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DC, and E is joined to B, and BE is bisected at F, and AD at H by perpendiculars through H and F, their perpendiculars are prolonged so as to intersect in 0, and OA, OD, OB, OE are joined, it can be proved that

ODC

OAB.*

ODE In the same way, Figures 2 and 2a may be drawn as an illustration. If in the triangle A B C, DO bisects B C at right angles, and 0 A bisects B AC, O A being sketched so as to bisect the angle B A C, then it may be proved that

A B C = A OC,

ACB = AO B. It is quite true, as Professor Culmann says in the preface of his work: "The constructing engineer will give preference to geometrical solutions wherever an accuracy of results up to three significant figures 1-1000th), which can be perfectly well obtained, is sufcient, for his drawing instruments are always at hand.”

Drawing is his habitual expression of thought, “his language,” and thanks to his topographical and To persons unfamiliar with this fallacy,

it should be explained that the error is in the drawing of the figure. Thus, in this problem, when properly drawn, the point O is always as shown in Figure la, so that OE always falls outside of D. Then we have

OE = OB,

OD OA, since both are drawn from a common point on bisecting perpendiculars to the extremities of the base line. Also

DE = AB by construction, hence in the triangles OAB and ODE we have the three sides of the one equal to the three sides of the other, and hence,

ADE = OAB ODC.
Similarly, in Figure 2, which will always fall as in 2a, the four points,
A BOC, fall in the circumference of a circle, since the equal angles at
A are subtended by the equal lines, OB and OC, which must then be
chords in the circumference through the four points. It then follows that

ABC AOC,
ACB = AOB.

-ED.

geodetical training, he is more accustomed to judge as to the exactness of sections, and to define lines and points with great precision than the architect or mechanician, but such accuracy in drawing is by no means naturally or intuitively acquired, and students require training in a course of graphical methods before he would appreciate their value. Moreover, such practice in actually performing the operations and becoming familiar with the solution is absolutely necessary, if it is to be expected that a student will really use these problems afterwards in his practical work, as such modifications become extremely puzzling owing to the want of a thorough acquaintance with the methods.

3. It is not only necessary that a student should be familiar with accurate drawing, but also that he should be familiar with graphical constructions as a means of solving problems. The plan ordinarily adopted in the teaching of statics, in conjunction with graphical methods themselves, seems expecting too much for the capacity of an ordinary student, and the author has attributed the difficulty of getting a class of even intelligent students to correctly solve problems out of the beaten track, to the difficulty involved in combining these two things. In the use of ordinary geometry or analytical methods there are separate classes for algebra, analytical geometry, trigonometry, and yet the ideas involved in them are no more difficult than those included in graphical constructions and methods, and graphic methods have the same claim to be considered as a separate branch of study.

The author, therefore, wishes to bring forward the following proposition: That in all engineering schools there may with advantage be introduced a separate course in graphical methods, which shall deal with such problems as have a practical bearing in mechanical science, and which do not involve applications to any concrete subjects, such as statics and dynamics, but which may familiarize the student by means of examples accurately worked out by himself with methods which he will be able afterwards to apply.

APPENDIX I.

Courses of Instruction in Graphical Methods in European Engineering and of forces in a plane-(3) Moments of forces situated in the same plane—(4) Infinitely small forces the lines of action of which are situated at an infinite distance—(5) Study of the equilibrium of bodies, solicited by forces situated in the same plane-Case where the body is held at one or two fixed points, or against one or two fixed surfaces—(6) Equilibrium of polygon-1st closed, 2d open-jointed and having its extremities fixed. Equilibrium and stability of an arch—(7) Study of the polygon of forces, and of the funicular polygon in their relations with conics—(8) Mechanical theory of reciprocal figures, and determination of efforts of tension and compression in a jointed system of bars——(9) Composition of forces in space-System of parallel forces and center of gravity-(10) Action of external forces on a solid-Determination of shearing effort and bending moment produced by fixed and movable loads.—Dangerous section.— (11) Maximum moments exerted by a movable load upon a beam, abutting its extremities—(12) Moments of the second order—Their graphic representation for a system of parallel forces of which the points of application are in the same plane–Moment of inertia and radius of gyration in relation to any axis—Conic of inertia-Ellipse of inertia and central ellipse-Systems of parallel forces of which the intensities are proportional to the products of elements of a plane area by their distance from an axis-Noyan (Kern) of a plane area-Construction of the central ellipse and of the noyan of inertia of plane figures (13) Method of funicular pencil-Application of composition forces to the calculation of a system of bars (14) Graphic integration-Definitions -Methods of the funicular polygon and funicular pencil-Applications.

Schools and Colleges.

ROYAL TECHNICAL HIGH SCHOOL, BRAUNSCHWEIG.
Time--Two hours weekly for one half year.

Graphic calculation-Chief aspects of graphic statics—Force and rope polygon-Relative properties of structures-Combination and resolution of forces Center of gravity-Moment of forces--Moment of inertia-Stress diagrams-Application of graphic statics to various examples. (Technical mechanics and descriptive geometry must be studied previous to the last course.)

ROYAL TECHNICAL HIGH SCHOOL, HANOVER. Time-Two hours weekly, and two hours weekly for exercise class, for one half year.

Transformation of areas-Force and rope polygon-The applications for finding the center of gravity and moment of inertia and also to beams, framework, retaining walls, and arches. (Elasticity must be studied previously or concurrently.)

SCHOOL OF ARTS, MANUFACTURES, AND MINES, LIEGE. 1. GRAPHICAL CALCULATION-Rule of signs and notation-The four first operations-Powers and roots—Calculating instruments-Areas with straight line perimeters—Transformation of curved figures and solids.

2. GRAPHIC STATICS—(1) Geometrical properties of the funicular polygon, and of reciprocal figures—(2) Composition of concurrent forces

ST. PETERSBURG.

Time- One hour weekly during one year, with exercises arranged.

I. GRAPHICAL STATICS-Graphical methods for finding a resultant of several forces acting in a plane-Scharnieur's Polygon-Geometrical relations between different Scharnieur's polygons for the same system of forces—Applications-Parallel forces—Finding of resultant and the sum of moments--Finding of the reactions in case of a beam supported at both extremities-Polygon of moments-Its application for finding the bending moments and sum of shearing forces for any section of the beam-Cases when the forces are distributed along the beam uniformly or otherwiseFinding of the angles made by the tangents to the line of resistance (elastic line) and the construction of the line of resistance-Determination of the supporting forces and bending moments of continuous beams

Beams with their ends encastre-Graphical determination of statical moments, centers of gravity, and moments of inertia of plane figuresKerns sections-Graphical methods for finding the stress in different members of trusses and girders and other systems with Scharnieur's combinations-Ritter's method—Construction of reciprocal diagrams after Clerk Maxwell—Illustrations-Duality of constructions in graphical statics and the reciprocal relations between the obtained figures—Theory of reciprocal polyhedra and its application to graphical statics.

arcs

ROYAL TECHNICAL INSTITUTE, MILAN. I. GRAPHIC STATICS.—Time.-Two hours weekly.-(1) Prod ucts powers, roots, logarithms, division of angles, auxiliary curves, amplification of diagrams, closed circuits—Theorem of Mobin's for the parallelogram-Reduction to lines of surfaces and volumes—Transformation and division of plane figures-Lines of compensation between of unit value, equal or different-Integral curve-Graphic tables and scales. (2) Composition and resolution of force in space and in a plaue-Infinitely small and infinitely distant force-Linear reduction of moments—Polygon of forces and their general properties–Different properties of the funicular polygon connecting systems of parallel forces, and of kindred lines—Equilibrium of systems of chains—Tensions of springs.' (3) Centre of parallel forces—Centre of gravity of points and corresponding geometrical theorems—General rules and auxiliary methods to find the centre of gravity of lines, areas, and homogeneous solids-Forces distributed along a line of resistance Construction of differential equations of funicular curves corresponding to a given area of load. (4) Composition of forces in space-Conjugate resistance--Polar (forcal) system—Tetrahedron of forces—Reciprocal polyhedra-Reciprocal figures according to Maxwell—Cremona–Framework, simple and compound, determined statically-General rules -Auxiliary methods for graphical calculation. (5) Moments of parallel forces--Conjoint antipolar system and case of the the hypothesis of parallel forces applied to the point of resistance-Moment of inertia, centrifugal moment, polar moment, modulus of a resistance of any plane figure, antipolar system-Geometrical properties and mechanical properties of a kernel-Geometrical constructions of the kernel of any simple figure (parallelogram, triangle, trapezium) and allied determination of the respective antipolar systems, central ellipse and kernel of a circle of an ellipse, of the segment of a parabola—General rules of auxiliary methods

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