QUESTIONS ON THE FIRST CHAPTER. Page 1. Define drawing. What are its sub-divisions? State the general characteristics of all objects. Which is most important? Why is so much stress laid upon correct outline? What is the scale? Page 2. What are the views given in projections? What are working drawings? How are dimensions indicated? How are the correct dimensions determined? How is the scale determined and where should it be placed? Why are dividers objectionable? When are decimal and duodecimal scales used? Page 3. Why are the vertical and horizontal scales sometimes different? What are profiles? What are cross sections? Page 5. In what direction is the light supposed to enter? What is the distinction between shade and shadow? Page 6. Give the rule for drawing shade lines. What are section lines? When may colors be used? Page 7. What are the requirements in drawings for contract work? for maps? Why are drawings necessary? What is the U. S. law concerning drawings and estimates ? CHAPTER II. "For which of you intending to build a tower, sitteth not down first, and counteth the cost, whether he have sufficient to finish it. Lest haply, after he hath laid the foundation, and is not able to finish it, all that behold it begin to mock him, saying, This man began to build, and was not able to finish.”—St. Luke xiv: 28, 29, 30. ESTIMATES AND MEASUREMENTS. That the engineer may be able to present the case properly in his advertisements as well as to determine for his Need of a prelimiown guidance in letting, the amount of work to be nary estimate. done, it will be necessary for him to prepare from the drawings or from surveys approximate estimates of quantities as well as of costs. In making up such estimates it must not be forgotten that there are other items of expense than mere cost of Elements to be emmaterials and labor. To these must be added the bodied in estimates, expense of engineering, superintendence, the tools, machinery, instruments, buildings, &c., known as the "plant," interest of money invested, a fair allowance for contractor's profit, wear and tear of tools and machinery, insurance if necessary, and a percentage for contingencies, such as damage to property or to the work itself from the elements or from accidents. Some of these quantities are functions of others which are very variable, as prices and animal power, systems of working, &c., so that such tables of statistics would be an incumbrance to this work, especially as they can be so readily referred to in numerous engineering hand-books, but the more constant data used in the calculation of quantities will be found serviceable for reference in this connection, and are therefore introduced in as condensed a form as possible. SECTION I. Formula for Computing Lines, Surfaces and Solids. LINES. Ratio of diagonal to side of square 1:1/2=1.414=10 nearly. Side of inscribed square R:: 1/2: 1. Side of inscribed equilateral triangle: R :: √/3:1. Side of inscribed regular hexagon=R. Side of inscribed regular decagon=0.618 R. For any part of triangle a:b or c:: sin A: sin B or sin C Ja, b, c sides. A B C angles opposite. Circle. Ratio of circumference to diameter=3.1415926+=355=π. Circumference=2 π R,= D where R=radius, D=diameter. chord of the arc and c' (the chord of half the arc)=c2+ver sin 2. Diameter(=2R)=Circumference_ Circum π Ellipse. Diameter. Perimeter=188 V1⁄2(a2+b2) nearly; a and b Length of Ellipse. being axes. Parabola. Length of arc cut off by a double ordinate= Length of Parabola. 21 y2 + x2, where y=the ordinate and r—the abscissa referred to the vertex as an origin. Hyperbola. Length of arc: 2 y:: (19 a2+21 b2) x+15 a b2: (9 a2+21 b2) x +15 ab2 where abscissa, 2 y=double ordinate, Length of Hypera and b transverse and conjugate axes respectively. bola. 2 Area of Triangle= included angle. SURFACES. where b base and h=height. 3 Area of Triangle=√ s(s—a) (s—b) (s—c): a, b and c=sides, s their half sum. I Area of Parallelogram=b h, where b-base, h=height. 2 Area of Parallelogram=a b sin C, a and b sides Parallelogram. C included angle. 3 Area of Parallelogram=2Vs(s—a) (s—b) (s-c) where c is the diagonal joining extremities of a and b and s the half sum. 1 Area of Trapezium= h, b and b'=parallel Trapezium. bases. b+b' 2 b+b' 2 2 Area of Trapezium= I sin C where length of one of the oblique sides; and C the angle between it and one of the bases. Any Regular Polygon= Area of Circle TC (a=length of one side. n Area of Ellipse a b; 2 a transverse, 2 b-con- Ellipse jugate axis. Surface of Right Cylinder, excluding bases=2π Rh Right Cylinder. Surface of Sphere=4 π R2. Sphere. Surface of Zone of two bases=2 R h, h=height. Zone. π Surface of Zone of one base=π c2 where c-the chord of the arc generating the zone. Surface of Right Cone R 1, slant height or Right Cone. element. Frustrum of Cone=ñ l' (R+r), l'=slant height Frustrum of Cone. VOLUMES. Of any Prism=B h, B=area of base, h=height. Prism. Of a Rectangular Parallelopipedon=aXbXc= Rect. parallelopip product of edges. Of a Cube=a3. edon. Cube. Of a Frustrum of any Cone=" (B+b+v/Bb), B Frustrum of Cone. and b areas of bases. 3 mid. Of a Frustrum of a Regular Pyramid, (same as Frustrum of Pyraabove.) Of an Ungula, when the section passes through Ungula. D2 dy Dd the opposite extremities of the bases=. Dh .2618 where D and d-diameters of lower and upper bases, h=height. Of the Wedge or cuneus=(21+e), where l= Wedge. length of back or base and b its breadth; e=the length of the edge and the height. b=breadth of road bed; h=perp. depth of cut at higher end; h' same at lower end; l=distance between sections and r= ratio of height to base of slope. π Of the Segment of a Sphere (R2 + r2 ) h + Segment of Sphere. 3 2 hs, where R and radii of the bases and h=height. If but one base, r becomes zero. Of the Spherical Pyramid= R s, s=area of Spherical pyramid. spherical polygon forming the base and R=radius of sphere. Of the Spheroid= fixed axis. π a2 b a=the revolving, b the Spheroid. Of the Frustra of Spheroids. (A) When the ends are cut off by planes perpen πι dicular to the axis of rotation=(a2+c2); a=revolving axis c=diameter of either end; length of frustrum. *For a much simpler formula recently discovered the student is referred to a work on "Formulæ for R. R. Earth Work," by Davis. For sale by D. Van Nostrand, New York, 1877. |