I LLINO I S UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN PRODUCTION NOTE University of Illinois at Urbana-Champaign Library Brittle Books Project, 2012. COPYRIGHT NOTIFICATION In Public Domain. Published prior to 1923. This digital copy was made from the printed version held by the University of Illinois at Urbana-Champaign. It was made in compliance with copyright law. Prepared for the Brittle Books Project, Main Library, University of Illinois at Urbana-Champaign by Northern Micrographics Brookhaven Bindery La Crosse, Wisconsin 2012 .... ... ... ,g. OL 12 1 .1 ..... . . ...... ..... .. .... N., ur 14, & vm 4 A J Z, 15. ............... 44 AL:-Z 12 7a, T O."A "I . .. ..... . Y. WK M." ".4p OV Fk 4ml. A vll .7. s 5i 2 gM Av Ae , S ZA tw Wt.. .... . . ..... t 4 e., I I -p- JS 'N--a EN ....... ...... . . . . . . . . . . Iq jb: . , . -TW F.N N 5kp W* 11 154 . .... .... . g NMI- Mmg "n n x., . ... ....... .. x -.. . .. . . . . . . . . . . 0'. "n; g ,Ox, q I Bisecting is dividing into two equal parts. A Perpendicular is a line falling at right angles on another line. RLTGELD HALL a GEL H1 HAL A c B b A B Prob. 1.-To bisect a given line. Let A B be the given line. From the points A and B as centers, with a radius greater than the half of A B, describe arcs intersecting each other above and below the line, as at points a and b. The line a b bisects A B at the point c. a A B Prob. .-To bisect a given are. Let A B be the given are. From the points A and B as centers, with a radius greater than half of the chord of the are, describe arcs intersecting each other, as at points a and b. The line a b bisects A B at the point c. D A B Let AB be the given line and C the given point. From the point C as center, with a radius less than half of A .B A B, describe an are, cutting A B at C the points a and b. From a and b as centers, with a radius greater than aC, describe arcs intersecting at the point c. Through a draw the line D C, which is the perpendicular required. AV - Perpendicular lines may be horizontal, vertical or oblique. Prob. 4.-From a given point to draw a perpendicular to a given line. Let A B be the given line and C the given point. From C as center, with a radius less than C A, describe an are intersecting A B at the points a and b. From a and b as centers, with a radius greater than half of a b, describe arcs intersecting at the point c. From C draw C D in the direction of c. C D is the perpendicular required. D Prob. 5.-To erect a perpendicular at the end of a given line. Let B be the end of the given line A B. Take any point without the line, as C, as a center, and with a rad- ius equal to C B, describe an are in- A B tersecting A B at the point a. Draw a line from a through C intersecting the are at the point b. Through the point b draw the line D B, which is the required perpendicular. C Nd A a B Prob. 6.-To erect a perpendicular at the end of a given line. (2d method.) Let B be the end of the given line A B. From B as center, with a rad- ius less than A B, describe an arc, cut- ting AB at the point a. From a as center with the same radius, set off on the are the point b, and from b with same radius, the point c. From b and c as centers, with same radius describe arcs intersecting at point d. Through A B the point d draw the line C B, which is the required perpendicular. A Parallel is a line which is equally distant at all points from another line. C lb A B Ae a B Prob. 7.-Through a given point to draw a parallel to a given line. Let A B be the given line and C the given point. From C as center, with a radius less than C B, describe an arc, intersecting A B at point a. From a as center and same radius describe are c C. Make a b equal to c C and draw a line through the points C and b, which is the parallel required. a - b -- A B C ID Prob. 8.-To draw a parallel to a given line at a given distance from it. Let A B be the given line and the given distance equal to C D. Take any two points near the ends of the line A B as centers, and with a radius equal to C D describe arcs above the line as a and b. Draw a line tangen- tial to the arcs and it is the parallel required. Prob. 9.-To bisect a given angle. Let A B C be the given angle and from B as center, with any radius, de- scribe an are cutting the sides of the angle at the point a and b. From a and b as centers, with a radius greater than half of a b, describe arcs inter- secting at the point c. Draw line B c, bisecting the angle. C >4 A --I- _ _ - i. II -I An Angle is the difference in the direction of two lines meeting at a point. Trisecting is dividing into three equal parts. ,Prob. 10.-To trisect a given right angle. Let A B C be the given right angle. From B as center, with a radius less than the shortest side of the angle, de- scribe an are, cutting the sides of the angle at the points a and b. From a as center and the same radius cut the are at point c, and from b as center, same radius cut the are at d. From B draw lines through the points c and d, trisecting the angle. D Prob. 11.-To construct an angle equal to a given angle. Let A B C be the given angle. Draw F E equal to C B. From B as center, with any radius, describe an are cut- ting the sides of the angle at the points a and b. From E as center, with the same radius, describe an arc, cutting F E at the point c. From c as center, with a radius equal to a b, intersect the are in d. Through the point d draw the line D E and the angle D E F will be equal to the angle A B C. All the Working lines should be drawn very light. Prob. 12.-To divide a given line into any number of equal parts, say six. Draw A b forming any angle with the given line A B. From B draw B a at the same angle with A B by Prob. 11. From A lay off on the line A b the required number of equal spaces less one, as, 1, 2, 3, 4, 5, and from B on the line B a set off the same number of spaces of the same length, Connect the points 5 1, 4 2, 3 3, 2 4, 1 5 by lines and they divide A B as required. D Prob. 13.-To divide a given line into parts that shall be proportional to a g'iven divided line. Let A B be the given line and C D the divided line. Place C D any dis- tance above and parallel to A B by Prob. 8. Draw A C and B D and pro- duce these lines until they intersect in a. Draw lines from a to A B through the points of divisions on C D and they will divide A B into parts pro- portional to C D. - -AL (it I 1 - - - - - - ic A Triangle is a plain figure bounded by three sides and containing three angles. Prob. 14.-To construct an equilateral triangle, its base being given. Let A B be the given base. From A and B as centers, with a radius equal to A B describe arcs, intersect- ing at the point C. Draw the line A C and C B and the triangle A B C is equilateral. Prob. 15.--To construct a triangle, its three ssaes oesng gzven. Let A B be the base and A C and B C the other two sides of the triangle. From A as center, with a radius equal to A C describe an are. From B as center, with a radius equal to B C, describe an are, cutting the former in C. Draw the lines A C and C B and A B C will be the triangle required. A I ? - -CI I -" w--- ~-- ------ ------------B B I Gy---- A Square is a rectangle having four equal sides and four right angles. An Oblong is a rectangle having four right angles and only its opposite sides equal. D A C Prob. 1.-To construct a square on a given line. At the point B of the given line A B erect a perpendicular B C, by Prob. 7, making it equal to A B. From the points A and C as centers, with a rad- ius equal to A B, describe arcs inter- secting at D. Draw the lines A D and D C and A B C D is the square re- quired. D - C A -r-JB Prob. 17.-To construct an oblong, two of its sides being given. Let A B and B C be the given lines. At the point B erect a perpendicular by Prob. 7, making it equal to B C. From A as center, with a radius equal to B C describe an are above A,and from C as center, with a radius equal to A B intersect the are at D. Draw the lines A D and DC and A B C I willbe the required oblong. A, A B p 0U -j B 10 A Rhomboid is a quadrilateral which has only its opposite sides equal and parallel and whose angles are not right angles. Prob. 22-To construct a rhomboid, two of its sides and the included angle being given. Let A B and A D be the given sides and A the included angle. At the point A construct an angle equal to the given angle by Prob. 11, and draw the line A D. From B as center, with a radius equal to A D describe an arce above B and from D as center, with a radius equal to A B, describe an are intersecting the former in C. Draw D C and B C, and A B 0 D is the rhomboid required. A A B C Prob. 23-To construct a rhomboid, two of its sides, and its long diagonal being given. Let A B and B C be the given sides and A C the diagonal. From A as center, with a radius equal to A C de- scribe an are above B, and from B as center, with a radius equal to B C, de- scribe an are intersecting the former in C. From A as center, radius B C, describe an arc above A and from O as center, radius A B, cut the are in D. Draw the lines A D, D C, B C, and A B C D is the rhomboid required. A I7? A- R u -C _ _ 11 A' 11 A diagonal is a line drawn from one angle to another, opposite to each other. Prob. 20-To construct a rhombus on its diagonals, a diagonal and the measure of one side being given. Let A C be the given diagonal and A B the measure of one side. From A and C as centers, with a radius equal to A B, describe arcs above and below the diagonal intersecting each other at the points D and B. Draw the lines A B, A D, D C, B C, and AB C D is the required rhombus. Prob. 21.-To construct a rhombus, one of its diagonals, and the meas- ure of the other, being given. Let A C be the given diagonal and B D the measure of the other. Bisect A C by a perpendicular, cutting A C at the point E. Then bisect B D and from E as center, with a radius equal to half of B D, describe arcs above and below A C, cutting the perpendicular at the points D and B. Draw lines A B, A D, D C, B C, and A B C D is the required rhombus. a -;u z(1 i -n 14 I 12 A Rhombus is a quadrilateral, having its sides equal, but its angles are not right angles. .A Prob. 18-To construct a rhombus, one side and an angle being given. Let A B be the given side and the angle A be the given angle. At the point A make an angle equal to the given angle by Prob. 11, and make A D equal to A B. From the points B and D as centers, with a radius equal to A B, describe arcs intersecting at C. Draw the lines D C and C B and A B C D is the rhornbus required. A '--I-- - R Prob. 19-To construct a rhombus, the base and a diagonal being given. Let A B be the given base and A C the given diagonal. From A as center, with a radius equal to A C describe an are above B, and from B as center, with a radius equal to A B, describe an are intersecting the former at C. From A and C as centers, with the same radius, describe arcs intersecting at D. Draw lines A D,D C and CB and A B C D is the rhombus required. A ,' - p A c( -- 13 The center of a triangle is also the center of a circle inscribed within a triangle. Prob. 24-To find the center of a given triangle. Let A B C be the given triangle. Bisect any two of its angles by Prob. 9, and produce the lines until they in- tersect, as at point D. The point D will be the center of the triangle. D Prob. 25-To construct a square on its diagonals, the diagonal being given. Bisect the diagonal A C by a per- pendicular as in Prob. 1, cutting A C at the point E. From E as center, with a radius equal to A E, describe arcs above and below the diagonal, cutting the perpendicular in D and B. Draw the lines A B, A D, D C, B C, and A B C D is the required square. A The altitude of a triangle is the perpendicular distance from the vertical angle to the base. Prob. 26.-To construct an isosceles triangle, its base and altitude being given. Let A B be the given base and C D the altitude of the triangle. Bisect A B and at the point D erect a perpen- dicular by Prob. 3, making it equal to C D. Draw the lines A C and C B, and A B C is the triangle required. D) A B Prob. 27.-To construct an isosceles triangle, its base and the vertical angle being f iven. Let A B be the base and C the ver- tical angle. Extend A B to D. From C as center, any radius, describe an are, cutting the sides of the given angle. From A as center, same radius, describe a semi-circle, and make the angle E A D equal to the angle C, by Prob. 11. Bisect the adjacent angle E A B by Prob. 9 and draw line F A. At point B make an angle equal to F A B, and produce the sides until they intersect at C. A B C is the triangle required. A 1? i - ) A -B. 1 17 An equilateral triangle is also equiangular, and each angle is equal to two-thirds of a right angle. Prob. 28--To construct an equilateral triangle, its altitude being given. Let A B be the given altitude. At the point B eiect a perpendicular by Prob. 6, and produce it indefinitely to the right and left of B. Through the point A draw a parallel to the perpen- dicular by Prob. 8, and from A as center, with any radius, describe a semi-circle cutting the parallel at a and b. From a and b as centers, with the radius of the semi-circle, cut the semi- circle at the points c and d. From A, through the points c and d draw lines to cut the base at C and D. CD A is the required triangle. Prob. 29.-To construct an equilateral triangle its altitude being given. (Second method.) Erect a perpendicular at point B and produce it to the right and left of B indifinitely. From A as center with any radius, describe an are cutting A B at point a. From a as center and same radius, describe another are cut- ting A B at point b. From b as cen- ter, same radius, describe arcs cutting the former at points c and d. From A draw lines through c and d until they meet the produced base at C and D. C D A is the required triangle. B A B 18 A Trapezoid is a quadrilateral, which has only two of its sides parallel. The altitude of a trapezoid is the perpendicular distance between the parallel sides. Prob. 30.-To construct a trapezoid equal to a given trapezoid. Let A B C D be the given trapezoid. Draw a line E F equal to A B. At the point E make an angle equal to the angle at A by Prob. 11, and make E H equal to A D. From F as center, with a radius equal to B C describe an are above F, and from H as center, with a radius equal to D C describe an are intersecting the former in G. Draw H G, G F, and E F G H will be equal to A B C D. D_ \/ A C / '-A B Prob. 31.-To construct a trapezoid, its base and the angles at the base and the altitude being given. Let A B be the base, C D the alti- tude and A and B the two angles at the base. Draw an indefinite line par- allel to A B and at a distance from it equal to the altitude C D. At A make an angle equal to the angle A and at B make an angle equal to the angle B by Prob. 11. Produce the lines until they intersect the parallel at F and E. A B E F will be the trapezoid re- quired. A SI) E 19 A trapezium is a quadrilateral which has no two of its sides parallel. Prob. 32-To construct a trapezium equal to a given trapezium. Let A B C D be the given trapezium. Draw E F equal to A B. From F as center, with a radius equal to B C, de- scribe an are above F and from E as center, with a radius A C, describe an are cutting the former in G. From E as center, with a radius A D, describe an are above E and from G as center, with a radius D C, intersect the are in H. DrawEH, H, FGand E F G H will be equal to A B C D. A B Prob. 33-To construct a triangle, its base and the angles at the base be- ing given. Let A B be the given base. At the point A construct an angle equal to the angle A and at the point B an angle equal to the angle B by Prob. 11. Produce the sides until they intersect at the point C, and the required tri- angle is formed. A u -- ~-E 20 Similar figures differ from each other only in magnitude; their angles are equal, each to each, and their sides proportional. Prob. 34--Upon a given line to con- struct an oblong similar to a given oblong. Let A B C D be the given oblong and A G be the given line. Draw the diagonal A C and produce it indefinite- ly. Through the point G draw a par- allel to D C by Prob. 8, and produce it until it meets the diagonal at F. Produce A B at the right of B and make it equal to G F. Draw F E and the rectangle A E F G is sirnilar to the rectangle A B C D. Prob. 35-To describe an equilateral triangle about a given square. Let A B C D be the given square. Produce A B to the right and left in- definitely. Then construct on the line D C an equilateral triangle by Prob. 14, and produce the sides G D and G C until they meet the produced line A B at E and F.-E F G is the requir- ed triangle. B' U A -- 'U Aw 'NO W -t, 4 Zoe 71 'IMP V ink s. M 0A R n n , pigg 5W ..... .... IV, - V M 2-r 1-, kk. Fk- mW 4 M - 4u Wm s, 35 j!: 4a A. F kkk Ai 49, This book is a preservation facsimile produced for the University of Illinois, Urbana-Champaign. It is made in compliance with copyright law and produced on acid-free archival 60# book weight paper which meets the requirements of ANSI/NISO Z39.48-1992 (permanence of paper). Preservation facsimile printing and binding by Northern Micrographics Brookhaven Bindery La Crosse, Wisconsin 2012