P'.
In this manner the section of any right prism whose base is given, intersected by a plane in any position, may be constructed.
PROBLEM   XXY.
To construct the section of a right quadrangular pyramid intersected by an  oblique plane; also to develope the section and the intersected surface in real size.
Let P Q P' Fig. 5, be the oblique plane intersecting a right pyramid, abed being the horizontal projection of its base, and n that of the vertex ; a' bf c' d! n' is the vertical projection.
In order to make the difference of construction of this diagram and that of Fig. 3 more apparent, the positions of the cutting plane and the base of the solid, are assumed the same in both.
It is evident, that the horizontal projection of the section will be different from that of the base ; the section is nearer the vertex, and will be smaller. As the edges of the pyramid are inclined to either plane of projection, the projection of the points where they intersect the cutting plane, must be found as in Problem X., Fig. 9, Plate VII.
The opposite slanting edges of a right quadrangular pyramid appear in horizontal projection as a straight line, and the projecting plane of that projection will contain both of these opposite edges ; consequently, the line of intersection of that projecting plane with the given cutting plane will intersect both of these edges in space ; their vertical projections will intersect also, giving at once two points of that projection. Let the construction of the projections mf7cf be considered, and that of ir V will become evident in the same manner.
The projecting plane of the horizontal projection dnb will intersect the horizontal trace of
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the cutting plane at g, and the vertical trace at h'; a line in space passing through h' and g will be the line of intersection of the two planes, and as g' is the vertical projection of g, the line hf g' is its vertical projection, which extended, will intersect the vertical projections d' n' and V n9 in m' and � respectively.
i' V being found in a like manner, connect the respective points by straight lines, and i* Tc' V m' will be the required vertical projection of the section. The lines Tc' V and V mf being hidden from view by the solid, are drawn in broken lines.
Ordinates to the ground line from these points, intersecting the corresponding horizontal projections of the slanting edges, give the horizontal projections i, 7c, I, m of the same points, which, connected by straight lines, give that of the section. This latter projection being found, the development of the section in real size, by means of the auxiliary plane P R P', is the same as in Fig. 3.
Fig. 6 is the development of the intersected surface. The slanting edges of a right pyramid being all alike, also the sides of the base, the developed surface of the slanting faces is easily constructed.
Find the real length of one of the slanting edges (see Problem XV.) and describe an arc with that length as radius ; space* off on the arc the length of a side of the base four times, connect the adjacent points by straight lines as also with the centre, and a figure as I C D A B C will be the result; this figure is evidently the developed surface of the four slanting faces of the pyramid. To construct the intersected portion of that surface, find the real length of that part of the intersected edges below the cutting plane, by the same problem (XV).
For instance: the hypothenuse of a right angled triangle, of which I c, Fig. 5, is the base, and the vertical ordinate of T its altitude, will be the real length of L C, Fig. 6.
The respective lengths so found, are laid off on the lines representing the edges in Fig. 6 from the corresponding ends toward the center, and CDABCLKJML will be the development of the intersected surface below the cutting plane, and NLMJKL the part above the same.
If either of these parts of Fig. 6 is cut out of paper, and bent at right angles in the lines of the slanting edges, bringing the edges marked L C or N L respectively together, the lines L M, M J, J K and K L will range, of lie in one plane.
The same result may be obtained by cutting and bending a paper of the shape of Figs. 2 or 4. The section of any plane sided solid, intersected by a plane in any position, may be obtained in the manner explained in this problem, by finding the projections of those points, where the edges are cut by the intersecting plane, separately.
INTERSECTION OF CYLINDERS AND CONES WITH PLANES.
The sections produced by the intersection of cylinders and cones with planes, are in some particular cases straight lines, but mostly curves of different shape, as the circle, ellipse, parabola, and 7iyperhola, called collectively the conic sections. The construction of the projections of any one of these curved sections, consists in finding the projections of a sufficient number of its points, and tracing the proper curve through the same by means of a curved ruler. To find the projections of such points, construct auxiliary planes so situated, as to intersect both the solid and the cutting plane ijn the required section ; then the line of intersection of the auxiliary and the cutting plane in space will contain one or more points of the section, and further, the projections of those points, where that line of intersection passes through the surface of the solid, are the points of projection required.
In most geometrical constructions, the position of a point is found by the intersection of two straight lines, or two circular arcs, or both, as the case may be ; because the straight ruler and a pair of compasses are the most convenient drawing tools ; therefore, to make the construction of the above sections most simple and convenient, the auxiliary planes ought to be so situated as to inter-
66
sect the curved surface of the solid either in straight lines or circular arcs, and then the projections of those points, where these straight lines or circular arcs intersect the line of intersection of the auxiliary and cutting plane, are at once the required points, and easily constructed.
The following facts, which are evident by the definitions of the cylinder and cone given, are important in regard to these constructions.
Concerning cylinders:
First.�On the curved surface of a cylinder, a straight line may be drawn parallel to its axis through any point of such surface.
Second.�All sections perpendicular to the axis are equal; if the cylinder be a right cylinder of revolution, these sections are circles equal to the base.
Third.�All planes parallel to the axis intersect the curved surface in two parallel straight lines.
Concerning cones:
First.�On the curved surface of a cone, a straight line may be drawn passing through the vertex and any point on such surface.
Second.�All sections perpendicular to the axis are similar curves, and if the cone be one of revolution, these sections are circles of various diameters ; th*e nearer to the base, the larger the circle.
Third.�All planes passing through the vertex, intersect the curved surface in two straight lines, making an angle whose vertex is in the vertex of the cone.
In view of the above stated facts, the auxiliary planes employed in the construction of sections of cylinders, ought to be parallel to the axis, and in the case of cones, they ought to pass through the vertex.
Planes are tangent to cylinders or cones, when they come in contact with their surface only in one straight line; it is evident that in the case of the cylinder, this line of contact is parallel to the axis.
When a cylinder or cone and any plane tangent to it, are intersected by another plane in any position, the line of intersection of the two planes, will also he a tangent to the sectional curve.
Remark.�In the following problems, the sections of right cylinders and cones of revolution will be considered only ; the operations in similar cases on elliptical ones are so much like these, that the reader who understands the theory of the section of circular cylinders and cones, will find ho difficulty in applying it in the construction of the others.
PROBLEM   XXVI.
The projection of a right cylinder of revolution being given, also the horizontal trace of two planes tangent to it; to construct their intersection with a plane perpendicular to the vertical plane of projection and inclined to the horizontal one.
Let P Q P', Fig. 7, be the traces of the cutting plane, the circle numbered from 1 to 12 the base of the cylinder, and the thick lines 7 / and /10 the traces of the tangent planes. These planes are, of course, perpendicular to the horizontal plane of projection, as the cylinder is a right one, and its axis also perpendicular to that plane of projection.
As the cutting plane passes through both of the apparent outlines of the cylinder in vertical projection, the section will be a whole ellipse ; the horizontal projection of this section is indentical with the base of the cylinder, and its vertical one coincides with the trace Q P', lMited by the apparent outlines of the cylinder in 4' 10'.
To develop the section in real size, assume vertical auxiliary planes, (which will, of course, be parallel to the axis of the cylinder,) parallel to the horizontal trace P Q of the cutting plane. Let one of these be considered ; the auxiliary plane passing through c, the centre of the cylinder, will intersect the cutting plane in a line parallel to the trace P Q. This trace being perpendicular to the vertical plane of projection, the line itself will also be perpendicular to the same, and appear as the point c] in its vertical projection ; this line will evidently pierce the surface of the cylinder in
67
two points, of which 1 and 7 are the horizontal projections. The lines of intersection of the planes passing through a, 5, d, e, will appear as the points a' V df e! in vertical projection, and pierce the surface of the cylinder in points, as 5, 3, 6, 2, &c. The planes passing through 4 and 10 being tangent, will intersect the trace Q P' in 4' and 10'.
Instead of revolving the cutting plane about its vertical trace, as was done in Fig. 1, to save space and lines, it may be revolved about the line 4 10�4' 10', passing through the center of the cylinder and parallel to the vertical trace Q P'. This operation will not alter anything in the relative position of the lines. The lines of intersection of the auxiliary planes with the cutting plane are in space evidently perpendicular to that fixed axis 4,10�4' 10', and, being also horizontal, the distances of their intersection with the surface of the cylinder from this axis, are in space equal to their horizontal projections el, 52, dS, &c. The perpendiculars to 4' 10' passed through the points a', 5', &c, are these lines of intersection in their revolved position, and, laying off on them from 4' 10' the respective ordinates in the horizontal projection, as 7' cf equal to 7 c, &c, the points so found, will Tbe points of the required section developed. 4' 10' is the major, and the perpendicular passing through the centre c', the minor axis of the ellipse, which is traced through the respective points.
The tangent plane 7 f is intersected by the cutting plane in a line parallel to 4, 10�4' 10', and consequently, 7' F is parallel to 4' 10' in the development, and a tangent to the ellipse at 7'.
The other tangent plane/10 is situated the same as any of the auxiliary planes ; therefore its line of intersection with the cutting plane, as F 10', is perpendicular to 4' 10' in the developed figure, and a tangent to the ellipse at 10'.
Fig. 8 shows the development of the intersected surface of the cylinder below the cutting plane. The length of the base line 1, 1, is equal to the circumference of the circle of the base. This length may be computed by the diameter, or found by Problem XXXIII (geometrical constructions), Fig. 12, Plate III. The points 2, 3, 4, . . . where the auxiliary planes intersect the surface, are laid off on this line, by taking a small distance between the dividers and spacing them off along the circumference of the circle ; or a narrow strip of paper may be bent to conform to the curve of the circle, several points marked, and then laid flat on the stretch-out line. From the points thus found, perpendiculars are erected, and the respective heights of the vertical ordinates of 4', a', b\ &c, laid off on them from the ground line.     It will be observed that pairs, such as
3 and 5, are equal to the height of a', 1 and 8 equal to the height of c', &c, while the points
4 and 10 are single heights.    A curve traced through these heights, as indicated by the thick line in the figure, will be the curve of intersection developed.
When figure 8 is cut out of paper, and bent in the shape of a cylinder, this curve will range as being in one plane.
PROBLEM  XXVIL
The horizontal projection of a right hollow cylinder of revolution in a vertical position and the horizontal traces of several planes tangent to the same being given, to construct the development of their intersection with an oblique plane.
Let P O P', Fig. 9, be the traces of the oblique cutting plane ; o is the centre of two circles described with o n and o mas radii, representing the given horizontal projections of the inside and outside surface of the right hollow cylinder; a c and h 7c are the given horizontal traces of two verticail planes, tangent to the cylinder at b and/.
The ground line Q R being a tangent to the circle at d, the vertical plane of projection is also a tangent plane to the cylinder. The auxiliary perpendicular plane P R. P' is constructed as in Fig. 3, perpendicular to the horizontal trace P Q. All straight lines on the surface of this right cylinder being perpendicular to the horizontal plane of projection, the auxiliary planes necessary for the development are assumed to be vertical and perpendicular to the first auxiliary plane PEF; their horizontal traces are then perpendicular to P R, or parallel to P Q, as indicated by the fine dotted lines through the points g, 5, c, &c.    The lines of intersection of these auxiliary
68
planes with the cutting p]ane are then horizontal, and the distances of their intersection with the surface of the cylinder from the plane PEP', are in space equal to those of the horizontal projections of the same points from the trace P R. If these perpendiculars are therefore produced, intersecting P P", and from these latter points of intersection, as b"f o"\ &c, perpendiculars erected and made equal to the length of the corresponding horizontal ordinates from P R, the two ellipses indicated by full thick lines may be traced through the points thus obtained, which will be the required developed section of the hollow cylinder. By this operation, the cutting plane containing the section, is evidently transposed into the horizontal plane of projection by means of the auxiliary plane P R P', according to Problem XXII., Fig. 10, Plate VIII. It is well to notice particularly the following facts:
First.�The line g e passing through the centre o, and parallel to the horizontal trace P R of the auxiliary plane, appears as Gr E, is parallel to P P", and contains the major axis of the ellipses, while I m, which passes also through the centre o and is perpendicular to P R, appears as L M in the development, is perpendicular to P P" and contains the minor axis of the same ellipses.
Second.�The lines I m and g e divide the cylinder into four equal quadrants in horizontal projection ; so do the lines L M and Gr E also divide the developed ellipses into four equal quadrants. The quadrant b m dm horizontal projection appears as B M D in the development.
Third.�b o and d o are radii of the cylinder; so appear B O and D O in the development, as passing to the centre of the ellipses. The same is the case with any other radius, as f o, which appears as F O in the developed section.
Fourth.�The section of the hollow cylinder is the narrowest in the minor axis, or in the direction perpendicular to the horizontal trace P R of the auxiliary plane.    N M is, equal to n m.
Considering the tangent planes, it is evident, the triangle a c c" is the intersected part of the perpendicular plane a c cr revolved into the horizontal plane of projection, and a c" the real length of the line of intersection of the tangent plane with the cutting plane. This line appears in equal length as A C in the developed section, and as a c is tangent to the larger circle at 5, so is A C tangent to the larger ellipse at B.    The same is the case with d h' and C H.
If from 7c, a line is drawn perpendicular to P R, intersecting P P" in 7c'", and the height, 7c 7c'", carried over to a line drawn from 7c perpendicular to h 7c, as indicated by the arc 7c'" 7c", also the point Ti" determined in a similar manner by the arc 7i' h", then the vertical plane 7i 7c�h" 7c" is limited by four planes ; viz: by the horizontal and vertical plane of projection in 7i 7c and h h", also by the vertical auxiliary plane PEP'inJ 7c", and the cutting plane P Q P' in 7i" Tc"; this latter line is equal to 7c"' H in the developed section, and, as Ti 7c is tangent to the larger circle in,/, so is 7c"f H tangent to the larger ellipse in F.
Thus it is evident, any "part of t7ie section with lines tangent to it in any direction, may be developed from the horizontal projection of any right vertical cylinder, if the inclination of the tangent lines in space is Tcnown and their horizontal projections are given. For instance : Let a c and c h be the horizontal projections of two lines tangent to the circle at b and d, and a c c" and c h�d 7i' their projecting planes as revolved about their respective horizontal projections, then, tangent a c would make the angle c a c" and tangent c h the angle hf Q h, with the horizontal plane of projection. It is evident, that both of these tangents are contained in a plane whose vertical trape is h' d, intersecting the ground line in Q, and as a is one point of the horizontal trace, Q a is thait trace. Both traces of the cutting plane being then known, the construction of the section of the quadrant b m d with its tangents b c and c d is done as usual, by constructing any auxiliary plane as P R P' perpendicular to the horizontal trace, and the section B C D M 1ST is obtained in real size, without any need of developing the whole section of the cylinder.
In case a development of the intersected part of the surface is required as in Fig. 8, it is easily done in the same manner as that figure ; the vertical ordinates between the horizontal trace P R and the revolved line of intersection P P" of the auxiliary plane, are the heights of the respective points above the base line.
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PK0BLEM  XXV III.
To construct an ellipse proportional to a given one,
A further discussion of Fig. 9 leads to the theory of constructing proportional ellipses. The two ellipses developed, representing the sections of the two concentric cylindrical surfaces of a hollow cylinder intersected by one and the same plane, are pkoportional ellipses.
If cords are drawn in both circles of the horizontal projection between the same radii, they are parallel, and the vertical projecting planes of these cords will intersect the cutting plane in parallel lines ; consequently, they will appear parallel in the developed figure- IsTow, as the points where these cords intersect the circles are also points of the developed ellipses, it follows that, when in any two proportional ellipses lines are drawn from any two points of the circumference of either to the common centre, then t?ie cords connecting the respective points of intersection of the same lines with either ellipse, will be parallel.
Let a e b and a' e' &', Fig. 10, be quadrants of two proportional ellipses ; from e draw a line to the centre o, intersecting the smaller ellipse in d; the cords a b and a' V will be parallel, as also a e and a' d, or eb and e' b''.
Further, the same demonstration shows, that the tangents A g and A' g' are parallel; these lines are tangents to two proportional ellipses at points in a line passing through the centre.
To illustrate the application of these facts, letafgdbea, given ellipse, and A g a tangent to the same at the point g; another ellipse is required, proportional to the given one and passing through a given point a' in the minor axis / also a tangent parallel to the given one.
From g, and some other point or points of the given ellipse, draw lines g o and f oto the centre, and connect the points a, f g and d by straight lines ; from the given point a' draw a line parallel to af intersecting/ o in/', which will be a point of the required ellipse.
From/ draw another line parallel to / g, intersecting g o in gf, which will be another point of the same ellipse ; dJ is also obtained by a line parallel to g d from gf; the curve is traced through the points so found, and gr will be the proper point of contact, through which to pass the required tangent A! gf parallel to A g.
PROBLEM   XXIX.
The projection of aright cone of revolution being given, to construct the intersection of the same with a plane perpendicular to the vertical plane of projection and inclined to the horizontal one.
Let P Q P', Fig. 11, be the given traces of the intersecting plane; the circle with o as a centre, is the horizontal projection of the given cone, and V 1' o! the vertical one. The vertical projection A Gr of the section, is contained in the vertical trace of the cutting plane passing through the cone, and limited by the apparent slanting outlines.
The horizontal projection of this section must be constructed, in or<3er to develope it in real size. Pass perpendicular auxiliary planes through the vertex o. It is most convenient, to pass one of them perpendicular to the horizontal trace of the cutting plane, as 1-7, and also one parallel to itj as 4 10 ; these two divide the circular base into 4 quadrants, which may be divided again into three equal parts each, making 12 divisions in all. This form of division gives six auxiliary planes^ each cutting two straight lines from the surface of the cone, which all intersect in the vertex o-o'; they are also so situated, that two and two of these surface lines appear in vertical projection as one line, except V o' and 7/ o', which evidently show the apparent outlines of the cone. A, V, d, d', d, ff and Gr are the vertical projections of the points, where these lines pass through the cutting plane, and the ordinates from these points perpendicular to the ground line, will intersect the respective horizontal projection of the same lines in the horizontal projection of the points ; for instance, d is the vertical projection, and c, h, the horizontal ones of those points, where the lines 9 0�9' 0' and 5 0�5' 0' pierce the cutting plane P Q P'.
Revolve the cutting plane, or the section it contains, around the line a g�A Gr (this being
70
the line perpendicular to the horizontal trace P Q), by drawing perpendiculars through the points each way, and set off on these the distances of the respective points from the line a g in horizontal projection ; for instance, make C d and H d equal to c n and h n.
The ordinate from the point d coincides with the horizontal projection of 4 o 10, therefore no intersection is produced ; to find the point d, we mast consider, that the distance o d is equal to the radius of a circle cut from the cone by a plane perpendicular to its axis, or in this case, parallel to the ground line, through the point d; the horizontal line d' m is evidently that radius, and therefore d o equal to it.
Fig. 12 shows the development of the intersected surface of the cone below the cutting plane.
To develope the surface of a right cone, (without reference to any intersection) describe an arc with a radius equal to its slant height, and make the length of the arc equal to the circumference of the base; the sector thus produced is the developed surface.
In Pig 12 the radius 1 0 is equal to V ()', Fig 11, and the length of the arc 1,1, is equal to the circumference of the circle 1, 2, 3, &c.
To construct the curve of intersection in this developed surface of the cone, divide the arc into 12 equal parts, as 1, 2, 3, &c, and draw lines to the centre O ; these lines represent the lines cut by the auxiliary planes through the surface of the cone. Find the length of the intersected parts of these lines, by drawing horizontals from the points A, b', o', d', &c, intersecting the apparent outline of the cone ; for instance, 10 d�10' d' is equal in real length to 7' m, and is laid off on the lines marked 4 and 10 in Fig. 12.
The curve G H A C Gf is traced through the points thus found. If this figure is cut out of paper and bent in circular form, the sides 1 Gf and 4 Gr coming together, the curve and the arc will each range or lie in a plane. Should the cutting plane be parallel to the base, as for instance, through the point G, Fig. 11, the intersected curve in the development is also a circular arc G G (indicated by a dotted arc), described from the centre O with o' G, Fig. 11 as radius.
The figure formed by both arcs, Fig. 12, and the sides 1 G, 1 G, is evidently the development of the surface of the frustum of a right cone, whose bases are parallel.
The construction of the section of a right cone intersected by an oblique plane, is similar to Fig. 11. After an auxiliary plane as P R P", is constructed as in Figs. 5 and 9, arrange the divisions of the circle of the base in the same relative position to the horizontal trace of the cutting plane, as they are in Fig. 11; and the figure of the section may be revolved about a line through its centre, as in that figure, or thrown back into the horizontal plane of projection as in Figs. 5 and 9.
THE   HELIX   OR   SCREW-LINE.
Before entering upon the special application of the foregoing problems to the practical construction of Staircases, the nature and construction of the helical curve ought to be considered.
Fig. 13 represents a screw in vertical projection, with a quadrangular, square edged thread or fillet.    The upper and lower surfaces ot this thread are evidently warped, winding, or twisted.
A straight line may be drawn through any point in such a surface, in a direction to the axis of the cylinder around which the thread is wound, and such straight lines will always be perpendicular to that axis. The exterior surface assumes the form of part of a cylinder, while the interior is in contact with a solid cylinder, called the core, and the distance between both of these cylinders having one common axis, is the depth of the thread.
If a point m is assumed to travel along either edge of the thread, it will describe a curve called a Cylindrical Helix.
When the point m in this movement arrives at a point ml, lying in a line drawn parallel to the axis through the point m from whence it started, it will have made one revolution, and the longitudinal distance m, m!, between the two points, is called the pitch of the helix or screw. This movement of the point is evidently in two directions: one, circular or around the axis, and the other, longitudinal or parallel to it, both being in an equal ratio.   To illustrate this more plainly,
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let the larger circle in Pig. 15 represent the horizontal projection of the exterior surface of the screw shown in vertical projection Fig. 13. Divide this circle into a convenient number of equal parts, say 16, and the pitch m m! also into the same number of equal parts. In the same space of time, that the point m is moving in a circular direction from one division to another, it moves also TV part of the pitch in a lateral direction, and so on through every division. This discussion of the nature of the helix shows plainly that it is a curve of double curvature, or, to use a popular expression, it is bent both ways. While in curves of single curvature, such as the circle or ellipse, all the points are contained in one and the same plane, in curves of double curvature not more than any three consecutive points lie in the same plane.
A square might move in the same manner as a point, and thus generate the screw thread.
It is evident, that in such helical movement, the square generating the thread and the axis of the screw, will be contained in a plane in all the different positions.
The exterior and interior surfaces of a square-edged screw thread, may be developed or unrolled upon paper, because they are parts of the surface of cylinders, and it is plain from the discussion of the helical movement of the point m, that the helical curve will be a straight line in the developed condition. So much of the curve, as is described by the point m during one entire revolution, is equal to the hypothenuse of a right angled triangle, of which the circumference of the base of that cylindrical surface in which the point moves, is the base, and the pitch, the altitude.
The upper and lower edges of the thread, will therefore appear as two parallel straight lines in the development.
THEORY   OF  HAND-RAILING.
Hand or Staie, Rails are frequently of a helical shape, especially in winding stairs.
Figs. 14 to 19 illustrate the theory of the application of the problems in descriptive geometry, to the practical construction of hand-rails.
Fig. 15 is the horizontal and Fig. 15 ^ the vertical projection of a hand-rail for a so-called winding flight of cylinder stairs, similar to one illustrated Fig. 3, Plate XXVII. The vertical pro-lection shows the rail as being in that condition, called blocked out, ready for moulding. The edges appear in full lines, in order to show more plainly the precise contour of the curves. There are assumed 15 risers ; No. 1 to be the first and No. 15 the landing riser. The number of steps in a flight of stairs, is always one less than that of the risers. The circle is divided into 16 equal parts, 14 of these represent the 14 steps, while the 15th and 16th are included in the so-called Landing, which is on a line level with the next floor.
This division is assumed with reference to that made in the explanation of the screw or helical curve, to show plainly the similarity between the two. The nosings of the steps in the cylinder, will all be points in a perfect helical curve, differing from that of the screw, Fig. 13, only in the pitch. The curve of the rail will differ from that of the nosings at each end, for the reason, that there must be an easement at the lower end to connect to the horizontal cap of a newel post, and also one at the upper end to connect to a level rail running along the landing on the next floor.
It will be noticed, this rail is made out of 5 pieces, connected by square joints at/', a', c', and gf; the three middle pieces are alike each other, as will be demonstrated, although they do not appear so in the vertical projection, and their curves correspond to that of the nosings, precisely. Therefore, the balusters supporting the rail to that extent, will be of equal length. Each one of those thre^ pieces extends over 4 steps, but the lowest piece mf f over only two, making the 14 steps in the stairs ; the upper piece g' rnf is the easement over the landing.
The centres of the joints of these pieces, are respectively at m,f, a, c, g and m, in horizontal projection Fig. 15, and m\f, a', cr, gr and m" in vertical projection, Fig. 15 ^. If a proper helical curve is conceived through these points, starting horizontally at m-m', and ending likewise level at m-<mn, it will be the curve running through the centre of a true rail. Such a curve is constructed separately, Fig. 14 and Fig. 14 ^ ; like letters refer to like points in the four figurfe.    It appears
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as a circle, Fig. 14 A, in horizontal projection, and will be contained in a cylindrical surface, of which this circle is the base, equal to the*middle one, Fig. 15.
To construct the vertical projection of this curve, Fig. 14 ^, space off the height of the risers from 3 to 15, on a vertical line through the centre o ; draw the horizontal lines through these points, intersecting the respective vertical ordinates from the divisions of the circle, Fig. 15, and the points of intersection thus found, are points in the curve. That part of the curve, passing through the centre of the rail from/7 to g\ is a regular cylindrical helix ; there being three equal pieces of rail, / this curve consists likewise of 3 equal parts, each part ascending the height of 4 risers, while it turns at the same time over the space of a quadrant of the circle, containing 4 steps. The first part of the curve from the point m' to/ is an irregular cylindrical helix ; while the regular curve ascends 2 risers in the same time it turns over 2 steps, this irregular one ascends not quite one riser although it turns over an equal space of 2 steps, from m to/. This curve may be said to start as a horizontal circle at m\ ascending gradually and emerging into the regular helix at/7.
Although the nosings of the first and second step lie in the same regular helix formed by the rest, this deviation from the regular curve by the rail is caused, as has been already remarked, by the necessity to intersect the miter-cap of the newel gracefully ; this difference is made up in the extra height of the newel post over that of the balusters. If the rail is set up vertically the usual length of the balusters above the nosing, the newel post in this case would have to be so much higher than that length, as the height m' 3 is less than the height of two risers.
The last part of the curve, from g' to m'\ is also an irregular helical one, similar to the first part; while it turns in a circular direction from the last riser at g to the point m, a distance equal to the space of two steps, it ascends not quite one riser. This curve may be said to start as the regular cylindrical helix at g\ descending gradually and emerging into a horizontal circle at mf'.
This deviation of the curve at that point, leads over to the level rail, and therefore the balusters used for the same, must be so much longer than those used on the steps, as the point m" rises above the point g\
The helix being a curve of double curvature, no part of the same will appear in any position as a strictly straight line ; and further, in any part of a helix of not more than one entire revolution, only three points of the curve line will be contained in any one plane. That part of the regular curve from af to c\ Fig. 14 ^\, approaches nearer to a straight line drawn through those two points in that position of its vertical projection, than in any other. A close inspection of the figure shows, that the curve first slightly rises above the straight line, crossing it at 9' midway between ar and c\ and then falling as much below it, as it rises above between the points af and 9'. That part of the curve from / to a\ although virtually a curve precisely alike to that from af to c' in space, shows as much curvature at e' from a straight line in that position of its vertical projection, as it does in the horizontal one, Fig. 14.
This illustration also shows that, if a plane is passed through any two points of a regular helix in such a manner, that it also contains a third one of the curve situated midway between the two first assumed, then the curve itself will deviate from that plane as much one way above, as it does the other way below its surface, and it is obvious, that the elliptical curve, produced by the intersection of a plane as above situated with the cylindrical surface containing the helix, coincides nearer with the helix, than any other of the plane curves.
It is also evident, that the greater the distance between the assumed points through whicli the plane is passed, the more the elliptical curve will deviate from the helical one, and the shorter that distance, the nearer both curves will coincide.
This is also true in regard to the tangents of the two curves. At first sight it might be supposed that a straight line as a' </, Fig. 14 ^\, is also the vertical projection of two lines tangent to the helical curve at those points, because it is the vertical trace of a plane passing through a1 and &, and also through the point 9', situated midway between a' and &.
Such is not the case, as will be demonstrated. This line a' & is also the vertical projection of two lines tangent to the elliptical curve at the points a-a' and c~c\ their horizontal projections being a b and c b.    As the elliptical curve passes through the point 9-9/, and its tangents are con-
73
tained in the same plane, of course these tangents will be of equal length, and b their point of intersection in space, must be contained in a horizontal line passing throug 97 and the axis of the cylinder; therefore the vertical projection of b coincides with 9'.
It has been mentioned, that all curves of single or double curvature may be assumed as being generated by the movement of a point according to certain conditions t the direction of this movement of the generating point changes at every consecutive point of the curve, and a straigM line passing through any one point of the curve in the direction of the movement which the generating point assumes at that point, is a taistgent to the curve.
Now, the helix being of double and the ellipse of single curvature, and the tangents of the latter in one plane intersecting each other, those of the former from the same points, cannot be contained in one and the same plane. The direction of the tangent of a vertical cylindrical helix, is in space the same as that of the developed curve from the point of contact.
To obtain the projections of such tangents, we must consider, that they are contained in a vertical plane which is also tangent to the cylindrical surface in a vertical line containing that point of contact; therefore the horizontal projection of such a tangent line will be a tangent to the circle of the base at the same point. Thus, for instance, a b and b c are the horizontal projections of two lines, which are tangents to the hellix at the points a-a' and c-c\ and so coincide with those of the tangents of the elliptical curve from the same points.
The horizontal projection of the tangent having been ascertained as above, the vertical projection of some other point besides that of contact, may then be found, through the development of a certain part of the curve itself, as follows : let ap be the stretch-out or development of the arc a e ; it is known, that the point e while being unrolled, moves in the horizontal plane of its position, and that therefore, the vertical projection^' oip, must be found in the horizontal line drawn through e' at its intersection with the vertical ordinate of p. The points q-q\ r-r' and s-s' being obtained in like manner, p' qr and r' s' are the vertical projections of lines, tangent to the helix at the points a-a' c-&. These two tangents are not contained in the same plane, nor any other two in any helix of less than one entire revolution.
Now, from all what has been said, it appears plain, that a true curved hand-rail, theoretically, may be said to be generated, by the helical movement of a rectangle (or, when in the moulded or finished state, by that of a plane figure of whatever shape the section of the rail may be) in such a manner, that said figure, with its lower side always horizontal, is in all positions perpendicular to the tangents of the proper helical curve passing through the centre of the rail.
This definition of the generation of the hand-rail, differs from that given of the screw-thread, because the former is composed of regular and irregular helices, while the latter is a regular helix throughout.
If the figure generating the rail were to keep the perpendicular position which it has at the beginning, as at m/ Fig. 15 ^ where the rail starts horizontal, the thickness of it would diminish as it ascends ; therefore, in order to produce a rail of equal thickness throughout, its position must be perpendicular to the curve at all points of the movement.
All are aware of the difficulty to produce in practice, any piece of work strictly according to its theory, in any mechanical business. This is the case in stair-building, wherein the professional mechanic who makes it a speciality, has to construct practically by hand and to set up in space, independent of other objects to build up to, more curves of various shapes interwoven with straight lines, than in any other occupation.
But it is equally true, that the nearer a staircase, and especially the hand-rail, is constructed according to the theory demonstrated above, the more pleasing and perfect the lines will appear to the eye of the observer from all positions.
PRACTICAL  CONSTRUCTION  OF  CURVED   HAND-RAILS.
Hand rails are made in sections out of plank of various thicknesses, according to the width
and thickness of the rail.   These sections have plane or so called butt-joints at the ends, which
io
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are as near perpendicular to the curve as practicable.   They are bolted together and held in propel relative position by means of rail-screws and dowels.    (See Fig. 14 and ^, Plate XI.)
It has been remarked before, a curved rail may be considered as occupying part of the space contained between the exterior and interior surfaces of a right vertical hollow cylinder, whose base shows either two concentric circles or ellipses, or whatever the shape of the so-called well-hole may be.
The plank out of which the sections are made has, of course, two parallel plane surfaces ,� therefore, the patterns called face-moulds, by which the twisted parts are sawed out, are obtained by the intersection of the respective cylindrical surfaces with planes passing through them in such position, that the elliptical curves and their tangents will, as near as possible, coincide with the proper helix and its tangents through the centre of the rail.
If we conceive a plane through the centre of the plank intended for a section of rail parallel to its plane surface, and the plank itself placed in such position, that said plane through its centre will be precisely in the position of the imaginary cutting plane by which the section has been obtained, it is obvious that the whole substance of the rail-piece will be contained in a plank requiring the least thickness possible, when in that position.
Thus, it will be seen by referring to Fig. 15 ^, that the rail-piece oJ b' d is contained between two broken lines drawn parallel to a! b' d, and which represent the vertical projection of the upper and lower surface of a plank out of which this piece can be made, when in that position. The distance between those lines is no more than the width of the rail, and if we consider that most all sections of rails, as represented in A, B, C, D, Plate XI., require the four corners to be taken off when moulded, it is evident, that twisted rail-pieces may be made of plank even less in thickness than the width of the rail, if placed in the right position. However, in all cases as represented in this work the rail may be moulded the usual patterns out of plank not thicker than the rail is in width.
The face-moulds ought to be placed in such position on the plank for the purpose of marking the same to be sawed out, that the grain of the timber runs as much as possible lengthwise, thus giving the most possible strength to it.
Fig. 16 represents the drawing of the face-mould for one of the three equal pieces of rail, Fig 15 ^ ; the position is chosen for the middle one, a' V <?', to show more plainly the operation. Like letters refer to like points in both figures.
a b and c d are the tangents in horizontal projection from the centre of the rail, which is a circle with o as centre ; the thick quadrants of circles from the same centre represent the inside and outside of the rail. The tangents will have the same inclination or pitch to the horizontal plane1 of projection in order to place the cutting plane containing them in the right position for a regulai helical rail-piece. From ato c are four steps ; therefore each tangent ascends the height of two risers, and thus their position in space is known. Assume a as being contained in the horizontal plane of projection; in order to develope the face-mould, we must find the intersection of the plane containing both tangents, with the horizontal plane of projection, or in other words, its horizontal trace. To do this, erect at b a line b b" perpendicular to a 5, making b b" equal to two risers ; then the triangle a b b" will be the development of the vertical projecting plane of a 5, and angle b a b" the angle of inclination or the pitch of tangent a b with the horizontal plane of projection.
Farther, erect at b and e perpendiculars to c 6, making b bf equal to b b" as indicated by the arc, and c d equal to four risers ; draw d hf intersecting c b produced in q ; then cb d' bf will be the development of the projecting plane of c 5, and q a point in the horizontal trace of the cutting plane ; now, a being contained also in that trace, the line a q is the horizontal trace sought. This trace being found, we proceed as in Problem XXVII. Lay the auxiliary plane a" d;/ for simplicity through the centre o, and of course, perpendicular to the horizontal trace or intersection a q produced ; then draw a line from c parallel to a a", or the same thing, perpendicular to a" d'\ making d" df equal to c d, or four risers, which is the whole height of the rail-piece from a to c. a" d" c" is the developed auxiliary plane thrown down into the horizontal plane of projection, and < d" a" d\ the angle of inclination of the cutting plane with that same plane of projection.
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For the development of the tangent lines, draw an ordinate from h the point of their intersection, which will in this case, pass through the centre 0, and farther, a sufficient number to determine intermediate points of the curve ; from the intersection of those ordinates with the line a" c", draw perpendiculars to the latter, making them equal to the corresponding horizontal ordinates from the line a" c"f; trace the proper curve through them, and a face-mould of elliptical outlines will be the result. If the drawing is exactly made, the developed tangent A B will be equal to a b" and C B equal to If cf, each of these lines being the real length of the respective tangents. In this case, where both tangents are alike in horizontal projection, and both inclined equally to the horizontal plane of projection, they are both equal. Joints are made through the points A and C perpendicular or square to the respective tangents, and also lines drawn on the face-mould through these points parallel to a" c'\ called the slide lines.
In this diagram, the quadrant of the circle over which the rail-piece passes, is completed in fine dotted lines to a semicircle, of which the horizontal trace a" d" of the auxiliary plane produced, is the diameter ; the development of the same is also carried out in like dotted lines, resulting in a semi-ellipse, which is evidently the section of the semi-cylinder, intersected by the same cutting plane, which passes through the three points, a, 5 and c. This is done to show at a glance the relation of the required part, which is drawn in full thick lines, to the whole of the semi-ellipse. It also shows how to apply the trammel to draw the elliptical outlines of the face-mould, instead of using the intermediate ordinates.     k
Referring to the figures on Plate IV., in regard to the different modes of constructing an ellipse, it will be seen, that when the major and minor axes are given, the trammel or a string and two pins, may be used to draw the ellipse. To apply either method in this case, the ordinates a" A, O B and c" 0 must be drawn as in the diagram, to find the position of the tangents as described; also the two points on the ordinate O B of one half of the minor axes of the two ellipses must be obtained. The major axis d' w" of the oatside ellipse is found, by drawing the vertical ordinates v v" and w w" from the intersection v and w of the outside circle with the horizontal trace of the auxiliary plane. As the point w" is evidently below the horizontal plane in which the point a is assumed, the vertical ordinate w w" fails the other side of a" d". The major axis for the inside ellipse is obtained similar.
As O is the centre of the ellipse, it is obvious that in practice only one of the points n" or w" is needed. The elliptical outlines are terminated for the face-mould required, by the joints square to the tangents at A and C.
The reader will find it more convenient in practice to use the first method of the ordinates, as about as many points have to be obtained to get ready for the trammel. This is however, a mere matter of taste, and both methods are frequently employed throughout this work to make the reader familiar with both of them. In many cases this finding of the necessary lines and points to set the trammel by, makes the drawings only more complicated, and in such cases, where the nature of the case makes the use of different lengths of balusters necessary, which may be ascertained by the use of vertical ordinates drawn through their centres (explained in the various special cases throughout the book), the method of developing the face-mould by means of such ordinates, is certainly preferable.
In practice, for the purpose of drawing such a face-mould only, without reference to any connecting piece, whenever the tangents are equal in their horizontal projection or the ground plan and are also equally inclined to the horizontal plane as is the case in Pig. 16, the intersection of the cutting plane with the assumed horizontal plane of projection need not be found by first developing the inclination of the tangents. It is obvious, that the trace a" d" of the auxiliary plane is parallel to the chord line a col that segment of the circle, from where the tangents are drawn ; therefore, this chord line may be assumed as such horizontal trace of the auxiliary plane at once, the known height of the wreath (in this case four risers) set up on the vertical ordinate from e, and thus the section developed.
But whenever another piece joins on such a section, the pitch or inclination of its tangents governs that of the connecting tangent of the adjoining piece, and must be developed in order to
76
draw that other face-mould. For instance, tangent kf f of the lower easement in Fig. 15 ^, until it intersects with horizontal tangent m/ Jc\ must have the same inclination as tangent f V of the next piece above it, by which it is governed. The drawing of the face-mould for such easements is illustrated particularly in several cases on Plates XXVI to XXIX.
When the inclination of the two tangents is different, then their development must be drawn, in order to find the trace of the cutting plane with the assumed horizontal plane of projection, which trace will then not be perpendicular to the chord line.
This whole process is thus lengthily explained, to initiate the reader into the philosophy of locating the inclination of tangents, cutting and auxiliary planes with their traces, in order to understand why and for what purpose such lines are drawn. After some practice, he will soon find where to leave out perhaps this or that line, without impairing the accuracy of the developed face-mould.
It is practiced by the author, as being the most convenient in many respects, to draw the ground-plan of a flight of stairs, or such parts of it as are necessary, in full size on a sheet of stiff brown paper (such as is sold by tjie yard or pound), instead of using a separate drawing-board for each flight, or the floor of the shop. The convenience of this will become more apparent in the practical construction of a stair-case or carriage as on Plates XXI to XXIII. After the stairs are built, this plan may be rolled up and laid aside, to refer to when the rail is wanted.
As all the necessary dimensions are laid down on such a plan, the face-moulds for the rail-pieces are drawn on the same sheet, a piece of board laid under each pattern, and the outlines, tangents and slide-lines pricked through by means of a sharp-pointed brad-awl. The shape is thus transferred to the board in an accurate and convenient manner ; after a little practice, stiff paper instead of board, will be found to answer all purposes in making the patterns of face-moulds, which are readily cut out with a knife.
The face-mould being thus made, it is laid on the plank to be used for the wreath-piece, which is required to be of about the same thickness as the rail is in width.
The reader will find it necessary in practice, to mark the narrowest place of the face-mould about one-eighth of an inch wider on the plank, to give more scope to shape the inside and outside vertical cylindrical surfaces ; but those places where the face-monld is much wider than the width of the rail, the mark on the plank may be curtailed each way, if it is enough to make it an objec* in order to save material; there are cases, that by the peculiar position of the cutting plane, a very wide face-mould is produced, and a piece of plank of a parallel width a trifle more than that of the rail, corresponding in shape to the centre-line of the mould, will be just as sufficient to make the proper wreath-piece, as if the plank were sawed the whole width of the face-mould.
These general remarks will become more evident in the different cases, as the reader proceeds in their study.
The wreath-piece being thus marked on one side of the plank, it is sawed out square through from the face, which may be done most economically at a mill, with a so-called up and down saw, or scroll-saw. After being sawed out, the side of the piece intended for the face or upper side, is faced off true and taken out of wind ; the joints, called butt-joints, are made accurately square to the tangent of the mould, as also square to the face of the plank.
These square joints ought to be made very particular, for the following reason : The relative position of the wreath-pieces is secured by these butt-joints; now, suppose a certain piece to be in its true position, the next piece connecting with it may be, say five feet long, but either joint is, say only one-sixteenth of an inch out of square in the width of it; the result would be, that the further end of the connecting piece would be thrown nearly an inch out of its true position. The butt-joints being made, the centres of the tangents, as A and C on the face-mould, are squared over the face of the joints ; a gauge is set to one half of the average thickness of the plank, and a mark made across the square line from the tangent just mentioned. The crossings of those marks on the joints are the centres of the rail. Then mark the slide-lines on both slides of the wreath-piece just as they are on the face-mould. In order to mark the outlines on the wreath-piece, by which its vertical cylindrical surfaces are shaped, the face-mould has to be moved a certain distance along the slide-lines from the butt-joint on each side�upward on the face-side and downward on the
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under side.    This sliding distance depends on the size of the plumb-bevel and the thickness of the plank.
The plumb-bevel is the angle made by the slide line in space, with any vertical line intersecting it. The upper angle of the right angled triangle, formed by the auxiliary plane in its revolved position, shows that same angle. For instance, angle a" c" c"\ Fig. 16, and marked p &, is the plum-blevel for the face-mould ABC. It is obvious without further explanation, that this angle is the same as that made by the slide-line of which a c is the horizontal projection, with a vertical line at c, or any other intersecting it.
To find this sliding distance, run a guage mark on any part of the wreath-piece where it is nearly straight, from the face along one side with the same opening as for the centre of the rail across the butt-joints (that is, one half of the average thickness); then set a bevel-square by the plumb-bevel; apply the same from the face on the side of the wreath-piece, to intersect the guage-mark run along as just mentioned, and from their point of intersection square over to the face. That distance on the edge of the face, between the bevel line and the square line just mentioned, is the distance to slide the face-mould along the slide-lines from the butt-joints. The face-mould being placed in proper position, its outlines are marked on each side of the wreath-piece, and the superfluous wood taken off by means of a round-faced plane on the inside, and a common plane on the outside, holding the plane as near as practicable in the direction of the plumb-bevel from one line to the other. This will produce the outside and inside cylindrical surfaces, between which the rail may be said to be contained. It will be found, that only the opposite corners of the block will have to be taken off to any extent, while the tool will strike the wood clear through only along the middle of the wreath, and there only slightly. The most of the corners may therefore be taken off by means of a hatchet and drawing-knife, and then the true shape finished with the planes. This does away with the slow process of sawing the sides of a wreath by means of a whip-saw on a rake, which requires two men to do it.
An experienced workman will shape in a few minutes, by means of a hatchet and drawing-knife, a wreath-piece so near, that only a few strokes with the proper planes are needed to give it the true shape inside and out. The same holds good in regard to shaping out the upper and lower sides, which are winding or warped. To square the rail on the butt-joints, or to apply the so-called BLooK-PATTEEisr, the spring bevels are required.
The spking-Bevel is the angle formed by the upper edge of the butt-joint, and the line of intersection between the plane of the butt-joint and the vertical plane containing its tangent.
This bevel may be obtained either by construction from the plan, or laid, out directly on the wreath-piece from the tangents on the face-mould, when the latter is in the proper moved position on the face-side of the block, as before described.
Both methods are correct, but on account of its simplicity, we shall consider the latter of the two first, explaining at the same time the application of the face-mould to the wreath by aid of the diagrams, Figs. 17 to 19.
Fig. 17 is a vertical front view of the wreath, drawn in position over the plan a c, Fig. 16, and is represented with a face-mould applied to both sides. To find the distance for sliding the face-mould, as has been mentioned before, apply the plumb-bevel a" c" d", Fig. 16, on some place of the wreath, where it is nearly straight; let q r p be that bevel so applied on Fig. 17 ; run a guage-mark with one-half of the average thickness of the plank from the upper or face-side across the lines squared over from the tangents on the butt-joints, as also across the plumb-bevel line rp, intersecting it at jp/ then A and C are the centres of the rail on the butt-joints, and p is the point from whence to square over to the face-edge to q; r q is then the distance to slide the face-mould along the slide line from the joint. Fig. 18 is a top view of the wreath, as it would appear, laid down flat on the work-bench with the face-mould applied to the upper side in the proper position ; the thick lines represent the face-mould, and the thin ones, the plank.
If the joints are made exactly square to the tangents on the face-mould, then marked by the same with a knife on the plank and made correctly by that knife-mark square through, there is no
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use to draw the tangent lines on the plank, although they are so indicated in the diagram, for the purpose of showing their relative position after the face-mould has "been moved.
From the points a' and d draw the slide-lines on the plank the same as on the face-mould, set the distance r q, found as in Fig. 17, from a' on the slide line upward, and place the corresponding point A' of the face-mould in the point so fdund, keeping at the same time the direction of the respective slide-lines coinciding to each other. It is obvious, that the other tangent point C will slide as much over d, as A' does away from a\ The face-mould being now in proper position on the face-side of the wreath, it is evident, that the tangent lines on the same are still perpendicular to the butt-joints of the wreath, and a line as a" A! squared over from the butt-joint to the point A', will be an extension of the tangent on the face-mould. The upper tangent C of the face-mould, crosses the edge of its butt-joint in c". Now, lines drawn from these points a" and d' across the butt-joints through the centre of the rail, are the spring-bevel lines. While the face-mould is in that position, the outlines of its sides are marked on the plank, to take off the superfluous wood, as has been explained before. On the underside of the plank, the face-mould is applied in a similar manner, with the difference, that it is moved along the slide-line from the upper butt-joint, the same distance as r q the other way, that is, downward, and its outlines also marked on the plank, to shape the cylindrical* sides by. It is obvious, that there is no need of marking the position of the tangents while the face-mould is in position on the underside; although it is well to call the attention of the reader to the fact that, if the plank is taken to an exactly parallel thickness, and the guage-marks run exactly through its centre, then the tangents of the face-moulds as applied to the under side, will agree with the spring-bevel lines on the lower edge 6f the butt-joints ; but otherwise, they will not agree exactly.
Now, for the very reason to avoid the unnecessary labor of taking the plank to an exact parallel thickness, the upper side of the wreath is faced off true, all squares applied and one half of the average thickness guaged on for the centres, from that side, and thus the spring-bevel is correctly obtained, independent of its thickness; it is also evident that, as a general rule, plank are sawed near enough to a uniform thickness, for the purpose of marking the outlines of the face-mould on the under side to form the cylindrical surface. If a face-mould were applied to each side of the wreath at the same time, they would appear in vertical view as in Fig. 17.
It will now be evident to the reader, that in such position, the respective tangent and slide-line points, as A!', A and A', also Q>", C and C, will be contained in vertical lines, and the respective tangents and slide-lines, in vertical planes.
This whole operation is represented very comprehensively in a perspective view in Fig 19 on a somewhat larger scale, in order to make the different lines more discernible.
The letters o, a, b, c represent the horizontal projection of the tangents and centre of the circles, and refer to like letters of the same points in Fig. 16. A B and B C are the imaginary tangents supposed to be contained in the cutting plane, which passes through the centre of the plank, while the. shaded surfaces ^JAB and c b C B (partly hidden by the plank) represent the vertical projecting planes of these tangents. The upper figure, drawn in thick lines, is the face-mould in position on the face or upper side of the plank ; A a' and C d are the lines squared over the butt-joints by the face-side of the plank from the points a' and &, where the tangents and slide lines meet on the joint of the face-mould. The face-mould being moved the proper distance on the slide-line, a" A' is then an extension of tangent A' B', and the line a" A, the spring-bevel line through the lower butt-joint; likewise, tangent C B' crosses the edge of the upper butt-joint at c", consequently c" C is the spring-bevel line for that joint.
The doubts expressed by some writers on the subject, as to the correctness of the spring-bevel obtained by this system, are unfounded, and are occasioned perhaps by an ignorance of the theoretical principles of the art, in all its bearings.
By a similar operation with the face-mould, the spring-bevels, in all cases where any occur, are found with despatch and correctness.
As the face-mould has to be placed in the proper position for marking its outlines on the
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wreath anyhow, it seems almost ridiculous, that any time should be spent in finding those bevels some other way.
For the purpose of making this work complete, the construction of these spring-bevels from the plan, will now be explained.
TO   FIND   THE   SPRING-BEVEL   BY   CONSTRUCTION.
Let it be considered, that the butt-joint of a wreath-piece is a plane, perpendicular to its tangent; therefore, all lines drawn on its surface must also be perpendicular to such tangent. Further, the tangent in space is the line of intersection of a vertical plane with the cutting plane; the upper edge of the joint represents therefore a line, contained in a plane parallel to the cutting plane, and is perpendicular to the tangent in space. The other side of the angle forming the spring-bevel, is a line contained in the vertical plane of the tangent, and also perpendicular to such tangent in space.
Now, here are two planes�the cutting plane and the vertical plane�intersecting each other, and also two lines, one in each of the two planes, perpendicular to that line* of intersection in one point; therefore, the angle formed by these two lines, which is the spring-bevel, is the diedral angle made by the two planes. The finding of the spring-bevel is, for this reason, a practical application of Problems XVIII and XIX.
Every wreath-piece, whose face-mould is produced by the intersection of an oblique cutting plane, will have a spring-bevel on each butt-joint. In that case, when both tangents are of equal length and inclination (as is the case in Fig. 16), the spring-bevel on both joints is the same.
In that case, when either one of the tangents assumes a horizontal position in space, the bevel on such joint will be simply the plumb-bevel of the developed auxiliary plane.
In all cases, where there is a spring-bevel on the upper joint of a wreath, by assuming the horizontal projection of the upper tangent as the base line of the vertical plane of projection, the diedral angle formed by the cutting plane and the vertical plane of projection, as demonstrated in Problem XVIII, Figs. 2, 3 and 4, Plate VIII, is the spring-bevel.                                   w
In all cases where there is a spring-bevel on the lower joint of the wreath-piece, it is found in the diedral angle formed by the cutting plane and the vertical plane containing the lower tangent, as demonstrated in Problem XIX, Figs. 5, 6 and 7, Plate VIII.
This method of finding the spring-bevel, although plain enough for any one who will study the theory of the problems thoroughly, may appear complicated to new beginners, and mistakes may occur. The author, being well aware, that few readers will take the necessary time for study if they can get along without, has introduced in Figs. 20 to 23 of Plate IX the development of the spring-bevels by construction, in conjunction with the plumb-bevels, for all the various positions which tangents may assume, in the drawing of face-moulds.
Fig. 20 represents the tangents as being of unequal inclination, but at right angles in their horizontal projection ; their wreath-piece must therefore reach over a quadrant of a cylinder. This position generally occurs in so-called quarter-circular, also in quarter-platform stairs.
Let R P" be the developed inclination of the lower tangent, R P being its horizontal projection, P' T' that of the upper one, with P T as horizontal projection ; then R Q is the horizontal trace (commonly called the intersection) of the cutting plane passing through both tangents.
To find the spring-bevel for the upper butt-joint, describe from P as centre (where the tangents intersect in their horizontal projection) an arc, just touching its tangent produced, and intersecting the horizontal projection of the same also extended, v&p ; from p draw a line to R, or the starting point of the lower tangent, and < P p R is the required spring-bevel
To find that of the lower butt-joint, the construction is precisely similar. From P as centre, describe the arc just touching its tangent, and intersecting the horizontal projection of the same in r ; from r draw a line to Q where the upper tangent intersects its own horizontal projection, and < P r Q is the spring bevel for the lower butt-joint (see Fig. 2 and 6, Plate VIII); < R P'" q is the
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plumb-bevel, and as that is always constructed in order to draw the face-mould, it will be explained in the different cases.
A little reflection will show, that if we conceive the triangle P p R turned up, using the side R P as a hinge, until P p stands square to Q P7, such triangle will occupy the same position, as the angle C c"d, Pig. 19, in the surface of that butt-joint.
R p will be a line contained in the cutting-plane, the same as ddr is one on the face of the wreath ; and P p will be square to the tangent P7 T7 contained in a vertical plane, the same as c" C is square to the tangent B7 C7, and contained in the vertical plane passing through that tangent. The same relation may be pointed out between the triangle P r Q, Fig. 20, and the angle A a" a' in the lower butt-joint, Pig. 19.
Fig. 21 represents the tangents as being again of unequal inclination, but their horizontal projections make an obtuse angle. This position occurs when the cylinders are so large, that it is not convenient to take a whole quadrant in one wreath-piece.
Let R P77 be the inclination and R P the horizontal projection of the lower tangent, and P7 T7 and P T those of the upper one ; the upper tangent extended, will intersect its own horizontal projection extended, in Q, and Q R is the horizontal trace of the cutting plane. As the horizontal projections of the tangents are not perpendicular to each other, as in Fig. 20, we must draw from P the point of their intersection, a line perpendicular to each of those projections, until they intersect the horizontal trace of the cutting plane, and the construction of the spring-bevels is then the same as in Fig. 20. To find the upper spring-bevel, draw P q perpendicular to P T the horizontal projection of the upper tangent, intersecting the horizontal trace in q ; from P as centre, describe an arc touching the upper tangent, and intersecting the horizontal projection of the same in p; then < P p q is the spring-bevel. The lower spring-bevel, < P r t, is found likewise (see Fig. 7, Plate VIII.).
Fig. 22 represents that case, where one of the tangents is horizontal and the other inclined ; their horizontal projections make an obtuse angle. (If the latter were at right angles, no spring-bevel would occur.)
Let R P77 be the inclined tangent, and R P its horizontal projection, also P7 T77 the horizontal tangent in position in space, and P T its horizontal projection.
The point R of the inclined tangent being contained in the assumed horizontal plane of projection, the horizontal trace R Q of the cutting plane, will be parallel to the horizontal tangent, or the same thing, to its horizontal projection. There will be a spring bevel on the lower butt-joint only. To find the same, the construction is similar as in the last figure. From P erect a line P s perpendicular to the horizontal projection P R of the inclined tangent, intersecting the horizontal trace R Q of the cutting plane in s ; also describe the arc from P as centre, touching the tangent R P77, and intersecting the horizontal projection of the same in r; then P r s will be the spring-bevel.
Conceive the triangle R P P77 in a vertical position over R P, and the triangle s P r revolved about s P until r P stands square to R P77, and it will readily be seen, that s b is the required bevel. The bevel on the horizontal tangent, will be the angle Q P777 P or the plumb-bevel. This occurs in such wreaths as the upper one g7 h7 m77, Fig. 15 A (see Fig. 5, Plate VIII.)-
Fig. 23 represents a case reverse to the last; here the lower tangent R Q is horizontal, and the upper one, Q P7, of which Q P is the horizontal projection, is inclined. Their horizontal projections form an obtuse angle ; if they were perpendicular to each other, no spring-bevel would occur.
R Q being the horizontal tangent, and contained in the horizontal plane of projection, is at the same time the horizontal trace of the cutting plane.
To find the spring-bevel for the upper butt-joint at P7, erect from P the perpendicular P s ; also describe the arc touching the tanget Q P', and s p P is the bevel.
That < s b7 is the required spring-bevel will readily be seen, if we conceive triangles Q P P7 and P sp revolved, until P p is square to Q P7 in space. The bevel for the lower joint is simply the angle P P77 �, or the plumb-bevel. This position of tangents occurs in turnouts and lower easements as m7 k7 f, Fig. 15 ^ (see Fig. 3 and 4, Plate VIII.).
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After the spring-bevels are found, a block-pattern of rectangular shape of the size of the rail is made, with a small hole in the center, and held by means of a brad-awl in the centre A, Fig. 19, of the butt-joint; its upper and lower sides are placed perpendicular to the spring-bevel line, and thus marked on the joint; the points for the dowel-pins are also pierced through at the same time. This gives at once a guide on each joint to shape the cylindrical vertical sides of the wreath, as well as those of the upper and lower winding surface.
A very good idea of this shaping process is given in Fig. 4, Plate XV., which is a perspective view of a wreath-piece, with the interior cylindrical side shaped out.
The block pattern as represented in Fig. 19, shows the corners as wanting on the upper and lower surface of the plank ; this is done to show the idea, that it is not necessary to use the plank thick enough to have them full, as they are taken off again more or less, when the rail is moulded. The other corners of the block pattern show full, and wood to spare; this shows that there is more wood than is wanted, as has been mentioned before, whenever the plank is sawed out as wide as the face-mould at such places, where it is wider than the width of the rail. In cases where many wreaths of one pattern are to be used, a separate pattern may be made to mark out the plank by, as is shown on a face-mould in Fig. 4, Plate XXVIII.
The development of a helix, as has been shown, is a straight line. The four edges of a rail are helices; the inside ones rise quicker than the outside, because they turn around a smaller cylinder, although having the same pitch. Therefore, to mark these edges for the purpose of shaping either the top or the underside of the rail, take a thin straight edge and bend it around laying close to the surface (after both vertical cylindrical surfaces have been shaped), with the edge calculated to mark by, in range of the respective sides of the block-pattern. These lines will produce the true shape of the helix required. When pieces are connected which lead to tangents of different inclination or pitch, the marking of the curve must be left to the judgment of the workman. However, if the cutting planes, in drawing the face-moulds, are located properly, the form of the wreath itself, after its cylindrical sides are shaped, will suggest at a glance where to take off the superfluous wood, and thus form a proper easing into other pieces.
It will be found advisable in practice, to shape out either the top-sides or under-sides of all the wreath pieces to where they connect with a level turn or straight piece, and bolt two and two together, to get a good looking and true shape over all the joints, and then guage the rail to a thickness. It is also advisable to do the same with the corners which are to be taken off, according to the pattern of the rail, in order to get true lines of the different members over the joints.
This point of the practical part is often neglected by stair-builders; nothing looks more insulting to the eye than those short humps and zigzag lines, so often met with in the strings and rails of staircases built by incompetent mechanics. The use of dry, well-seasoned lumber, correct drawings and tools (especially the try-square by which the butt-joints are made), and their accurate application, will ensure a correct hand-rail.
While bolting up the wreaths, particular attention must be paid that the spring-bevel lines on the butt-joints coincide, or are at least parallel, inasmuch the auger-bit will run sometimes while boring the dowel-pin holes, and thus throw the piece out of proper position. The spring-bevel lines are the only guide in such cases.
A rail thus correctly blocked out and moulded, may be glued all up for the length of a story in one piece, sand-papered and finished at the shop, and set right up from the newel in the building without trouble. This has been practiced in the author's establishment for fourteen years, and the first mistake has yet to occur. If the hand-rails are to be shipped out of town, they are glued up to such lengths as are convenient for shipment, but all joints are made at the shop, as far as the dimensions furnished by outsiders will warrant.
All the necessary instruction for cutting straight pieces between easements, wreaths, &c, will be given in the several cases throughout the book.
In this lengthy discussion on the general principles of hand-railing, enough has been said to enable the reader to " see through it" understandingly.
The explanation necessary in the description of all the different cases occuring in hand-
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railing, as laid down in the following plates, may then be reduced to the proper location of tangents, ascertaining the lengths of balusters, sliding of face-moulds in peculiar cases, &c. To introduce a certain brevity in these descriptions, the following considerations and terms will be a guide, to which reference may be had in all cases:�
The system of notation used in the problems of descriptive geometry has been adhered to, as it is the most convenient to comprehend readily the different locations of points and lines.
In order to distinguish at a glance, lines appertaining to the construction of the stair part, from those used to develop the face-moulds, whenever both occur in the same diagram, the former, such as indicate steps, risers, strings, cylinders, &c., are drawn in continuous fine lines_________
Of the lines appertaining to the construction of handrails, " all their outlines and centre-lines in horizontal projection or ground plan, also all tangents given or required in their various positions, and all traces of cutting planes, together with the face-moulds, their slide-lines, tangents and joints, as developed in finished shape," are drawn in thick continuous lines ___
All auxiliary tangents, also all traces of auxiliary planes and the lines of intersection of the latter with the cutting planes, are drawn in thick broken lines _ _ _� .__, __
All other auxiliary lines are drawn in fine broken lines------_______________
Of the ordinates, all those by which the principal points in the location of tangents and planes are obtained, are drawn in thick dotted lines mmmwmmiimammmnam
All other ordinates are drawn in fine dotted lines_______
The same letter is used to indicate the same point in all its various positions.
All points in horizontal projection or ground plan are indicated by small letters unaccented, while the same points in the developed position (such as the finished face-mould) are indicated by the same letters in capital respectively, and for all other positions of the point, small letters with one or more accents are used, as the case may require.
As all curved handrails or parts of them, may be considered as solids bounded vertically by two concentric cylindrical surfaces, all the face-moulds necessary for their construction may be developed from their horizontal projection alone, the location of the tangents being known. (See Fig. 9, Plate IX.)
The location of these tangents is indicated, by supposing their vertical projecting planes, as being revolved back into the horizontal plane of projection. The horizontal traces of these tangents give the horizontal trace or intersection of the cutting plane in which they are contained, and, as this horizontal trace alone is needed to locate the vertical auxiliary plane by which the face-mould is developed, we may ignore in the diagrams the vertical plane of projection and a ground-line ; but, considering the drawing of each separate face-mould as a separate diagram, we may assume the vertical projecting plane of one of the two tangents as the vertical plane of projection, and its horizontal trace as the ground-line ; then the location of such tangent in space, would become the vertical trace of the cutting-plane ; for instance: if we consider the vertical projecting plane of tangent b c�V c', Fig. 16, Plate IX, as the vertical plane of projection, and the horizontal projection b c of that tangent as the ground-line, then V cf would be the vertical trace of the cutting-plane a q c'.
The similarity of these diagrams with those used in the problems of descriptive geometry, where a ground-line is always assumed as parallel to the front of the draughtsman, will then become at once apparent.
Now, for reasons above stated, the various lines will be named as follows.
In speaking of a tangent, naming it by the unaccented letters, as: tangent a b, Fig. 16, Plate IX, the horizontal projection or ground-plan of such tangent is meant; while, tangent a b", means the same tangent in special reference to the position in space as indicated.
Trace or intersection of a cutting plane, means its horizontal trace, or its intersection with the assumed horizontal plane of projection.
Seat line, means the horizontal trace of the auxiliary plane by which the face-mould is developed ; while plumb-bevel line is the line of intersection of that auxiliary plane with the cutting plane, in revolved position.
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In speaking of ordinates, those indicating the perpendicular distances of any given points from the seat line, in their horizontal projection, are called horizontal ordinates. Vertical ordinates, are the perpendicular heights of the same points above the assumed horizontal plane of projection, and are therefore on paper, the horizontal ordinates extended between the seat line and plumb-level line. Developing ordinates, are those drawn perpendicular to the plumb-bevel line from the point of intersection of the vertical ones. The developing and horizontal ordinates are alike respectively.
For instance, in Fig. 16, a q is the intersection of the cutting plane, passing through the tangents a b" and b' c'.
a" c"' is the seat line, and a" c" the plumb-bevel line of the face-mould ABC.
c d" is the horizontal, and d" c" the vertical ordinate of the point c, while c" C is the developing ordinate of C.
Plumb-bevels are indicated by <p b.
Spring-bevels are also indicated in some cases ; < s b means that of the lower, and < s V that of the upper butt-joint.
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GENERAL PRINCIPLES OF PLANNING STAIRS, DEFINITION OF TERMS, ETC.
, Stairs may be divided into outside and inside stairs.
Outside stairs, called stoops, lead from the ground to the principal floor of a building.
In cities, they are generally constructed of stone, with railings of either stone or iron, while in the country, wood is mostly used.
Inside stairs, also called staircases, afford the means of communication between the different stories of a building, and are, according to its character, constructed of either wood, stone or iron. With the exception of those in the few fire-proof public buildings, stores and private mansions as yet erected in this country, staircases are built of wood.
This work treats particularly of the construction of wooden staircases, although the general principles are applicable to either material.
In the erection of iron staircases, wooden patterns have to be made to cast the different parts by, and are therefore subject to the same rules, as far as their plan and shape are concerned, as wooden staircases, with such modifications as the nature of the material involves to put the different parts together.
Staircases are divided into Principal or Main or Front-stairs and Rear or Private Stairs.
Both ought to be convenient, but especially the main stairs ; they ought to be well lighted and so planned and located, as to attract the attention of visitors on entering a building, as also afford easy access to the principal rooms.
Main stairs are mostly located in the main hall of the building, but sometimes in a separate space aside from immediate view ; in country residences, a separate tower is frequently designed purposely for the same.
Stairs located in halls from which parlor doors open either on one side or both, ought to be so planned, that the newel post stands back from such doors, or at least as far as their centre.
When the hall is so short that so much room cannot be gained, they must at least be so built that the doors will swing clear of the newel.
The width of stairs is in a measure governed by the width of the hall, which ranges in most city houses from 7 to 10 feet; on account of the cylinder, that of the stairs will range from about 3 to 4 feet. However, there are stairs as narrow as 2 feet 6 inches in small houses, and as wide as 6 feet in large, elegant private mansions.
For public halls, stairs ought to be not less than 6 feet wide, running straight, with a so called platform or resting place intervening.
Although the general plan of a staircase is laid out in the set of plans prepared by the architect, due consideration is not always given to this subject, and much depends on the personal experience of the stairbuilder, to make the best use, considering all circumstances, of the space allotted for the purpose.
Aside from the width and location, the convenience of a flight of stairs depends upon its form, rise and run.
In regard to form, stairs may be divided into straight, platform, and winding stairs.
In straight stairs, the string pieces run on a straight line each side, and all the steps are cut square to the edge at both ends.    Plate X.
Platform-stairs consist of two or more straight branches, with platforms or resting places intervening.
If the branches form a right angle on the ground plan, the platform consists of a square, usually about as large as the width of the stairs; they are termed quarter platform or quarter space.    Fig. 1, Plate XVI.
If the second branch returns from the platform in a direction toward the foot' of the first branch parallel to it, then the platform runs clear across the hall; such stairs. are termed half jplatform or half spaoe, Fig. 1, Plate XIV.
85
When the branches run neither at right angles nor parallel, they are termed angular plat, form, or angular space.    Fig. 2, Plate XVI.
Winding stairs are such as either wholly or partly turn around on a circular or other curve.
Such stairs as turn around one quadrant of a circle or ellipse, are termed quarter space winding. Figs. 1 and 2, Plate XXV ; also Fig. 3, Plate XXVIII. Such as turn a semicircle or ellipse, halfspace winding.    Figs. 1 & 2, Plate XXVII.
Either of these two may be connected with straight parts, without changing particularly the above terms applied.
Stairs which do not contain any straight steps, are generally termed full winding stairs. Fig. 3, Plate XXVII, and Fig. 1, Plate XXIX.
There are also winding platform stairs. Fig. 6, Plate XXIX. Stairs, which run on a platform as in this figure, and then branch off each way, either straight or winding, are termed
DOUBLE STAIRCASES.
Staircases may assume an endless variety of forms in the ground plan, such as the letter S or that of a heart, &c. ; but as all are composed of parts of circles or ellipses, there is no particular difficulty in their construction when a workman understands all the different cases laid down in this work.
The study of the problems of descriptive geometry in conjunction with these, will enable him to build any form of stairs, such as that of a conical helix, &c.
The rise and run of a staircase ought to be in a certain proportion, to make their ascent and descent as convenient as possible in the space given.
The rise is the vertical distance from the top of one step to the top of the next, and ought to be alike throughout each one flight.
The run is the horizontal distance from the face of one riser to that of the next; this also ought to be alike throughout the same flight.
An exception is usually made in what is termed a turnout.
The turnout, or invitation of a stair-case, is the curve given to the rail and steps at the bottom or starting of the straight part, Fig. 1, PI. XVIII., and is done to give it a graceful appearance. In winding stairs, a continuance of the curve of the cylinder combined with an increased width of steps, give the bottom the appearance of a turnout; Fig. 3, PI. XXVI., and Fig. 2, PL XXVII.
When the radius of the circular frontstring is large, as in Fig. 1, Pl^XXVIII., a smaller circle or other curve is employed at the bottom to produce the same effect.
In these turnouts, the width of the first two to four steps is increased along the backstring ; this produces a curve or swell in the edge of the steps, and also an increase of the run.
With this exception, the run of each one flight ought to be equal throughout.
In winding stairs the run of the steps is generally spaced off on the proper curve at about 18 to 20 inches from the railing, and made equal to those of the straight ones whenever they connect with any.
The custom of spacing off the run of winding steps through the middle of the stairs irrespective of their width, is erroneous ; a person having hold of the hand-rail will naturally walk at about the distance of 18 to 20 inches from the same, and for this reason the run on such a line ought to correspond as near as practicable to that of the straight steps.
Various rules exist to ascertain the proportions of rise and run ; but all of them only hold good within certain limits.
One of these rules is founded on the principle, that it takes twice as much effort to raise the body a certain height, as to walk the same distance forward on level ground.
Now, the usual convenient step is assumed to be 25 inches; therefore, if twice the rise is subtracted from 25 inches, the remainder ought to be the run of the steps in proportion to such rise.
For instance : rise to be 7 inches
2x7=14
25�14=11 inches run.
86
According to this rule the run for a 6 inch rise would be 13 inches, and for an 8 inch rise, 9 inches.
Another rule assumes the proportion of 6 inch rise by 12 inch run as the standard. The product of these is 72.
Now, to find the run for any other rise, divide 72 by the rise in inches, and the quotient is the run.
This rule would give for a 7 inch rise 10f" run, and for an 8" rise, a 9" run.
However, these proportions are not very definite, and may be varied by various considerations.
For private dwellings, about a 7" to Tj-" rise, with from 9^ to ll" run, make about as convenient stairs, as can be built.
The lower the rise, the more steps a person has to take to reach a certain height, and therefore it ought to be not less than 7".
If the height of the story is, say from 12 to 14 feet, a person will arrive at the landing of a flight of stairs having an 8" rise by 9-J-" run with less muscular effort, than on one having a 5" rise by 14" run.
The reason of this is obvious ; on the narrow step the foot has about room enough to plant itself firmly, and, during the aot of raising the body, the weight is supported by the leg while in an almost vertical position; but on the wide step the foot has to be moved forward so far, that during the act of raising the body the leg must support the weight while in an inclined position, which takes more power.
From this it would appear that a 5" rise in combination with a 9J run, would make easy stairs. Such is not the case however. The natural elevation of one foot above the other while contracting the leg is about 8 inches ; in swinging the elevated foot forward it descends slightly, and will strike gently on the next step when the riser is about 7 to 7�", while, when the rise is only 5 inches, the foot will again descend so much in order to strike the step, that the useless expenditure of power is felt.
On the other hand, when the rise is increased much more than 8 inches, the extra or unnatural effort which has to be made to elevate the foot high enough in order to plant it on the next step, will make the stairs "hard."
In public buildings a low rise with broad run is frequently used, especially in outside step? leading to the poriigo.
They give a grand appearance to the entrance, but if the number of steps exceed 10 or 12 in succession, their inconvenience becomes perceptible.
Whenever such are used in the interior of a building, platforms ought to be introduced at about every 10 steps, in order to relieve the muscles.
Platform stairs are the most convenient form of stairs in any building for various reasons ; they ought to be introduced whenever practicable.
It is evident, that the number of risers and steps depends upon the height of the story from top to top of floor, and the length of the horizontal space to be occupied by the stairs from the first to the last riser.
This horizontal length is termed the whole run, of the stairs.
To ascertain the number of risers to be employed when the height of the story and the whole run are given, the following rule may be used.
Add the height to one half of the whole run and their sum in feet is the number of risers.
For instance, height of story 12 feet, whole run 16 feet, number of risers=12+^===20 risers.
Divide 12 feet or 144 inches into 20 parts, which gives for each riser 7-J- inches.
Divide 16 feet or 192 inches into 19 parts, which gives for each step 10T2^ inches run.
Another case : Height of story 13 feet; whole run 30 feet; number of risers�=13+^~=13+1P �=28 risers.
13 feet divided by 28 gives each riser==5f".
30 feet divided by 27 gives each run�13�".
87
A third case: Height of story 9 feet 8 inches ; whole run 8 feet; number of risers=9x 8"+4' �13f risers.                                                                                                                            ,
As the number of risers must be a whole number, 13 or 14 may be employed in this case.
Assuming 14 risers, we divide 9 feet 8 inches by 14, which gives each riser==8f" ; and 8 feet divided by 13 gives each run===7Ty\                                                  #
Now this is too narrow for a run, although adding \\ inches for the nosing of the step, would make its width nearly 8| inches.
The nosing increasing thus the width of the step, may be of some advantage in going up stairs, but in coming down it will be of no advantage, as the heel of the foot has to clear it.
Narrow steps are very dangerous in going down stairs, and therefore it will be better in this case, to use only 13 risers making them about 9 inches, and 12 steps giving 8 inches run, and including nosing, 9J inches width of step.
In all cases, if the proportion of rise and run of a flight of stairs can be brought within the limits of the following schedule, they may be considered as being as convenient as can be made in the space at disposal:
Rise.			Run.	
5	inches	from 13	to	15   incl
5|	u	"     12	a	14       "
6	it	"     Hi	u	13       "
6*	a	"     10i	U	12       u
7	a	"     10	a	Hi      "
n	a	"  n	u	10i      fc<
7f	(C	"     9}	u	101      u
8	a	"     9	u	10        u
8i	a	"     8J	u	9|        "
8*	a	"     8J	a	9          "
9	u	"     8	a	8i        "
9*	a	"     7|	a	8i        "
10	u	"   n	u	8
Another important point in the planning of stairs is the headroom.
This is the vertical distance between the nosings of one flight, and the face of the plastering on the underside of the flight above.
This distance ought to be 8 to 9 feet, but certainly not less than 7 feet.
When stairs go up only one story, then the headroom is the vertical distance between the lower edge of the faceboard enclosing the opening across the stairs, and the step underneath the same.
In such cases 7 feet may be considered sufficient for most stairs, but ought to be more in wide first-class ones.
When headroom is low, 4 to 6 inches may be gained by leaving out the faceboard across the opening, and substituting an easement in the plastering.    In such cases 6 feet headroom will do.
It sometimes happens that the framing of the opening cannot be altered to make headroom enough ; then the run of the stairs must be contracted to accomplish that purpose.
Staircases may be either enclosed or open.
Enclosed stairs, also termed box-stairs, run up between two walls or partitions.
Open or cylinder stairs, run up along a wall on one side, and receive a railing with balusters on the other.
When open stairs have a railing on both sides without any partition underneath, they are self-supporting.
That part of double staircases, starting in the centre of the hall andrlanding on a platform is usually self-supporting.                                                         -  -
The construction of open stairs only will be treated'of in this work; that of closed stairs being governed by the same rules, the workman who can build Oopen stairs properly, will find no difficulty in planning and building closed ones.
PLATE    X.
STRAIGHT     CYLINDER    STAIRS.     .
Figs. 1 and 2 show the ground-plans, and Pig. 3 the side elevation of two flights of continued straight cylinder-stairs, one above the other.
They are of plain finish, with a railing, newel post and balusters suitable for common dwellings.
The same letters refer to the same parts in the three figures.
The different parts of a flight of stairs besides the railing, newel and balusters, consist of steps, risers, strings, cylinder and face-hoard.
The steps, or treads, numbered 1, 2, 3, &c, are usually made of 1�" pine-plank. First-class stairs receive iy steps, while in public buildings 2" hard wood plank is used ; they project over the vertical riser boards as many inches as they, the steps, have thickness- This projection is called the nosing, and is usually worked in the shape of a half-round, sometimes an Ogee, with a suitable cove underneath.
The face of the risers is indicated by dotted lines, running parallel to the nosing of the steps in Figs. 1 and 2, and by the letter A in Fig. 3.
Step, riserboard and cove are grooved nailed and glued together, as in Fig. 6, PL XI.
The last step is particularly called the landing, and is level with the floor.
The plank running along the wall into which the steps and risers are housed, is called the back or wallstring, B B'. The Frontstring C C is a plank, sawed out square through the horizontal way, to have the steps rest on it, while the risers are mitered to the same.
The cylinder D D' forms the curve at the top of the first and the bottom of the next flight, to connect thefrontstring with the so-called eaoeboard or skirting, E E'. The latter finishes the stair-opening along the floor timbers, dropping about �" below, to receive lath and plaster.
Cylinders, besides forming a pleasing curve, admit light through their opening, and are of particular benefit where halls are lighted by sky-lights.
At the bottom oi the frontstring, an easement F is carried square down to the floor, against which the baseboard is fitted and returned in itself.
The usual finish in common stairs at the lower edge of the frontstring, is a f" inch bead returned toward the back side, to give the mason a chance to run a quirk against it; this bead is carried around the lower edge of the cylinder and faceboard.
The finish on the backstring is always the same as that of the base Gr Gr' around the hall, and to which it is connected by easements H H' bottom and top. ' The same finish must be continued, of course, until it strikes a door-casing on the next floor.
The framing of the timbers around the opening is indicated by dotted lines on the ground-plan.
Usually the timbers run crosswise through the building, and therefore at right angles to the line of the stair-opening.
Those two timbers which form the ends of the opening, are called the headers, J J', and the timber K K7 running lengthwise of the opening between them and into which the other floor-timbers L 1/ are framed, is called the trimmer. Those three timbers ought to be somewhat thicker than the rest.
The ends of the steps resting on the frontstring, are sawed off square and flush with the face of it, excepting the nosing, which is mitered in order to receive the return nosing.
TIJL
JMJ^
&
� ^
CE
k
l^ig
II.
7==r   JH'
J>
T^tr:
K'

it/"
89
Qn each straight step ther,e are two dove-tails cut in, to receive the corresponding dove-tails of the balusters, with the exception of the first step on which the newel post stands; this has only one dove-tail.
To build a flight of stairs, for instance for the first story as represented on this plate, the following dimensions must be taken:
1st. Measure the height, a e of the first story from top to top of floor.
2d. Plumb down from e9 the face of the header against which the stairs land, and measure the horizontal distance 5 a, to where the face of the first riser may be placed, taking into consideration the situation of doors. &c, as before mentioned.
3d. The width of the opening between the face of the trimmer and wall.
4th. The length of that opening between the face of the headers.
5th. The depth of the second floor timbers; if these are cross-furred, measure from top of timber to underside of furring.
6th. Ascertain the plinth of the base together with the shape of the moulding.
7th. The thickness of the flooring at the landing.
This plan is drawn on a scale of -J" to the foot.
The height of the story is 9/ between joists ; these are 8", making in all 9' 8" from top to top of timbers; as the flooring is usually the same thickness on both floors, the-height from top to top of floor will be the same, 9' 8".
The distance from the header to the first riser is, say 13'.
The width of the opening from face of trimmer to the studs is 3' 10". Estimating the width of the stairs, we must allow 1" for lath and plastering, also 1" or 1J" for the faceboard as the case may be, leaving 37 8" for the stairs and cylinder; the proper size of the cylinder would be 9" in this case, leaving 2' 11" for the stairs from face of frontstring to back side of wallstring.
The thickness of the base must also be considered in this calculation ; supposing the base to be 1�" thick and the wallstring the same, then 2' 11" will be the right width of the stairs; but if the base (of course meaning from face of wall) is only f" and the wallstring 1�" thick, then the stairs must be the difference, or f" wider, in which case the plastering will partly set over on the edge of the wallstring.
Using a 9" cylinder, and placing it even with the last riser on the front string, we have to deduct about 6" from the horizontal distance h a: 4|" for the hollow of the cylinder and \\" for the thickness of the wood, leaving 12' 6" as the whole run between the first and last riser.
Supposing the opening to be long enough to offer no obstacles in regard to headroom, by employing the rule given on page 86 to find the number of risers, we have:
Whole height,.....9' 8"
One-half of whole run,    .    .    6' 3"
Their sum,.....15' 11"
This being so near 16', we may employ 16 risers, making them 7�" ; and dividing 12' 6"=150" by 15, gives 10" run for each step.
The timbers being laid 12" from centers, cutting out 11 of them would give 11' 9" length of trimmer or opening.
To calculate the head room in this case and supposing there were no stairs above, we must take out 6" for the cylinder of the landing and 1�" for the faceboard across the opening, leaving clear between the last riser and said faceboard 11' If" ; this distance divided by 10" or the run, would allow thirteen steps and a few inches over; therefore a person going down stairs, while arriving under the face-board would have gone down thirteen risers : 13 x 7�"=94i". Out of this height take 8" for the timber, 1J" for flooring and �" for plastering, in all 10", which leaves 84�"= T ��" headroom.
To make the calculation for the second flight, we may suppose the height of the story to be the same as that of the first; the width of the opening is, as a matter of course, the same, and the header
12
90
against which the stairs land, is usually plumb or vertical over the header of the first story landing; therefore the horizontal space occupied by the second flight is the same as the length of the opening, namely, ll7 9".
The width of the stairs is the same, unless there is a difference in the thickness of the base.
For instance, if the base is 1J" thick on the first floor, and f-" on the second, the face of the wall being straight and plumb, the second story stairs must be |/; wider than those of the first story, in order to bring both front strings plumb over each other.
The cylinders will also be the same as in the first story, and as there is one at each end, we must allow 6" for each, or 12" for both. Placing them at the face of the first and last riser in the front string, this arrangement will leave then 107 9" as the whole run between the first and last risers.
Employing the same rule again we find : number of risers, 9' 8" + 5' 4J-"�45' i//==15 risers. Dividing 9' 8" by 15, gives 7J-J-" for each rise.
As carpenters are always in the habit of calculating by the even fractions of inches, as halves, quarters, eighths, sixteenths, &c, we may reduce the remaining 11" into 44 quarters of an inch.
Now, if we had 45 quarters, the rise would be 7f"5 and as this would carry the landing only i" higher than it ought to be, by taking the rise a trifile scant of 7f " it will be as near as the workman could get at it by means of his rule ; or, to be precise, the rise will be -fe" less than 7|".
To find the run of the steps, we divide 10' 9" into 14 parts, giving 9T3^//; now, if we make the run 9^-", we would have ���" to spare in the whole run.
The headroom calculated as above, will be a trifle contracted on account of the depth of the front string, which receives a slight easing into the cylinder, as at D', Fig. 3, looking at it sideways ; but, as the upper flight is steeper than the lower one, this headroom increases further up the stairs, and will be sufficient.
In case more headroom were wanted, the first story stairs may be made with 15 risers only, or another timber cut out at the foot of the second flight.
Some owners would prefer a low rise and less headroom, while others would rather have the first story rise somewhat higher and more headroom.
If the stair builder be able to figure all this upon the spot, and thus present the case in itg different bearings, he will be more likely to please the parties interested.
PI .XI.
PLATE   XL
DETAILS IN THE CONSTRUCTION OF STRAIGHT CYLINDER-STAIRS
AND RAILS.
The figures on this plate are drawn on a scale of one inch to the foot, except Pig. 14 and ^9 iip �j, �p their scale being 2" to the foot.
Pig. 1 shows the wallstring ; the lower part has the top of the step and the face of the riser marked on it, while the upper part shows the mortised grooves, to receive the steps and risers.
To lay this string out, make a so-called pitch-board.
The pitch-board is a right angled triangle, made of a piece of dry seasoned board, whose sides are equal to the rise and run of the stairs ; the hypothenuse is planed off accurate, and the grain of the wood ought to run about parallel with it, that the sharp corners may not be easily damaged.
Such a pitch-board is made on purpose for every flight, (unless there is more than one flight of the same rise and run) and after the strings are laid out, it is laid aside for future use, to get out the rail by.
Joint the upper edge of the plank intended for the string, usually 11" to 12" wide, and run a guage mark A D the whole length, leaving about 1�" between that and the lower edge of the plank ; there is no need of jointing this lower edge, although the plank ought to be of about a parallel width.
Place the hypothenuse of the pitch-board on this guage mark as at A and B, and make a knife mark along the sides, which will indicate the face of riser and top of step ; repeat this operation for as many steps as are wanted, always being particular to make the end points of the pitch-board coincide.
The grooves are laid out by a tapering pattern, with the shape of the nosing and cove at the narrow end, laying its top edge even with the mark of the step.
At its narrowed part, it is the thickness of the step, and at the widest about ��" is added.
A similar pattern is made for the risers, being their thickness at the narrow end ; both patterns are of the same taper, in order to have the wedges fit all alike ; these are all got out of a cor* responding taper running to a point, and are driven into the grooves under the steps and behind the riser boards, as is shown in Fig. 6.
These grooves are cut to the depth of one half of the thickness of wall-string.
A piece is glued on at the bottom of the string to form the easement E, connecting with the base ; at the top end, the part P below the landing, is sawed out far enough to admit the backside of the cylinder against the header.
When the string is set up, the joint at A sets on the top of the lower, and that at P hooks over on the top of the upper floor.
Pig. 2 shows the front string ; it is drawn in its relative position to the wall-string, the steps being indicated by dotted lines.
To lay out this string, joint off the lower edge of the plank, and run a guage mark 6" or more (according to the width of the timber to be used for the furring), and lay on the pitch-board, the same way as on the wall-string.
It will be noticed that the joint at A, is cut 1�", (or whatever thickness of step) above the point of the pitch-board ; this will bring the top of step horizontal, when set up.
At D is a shoulder, cut on a vertical line, to receive the cylinder.
A piece Gr is glued on the lower end, to form the easement.
Pig. 5 shows the cylinder ; it is made of 3 equal staves, of 2" plank; the joints run toward the centre, and therefore the bevel is equal on all the edges.
TJie side connected to the front-string, is taken off to one half of the thickness of it, and fitted to the shoulder at D.    Pig. 2.
92
The other side, connecting with the faceboard, is made to conform to the thickness of the ^same.
To get out these staves, face off the stuff and joint one edge square : set one end on the plan of the cylinder, mark the beveled edges a &, and run on the guage-marks ; bevel the edges accurate, and work the hollow side by a proper pattern.
These staves are simply rubbed together with good strong glue, and after the cylinder is all finished may be nailed to make them more secure.
Fig. 6 shows different modes of putting steps and risers together.
The one at A is the usual one.
The cove is plowed and glued into the underside of step, as a square cornered strip ; the riser is also glued and nailed against it: short blocks as a are glued behind both, which makes in all a very solid connection, not likely to squeak when the stairs are used.
The lower edge of the riser is mostly nailed against the back edge of the step as at D ; sometimes the step is plowed into the riser as at E ; this latter mode is used to prevent the showing of the joint, in case the steps should shrink. B and 0 show the risers as being rabbited, plowed and glued into the steps.
The steps, risers and coves being glued together, the nosings and coves are worked as shown in the figure.
The method of simply nailing the coves under, as in B, is not a very good one ; they will in time work loose and show joints.
Fig. 3 represents a step, mitered and dove-tailed ; the dotted line at the left, indicates the face of the wall-string, and the one parallel to the front edge, the face of the riser.
The right-hand end is sawed off square, to set flush with the face of the front string, except the miter at a, to receive the return nosing.
The centres of the dove-tails are set one-half of the run of the step apart; the one next to the nosing must be so placed, that the base of the baluster will project as much over the face of the riser, as it does over the edge of the step on the front-string.
This projection varies with the size of the balusters.
As the centre of the baluster is also the centre of the rail plumb over it, and a shoulder is also cut on the dove-tail of the baluster toward the stairs, the depth of the dovetail is counted from the centre of baluster; for 1-J-" balusters, their centre sets y from the edge of the step, giving i" projection each way.
2" balusters f" giving I" projection. W     do       |"     do     y       do 3"       do       1"    do     \"       do
Supposing the balusters used to be 2", then the centre of the dove-tail b in the step, must be |" from the edge of the step, and also the same from the face of the riser ; the whole depth of it will be 1�", counted from the edge of the step. The run being 10", the dove-tails will be 5" from centres.
A side view of these dove-tails is given in Fig. 8.
Fig. 4 represents the landing ; the dotted line is the face of the last riser, while the dotted semicircle indicates the hollow side of the cylinder Fig. 5.
The nosing in the cylinder may be worked separate, or on the solid with the cove cut under ; the latter is preferable for small cylinders, and in this case 3 balusters, a, 6, c, will have to be put in with pins or tenons.
White pine stairs may as well be put up before the plastering is done, which has som decided advantages.
All partitions may be set, and all the plastering finished right up to strings, cylinders and face-boards, and thus considerable time gained.
Hard wood stairs, such as chestnut, walnut, butternut, oak, &c., ought to be put up on carriages, as the plastering would stain the wood.
The putting up of the carriage involves extra cost; therefore when only one flight of hard,
93
wood stairs is to be put up, all the other plastering may be done, and the stairs put up on top of the hard finish, as thus only the plastering on the underside of the stairs would remain to be done. A flight of stairs can be built and set up in the building without a carriage, much quicker than the same work can be done after the carriage is up; to say nothing about the labor on the carriage in the first place.
All pine stairs which may be put up before plastering, can be put together and trimmed in the shop all complete, winding stairs as well as straight ones.
They are then taken to the building, set up in proper position, furred, and the face-boards cased off.
To put a flight of stairs together in the shop after the different parts are all prepared, wedge the first and last step (not the landing) into the wall-string, and nail them also to the front-string ; the last one for good, but the first one temporary. This operation may be done most conveniently by placing both strings on a couple of saw benches in such a manner, that the two steps will range; then put in the steps from the top down, nailing every step and riser properly on the front-string. Having put them all in, turn up the stairs, placing the backside of the wall-string on a stout straight plank, lay a board or strip on the face-string covering the points of the risers, and cut braces between the latter board and the ceiling of the shop, thus pressing the steps firmly into the grooves of the wall-string.
Then glue well the wedges, driving them as well as the steps and risers, until they make a good fit on the face of the wall-string against the edges of the grooves ; nail the risers well to the back-edge of the steps, and release the braces.
A flight of stairs put together in this manner, will be as solid as a truss.
If the stairs are partly winding, the straight part is put together first, and the winders up to it.
The return-nosings may then be fitted, the bottom riser trimmed off even with the strings, the last riser put in, and the landing piece temporarily nailed on,
When setting the stairs up in place, the notch on the top end of the wall-string will hook on the second floor, while the back-side of the cylinder will rest against the header.
The bottom riser and both strings will set on the first floor, and may need some scribing on account of its frequent unevenness.
The stairs being properly secured, the landing piece is cut in, which ought to be wide enough to reach about one inch on the header, as shown on Plate X.
In regard to setting up an upper flight, it will be convenient to unscrew the bottom cylinder, until the stairs are secured.
Fig. 7 is a section through the stairs perpendicular to the strings.
Timbers A and B are put under the stairs to stiffen them, also to lath to. For stairs from 2'8;/ to 3' wide, two timbers are sufficient; one is nailed against the back-side of the front-string, and the other is placed about in the centre, fastened to the steps and risers by rough board-brackets, as shown in Figs. 2 and 4, Plate XIV.
It is evident, the under-side of these timbers must set back about f" from the edge of the string, to receive lath and plaster L L.
A furring strip C is nailed against the wall in range with both timbers to lath to.
The timber under the first flight will rest against the header and on the floor ; under the second flight, they rest also against the header at the top, but are secured to the header at the bottom by means of hang-irons or iron bolts, as shown in Figs. 2 and 3, Plate XIV.
Fig. 8 represents the newel with cap and easement from the straight rail, also two balusters on one step.
Newel posts are usually cut 3r 6" long ; of this length 2' 4" is taken up by the turning and shaft from the under-side of cap to the top of the base, and 1" for the pin, which leaves 1' V for the base.
Balusters are usually cut 2' 4" and 2' 8" long ; the common average of risers is assumed to be 8", hence a difference of 4" in their length, as it takes one short and one long one on every
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straight step. Of this length 1J" is taken for the dove-tail, 3" for the base, 16" for the shaft and turned members on the short, and V 10" on the long ones, leaving a straight pin of 5f in length at the upper end of each baluster.
The newel post is usually so set that its centre comes in range with the face of the first riser one way, and with the centre of the balusters the other.
The base of the newel ought to show as much above the first step as that of the balusters, say 3" ; adding this to the turning and shaft which is 2' 4", will make in all 2' 7" from top of step to under-side of miter-cap ; to allow the balusters to enter into the rail, the latter must be at least 6" lower plumb over the face of the risers, than the post; this difference is taken up by the so-called EASEMENT.
Fig. 9 shows how to lay out the pattern for such an easement in a practical manner.
Fig. 10 is a top view of the same ; as such an easement appears straight in the ground plan, it is termed a straight easement, in distinction from the turnout (Plate XVIII), which is curved both ways.
To lay out the pattern, joint one edge of a piece of board about 7" wide and V 6" long; run on a guage-mark a C K, one-half of the thickness of the rail; place the hypothenuse of the pitch-board on this line, and mark the rise a b and the run b C.
On the line a b set up 6" to o, and describe from this point a circle to represent the largest diameter of the cap ; from o draw also a line o J perpendicular to the rise a b ; from the point F^ where the circle intersects o J, set off to Q one-half of the width of the rail; then a line through O parallel to a 5, will be the vertical line of the miter point g, Fig. 10. h and I are the points where the outsides of the rail intersect the cap.
This rule of laying out the miter joints ought to be followed in all cases.
It is the only way to make the members of the cap and rail agree, and, in fact, if the distance gf Fig. 10 is more or less than one-half of the width of the rail, the members will not agree, as can easily be proved by the principles of intersection of solids in descriptive geometry.
If the cap is turned on its outer edge anything near to the shape of one-half of the rail, the easement, mitered in and worked by this plan will be so near the shape of the rail, that any slight variation on account of careless turning will hardly be perceived.
To find the points h and I on the pattern, there is no need of drawing a figure like Fig 10, but they may be found in Fig. 9 at once, by setting off one-half of the width of the rail parallel to the centre line G J, intersecting the circle at m and n; the vertical line m n is then the mark to which the miter is to be taken off each side of the easement.
If we conceive the line a 5, Fig. 9, to be contained in a vertical line from the face of the first ris�r, then the point C will be plumb over the face of the second ; Gr J will be the horizontal and J K the inclined tangent of the easement pattern. From these tangents set off one-half of the thickness of the rail for the underside, and mark the curve by means of a slender rod.
Having trimmed off the under side, the pattern is guaged to parallel width, as shown in the figure.
Remark. �The usual method of intersecting lines employed in such and similar cases to draw a curve tangent to two lines, is an unnecessary loss of time.
Setting up'6" on the line of the riser from the point a to the centre o of the cap and making the newel 2' 7" high from top of step, will bring the under side of the straight rail 2' f" high above the top of step on a line plumb over the face of the riser, as is indicated by figures in Fig. 8.
The centres of the short balusters set back from the riser, according to their size, from #' to 1", which again causes a variation in their length to where the pins enter the rail; the lifference in the pitch of the stairs also causes such a variation.
The above calculation gives about -J" to spare in their length, which in a flight of the rise and run as assumed in this example, would have to be cut off of each baluster, while they would be just about long enough on a steep flight, as the second story, PL X.
In order to have the top members of all the balusters follow the curve of the rail, the shaft
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of the baluster next to the newel post, must be about 2" longer than that of a common long one, as indicated in Fig. 8.
For first class stairs, the difference in the length of the shafts of the short and long balusters, ought to be evidently one half of a riser, in order to have the top members parallel to the rail. A set of them must therefore be turned on purpose.
It will be seen by Fig. 8, that the short baluster on the second step, ought to be also a trifle longer than the rest; but very seldom any attention is paid to it.
In case a newel should not be more than 2' 6" to the underside of cap, set up only fi" instead of 6" on the line of the riser, which makes the easement somewhat quicker, but the rail will be the same height above the steps.
In all cases, whatever is set up to the centre of the cap, add to 2f 1", which gives the height of the newel-post, with the use of the usual length of balusters.
It is evident, that from the square line through the point C which indicates one step, any number of steps may be counted off by the hypothenuse of the pitch-board along the straight rail.
Fig. 11 shows the plan of the rail around the cylinder, and the development of the face-mould for the twist.
Let the fine line indicate the edge of the step, and the semicircle the inside of the cylinder ; draw the centre of the rail or its tangents a c, c e and ef through the centre of the balusters, and its outlines one half of its width each way from the centre.
It is evident, that the point d must be as much from the cylinder as b or/, and that the circles of the outlines are struck from the centre of the cylinder.
There is a joint made at d, and the piece d ef is a level quarter-turn, with as much straight wood added along the face board, as is convenient; the joint at d is -made square to tangent de, which tangent is of course, perpendicular to a c and ef.
The twist is got out of a piece of plank of about the same thickness as the rail is wide, and the drawing of the face-mould is an application of Problem XXVI.    Fig. 7, PL IX.
The point at b indicates the last rise ; and a b B is the pitch-board ; its hypothenuse a B is then the inclined tangent through the centre of the straight rail. From c erect the perpendicular c C intersecting the tangent extended in C ; draw also the necessary ordinates to obtain the outside points of the rail, and some intermediate ones in the curve. At the intersection of these with the tangent, as also through the points a, B and C, draw lines perpendicular to the inclined tangent, and make them equal to the respective horizontal ordinates ; as 0 D equal to c d; on the straight part set off one half of the width of the rail each way from the tangent, and the joint at D is made square to the tangent C D, or the same thing, parallel to a C.
Instead of using ordinates, the curved portion of the face-mould may be drawn by a string and two brad-awls as indicated in Fig. 1, PL XII, or by the trammel, Fig. 5 of the same plate.
Fig. 12 is a perspective view of the twist in position, with the face-mould applied to its face side.
To get out this twist, face off the top side and make the straight outside of the plank square with it; lay on the face-mould, its straight outside even with the square edge just mentioned, and mark the joints. Make both of them square to the face of the plank and mark a square line through the centre of the tangents, as d D and a A ; also guage on one half of the average thickness of the plank.
There will be no spring bevel to such a twist; at the lower joint A, the top and bottom side of the block pattern will be parallel to the face of the plank. At the upper joint apply the hypothenuse of the pitchboard to the top edge of the plank, and mark the plumb-line D D' through the centre D by the riser; apply the block pattern at the centre perpendicular to this plumb-line, as shown, and the lower side of it will then be horizontal when in position.
Apply the face-mould, sliding it along the outside of the plank and the joint D, until the tangent line at D' coincides with the plumb bevel-line through the centre ; then the outlines will also be in range with the outlines of the block pattern. Mark the inside curve while in this position ; apply the face-mould also to the underside of the twist, sliding downward in a similar manner, and mark the outside curve of the face-mould.
96
The superfluous wood is then taken off vertically inside and out, as shown in Fig. 4, PL XII., and the top side shaped as indicated by the broken line in the same figure.
Remaek.�The elliptical curve produced by the intersection of the cutting plane with the vertical cylindrical surface through the center of the rail, is in theory the real easement in these cases. In practice, the inside top edge of the twist must rise above such a curve in the middle as at m, and the outside top edge must drop below at the upper end where it comes around to a level. Both are therefore irregular helical curves.
It is evident that the twist, Fig. 12., contains the length of one step between the point B and the square joint at A. However, such twists may contain more or less straight wood to suit convenience ; the joints being made square from the outside, the straight wood is then counted from the square line through B.
In order to cut the straight rail to a length, we have the length 'of one step in the easement and cap to the point C Fig. 9 ; B, Fig. 12, is the last riser in the twist; consequently the straight part must contain the length of 14 steps (run off by the hypothenuse of the pitchboard) between those two points (for the first story flight represented on PI. X).
That rail which goes up another story, must have a level quarter turn with a twist as shown in Fig. 13, to correspond with the cylinder at the bottom of the second flight; therefore the level part of the rail is cut the length of the faceboard between the turns, including the straight wood contained in them.
The bottom and top twist of the second flight are made the same way as the one described, with the difference of applying the second story pitchboard instead of the first.
As the second story is steeper than the first, and the cylinders are the same, it is evident that the distance of B C, Fig. 11, must be more on the steeper flight; there is no need of drawing another face-mould on that account, but such distance, whatever it is, may be simply marked on the facemould, and the straight wood is then counted from that point on the twist.
Supposing E, Fig. 13, to be such point (which will be the same on both bottom and top twist) then the length of fourteen steps between those two points, will be the length of the straight rail. It will be seen that the tangent rises about half of a rise above the last riser, because the distance b c, Fig. 11, is nearly equal to half of the run of the steps ; therefore it takes long balusters along the level floor. The tangent of the bottom turn of the second flight rises the same way, one half rise or more, before it reaches the first riser; now this rising in both turns makes up for the first riser of the second story, and therefore the rail on the second flight will be the same height above the steps, as on the first. The precise height which the tangent B C, Fig. 11, rises, may be ascertained on the pitchboard.
The cylinders should be so arranged, that the distance from the last riser to the centre of the rail is precisely or nearly, one-half of the run of the steps.
Fig. 14 shows the manner of doweling and bolting the rail-pieces together by means of the so-called hand-rail bolts.
There are two kinds; with two nuts and with one nut. 1  The two-met bolts have a square one at one end, and a ratched one on the other; these are used as represented in the figure, whenever the grain of the wood runs lengthwise with the rail. The nuts are introduced through mortices on the underside of the rail, and the ratched nut is drawn up by a nail-set or small cold chisel.
The one-nut or single nut bolts, have a coarse screw thread at one end, which is turned into the twist or wreath whenever it presents the grain of the wood crosswise enough to warrant their use ; the other end has a ratched nut, the same as the two-nut bolts.
Figs. Al@ and D are sections of moulded rails which are much in use; M and D are called toad or hunchbacks.
To put up a rail, set the newel-post first.
This is fastened by means of a pin let into the step, or by a bed-bolt, while the balance of the base is let down through the floor
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String up the rail, and hold it temporarily in position by a piece of board notched out and projecting from over the level floor; plumb and bore the rail for those balusters which are mortised or bored into the landing piece, and put them in; then plumb and bore the- rail for every baluster from the centre of the dovetails.
Along the level parts of the floor set the balusters from 4�" to 6" apart from centres, according to size.
Whenever the end of the rail strikes the plastering,  a round fancy turned platter, from
5" to 8" in diameter by F thick, ought to be fastened to the end of the rail and then the platter
screwed to the wall, to prevent the cracking of the latter.
While the stair part is being put up, or any time before the plastering is done, a piece of studding ought to be fastened into the partition or the brick work wherever the rail will strike, in order to fasten the platter by means of the screws.
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PLATE   XII.
STRAIGHT CYLINDER-STAIKS AND RAILS�Coiminjm
Fig. 1 shows the construction of a large cylinder, with the disposition of the dove-tails and the last riser, also the face-mould for the twist.
Fig* 2 is the plan of the staves, and Fig. 3 a section through the cylinder, showing also the brackets, fillets and finish on the lower edge of the front-string.
Scale 2" to the foot.
The cylinder is 18J" diameter and the fillets �" thick, leaving the opening between the edge of the steps and the edge of the floor, or in other words, the distance between fillets, 18". The run of the stairs is 10i" by 7jr" rise and the size of the balusters 2�" ; therefore, the centre of the rail sets back f" from the face of fillet. In order to have the rail rise one half of a riser above the landing, the distance from the face of the last riser c to the centre of the rail d, must be one half of the run, in this case 5�". The last step will rest partly on the cylinder, and must have a corresponding curve, from the joint of the cylinder to the point/.
The front-string is mitered as usual, but as the edge of the step projects the thickness of the fillet, the miter* of the riser will also pass by the string the same; the fillet along the riser is beveled and thus forms a miter-joint with the edge of the same. This �" is equalized on the front-string on the last step, between the miter-point and the j bint of the cylinder, that is: the distance from the miter-point a Fig. 2 to the joint b of the cylinder, is i" less than that between the face of the riser a Fig. 1 and the joint b of the cylinder. The horizontal fillet shows as much below the cove, as the vertical ft] let against the riser.
The nosings in the cylinder are worked separate, and the centre of the dove-tails all show toward the centre of the cylinder, to make the bases of the balusters, (which are either square or octagon,) stand normal to the curve.
To draw the face-mould, apply the pitch-board A d C, placing the rise c'Cona line with the face of the last riser, and proceed as on the previous plate to find the points D and E. Instead of finding the curves by ordinates, the method of using the string and brad-awls is here indicated. O is the centre of both ellipses, O V and O W one half of the minor, and O G and OH one half of the major axes. The line of the latter is parallel to the tangent AD; n, nf and m, mf are the focii. The staves as represented in Fig. 2 are five ; four equal ones, and one odd one from the joint b to the miter/. It will be noticed, that the bevel bfs of the joint from the face of the odd stave, is different from that of the others ; the thick line at/ indicates the miter of the last riser, and the one at a that of the second last.
The finish of these stairs being calculated as first class, there is a moulding p Fig. 3, put all around the lower edge of the string, to cover up the joint of the plastering.
This moulding ought to be worked out of a solid piece around the cylinder, and not cut around in short, straight pieces, as is frequently done ; r is the cross-furring.
Fig 4 is a perspective view of the twist, with the vertical sides worked off.
B is the line where the curve commences, and 0 the line to count the steps from, in case that the twist should not be just the length of one straight step.
Fig. 5 shows the plan of a 12" cylinder, with the requisite face-mould. The last riser is placed in the joint of the cylinder, although this makes the distance b c to the centre of the rail, more than one half of the run. Now in cases, where the difference is not too much, and the room can be had (as this takes just that difference more in the whole run), this may be done, to save the extra labor to cut partly into the cylinder with the last step.
To draw the face-mould, proceed as in the other cases.    Let a b B be one-half of the pitch-
PI XII.
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"board, which gives one half of the length of a step of straight rail in the face-mould. Assuming the run as lOf" and the rise 7�", and the distance from 5 to c being 6|" which exceeds one half of the run by If," the tangent B C will rise 4-|"above the horizontal line B c'.
The long "balusters "being only 4" longer than the short ones, this rising in the twist ought not to exceed that 4" too much. Guaging the thickness of the rail from the face-side on the straight part as represented in Fig. 6, instead of working from the centre as represented in Fig. 4, while the centre of the rail on the upper joint is placed as usual: the rail will drop �" on the level, "bringing it thus down to the proper height, and "besides, will make an easier curve on the inside of the twist.
This variation may "be found as follows : place the face-mould in proper position on the face side of the plank as shown in Fig. 6, and mark the plumb-line b V of the cylinder joint B', down on one side.
By working from the centre of the plank, the centre of the rail will be at b ; "but "by taking the face of the plank as the upper side of the rail, the centre of it will be at V and the centre of the upper joint will be the distance b b' lowered from where it would be in the first case ; this difference amounts in this instance to f "
Working a rail from the face-side, gives also more straight wood in the twist; while we should count from the point 5, for the length of the straight rail when working from the centre of the plank, we must count from b' for that same purpose if working from the face-side ; thus giving the length e b' more straight wrood.
It is evident this method may also be employed for bottom twists.
Both top and bottom twists are frequently made in this manner, on account of the easier appearance of the inside curve ; but the outside curve looks rather awkward, because there is too much level part in it.
The quick spot, so often noticed on the inside of small cylinder turns near the centre joint, may be avoided by leaving some wood on the twist below the block-pattern, and using for the level turn a piece somewhat thicker than the rail; bolt the top edges even, and ease over on the underside into the level turn.
HOW TO  FIND  THE  LENGTH  OF  THE  ODD  BALUSTEES  FOR  TWISTS AND
WREATH-PIECES.
In winding stairs, where the centre-curve of rails form perfect helices corresponding to the nosings, the balusters are of the same length, as has been demonstrated in the explanation of Fig. 14, Plate IX. In straight cylinder stairs, where the cylinders are small, long balusters of the usual length as used on the steps, come near enough for all practical purposes in the cylinder; but in larger cylinders, as those represented in Figs. 1 and 5, Plate XII, balusters situated as No. 4, require to be of a length between a short and a long one, in order to show the top members as being contained in a curve corresponding to the centre of the under-side of the rail.
To understand the idea of finding these lengths, let the nature of a central curve conceived as running through the centre of the rail, and also the curve of the centre of the under-side of the same, be considered.
The central curve of twists and wreath-pieces which connect with straight wood at either one or both ends, nearly coincide with the elliptical curves produced by the intersection of the cutting-plane with a vertical cylindrical surface, passing through the centre of the twist; the central curve will therefore be contained in a plane. The centre of the under-side of the rail, will form a curve parallel to the central one in space, but will not be contained in a plane precisely^ unless both tangents have the same inclination.
However, this variation can be taken into consideration, and the lengths of the various balusters may be ascertained, by comparing the lengths of the vertical ordinates passed through their centres.
We shall now proceed to find the length of baluster No. 4, Fig. 15 Plate XII, while those
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which occur in the different forms of platform and winding stairs will be explained in each separate case.
Fig. 7 shows the development of both curves of the twist, Fig. 1. The thick broken line ABCEis the central curve, while the continuous thick line a b c e is the centre line of the underside of the rail, represented as laying on the edge a of the last step, and c of the landing. 1, 2, 3, 4 and 5 are verticals through the centre of the balusters. The development of the central curve A B C B, is obtained by taking a b Fig. 1, and the stretchout of the quadrant b e as a base line, using the corresponding vertical ordinates over A d\ as heights.
The curve ab c e Fig. 7, is drawn parallel to it, the perpendicular distance between them representing one-half of the thickness of the rail, which is in this case 1�".
This causes a variation in their vertical distance : E e, where they join the level curve of the quarter-turn, is lj", while 0 c on the inclined portion is If".
Balusters Nos* 1 and 3 are evidently short ones of the regular length, while No. 2 is a regular long one, being one-half of one rise, or 3�" longer than a short one.
The vertical line 3 3a shows, that the under-side of the rail rises ��" over the top of the step, when it reaches the centre of that baluster; as both curves are there nearly straight, the central curve rises at the point 3b the same distance of ���" above a horizontal line through C. There are fine dotted lines drawn horizontally through 3a and 3b, to indicate the heights which the curves rise above the point of a short baluster.
The heights to the lower curve will naturally indicate how much longer Nos. 4 and 5 are than a short baluster, provided the vertical distances of the top members below the under-side of the rail were kept equal; No. 4 would be 2^" longer, and No. 5 as well as all the rest under the level portion of the rail, 3i" longer than a regular short one. The heights to the central curve are less ; No 4 is only 2�", and No. 5, 3\" more than No. 3, It will be noticed, that these heights are the same as those of the corresponding vertical ordinates over the line drawn through 3b Fig. 1 ; there, 4b is 2 i" higher than 3b, and D which represents No. 5, is �" higher than 4b.
This variation of the curves over the same points, is caused by the diminution in their vertical distance, as they approach the level portion. Now, practically, if the top members are vertically all equally distant below the under-side of the rail, they will appear closer to it along the inclined portion on the steps, than they do on the level portion, as at 5 and beyond.
Therefore, whenever the twist rises from centre to centre one-half of a rise above the landing, as is usually the case, the long balusters on the level portion are made at least as long as those on the steps.
In this case, instead of making them 3J" longer than a short one as indicated at e9 they are made 3f" longer, or one-half of a rise whatever that may be.
To be exact in the matter, this difference of �" ought to be proportionally increased in No. 4. Being situated about midway between 3 and 5, this portion of the difference would be -jr", making No. 4 instead of 2&" (as indicated by the lower curve) 2^/' longer than a short baluster ; or the same thing, 1A shorter than a long one.
In practice, in nine cases out of ten, the usual 2' 8" and 2' 4" balusters are used ; but, whatever may be the difference in their length, a proportional length for the odd balusters may be obtained from their vertical ordinates or the pitch-board, without drawing a development of the curves, by the following general rule :�
Find the difference in the vertical ordinates of the last regular short baluster and the level joint by the plan or pitch board; then subtract this difference from the difference between a long and short baluster as they are used, and divide the remainder in as many parts as there are spaces of balusters between the last regular short and the first regular long one on the level.
To ascertain how much longer an odd baluster is to be than the regular short one, add the difference in the length of their vertical ordinates, and also one part of the remainder before-mentioned, together to the length of the regular short one.
For an example, apply this rule to the present case: Let the long balusters be 3f" longer than the short ones.   The difference of the vertical ordinates, Nos. 3 and 5 Fig. 1 is 3�" ; taking 3�"
101
from 3f, leaves a remainder of �".    There being two spaces between Nos. 3 and 5, gives ^/V for each space. *
To ascertain the length of baluster No. 4, add T5�'' to 2J" which is the difference of the vertical ordinates of 3 and 4, giving 2TV Then No. 4 is to be %^'r longer than No. 3, which agrees with the result obtained in laying out the developed curves.
This same length may also be obtained by taking the length of the regular long one, No. 5, as a basis. The difference between the vertical ordinates of Nos. 4 and 5 is $" ; add to this one-half of the remainder, or ^y, making together Ijfa" ; subtract this from the length of No. 5, which gives the length of No. 4, agreeing again with the length found before. In case the long balusters used are 2' 8", which is 4" longer than the regular short ones, one-half of the remainder above-mentioned would be tY' instead of &" ; which would give, for the length of No. 4, 2^" longer than a short, or 1-^" shorter than a long one.
In cylinders of large diameter, say 3', there may be 2 or three odd balusters needed, making 3 or 4 spaces. Their lengths are found the same way ; divide the remainder in 3 or 4 parts, as the case may be, and add to the difference of the vertical ordinates as before. The length of baluster No. 4, Fig. 5, is found in a similar manner.
It is evident the central curve will not be contained in a plane in this case, but appears as a curved line instead of a straight one, if viewed in the direction of the cutting plane. The difference in the vertical ordinates of No. 4 and d (to the curve) is in this case f. The rise and run being the same as in Fig 1, the regular long balusters and the remainder are the same in this case as in the previous one. Therefore, add one-half of the remainder or ^.", to the difference in the vertical ordinates which is f", and deduct their sum, y", from the length of the regular long one, which gives the length of No. 4,
PLATE   XIII.
PECXJLIAK   CASES   OP   STRAIGHT   CYLINDER   STAIRS.
When cylinders are so small that the distance from the joint to the centre of rail will no: amount to one-half of the run of a step, the rail will not rise high enough if the last riser is placed in the joint; in such cases enough straight wood must be glued on the joint of the cylinder, to make up the difference.
Figs. 1, 2 and 3 represent this case in a 6" cylinder ; Scale V to the foot.
Fig. 1 is the ground plan of the last step and the cylinder ; run 10", cylinder 6", balusters H"; the distance c d being 3-J", it needs 1�" of straight wood to make 5" or one-half of the run. The face-mould a B C E is drawn as usual; a b B is the pitch-board. Cutting the rail to a length, count the number of straight steps from B.
Fig. 2 shows the construction of the cylinder ; it is made in two equal staves out of 2" plank ; the backside is taken off to set square against the header. The piece b c is glued up and down on the joint of tlie stave at c ; the miter of the last riser is shown at 5.
Fig. 3 is a top view of the landing piece.
Fig. 4 exhibits the ground-plan of a hall, with a flight of straight cylinder stairs, which takes up as little room as possible. The proper form of the stairs would be quarter-space winding, or at least a quarter platform at the top; but when the owner would rather put up with some inconvenience than go to the extra expense of building either, this plan may be adopted: Scale, i" to the foot; length of hall, 15' 1" ; height of story, 9' 6" between joists; depth of timbers, 7" ; width of front door 2' 11" ; width of stairs 2' 9".
To allow the front door to swing clear of the nosing of the first step, the face of the first riser must be at least 3'1" from the inside of the front wall; the landing at the top ought to be at least as much as the width of the stairs. Taking both the spaces at the foot and the landing, which amount to 5'10", out of 15' 1", will leave 9' 3" for the stairs between first and last riser. The whole height of story being 10' 1", will give 14 risers 8f ^" ; 13 steps of 8J" run will take up 9' 2-J-', leaving -J-" to spare of the space given, which may be added to the platform.
The first step is curtailed toward the front string, allowing the newel-post to set back in order to clear the door.
Fig. 5 shows the construction of the easement pattern, and Fig. 6 that of the wreath-piece at the top.    Scale, 2" to the foot.
The easement must of necessity be rather quick, and the newel post higher than usual, in order to have the hand-rail the usual height.
The easement pattern is drawn like Fig. 9, Plate XI. ; a b C is the pitch-board, and a C K the centre line of rail; set up in this case at least 9" to the centre of the cap ; placing the centre of the post one-half of the run, or 4J" from the face of the second riser, the centre o of the cap must be in the same vertical line ; describe the circle ; make F G equal to one-half of the width of the rail, and proceed as in Fig 9, Plate XI. Gt will be the miter-point, Gf J the horizontal, and J K the inclined tangent The point C will be over the face of the second riser, and therefore the point to count from when cutting the straight rail to a length.
Setting up 9" from the point a to the horizontal line of the centre of the cap, will make the newel-post 2" 10" high from the top of the curtailed step to the under side of the miter cap.
Let//", Fig. 6, be the face of the last riser, and/ a equal to one step or run ; then ab, b d, d e, are the tangents in horizontal projection at right angles. To locate their position in space, *vpply the pitchboard aff\ the run coinciding with tangent a b ; then aff is the tangent of the straight rail, intersecting d b extended in 6" / draw the perpendiculars b b\ c c\ and d d'\ and carry the height b 5", (which the first tangent rises) around to b\ as indicated by the dotted arc.
P1XIIL
Pig. I.
Fig.n.
Titf. m.
K|
103
The tangent d e is to connect with the level straight rail, and is therefore horizontal in space. The whole twist being made in two pieces with a joint at c, we must obtain the whole requisite height in the two tangents a b and bd; a being the starting point and contained in the assumed horizontal plane of projection, we have to raise one rise to the landing, and one-half of a rise above, to attain the proper height of the level rail.
This height of 1J risers is set up on the perpendicular d df; the first tangent rises to V, consequently the cross tangent must occupy the position of bf df in space, which extended intersects its horizontal projection in O, and O a is the intersection of the cutting plane in which both inclined tangents are contained. The first wreath-piece rises the height c d, and the second the balance of the whole height, from centre to centre of butt-joint each respectively.
To develop the face-mould of the first wreath, draw a seatline a" c" perpendicular to the intersection a O, also c dr' parallel to it through c; set up the height c d of the wreath-piece from the seatline at c" to d", then a" d" will be the plumb-bevel-line, and angle a" d" d' the plumb-bevel.
There will be two balusters, marked 1 & 2, on the last step, placed the same as on the straight steps from the face of the riser ; the cylinder being small, it will look best to place one in the centre of it at c ; the next one on the level floor at e, and from there about 5" from centres. It is evident baluster No. 1 will be a common short one, and No. 2., as also the one at e, common long ones; but the one at c will be of an odd length, which may be ascertained by comparing the length of the vertical ordinates through their centres, as will be shown further on.
In order to develope the face-mould, draw an ordinate through b the point of intersection of the tangents in horizontal projection, also one through 2 on account of ascertaining the length of the odd baluster, and some more through the curved portion of the wreath to ascertain a sufficient number of points in the elliptic curve of the face-mould, all parallel to a O. Draw the developing ordinates perpendicular to the plumb-bevel-line from the intersection of the vertical ordinates, and make the former equal to the respective horizontal ones ; as : a" A equal to a a", b'"r B equal to b &'", and d" C equal to c &'. Braw the tangents A B and B C, and make the butt-joints at A and C square to them.
The straight outlines of the facemould join the curve part at P and Q in like manner as the straight part of the rail joins the curve at^> and q in the horizontal projection ; it is also evident that, if the drawing is accurate, these straight lines from P and Q must be parallel to, and equidistant from the tangent A B. The curved portions are traced through the points so found. From the tangent points A and C, draw the sidelines Am and Gn parallel to the plumb-bevel-line a" &"> which finishes the facemould.
The face-mould of the upper twist is obtained also by ordinates, as shown; the situation of the cutting plane is similar to that in the common cylinder twist, Fig. 9, PL XI, or Figs. 1 and 5, PL XII; the face-mould differs from those just mentioned only in regard to the straight wood. In the present case, some straight wood, as Gr K, is added on the horizontal tangent d' E or the widest part of the face-mould, while in the former cases the straight wood is connected with the inclined tangent at the narrower part. The elliptical curves of this face-mould may also be drawn by means of the trammel or the string, if deemed convenient.
Fig. 7 is a perspective view of the lower wreath-piece with the face-mould A' B' C on the face-side in proper position ; Ti d" g is the plumb-bevel applied to some part of the plank, which is nearly straight; g is the centre, and Ji d" the sliding distance, which is laid off on the face-mould as A m and 0 n9 Fig. 6. This produces a spring-bevel at both butt-joints, and as the inclination of the tangents is different, the spring-bevels are different; but each is given by the tangents of the face-mould when in position, as shown in Fig. 7; a a' A is the spring-bevel for the lower, and c d 0 that for the upper joint. The face-mould being in that position, the point n Fig. 6 evidently coincides with c in Fig. 7.
Marking the plumb-bevel line d" g on the side of the sawed plank, as shown in Fig. 7, gives a direction for holding the round-faced plane while shaping out the vertical cylindrical surface, although it is not absolutely necessary to do so for the purpose of finding the sliding distance.
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If deemed more convenient, take one-half of the average thickness of the plank (the same as is ganged on for the centre of the rail on the butt-joints) and draw a short line parallel to the plumb-bevel line, Fig. 6, intersecting the perpendicular c" cf" as in g; square over from g to the plumb-bevel line as in \ and Ti c,n will be the same sliding distance.
Fig. 8 is the perspective view of the upper twist with the face-mould applied in position on the face-side. The tangent K' D' having a horizontal position, there is no spring-bevel at K. The pitch-bevel of the inclined tangent, < & dl d, Fig. 6, is the bevel to apply through the centre K ; the block-pattern at the lower joint C will be parallel to the face of the plank. The face-mo aid will slide parallel to the butt-joint at K and along the tangent line c C D\ and the position of the block-pattern at the joint K will govern the sliding distance of the face-mould on the upper and underside of the plank, as will be seen by the diagram.
To cut the rake part of the straight rail to length, it is plain that, as the last step is contained in the wreath, Fig. 7, and the point C Fig. 5 is plumb over the second riser, we must have the length of eleven pitch boards between the point C of the lower easement, and A the lower joint of the wreath, making the thirteen steps in all. Whatever the length of the level rail on the second floor may be, it is counted from the line E of the top twist, Fig. 6.
It is evident, that any portion of straight wood may be added on the straight part of the wreath Fig. 7, or taken from it, and a corresponding allowance made on the rake part of the straight rail.
The curves of the top and bottom side of the cylindrical twist, are governed by the line of the straight part at the bottom and top, and the block-pattern at the centre joint. Any intelligent workman will see where to take off the superfluous wood, after the vertical cylindrical surfaces are shaped.
m.xjv.
PLATE   XI"V.
PLATFORM   STAIRS.
SCALE  EOR FIGS.   1,   2,   4 AND  5,   �"  TO  THE FOOT.
Fig. 1 shows the plan of a flight of half-space platform stairs ; the "broken lines indicate the framing of the timlbers. The hall being T wide, and the cylinder 9", leaves 3' 1-J-" width of stairs from face of wall to face of string. Both branches are usually made of equal width, and the platform of the same width to inside of cylinder.
The height of the story is 11/ 6" between timbers, which being 8", makes the whole height 12' 2". There is 16' of space length-wise from the first riser to the rear wall; taking out 3' iy for the platform and 4-J" for the depth of the cylinder, leaves 12' 6" to be taken up by the steps. Dividing the whole height into 19 risers, gives 7H" for each, and dividing the run of the first branch or 150" into 15 steps, gives 10" run, which is a good proportion to the rise.
As the platform forms one riser, it will be 16 risers high to the top of it, or 10' 3".    The timber being 8", cross-furring 1", floor and plastering 2", making in all 11", will leave 9' 4" in the clear underneath the ceiling of the platform ; this is a convenient height for a door to pass to the rear ' part of the building.
The platform being 16 risers high, and the last landing forming also one rise, it will require 2 steps between the platform and the 2d floor ; these are generally of the same run as those in the lower branch.
The flooring on the platform is usually the same thickness as the rest, so that the height from top of the main timbers to top of platform timbers, is the same as the real height of the 16 risers, or 10' 3",
The first platform timber, which is somewhat thicker than the rest, ought to be so placed, that the back of the cylinder will rest against it; the landing pieces are then of sufficient width to lap about 1" or more on the same. The distance in this case will be 3'1" from the rough wall or the studding, to the face of first platform timber.
The second story being generally lower than the first, it will not require so many risers, and the difference is usually omitted on the bottom of the flight. This is, however, governed by the disposition of doors and rooms on the second floor ; a few more steps may be placed in the return branch of the second flight, which would give so much more room at the foot of the lower branch.
The head-room must be considered in these arrangements, and may be found by an elevation as shown in Fig. 2. This figure in conjunction with Fig. 4, is a section through the long branch of the first flight, platform, and lower part of the second flight.
The construction, putting up, and furring of such stairs, is done in the same way as straight ones, as is explained in Plate X and XI.
Fig. 5 is a cross-section through the second floor, showing also a front view of the cylinder and landing of the platform, together with the timber under the short branch.
Fig. 3, which is drawn on a scale of 1" to the foot, is an elevation of the short branch with the face-string removed, to show the fastening of the supporting timbers.
Whenever practicable, those timbers of the second floor which form the well-hole, ought to run through from the room over the partition, and thus support themselves. For some reason or through neglect, this is not always done, and they have to be held up by the furring timbers A, framed and fastened against the platform timber B by an iron bolt. In such cases the first platform timber ought to be particularly strong and bridged against the others, as shown in Fig. 2.
The trimmer D, Fig. 3, and the first timber C, which support the landing, are halved and spiked together, C is placed a few inches back from where the last riser will come, in order to allow
14
106
a fork to be cut on the upper end of A, as shown in the diagram. When framed and fastened in this manner, the flooring will not settle, but be firm and solid. In regard to the twists over the bylinder, they are got out the same as those for straight stairs, as shown on plate XI; instead of connecting with a level quarterturn, two of the twists are bolted together.
E F Fig. 3 shows the twist running up the short branch, and containing the straight wood for one step. The pattern for the easing over the landing, is shown in G H K. Tangent G H rises the height 7c K or one half of a rise from centre to centre ; whenever there are only 2 or 8 steps from the platform up, instead of cutting a short straight piece asFG- between the twist and the easement, this straight wood is left on the latter, in order to have but one joint. But whatever the number of straight steps, they are counted between the point J which is over the face of the bottom riser, and Gr over the landing riser. The straight wood along the level floor is also counted from Gr, deducting the distance G #, or whatever is contained in the easement.
In regard to placing the risers in the cylinder, the same rules apply as for straight cylinder-stairs, in order to raise one half rise in each twist.
It is evident, if the distance from the face of risers Nos. 16 and 17 on the platform, to the centre of rail in the cylinder, be one half of the run of the steps, the underside of rail will be the' same height on the short branch as on the long one, and consequently the balusters will be of the same length.
The second baluster on the second-floor landing ought to be somewhat shorter than a regular long one, to look well. Its length is easily ascertained by the easement pattern, as indicated in the figure.
When cylinders are of the size as in this case, that it requires two long balusters on the platform, it will look better to have the first one about ��" shorter, and the second i" longer than a regular long one�
Platform stairs of this form are frequently used, whenever the rear part of a house is somewhat lower than the main part, and then the platform is generally on a level with the rear floor. In such cases, the division of the risers in the first branch is, as a matter of course, governed by that height, and that of the return branch by the height of the second floor of the main building above that of the rear part; the risers in both branches may therefore vary considerably, and the twists must be laid out accordingly.
PI. XV.
PLATE   XV\
HALF-SPACE PLATFOKM STAIRS.
Fig. 1 shows the disposition of the risers in a larger cylinder (similar to one used in Fig. 1, PL XII., for straight stairs), scale 2" to the foot. The cylinder is 20" diameter, run of step lOf" ; the centre of the rail sets back f" from the edge of step ; therefore the risers on the platform must be placed 4f-" from the inside of the cylinder, leaving 5f" of the same contained in the straight step.
At A and B are shown the miter of the nosings in the cylinder. They are obtained by describing an arc from the centre O of the cylinder, with a radius which is as much less than the radius of the cylinder, as the nosing of the step projects over the riser. Where these arcs intersect the front edge of the nosing, as in A and B, the miter points will be found. The joints are then drawn to the miter points of the fillet and riser, or to the front-string and riser when fillets are not used.
The twists are got out on the same principle as those in Fig. 1, PI. XII. ; balusters Nos. 1, 3 and 7 will be regular short; and Nos. 2, 5 and 8 regular long ones. Nos. 4 and 6 will be odd lengths ; that of No. 4 is found as previously shown, and No. 6 will be as much longer than a regular long one, as No. 4 is shorter.
Cylinders of this size may be bent and keyed-up over a drum as shown on PL XXIY., instead of being staved up as shown in Fig. 2, PL XII.
Figs. 2 and 3, PL XV., show the plan of a platform-cylinder, in which the risers are so arranged, that the stretch-out of the cylinder-line, together with the straight parts of the front-string between the riser points, are equal to the run of the steps each.    Scale 2" to the foot.
A cylinder so arranged may be bent over a drum, and the lower edge of the board of which it is made, will on the stretch-out form a straight line in connection with the lower edge of the straight part of the front-string. When bent, this straight lower edge will form the natural easement of the cylinder. This arrangement is frequently adopted on account of saving room in the whole run of the stairs, as it brings both risers in the cylinder near the timber of the platform, and is therefore used in staved cylinders as well.
To lay out such a cylinder, the stretch-out is found by Prob. XXXIII., Fig. 12, Plate III. Draw a line tangent at the centre g, Fig. 2, Plate XV, and find the point m, in the intersection of the centre line g o extended arid an arc described from the joint q with the diameter of the cylinder as radius ; then g i is the stretch-out of the quadrant g s q. On this stretch-out line, set off from g to h one-half of the run of a straight step, and draw 7i m intersecting the cylinder at s, which will be the miter point of a riser. From Tito n set off one whole run and transfer the remainder i n, on the straight string-part from the joint q to t, which is the miter of the next riser.
The corresponding points on the other half of the cylinder are transferred over, and the opposite straight risers will then form a straight line.
Draw those risers which form the platform, the distance of a run parallel to them, and make a proper easement toward the miter-point s, as shown in the diagram.
Fig. 3 shows the disposition of the dovetails and the miter-joints of the nosings. There are two dovetails on each step, and the same on the landing ; the balusters will all be of the regular lengths.
The miter-joints are found by the same rule as laid down for Fig. 1. Draw easement lines representing the front edge of the nosings parallel to the curve of the risers, and the intersection of the dotted arcs give the miter-points.
The diagram shows that the stairs are right-hand, while those of Fig. 1 are left-hand. The riser on the platform landing is sawed in or dadoed on the back side, to produce the proper curve;
108
while the first one from the platform up, may "be worked off on the solid or bend also, to suit convenience.
Eemark.�The tool with which the operation of dadoing is performed is called a dado (as I find it spelled on a printed list of tools).
Dadoing, dadoed, are expressions used by carpenters and stair-builders all over the country, although the definition of dado, as given in " Webster," can have no reference to its meaning in this sense.
Fig. 2 also shows the location of the tangents and the development of the face-mould, in several ways.
The rise and run being assumed the same in both branches of the stairs, the face-mould is the same for both wreath-pieces, which will be similar to the one, Fig. 7, Plate XIII.
To find the location of the tangents, we will assume the first butt-joint over the face of the straight riser at a, which will bring the corresponding joint of the second wreath-piece over the opposite riser at/.
Draw the tangents in horizontal projection' at right angles through the points a c and /, extended ; apply the pitch-board at a, and draw the pitch-line or tangent a b" ; then this tangent will rise in space the height b b", which carry around to bf, as indicated by the arc.
Both wreath-pieces combined have to rise the height of 3 risers between a and /, from centre to centre of joints. As the upper joint is to connect with another straight rail-piece, set up from,/to,/' on the riser-line that height of 3 risers, and draw the line of the hypothenuse of the pitch-board from/7 down to d" ; carry the height d d" of that point around to df as indicated by the arc, and it becomes evident that the cross tangent must have the position V dr in space. The joint being in its centre at d each wreath-piece will rise \\ risers.
The face-mould may be developed on either side, and the result will be the same. Both are shown in the diagram, to convey the right idea to the student5 s mind. In practice, the height of li risers is set up from c to d at once, and the tangent V d drawn, if the face-mould is developed on that side. If on the other side, then the height of 1J risers is set up at once from/ on the riser line, and the tangent 1\ u drawn down by the pitch-board, intersecting its horizontal projection at u, and then u c is the horizontal trace or intersection of the cutting plane. This intersection being found, there is no need of finding the inclination of the cross-tangent at all for the purpose of developing the face-mould,
In the other case, when the face-mould is developed as on the left side of the diagram^ this inclination of the cross-tangent must be constructed in order to find the horizontal trace of the cutting-plane.
As this inclination of d V is so flat that its intersection with c b extended would fall outside of the diagram, an application of Prob. VII., Fig. 6, PL VII. may be made, and thus the trace found ; this is represented separately in Fig. 3, to avoid a confusion of lines.
Let V Jcf, Fig. 3, be the tangent d b'', Fig. 2., extended ; then this line in Fig. 3 will represent the vertical trace of the cutting-plane, b k the ground line, and a one point of the horizontal trace. To find some other point of the same trace, carry the distance b a around to a", and draw a" b'; then, from any point Jcf of the tangent, draw a line ~kr I" parallel to a" b', intersecting the ground line in I". From ~k! draw also a line perpendicular to Jc b extended, intersecting the arc described from Jc as centre and 7c I" as radius in I; then a I will be the horizontal trace or intersection of the cutting-plane.
This intersection being found, draw the seat-line a" d', Fig. 2, perpendicular to it through any point; for convenience, through b, and erect the perpendicular c d' d" through the centre of the joint at c; make df d" equal to c d, and proceed to develop the face-mould as in Fig. 6, PL XIII. From this demonstration it will be seen, that the method of finding the intersection of the cutting-plane as shown on the other side over c df, is the most convenient, and is always used when both rise and run are the same respectively on both branches of the stairs, as in this case.
This demonstration has only been introduced to make the theory of the subject entirely plain to the student.
109
The curved outlines of the face-mould FDC, are shown as being drawn by a trammel, or the string and brad-awls. In this case either may be employed, as there are no lengths of odd balusters to be ascertained.
The intersection uc being found, draw the seat-line c"f" perpendicular to it; for convenience, through the point d, as then the vertical ordinate will give the point D at once. Dmwff'f"1 through the point/parallel to the intersection ; make/"/"' equal to 1| risers, and/'" c" will be the plumb-bevel-line. Develop the tangents P D and 0 D as usual, making the butt-joint lines perpendicular to them.
To find the major and minor axes of the semi-ellipses, it is the safest to draw a line jp o through the centre o of the cylinder, parallel to the seat-line, and describe the semi-circle, p cr e, which represents the centre line of the rail in the ground-plan. Prom the points^ and o, draw the ordinates perpendicular to the seat-line intersecting the plumb-bevel line mp" and <?" ; then draw the developing ordinates jp" P and o"f O, and make them equal to the horizontal ordinate o o"< As the line op is parallel to the seat-line, so will OP, which is the line of the major axes of the ellipses, be parallel to the plumb-bevel line, while o'n O is the direction of the minor axes, and O the centre. Making o!n R equal to o" r, gives in R the vertex of the minor axis of an ellipse, of which P is the vertex of the major axis. This ellipse passes through the centre of the face-mould at its narrowest part, as also through the point 0 : therefore, set off from R one-half of the width of the rail each way, which points will give the minor axes of the ellipses required ; the corresponding points on the major axes may be found by drawing lines parallel to R P, as indicated in the figure, and as demonstrated theoretically in Prob. XXVIII., Fig. 10, Plate IX., for the construction of proportional ellipses.
One-half of each major and minor axis being then known, the trammel may be set, or the foci w v found.
A line drawn through O, and the point E of the tangent, (which point represents the joint of the cylinder), will intersect the ellipses in those points from which the straight lines are draw^n parallel to the tangent, in order to finish the outlines of the face-mould. The point E is found by means of the ordinate from 6, the joint of the cylinder.
After the tangents and the points E and 0 are found, lines may be drawn parallel to R E or R C from the vertexes of the minor axes, as indicated in the figure, and thus a point outside of the minor axis, found for each ellipse. The trammel may then be set according to Prob. 36, Fig. 16, Plate III.
Both tangents being inclined, there will be a spring-bevel on each joint, and slide-lines must be drawn from C and F parallel to the plumb-bevel line. The sliding distance is found as in Figs. 6 and 7, Plate XIII
Fig. 4, Plate XV., represents the upper wreath piece with the face-mould applied, in perspective. The joints are made square from the face, and the face-mould is applied, as has been explained in the same figures, Plate XIII. This view shows the inside cylindrical surface shaped out. Its form suggests at once the lines by which to shape the upper surface of the wreath. At the center joint, C, some more wood than the block-pattern ought to be left on the inside, as that joint will not appear square to the curve when both wreaths are bolted together. If taken down to the block-pattern, the rail will be too thin at that point.
PLATE   XVI.
QUARTER-SPACE   PLATFORM   STAIRS.
Figs. 1 and 2 of this plate show the ground plans of quarter-space platform stairs, consisting of three "branches each.    Scale J" to the foot.
The form of stairs represented in Fig, 1 is frequently employed in school-houses, asylums, and similar institutions, and is generally located in a separate rectangular space, on one side of the main hall or corridor.
In order to show the student how to proceed in laying out the rise and run of this class of stairs, we shall assume the following dimensions, as given :
The "branches run at right angles to each other.
Depth of room, 10' 6" from face of hall wall to the rear wall; length in clear, 14' � height of story, 14/ 2", from top to top of floor.
The second platform is required to have a height of 10' 2", in order to be on a level with the floor in a rear building.
The timber A inclosing the opening of the second floor, rests on the hall partition, and gives therefore only 10' 2" "between its face and the rear wall.
The stairs are required to be 3' 6" wide from face of wall to face of string.
One of the principal points to be observed in laying out the ground plan of quarter-space platform stairs is, to have the distance from the face of riser on the landing, to the centre of rail or where the tangents intersect, equal to one-half of the run of the corresponding steps. This is done, that the tangents of the wreath-piece (which are governed by the pitch-board), may intersect in space. For instance : let 5, e and n, be the points where the lines through the centre of the rail will intersect ; the run of the steps is 9�" in the first and second branches, and 10�" in the third ; then the distance from riser No. 9 to 5, as also No. 10 to 6, and No. 18 to e, will have to be 4f", while those from No. 19 to <?, and No. 26 to n, are to be 5J". This arrangement on the last landing is necessary in order to have the rail rise a half riser above the landing.
The balusters to be used are, say 2�"; the centre of rail will set back �" from the edge of step ; the stairs being 3' 6" wide, will leave 3' 5i" between that centre and the face of wall.
There being two platforms, they will take twice 3' 5i" or & 10J-" out of the run lengthwise, which is 14', leaving T !���" whole run on the second branch from centre to centre of rail.
If the stairs were to start at the corner of the hall, the run of the first branch to the point b would be 10' 6", minus 3' 5$", leaving 7'f".
Adding the run of both branches, will give 14' 2J" whole run to the second platform, which is to be as required, 10' 2" high. Employing the rule given, page 86, to find the proper number of risers, we have�
Half of run.        .       .       7; if Whole height     .        .      10' 2"
Their sum     .        .      17' 3-�-" or 17 risers.
In buildings of the description mentioned, a low riser is preferred, even if the run of the steps becomes a trifle less on that account; in this case, therefore, divide 10' 2" into 18 risers instead of 17, making each riser 6| or approximate 6f and -^V'.    The number of steps is then 17, and as the



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first platform counts one, it leaves 16 ; the run of both branches being nearly the same, places 8 steps in each. As there is to be half of a run at each end of the second branch, divide the space between the points h and e, which is 7' 1J-", into 9 parts, giving 9�" run for each of the 8 steps.
For regularity, make the run of the steps in the first branch the same as in the second, 9f", which will take up & 81" from the first riser to the point 5.
The height from the second platform to the second floor is 4'; the run from e to n is & 8$-". Although this run is 6' 8f" to the face of the timber, the face-board being 1" thick, makes it %" less to the centre of the rail. The number of risers in this branch will be 8 of 6" each, and as there is half of a run at each end from the riser to the centre of rail, we must divide & 8J" also into 8 parts, giving 10J" run for each of the 7 steps.
The size of the cylinders ought to be such, that its part in the platform amounts to about half of a step, in order to place one baluster on the same, and that they may be about equal distance apart.    The cylinders in this case are assumed to be of 8" radius.
It may be well to remark in this connection that, if the division of the steps could not be arranged conveniently to have the risers at the platform, such as Nos. 9 and 10, equally distant from the point 5, they may be arranged in such a manner, that the sum of their distances from that point, amounts to one run ; the tangents of the wreath-piece would then also intersect in space. However, the way as laid out in this diagram will look the most symmetrical, and that is an important point in stair-building.
Fig. 2 represents a form of stair-case much used on steamboats, and in buildings where the lots run on an obtuse or acute angle to the street-line.
The laying out of such stairs is governed by the same principles as the former. It must be observed, that the distances from risers Nos. 7 and 8 to &, are each equal to one half of the run of the step.
The size of the cylinder in that angle may be such, that the intersection of the risers 7 and 8 extended, will be its centre. Two baluster^ are placed on that platform, although they will be a trifle closer together than on the straight steps.
In the acute angle at e, the distance from risers Nos. 14 and 15 to e on a straight line, ought to be also equal to one-half of a run each respectively. It would not look well to place a cylinder so small in that sharp angle, as the intersection of risers 14 and 15 on a straight line would indicate. Therefore both of these risers are curved, and a cylinder of such size is used, as to allow the thirteenth step, the platform, and also the fifteenth step, to appear of about half of the width of the straight steps, which will allow at the same time one baluster on each of them. This is shown on a larger scale in Fig. 6.
The riser and run of the first and second branch are assumed to be the same respectively, while in the third branch the run is larger and the riser less ; on account of this variation, the joint of the cylinder in Fig. 6 does not come in the face of the straight risers in both branches. The distance from the thirteenth riser to e9 is less than that from e to the sixteenth riser, although each is equal to li| run of each branch respectively.
These two staircases, Figs. 1 and 2, comprise all the different cases which may occur in regularly built quarter-space platform stairs, and Figs. 3 to 6 show the development of the face-moulds of the wreath-pieces, required for the four different angles and positions of tangents.
The twists for the second floor landings are of course treated as those for straight stairs, mentioned before.    Scale of these figures 1" to the foot.
Fig. 3 represents the case of the first platform, Fig. 1.
For convenience, place the joints of the wreaths at a and c, thus including one straight step of each branch as also the platform in the wreath, which will rise three risers from centre to cer, tre of joint.   As both tangents have the same inclination and are of equal length, there is no need U
112
drawing their positions, & V and a bf\ which are therefore only indicated in fine broken lines. The intersection of the cutting-plane a a" is in all such cases parallel to the line o b, which is passed through o the centre of the cylinder, and b the intersection of the tangents.
Draw a seat-line a" b c" perpendicular to b o, and the ordinates as a a", e c" c'" parallel to it; from c/; set up the height of three risers to cn\ draw a" c"\ which will be the j.lamb-bevel line, and develop the face-mould ABC with the slide-lines, as usual. The spring-bevels will be alike on both joints. If the ordinates are passed through the centres of the balusters, as in the diagram, it will be found that the baluster on the platform requires to be about f " longer than a regular short one.
Fig. 4 represents the case of the second platform, Fig. 3. The wreath is again assumed as containing one step of each branch besides the platform. The inclination of the tangents being unequal, the position of the upper one er f, must be drawn, intersecting its horizontal projection ef in q, in order to find the intersection d q of the cutting-plane.
To locate /' e' in space, we know tangent d e rises 1| risers of 6-J", while e /rises 1| risers of 6". Therefore, set up the sum of 1| risers of each, on the riser-line from /to/' and apply the tipper branch pitch-board at/7, as shown in the diagram ; its hypothenuse locates/7 e'.
As tangent d e rises one-half of a riser of the 6$" riser from the 18th riser to the point e, and the 19th riser is only 677, it is obvious: that the centre of the upper joint will be about f77 or one-half of the difference in the risers, higher than it ought to be. In order to have the rail of precisely the same height over the face of the risers in both branches, this slight variation may be adjusted by lowering the block-pattern on the upper joint along the spring-bevel line, if deemed enough of an object to do so.
It is evident, that it would be erroneous to undertake to rectify this variation by lowering the point f while locating the tangent; this would give the intersection d q a, wrong position, and also the tangents on the face-mould, which would again cause the butt-joints to come wrong. The baluster on the platform will be about fn longer than a regular short one. The spring-bevels will not be alike on these joints.
The straight rail between the first and second wreath-piece over the second branch, will be evidently the length of 6 pitch-boards. That on the first branch will depend also on the length of the easement on the cap, and that of the third branch on the length of the twist at the landing.
Fig. 5 shows the development of the face-mould of the wreath over the obtuse angle, or the first platform, Fig. 2. The wreath is assumed, for convenience, to contain one step of each branch besides the platform. The inclination and length of both tangents being equal, the construction is essentially the same as that of Fig. 3. As there is no need of drawing the positions of a b'f and c' V, they are only indicated by fine broken lines in this diagram. The seat-line al' c" is drawn on the outside line of the rail, to avoid confusion, and is parallel to the cord-line a c, or the same thing, perpendicular to o b. The wreath rises the height of three risers from centre to centre of joint; set up this height from c" to c777, and draw the plumb-bevel line c'n a'1'.
The height of the three risers may be set up from c on the perpendicular, and the cord a o used as a seat-line, which would produce the same result-   The same may be done in Fig. 3.
Fig. 6 shows the development of the face-mould for the wreath over the acute angle, Fig. 2. The inclination of the tangents is assumed to be unequal. The riser is less and the run of the steps more in the third branch than in the second. The butt-joints are assumed as being in the centre of each respective step ; consequently, tangent d e" rises two risers of the second branch, and tangent e'f two risers of the third.
In order to find the intersection of the cutting-plane, the operation is similar to Fig. 4, Draw a line//' from/ perpendicular to ef, and set up the height of four risers, two of each branch ; then apply at/7 the pitchboard of the third branch, and its hypothenuse gives the
113
position of e'f, which intersects its horizontal projection extended in q, and q d is the intersection. Draw the seat-line/ d" perpendicular, and the ordinate/F parallel to the same. Set up the height //' from/to F, and develop the face-mould F E D as usual. F d" is the plumb-bevel line, and as the point F is contained in the same, it will form at once the side-line from F ; draw another one from D parallel to it, which completes the face-mould. For the same reason, as stated in Fig. 4, the block-pattern will have to be lowered about �" or one-half the difference between the risers on the upper joint, in order to bring the rail of the same height as on the second branch.
Balusters Nos. 2 and 5 are regular short ones, and if the ordinates are passed through the centre of the balusters, as has been done in the diagram, it will be found that Nos. 3 and 4 are to be somewhat longer than the regular short ones.
To cut the straight rail, it is evident that it will take the length of 3% pitchboards between the joints C and D over the second branch, Fig. 2.
The application of the face-moulds in these cases is the same as Fig. 4, Plate XV.
15
PLATE    XYIIo
PLATFOEM   STAIRS.-Contikod.
Fig. 1 represents the ground plan of a first-class "half-space platform staircase,59 with a turnout at the foot, and the risers around the cylinders so arranged, that the same may be bent with the lower edge straight on the stretchout, as has been explained in Fig. 2, PL XV".
Fig. 2 shows a plan in which the riser points around the cylinder are arranged precisely the same as in Fig. 1, with the difference, that two quarter-space platforms are introduced, and three siteps between them, making in the whole the same number of risers as in the other. These arrangements are mere matter of taste, as both staircases occupy precisely the same ,space. It is advisable however, to avoid as much as possible the introduction of a few steps between platforms; they cause a certain inconvenience in their use.
The facemoulds for the wreaths will be the same in both cases, as long as the rise and run are the same ; their construction is shown in Fig 3 and Fig. 3 ^.
The turnout part of Fig. 1 is separately explained on PL XVIII., and the scroll of Fig. 2 on PL XIX.
It will be noticed, that the straight risers 10 and 17 of both figures, are opposite each other in a line. To find their position, construct the stretchout c % Fig. 3, on one side from the centre of the cylinder, and set up three and one-half runs to n ; take i n and set it off on the straight string from g to t, which latter point is then the riser point required. (See Problem 33, Fig. 12, PL III., also Fig. 2, PL XV.) As long as rise and run are the same in both branches, the construction of the facemould is as Fig. 2, PL XV.    One facemould will answer for both wreaths.
Whenever either rise or run, or both, are different in the two branches, a separate facemould must be drawn for each wreath.
It is evident also, that in such a case, the lower edge of the bent cylinder will not form an exact straight line on the stretchout, although the run may be alike.
When such stairs continue more than one story, the landing of the first flight and the foot of the next one, ought to be constructed in ground plan the same as around the platform in Fig. 1, and as indicated in the landing of Fig. 2.
The rise of different stories generally varies. To illustrate the drawing of the facemoulds in such cases, the rise of the first branch in Fig. 3, is assumed to be 6f" and that of the second one 7�'\
As the inclination of the tangents varies, their position in space must be constructed. For convenience, assume the lower joint at a, and the upper one at e, with the centre-joint at c ; both wreaths rise, therefore, seven risers : three of 6f" and four of 7-J;'; these are set up on the riserline from e to e\ as indicated in the diagram. Apply the second pitchboard at e' and draw e' d'r; extend the three tangents as usual and carry the height d d" around to d\
By applying the first pitchboard at a, we find that tangent a hf/ rises the height b 5", which is carried around to V ; it becomes then evident, that the cross-tangent must assume the position V d', with the centre joint at c'.
The tangents being thus located, the intersection a s of the first cutting plane, is found by applying Problem VII., Fig. 6, PL VII., (as has been applied in Fig. 3, PL XV.) The point s in this diagram, is found by describing the arc c' cf/ from c as centre, and drawing from c" a line c" s parallel to a 5", intersecting & m in s.
The trace being thus found, draw a convenient seatline c a\ and develope the facemould as usual.
To avoid a confusion of lines, this development has been transferred as Fig. 3 /^. Another seatline a'" c"r parallel to the first one, has been drawn, with the intersection a s extended to a"', and g c'" parallel to a a!". Make c'" C equal to c c\ and draw the plumbbevelline C a'". Then, the developing ordinates are made equal to the horizontal ordinates from the first seatline, as arn A equal to a a\ &c.
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115
To draw the second facemould, we assume e, (the lower joint of the second wreath) as the starting point; or in other words, as Tbeing contained in the horizontal plane of projection. As the first wreath rises the height c d, the second wreath must rise the balance of the whole height e d; therefore, to locate the upper tangent d e in position for the second wreath, take c d and lay off the same from d down to df on the riserline ; draw d' d'" parallel to d d'\ intersecting e d extended in u, and u c is the required intersection of the second cutting plane.
Draw a convenient seatline as d" dfn perpendicular and e d'" parallel to c u; make d" d"' equal to the height e d\ draw the plumlblbevelline d'" d'" and complete the facemould with slidelines as usual.
The balusters are all about the regular lengths in these cases; the ordinates may therefore be placed as convenient.
P JL _A_T E    XVIIL
THE   TURNOUT.
Fig* 1 is the ground plan of a turnout, with the development of the facemould.
Scale for Figs. 1, 1 ^, 1 U, B, 4 and 6, one inch to one foot.
Steps No. 1, 2 and 3, are wider on the lbackstring than No. 4, which is a straight one or flyer.    This arrangement causes a curve in the risers and takes up more run.
The frontstring is Tbent on an elliptical curve, and the run of the curved steps is generally laid out on the stretchout, the same as the run of the flyers. This string is dadoed as far as the curve reaches, vertically on the Ibackside, (of course, after the Tbottom easement has been glued on its lower edge,) "bent over a drum of the shape of the curve, and keys are tightly fitted and glued in.    After it is thoroughly dry it will stand the shape as desired.
The increase in the width of the steps is a matter of taste ; where there is plenty of room, it may be in about the same ratio as shown in the diagram.
The flyers being 10" run, No. 3 is made Hi", No 2, 13" and No. 1, 16". The turnout of the newelpost from the line of rail is in this case 9" from centres. The rail will assume an imposing curve, which, in conjunction with the curved steps, adds materially to the appearance of the staircase, although there is no increased convenience.
Where the room is rather contracted, a turnout similar to the one shown in Figs. 1 and 3, PI. XXV., may be adopted.
'The curve ought to be always part of an ellipse or parabola; a circle, employed for the turnout curve of the frontstring and rail, will look abrupt where it connects with the straight part.
The dovetails are cut normal to the curve, in order to show the face of the balusters parallel to it.
The buttjoint of the easement is made at some convenient point in the rail, so that it contains part of the straight wood, as in this case at a over the face of the fourth riser. The newelpost is made from 5" to 7" higher than a short baluster, as was done in the straight easement, Fig. 8, PL XI.
Whatever it set up in the drawing of the facemould, is added to 2' 1" for the height of the newel from top of first step to underside of cap ; the rail and the balusters will come the same as has been explained in PI. XI.
To draw the facemould, Fig. 1, set up the number of risers, in this case three, (omitting the first one, as we count from top of first step) on the fourth riser line from the point a, and count back to a! whatever is calculated as easement: in this case 5". This leaves the distance a a' as the height, which the turnout twist will have to rise from centre of miter at c (which is also the centre of the cap) to centre of buttjoint at a. Place the pitchboard at a' and draw the tangent line a! &, intersecting its horizontal projection-in &.
The centre o of the cap, ought to be situated in the continuance of the curve through the centre of the rail in the groundplan, and at such a point, that one baluster may be placed on the first step without interfering with the base of the newelpost. Describe from o a circle indicating the largest diameter of the cap, as also an arc with such a radius, which is one half of the width of the rail less than that of the cap, intersecting the centreline at c; this point is then the miter point. Set off also one half of the width of the rail at different points each way from the centreline, and draw the outside curves of the rail, which will intersect the circle of the cap in / and g; these give the other two points of the miterjoints.
From c draw a line to b extended, which is the intersection or horizontal trace of the cutting plane ; this trace being found, the development of the facemould is similar to the others.
As the variation in the length of the balusters may be found by the vertical ordinates, draw these through their centres parallel to c b as done in the diagram ; an ordinate must also be passed through the points / and g each, to ascertain their position in the facemould.    Draw the seatline
Pi. xvm.
117
a b" for convenience through a, make the principal ordinate a A equal to a a', and draw pluxnb-bevelline A b" somewhat extended.
Make all the developing ordinates equal to their respective horizontal ones, as usual. In cases as that of the ordinate from /, which intersects the plumbbevelline at/" before it reaches the seatline in/', it is evident that the developing ordinate f"� is drawn from the point /", and made equal to ff. The ordinate through the baluster No. 1, is only drawn for the purpose of getting the vertical one, to ascertain the length of the baluster.
AB is the inclined tangent, FC and GC are the miterjoints of the facemould; a separate slideline D E, parallel to the plumbbevelline A b is drawn somewhere on the lower part.
The intersection is in this case the horizontal tangent of the twist, but as neither of the miterjoints is perpendicular to it, (unless accidentally so,) only the buttjoint at A is made square through from the face of the plank; as the miterjoints are vertical surfaces when in position, they are found by sliding the facemould the proper distance, (which sliding distance is found as in Fig. 7, PL XIII.) along the slidelines from the buttjoint each way, as usual. This is further explained in Fig. 2, which represents the twist viewed from the underside, with a facemould applied to each side of the plank to show the operation, although in practice only one is used.
After the face of the plank, (which in these cases ought to be of parallel thickness,) is dressed off and the buttjoint made, mark the slidelines on both sides, as also the square line a A, Fig. 2, from the tangent across the joint. Find the sliding distance 7i Z, Fig. 1, and set it off on the slideline of the facemould from a to A', Fig. 2, place the latter in proper position, and the tangent will cross the edge of the buttjoint in a! ; then a' A is the spring bevel. While in that position, mark also the .outlines of the facemould with the miterjoints; (7 is then the miterpoint. Repeat the operation on the underside of the plank, where the facemould will occupy the position as represented, C" being the lower miterpoint. The miterjoints are then made and the vertical sides of the twist taken off, as shown in the figure. The centre C, between the points C and C", is the centre of the cap, which, when mitered by the plan f c g, Fig. 1, square from its underside, will be found to fit to the miterjoints of the twist.
When marking the plank to be sawed at the mill, some surplus wood must be left ,on the outside of the miterjoints, as indicated by the dotted lines in Fig. 1, to allow for their sliding, although no wood is wanted beyond a line through C parallel to the slideline.
If the facemould is much wider than the width of the rail, it may be curtailed some on each side while marking the plank, to save material; the thickness required, need be no more than the width of the rail.
All the superfluous wood is taken off on a line with the under side of the straight rail on the upper end, and also all below the under side of the cap on the lower end of the twist. The workqian will then not be left in any doubt, how to shape the underside, to make a good looking easement.
Comparing the lengths of the vertical ordinates, it will be seen that baluster No. 1 is 2J" and No. 3, f" longer than No. 5, which is a regular long one ; and that No. 2 is -f" longer than the regular short one No. 4. In common turnouts, where only two steps are curved, the baluster next to the newel is generally made 2" longer than a regular long one, and the rest are of the regular lengths, as is done in Fig. 8, PL XL
Some stair-builders, in getting out a turnout twist, make a joint on the facemould through the miterpoint C, Fig. 1, parallel to the plumbbevelline, or the same thing, perpendicular to the developed intersection B C, as indicated by the dotted line, instead of making the miterjoints at once.
This joint is then made square through from the face of the plank, the point C squared over and the centre guaged on, as on any other buttjoint. The plumbbevel is then marked through the centre, which becomes the line C" C C in Fig. 2. The miterjoints are then made afterwards, and the result is the same in the end. All the difference is, it involves the extra labor of making that buttjoint through C.
Fig. 1 ^ shows the elevation of the steps, easement on the wallstring, and finish of the front-string.    It is trimmed with fillets and carved brackets.    The space of the lower easement is filled % by a sunken panel with carved mouldings.    H is a section through the lower edge of the string, showing the carriage-timber, the plastering, and a half-round moulding to cover the joint.
118
The vertical fillet below the first step, ought to show as much outside of the newel-post as on the riser.
Fig. 1 (� shows the panel-work of the wainscoting along the wallstring. The panel-heads are elliptical in this design. The cap consists of a half rail, corresponding to the handrail of the stairs.    K shows a section through the centre of a panel of the level part.
As the wainscoting ascends along up the stairs, it is gradually contracted in its several parts, conforming to the curve of the wall-string. This is done to have the cap of about the same height as the handrail. However, it is a matter of taste, as some builders prefer to give the wainscoting the same vertical height, both on the level and inclined portions. The horizontal width of the panels is usually alike in all of them.
It will be noticed, that the position of the elliptical heads change, or rather the proper ellipses change, for every panel between the level one No. 1 and the regular inclined one No. 6.
The lower mouldings of these panels also change in conformity to the curve of the wall-string.
Fig. 5 shows the construction of these ellipses. After the panels are properly laid out, the inclination of the curve over each panel, gives the inclination of the major axis of the proper ellipse. The minor axis for these ellipses are all alike.
Both axes being then known, we may construct each separate ellipse by Problem 43 of the Geometrical Constructions. The points of contact as a and d, found by this construction, are the proper points where to cut the joints, in order to intersect with the vertical moulding. It will be seen, that two heads are thus cut from one elliptical moulding for the regular inclined panels.
To be particular, a separate ellipse ought to be turned for each of the intervening panels between No. 1 and 6. As the panels are laid out for the rabbet of the moulding, so are the ellipses constructed from those points, and those ellipses, which form the inner and outer edge of the moulding, are parallel to them.
A line from the point d parallel to the next part of the curve of the cap, gives the joint between the stile and the toprail of the framing.
It is obvious, that it requires good judgment to lay out such a piece of panel-work properly.
Fig. 3 and 3 ^ show the intersection of the mouldings, when the panel-heads are segments ol circles, and Fig. 4 and 4 ^, when they are half circles.
Fig. 6 are designs for a carved newelpost, also carved balusters with octagonal shaft, square base and top.    The design represents one regular short and one long baluster.
Figs. 7 and 8 are sections of rails, suitable for heavy work.    Scale 2" to a foot
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PLATE    XIX.
SCROLLS.
This plate represents the application of the scroll in place of a newel-post at the foot of staircases.
Scale for all the figures, 1" to the foot.
They are rather out of date and not often used.
The curved part of the first step is usually made of a solid block and veneered. The front edge may run either straight for the width of the stairs or be curved. Both cases are represented in Fig. 2.
The fine continuous lines are the risers and strings ; the broken lines the edge of the nosings. Either front line may be made to conform to the scroll.
The thick continuous line represents the centre of the rail and balusters, which are placed about the same distance from the centres as on the steps.
To lay out a scroll, it is best to draw the shape of the rail part first, as represented in Fig. 1, and conform that of the step to suit, describing the corresponding arcs from the same centres.
Determine the diameter of the scroll a b, Fig. 1, from outside to outside of rail, and describe it as in Problem 48, Fig. 4, PL IV. The diameter is. in this case divided into nine parts. Rail 4" wide.
The inside line of the rail will meet the outside one in some point as in c. The shape of the rail is worked on the outside all around, intersecting the inside, and terminating in a full circle, called the eye, as indicated by the dotted lines. It depends mostly on the judgment of the workman to make a proper finish. The centre of this eye, which is also the centre of the scroll of the step, is supported by a wrought iron turned column, bolted through the step. A baluster may be used in its place, however,,
The facemould B E F is drawn as usual for a twist of straight cylinder stairs, explained on PL XI. ; a joint being made at/, the balance of the scroll is horizontal. A side view of the twist is shown; the thickness of the rail being worked from the underside, the straight rail is cut to length from the point E'.
By comparing the vertical ordinates of the balusters, it will be found that those under the level part will be 1", and the first one under the twist If" longer than a regular short one ; the others are of the regular lengths.
To show the different shape the scroll assumes through a division of the $ame diameter into a different number of equal parts, Figs. 3 and 4 are drawn.
Fig. 3 represents a 4" rail, the diameter being divided into seven equal parts; it is more open than Fig. 1.
Fig. 4 represents a 3" rail, the diameter being divided into eleven equal parts. If this division for instance, should be used for a 4" rail, the scroll would close already in the first quadrant.
The twist connected with the straight rail by a quadrant of a circle, has rather an abrupt appearance. To avoid this an elliptic curve may be used, as shown in Figs. 5 and 6, which, in combination with swelled steps, gives the whole the appearance of a turnout.
The ground plan of such a scroll is drawn by dividing its diameter into an even number of parts, as eight or ten, adding one of them to the elliptic part of the curve, and leaving an odd aumber to describe the scroll from with compasses, as before.
Thus, the diameter of Fig. 5 is divided into eight parts, making o n equal to 5, which is then one half of the minor axis of the ellipse, and/r equal to three of these divisions. The construction will be further explained by Fig. 6, the diameter r n of which is divided into ten equal parts. Make rj equal to four, j q and jb each to five of these divisions; construct the rectangle J ben and draw en extended, which will be the outside line of the straight rail to which the scroll is to connect. Draw the diagonal Je,   also from 6 a line bo perpendicular to it,  intersecting the straight
120
rail-line in c, and draw c o parallel to j n; then b o will be one half of the major, and c o one half of the minor axis of the elliptical curve, which may be drawn by any of the methods explained. In this figure, the construction by compasses according to Problem 44, Fig. 1, PL IV., is employed. The respective points are indicated by the same letters as in the Problem mentioned.
qr, Fig. 6, being equal to nine and j r to four parts of the diameter, the construction of the rest of the scroll by circular quadrants is the same as in Figs. 1, 3 and 4. An inspection of Fig. 6 shows, that radius j 5, (which would be the proper one for the first quadrant were the scroll described entirely with such and taking rq as its diameter,) is also the radius of the arc which forms part of the ellipse from the vertex 6, thus forming a perfect easing around that point.
As the ellipse constructed by the above problem from five centres, very nearly coincides with the ellipse proper of the same axes,- it is evident, that these axes as above found, are the proper ones to construct the ellipse by, with either the trammel or other means, and conform to the rest of the scroll.
Fig. 5 shows the shape of the turnout steps, similar to Fig. 1, PL XVIII. The scroll part of the first step is made solid, while the curved part of the frontstring may be bent over a drum as far as the joint through c. The run of the second and third step is the same on the stretchout as that of the flyers.
To draw the facemould for the wreathpiece, make a buttjoint somewhere that it may contain part of the straight wood, as at a, which is one half of the third step. Draw the tangents a b and 6 d through the centre of the rail, and make a buttjoint at c.
If it is desired to make the balusters under the level part of the scroll the same length as the regular short ones, set up the height of two and one half risers from a on the perpendicular to a!, and draw the inclined tangent a' V by the pitchboard; this tangent will be the distance b b'f vertically over the point 5, therefore carry this height over to b\ and draw bf d, which is the inclination of the cross tangent.'
The buttjoint being at c9 it is evident that the wreathpiece rises from centre to centre the whole height a a\ less the distance c d. Therefore, to find the intersection of the cutting plane, toake ar a" equal to c d and draw from a/; the tangent a" u parallel to a' 5", intersecting its horizontal projection in u, and uc will be the intersection sought. The development of the facemould is then drawn as usual; u a!" is the seat, and u a"" the plumbbevelline; a!" a"" equal to a a". The joint at C being a buttjoint, is made square to the tangent, and also square to the face of the plank.
To make the balance of the scroll, it is cheapest to saw it out on the gigsaw just the shape of the ground plan, using a plank thick enough to allow for the short upward curve connecting with the wreathpiece as shown in Fig. 7.
The joint d s is made on a bevel from the underside to be square to the tangent b' d.
If it is desired to use longer balusters for the horizontal part of the scroll, set up the whole height that much less than two and one-half risers, as the balusters are longer than regular short ones. If, for instance, the said balusters were the distance b b" longer than regular short ones, the tangent b d would be horizontal instead of being inclined, and the easement would be a twist as in Fig. 1, instead of a wreathpiece with springbevels at both joints.
A turnout and scroll, constructed as in Fig. 5 and 7, make about as good an appearance as can be given to them.
PI XXI
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PLATE    XXo
IRREGULAR   PLATFORM   STAIRS.
The figures on this plate represent the ground plans of irregular built platform stairs, and the drawing of the facemoulds of the rails.
Scale of Figs. 1, 2 and 7, i" to the foot; Figs. 3 to 6 and 8, 1" to the foot.
Fig. 1 is the ground plan of a flight of quarter-space platform stairs, as built in former times, with small posts in the corners, the rail running up to them on a miter.
Fig. 2 is one of the same style, half-space platform. They are called dogleg stairs, and are frequently met with in old houses, many of which are remodelled at the present time to conform more to the modern style of building.
The stairs are then also altered, as such spaces as represented in Fig. 1, give a good opportunity to build a convenient flight of quarter-space winding stairs, illustrated on plates following.
Some owners do not wish to go to the expense of putting up an entirely new flight, but prefer to have a modern rail put on the old stairs only. As the new railing ought to have turns instead of the corner-posts, the stairs ought to be altered also to conform to them.
Fig. 3 shows how this may be'done without great expense, in the case of Fig. 1. A cylinder is made for each corner, of a radius equal to the run of a step. The platform, landing, and adjoining steps, are curtailed in proportion as shown in the figure, giving one baluster on each step around the platform, and two on the last step. Of course, a new stringpiece must be made between the two cylinders, on account of the easings, but the first cylinder can be fitted to the old front-string of the long branch.
Fig. 4 shows the alteration of Fig. 2.
A half cylinder is placed in the opening, and the platform with the adjoining steps curtailed as shown. The frontstring of the second branch is generally made new, on account of removing the upper post and easing it over into the faceboards as shown on Fl. XIII. The stairs become then of a shape similar to Fig. 2, PL XV., around the cylinder.
If in these cases, rise and run are equal respectively in both branches, the facemoulds for the rail are drawn as in Fig. 2, PI. XV. ; if unequal, as in Fig. 3, PL XVII.
The altered parts are drawn in full size, as shown in Figs. 3 and 4, on paper,, and the shape of the steps pricked through on the plank ;  the same paper is then used to draw the facemoulds.
Fig. 5 shows the development of the facemoulds for the rail over the platform as altered by Fig. 3.
For convenience, place the lower buttjoint at a the centre of the ninth, and the upper one at e the centre of the thirteenth step, the whole height being then four risers from centre to centre of joint. It is evident, that one wreathpiece would not bring the rail high enough on the second branch ; it must therefore, be made in two pieces, making another buttjoint at <?, the centre of the quadrant of the circle.
a b and e d being the tangents on a line with the straight rail, draw another tangent d b through c perpendicular to o c. We have assumed in this case, the risers of the two branches as unequal; that of the long one being 6f" and that of the short one 8". This requires a facemould for each piece, and shows at once how to get over the most difficult cases. If rise and run were equal respectively in both branches, it is evident both pieces would be alike, only the ends reversed and one facemould required.
To draw the facemoulds, erect at b a line b b" perpendicular to a 5, also b 5', c c' and d df perpendicular to b d, and d dn and ee" to d e ; place the pitchboard of the first branch at a and draw the inclined tangent a 6", which rises then the height b 5", and carry that height over to b\ 16
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The whole height being four risers, there are one and one-half risers of 6f" and two and one, half risers of 8" each, which are set up from e to e" as indicated in the diagram. Place the pitchy board of the second branch at e", draw the upper inclined tangent e" d", and carry the height d d" over to df; it then becomes evident, that the middle tangent must occupy the position bf d' in space, in order to carry the rail the required height, with & as the centre of the middle buttjoint.
The first piece rises the height c cf \ to find the intersection of its cutting plane, extend V & intersecting its horizontal projection c b in u, and draw u a, which is the intersection required. Draw a seatline a" c" perpendicular, and the ordinate e er" parallel to the intersection, make c" c'" equal to c d and a" c,rr is the plumb-bevel line.
The second piece rises the balance of the whole height. To find the intersection of its cutting plane, make e" ef equal to c & or the height of the first piece, and draw from e' the tangent e' q parallel to e" d", intersecting e d extended in q ;  q c is the intersection.
Draw a seatline e c"" through e as the most convenient, and make e E equal to e e!.
The springbevels are obtained in the usual manner, by sliding the facemoulds.
Fig. 6 shows the drawing of the facemoulds over the landing ; ef is the tangent of the inclined straight rail, and hf on a line with the horizontal one ; as there is only one straight step in the second branch, place the lower buttjoint at e the centre of it, which corresponds to the same point in Fig. 5.
To raise the level rail the usual height of one-half of a rise above the landing, the crook must rise two risers from e to A, or wherever it connects with the level part. Place the pitchboard at e and draw the inclined tangent ef", which rises the height //''; carry this height around to/', and draw another tangent/' 7c, intersecting the horizontal one in some point beyond the joint of the cylinder, as in /*;. This causes a ramp easement in the rail along the straight part of the landing, of which hf 7c and 7c g1 are the tangents, connecting with the upper buttjoint of the wreath at g' and with the level rail at hf.
The height Ji hf is equal to two risers.
This leaves the distance g gf as the height of the wreath from centre to centre of buttjoint. The intersection e en of the oblique cutting plane, may be found as in Fig. 3, PI. XV., in case the intersection of tangent gf f with gf both extended, should fall outside of the paper.
Draw the seatline e" g"% for convenience through the point /, perpendicular and the ordinate g g'" parallel to the intersection e e" ; make g" g'n equal to g g' and draw e" g/;\ which is the plumb-bevel line.    The development of the facemould E F Gr with the slidelines is then as usual.
Comparing the vertical ordinates through the centres of the balusters, gives the variation in their length, as indicated in both Figs. 5 and 6.
There is another very simple construction, to find the intersection of the cutting plane when one of the tangents is very fiat, as/ g' in Fig. 6. It depends on the same principles as that of Problem VII., Fig. 6, PL VII., and is as follows:
Locate the tangents ef" and/' g' as before. The directing ordinate g g'n is parallel to the intersection e e". The point g being given, the point m where this ordinate intersects the horizontal projection ef of the lower tangent extended, is found by erecting a line e n perpendicular to ef; make e n equal to the height g g' of the wreath, apply at n the pitchboard reversed, drawing the line nm by the hypothenuse, which will intersect ef extended in m. Then gmis the directing ordinate, to which the seat e" g" is perpendicular and e e" parallel. Angle en on is the same as angle //" e.
Fig. 7 represents one of those cases, when platform stairs are built, either by neglect or necessity, in the most irregular manner possible.
Although stairs should not be built in this manner, the stair-builder wi 11 sometimes be called upon to get out rails for similar cases.
Fig. 8 shows the development of the facemoulds. The rise of the first branch of the stairs is assumed to be 8" by 9J" run; the step from the platform to the landing is 10J" run by 7f" rise ; the risers around the platform are placed without any regard to regularity.
There will be two wreathpieces, also a ramp-easement to the level rail. The location of the tangents is similar to that in Fig. 3, PL XVII.
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For convenience, make the lower buttjoint at a, the face of the tenth riser, and the upper one at e or the face of the landing riser.                     *
Draw the tangents a b and d e on a line with the straight rail, and b d through the proper point c perpendicular to the same. Extend these tangents as shown, and apply the pitchboard of the first branch at a; draw by its hypothenuse tangent a J", rising the height b 5", which carry around to V.
The whole height of both wreaths is three risers, one of 8" and two of 7f", which set up on the last riser line from e to df as indicated.
Apply the pitchboard of the last step at e" and draw the upper inclined tangent e" d" ; carry the height d d" over to d\ and b' d' will be the position of the cross tangent, with d as the centre of the middle buttjoint
Find the intersection a a" of the cutting plane of the first wreathpiece, or the parallel ordinate o d" by the methods mentioned in Fig. 6, and locate the seatline a" c"; c" d" is equal to c c\ which is the height of the first wreathpiece.
To find the intersection of the second cutting-plane, set off the height c d of the first wreath, from d' to d, then e d is the remaining height of the second ; draw d u parallel to d' d" and u c is the intersection. Locate the seatline d" d"f as convenient, make d" d"f equal to the height e d, and d,n dnt is the plumb-bevel line.
To draw the ramp-pattern, extend tangent d u, intersecting the horizontal tangent/' g in g. This horizontal tangent is located one-half of a rise higher than the point d ; d g and gf are then the tangents of the easement from the wreath to the level rail.
Jiemarfc. �In most cases the rail is raised one-half of a rise above the landing, because they vary not much from 8" on an average. This height may be made, however, to suit convenience; some persons prefer a high level rail, others a lower one.
When the risers are 6" for instance, it would be well to add 4" instead of 3" to the height; while, when the risers are 10", add also only 4" instead of 5".
PLATE    XXI.
WINDING   STAIR-CASES.
This plate represents a flight of double quarter-space winding stairs, or commonly called stairs winding top and bottom.
Fig. 1 is the ground plan. Figs. 2 to 4 are the wall-strings. Scale i" to the foot. Figs. 5 to 7 are details.    Scale 1" to the foot.
This form of staircases is frequently employed in school-houses and churches.
The steps are of hard wood 1J" to 2" thick ; the frontstring, cylinders, and opening of the well-hole are ceiled with 1�" or ��" narrow matched-boards, dressed both sides, receiving a cap of railing which is grooved on the underside to the depth of about i", corresponding to the thickness of the ceiling-boards.
The rail is moulded and the joints bolted together the same as a railing with balusters.
The ceiling-boards may be put up and the lower edge of them finished to receive the plastering in the form of a regular cylinder, and the rail fitted and fastened to them after the plastering is all done.
This requires some experience however. The safer way for the beginner will be, to put a ground on the frontstring, cylinder and around the opening for plastering ; when the masons are done, string up the rail as usual, and fit the ceiling-boards singly underneath, trimming them the suitable shape on the lower end to conform to the plastering.
A moulding is then put on the lower edge to cover the joint of the plastering; on cheap jobs the plastering is pointed up against the ceiling-boards.
To illustrate the "planning and laying out of these and similar stairs, we will assume the following dimensions as being given :
Length of space between studs, 14' 2" ; width of the opening in the second floor, 6' between studs and face of trimmer B ; length of the same, 12'; at A, the foot of the stairs, is a door-casing 5' 6" from the studding. Stairs to be 4' wide to outside of ceiling ; height of story, 12' 6" from top to top of floor;  depth of timbers, 10".
Allowing -J" for lath and plastering, the length of space will be 14' �", and the width of the opening 5' Hi". Using f" ceiling boards, the stairs will have to be 3' H-J" wide from face of wall to the face of frontstring, making in all the 4' required. This leaves 2' space between face of trimmer and frontstring, for the cylinder from outside to outside of ceiling boards.
To lay out the plan as Fig. 1, draw the outlines of the walls and trimmer ; find the centres m and n of the cylinders, which are equally distant from the cross and long partitions, and describe the semicircle fj of the upper, as also the quadrant e 1c of the lower cylinder. About 20' from these circles draw the dotted quadrants a b and c d, connecting them by a straight line b c; these will form the line to space off the run of the steps.
To use every inch of the space, let the last riser rest against the face o'f the trimmer, and set
the face of the first riser 1-J-", (or whatever the projection of the nosing) back from the door-casing.
This leaves the length a b c d as the whole run from face of first to face of last riser ; which length
is measured near enough for the purpose, by taking, say one foot, in the compasses, and spacing
it off, giving in this case 16'.    The height being 12' 6", employ the rule given Page 86, for finding
the number of risers:
Height,......12' 6"
One half run, . .    8'
Number of risers,   20^-Employing twenty risers, gives Tjfc" for each, and dividing 16' or 192" into 19 steps, gives lOft" run for each.
In laying out winding stairs in which part of the steps are straight ones, one or two of the winders are located in the straight part of the front string, to give a proper easement to the cylinders
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and the rail, as also to balance the direction of the nosings of the steps. Wherever a division of the steps on the run-line happens to come suitable for that purpose, make it a square riser ; for instance : the face of the seventh riser comes in a position to make it the riser of the first square step, leaving about 5" between it and the joint of the cylinder at e, while the face of the thirteenth riser happens to come within 9J" of the joint/ of the upper cylinder, thus giving six straight steps. Practically the run of the straight steps is made the nearest even fraction of inches ; so instead of making it 10^-" in this case, let it be 10", and divide the trifling difference among the winders from the thirteenth to the last riser.
T&e division of the risers in fhe cylinder is made to suit convenience in its construction, as well as to have the direction of the risers well balanced.
The winders next to the straight steps, as No. 6 and 13, ought to be about one half the width of those on the frontstring. It is convenient to have one of the risers come in the joint of the cylinder, as No. 6, in this case. The quadrant elc is divided into four equal parts, making the winders there also about one-half the width of a straight step, and four equal staves in the cylinder, which facilitates the work. The stave of the first step is made square to the riser, as the newel post will mostly cover it. The lines of the risers are then drawn from the divisions of the cylinder, through the respective points of division on the dotted line of the run ; this will bring 3, 4 and 5 as running nearly to the centre of the cylinder, while 1 and 7 run square to the line of the stairs, and 2 and 6 have a balancing direction between them. At the upper cylinder, the distance from the thirteenth riser to the joint of the cylinder/, happens to be 9J" ; this is hardly enough to divide into two winders. Considering that there is room enough to make them about half the width of the straight steps in the cylinder, let the winder 13 be then about 6", and 14 about 5", which brings the 15th riser to a point g in the cylinder. Then divide the balance g j of the same into seven equal Staves, giving the landing two, and each winder one of them. Draw the riser lines as was done in the lower part, which will bring 16 and 17 in a direction toward the centre of the cylinder, while. 14, 15, 18 and 19 graduate toward the square ones ; the last riser is curved from a line parallel to the timber to the point h.
Giving two staves to the landing, allows a proper easement in the cylinder to the level ceiling before it reaches the straight part.
The practice of some*stair-builders to carry these easements over into the straight part, causes a bad looking spot in the plastering and the cylinder, and ought to be avoided.
To put more than two winders into the straight part of the frontstring, is also erroneous ; it certainly does not add to the grace of the easement as is supposed, but decidedly destroys the convenience of the stairs, as might be illustrated by a reference to Fig. 6, PL XXIII. The dotted lines in that Figure represent the location of the risers as laid out in this plate.
Now, supposing the twelfth riser were taken as the last square one instead of the thirteenth, and the risers located as indicated by the full drawn lines, thus bringing three winders into the frontstring ; then three or four of those risers assume a position so much distorted to the walking-line of the stairs, that it actually becomes dangerous to use them. Coming down, the person's right heel, and going up, the left toe would strike the nosing.
Winding stairs can never be as convenient as straight ones, and their inconvenience ought therefore not be increased by an erroneous construction.
Figs. 2, 3 and 4, of this plate, show the wall-strings. The laying out of the steps and risers becomes plain from the diagrams.
The easement lines of the top edges are about equally distant from the nosings ; the main point, however, is to get good easy lines without being confined to an exact equal distance from the nosings ;  sufficient wood must be left on the lower edge to give a good support to the wedges.
The long string, Fig. 3, may be made with joints as shown in Fig. % PI. XXIII.; or a long plank about 13" wide may be used, a piece glued on the top edge of the lower part, and also one on the lower edge of the upper part, as indicated by a dotted line. These wall-strings are 1�" or 1�" thick.
Fig. 5 shows the shape of the steps and risers where they enter into the wall-string on an acute or obtuse angle.    The broken line in the centre of the string indicates the depth of the
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mortices, and the edges of the steps. Where a step and riser enter into the string as No. 16, the sharp angle backside of the riser is taken off square to the line of the string as shown at y, in order to allow the wedge to strike square against it. The mortice of the nosing in the string will evidently have an elliptic form at p instead of a half round.
Whenever a step and riser enter into the string as No. 17 and 18, the sharp angles of both the nosing and riser on the faceside are taken off square to the line of the string, as far as they enter into the same, as shown at s and t. This is done to avoid the difficulty of morticing the string sufficiently under and preserve the edges to make a good joint
It is evident this part requires accurate workmanship. The short strings have a tongue r cut on the end, which enters a corresponding vertical groove in the long one.
Fig. 6 is the ground plan of the steps, risers and strings of the lower corner.
Fig. 7 represents the corresponding part of the wall-string, showing more distinctly the elliptic form of the nosings.
The mortices of the winders are cut on the same taper lengthwise, as those of the straight steps, thus using wedges of the same taper, only longer. It will also be noticed, that the mortices of the risers increase in width as they enter more or less on a sharp angle.
In practice, the plan of the stairs may first be drawn on a small scale; then the winding parts, with the cylinders, &c, are laid out in full size, as shown in Fig. 1, on a sheet of thick brown paper; the lower parts, up to the seventh riser, may be drawn on one side, and the upper part from the thirteenth step to the landing, on the other side of it, requiring in this case a sheet about 6' square.
To get out a step, lay this plan on the plank, and prick the points of the outlines through with a sharp pointed brad-awl; the shape is thus drawn on the plank very accurate and convenient. The strings and cylinders are also laid out from this full-sized plan, and after the stair parts are built, it is rolled up and laid aside until the rail is wanted. The facemoulds for the rail are then developed on the same sheet, and pricked through on pieces of boards in the proper shape.
All the dimensions required are thus kept together on the same sheet for each separate flight of stairs.
pi xxn
PLATE     XXII.
CAKKIAGE    OF   WINDING-STA.IRS.
This plate represents the construction of the carriage for winding-stairs.
Scale for Figs. 1 to 3, J" to the foot; for Figs. 4 and 5, 1" to the foot.
As an example, the plan of PL XXI. is chosen.
Stairs of from 3' 6" to 4/ wide, as in this case, ought to have two strings between the front and wall-string. The latter may be made of 1^" or 1-J" plank, as it is spiked directly against the partition studs or brick wall, while the frontstring and the two middle ones are of 2" plank.
The object of the carriage is, to have all the plastering done before the finished steps, risers, and strings are put up, to prevent their being stained or otherwise injured by the mason work ; also to avoid the swelling of the materials, which is more or less occasioned by the dampness of the plastering. According to the way the stairs are to be finished, the rough carriage wall-string may be needed or not.
In school-houses, churches, and similar buildings, a wainscoting consisting of matched ceiling-boards is carried up against the wall, and may be fitted right down on the steps and against the risers ; this does away with the finished wall-string into which it is otherwise housed. In such a case, the rough carriage wall-string is needed to support the ends of the steps.
Whenever a finished wall-string is used, it may be put up in two different ways.
In one way the rough wall-string is also needed, and the steps and risers run against the wall as in the case of receiving ceiling-boards ; a groove about ���" wide and about as deep, is cut accurate in each of the steps and risers on a line of the face of the wall-string. The latter is sawed out to correspond with the run and rise, and a rabbet cut all around on the backside of the sawed out parts, leaving, a tongue which enters into the groove of the steps and risers mentioned.
It is evident that these grooves must be cut in each step and riser before they are finally nailed, and that it is well to scribe the different string-pieces before rabbeting them.
The steps receive a tongue on their back edge, whicn enters into a corresponding groove at the lower edge of the riser.    They are fitted and nailed, commencing at the top of the flight.
This system may be applied to circular or elliptical back strings, as well as straight ones. Unless the wall-string is thoroughly seasoned it is liable to shrink out of the groove.
The other system is, to mortice out the finished wall-string as represented in the previous plate, with the difference of turning the taper of the riser-groove upward and leaving out the rough carriage wall-string, in order to drop the finished one down in proper position when it is fastened to the wall. Then, commencing at the bottom of the flight, each step and riser is fitted, glued, wedged, and nailed singly, resting at the same time on the other carriage strings.
The steps are left �" wider at the back edge, full thickness, and the lower edge of the risers rabbeted to correspond; they overlap thus on the steps, are driven down solid by means of the wedge in the wall-string, and are nailed to the back edge of the steps ; it is on this account that the taper of the riser-groove is turned upward, in order to allow the wedge to be driven downward. Each step may thus be fitted and nailed very solid, and all squeaking avoided ; the cove is glued and nailed under the nosing, putting the glue on the upper side only, so that, in case there should be shrinkage any where, no joints will be opened.
The carriage represented on this plate, is laid out precisely to correspond with the plan on PL XXI. were it calculated for the second story. This has been done to show the construction of the lower portion, to form the proper furring for lathing in such cases. Like letters refer to like parts in all the figures. A is the header and B the trimmer of the second floor, while B' is the trimmer of the third.
The continuous riser lines in Figs. 1, 4 and 5, which are ground plans, represent the faces of the finished risers, while the dotted lines are their backsides, A carriage constructed by this system, does not require any rough risers.
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la, lb, lc, 2a, 3a, &c, are the different string-pieces, for which rough, sound pine plank are used. It will be seen, that by laying them all out from the sides toward the cylinder, as indicated by the thicker lines, any variation in their thickness will make no difference.
The riser-marks are laid out, of course, by the lines indicating their back sides, and at the top and bottom step, a proper allowance is made for any variation between the thickness of the flooring and that of the finished step, in order to have the risers from top of floor to top of first step, and from top of last step to top of landing, the same as the rest when finished. Of the bottom pieces, that marked 4a, abuts against the trimmer B, while 2a and 3a rest on some pieces of timber put in for that purpose, la being fastened to the partition. At the upper end, lc is fastened to the partition also, while 2C 3C and 4C are fastened to the upper trimmer B; by means of iron bolts or stout screws, as shown in Fig. 2. The different pieces are all framed into each other by tenons and corresponding mortices, and ought to be well spiked.
Fig. 2 also shows how the easements are made on the lower edges, forming at once a true furring for lathing on the upper portion. The easings are the longest in the wall-strings, and grow quicker gradually toward the front one. The easings of the lower portion are all alike from the straight steps down to the lower edge of the header. To obtain that shape, the frontstring with the cylinder, (of which Fig. 3 is the development)" must be laid out first, and the others conformed to it. It is important that the header A sets back from the cylinder at least 10" to 12", as otherwise the lower easement of the carriage would be too abrupt. The space between the header and the cylinder is rounded to in a proper curve.
In cases where the wall-strings la, lb and lc are not needed, pieces of timber must be fastened properly in the wall or partition, to which to secure the pieces 2a and 2C; the rest of the carriage being put up, pieces of board are fastened all around against the wall to lath to, in place of the wall-string; they must however, be narrow enough, not to interfere with the finished wall-string.
Bridgings are nailed in the proper places as indicated by the crossed lines in Fig. 1, which must also be kept so low as not to interfere with the risers.
A carriage constructed in this manner is very firm, and not liable to settle by any weight which is ever likely to be placed xupon it. The grain of the wood runs in all the pieces as much lengthwise as possible, and no settlement will take place on account of shrinkage. The carriage of stairs of any form enclosed on one side by walls, may be constructed by this system.
Fig. 4 shows the construction of the upper cylinder, which is let into the front-string and the trimmer.
The stave of the step No. 14 is the thickness of the riser wider, while the last one in the trimmer is the same thickness narrower, than if the risers were mitered into the cylinder ; the other staves are all alike.
Fig. 5 shows the lower cylinder constructed somewhat differently, which may also be done in the upper, if deemed advisable. Instead of hollowing the staves into a complete cylinder, straight pieces are thrown in, their faces conforming to the circle of the cylinder. By this means the curve is near enough correct to lath and plaster to, while the finished cylinder covers up all deficiencies afterward.
Temporary rough steps are placed on the carriage, as also grounds all around the strings, cylinders, and openings.
Tixxnr
PLATE    XXIII.
WINDING   STAIRS   WITH   CIRCULAR   WALL-STRINGS.
Fig. 1 represents the ground plan of a flight of double quarter-space winding stairs, with circular wall-strings, rail and "balusters, planned as a second story flight to suit the dimensions given for the plan on PL XXL
The cylinder is assumed in this case to be 2' in diameter finished, which will leave the width of the stairs 1" less, or 3" 10-J-" from outside to outside of strings. One inch is to be allowed for the face-board.
The planning of this stair-case is in the whole similar to that of PL XXI. The usual allowance is made for lath and plaster against the enclosing partitions ; the thickness of the wall-strings is generally 1�". They are plowed to receive the tongue of a suitable moulding, the backside being calculated to come flush with the face of the plastering.
Draw the semicircles for the cylinders, the straight lines of the front and wall-string, also the run line abed; the circular quadrants o r and s v of the wall-string are described from the same centres as the cylinders.
Space off the run, and draw the lines suitable for the first and last square risers, as 7 and 13; divide the winders in the cylinders as mentioned in the plan PL XXL, and draw risers Nos. 1 and 20.
In order to facilitate the laying out of the circular wall-string, the winding riser lines are not all drawn from their points of division in the cylinder through thoir respective points on the run line, as was done in PL XXL where the wall-strings form a corner; but in this case, only those riser-lines, as Nos. 2, 6, 14 and 19, (next to the first, last, and the square ones) are drawn in that manner, intersecting the face of the wall-string inp, q, t and u*
In the lower part, the arc_p q is divided into four, and in the upper one, the arc ttt, into five equal parts; the ri ser-lines are then drawn from the points so found to their respective points in the cylinder.
This manner of dividing the winding riser-lines will make the width of steps Nos. 2 to 5 and 14 to 18, alike respectively on the stretchout of the circular wall-string, while that of Nos. 6 and 13 (which are partly in the circular, partly in the straight part) is somewhat less. This division alters the run of the steps on the dotted line only slightly, if at all, and tends to form an easement in the wall-string.
Fig. 2 is the wall-string as laid out on the stretchout. It is generally made in three pieces: the middle one contains all the square steps, while the first and last contain the winders; they are connected by shoulder-joints, as shown at 7 and 13, and Fig. 2a.
The straight part is laid out as explained on PL XL For the circular parts, the lower edge of the plank is jointed straight, a guage-mark run on about 2" from the same, and the steps and riser-lines marked by a pitch-board made for the purpose: one side of this pitch-board is equal to tlie stretchout distance from face to face of riser of those winding steps, which are alike in width, and the other side is the height of the riser. It is evident, that by laying the hypothenuse of this pitch-board of the equal steps on the guage-mark, and marking the step and riser successively, they will be in proper position after the plank has been bent over the drum. The horizontal length of the stretchout of each step, from face to face of riser, is measured practically, by means of a slender rod, which is bent in the curve of the ground plan. The length of the stretchout of the odd steps, as Nos. 1, 6, 13 and 19, is laid off respectively adjoining the equal ones, and the shoulder-joints cut on a line with the risers. The mode of making these joints as represented in the diagram, is the strongest, and shows a joint only above the nosing. Some workmen cut a square joint instead of a vertical one, but it is certainly no stronger, and not as easily laid out.
This will bring the three pieces in the position as shown. The easements of the top edge are laid out as shown on the Figure: the line of the lower one rises above the seventh step, and passes about �" below a straight line parallel from the nosings of the winding steps ; while the upper one drops on the thirteenth step, and rises above the parallel line over the winders. Of course, the proper easements are made to connect to the base at top and bottom, whenever necessary. 17
130
The vertical lines as o, r, s and % (which represent the same points in Fig. 1,) are laid out on the string, by taking the proper distances between those points and the adjoining risers from the ground plan.
The part between those lines is dadoed to within about J" of the face side, bent and keyed up over a drum as shown in the perspective view, Fig. 4, next plate.
The keys are shown in the upper circular wall-string, Fig. 1; they may be 1-J-" apart in a string from 3' to 3' 6" radius, 2" in one from 3' 6" to 4', and 2i" in one from 4' to 5' radius. The surplus wood of the keys (which must be well glued) is taken off, and a strip of strong canvass about 6" wide, is glued over the lower edge of the backside. When thoroughly dry, the string will retain very near the shape, but as it is liable to spring back some, it is well enough to make the radius of the drum one or two inches smaller than the ground plan. The strings are morticed as usual on a taper, to allow wedges to be glued in, when the stairs are pat together ; the canvass is glued on the backside for the purpose of strengthening the lower edge, where the face part is cut out.
The cylinder parts of the front-string are laid out in a similar manner by a separate pitch-board, one side of which is equal to the stretchout of the equal winders from face to face of risers.
When a rough carriage is used for the stairs, the finishing cylinders and frontstring are made of 1" boards. The cylinders may be dadoed and keyed up the same as the backstring, cutting the grooves to within -J" of the face and about 1" apart; or the whole of the wood on the circular part may be removed, leaving the thickness of a stout veneer, which is bent also over a drum and staved up on the backside, as represented in Fig. 3, next plate.
This small drum for the cylinder is made of the same radius as the plan : the wood being cut down to within ��-", it is not liable to spring back as the wall-string is cut into within J" or ���" in order to retain the strength, as also not to show any keys outside of the moulding.
The keys which will show on the lower edge of the cylinder, are covered by the moulding or half round which is usually put on in such cases, to cover the joints of the plastering.
The circular parts of such a staircase are drawn full size on paper, as has been explained on PI. XXI., and the shape of the steps is pricked through on the plank.
Fig. 3 shows for instance, the shape of the sixteenth step, indicated by the thick line.
If the stairs are built of pine and put up before plastering, the circular partition studs are set by the strings ; about 1" furring is put between them, to allow for lathing and plastering as shown in Fig. 1.
C, Fig. 1, represents a niche, of which Fig, 4 is a front view. These niches are usually 20" in diameter when finished, by about 47 6" high to the top of the arch.
Their frame-work is very simple ; use two of the circular partition studs D E, as the vertical sides, and fasten two boards F F with the semicircle of the niche marked on them, in the proper position between said studs DE, and cut the shorter pieces HHH between the boards. The semicircular dome is then formed by nailing quarter circular ribs Gr Gr in the upper corners. Barrel hoops are used for lathing.
If the stairs are to be put up on carriages, these are built as shown on PL XXII., running into the corners, and the circular partitions set on and between the same.
Stairs of this class have usually a turnout, as shown on PL XVIII., at the foot of the first story flight.
Fig. 5 shows another method of building the carriage. The straight parts of the strings are sawed out as usual, with rough risers A A nailed against them, which are fastened into the partition or brick wall. A piece of timber B is run up diagonally underneath, fastened into the wall at both ends. Brackets C C connect the rough risers and the timber, which thus combined support the whole structure. D T> are so called stress blocks, which form the furring underneath and give support to the steps.
The frontstring and cylinder are constructed as in the other method, PL XXII.
The plan of this carriage is drawn to correspond with the stairs Fig. 1. The dotted lines indicate the finishing risers, which are mitered to the finishing front-string and cylinders.
Fig. 6 shows a faulty plan of winding stairs which has been explained in PL XXI.
PI. XXIV.
PLATE    ZXXIV^
Pig. 1 shows the development of the face-moulds of the rail for the double quarter-space winding-stairs of PL XXIII.    Scale f " to the foot.
To avoid a confusion of lines, the risers Nos. 1, 7, 13 and 20 are drawn in full lines, while the others are only indicated ; the width of one straight step is shown between Nos. 7 and 13 to facilitate the drawing of the ramp-easings.
The tangents ah, di and I m in the ground plan are parallel to the straight parts of the string and face-board, while hd and i I are at right angles to them.
To locate the tangents of the cutting plane for the lower wreath-pieces, set up the height of 6J risers from /to f on the seventh riser line (which is, of course, perpendicular to tangent/(f). This is the whole height of the rail from centre to centre up to that point; long balusters are used on the level floor, consequently we must raise half of a rise to make up the balance of the first one; there are five winders in the cylinder and one in the straight part of the front string, making in all 6|- risers.
From/7 draw down the tangent/' df agreeable to the pitch of the sixth winder, intersecting ~b d extended (which is also perpendicular to/d,) in d'; carry the height dd' around to d" as indicated by the arc, and draw d" 5, which is then the inclination of the cross tangent.
Tangent ah is horizontal in space, because it represents the centre line of the level rail extended.
Prom f draw also the tangent/' g\ which represents the hypothenuse of the pitch-board of the square steps or flyers.
A butt-joint is made at d vertically over the centre c of the cylinder, and also one at some point d over the sixth winder, giving some straight wood to the wreath, and forming the ramp-easement of which d f and /' <f are the tangents. This ramp forms then the connection between the straight rail over the flyers and the crook of the cylinder.
However, this joint d may be placed over the joint of the cylinder, as shown in similar cases on Pis. XXV. and XXVI.
The first crook will rise in this case the height cd from centre to centre of the butt-joint, and its face-mould A 5 d is developed and applied to the plank the same as the similar one Pig. 8, PL XIIL    From 2" to 3" of straight wood are added beyond the joint of the cylinder.
The second crook, of which cd and d e are the tangents in horizontal projection, will be a wreath with spring-bevels at both joints.
To find the intersection c u of its cutting plane, take the height c d of the first crook and set it from d down to d!; draw d' u parallel to d d' intersecting e d extended in % when u c will be the intersection wanted. Draw the seat d1 d" perpendicular, and ordinate e d,,r parallel to the intersection ; make d" dnt equal to edf, which is the height of the second crook, and draw the plumb-bevel line d"' d'V The development of the face-mould is then as usual, and its application to the plank similar to Fig. 7, PL XIIL The ordinates are drawn through the centre of the balusters, which are numbered for reference, to obtain their different lengths.
No. 1 is a regular long one; No. 2 is i" and No. 3 3i" longer. Nos. 5 and 9 are regular short ones ; No. 4 is 3" longer, Nos. 6 and 8 a trifle shorter, while No. 7 is about �" shorter than a regular short one.    No. 10 ought to be about y shorter on account of the easement in the ramp.
The location of the tangents for the upper wreath-piece is precisely similar to that of. the lower ones.
Draw the tangents 7i i9 i I and I m parallel and at right angles respectively, and extended, as represented in the diagram ; also the line Jc k' through the centre of the cylinder, parallel to i i' and IV. A butt-joint is made again over 7c, and for convenience one at h over the face of the fourteenth riser. To locate the tangents in this case, draw at Ti the pitch-board of the thirteenth step (which is the first winder above the flyers), as indicated by the rectangular triangle 7i g g", and from g"
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also the pitch-line of the flyers as g"f"; //" being then equal to two risers, and Ji g'\ g" f" are the tangents of the ramp pattern. Extend g" h, intersecting I i produced in i"9 and carry the height i i" around to iK as indicated by the arc.
Tangent Im is horizontal in space and raised half of a rise above th,e landing. The whole height of the two crooks from the point h to this horizontal tangent I m will be 6J risers, which height is set up from I to V ; then a line V i' will be the position of the cross tangent in space.
This tangent V i' extended intersects its horizontal projection I i in q ; then q Ji is the horizontal trace or intersection of the cutting plane for the wreath reaching from h to Jc.
The seat-line Ji" Jc" is again drawn perpendicular and the ordinate Jc Jc"' parallel to the intersection Ji q ; make Jc" Jc'" equal to Jc Jcf and Jc'" Ji" is the plumb-bevel line of the face-mould H J K, which is developed as usual.
This wreath will also have a spring-bevel at each joint, and the face-mould is applied as in Pig. 7, PL XIII.
The upper twist, for which Jc' V is the inclined tangent, is drawn and the face-mould applied as in Fig. 8, PI. XIII.    About 3" of straight wood are added beyond the joint of the cylinder.
A comparison of the vertical ordinates drawn through the balusters will give the variation in their length.
N6. 12 is about -J" longer than a regular short one on account of the easement, while No. IS is a short one of the usual length.
To find the different lengths of the others, draw a line through the upper point of the vertical ordinate No. 13, parallel to the seat line, and set up from that line on every vertical ordinate the respective number of risers the steps ascend, and add the balance to the length of No. 13, which is known to be a regular short one.
For instance: No. 16 is situated three risers higher than No. 13 ; from the point 16a, (which is the height.of the vertical ordinate of No. 13 above the seat line,) set up three risers to 16b, leaving a remainder 16b 16c or about 4", which is to be added to the length of No. 13. Consequently, No. 16 will be 4" longer than a regular short one.
By a similar process we find, that No. 14 is about J" longer; No. 15, 2"; No. 17, 4"; No. 18, 2/;; and No. 19, 1" longer than a regular short one; while No. 20 is %" shorter than a regular long one.
It has been mentioned that in practice, generally 2' 4" and 2' 8" are the regular standard lengths, and'no attention is paid to any slight variation in the height of the rise from 8", in regard to those lengths; therefore, such odd lengths as No. 20, where the observer's eye will take the range of the! long ones, are compared with the latter.
For the same reason, most turners will more readily comprehend a stair-builder's order for a lot of baliisters varying in the length of the shaft, by naming the entire length of such balusters. For instance, in this case the odd balusters required around the upper ramp and cylinder,
would be:
No. 12, 2' 4J" ; No. 15, 2' 6".
Nos. 16 and 17, 2' 8" or regular long ones. No. 18, 2'6"; No. 19, 2' 5". No. 20, 2' 7%".
No. 14 varies so little from a regular short one, that practically no account need be made of it in that position, but a regular short one is used.
There arfe six square steps between the winders in this case; it will take the length of four pitch-bbards between the joints g' and f'\ as one square step is contained in each ramp pattern.
Rema/rJc.�The inclination of such tangents as d'f or Ji i" is not always governed alone by the pitch, of the winder next adjoining the square riser, because the width of such a winder may be more or less, just as the division of the steps happens to come while planning the stairs, as explained in PI. XXI. For instance, to consider the case of tangent d'f', the two points b and /', or n instead off, are given ones, but the height of the point d' may be higher or lower, and the length of the balusters in the cylinder regulated by it.
It is desirable to have those balusters as near as possible of equal length; if it is found, after
133
drawing the seat and plumb-bevel line with a few ordinates through such balusters as Nos. 5, 7 and 9, that No. 7 would be short (in this case �"), then the point dl may be raised, and another intersection found.
Raising the point d\ will also raise d half that much, and make balusters 4 and 5 also somewhat longer; now, as No. 4 is already about 3" longer than a regular short one, it appears that the position of the cutting plane, or the height of the point d\ is about right as drawn in the diagram for this special case.
If however, the first riser should be located 2" or 3" nearer the point b and the other risers in proportion, which is frequently the case (see next plate), then although the height of point fr would remain precisely the same, the inclination of tangent d' f would have to be changed by rising up the point d\ or balusters Nos. 5, 6 and 7 would be too short.
Considering the position of the cutting plane of the wreaths in the upper cylinder, we find by the vertical ordinates, that the balusters on the winders are mostly longer than regular short ones. This position may be altered to make the balusters shorter, by lowering the point f, which would then flatten the inclination of tangent hj" and bring its intersection with tangent f" g" of the straight rail, at some point over the square steps.
Having the rail somewhat higher in the upper, and rather low over the bottom cylinder, will favor the easement of the first and last wreath-pieces connecting with the level rails, which easements would otherwise look rather abrupt.
As long as we undertake to get over the half cylinder of quarter-space stairs in two wreath-pieces, we must use more or less balusters of various lengths, because the elliptic-plane curve of the wreath very seldom coincides with the helical curve of the nosings
But it is evident, that by employing this system of drawing face-moulds by ordinates, we can always tell by comparing the vertical ones, whether the rail will come too low or too high over the nosings, and hence locate the tangents, so that the cutting planes will as near as possible coincide with the helical curve of the nosings, whenever practicable. (See Fig. 15, PI. IX., also compare what is said about locating tangents and the intersection of cutting planes in Fig. 6, PL XXVI.)
Fig. 2 shows the development for the lower wreath mitering into the newel-cap. The rail to receive ceiling boards instead of balusters, as in the case of the plan PI. XXI., of which this figure is a part copy on a scale of f " to the foot.
The fine continuous lines indicate the face of the string, cylinder and risers.
It will be noticed that the centre line of the rail and its tangent geb, are drawn half of the thickness of the ceiling boards toward the centre of the cylinder, instead of toward the stairs, as is done in case of balusters.
The ramp joints e' gf are placed over the same points as in the lower part of Fig. 1
Starting from a newel-cap and the newel standing on the first step, we omit the first rise in counting the height of the crook, and also a certain distance must be allowed to obtain an easement from the cap. This distance ought not to be less than one rise, instead of from 5" to 7", as is sufficient for an easement starting from straight steps or a turnout.
Allowing the height of one rise in this case for such easement, there remains the height of five risers, from the centre of the cap to that point f of the rail, which is plumb over the first square riser line, or No. 7.
To locate the tangent of the wreath, set up this height of five risers on the seventh riser line from f to f\ and draw the pitch of the sixth winder, (which is the one next to it,) from/' downward, intersecting g e extended in u.
The centre n of the cap ought to be situated somewhere in the centre line of the rail, if such were extended ; and also so that the newel will clear the second riser. The miter point b is found as explained in the case Fig. 1, PL XYIII. Draw the line ub, which is the intersection of the cutting plane.
The seat-line bN e,f is drawn wherever convenient perpendicular, and the ordinate e ent through the centre of the joint at e, parallel to u b. The height e" e"' of the wreath is equal to e e\ and bn ef" the plumb-bevel line. The development of the face-mould EABC is as usual. The developed ordinate b" U is equal to its horizontal one b" u.    The Dosition of the inclined tangent
134
E U on the face-mould may also be obtained by any other ordinate intersecting it, as the point K from Jc.
Using ceiling boards instead of balusters, and there being no occasion to find their odd lengths, the trammel may be employed to describe the curve lines, if deemed convenient, as indicated in the figure.
The application of the face-mould to the plank is the same as in the case of the turnout, PL XVIII.    The vertical miter-joints are found by sliding the face-mould on each side.
The tangents for the ramp are er f and fr g'
The upper parts of the rail for the plan PL XXI., are drawn similar to the upper part of Pig. 1 on this plate.
Whenever rails are to be fitted on ceiling which is already put up, it is not advisable to cut the straight rail to an exact length, as the ceiling is rarely put up with nicety enough to warrant the same. It is better to fit the crooked parts to their respective places, cut the straight rail between them, and then bolt up the whole in one piece from floor to floor.
Fig. 3 shows a semicircular drum, over which cylinders of about 16" or more in diameter, may be bent and keyed, or staved.
It is made of 1�" plank, the strips of parallel width and jointed together. The two pieces A and B have a straight surface and are at right angles to the diameter, and therefore parallel.
All vertical lines, as those of the risers for instance, must run parallel to the joints of the strips when the board is placed in position on the drum. The straight parts are fastened by hand screws.
Fig. 4 shows a quarter circular drum for forming circular wall-strings. The strips are about \\" thick by 2y wide, and fastened parallel about 2" apart. The pieces A and B form the straight wood of the string boards, and their surfaces are evidently at right angles.
Applying the dadoed string-piece to the drum9 attention must be paid that the riser and other vertical lines run parallel to the strips. These drums are made of various shapes to suit the form of the staircase.
PI. XXV.
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PLATE    XXV.
QUARTER-SPACE   WINDING   STAIRS.
Figs. 1 to 4 of this plate exhibit ground plans of quarter-space winding stairs as usually constructed in cheap dwellings.    Scale -J" to the foot.
Figs. 5 and 6 show the face-moulds,    Scale 1" to the foot
The halls of the first story are 13' 6" long by T wide, and represent those of a double house, in which the centre wall runs on an angle, a;nd the rear partition parallel to the front. The wall strings form thus an acute angle in Fig. 1, and an obtuse one in Fig. 2.
In Fig. 1 the cylinder is 1', which is divided into six equal staves ; two of them are taken up for the landing, and one for each winder.
The first riser is set back far enough to clear the front door, and the last riser runs parallel to the timber ; this gives 9" run for each step on the division line, and places one winder in the front-string,
In Fig. 2 the cylinder is V 6" and is divided into seven equal staves, two of which are again taken for the landing. The last riser is curved, so that it strikes against the timber. The steps are paced off on the run line from the last riser, the same run of 9" as in Fig. 1, thus placing two winders in the front-string.
It will be noticed that there is considerable more room at the foot of the stairs in Fig. 2 than there is in Fig� 1, although there are the same number of steps of the same run in both flights.
A line through the centre of cylinder is of the same length, 13' 6" in each hall, and while stairs Fig, 1, take up 10' 9" to the face of the riser, leaving only 2' 9" between that and the centre of the door, flight Fig. 2 takes up only 9' 10", and leaves 3' 8" between the same points.
This gain of 11" in the room at the foot of stairs in Fig. 2 is occasioned by using a cylinder of 18" diameter and bringing the last riser against the timber, instead of a 12" cylinder as in Fig. 1? and running the last riser parallel to the timber from the joint of the stave, which leaves 3" of lost space between them.
Fig. 3 is the plan of the second story flight winding top and bottom, and corresponding to Fig. 1. The position of the winding riser lines is the same top and bottom, with three square steps in the centre, and of the same run as in the first story. The plan of the top winding part is therefore the same in both stories. The length of the space taken up by the stairs on a line through the the centre of the cylinders is 10' 3".    The number of steps is one less than in the first story.
Fig. 4 is the plan of the second story flight, corresponding to the first story one Fig. 2� The size of the cylinders and the run on the division line, as also the shape of the first and last risers, are the same in both stories. The number of steps is one less in the second than in the first flight. The length of the space taken up on a line through the centre of the cylinders is 8' 6" for the second flight, thus leaving a small bedroom over the hall. There being but little straight wood in the front-string between the cylinders, the steps are all winders, and although the division of the risers in the cylinders is alike both top and bottom, it will be noticed that their position is different from the top part of the first story flight.
The partitions A and B, Figs. 1 and 3, are plumb over each other, leaving a passage of W on the second floor, the stairs being also 3' wide ; but the partition B of the second and third floor in Fig. 4 must set back 6" from over the partition A Fig. 2 of the first floor, on account of the larger cylinder, in order to leave the same passage of 3' on the second and third floors, making the hall T 6" wide.
The plans of the stairs as represented in these figures, are chosen of this peculiar shape, to exhibit comprehensively, how a variation in the dimensions alters their form; also to illustrate how, when the hall is short, an important advantage may be gained by making the cylinder somewhat larger and using every inch of the available space.
136
It is evident that if the partitions are square instead of on an angle, the comparison between the different cases will be the same.
Fig. 5 shows the development of the facemould for the turnout and the upper crooks for the flight Fig. 1. The rail is assumed to be a round one, and therefore only the curve of the centre line is drawn on the ground plan ; this centre line is developed with the tangents, and one half of the diameter of the rail set off each way, which gives all the face-moulds of a parallel width.
All the joints are made square to the tangents and the face of the plank, except the miter at the cap, which is found by sliding the face-mould, or by the plumb-bevel, as mentioned, Fig. 1, PL XVIII.
The spring-bevels may be found by sliding the face-moulds as usual, or by construction as indicated, according to the problems on PI. IX.
All the crooks for a round rail are sawed accurately by the facemoulds, out of a plank of the same thickness as the diameter of the rail; they are made octagon all around and finely rounded off.    The shape of the face-moulds are the natural easements.
The location of the tangents and the development of the centre line of the face-mould for the turnout, are the same as explained in PI. XVIII.
In this case the joint is made at e, which is half of the third step. The newel-post setting back close to the second riser, 8" or one rise is taken in this diagram for the easement, which will make the newel then 2' 9" high from top of step to underside of cap, leaving li risers to be set up on the perpendicular e e'. Apply the pitch-board at e\ and draw the tangent e' d; b d is then the intersection.
The miter a 5 c is found as explained, PL XVIII. For explanation of this spring-bevel, see Fig. 18, PL IX.
Of course, the points A, B and C are found by ordinates which give the joints; these and the outside curves parallel to the centre curve, determine the shape of the face-mould.
In the crooks for the cylindrical part, a butt-joint is made at 7i the joint of the cylinder, and one at 7c the centre of the same. To locate the tangents, we must consider there are four winders in the cylinder ; the rail must rise therefore the height of 4|- risers from the joint 7i to the horizontal tangent of the level rail at Z, which is as usual raised 4" above the landing ; set up this height from I to V on the perpendicular. If we make the distance i q equal to i 7i> and draw a line from V to q, and let this line be the position of the cross tangent, we find that ii' is just one-third of the whole height, and 7c 7c' two-thirds ; because the three distances qi, ilc and It I are equal, and the perpendiculars parallel; 7i q being then the intersection, and the inclination of tangent Ji i in space the same as i' I'. In that case, the position of these tangents will be similar to those in Fig. 15, PL IX., and the intersection Ti q perpendicular to the cord line 7i It. This cord line may then be used at once as the seat line, and the face-mould developed. In this diagram a separate seat line H Jc" is drawn to avoid a confusion of lines. K 7c" is equal to 7c 7cr or three risers. The spring-bevels (the construction of which is shown in the diagram) are alike on both butt-joints in this wreath, because both tangents have the same inclination to the horizontal plane of projection.
The face-mould of the upper crook, which rises just one-third of the whole height, is drawn further down in this diagram to save room. LM is equal to Im, and perpendicular to 7c' V ; K L equal and parallel to 7c' I'.
*To draw the ramp pattern, the inclination of the wreath tangent may be transferred by carrying the height i i' over to i'\ then i" 7i is such tangent; make g g' equal to one, and ff equal to two risers ; f gf represents the hypothenuse of the pitch-board, or the tangent of the straight rail. Extend both/7 g' and 7i i" intersecting in n'; then/'7 n' and 7i n' are the tangents of the ramp pattern, containing one square step to the point/7. To cut all the joints, it is evident that it takes for the flight Fig. 1 the length of 5^ pitch-boards of straight rail, between the joints e" and/7.
Locating the tangents in this manner will make all the balusters on the winders regular short ones, except that on the landing, which is about 17/ shorter. The baluster No. 1 of the turnout is about l77 longer than a regular short one.
In some cases the face-moulds of the cylindrical twists of the first story may be u�ed for the second, top and bottom, if the stairs are planned alike around the cylinder as in Figs. 1 and 3, and
137
the risers of the two flights do not vary too much ; the variation is made up in a separate ramp pattern asp" n" 7i". For instance : the risers of the first story, Fig. 1, are assumed to be 8" ; now let those of the second story flight Fig. 3, be 8J". The cylindrical twists pass over four steps and the landing, thus containing the height of 4� risers or 36" in the first flight, for which the face-moulds were drawn. The same number of risers in the second flight contains the height of 37i", consequently there is a difference of li".
To construct a suitable ramp pattern for the purpose of using the first story face-moulds in the second, lay out the elevation of the winder in the straight string and one square step with the second story risers, as shown by h"' g" and g" f" in this diagram. Extend the riser line through /V" svnd make Ti"! Ti" equal to the difference of 1�". Draw from 7h" a tangent h" n" parallel to hn'j intersecting the hypothenuse f" g" in n"; then h" n" and n"f" are the tangents of the ramp pattern, which will bring the rail the proper height again over the square steps. It is evident that this same ramp pattern used reversedly for the lower part, will bring the twists again in right position, so that the whole height from level to level rail will be as required. In consequence of this, the baluster on step 5, Fig. 3, will be 1" shorter, and the one on the corresponding upper step or 3STo. 10, will be 1" longer than a regular short one, and some of the others will vary in proportion.
Where there are so few square steps between the winders, as in Fig. 3, enough straight wood is added to tangent f" n" to make up the whole in both ramp pieces, instead of cutting a short piece of straight rail between them. In this case make p" f" equal to half of the hypothenuse of the square step pitch-board.
Fig. 6 shows the cylinders of Fig. 4, with the location of the tangents and the centre line of the face-moulds, which are to be used for top and bottom cylinder.
Stairs of this description are planned with a view of using only one set of face-moulds.
The winders in the centre are made about 1" to 1�" wider than those in the cylinder.
The location of the tangents is similar to Fig. 5, but there will be no ramp, as the joints come together at a.
The whole height of the two crooks is seven risers, which is set up on the perpendicular from d to d;. The tangent in this case must be so located, that all balusters are as near as possible regular short ones ; because whatever they vary one way in the upper cylinder, they will vary the other way in the lower one. To locate the point d the proper height, we must consider that c, or the centre of the cylinder, is about quarter of the width of a winder distant from the twelfth riser. Now, if a regular short baluster is used on that winder, and the rail follows the line of the nosings, it must descend 2J risers from the point d' down to <?', namely: half of a riser for the landing, one for step 13, and three-fourths of a riser for the balance of the twelfth step. Make d; n equal to 2J risers, and square across, intersecting c d in d ; then d' d is the position of the cross tangent and a q the intersection. The development is then as usual. Comparing the vertical ordinates of the balusters, they will be found to be nearly all regular short ones.
If b b" is made equal to b b', then a b" is evidently the pitch of the tangent passing over the straight part of the string. This is drawn in the diagram, for the purpose of finding by construction, the spring-bevel s b belonging to that joint A. If that spring-bevel were found by sliding the face-mould, it is evident there would be no need of finding a b" in this case. In the construction of round rails, if the spring-bevels are found as indicated by construction, there is no occasion for using any slide lines on any of the wreaths, where both joints are made square from the tangents.
It is well to mention in connection with this figure, that in case there should be more than three winders contained in the straight string, the pitch of such winders must be taken as the inclination of tangent a 6", and then that of the cross tangent b' d' will be governed by the same.
18
PLATE   XXVI.
HALP-CIRCULAE   AND   QUARTER-CIRCULAR   STAIRS.
The figures on this plate are ground plans of various forms of winding stairs, also the development of the face-moulds.
Scale for Figs 1, 2, 3 and 7, J" ; and Figs. 4, 5, 6 and 8, 1" to the foot.
Fig. 1 is a flight of half-circular stairs, with some square steps at each end. The division of the riser-points in front and back-string, is the same as explained in Plate XXIII. The semi-cylinder is divided into eight equal parts, and those eight steps from six to thirteen, are equal on the stretch-out in the wall-string. The last riser is placed in the upper cylinder to make the rail rise one-half of a riser above the landing in a common twist.
Fig. 2 is the plan of quarter-space winding-stairs, starting with one straight step at the foot, and landing the same as a flight of straight cylinder stairs.
Fig. 3 shows some curtailed steps at the foot, and the cylinder is formed to give them the appearance of a turnout.
Fig. 4 shows the drawing of the ramp D gf\ and the face-mould D A B C of the wreaths for the flight Fig. 1.
The rise and run of the square steps, as also the cylinder part being the same from the centre c each way, one face-mould and ramp-pattern will answer for both winding portions.
This figure shows a modification of drawing the face-mould, as represented in Fig. 15, Plate IX. There are four equal winders in the quadrant of the circle ; the inclination of tangents a b and b c is made equal, and then carried into the ramp-pattern to connect with the hypothenuse of the pitch-board of the square steps. This will make the balusters on the winders all regular short ones.
To draw the face-mould in this manner, use the cord a c as the seat-line ; draw the line o b extended, which will be perpendicular to the seat, and bisect the same in i. Make i i' equal to two risers, then a i' will be equal to one-half of the plumb-bevel line of the wreath, and p b the plumb-bevel. To avoid confusion, draw a line A C through 5, parallel to a c, and make it from b each way equal to a i'; then A C will be the plumb-bevel line in full length.
Divide one-half of the cord a i into a suitable number of equal parts, as two, a 7i and 7i i in this diagram; then divide each half, A b and b C, of the whole plumb-bevel line into the same number of equal parts. Through each of these points of division of the seat and plumbbevel line draw the perpendicular ordinates, and make them equal respectively, as 7^/ 7if equal to 7i 7e, and b B to ib. A B and B G are the tangents, and the other points indicate the curves of the face-mould. Some straight wood, as A D, may be added at one end, to connect with the ramp at D. To find the inclination of the tangents in space, make b br also equal to two risers, then a b' is one of these tangents, which extended, will intersect the hypothenuse/' er of the square step pitchboard at g. Make a D equal to the straight wood A D added on the face-mould, then D g and g f are the tangents of the ramp-pattern, containing one square step; ff is equal to two risers. If the drawing be correct, tangents A B and B C will be equal to a bf.
Fig. 5 shows the development of the face-mould and ramp-pattern for the flight, Fig. 2. The point (f belongs to Fig. 6.
The construction is the same as in Fig. 4, with the difference, that the use of the trammel is indicated to describe the center line of the face-mould intended for a round rail. In this case, the cord-line is extended ; A i and C i each are equal to a i\ then A b and b C are the tangents of the face-mould.   Aline through o, parallel to a c, is the direction of the major axis; m V is one-half of
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its length, while o I is one-half of the minor axis. The curve of the semi-ellipse will, of course, pass through A and 0 (see Fig. 15, Plate IX.)
There being only one square step at the foot of the stairs, the easement for the miter-cap will contain the whole of the straight wood connecting with the wreath. The miter-cap easement and ramp pattern are drawn in this diagram, to use balusters 2' 6" long on the winders. Some parties prefer to do so in stairs of this description.
For this purpose the points a and o of the wreath are raised 2" above the height of regular short balusters. This extra height is added to the lower easement and taken off the upper ramp, which will bring the newel-post and the balusters on the square steps of the usual height.
To draw the easement of the miter-cap, make a a' equal to 2", and draw a; E parallel to a b', or the inclination of the tangent; draw an elevation of the square step and first winder, as indicated by the dotted lines, and set up as usual, 6" from the top side of the step to the centre of the miter at D, to obtain an easement. Draw D E parallel to d a, then a' E and D E are the tangents of the pattern. To draw the upper ramp make e & equal to two risers, and b & is the inclination of the tangent; niake^ n' equal to four risers, and draw the hypothenuse n' h' of the square-step pitch-board; set up 2" from d to c", and draw c" p parallel to b c', intersecting the centre line of the straight rail at^>; then c" p and p n' are the tangents of the ramp-patterri, containing one square step.
Figs. 6 to 8 show the drawing of the face-moulds for the wreaths and ramp of the flight, Fig. 3.
The quadrant of the cylinder is again divided into four equal staves, and those of steps E~os. 1, 2 and 3 are of the same width.
In Fig. 6 a short ramp is used to ease from the square steps into the wreath ; the latter also contains the easing into the miter-cap.    The balusters will be more or less of various lengths.
Fig. 7 shows the ramp, wreath and easing of the cap contained all in one piece ; this method makes an abrupt easement from the square steps into the wreath part, and the balusters on the upper winders are so very long that it never should be used.
Fig. 8 shows the easement into the miter-cap separate, which, in connection with the face-mould and the ramp pattern of Fig. 4, will make the best looking rail for this flight, although the crooked part will consist of three pieces. The balusters will be regular short ones., except on the second, third and ninth steps.
To draw the rail patterns as indicated in Fig. 6, we have to find the intersection of a cutting-plane in such a position, that the rail will follow nearly the line of the nosing.
Setting up one riser for an easement at the newel-cap, the point b of the miter is given ; the rail will rise from that point to g or the first square riser the height of 7 risers, which gives a second point in the cutting-plane. A third one may be assumed nearly in the middle of the curve as at e, and its height is to be that of a regular short baluster ; this point e being situated half-way between the 5th and 6th risers, the rail will rise from b to e the height of 3| risers, which gives the third point in the cutting-plane. To find its intersection, draw a line from g to e extended and erect at both points perpendiculars g g' and e e'; make g gr equal to 7 and e e' to 3|- risers. Draw g' e\ intersecting g e extended in q, then b q will be the intersection wanted (see Problem IV, Fig. 1, PL VII).
The inclination of the tangents passing through the wreath and the ramp must next be found.
If there were room on the paper, we would set up the height of the 7 risers from g on the 9th riser-line, and then a line drawn from the point so obtained to u0 which is the intersection of bq and h f both extended, would be the tangent required at once. For the reason stated, the 10th riser-line is extended below h, 3 risers to h', also a line 7if uf is drawn parallel and u u' perpendicular to li Us both intersecting at u'.
From li set up an additional 5 risers to A", making in all from 7i' to 7^" 8 risers ; at h" apply the square-step pitch-board, h" g" being its hypothenuse ; then the point g" will be the height of 7 risers, perpendicular above the line vf h'.    A line u' g" is the inclined tangent required.
Placing the butt joint at/", the distance f f will be the height of the wreath, and/' g" and g" h" the tangents of the ramp.                                                                                         .            A?,
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The intersection and the height being found, the drawing of the face-mould is then as usual.
The seat-line b' f" is perpendicular and the ordinate f f" through the point/, parallel to b q; f'f'" is equal to f f" and V f"" is the plumb-bevel line.
The miter-joints ABC are formed on the plank by sliding the face-mould, which is applied as explained in PI. XVIII.
Drawing an ordinate through the centre of the baluster on the 8th step, marked 8, parallel toff", we find that this baluster is the distance 8a 8b longer than a regular short one. To compare this length with the others on the winding part, set the distance 8a 8b from the plumb-bevel line down on the vertical ordinate through 8, then the point marked 8=2i+, indicates the relative length of that baluster. From that point pace down equal riser distances, marked 7, 6, 5, 4, &c, and square across, intersecting the vertical ordinates respectively. These points of intersection, as they fall above or below the plumb-bevel line, indicate how much longer or shorter those balusters are than regular short ones.   For instance, No. 6, is 2i" longer and No. 4, 2J" shorter than a regular short one.
It is evident, that the cutting plane may be located through the point mr instead of g'\ thus lowering the upper part of the wreath, and raising the lower, which will bring the balusters in the cylinder more nearly of equal length, and the ramp pattern more like that of Fig. 4.
The way the cutting plane is located in the diagram will favor the easement into the miter cap.
Fig. 7 shows how to draw a face-mould which will contain the whole of the crooked part in one piece. On the perpendicular h h' set up the whole height of the eight risers ; apply the pitch-board at hr and draw the tangent hr gr of the straight rail, intersecting its horizontal projection in u ; then u b is the intersection.    The face-mould H B is then drawn as usual.
A comparison of the vertical ordinates through the balusters, shows that some of them are very long ; the one on the seventh step for instance, would be 2' lOf", which would not look very well in that position.
Another way of drawing the face-moulds for this rail is, to draw one for the quadrant of the circle from the fourth to the eighth risers, giving both tangents an equal inclination as in Fig. 4. This will make the ramp also the same as in that figure, and bring the butt-joint C downward over the fourth riser, and f upward over the tenth riser Fig. 3. Then draw a separate face-mould to connect the miter cap with the joint C, passing over the three first steps as shown in Fig. 8, which is part of the ground plan of Fig. 6.
Tangent c q Fig. 8 represents tangent b c Fig. 4 extended (assuming the latter figure reversed). Taking one riser for an easement, the rail will rise two risers from the centre of the miter cap to the centre of the butt-joint over c Fig. 8. Make c c\ which is perpendicular to c q, equal to two risers and angle c c' q equal to angle bbf a Fig. 4, and draw the inclined tangent d q, intersecting its horizontal projection in q ; then q i is the intersection of the cutting plane. In this particular case, as o d Fig. 8, happens to be equal to b br Fig. 4, each being two risers: c q of the former will be equal to a & of the latter figure ; but this is only accidental, and the rule applicable in all cases is, to locate the point </, and transfer the angle of the inclined tangent as above mentioned.
The intersection i q being found, draw a seat-line c q' perpendicular and the ordinate c C parallel to the same ; make c O equal to c c\ then C qf is the plumb-bevel line.
The application of the face-mould C K J H is the same as that of all others containing the miter, and is explained in Plate XVIII.
To find the length of the balusters on the second and third steps, draw also an ordinate through No. 4, or the one on the fourth step, which we know to be a regular short one in this case. From the upper point marked 4, pace down the two riser points marked 3 and 2, square across to the vertical ordinates as mentioned in Fig. 6, and it will be found, that No. 2 is 3J" and No. 3, If" longer than No. 4.
Remark.�The various ways of drawing the face-moulds for this rail as explained on this plate, give the reader a good insight into the theory of locating the tangents and cutting planes.
He will see, that there need be no guessing about the inclination to be given to such tangents as v! gf\ Fig. 6, as laid down in other books on Hand-railing, but that he may know beforehand jwt where the rail is coming.
PI. XX VII.
PLATE   XXVIL
CIRCULAR   STAIRCASES.
Figs. 1 to 3 are plans of half and full circular staircases. Scale J" to the foot. Figs. 4 and 5 are details in the construction of Fig, 3; Figs. 6 to 9 show the development of the face-moulds for the rails.    Scale f" to the foot.
Fig. 1 exhibits the plan of a half circular staircase similar to Fig. 1, Plate XXVI., the landing riser being on a line with the diameter of the cylinder, and the face-board at right angles to the same.
This form of staircase is sometimes employed in the rear end of a hall which runs through the whole depth of the main part of the building ; it allows a door underneath to pass to the rear.
Fig. 2 is the plan of a flight of half circular stairs, which makes a very good appearance and takes up comparatively but little space ; it is employed to good advantage under similar circumstances as Fig. 1. The hall in this case is 10' wide, and the semi-circle of both wall and front-string is divided into 12 equal parts, giving about 9J" run on the walking line.
Steps ISTos. 1, 2, 15 and 16 are of the same width as the others in the cylinder, and on the wall-string equal to the run. This causes them to be curved on the nosing line, and gives the appearance of a turnout at the foot of the stairs, and a landing square to the line of the hall.
If there be room enough at the foot of the stairs, steps INTos. 1 and 2 may be made of nearly the same width as the others on the wall-string, which would improve their appearance.
Fig. 3 is the plan of a flight of full circular stairs ; the steps are all alike on both cylinder and wall-string respectively.
This form of staircase is suitable for private use in stores, as it may be placed in one corner and takes up but very little room ; it receives then a railing on both inside and outside. Two points of support besides the starting and landing, are had by means of iron brackets A A driven into the wall.
The timbers B B which form the platform, ought to be halved over the- trimmer and run into some timber back of the same, in order to support themselves. A small cylinder C is connected with the outside string, and the face-board and railing returned against the wall.
CONSTRUCTION OF SELF-SUPPORTING STAIRS.
If it be desirable to make this or any other form of circular stairs self-supporting, the following construction may be employed :
Fig. 4 shows in ground plan the arrangement of the frame work of self-supporting circular stairs of the form Fig. 3, the steps removed. The inside cylinder D, when not more than 2' or 3' in diameter, may be staved up or bent over the drum; when staved, the fibres of the wood run vertical as in any other cylinder, but the staves ought to be screwed together besides being glued, to afford greater security.
fc         E E are the riser-boards, mitered at both ends, and nailed to the stress-blocks F F, which
form the truss-work of the outside strings. These stress-blocks are made of well-seasoned hard wood plank, about 3" thick ; the fibres of the wood run about parallel to the hypothenuse of the pitch-board, as shown in Fig 5. They are of such a form that the face of the vertical butt-joint corresponds to the back side of the riser-board, and the lower edge forms the furring for lathing ; they are bolted together by stout bolts, Gr Gr, similar to the manner of bolting rail pieces. The outside of them is rounded to conform to the curve, and the whole covered by a veneer about ��" thick, which is mitered to the risers and drops below the lower edge of the stress-blocks to receive
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plastering. To add to the security of the whole structure, furring strips K are nailed and bent around the inside.
The underside of such stairs is frequently paneled with some fancy wood, instead of being lathed and plastered. It is evident they can never be made strong enough to support very great weight, and as they are in most cases a matter of luxury and elegance, the extra cost of the panel-work is seldom taken into consideration, and ought to be employed in preference to plastering, as the latter is liable to work loose in the course of time.
A very handsome-looking affair is made of such stairs, when the inside and outside strings are dressed both sides to the proper curve, scroll-work sawed on the lower edge, and panels cut into the sides ; steps and risers are then dressed both sides, as the whole structure is seen from the underside as well. A good solid rail, with balusters well glued top and bottom, adds materially to their strength. When the diameter of the stairs is small, a solid spindle from 6" to 8" in diameter is sometimes used, instead of the hollow inside cylinder. The steps and risers are fitted into the spindle, dove-tailed, glued and wedged. The outside string is either constructed in a manner similar to Fig. 4 of lighter material, or a wrought iron one is used, made of y by 1" iron, and the steps bolted down.
The spindle is fastened to the platform timbers B B, and runs high enough above the floor to receive the railing which incloses the opening of the landing.
Fig. 6 shows the ramp pattern of the rail over the landing of the stairs, Fig. 1. The face-mould of the two wreath-pieces reaching over the cylinder, is drawn as in Fig. 4, Plate XXVI. The second wreath, instead of having the straight wood from A to D added as is done in the lower one, has a butt-joint made at A in that figure, which will reach over the second quadrant c d f of the cylinder Fig. 6 of this plate. Let d F be the inclination of the upper tangent, then extend the same, intersecting the horizontal one H Gr, which is at the height of half of a riser above the point F, and F Q and H G- are the tangents of the ramp pattern. Make both tangents of equal length and the easement may be described with compasses, as indicated.
If the stairs are steep and the cylinder small, such a termination of the wreath-part will look abrupt; this ought to be avoided if possible, by throwing the landing riser back into the face-board ; a separate face-mould is then drawn for the second wreath, in which the upper tangent d F receives a flatter position, which makes the easement of the ramp longer.
This may also be done when the last riser is placed as in Fig. 1, by using balusters shorter than the usual length on the thirteenth step and the landing.
Fig. 7 exhibits the drawing of the face-moulds for the flight Fig. 2.
There are sixteen steps in the flight; the most convenient way to divide the butt-joints in this case, will be to make two equal wreath-pieces of six steps each, leaving three for the easement into the miter cap, and one step for that from the winders into the level portion. This division will place one joint at c over the fourth riser, one at d over the tenth and one at e over the sixteenth riser. The face-mould for the two equal wreaths must be drawn first, as the inclination of the other tangents is governed by the same. As both of those wreaths are parts of true helical twists, their tangents will be of equal length and inclination. All of the twelve steps being of equal width in the cylinder, the face-mould is drawn in the diagram for convenience, over those six steps from 7i the sixth riser, to 7c, the twelfth riser.
The tangents 7i i and 7c i are drawn perpendicular to the radii of the circle, and intersect in i.
Both tangents being of equal length and inclination, the chord-line 7b 7c is taken for a seat-line (see Fig. 15, Plate IX, and Figs. 4 and 5, Plate XXVI). It is evident the line t1\ perpendicular to and bisecting this chord, will pass through i. To find the inclination and real length of the tangent, extend i 7i and make 17ir equal to 7i i or their length in the ground-plan. We know each tangent rises the height of three risers ; therefore set up this height on the perpendicular from i to l\ and if 7i' will be the real length, and angle t7i' f the inclination of each tangent. Angle i I' 7i is the plumb-bevel, and 7i t' one-half of the length of the plumb-bevel-line. To avoid confusion, draw a line H K through some point as 9, parallel to the chord-line, and make 9 H and 9 K each equal to 7i if.    Extend the perpendicular 11\ and make 9 J equal ioti; then if the drawing is accurate, II
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J and K J will be equal each to f hr and constitute the tangents of the face-mould ; H K is the slide-line.
To draw the curve-line of that face-mould, divide either half of the chord, as t 7c, into any convenient number of equal parts, say four, and draw ordinates through the points of division perpendicular to said chord ; then divide 9 H and 9 K each into the same number of equal parts, draw also perpendiculars to H K through the points of division, and make them equal to the respective ones from the seat-line t Jc.   The curves are traced through the points thus obtained.
The face-mould for the miter-cap wreath is drawn as Fig. 8, Plate XXVI. In this case there are three steps contained in it; allow one rise for the easement, set up two risers from c to d on the 4th riser-line, which runs to the centre, and draw tangent c b perpendicular to the same. Transfer the inclination of the next following tangent as h' f from d down ; that is, make <c d b equal to < tf hf, then d b is the inclined tangent, and a b the intersection of the cutting-plane. Draw a seat-line bn c" wherever convenient perpendicular, and the ordinate c d" through the point c parallel to the intersection. Make df d" equal to c d, and d" b" is the plumb-bevel line ; develop the face-mould ABC with the slide-lines as usual, and apply to the plank as explained in PL XVIII.
The face-mould E F G- of the upper wreath is drawn in the following manner : Draw tangent ef perpendicular to the radius from e and transfer the inclination of the preceding tangent; that is, make <f ef equal to < t h' t\ The whole height of the wreath will Ibe If risers from centre to centre of joint, as it contains the last step, and rises as usual, half of a riser above the landing. This height must "be gained in the inclined tangent ef ; therefore draw a line at the distance of If risers parallel to ef, intersecting the tangent ef in/7, and let fall from/7 the line/7/perpendicular to ef ff is evidently the point where the inclined tangent meets the horizontal one, which may be drawn from/to g, the joint of the cylinder, in order to take in all of the circular level part in this last wreath. If the face-board or the cylinder were to continue on the circle, this horizontal tangent ought to be drawn from/ tangent to the centre-line of the rail, as is done in a similar case, Fig. 9 ; "but as the point of contact would come so near to the joint g, the tangent is drawn in this case to that point, to avoid the trouble of having a short circular part on the level straight rail. Asfg is the horizontal tangent, the intersection will pass through e parallel to it; therefore to develop the face-mould, extend g f draw e d parallel and the seat-line d f" perpendicular to it. Make/77/777 equal to//7 and d f"! is the plumb-bevel line, from which draw the developing ordi nates as usual.
In this case, where the butt-joint through g does not stand square to the horizontal tangent f g in the ground-plan, the butt-joint on the face-mould at & will also not be square to the tangent F G ; nor will said butt-joint be made square from the face of the plank on the wreath, but must be found by sliding the face-mould on both sides, similar to finding the miter-joints of the miter -cap 'wreath.
The joint through G is found on the face-mould by taking the lengths of the horizontal ordinates passing through the points of intersection of the joint through g and the outlines of the rail, and transferring those lengths to the respective developing ordinates from the plumb-bevel line. The joint at E is as usual square to the inclined tangent, and consequently is made also square from the face of the plank.
The shape of the face-mould is somewhat peculiar, as some of the ordinates intersect the same outline of the rail in the ground-plan twice. When blocking this wreath, after the vertical inside and outside cylindrical surfaces are shaped, saw off all the wood above the upper line of the block-pattern on the vertical joint square from the same, and then there will be no doubts left how to shape the top-side of the wreath and get a good easement. On account of the peculiar situation of the cutting-plane, which in this case intersects the vertical cylindrical surfaces of the rail partly in points higher than the highest point of the level rail, it is well to use a plank about �J" thicker than the width of the rail for this wreath.
Fig. 8 represents the stairs Fig. 3 with the development of the face-moulds for continued rail, and illustrates in particular the drawing of the same for the upper wreath of the first flighty and the lower one of the second.
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The central part of the rail is represented as "being made of three equal pieces, containing four and two-third steps each, making fourteen steps from the second to the sixteenth riser.
There being sixteen steps in all, this arrangement will leave one step, as also one-half of the platform between the landing riser of the first flight and the bottom riser of the second, to be contained in each respective landing and starting wreath. If the rise were alike in both flights (the ground plan being the same), then both wreaths would be alike with their ends reversed, and only one face-mould required.
For instruction, the rise is assumed to be different for the two flights in this diagram, and after going through with the explanation in such a case, the process of drawing the face-mould when the risers are alike, will also be evident.
The whole construction is similar to that on Plate XVII. Let/8 g\ Fig. 8 of this plate represent the inclination of the preceding tangent of the first flight, and a bf that of the succeeding one of the second flight. Extend both tangents/ g and a b in the ground plan, and draw one 7c 7i through 2, the centre of the platform, of course perpendicular to the radius of that point. Then g h and li i will be the tangents of the top wreath of the first flight, and a 7c and 7c i those of the starting wreath of the second, in their horizontal projection or ground plan. From 7i draw 7i 7i' perpendicular to g 7i, and from 7c and a draw 7c 7c; and a a! perpendicular to a 7c. Draw also h 7i", i i\ and 7c 7c" perpendicular to 7i 7c.
From g draw the inclined tangent g 7ir parallel tof gf intersecting the perpendicular in 7i\ and carry the height 7i 7i' round to the next perpendicular in 7i". Both wreaths must contain from g to a the height of three risers ; one of the first flight and two of the second.
This height of the three risers set up on the perpendicular from a to a\ and draw ar 7c' parallel to a b\ intersecting the next perpendicular in 7c\ and carry the height 7c 7cf round to 7c", as indicated by the arc. A line Tc" 7i" drawn through these two points is evidently the position of the connecting tangent with the centre of the butt-joint at ir; the top wreath of the first flight contains the height i i\ and the starting wreath of the second flight the height a a! minus i i'. To find the intersection of the cntting plane of the former, extend i' 7i" and i 7i intersecting each other in q; q g is then the intersection. To develop the face-mould Gr H J, draw the ordinate i i'" parallel and the seat i" g" perpendicular to the intersection ; make i" %'" equal to i ir and %'" g" is the plumb bevel line. Both butt-joints are square to the tangents, and square from the face of the plank.
The face-mould for the starting wreath of the second flight is not carried out in the diagram, as it is self-evident after the tangents are thus located.    (Compare the process on Plate X"ViI.)
Fig. 9 represents a similar case where the platform is so wide, that the connecting tangents attain a horizontal position, and both the top wreath of the first flight, and the starting wreath of the second, are connected by a level piece of raiL
To illustrate this case we have omitted the first and last step of the ground plan Fig. 3, retaining only fifteen risers. The central part of the rail of each story is assumed to consist of three equal pieces containing four steps each, making twelve steps, which leaves one step for each of the top and starting wreaths. The proceeding in locating the tangents is similar to that in Fig. 7 for the top wreath E F Gr.
Let/* gr be the inclination of the preceding tangent of one of the equal pieces in the first flight, draw g 7if parallel to it until it attains the perpendicular heiglit 7i 7ir from g 7i, which height is equal to one rise of the first flight and half a rise of the next; from the point 7i so found, draw h i tangent to the centre line of the rail, i being the point of contact in which place a butt-joint.
To determine the location of the tangent and butt-joint of the starting wreath of the second flight, let a b; represent the next succeeding tangent of one of the equal pieces in that second flight; the height of this starting wreath is evidently one and a half rise of the second flight. Set up this height on the perpendicular from a to a! (which perpendicular is in this case a riser-line, as they all run to the center) and draw a' m parallel to a 5', intersecting aminm.
From this point m thus found, draw m I tangent to the centre line of rail, I being the point of contact and the center of the butt-joint.
The intersection of the cutting plane of the top wreath is parallel to i 7i> and the development
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of the face-mould is then as that of the top wreath in Fig. 7, with this difference, that the joint at i being perpendicular to tangent i Ji in ground plan, will also be square to the tangent on the face-mould, and made square from the face of the plank.
Tangent Im of the starting wreath being horizontal, is also the intersection of the cutting plane of that wreath. The development of its face-mould, of which only the tangents L M and A M are developed in this diagram, is similar to that of the miter-cap wreath in Fig. VII., with this difference, that the butt-joint at L will be square to the tangent and square from the face of the plank, because it is square to tangent m I in ground plan. This construction of the face-mould requires a short level piece of rail, of which I 7c and i Tc are the tangents. If this piece is not too long, it may be done away with, by placing a joint at Tc and extending both wreaths similar to the top wreath. Fig. 7 ; in that case the joints connecting the horizontal tangents would not be square to them, and would have to be found by sliding the face-moulds on both sides of the plank.
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PLATE   XXVIII.
Fig. 1 is the ground plan of a quarter circular staircase, and Fig. 3 that of a quarter space elliptical one.    Scale �" to the foot.
Figs. 2 and 4 exhibit the method of drawing the face-moulds.    Scale f" to the foot.
Fig. 1 represents a form of staircases frequently employed in steamboats, where they are inclosed by panel-work, and receive a railing without balusters fastened to the panel-work by iron brackets.
The same form may also be employed in the corner of a large square hall, receiving rail, newel-post and balusters on the front-string, as usual.
The radius of the wall string is T 6", and that of the front string 4'. The steps Nos. 4 to 13 are alike on back and front string, the riser lines all struck from the centre O. The first and second riser, also the fiftteenth, sixteenth and seventeenth are curved. The steps are all of the same width around the cylinders on the stretch-out; the three last steps diminish in width on the wall string, in order to keep the last riser within certain limits and to effect a square landing,
At T or the fifteenth riser, a smaller curve is connected, described from R as centre, to form the cylinder and opening of the second floor; the face-board is run from this cylinder straight across, which gives sufficient headroom. A smaller cylinder is also connected at the bottom of the flight, to give it the appearance of a turn-out.
Fig. 2 shows how to draw the face-moulds for the flight Fig. 1.
Divide the butt-joints to make the wreaths on the larger circle of about equal length, placing them at the points c, 6, g and i ; the location of the joint at I depends upon the inclination of the tangents, which will be seen as we proceed. This division of the joints gives two wreaths precisely alike: from c to e, and from e to g, each rising four risers, with both tangents equally inclined, and all balusters of regular length. The drawing of the face-mould is not carried out, as it is the same as the similar one in Fig. 7, Plate XXVII.
The construction of the face-mould A B C of the turn-out, is also the same as that of the one in that figure. As it happens in this case that the inclined tangent b c rises two risers (setting up one riser for an easement into the miter-cap), which is the same as each tangent rises in the wreath connected with it: c b is equal to c d, and therefore the intersection a b may be drawn at once, without first transferring the angle of inclination of tangent c d, as was necessary in Fig. 7, of the previous plate.
Although the wreath from g to i contains also four risers, a separate face-mould must be drawn for it, as it contains part of the smaller circle. The proceeding is as follows : Draw tangent g Ti perpendicular to radius g O, and i h to i R, intersecting each other in 7i. From li erect h h! perpendicular to g 7i, and from i and h each, i i' and Ti Ti" perpendicular to i Ti.
f gf is the inclined tangent of the connecting wreath below, g g! being equal to 2 risers. Draw g 7i' parallel to f g\ intersecting the perpendicular from Ti in hr; then li 111 is evidently the height of that tangent where ,it meets the next one. Transfer this height to h" as indicated by the dotted arc, and make i if equal to four risers, which is the whole height of the wreath; draw i' Ti" extended, intersecting i 7i produced in q, then q g is the intersection ; draw the seat g" i" perpendicular, and the ordinate i i'" parallel to g q, make i" i'n equal to i i! and draw the plumb-bevel line inr g".    Develop the face-mould G H J as usual.
To construct the face-mould J K L of the last wreath, which connects with the horizontal circular part at Z, draw i Tcr parallel to h" i\ which is the inclination of the preceding tangent. The height of this wreath is 1-J- risers ; set up on the perpendicular i i' this height from i to p, and
Pi.:xxyiiE.
Y	Y	
C		
	L\_	
147
square across, intersecting the tangent i 7cf in 7c'; then 7cf is evidently the point in space, where the inclined tangent would have to meet a horizontal one, to connect with the level circular part. Let fall from 7cf a perpendicular to i 7c, intersecting in 7c j then this point 7c is the horizontal projection of 7c\ and a line 7c I, drawn from 7c tangent to the circle (I being the point of contact), is the horizontal tangent of the wreath.    Draw a seat-line 7cff %"" perpendicular, and the ordinate i iv parallel to I Jc.
The height of the wreath-piece is set up in this diagram from i"" to zv, and the plumb-bevel line drawn from i" down to 7cf', in order to throw the figure of the face-mould J K L away from the lines of the previous face-mould, and avoid confusion. The result is the same, whether we draw the plumb-bevel line inclined this way or the other, as long as the height and the seat-line are the same.
The joint at L is a butt-joint in this case, as it is perpendicular to the tangent at I in the ground-plan. It is parallel to the slide-line on the face-mould, and the bevel to be applied through its centre, is therefore the plumb-bevel, while the spring-bevel at J is obtained by sliding the face-mould in the usual manner.
The lines I m and n m are perpendicular to the radii R I and II n respectively, and are therefore the tangents of the face-mould for the horizontal circular rail-piece.
Fig. 3 is the ground-plan of a flight of quarter-space winding-stairs ; the cylinder consists of the quadrant of an ellipse combined with that of a circle.    The wall-string is also elliptical.
The planning and laying out of such stairs, is in principle the same as explained on Plate XXIII.
To describe the cylinder, take two-thirds of the opening for one-half of the minor axis O P of the ellipse, one-third for the radius R T of the circle, and one and a half for one-half of the major axis O S.    For instance : in this diagram the opening is 2'; then O P is V 4", R T, 8" and OS 3'
The balancing of the winding steps is somewhat different from that in a circular cylinder. The run of the square steps, as also on the walking line of the winders, is in this diagram 10". The first winder No. 10, is 9", and No. 11, 6J" wide in the cylinder, while the rest are all about 5�". In the wall-string, No. 10 is 11", No. 11, 13", while all the rest are 14J" wide on the stretch-out This manner of division keeps the riser lines nearly normal to the curve of the ellipses, which is essential to the convenience of such stair-eases*
In order to have the balusters appear a proportional distance apart, the one on the 11th step sets back about 2i" from the face of the riser ; the rest are placed as usual.
If it is not convenient to make an elliptic drum to bend the wall-string on, a circular one may be used as indicated by the dotted line, which will make some of the steps a trifle longer, but would be no serious fault in the general appearance of the stair-case.
To build a staircase of this form requires some more work than one with a circular cylinder, as all the staves are of different shape, but it gives a grand and easy appearance to all the curves on the front string and raiL    There is no material difference in the labor on the rails.
The construction of the face-mould is essentially the same as those of the upper cylinder on Plate XXIV., with the difference of the ramp, which in this case consists of a wreath-piece. The turn-out shown in this plan is treated as in Plate XVIII.
Fig. 4 shows the construction of the face-moulds for the elliptical part.
A joint is placed at a the centre of the last square step, in order to include part of the straight rail; another at c over the twelfth riser, and the third at e, which is the junction of the ellipse and the circle.
To construct the tangents in horizontal projection, draw a line b d tangent to the ellipse through the point c (see Probs. 37 and 88, Figs. 13 and 16. Plate III). This line will intersect the tangent of the straight rail at 6, and the tangent/ d (which is perpendicular to the major axis at c), in d ; then a b and b c will be the tangents of the first wreath, o d and d e those of the second, and ef and,/ g those of the third one.
Draw the ordinate lines b bf perpendicular to a &, then b b", c & and d d' to 5 d> also d d", eef and f f perpendicular to df.
The first wreath rises 2% risers ; apply the square step pitch-board at a, and draw the inclined
148
tangent a b' of the straight rail, intersecting the perpendicular at V; this height transfer to J", as indicated by the arc. Make c c' equal to the height of the wreath, that is 2-J- risers, and draw & 6," Miich is then the inclination of the next tangent, intersecting its horizontal projection in q, andq a is the intersection of the cutting plane ; a" cf,f is the seat and d" c"" equal to c c\ which gives the plumb-bevel line a"c"".
The development of the face-mould A B C is carried in this direction in the diagram, to avoid Confusion.
To locate the tangents of the second wreath, draw c dr parallel to the preceding tangent V d, intersecting the perpendicular in d\ and cany this height d d' around to d d". Now, in order to meet the horizontal tangent g f of the level rail at the proper height, we set up 6^ risers from fto f, which iS ihe entire height of both wreaths from the centre of the joint at e to the horizontal tangent mentioned; then f d' is evidently the inclination of the cross tangent, intersecting its horizontal projection at u ; and u <? the intersection of the cutting plane.
Draw the seat-line d' d perpendicular, and the ordinate e d" parallel to u c ; make e" e,n equal tof eb\ which is the height of the wreath from centre to centre of butt-joint, and e"f c" is the plumb-bevel line of the face-mould CDE
The face-mould of the third crook-has e' f for the inclined, and/*7 Gt for the horizontal tangent. :   ;    All of these face-moulds are applied as Figs. 7 and 8, Plate XIII., respectively.
By comparing the vertical ordinates of the balusters, they will be found to be mostly of the regular lengths.
A piece of plank of the shape as indicated by the fine curve-lines in the face-mould A B C, is sufficient to make that wreath as well, as if it were the whole width of it.
Remark.�Some stairbuilders would undertake to make the whole of this wreath-part in two pieces instead of three, by extending the tangent a b of the straight rail until it intersects the cross tangent which passes through e ; but the rail would be found to come so low on the winders, that such balusters as those on the fourteenth and fifteenth steps, would be over 5" shorter than the regular short ones.
Pl.iXEL.
PLATE     XXXX.
ELLIPTICAL     ST A I R CASES.
Figs. 1,2 and 6 represent various forms of staircases with elliptical cylinders. Scale J" to the foot. Figs. 3, 4 and 5 represent drawings of face-moulds for the same. Scale f " to the foot. The form of staircase Fig. 1 maybe used under similar circumstances as mentioned in Fig. 2, PL XXVII. The wall-string is of semi-circular form, divided into sixteen equal parts, one for each step, with the exception of Eos. 1 and 2, which are somewhat wider. The riser lines, from Nos. 4 to 14, run to the centre of the diameter of the wall-string, as also the straight parts of the curved risers Nos. 3, 15, 16 and 17. The cylinder has an elliptical form ; the major axis being 4" and the minor 2' &'. The vertex of the minor axis is contained in the centre of the semicircle of the hall, which is 11/ wide, thus leaving the stairs from the third to the fourteenth step of about an equal width of 3'.
Of course the wall-string may be made also elliptical instead of circular, in which case the ellipses must be parallel, and that of the wall-string merge into the straight wall, in the line of the major axes.
The form shown in the diagram is chosen to illustrate an advantageous combination of a circular wall-string with an elliptical cylinder. The width of the steps in the cylinder are nearly the same on the stretcliout, so that the cylinder may be bent over a drum. By this arrangement the curve at the landing will have a graceful appearance both in the cylinder and rail, and the bottom part will appear as a turnout.
This form of staircase is susceptible of variety in ornamentation, as shown in the frontispiece, which is a perspective view of this flight, as seen through the vestibule door, the hall being lighted by a gasburner over the newel-post.
The work is finished in gothic style ; the wall-string has a paneled wainscoting with a half rail as a cap. The cylinder is ornamented with sunken panels, and has a cap a trifle smaller than the hand-rail, on which the balusters stand instead of on the steps.
Below the panels runs a raised moulding ornamented with gothic work, sawed out and coved on the solid. The lower edge of the cylinder drops below the plastering, and is finished with a cluster of three beads. In such a case, it is best to stave the cylinder solid, dressed to the proper curve vertically both sides. The space between the cap and the gothic moulding, as also the sunken panels, may be veneered with different kinds of wood.
The balusters are sawed out in lancet form and coved.
The newel-post is octagon, paneled and surmounted by a suitable bronze figure holding a gasburner.
Different varieties of wood may be employed for the different parts, according to taste of the parties concerned.
Fig. 3 of this plate shows the development of the facemould for the elliptical cylinder of Fig. 1. The rail is drawn as consisting of three pieces. The central one, of which de an&f'e are the tangents in horizontal projection, comprises seven steps, from the centre of the fifth to the centre of the twelfth step. The location of these tangents must first be ascertained before either the first or third wreath can be drawn. The process is in all particulars similar to that in Fig. 7, PI. XXYII. Use the chord df, Fig. 3 as the seat line, and erect i if perpendicular in its centre, which will pass through e the intersection of the tangents. Each of these tangents rises the height of 3i risers, which height set up from i to i' and d i' is one half of the plumb-bevel line with < i if d as plumb-bevel of the wreath.
To avoid confusion of lines, draw a line D F through any convenient point as i" parallel to the seat df, and make i"T> and i"F each equal to i'd, as also i"E equal to ie; then D F will represent the full length of the plumb-bevel line, and D E and F E will be the developed tangents of the face-mould, with the butt-joints perpendicular to the same ;  D F is also the slide line.
To determine the curved outlines of this face-mould, divide one half of the seat line into any number of equal parts, as also the plumb-bevel line each way from the middle into the same uumber of equal parts, and transfer the length of thk respective,horizontal and developing ordinates, as was done in Fig. 4, PL XXVI., and Fig. 7, PI. XXVII., and also .indicated in this diagram.
150
The cap of the newel-post is represented as octagon, and must be moulded on the edges the same as half of the rail, in order to get a proper miter as shown in a be. The newel must be so placed that the outlines of the rail intersect two edges of the cap in points ~b and c in such a manner, that those points are equally distant from the next corners of the cap, and the point a, where the moulded edges intersect at the top of the cap, is the miter point of the wreath.
Sometimes the newel, which in such cases consists of an octagon column without any turned members, (as represented in the frontispiece) is made so high, that the rail intersects the newel below the cap ; then the newel is set in such a position, that the rail abuts against one of its fiat sides, normal to the curve. It is not absolutely necessary in such cases, that the edges of the cap are moulded to correspond to the rail, but the top part of the newel may be built to suit any proper fancy of the designer. Of course, the employment of such newels is not confined to any particular form of staircases, but may be used whenever deemed suitable.
The rail is represented in this diagram as intersecting the cap, as mentioned above, and the development of the face-mould is in all respects similar to that in Fig. 7, PI. XXVII.
Extend tangent ed and erect the perpendicular dd\ There are 44 steps contained in this wreath, and as one riser is omitted as usual for the easement, the whole height is 3� risers. Now as this happens to be the same height as that of each one of the tangents of the central wreath, make d q equal to d e, and a q will be the intersection of the cutting plane of the first wreath. If 3i risers are set up from d to W on the perpendicular, and as d q is equal to d e, the line 6/ q is evidently the length and inclination of both the inclined tangent of the first wreath and that of the succeeding one of the central wreath. The intersection a q being found, the development of the face-mould CABD is as usual.
The development of the face-mould for the upper wreath, of which P G and H G are the tangents, is also similar to that of the respective ones in Figs. 7 and 9, PL XXVII.
Extend tangent ef, Fig. 3, and erect the perpendicular//' extended ; make fff equal to 3^ risers, and ef will be the inclination of the preceding tangent. From / set up the height of five risers on the perpendicular to 5, which is the whole height of the wreath, including half riser above the landing. From this point 5, draw a line 5 g' parallel to / g, and also a line / gf from / parallel to ef'\ both lines intersecting in gr \ then f g' is evidently the inclined tangent of the top wreath, and gf the point where it will meet the horizontal one. From the point gr let fall a perpendicular to / g intersecting it in g, and fg is this same inclined tangent in ground plan. From g draw a line g li tangent to the elliptical centre line of the rail (according to Problems on PL III.); 7^, the point of contact, is the proper place for the butt-joint connecting with the horizontal rail piece.
The intersection //" of the cutting plane is parallel to li g, and the seat g" f" perpendicular to it. Make g" g'" equel to g g\ and/" g,u is the plumb-bevel line. The joint through H is a butt-joint square to the tangent, and also from the face of the plank, because it appears square to tangent Jig in ground plan.
The balusters, from JSTos. 3 to 15, will be all regular short ones, No. 2 somewhat longer, and 16 and 17 somewhat shorter. Nos. 18 and 19 will be somewhat shorter than a regular long one, beginning at 7i. This variation in length may be ascertained by comparing the vertical ordinates through their centres, as indicated in the diagram and explained in full in Fig. 6, PL XXVI.
It will be noticed, that in full circular stairs, as those in Figs. 7, 8 and 9, PL XXVII., and Fig. 2, PL XXVIII., the butt-joints are so distributed, that only one step is contained in the landing wreath, while those in Fig. 3 of this plate contain as many as four and a half steps.
To take more than one step in the last wreath on a circular cylinder would cause an abrupt easement into the level portion of the rail, for the reason, that the more steps are contained in it, the sooner will the horizontal tangent come in contact with the circle beyond the landing ; that is, the closer will the butt-j oint be to the last riser, and consequently will give less chance for an easement.
But, this is different in an elliptical cylinder, where the last risers run well toward the elongated part of the curve ; the horizontal tangent will come in contact with it far enough beyond the last riser to afford a good easement, even if quite a number of steps are contained in the top wreath, as in this figure.
Fig. 2 is the ground plan of a flight of platform stairs with elliptical cylinder, the major axis being very long in proportion to the minor one, in order to permit the use of straight wall-strings.
151
Fig. 4 exhibits the drawing of the face-moulds for the cylinder Fig. 2. To cover at once all the possible cases, the rise is assumed to be different in the two branches.
The tangents of the wreaths passing over the elongated central portions of the curve, are equal in length and inclination in each branch respectively, and their face-moulds are constructed as that in Fig. 3. There are evidently needed two short wreaths to pass over the platform, and to connect the long ones of the first and second branch. The location of the tangents and development of their face-moulds are in all particulars similar to Fig. 8, PL XXVII.
Extend the tangents a b and gf of the long wreaths, draw the cross tangent c e perpendicular to the major axis, and erect at the points c, d, e and/ the respective perpendiculars. Let b a' be the tangent of the long wreath of the first branch, rising three and a half risers, and f gf that of the second, rising two and a half risers respectively, b af extended will intersect the perpendicular c & in c\ which height carry round to c". The whole height of both short wreaths comprises one rise of the first branch, and two of the second, which height is set up from / to /', and ff ef drawn parallel to f g!, intersecting the perpendicular e e' in e'. The height e e' is carried round to er\ and e" c" is evidently the position of the cross tangent, with the butt-joint at d".
To draw the face-mould DBF for the second of these wreaths, which is carried out in the diagram, take the height d d" of the first one, and set down from /' to f". Draw /" q parallel to f e\ intersecting its horizontal projection extended in q, then d q is the intersection of its cutting plane. The face-mould for the first wreath (which is not carried out in this diagram) is evident enough after the tangents are thus located. The reader is referred to Fig. 8, PL XXVII., where a similar one is carried out.
H K J is the face-mould for the top wreath, where the rail is assumed to connect with the level part.    Fig. 2 of the previous, and Fig. 3 of this plate represent similar cases.
Let g 7if represent the inclination of the preceding tangent, and draw 7i ir from h parallel to it; also extend tangent g li in the ground plan. This top wreath contains one step, and in order to reach the proper height of the level rail, will have to rise one and a half risers. Draw a short line at this distance parallel to i 7i, intersecting li il in if; from this point %' let fall a perpendicular to h i, intersecting in i ; then i is the point from which to draw i 7c tangent to the central curve of the rail in ground plan, 7t being the point of contact, and the proper place for the butt-joint connecting with the level rail.
The intersection TiTi" is parallel, and the seat ill" perpendicular to Jci. Set up the height of one and a half risers from i to i'\ and i" h" is the plumb-bevel line. To avoid confusion of lines, another line 7i'n J is drawn parallel to 7in i" in the diagram, and the developing ordinates made equal from that, to their respective horizontal ones from iTi". Both butt-joints are square to the tangents.
Fig. 5 represents that case of an elliptical front-string or cylinder similar to Fig. 2, in which the starting and landing risers are placed in such a position, that their distance (as 17 If) from the centre of the rail, square through the vertex c of the major axis, amounts to half of the run of the steps. The rail over such a branch of stairs may be made in two equal pieces, placing ox\e joint in the centre at a ; or it may be made in one piece if deemed convenient.
The drawing of the face-mould ABC and its application to the plank is very sini ple} and similar to the various cases mentioned in straight cylinder stairs, a b' is the inclination and length of the tangent, which rises in this case three and a half risers.
Fig. 6 represents the ground plan of a double staircase of the form usually employed on steam-boats. The outlines consist of a combination of various circular and elliptical curves. For all the face-moulds needed for such rails, similar figures will be found in the latter part of this4 work.
The line of the back-string passing through c, as also that of the face-board b b\ are circular arcs described from the same centre d. The lines d b and d b1 extended, will intersect the extended line of the fourth riser, which is straight, in a and a\ The parts of the front-string contained between those lines, consist of circular arcs described from those points a and af as centres. The rest of the curves of the front-string are parts of suitable ellipses.
To distribute the butt-joints of the rail, one may be placed over the fourth and one over tli<r eleventh riser, leaving three steps to be contained in the turnout, and two in the top wreath, whic) will then form a good easement into the level rail.
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