Fig. 74.
Fig. 74.
Great care must be taken in filling in the zones formed by the contour lines, that the drawing when finished do not present the appearance of separate layers or bands; for such an appearance is not only quite opposed to artistic effect, but it conveys a false notion of the character of the ground. The successive zones are not separate portions of the surface, but each is a continuation of the one adjoining it. The great principle to be observed in this, as in all matters of hill shading, is that changes of slope are gradual. When the contours are only pencilled in as guide lines to be afterwards erased, the above-mentioned defect may be avoided by drawing the hachures over them, without reference to exact spacing. But when, as is usually the case in regular surveys, the contours are inked in in dotted lines, the only means of avoiding it is to space the hachures on each side of a contour line at the same distance apart.
The student of map drawing should practise assiduously thissystem of shading in detached portions before undertaking the delineation of a complete hill. For such exercises, either a soft, medium-pointed steel pen, or a quill may be used.
—The foregoing system of shading is known as theHorizontal, and is now generally employed in this country for all kinds of surveys. There is, however, another system much used abroad, and frequently adopted here for engraved maps. In this system, which is known as thevertical, the shading lines are made to radiate from or converge into the curved parts of a hill, according as they project or re-enter. Such lines are calledlines of greatest descent; they are supposed to describe the same course that water would describe if allowed to trickle in streams down the slopes, and hence they exhibit both thedirectionand thedegreeof the slope. Having the horizontal sections given, we may obtain a complete knowledge of the direction in which the ground slopes by drawing perpendicular to them any number of lines of greatest descent; the degree of declivity is expressed by purely conventional means. The means adopted for this purpose are of two kinds. One depends upon the principle of vertical illumination, in which the maximum quantity of light is reflected upwards to the eye by a horizontal surface, and a minimum by a surface inclined 45° to the horizon. This is the English and German convention, and it lays more stress upon the proportions of black to white in indicating the degree of slope, than upon the distance between the shading lines. The other convention, which is the French, on the contrary, makes its expression depend more upon the distance between the lines of greatest descent than upon the shade of colour produced, though in this also the tint is graduated from dark to light, according to the degree of declivity.
A scale of shade is used for this system, founded upon the same principles as that already given for the horizontal system. The scale adopted is due originally to Major Lehmann, of the Saxon Infantry; but it has received some modification to adapt it to the requirements of practice.Fig. 75shows Lehmann’s scale. It is constructed for every 5°, from a level up to a slope of 45°, which is the steepestslope at which earth will stand. Each division of the scale corresponding to a given slope is subdivided into nine parts, to show the proportions of black to white. For a level, the whole of these spaces are left white; for a slope of 5°, the proportion is one black to eight white; for a slope of 10°, two black to seven white; and so on up to 45°, for which slope we have all black. The longitudinal divisions of the scale below that against the outer edge A B contain the same proportions of black to white, but equally distributed to show the mode of applying it. Thus, in the divisiono p r s, corresponding to a slope of 5°, the single black space is, in E F G H, divided into two equal parts and distributed; in G H I K, these two parts are again equally divided and distributed; and so on throughout the other longitudinal divisions. If now the scale be cut off along the line L M, the part L M C D will constitute a scale, the graduated edge L M of which will furnish us with a means of marking off the distance between the centres of the shading lines.
Fig. 75.Lehmann’s Scale of Shade.Larger illustration(48 kB).
Fig. 75.Lehmann’s Scale of Shade.
Larger illustration(48 kB).
To find the proportion of black to white in the foregoing scale for any given slope:—Subtract the given inclination from 45° for a denominator, and put the given inclination for a numerator. In the scale, as drawn in the figure, the variations are by 5°; but it is obvious that a scale may be drawn in the same manner to mark smaller variations, if thought desirable.
In applying this method in the United States’ Coast Survey, it was remarked that “this scale of shade does not represent slopes greater than 45°, thereby limiting the graphic capabilities and effect of the map. It also makes the slopes too dark as they approach the inclination of 45°, and does not well represent slopes of less than 5°, which latter it is often desirable and necessary to express distinctly.” The following modification was thereforemade:—
By this scale, the slighter slopes are represented distinctly. For slopes less than 16°, the shades are darker than in Lehmann’s scale; this makes their difference more noticeable. Above 25° the shades are lighter.
A further modification, which for ordinary purposes possesses the advantages of simplicity and facility of application, has been made in England, and very generally adopted. This modification consists in fixing with accuracy only three proportions of black to white for three medium slopes, asfollows:—
A scale of shade may at once be constructed from this Table, by assuming the thickness of the shading line for the medium slope of 221⁄2°, which thickness must be suited to the scale, and to the degree of fineness and finish it is intended to give the drawing. Generally,if the lines have such a relation to the scale of the drawing as to present a well-connected appearance, it will be found that fewer shading lines and a rather coarse texture will conduce more to clearness of expression than a finer texture, which tends to produce a dryness of style. In shading to this scale, it should be applied to the drawing wherever the slope corresponds to one of the three on the scale. Intermediate slopes are indicated by graduating the thickness of the shading lines. In all cases a good deal must be left to correctness of eye and skill of hand.
In the French method, as we have said, the inclination is expressed by the distances between the centres of the lines of greatest descent. The limits of the slopes that can be represented by this method are,1⁄1or 45° for the greatest and1⁄64or 0° 53′ 43″ for the smallest. The largest scale that will admit of conveniently drawing the lines of greatest descent is1⁄600full size, or about 83⁄4feet to a mile. The vertical distance between the horizontal sections is generally taken as 1 yard. Hence to a scale of1⁄600the least width of zone will be6⁄100inch, and the greatest6⁄100× 64 = 384⁄100inches.
The distance between the shading lines is reckoned from centre to centre, and is determined by the rule:—To the distance between the upper and the lower curves of any zone add3⁄10of an inch; a sixteenth part of this sum will be the proper interval for the shading lines. The distance is measured along the line of greatest descent. Thus, if the inclination be1⁄60and the scale1⁄600, the width of zone will be ·06 × 60 = 3·60 inches, and by the rule we have3·60 + ·316=3·916= 0·244 inch. Another rule is:—To a fourth of the distance between the upper and the lower curves of any zone, add75⁄1000of an inch; a fourth part of the sum will be equal to the interval.
The thickness or breadth of the lines is made to vary directly as the inclination to assist in expressing the declivity. This thickness is determined by the following rule. For a slope of1⁄1the thickness of the shading lines is equal to2⁄3of the distance between their centres,and this thickness will diminish with the inclination down to1⁄64, where the lines will be as fine as they can be drawn. In a slope of1⁄1this rule will always make the breadth of the shading lines twice that of the white space contained between them.
To represent declivities by the vertical system of shading a considerable amount of practice is required. This practice should be commenced by drawing repeatedly the scale of shade, and gradually applied, as proficiency is attained, to the varying inclinations of a hillside. Having the horizontal sections of the hill given, the degree of slope should be written upon it in pencil in as many places as is necessary. The distances between the centres of the shading lines may then be marked off upon the upper curve of the zone from the scale of shade, and the lines of greatest descent drawn through the points thus determined. The exact proportion of black to white being then adopted, the colour will express the degree of the slope, and the line of greatest descent will show its direction.
The principle of making the shading lines longer on a gentle slope than on a steep one should be adhered to generally; but in this matter much must be left to the judgment and the skill of the draughtsman. Frequently on slight inclinations it will be desirable to divide and subdivide the zone by medial lines, as shown inFig. 76, and on very steep slopes the shading lines may be drawn over two or more zones. For ordinary scales the extremes of length may be fixed at1⁄6of an inch on the steepest slopes, and3⁄4of an inch on the gentlest.
Fig. 76.
Fig. 76.
It is not necessary to repeat the process of construction for every line, such a mode of proceeding would be too laborious and slow. It will be sufficient to determine the lines in this exact manner at those parts where the greatest changes of slope occur. Thus a group should be constructed in each zone where the slope is greatest and anotherwhere it is least, after which a few intermediate ones may be put in. The vacancies may then be filled in, taking care to graduate the changes in passing from group to group. By this means we do not, of course, get a mathematically exact representation of the surface, but it is sufficiently accurate for practical purposes.
When the preparatory pencil lines have been drawn in and the spaces for the shading lines laid off by dots, the shading should be commenced at the steepest part of the upper zone. The lines should be drawn firmly from curve to curve, taking care to make each row terminate evenly at the lower edge; they must always be drawn downwards and from left to right, proceeding in this direction round the zone till the point of setting out is reached, where the joining must be carefully effected. This can always be done most neatly where the lines are thickest, as we have previously pointed out. The succeeding zones should be filled up in the same manner. As changes must be gradual in every direction, care must be taken to make the contiguous zones blend into each other. When it is required to pass from a light zone to a darker one beneath it, the lower ends of the lines in the light zone should be thickened a little, so as to meet the upper ends of the lines in the dark zone with nearly the same colour. The upper ends of these latter lines should also be slightly lightened. The lines of one zone must not be continued into those of the next. Even on a uniform slope such a prolongation of the lines would produce a hard appearance, which should be avoided. But in the case of a conical hill, like that shown inFig. 77, it would give rise to an error in principle; for soon after leaving the summit we should have too few lines of descent. When the hill has been covered with shading lines, the base and the summit must be softened off by tapering the lower end of each line at the base, and the upper end of each line at the summit. To give the taper to the latter, the drawing should be turned upside down.
Fig. 77.
Fig. 77.
When the curves are parallel or nearly so, the shading lines arestraight, and also nearly parallel. But when the curves depart widely from each other, the shading lines will themselves have a slight curvature, for being lines of greatest descent, they must be normal to the curves. In such cases, a number of normals should be put in at short distances with the pencil, as shown inFig. 78, to serve as guides to the shading lines. The foregoing directions for shading a hill apply equally to the shading of a hollow, the shading lines in which are converging.
Fig. 78.
Fig. 78.
Occasionally short slopes steeper than the “natural slope” of 45° will be met with. Such being exceptions to the law of slopes, are marked in an exceptional manner. When the surfaces of these slopes are of earth, they are shown by black lines drawn parallel to the horizontal curves, and when of rock, by black lines drawn in all directions, not intersecting, but abutting abruptly upon each other in short heavy masses, as shown inFig. 78.
—Frequently in topographical drawings, and still more frequently in mechanical drawings, colour is resorted to to produce the effect of shading lines. As the principles according to which colour is applied for this purpose are the same as those which determine the use of shading lines, there remains little to be said on this matter beyond describing the modes of applying the colour.
—In representing slopes, the tint employed to give the effect of that produced by the ink lines already described is composed of indigo and burnt sienna, and is applied as a flat-wash. A little lake is added to neutralize the greenish hue of this tint when it is to be laid over sand or cultivated ground. The different degrees of intensity required to express the inclination are produced by repeating the wash over those parts which are darker than the rest. To accomplishthis neatly, the darker portions must be washed in first, so that the final washings may cover the whole surface, and the edges of each successive wash must be softened off or blended into the next with a brush and clean water. In shading hills, the paper along the crest of the slope should be first moistened with the water-brush, and before it dries, the laying on of the colour should be begun on the moistened part, and proceeded with down the slope. The effect of representing hills by this method, which is a very expeditious one, is much improved by adding light shading lines with the pen, either in pale ink, or a mixture of indigo and burnt sienna. The ground is always covered with its appropriate sign before the shading tint is laid on.
—In shading cylindrical surfaces and drawings generally, three methods are employed. One of these is known assoftening off, and is employed on finished drawings of machinery. For shading by this method, a brush called a softener is required; this has a brush at each end of the handle, one being larger than the other. Having moistened the paper, and filled the smaller brush with colour and the larger one with water, a narrow strip of colour is laid along the darkest part of the cylinder, and immediately after, while the colour is quite moist, the water-brush is drawn along one edge of the strip and then in like manner along the other, so as to cause the colour to flow over that portion of the surface which has been damped. The brush is then wiped upon a cloth and drawn lightly down the edge to take up the superfluous water. The colour should be light to begin with, and the quantity to be taken in the brush must be determined by experience. The same remark applies to the water-brush, for if too little be used the colour will not spread sufficiently, and if too much, the colour will be diluted and rendered uneven. These operations of laying on the colour and softening off are continued until the cylindrical appearance has been produced. Each succeeding coat should be laid on before the preceding one is quite dry, as the colour will spread more evenly over a damp surface. The previously applied coat must, however, have beensufficiently absorbed not to wash up, or a clouded appearance will be the result.
Another method, known as the French, consists in applying a narrow strip of colour to the darkest part, and overlaying this with other strips, each wider than the one previously laid on. To regulate the breadth of the strips, a number of meridian lines are drawn upon the cylinder. When shaded in this manner, the cylinder presents the appearance of a polygon rather than that of a cylinder.
The third method, by reason of the facility it affords of producing effect, is very suitable for large drawings and diagrams for illustrating papers and lectures. In shading according to this method, a thick line or a narrow strip of very thick and black Indian ink is laid on the darkest part of the cylinder with the point of the brush. The breadth of the strip will be regulated by the diameter of the object, and it should be previously lined out in pencil. When dry, a damp brush is passed over it so as to remove the sharp edges of the strip, and to cause the ink to run slightly over the moistened surface of the paper. The flat colour washes are then applied as required, the washes being carried over the black strips, which will be further reduced in tone by a portion of the ink mixing with the colour.
In shading, it will be found convenient to keep the light side of the object next to the operator, as it is easier to wash towards the body than from it with the water-brush. The brush should be held in as nearly a vertical position as possible, as it is more easy, when that position is observed, to keep within the boundary lines.
Line
—The lettering of a plan, map, or drawing of any kind, occupies a prominent and conspicuous position, and may be considered as forming an essential part of the drawing. It is, therefore, obvious that the character of the lettering, and the degree of finish introduced into its execution, will have an important influence on the general appearance of the drawing. Nothing detracts more from the value of a map, considered as a work of art, than a bad style of lettering, while, on the other hand, a well-chosen and well-executed style is both pleasing to the eye, and produces on the mind an impression of accuracy in the more important features of the work. Hence it is not merely desirable, but necessary, that the draughtsman should acquire the ability to form letters correctly and neatly, especially if he be engaged on topographical drawings, into which lettering enters very largely.
The formation of letters requires great attention and long practice. It is not a matter in which much assistance is to be derived from descriptions or written instructions of any kind; practice alone from good models will give the requisite skill. The difficulty of forming the letters correctly and of uniform dimensions may, however, be considerably lessened by using guide lines drawn in pencil, to be afterwards erased. Such lines are calledconstruction lines, and the mode of employing them is shown inPlate 4. A careful study of this Plate will give the student a clear understanding of the use of these lines, which could not be imparted by pages of description. A reference to the letters B, E, and T, in connection with the construction lines willshow most readily the nature and the degree of assistance afforded by the latter.
In making capitals, each letter must be sketched in pencil; the outline must then be drawn in ink with a firm and steady line, and afterwards filled up with the pen. In forming the small roman and italic letters, three construction lines are drawn, the lower two to limit the height of the ordinary letters, and the upper one to limit the height of such letters asdandl, and the capitals. The heavy parts of these letters are made at once by a bold pressure of the pen. The curved portions should be carefully distinguished from the straight. The lettersa,c,g,o,s, &c., for example, are composed wholly of curved lines. They should be drawn symmetrically, and their width should be only a little less than their height. The round portion of thegshould not quite reach to the lower line. A perfect regularity should be maintained throughout the letters, as the beauty of their appearance depends greatly on this. Care must also be taken, in italic writing, to keep the inclination the same everywhere. Manuscript lettering should be more extended than the clear roman or italic type, for crowding greatly mars its appearance.
The character of the letters employed should be in accordance with that of the drawing upon which they are to appear. Thus for engineering and mechanical drawings, there is nothing more suitable generally than the plain block letter. But on drawings of a more artistic and ornamental character, a more elaborate form of letter may and should be used. And of these elaborate forms, there will always be one more suitable than the rest to the particular character of the drawing. The choice of this form is a matter to be left entirely to the judgment and the taste of the draughtsman.
Another matter on which the draughtsman will have to exercise his judgment is thesizeof the letters employed. This must manifestly be in accordance, first, with the character of the object denoted, and, second, with the scale of the drawing. With regard to the former of these conditions, it is obvious that propriety will demand a larger letter for the city than the town, for the town than the village, forthe village than the farm, and for the mansion than the gate-lodge. This propriety of relative importance must be everywhere observed. The different types of lettering are arranged in the order of importance as follows:—1, The upright capital; 2, the inclined capital; 3, the upright roman, or ordinary small type; and 4, the small italic. The draughtsman will have to exercise his judgment in suiting the size to the scale of the map, but the following Table may be taken as a generalguide:—
The thickness of the capital should be one-seventh of the height.
As far as practicable, the lines of lettering should be parallel to the base of the drawing. Frequently, however, cases will occur in which it will be desirable to letter in other directions and in curved lines. In writing along a curved or very irregular outline, the course of a river or the boundary of an estate, for example, an agreeable effect is produced by making the lines of lettering conform in some degree with the outlines against which they are written.
The arrangement of the letters in titles and the effective disposition of the words are also matters requiring great care and some taste. The design and the execution of the title afford another opportunity of enhancing the beauty of a drawing by a display of striking arrangement and appropriate ornamentation.Plates 7and8show some useful models for plans, andPlate 25contains some specimens of flourishes which may frequently be introduced with pleasing effect. The form which the title shall assume and the space which it shall occupy must be determined before beginning to put it upon the drawing. To avoid erasures, it is well to sketch roughly upon a piece of paper, a trial title, emendations in which can be easily made. When found satisfactory, draw a vertical centre line, whichshould pass through the middle letter of each line. Apply this centre line to the centre line of the title on the drawing, and lightly mark in with the pencil the position of each letter. When this method is not adopted, the middle letters should be put in first upon the centre line, and the others afterwards inserted from left to right, and from right to left.
In maps, the title may be placed outside the border if it consist of one line only, but if it occupy more than one line, it should be placed within the border. Generally, it should be placed in one of the corners of the map, and its size should bear some proportion to that of the map. The letters composing the name of the locality, which is usually the most important word, should not exceed in height three-hundredths of the length of the short side of the border. The letters of the other words will vary in size according to their relative importance.
—Plain borders usually consist of two lines, the outer one heavy, and the inner one light. The heavy line should be equal in breadth to the blank space between it and the light line, and the total breadth of the border, that is, of the two lines and the space between them, should be one hundredth part of the length of the shorter side. Ornamental corners may be made to embellish a drawing considerably, and they afford some scope to the fancy and the taste of the draughtsman. Several examples of borders and ornamental corners will be found in the accompanyingPlates.
—The meridian or north and south line is an indispensable adjunct to every topographical drawing. When the extent of country represented is considerable, it is usual to make the top of the map the north, and in such a case the side border lines are meridian lines. Frequently, however, in plans, the shape of the ground does not admit of this arrangement, and then it becomes necessary to mark a meridian on some part of the map. This line is usually made a conspicuous one, and its north extremity is often ornamented with some fanciful device. The ornamentation of the meridian line should be in keeping with the rest of the map.Plate 9contains several examples which may be adopted or modified as deemed desirable.
To all drawings which do not show the full size of the objects represented, it is necessary to affix the scale according to which the objects are drawn. Such a scale is called a scale of lengths or distances, because, by means of it, the distance from one point to another is ascertained. The scale of distances does not contain very minute subdivisions, and consequently is not suitable for use inconstructingthe drawing. For the latter purpose, another scale, similarly but more minutely divided, is employed, and is known as the scale of construction. A familiarity with the modes of constructing both of these scales should be early acquired by the young draughtsman.
—One means of denoting the scale of a drawing is furnished by what is called itsrepresentative fraction, the denominator of which shows how many times greater the actual length is than that in the drawing. Thus a scale of1⁄24shows that 1 inch on the drawing represents 24 inches on the object; in other words, that the object is twenty-four times larger than the drawing. But in addition to this representative fraction, it is usual to affix a graduated straight line, termed ascale, for the purpose of conveniently measuring distances upon it. It is manifest that the unit of length in this scale must bear the same ratio to the real unit of length that a line in the drawing bears to the line which it represents. Thus if the representative fraction be1⁄24, 1 inch on the scale will represent 2 feet.
Scales of distances are usually of such a length as to be a multiple of 10 linear units of some kind, as 100 miles, 50 chains, 20 feet; and this length should also be such as to allow of long lines being taken off at one measurement. To construct the scale, two light lines should be drawn at a suitable distance apart, and below the lower of these lines and at a distance from it equal to one-third of the space between them, a third and heavy line should be drawn. The primary divisions may then be made with the compasses in the following manner. Supposing the number of divisions to be five, open thedividers to what appears to be the fifth part of the line, and step this distance along the line; if the fifth step exceed or fall short of the end of the line, close or open the dividers1⁄5of the distance, and repeat the trial. This is the quickest and, for large divisions, the most accurate method of dividing a line. To render the divisions more distinct, draw a heavy line between the two light lines in alternate divisions. The left-hand division must be subdivided into the units or lesser measures of which it is made up. For example, if the primary divisions are each of 10 feet, the subdivisions will be feet; if they represent feet, the subdivisions will be inches, and so on. The subdividing should be performed in the following manner. Having erected a perpendicular of indefinite length from the left-hand extremity of the scale, take with the compasses from any scale the number of divisions into which it is required to divide the part. With this distance in the compasses, strike, from the first primary or zero division, an arc cutting the perpendicular, and join the point of intersection to the centre from which the arc is struck. Thus we shall have a right-angled triangle formed of the first primary division of the scale, the perpendicular and the radius, the latter being the hypothenuse (seeFig. 79). Mark on the hypothenuse the divisions to which it was made equal, and from the points of division let fall perpendicular lines upon the scale. These will divide the latter into the required number of equal parts. The length of the hypothenuse should be so chosen as to make an angle not greater than 50° with the base.
Fig. 79.Larger illustration(19 kB).
Fig. 79.
Larger illustration(19 kB).
The total length of the scale will be determined by the greatest length which it is required to read off at once, and in the following manner. Thus, let it be required to construct a scale of1⁄24, =1⁄2inch tothe foot, to show 12 feet. Here ·5 inch :xinches :: 1 inch : 12 inches; whencex= 12 × ·5 = 6 inches. This distance of 6 inches must, therefore, be set off upon the lines intended for the scale, and divided in the manner described above. Again, to construct a scale of1⁄10560, = 6 inches to a mile, to show 100 chains. Since 6 inches represents 5280 feet or5280⁄60= 80 chains, the proportion becomes 6 :x:: 80 : 100; whencex=600⁄80= 71⁄2inches. If the scale is1⁄3960= 16 inches to a mile, = 5 chains to an inch, and the distance to be shown is 30 chains, we have 1 :x:: 5 : 30; orx=30⁄5= 6 inches. In a scale of 10 yards to the inch, for example, the representative fraction is 10 × 3 × 12 =1⁄360. So, on the contrary,1⁄360=360⁄36= 10 yards to the inch. Sometimes it is required to construct a comparative scale, that is, a scale having the same representative fraction, but containing other units. Thus suppose, for example, we have a Russian plan on which is marked a scale ofarchinesmeasuring a length of 50 archines. It is required to draw upon this plan a comparative scale of yards, upon which a distance of 50 yards may be measured. The Russian archine = ·777 yard. Hence we have the proportion 50 :x:: ·777 : 1, whencex=50⁄777= 64·35 archines. Measure off this length from the Russian scale, and upon it construct the English scale in the manner already described. This scale may then be used to measure distances on the plan.
Amongst Continental nations, decimal scales are usually employed, which are far more convenient in practice than those involving the awkward ratios of miles, furlongs, chains, yards, feet, and inches. The decimal scale has also been adopted for the United States’ Coast Survey, the smallest publication scale of which is1⁄30000; this is also the scale of the new map of France.
In choosing a scale, regard must be had alike to the purposes for which the drawing is intended, and to the nature and the amount of detail required to be shown. Thus a larger scale is required in plans of towns than in those of the open country; and the smaller and more intricate the buildings, the larger should the scale be. Also a plan to be used for the setting out of works should be to a larger scale than one made for parliamentary purposes.
The following Tables, given by Rankine in his ‘Civil Engineering,’ enumerate some of the scales for plans most commonly used in Britain, together with a statement of the purposes to which they are best adapted.
The vertical scale, or scale of heights, is always much greater than the horizontal scale or scale of distances, and the proportion in which the vertical scale is greater than the horizontal, is called theexaggerationof the scale. This exaggeration is necessary, because the differences of elevation between points on the ground are in general much smaller than their distances apart, and would therefore, without exaggeration, be unapparent, and also because, in the execution of engineering works, accuracy in levels is of more importance than accuracy in horizontal positions.
—Scales of construction are intended to afford means of measuring more minute quantities than scales of distances. Of the former there are two kinds, known respectively as theDiagonaland theVernierscale. The diagonal is the more frequently employed. Its construction involves no peculiar difficulty, as it consists simply of an ordinary scale of distances, with the addition of a number of parallel lines crossed by other parallel lines drawn diagonally from the smaller points of division. An example will best show the construction and mode of using this scale. Suppose it to be required to construct a scale of 10 miles to the inch, showing furlongs diagonally; the scale to measure 50 miles. Here 1 : 10 ::x: 50, whencex= 5 inches. Divide this length of 5 inches into five equal parts, and the first part into tenths to show miles, in the manner already described for scales of distances. Then, since it is required toshow furlongs or eighths of a mile, eight equidistant parallel lines must be drawn above the scale, at a convenient interval apart, as shown inFig. 80. Produce the primary points of division to meet the top parallel; and from the last secondary point of division draw a line to the point in which the extreme primary division meets the top parallel. Draw from the other points of division, lines parallel to this one, and the scale will be complete. It will be seen that the inclined lines are the diagonals of the rectangular figures formed by the top and bottom parallels and vertical lines drawn from the smaller points of division.