PLANE GEOMETRICAL DRAWING ROBER'I" HARRIS fytmll Uttivmitg | BOUGHT WITH THE INCO FROM THE SAGE ENDOWMENT THE GIFT OF Mitnvu m. Sag* 1S9X ME fund' ./l^fo/k^. /i.S'^.^A-^ Cornell University Library The original of tliis bool< is in tine Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924031272705 PLANE GEOMETRICAL DRAWING GEORGE BELL & SONS, LONDON : YORK STREET, COVENT GARDEN, BOMBAY : S3 ESPLANADE ROAD, AND NEW YORK : 66, FIFTH AVENUE, CAMBRIDGE ; DEIGHTON, BELL & CO. NOTE-BOOK ON PLANE GEOMETRICAL DRAWING WITH A CHAPTER ON SCALES, AND AN INTRODUCTION TO GRAPHIC STATICS BY ROBERT HARRIS, ART MASTER AT ST I'AULS SCHOOL; LATE EXAMINER TO THE BOARD OF INTERMEDIATE EDUCATION, IRELAND; AUTHOR OF THE "ART SERIES OF DRAWING-BOOKS FOR USE IN PUBLIC SCHOOLS " A-AH-' EDITION, REVISED AND ENLARGED. LONDON GEORGE BELL & SONS 1895 PREFACE. This note-book on Practical Geometry has been written primarily to meet the requirements of students preparing for the Entrance Examinations of the Eoyal Military Academy at Woolwich, and the Eoyal Military College at Sandhurst. It is hoped that the book may also be of use to Candidates for the Engineering College, Cooper's Hill ; the Indian Public Works and Telegraph Department, the Eoyal Marine Light Infantry, the Art Master's Certificates of the Third Grade, Science Subject I. of the Science and Art Department, the Oxford and Cambridge Local Examinations, &c. Solutions of problems set at Woolwich and Sandhurst are given, together with a chapter on Scales, and an introduction to Graphic Statics ; with methods of applying instruments used in Military Surveying and Engineering. It has been my object to give as much practical information on the subject of Plane Geometrical Drawing as the limits of one volume would allow. A knowledge of Practical Geometry is the basis of all the mechanical and decorative arts; and many of the problems given will themselves suggest the application ' for practical purposes. Acknowledgment must be made of the valuable assistance received from Bradley's Elements of Geometry, Lacroix' Elements de Oeometrie, Sonnet's Geometrie Theorique et Pratique, and other authorities, which have been consulted in the progress of the work. CONTENTS. PAGE Intkoductoey Hints 1 Definitions ... 3 Practical Gbometky 7 Cycloidal Curves 113 Examples (Pkaotioal Gbometky) 119 Concerning Scales 144 Plain Scales . 145 Diagonal Scales . . 147 Comparative Scales . . 150 Scale of Chords 152 The Sector ... 153 Marquois Scales 155 Vernier Scales 156 Examples (Scales) . 159 Geometrical Pattern Drawing . ... 175 Graphic Arithmetic 185 Graphic Statics 203 Examples ... 228 GEOMETRICAL DRAWING. INTEODUCTOEY HINTS. Neatness and accuracy are essential for the ■working of the problems, and care should be taken to see that your instruments are in good order. Avoid drawing unnecessary Hnes, and all which are drawn should be made long enough not to require subsequent pro- longation, Draw all lines from left to right, and avoid pressing against the edge of the ruler or other instrument, as it produces an uneven line. Construction lines should be drawn as fine as you con- veniently can, either continuous or dotted. In using the compasses, press as lightly as possible, to avoid making holes in the paper. IMake all construction on as large a scale as is practicable, in ■order to ensure the utmost attainable accuracy. The instruments necessary for Practical Geometry are : — Compasses. — One large pair, having a movable leg to insert a pencil or pen for drawing circles of long radii, and one smaller pair called dividers. Bow-pencil and Bow-pen. — For drawing circles of short radii. Ruling pen. — For inking-in straight lines. Six-inch ruler. — This should be carefully selected — one A ^ GEOMETRICAL DRAWING. having decimal scales, scale of chords or protractor for measur- ing angles, and inches accurately marked, is the best. Set squares. — Two — one a 45° set square, called so because two of its angles are 45° and the third angle 90" ; the other a 60° set square, called so because one angle is 60°, another is 30°, and the tliird angle 90°. These are used for drawing lines perpendicular and parallel to each other. DEFINITIONS. 1. A point marks position only. 2. A line has length without breadth. 3. A right or straight Hne is the shortest distance between two points. 4. A curved line is always changing its direction, and no part of it is straight. 5. A vertical line is upright. 6. A horizontal line is perfectly level. 7. An oblique line is neither horizontal nor vertical. 8. Parallel straight lines are two or more lines drawn in the same plane, such that if produced indefinitely either way they would never meet. 9. Parallel arcs of circles are concentric — i.e., arcs of different circles having the same centre. 10. A perpendicular. When one straight line stands upon another straight line and makes the adjacent angles equal, these angles are called right angles ; and the line standing upon the other is said to be perpendicular to it. 11. A plane rectilineal angle is the inclination of two straight lines to each other, which meet in a point. 12. There are three kinds of angles — a right angle, contain- ing 90° (see definition 10) ; an obtuse angle, which is greater or blunter than a right angle; an acute angle, which is less or sharper than a right angle. "When an angle has opened out until the arms are in the same straight line, it may then be considered as an angle equal to two right angles, or 180°. Such an angle is sometimes called a straight angle. 13. Triangles are plane figures bounded by three straight lines, and have therefore three angles. Any one side may be called the base. The sum of the angles of any triangle equals two right angles, or 180°. 4 GEOMETRICAL DKAWING. 14. An equilateral triangle has three equal sides, and therefore three equal angles. Each angle contains 60°. 15. An isosceles triangle has two equal sides, and therefore two equal angles. An equilateral triangle is also isosceles. 16. A scalene triangle has three unequal sides and three unequal angles. 17. A right-angled triangle has one angle equal to a right angle. The side opposite the right angle is called the hypo- tenuse. A right-angled triangle may be isosceles. . 18. An acute-angled triangle has three acute angles. 19. The apex or vertex of a triangle is the corner opposite the base. 20. The altitude of a triangle is a line drawn from the apex at right angles to the base or base produced. 21. A rectilineal figure is a figure formed by three or more straight lines. 22. Quadrilaterals are figures formed by four straight lines and have therefore four angles. The sum of the angles of quadrilaterals is four right angles, or 360°. 23. Quadrilaterals are of six kinds — four of which are parallelograms and two trapezia. 24. Parallelograms have their opposite sides equal and parallel, and their opposite angles equal. They are a square, an oblong or rectangle, a rhombus, and a rhomboid. 25. A square has four equal sides and four equal angles. Each angle is a right angle, 90°. 26. An oblong or rectangle has its opposite sides equal and all its angles right angles. 27. A rhombus has four sides equal and its opposite angles equal. (It has two angles obtuse and two angles acute.) 28. A rhomboid has its opposite sides equal and its opposite angles equal. (It has two angles obtuse and two angles acute.) 29. There are two kinds of trapezia — a trapezium and a trapezoid. 30. A trapezium has no two of its sides parallel nor oppo- site sides equal, but may have two of its opposite angles equal. 31. A trapezoid has two only of its sides parallel, and may have adjacent angles equal. 32. A diagonal of a quadrilateral or any rectilineal figure is a straight line joining two of its opposite angles. 33. A diameter of a rectilineal figure having an even number DEFINITIONS. 5 of sides is a straight line drawn through the centre, parallel to or at right angles to two of its sides. 34. A polygon is a figure having more than four sides and more than four angles. It is also called multilateral (many- sided) for a similar reason. There are two kinds — regular, having equal sides and equal angles; and irregular, having unequal sides and unequal angles. 35. A polygon with 5 sides is called a pentagon. 6 ,, a hexagon. a heptagon. 9 10 11 12 an octagon, a nonagon. a decagon, an undecagon. a dodecagon. A polygon may have any number of sides more than four, but for general use the above are sufficient. 36. A circle is a figure contained by a line which is called the circumference, and is such that all straight lines drawn from a certain point within it, called the centre, to the circum- ference are equal. 37. A radius is a straight line drawn from the centre of a circle to the circumference. 38. A diameter is a straight line drawn across the circle and terminated by the circumference, but passing through the centre. 39. A chord is any straight line drawn across the circle ter- minated by the circumference, but not passing through the centre. 40. An arc is a portion of the circumference of a circle. 41. A segment is a portion of a circle bounded by a chord and that part of the circumference which it cuts off. 42. A sector is a portion of a circle bounded by two radii and that part of the circumference intercepted by them. 43. The circumference of a circle subtends at the centre an angle equal to four right angles (360°). A sector containing a quarter of a circle is called a quadrant. , 44. A semicircle is half a circle ; it is bouiaded by a diameter and half the circumference. Two straight lines drawn from any point on the circumference to the ends of the diameter are at right angles to each other. 45. In a segment less than a semicircle lines drawn from any point on the arc to the ends of the chord contain an angle 6 GEOMETRICAL DEAWING. greater than a right angle, and in a segment greater than a semicircle an angle smaller than a right angle. 46. A tangent is a straight line lying in the plane of the circle which meets the circumference in one point, and only one. This point is called the point of contact. 47. If a radius of a circle be continued outside the circle, the whole line is a normal to the circle — i.e., a perpendicular to the circumference. 48. Similar figures are not necessarily equal in area ; but are equi-angular and corresponding sides are proportional. 49. One rectilineal figure is said to be inscribed 'within another when each angular point of the former figure lies upon a side of the latter. 50. A right figure is inscribed within a circle when all the angles of the figitte lie on the circumference of the circle. 51. A right figure is described about a circle when each side of the figure is tangential to the circle. 52. An ellipse is a figure bounded by a continuous curve, and is such that the sum of any two straight lines drawn from two fixed points within it, called the foci, and meeting together on the curve is constant — i.e., always the same. 53. Tangents and normals can be drawn to an ellipse. 54. The area of a plane figure is expressed by the number of square inches or square feet, &c., its surface contains. 55. A surface has only two dimensions, length and breadth. 56. A plane surface is flat, and is called a rectilineal surface when bounded by straight lines, and curvilineal when bounded by curved lines. 57. The perimeter of a figure is the sum of its sides. Thus, the perimeter of a triangle is a line equal in length to the sum of the three sides. 58. Periphery is the measurenient round curved surfaces. 59. A spiral is a curve traced out by a point which con- timjally recedes from its initial position, and at the same time moves round it. The initial position of the point is called the pole of the spiral. 60. An involute. Suppose a perfectly flexible line to lie along a curve so as to coincide with it ; if it be then gradually unwound, each point in the line will describe an involute of the curve. PRACTICAL GEOMETRY. (1) To bisect a given straight line AB. Fig. 1. With centres A and B, and more than half the line a radius, describe arcs cutting each other in points 1 and 2. A straight line joining 1, 2 bisects AB. (10 Euc. I.) An arc is bisected in the same manner as Problem 1. (2) To bisect an angle ABC. Fig. 2. With B as centre, describe an arc cutting AB and BC in points 1 and 2. With 1 and 2 as centres, and any radius, 8 GEOMETRICAL DRAWING. describe arcs cutting each other in point 3. A straight line drawn through points B and 3 bisects the angle. (9 Euc. I.) (3) To erect a pei^jendioular at point C, to a given straight line AB. From C, with any radius, describe an arc 1, 2. From 1, with radius CI, set off on the arc 1, 2, points 3 and 4; with 3 and 4 as centres describe arcs meeting in point 5. Draw a line from C through point 5. This line is the perpendicular required. (11 Euc. I.) (4) To draw a perpendicular to AB from a point O lying away from the line. _i4. y V Fig. 4. From C describe an arc cutting AB, or AB produced in points 1 and 2. With 1 and 2 as centres, and any radius PRACTICAL GEOMETEY. » describe arcs meeting in point 3. The line 03 is perpendicular to AB. If point C is beyond the extremity of the line AB, take any two points as centres in AB, as 1 and 2, and describe arcs through point C, intersecting below the line AB in a point 3. A line drawn through points C and 3 will be perpendicular to AB. (12 Euc. I.) (5) To draw a line parallel to AB, and at a givea distance ftom it. Fig. 5. Take any two points 1 and 2 in AB as centres, and describe- arcs with a radius equal to the given distance. The line CD dr^wn tangential to the arcs is parallel to AB, and at the required distance from it. (6) To draw a line parallel to AB, and passing through, a given point C. 3 S Fig. 6. With C as centre, and any radius, describe an arc 1, 2. With 1 as centre, and the same radius, describe an arc C3.. Make the arc 1, 2 equal to the arc 03. The line drawn from C through point 2 is the parallel required. (31 Euc. I.) 10 GEOMETRICAL DRAWING. <7) To draw a line from a given point O, to meet a given straight line AB, and making a given angle X with AB. Fig. 7. By the last problem, draw a line through C parallel to AB. With the corner of the given angle X as centre, describe an arc 1, 2 ; and with C as centre, and the same radius, describe the arc 3, 4. Measure the length of the arc 1, 2 in the angle X, and set off from 3 to 4 on the arc 3, 4. The line from C •drawn through point 4 makes the required angle with AB. (29 Euc. I.) <8) Method of dividing a line AB into a prime number of equal parts. Fm. 8. Draw the line AC, making any convenient angle with AB. and set off the required number of equal parts, as 1, 2, 3, &c.. PRACTICAL GEOMETRY. 11 Join the last division (10) and B. Draw lines from 9, 8, &c., parallel to lOB. AB is divided into the required number of equal parts. (Set squares may be used.) (2 Euc. VI.) <9) To divide a given line AB into two segments which, shall have to each other a given numerical ratio of 7 : 5. Divide the line AB into 7 + 5 parts, by Problem 8. The 7th or 5th point of division will obviously divide the line in the given ratio. (10) To divide a space contained between two parallel lines AB and CD into any number of equal parts, by means of lines parallel to AB, — in this case 10 equal parts. Fig. 9. Draw AE, making any convenient angle with AB, and set oif on this line the required number of equal parts, which may be any size — viz., 1 to 10. With centre A, and radius A 10, describe an arc cutting CD in F. Join AF. From A describe arcs with radii A9, A8, &c., cutting AF in points G P. Through these points draw lines parallel to AB. (Set squares may be used.) <11) To bisect the angle contained by two converging straight lines AB and CD. Draw any line 1, 2 across the converging lines, and make 12 GEOMETKICAL DRAWING. the angle M equal to the angle N. Bisect the angle 3, 2, 1 by J — ~_ 4 3 line 2, 4. Bisect the line 2, 4 by line EF. EF bisects the 9,ngle between AB and CD. (12) From a point A taken anywhere between two Fig. 11. converging straight lines BC and DB, to draw^ PRACTICAL GEOMETRY. 13 a line making equal angles with the two given straight lines. From any point F in one of the converging Hnes, as BC, •draw a line FG, parallel to DE. Bisect the angle CFG by a line meeting DE in H. A line drawn from A parallel to FH is the line required. <13) To draw a line through a given point B that would pass through the point in which two hnes AB and CD would meet if produced. Fie. 12. Draw any straight line 1, 2 through point E, and line 2, 3 parallel to 1, 2. From point 2 draw a line 2, 4, making any angle with 1, 2 and equal in length to 2, 3. Join 4, 1. From E draw EF parallel to 1, 4. Set off 3 G, equal to 2 F. From E draw a line through G. The line 2, 3 is divided in the same proportion as the line 1, 2. <14) To construct angles of a given number of degrees, as 15°, 30°, 45°, 60°, 90°, 120°, and 135°. At A draw a line AC at right angles to AB, and trisect the right angle. Thus, from C mark off CE with radius AC, and from F set off FD with radius AC. The angle CAB = 90°, DAB = 60°, EAB = 30°. Bisect 30° we obtain 15°. Bisect 14 GEOMETKICAL DRAWING. 90°, 45° is obtained; 90° + 30° = 120°, and 90° + 45° = 135° Fig. 13. By further sub-dividing, angles containing otlier numbers of degrees may be constructed. (15) To construct a scalene triangle the length of the sides AB, 3", BO, 2^", CA, 2", being given. Draw a line AB, 3" long, for the base. With. B as centre, and radius BC, describe an arc ; with A as centre, and radius CA, describe another arc. The two arcs intersect in point C. Join the points AC CB. (22 Euc. I.) (16) To construct an equilateral triangle, the base AB being given. With A and B as centres, and radius AB, describe arcs intersecting each other in C. Join AC CB. (1 Euc. I.) (17) Given any triangle ABO to find the altitude. With the apex C as centre describe an arc cutting AB, the base, or the base produced in points 1 and 2. With 1 and 2 as centres describe arcs intersecting in point 3. Join C3. This line meets the base or base produced in point D. CD is the altitude. PEACTICAL GEOMETRY. 15 (18) To construct an equilateral triangle, the altitude CD being given. From C draw lines making 30° with CD. Througk D draw a line at right angles to CD. The lines from C and the line through D meet each other in points A and B. ABC is the equilateral triangle required. (19) To construct a triangle, the altitude CD and the two angles at the base, A and B being given. Fig. 15. Through C and D draw lines at right angles to CD. From 16 GEOMETRICAL DRAWING. ■C draw lines CA and CB, making angles X and Y equal to the given angles A and B. ABC is the required triangle. >(20) To construct an isosceles triangle, the base AB, 1" long, and the altitude CD, 2^" long, being given. Draw the base AB and bisect it by a line at right angles in point D. From D set off the altitude 2^" to point C. Join ■CA CB. ABC is the triangle required. (21) Given one side AC, and one of the angles N, at the base, of an isosceles triangle, to con- struct it. Draw the side AC. From A draw a line AD, making an angle with AC equal to IST. With C as centre, and radius CA, describe an arc intersecting the line AD in point B. Join CB. ABC is the triangle required. (22) To construct an isosceles triangle, the base AC, and the angle at the apex B, being given. With centre B, and any radius, describe an arc 1, 2. Join 1, 2 by a straight line, and make the angles at A and C equal PRACTICAL GEOMETRY. 17 to the angle B, 2, 1. The angle at the apex is equal to the given angle. (23) To find the centre point in any triangle equi- distant from the sides. Bisect two angles of the triangle by lines meeting in point 0. Point O is equidistant from the sides of the triangle, and is a centre of the triangle. (24) To find a point equidistant from the corners of , a triangle. Bisect two of the sides by lines meeting in point 0. Point O is equidistant from the corners of the triangle and is a centre of the triangle. (25) To draw a triangle having a given perimeter AB, and similar to a given triangle CDE. Fig. 17. On AB draw a triangle similar to the given triangle CDE by making the angles M and N equal to M and N. Find the centre of the triangle ABP. From {draw lines OF OG parallel to AP BP. FGO is the triangle required. B 18 GEOMETEICAL DRAWING. (26) Within a triangle to place a similar triangle, having a base of a given length, also a common centre. Fig. 18. Pind the centre D of the triangle ; join D and tlie corners of the triangle ; set off AE on AB equal to the . given base. From E draw a line parallel to AD, meeting DB in point F. Draw FG and FH parallel to AB and BC. Join GH. FGH is the triangle required. Note. — By this method triangles triangles, parallelograms within or polygons within or about polygons. may be placed about about parallelograms, (27) To draw a right-angled triangle, two sides AB, 2^", and BO, 1", being given. Bisect AB in O and describe a semicircle on AB. From B set off on the semicircle BC. Join GA. The triangle ABC is the right-angled triangle required. (28) The hypotenuse AB, 3", of a right-angled tri- angle is given, and is twice the length of the base, construct the triangle. Draw the hypotenuse AB and bisect in 0, describe a semi- PEACTICAL GEOMETRY. 19 circle on AB. From B set oif on the semicircle a length equal to BO in point C. Join CA CB. ABC is the triangle required. (20) To draw a right-angled triangle, the hypotenuse and the perpendicular let fall upon it from the opposite angle being given. On a Une AB equal to the hypotenuse describe a semi- circle. Parallel to the hypotenuse, at a distance from it equal to the perpendicular, draw a line meeting the semicircle in C ABC is the triangle required. (30) Within a triangle to inscribe another given triangle. one similar to Fia. 19. On BC construct a triangle BCD similar to the second given triangle. Join DA by a line meeting BC in E. Trom E draw lines EF and EG parallel to CD and BD. Join FG. EFG is the triangle required. 20 GEOMETEICAL DKAWING. (31) To construct a triangle having a given perimeter AB, and its sides in a given ratio, as 2 : 3 : 4. Draw the perimeter AB and divide it into 9 equal -parts, 2 + 3 + 4 = 9. From A to the 4th division will be one side, from the 4th to the 7th division a second side, from the 7th division to B the third side of the triangle. The problem resolves itself into this : Given the three sides of a triangle, to construct it. (22 Euc. I.) (32) To draw a triangle having a given angle, the base and the sum of the remaining sides. Fig. 20. Draw a triangle ABC, making the angle BAG equal to the given angle and the line AC equal to the sum of the remaining sides. Make the angle CBD equal to ACB. ABD is the triangle required. (33) To draw a triangle whose perimeter is 3" and its angles as 2 : 3 : 4. Determine the angles of the triangle thus : 180x2 .^o 180x3 en" 180x4 ^n- On AB as base which is equal in length to the perimeter, construct a triangle having the required angles. From C, the PKACTICAL GEOMETKY. 21 centre of tMs triangle, draw lines CD and CE parallel to the sides of the triangle. CDE is the triangle required. (29 Euc. I.) Fig. 21. (34) To construct a triangle whose perimeter is 5" Altitude 1^", and the vertical angle 45°. A Tig. 22. 22 GEOMETRICAL DEAWING. Draw lines AB, AC, making 45° -with each other. With A as centre, and l^" radius, describe an arc DE. Measure on the line AB a length equal to half the peri- meter as AF, draw FG at right angles to AB, and meeting the line AH, which bisects the vertical angle in point G. "With G as centre, and radius GF, describe the arc KL. A hne tangential to the arcs KL and DE contains the base of the required triangle ; this tangent meets the lines AB and AG in points M and K The triangle AMN is the one required. (35) To inscribe a square within a triangle. _4 Fig. 23. Draw AB, the altitude of the triangle ; make AC equal to AB and at right angles. Join CD by a line cutting AF in point E, a corner of the required square. Draw the sides of the square parallel to the altitude and the base of the triangle respectively. (36) To inscribe an oblong having a given side within a given triangle. Set off AD on AB equal to the side of the oblong. Draw a line from D parallel to AC, meeting BC in point E. PRACTICAL GEOMETRY. 23 Draw EF parallel to AB, and Hnes FH, EG perpendicular to AB. GEFH is the oblong required. Fig. 24. (37) One angle of a triangle is 60°, the base 1-5", and the sum of the two sides 2 "5". Draw the triangle. Fig. 25. Draw a line AB equal to the sum of the sides, also a line BG making half the required angle with AB. With ,V as centre, and radius equal ' to the given base, describe an arc cvitting BC in D and E. Make the angle FEB 24 GEOMETRICAL DEAWING. equal to the angle FEE. Draw DG parallel to EF. the triangles AGD and AEF fulfil the conditions. Both (38) Given a circle, to draw a chord so that the angle subtended by it at the circumference of the circle may be equal to a given angle. Fig. 26. Draw a tangent from any point A in the circumference. Make the angle DAC equal to the given angle. AC is the chord required. (39) To construct on a given line AB as a chord, an arc of a circle, which shall be capable of a given angle. Make ABD equal to the given angle. Draw a hne from B, at right angles to BD, to meet the line bisecting AB at right PEACTICAL GEOMETKY. 25 angles in point C, the centre of the arc reauired. (33 Euc. III.) Fig. 27. (40) To draw a triangle when the verticle angle, base,, and sum of the sides are given. Fig. 28. 26 GEOMETRICAL DKAWING. On a line AB equal to the base as a oliord, draw the segment of a circle to contain half the given vertical angle. With A as centre, and radius equal to the sum of the sides of the triangle, describe an arc cu.tting the segment in C. Join AC and EC. Make the angle DEC equal to DCB. ABD is the triangle required. <41) To construct a square on AB as one side. At one end B of the given line erect a perpendicular EC. Ma^e EC equal in length to AB. With A and C as centres, and radius equal to the given side, describe arcs meeting in point D. ABCD is the square required. (46 Euc. I.) (42) Given the diagonal AC of a square, to construct it. Bisect AC in 0. With'O as centre, and radius OA, describe a circle intersecting a line bisecting AC at right angles in points B and D. ABCD is the square required. <43) To construct an oblong or rectangle, two ad- jacent sides AB and AD being given. At A erect a perpendicular to AB equal in length to AD. With B as centre, and radius AD, describe an arc ; with D as centre, and radius AB, also describe an arc : the arcs intersect in point C. ABCD is the oblong required. <44) Given one side and a diagonal of an oblong, to construct it. Fig. 29. PRACTICAL GEOMETKY. 27 Bisect the diagonal AB in point 0. With as centre, and radius OA, describe a circle. Set off from A and B, AC and BD equal to the given side. Join AD and BC. ABCD is the oblong or rectangle required. i(45) To draw a rhombus, one of the sides AB and one of the acute angles being given. From a point A draw two lines meeting together and con- taining the given angle. From A make AB and AD equal in length to each other. With B and D as centres, and radius AB, describe arcs intersecting in point C. ABCD is the rhombus required. (46) To draw a rhombus, a diagonal BP and one of the sides GH being given. Take points E and S' as centres, and describe arcs with Tadius equal to GH, meeting in M and E". EMFIf is the rhombus required. (47) To draw a rhomboid when AB and BP are the lengths of two adjacent sides, and S is one of the angles. D Make the angle BAD equal to S. Mark off AD equal to EF. With B as centre, and EF as radius, describe an arc ; with D as centre, and radius AB, also describe an arc. The arcs intersect in point C. ABCD is the rhomboid required. 28 GEOMETEICAL DKAWING. (48) To draw a rhomboid, a diagonal AC and two adjacent sides AB and AD being given. (See last figure.) With A and C as centres, describe arcs with radius AB, above and below AC ; also, with A and C as centres, and radius AD, describe arcs intersecting the first two arcs in E and F. ABCF is the rhomboid required. (49) To construct a trapezium with pairs of equal adjacent sides, when the adjacent sides are 2" and 1" and the included angle 120°. , Draw two lines AB and BC making 120° with each other, and 2" and 1" in length respectively. Join AC. On the oiDposite side of AC make a triangle ADC equal to the triangle ABC. ABCD is the trapezium required. Note. — The diagonals of a trapezium with pairs of equal adjacent sides are always at right angles to each other. (50) To inscribe a square within a trapezium having pairs of equal adjacent sides. Fig. 31. Draw the diagonal AB ; make AC equal to AB and at right angles. Join CD by a line cutting AE in F, a corner of the required square. The sides of the square are drawn parallel to the diagonals. PKACTICAL GEOMETKY. 29 (51) To place a rectangle within a square ABCD, the length of one side being given. Draw the diagonals of the square AC and BD. From C on AC set off CM equal to the given length of side. Bisect CM hy a line meeting CD and CB in E and P. Draw FG and EH parallel to AC. EFGH is the required oblong. (52) To find the centre of a given circle. Draw any two chords. Bisect the chords by lines at right angles, intersecting in point the centre of the circle. (1 Euc. III.) (53) To draw a tangent to a given circle or arc, at a given point of contact B. Find the centre C of the circle or arc by the last problem. From C draw the radius CB and produce it, making BA equal to CB. "With A and C as centres describe arcs intersecting in E. A hne drawn through E and B is the tangent to the circle. (16 Euc. III.) _^o;e._The Hne AC is a normal to the circle. 30 GEOMETRICAL DRAWING. (54) To draw a tangent to an arc, without using the centre from which the arc is described. Fig. 33. AB is tlie arc and C the point of contact. With C as centre, and any radius, describe a circle, cutting the arc in 1 and 2. With centres 1 and 2, and any radius, describe arcs meeting each other in point 3. Draw a line from 3 through C, cutting the circle in points 4 and 5. With 4 and 5 as centres, describe arcs meeting in point 6. A line drawn from 6 through point of contact C is the tangent required. (55) Prom a point B outside a circle, to draw a line cutting the circle in two points, x and y, so that the chord xy shall be equal to a given straight line AD. Set off the length of AD anywhere in the circle, as EF. With the centre C of the circle as centre, and radius CB, describe an arc meeting EF produced in point. G. With B as centre, and radius FG, cut the circumference of the circle in PEACTICAL GEOMETRY. 31 X. Draw a line from B through x, meeting the ch-cumference in y. The chord required is xy. Fig. 34. (56) Draw a straight line AB, 4" long; trisect it at points C and D ; at C erect a perpendicular CB, I" long. Draw the arc of a circle to Fig. 35. pass through AB and B without finding or making use of the centre, and determine at least six other points in the arc. 32 GEOMETEICAL DKAWING. With A and B as centres, and radius AB, describe in- definite arcs. Draw lines from A and B through E, to cut the arcs in F and G. Set off from these points equal chords each way along the arcs, as 1, 2, 3, &c. Draw lines from A and B through these points of division. Bl cuts Al in point R, in the supposed circle, and B2 cuts A2 in another point Q, &c. The problem is completed by taking the points in their proper order, and drawing* the curve freehand through the points. (57) To draw a straight line tangential to a circle, from a point outside the circumference. Fig, 36. A is the circle, and B the given point. Join B and C, the centre of the circle. Bisect BC4n D. On CB describe a circle, cutting the circumference in E and C. From B draw lines through the points of contact E and F. BE or BF is the tangent required. (58) To draw a tangent to two equal circles; o, exterior ; h, interior. a. Join the centres of the circles A and B, erect perpen- diculars at A and B to the line AB, meeting the circumferences PEACTICAL GEOMETRY. 33 ■of the circles in H and I. Through these points of contact draw the tangent MN. I. Join the centres of the circles A and B. Bisect AB in Fig. 37. C, and describe semicircles on AC and CB ; these semi- circles cut the circumferences of the circles in D and E. Through these points of contact draw the tangent FG. (59) To draw an exterior tangent to two unequal circles. Fig. 38. Join the centres A and B. Bisect AB in C, and describe a # c 34 GEOMETKICAL DRAWING. semicircle on AB. Set off DE equal to the radius of the circle B. Make AF equal to AE. Draw a line from A through point F on the semicircle, cutting the circum- ference of the circle A in G. From B draw a line parallel to AG, cuttiag the circumference of the circle B in H. Through G and H, the points of contact, draw the tangent required. (60) To draw an interior tangent to two unequal circles. Join the centres A and B, and describe a semicircle on AB. CD equals the radius of the circle B. "With A as centre, and radius AD, describe an arc, meeting the semicircle in E. Draw AE, cuttiag the circumference of circle A ia F. From the centre of the circle B draw a line parallel to AE, meeting the circumference of circle B in G. Through F and G, the points of contact, draw the tangent required. (61) To draw a circle of a given radius E, which shall be tangential to two converging straight lines, AB and CD. PRACTICAL GEOMETRY. 35 Draw EG and HK parallel to AB and CD respectively, at ^ ^ M Fig. 40. distances equal to the radius E. These Unes intersect in L, the centre of the required circle. (62) Griven two circles tangential to each other exter- ''46\ 721 Fig. 41. 36 GEOMETRICAL DRAWING. nally, to circumscribe a triangle whose angles shaU be 46°, 62°, and 72°. Draw a line AB tangential to the two circles by Problem 59. From any points in A and B draw lines making 46° and 72° with AB. Lines drawn parallel to these, each tangential to a circle, complete the triangle ABC required. (63) To inscribe a circle within a given triangle ABC. Fig. 42. Bisect the angles BAG and CBA hy lines meeting in D, the centre of the triangle. Draw DE at right angles to AB. With D as centre, and radius DE, describe the circle required. (4 Euc. IV.) (64) To describe a circle to touch any three given straight lines. Bisect two of the angles formed by the lines or the lines produced, and proceed as in the last problem. (65) To draw a circle of a given radius to touch a given straight line AB, and a given circle C. Draw FG parallel to AB, at a distance from AB equal to PRACTICAL GEOMETRY. 37 the given radius. Draw the line CD, and set off DE equal to the radius of the required circle. An arc described from C Fig. 43. with radius CE, meets FG in point O, the centre of the required circle. (66) To draw a circle of a given radius D to touch a given straight line AB, and to enclose a given circle 0. A B Fig. 44. Draw any diameter across the circle C, and measure from 38 GEOMETEICAL DRAWING. one end of it a distance 1, 2 equal to the given radius. Draw a line 3, 4 parallel to AB, at a distance from ,AB equal to D. With O the centre of the circle C as centre, and radius 02, describe an arc meeting the line 3, 4 in point 5, the centre of the required circle. (67) To draw a circle of a given radius tangential to two giv6n circles O, 0. Fig. 45. Draw the radius CE in each circle and produce them. Make ED equal to the radius of the required circle; arcs described from C and C, with radius CD, intersect in 0, the centre of the required circle. (68) To draw a circle of a given radius, tangential to and enclosing two given circles. From any points, as A and B in the given circles, draw indefinite lines through the centres. Set oif from A and B the radius of the required circle, as AE and BF. From C PEACTICAL GEOMETKY. 39 as centre, and radius CE, and from D, with radius DF, describe arcs intersecting in G, the centre of the required circle. Fm. 46, (69) To draw a circle to touch a straight line AB alid a given circle C in a given point P. _A : M M- Fig. 47. Draw the radius CP. Through P draw a tangent to the 40 GEOMETKICAL DEAWING. circle, meeting AB in point M. MN is drawn bisecting the angle AMP, and meeting CP produced in H, the centre of the required circle. If a Hne be drawn from point M bisecting the angle BMP,, and meeting PC produced, the centre of a circle is found thai will enclose circle C, be tangential to it at point P, and tangen- tial to the hne AB. (70) To draw a circle to pass through a given point P, and tangential to two given converging straight lines. Fig. 48, Draw BD bisecting the angle CBA. With D as centre,, and radius DE, describe a circle tangential to the two con- verging lines. Through P draw a Ime PB, cutting the cir- cumference of this circle in point F. From P draw a line parallel to DF, meeting BD in point G, the centre of the required circle ; GH is the radius drawn perpendicular to AB. (71) To draw a succession of circles to touch two straight lines and each other in succession. The first circle C is drawn by Problem 61. PRACTICAL GEOMETRY. 41 Draw the radius FG at right angles to AB, and HK at right angles to the bisecting line LF, meeting DE in point K. Make KM equal to HK, and draw MN at right angles Fig. 49. to DE, meeting LF in N, the centre of the second circle. This construction repeated gives the centre of a third circle, and so on for others, either to the right or left of the first circle. (72) To inscribe two circles having given radii A and B, v/ithin a given circle O, tangential to each, other and the given circle. Fig. 50. 42 GEOMETRICAL DRAWING. Draw a diameter DE, and produce it beyond E. From E set off EF on EC within the circle equal to A ; and from F as ■centre, FE as radius, inscribe the first required circle. From D, with radius B, set off DG- on DC within the circle ; and from E, same radius, set off EH outside the circle. Then from the centre of the given circle C, with radius CG, describe an arc; and from F, with radius FH, describe another, cutting that from G in K, which is the centre, B being the radius of the second required circle. Note. — The sum of the diameters of the inscribed circles must not be greater than the diameter of the given circle, (73) To describe a circle to touch a given circle in a given point A, and to pass through a point B outside the circle. Join AB, and bisect AB by a line at right angles, meeting a line drawn from A through the centre of the given circle in point C. With C as centre, and radius CA or CB, describe the circle. <74) To draw a circle to touch a given circle O, and a given line AB in point P. A E f Fig. 51. PEACTICAL GEOMETRY. 43 Draw PR at right angles to AB. With E as centre, and radius EP, describe a circle cutting the given circle in points D and E. Draw a line through points D and E, meeting AB in F. From F draw a tangent to the circle C, having the point of contact S. Draw the line CS, meeting PR in T, the centre of the circle required. (75) To draw a circle to pass through two given points P and Q, to touch a given circle C. Fig. 52. Draw AB bisecting at right angles PQ. Take any point on AB as centre, and describe a circle passing through P and Q, cutting the circumference of the circle C in points D and E. Draw a line through D and E to meet P.Q produced in F. From F draw a tangent to the circle C. From the centte of the circle C draw a line through the point of contact to meet AB in G, the centre of the circle required. (76) To draw a circle passing through two given points O and D, and tangential to a given straight line AB. 44 GEOMETRICAL DKAWING. Join DC, and produce the line to meet AB in E. EF is a mean proportional between EC and ED. Draw FO at right Fig. 53. angles to AB, meeting the line GH, drawn bisecting CD at right angles in point 0, the centre of the circle required. Note,~-'Eoi mean proportional, see Problem 128. (77) To describe a circle tangential to a given circle A and a given straight line BO, and passing through a given point P. fie. 54. Through the centre of the circle A draw a Une perpendicular to BC, cutting the circumference of the circle A in points D PKACTICAL GEOMETRY. 45 and E, and the line EC in F. Draw a line from T> through point P, and produce it indefinitely. Find a fourth propor- tional DG, greater to the three lines DF, DE, and DP. A circle drawn passing through the points G and P, tangential to EC, by Problem 76, is the circle required. Note. — For fourth proportional, see Problem 135. <78) To describe a circle passing through a given point A, to touch and enclose two given circles. Fig. .1.5. Let A be the given point, B and C the centres of the given circles. Draw the common tangent PQ through the external centre of similitude 0. Describe a circle passing through PQA cutting OA in A' and the circle C in Q'. Draw a Hne through QQ' meeting AA' in point E. Draw the tangent ED to the circle C. From draw a line through D, meeting the second circle in the farther point E. Points D and E are poiats of contact of the required circle. Lines EB and DC meet in point M, the centre of the required circle. 46 GEOMETKICAL DKAWING. If the other tangent from E be taken, a, circle is obtained touch- ing the given circles externally and passing throjigh point A. If the tangent passing through the internal centre of simili- tude (instead of the tangent PQ), circles are obtained touching one of the given circles externally and the other internally. (79) To describe a circle enclosing any three given circles A, B, and C tangentially. Fig. 56. Join the centres of the two largest circles A and B, produce this line to meet their circumferences in D and E. From D and E set off DF, EG equal to the radius of the circle C. Join the centre of the circle C with points F and G. Bisect CF and FG by lines meeting in. 0, the centre of the required circle. The radius OH is found by drawing a line from 0, through the centre of the circle A, meeting the circumference in H. N.B. — The above problem is an approximation. It may be accurately worked by describing circles with FB and AG as radii, point C the centre of the smaller circle being taken as a given point and applying problem 78. It may be also accurately worked by finding the line of similitude, poles, and radical centre of the circles. PKACTICAL GEOMETKY. 47 (80) To inscribe a circle within a trapezium ABOD, having pairs of equal adjacent sides. Fig. 57. Bisect the angles EGA and DEC by lines meeting in E, the centre of the trapezium. Draw EF at right angles to EC. "With E as centre, and radius EF, describe the circle required. (81) Within a sector to inscribe a semicircle. A 48 GEOMETRICAL DEAWING. Bisect the angle BAG by a line AD ; at D draw a tangent by Problem 53. Bisect the right angles formed by the tangent and AD by Hnes meeting BA and AC in points E and F. Join EF by a line meeting AD in G. G- is the centre of the required semicircle, and GD is the radius. A semicircle may be placed mthin an isosceles triangle by the same method. (82) Within a sector to inscribe a circle. A Bisect the angle BAG by a line AD ; at D draw a tangent DE, by Problem 53. Produce AC, meeting the tangent in E. Bisect the angle DEA. F is the centre of the required circle, and FD is the radius. (83) Within a sector to inscribe a square. Draw AD at right angles to AB, and equal in length. Join DC by a line cutting the arc of the sector in point E. Draw EF parallel to AB. EF is a side of the required square. PEACTICAL GEOMETRY. 49 Fig. 60. <84) To inscribe a square within a given circle. Draw two diameters of the circle at right angles to each other, meeting the circumference of the circle in points AC and BD. Join the points. <85) To describe a square about a given circle. Find points ABCD as in the last problem. With these points as centres describe arcs with a radius equal to that of the circle, and intersecting each other in points EFGN. Join the points. (86) To inscribe two equal circles within a square. Draw a diagonal of the square, thus dividing the square into two equal isosceles triangles. Witliin each triangle place a circle. D 50 GEOMETRICAL DRAWING. (87) To inscribe the largest possible semicircle within. a square ABCD. Fig. 61. Draw the diagonals AC and DB. Describe a semicircle on DB. Draw EF at right angles to AD. Join P and C. Through point G drawGH parallel to FE. H is the centre and HG the radius of the semicircle required. (88) To inscribe four equal semicircles, having their Fig. 62. diameters adjacent, within a square, arc to touch one side of the square. Each PEACTICAL GEOMETRY. 51 Draw the diagonals which divide the square into four equal triangles. Inscribe a semicircle within each triangle. (89) To inscribe four equal semicircles, having their diameters adjacent, within a given square. Each arc to touch two sides of the square. Draw the diagonals AC, BD, arid the diameters EF, GH, of the given square. Bisect DG in K, and EA in L. Join KL by a line intersecting EF in M. With as centre, and radius OM, set off 01!^, OP, OQ. Join these points by lines (which are the adjacent diameters), meeting the diagonals of the square in points 1, 2, 3 and 4, the centres of the required semicircles. (90) To inscribe four equal circles within a given square, each circle touching two others and two sides of the square. Draw the diagonals and diameters of the square, meeting in E ; also the remaining diagonals of the four smaller squares, meeting the diagonals of the given square in points 1, 2, 3 and 52 GEOMETKICAL DRAWING. 4 Join 1, 2, meeting GE in point 5. centres, and 1, 5 is the radius required. 1, 2, 3 and 4 are the O i_ 5 Fig. 64. (91) To inscribe four equal circles within a given square, each circle touching two others and one side of the square. _D C if'lG. 86. Draw the diagonals and diameters of the given square, meeting in E. Bisect one of the angles of 45°, as DCE, the line bisecting meeting nearest diameter in point 1. From E set oflf El on each semi-diameter in points 2, 3 and 4, the centres of the required circles. 1, D, is the radius required. PRACTICAL GEOMETRY. 53 (92) To inscribe eight equal circles within a square. Draw the diagonals and diameters of the square, thus divid- ing the square into eight equal isosceles triangles. Within each triangle place a circle., (93) To inscribe three equal circles within an equi- lateral triangle, each circle touching two others and one side of the triangle. Fig. 66. Find the centre of the triangle, and draw lines from the corners of the triangle to the centre, thus obtaining three equal triangles ; in each triangle place a circle. (94) To inscribe three equal circles within an equi- lateral triangle, each circle touching two others and two sides of the triangle. (See Fig. 67, p. 54.) Find the centre of the triangle. Through the centre draw hnes from the corners of the triangle, meeting the opposite sides, thus obtaining three equal trapezia. "Within each trapezium inscribe a circle. 54 GEOMETRICAL DKAWING. i'lQ. 07. (05) To inscribe three circles in any triangle, the circles touching each other and one side of the triangle. Fig. 68. Find D, the centre of the triangle. The lines DA, DB and DC divide the given triangle into three elementary triangles. Wilhin each triangle inscribe a circle. PRACTICAL GEOMETRY. 55 (96) To inscribe three circles in any triangle, each circle touching each other and two sides of the triangle. Fig. 69. Find D, the centre of the triangle. Draw lines DE, DF and DG at right angles to the sides of the triangle, dividing the triangle into three trapezia. Within each trapezium inscribe a circle. (97) To inscribe six equal circles within an equilateral triangle. Fig. 70, 56 GEOMETRICAL DKAWING. - Proceed as in Problem 93 for the first three. From centres 1, 2 and 3, and with a radius equal to 3, 1, set off points 4, 5 and 6. These points are the centres of the remaining three circles required. (98) To inscribe six equal circles within an eqiiilateral triangle, each circle touching one side of the triangle. Fig. 71. Draw lines bisecting each angle, thus dividing the equi^ lateral triangle into six equal triangles. Within each triangle place a circle. (99) To inscribe a triangle within a circle, having angles of a given number of degrees, — in this case 45', 60°, and 75°. At any point A in the circumference of the given circle draw a tangent DE. The angle DAB = 45°, and the angle EAC = 60°. Join BC. The angle at B = 60°, at C = 45°, at A =75°. Note. — By the same method as shown in tliis problem. PEACTICAL GEOMETRY. 5f a triangle may be inscribed within a circle similar to a given triangle. (4 Euc. IV.) (100) To describe a triangle about a circle, having^- Fig. 73. angles of a given number of degrees, — in thiat case 60°, 45°, and 75°. 58 GEOMETRICAL DRAWING. Make the angle BAG = 180° - 60° = 120°. „ DAG = 180° -45° = 135°. The sides of the triangle are tangents to the circle at points B, C and D. (3 Euc. IV.) Note. — By the same method as shown in this prohlem a triangle may be described about a circle similar to a given triangle. <101) To inscribe three equal semicircles having their diameters adjacent within an equilateral triangle, each semicircle touching one side of the triangle. Pind the centre of the triangle, and draw lines from the corners of the triangle through the centre, meeting the sides in A, B and C, thus dividing the triangle into three equal "triangles. In each triangle place a semicircle by bisecting the Tight angles at A, B and C. (102) To inscribe three equal semicircles having their diameters adjacent within an equilateral triangle, each semicircle touching two sides of the triangle. PRACTICAL GEOMETRY. 59 Find points A, B and C as in Problem 93. Join AB, and describe a semicircle on AB, with point 1 as centre. From 1 draw a line at right angles to the side of the triangle, meeting the semicircle in point 2. Draw a line from 2 to the centre of the triangle, meeting the side in point 3. From 3 draw a line 3, 4 parallel to 1, 2. Through 4 draw a line 5, 6 parallel to AB. On 5, 6 construct an equilateral triangle. The sides are the adjacent diameters. Points 4, 7 and 8 are the centres of the required semicircles. (103) To construct a regular polygon on a given line AB. T X M ^^^5i-— _ — — ^^==^ N Fig. 75. Bisect AB by a line MN at right angles. With A as centre, and radius AB, describe an arc B6; point 6 is the centre of a hexagon, having AB as one side. Divide the arc B6 into six equal parts, in points 5, 4, 3, 2 and 1. With 6 as centre describe the arc 5X. Take X as centre, ■and with radius XA, describe a circle. AB is the side of a pentagon inscribed within the circle. To construct a heptagon, 60 GEOMETKICAL DRAWING. proceed as in the pentagon, but describe the arc 5Y above.. Y is the centre, and YA the radius of a circle, that AB will measure seven times round its circumference. It will be evident any regular polygon may be thus constructed. N.B. — The above problem is an approximation, but useful for practical purposes. (104) To divide the circumference of a given circle A^ into equal arcs by means of the compasses only. The radius AB, set off from B, divides the circumference! into six equal parts, and therefore into three and two. With the chord of J the circumference, describe arcs from B and E as centres, the arc^ intersecting in H and K. AH and AK are chords of ^ the circumference, and arcs described from H and K as centres, with radius AB, again bisect the quadrant ; from points G and as centres, with AH and AK as radii, describe arcs intersecting in L. The radius AE is medially divided in L; AL is the chord of -^ of the cir- cumference, and by taking alternate points, we have the circumference divided into five equal parts. PRACTICAL GEOMETRY. 61 <105) Given the perimeter of a regular polygon, to construct it. Divide the perimeter into as many equal parts as the polygon has sides. On one of the parts construct the polygon, by Problem 103. (106) To construct a regular hexagon on a given line AB. With A and B as centres, and radius AB, describe arcs meeting in point 0. With as centre, and radius OA, •describe a circle. Set off AB round the circumference, and join the points. <107) To find the angles the sides of a regular polygon make with each other. li y — number of sides, and x = number of degrees between the sides, ••• 22/ -4 X 90 = a;. y For example, find the angle between the sides of a regular pentagon. ^^ ~ ^ X 90 = AJL^ = 108° between the sides. 5 5 (108) To inscribe any regular polygon within a circle. (See Fig. 77, p. 62.) Divide the diameter AB into as many equal parts as the required polygon has sides, — in this case five. With A and B as centres, and radius equal to the diameter, describe arcs intersecting in M. From M draw a line through the second division on the diameter, to meet the circumference of the circle in point C. AC is a side of the polygon required, — ^in this case a pentagon. N.B.— The above problem is an approximation, but useful for practical purposes. 62 GEOMETRICAL DRAWING. (109) To inscribe a regular heptagon within a circle. Divide the diameter into seven equal parts, and proceed as with the pentagon in. Problem 108. (110) To inscribe a regular hexagon within a circle. Set off the radius of the circle round the circumference, and join the points. (111) To inscribe a regular octagon within a circle. Draw a diameter AE and bisect it by a line CG at right angles. Bisect the right angles formed by these lines. These lines meet the circumference in BDF and H. Join the points. (112) Within a given circle to inscribe any number of equal circles, each circle touching two others and the circumference of the given circle, — in this case five. PRACTICAL GEOMETRY. d'S Divide the circumference of the circle into the same number of equal parts as circles you wish to inscribe, by Problem 108. Join the centre and the points in the circuni- FiG. 78. ference of the circle. In other words, we have the circle divided into the same number of equal sectors as circles required. In each sector inscribe a circle. (See Problem 82.) (113) Semicircles having their diameters adjacent «4 GEOMETRICAL DKAWIKG. may be inscribed within a circle by plac- ing a semi-circle in each sector. (See Pro Hems 81 and 112.) <114) To inscribe any number, as three, equal circles within a given circle, equidistant from each other and the circumference of the given circle. Fig. 80. Divide the circle into three times the number of equal sectors as circles required, — in this case nine. Join points 1 and 3, 4 and 6, 7 and 9. In each of these triangles inscribe a circle. (115) To describe any regular polygon about a circle, — in this case a regular hexagon. Divide the circle into the same number of sectors as the polygon has sides. A hne bisecting one of the angles meets PEACTICAL GEOMETRY. 65 the circumference of the circle in point A. At point A draw a tangent to the circle, meeting two of the radii in B and C. With the centre of the circle as centre, and radius OB, describe a circle intersecting the radii continued in points DEFG. Join the points. (116) "Within a polygon to place another having a given side. Divide the polygon into equal isosceles triangles, and proceed as in Problem 26. Note. — By this method any number of concentric polygons (polygons having the same centre) may be drawn. (117) To inscribe a square within a regular polygon, — in this case within a pentagon. C Fig. 81. Draw AD at right angles to the diagonal AB, and equal m. length to it. Join DC by a line intersecting AG in point E. Draw EE parallel to GH. EF is a side of the required square. E 66 GEOMETRICAL DRAWING. (118) To inscribe a square in any regular polygon having an even number of sides. Draw a diagonal AB, and bisect it by a diameter CD. Bisect the right angles so formed by lines meeting the sides of the polygon in points EFGH. Join the points. (119) To inscribe as many equal circles within a polygon as the polygon has sides, each circle touching two others and one side of the polygon. Divide the polygon into equal isosceles triangles, and in each triangle inscribe a circle, by Problem 63. 120) To inscribe as many equal semicircles within a polygon as the polygon has sides, each semicircle touching one side of the polygon and having their diameters adjacent. Divide the polygon into equal isosceles triangles, and in each triangle place a semicircle, by Problem 81. (121) To inscribe as many equal circles within a polygon as the polygon has sides, each circle touching two others and two sides of the polygon. Divide the polygon into equal trapezia, and within each trapezium inscribe a circle, by Problem 80. By joining two centres the radius is obtained. (122) To inscribe the largest possible equilateral tri- angle in a regular polygon. ■ From one of the angular points A draw a line AB, bisecting the angle between the adjacent sides. From A draw lines PRACTICAL GEOMBTKY. 67 making 30° witli AB, meeting two sides of the polygon in points C and D. ADC is the triangle required. <123) To inscribe the largest possible equilateral tri- angle in a regular hexagon. By drawing lines joining the alternate angles the required equUateral triangle is inscribed within the hexagon. (124) "Within a square to inscribe the largest possible equilateral triangle. Draw a diagonal AC. From A draw lines making 30° with AC, and meeting two sides of the square in points D and E. ADE is the triangle required. (125) In a given square to inscribe the largest possible isosceles triangle, having a given base. Let ABCD be the given square. Draw the diagonal AC. From A set off AE, equal to the base of the required triangle. Bisect AE by a line at right angles, intersecting AB and AD in points F and G. The triangle FOG is the one required. 68 GEOMETEICAL DRAWING. (126) In a given square to inscribe a square, one corner of whioli shall coincide with a given point B in a side of the given square. Fig. 83.', Draw a line from E, the given point, throiigli the centre of the square, meeting AB in F. EF is a diagonal of the required square. With O as centre, and radius OE, describe a circle intersecting the other sides of the given square ia poiatsJC and D. (127) Within a square to inscribe a regtdar octagon. B S Fig. 84. PRACTICAL GEOMETRY. 69 The alternate sides of the octagon to coin- cide with the sides of the square. With the comers of the square as centres, and radius equal to half the diagonal, describe arcs meeting the sides of the square in points 1 8, the corners of the required octagon. (128) To find a mean proportional to two given straight lines AB and BO. Fig. 85. Draw a line AC, equal in length to the sum of AB and BC. Bisect AC in D, and describe a semicircle on AC. Draw BE perpendicular to AC. This ordinate BE is a mean between the segments AB and BC. .-. AB : BE : : BE : BC. (13 Euc. VI.). and describe all the circles of 2" diameter which will touch both. *(63) ABCDEF is a regular hexagon. G is the middle point of CD, and AG equals 2"25". Draw the hexagon. *(64) The sides of a triangle are 2", 2^", and 2|" respectively. Trisect the triangle by straight lines drawn from the middle point of the longest side. 124 GEOMETKICAL DRAWING. (65) Inscribe a trefoil of adjacent semicircles in a circle of 1|" radius. Also a cuspidate trefoil in the same circle. *(66) Describe an isosceles triangle having each of the equal sides 2" long and the vertical angle half either of the base angles. What is the magnitude of the vertical angle 1 *(67) A triangle on a base of 2" has an area of 3 square inches. What is the altitude ? (68) ABC is any triangle. On the side BC construct an equal isosceles triangle. (69) Describe an equilateral triangle about a given circle. *(70) Describe a circle of 2" radius. Suppose this to be the face of a watch, and show accurately the position of the two hands at 25 minutes past 8. (71) Two circles of radiL 2" and 1" have their centres 4" apart. Describe a circle of 1'5" radius which shall touch both. *(72) ABCD is a trapezium. The angle at A = 120° ; AB is 2", AD is 1", BC is 2", DC is f ". Bisect this trapezium by a straight line drawn from A. Make a square equal to half the trapezium. *(73) ABC is a right-angled triangle, having the right angle at A. Construct a square which shall be equal in area to the sum of the squares described on BC, CA and AB. Also draw a square which shall be equal in area to the difference of the squares described on AB and AC. (74) Describe by the method of intersecting ares an ellipse having its semi-minor axis 1"54" and its semi-major axis 2"86". (75) The hypotenuse of a right-angled triangle is 2'75", and the other two sides are to each other in the ratio of 3:5. Construct the triangle. *(76) Draw two straight Hnes 2" apart and parallel to each other. Take a point P between them, and distant "5" from one of them. Describe a circle that shall touch the two straight lines and pass through the point P. (77) Describe a circle of 1"5" radius, and in it inscribe three equal circles, each touching the other two and the original circle. EXAMPLES. 125 ^(TS) Draw a straight line AB, and take a point P 1" above it. Draw a curve every point of which shall be equally distant from AB and the point P. What is the name of the •curve ? (79) Draw a straight line AA, 6" long, and take a point P 2"25" above it ; from the point P draw all the Hnes which make angles of 38°, 56° and 74°, with the line AA. *(80) Construct four concentric, similarly situated hexagons 0"3" apart ; the side of the largest to be 2"25". (81) Between two concentric arcs of circles of 2"4:" and 3"7" radius describe four circles touching the given circles and each other successively. ^, (82) ABC is a triangle, and D a point in AC. Construct an isosceles triangle standing on AD as base, and equal in area to the given triangle ABC, (83) Inscribe four equal circles in a regular octagon of 2" side. *(84) Draw a third proportional to two straight lines whose lengths are IJ" and If. What is its length? *(85) Describe three circles of radii 1", 2", and 2J", touching each other. *(86) Draw a straight line 4:"37" long. Divide it into five equal parts, and through the points of division draw parallels '65" apart. (87) ABC is a triangle, and D is a point in AB such that AD is f- AB. Draw a straight line through D which shall bisect the triangle. (88) With a radius of 2"5" describe a quadrant of a circle, and in it inscribe a circle. What is the radius of the. circle 1 *(89) Construct a triangle having its sides 3'25", 2", and 2"5". Divide it into four equal areas by straight lines drawn parallel to the shortest side. (90) Make a square the area of which is -| of the area of another square whose diagonal is 2", What is the length of the side of the latter square ? 126 GEOMETRICAL DRAWING. *(91) Describe a regular octagon having each side 1" long. Describe another octagon, one-third of the area of the former- one. *(92) ABC is a triangle such that AB is 2" long, BC 2^", and CA 2f ". P is a point within the triangle, and AP is |". Draw straight Hnes from P which shall trisect the triangle. (93) ABC is an isosceles triangle, having AB equal to AC, and BC is 2" long. Describe a. rectangle onBC, the area of which is equal to that of the triangle. *(94) Construct a square having its sides equal to ^^3", and in it inscribe four circles, each touching two other circles and one side of the square. *(95) ABC is a triangle. Find a point P in BC, such that the perpendiculars from it on AB, AC are equal. How would you solve this problem if the directions BA and CA are given, but the point A is inaccessible 1 *(96) Describe an ellipse with a paper trammel, having its major axis a line ^5" long, and for its minor axis a Hue JS" long. (97) Describe a hexagon having each side 1" long. Eemove one of the six equilateral triangles composing it ; reduce the remaining figure to an equal square. *(98) From a point A draw two straight hnes containing an angle of 30°, and across them draw another straight line PQ, 2" long, such that AP : AQ : : 2 : 3. (99) Describe a circle of 1" radius. Within the circle inscribe a triangle whose sides are in the proportion of 6 : 19 : 21. (100) Divide a straight hne 6" long in the proportion of 2:3:4:5. Make a quadrilateral whose sides are equal to these parts respectively, and reduce the quadilateral to a triangle of equal area. *(101) Two angles of a triangle are 30° and 40° respectively. Construct a similar triangle having a perimeter of 4". *(102) Make a triangle equal in area to that constructed by the last question, but having its angles 30°, 60°, and 90°. EXAMPLES. 12T *(103) The base of a triangle is 1", its altitude 1|", and its vertical angle 30°. Construct it. *(104) The base of a triangle is 1", one of its sides is IJ", and its vertical angle is 30°. Construct it. (105) Inscribe three circles in an equilateral triangle, each circle touching the two other circles and one side of i^the triangle. (106) Inscribe three circles in any given triangle, each circle- touching the two others and two sides of the triangle. (107) Take a straight line 3"56" long, and divide it in extreme and mean ratio. What are the lengths of the two- segments ? (108) About a circle of 2J" diameter construct a triangl& having angles of 30°, 45°, and 105°. *(109) Construct a square on a vertical line 3" long as a diagonal. "Within the square insert a quatref oil of equal semi- circles with adjacent diameters, the arcs touching two sides of the square. *(110) Within a square of 2" side inscribe the largest possible equilateral triangle. *(111) From any point in the circumference of a circle of 1 J" radius draw a chord which will cut off a segment containing an angle of 30°. (112) On a given line 2" long construct an equilateral triangle ; also on the same Kne a scalene triangle of the same area, and having an angle of 20°. *(113) Within an equilateral triangle inscribe a trefoil of tangential arcs of circles. *(114) Draw a circle of 1" diameter. About this circle draw six equal circles touching one another and the given circle. (115) Inscribe an equilateral triangle within a pentagon of IJ" sides. *(116) Within a circle of 3" diameter inscribe two circles of 1" and IJ" diameter. The three circles to be tangential to each other. 128 GEOMETKICAL DKAWmG. *(117) Given a circle and a point outside. Draw a circle to pass through the given point and to enclose the given circle tangentially. (118) Draw a triangle having one of its angles 52°, and the two sides containing it 1'8" and 2"6". Knd the centres of the inscribed and circumscribing circles. *(119) Draw a rectangle 2'2" long and "9" wide. Divide it by lines parallel to the shorter sides into five rectangles, in such a way that the second rectangle shall be twice the area of the first, and each of the others as large again as the one it succeeds. (120) Within a circle of 3" diameter place five equal circles equal to and touching each other and the periphery of the larger circle. *(121) Construct a right-lined polygon of seven sides from the following conditions: AB = l-5", EC = 1-8", CD = 2-1", DE = 2-4", EF = 2-28". FG to be made equal GA. Angle GAB = 120°, ABC =130°, BCD =140°, CDE=130°, DEF = 120°. Eeduce this figure to a triangle of equal area having its vertex in E and its base in GA produced. *(122) Construct an irregular polygon from the given dimensions, and reduce it to a square of equal area. AB 2J", EC 2", AD 3-1", AE and DE each 2^". Angle BAG 27°, DAC 25°. (123) About a circle of 1" diameter place four Others equal to each other, and each touching two others and the given circle. *(124) Draw three equilateral triangles having their sides respectively in the ratios of 2, 3, and 5, the least being 1" in length. *(125) Construct a parallelogram the adjacent sides of which measure 2'5" and 1"65", and the included angle 115°. Construct a square equal in area to the parallelogram. *(126) Plot an angle of 57° and bisect this angle ; again bisect the angles so obtained. (127) A parallelogram has one side 2* 98" long, and its diagonals are 2"23" and 4"5" long. Construct it. EXAMPLES. 129 *(128) Upon a straight line 3*5" long, and upon the same side of it describe two segments of circles containing respect- ively 145° and 65°. Draw an angle in each segment. (129) Construct the segment and sector of a circle, each having the same chord, 3"4", and the same radius, 2'3". Measure and write down the magnitude of the angle in the segment. *(130) Determine by construction a third proportional to two given lines, 2^" and 2|" long. Find a fourth proportional to the three Hnes. *(131) A line AB is 2'7" long. At A a line AC, 1" long, makes an angle of 110° with AB. At B a line BD, 2" long, makes an angle of 130° with AB. Prom C, with a radius of 2y, and from D, with a radius of 2f", describe arcs meeting in E. Eeduce this figure to a triangle of equal area and having one of its angles 90°. (132) Draw a regular hexagon having a perimeter of 9", and within it draw a second having its sides equidistant from and parallel to the first, but forming a figure of only half the area of the outer hexagon. *(133) Construct an isosceles triangle of which the base measures 2"35" and the angle subtended by the base 41°. Measure and write down the length of the sides, and divide one of them into nine equal parts. *(134) Draw a circle of 5" diameter. In it place the largest possible octagon; in the octagon the largest possible square ; in the square the largest possible circle ; and in the circle the largest possible equilateral triangle. (135) Draw a series of circles, diameters 1", r25", r5", and 1"75", touching each other successively, and all touching a straight line. (136) Show how to find the centre, foci, and diameters of a given ellipse, and draw a tangent at any point in the curve. *(137) Construct a square containing an area of 4'84 square inches, and to a line 2"6" long apply a parallelogram of equal area with the square and having one of its angles 65°. I 130 GEOMETRICAL DK AWING. *(138) The base AB of a right-angled triangle is 3' 6" long, and the perpendicular drawn from the right angle at A to the hypotenuse is 2". Construct the triangle. *(139) Two points, A and B, are 2" apart. From each of them draw lines 2^" long, intersecting each other at right angles. *(14:0) Upon a straight line 3" long construct a square. Bisect the liae, and upon one of the halves construct a second square inside the first one. Eeduce the space by which the first square exceeds the second to a triangle of equal area. *(141) Draw two parallel lines l^" apart and 2" long. Between these insert nine other lines parallel to them and dividing the space between them into equal parts. Ink in the lines with thick, thin, and dotted lines alternately. *(142) Draw a triangle of which two sides measure 3*2" and 2"6", and the eftclosed angle 108°. Draw an equilateral triangle of equal area. *(143) Draw a polygon ABODE, of which the side AB is If" long, BC 2-3", AE 1-85", DE IJ". The angle ABC is 111°, BAE 113°, AED 122J°. Measure and write down the length of the remaining side and angles, and reduce the polygon to a triangle of equal area, and having its base in AB pro- duced. *(14:4) Two circles, A and B, each having a diameter of 1", have their centres 1"66" apart. Draw a straight line between these circles that shall be tangential to both. (145) Construct a square and equilateral triangle, each having an area of 3"8 square inches. *(146) Draw two circles with radii of r25" and 1'75" re- spectively, to touch each other externally, and an arc with a radius of 3 '5" touching externally the two circles. *(147) Construct an equilateral triangle, having an area of 3f square inches, and place in it a square having one of its sides resting on one side of the triangle. (148) Find a point C, 1" from B, and 2-5" from A, the ex- tremities of a straight line 2'45" long. Point C is the centre of a circle of "75" radius. Draw a second circle of 1" radius tangential to the circle and straight line. EXAMPLES. 131 (149) Draw any two straight lines at any angle with each ■other, but not meeting. Draw a circle that shall touch them both. *(150) Draw an ellipse haviag axes of 5" and 3" respectively. Assuming its centre to be imknown to you, show by what geometrical method you would ascertain it, *(151) Draw three circles, each touching two others, their radii being 1"39", •45", and "SS". By means of tangents place them in a scalene triangle, and give the length of its three *(152) Draw an equilateral triangle equal in area to the sum of two equilateral triangles, one having a perimeter of 3-31", and the other of 10-17". (153) On a base 2" long describe an isosceles triangle of 4" side, and divide it into four triangles equal to each other and similar to the original triangle. *(154) Construct a square of 2'87" side, and in it inscribe an isosceles triangle, having its base equal to r56", and its apex in one angle of the square. In this triangle, and in each of the three triangles that are external to it, place a circle. (155) About a pentagon of 1" side construct a pentagon of 1'9" side, its sides being parallel to and equidistant from those of the first. *(156) Pind a point P in a line AB (3" long) produced, so that AP : AB : : 7 : 4. *(157) Draw a square of 2'5" side, and divide the square into three equal parts by hues drawn from one corner. *(158) Construct a regular pentagon whose diagonal shall be 3". The diagonal is from A to C in the figure ABCDE. *(159) Divide a square of 3" side into three equal areas by lines drawn parallel to a diagonal. (160) Draw an isosceles triangle of 3'5" area, the vertical angle being 35°. *(161) The semi-conjugate diameters of an elUpse are 2'5" and 1'3" in length, and contain an angle of 60°. Draw the curve and determine the axes. 132 GEOMETRICAL DRAWING. *(162) Draw a triangle two of whose sides are 2"5" and 3'25", the angle opposite the shorter one being 40°. Draw also- the circumscribing circle. (163) In a square of 3" side inscribe an octagon, so that the alternate sides of the octagon shall coincide with the sides of the square. (164) The side of a pentagon is 1"5". Draw an equilateral triangle of the same area as the pentagon. *(165) A parallelogram whose sides are 4" and 1'2", and included angle 50°, is given. Determine a rhombus of equal area, and haying the same included angle. *(166) Three points, P, Q, K, are thus situated : PQ is 2",. QR 2"5", PR 3'75". Determine points in the circle which would pass through them, the centre being supposed inaccessible. *(167) Draw a triangle having its base 2", the angle opposite- the base 42°, and its altitude 1'75". (168) Two circles are given, radius 1" and "5" respectively,, their centres being 2"5" apart. Draw a circle of 1'5" radius to- touch both, but to contain the smaller one. *(169) To a circle of 1"25" radius draw two tangents which shall contain an angle of 60°. *(170) The perimeter of a rectangle is 9". Construct it so that its area shall be 4"25 square inches. *(171) Construct a square whose area shall be J of that described on a line 2 '75" long. *(172) Divide a line AB, 3'5" long, in a point P, so that tile- rectangle contained on AB and AP may be equal to the square- upon BP. *(173) Draw a triangle whose sides shall be as 2 : 3"5 : 4, and whose circumscribing circle has a radius of 1"5". *(174) Draw a pentagon whose area shall be 5*5 square inches. *(175) Divide a line 3'5" long into two segments, so that the area of the rectangle contained by those segments may be 1;^ square inches. ^ EXAMPLES. 133 *(176) A point P is 1"75" distant from the centre of a circle ■of 2"5" diameter. Draw through P a straight line cutting the ■circle in Q and E, so that QK shall be twice PQ. *(177) Draw the curve described by a poiat on the surface of a disc, and |-" from its edge, while the disc rolls along a ■straight line. Diameter of disc 2 '5". *(178) A and C are two points, 3"25" apart. B is a third point, 1*25" from A, and 2"75" from C. Given that A and C are the foci of an ellipse, and that B is a point on the curve, draw the curve. (179) Describe a circle enclosing and touching three other circles of 0'6", 0"8" and 1" radius respectively, each of which touches the other two. *(180) Construct a triangle having a base of 3"5", a perimeter of 10", and an area of 4'5 square inches. *(181) From a given isosceles triangle, cut off a trapezoid which shall have the same base as the triangle, and its remain- ing three sides equal to each other. *(182) Construct a triangle whose perimeter is 7" and whose sides are as 2 : 4 : 5. (183) Given the base, altitude, and vertical angle, to con- struct the triangle. *(184) A Hne AB is 3", and AC is 2". Find a point D in it such that AB x CD = AD x BC. Also find a point D in AB produced, such that AB + CD = AC -i- BD. *(185) Construct an irregular five-sided figure ABODE imder the following conditions : — Sides. Diagonals. Angle. AB = 2-2" AC = 2-6" BCD = 75' BC=r6" BE = 2-4" CD = 2-2" DE=l-5" *(186) Describe a series of circles in succession, tangential to two given converging lines, — the diameter of the smallest •circle being 1". 134 GEOMETRICAL DEAWING. *(187) Determine by construction ^3 of a line 1" in length, ■with the compasses only. *(188) Divide the area of a circle of 3" diameter into four parts equal in area and periphery by means of semicircles. (189) The sides of a triangle are as 2-5:3: i, and the radius of the inscribed circle is "75". Construct it. *(190) A point P is 1" from the circumference of a circle of 1" radius. Draw a line PE cutting the circle in QE, so that QK may be 1-25" long. *(191) A line 3" long moves with its extremities in two lines at right angles. Determine the curve traced out by a point in the line at 1" from its end. *(192) A circle of 1" radius rolls along a straight line ; draw half the curve traced by a point in the circumference of the circle. (193) Describe a circle of 1" radius. Inscribe five circles iit it, and describe ten circles about it. *(194) Plot an angle of 73° with the protractor. Divide it into four equal parts, and in each part inscribe a circle of one inch diameter. *(195) ABC is a triangle having the angle at A 60°. AB is 2", and AC 3" long. P is a point within the triangle, and AP is 1". Trisect the triangle by straight lines drawn from P. (196) About a given triangle ABC describe another, whose sides shall be parallel and equidistant from those of the former,, one side DE being given. *(197) On a Hne AB, 1"7" long, as base, construct a regular octagon. On Une AB as base, draw within the octagon an equilateral triangle. Draw lines from all the angles of the- octagon to the apex, and in each of the eight resulting triangles place a circle. *(198) On a chord 2' 6" long describe the segment of a circle to contain an angle of 128°. Measure and write down the length of the radius of the circle, and the size of the angle of the sector having the same chord as the segment. EXAMPLES. 135 *(199) Draw a circle of 2" diameter; show a geometrical means of finding the centre ; and from B, a point 3" from the centre, draw a tangent. Letter the point of contact C, and divide BC into three equal parts in D and E. On DE, one of these parts, construct a regular pentagon. *(200) Draw a regular heptagon of li" side, and within it place seven equal circles, each touching two others and two sides of the polygon. *(201) Two points are 2" and 2"5" from the centre of a circle of 1" radius, and 3" from each other. Draw the circle which, passing through these two points, shall touch the given circle. (202) Draw two tangents to a circle of 1" radius to meet at an angle of 60°. *(203) Construct an irregular seven - sided polygon ABCDEFG under the following conditions : — Sides. Diagonals. Angles. AB=l-25" BP (in BE) = 2-3" BPD = 113° ) t,-pt,_ .-o AE = 3-72" PD =2-15"BPG = 67° /-^^^^-o^ BE = 3-5" PG = 2" BC = 2" PF =1-58" DC = 2" *(204) A hue AB is 4", and AC is 24". Find a point D in it, (a) Such that AB x CD = AD x BC. (&) Find a point D in AB produced, such that AB'CD = AC-BD. *(205) Construct a triangle under the following conditions : — One side AB = 2-25". The perimeter 8", and AB : AC : : BC : AC. *(206) A straight line is 2" from the centre of a circle of 1" radius. Draw the curve, every point of which is equidistant from the line and from the circumference. The arc to be continued till it meets the diameter parallel to the line. *(207) A point F is 2" from a line. Draw half the curve, the distance of every point of which from the given point is to its distance from the line as 2 : 3. 136 GEOMETRICAL DEAWING. *(208) Show the construction for determining the arc of a circle passing through three given points, when they are so nearly in a straight hne that the one usually adopted would fail. For instance, take the three points A, B, C, and make AB = 2|", BC = If", AC = 4". *(209) Draw the hypooycloid produced when the directing circle is of 1"75" radius, and the generating circle is of '58" radius. *(210) Draw an ellipse whose axes are 3-5" and 2"5", and from any point outside it (not in the prolongation of either axis) draw tangents to it. *(211) A line 1^" long represents a square of 2" side. Determine the length of a line which would represent on the same scale a hexagon of If" side. (212) Construct a triangle having given the base 2^", one of the angles at the base 55°, the sum of the remaining two sides 5^". *(?13) If a line J" long represents the area of an equilateral triangle of 2" side, obtain a length by construction which will represent the area of an octagon of l\" side. (214) Draw any irregular four-sided iigure, no side less than l^". Construct a similar figure whose sides are 1^ times those of the first figure. *(215) The sum of the diagonal and one side of a square is 6". Construct the square. *(216) Construct a hexagon equal to an octagon of 1" side. (217) Describe a circle enclosing and touching three other circles of 0-63", 0'82", and I'l" radius respectively, each of which touches the other two. (218) Show the method of finding the major and mitior axes of an elhpse, and draw a tangent to it at a point P in the curve. *(219) Construct a triangle having a base of 4'25", a peri- meter of 12", and an area of 5*5 square inches. *(220) The Hne joining one corner of a square with the centre of the opposite side is 3" long. Draw the square. EXAMPLKS. 137 *(221) Two lines A and B converge to a point without the paper. Take any point P and draw a third line through P •converging to the same point. *(222) Draw a four-sided figure ABCD : AB=l-2" CD = 2-2" BD = 2-4" BC = r5" AD = 2-5" Draw a rhombus equal in area to the quadrilateral ABCD. (223) Divide a square of 3" side into four equal areas by lines drawn parallel to a diagonal. (224) Draw a circle of 1" radius, and take a point P 1'5" irom the centre. From P draw a hne to cut the circle in Q and E, so that PQ shall be J of the whole Une PE. *(225) Construct a triangle having its vertical angle 50°, its base 2", and the Hne bisecting the vertical angle dividing the base in the ratio of 2 : 3. *(226) Draw the curve traced by a given point C in a given straight line AB, which moves with its extremities in two straight lines, DE and DF, placed at right angles. (227) Draw a line through a given point A situated between "two given converging lines BC and DE, making equal angles with the two given lines. Also (a) Draw straight lines from any two given points, A and B, outside a given straight line, CD, meeting CD so as to make ■equal angles with the given line ; (6) to meet CD, so that they may be equal in length ; and (c) to find a point. A, in the base of any given triangle, BCD, from which the two lines drawn parallel to the sides to meet them are equal. (228) (a) Inscribe a regular octagon within a given square, having its alternate sides coinciding with the sides of the square ; and (6) its alternate angles touching the sides of the given square. *(229) Inscribe within a given square of 2J" side another square having its angles in the sides of the first but f its area. (230) Draw a semi-elliptical arch within a rectangle whose sides are AB, 3", and CD, 2J", AB being a conjugate axis. 138 GEOMETEICAL DEAWING. (231) Within a square of 2^" side inscribe a rectangle, the- length of two sides being each 2" in length. The corners of the rectangle to be in the sides of the square. *(232) The angle between two sides of a rhombus is 60°, the length of the sides 3". Draw an elhpse that shall touch the four sides of the rhombus. *(233) About a rectangle 3" by IJ" draw an ellipse pass- ing through the corners of the parallelogram. *(234:) Divide a circle into three equal areas by concentrie annuli. *(235) Divide a circle into three parts equal in area and periphery by means of semicircles. *(236) Draw a triangle having sides of 200 feet, 250 feet and 300 feet, scale 1"=100 feet. Divide this triangle intO' three equal areas by lines drawn from the apex. *(237) The sides of three different squares measure 1", l^", and 1 J". Draw a square equal in area to the sum of the three squares. *(238) Draw a square equal in area to the difference betweea two given squares of 2" and 1|" side. *(239) Describe a quatrefoil of semicircles within a square, each semicircle touching two sides of the square. *(240) Within a circle of 3" diameter inscribe five equal semicircles having their diameters adjacent. (241) About a circle of 2J" diameter describe a regular heptagon. *(242) Draw a regular hexagon of 1" side, and within it place another hexagon having the same centre and each side f ". *(243) Draw a regular pentagon of 1" side, and about it place another pentagon having the same centre and each side H". (244) Within a regular heptagon inscribe the largest possible equilateral triangle. (245) The perimeter of a pentagon is 7|". Draw the pentagon. EXAMPLES. 1S9> *(246) Construct a regular decagon of Ij" side, and divide- the decagon into equal isosceles triangles. Within each isosceles. triangle inscribe a square. *(247) Within, an equilateral triangle inscribe three equal semicircles, each semicircle touching two sides of the triangle. (248) Divide an equilateral triangle into four equal equi- lateral triangles, and inscribe a circle within each triangle. *(249) Draw a triangle whose sides, are AB 2|", EC 2",. and AC IJ". Consider points A and B as the foci of an elhpse, and a point C in the curve given. Draw the ellipse. (250) Draw an equilateral triangle having its perimeter 6", and divide it into four equal equilateral triangles. (251) The sides of a triangle are 200, 250 and 300 yards- long. Divide the triangle into two equal areas by a line- drawn from one corner. Scale 100 yards = 1". (252) The sides of a triangle are 2 feet, 2J feet, and 3 feet, long. Place another triangle within it having a base 1^ feet,, both triangles having a common centre. Scale 1" to 1 foot. *(253) The perimeter of a trapezium is 8", and its sides are- as 2, 3, 4 and 5. Draw the trapezium. *(254) Within an oblong place a second oblong having the- same centre and its long sides f the length of the long sides of the given one. (255) Two circles of radii 1" and ^" have their centres 2'" apart. Describe a circle of f" radius which shall touch, both. (256) Draw three circles having diameters 1", 2" and 2J" touching each other. *(257) Draw a sector of 2" radius and having an angle of 100°. Within, the sector inscribe a circle. *(258) A trapezium has two sides, each 1^" long, and the enclosed angle is 80°. The other two sides are each 2^" long.. In this trapezium inscribe a circle. (259) Draw a square ^ the area of a given square. (260) Draw a triangle J the area of a given triangle. 140 GEOMETKICAL DflAWING. (261) Draw an oblong the area of a given oblong. *(262) The diameters of two circles are 3" and 2". Draw a ■circle equal to the difference in area between the circles having the given diameters. *(263) Find lines equal to 73"and jW, the unit being 1". *(264:) Construct a triangle having one side 3" long, angles adjacent to that side being 30° and 45°, and within it inscribe ;a square and about it describe a circle. In this circle inscribe a pentagon, one of its angles touching the circumference at the same point as one of the extremities of the longest side of the triangle. *(265) Describe a circle of 1" radius. Divide the circum- ference into sixteen equal parts, and draw radii from the points •of division, stopping them at a circle of '2" radius having the same centre. *(266) Draw a circle having a diameter of 4J". In this inscribe an octagon, and in the octagon a square. State the length of one side of each of these. *(267) Within a hexagon of 2^^" side draw a hexafoU of semicircles to which the sides of the hexagon shall be tangents. On the diameter of each semicircle as a diagonal construct a square. *(268) In a circle 3" diameter place a quatrefoil of semi- circles of which the diameters shall be adjacent, and from the centre of each semicircle and of the circle draw circles of 'SS" radius. *(269) Draw a third proportional AC to AB and AD, li" and \y long respectively. Secondly, divide a line EF, 2" long, into extreme and mean ratio. *(270) At point A in a line AB draw three Unes making angles of 33°, 79°, and 113° respectively with AB. In each ■angle inscribe a circle of 1|" diameter. *(271) Draw two Unes, AB and AC, each 4" long and making an angle of 32° at A. Find a point D on either of the lines 2^" from A. Draw a circle tangential to the given lines and touching one of them in D, and from the same centre two other circles of '6" and •27" radius respectively. EXAMPLES. 141 *(272) Draw three equal circles in contact with each other- and inscribed in a circle of 1"5" radius. Circumscribe the outer circle by an equilateral triangle, the sides touching the circle in the points of contact with it of the inscribed circles^ Circumscribe this triangle again by a regular hexagon, its alternate angles coinciding with those of the triangle. In each of the three resulting isosceles triangles lying outside the equilateral triangle place a square having one of its sides resting on the longer side. *(273) Draw a rhombus having a side of 2" and an angle of 60° given. In this place a square having its angles resting on the sides of the rhombus. On each side of this square draw another as large as the bounding lines of the rhombus will permit. *(274) Draw a straight line AB, 4'8" long, and from it& extremities A and B, without producing it, erect perpendiculars AD and BC, each equal 3 "2". Join CD, and divide the parallelogram so formed into six equal squares. In each square inscribe a circle. *(275) Upon a base BC, 2" long, describe a triangle ABC, having the side AC 1"75", and the angle ABC 55°. Upon AB construct a square ABEF, and on AC a square ACGH. Join FH, and reduce the whole figure FEBCGH to a triangle of equal area. *(276) Draw six equal circles each touching those adjacent to it, and so placed that the centres will all be found on the bounding lines of an equilateral triangle of 2|" side. *(277) Construct a square having an area of 1", ^nd a second square in the proportion as 3 is to 1 of the first. Place the small square within the large one, so that its sides are parallel to and equidistant from it. *(278) On a chord 2"4" long describe the segment of a circle to contain an angle of 54°. With a radius of 1'15"^ inscribe a circle to touch the chord and the arc of the segment. *(279) In a hexagon having a perimeter of 7^" place a square, each of its angles resting on a side of the hexagon. Divide this square into four smaller ones, and in each place an equilateral triangle, the four apices to meet in one point. 142 GEOMETKICAL DRAWING. *(280) Draw a straight line AB, 4" long, and from its ■extremities erect perpendiculars AC and BD, 2'5" and 1*8" long respectively. Find a point in AB equidistant from C and D. *(281) Describe a circle — (1) to touch a given circle and a given straight line at a given point D. (2) Describe a ■circle passing through two given, points AB, and tangential to a, given straight line CD. (3) Describe a circle tangential to two intersecting lines and passing through a given point X lying between them. *(282) Draw two concentric circles, having radii of 1" and 2". In the space between these place seven circles at equal distances consecutively from each other, having a radius of -j^", ■and their centre half-way between the two larger circles. *(283) With a centre C, and radius of 2", describe a sector of a circle having the angle at C 60°. In the sector inscribe a square having one of its corners in the arc AB. *(284) Construct an isosceles triangle having a base of 2" and a vertical angle of 37°. Obtain a similar triangle of half the area. *(285) Draw five concentric squares 0"15" apart, the smallest having a side of '9". Ink in the squares with lines that increase in strength from the smallest to the largest. *(286) Describe a circle having a radius of 2'12". By means of four concentric circles divide its area into five equal parts. From any point in the outer circle draw a tangent to the innermost. *(287) Draw a rectangle having sides 2"48" and 5*12" long. Trisect each angle, and produce the lines until they touch the bounding lines of the figure. Letter all points, and give the lengths of the trisecting lines. *(288) Construct a regular pentagon and an equilateral triangle, each of 4 square inches area. *(289) "Within a circle of IJ" radius place a trefoil of tangential arcs of equal circles, and from the centres of these draw three circles having diameters of I'l", •74", and "SS", increasing each smaller circle in thickness of line when inked in. EXAMPLES. 143 *(290) Upon two bases of 2" and 2|" respectively construct isosceles triangles having angles at their bases of 50° and 75° respectively. Construct a square having an area equal to their united surfaces. *(291) Draw a circle of 1" radius and divide its circum- ference into 6 equal arcs. Describe an arc of a circle of 1" radius from each point of division, stopping these arcs at the circumference of the first, and each passing through the centre. *(292) Two points are each 3f" from the centre of a circle of 1^" radius and 2f" from each other. Describe a circle that shall pass through the two points and touch the given circle. *(293) Describe a square of 4" side, and obtain two other squares together equal in area to the first, and having a ratio of 1 to 2. *(294r) Determine the side of a square of 7" area. Draw the square, and trisect it by lines drawn from one of its angles. *(295) Given 1" as the unit, obtain by geometrical construc- tion ^fT *(296) Construct a square of 5" area, and in it inscribe a second square having its four corners in the four sides of the first, but its area J less. *(297) Draw a straight line AB, 6" long, bisect it in C, and erect a perpendicular CD 1". Draw the arc of a circle that shaU pass through the points A, B and D without using the oentre of the circle. *(298) Draw two straight lines inclined to each other at an angle of 35°, and describe a circle of |-" radius to touch both of them. Draw a second circle touching the first and the two straight hnes. *(299) Draw the curve of an eUipse by any method you please, providing the major axis is ^9" and the minor axis *(300) Given a scalene triangle, describe three circles with their centres at the angles, each touching the other two. 144 GEOMETEICAL DRAWING. CONCERNING SCALES. It is necessary, in tTie study of military surveying, engineering, arcliitecture, &c., that the student should understand the con- struction and use of Scales. The construction of Scales requires judgment and ingenuity, and the application of them a certain amount of neatness and precision. A Scale is a Hne of standard length divided into a definite number of equal parts, and represents in convenient dimen- sions, without distortion, either large or small measure- ments. A drawing is said to he made to scale when its parts bear a certain fixed proportion to the parts of the object it represents. Let A = B be a line that actually represents 100 yards. AB will be found to measure 1" — i.e., a length of 100 yards is represented by a line 1" in length ; therefore the line is said to be drawn to a scale of 100 yards to 1". We must now find the proportion of the drawing to the original object — ^briag 100 yards to inches. The number of inches in 100 yards = 100 x 36 = 3600 inches. It is therefore evident that every line to the original object is 3600 times the line which represents it in the drawing ; or every line in the draw- ing is aa'bo th part of the corresponding line of the object — i.e., the representative fraction of the scale is ■^^^. The student must clearly understand what is meant by the Kepresentative Fraction, to find which he must reduce the number of imits represented by 1" to inches. The representative fractions are used to express the ratio of any line in a drawing made to scale, to the corresponding given line of the original ; thus, if the fraction of a scale is ^th, it shows that the drawing is ^th of the real size of the object it represents, or 1" represents 20". CONCERNING SCALES. 145 Therefore : If 1 inch represents 1 foot, the fraction of the scale is ys 4 feet, „ 1 yard, 1 pole, „ 1 furlong, 50 yards, „ 1 mUe, ,, 150 yards, „ , „ ■' ' " 150x36 1080 The representative fraction is, therefore. Any line in the drawing in inches. What that line represents in inches. 1 )» JJ 48 1 n J» 96 1 jy >J 24 1 1 36 X, H 198 1 1 36x5| x40 7920 1 1 50 X 36 ~ 1800 1 1 1760 X 36 ~ 63360 5 1 PLAIN SCALES. A Plain Scale is the simplest form of a scale of equal parts, set off on a straight line, as feet and inches, or iaches and tenths. A PLAIN SCALE OF FEET AND INCHES. tgtrffiinii I I - I I I ■ I 1 1 fiAi«l236 3 1 Z 3 ■ » 5 e 7 B feet Fig. 1. 146 GEOMETRICAL DBAWING. (1) To construct a plain scale of feet and inches long enough to naeasure 12 feet from, when 1" represents 3 feet. .•. 1 X '^■^ =i", length of scale. ,, i i i ii I I r I I I I I 1 I — I Jnchcslxaeso 1 Z3t567 8. aw Mfett Fig. 2. (fths Scale.) 1 1 The fraction of the scale is - — — _ --. 3x12 36 Draw a hne 4" in length and divide it into twelve equal parts — each part represents 1 foot ; subdivide the first primary division to represent inches. Ifote. — The bottom Hne of a scale is shown thicker than the top line ; primary divisions are figured from left to right, subdivisions from right to left. Heavy Hnes in a scale are merely for contrast, (2) Draw a plain scale of feet to measure 25 feet from, when 3 yards is represented by 2". What is the fraction of the scale? .'. 2 X Y = ^ = 5*6 or 5*56" nearly, length of scale. Draw a hne 5'56" long and divide it into twenty-five equal parts. The method of drawing all plain scales is similar to Fig. 1. 2 1 The fraction of the scale = 3 X 36 54" Note. — Since in the representative fraction is expressed the number of inches represented by 1", the scale may be drawn when this fraction is given. (3) A scale is required to measure 20 feet from, when the representative fraction is j^, .1x20x12 _„ , ,, . , . • Jo = ^ ' length of scale. DIAGONAL SCALES. 147 For drawing the scale, proceed as in former plain scales. (4) A scale is required to measure 20 miles from, the representative fraction being -^-g-itru- 1 X 20 X 1760 X 36 253440 - = 5", length of scale. DIAGONAL SCALES. It -vyill be seen that the solution of some of the problems on scales requires a decimal notation. A specimen of a diagonal scale is given, from which the student can observe the method of measuring lines and distances with greater accuracy than could be done by means of a simple scale. Let AB be the unit of length to be divided into X parts, this being resolved into two factors, Y, Z. AB is divided into Y parts, the Z lines at any convenient equal distances being drawn parallel to AB; two lines perpendicular to AB are drawn, one through each extremity. The distance between these on the farthest parallel being again divided into Y parts, the points of division are joined by oblique lines, as shown in the figure, and thus each of the Y divisions of AB is in fact subdivided into Z equal parts, or the whole hne into X parts. (Let AB represent 100 in this case.) •. Y X Z = X parts— i.e., 10x10= 100. A DIAGONAL SCALE. 1 M 1 1 1 ft ± fi i ^i _r 1 1 ^ 7 _ ± _±± \\\ A 8 6 t Z B 100 Fig. 3. 200 300 The distance between the dots on the 9th parallel shows 159. „ 5th „ „ 255 148 GEOMETEICAL DKAWING. A diagonal scale must be carefully made, or it is useless. On ordinary boxwood scales a decimal scale is generally given on one side, of inches or of half inches. By the appUca- tion of distances measured from the scale, fig.- 3, tenths or hundredths of an inch may be measured. Considerable practice is required to get familiar with a diagonal scale. (5) Draw a diagonal scale of 5 feet to 1", to show inches. Assume 25 feet. 1x25 = 5", length of scale. huchesK " to I 1 \ \ 1 \ I \ ? \ \ 5 f- 3 2 J 10 IS 20 25 'ieei,' Fig.' 4. (iths Scale.) Note. — Diagonal scales are used for measuring off more minute distances than can be done by an ordinary scale. Metlwd of dramng the diagonal scale. — Draw a Hne 5" long and divide it into five equal parts, and the first primary division into five equal parts. Each subdivision will represent 1 foot. Draw twelve parallel lines the same distance apart as shown ; also draw perpendiculars through the points of division. Across the subdivision on the extreme left draw a diagonal — this increases from left to right 1" — on each parallel line. Eetween the dots on the 5th parallel we measure 14ft. 5in. 9th „ „ 19ft. 9in. It is obvious that any measurement from 1" to 25 feet may be taken from the scale. DIAGONAL SCALES. 149 All diagonal and decimal scales are drawn on the same principle. Note carefully the manner the diagonal scale is figured, and below the required parallel lines draw a thick line, and write feet and inches, or whatever the scale represents, on the extreme left and right of scale. <6) It is required to construct a diagonal scale of 8 miles to an inch, to show furlongs. Assume 45 miles. .1x45 Eepresentative fraction. = 5-62", length of scale. 1 1 8 X 1760 X 36 506880' Draw a line 5 •62" long and divide it into nine equal parts, and the first primary division into five equ.al parts. Each part will represent one mile. Draw eight parallel lines the same •distance apart (8 furlongs = 1 mile), and draw perpendiculars through the divisions, and draw a diagonal across the subdivision ■on the extreme left. <7) Draw a diagonal scale of 1 furlong to 1", to measure yards. Make your scale any convenient length, say 5" ; each inch represents one furlong. A furlong equals 220 yards, which, resolved to factors, = 11 x 20. So divide the first inch on the scale into twenty parts ; each part wUl represent 1 1 yards. To measure single yards, draw eleven parallel hnes, and proceed as in the other diagonal scales. Eepresentative fraction, ^^^ = ^^. <8) The distance between A and B is 5 miles, and is represented on a plan by 1-25". Construct the scale and show furlongs, when 20 miles is the number assumed. Draw a line 5" long and divide it into twenty equal parts. 150 GEOMETRICAL DRAWING. Subdivide diagonally the first division into eight equal parts,, to represent furlongs. The representative fraction of the scale is 1 ^Xl76Ox36x^00 253440 4 (9) The representative fraction of a scale is ^^.. Draw the scale and show 250 yards. 960 ® (10) Construct a diagonal scale showing inches, tenths, and hundredths. COMPAEATIVE SCALES. A scale constructed from a given scale to read in some other different measure is called a Comparative Scale. (11) On a map the distance between two points is 4", and represents 5 miles. Required a com- parative scale of furlongs. 5 miles : 4" : : 1 furlong : a-. 4x1 Or —Tfr- = ^ of an inch to represent a furlong. I M 1 1 1 1 1 i 01X3*5878810 20 30 10 flirUinffi Fig. 5. (Jths Scale.) COMPAEATIVE SCALES. 151 Therefore draw a line 4" long and divide it, into forty equal parts. Each division will represent 1 furlong, and is compara- tive or corresponding to the given scale. (12) I examine a French plan of a building, and find a scale of decimetres, 10 to an inch. I desire to measure EngUsh feet. Draw a comparative scale showing 20 feet. (A deci- metre =0-327 feet.) 1" represents -327 x 10 = 3-27 English feet. We have also 3-27 feet : 1" : : 20 feet : a; = 6-11", length of scale. f. 20 2000 ^' 3^2T = 32y = ^'1^ • Draw a line 6-11" long and divide it into twenty equal parts. Each division will represent 1 foot, and will he com- parative to the given scale of decimetres. (13) A plan is drawn to a scale of 88 feet to an inch. Draw a comparative scale by which any distance from 5 yards to 150 yards may be measured on the plan. 1 X 150 X 3 ,. oQ =0-11 m length. Draw a hne 5-11" in length and divide it into fifteen equal parts. Each part represents 10 yards. Bisect the first divi- sion of 10 yards, so as to ohtaia 5 yards. (14) On a scale 60 Russian versts measure 7-5". Supply a comparative scale of BngUsh miles, taking a verst as 1167 yards. Show 50 miles. 50x1760x75 ^ j , j^ ^ j representing 600x1167 ^ . i' 5 50 English miles. 152 GEOMETRICAL DKAWING. SCALE OF CHOEDS. A Scale of Chords is one by which, in lieu of a protractor, angles of any proposed number of degrees can be constructed, or angles already laid down measured. This scale is usually found on boxwood and ivory scales, and is indicated by the sign C, or CHO, at its commencement. (15) To construct a scale of chords. Construct a quadrant ABC, and divide its arc BC into nine equal parts of 10° each; thus, trisect the arc with the radius of the arc as radius. Divide each angle of 30° into Fig. 6. three equal parts by trial. Draw the chord of the arc BC, and from C as centre, to the points of division on the arc as radii, describe arcs cutting the chord BC, and number each point of division in tens of degrees, from to 90, thus transferring THE SECTOE. 153 the degrees in the arc to the straight line, from either of which the same measurements may be taken. Method of using the scale of chords. — Whatever angle is required, always take the distance from to 60 as radius, and Fig. 7. with centre C describe an arc LM. For an angle of 50°, take the distance from to 50 on the scale of chords, and set it off from L to P. Draw the Hne CP. The angle PCL is 50°. THE SECTOE. A Sector is used to divide lines into segments which shall have to each other a given numerical ratio. (10 Euc. III.) ■Constructions are greatly facilitated by means of a sector. This is a jointed ruler, on each leg or limb of which are lines radiating from the centre of the hinges ; the two marked L (Line of Lines) are those most required. These are divided into equal parts commencing from that centre, each division teing again decimally subdivided on the instrument. It is obvious that by the principle of similar triangles the 154 GEOMETRICAL DRAWING. distances indicated by the dotted lines will be in the ratio of 3 : 6 : 10. For example, the distance 3 and 3 has the same ratio to 6 and 6 as the number 3 is to 6, or one-half. If the Fig. 8. dotted line AB measure 2^", the dotted line 6, 6 will measure I of 2J", or 1|" ; the line 3, 3 will measure f^ of AB, or f of an inch. (16) To find f of a line 3-3" long. Take 3 "3" in your compasses and place one point of the compasses in the 9th point of the Hne L, on one limb of the sector, and open the instrument till the other point of the compasses falls into the corresponding point 9 of the other limb ; then the transverse distance between 5 and 5 will be the length required. (17) To find I of a line 4*22" long. Take 4"22" in your compasses and place one point of the compasses in the 5 th point of the line L, on one limb of the sector, and open the instrument till the other point of the MAEQUOIS SCALES. 155 compasses fall into the corresponding point 5 of the other limb ; then the transverse distance between 3 and 3 will be the length required. Again. On a map a distance of I' 6 represents a space of 80 roiles. Complete the scale to 100. Take the given space 1"6 in the compasses and place it between 80 — 80. Open the compasses from 100 to 100, which gives the whole length of the scale. In using the sector care must be taken not to alter the angle- at which it is set, while ascertaining the length of a hne in a given ratio, and only use the divisions on the lines which pass, through the centre of the joint. MAEQUOIS SCALES. Marquois Scales, called so from the name of the inventor, consist of' two rectangular rules 1 foot long, and a right-angled triangle or set square ; the set square is so constructed that the- hypotenuse is three times the shortest side, and near the centre of the hypotenuse is an index d. Besides being able to draw lines parallel to one another, we can also draw them at any required distance apart; this is one property of the Fig. 9. Marquois Scales which makes them so valuable to a military student. The principle embodied in the construction of Marquois Scales, each of which consists of two parts, an outer or artificial scale, and an inner or natural scale. The drawing is made to the natural scale, and the numbers written below it signify the number of units, whether of yards, chains, &c., represented by 1". Take, for example, the scale of 30. It will be seen that the- 156 GEOMETRICAL DRAWING. primary divisions are marked from left to right, and the first division subdivided into ten parts. Taking these subdivisions ■as single units of measure, three primary divisions will contain ■30, and the scale wiU be called a scale of 30 units to the inch. The primary divisions upon the artificial scale are marked from zero both ways, and numbered 10, 20, &c. These divi- sions are three times those on the natural scale, thus giving the scales the same ratio that the longest side of the triangle has to the shortest side. It is required to draw a line parallel to a given line at a ■distance of 17 yards from it. We will use the scale 30, which is 30 yards to 1". Let the bevelled edge of the set square coincide with the given line, so that the zero on the artificial scale is against the index d of the set square; hold the rule fast with the left hand, and slide the set square either to the right or left, until the index d is at 17, and draw the line. It must be understood that the subdivisions in the first primary division on the natural scale may represent any number of units. It is obvious the scales may be varied to any extent. The proof of the construction depends on 2 Euc. VI. Again, lines horizontal, vertical, and diagonal may be drawn by the Marquois Scales. By moving the index we can rule lines which are not only parallel, but equidistant also, or the •distance may be specified. If the parallels are required to be Yi'th of an inch apart, use the 60 scale, and move the index over five graduations at a time. If the parallels are required to be ^th of an inch apart, use the 100 scale, and move the index over five graduations at a time. VEENIEES. Verniers are small movable scales attached to the index of ■astronomical and surveying instruments, barometers, &c., and so constructed that when the index moves, the divisions of the vernier successively coincide with the graduations of the fixed scale of the instrument. For instance, if it were required to read to -j^j- of a milli- metre a barometer whose scale was divided into millimetres, the vernier attached to the index would have 10 divisions equal VERNIERS. 157 to 9 divisions of the scale. Hence it follows that if (say) the 4th mark ahove the index point coincides with a division of the scale, the index will be ^ of a millimetre above the scale division next below it. In some cases it is more convenient to- make 10 divisions of the vernier equal to 11 of the scale.. With a scale divided to J of an inch, by dividing a vernier of 6 or 6J" into 25 equal parts the' position of the index may be determined to ^dtt o^ ^^ inch. Again, with a circular scale of half degrees, by dividing a vernier of 29 or 31 half degrees into 30 equal parts the index reads to minutes on the arc. They are seldom used in drawing, but they may in some cases be used instead of diagonal scales. A Vernier Scale is constructed to measure distances according as the primary divisions represent hundreds, tens, or units ; therefore, between the same points on the scale we may measure 116, 11"6, or 1"16. The method of drawing the scale is to set off on a line of any convenient length a number of inches, which is called the natural scale, AB. The Vernier Scale is XY, drawn under the natural scale. A O Z * 6 8 ]p 2 ■>■ e B 2p X. ■^ fi H SP 2 » ft 8 W ft rl 'l'j I ! I iT l wee iii|iiri1 1 n 1 1 ri riTiTM n iTn Fig. 10. (|ths Scale.) Take a Hne AB, of any convenient number of inches in length, and divide each inch into ten equal parts. At the primary divisions mark 0, 10, 20, 30, 40, &c., and the sub- divisions 2, 4, 6, 8, &c. Take eleven subdivisions from AB, the natural scale, and set off from X to Y and divide into ten equal parts. The scale XY is the Vernier Scale. It is obvious that these subdivisions on the Vernier Scale exceed those on the natural scale --ny of a subdivision, or -j-J-^ of a primary division. Taking each subdivision on the natural scale as 10 units, then from X to the division 6 will be 66 units, equals 60 + 6— that is, 60 units + 3^ of ten units. 158 GEOMETRICAL DRAWING. The distance between the dots represents 116, 11 "6, or 1"16, according as the primary divisions represent hundreds, tens, or units. Let it he required to measure off 116. Place the leg of the compasses on the division 6 of the Vernier Scale, and the other leg on the division 5 on the right of the division 10. From X to 6 on the Vernier Scale is 6x11 = 66; and from X to the division 5 = 50, which, added to 66, gives 116. EXAMPLES. 159 EXAMPLES. Questions marked (*) have been set at examinations. (1) Construct a scale to show 20 feet, when 1" represents 4 feet. (2) Construct a scale to show 70 yards, when 1" represents ■9 yards. *(3) The distance between two places is 4 miles, and is represented in a plan by 1"5". Construct the scale and show furlongs, when 20 miles is the number assumed. *(4) Construct a scale of 10 miles to 1", to measure distances of 1000 yards. *(5) Draw a scale of 1 mile to 1", to show furlongs. (6) Construct a scale of I foot to 1", to measure inches. Show 7 feet. *(7) Draw a scale of 1 league to 1", to measure miles. Show 8 leagues, and give the representative fraction. *(8) The distance between L and C is 30 mUes, and measures on a map 18"3". Draw a scale to the map showing miles and furlongs. State the representative fraction. *(9) The liaes of a drawing are -/^ of the real lines which they represent. Provide a scale of feet for the drawing, showing at the least 50 feet. *(10) On a map a furlong is represented by 1*25". Draw a scale of poles for this map, and show a line on your paper 227 poles long. (5^ yards = 1 pole.) *(11) Make a plain scale to a drawing when 40 feet are represented by 1". It must be so marked off as to be practically useful, and show 10 feet as the smallest division. 160 GEOMETEICAL DEAWING. (12) Construct a scale of -^^ to show feet and inches hy diagonal divisions. *(13) Draw a plain scale of 12'5 yards to 1". It must be long enough to measure 70 yards. (14) Draw a scale of 12 yards to 1", showing feet by the method of diagonals. *(15) Draw a decimal scale of f" to 1 foot, and show a line equal in length to 5 "82 feet. *(16) A map is constructed so that 1" represents 45*5 yards. Draw a scale for it from which paces can be measured, the pace being 32". Assume 250 paces. *(17) The measurement on a map between A and B is 5", representing 8"25 miles. Draw a scale for the map, and show a Kne on the scale equal to 10 miles 5 furlongs. *(18) Construct a scale to read feet and laches, 8 J feet beiag equivalent to 5'27". Give the representative fraction. *(19) Draw a plain, scale of yards of ■^rw'j' ^^^ yards being the greatest and 10 yards the least dimension shown. (20) Draw a diagonal scale of mfetres, comparative to No. 19, and show single mfetres. (A mfetre= 1'0936 yards.) *(21) An Englishman wishing to examine a Spanish plan, finds only a scale of Spanish pahns, 20 to an inch. Supply him with a corresponding scale of English feet, taking the palm as •684 Enghsh foot. Show 60 feet. *(22) On a scale 60 Eussian versts measure 7"5". Supply a comparative scale of English miles, taking a verst as 1167 yards. Show 50 miles. *(23) D^aw a scale of y^ to measure Belgian feet. (One Belgian foot = -90466 English foot.) *(24) Draw scales of ^ ^ g^^^ ^ ^ to show English mUes and Eussian versts. (A verst = 1166'68 yards.) (25)- Draw a diagonal scale of j^-g, showing poles, yards and feet. Assume 5 poles. *(26) On a certain drawing 1250 yards of real magnitude are represented by 15'5". Eurnish a scale for the drawing, and also a comparative scale of French mfetres. (One mfetre = 1-0936 yards.) Show 500 yards. EXAMPLES. IGl (27) Draw a scale of miles ^^nrbinrj showing furlongs by the diagonal method, *(28) A map is constructed so that 1" represents 60 yards. Draw a scale for it from which paces can be measured, the pace being 32". *(29) Draw a scale of miles g^aeO) showing furlongs by the ■diagonal method. *(30) Draw a scale of miles, furlongs, and yards, whose representative fraction is -g^^-g^, showing 20 yards as the smallest reading. *(31) Draw a plain scale of 665 paces to the inch, 100 paces being the least, and 5000 paces the largest dimensions shown. Give the calculation and mark the representative fraction, assuming a pace to measure 32". *(32) A distance of 37;^ miles is represented on a map by 4"15". Draw a scale for the map by which single miles may be measured, showing 50 miles. Convert the scale so drawn into a diagonal scale to read furlongs. Give the representative fraction. (33) Construct a scale of 15 miles to 1" to measure distances of 1000 yards. (34) Draw a diagonal scale of 9 feet to 1" to show inches. Assume 50 feet. (35) Construct a diagonal scale showing inches, tenths and hundredths. *(36) A length of 100 yards is found to measure 3'6" on a drawing. What is the fraction of the scale? Construct a scale to read yards, making it not less than 4" long. *(37) Constru.ct a diagonal scale to read metres with a representative fraction of -^-i^. (A metre is 3"28 feet.) (38) On a plan 5 square inches represent 720 square yards. Draw a scale for the plan. *(39) Draw a scale of chains j^nrrD showing poles diagonally. (1 chain = 66 feet = 4 poles.) *(40) A drawing is on a scale of 4 chains to 1". Make a diagonal scale for it about 6" long to read yards. (A chain = 22 yards.) L 162 GEOMETEICAL DEAWING. *(41) Give the representative fraction of a scale on which 3J" represent 2247 feet. Construct a scale of ^, reading feet and inches. *(42) Construct a scale of jJj to read decimetres, and show 10 mfetres. (A decimetre = 0-328 feet.) *(43) Construct a scale of -^g, showing yards. The scale is to he properly figured, and not less than 7" long. (44) Construct a scale of ^-J^ to show 50 yards. Construct a comparative scale of mfetres. (A decimetre = 0*328 feet.) *(45) The plan of a building is a square of 3" side, the diagonal of which represents 100 feet. Make a scale from which feet and inches may be measured. *(46) A plan is drawn to a scale of 88 feet to an inch. Draw a comparative scale by which any distances from 5 yards to 150 may be measured on the plan. The scale to be properly iigured. (47) Construct a diagonal scale of 45 feet to 1" by which single feet may be measured, and write down the repre- sentative fraction. Show 250 feet. *(48) On a plan 6"5" represent an English mile. (a) Draw a plain scale of yards to suit the plan, showing 1500 yards, and divide it to show distances of 50 yards. (6) Draw a comparative scale of Spanish yards. (A Spanish yard = "927 of an English yard.) Each scale to be properly figured, and all calculations shown. (49) On a certain drawing 1250 yards of real magnitude are represented by 15 "5". Furnish a scale for the drawing. *(50) Make a diagonal scale to read miles, furlongs and chains, if 3 miles are shown by 3^". Assume 7 miles. Take off 3 miles 7 furlongs 6 chains, and state the representative fraction. *(51) On a scale 10 furlongs are represented by 2^". Draw the scale, measure 2 mUes 2 furlongs by it, and state the repre- sentative fraction. EXAMPLES. 163 (52) Draw a diagonal scale to measure tenths and liundredtiis, "7 being represented by a line 1;^" long. *(53) If a line 5" long represents 4 miles 5 furlongs, how long is the line which represents 3 miles 3 furlongs 3 chains 1 Give the representative fraction. *(54) Draw a plain scale of miles and furlongs in which IJ furlongs equal ^ of an inch. Show 8 miles. Give the repre- sentative fraction. *(55) The distance between two places is 35 miles, and measures on a map 4'4". Draw a diagonal scale to suit the map, showing 50 miles. What is the distance represented by 2-7"? *(56) Construct a diagonal scale of feet and inches on which 17 feet would be represented by IJ". Give the represent- ative fraction, and show 50 feet. *(57) Draw a line 4"77" long. Let this represent a length of 3 feet 7". Produce it to a length of 5 feet 8". (58) If 2 miles 3 furlongs are represented on a map by a line 4'37" long, how long would a line be representing 3 miles 5 furlongs? (59) On a map 1" represents f of a rmle. Draw the scale, and show 3 miles 5 furlongs. What is the representative fraction ? *(60) Draw a scale of feet to measure distances from 1 foot to 50 feet, 5J feet being represented by •52". Draw this scale by the diagonal method, showing inches. Mark off on the scale a distance of 42 feet 5". (61) A distance of 19 miles is represented by a line 2"37" long. Show 45 miles. *(62) Draw a scale of ^^200 ^° show furlongs. (63) Draw a line 2"14" long. Let this represent a line 4 feet 4" long, and from one end mark off a distance of 1 foot 9". *(64) A distance of 3" on a plan represents 1*9 miles. Construct a plain scale of miles and furlongs to suit the map, showing 4 miles. 164 GEOMETRICAL DRAWING. *(65) The distance between A and B on a map is 41 miles, and measures 1'7". Construct the scale by the diagonal method, showing single mUes. The scale must be long enough to measure 100 miles. (66) If a map of the world be drawn to scale of 180 miles to the inch, what is the representative fraction? Draw the scale. (67) Draw a diagonal scale of f of a mile to the inch, to measure miles, furlongs, and chains. ' Draw a line by this scale 2 miles 5 furlongs 3 chains long. *(68) Draw a diagonal scale of 850 paces to an inch, to measure distances from 10 to 4000 paces. State the repre- sentative fraction, assuming the pace to measure 32", *(69) Draw a diagonal scale to measure yards, feet, and inches, 2 feet being represented by 1". Show distances of 2 yards 2 feet 2 inches, and 1 yard 1 foot 1 inch. (70) In a scale f" represents 5 furlongs. Draw the scale, and measure 2 miles 5 furlongs from it. (71) Draw a scale of miles, furlongs, and chains, 2J miles being represented by 3f ". (72) Draw a diagonal scale of 120 feet to 1", to measure single feet. Show 400 feet. *(73) Draw a scale of feet to measure 400 feet, least dimen- sion being 10 feet ; 60 feet being represented by •75". Give the representative fraction. (74) A line 1"75" long represents 5 furlongs. Draw a scale of miles, furlongs, and chains. *(75) Construct a scale in which 5 furlongs are equal to ■9". Show miles and furlongs and chains diagonally. *(76) A distance of 729 yards is represented on a plan by 10'8". Construct a diagonal scale of yards for the plan, showing 500 yards, by which single yards may be measured. Show all your calculations, figure your scale properly, and write above it the representative fraction. Show the points you would take in order to measure off a distance of 127 yards. EXAMPLES. 165 *(77) The measurements of an irregular six-sided field, A, B, C, D, E, F, are taken and found to be as follow ; — Side AB = 250 yards. Diagonal EC = 260 yards. „ BC=190 „ „ BE = 300 „ „ CD = 140 „ „ CF=360 „ „ DE = 220 „ Angle ABC =110°. „ EF=150 „ Draw a scale of yards having a representative fraction of 2 8^0 ) ^^^ draw a plan of the field to that scale. Write down the length in yards of the side AF and the diagonal AC, and calculate the area in yards of the whole field. *(78) A map is drawn to a scale of 12" to a mile. Draw a comparative scale of chains for the map, showing 50 chains. Show all youi calculations, figure your scale properly, and write above it the representative fraction. (One chain = 66 feet.) *(79) Draw a diagonal scale of YirVa *° measure single feet. Show 500 feet. Figure your scale properly, and show your calculations. Show by two small marks on the scale the points you would take in order to measure off a distance of 373 feet. *(80) A map is drawn to a scale of 6" to an English mile. Draw a plain scale of Spanish yards for the map, showing 2000 Spanish yards, and divided to show distances of 50 Spanish yards. Show all your calculations, figure your scale properly, and write down the representative fraction. (One Spanish yard = -9277 English yard.) *(81) Draw a scale of -g^^ to measure feet. Show 200 feet, and divide the scale to show distances of 5 feet. *(82) A distance of 1"35 miles is represented on a map by 2"15". Construct a diagonal scale of chains for the map by which single chains may be measured. Show 250 chains. Show all your calculations, figure your scale properly, and write above it the representative fraction. Draw a line, and from this scale measure a distance of 137 chains. (One mile = 80 jjhains.) *(83) The distance between two towns is 27J miles, and measures on a map 3'25". Construct a diagonal scale of miles 166 GEOMETRICAL DEAWING. and furlongs for the map, showing 50 miles. Show all your calculations, figure your scale properly, and write ahove it the representative fraction. By means of the scale draw a Kne 23 miles 5 furlongs long. *(84) On a plan 1250 yards are represented by 15"5". Draw a comparative scale of French mfetres for the plan, show- ing 500 metres, and divided to show distances of 10 mfetres. (One mfetre= 1-0936 yard.) *(85) Draw a scale of feet, having a representative fraction of -^^x^, by which distances of 25 feet up to 1000 feet may be measured. Show all your calculations, and figure the scale properly. *(86) On a Eussian map 10^ versts are represented by 1"25 English inches. Draw a scale of English miles for the map, showing 40 miles. (One verst = "GGl of an English mUe.) *(87) It is required to ascertain the distance in mfetres and decimetres from a point A, of three points, C, D, and E, all on the same side of a straight line-AB, 24 j mfetres long. The angles are measured and found to be as follow : BAG = 100°, BAD = 80°, BAE = 55°, ABC = 47°, ABD = 65°, ABE = 85°. Construct a scale of 7 mtoes to an English inch, show deci- metres by the diagonal method, and by means of your scale draw a plan showing the position of the points C, D, and E. Scale and write down their distances in metres and decimetres from the point A. A protractor may be used. (One metre = 10 decimetres.) *(88) A line AB, 3" long, represents on a plan a distance of 5 furlongs 15 poles on the ground. Construct a diagonal scale of furlongs and poles for the plan, showing 1 nule. Figure your scale properly, and write above it the representa- tive fraction. Show your calculations. (One mile = 8 furlongs = 320 poles.) *(89) On a plan a square inch represents 4 acres. Draw a scale of chains for the plan, showing 30 chains. (One acre = 4840 square yards ; one chain = 66 lineal feet.) *(90) The distance between two places is 13 miles, and measures on a map 2^". Draw a scale of leagues and miles to EXAMPLES. 167 suit the map, showing 10 leagues. Also draw a comparative scale of yards by which distances of 1000 yards may be measured. *(91) The length of an ordinary pace is 32" ; in " stepping short" it is 21". Draw a scale of -g-jnTij *° show 600 ordinary paces, and also a comparative scale of short paces. *(92) Construct a diagonal scale of chains and links for a plan on which 27 chains are represented by 31". Show 5 chains. *(93) Draw a plain scale of feet and inches to measure 7 feet, 1 j feet being represented by 1"3". Draw also a diagonal «cale comparative to the above, to show metres, tenths of metres, and hundredths of metres. *(94) Construct a scale on wliich 80 feet are represented by «". Show 50 feet. *(95) Draw a diagonal scale of 120 feet to 1", to measure single feet. Show 700 feet. *(96).Find the representative fraction of a scale in which 210 yards are represented by 1'75". Construct a scale of 13 yards to 1", and construct a diagonal scale of -^ touo reading furlongs. *(97) The representative fraction of a scale is Yxjig-o-. Con- struct a scale of chains to show 1 mile. The smallest unit 1 chain. *(98) Draw a scale of 2 chains to 1" to measure 264 yards. *(99) On a map 67J miles are represented by 9'3". Draw a scale of miles for the map, showing 30 miles. Show furlongs by the diagonal method. Show all your calculations, figure your scale properly, and write above it the representative fraction. Show the points you would take in order to measure off a distance of 17 miles 3 furlongs. *(100) A distance of 37 EngHsh miles is represented on a map by 4'3". Draw a scale of Irish miles for the map, showing 40 Irish miles. (An Irish mile = 2200 yards.) 168 GEOMETEICAL DKAWING. (101) On a plan 6"5" represent an English mile, or 1760 yards. (a.) Draw a plain scale of yards to suit the plan, showing 1500 yards, ari,d divide it to show distances of 50 yards. (&.) Draw a comparative 'scale of Spanish yards. (A Spanish yard = '927 of an English yard.) Each scale to be properly figured, and all calculations- shown. (102) A ship sails so that the sum of its distances from two lighthouses is always 4750 yards. The Hghthouses are 4000 yards apart. Trace the ship's course for not less than 3000 yards, starting from a point 1200 yards from one of the Hght- houses. Scale ^ildo- (103) The distance between two places is known to be 13 miles, and measures on a map 2;|". Draw a scale of leagues and miles to suit the map, showing ten leagues. Draw a comparative scale of yards by which distances of 1000 yards may be measured. (104) Draw a diagonal scale to read leagues, miles, furlongs. Eepresent one league by 1". Show 6 leagues. What is the representative fraction ? (105) On a plan a rectangle 2" x 2J" represents 720 square yards. Draw a scale for the plan. (106) Draw scales of j%^ to represent English feet, French metres, and Greek cubits, if 1 mfetre = 3*27 feet, and 1 cubit = "45 mfetres. (107) Make a scale of chains, 1 mile being shown, and the smallest unit being 1 chain. The representative fraction (108) Draw a scale of paces to measure distances between 1100 and 20 paces, scale f" to a furlong. Erom the scale- draw a Hne 740 paces long. (1 pace = 30".) *(109) Draw a plain scale of 665 paces to the inch, 100 paces being the least and 5000 paces the largest dimension shown. Mark the representative fraction, assuming a pace to. measure 32". EXAMPLES. 169 *(110) A military sketch is to be drawn to the scale of 2^" to the mile. Construct a scale of paces — a pace equals 33" — capable of measuring any distance between 4000 and 100 paces. (Ill) A ladder 79 feet long is placed against a building 77 feet high so that the top of the ladder just reaches the top of the building. Draw a diagram to a scale of -j^i and measure the angles with a protractor. The scale must also be drawn. *(112) The distance between two towns is 19 English miles, and measures on a map 2-7". Draw (a) a scale by which single miles can be measured. Show 40 miles. (&) A com- parative scale to show 10 Austrian miles, if one Austrain mil© = 3-3312 English miles. (113) There are 69^ statute miles to a degree. Draw a scale of statute miles on the scale of 6" to a degree, and give the representative fraction. (114) Draw a diagonal scale of 120 feet to the inch, t» measure single feet. Show 700 feet. *(115) A given scale is one of Russian versts, and 1 verst = ■6628 EngHsh miles. Draw a comparative scale of English miles showing furlongs diagonally. What is the representative fraction ? *(116) Draw AB 37 feet long. From A draw (on the same side of AB) AC, AD, making the angles BAG,- BAD, 56° and 115° respectively. Make AD 46 feet long. From B draw a perpendicular to AB, cutting AC in C. Join CD. Write down lengths of BC and CD in feet, and the magnitude of the angles BCD, ADC. Scale 15 feet to the inch. Assume any two points, OP, outside AD, and from each of them draw lines making angles of 63° with AD. (117) If 60 Eussian versts are represented by 7 '5", and a verst=1167 yards, draw a comparative scale to read miles. Show 40 miles. ■"■(lis) On an Austrian map a distance of 2^ Austrian miles is represented by 1'15". Draw a scale of English miles for the map. Show 30 English miles. Figure your scale, give repre- sentative fraction, and show all necessary calculations. (One Austrian mile = 3'3312 EngHsh miles.) 170 GEOMETKICAL DRAWING. *(119) A line which measures 2^" on a plan represents 32| chains on the ground. Draw a scale of yards for the plan, showing 1500 yards; divide it to show distances of 50 yards. Show your calculations, figure the scale, and give representative fraction. *(120) Two men start from the same point A, to walk by two different routes to B, where the routes cross. One walks due north from A for the distance of 2300 yards and turns off to the right through an angle of 45°. The other walks due east from A for a distance of 3700 yards, and then due north to B. Draw a scale of yards having a representative fraction of 2 al-jjo-. Draw a plan of their walk to that scale, and write down the total distance traversed by each. *(121) On a map a distance of 3'27" represents 2 chains 78 links. Draw a diagonal scale by which distances of 5 links may be measured. Show 5 chains. Figure your scale, show calcu- lations, and give representative fraction. By means of the scale draw a line 3 chains 65 links long. (1 chain = 100 links = 66 feet.) (122) A French plan has a scale of decirrifetres, 10 to the inch, and a decimetre = "327 English feet. Make a comparative scale to read feet. Show 20 feet. *(123) Draw a scale of yards having a representative fraction •of •j-^'j-2, and by means of the scale draw a plan of a five-sided field ABODE from the following data :— &"cZes— AB = 85 .yards; BO = 130 yards; CD = 105 yards; DE = 170 yards; AE=135 yards. Diagonals— BD^ 17 5 yards; AD = 200 yards. Compute the area of the field in square yards. *(124) On a given map 325 chains are represented by 3|". (a.) Construct a plain scale of chains showing 400 chains, and divided to show distances of 10 chains. (&) Construct a comparative scale of yards for the same map, showing 3000 yards, and divided to show distances of 200 yards. *(125) Draw a scale of yards having a representative frac- tion of a 3^0. Draw a straight line AB representing 240 yards of this scale, and besect it in C. On the same side of AB con- EXAMPLES. 171 struct the angles ACD = 45°, ACE = 75°, and ACF= 120°. Make CD = 250 yards j CE=165 yards; CF = 285 yards, and join ADEFB. Reduce the figure to a triangle of equal area, having its apex at E, and its base in AB produced. *(126) Construct a plain scale of nautical miles, showing cables, and least divisions 10 fathoms. Eepresentative fraction li^ooo - (One nautical mile =10 cables =1000 fathoms = 6086 feet.) *(127) A man walks from a point A, 3^ miles in a straight line due north to B, when he turns to the right through an angle of 45° and walks in the new direction 1|- miles to C, where he again turns to the right through an angle of 75° and ■walks 2| miles to D. From D he walks due south for 2^ miles to E. Draw a scale of f mile to 1", and draw a plan to scale. How far is it from E to A ? A protractor is not to be ^ised. *(128) Two lines AB, 3", and CD, 3^", are drawn to scale, and represent distances of 1320 yards and 3^ miles respectively. ■Give the representative fraction of the scale to which each line is drawn. Show your calculations. *(129) A distance of 31 miles 5 furlongs is represented on a map by 11". Draw a diagonal scale of miles and furlongs for the map, showing 15 miles. Give the representative fraction. Draw a line 7 miles 5 furlongs in length by scale. *(130) Construct a scale of chords to read to 5°. (Eadius 4".) By means of the scale plot angles of 37° and 78°. *(131) Draw a scale of y-y^ to read mfetres. Show 20 metres. (1 mfetre=10 decimetres; one decimfetre = "327 ^nghsh feet.) *(132) Construct a diagonal scale of yIt *° show single jards. Length of scale 150 yards. Draw a line 77 yards in length. *(133) Given representative fraction = 975. Construct a •diagonal scale to measure 5 feet, and to show |^ths of an inch. Show by two small dots on the scale a distance of 3 feet i^". 172 GEOMETRICAL DRAWING. *(134) A Spanish plan is drawn to a scale of 10 palms to- the inch. Draw a scale showing English feet for the plan, and giving a length of 35 feet. Show your calculations, and give the representative fraction of the scale., (1 Spanish palm = '685 English feet.) *(135) Two men start from a point A to reach a point B by two diflferent routes, which leaving A unite at B. One of thfr men walks due north for 750 yards, and then turns off to the right through an angle of 55° ; the other man walks due east from A for 600 yards, then due north to B. First calculate and then draw a scale with a representative^ fraction of xs4^-g-- Draw a plan of the routes taken by the- two men, and write down the total distance each traverses. *(136) Construct a plain scale of metres, showuig altogether 3| kilometres. Least divisions 50 metres. Representative- fraction xwoTT- 0- kilometre = 1000 metres = "62 of a mile; 1 raile = 1760 yards.) *(137) The distance between two points on a map is 23"45". The actual distance is 24 miles 750 yards. Construct a diagonal scale for the map, showing miles, furlongs, and chains. Show by two small dots on the scale a distance of 2 miles 3 furlongs- 9 chains, and state the representative fraction. (1 chain = 22 yards.) *(138) An Enghsh map is drawn to a scale of 6" = 1 mile. Construct a scale for use with this map in Erench measure One mfetre = 39'37" (nearly). Give the representative frac- tion, and indicate on the scale a distance of 1470 metres. *(139) In a certain country 10 units of area = an acre.. Make a scale for use in this country which wUl read to tenths and hundredths by the diagonal method. The representative fraction is yts- Indicate by two marks on the scale 3"78 units- of length. *(140) Draw a scale whose representative fraction is -g-o^ry. Show furlongs, and represent 50 miles on the scale. Draw a;- line AB under the scale 7 furlongs in length. *(141) A mUe is represented on a map by 8". Draw a plain scale of chains for the map, showing 50 chains, and give the representative fraction. EXAMPLES. 173 *(142) Draw a diagonal scale of yards to read feet and inches, the representative fraction being -j-^. ' Show 10 yards. *(143) A vertical pole, 100 feet high, casts a shadow on the ■ground, the sun's rays being incUned to the horizontal at an angle of 38°. If the pole is represented by a line 3" long, con- struct a scale to measure the shadow, and find its length in feet T)y means of the scale. *(144) Construct a scale of yards, to read feet and inches. Eepresentative fraction ^r. Show, by marks upon your scale, a length of 1 yard 2 feet 7". *(145) Construct a diagonal scale of miles, furlongs, and ■chains. Take If" to represent 8 furlongs. Give the repre- sentative fraction. Make the scale to read 4 miles. Show by two marks upon your scale the points you would take to measure oif a length of 2 miles 1 furlong 7 chains. *(146) The distance between two towns is 27^ nules, and measures on a map 3"25". Construct a diagonal scale of miles and furlongs for the map, showing 50 miles. Give the calcula- tions and representative fraction. By means of the scale draw a line 23 miles 5 furlongs long. *(147) On a plan 1250 yards are represented by 15'5". Draw a comparative scale of French metres for the plan, show- ing 500 metres, and divided to show distances of 10 metres. <1 metre = 1-0936 yards.) *(148) A map is drawn to a scale of 6" to an English mile. Draw a plain scale of Spanish yards for the map, showing 2000 Spanish yards, and divided to show distances of 50 Spanish yards. Show all your calculations, figure your scale properly, and write above it its representative fraction. (1 Spanish yard = -9277 English yard.) *(149) A man starts from a point A and walks 3 miles 1 furlong to the right in the direction of B. He then turns to his right at right angles and walks f of a mile. He then turns 60° to his left and walks 3 miles 5 furlongs. He then turns back 130° to his right and walks 4^ miles. He then turns 70° to his right and walks 2 miles 3 furlongs, arriving at C. Plot 174 GEOMETRICAL DRAWING. his journey, and measure and write down the distance from C to A. A scale must be drawn f of an inch to a mile. *(150) Construct a diagonal scale showing Russian sagenes and feet. Make the scale to show 100 sagenes. Mark by two- dots on the scale a length of 47 sagenes 3 feet. Eepresentative fraction ytstj- (-^ Eussian foot is equal to an English foot ; 1 sagene = 7 feet.) GEOMETRICAL PATTEEIT DRAWING. 175 GEOMETEICAL PATTERN DRAWING-. The student should now be able to combine the figures he has learned to draw, so as to form geometrical patterns. To ensure accuracy, care must be taken to make a framework of the Hues of construction. It is important always to draw centre lines of construction wherever needed. The student should draw them not only on the same scale as that given, but twice or three times as large, or in any other proportion which may appear suitable. All pattern drawings should be inked in. (1) First, dra-w the horizontal and vertical lines of construction in squares ; afterwards darken the portions of the lines which form the pattern. Fig. 1. (2) Copy this Greek fret to any scale required. Proceed as in Fig. 1. IlG. 2. 176 GEOMETRICAL DKAWING. (3) Draw the given figure full size, adhering strictly to the figured dimensions. It is based on an equilateral triangle. Fig. 3. (4) In the given figure the five points of the star are situated at the angles of a regular pentagon. Draw the figure from the dimensions attached. Fig. i. GEOMETKICAL PATTERN DKAWING. 177 Fiyst obtain the five points of the star by construction within the circle of the required diameter. Join the alternate points, and proceed as in the given figure. (5) To draw a foiled figure about a given regular polygon, — say a cuspidate cinquefoil about a regular pentagon. Fig. 5. Take the corners of the pentagon as centres, and with a radius equal to half the side of the pentagon describe arcs. These will form the foiled figure required. Note. — When two circles pass through a common point in the line joining their centres they touch at that point. The figure thus constructed will be cuspidate. (6) Draw the three squares as shown. (Fig. 6, p. 178.) With A and B as centres describe arcs meeting the inner square, with radius AC. The quatrefoil in the centre is drawn round the small square. M 178 GEOMETRICAL DRAWING. Fig. 6. (7) Draw the geometrical pattern shown, making it half as large again as the copy. Fig. 7. First, draw the parallel Hnes for the sides, also a centre line. The arcs are cuspidate trefoils about equilateral triangles. (8) The following figure is based on the construction of regular hexagons. First, draw seven parallel lines at equal distances; then construct the large hexagons on the bottom line. The rest of the work will be seen from the figure. GEOMETKICAL PATTERN DKAWING. 179 Fig. 8. (9) Within a given hexagon draw a hexafoil of semi- circles to -which the sides of the hexagon shall be tangents. On the diameter of each semi- circle as a diagonal construct a square. Fro. 9. Draw lines from the corners of the hexagon to the centre, also lines from the centre bisecting the sides of the hexagon. (10) The following figure is to be drawn to twice the scale. 180 GEOMETBICAL DEAWING. The construction lines as shown are sufficient for the work- ing of the figure. Fig. 10. (11) The given figure represents a windo"w of the GEOMETKICAL PATTERN DKAWING. 181 English decorated style. The construction is suffloiently shown and dimensions are given. Draw the window to a scale of IJ" to 1 foot. (12) In a circle of 1^" radius draw the figure shown. Fig. 12. (13) Draw a figure similar to the given one, but half as large again. Fig. 13. 182 GEOMETKICAL DKAWING. (14) Draw the given figure full size from the dimen- sions given in the accompanying diagram. Fig. 14. (15) Draw the given figure fuU size, adhering strictly to the figured dimensions. Fig. 15. GEOMETRICAL PATTERN DRAWING. (16) Draw the given figure twice the size. ■183 Fig. 16. <17) Copy as accurately as you can the accompanying figure. AH the curves are portions of circles. Fig. 17. 184 GBOMETEICAL DRAWING. (18) Copy accurately the figure given below. Fis. 18. GKAPHIC AEITHMETIC. 185 GEAPHIC AEITHMETIC. Graphic Aeithmetic is the employment of lines representing numbers. Lines are the results, and represent the required product, dividend, &o., on the same scale of units as the original lines. The results are obtained by means of geometrical proportions, instead of the ordinary process of Arithmetic. A line may represent any number of units. For instance, a line 2" long may represent 20 units : therefore ^ of the line represents 1 unit, -^ of the line 2 units, ^ of the line 5 units, J of the line 10 units, f of the line 15 units ; or any fraction of the unit may be shown. (1) If a line 2" long represents 5, what is the unit ? Divide the Une into 5 equal parts. Each part thus obtained is the unit. (2) A line 2" long represents 5. Produce it to re- present 8. Divide the line 2" into 5 equal parts. Produce the line and add 3 of the equal parts. The whole line represents 8, or 3 + 5.. (3) Show f when the unit is 2". Divide the unit into 5 equal parts. 3 of these parts represent I of 2" the unit, or|-f = |. (4) The unit being |", represent 5 + 2-3 + 1-4. Draw an indefinite line, and from one end A measure 5 units AB. From B add 2 units EC. From C set back 3 units CD. Add 1 unit DB. From B set back 4 units BE. AE will be found to measure 1 unit or |" for 5 + 2 - 3 + 1 - 4 = 1. 186 GEOMETKICAL DRAWING. <5) Find f when the unit is 1". To divide one line by another the quotient is a fourth pro- portional to the divisor the dividend and the unit. (6) To determine the product of any number of Unes, a X 6 X c, a = 2", 6 = If, c = |". Unit 1". Take any line AB, and set off from A a length Al, equal to the given unit, — in this case 1", — and erect a perpendicular IC. Set off distances on IC from 1, equal to the lengths - oi given lines, as la = a, 16 = 6, &c. Draw lines from y^ 2 Fig. 1. 3 B A indefinitely through these points. Along AB set off A2 = 6, and raise a perpeindicular 2a!, cutting AD in x. .". 2a; = ax6. From A set off 1x on AB, as A3, and at 3 erect a perpen- GKAPHIC ARITHMETIC. 187 dicular meeting AE in y ; 3y = 2xxc = axbxc. The line 3y measured from the same scale of units as the original line is the product required. axbxc — 3y, or2xl|x^= 1-|". Note. — If lines are substituted instead of numbers, the square or any power of a Hne may be found by construction. (7) Find the square of a line IJ" long when the unit is 1". A third proportional to the unit and the line is the square ■of the line. (8) Find the cube of a line 1^" long. Unit 1". First find the square of the line as in the last problem. A third proportional to the line and the square of the line is the oube of the line. <9) Find the fourth power of a Une li" long. Unit 1". Obtain the square and the cube of the line by Problems 7 and 8. A third proportional to the square and cube of the line is the fourth power of the line. <10) Find the fifth power of a line H" long. Unit 1". A third proportiona,l to the cube and the fourth power of the line is the fifth power of the line. Note. — The square or any power of a fraction may be found in a similar manner. The square root of any number is the mean proportional between that number and unity. (11) A line CD represents 9. Obtain by construction a line which will represent the square root of CD. Divide CD into 9 equal parts, and find a mean propor- tional DE to CD and ^ of CD. DE is the squai'e root required. 188 GEOMETKICAL DRAWING. (12) Division. — To divide one line by another, the dividend is a fourth proportional x to the two lines and the given unit. (13) Addition of Fractions. Reduced to a common denominator = Obtain the value of Jjj + f + 1. 3 + 20 + 6 30 Draw two lines at right angles to each other, as BAG, and make AD = to th^ common denominator ; and on the same line set off Aj, Ag, Ag, equal to the respective denominators 10, 3, and 5. Along AB set off An, An', equal to the numerators. Join the point n on AB with 1 and 3 on AC, also n' and 2. Draw lines from D parallel to n'2, w3 and nl meeting AB Am + Am' + Am« m m, m , m^. TV + t + 30 If X be the sum of the lines Am, Am', Am^, a fourth pro- portional to the denominator x and the given unit will be a line representing the sum of the given fractions. GRAPHIC ARITHMETIC. 189 (14) If A (a line 1" long) represents Jl, what is the unit? Fig. 3. Find iJ7 from any scale of units, as BC. From B on BG continued set off BD equal to A. Draw a line from D parallel to CE, meeting BE continued in F. BF is the unit required. (15) Find by construction the quantities —j-, Jj^ the unit being a length of 2". /3~ /30 Simplified are -^j -^!^' and, further simplified, J of ,^3, ^ of ^30. J of ^3, the unit being 2", is found by describing a circle of 2" radius, and proceed as in Problem 131 page 70. ^ of ^30, the unit being 2". The mean proportional between a line 10" and ^ of 10" is the length of v'fT (16) Determine by construction the line — «= Va2-&2 + c2, a=li", e=ll", and 6 = f". Draw a and c at right angles to each other and equal to the given lengths, and on Ja^ + ^2 describe a semicircle; set off 190 GEOMETRICAL DRAWING. the length of h within the semicircle, from the extremity of a to point M. Draw a line from M to the extremity of c, which is X, the line required. (17) Obtain by construction a line which represents A + 2^f. Unit 0-5". 4 _ 4/3 4 /6~ 4 5 + 2 Vf simplified is | + 2^=g + 2 x^ = g+ ^6. Find a line ^ of '5". Also ^6, which is a mean proportional to six times '5" and "5" — i.e. 3" and "5". A line equal to the sum of their lengths is the one required. 2a (18) Find by construction x= —^, a being 2" in length. 3 X 7 Fig. 5^ (5 Scale.) vr GRAPHIC ARITHMETIC. 191 Draw lines 1, 2 and 1, 3 at right angles to each other, 1, 2 equal to the length of ^3, and 1, 3 equal to 2". Join points 2, 3, and from 3 draw a Une 3, 4 at right angles to 2, 3, meet- ing 1, 2 produced in 4. The line 1, 4 = a;. (19) Determine a line whose length shall represent J^, taking |" as the unit. Draw a line AE equal in length to the unit, and produce it to B, so that EB shall represent 3^. A mean proportional ED between AE and EB is the square root of BE or ^|. (20) If a given line AB, 1^" in length, represents ^, determine the unit. Divide AB into 7 equal parts. 4 of these parts represent the unit. (21) If the given line A represents the unit, what numbers do the lines B, C and D severally 192 GEOMETRICAL DRAWING. represent ? Determine also a line representing B CxD- Draw two lines at any angle, and mark off on one of them distances equal to A, B, C and D. From point on the other line set out i" from 0, and join point A. Draw parallels to A i" through C, D and B. The lengths OC, OD', OB', when referred to an ordinary scale of ^", will show the numbers represented by the given lines. Draw AB equal to the unit, and at B set out BC perpen- dicular and equal to given line C. Draw AC indefinitely. Produce AB indefinitely towards G. Measure AD equal to D, and draw DF perpendicular to meet AC in F. DF repre- sents CxD. Make DE equal to the unit, and DG equal to B. Join FG, and draw EH parallel. DH = _^ (22) If A represents the product of B and C, what length of line will represent the product of A and C, To find the unit draw DE and EF at right angles. Make DE equal to B and EF equal to A. Draw a line from D GRAPHIC ARITHMETIC, 193 through point F. Make EG equal to C and draw GH, HK parallel to DE and EF respectively. DK is the unit. Find the product of A and C when DK is the unit. Make DL equal to A. A perpendicular to DL from L, to meet DF produced in M, gives LM, the product of A and C. (23) If the area of an equilateral triangle of 2^" side is represented by 3 J", what is the unit? Draw a rectangle ABCD equal in area to the triangle : there- fore AB X AD is the area of the figure. Find a imit so that AB X AD shall he a line SJ" long. At B draw BE perpendicular to AB and equal to 3^". A line joining A and E meets CD in F. Draw a line from F parallel to AD, meeting AB in point G. AG is the luiit required. (24) N and M are two lines. Determine J^, the unit being 0-5". (iV=lf" and M=4" in length.) A fourth proportional x to the two lines N and M and the given unit is the quotient. The root of the quotient measured in units is found by Problem 131, page 70. Thus J'^ is determined. The value of a; in the following equations may be determined in the same manner, 1" being the unit. ^ ,2 ,10 /78 ^ /V3 (25) Find the angle between two lines whose equa- tions are x + y = \. y = x + 2. Draw two lines at right angles to each other, as X and Y. a; + y=l. .-. from a scale of units set off 1 unit to the right of O on X as m, and 1 unit above on Y as m'. Draw a hne through m and m'. For the second equation set off 2 units from O to the left on X as n, and 2 units above on Y in n'. A line drawn through n, n' meets the line drawn through tn, m' in A. The angle at A is the one required. 194 GEOMETRICAL DKAWING. Fia. 8. (26) Find the length of the perpendicular from point 1, - 2, on the line x + y-4 = 0. Via. Q. GRAPHIC ARITHMETIC. 195 Draw two lines at right angles to each other representing X and Y, and meeting in point O. From any scale of units set off + 4 on hne X, as OA, and 4 units on Y, as OB. The line drawn through AB is a; + ?/ - 4 = 0. From the same scale of units set off from O on X 1 unit, and from on Y 2 units below. Lines drawn from 1 and 2 on X and Y, parallel to X and Y, meeting in C, give the point 1,-2. The perpendicular CD to AB measured from the same scale of units is the length required. (27) Determine the locus of the curve whose equa« tion is : (2/ - If + (a; - J^f = ^2. Unit 1". Fig. 10. ftbs Scale.) 196 GEOMETRICAL DRAWING. Assume a point A for origin, draw lines Ax, Ay at right angles, set off AB = 727 which is 1-414, and AC = 2". Draw BD parallel to AC, and CD parallel to^B. From D as centre, and radius 1^, describe a circle ( ^2 is found by taking a mean proportional between ^2" and 1"). If point E be taken in the circumference representing oey, the line EF drawn parallel to Ay, meeting CD produced in F, it may be seen that EF = 3/-2,_and DF = a;- ^/2, and that (EF)2 + (DF)2 = (DE)2, or {x - J2Y + {y- 2f = J2. (28) Draw the curve whose equation is : y^ + i)i?-6y+- 4a; -3 = 0. Unit -5". Jio. 11. Simplified we have- y^-6y+9 + x^ + ix + 4: = 3 + i + 9; .: {y-3y+{x+2f = lQ. What is required is a circle having a radius equal to i units. Draw two lines Ax, Ay at right angles to represent x and y, set off from A on Ax to the left 2 units, and set off from A GRAPHIC ARITHMETIC. 197 on Ay 3 units. Draw a line from 3 parallel to Ax, and a line from 2 on Ax, parallel to Ay, meeting in 0, the centre of the ■circle required. The radius is equal to 4 units. Add 1 unit to line 02 below Ax. This hne is the radius required. <29) Determine the intersection of the circle y^+x^ = 16 with the lines y + x = \, y + x= -3. Draw two lines at right angles to each other representing X and Y, and meeting in point O. With as centre, and radius equal to 4 units, from any scale of units describe the circle. From on X and T set off 1 unit as OB, OA. A line drawn through A and '&isy + x=\. From on X and T set off 3 units as OC, OD. A line drawn through C and D is 2/ + a; = - 3. The lines intersect the «ircle in points 1, 2, 3, and 4. 198 GEOMETEICAL DRAWING. (30) Find the angle between the lines x + y ^3 = and x-y JB = 2. Draw two lines at right angles to each other representing X and Y, and meeting in point O. From any scale of units set off on X to the left of O ^37 and on T above X 1 unit as 01. Complete the Fie. 13. parallelogram, and draw the diagonal OC. The line 0C = a; + y>/r=0. From the same scale of units set off 2 units to the right of O on X as A, and AB = J^. BD = 1 unit. Complete the parallelogram ABDE, and draw the diagonal AD. The line AD is a; -2/ ^3 =2. The lines OC and AD intersect in G. The angle AGO is the angle required, and is one of 120°. GRAPHIC ARITHMETIC. 199 (31) Draw lines represented by the equation The equation simplified is {y - 3a;) {y -x) = 0. Draw two lines at right angles to each other representing X and Y, and meeting in point 0. From any scale of units set off on Y above 3 units as OA, and from on X 1 unit as OB. Complete the parallelogram ACB. The diagonal 00 is y - 3a; = 0. From the same scale of units set off on X OB equal to - 1 unit, and on Y OD equal to 1 unit. Complete the parallelogram ODEB. The diagonal OE equals y-x = Q. Note. — A parabola gives the square of all quantities. (32) Draw a curve which will give the square of all quantities fi:om O to 6, taking unit. 1 " as the 200 GEOMETRICAL DRAWING. Draw any straight line gl, and mark upon it a starting point 0. Set off six equal distances to the left of this point, each 1 unit in length, as a,e, d,, e, f, g, and any other six equal distances to the right, as 1, 2, 3, 4, 5, 6. At each of these points draw lines at right angles to gh. Set off one unit •^" op below gh. Join points p and g, and draw gG at right angles to gp, meeting the perpendicular through o in G. Join pc, and Fig. 15. draw a line cC at right angles to cp. Points BDEF are found in a similar manner. Horizontal lines drawn through these points will meet the vertical lines drawn through points 1, 2, 3, &c., in points on the required curve, which is a parabola. We may apply the problem by finding the square of 2|-. Take a point along oh to correspond to 2J of the distances set off from to the right. A vertical line through the point will meet the curve ; its length will measure (2|)2 or 6J, i.e., 6J times the given unit -j^". (33) Construct the curve whose equation is y^=2a;- 1. Unit 1". The equation simplified is y^ = 2{x - 1^). The curve there- fore is a parabola. GRA.PHIC AEITHMETIC. 201 Draw the axis AB. The focus E will measure from A the unit 1". Through A draw the directrix CG at right angles to AB. The vertex D is half a unit from the focus, measmed ■on the axis. Draw EF at right angles to AB. EF equals Fig. 16. the unit 1". The curve is completed by Problem 195 in -Practical Geometry. Note. — The area of any rectilineal figure may be represented by a Hne. To obtain this Une the figure must be reduced to a triangle of equal area. (34) To represent by a line the area of the given triangle ABC. Take any angle of the triangle ABC as centre (in this case B), and describe an arc with radius equal to 2 units of the scale to which the given figure is drawn. Draw a tangent from A to this arc, from C draw a line parallel to AB meeting the tangent in D. The line DA represents the area of the figure 202 GEOMETRICAL DRAWING. measured on the same scale of units to which the origiaal figure is drawn. This process is called the reduction of a triangle to a given Fio. 17. base, the base in this case being 2 units. Eeduction of areas, to a given base is necessary for finding the centres of gravity of plane figures. (35) To find a line to represent the area of an7 rectilineal figure. First reduce the figure to a triangle of equal area, and pro- ceed as in the last problem. GRAPHIC STATICS. 205 GEAPHIC STATICS. (1) To find the resioltant of two velocities along the same line. The resultant of two or more forces acting in the same^ direction is the sum of those forces. If in opposite directions,, the difference is the resultant. (2) To find the result of two velocities in diflferent directions. This is found by what is known, as the parallelogram of forces. From point draw one of 'the forces OA, representing the magnitude and direction of the force, and from A draw AC, representing the other force in magnitude and direction. Complete the parallelogram OACB. Join OC. OC represents, the resultant. (3) Resolve each of the given forces P and Q (P = 7 lbs., Q = 9 lbs.) along and perpendicular to 204 GEOMETRICAL DRAWING. the given line AB, and write resultant force in each direction. down the From any scale of units set off QR equal to 9 units, and from the same scale set off PS equal to 7 units. Complete the rectangles of which QE and PS are the diagonals. Fig. 2. ■Continue two sides of the rectangles to meet AB in points 1, 2 and 3, 4, and the other two sides to meet a line XT drawn perpendicular to AB in points 5, 6 and 7, 8. The force perpendicular to AB is the sum of the lines 5, 6 and 7, 8 measured from the scale of units. The force acting along AB is the line 1, 2 forcing to the left — the line 3, 4 forcing to the right, measured from the scale of units. GRAPHIC STATICS. 205 (4) To find the position a weight free to move will take on a cord. Fig. 3. A and B are the points of suspension. Set oif AC equal to the length of the cord ; bisect BC in D. Draw a horizontal Une from D meeting AC in E. Point E is the required position of the weight. (5) A sphere O, of a given weight, is placed on a. Fig. 4. 206 GEOMETKICAL DBAWING. smooth inclined plane AB, and is sustained by a force P which acts along AB in the vertical plane, which is at right angles to AB. Find F, The effect of the inclined plane is to produce a reaction E, at right angles to the plane, on the sphere. If we introduce this force, the plane we may imagine removed^ Therefore draw a parallelogram CWSR, having its sides parallel to the direction of E and W. The diagonal SC is the direction and magnitude of the force F required. (6) The direction and magnitude in lbs. of five forces, acting at one point A, are given. Determine the direction and magnitude of their resiiltant. J9 '» Fig, 5, Draw a polygon of forces — i.e., lines parallel to the directions of the forces, and equal in length to as many units as magnitude GRAPHIC STATICS. 207 in lbs. The line joining the end of the first direction of forces and the end of the last is their resultant, and the direction of this force is from the starting point a to the finishing point /. The magnitude of the force is the number of units the resultant xieasures. Note. — If all the forces act in parallel directions, then the polygon of forces is a straight line ; and in the case of roof and bridge trusses, this polygon is spoken of as the Line of Loads. (7) Draw six lines ao, ho, co, do, eo, fo, radiating from a point 0, any two adjacent lines including an angle of 60°. These six lines are respectively the lines of action of forces of 80, 100, 90, 60, 120, and 50 lbs. all acting towards o except those along od and of, which act away fi-om o. Determine and write down the magnitude and direction of the result- ant of the forces. Fig. 6. ao and od act in the same straight line, and are therefore equal to a single force acting from o towards tZ of 80 + 60 = 140 lbs. oe and of are equal to 50 + 90 = 140 lbs. acting from o towards /. oh acts in the opposite direction to oe, and are equal to 120 - 100 = 20 lbs. acting from o towards h. 208 GEOMETRICAL DRAWING. The resultant of the two forces is 140 lbs. acting from a through e at 60°, and forms an equilateral triangle ofe. Therefore the resultant, allowing for the 20 lbs., is 120 lbs. acting from o to e. In the drawing any unit may be taken to represent 10 lbs. The resultant of any number of forces acting in a plane, and if their directions do not pass through one point, a funicular polygon may be employed to find the resultant. A funicular polygon is important in the subject of Statics, and is so called because it is the form a cord would assume when subjected ta the forces shown. (8) Pour forces AB, BC, CD, DB act in a body. To find the resultant. Fig. 7. The polygon of forces will give the magnitude and the direc- tion of the resultant, and the funicular polygon a point on its line of action. First draw the polygon of forces, which gives the resultant ae in inagnitude and direction. Assume any point o as pole, and join oa, 6b, &c. Draw lines in the spaces B, &c., as lines 1, 2 2, 3 &c., parallel to ob, oe, &c. From point 4 draw 4F parallel to oe. Point F is on the line of action of the resultant forces. 1, 2, 3, 4 is the funicular polygon. GRAPHIC STATICS. 209 Note. — The process of finding a number of forces which shall have a given force for their resultant is called, Resolution of a force into its components. (9) Given a force AB, resolve this force into two components in the directions x and y. Fig. 8. From A and B draw lines parallel to x and y meeting in C. AG and CB represent the components of AB. (See Problem 2.) (10) A given force AB may be resolved into two components along two parallel lines of action X and y. c X Fie. 9. If the force AB be reversed, this force and the components of AB in the original direction will be in equilibrium. Draw 210 GBOMETKICAL DRAWING. the polygon of the forces, ha representing the original force reversed, and complete the polygon. Draw also a f unicTilar polygon, so that the sides in the spaces A and B are parallel to oa and oh, and meeting the lines of action in points C and D, Draw oc parallel to CD. Then ae, ch represent the components along x and y. (11) A uniform rod AB, weighing 53 lbs., is pivoted at A. If a force P of 32 lbs, is applied at O, where must a parallel force of 41 lbs. be applied to maintain equilibrium? > 1 f Si f \ *- " \ / 4 A\ i € o / D Fig. 10. Four forces keep the bar in equihbrium — the weight of the har acting downwards at the centre 0; the given force P acting downwards at C; the reaction at the pivot A act- ing upwards ; and 41 pounds acting upwards at an unknown point D. Draw the polygon of forces to any scale of units — ah 32 lbs. or units, 6e 53 units, ct? 41 units. Complete the funicular polygon 1, 2, 3, 4. Point 4 is the line of action of the weight 41 lbs. A perpendicular through point 4 meets the rod AB in point D required. GRAPHIC STATICS. 211 (12) Resolve the given force P, 240 lbs., along the given lines aa, hh. Write down the magni- tudes of the resolved components, and indicate their directions by arrows. Fie. 11. Take any scale of units and draw cd parallel to P, and equal to 240 units. Complete the triangle cde, de parallel to hh, and ce parallel to aa. Reverse the direction of the force P on ed, and f oUow the triangle round as shown. The arrow-heads show the direction of the two components, de and ec measured from the same scale of units will give the magnitudes of the forces ' C , -j>^ ' E b c f mm a r — 4 w e \ ^ ^ K Fig. 13. The loads are at AB, BO, &c., EF and FA the reactions. Draw the polygon of forces, also the funicular polygon in the spaces A, B, &c., parallel to oa, &c., and join GH. JDraw of GKAPHIC STATICS. 213 parallel to GH, then ef and fa are the reactions on the right and left walls. Note. — The moment of a force about a point is equal to double the area of the triangle whose base represents the magnitude of the force, and whose height is equal to the perpendicular dis- tance of the point from, the direction of the force. (15) To represent the moment of a force about a given point. Fig. 14. P represents the force about point 0. Draw ab parallel to P, and representing the force to any scale of units. Let o be a point taken at a unit distance from ah ; draw oa and oh. Take any point M on the line of action P, and draw ME" and MQ parallel respectively to oa and oh. From O draw a line KQ parallel to P. The length of NQ represents the moment of P about 0. The line NQ measured in units on the scale adopted is the moment required. .•. 2/xa;'=2! = P X lo. (16) To find the resultant moment about a gi-«-en point P of any system of parallel forces, AB, BO, CD, DE. Draw the polygon of forces, ae, representing the resultant in magnitude and direction. Also draw the funicular polygon 1, 2, 3, 4, 5. The sides 4, 5 and 1, 5 intersect in point 5 ; a point 214 . GEOMETRICAL DRAWING. in the line of action of the resultant ae. A line drawn through P -a e Fig. 15. P parallel to ae meeting lines 1, 5 and 4, 5 in points x and y. Then xyxw ia the required moment. In practice should be taken, so that the product iv ■>< xy may be easily determined. Some multiple of 5 will be found the most convenient length of the polar distance. If O be taken at unit distance from ae then w y. xy = xy, and the required moment will be found by measuring xy. Note. — The moments of a system of forces can be represented by a series of rectangles ; and if reduced to equivalent rectangles all of which have equal bases, a scale can be constructed from which the heights of the rectangles read off, give the value of the several moments according to the scale of units adopted. (17) Eeduction of moments to a common base. In the figure a^ Pi, a^ P^ . . . . represent the magnitude, direction, and position of forces whose moments about any point in their plane are to be reduced to a common base. Suppose that the common base to which the moments are to be reduced is H. Draw any line XY through 0, and from a^, a^, &c., draw lines a^^, a%^, &c., parallel to XY, and equal to the given base H. Join Jipi, b'^'2'^, &c. Through draw Oci, Oc2, &c., paraUel to 6ipi, W^'^, &c., and meeting aipi, V'2\ &c., produced in c^, (?, &c. From c^, c^, &c., drop perpendiculars GRAPHIC STATICS. 215 cW, c^dP, &c., on the line XY. These perpendiculars, c^d?- = in}, cH^ = m^, &c., are the.requrred heights of the rectangles, having H as the base. Fig. 16. The scale of force is 100 lbs. to 1". The linear scale is 80 feet to 1". The scale of moments must be so drawn that y units on the scale of force equals i/ x 100 units on the scale of moments; thus on the latter scale 1" represents 1000 foot lbs. I 'exile of Forcer 100 wo 300, Scale oF MoTwents 1000 woo aooo Via. 17. (18) To find the resviltant moment about a given point P of any system of forces AB, BO, CD, DB. 216 GEOMETKICAL DEAWING. Draw the polygon of forces, ae representing the resultant in magnitude and direction. Also draw the funicular polygon 1, 2, 3, 4, 5. The sides 4, 5 and 1, 5 intersect in point 5 ; a point in the line of action of the resultant ae. A Una drawn through P parallel to ae, meeting lines 1, 5 and 4, 5 in points X and y. Then xy>iw is the required moment. A couple consists of two concurrent parallel forces which act in the same direction, or two parallel forces which act in opposite directions, which are said to be non-concurrent. If the forces forming a couple were in the same straight line they would Be in equilibrium. The two forces applied by the thumb and fingers to the handle of a common tap form a couple. The twist appHed to a screw-driver without pressure is a couple. A couple tends to cause rotation ; and when in the direction of the hands of a watch, the couple is called negative, when in the opposite direction it is called positive. The arm of a couple is the perpendicular distance between the lines of action of the forces. The moment of a couple is equal to one of the forces multiplied by the arm of the couple. Couples may be solved graphically by the funicular polygon, the polygon of forces, and Graphic Arithmetic. GKAPHIC STATICS. 217 (19) Weights are suspended from the given points along a weightless rod, as AB, BO, CD, DB, BP, FG. To determine the point about which the rod will balance. Fig. 19. Draw the polygon of forces, also the funicular polygon 1, 2, 3, 4, 5, 6, 7. The sides 1, 7 and 6, 7 intersect in point 7, a point in the line of action of the resultant. A line drawn from 7 parallel to the hne of loads meets the rod in point 0, about which the loaded rod will balance. (20) Let the vertices of a string or chain be denoted by the ntmibers 1, 2, 3, 4, 5 and 6, and let the forces Pg, &c., act at the vertices. The tension on the portion of the string 1, 2 being known, to find the tension on the remaining parts. Take a point 0, and draw from it a line A parallel to the string (1, 2) and proportional to the tension T12 (that is, draw from a line parallel to 1, 2, and 12 units in length). From the end of A draw the line B parallel to Pj, and a line C from O parallel to the portion of the string 2, 3, meeting the line B. The line C measured in units is the tension on the portion of the string 2, 3. By completing the 218 GEOMETKICAL DRAWING. polygon of forces, the tension on the successive portions of tha string may be found, o Fig. 20. (21) The following skeleton diagram shows a wharf crane, carrying a weight W, 6 tons. Show by B Fig. 21. GRAPHIC STATICS. 219' construction the different stresses on the tension rod AB, and the jib BC. At the jib-head B three forces meet and are in equilibrium — the downward pull of the weight, the resistance of the tension rod, and the thrust of the jib. Therefore draw a line ab 6 units in length parallel to the downward pull of the weight. The length of the other sides of the triangle ah, bo, will represent the tension upon the tie-rod and the compression upon the jib. (The sides of the triangle are drawn parallel to- AB and CB.) (22) In the given figure BO represents the position of a pole and AO a tension guy. Show by con- struction the method of ascertaining the strains upon the pole and the guy, W being 6 tons. © Fig. 22. The tension rod in this case is caUed a " guy," and is generally made of rope. The holdfast of the guy is a strong picket driven into the groimd, and the pole is supported by a, foot plate, placed upon timber at B. Draw the vertical line XY 6 units in length to represent "W. Draw YZ parallel to BC, the sheer leg, meeting a line drawn from X parallel to the guy AC. The length of XZ measured to the same scale" of units gives the tension on the guy AC, and YZ measured in units the thrust on the sheer leg. 220 GEOMETKICAL DRAWING. <23) If a load of 2 tons acts on the ridge of a roof truss, to find the stresses on its members and to determine which are in compression and tension. Fig. 23, Draw ah 2 units in length to represent 2 tons. The roof is evenly loaded, the reactions are equal. Bisect ab in c ; be, ca represent the reactions BC and CA. Draw ce parallel to CE, he parallel to BE. bee is the triangle of forces for half the truss. be and ce measured from the same scale of units will give the magnitude of the stresses. The triangle acd in the same manner will give the other half the truss. Notice carefully the direction of the Hnes in the triangles of forces — be upwards .'. BE in compression; ec downwards .". CE in tension; and so on clockwise round the truss. <24) Equal weights WW, each 500 lbs., are sus- pended from the points a and 6 of the given roof truss. The truss is otherwise unloaded. Determine by construction and write down the stresses in the several bars. The weight distributed on each wall is 500 lbs. plus the GEAPHIC STATICS. 221 weight of the truss. From the question the weight of the truss need not be considered. The whole structure is kept in Fig. 24. equilibrium by four exterior forces : the reactions on the wall M and Q, and the weights Wj and Wg. These four exterior forces are balanced by the interior forces Fig. 25. of compression and tension, the amounts of which can be found graphically. If one force be known, and the lines of action of another two, and all these acting at a point in one plane, then the nature 222 GEOMETRICAL DRAWING. and amounts of these forces may be found by the triangle of forces. Therefore, draw a vertical line AS, to represent the vertical thrust on the wall, — in this case 50G units. A hne ST drawn from S parallel to the rafter X, meeting a Hne AT drawn at right angles to AS, measured in units, is the oblique thrust upon the rafter X. A line drawn from T parallel to the tie-rod Y, meeting the horizontal line drawn through S, measured in units, represents the oblique tension on the tie-rod Y. Complete the parallelogram ZTAS. SZ, measured in units, represents the horizontal tension on the parts a S,bS' of the tie-beam SS'. (25) To find the centre of gravity of a triangle ABO. Eisect AC in E, join BE ; the centre of gravity lies in the line BE. Bisect EC in D, join AD ; the centre of gravity lies in the line AD. The lines AD, BE intersect in G, the centre of gravity of the triangle. EG measures one third of BE, DG one third of AD. (26) To find the centre of gravity of any quad- rilateral ABOD. Draw the diagonal BD, and by the last problem find the centres of gravity 1 and 2 of the triangles ABD and BCD. Joia 1 and 2 ; the centre of gravity of the quadrilateral lies in this Une. Draw the diagonal AC, and find the centres of gravity 3 and 4 of the triangles ABC and ACD ; join points 3 and 4. The hnes 1, 2 and 3, 4 intersect in point G, the centre' of gravity required. Note. — The centre of gravity of a uniform straight line is its middle point. (27) To find the centre of gravity of a broken line AB, BO, OD, DB. The weight of the different parts AB, BC, &c., act through the middle points of the lines and in proportion to their lengths. Draw the polygon of forces and take any pole o. ae GRAPHIC STATICS. 223 is the direction and half the magnitude of the resultant. Draw the funicular polygon 1, 2, 3, 4, 5, point 5 being on the line of action of the resultant. Draw 5G parallel to ae. By Fie. 26, placing the polygon of forces in another position, say at right angles to the first, we may obtain a second funicular polygon. This polygon is not shown in the figure, to save confusion; but the direction is shown, which passes through the point of action of the second funicular polygon. The resultants intersect in G, the centre of gravity. (28) To find the centre of gravity of a broken line AB, BO, CD, the lines forming a part 6f a regular polygon. If AD be joined and bisected by EF at right angles, EF is an axis of symmetry, and will contain G, the centre of gravity. Proceed as in the last problem. Only one funicular polygon will be required. (29) To find the centre of gravity of an arc ABO. If AC be joined and bisfected by OB at right angles, OB is an axis of symmetry, and will contain the centre of gravity. 224 GEOMETEICAL DKAWING. Draw a tangent BD at B, equal in length to the arc AB. Draw a line from D to the centre of the arc. Draw a line through A parallel to OB, meeting DO in point E. A line from E drawn parallel to AC meets OB in point G, the centre of, gravity. (30) To find the centre of gravity of a sector AOO. If we consider the sector AOO (see last problem) consisting of a number of triangles, the line AO may .he considered as one, — its centre of gravity being F, f of AO, measured from 0. An arc described with as centre, and radius OF, contains all the centres of gravity of all the triangles in the sector. The centre of gravity G of this arc is the centre of gravity of the sector AOC. (31) Determine by construction the different stresses in the given roof truss, the total load on the truss being assumed as 18,000 lbs. The load may be considered as concentrated at a, e, and b. It is best to work from the reaction of the support— which is equal to the weight upon it, and is half the vertical loads act- ing at c, b, c', or 9000 lbs. To find the strain on the lower part of the principal rafter GRAPHIC STATICS. 225 and tie-rod. From a draw a vertical line, and from any scale of units set off ah equal to the reaction of the support — viz. 9000 Ihs. Draw hi parallel to ac, meeting ae in i ; then hi represents the thrust in ac, and ai the tensile strain in ae, measured from the same scale of units. To find the strains in members round point c. Draw a vertical line and mark off i'i' equal to the load carried at c (6000 lbs.) Draw h'i' parallel to ae and equal to ai. Draw b'f 6000 Ihs Fig. 28. parallel to he and /'A' parallel to ee. Then Vi! equals thrust in ac, i'U equals 6000 lbs., b'f equals thrust in he, and h'f equals thrust in ec. To find the strains in the members round joint e. Draw h^a^ parallel to ea and equal to ai ; draw ajA parallel to ec and equal to fh'. These are the known strains. Then draw fjc^ parallel to eh and kj}^ parallel to ee ; fjf^ '^^iU be the tension in the bar eh and It^^ the tension in ee. (32) Find by construction the thrusts on the rafters and struts, also the. strains on the tie-rod and the King and Queen rods of the given truss. Ik is the King rod, e/ is a Queen rod ; af and ek are struts ; P 226 GEOMETRICAL DKAWING. ae is a bay formed by the intersection of the stmts and rafters. There are six bays in the diagram. Total load on Trviss Number of Bays = W, the load carried by each bay. To find the thrnst in ma and tension in mf. Draw CA parallel to reaction at m, and equal to it from any scale of units. Draw AB parallel to am and BC parallel to mf. Then AB equals the thrust in am, and BC the tensile strain in mf. To find the strains in members round the joint at a. We know the thrust in am and the load a. Draw these known forces DE parallel to ma and equal to BA, also EF equals the load at a acting vertically. Draw lines parallel to ea and af, as FGr and GD. These give the thrusts in ea and af. To find the strains in members round joint /. The strains in fm and fa are known. Draw these as HI equals BC and H J equals DG. Then draw JK parallel to fe to meet HI in K. Then JK equals strain in fe and KI strain in fit. In the same way we deal with the other joints. We know from the preceding diagrams all the forces acting at the joints GRAPHIC STATICS. 227 except two. These can easily be found as their directions are known. To illustrate this, take the members round the joint at e. We know the strain in fe, in ae, and the load acting downwards at e. We require the strains in el and ek. Form a portion of a polygon with the known forces taken in order round the joint, then the closing lines of the polygon (jDarallel to the other members el and eJt) will be the strains in those members. 228 GEOMETRICAL DEAWING. EXAMPLES. (1) What is Graphic Arithmetic? (2) Divide the circumference of a circle, by means of the compasses only, into five equal parts. (3) If a line AB 3"4" long represents 17, what does a line CD 2" long represent ? State the unit. (4) Find the product of lines A x B x C x'D x E. A = 1", B-l", C = 14", D = 2", E = J". (5) Two lines AB = 2^" and CD = If". Divide AB by CD, and write down the length of the quotient. AB (6) ^ X EE. AB = 3", CD = If", EE = If. (7) Obtain the value of -f + ^ + -^ + Jg-. (8) Obtain the value of J + 1 + li ■)- ^. (9) Determine the roots ^3, ^4, ^7, ^9, the unit being 1". (10) Determine the roots ^^5, ^6, ^8, with the pencil com- passes only, the unit being 1". (11) If a hne represents 11, find the square root by con- struction. (12) If XY represents 9, find the square root of XY by con- struction. (13) Find by construction the quantities -^. ^, the unit being ^ . (14) Deter mine by cons truction the line — af= V«^ X &2 - c2. a = 2|", h = 1^", c = 2". EXAMPLES. 229 3a (15) Find by construction x = /==, a being 2" in length. 2a (16) Find by construction x = -^, a being 2" in length. (17) Obtain by construction tlie following lines, the unit be- ing -5" :— (18) Draw a triangle having sides 3", 2", 2f". Determine a line representing the area of the triangle! (19) Find the angle between two lines whose equations ure •»-2/=l, y=x;-2. (20) Find a point 2-3, and from the point draw a line to meet x + y-3 = 0, the angle between the lines being 30°. (21) If a line 1" in length represents ^1^, what is the unit? (22) Find the angle between the lines x + y-0, and x, -y = 'J. (23) Find the angle between the lines x + yj2 = i^ and x-y ^2 = 0. (24) Determine the lines y + x='2. y + x= - 2. // + x=l. y + x= -3. (25) What is meant by the moment of a force about a point ? (26) Three forces acting at a point are in equilibrium ; two of the forces are 7 lbs. and 5 lbs. respectively, and the angle between these forces is a right angle ; find the other force ? (27) Two parallel forces of 7 and 10 lbs. act at a distance of 1 foot. Find the resultant, (a) when the forces are in the same direction ; (6) when they act in different directions. (28) If equal parallel forces aU in the same direction act at four of the angular points of a regular pentagon, find a single equivalent force to them. (29) Find the centre of gravity of a rhomboid, 3" x 1^", and included angle 60°. (30) Four forces act away from a point of 12, 9, 17 and 10 units respectively. Find the force that will produce equiHbrium. 230 GEOMETRICAL DKAWING. (31) Four equal weightless rods are hinged together to form a rhombus ABCD, and the hinges A and C are connected hy a string. If the rhombus be suspended from A, and equal weights of 1 cwt. each be suspended from B and D, find the tension on the string. (32) The sides of a triangle are 2", 3" and 4". Find the area in square inches. (33) A uniform beam AB, weight 100 lbs., is supported by two strings AC, ED, the latter being vertical. It is maintained in this position by a horizontal force P applied at B. Find the value of P in lbs. (34) Find the centre of gravity of an arc of a circle. (35) If four forces in one plane be in equilibrium, and the lines of action of all be given, but' the magnitude of only one, sJiow how the magnitudes of the other three may generally be determined. (36) Find the centre of gravity of a quadrilateral figure which has two sides parallel. (37) Enunciate the Polygon of Forces. (38) Find the centre of gravity of a broken line ABODE. (39) Determine the Une of action, and write down the magni- tude of the resultant of five given parallel forces acting in one plane. (40) A line 3" long represents the sum of the areas of a pentagon, square, and equilateral triangle of 1" side. Deter- mine the length of the unit. PKINTED BY NEILL AND COMPANY, EWNBDRGH. 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