c c < c m *p? c c . Geometrically lines have but one dimension, .length, and the direction of a line is the direction from point to point of the points of which the line is composed : in drawing, lines are visible marks of pencil or pen upon paper. FIG. 3. A straight line is such as can be drawn along the edge of the ruler, and is one in which the direction is the same throughout. In drawing a straight line through two given points, place the edge of the ruler very near to and at equal distances from the points, as the point of the pencil or pen should not be in contact with the edge of the ruler (Fig. 3). Lines in geometry and drawing are generally of limited extent. A given CONSTRUCTION OF GEOMETRICAL PROBLEMS. 3 FIG. 4. or known line is one established on paper or fixed by dimensions. Lines of the same length are equal. To draw Curved Lines. Insert the pencil-point in the compasses, and open them to a suitable extent. With the needle or sharp point resting on the paper describe a line with the pencil around this point ; the line thus described is usually called a circle more strictly it is the circumference of a circle the circle being the space inclosed. A portion of a circumference is an arc. The point around which the circumference is described is the center of the circle (Fig. 4). If a line be drawn from the center to the circum- ference it is called a radius. As it is the length embraced between the points of the compasses, it is often called by mechanics the sweep. If a line be drawn through the center, and limited by the circumference, it is called the diameter, and is equal to two radii. A radius is a semi-diameter ; a diameter is the longest line that can be con- tained within a circumference. Lines limited by the circumference, and which are not diameters, are chords. It will be observed that arcs are lines which are continually changing the directions, and are called curved lines, but there are other curved lines than those described by compasses, of which the construction will be explained hereafter. Besides straight and curved lines there are often lines, in drawing, which can neither be drawn by rulers or compasses, as lines representing the direc- tions of brooks and rivers, the margins of lakes and seas, points in which are established by surveys, defined on paper, and connected by hand-drawing. These may be called irregular or crooked lines. Where it is necessary to distinguish lines by names, we place at their extremities letters or figures, as A B, 1 2 ; the line A B, or 1 2. But in lines other than straight, or of considerable extent, it is often necessary to introduce intermediate letters and figures, as a a a. In the following problems, unless otherwise implied or designated, where lines are mentioned, straight lines are intended. If we conceive a straight line to move sideways in a single direction, it will sweep over a surface which is called a plane. All drawings are projections on planes of paper or board. Two lines drawn on paper, and having the same direction, can never come any nearer each other, and must always be at the same distance apart, however far extended. Such lines are called parallel lines. 4: CONSTRUCTION OF GEOMETRICAL PROBLEMS. PEOB. L To draw a line parallel to a given line, and at a given distance from it (Fig. 5). Draw the line A B for the given line, and take in the compasses the dis- tance A C the distance at which the other line is to be drawn. On A, as a Jj FIG. 5. FIG. 6. center, describe an arc, and on B, as a center, another arc ; draw the line C D just touching these two arcs, which will be the parallel line required. PKOB. II. To draw a line parallel to a given line through a given point outside this line (Fig. 6). Draw the given line A B, and mark the given point C. With C as a centei, find an arc that shall only just touch A B ; and with B as a center, and the same radius, describe an arc D. Draw through the point C a line just touching this last arc, and the line C D will be the parallel line required. Two lines in the same plane, not parallel to each other, will come together if extended sufficiently far. The coming together, cutting, or intersection of two lines, is called an angle (Fig. 7). If but two lines come together, the angle may be designated by a single letter at the vertex, as the angle E. But, if three or more lines have a common vertex, the angles are designated by the lines of which they are composed, as the angle D B C of the lines D B D FIG, 8 and B C ; the angle A B C of A B and B C ; the angle A B D of A B and B D. The letter at the vertex is not repeated, and must always be the central letter. Describe a circle (Fig. 8). Draw the diameter A B. From A and B as centers, with any opening of the compasses greater than the radius, describe two arcs cutting each other as at D. Through the intersection of these arcs and the center C, draw the line D E. D E makes, with the diameter A B, four angles, viz., A C D, D B, B C E, and E C A. Angles A* CONSTRUCTION OF GEOMETRICAL PROBLEMS. are equal whose lines have equal inclination tfc^ach other, and whose lines, if placed one on the other, would coincide. By construction, the points C and D have, respectively, equal distances from A and B ; the line D C can not, there- fore, be inclined more to one side than to the other, and the angle A C D must be equal to the angle BCD. Such angles are called right angles. It can be readily proved that all the four angles, formed by the intersection of D E with A B, are equal, and are right angles. The angles A C D and D C B, on the same side of A B, are called adjacent angles ; as also DOB and B C E, on the same side of D E. When a line, standing on another line, makes the two adjacent angles equal, the angles are right angles, and the first line is perpendicular to the other. If the second or base line be parallel with the surface of still water, it is called an horizontal line, and the perpendicular line is called a ver- tical line. Draw the line C F. It will be observed that the angle F C D is less than a right angle, and it is called an acute angle ; the angle F C A is greater than a right angle, and it is called an obtuse angle. It will be observed that, no matter how many lines be drawn to the center, the sum of all the angles on the one side of A B can only be equal to two right angles, and, on both sides of A B, can only be equal to four right angles. It will be observed that the angles at the center include greater or less arcs between their sides, according to the greater or less inclination of their sides to each other ; that the right angles intercept equal arcs, and that, no matter how large the circle, the pro- portion of the circle intercepted by the sides of an angle is always the same, and that the arcs can therefore be taken as the measures of angles. For this purpose the whole cir- cumference is supposed to be divided into three hundred and sixty degrees (360), each degree subdivided into sixty minutes (60'), and each minute into sixty seconds (60*). Each right angle has for its measure one quarter of the whole circumference (-^p-), or 90, and is called a quadrant. PROB. III. To construct an angle equal to a given angle (Fig. 9). Draw any angle, as C A B, for the given angle, and the line a b as the base of the required angle. From A, with any suitable radius, describe the arc B C, and from a, with the same radius, describe the arc b c. Measure the length of the arc B C, or rather the chord, that is, the distance in a straight line from B to C, and lay off the same distance on the arc b c. Draw the line a c, and the angle cab will be equal to C A B. PROB. IV. To construct an angle of sixty degrees (Fig. 10). Lay off any base-line, and from A, with any radius, describe an arc, and Fio. 9. 6 CONSTRUCTION OF GEOMETRICAL PROBLEMS. from B, with the same radius, describe another arc cutting the first, as at C. Draw the line C A, and the angle CAB will be an angle of sixty degrees. The reason of this construction will be readily understood if, on the cir- \ FIG. 10. ;# Fio. 11. cumference of any circle, chords equal to the radius are stepped off succes- sively. Six will exactly complete the circle, making 360, or each 60, and the angle corresponding will be 60. PKOB. V. To draw a perpendicular to a line from a point without the line (Fig. 11). Draw a line, and mark the given point outside it, A. From A as a center, with a suitable radius, describe an arc cutting the line at G and F. From G and F, as centers, describe arcs cutting each other. The line drawn through the point A, and the point of intersection E, will be perpendicular to the line G F. The radial line A E divides the chord G F and the arc G E F into two equal parts ; and, conversely, the line perpendicular to the middle point of a chord of a circle is radial passes through the center of that circle. PROS. VI. To draw a perpendicular to a line from a point within that line (Fig. 12). 1st Method. Draw a line, and take the point A in the line. From A, as a center, describe arcs cutting the line on each side at B and C. From B and Nr ,'D A FIG. 12. C l Fia. 13. C, as centers, describe intersecting arcs at D. Draw a line through D and A> and it will be perpendicular to the line B C at A. CONSTRUCTION OF GEOMETRICAL PROBLEMS. \C/ 2d Method (Fig. 13). Draw the line, and mark the point A as before. From any center F, without the line, and not directly over A, with a radius equal to F A, describe more than a half-circle cutting the line, as at D. From D, through the center F, draw a line cutting the arc at E. Draw A E, and it will be the perpendicular to the line A D. It will be observed that the line D E is the diameter of a circle, and that the angle DAE, with its vertex at A in the circumference, would embrace with its sides half a circle, had a full circle been described. It has been shown that angles at the center of a circle have for their measure the arc embraced by their sides. It is easily demonstrable that angles, with their vertices in the circumference, have for their measure half the arc embraced by their sides, and, consequently, angles embracing half a circumference are right an- gles, and their sides are perpendicu- lar to each other. PROB. VII. To draw a perpen- dicular to the middle point of a line (Fig. 14). From the extremities A and B of the line, as centers, describe in- tersecting arcs above and below the line. Through these intersections draw the line D. It will be per- pendicular to the line A B, and bi- sect or divide it into two equal parts. If the line A B be considered the chord of a circle, its center would lie in the line C D. This construction is sometimes used merely to divide a line into two equal parts, or bisect it ; but if we have dividers or compasses, with both points sharp, it can be more readily done with them (Fig. 15). Place one point of the dividers on one end of the line, and open the dividers to a space as near as may be half the line. Turn the dividers on the central point ; if the other point then falls exactly on the opposite extremity DIE FIG. 14. FIG. 15. of the line, it is properly divided ; but, if the point falls either within or with- out the extremity of the line, divide the deficit or excess by the eye, in halves, and contract or extend the dividers by this measure. Then apply the dividers as before, and divide deficit or excess till a revolution exactly covers the length of the line. By accustoming one's self to this process, the eye is made accurate, and one estimate is sufficient for a correct division of any deficit or 8 CONSTRUCTION OF GEOMETRICAL PROBLEMS. excess. By a similar process it is evident that a line can be divided into any number of equal parts, by assuming an opening of the dividers as nearly as possible to that required by the division, and, after spacing the line, dividing the deficit or excess by the required number of parts, contracting or expanding the dividers by one of these parts, and spacing the line again, and so on till it is accurately divided. PKOB. VIII. To bisect a given angle (Fig. 16). Construct an angle, and from its vertex A, as a center, describe an arc cutting the two sides of the angle at B and C. From B and C, as centers, describe intersecting arcs. Draw a line through A and the point of intersec- tion D, and this line will bisect the angle. -B 6 I) FIG. 16. FIG. 17. PROB. IX. To Used an angle when the vertex is not on the paper (Fig. 17). Draw two lines, A B and E C, inclined to each other, but not intersecting. Draw two lines intersecting each other, a b and a c, inside and parallel to A B and E C. Bisect b a c by the line a d, and this line will also bisect the angle whose vertex is not on the paper. PROB. X. Through two given points to describe an arc of a circle with a given radius (Fig. 18). From B and C, the two given points, with an opening of the dividers equal to the given radius, describe two arcs crossing at A. From A, as a center, with the same radius, describe an arc, and it willbe the one required. FIG. 18. FIG. 19. PROB. XL To find the center of a given circle, or of an arc of a circle. Of a circle (Fig. 19). Draw the chord A B. Bisect it by the perpen- dicular C D, whose extremities lie in the circumference, and bisect C D. Gr, the point of bisection, will be the center of the circle. CONSTRUCTION OF GEOMETRICAL PROBLEMS. Of an arc, or of a circumference (Fig. 20). Select the points A, B, and C in the circumference, well apart. With the same radius from A and B as centers, and then from B and C as centers, describe arcs cutting each other ; draw the two lines D E and F G through their intersections. The point 0, where these lines meet, is the center required. PKOB. XII. To describe a circle passing through three given points (Fig. 20). Proceed, as in the last problem, to find the center 0. From 0, as a center, with a radius A, describe a circle, and it will be the one required. FIG. 20. PKOB. XIII. To describe a circle passing through three given where the center is not available. 1st Method (Fig. 21). From the extreme points A and B, as centers, describe the arcs B G and A H. Through the third point, C, draw A E and B F, cutting the arcs. Divide the arcs A F and B E into any number of equal parts, and set off a series of equal parts of the same length on the upper por- tions of the arcs beyond E and F. Draw straight lines, B L, B M, etc., to the points of division in A F, and A I, A K, etc., to the points of division in E G ; the successive intersections N, 0, etc., of these lines are points in the circle required between the given points A and C, which may be filled in accord- ingly. Similarly, the remaining part of the curve, B C, may be described. Zd Method (Fig. 22). Let A, D, and B be the given points. Draw A B, A D, and D B. Draw e f parallel to A B. Divide D A into a number of equal parts at 1, 2, 3, etc., and from D describe arcs "' through these points to meet ef. Divide the arc A e into the same number of equal parts, and draw straight lines from D to the points of division. The intersec- tions of these lines successively with the arcs are points in the circle, which may be filled in as before. Note. The second method is not perfectly true, but sufficiently so for arcs less than one fourth of a circle. 10 CONSTRUCTION OF GEOMETRICAL PROBLEMS. To describe the arc mechanically. Let a, c, I be the three points of a curve ; transfer these points to a piece of stout card-board, and draw the lines a c and c I, and extend them beyond a and I. Cut out the card-board along these FIG. 23. lines. Insert upright pins on the points a and I of the drawing, and placing the edges of the cut card-board against them, and maintaining the contact of the edges of the card-board with the pins, slide the card each way. Dot the positions of the vertex of the angle c, and the dots will be points in the curve. PROB. XIV. To draw a tangent to a circle from a given point in the cir- cumference. 1st Method (Fig. 24). Through the given point A draw the radial line A C. The perpendicular F G- to that line will be the tangent required. FIG. 24. FIG. 25. 2d Method (Fig. 25). From the given point A set off equal arcs, A B and A D. Join B and D. Through A draw A E parallel to B D, and it will be the tangent required. This method is useful when the center is inaccessible. PROB. XV. To draw tangents to a circle from a point without it (Fig. 26). From the given point A draw A to the center of the circle. From D, the FIG. 26. FIG. 27. intersection of A C with the circle, describe an arc, with a radius D C, cutting the circle at E and F. Draw A E and A F, and they will be the tangents required. CONSTRUCTION OF GEOMETRICAL PROBLEMS. 11 To construct within the sides of an angle a circle tangent to these sides at a given distance from the vertex (Fig. 27). Let a and b be the given points equally distant from the vertex A. Draw a perpendicular to A C at a, and to A B at I. The intersection of these perpendiculars will be the center of the required circle. In the same figure, to find the center, the radius being given, and the points a and b not known. Draw lines parallel to A C and A B, at a distance equal to the given radius, and their intersection will be the center required. PROB. XVI. To describe a circle from a given point to touch a given circle (Figs. 28 and 29). D E being the given circle, and B the given point, draw a line from B to the center C, and produce it, if necessary, to cut the circle at A. From B, FIG. 28. FIG. 29. as a center, with a radius equal to B A, describe the circle F G, touching the given circle, and it will be the circle required. The operation is the same whether the point B is within or without the circle. It will be remarked that, in all cases of circles tangent to each other, their centers and their points of contact must lie in the same straight line. PROB. XVII. To draw tangents to two given circles. 1st Method (Fig. 30). Draw the straight line ABC through the centers of the two given circles. From the centers A and B draw parallel radii, A D FIG. 30. and B E, in the same direction. Draw a line from D to E, and produce it to meet the center line at C ; and from C draw tangents to one of the circles by Problem XV. Those tangents will touch both circles as required. 2d Method (Fig. 31). Draw the line A B connecting the two centers. Draw in the larger circle any radius, A H, on which set off H G, equal to the 12 CONSTRUCTION OF GEOMETRICAL PROBLEMS. radius of the smaller circle. On A describe a circle with the radius A G, and draw tangents, B I and B K, to this circle from the other center, B. From A FIG. 81. and B draw perpendiculars to these tangents. Join C and D, also E and F. The lines D and E F will be the required tangents. Note. The second method is useful when the diameters of the circles are nearly equal. PKOB. XVIII. Between two inclined lines to draw a series of circles touching these lines and touching each other (Fig. 32). Bisect the inclination of the given lines A B and C D by the line N ; this is the center line of the circles to be inscribed. From a point, P, in this line, draw P B perpendicular to the line A B ; and from P describe the circle B D, touching the given lines, and cutting the center line at E. From E draw E F perpendicular to the center line, cutting A B at F ; from F describe an arc, with a ra- dius, F E, cutting A B at G ; draw G H parallel to B P, giving H the center of the second touching circle, described with the radius H E or II G. By a similar process the third circle, I N, is described. And so on. Inversely, the largest circle may be described first, and the smaller ones in succession. Note. This problem is of frequent use in scroll-work. PROB. XIX. Between tivo inclined lines to draw a circular arc to fill up the angle, and touching the lines (Fig. 33). Let A B and D E be the inclined lines. Bisect the inclination by the line F C, and draw the perpendicular A F D to define the limit within which the circle is to be drawn. Bisect the angles A and D by lines cutting at C, and from C, with radius C F, draw the arc H F G, which will be the arc required. PROB. XX. To fill up the angle of a straight line and a circle, with a cir- cular arc of a given radius (Fig. 34). On the center C, of the given circle A D, with a radius C E equal to that of the given circle plus that of the required arc, describe the arc E F. Draw FIG. 32. CONSTRUCTION OF GEOMETRICAL PROBLEMS. 13 G F parallel to the given line H I, at the distance G H, equal to the radius of the required arc, and cutting the arc E F at F. Then F is the required H I FIG. 34. center. Draw the perpendicular F I, and the line F C, cutting the circle at A ; and, with the radius F A or F I, describe the arc A I, which will be the arc required. PKOB. XXI. To fill up the angle of a straight line and a circle, with a circular arc to join the circle at a given point (Fig. 35). In the given circle B A draw the radius to A, and extend it. At A draw a tangent, meeting the given line at D. Bisect the angle A D E, so formed, with a line cutting the radius, as extended at F ; and, on the center F, with radius F A, describe the arc A E, which will be the arc required. PKOB. XXII. To describe a circular arc joining two circles, and to touch one of them at a given point (Fig. 36). Let A B and F G be the given circles to be joined by an arc touching one of them at F. Draw the radius E F, and produce it both ways ; set off F H equal to the radius, A 0, of the other circle ; join C to H, and bisect it with the perpen- dicular L I, cutting E F at I. On the center I, with radius I F, describe the arc F A, which will be the arc required. CONSTRUCTION OF GEOMETRICAL PROBLEMS. PROB. XXIII. To find the arc which shall be tangent to a given point on a straight line, and pass through a given point outside the line (Fig. 37). Erect at A, the given point on the given line, a perpendicular to the line. From C, the given point outside the line, draw C A, and bisect it with a per- pendicular. The intersection of the two perpendiculars at a will be the center of the required arc. a. FIG. 37. FIG. 38. PROB. XXIV. To connect two parallel lines by a reversed curve composed of two arcs of equal radii, and tangent to the lines at given points (Fig. 38). Join the two given points A and B, and divide the line A B into two equal parts at C ; bisect C A and C B by perpendiculars ; at A and B erect perpen- diculars to the given lines, and the intersections a and b will be the centers of the arcs composing the required curve. PROB. XXV. To join two given points in two given parallel lines by a reversed curve of two equal arcs, whose centers lie in the parallels (Fig. 39). Join the two given points A and B, and divide the line A B in equal parts at C. Bisect A C and B C by perpendicu- lars ; the intersections a and b of the parallel lines, by these perpendiculars, will be the centers of the required arcs. PROB. XXVI. On a given line, to construct a compound curve of three arcs of circles, the radii of the two side ones being equal and of a given length, FIG. '/b H / D FIG. 40. CONSTRUCTION OF GEOMETRICAL PROBLEMS. 15 and their centers in the given line ; the central arc to pass through a given point on the perpendicular, bisecting the given line, and to be tangent to the other two arcs (Fig. 40). Let A B be the given line, and C the given point. Draw C D perpen- dicular to A B ; lay off A a, B b, and C c, each equal to the given radius of the side arcs ; draw a c, and bisect it by a perpendicular ; the intersection of this line with the perpendicular C D will be the required center of the central arc e C e r . Through a and b draw the lines D e and D e' ; from a and b, with the given radius equal to a A or b B, describe the arcs A e and B e f . From D, as a center, with a radius equal to C D, and, consequently, by construction, equal to D e and D e', describe the arc e C e'. The entire curve A e C e' B is the compound curve required. t It will be observed in all the preceding problems that, when a line is tangent to a curve, the center of that curve must be in the perpendicular to the line at its tangent point ; and that, when two curves are tangent to each other, their centers must be in the same radial line passing through the point of tangency. PROBLEMS ON POLYGONS AND CIRCLES. Three lines inclosing a spa^ce form a triangle (Fig. 41). If two of the sides are of equal length, it is an isosceles triangle ; if all three are of equal length, FIG. 41. FIG. 42. it is an equilateral triangle. If one of the angles is a right angle, it is a right- angled triangle, and if no two of the sides are of equal length, and not one of the angles a right angle, it is a scalene triangle. PROB. XXVII. To construct an isos- celes triangle (Fig. 42). FIG. 43. FIG. 44. Draw any line as a base, and, from each extremity as a center, with equal radius, describe intersecting arcs. Draw a line from each extremity of the base to this point of intersection, and the figure is an isosceles triangle. PROB. XXVIII. To construct an equilateral triangle (Fig. 43). 16 CONSTRUCTION OF GEOMETRICAL PROBLEMS. Draw a base line, and from each extremity as a center, with a radius equal to the base line, describe intersecting arcs. Draw lines from the extremi- ties of the base to this point of intersection, and the figure is an equilateral triangle. PKOB. XXIX. To construct a right-angled triangle (Fig. 44). Construct a right angle by any one of the methods before described. Draw a line from the extremity of the one side to the extremity of the other side, and the figure is a right-angled triangle. It will be evident, in looking at any right-angled triangle, that the side opposite the right angle is longer than either of the other or adjacent sides ; this side is called the hypothenuse. PROB. XXX. To construct a triangle equal to a given triangle. Let ABC (Fig. 45) be the given triangle. 1st Method (Fig. 46). Draw a base line, and lay off its length equal to FIG. 45. FIG. 46. A B ; from one of its extremities, as a center, with a radius equal to A C, describe an arc ; and, from its other extremity, with a radius equal to B C, describe an arc intersecting the first. Draw lines from the extremities to the point of intersection, and the triangle equal to A B C is complete. 2d Method (Fig. 47). Draw a base line, as before, equal to A B. At one C FIG. 47. FIG. 48. extremity construct an angle equal to C A B, and at the other an angle equal to C B A. The sides of these angles will intersect, and form the required triangle. 3d Method (Fig. 48). Construct an angle of the triangle equal to any angle of A B C, say the angle A C B. On one of its sides measure a line equal to C A, and on the other side one equal to C B ; connect the two extremiities by a line, and the triangle equal to A B C is ~~ complete. FIG. 49. From the above constructions it will CONSTRUCTION OF GEOMETRICA 17 FIG. 50. foe seen that, if the three sides of a triangle, or two sides and the included an- gle, or one side and the two adjacent angles are known, the triangle can be constructed. Construct a triangle, ABC (Fig. 49). Extend the base to^E;"8m^Sraw B D parallel to A C. As A C has the same inclination to C B that B D has, the angle C B D is equal to the angle A C B. As A C has the same inclina- tion to A E that B D has, the angle D B E is equal to C A B. That is, the two angles formed outside the triangle are equal to the two inside at A and C ; and the three angles at B are equal to the three angles of the triangle, and their sum is equal to two right angles. There- fore, the sum of the three angles of a trian- gle is equal to two right angles. On one side of a triangle (Fig. 50) con- struct a triangle equal to the first, with op- posite sides parallel. The exterior lines of the two triangles form a four-sided or quadrilateral figure, of which the opposite sides are equal and parallel, and the opposite angles equal. This figure is called a parallelo- gram, and the line C B, extending between opposite angles, is a diagonal. On the hypothenuse of a right-angled triangle (Fig. 51) construct another equal to it, and the exterior lines form a parallelogram, which, as all the angles are right angles, is called a rectangle. If the four sides are all equal, it is called a square. A parallelogram in which all the sides are equal, but the angles not right angles, is called a rhombus (Fig. 52) ; if only the opposite sides are equal, it is called a rhom- boid ; if only two sides are parallel, the figure is a trapezium (Fig. 53). Describe a circle (Fig. 54). Draw a diameter, and erect on its center C the perpendicular C F. Draw at any angle with the diameter the line C A. Draw D H and A B perpendicular to the diameter, the first from the intersection of the line C A with the circumference, the other from the extremity B of the FIG. 51. FIG. 52. FIG. 53. diameter. Draw D G and E F perpendicular to the radius C F, one from the point D, the other from the extremity of the radius C F. The angles DOG and D C H are complements of each other ; that is, together they form a right angle, as it completes with it a right angle. The line D H is the 2 18 CONSTRUCTION OF GEOMETRICAL PROBLEMS. sine of the angle D C H and the cosine of D C G. D G is the sine of the angle D C G and the cosine of D C H. A B is the tangent of the angle .DOB and the cotangent of D C G. E F is the tangent of the angle DOG and the cotangent of D C H. A C is the secant of the angle D C H and the cosecant of D C G. C E is the secant of the angle DOG and the cosecant of D C H. H B is the versed sine of the angle D C H, and G F of D C G. It will be observed that the angles of the triangle D H are equal to those of A C B, and that, if we suppose C A to be the radius of a larger circle, the arcs, and consequently the half -cords or sines D H and A B, will be propor- tionate to the radii ; that is, D H will A be to A B as C D is to C A. Triangles which have equal angles have their sides proportional, and are called similar. This is demonstrable of other triangles than the right-angled ones in the figure. Take any figure (Fig. 55) of more than three sides bounded by straight FIG. 54. FIG. 55. lines, and from any angle draw lines to the opposite angles. The figure will be divided into as many triangles as there are sides, less two, and the sum of the angles of the figure will be equal to as many times two right angles as there are sides, less two. If another figure were made with similar triangles, similarly placed, the two figures would be similar. Polygons, or many-sided figures, are similar when their angles are equal to each other and similarly placed, and their homologous sides, or sides including these angles, proportional. FIG. 56. FIG. 57. FIG. 58. FIG. 59. On this principle of similarity of figures the science of drawing is based. With a scale of equal parts, one inch on paper, for instance, representing a CONSTRUCTION OF GEOMETRICAL PROBLEMS. 19 foot, a yard, or a mile, in nature, the figure drawn on that scale will represent the object accurately in reduced form ; and measurements may be made in de- tail by the scale as well as from the natural object in the shop or on the estate. Polygons, with their sides and angles equal, are called regular polygons (Figs. 56, 57, 58, 59). Regular polygons are easily constructed by means of circles, whose circum- ferences are divided into the number of sides required, with chords drawn representing the sides. As the circle is then outside the polygon, the circle is said to be described about it, while the polygon is in- scribed within the circle. If the polygon is described about the circle, its sides are tan- gent to it. PROB. XXXI. To describe a circle about a triangle (Fig. 60). Bisect two of the sides A B, A 0, of the tri- angle at E, F ; from these points draw perpen- diculars cutting at K. From the center K, with K A as radius, describe the circle ABC, as required. PROB. XXXII. To inscribe a circle in a triangle (Fig. 61). Bisect two of the angles A, 0, of the triangle A B C, by lines cutting at D ; from D draw a perpendicular D E to any side, as A ; and with D E as radius, from the center D, describe the circle required. When the triangle is equilateral, the center of the circle is more easily found by bisecting two of the sides, and drawing perpendiculars, as in the previous problem. Or, draw a perpendicular from one of the angles to the opposite side, and from the side set off one third of the length of the perpendicular. FIG. 60. FIG. 62. PROB. XXXIII. To inscribe a square in a circle ; and to describe a circle about a square (Fig. 62). To inscribe the square. Draw two diameters, A B, D, at right angles, and join the points A, B, 0, D, to form the square as required. To describe the circle. Draw the diagonals A B, C D, of the given square, cutting at E ; on E as a center, with E A as radius, describe the circle as required. In the same way, a circle may be described about a rectangle. 20 CONSTRUCTION OF GEOMETRICAL PROBLEMS. PROB. XXXIV. To inscribe a circle in a square ; and to describe a square about a circle (Fig. 63). To inscribe the circle. Draw the diagonals A B, C D, of the giver square, cutting at E ; draw the perpendicular E F to one of the sides, and with the radius E F, on the center E, describe the circle. To describe the square. Draw two diameters A B, C D, at right angles, and produce them ; bisect the angle D E B at the center by the diameter F G, and through F and G draw perpendiculars A C, B D, and join the points A, D, and B, C, where they cut the diagonals, to complete the square. PROB. XXXV. To inscribe a pentagon in a circle (Fig. 64). Draw two diameters, A C, B D, at right angles ; bisect A at E, and from E, with radius E B, cut A C at F ; from B, with radius B F, cut the F FIG. 63. B FIG. 64. FIG. 65. circumference at G and H, and with the same radius step round the circle to I and K ; join the points so found to form the pentagon. PROB. XXXVI. To construct a regular hexagon upon a given straight line (Fig. 65). From A and B, with a radius equal to the given line, describe arcs cutting at g; from g, with the radius g A, describe a circle ; with the same radius set off from A the arcs A G, G F, and from B the arcs B D, D E. Join the points so found to form the hexagon. PROB. XXXVII. To inscribe a regular hexagon in a circle (Fig. 66). J> FIG. 66. FIG. 67. Draw a diameter, A B ; from A and B as centers, with the radius of the circle A C, cut the circumference at D, E, F, G ; draw straight lines A D, D E, etc., to form the hexagon. CONSTRUCTION OF GEOMETRICAL PROBLEMS. 21 The points of contact, D, E, etc., may also be found by setting off the radius six times upon the circumference. PROB. XXXVIII. To describe a regular hexagon about a circle (Fig. 67). Draw a diameter, A B, of the given circle. With the radius A D from A as a center, cut the circumference at C ; join A C, and bisect it with the radius D E ; through E draw F G parallel to A C, cutting the diameter at F, and with the radius D F describe the circle F H. Within this circle describe a regular hexagon by the preceding problem ; the figure will touch the given circle as required. PROB. XXXIX. To construct a regular octagon upon a given straight line (Fig. 68). Produce the given line A B both ways, and draw perpendiculars A E, B F ; bisect the external angles at A and B by the lines A H, B C, which make A B FIG. 68. FIG. 69. equal to A B ; draw C D and H G parallel to A E and equal to A B ; and from the centers G, D, with the radius A B, cut the perpendiculars at E, F, and draw E F to complete the octagon. PROB. XL. To convert a square into a regular octagon (Fig. 69). Draw the diagonals of the square intersecting at e; from the corners A, B, C, D, with A e as radius, describe arcs cutting the sides at g h, etc. ; join the points so found to complete the octagon. PROB. XLI. To inscribe a regular octagon in a circle (Fig. 70). FIG. 70. FIG. 71. Draw two diameters, A C, B D, at right angles ; bisect the arcs A B, B C, etc., at e, /, etc.; and join A e, e B, etc., for the inscribed figure. 22 CONSTRUCTION OF GEOMETRICAL PROBLEMS. PKOB. XLII. To describe a regular octagon about a circle (Fig. 71). Describe a square about the given circle A B ; draw perpendiculars h k, etc., to the diagonals, touching the circle. The octagon so formed is the figure required. Or, to find the points h, k, etc., cut the sides from the corners of the square, as in Prob. XL. PKOB. XLIII. To inscribe a circle within a regular polygon. When the polygon has an even number of sides, as in Fig. 72, bisect two opposite sides at A and B, draw A B, and bisect it at C by a diagonal D E drawn between opposite angles ; with the radius C A describe the circle as required. When the number of sides is odd, as in Fig. 73, bisect two of the sides at A and B, and draw lines A E, B D, to the opposite angles, intersecting at C ; from C, with C A as radius, describe the circle as required. FIG. 72. FIG. 73. PROB. XLIV. To describe a circle without a regular polygon. When the number of sides is even, draw two diagonals from opposite angles, like D E (Fig. 72), to intersect at C ; and from C, with C D as radius, describe the circle required. When the number of sides is odd, find the center C (Fig. 73) as in last problem, and, with C D as radius, describe the circle. The foregoing selection of problems on regular figures is the most useful in mechanical practice on that subject. Several other regular figures may be constructed from them by bisection of the arcs of the circumscribing circles. In this way a decagon, or ten-sided polygon, may be formed from the penta- gon by the bisection of the arcs in Prob. XXXV, Fig. 64. Inversely, an equilateral triangle may be inscribed by joining the alternate points of division found for a hexagon. The constructions for inscribing regular polygons in circles are suitable also for dividing the circumference of a circle into a number of equal parts. To supply a means of dividing the circumference into any number of parts, including cases not provided for in the foregoing problems, the annexed table of angles relating to polygons, expressed in degrees, will be found of general utility. In this table, the angle at the center is found by dividing 360, the number of degrees in a circle, by the number of sides in the polygon, and by setting off round the center of the circle a succession of angles by means of CONSTRUCTION OF GEOMETRICAL PROBLEMS. 23 the protractor, equal to the angle in the table due to a given number of sides : the radii so drawn will divide the circumference into the same number of parts. The triangles thus formed are termed the elementary triangles of the polygon. TABLE OE POLYGONAL ANGLES. Number of Sides of Kegu- lar Polygon ; or number Angle at Number of Sides of Angle at of equal parts of the cir- Center. Kegular Polygon. Center. cumference. No. Degrees. No. Degrees. 3 120 12 30 4 90 13 27* 5 72 14 25f 6 60 15 24 7 51-f 16 22J 8 45 17 21* 9 40 18 20 10 36 19 18|f 11 32* 20 18 CONSTRUCTION OF THE ELLIPSE, PAEABOLA, HYPERBOLA, CYCLOID, EPICY- CLOID, INVOLUTE, AND SPIRAL. An ellipse is an oval-shaped curve (Fig. 74), in which, if from any point, P, lines be drawn to two fixed points, F and F', foci, their sum will always be the same. The line A B passing through the foci is the transverse axis, and the perpendicular C D at the cen- ter of it is the conjugate axis. PROB. XLV. To construct an ellipse, the axes being known (Fig. 75). 1st Method. Let the two axes be the lines A B and C D. From as a center, with a radius equal to E B (half the transverse axis), describe an arc cutting this axis at two points, F and F', which are the foci. Insert a pin in each of the foci, and loop a thread upon them, so that, when stretched by a pencil inside the loop, the point of the pencil will coincide with C. Move the pencil round, keeping the loop evenly stretched, and it will describe an ellipse. This construction follows the definition above given of an ellipse, that the sum of the distances of every point of the curve from the foci is equal. It is seldom used by the draughts- man, as it is difficult to keep a thread evenly stretched ; but for gardeners, laying out beds or plots, it is very convenient and sufficiently accurate. CONSTRUCTION OF GEOMETRICAL PROBLEMS. 2d Method. Carpenters, almost invariably, lay out an ellipse by means of a trammel (Fig. 76), which consists of a rectangular cross, E Gr F H, with guiding grooves, in which metal rods, attached to slides on a bar, are fitted so as to move easily and uniformly. In describing the ellipse, the tram- mel is placed with its grooves on the lines of the axes. Ad- just the metal points, Tc and Z, which slide in the grooves, so as to have between them a dis- tance equal to half the conju- gate axis, and make the dis- tance from k to m (the position on the bar of the pencil or marker) equal to half the transverse axis. Kevolve the bar, keeping the points Tc and I always in the grooves, and the pencil will describe an ellipse. Xeat instruments of this sort are made for the use of the draughtsman, but, for of- fices where curves of this sort are required but little, a substitute for the tram- mel can be had in a strip of paper (Fig. 77), by marking the straight edge at a and b and c, the distance c a being made equal to half the trans- verse axis, and the distance c b to half the conjugate FIG. 77. CONSTRUCTION OF GEOMET 25 axis. Revolving the strip of paper, keeping b on the line of the transverse axis, and c on the line of the conjugate axis, and dotting the positions of a at short intervals, enough points of the curve will be determined through which the ellipse may be drawn readily. PEOB. XLVI. To describe an ellipse approximately, by means of cir- cular arcs. First, with arcs of two radii (Fig. 78). Take the difference of the transverse and conjugate axes, and set it off from the center to / a and c, on A and C ; draw a c, and set off half a c to d; draw d i parallel to a c, set off e equal to d, join e i, and draw e m, d m, parallels to d i, i e. On cen- ter m, with radius m C, de- scribe an arc through C, and from center i describe an arc through D ; on centers d y e, also, describe arcs through A and B. The four arcs thus described form approximately an ellipse. This method does not apply satisfactorily when the conjugate axis is less than two thirds of the transverse axis. Second, with arcs of three radii (Fig. 79). On the transverse axis A B, draw the rectangle B G, equal in height to C, half the conjugate axis. Extend C above and below the rectangle. Draw Gr D perpendicular to A 0, intersecting C extended at D. Set off K equal to C, and on A K as a diameter describe the semicircle A N K ; draw a radius parallel to 0, 26 CONSTRUCTION OF GEOMETRICAL PROBLEMS. intersecting the semicircle at N and the line G E at P ; set off M equal to P N, and on D as a center, with a radius D M, describe an arc ; from A and B as centers, with a radius L, intersect this arc at a and b. The points H, #, D, b, H', are the centers of the arcs required ; produce the lines a H, D a. D b, b H', and the spaces inclosed determine the lengths of each arc. This process works well for nearly all proportions of ellipses. It is em- ployed in striking out vaults and stone bridges. PROB. XLVII. To draw a tangent to an ellipse through a given point in the curve (Fig. 80). From the given point T draw straight lines to the foci F, F'; produce F T beyond the curve to c, and bisect the exterior angle c T F' by the line T d. This line T d is the tangent required. PROB. XLVIII. To draw a tangent to an ellipse from a given point with- out the curve (Fig. 81). From the given point T as center, with a radius equal to its distance from the nearest focus F, describe an arc ; from the other focus F', with the trans- ;** \K verse axis as radius, cut the arc at K, L, and draw K F', L F', cutting the curve at M, N ; then the lines T M, T N, are tangents to the curve. The Parabola. The parabola may be defined as an ellipse whose transverse axis is infinite 5 its characteristic is that every point in the curve is equally distant from the directrix E N, and the focus F (Fig. 82). PROB. XLIX. To construct a parabola when the focus and directrix are given. CONSTRUCTION OF GEOMETRICAL PROBLEMS. 27 1st Method (Fig. 82). Through the focus F draw the axis A B perpendicu- lar to the directrix E N, and bisect A F at e, then e is the vertex of the curve. Through a series of points, C, D, E, on the directrix, draw parallels to A B ; connect these points, C, D, E, with the focus F, and bisect by perpendiculars the lines F C, F D, F E. The intersections of these perpendiculars with the par- allels will give points, C', D', E', in the curve, through which trace the parabola. 2d Method (Fig. 83). Place a straight-edge to the directrix E N, and apply to it a square LEG; fasten at G one end of a cord, equal in length FIG. 82. FIG. 83. ri to E G ; fix the other end to the focus F ; slide the square steadily along the straight-edge, holding the cord taut against the edge of the square by a pencil, D, and it will describe the curve. PKOB. L. To construct a parabola when the vertex, the axis, and a point of the curve are given (Fig. 84). Let A be the vertex, A B be the axis, and M the given point of the curve. Construct the rectangle A B- M 0. Divide M into any num- ber of equal parts, four, for in- stance ; divide A C in like man- ner ; draw A 1, A 2, A 3 ; through 1', 2', and 3', draw lines parallel to the axis. The intersections I, II, and III, of these lines are points in the required curve. The Hyperbola. An hyperbola is a curve from any point P, in which, if two straight lines be drawn to two fixed points, F, F', the foci, their difference shall always be the same. CONSTRUCTION OF GEOMETRICAL PROBLEMS. PKOB. LI. To describe an hyperbola (Fig. 85). From one of the foci F, with an assumed radius, describe an arc, and from the other focus F', with another radius exceeding the former by the given difference, describe two small arcs, cutting the first as at P and p. Let this operation be repeated with two new radii, taking care that the second shall exceed the first by the same difference as before, and two new points will be determined ; and this determination of points in the curve may thus be con- tinued till its track is obvious. By making use of the same radii, but trans- posing, that is, describing with the greater about F, and the less about F', we have another series of points equally belonging to the hyperbola, and answer- ing the definition ; so that the hyperbola consists of two separate branches. FIG. 85. FIG. 86. The curve may be described mechanically (Fig. 86). By fixing a ruler to one focus F', so that it may be turned round on this point, connect the other extremity of the ruler R to the other focus F by a cord shorter than the whole length F 7 R of the ruler by the given difference ; then a pencil P keep- ing this cord always stretched, and at the same time pressing against the edge of the ruler, will, as the ruler revolves around F', describe an hy- perbola, of which F F' are the foci, and the differences of distances from these points to every point in the curve will be the same. PEOB. LII. To draw a tangent to any point of an hyperbola (Fig. 87). Let P be the point. On F' P lay off P p, equal to F P ; draw the line F p ; from P let fall a perpendicular Fio 87 on this line, P p, and it will be the tangent required. The three curves, the ellipse, the parabola, and the hyperbola, are called conic sections, as they are formed by the intersections of a plane with the sur- face of a cone. See CONSTRUCTION OF THE CONIC SECTIONS. CONSTRUCTION OF GEOMETRICAL PROBLEMS. If the cone be cut through both its sides by a plane not parallel to the base, the section is an ellipse ; if the intersecting plane be parallel to the side of the cone, the section is a parabola ; if the plane have such a position that, when produced, it meets the opposite cone, the section is an hyperbola. The opposite cone is a reversed cone formed on the apex of the other by the con- tinuation of its sides. The Cycloid. The cycloid is the curve described by a point in the circumference of a circle rolling on a straight line. PKOB. LIII. To describe a cycloid (Fig. 88). Draw the straight line A B as the base ; describe the generating circle tan- gent at the center of this line, and through the center draw the line E E parallel to the base ; let fall a perpendicular from C upon the base ; divide the semi-circumference into any number of equal parts, for instance, six ; lay off on A B and E distances C I/ V 2'. . ., equal to the divisions of the circumference ; draw the chords D 1, D 2. . . ; from the points 1', 2', 3'. . .on the line C E, with radii equal to the generat- ing circle, describe arcs ; from the points 1', 2', 3', 4', 5', on the line B A, and with radii equal successively to the chords D 1, D 2, D 3, D 4, D 5, describe arcs cutting the preceding, and the intersections will be points of the curve required. 2d Method (Fig. 89). Let 9' be the base-line, 4 9 the half of the generating circle ; divide the half circle into any number of equal parts, say nine, and draw the chord 1, 2, Fl0 ' 89 * 3, etc. ; lay off on the base 1', I' 2', 2' 3' , equal respectively to the length of one of the divisions of the half circle 1 ; draw through the points 30 CONSTRUCTION OF GEOMETRICAL PROBLEMS. 1', 2', 3' lines parallel to the chords 1,0 2, 3 ; the intersections I, II, III of these lines are centers of the arcs a, al), I c , of which the cycloid is composed. The Epicycloid. The epicycloid is formed by a point in the circumference of a circle revolv- ing either externally or internally on the circumference of another circle as PKOB. L1V. To describe an epicycloid. Let us in the first place take the exterior curve. Divide the circumfer- ence A B D (Fig. 90) into a series of equal parts 1, 2, 3 , beginning from the point A ; set off in the same manner, upon the circle A M, A N, the divis- ions 1', 2', 3' equal to the divisions of the circumference A B D. Then, as the circle A B D rolls upon the circle A M A N, the points 1, 2, 3 will coincide successively with the points 1', 2', 3'; and, drawing radii from the point through the points 1', 2', 3', and also describing arcs of circles from the center 0, through the points 1, 2, 3 , they will intersect each other successively at the points c, d, e Take now the distance 1 to c, and set it off on the same arc from the point of intersection i, of the radius A C ; in like manner, set off the distance 2 to d, from b to A 2 , and the distance 3 to e, to A 8 , and so on. Then the points A 1 , A 2 , A 3 , will be so many points in the epicycloid ; and their frequency may be increased at pleasure by shortening ^^\. CONSTRUCTION OF GEOMETRICAL PROBLEMS. 31 the divisions of the circular arcs. Thus the form of the curve may be deter- mined to any amount of accuracy, and completed by tracing a line through the points found. As the distances 1 to c, which are near the commencement of the curve, must be very short, it may, in some instances, be more convenient to set off the whole distance i to 1 from c, and in the same way the distance b to 2 from d to A 2 , and so on. In this manner the form of the curve is the more likely to be accurately defined. 2d Method. To find the points in the curve, find the positions of the center of the rolling circle corresponding to the points of contact 1', 2', 3', etc., which may be readily done by producing the radii from the center 0, through the points 1', 2', 3', to cut the circle B C. From these centers describe arcs of a circle with the radius of C A, cutting the corresponding arcs described from the center 0, and passing through the points 1,2, 3, as before. The intersections of these arcs at A 1 , A 2 , A 3 , . . . .give points of the curve. When the moving circle A B D is made to roll on the interior of the cir- cumference A M, A N, as shown (Fig. 91), the curve described by the point x \ FIG. 91. A is called an interior epicycloid. It may be constructed in the same way as in the preceding case, as may be easily understood, the same figures and letters of reference being used in both figures. The Involute. The involute is a curve traced by the extremity of a flexible line unwind- ing from the circumference of a circle. 32 CONSTRUCTION OF GEOMETRICAL PROBLEMS. PROB. LV. To describe an involute. Divide the circumference of the given circle (Fig. 92) into any number of equal parts, as 0, 1, 2, 3, 4, ; at each of these points draw tangents to the FIG. 92. given circle ; on the first of these lay off the distance 11', equal to the arc 1 ; on the second lay off 2 2', equal to twice the arc 1 or the arc 2 : establish in a similar way the points 3', 4', 5', as far as may be requisite, which are points in the curve required. It may be remarked that, in all the problems in which curves have been determined by the position of points, the more numerous the points thus fixed, the more accurately can the curve be drawn. The involute curve may be described mechanically in several ways. Thus, let A (Fig. 93) be the center of a wheel for which the form of involute teeth is to be found. Let m n a be a thread lapped round its circumference, having a loop-hole at its extrem- ity, a; in this fix a pin, with which describe the curve or in- volute a b h, by unwinding the thread gradually from the circumference, and this curve will be the proper form for the teeth of a wheel of the given diameter. The Spiral The spiral is the involute of a circle produced beyond a single revolution. CONSTRUCTION OF GEOMETRICAL PROBLEMS. 33 PROB. LVL To describe a spiral (Fig. 94, and Fig. 95 of the primary on a larger scale). Divide the circumference of the primary into any number of equal parts, say not less than eight. To these points of division o, e, f, i, etc., draw tangents, and from these points draw a succession of circular arcs ; thus, from o e lay FIG. 94. FIG. 95. off o g, equal to the arc a o reduced to a straight line, and connect a and g by a curve ; from e, with the radius e g, describe the arc g h ; from / the next arc, and so on. Continue the use of the centers successively and repeatedly to the extent of the revolutions required. Thus the point a in the figure is used as a center for three arcs, b I, c m, d n. USE or TKIAKGLE A:NT> SQTJABE. Right-angled triangles constructed of wood, hard rubber, or metal, are very useful in connection with a straight-edge, or ruler, in drawing lines parallel or perpendicular to other lines. To draw lines parallel to each other, place any edge of the triangle in close contact with the edge of the ruler. Hold the ruler (Fig. 96) firmly with the 3 34 CONSTRUCTION OF GEOMETRICAL PROBLEMS. thumb and little finger of the left hand, and the triangle with the other three fingers ; with a pencil or pen in the right hand, draw a line along one of the free edges of the triangle ; withdraw the pressure of the three fingers upon the FIG. 96. triangle, and slide it along the edge of the ruler, keeping the edges in close contact ; a line drawn along the same edge of the triangle, as before, will be parallel to the first line. If the edge of the hypothenuse of the triangle be placed in contact with the ruler, lines drawn along one edge of the triangle will be at right angles to those drawn along the other. FIG. 97. PROB. LVII. Through a given point to draw a line parallel to a given line (Fig. 97). Place one of the shorter edges of the triangle along the given line A B, and CONSTRUCTION OF GEOMETRICAL PROBLEMS. 35 bring the ruler against the hypothenuse ; slide the triangle up along the edge of the ruler until the upper edge of the ruler is sufficiently near to the given 2k 'V 4 -\ E D r^ FIG. 98. FIG. 99. point C to allow a line to be drawn through it. Draw the line, and it will be parallel to A B. If the triangle be slid farther up along the edge of the ruler, and a line be drawn through C along the other edge of the triangle (Fig. 98), this line will be perpendicular to A B. If the triangle be slid still farther up along the edge of the ruler, and a third line be drawn touching A B, the figure con- structed will be a rectangle ; and if E D be laid off on A B, equal to C E, the fig- ure inclosed is a square (Fig. 99). It will be seen that the triangle and ruler afford a much readier way of draw- ing parallel lines, and lines at right an- gles, than the compasses and ruler, and may be used in solving the following problems : The area of a figure is the quantity of space inclosed by its lines. Construct a right angle (Fig. 100). Divide the base and the perpendicular by dividers into any number of equal spaces ; for instance, eight on the one and five on the other. Construct a rectangle with this base and perpendicu- lar, and through the points of division lay off lines parallel to the base and perpendicular. The rectangle will be divided into forty equal squares, and its measure in squares will be the divisions eight in the base, multiplied by the five in the perpendicular. If the division were inches, then the area of FIG. 100. FIG. 101. B FIG. 102. this rectangle would be forty square inches ; if feet, then forty square feet. If there were but five divisions in the base and five in the perpendicular, the surface would be twenty-five squares. Therefore, a rectangle has for its measure the base multiplied by its adjacent side or height. 36 CONSTRUCTION OF GEOMETRICAL PROBLEMS. Draw a diagonal, and the rectangle is divided into two equal triangles. Each triangle must therefore have for its measure the base multiplied by half the perpendicular, or, as is usually said, by half the altitude. Take any triangle (Fig. 101), and from its apex draw a line perpendicular to the base. The triangle is divided into two right-angled triangles, which must have for their measure A D x C D, and D B x ^ C D, and the sum of the two must be A B x C D. If the perpendicular from the apex falls outside the triangle (Fig. 102), then the triangles B D C and ADC will have for their measure B D x J C D and A D x C D ; and as the origi- nal triangle A B C is the difference of these two triangles, its measure must be A B x C D. Every triangle must have for its measure the base multi- plied by half the altitude, and it makes no difference which side is taken as the base. Construct the right-angled triangle A C B (Fig. 103), and let fall the FlG - 103 - perpendicular C D. As will be seen by the equality of the angles compos- ing the triangles, the perpendicular divides the original triangle into two right- angled triangles, similar to each other and to the original triangle. Therefore FIG. 104. A D is to C D as C D is to B D, or, expressed by signs, A D : C D : : C D : B D ; therefore, by the Rule of Three, A D x B D = C D 2 ; that is, C D is a mean proportional between A D and B D. So that the perpendicular let fall CONSTRUCTION OF GEOMETRICAL PROBLEMS. from the vertex of a right angle upon the hypothenuse of the triangle, is a mean proportional between the two parts of the hypothenuse into which it is divided by the perpendicular. In comparing the two triangles with the original triangle, A C is a mean proportional between A D and A B, and B C is a mean proportional between B D and A B ; that is, A C 2 =A Dx A B BC 2 =BDxAB adding the two, A C a +B C 2 = (A D+B D)xA B and as A D + B D = A B, we have A C a + B C 2 = A B 2 ; that is, the square on the hypothenuse is equal to the sum of the squares on the other two sides. Construct squares on the three sides of a right-angled triangle (Fig. 104). b FIG. 105. FlCr. 106. PROB. LVIII. To construct a square equal to one half of a given square (Fig. 105). FIG. 107. FIG. 108. Construct the given square, and draw diagonals in it. The square, abed, constructed on one half of one of these diagonals will be equal to one half the given square. 38 CONSTRUCTION OF GEOMETRICAL PROBLEMS. PKOB. LIX. To construct a square equal to double a given square (Fig. 106). Construct a square on one of the diagonals in the given square, or en- close the square with parallels to the diagonals of the square. PEOB. LX. To construct a square equal to three times a given square (Fig. 107). Extend the base of the given square, and lay off on it the length of its diagonal. Draw a line from the point at which this diagonal ends to the ex- treme angle of the square, and upon this line erect a square, which will be the square required. For a square four times the size of a given square, make the base double that of the given square. PKOB. LXI. To construct a square equal to five times a given square (Fig. 108). Extend the base of the given square, making the extension to d equal to the base of the given square. From d draw a line to a, and on this line con- struct a square, abed, which will be the square required. FIG. 109. Assuming the side of the given square in Figs. 105, 106, 107, and 108 to be the radius (or diameter) (Fig. 109) of a given circle, then the side of the square to be constructed half, twice, three, four, or five times the size of the given square will be the radii (or diameters) of the circles half, twice, three, four, or five times the size of the given circle. CONSTRUCTION OF GEOMETRICAL PROBLEMS. 39 PKOB. LXII. To determine how much is added to a given square by extending its base and constructing a square thereon (Fig. 110). p c FIG. 110. H K J Let a represent the length C D of the base of the given square. Its square will be a X a or a? . Extend the base C D by a certain length, D G, represented by I. Then the new square (a + b) x (a + b) will be made up of the old square, or a 2 and two rectangles, D G E H and P E K L, or 2 (a x b) or 2 a b and one square, E H K J, or b x b or b 2 PROB. LXIII. To determine how much is taken from the area of a given square, by reducing its base and constructing a square (Fig. 110). Let a represent the length C G of the base of the given square. Reduce C G by a certain length, G D, to be represented by b. Then the new square (a b)* will be the old square, or a 9 diminished by two rectangles, D G J K and P L J H, or 2 a b excepting one square, E H J K, or b x b or 4- b* The last two constructions, in default of a table of squares, may often be found of use. DRAWING INSTRUMENTS. THE simple drawing instruments, already illustrated and applied in the construction of the preceding problems, together with scales of equal parts, a protractor and a drawing pen, are all the instruments essential for topo- graphical or mechanical drawing. It is often convenient, for facility in work- ing, to have compasses of varied sizes and modifications, and these, together with an assortment of rulers, triangles, squares, scales, and protractors, adapted to varied work, are included in boxes of drawing instruments as furnished by dealers. The smaller rulers and triangles, as furnished, are generally of hard rubber, and the larger of wood. As it is often incon- venient to carry long rulers, and difficult to procure them ready-made, the draughtsman may have to depend on a carpenter or joiner for them. Eulers should be of close-grained, thoroughly - seasoned wood, such as mahogany, maple, pear, etc. They should be about -J of an inch thick in the square or slightly rounded edges, 1 to 2% inches wide, according to their length. As the accuracy of a drawing depends greatly on the straightness of the lines, the edge of the ruler should be perfectly straight. To test this, place a sheet of paper on a perfectly smooth board ; insert two very fine needles in an upright position through the paper into the board, distant from each other nearly the length of the ruler to be tested ; bring the edge of the ruler against these needles, and draw a line from one needle to the other ; reverse the ruler, bringing the same edge on the opposite side and against the needles, and again draw a line. If the two lines coincide, the edge is straight ; but, if they disagree, the ruler is inaccurate, and must be re-jointed. When one ruler has been tested, the other can be examined by placing their edges against the correct one, and holding them between the eye and the light. Triangles may be made of the same kinds of wood as the ruler, and some- what thinner, and of various sizes. They should be right-angled, with acute angles of 45, or of 60 and 30. The most convenient size for general use measures from 3 to 6 inches on the side. A larger size, from 8 to 10 inches long on the side, is convenient for making drawings to a large scale. Circular openings are made in the body of the triangle for the insertion of the end of the finger to give facility in sliding the triangle on the paper. Triangles are sometioies made as large as 15 to 18 inches on the side ; but in this case they are framed in three pieces of about 1J wide, leaving the center of the triangle open. The value of the triangle in drawing perpendicular lines depends on the accuracy of the right angle. To test this (Fig. Ill), draw a line with an DRAWING INSTR 41 accurate ruler on paper. Place the right angle of the triangle near the center of this line, and make one of the adjacent sides to coincide with the line ; now draw a line along the other adjacent side, which, if the angle is strictly a right angle, will be perpendicular to the first line. Turn the triangle on this perpendicular side, bringing it into the posi- tion ABC'; if now the sides of the triangle agree with the line B C' and A B, the angle is a right angle, and the sides straight. If they do not agree, they must be made to do so with a plane, if right angles are to be drawn by the triangle. The straightness of the hypothenuse or longest side can be tested like a common ruler. The T square is a thin " straight edge " or ruler, a (Fig. 112), fitted at one end with a stock, b, applied transversely at right angles. The stock being so formed as to fit and slide against one edge of the drawing-board, the blade reaches over the surface, and presents an edge of its own at right angles to FIG. 111. FIG. 112. that of the board, by which parallel straight lines may be drawn upon the paper. The stock should be long enough to give sufficient bearing on the edge of the board, and heavy enough to act as a balance to the blade, and to relieve the operation of handling the square. The blade should be sunk flush into the upper half of the stock on the inside, and very exactly fitted. It should be inserted full breadth, as shown in the figure ; notching and dove- tailing is a mistake, as it weakens the blade, and adds nothing to the secu- rity. The upper half of the stock should be about \ inch broader than the lower half, to rest firmly on the board and secure the blade lying flatly on the paper. One half of the stock, c (Fig. 113), is in some cases made loose, to tarn FIG. 113. upon a brass swivel to any angle with the blade a, and to be clenched by a screwed nut and washer. The loose stock is useful for drawing parallel lines 42 DRAWING INSTRUMENTS. obliquely to the edges of the board, such as the threads of screws, oblique- columns, and connecting-roads of steam-engines. In many drawing-cases will be found the parallel ruler (Fig. 114), consist- ing of two rulers connected by two bars moving on pivots, and so adjusted that the rulers, as they open, form the sides of a parallelo- e^ G^ | gram. The edge of one of the rulers being retained in a position coinciding with, or parallel to, a given line, the \ \> I other ruler may be moved, and lines drawn along its edge must also be parallel to the given line. This instrument is only useful in drawing small parallels, and in accuracy and convenience does not compare with the triangle and ruler, or T square. An improvement on the above parallel ruler has been patented by Lieuten- ant-Commander Sigsbee, U. S. N. (Fig. 115), in which the blades are* made FIG. 115. with hinges, by which, holding one blade on the paper, the other may be raised over creases or torn edges of the paper, or over thumb-tacks. One blade can be raised, if necessary, at right angles to the other, still preserving the parallel- ism of lines that may be drawn along these edges. Small cushions of rubber inserted in the blades, pressed by the fingers, prevent the slipping of the blades. FIG. 116. SWEEPS AND VARIABLE CURVES. For drawing arcs of a large radius, beyond the range of ordinary com- passes, and lines not circular but varying in curvature, thin slips of wood, DRAWING INSTRUMENTS. FIG. 117. termed sweeps (Figs. 116 and 117), are usually employed. These two forms are of very general application, but others of almost every form, and made of hard rubber, can be purchased. Whatever be the nature of the curve, some portion of the sweep will be found to coincide with its commencement, and it can be continued throughout its extent by applying, successively, such parts of the sweep as are suitable, care being taken that the parts are tangent to each other, and that the continuity is not injured by unskillful junction. No varnish of any description should be applied to any of the wooden instruments used in drawing, as the best varnish will retain dust, and soil the paper. Use the wood in its natural state, keeping it care- fully wiped. Various other materials besides wood have been used, as steel for the blades of the T square and the ruler ; the objection is the liability to soil the paper. Glass is frequently used for the ruler and the triangle, and retains its correctness of edge and angle, but it is too heavy, and liable, of course, to fracture. Thin splines are also to be had, which, held in position by leaden weights, serve admirably for a guide to the pen in describing curves (Fig. 118). For the same purpose a thin, hard rubber ruler, with soft rubber backing, answers well, and, as it can be readily rolled up, is extremely portable. The weights above shown are very convenient in holding the drawing-paper on the board, but the drawing-pins (Fig. 119), steel points, or tacks, with large, flat heads, are in general use. Elliptic and parabolic curves are furnished in sets, but the draughtsman can readily make a model out of thick card-board, with which he can draw a very uniform curve. For the drawing of ellipses, very neat trammels or com- passes, with elliptic guides or patterns, may be purchased. The drawing-pen (Fig. 120) is used for drawing straight lines. It consists of two blades with steel points fixed to a handle ; and they are so bent that a sufficient cavity is left between them for the ink, when the ends of the steel points meet close together, or nearly so. The blades are set with the points more or less open by means of a mill- headed screw, so as to draw lines of any required fineness or thickness. One of the blades is framed with a joint, so that by taking out the screw the blades may be completely opened, and the points effectively cleaned after use. The ink is to be put between the blades by a common pen, and in using the pen it should be slightly inclined in the direction of the line to be drawn, and care should be taken that both points touch the paper ; and these observations equally apply to the pen-points of the compasses before described. The drawing-pen should be kept close to the ruler or straight edge, and in the same direction during FIG. 118. 44 DRAWING INSTRUMENTS. the whole operation of drawing the line. Care must be taken in holding the straight edge firmly with the left hand, that it does not change its position. For drawing close parallel lines in mechanical and architectural drawings, or to represent canals or roads, a double pen (Fig. 121) is frequently used, with an adjust- ing screw to set the pens to any required small distance. This is usually called the road-pen. Border-pens, for drawing broad lines, are double pens with an intermediate blade, and are applicable to the drawing of map-borders. The same work may be done by drawing the outer lines with the common drawing-pen, and filling in with a goose-quill, cut as shown in Fig. 122. In drawing with this pen, incline the drawing-board so that the ink will follow the pen. The curve-pen (Fig. 123) is especially designed for the ready drawing of curved lines. The dotting-point (Fig. 124) resembles a drawing-pen, except that the points are not so sharp. On the back blade, as seen in the engraving, is a pivot, on which may be placed a dotting-wheel, , resembling the rowel of a spur ; the screw ~b is for opening the blades to remove the wheel for cleaning after use, or replacing it with one of another character of dot. The cap c, at the upper end of the instrument, is a box containing a variety of dotting- wheels, each producing a different-shaped dot. These are used as distinguishing marks for different classes of bound- aries on maps ; for instance, one kind of dot distinguishes county boundaries, another kind town boundaries, a third kind distinguishes that which is both a county and a town boundary, etc., etc. In using this instrument, the ink must be inserted between the blades above FIG. 120. FIG. 121. FIG. 122. the dotting-wheel, so that, as the wheel revolves, the points shall pass through the ink, each carrying with it a drop, and marking the paper as it passes. It sometimes happens that the wheel will revolve many times before it begins to deposit its ink on the drawing, thereby leaving the first part of the line altogether blank, and, in attempting to go over it again, the first-made dots are liable to get blotted. This evil may be mostly remedied by placing a piece of blank paper over the drawing to the very point the dotted line is to com- FIG. 123. DRAWING INSTRUMENTS. mence at, then begin with drawing the wheel over the blank paper first, so that, by the time it will have arrived at the proper point of commencement, the ink may be expected to flow over the points of the wheel, and make the dotted line perfect as required. The best pricking-point (Fig. 125) is a fine needle held in a pair of for- ceps, and is used to transfer drawings by pricking through at the points of a drawing into the paper placed beneath. When drawings are transferred by I FIG. 124. FIG. 125. FIG. 126. tracing a prepared black sheet being placed between the drawing and the paper to receive the tracing the eye-end of the needle forms a good tracing- point. Compasses, in addition to pencil-points, as before shown, are fitted with movable ink-points and lengthening bars, so that larger circles may be drawn. Compasses should have joints in the legs, so that the points, pencil, and pen may be set perpendicular to the planes in which the circles are described (Fig. 126). Compasses of this general form may be had in sizes of 3 to 7 inches. For the measurement and laying off of small spaces, and the describing of small circles, there are small bow- compasses (Fig. 127). These are sometimes made with jointed legs. For the measurement or laying off of distances the plain dividers are convenient, but for ready and close adjustment the hair dividers (Fig. 128) are most suit- able. The only difference is that, in the hair dividers, FIG. 127. DRAWING INSTRUMENTS. one of the points is attached to the body by a spring, and by means of the screw b it can be moved toward or from the fixed point a very small amount more accurately than by closing or opening the dividers. In dividing a line into equal parts especially, it enables one to divide the excess or deficit readily. Large screw dividers (Fig. 129) are used for the same purpose, but they belong rather to the mechanic than to the draughtsman. For convenience of carrying in the pocket, there are portable or turn-in compasses (Fig. 130). FIG. 128. FIG. 129. For setting off very long lines, or describing circles of large radius, learn compasses are used (Fig. 131). These consist of a mere slip of wood, A FIG. 130. which is readily procured ; two brass boxes, B and 0, which can easily be attached to the beam, and connected with the brass boxes are the two points of the instrument, G and H. The object of this instrument is the nice adjust- ment of the points G and H at any definite distance apart ; at F is a slow- motion screw, by which the joint G may be moved any very minute quantity after the distance from F to G has been adjusted as nicely as possible by the hand alone. The important parts of this instrument can be carried in a very small compass. There are beam compasses in which the beam is graduated, and in which the boxes corresponding to B and 0, in Fig. 131, are fitted with vernier or reading plates, to afford the means of minutely subdividing the divisions on the beam. DRAWING INSTRUMENTS. 47 Proportional dividers (Fig. 132), for copying and reducing drawings, are found in most cases of instruments. Closing the dividers and loosening the screw 0, the slide may be moved up in the groove until the mark on the slide or index corresponds with the required number; then clamping the screw, the space inclosed between the long points, A B, will be as many times that between the short points, E D, as is shown by the number opposite the in- dex. If the lines are to be reduced, the distances are measured with the long points, and set off by the short ones ; if the lines .are to be enlarged, then vice versa. It often happens that the length of the points becomes re- FKI. 131. FIG. 132. duced by use or accident, and the division on the instrument then becomes useless, but the purpose may be served by trial on paper, moving the slide up or down until a measured line is reduced or enlarged, as required* SCALES. Practically, a two-foot rule, with its division into inches, half inch, quarter inch, eighth inch, and sixteenth inch, may be made use of as a scale of equal parts, the inch or any of its parts being taken as the unit to represent a foot, a yard, or a mile ; but among drawing instruments, scales especially adapted to the purpose are found in great varieties of form, division, and material. Fig. 133 represents the usual scale to be found in the common boxes of drawing instruments. It contains, on its two sides, simply divided scales a diagonal scale on the reverse side and a protractor along the edges. The simply divided scales consist of a series of equal divisions of an inch, which are numbered 1, 2, 3, etc., beginning from the second division on the left hand ; the upper part of the left division in each is subdivided into 12 equal parts, and the lower part into 10 equal parts. In Fig. 134 the scales are marked 30, 35, 40, etc., and the subdivisions of tenths can be considered as units, one mile, or one chain, or one foot, then each primary division will 48 DRAWING INSTRUMENTS. represent ten units, ten miles, ten chains, or ten feet, and the scale is said to be 30, 35, 40 (according to the scale selected) miles, chains, or feet to the inch. Thus, suppose that it were required, on a scale of 30 feet to the inch, to lay off 47 feet. On the scale marked 30, place one point of the compasses or dividers at 4, and bring the other point to the seventh lower subdivisions, counting from the right, and we have the distance required. Each of the primary divisions may be regarded as unit, one foot for instance ; then the upper sub- divisions are twelfths of a foot or inches, and the lower subdivisions tenths of an inch. In Fig. 133 the scales are marked at the left, 1 inch, f , , ; the primary divisions are 1 inch, f, -J, and i of an inch. These scales are more generally used for drawings of machinery and of architecture, while those of Fig. 134 are for topographical drawings. The applications of these scales are similar to those already described. When the primary divis- ions are considered inches, then the drawings will be each full, f, -J, or \ size, according to the scale adopted. On the selection of the scale. In all work- ing architectural and mechanical drawings, use as large a scale as possible ; neither de- pend, even in that case, that the mechanics employed in the construction will measure correctly, but write in the dimensions as far as practicable. For architectural plans, the scale of J- an inch to the foot is one of very general use, and convenient for the mechanic, as the common two-foot rule carried by all mechanics is subdivided into ^ths, ^ths, and sometimes sixteenths of an inch, and the dis- tances on a drawing to this scale can therefore be easily measured by them. This fact should not be lost sight of in working drawings. When the dimensions are not written, make use of such scales that the distances may be measured by the subdivisions of the common two-foot rule ; thus, in a scale of i or ^ full size, 6 inches or 3 inches rep- resent one foot ; in a scale of an inch to the foot or twelfth full size, each i an inch represents 6 inches, i of an inch, 3 inches ; but when or T V an inch to the foot, or any similar scale, is adopted, it is evident that these divisions can not be taken by the two-foot rule. The scale should be writ- FIG. 133. DRAWING INSTRUMENTS. 49 ten on every drawing, or the scale itself should be drawn on the margin. In topographical and geodesic drawings the latter is essential, as the scale adopted frequently has to be drawn for the specific purpose, and the paper ^ t t [ 8 Jo i : i, I j. i 50 ;ip i | i i I L -? -, L -1 - 4 ; * it 5 -i^p- i 2 i s 1 ,1 I , it frfl "p 1, [ IG Is 1 l\0 [ 35 -^p 5 2 j L 1 L 30 * 1 1 1 ^ FIG. 134. itself contracts or expands with every atmospheric change, and the measure- ments will therefore not agree at all times with a detached scale ; and, more- over, a drawing laid down from such a detached scale, of wood or ivory, will not be uniform throughout, for on a damp day the measurements will be too short, and on a dry day too long. Mr. Holtzapffel has sought to remedy this inconvenience by the introduction of paper scales ; but all kinds of paper do not contract and expand equally, and the error is therefore only partially cor- rected by his ingenious substitution of one material for another. tn 1 ' 1 2 1 3 4 \ 6 1 6 7 | 96 3 | X n yi Si 8 tj e 9j i S t 01 I 1 Sit 8;l fr I S m III 1 \ 1 1 1 1 1 1 1 1 1 1 1 FIG. 135. Plotting scales (Fig. 135) are scales of equal parts, with the divisions on a fiducial edge, by which any length may be marked off on the paper without using dividers. There are also small offset scales, for use of which see " Topo- graphical Drawing." Sometimes these scales are made with edges chamfered on both sides, and graduated to four different scales. Sometimes the section of the scale is tri- angular (Fig. 136), with six scales on the different edges. Both of these scales are convenient as portable instruments. To avoid the objection that having A FIG. 136. many scales on one ruler leads the draughtsman into error by the confusion of the scales, the triangular has a small slip of metal, A, readily put on, which covers partially the scales not in use. 50 DRAWING INSTRUMENTS. To divide a given line into any number of equal parts (Fig. 137). Let A B be the line, and the number of parts be ten. Draw a perpendicu- lar at one extremity, A, of the line ; with a plotting scale place the zero at the other extremity, B, of the line ; make the mark 10 on the scale coincide with the perpendicular ; draw a line along the edge of the scale, and mark the line at each division of the scale 1 to 9 ; draw perpendiculars through these marks to the line A B, and they will divide A B into ten equal parts. The construction is based on the principle of the proportions of parts between similar triangles, and it is evident that if the perpendicular at 1 be taken as a unit, that at 2 will be two units, and so on. This way of dividing a line will often be found convenient in practice. The lines may be at any angle to each other, and the lines connecting the divisions must be parallel to the line completing the triangle. The above figure illustrates the construction of diagonal scales. The simply divided scales give only two denominations, primaries and tenths, or twelfths ; but more minute subdivision is attained by the diagonal scale, which consists of a number of primary divisions, one of which is divided into tenths, and subdivided into hundredths by diagonal lines (Fig. 138). This scale is constructed in the following manner : Eleven FIG. 137. Fm. 138. parallel lines are ruled, inclosing ten equal spaces ; the length is set off into equal primary divisions, as D E, E 1, etc. ; the first D E is subdivided, and diagonals are then drawn from the subdivisions between A and B, to those FIG. 139. between D and E, as shown in the diagram. Hence it is evident that at every parallel we get an additional tenth of the subdivisions, or a hundredth of the DRAWING INSTRUMENTS; stlfli 51 primaries, and can therefore obtain a measurement with great exactness to three places of figures. To take a measurement of (say) 168, we place one foot of the dividers on the primary 1, and carry it down to the ninth parallel, and then extend the other foot to the intersection of the diagonal, which falls from the subdivision 6, with the parallel that measures the eight-hundredth part (Fig. 139). The primaries may, of course, be considered as yards, feet, or inches ; and the subdivisions as tenths and hundredths of these respective denominations. The diagonals may be applied to a scale where only one subdivision is required. Thus, if seven lines be (Fig. 140) ruled, inclosing six equal spaces, 7/V s / V 9/ -V- 7 \2 / \1 1 \ 0/2 FIG. 140. and the length be divided into primaries, as A B, B 0, etc., the first primary, A B, may be subdivided into twelfths by two diagonals running from 6, the middle of A B, to 12 and 0. We have here a very convenient scale of feet and inches. From C to 6 is 1 foot 6 inches ; and from C on the several parallels to the various intersections of the diagonals we obtain 1 foot and any number of inches from 1 to 12. Vernier scales are preferred by some to the diagonal scale already de- scribed. To construct a vernier scale (Fig. 141) by which a number to three places may be taken, divide all the primary divisions into tenths, and number 10 2 4 6 8 f f l f || I I I I I I I I I I I I I I I I I I I I I I I i I I I I I I I I I I I I I ._ I I I I I I I 100 fr a 6 J 4 2 FIG. 141. these subdivisions 1, 2, 3, from left to right. Take off now with the com- passes eleven of these subdivisions, set the extent off backward from the end of the first primary division, and it will reach beyond the beginning of this division, or zero point, a distance equal to one of the subdivisions. Now divide the extent thus set off into ten equal parts, marking the divisions on the opposite side of the divided line to the lines marking the primary divisions and the subdivisions, and number them 1, 2, 3, etc., backward from right to left. Then, since the extent of eleven subdivisions has been divided into ten equal parts, so that these ten parts exceed by one subdivision the extent of ten subdivisions, each one of these equal parts, or, as it may be called, one division of the vernier scale, exceeds one of the subdivisions by a tenth part of a sub- division, or a hundredth part of a primary division ; thus, if the subdivision be considered 10, then from to the first division of the vernier will be 11 ; to the second, 22 ; to the third, 33 ; to the fourth, 44 ; to the fifth, 55, and so on, 66, 77, 88, 99. 52 DRAWING INSTRUMENTS. To take off the number 253 from this scale, place one point of the dividers at the third division of the vernier ; if the other point be brought to the pri- mary division 2, the distance embraced by the dividers will be 233, and the dividers must be extended to the second subdivision of tenths to the right of 2. If the number were 213, then the dividers would have to be closed to the sec- ond subdivision of tenths to the left of 2. To take off the number 59 from the scale, place one point of the dividers at the ninth division of the vernier ; if the other point be extended to the mark, the di- viders will embrace 99, and must therefore be closed to the fourth subdivision to the left of 0. These numbers, thus taken, may be 253, 25 '3, 2-53 ; 213, 21 -3, 2'13 ; 59, 5 -9, .59, according as the primary divisions are taken as hundreds, tens, or units. The construction of this scale is similar to that of the verniers of theodolites and surveying instru- ments ; but, in its application to drawing, is not as simple as the diagonal scales (Figs. 138, 140). The sector (Fig. 142), now seldom used, consists of two flat rulers united by a central joint, and open- ing like a pair of compasses. It carries several plain scales on its faces, but its most important lines are in the pairs or double scales, running accurately to the central joint. The principle on which the double scales are con- structed is that similar triangles have their like sides proportional (Fig. 143). Let the ^ ^C lines A B, AC, represent the legs of the sector, and A D, A E, two equal sections from the center ; then, if the points B and D E be connected, the lines B C and D E will be parallel ; therefore, the triangles A B C, A D E, will be similar, and, consequently, the sides A B, B C, A D, D E, propor- tional that is, as A B : B C : : A D : D E ; so that if A D be the half, third, or fourth part of A B, then D E will be a half, third, or fourth part of B C ; and the same holds of all the rest. Hence, if D E be the chord, sine, or tangent of any arc, or of any number of degrees to the radius A D, then B C will be the same to the radius A B. Thus, at every opening of the sector, the trans- DRAWING INSTRUMENTS. 53 verse distances D E and C B from one ruler to another are proportional to the lateral distances, measured on the lines A B, A C. It is to be observed that all measures are to be taken from the inner lines, since these only run accurately to the center. On the scale in common boxes of drawing instruments, the edge of one side is divided as a protractor, for the laying out of angles, whose use will be readily understood from the description of the instrument, when by itself. It consists of a semicircle of thin metal or horn (Fig. 144), whose cir- cumference is divided into 180 equal parts or degrees (180). In the larger protractors each of these divisions is subdivided. Application of the protractor (Fig. 144). To lay off a given angle from a given point on a straight line, let the straight line a b of the protractor coin- cide with the given line, and the point c with the given point ; now mark on the paper against the division on the periphery coinciding with the angle required ; remove the protractor, and draw a line through the given point and the mark. For plotting field-notes expeditiously, drawing paper can be obtained with large, full circular protractors printed thereon, on which the courses can be readily marked, and thus transferred to the part of the paper required by a parallel ruler, or by triangle and ruler. These sheets are of especial use in plotting at night the day's work, as, on account of the large size of protractor, angles can be laid off with greater accuracy than by the usual protractor of a drawing-instrument case, with less confusion of courses, and more expe- ditiously. For accurate plotting of angles, the circular protractor (Fig. 145) is one of the best. It is a complete circle, A A, connected with its center by four radii, a a a a. The center is left open, and surrounded by a concentric ring or collar, &, which carries two radial bars, c c. To the extremity of one bar is a pinion, d, working in a toothed rack quite round the outer circumference of the pro- tractor. To the opposite extremity of the other bar, c, is fixed a vernier, which subdivides the primary divisions on the protractor to single minutes, 54 DRAWING INSTRUMENTS. and by estimation to 30 seconds. This vernier is carried round the pro- tractor by turning the pinion d. Upon each radial bar, c c, is placed a branch, ee, carrying at their extremities a fine steel pricker, whose points are kept above the surface of the paper by springs placed under their supports, which give way when the branches are pressed downward, and allow the points to FIG. 145. make the necessary punctures in the paper. The branches e e are attached to- the bars c c with a joint which admits of their being folded backward over the instrument when not in use, and for packing in its case. The center of the instrument is represented by the intersection of two lines drawn at right angles to each other on a piece of plate glass, which enables the person using it to place it so that the center or intersection of the cross-lines may coincide with any given point on the plan. If the instrument is in correct order, a line connecting the fine pricking points with each other would pass through the center of the instrument, as denoted by the before-mentioned intersection of the cross-lines upon the glass. In using this instrument, the vernier should first be set to zero (or the division marked 360) on the divided limb, and then placed on the paper, so that the two fine steel points may be on the given line (from whence other and angular lines are to be drawn), and the center of the instrument coincides with the given angular point on such line. This done, press the protractor gently down, which will fix it in position by means of very fine points on the under side. It is now ready to lay off the given angle, or any number of angles that may be required, which is done by turning the pinion d till the opposite vernier reads the required angle. Then press downward the branches e e, which will cause the points to make punctures in the paper at opposite sides of the circle ; which being afterward connected, the line will pass through the given angular point, if the instrument was first correctly set. In this manner, at one setting of the instrument, a great number of angles may be laid off from the same point. The pantagraphs are used for the copying of drawings either on the same scale, on a reduced scale, or on an enlarged scale, as may be required. The DRAWING INSTRUMENTS. 55 form of pantagraph as shown in Fig. 146 consists of a set of jointed rulers, A, B, and another, C, D, about one half the length of the former. The free ends of the smaller set are jointed to the larger at about the center. Casters are placed at a a, etc., to support the instrument and to allow an easy move- ment over the paper. The rulers A and C are divided with a scale of propor- tional parts, marked i, -J, etc. These arms are also provided with movable indices, E, F, which can be fastened at any division by clamp screws. Each index is provided with a socket adapted to carry either a pencil or a tracing point. Fig. 146 represents the instrument in the act of reducing the plan H to h, one half the size. The tracing point is placed in the socket at E, the pencil at F, and the fulcrum at G. The indices, E, F, are clamped each at on the scales. If the instrument is correct, the points E, F, G, are in a straight line. Pass the tracing point delicately over the plan H, and the pencil point F will trace h, one half the original size. If the object had been to enlarge the drawing to double its scale, then the tracer must have been placed at F, and the pencil at E. And if a copy be required, retaining the scale of the original, then the slides E and F must be placed at the divisions marked 1. The fulcrum must take the middle sta- tion, and the pencil and tracer those on the exterior rules A and B of the instrument. Another form of this instrument is shown in Fig. 147. FIG. 147. The camera lucida is sometimes used for copying and reducing topograph- ical drawings. A description of the use of this instrument will be found under the head of topographical drawing. The drawing table and drawing board. The usual size of the drawing table should be from 5 to 6 feet long and 3 feet wide, of 1|- or 2-inch white pine plank well seasoned, without any knots, closely joined, glued, doweled, and clamped. It should be fixed on a strong, firm frame and legs, and of such 56 DRAWING INSTRUMENTS. a height that the draughtsman, as he stands up, may not have to stoop to his work. The table is usually provided with a shallow drawer to hold paper or drawings. Drawing tables are made portable by having two horses for their supports, and a movable drawing board for the top ; this board is made similar to the top of the drawing table, but of inch boards, and barred at the ends. Various woods are used for the purposes, but white pine is by far the cheapest and best. Drawing boards should be made truly rectangular, and with per- fectly straight sides for the use of the T square. Two sizes are sufficient for common purposes, 41 X 30 inches to carry double elephant paper with a mar- gin, and 31 X 24 inches for imperial and smaller sizes. Boards smaller than this are too light and unsteady in handling. Small boards are occasionally made, as loose panels fitting into a frame, flush on the drawing surface, with buttons on the back to secure them in position. The panel is mostly of white pine, with a hard-wood frame. DKAWIKG PAPER. Hand-made drawing paper is usually made to certain standard sizes about as follows : Demy ........... 20 inches by inches. Medium Eoyal 22| 24 ' 17* ' 191 Super Royal Imperial 27i 30 j. t/ ^ ; 19^ ' 22 Elephant 28 ' 23 Columbier 35 inches by 23^ inches. Atlas 34 " 26 " Double Elephant. 40 27 Antiquarian 53 '* 31 Emperor 68 " 48 " But of late machine-made papers are the most used, and are furnished in rolls of widths up to 58 inches, and wider can be obtained by order. Whatman's white paper is the quality most usually employed for finished drawings ; it will bear wetting and stretching without injury, and, when so treated, receives color readily. For ordinary working drawings, where damp- stretching is dispensed with, cartridge paper, in rolls of a coarser, harder, and tougher quality, is preferable. It bears the use of India-rubber better, receives ink on the original undamped surface more freely, shows a fully better line, and, as it does not absorb very rapidly, tinting lies better and more evenly upon it. For delicate small-scale line-drawing, the thick blue paper, such as is used for ledgers, etc., imperial size, answers exceedingly well ; but it does not bear damp-stretching without injury, and should be merely pinned or waxed down to the board. With good management, there is no ground to fear the shifting of the paper. Good letter paper receives light drawing very well ; of course, it does not bear much fatigue. Drawings destined for rough usage and frequent reference should be on sheet or roll drawing paper, backed with cotton cloth, which can be purchased at the stationer's. Tracing paper is a preparation of tissue paper, transparent and qualified to receive ink lines and tinting without spreading. When placed over a drawing already executed, the drawing is distinctly visible through the paper, and may be copied or traced directly by the ink instruments ; thus an accurate copy may DRAWING INSTRUMENTS. 57 be made with great expedition. Tracings may be folded and stowed away very conveniently ; but, for good service, they should be mounted on cloth, or on paper and cloth, with paste. Tracing paper may be prepared from thick tissue paper by sponging over one surface with a mixture of one part raw linseed oil and five spirits of tur- pentine ; five gills of turpentine and one of oil will go over from forty to fifty sheets of paper. Tracing cloth is a similar preparation of linen, and is preferable for its toughness and durability. Tracing paper and cloth are usually to be had in rolls, and tracings on cloth are now preserved as originals, and copies are made from them by some sun process. Mouth Glue, for the sticking of the edges of drawing paper to the board, is made of glue and sugar or molasses ; it melts at the temperature of the mouth, and is convenient for the draughtsman. Drawing paper may be fixed down on the drawing board by the pins at the corners, by weights, or by gluing the edges. The first is sufficient when 110 shading or coloring is to be applied, and if the sheet is not to be a very long time on the board ; and it has the advantage of preserving the paper in its natural state. For shaded or tinted drawings, the paper must be damped and glued at the edges, as the partial wetting of paper, loose or fixed at the corners merely, by the water-colors, distorts the surface. Damp-stretching is done as follows : The edges of the paper should first be cut straight, and, as near as possible, at right angles with each other ; also, the sheet should be so much larger than the intended drawing and its margin as to admit of being afterward cut from the board, leaving the border by which it is attached thereto by glue or paste, as we shall next explain. The paper must first be thoroughly and equally damped with a sponge and clean water, on the opposite side from that on which the drawing is to be made. When the paper absorbs the Water, which may be seen by the wetted side be- coming dim, as its surface is viewed slantwise against the light, it is to be laid on the drawing board with the wetted side downward, and placed so that its edges may be nearly parallel with those of the board ; otherwise, in using a J square, an inconvenience may be experienced. This done, lay a straight flat ruler on the paper, with its edge parallel to, and about half an inch from, one of its edges. The ruler must now be held firm, while the said projecting half- inch of paper be turned up along its edge ; then a piece of solid or mouth glue, having its edge partially dissolved by holding it in boiling or warm water for a few seconds, must be passed once or twice along the turned-up edge of the paper, after which, by sliding the ruler over the glued border, it will be again laid flat, and, the ruler being pressed down upon it, that edge of the paper will adhere to the board. If sufficient glue has been applied, the ruler may be re- moved directly, and the edge finally rubbed down by an ivory book-knife, or by the bows of a common key, by rubbing on a slip of paper placed on the draw- ing paper, so that the surface of the latter may not be soiled, which will then firmly cement the paper to the board. This done, another but adjoining edge of the paper must be acted upon in like manner, and then the remaining edges in succession ; we say the adjoining edges, because we have occasionally ob- 58 DRAWING INSTRUMENTS. served that, when the opposite and parallel edges have been laid down first, without continuing the process progressively round the board, a greater degree of care is required to prevent undulations in the paper as it dries. Sometimes strong paste is used instead of glue ; but, as this takes a longer time to set, it is usual to wet the paper also on the upper surface to within an inch of the paste mark, care being taken not to rub or injure the surface in the process. The wetting of the paper in either case is done for the purpose of expanding it ; and the edges, being fixed to the board in its enlarged state, act as stretchers upon the paper, while it contracts in drying, which it should be allowed to do gradually. All creases or undulations by this means disappear from the surface, and it forms a smooth plane to receive the drawing. To remove the paper after the drawing is finished, cut oif inside the pasted edge, and remove the edge by warm water and the knife. With paneled boards, the panel is taken out, and the frame inverted ; the paper, being first damped on the back with a sponge slightly charged with water, is applied equally over the opening to leave equal margins, and is pressed and secured into its seat by the panel and bars. MOUNTING PAPER AND DRAWINGS, VARNISHING, ETC. When paper of the requisite quality or dimension can not be purchased already backed, it may be mounted 011 cloth. The cloth should be well stretched upon a smooth flat surface, being damped for that purpose, and its edges glued down, as was recommended in stretching drawing paper. Then with a brush spread strong paste upon the canvas, beating it in till the grain of the canvas be all filled up ; for this, when dry, will prevent the canvas from shrinking when subsequently removed ; then, having cut the edges of the paper straight, paste one side of every sheet, and lay them upon the canvas sheet by sheet, overlapping each other a small quantity. If the drawing paper is strong, it is best to let every sheet lie five or six minutes after the paste is put on it, for, as the paste soaks in, the paper will stretch, and may be better spread smooth upon the canvas ; whereas, if it be laid on before the paste has moist- ened the paper, it will stretch afterward and rise in blisters when laid upon the canvas. The paper should not be cut off from its extended position till thoroughly dry, which should not be hastened, but left in a dry room to do so gradually, if time permit ; if not, it may be exposed to the sun, unless in the winter season, when the help of a fire is necessary, provided it is not placed too near a scorching heat. In joining two sheets of paper together by overlapping, it is necessary, in order to make a neat joint, to feather-edge each sheet ; this is done by care- fully cutting with a knife half way through the paper near the edges, and on the sides which are to overlap each other ; then strip off a feather-edged slip from each, which, if done dexterously, will form a very neat and efficient joint when put together. For mounting and varnishing drawings or prints, stretch a piece of linen on a frame, to which give a coat of isinglass or common size, paste the back of drawing, which leave to soak, and then lay it on the linen. When dry, give it at least four coats of well-made isinglass size, allowing it to dry between each DRAWING INSTRUMENTS. 59 coat. Take Canada balsam diluted with the best oil of turpentine, and with a clean brush give it a full flowing coat. MANAGEMENT OF THE INSTRUMENTS. In constructing preparatory pencil-drawings, it is advisable, as a rule of general application, to make no more lines upon the paper than are necessary to the completion of the drawing in ink ; and also to make these lines just so dark as is consistent with the distinctness of the work. With respect to the first idea, it is of frequent application : in the case, for example, of the teeth of spur wheels, where, in many instances, all that is necessary to the drawing of their end view in ink are three circles, one of them for the pitch line, and the two others for the tops and bottoms of the teeth ; and again, to draw the face view of the teeth that is, in the edge view of the wheel we have only to mark off by dividers the positions of the lines which compose the teeth, and draw four pencil lines for the two sides, and the top and bottom of the eleva- tion. And here we may remark the inconvenience of that arbitrary rule, by which it is by some insisted that the pupil should lay down in pencil every line that is to be drawn before finishing it in ink. It is often beneficial to ink in one part of a drawing before touching other parts at all ; it prevents confusion, makes the first part of easy reference, and allows of its being better done, as the surface of the paper inevitably contracts dust and becomes otherwise soiled in the course of time, and therefore the sooner it is done with the better. Circles and circular arcs should, in general, be inked in before straight lines, as the latter may be more readily drawn to join the former than the former the latter. When a number of circles are to be described from one center, the smaller should be inked first, while the center is in better condition. When a center is required to bear some fatigue, it should be protected with a thickness- of stout card glued or pasted over it, to receive the compass-leg. India-rubber is the ordinary medium for cleaning a drawing, and for cor- recting errors in the pencil. For slight work it is quite suitable ; that sub- stance, however, operates to destroy the surface of the paper ; and, by repeated application, it so ruffles the surface, and imparts an unctuosity to it, as to spoil it for fine drawing, especially if ink shading or coloring is to be applied. It is much better to leave trivial errors alone, if corrections by the pencil may be made alongside without confusion, as it is, in such a case, time enough to clear away superfluous lines when the inking is finished. For cleaning a drawing, a piece of bread two days old is preferable to India- rubber, as it cleans the surface well and does not injure it. When ink lines to any considerable extent have to be erased, a small piece of damped soft sponge may be rubbed over them till they disappear. As, however, this process is apt to discolor the paper, the .sponge must be passed through clean water, and ap- plied again to take up the straggling ink. For ordinary small erasures of ink lines, a sharp rounded pen-blade, applied lightly and rapidly, does well, and the surface may be smoothed down by the thumb-nail. In ordinary working draw- ings, a line may readily be taken out by damping it with a hair-pencil and quickly applying the India-rubber ; and to smooth the surface so roughened, a light application of the knife is expedient. In drawings intended to be highly 60 DRAWING INSTRUMENTS. finished, particular pains should be taken to avoid the necessity for corrections, as everything of this kind detracts from the appearance. In using the square, the more convenient way is to draw the lines off the left edge with the right hand, holding the stock steadily but not very tightly against the edge of the board with the left hand. The convenience of the left edge for drawing by is obvious, as we are able to use the arms more freely, and we see exactly what we are doing. To draw lines in ink with the least amount of trouble to himself, the me- chanical draughtsman ought to take the greater amount of trouble with his tools. If they be well made, and of good stuff originally, they ought to last through three generations of draughtsmen ; their working parts should be care- fully preserved from injury, they should be kept well set, and, above all, scru- pulously clean. The setting of instruments is a matter of some nicety, for which purpose a small oil-stone is convenient. To dress up the tips of the blades of the pen or of the bows, as they are usually worn unequally by the customary usage, they may be screwed up into contact in the first place, and passed along the stone, turning, upon the point in a directly perpendicular plane, till they acquire an identical profile. Being next unscrewed and exam- ined to ascertain the parts of unequal thickness round the nib, the blades are laid separately upon their backs on the stone, and rubbed down at the points, till they be brought up to an edge of uniform fineness. It is well to screw them together again, and to pass them over the stone once or twice more, to bring up any fault ; to retouch them also on the outer and inner side of each blade, to remove barbs or fraying ; and, finally, to draw them across the palm of the hand. The China ink which is commonly used for line-drawing ought to be rubbed down in water to a certain degree, avoiding the sloppy aspect of light lining in drawings, and making the ink just so thick as to run freely from the pen. This medium degree may be judged of after a little practice by the ap- pearance of the ink on the palette. The best quality of ink has a soft feel when wetted and smoothed ; free from grit or sediment, and musky. The rubbing of China ink in water tends to crack and break away the surface at the point ; this may be prevented by shifting at intervals the position of the stick in the hand while being rubbed, and thus rounding the surface. Nor is it advisable, for the same reason, to bear very hard, as the mixture is otherwise more evenly made, and the enamel of the palette is less rapidly worn off. When the ink, on being rubbed down, is likely to be for some time required, a considerable quan- tity of it should be prepared, as the water continually vaporizes ; it will thus continue for a longer time in a condition fit for application. The pen should be leveled in the ink, to take up a sufficient charge ; and, to induce the ink to enter the pen freely, the blades should be lightly breathed upon before immer- sion. After each application of ink, the outsides of the blades should be cleaned, to prevent any deposit of ink upon the edge of the squares. To keep the blades of his inkers clean is the first duty of a draughtsman who is to make a good piece of work. Pieces of blotting or unsized paper and cotton velvet, wash-leather, or even the sleeve of a coat, should always be at hand while a drawing is being inked. When a small piece of blotting paper is DRAWING INSTRUMENTS. 61 folded twice so as to present a corner, it may usefully be passed between the blades of the pen now and then, as the ink is liable to deposit at the point and obstruct the passage, particularly in fine lining ; and for this purpose the pen must be unscrewed to admit the paper. But this process may be delayed by drawing the point of the pen over a piece of velvet, or even over the surface of thick blotting-paper ; either method clears the point for a time. As soon as any obstruction takes place, the pen should be immediately cleaned, as the trouble thus taken will always improve and expedite the work. If the pen should be laid down for a short time with the ink in it, it should be unscrewed to keep the points apart, and so prevent deposit ; and, when done with alto- gether for the occasion, it ought to be thoroughly cleaned at the nibs. This will preserve its edges and prevent rusting. For the designing of machinery, it is very convenient to have some scale of reference by which to proportion the parts ; for this purpose a vertical and horizontal scale may be drawn on the walls of the room. EXERCISES WITH THE Before proceeding to the construction of finished drawings, skill should be acquired in the use of the drawing-pen, supplemented often by the steel pen. Beginning with lines, outlines of figures, alphabets, and the like, the draughts- man should strive to acquire the habit of readily drawing clean, uniform lines, without abruptness or breaks, where straight lines connect with curved ones. Draw straight lines of different grades : as, fine - medium - - -- coarse ^ ^^ at first, lines of indefinite length, taking care that they are drawn perfectly straight and of uniform width or grade ; then draw lines of definite length between assumed points, taking care to terminate the lines exactly at these points. Lines as above are full lines, the grades depending on the effect which the draughtsman wishes to give. Draw dotted lines, broken lines, and broken and dotted lines, of different grades : Draw fine lines at uniform distances from each other DRAWING INSTRUMENTS. To give uniform appearance, the lines must be of uniform grade and equally spaced. Practice in lines of this sort is important, as they are much used in drawing to represent sections, shades, and conditions, as soundings on charts, density or characteristics of population, areas of rain, temperature, and the like. Draw lines as in Fig. 148. These lines are diagonal with the border-lines, and FIG. 148. are used to represent sections of materials. In the figure, lines differently in- clined represent different pieces of the same material. Sections of different materials may be represented in different kinds of lines, as in Figs. 149, 150, 151. FIG. 149. FIG. 150. FIG. 151. These particular ones are used to represent sections of wrought-iron, steel, and cast-iron ; but they may be used to represent different colors, the location of different mineral or agricultural products, etc. To represent cylindrical surfaces (Fig. 152). Draw a semi-circumference, and mark on it a number of points, at equal distances apart, and through these points draw lines perpendicular to the FIG. 152. FIG. 153. diameter across the surface to be represented. It is not absolutely necessary that the central space should be equal to the others ; it will be more effective to leave out two of the lines, and make it to this extent wider. DRAWING INSTRUMENTS. 63 To construct a mass of equal squares (Fig. 153). Lay off a right angle, and on its sides mark as many points, at equal dis- tances apart, as may be necessary ; through these points draw lines parallel to the sides. Or, construct a rectangle ; mark on its sides as many points, at equal distances apart, as may be necessary ; through these points draw the lines. To construct the squares diagonally to the base (Fig. 154). Mark on the sides of the right angle as many points, at distances apart equal to the diagonal of the required squares, as may be necessary. Con- nect these points by lines as shown, and through the same points draw lines at right angles to the others. Or, as above, construct a rec- tangle, and mark on its sides points at distances apart equal to the di- agonal of the required squares. To cover a surface with equi- FlG 154 lateral triangles (Fig. 155). Construct an angle of 60, and mark on its sides points at distances apart equal to the side of the triangle. Connect these points ; and through these points draw lines parallel to the sides of the angle. Figures composed of two triangles, with the same base, are called lozenges. Six triangles may be arranged as a hexagon. The whole surface may be arranged in lozenges or hexagons. To cover a surface with octa- gons and squares (Fig. 156). Lay off the surface in squares having sides equal to the width of the octagons. Corner the outer squares to form octagons, as by Prob. XL., page 21. Extend the sides of these octagons across the other squares, and similar corners will be cut off, and the octagons and squares required will be com- FlG 155 plete. With the aid of paper thus covered with squares, triangles, and lozenges, various geometrical designs may be readily constructed, pleasing in their effect, and affording good practice to young draughtsmen. In the examples given of designs constructed on squares or triangles, if it is desired to increase or diminish the size of the original designs, it is only neces- sary to make the sides of the squares or triangles larger or smaller, and taking 64 DRAWING INSTRUMENTS. relatively the same squares for the construction of the figures. In transferring designs and drawings from books or plates, on which squares can not be drawn, it is very convenient to have a square of glass, with squares upon it, which may be laid on the drawing, and thus serve the same purpose as if squares had been C F drawn. The glass may be readily prepared by painting one of its surfaces with a thin coat of gum, and drawing squares upon it with the drawing-pen ; if every fifth or tenth line be made fuller or in a different color, it will be still more convenient for reference. Fig. 157 is the front view and side of an acanthus-leaf, of which the sur- faces are covered with squares, somewhat larger than would be recommended FIG. 157. FIG. 158. in practice, but sufficient to illustrate the principle, which may be done by the learner on the same or other sized squares. If the same size, the intersec- tions of the lines of the figure with those of the squares are easiest transferred by a straight-edged slip of paper, placed along a line, and making all the inter- sections at once, and then transferring the marks to the copy. DRAWING INSTRU 65 Fig. 158 is the side-view of the acanthus-le^f, in a reversed position from the original (Fig. 157) ; that is, right-handed, while the original is left-handed. It will readily be understood how this may be done by observing the letters on the side and the numerals at the top of the squares. Fig. 159 represents the construction of Gothic letters and numerals on a system of squares. These letters are formed mechanically by the drawing-pen and dividers. Fig. 160 are Italic letters, drawn on rhombs, in which the upright lines are inclined to horizontal. On pages 66, 67, 68, 69, are specimens of type taken from the printer's font, which can be readily transferred to a drawing, by covering them with a bit of glass or horn, laid off in squares, as described above. Printers' let- ters are in general well proportioned, but it is customary often to distort letters, to call attention to them, or to adapt them to the position in which they are to be placed. Spaces between the letters are in printing uniform, but in drawing, when such letters come together as F and A, L and T, one wide at top and the other at bottom, the spacing between them may be reduced a little. The acquisition of a ready hand in lettering enables a draughtsman to give a finish to a good drawing or map which might other- wise be spoiled by poor lettering. FIG. 159. 66 DRAWING INSTRUMENTS. LARGE ROMAN. ABC DE FGH IJ KLMN OP QRST TJV WX YZ SMALL EOMAK'. abc de fgh ij klmn op qrst uv wx yz 1234567890 DRAWING INSTRUMENTS. 67 ENGLISH GOTHIC. ABC DE FGH IJ KLMN OP QRST UV WX YZ 1234567890 ITALIC. ABC DE FGH IJ KLMN OP QRST UV WX YZ abc de fgli ij klmn op qrst uv wx yz TUSCAN. ABC DE FGH IJ KLMN OP QRST UV WX YZ 1234567890 ABC DE FGH IJ ELMNOPQEST U7WZYZ abc de fgh ij klmn op qist uv wx yz 68 DRAWING INSTRUMENTS. TELEGRAPH. OEK AMEKTED. UV WX de uv wx OLD ENGLISH. Jfi QV CD 1 e;Ci- ak to fg| ij klrnn 0p qrst DRAWING INSTRUMENTS. 69 ENGLISH CHURCH TEXT. aic t aht it fgji ij klmn up qtst un rax MEDIEVAL 13 Jf afir bp fg| ij hlmn op qrsf- uti tof BI F& II ij felmtx xxp qrst utr Paper printed in squares is used by designers of figures for calicoes, silks, and woolens. For the engineer, there is a class of papers called cross-section papers, sold in sheets or rolls, and of various scales, originally intended, as the name implies, for cross-sections of railway or canal cuts, but now extensively employed by the architectural and mechanical designer for the rough sketches of works either executed or to be executed ; by the sanitarian for the plotting of death-rates ; for thermometric and hygrometric readings ; by the broker and merchant for the graphic representation of the prices of gold, stocks, or articles of merchandise, during a term of years ; by the railway superintendent for the movement of trains ; and for multitudes of other uses. These may hardly be considered in the light of drawings ; but, as they involve the drawing of lines, shading of spaces, and lettering, and as there is no head of drawing under 70 DRAWING INSTRUMENTS. which this use of cross-section paper can be classed, it seems proper to give here a few illustrations, which will show its general application. Fig. 161 shows a graphical method of determining the equivalent values of the metric system of measurements in United States units, or vice versa. The vertical scale represents the metric units, and the horizontal the common or oo TJI to so UNITED STATES UNITS. FIG. 161. United States units. The method of using the diagram can be best shown by taking one or two examples. What is the equivalent value of seven kilometres in miles ? Read upward on the metric scale to 7, then read on that horizontal line to the point of in- tersection with the line designated "MILES & KILOMETRES," that is, at the point on the United States scale of units representing 4 '35 ; therefore, seven kilometres are equal to 4*35 miles. What is the value of five pounds in kilogrammes ? The process is the same as the foregoing, except that, to change United States units into the metric units, first read horizontally, then upward. The result will be in this case that five pounds is found equal to 2*25 kilogrammes. The divisions may represent single units, ten units, one hundred units, etc. ; that is, if we had wished to find the equivalent of 500 pounds, it would have been 225 kilogrammes. DRAWING INSTRUMENTS. 71 Fig. 162 is a diagram illustrating graphically the difference charged on a ton of merchandise per mile } on the New York Central and Hudson River Railroad and the Erie Canal, for every year between 185? and 1880 ; the values being FIG. 162. published in the Report of the United States Bureau of Statistics for 1880. The higher values in every case represent the railroad rates and the lower the canal rates. The black band shows the difference between these values. In 1865, for instance, the railroad rates were 3 '30 cents, and the canal 1'02 cents, the difference being 2 -28 cents. Fig. 163 is made up from the time-table of the New York, New Haven, and Hartford Railroad, showing the movement of trains, two from New York and two from New Haven, the abscissas (horizontal lines) being cut off on a scale of miles for each station, the ordinates (vertical lines) being a scale of hours. DRAWING INSTRUMENTS. I : * T ! 5 f r \ * 1 T H -r \ 2 5 $ I ? ; 5 ! s ; s ii 3 ; i S ^ \ 5 ; ^ 1 * u f s s i s s s 3 S \ \ ^ S i- 5 i. ; I: 5 - *- fl Hre. -- _^-i ^ ^^ .S ^"^ 10 / ^^ ^r / ^/ V S / ^>r ^ -/* >X_ -^ -?s ^^ 4* ^/ s* s ^,r ! \ ^^^ s \ S^ >^- __^>. -^ .X ^'x^ y/ ^ SS N S ^ ^ X ^ ! p 5 1 ^ *} i t c a n 3 C ^ 3 "; : 2 ; ^. 3 i ; 1 ; ] I 1 I t n ?IG. 1 J 1 j 63. i - s s , J 5 p i 2 i I J O cc Fairfield M 3 j : a 5 N " f 3 C 2>- 1 B ? 11 > 3 { I ! 3 5 i & New Haven Hrs. 10 Fig. 164 shows the method of finding the average of a number of observa- tions. The figure represents the path of a float in a wooden flume or channel, L ^ -- ^ _{ i ~r> ( b \s t >v> x , / v ) / u b- "x x U A r % n /n n x, y^ TV r>. s s o FIG. 164. taken from the last edition of Francis's "Lowell Hydraulic Experiments." The cut was copied directly on the wood, and is therefore reversed. The DRAWING INSTRUMENTS. 73 width of the cut represents the width of the flume, each abscissa being one foot ; the ordinates are the speeds of float in divisions of 0*1 foot per second ; the o o on the cut are meant to represent the floats in their observed path and speed ; and the curved line the average velocity in the different threads of the stream. Fig. 165 is from Clarke's " Railway Machinery." The abscissas represent the speed in miles per hour ; the ordinates the pounds per ton resistance of a 100- ton train. - > N, - "^ > + > x, ; K ^ ^ 1 - > FIG. 165, Fig. 166 is a diagram illustrating the daily mortality during the month of November, 1873, in New York City. The figure is a copy of a portion of the chart published in the Report of the Metropolitan Board of Health for that year. The lower irregular line shows the daily mortality. The upper single irregular line shows the daily average temperature. The terminal cross-lines at the ends of perpendicular bars show the daily range of temperature. The double irregular line shows the daily humidity, saturation being 100 on the scale of temperatures. The black bands in the upper portion of the diagram give the daily rain- fall in inches. This method of representing the rain-fall will do for this chart, but, for most meteorological purposes, is insufficient. The time of the commencement and end of the rain-fall should be given where any effect due to the rain is to be detected. These few diagrams illustrate the method of graphical representation, so that any one should with little difficulty be able now to make them for such cases as he may see fit. On pages 75, 76, 77, are some designs, showing other uses to which squared or quadrille paper can be put. The execution of such ornamental designs is greatly facilitated by the use of this paper. The figure on page 77 illustrates how color may be represented in a design, by different grades and directions, of black lines and white spaces. DRAWING INSTRUMENTS. NOVEMBER, 1873. FIG. 166. DRAWING INSTRUMENTS. 75 76 DRAWING INSTRUMENTS. x x X X _ X X x x x x x X x x x x X x X x x x 5 x x x x x x x x x x x x x x x x x x x x s x x X x ^ X x x X x x x X x x. x x x X x x X x x X x x X x , x x X, x X x X X x x x x x x x x x x x x x X x x x x x X x A x x ^\ x /s x X X X X x x H X X X x x x X X X x x V x X V X V x x X ,x < > < > < > < > < > x x x x x /\ x X /\ x ^ x x ^ X x X x x. X X x X x x X x ] x x x V x X V X V X X X x y& x X x x x x X x ^x \x\ X X X X x X x x x x X X Y X x x X x X x X X X x