Chapter 9 — Maps and Scales

A map is a representation of the whole or a part of the earth's surface, drawn to scale on a flat surface, showing selected features by conventional signs and symbols. Three ideas are packed into this definition. First, a map is drawn to scale: the distances on the paper bear a fixed ratio to distances on the ground, so that the map can be measured. Second, a map is drawn on a flat surface, and since the earth is a sphere, every map involves some distortion, which is why map projections are needed. Third, a map shows selected features by symbols: it cannot show everything, so the map-maker chooses what matters for the purpose and shows it by lines, colours, dots and signs that a legend explains.

Map, globe, plan and sketch. A globe is a model of the earth on a sphere; it shows shapes, areas and directions correctly all at once, but it is bulky, cannot be folded into a book, shows only half the earth at a time and cannot show a small area in detail. A map can be drawn for any area at any scale, carried in a pocket, printed in thousands and compared side by side, and this is why maps are the working tool of geography. A plan is a map of a very small area, such as a house, a school compound or a village, drawn to a very large scale so that every wall and path appears. A sketch is a rough drawing of an area made from observation or memory, without an accurate scale; it shows relative positions but cannot be measured.

History. The oldest known map is a clay tablet from Babylon about 2,500 years old; the Greeks — Anaximander, Eratosthenes, who measured the earth's circumference, and Ptolemy, whose Geographia gave latitude and longitude for 8,000 places — laid the scientific foundation. Indian astronomers such as Aryabhata and Varahamihira wrote on the shape and size of the earth. Arab geographers such as Al-Idrisi kept the science alive, and the age of exploration from the fifteenth century produced the great European atlases, Mercator's projection of 1569 and the word atlas itself. The Survey of India, founded in 1767, mapped the subcontinent through the Great Trigonometrical Survey begun by Lambton in 1802 and continued by George Everest, and it still produces India's topographical maps. Today satellites, aerial photographs, the Global Positioning System and computers (Geographic Information Systems) make maps faster and more accurate than ever.

Uses. Maps are used by travellers, soldiers, planners, farmers, engineers, sailors and pilots, weather forecasters, geologists and disaster managers; the student of geography uses them to locate places, measure distances, understand relief and drainage, compare regions and present data. Learning to read a map, said one geographer, is learning to read the landscape at a glance.

A map is only useful if the reader can understand it without the map-maker standing by. Certain elements must therefore appear on every proper map; geographers remember them by the phrase title, scale, direction, legend, grid, source.

1. Title. The title tells what the map shows and of which area: West Bengal — Rainfall; India — Political; Kolkata — Road Map. It is placed at the top, and a date is added if the information changes with time, for example a population map of 2011.

2. Scale. The scale states the ratio between distance on the map and distance on the ground, in words, as a fraction or as a drawn bar. Without it no distance or area can be measured and the map is only a picture. The scale is usually placed below the map.

3. Direction. A map must show which way is north, by a north arrow or the letters N, S, E, W, so that all other directions can be read. By convention north is at the top of the map unless the arrow shows otherwise; the geographic north (true north) and the magnetic north to which a compass points differ by a small angle called the magnetic declination, and topographical sheets show both.

4. Legend or key. The legend explains the conventional signs, symbols and colours used: a blue line for a river, a red line for a road, a black cross for a church, a triangle for a peak, brown contours for relief, green for forest. Without the key the map cannot be read; with it any map can be read, in any language.

5. Grid or reference system. A network of lines lets the reader state the exact position of a feature: on world and country maps the lines of latitude and longitude; on topographical sheets the numbered grid squares (eastings and northings); on street maps letters and numbers along the edges. The grid also lets two maps be joined and a place be found quickly.

6. Source, date and author. The map should say where the information came from — a survey, a census, a satellite image — and when it was made, so that the reader can judge how reliable and how current it is. The name of the surveyor or agency, the projection used and the sheet number are printed in the margin of official maps.

Two further elements give a map its quality. Neatness and generalisation: the map must be clean, the lettering legible, and the detail suited to the scale — a small-scale map must leave out what a large-scale map shows. Projection: the method by which the curved earth has been drawn flat, since it decides which properties — shape, area, distance, direction — are correct and which are distorted.

A student drawing a map for an examination should therefore always add the title, a north arrow, a scale, a key and a neat border. The examiner looks for these elements before the content.

Maps are classified in two chief ways, by scale and by purpose. By scale, that is by the amount of ground shown on a given size of paper and hence by the amount of detail, maps fall into four groups.

1. Cadastral maps. The word comes from the French cadastre, a register of property. These are maps of the largest scale, drawn to show the boundaries of individual plots of land, fields, houses and roads for the purpose of recording ownership and collecting revenue. The village maps (mouza maps) of West Bengal prepared by the Land Records department are cadastral maps, at scales such as 16 inches to a mile (1 : 3,960) or 1 : 4,000, showing every plot with its number (dag number). City surveys and municipal maps at 1 : 1,000 or 1 : 2,000 also belong here. They are used for land registration, property tax, town planning and settling boundary disputes.

2. Topographical maps. These show the topography, that is the physical and cultural features of an area in detail — relief by contours, rivers, forests, roads, railways, villages, temples, wells and boundaries — at scales from 1 : 25,000 to 1 : 250,000. The Survey of India publishes them for the whole country in the series of 1 : 50,000 (the basic sheet, 1 cm to 500 m), 1 : 25,000 for special areas and 1 : 250,000 for the degree sheets. They are the maps of soldiers, engineers, surveyors, foresters, trekkers and geography students, and reading them is a separate chapter of the syllabus.

3. Wall maps. These are large sheets meant to hang on a classroom wall and be read from a distance, showing a country, a continent or the world at scales such as 1 : 1,000,000 to 1 : 20,000,000 with bold colours, thick lines and big lettering. Detail is small; only major rivers, cities, mountains and boundaries appear.

4. Atlas maps. An atlas is a bound collection of maps, named after the Titan Atlas who held the sky, whose figure Mercator printed in his 1595 collection. Atlas maps are of the smallest scale — a world map at 1 : 100,000,000, India at 1 : 15,000,000 — so that a whole country or continent fits on a page. They show broad patterns of relief, climate, population and products, and are the maps a student uses most.

Chorographical maps, showing regions at intermediate scales between topographical and atlas maps, are sometimes added as a fifth class. The order of the four classes is the order of decreasing scale: cadastral (largest) → topographical → wall → atlas (smallest). As the scale decreases, the area covered grows, the detail shrinks, and more features must be generalised — a winding river becomes a smooth line, a town a dot.

By purpose or content, maps are grouped according to what they are made to show. The two broad families are general reference maps, which show many kinds of features together for locating things, and thematic or special-purpose maps, which show the distribution of one theme.

Physical or natural maps show the features of nature. Relief maps show the height and shape of the land by contours, layer colouring (green for lowland, yellow and brown for higher, white for snow), hachures or hill shading. Geological maps show the rocks beneath the surface. Drainage maps show rivers and lakes. Climatic maps show temperature by isotherms, pressure by isobars, rainfall by isohyets and winds by arrows. Soil and natural vegetation maps show the distribution of soil types and forest types. Astronomical maps show the stars and planets.

Cultural maps show the works of human beings. Political maps show boundaries of countries, states and districts with capitals and cities, usually in flat colours; they are the commonest atlas maps. Historical maps show boundaries and events of the past — the Mauryan empire, the partition of Bengal. Population maps show the distribution and density of people by dots or shading. Economic maps show agriculture, minerals, industries, trade and transport. Transport maps show roads, railways, airways and shipping routes. Military maps, tourist maps and town plans serve particular users.

Thematic maps by method of drawing. The way a distribution is shown gives further types. A dot map uses one dot for a fixed number, such as one dot for 10,000 people, to show population or cattle. A choropleth map shades areas such as districts in grades of one colour according to a value like density or literacy. An isopleth map joins points of equal value by lines — isotherms, isobars, isohyets, contours. A flow map uses arrows of varying width for movement of goods or people. A pie or bar map places small graphs on the areas they describe.

Special modern forms. Weather maps, issued daily by the India Meteorological Department, show pressure, wind, rain and cloud. Satellite image maps and digital maps on phones (such as online maps with GPS) are the newest forms, and Geographic Information Systems combine many map layers for planning.

Choosing the right kind of map is part of geography. To find the capital of a state one opens a political map; to find why Purulia is dry, a rainfall map; to plan a railway, a relief map; to see where the jute grows, an economic map; to study the 2011 census, a choropleth of density. The same area appears very different on each, because each map tells one story about it.

The earth's surface is enormous and paper is small, so every map shrinks the ground by a fixed proportion. This proportion is the scale of the map: the ratio between a distance measured on the map and the corresponding distance measured on the ground. If 1 cm on the map stands for 1 km on the ground, the scale is 1 cm to 1 km, and since 1 km is 100,000 cm, the map is one hundred-thousandth the size of the ground. Scale is what turns a picture into a map: with it, any distance or area can be measured.

Scale can be expressed in three ways.

1. Statement scale (verbal scale). The scale is stated in words with units: 1 cm to 5 km, 1 inch to 4 miles, 2 cm represent 1 km. It is the simplest to understand and the commonest on Indian maps, but it is tied to particular units — a reader who does not use inches cannot use an inch scale — and it becomes wrong if the map is enlarged or reduced by photocopying.

2. Representative fraction (R.F.) or numerical scale. The scale is written as a fraction or ratio in which the numerator is always 1 and both terms are in the same unit: 1 : 50,000 or 1/50,000, meaning one unit on the map represents 50,000 of the same units on the ground — 1 cm to 50,000 cm, 1 inch to 50,000 inches, 1 anything to 50,000 of the same thing. Because no unit is named, the R.F. is understood in every country and every language, and it makes comparison of maps easy: the smaller the denominator, the larger the scale. Its disadvantages are that it too becomes wrong when the map is photocopied at a different size, and that the reader must do arithmetic to find a distance in kilometres.

3. Graphical, linear or bar scale. A straight line is drawn on the map, divided into equal parts, and each part is labelled with the ground distance it stands for — 0, 5, 10, 15 km. To find a distance the reader lays a strip of paper or a divider between two points and reads it off against the bar, with no calculation. Its great advantage is that it enlarges or shrinks with the map, so it remains correct after photocopying or projection on a screen; it is also independent of units for the reader who simply compares lengths. Its limitation is that it must be carefully constructed, and it cannot express a scale in words for a person who only wants the ratio.

Most good maps carry all three: the statement for quick understanding, the R.F. for comparison and calculation, and the bar for measurement. The three are simply the same fact in three languages, and the next topics show how to translate between them.

Converting scales is mostly a matter of units, so the units must be known exactly. Two systems are met on maps: the metric system, used in India since 1957 and on all modern Survey of India sheets, and the British or F.P.S. system, used on older maps and still common in examination problems.

Metric units of length.

• 1 kilometre (km) = 1,000 metres (m)
• 1 metre = 100 centimetres (cm)
• 1 centimetre = 10 millimetres (mm)
• Therefore 1 km = 1,00,000 cm (one lakh centimetres) = 10,00,000 mm.

The single most useful figure in this chapter is that a kilometre has 100,000 centimetres. Every conversion between a centimetre-to-kilometre statement and an R.F. passes through it.

British units of length.

• 1 mile = 8 furlongs = 1,760 yards = 5,280 feet
• 1 furlong = 220 yards = 660 feet
• 1 yard = 3 feet; 1 foot = 12 inches
• Therefore 1 mile = 63,360 inches (5,280 × 12).

The figure 63,360 is the British counterpart of 1,00,000: every inch-to-mile scale becomes an R.F. by multiplying miles by 63,360. The old one-inch sheets of the Survey of India (1 inch to 1 mile) thus have an R.F. of 1 : 63,360, and the quarter-inch sheets (1 inch to 4 miles) 1 : 253,440.

Conversion between the systems.

• 1 inch = 2.54 cm; 1 foot = 30.48 cm; 1 yard = 0.9144 m
• 1 mile = 1.609 km (about 1.6 km); 1 km = 0.621 mile (about 5/8 mile)
• 1 nautical mile = 1.852 km, used at sea and in the air

Units of area.

• 1 square km = 100 hectares = 1,000,000 sq m
• 1 hectare = 10,000 sq m = 2.47 acres
• 1 sq mile = 640 acres = 2.59 sq km
• Traditional Bengal units: 1 bigha ≈ 1,333 sq m (about one-third of an acre), 20 katha = 1 bigha

Care with units in scale problems. The rule of the representative fraction is that both terms are in the same unit and the numerator is 1. So to write a statement as an R.F., the ground distance must be brought into the unit of the map distance — kilometres into centimetres, miles into inches. The commonest examination mistakes are forgetting a zero (1 km = 1,00,000 cm, five zeros), mixing inches with centimetres, and leaving the numerator larger than 1. Indian numbering (lakh, crore) and international numbering (hundred thousand, million) are both accepted: 1 : 1,00,000 and 1 : 100,000 are the same scale.

The representative fraction is found from a statement scale by expressing the ground distance in the same unit as the map distance and then reducing the fraction so that the numerator is 1. The method has three steps.

Step 1. Write the statement as map distance : ground distance.
Step 2. Convert the ground distance into the unit of the map distance (km → cm by multiplying by 1,00,000; miles → inches by multiplying by 63,360).
Step 3. Divide both terms by the map distance so that the first term becomes 1.

Worked example 1 — metric. Statement: 1 cm to 5 km.
1 cm : 5 km = 1 cm : 5 × 1,00,000 cm = 1 : 5,00,000.
So the R.F. is 1 : 5,00,000.

Worked example 2 — map distance not 1. Statement: 4 cm to 1 km.
4 cm : 1 km = 4 cm : 1,00,000 cm. Divide both by 4: 1 : 25,000.
So the R.F. is 1 : 25,000.

Worked example 3 — British. Statement: 1 inch to 4 miles.
1 inch : 4 miles = 1 inch : 4 × 63,360 inches = 1 : 2,53,440.
So the R.F. is 1 : 2,53,440.

Worked example 4 — mixed British. Statement: 2 inches to 1 mile.
2 inches : 63,360 inches = 1 : 31,680.
So the R.F. is 1 : 31,680.

Worked example 5 — metres. Statement: 1 cm to 500 m (the Survey of India 1 : 50,000 series).
500 m = 500 × 100 cm = 50,000 cm, so the R.F. is 1 : 50,000.

Worked example 6 — a fraction of a kilometre. Statement: 5 cm to 2 km.
5 cm : 2,00,000 cm = 1 : 40,000.

Checking the answer. The R.F. should have no units and a numerator of exactly 1, and its denominator should look reasonable: a village plan will have a denominator in the thousands, a topographical sheet in the tens of thousands, an atlas page in the millions. If the denominator comes out as a fraction or decimal, the arithmetic has slipped. A quick mental check: 1 cm to 1 km is 1 : 1,00,000; a statement with more kilometres per centimetre gives a bigger denominator, and one with more centimetres per kilometre gives a smaller one.

The reverse — R.F. into statement. To turn an R.F. into a statement, take the denominator as the number of centimetres (or inches) that 1 cm (or 1 inch) represents and convert it into convenient units. Thus 1 : 2,00,000 means 1 cm to 2,00,000 cm = 1 cm to 2 km. And 1 : 1,26,720 means 1 inch to 1,26,720 inches = 1 inch to 1,26,720 ÷ 63,360 = 2 miles. If the denominator does not divide evenly, choose the convenient form: 1 : 2,50,000 means 1 cm to 2.5 km, and 1 : 1,000,000 means 1 cm to 10 km (or 1 inch to about 15.78 miles).

Because an R.F. has no units, the same R.F. can be read in centimetres and kilometres or in inches and miles. Examination questions often ask for an R.F. given in one system to be stated in the other, or for a metric statement to be changed into a British one. The key facts are 1 inch = 2.54 cm and 1 mile = 1.609 km (or 63,360 inches and 1,00,000 cm respectively).

Worked example 1 — R.F. to metric and British statements. R.F. 1 : 1,000,000.
Metric: 1 cm represents 1,000,000 cm = 10,000 m = 10 km. Statement: 1 cm to 10 km.
British: 1 inch represents 1,000,000 inches = 1,000,000 ÷ 63,360 miles = 15.78 miles. Statement: 1 inch to 15.78 miles (about 15.8 miles).

Worked example 2 — R.F. 1 : 63,360 (the old one-inch sheet) in metric. 1 cm represents 63,360 cm = 633.6 m = 0.6336 km. So the statement is 1 cm to 0.634 km, or more conveniently about 1.58 cm to 1 km.

Worked example 3 — metric statement to British statement. 1 cm to 2 km. How many miles to the inch?
1 inch = 2.54 cm, so 1 inch represents 2.54 × 2 km = 5.08 km. And 5.08 km = 5.08 ÷ 1.609 miles = 3.16 miles. Statement: 1 inch to 3.16 miles.
Check by R.F.: 1 cm to 2 km is 1 : 2,00,000; 2,00,000 ÷ 63,360 = 3.16 miles. The two routes agree.

Worked example 4 — British statement to metric statement. 1 inch to 1 mile. How many kilometres to the centimetre?
1 inch = 2.54 cm represents 1.609 km, so 1 cm represents 1.609 ÷ 2.54 = 0.6336 km, that is about 634 m. Statement: 1 cm to 0.634 km.

Worked example 5 — finding the R.F. from a map and a known distance. On a map, two towns 12 km apart are 6 cm apart. What is the scale?
6 cm represent 12 km, so 1 cm represents 2 km, and the R.F. is 1 : 2,00,000.

Worked example 6 — using latitude. On a map, the parallels of 22° N and 23° N are 5.5 cm apart. Since one degree of latitude is about 111 km, 5.5 cm represent 111 km, 1 cm represents 111 ÷ 5.5 = 20.18 km, and the R.F. is about 1 : 20,00,000. This is how the scale of a map that has lost its scale is recovered.

Rounding. Answers involving 2.54 and 1.609 do not come out even; give them to two decimal places and say so. Examiners accept 1 : 1,000,000 as 1 inch to 15.78 miles, 15.8 miles or roughly 16 miles.

Because scale is a fraction, students often confuse large and small. The rule is simple: the scale is a fraction, and a fraction with a smaller denominator is a larger number. 1/10,000 is larger than 1/1,000,000, so a map at 1 : 10,000 is a large-scale map and one at 1 : 1,000,000 is a small-scale map. In words: a large-scale map shows a small area in large detail; a small-scale map shows a large area in small detail.

Large-scale maps. Cadastral maps, town plans and topographical sheets from 1 : 500 up to about 1 : 1,00,000 are large-scale. On them a single village, a town or a few hundred square kilometres fill the sheet. Every road, stream, well, building block, field boundary and contour can be shown; the shapes of features are nearly true and little generalisation is needed. They are used for land records, engineering, town planning, forestry, military operations, trekking and detailed geographical study. Their disadvantage is that many sheets are needed to cover a region — the Survey of India needs about 5,000 sheets at 1 : 50,000 to cover India — so they are expensive to make and keep up to date.

Small-scale maps. Wall maps and atlas maps at 1 : 1,000,000 and smaller are small-scale. A whole state, country, continent or the world fits on one sheet. Only major features can be shown — main rivers, big cities, mountain ranges, boundaries — and these must be simplified: a meandering river becomes a smooth curve, a city becomes a dot, a coast loses its small bays. They are used for showing broad patterns and distributions — climate, population, products, political divisions — and for general reference and teaching. Their advantage is breadth and cheapness; their limitation is that measurements on them are approximate and small features are invisible.

Comparison by area. The area a sheet covers grows with the square of the change in scale. A sheet 40 cm square at 1 : 50,000 covers 20 km × 20 km = 400 sq km; the same sheet at 1 : 5,00,000, a scale ten times smaller, covers 200 km × 200 km = 40,000 sq km, a hundred times the area. This is why the small-scale map must leave so much out.

Medium scale is a term for the range between, roughly 1 : 1,00,000 to 1 : 1,000,000, used for district and regional maps.

Choosing a scale. The purpose decides. To find your way through Kolkata you need a street map at about 1 : 20,000; to plan a bus journey from Kolkata to Siliguri, a state road map at about 1 : 10,00,000; to compare India with China, an atlas map at 1 : 50,000,000. Asking for the largest scale is not always right — it would take a whole shelf of topographical sheets to see the shape of West Bengal.

An examination question that gives two scales and asks which is larger is answered by comparing denominators: smaller denominator, larger scale, smaller area, more detail.

A linear scale or bar scale is constructed so that the reader can measure distances directly. The construction follows fixed steps, and examinations ask for it with a given R.F.

Steps.

• 1. From the R.F., find the ground distance that 1 cm represents.
• 2. Decide the total length of the bar — usually a convenient length between 10 and 15 cm — and the round number of kilometres it should show.
• 3. Calculate the map length that this round number of kilometres needs.
• 4. Draw the line of that length, divide it into equal primary divisions and label them from 0 at the left in kilometres.
• 5. Extend the line one primary division to the left of zero and divide this part into smaller secondary divisions (metres or hundreds of metres), labelled from right to left. Distances are then read as whole kilometres to the right of zero plus a fraction to the left.
• 6. Write the R.F. and the statement scale above or below the bar.

Worked example 1. Construct a linear scale for R.F. 1 : 50,000 to read up to 5 km and in units of 100 m.
1 cm represents 50,000 cm = 0.5 km, so 1 km needs 2 cm and 5 km needs 10 cm. Draw a line 10 cm long and divide it into 5 equal parts of 2 cm; label 0, 1, 2, 3, 4, 5 km from the left. Extend 2 cm to the left of 0 and divide it into 10 parts of 2 mm, each representing 100 m; label 1,000, 500, 0 from the left. A distance of 3 km 400 m is then read as 3 primary divisions to the right of zero plus 4 secondary divisions to the left.

Worked example 2. R.F. 1 : 2,00,000, bar to show 20 km.
1 cm represents 2 km, so 20 km needs 10 cm. Draw 10 cm, divide into 4 parts of 2.5 cm each representing 5 km; label 0, 5, 10, 15, 20 km; extend 2.5 cm to the left divided into 5 parts of 1 km each.

Worked example 3 — British units. R.F. 1 : 63,360, bar to show 6 miles.
1 inch represents 1 mile, so 6 miles needs 6 inches. Draw 6 inches, divide into 6 parts of 1 inch; extend 1 inch to the left and divide it into 8 furlongs.

Worked example 4 — awkward R.F. R.F. 1 : 1,50,000. 1 cm = 1.5 km. For a bar showing 12 km: 12 ÷ 1.5 = 8 cm. Draw 8 cm, divide into 4 parts of 2 cm (3 km each), label 0, 3, 6, 9, 12; extend 2 cm to the left divided into 3 parts of 1 km. The trick is to choose a total distance that the ratio divides evenly.

Reading a distance. Place the edge of a paper strip along the two points on the map, mark them, then lay the strip on the bar with the right-hand mark on a whole-kilometre division and read the left-hand mark on the secondary divisions. A pair of dividers does the same more precisely.

The linear scale is the only scale that stays correct when the map is enlarged or reduced, which is why every published map carries one, and why an examination map without one loses marks.

A map is often needed at a different size from the one available — enlarged so that more can be written on it, or reduced to fit a page. When the map is enlarged or reduced, its scale changes in the same proportion, and the new scale must be calculated and written on the new map.

How the scale changes. If a map is enlarged n times, every length becomes n times longer, so 1 cm on the new map represents only 1/n of what it did before; the denominator of the R.F. is divided by n. If it is reduced to 1/n, the denominator is multiplied by n.

Worked example 1. A map at 1 : 1,00,000 is enlarged 4 times. New R.F. = 1 : 1,00,000 ÷ 4 = 1 : 25,000. Check: on the old map 1 cm = 1 km; on the new map that kilometre is 4 cm long, so 1 cm = 250 m = 25,000 cm.

Worked example 2. A map at 1 : 50,000 is reduced to half. New R.F. = 1 : 50,000 × 2 = 1 : 1,00,000.

Worked example 3. A map at 1 inch to 1 mile is reduced to one-third. New scale: 1 inch to 3 miles, R.F. 1 : 63,360 × 3 = 1 : 1,90,080.

Worked example 4 — area. A map is enlarged 3 times. Lengths become 3 times, so areas become 9 times; a map 10 cm by 8 cm (80 sq cm) becomes 30 cm by 24 cm (720 sq cm). The area on the paper changes with the square of the enlargement; the ground area shown does not change at all.

The square method (method of squares). This is the simplest way to enlarge or reduce a map by hand.

• 1. Draw a grid of equal squares — say 1 cm — over the original map, numbering the rows and columns.
• 2. On a fresh sheet draw the same number of squares, but of the new size: for an enlargement of 2 times, 2 cm squares; for a reduction to half, 0.5 cm squares.
• 3. Copy the details of each original square into the corresponding new square, keeping the positions where lines cross the sides of the squares in the same proportion.
• 4. Rub out the grid on the new map, add the title and north arrow, and write the new scale.

Other methods. The similar triangles method draws rays from a point through the corners of the map and marks off proportional distances. The pantograph is a jointed instrument whose tracing point follows the original while a pencil at another joint draws an enlarged or reduced copy. The eidograph and the camera lucida were older instruments; today a photocopier or a computer does the job in a second — but the printed R.F. must still be corrected, which is why the bar scale, which enlarges with the map, is so valuable.

Combining changes. If a map at 1 : 20,000 is enlarged 5 times and then reduced to half, the net change is 2.5 times enlargement: new R.F. = 1 : 20,000 ÷ 2.5 = 1 : 8,000.

The whole point of scale is that the map can be measured. Two measurements are commonly asked: the distance between two places, straight or along a road or river, and the area of a region.

Straight-line distance. Measure the length between the two points with a ruler or dividers, then apply the scale. Formula: ground distance = map distance × denominator of R.F. (in map units), then convert. Example: two towns are 7.5 cm apart on a 1 : 2,00,000 map. Ground distance = 7.5 × 2,00,000 cm = 15,00,000 cm = 15 km. With a statement scale of 1 cm to 2 km the answer is simply 7.5 × 2 = 15 km.

Curved distance along a road, railway or river. Three methods:

• Thread method. Lay a thread along the curve, following every bend, mark its ends, straighten it and measure against the ruler or the bar scale.
• Paper strip method. Lay the straight edge of a strip of paper along the first short straight section, mark, pivot the strip at the mark to follow the next section, and so on, then measure the total.
• Dividers. Set a small opening, such as 0.5 cm, and step along the curve counting the steps; the last partial step is measured separately.
• Opisometer. A small wheel on a handle that is rolled along the curve and reads the length on a dial; the rotameter is its modern name.

Example: a river measured by thread is 23 cm long on a 1 : 50,000 map; its ground length is 23 × 50,000 cm = 11,50,000 cm = 11.5 km.

Area. Area on the ground = area on the map × (denominator)², because both length and breadth are scaled. On a 1 : 50,000 map, 1 cm represents 0.5 km, so 1 sq cm represents 0.5 × 0.5 = 0.25 sq km. On a 1 : 1,00,000 map, 1 sq cm = 1 sq km; on 1 : 2,00,000, 1 sq cm = 4 sq km. Methods of finding the map area of an irregular shape:

• Square (grid) method. Place a transparent sheet ruled in 1 cm squares over the area; count the whole squares inside, then count the partial squares and take half of them as whole (or estimate each), add, and multiply by the ground area of one square. Example: a lake covers 18 whole squares and 12 partial squares on a 1 : 50,000 map: 18 + 12/2 = 24 sq cm × 0.25 = 6 sq km.
• Strip method. Divide the area into parallel strips of equal width, treat each as a rectangle with the mean length, add the areas.
• Geometrical method. Divide the area into triangles and rectangles and use the formulas.
• Planimeter. An instrument whose tracing point is run round the boundary and whose dial reads the area.

Worked example. A forest on a 1 : 25,000 map covers 30 whole squares of 1 sq cm and 16 partial squares. Map area = 30 + 8 = 38 sq cm. At 1 : 25,000, 1 cm = 0.25 km, so 1 sq cm = 0.0625 sq km. Ground area = 38 × 0.0625 = 2.375 sq km = 237.5 hectares.

Direction and bearing are the third measurement: with the north arrow and a protractor, the bearing of one place from another is read clockwise from north, from 0° to 360°; Siliguri lies at a bearing of about 350° from Kolkata, that is nearly due north, slightly west.

Beyond the plain linear scale, surveyors and map-makers use a few special scales that the syllabus introduces.

Diagonal scale. A plain linear scale reads two units — kilometres and hundreds of metres. A diagonal scale reads three — kilometres, hundreds of metres and tens of metres — by using the principle of similar triangles. Above the primary and secondary divisions of a linear scale, ten equally spaced horizontal lines are drawn, and in the left extension diagonals are drawn from each secondary division on the bottom line to the next secondary division on the top line. Because the diagonal rises evenly, its distance from the vertical at the first horizontal line is one-tenth of a secondary division, at the second two-tenths, and so on. A distance of 3 km 470 m is read by moving 3 primary divisions right of zero, 4 secondary divisions left along the bottom, and then up the 4th diagonal to the 7th horizontal line. It gives a precision ten times that of the plain scale and is used on engineering drawings and large-scale surveys.

Comparative scale. When a map may be read by people using different units, or when distances must be compared with another map, two bar scales with the same R.F. but different units are drawn one above the other on a common zero — one in kilometres, one in miles. A reader measures once and reads either unit. For R.F. 1 : 1,000,000 the kilometre bar has 1 cm per 10 km, and the mile bar has 1 inch (2.54 cm) per 15.78 miles, or more conveniently 6.34 cm per 100 km against 10.16 cm per 100 miles, so that 100 miles = 160.9 km can be checked directly.

Time scale. A time scale is a bar scale in which the divisions are labelled not in distance but in the time taken to cover it at a given speed — the time a walker, cyclist or train needs. If a map is 1 : 1,00,000 and a person walks at 5 km per hour, then 1 cm (1 km) represents 12 minutes and a 5 cm division represents 1 hour; the bar is labelled 0, 1, 2, 3 hours. Such scales are drawn on trekking maps, military maps and railway maps. The construction is: distance per unit of time × R.F. = map length per unit of time.

Worked example. Construct a time scale for a map of R.F. 1 : 2,00,000 for a bus travelling at 40 km per hour, to show 3 hours. In 1 hour the bus covers 40 km; on the map 40 km = 40 × 1,00,000 ÷ 2,00,000 = 20 cm. That is too long, so show 1 hour = 20 cm only if the paper allows; otherwise show half-hours: 30 minutes = 10 cm. Draw a bar of 30 cm for 3 hours divided into six parts of 5 cm (30 minutes each), or, for a 15 cm bar, label it 0 to 1½ hours.

Vernier scale and scale of chords. The vernier is another device for reading a fraction of the smallest division, used on instruments rather than maps; the scale of chords measures angles with a compass. They are mentioned for completeness.

Scale on projections. A map of a large area cannot have one exact scale everywhere, because the earth is curved; the scale printed is true along certain lines (the standard parallels) and differs elsewhere, which is why measurement on an atlas map of the world is only approximate. On topographical sheets of small areas this variation is negligible.