To construct a square that shall be a Multiple of any given square.

—Let A B C D (Fig. 16) be the given square, and let it be required to construct a square that shall contain 2, 3, 4, &c., times its surface. Draw the diagonal B C, then the square described on B C will be double the square A B C D. Lay off D E, equal to B C, and draw C E; then the square described on C E will be three times the square A B C D. In the same manner lay off D F, equal to C E, and the square described on C F will be four times the square A B C D; and so for any multiple of the square A B C D.

To construct a square that shall be equal to ¹⁄₂, ¹⁄₄, &c., of any given square.

—Let A B C D (Fig. 17) be the given square. On A B, as a diameter, describe the semicircle A G B, and erect at the centre E the perpendicular E G. Draw G B, which will be the side of a square equal to one-half of A B C D. Lay off B F, equal to one-fourth of A B, and erect the perpendicular F H; then the square described upon H B will be equal to one-fourth of A B C D. In the same manner a square may be constructed equal to any part of A B C D.

To construct a square that shall be in any Proportion to a given square.

—Let A B C D (Fig. 18) be the given square. It is required to construct a square which shall be to A B C D as 2 is to 5. Upon the side A B as a diameter describe the semicircle A F B, and divide the line A B into five equal parts. At the second point of division erect the perpendicular E F and join A F; the square described upon A F will be to the given square A B C D as 2 is to 5.

To construct, upon a given base, a Rectangle, which shall be similar to a given rectangle.

—Let A E F G (Fig. 19) be the given rectangle. It is required to construct upon the base A B, one that shall be similar to A E F G. Produce A E and lay off the given base from A to B; draw the diagonal A G and produce it indefinitely. Erect a perpendicular to A B at B, and from the point D where it intersects the diagonal produced, draw D C perpendicular to A F produced. Then A B C D will be similar to A E F G. All rectangles having their diagonals in the same line are similar.

To describe a regular Pentagon on a given line.

—Let A B (Fig. 20) be the given line. Bisect A B at C, draw C F perpendicular to A B, and make C D equal to A B. Draw A D and produce it indefinitely; make D E equal to half A B. From A as a centre, with A E as a radius, describe an arc cutting the perpendicular C D in F; and from A F and B as centres, with radius A B, describe arcs cutting each other in G and H; join A G, B H, F G and F H; then A G F H B will be the pentagon required.

To describe a regular Hexagon.

—With a radius equal to the length of one side of the required hexagon, describe a circle (Fig. 21), and set off the same radius round the circumference of the circle, which will be thus divided into six equal parts. Join the points thus found, and the required hexagon will be completed as A B C D E F.

To draw a Parabola, the base and height being given.

—Let C A (Fig. 22) equal half the base, and C D the height. From the point D draw D E parallel and equal to A C, and from the point A draw A E parallel and equal to C D. Divide D E and A E similarly, making the end E of A E correspond to the end D of E D. Through 1, 2, &c., in DE draw 1, 1; 2, 2, &c., parallel to D C. Join D to the several points 1′, 2′, &c., in A E. The parabola will pass through the points of intersections of these lines with the verticals drawn from D E to C A.

To draw an Ellipse.

—I. By means of a piece of string and pins. Place the diameters A B and C D (Fig. 23) at right angles to each other, and set off from C half the major axis at E and F; then will E and F be the two foci in the ellipse. Fix a pin at E and another at F; take an endless string equal in length to the three sides of the triangle E F C and pass it round the pins, stretch the string with a pencil G, which will then describe the required ellipse. II. From the centre O (Fig. 24) describe a circle of the diameter of the minor axis of the required ellipse. From the same centre, describe another circle with a diameter equal to its major axis. Divide the inner circle into any number of equal parts as 1, 2, &c., and through these points draw radii cutting the outer circle in 4, 3, &c. From 1, 2, &c., draw horizontals, and from 3, 4, &c., draw perpendiculars cutting each other in E F, &c.; the curve traced from C through the points C E F A, &c., will complete the curve of the required ellipse. III. Let A B (Fig. 25 a) be the major and C D the minor axis of the required ellipse. On any convenient part of the paper draw two lines F G, F H (Fig. 25 b) at any angle with each other. From F with the distance E C or E D, the semi-axis minor, describe an arc cutting the lines F G, F H, in I and K; and from F with the distance E A or E B, the semi-axis major, describe the arc L M. Join I M, and from L and K draw lines parallel to I M, cutting F G, F H, in N and O. From A and B (Fig. 25 a) set off the distance F N (Fig. 25 b) in points N′, and from these points as centres, with F N as radius, describe an arc of about 15° on each side of the major axis. From C and D (Fig. 25 a) set off on the minor axial line the distance FO (Fig. 25 b) in points O′, and from these points as centres, with radius FO, describe arcs of about 15° on each side of the axis C D. To obtain any number of intermediate points take a slip of paper (Fig. 25 a) and mark upon one edge, with a sharp-pointed pencil, 1, 3, equal to the semi-axis major, and 2, 3, equal to the semi-axis minor. If the slip of paper be now applied to the figure and moved over it in such a manner that the point 2 is always in contact with the major axis, and the point 1 in contact with the minor axis, the outer point 3 will describe a perfect ellipse, any number of points in which can be marked off as the “trammel” is moved into successive positions.

For this last method, which in practice is by far the best, we are indebted to Binns’ ‘Orthographic Projection.’

To construct a Semi-Elliptical Arch.

—The span A B (Fig. 26) and rise C D being given, divide C A and C B into any number of equal parts. Through the point D, draw E F parallel to A B, and from the points A and B erect the perpendiculars A E and B F. Divide A E and B F similarly to C A and C B. Produce C D and make C G equal C D. From D draw lines to the points 1, 2, 3, &c., in the lines A E and B F; also from G draw lines through the points 1, 2, 3, &c., in the line A B, and produce these lines until they cut those drawn from D to the corresponding numbers in A E and B F. Through the points thus obtained draw the curve of the ellipse.

To draw the Gothic Equilateral Arch.

—From the points A and B (Fig. 27), with radius A B equal to the span, describe the arcs B C and A C. By joining C to A and B we obtain an equilateral triangle from which this arch derives its name.

To draw the Gothic Lancet Arch.

—In this arch, the centres E and D (Fig. 28) from which the arcs are struck, are situate outside of and in a line with the points of springing A and B; thus it is constructed on an acute-angled triangle, as will be seen by joining C to A and B.

To draw the Gothic Obtuse Arch.

—This arch, called sometimes the Drop-Arch, is constructed on an obtuse-angled triangle; the centres E and D (Fig. 29) being situate below and within the points of springing A and B.

To draw the Gothic Tudor Arch.

—On the line of springing A B (Fig. 30), take any two points as F and G, so that A F is equal to G B. Draw F E and G D cutting each other on the bisecting line through C; from F and G, with radius F A or G B, describe the short arcs, and from E and D, with radius E C or D C, describe the arcs meeting in C.

To draw the Moorish Horse-Shoe Arch.

—The centres E and D (Fig. 31) from which the arcs forming this arch are struck, are situate above and within the points of springing A and B. One of the most graceful forms of this arch is obtained when the height of the points E and D above the line of springing and their distance from the bisecting line through C are equal to one-third of the span A B.

To draw the Gothic Ogee Arch.

—The most pleasing form of this arch is that constructed on an equilateral triangle, in the following manner. Having drawn the equilateral triangle A B C (Fig. 32), draw F G parallel to A B. Bisect the sides A C and C B and produce the bisecting lines to F G and H, which will complete the triangle F G H similar and equal to the triangle A B C. From H, with radius H A or H B, describe the arcs A E and B D, and from F and G, with the same radius, describe the arcs E C and C D.

To draw the Roman Cyma Recta and Cyma Reversa.

—Join A B (Fig. 33) and bisect A B in C. From the points C and B, with the distance B C, describe arcs cutting each other in E; and from A and C, with the same radius, describe arcs cutting each other in D; from D, with the same radius, describe the arc A C, and from E describe the arc C B. The projection of the upper end of the curve over the under, as F B, is generally equal to the height, A F, of the moulding. The same description applies to the Cyma Reversa (Fig. 34) letter for letter.

To draw the Gothic Trefoil.

—Having drawn the equilateral triangle A B C (Fig. 35), bisect the angles and produce the bisecting lines D E F which will bisect the sides of the triangle in G H I. From A B and C as centres, with radius A H or A I, equal to half the side of the triangle, describe the arcs K L M, and those concentric with them, and from the centre O of the triangle describe the outer circles and concentric arcs, which will complete the figure.

To draw the Gothic Quatrefoil.

—Draw the square A B C D (Fig. 36); bisect the sides of the square at I K L M and produce the bisecting lines to E F G H. From the angles A B C D of the square as centres, with radius A I or A M equal to half the side of the square, describe the arcs P N R S, and draw the outer concentric arcs. The circles, completing the figure, are drawn from the centre O of the square.

To construct the Gothic Cinquefoil.

—Having drawn the regular pentagon A B C D E (Fig. 37), bisect the angles and produce the bisecting lines to F G H I K, which will cut the sides of the pentagon in a, b, c, d, e. From A B C D and E as centres, with radius A a or A b, equal to one-half of the side of the pentagon, describe the arcs L M N P R, and draw the outer concentric arcs and those concentric with them. The circles are drawn from the centre O of the pentagon, as in the preceding example.

Section III.—Lines, Dots, and their Combinations.

All kinds of drawings are made up of lines and dots; these are the constituent parts, the materials which the draughtsman has to employ. It is therefore essential that he should make himself acquainted with their various forms and uses, and familiar with those means of producing them which experience has shown to be the best, before commencing the study of the principles by which the representation of an object is delineated. And moreover, it is desirable that he should acquire a familiarity with the operations required in the delineation of isolated objects, previously to making any attempt to place them in combination for the purpose of producing a complete drawing. The student will, therefore, do well to study carefully and to practise diligently the forms and examples given in this Section.

Straight and Curved Lines.

—All straight lines, however short, should be ruled, whether they be drawn with the pencil or the pen. Pencil lines, which are intended to serve merely as guides to the pen, should be drawn lightly, as otherwise it will be difficult to rub them out without injuring the ink. They should also be drawn a little beyond the point at which the line is required to terminate, because the intersection of the lines at that point makes it more distinctly visible, and there is, consequently, less danger of passing beyond that point or of stopping short of it when inking in. It is very important not to stop short of the required length when ruling a straight line with a pen, for it is extremely difficult to lengthen the line subsequently without leaving the join visible. An accurate line cannot be drawn unless the point of the pencil or the pen be kept close up to the rule, and to do this the top should be inclined a little outward. Before inking in a line that has been drawn in pencil, the indiarubber should be passed lightly over it, to remove the particles of lead adhering to the paper, for if these particles are allowed to remain, they get between the nibs of the pen and prevent the ink from flowing freely. The chief difficulties in ruling a straight line with the pen are, to keep it of a regular thickness throughout, and, when numerous parallel lines have to be drawn, to keep them at equal distances apart. To draw an even line, a first requisite is that the pen be in good condition. Frequently it will be found when drawing fine lines that the pen ceases to mark before the end of the line is reached, and as we have already said, it is very difficult to join a line without leaving visible traces of the operation. To remedy this defect, the pen must be reset as described in Section I. If a very hard pencil has been used, or if the pencil has been pressed heavily upon the paper, the pencil line will lie in a groove in the paper, and as the nib of the pen will not touch the bottom of this groove, the line drawn will be ragged. Another cause of unevenness is unduly pressing the pen against the rule; this pressure closes the nibs, and besides producing an irregularity in the thickness of the line, is very apt to cause a blot by forcing out the ink, which adheres to the rule when brought into contact with it. To prevent this, care should be taken to press the pen very lightly against the edge of the rule. A pen is manufactured by Stanley, of Holborn, London, which has the back nib much stiffer than the other, so that all danger of defect from this cause is removed by the construction of the instrument. To ensure a good line, the pen should rest lightly upon the paper, and the handle of the pen should make the same angle with the paper from the beginning to the end of the line. A considerable amount of practice is required to accomplish this, and to acquire the habit, the same attention should be given to the pencil as to the pen. The ability to draw a number of parallel lines at equal distances apart without measuring requires considerable training of the eye, and this training can be obtained from practice alone. This ability must be acquired before anything further is attempted, and the student who spends a good deal of time in its acquisition may have the satisfaction of knowing that while he is going through this somewhat monotonous practice, besides exercising himself in drawing accurate lines, he is acquiring a correctness of eye and a power of hand that will be of incalculable service to him later.

The straight line, besides being used for the outlines of regular objects, is employed conventionally for various purposes. When it is required to show an object in section, the part in section is covered with straight and parallel lines drawn at an angle of 45° and at equal distances apart, as in Fig. 38. To represent standing water, as ponds and lakes, horizontal straight lines are drawn parallel to each other and at equal distances apart over the surface, as shown in Fig. 39.

Curved lines, when arcs of circles, are drawn by the compasses. Other curves are drawn by hand through points previously found. To draw the curve correctly through these points, unless they be very numerous, a knowledge of the nature of the curve is necessary, which the draughtsman should in all cases endeavour to obtain. When the curved line is long, it is usually inked in with the drawing pen, with the aid of an instrument called the French curve, or cardboard moulds cut for the purpose; but for short lines an ordinary fine-pointed steel-pen point, or better, a good quill is used. In general, all lines drawn by hand should be drawn towards the body, as a better command of the pen can be obtained in that direction than in any other. In inking in curves by this means, the draughtsman should proceed continuously along the pencil-drawn line by partly repeated touches with the pen point, so held that the divided points of the pen may follow partly in the same track. Each touch should be made about one-thirtieth of an inch in length, and it should begin and end fine. Each succeeding touch must begin half its length back, so that the line is advanced by one-sixtieth of an inch. In map drawing all irregular lines are drawn in this way. Tracing maps will afford the student excellent practice in this mode of using the pen.

Lines of uneven thickness.

—Though generally a line is required to be of even thickness throughout, cases sometimes occur in which a variation in the thickness may be made to express some feature or quality of the landscape. The usual application of this kind of line is to mark the outline of rivers, lakes, and ponds, as shown in Fig. 40. The drawing of such a line presents no difficulty; the increased thickness is produced by going over those parts of the line again with the pen. Care must, however, be taken not to make a sudden increase in the breadth of the line, but to begin and end imperceptibly.

The Broken Line.

—The broken line, shown in Fig. 41, is of frequent occurrence in all kinds of drawings. In architectural and engineering drawings it is usually employed in roofs, as in Fig. 42, and for water in sections, as in Fig. 43. It is also used in combination with other lines for various purposes. In drawing a succession of broken lines, care must be taken not to allow the break in one line to be immediately over that in another. The effect may be varied considerably by increasing or diminishing the extent of the break. As in section lining, the lines should be at regular intervals apart, and be all of the same degree of fineness. Broken lines are sometimes used upon the face of stone buildings, instead of marking in the joints and etching or colouring. In such a case the breaks are long, and the lines widely spaced.

The Dotted Line.

—Of still more frequent occurrence is the dotted line. There are two kinds of dotted lines, distinguished by the shape of the dot, and known as the long and the round dotted line. These are shown in Fig. 44, as well as a combination of the two.

The round dotted line is of very general application. In architectural and mechanical drawings, it is used to distinguish hidden parts, and to mark the path of a moving piece in a machine. In plans, it is used to show the position of proposed works, to denote the walks through pleasure grounds and gardens, to indicate lines chained over in surveying, and frequently for other purposes, at the pleasure of the draughtsman. The long dotted line is employed to mark the boundaries of a township, the navigable channel of a river or creek, and in large-scale maps to show farm and bridle roads, footpaths, and the divisions of land among different tenants. The combination of the long and round dotted lines is used for the boundaries of a parish. Another combination of two round and one long dots, or sometimes of three round and one long, is used to denote proposed railways, canals, roads, and other similar works.

To draw a good dotted line requires some care. The difficulty lies in keeping the dots at equal distances apart, and in making them equal in size; and unless both these conditions are fulfilled, the line will not present a pleasing appearance. To obviate this difficulty, an instrument is sold by mathematical instrument makers, called the dotting or wheel pen. But it requires very great care in using, as otherwise it frequently happens that the ink escapes from it and spoils the drawing. For this reason, its use has been generally abandoned by draughtsmen. But if the instrument were better constructed and carefully handled, it might be made to do good service.

Combinations of Straight, Broken, and Dotted Lines.

—Combinations of the foregoing lines are used for various purposes. Some draughtsmen employ alternate, full, and dotted lines, to denote wood in section, as in Figs. 45 and 46; when wood is used in combination with iron or other metal, this is a very good way of distinguishing it. Wood-graining, though not made up of straight, broken, or dotted lines, yet partakes somewhat of the nature of all three kinds, and may therefore be introduced here. Oak-graining is shown in Fig. 47, and fir-graining in Fig. 48. The former is executed with the drawing pen, and requires some care and practice; the latter is most readily done with a common pen or a crow-quill. End wood is grained as shown in Fig. 49. The spring bows are very suitable for drawing in the circles, as a certain degree of turn to the nut will open the ink leg to the required distance after drawing each circle. A few broken wavy lines, called shakes, radiating from the centre, produce a good effect. When several pieces of end wood come together, the centres in each should not be in the same relative position.

Cultivated land is represented by alternate broken and dotted lines, suggesting furrows, as shown in Fig. 50. For the sake of variety, these lines are put in in sets, and in different directions, one set being usually parallel to one side of the enclosure. The lines are first ruled in continuously with the pencil, and the broken and dotted lines afterwards drawn in over them by hand. The portions of the broken lines must in this case be short, and the breaks still shorter. The dots must be fine and close together; they are made by touching the paper with the point of the pen, and immediately lifting it off without dragging it over the paper. All round dots must be made in this way.

The Wavy Line.

—The wavy line is very important in topographical drawings, as it is employed to represent running water, and frequently large bodies of standing water to which motion is communicated by the wind, as lakes and the sea. These rippled lines are intended to represent the ripples in the water, a purpose which they fulfil in a very pleasing manner. They must, however, be well executed, or the pleasing effect will not be produced. The operation of drawing these lines is usually regarded by the draughtsman as a tedious and an uninteresting one. But such ought not to be the case, for there is ample scope in it for the exercise of the taste and the judgment, and in proportion to the taste displayed and the judgment exercised, will be the effect of the work when executed.

Fig. 51 shows the manner of employing these lines. In representing water by this means, the lines should be drawn from the shores towards the middle of the stream or lake, and never from the middle outwards, for if the latter mode of proceeding be adopted, the proper graduation of the spaces between the lines becomes impossible. The shore line, or outline of the water, should be a moderately thick line, and of uniform thickness throughout. The first shading line may be of nearly the same thickness as the shore line, and it must be drawn as near to it as possible. Also this shade line, as well as all subsequent ones, must follow exactly all the windings of the shore line; this is essential to a correct expression. To effect this with accuracy, care should be taken to make the space between the shore and the shade line a fine white line. The second shade line must be drawn a little finer than the first, and at a slightly increased distance from it. This gradual diminution of the thickness of the lines, and increase of the spaces, must be continued to the middle of the current. The last line in the middle of a piece of water must always return to itself. When the shading lines meet the margin of the drawing, they should terminate in it, that is, they should be drawn out to the margin as though they had been continued beyond and cut off.

These lines require to be drawn clean, and to do this the hand must be kept steady. This steadiness may be obtained by taking a very short hold of the pen, and resting the middle finger upon the paper. The lines, as we have already said, should be drawn towards the body, the drawing being turned about as required to facilitate this, and the last line drawn must be always kept on the left of the one being drawn. By this means the last line and the point of the pen are kept constantly in sight. It is also important that the lines should be completed successively, rather than that several should be carried on at once, because if the latter mode of working be adopted, the eye is apt to become confused by the different intervals, and an uneven distribution of the lines is the result. A principle to be attended to is that every line shall return to itself, spirals being altogether inadmissible. The distance of the lines apart and their thickness are expressive of the character of the object; thus, in a small pond, for example, they will be fine and close together; in a large pond or a lake they will be thicker and more widely spaced; and in the open sea they will be made to present a bold appearance by increasing still more their thickness and the distance between them.

Grass-land.

—Various combinations of lines and dots are used, conventionally, to represent certain natural features of common occurrence. As far as convenient execution will allow, these signs are made to resemble the objects denoted. Thus the sign for grass-land consists of groups of short lines, arranged like tufts of herbage, as shown in Fig. 52. Each tuft is composed of five or seven lines converging towards a point situate below the base, the middle line being the longest, and the outside ones mere dots. In drawing these groups, the base must be kept quite straight, and parallel to the base of the drawing whatever the shape of the enclosure may be. Beginners usually experience considerable difficulty in keeping the base straight, the tendency being to make it curved. Great care is needed to distribute the groups evenly over the paper, and to avoid the appearance of being in rows, for the latter arrangement is destructive of that natural aspect which this sign otherwise possesses.

Swamps and Marshy Ground.

—As the surface of marshy ground consists of water and grass, a combination of the signs for these objects is employed to represent it. An illustration of this is given in Fig. 53. The lines representing the water should always be ruled parallel to the base of the drawing, and they should be grouped in an irregular manner so as to leave small islands interspersed throughout the locality. These islands should be covered with grass, and to show them out more distinctly, there should be nothing but water immediately around them. The division between the land and the water should be sketched in lightly before proceeding to rule in the lines. Sometimes dotted lines are used for the water, but full lines are to be preferred. The addition of a tree here and there improves the appearance of a drawing. A distinction is frequently made between a swamp and a marsh by watering the former more extensively than the latter. In drawing in marsh land, care should be taken to make the fineness of the lines in accordance with the scale of the map, as otherwise an offensive appearance will be produced. This caution applies equally to all signs.

Sand and Gravel.

—Sand and gravel are represented by dots, the dots being made larger for the latter than for the former, as shown in Fig. 54. Simple as the operation of filling in these dots is, it is one that requires some degree of care. Beginners are apt to mar the appearance of their drawings by inattention in this respect. The dots should be made in the manner already described when speaking of the dotted line, that is, the point of the pen should be brought slowly down upon the paper, and lifted without dragging it; and no dot should be made without a deliberate intention respecting its position. All arrangement in rows must be carefully avoided. In sand-hills, the slopes should be made darker than the level parts by placing the dots closer together. Mud in tidal rivers may be represented by very fine dots placed close together.