Geometry, and therefore chain lines are used by some draughtsmen in preference for construction. Otherwise fine dotted lines may be used for construction, and thicker ones for concealed lines. Army Candidates are advised to read the Text-books for Sandhurst and Woolwich in Military Topography and Fortification. It is not necessary to read these books beyond the chapter in scales in Military Topography, and Solid Geometry and Indices in the book on Fortification, and to look at further plates for style and colouring. In drawing a straight line through two points, adjust the ruler so that the pencil point passes through both points when placed against the edge of the ruler, before drawing the line. When many lines pass through a point, begin to draw from this point. When a figure consists of curves and straight lines, ink in the curves before the straight lines. When inscribing a circle in a figure or circumscribing a figure by a circle, the centre required being found, make sure before describing it that the circle does pass through all the points required, by testing with the compass. If it does not do so, usually the defect can be remedied by very slightly altering the centre, or radius, or both before drawing the circle. French curves are useful to construct conic sections and cycloids. [For Elementary Science Course read specially (1) to (42), (47) to (109), (111) to (159), (163) to (176), and omit the article on Marquoise Scales and Sector: for Cambridge Local Examinations (1) to (114), (123) to (155).] SIMPLE CONSTRUCTIONS. A POINT has no parts and no magnitude but position. [Euclid, Defn. Bk. 1.] When a point is given its position is marked by a dot surrounded by a small circle. A line is length without breadth and may be either straight or curved. A straight line lies evenly between its extreme points. Though the geometrical definition of a line states that it has no breadth, yet in representing a line in drawing it must have a certain breadth or thickness to make it visible; besides which by drawing lines thin, thick, dotted or chain certain sets may be distinguished from one another. For example lines required for construction are drawn thin, or dotted, but lines marking the solution of the problem rather thicker. A superficies or surface has length and breadth but no thickness. A plane surface, or more shortly a plane, is such that any two points being taken in it the straight line between them lies wholly in that surface. The surface of a perfectly flat piece of paper is therefore a plane, and is called the plane of the paper. When we speak of Plane Geometry we mean the construction of figures which lie entirely in the plane of the paper, as distinguished from solid bodies which are represented by methods of projection. When figures are contained by straight lines they are called rectilineal figures. A triangle is said to be a plane rectilineal figure because it is contained by three straight lines all lying in the plane of the paper. A circle is a figure in a plane contained by one curved line such that all points in this line are equally distant from a point within it called the centre. A circle then is a plane figure, but not a plane rectilineal figure. An angle is the inclination of two straight lines to one another. When a straight line meets another straight line so as to make the adjacent angles on opposite sides of it equal to one another, each of these angles is called a right angle, and the first straight line is said to be a perpendicular to the second. In In constructing the figures of Euclid's propositions straight lines are drawn of indefinite length and circles are completely constructed, when a very short arc would be sufficient. Geometrical Drawing we do not produce lines or circular arcs further than is necessary. It is not necessary to give any proof or description of your work in Geometrical Drawing, nor is it necessary in the solution of problems to indicate points by letters unless this is done in the question. When letters are necessary for the description of a figure they should be neatly printed, and words as in Italic printing. The word "equals" is represented by =; feet by one dash, and inches by two dashes, thus 6′ 4′′ means six feet, four inches. The signs "plus" and "minus,” + and denote addition and subtraction. Take the two quadrilateral (or four-sided) rectilineal figures ABCD, EFGH. In the first of these let AB-AD and CB-CD, and in the second EH-FG and GH=EF. Let the diagonals AC, BD meet in K. [As an illustration of the way of drawing straight lines BD, GH, and EF have been drawn thick; AC, dotted; EH and FG, chain.] If the triangle ADC were laid upon ABC it would exactly coincide with it so that the angle DAC=BAC; also the triangle AKD, AKB are equal, DK=KB, and the angle AKD=AKB. This may be shewn to be the case by Euclid's method of superposition or laying one triangle upon another. We also find the triangle EFH=GHF, so that the angle EFH=GHF. When a straight line HF meets two other straight lines GH and FE in this manner, EFH and GHF are called alternate angles, and if these angles are equal, however far GH and EF are produced, they will not meet in either direction and these straight lines GH and FE are said to be parallel. From these figures we draw the following inferences. If we want to bisect a given angle DAB we must find a point such that CD=CB and AD=AB, then join AC. If we want to bisect a given straight line BD we must find a straight line passing through two points A and C such that AD AB and CD=CB. If an angle is a right angle as AKB, it must be such that if BK were produced to ↳ making KD=KB then AD=AB. If a triangle as ABD has two sides AD and AB equal the angles ADB and ABD are equal, and vice versa. If a four-sided rectilineal figure has its opposite sides equal they are also parallel. (1) To bisect a given angle DAB. Mark off two equal distances AD, AB. With centres D and B and any convenient equal distances make arcs to intersect in C. Join AC. AC bisects the angle DAB. (2) To bisect a given straight line BD. With centres B and D and any convenient equal distances make arcs to intersect in A and C. The straight line AC bisects BD in K. (3) To draw a perpendicular to DB from a given point K in it. Take two points B and D such that KD=KB. With centres D and B and at any convenient equal distances make arcs to intersect in A. Join AK. AK is perpendicular to BD. (4) From any given point A without a straight line BD to draw a perpendicular to it. With any convenient distance and centre A make arcs to meet BD in B and D. With B and D as centres and any equal distances make arcs to intersect in C. Join AC. AC is perpendicular to BD. (5) To construct an equilateral triangle on a given base BD. Draw BD of the length given in the question. With centres B and D and distances equal to BD make arcs to intersect in A. ABD is an equilateral triangle. An equilateral triangle is also equiangular, each of its angles being equal to one-third part of two right angles. (6) To construct an isosceles triangle on a given base BD and with the other sides of given lengths. [An isosceles triangle has two equal sides, and therefore two equal angles.] Draw BD of the length given in the question. With centres B and D make arcs at a distance equal to the given length of each of the other two sides. Let these arcs meet in A. Then ABD is the required triangle. An acute angle is less than a right angle. |