## Numerical scale

scale is a scale, which is expressed by a fraction. Its numerator equals 1 (=1) and its denominator shows how many times the horizontal distance line in the image area is reduced while representing a horizontal line on a plan or map.Numerical scale – is an unnamed value. . It is written in the form: 1:1000, 1:2000, 1: 5000, etc. Numbers 1000, 2000 and 5000 are called the denominator of scale M there.

Numerical scale shows, that one unit of length of the line on the plan (map) contains the same number of units of length of line, located on the ground. For example, one unit of the length of the line on the plan of scale 1:5000 contains exactly 5000 units of the same length on the ground. It means, that one centimeter on the plan 1:5000 corresponds to 5000 centimeters or 50 meters on the ground.

In some cases, the linear scale can be used for solving certain tasks. A linear scale is a special graphical representation of numerical scale (see Fig. 1). The segment AB is a basis of the linear scale. Usually it equals 2 cm. It is transferred into the appropriate length on the ground and signed. The leftmost base scale is divided into 10 equal parts.

Fig.1

Example (Fig. 1): This linear scale is used for work on topographic plan of scale 1:2000. Its basis equals 2 cm which is equivalent to 40 meters on the ground for this scale. The smallest part of the basis equals 2 millimeters or 4 meters on the ground.

The segment CD is taken from the topographic plan of scale 1:2000. It consists of two bases and two smallest parts which, as a result, correspond to 2*40m+2*2m = 84 m. (Fig. 1)

More accurate graphical definition and construction of line lengths can be done using another graphics - on the transverse scale (see fig. 2).

## The transverse scale

– scale is a graph for the most accurate measurement of distances on the topographic plan (map).The base of scale AB equals 2 cm. The smallest segment of base is CD=0.1*AB=2mm. The smallest part of transverse scale’s base is cd=0.1*CD=0.1*AB=0.2 mm. This relation follows from the similarity of triangles BCD and Bcd (see fig. 2).

Thus, base of transverse scale will correspond to 40 m for the numerical scale 1:2000, the one-tenth of base will equal 4 m, and the hundredth of AB will equal 0.4 m.

Example: the segment AB (Fig. 2), taken from the plan of 1:2000 scale, corresponds to the length 137.6 m. It equals 3 bases of transverse scale (3x40 = 120), four bases and the small segment of base (4x4 = 16 m ) and one smallest segment of the base of scale equal 0.4*4 or 1.6 m.

Let’s consider the most important characteristic of the concept of "scale".

The precision of scale is the horizontal segment on the ground, which corresponds to segment 0.01 cm on the plan of this scale. This characteristic depends on resolving power of naked human eye. Resolving power allows to consider minimal distance, equal to 0.1 mm, on the topographic plan. This value will be equal to 0.1 mm*M on the terrain, where M is a denominator of scale.

Fig.2

Transverse scale allows to measure the length of the line on the plan (map) scale of 1:2000 accurate to precision of this scale.

Example: 1 mm of 1:2000 plan contains 2000 mm of the terrain and 0.1 mm on the terrain contains, accordingly, 0,1xM (mm) = 0.1 x 2000 mm = 200 mm = 20 cm or 0.2 m.

The value of measurement of the line length on the plan should be rounded to the precision of scale. Example: The length of the line equals 58,37 meters. It is built in scale 1:2000. The length of the line is rounded up to 58,4 m, because precision of scale equals 0,2 meters. If line is built in scale 1:500, its length is rounded up to 58.35 (precision of scale 1:500 equals 0,05).

Fig.3