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.
[Illustration: FIG. 79.]
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 |Fraction| Use.
of Scale. |of real |
| Dimen- |
| sions. |
----------------------------+--------+--------------------------------
1.--1 inch to a mile | ¹⁄₆₃₃₆₀| Scale of the smaller Ordnance
| | maps of Britain. This scale is
| | well adapted for maps to 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 deposited plans of
| | proposed works.
3.--6 inches to a mile | ¹⁄₁₀₅₆₀| Scale of the larger Ordnance
| | maps of Great Britain and
| | Ireland. This scale, being just
| | large enough to show buildings,
| | roads, and other important
| | objects distinctly in their
| | true forms and proportions, and
| | at the same time small enough
| | to enable the eye of the
| | engineer to embrace the plan of
| | a considerable extent of
| | country at one view, is on the
| | whole the best adapted for the
| | selection of lines 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 “enlarged plans” 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 engineering works, and
| |}for enlarged parliamentary
| |}plans.
8.--5 chains to an inch, | ¹⁄₃₀₆₀ | (Scale 8 is that prescribed in
or 16 inches to a mile.| | the Standing Orders of
| | Parliament for “cross sections”
| | 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
| | mentioned are 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 | ¹⁄₁₀₈₀ | Scale of the Ordnance plans of
60 inches to a mile. | | 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 | ¹⁄₅₂₈ | Scale of the Ordnance plans of
120 inches to a mile. | | the more intricately built
| | towns.
16.--126·72 inches to a | ¹⁄₅₀₀ | Decimal scale having the same
mile | | properties.
17.--30 feet to an inch | ¹⁄₃₆₀ |}
18.--20 feet to an inch | ¹⁄₂₄₀ |}Scales for special purposes.
19.--10 feet to an inch | ¹⁄₁₂₀ |}
----------------------------+--------+--------------------------------
VERTICAL SCALES.
------------+--------+------------------+-------------+---------------
Ordinary |Fraction| Horizontal Scales|Exaggeration.|
Designation| of real| with which the | | Use.
of Vertical| Height.| Vertical Scale is| |
Scale. | | usually combined.| |
------------+--------+------------------+-------------+---------------
| | | from |
1.--100 feet| ¹⁄₁₂₀₀ |¹⁄₁₅₈₄₀ to ¹⁄₁₀₅₆₀| 13·2 to 8·8 | Smallest scale
to an inch | | | | permitted by
| | | | the Standing
| | | | Orders of
| | | | Parliament for
| | | | sections of
| | | | proposed
| | | | works.
2.--40 feet | ¹⁄₄₈₀₀ |¹⁄₄₈₀₀ to ¹⁄₃₉₆₀ | 10 to 8·25 | Smallest scale
to an inch | | | | permitted by
| | | | the Standing
| | | | Orders of
| | | | Parliament for
| | | | cross sections
| | | | showing
| | | | alterations of
| | | | roads.
3.--30 feet | ¹⁄₃₆₀ |¹⁄₃₉₆₀ to ¹⁄₂₃₇₆ | 11 to 6·6 |}Scales
to an inch | | | | suitable for
4.--20 feet | ¹⁄₂₄₀ |¹⁄₃₉₆₀ to ¹⁄₂₃₇₆ | 16·5 to 9·9 |} working
to an inch | | | | 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.
[Illustration: FIG. 80.]
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_ + 1)/_n_
units; and the difference between one division of the latter and one of
the former will be ((_x_ + 1)/_n_) - 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 _x_th divisions
from this on both scales will be the required length of _x_/_n_ of a
unit. The same reasoning will evidently hold good if a length equal to
_n_ - 1 be taken.
[Illustration: FIG. 81.]
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.