Section II.—Scales.

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 in constructing the 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.

Scales of Distances.

—One means of denoting the scale of a drawing is furnished by what is called its representative fraction, the denominator of which shows how many times greater the actual length is than that in the drawing. Thus a scale of ¹⁄₂₄ shows 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 a scale, 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 be ¹⁄₂₄, 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 the dividers 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 dividers ¹⁄₅ of 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 (see [Fig. 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).

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 of ¹⁄₂₄, = ¹⁄₂ inch to the foot, to show 12 feet. Here ·5 inch : x inches :: 1 inch : 12 inches; whence x = 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 of ¹⁄₁₀₅₆₀, = 6 inches to a mile, to show 100 chains. Since 6 inches represents 5280 feet or ⁵²⁸⁰⁄₆₀ = 80 chains, the proportion becomes 6 : x :: 80 : 100; whence x = ⁶⁰⁰⁄₈₀ = 7¹⁄₂ inches. If the scale is ¹⁄₃₉₆₀ = 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; or x = 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 of archines measuring 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, whence x = 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 is ¹⁄₃₀₀₀₀; 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.

Horizontal Scales.
Ordinary Designation of Scale.		Fraction of real Dimensions.	Use.
1.—	1 inch to a mile	¹⁄₆₃₃₆₀			Scale of the smaller Ordnance maps of Britain. This scale is well adapted for mapsto be used in exploring the country.
2.—	4 inches to a mile	¹⁄₁₅₈₄₀			Smallest scale permitted by the Standing Orders of Parliament for the depositedplans of proposed works.
3.—	6 inches to a mile	¹⁄₁₀₅₆₀			Scale of the larger Ordnance maps of Great Britain and Ireland. This scale, beingjust large enough to show buildings, roads, and other important objects distinctly in their true formsand proportions, and at the same time small enough to enable the eye of the engineer to embrace the planof a considerable extent of country at one view, is on the whole the best adapted for the selection oflines for engineering works, and for parliamentary plans and preliminary estimates.
4.—	6·366 inches to a mile	¹⁄₁₀₀₀₀			Decimal scale possessing the same advantages.
5.—	400 feet to an inch	¹⁄₄₈₀₀			Smallest scale permitted by the Standing Orders of Parliament for “enlargedplans” of buildings and of land within the curtilage.
6.—	6 chains to an inch	¹⁄₄₇₅₂		-	Scale answering the same purpose.
7.—	15·84 inches to a mile	¹⁄₄₀₀₀			Scales well suited for the working surveys and land plans of great engineeringworks, and for enlarged parliamentary plans.
8.—	5 chains to an inch, or 16 inches to a mile.	¹⁄₃₀₆₀
					(Scale 8 is that prescribed in the Standing Orders of Parliament for “crosssections” of proposed railways, showing alterations of roads.)
9.—	25·344 inches to a mile	¹⁄₂₅₀₀			Scale of plans of part of the Ordnance survey of Britain, from which the maps before mentionedare reduced. Well adapted for land plans of engineering works and plans of estates.
10.—	200 feet to an inch	¹⁄₂₄₀₀			Scale suited for similar purposes. Smallest scale prescribed by law for land or contract plans in Ireland.
11.—	3 chains to an inch	¹⁄₂₃₇₆			Scale of the Tithe Commissioners’ plans. Suited for the same purposes as the above.
12.—	100 feet to an inch	¹⁄₁₂₀₀			Scale suited for plans of towns, when not very intricate.
13.—	88 feet to an inch, or 60 inches to a mile.	¹⁄₁₀₈₀			Scale of the Ordnance plans of the less intricately built towns.
14.—	63·36 inches to a mile	¹⁄₁₀₀₀			Decimal scale having the same properties.
15.—	44 feet to an inch, or 120 inches to a mile.	¹⁄₅₂₈			Scale of the Ordnance plans of the more intricately built towns.
16.—	126·72 inches to a mile	¹⁄₅₀₀			Decimal scale having the same properties.
17.—	30 feet to an inch	¹⁄₃₆₀		-	Scales for special purposes.
18.—	20 feet to an inch	¹⁄₂₄₀
19.—	10 feet to an inch	¹⁄₁₂₀

Vertical Scales.
Ordinary Designation of Vertical Scale.		Fraction of real Height.	Horizontal Scales with which the Vertical Scale is usually combined.			Exaggeration.			Use.
						from
1.—	100 feet to an inch	¹⁄₁₂₀₀	¹⁄₁₅₈₄₀	to	¹⁄₁₀₅₆₀	13·2	to	8·8			Smallest scale permitted by the Standing Orders of Parliament for sections of proposed works.
2.—	40 feet to an inch	¹⁄₄₈₀₀	¹⁄₄₈₀₀	to	¹⁄₃₉₆₀	10	to	8·25			Smallest scale permitted by the Standing Orders of Parliament for cross sections showing alterations of roads.
3.—	30 feet to an inch	¹⁄₃₆₀	¹⁄₃₉₆₀	to	¹⁄₂₃₇₆	11	to	6·6		-	Scales suitable for working sections.
4.—	20 feet to an inch	¹⁄₂₄₀	¹⁄₃₉₆₀	to	¹⁄₂₃₇₆	16·5	to	9·9		-	Scales suitable for working sections.

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 the exaggeration of 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.

—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 the Diagonal and the Vernier scale. 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, whence x = 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 to show furlongs or eighths of a mile, eight equidistant parallel lines must be drawn above the scale, at a convenient interval apart, as shown in [Fig. 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.

Fig. 80.

[Larger illustration] (31 kB).

To use this scale, suppose a length of 24 miles 5 furlongs is required. Place one leg of the dividers upon the point in which the fourth diagonal intersects the fifth parallel, and extend the other to the point in which the primary division marked 20 intersects the same parallel. In like manner, if the distance required be 33 miles 3 furlongs, it must be taken from the intersection of the third diagonal with the third parallel, to the intersection of the primary division marked 30 with the same parallel.

It is obvious that if a scale of feet showing inches diagonally be required, twelve equidistant parallel lines must be drawn instead of eight as in the foregoing example where furlongs are required. The diagonal scale possesses the important advantages of accuracy and distinctness of division which render it very suitable as a scale of construction. Another practical advantage is that it is less rapidly defaced by use than the other kinds, in consequence of the measurements being taken on so many different lines.

The construction of the vernier scale is similar to that of the graduated arcs of surveying and astronomical instruments. The principle of the vernier is as follows. If a line containing n units of measurement be divided into n equal parts, each part will, of course, represent one unit; and if a line containing n + 1 of these units be also divided into n parts, each part will be equal to n + 1n units; and the difference between one division of the latter and one of the former will be x + 1n - 1 = ¹⁄_n of the original unit. Similarly, the difference between two divisions of the one and two of the other will be ²⁄_n of a unit, between three of the one and three of the other, ³⁄_n, and so on. Hence, to obtain a length of ^x⁄_n of a unit, we have only to make a division on one scale coincide with one on the other scale; the space between the two corresponding xth divisions from this on both scales will be the required length of ²⁄_n of a unit. The same reasoning will evidently hold good if a length equal to n - 1 be taken.

Fig. 81.

[Larger illustration] (23 kB).

To show how the foregoing principle is applied in practice, we will take an example. It is required to construct a scale of ¹⁄₁₀₀ to show feet and tenths of a foot. Construct a scale in the ordinary way, and subdivide it throughout its whole length, as shown in [Fig. 81]; then each division will show one foot. Above the first primary division, draw a line parallel to the scale and terminating at the zero point. From the zero point, set off on this line towards the left a distance equal to eleven subdivisions, and divide this distance into ten equal parts. Now, as eleven divisions of the plain scale have been divided into ten equal parts on the vernier, each division on the latter will represent ¹¹⁄₁₀ = 1·1 of that on the former; and as the divisions of the plain scale represent feet, those of the vernier will represent 1·1 foot. Consequently, the distances from the zero of the scale to the successive divisions on the vernier are 1·1, 2·2, 3·3, 4·4, 5·5, 6·6, 7·7, 8·8, 9·9, and 11 feet. It will be seen that the divisions of the two scales are made to coincide at the zero point.

The mode of using this scale will be seen from the following example. Let it be required to take off a distance of 26·7 feet. From zero to the 7th division of the vernier is, as we have seen, 7·7 feet. Therefore, to ascertain how far to the right of zero we must go to obtain the distance of 26·7 feet, we must subtract 7·7 from that distance, which gives 19. Thus to take off the distance, one leg of the dividers must be placed on the 7th division of the vernier, and the other on the 19th division of the plain scale. If the distance to be taken were 27·6 feet, one leg of the dividers would have to be placed on the 6th division of the vernier, and the other on the (27·6) - (6·6) = 21st division of the plain scale.

To construct a scale to show feet and inches, make the vernier equal to thirteen divisions of the plain scale and divide it into twelve equal parts. Each of these divisions will then represent ¹³⁄₁₂ = 1¹⁄₁₂ of a foot.

Scales of construction may be purchased upon box-wood or ivory, but where great accuracy is important, it is best to lay down the scale upon some part of the drawing, as in such a case it expands and contracts with the drawing under the influence of moisture.

Examples of scales of distances will be found on [Plates 8] and [9].