Notes on Linear Measurements (Chapter 3)

3.1 Different Methods

Linear measurements can be made by three main approaches, with merit depending on required precision:
- Direct measurements: distances measured on the ground with a chain, tape, or similar instrument.
- Optical methods: observations through a telescope with calculations to determine distances (tacheometry, triangulation).
- Electro-magnetic methods: distances measured with instruments relying on propagation, reflection, and reception of radio, light, or infrared waves.
References to related chapters:
- Optical measurements: see Chapter 22 on Tacheometric Surveying.
- Electromagnetic distance measurement (EDM): see Chapter 24.

3.2 Direct Measurements

Direct measurement methods (in increasing formality/precision):
1. Pacing
2. Passometer
3. Pedometer
4. Odometer and speedometer
5. Chaining

3.2.1 Pacing

Used mainly for rough, preliminary surveys or quick checks.
Method: count paces between two points; distance is computed using the average length of a pace.
Pace length variability due to individual, ground, slope, and speed.
Prefer using a pace close to natural step on level ground; accuracy roughly ~1 in 100 on level, unobstructed ground.
Rough or sloped/rough ground reduces accuracy.

3.2.2 Passometer

A wristwatch-like instrument carried in pocket or on leg; motion-activated.
Automatically records the number of paces, reducing manual counting fatigue.
Distance estimate = (number of paces) × (average pace length).

3.2.3 Pedometer

Similar to a passometer but adjusted to the carrier’s pace length; records total distance for any number of paces.

3.2.4 Odometer and Speedometer

Odometer: counts revolutions of a wheel; distance = (number of revolutions) × (wheel circumference).
Readings are accurate on smooth ground; undulations/uneven ground reduce accuracy.
Speedometer (in vehicles) can be used for approximate measurements along a route.

3.2.5 Chaining

The most accurate direct method for line-length measurement.
For ordinary precision, a chain suffices; for higher precision, a tape or a special bar is used.
The distances established by chaining form the basis of all surveying. No matter how accurately angles are measured, the survey’s precision is limited by chaining accuracy.

3.3 Instruments for Chaining

Core instruments used for determining line length by chaining:
1. Chain
2. Arrows
  A. Ranging rods
3. Plasterer’s laths and whites
4. Plumb bob
5. Pegs (mentioned in context of setting chain points)

3.3.1 Chain

Chains are made of straight galvanised mild steel links joined by rings; ends have brass handles with swivel joints to prevent twisting.
A link length is the distance between centres of two consecutive middle rings; the chain length is measured from the outside of one handle to the outside of the other.
Common chain types:
- (i) Metric chains
- (ii) Gunter’s chain (Surveyor’s chain)
- (iii) Engineer’s chain
- (iv) Revenue chain

Metric chains

After metric units were adopted in India, metric chains became common.
Typical lengths: 5 m, 10 m, 20 m, 30 m.
IS: 1492-1970 covers metric surveying chains.
Reading aids: tallies at every metre for 5 m and 10 m chains; tallies at every five metres for 20 m and 30 m chains; brass rings at metre marks; some markings designated with ‘m’ to distinguish metric.
Details (illustrated in figures in the text): construction features such as a groove on the handle to hold arrows, and the arrangement of tallies and markers.

Gunter’s Chain (Surveyor’s Chain)

Length: 66 ft (approx. 20.1169 m) with 100 links; each link ≈ 0.6 ft (7.92 in).
Historical importance: 10 chains = 1 furlong; 80 chains = 1 mile; 10 chains = 1 furlong; 1 acre = 10 square chains.

Engineer’s Chain

Length: 100 ft; 100 links; each link = 1 ft.
Brass tags every 10 links; distances recorded in feet and decimals.

Revenue Chain

Length: 33 ft; 16 links; each link = 2 ft.
Used mainly for cadastral surveying of fields.

Steel Band (Band Chain)

A long, narrow steel strip (blue steel): width 12–16 mm; thickness 0.3–0.6 mm.
Metric steel bands lengths: 20 m or 30 m.
Divided by brass studs every 20 cm; numbered every metre.
Ends may be reinforced with leather/plastic; first and last links subdivided for finer readings.
Advantages: lighter, less prone to kinks; more stable in length; easier to handle and more accurate than chains in some cases.
Disadvantages: can break and be difficult to repair in the field.
In practice: tests assert that steel bands are preferred for accuracy; keep them away from rough handling; avoid heat/humidity damage.

3.3.2 Testing and Adjusting Chain

Chains elongate or shorten with use due to bending, stretching, wear, or deformation.
Regular testing against a standard gauge is required; in the field, if no permanent gauge exists, erect a fixed gauge using pegs and stones as shown (Permanent Test Gauge).
Tolerances (with 8 kg pull at 20°C):
- 20 m chain: ±5 mm
- 30 m chain: ±8 mm
- Per-metre accuracy: ±2 mm
Adjustments when the chain is too long (lengthened) or too short (shortened):
- If long: shorten by adjusting ring joints, reshaping rings, removing rings, replacing worn rings, or adjusting end links; adjust symmetrically to keep the central peg position unchanged.
- If short: lengthen by straightening links, flattening rings, replacing rings with larger ones, or inserting additional rings; adjust symmetrically.

3.3.3 Tapes

Tapes are used for more accurate measurements and are classified by material:
- Cloth/Linen tape
- Steel tape
- Metallic tape
- Invar tape

Cloth or Linen Tape

Made of closely woven linen, 12–15 mm wide, varnished to resist moisture.
Characteristics: light, flexible; package lengths commonly 10 m, 20 m, 25 m, 30 m; other lengths include 33 ft., 50 ft., 66 ft., 100 ft.
End has a small brass ring; reading includes the ring length.
Limitations: easily affected by moisture (shrinks), subject to stretch, tends to twist or tangle, not very strong.
Maintenance: clean and dry before winding.

Metallic Tape

Made of varnished, waterproof linen interwoven with brass/copper/bronze wires; low stretch.
Useful for cross-sections and some topographic methods where small length errors are tolerable.
Lengths: 2, 5, 10, 20, 30, 50 m.
Ends: a brass ring attached to outer end, reinforced with leather or plastic for ~20 cm; tapes of 10, 20, 30, 50 m are provided in metal or leather cases with a winding device.

Steel Tape

Highly accurate graduation; more robust than cloth/metallic tapes.
Typical width: 6–10 mm.
Lengths: 1, 2, 10, 20, 30, 50 m.
10, 20, 30, 50 m lengths have a brass ring at the outer end; wound in leather or corrosion-resistant cases; longer than 30 m typically on a metal reel.
Care: delicate; wipe clean, dry after use; oil lightly to prevent rust.

Invar Tape

An alloy (nickel about 36% + steel) with very low thermal expansion.
Advantage: base lines can be measured with high speed and accuracy due to minimal expansion with temperature.
Disadvantages: creep (length increases a bit over time), coefficient of expansion changes over time; softer and more easily deformed than steel tapes; more expensive.
Uses: high-precision linear measurements such as base lines.
Availability: commonly 20, 30, and 100 m lengths; usually around 6 mm wide; must be kept on large-diameter reels to avoid bending damage.

3.3.4 Arrows

Marking pins used with chain surveys.
Made of hardened steel wire, ~4 mm diameter (8 s.w.g); black enamelled.
Typical lengths: 25–50 cm, with 40 cm common.
One end sharp; other end bent into a loop for carrying.
Usually supplied in sets of about 10 arrows with a chain.

3.3.5 Pegs

Wooden pegs mark station positions.
Typical dimensions: ~2.5–3 cm square cross-section, ~15 cm long, tapered at the end; driven into ground with a wooden hammer; approx. 4 cm protruding above soil to mark the point.

3.3.6 Ranging Rods

Lengths: 2 m or 3 m (2 m is more common).
Base: iron point; painted in alternating bands (black/white, red/white, or black/red/white) with each band ~20 cm wide; helps visibility at distance (up to ~200 m).
Cross-section: circular or octagonal; 3 cm nominal diameter; made of seasoned timber.
Flags: long lines require a red/white/yellow flag (30–50 cm square) near the top for visibility on long lines.

3.3.7 Ranging Poles

Similar to ranging rods but longer and thicker for very long lines.
Typical length: 4–8 m; diameter 6–10 cm.
Base: foot sunk into the ground; kept vertical with a plumb bob; usually not painted, but clearly marked with a flag.

3.3.8 Offset Rods

Used to measure rough offsets near a survey line.
Length: ~3 m; round wooden rod with a pointed iron shoe on one end and a notch or hook on the other end to facilitate pulling/pushing the chain.
Has two narrow slots at eye level aligned perpendicularly for aligning the offset line.
Butt rods: alternative offset device used by building surveyors; consists of two laths (1 yard or 1 m) riveted together with a spring catch; painted black; divisions in feet and inches marked in white and red.

3.3.9 Plasterer’s Laths and Whites

Laths: straight wooden slats about 0.5–1 m long used to mark intermediate points; easy to carry and sharpens with a knife.
Whites: sharpened thin sticks used for ranging; tipped with a notch or split at the top; paper bits may be inserted to improve visibility in grass.
Uses: cross-sectioning, temporary contour marking, extending lines across depressions or hedges.

3.3.10 Plumb Bob

Used to transfer points to ground, ensure vertical alignment of ranging poles, transfer lines from a line ranger to ground adjustments, and act as centering aid in other instruments.

3.4 Ranging Out Survey Lines

When measuring a line, the chain or tape must be stretched straight along AB (the terminal stations).
If the line is shorter than the chain, it’s straightforward; if it is longer, intermediate points must be fixed in line before chaining begins. This process is called ranging.

3.4.1 Direct Ranging

Used when the two ends of the survey line are in view.
Methods:
- By eye
- By optical instrument (line ranger or theodolite)

Ranging by Eye

Setup: Points A and B are the ends; a ranging rod is placed at B; the surveyor at A holds a rod at about half a metre length.
Assistant moves a second ranging rod to align with AB, at a distance not greater than one chain length from A.
The surveyor at A signals the assistant to move transversely until the line from A to B is in line with the observer; repeat for additional intermediate points.
Signals (code) table summarizes hand/arm motions and corresponding assistant actions (rapid sweep, slow sweep, arm positions, etc.).

Ranging by Line Ranger (Optical Line Ranger)

Line ranger: combination of two plane mirrors or two right-angled prisms (isosceles) stacked; diagonals are mirrored to reflect inputs from A and B onto the observer.
A line ranger is held near AB; a plumb bob transfers the point onto ground.
Procedure:
- Place ranging rods at A and B; observer views images of A and B via the line ranger.
- Move the line ranger sideways until the two images line up vertically; the point P is then transferred to the ground with the plumb bob.
Benefits: only one person is needed to range an intermediate point; the line ranger fixes the point without moving to either end.

3.4.2 Adjustment and Alignment of the Line Ranger

One mirror or prism is usually adjustable.
To test perpendicularity of reflecting surfaces, range three poles with a theodolite; the line ranger is placed over the middle pole; if the end images coincide, the instrument is properly adjusted; otherwise, adjust using the movable prism via the adjusting screw until coincidence is achieved.

3.7 Errors Due to Incorrect Chain

A chain that does not match its nominal true length introduces systematic errors in measured distances.
Key concepts:
- Cumulative errors: accumulate in the same direction and tend to increase over the measurement.
- Compensating errors: can occur in either direction and may offset some effects.
- Errors can be positive (result too great) or negative (result too small).

3.7.1 Common Sources of Error

Erroneous length of chain or tape (cumulative, can be positive or negative).
- If the chain is longer than its nominal length, measured distances tend to be too small (negative correction); if shorter, measured distances tend to be too large (positive correction).
Bad ranging (cumulative, positive): stretching the chain off-line increases the measured length; the error is especially serious in offsetting tasks.
Careless holding and marking (compensating, variable): inconsistent handling may introduce a systematic error that can partly offset with care.
Bad straightening (cumulative, positive): an irregular horizontal curve increases the measured distance.
Non-horizontality (cumulative, positive): on sloped ground, measurements tend to be longer than the true ground distance.
Sag in chain (cumulative, positive): chain sags over irregular terrain, increasing the measured length.
Variation in temperature (cumulative, ±): chain length changes with temperature; higher temperatures lengthen the chain (negative error in measured distance), lower temperatures shorten it (positive error).

3.7.2 Corrections Related to Sag and Temperature/Pull (Illustrative Examples)

Sag correction (illustrative form): the sag correction can be estimated as
$C_{ ext{sag}} ext{ (approx)} o rac{4 w h}{24 P^{2}} = rac{w h}{6 P^{2}},$
where w is the weight per unit length, h is the sag, and P is the pull. An example in the text yields a small correction, e.g., about 0.00208 m in the given case.
Example 3.12 (temperature and pull corrections for a steel tape):
- Temperature correction (additive):
 $ext{Correction}{T} = L imes \beta imes rac{ ext{(T}{ ext{current}} - ext{T}_{ ext{standard}})}{1} = 20 imes 6.2 imes 10^{-6} imes (80 - 55) = 0.0031 ext{ m},$
 where
 $\beta = 6.2 imes 10^{-6} ext{ per } ^ ext{°F}$
- Pull correction (Cp):
 $C{p} = rac{(P - P{0}) L}{A E},$
 where the tape cross-sectional area is determined from its weight and dimensions; for the example, A ≈ 0.051 cm² and the additive correction approximates to 0.00112.
- Sag correction (C{ ext{sag}}): $C{ ext{sag}} = rac{4 w h}{24 P^{2}} = rac{w h}{6 P^{2}},$
 with the example giving C_{ ext{sag}} ≈ 0.00208 m (subtractive).
Total correction in the example:
$ext{Total correction} = +0.0031 + 0.00112 - 0.00208 = +0.00214 ext{ m}.$

3.7.3 Degree of Accuracy in Chaining

Several factors influence accuracy:
- Fineness of graduations on the chain or tape.
- Terrain type and ground conditions.
- Time and money available for work.
- Weather and temperature conditions.
Accuracy can be expressed as a ratio 1:n, indicating 1 unit of error in n units of distance.
Typical accuracies under different conditions (as cited in the text):
- For measurements with invar tape, spring balances, thermometers, etc.: 1 in 10,000
- For ordinary measurements with steel tape, plumb bob, chain pins, etc.: 1 in 1,000
- For measurements made with tested chain, plumb bob, etc.: 1 in 1,000
- For measurements made with chain under average conditions: 1 in 500
- For measurements with chain on rough or hilly ground: 1 in 250

3.12 Precise Linear Measurements

In high-precision linear measurements, errors must be reduced beyond ordinary chaining. The method of linear measurements is categorized into three orders:
- Third order measurements: commonly used in chain surveying and related minor surveys (described in prior sections).
- Second order measurements: used in the measurement of traverse lines with directions measured by theodolite.
- First order measurements: used in triangulation surveys for determining base-line lengths.
Connections to foundational principles and real-world relevance:
- Direct measurement accuracy foundations: instrument calibration, stable ground, and proper handling.
- Optical and EM methods complement direct measurements by enabling longer distances or otherwise inaccessible measurements.
- Understanding and mitigating errors (systematic, cumulative, compensating) is essential to producing reliable survey results.
Practical implications and considerations:
- The choice of method depends on required accuracy, terrain, time, and equipment availability.
- For critical base-lengths and primary data, instrument integrity (chain/tape, thermally stable materials, and proper test procedures) is essential.
- Invar-based baselines offer high precision but require careful handling to avoid creep and expansion variations over time.
Ethical and professional implications:
- Accurate chain measurements underpin reliable land records, infrastructure projects, and legal land descriptions.
- Misreporting measurements or ignoring known instrument limitations can lead to costly downstream errors and legal disputes.
Notation and key formulas used in these notes (summarized):
- Chain and tape types and their properties, tolerances, and maintenance requirements.
- General relation for a measured length when the chain used has a non-nominal length:
 $L{ ext{true}} = l{ ext{measured}} imes rac{L'}{L},$
 where $L$ is the nominal chain length and $L'$ is the actual (used) chain length.
- Example 3.1 demonstrates this with $l{ ext{measured}} = 250$ m, $L = 20$ m, $L' = 20.10$ m, yielding $L{ ext{true}} = 250 imes rac{20.10}{20} = 251.25 ext{ m}.$
- Example 3.12 illustrates temperature and pull corrections for a steel tape, including:
- Temperature correction: $ext{Correction}{T} = L imes \beta imes (T{ ext{current}} - T_{ ext{standard}}),$
 with $eta o 6.2 imes 10^{-6} ext{ per }^ ext{°F}$ and $L = 20 ext{ m}$ giving $0.0031 ext{ m}.$
- Cross-sectional area $A$ derived from weight and density to compute pull correction:
 $Cp = rac{(P - P0) L}{A E},$
 example yields $A oxed{ ext{(≈ 0.051 cm}^2)}$ and $C_p oxed{≈ 0.00112}$ (additive).
- Sag correction (illustrative):
 $C_{ ext{sag}} = rac{4 w h}{24 P^2} = rac{w h}{6 P^2},$
 leading to a value such as $0.00208 ext{ m}$ (subtractive).
- Total correction in the example: $ext{Total} = +0.0031 + 0.00112 - 0.00208 = +0.00214 ext{ m}.$