When developing the notes for “Simple Land Surveying Instruments”, calculations and experiments were carried out, and the design of old naked-eye instruments studied. For brevity and ease of understanding, much of this data was omitted from the notes, but has been collected here for reference.

Since earliest times, people have measured the earth, for the purposes of land ownership, agriculture, building, civil engineering, navigation, and astronomy, to name a few. Optical instruments with open sights - a backsight and a foresight - were developed, based on the self-evident propositions that light travels in straight lines, and two fixed points are needed to define a straight line.

The ancient Greeks developed surveying to the point where they could dig a tunnel starting from both ends, and meet in the middle. The later Roman aqueducts boasted a gradient of 1 part in 20,000 - an astonishing optical resolution of approximately 10 arcseconds.

Naked eye surveying revived in the Middle Ages and reached its peak in the 18th. century. Telescopic instruments came into use in the 19th. century for general surveying, and much later - mostly after world war 2 - in the building industry. Why did it take so long for the telescope to be adopted ? Mainly because naked eye instruments were adequate, and there was little point in adopting more complex and expensive technology.



Naked-eye Sighting Methods. A theodolite,for example, is good for setting out or measuring in undulating country, because its telescope can swing through a vertical angle without changing the horizontal bearing, enabling the line to be surveyed over hill and dale. But telescopes are unsuitable for do-it-yourself instrument construction; their narrow fields of view (the higher the magnification, the smaller the visual field) need accurately engraved degree scales and precisely machined mechanical movements. Making such mechanisms with hand tools is almost impossible, but open sights have some optical properties which cannot easily be duplicated by telescopes, and these can at least partly offset the telescope’s advantages. For example, a good substitute for surveying in hilly terrain is to use long crosshairs and slits, which give a much wider field of view than a telescope, and when oriented vertically, will cope with the most severe undulations.

Open Sights. Early instruments may have sighted a target by looking along a surface or edge of a board or pole, which is an art requiring practice, and can be quite inaccurate. If the observer’s eye is closer to one end than about 250 mm., that end will be out of focus and appear blurred; if the eye is further away, it is hard to keep it on the sight line. And depending on the intensity and direction of the light, if the board or pole is similar in colour to the target it is hard to see exactly where the front end is pointing. These problems were overcome in prehistoric times by adding sighting vanes, similar to the way sights are placed on a rifle barrel.
The maximum resolution of the human eye depends on the ambient light level and the colour and contrast of the target pattern, but is usually taken to be at least 1 arcminute. This represents a tolerance of plus or minus 3 mm. at a target distance of 10 M., a figure more than adequate for small building or civil engineering work.

Advantages of open sights include:-

1. More than one set of sights, as in the Quick Cross or Crevel, where two sets are mounted on sight lines at right angles to each other, and aligned after construction, by a process similar to an end for end test. This allows the setting out of offset lines at right angles, and avoids the need for accurate mechanical movement or a degree scale.

2. Bidirectional sights. These can be looked through in opposite directions, to give an accurate projection of a straight line both ways. Again, the need for precise movement or a scale is avoided.

3. Wide fields of view at right angles to the direction of measurement or laying out - i.e. wide vertical fields for horizontal measurements or laying out, and wide horizontal fields for vertical measurements, including taking levels and gradients. The Dumpy Levels (particularly the Wide Angle Dumpy Level, with a 90º field), the Crevel, and the Dumpy Plummet, all have this feature, which allows the observer to simply look up and down or from side to side, without the need for mechanical movement.

4. Large depth of field. Pinhole, slit, and V-notch backsights (see below) all limit the effective diameter of the iris of the eye, in the same way as a camera iris diaphragm limits the lens aperture. This shortens the eye’s minimum focal length - normally about 250 mm.; the narrower the slit or V-notch, or the smaller the diameter of the pinhole, the closer the eye can see sharply. The graph plots minimum focal length against slit width, for one observer only; results for pinholes and V-notches are similar.

So instruments can be made in which everything in the field of view, from the minimum focal length - typically 160 to 180 mm. - to infinity, appears sharp, and no focusing adjustment is needed; foresights can be mounted at this minimum distance, giving small instrument dimensions.

5. No parallax error. The line of sight for normal seeing is through the center of the iris of the observer’s eye; if the eye moves, the line of sight moves also, and objects at different distances from the observer appear to move relative to each other. For objects as far apart as, say, an instrument crosshair and a distant scene, the movement is substantial. But when the line of sight is held stationary by a small aperture backsight, this movement is almost or completely imperceptible (depending on the distance of the observed object).

The foresight can take several forms:-

1. A short narrow slit or V-shaped notch in a metal vane, mounted at right angles to the line of sight. The usual pattern for bidirectional sights, the slit or notch can be used as both foresight and backsight.

2. A long slit, or series of slits separated by larger holes, in a long vane, for wide angle use. These are also bidirectional.

3. A crosshair made of thin nylon or cotton thread, stretched horizontally or vertically across the line of sight, for wide angle use.

4. In measuring instruments, a linear or degree scale at right angles to the sight line. In this case a wide angle view is obtained, with the scale used to either indicate or measure the direction of the sight line.

Backsights take two fundamentally different forms:-

a. In focus, as in mormal rifle sights, in which the sight is mounted just beyond the ‘near point’ - at least 250 mm. - from the observer’s eye, and is always in focus. This type requires that the observer’s eye is stationary, to avoid parallax error;with a firearms this is accomplished by the marksman resting his cheek on the rifle butt. This type is not used in these instruments.

b. Out of focus (sometimes called a ‘peepsight’), in which a backsight is mounted as close to the observer’s eye as possible, and is responsible for the advantageous properties listed in section 4 above. All backsights described in these notes are of this type, as follows:-

1. A small round hole (‘pinhole”) drilled in a metal vane, mounted at right angles to the line of sight, or a small aperture made by the intersection of a shallow V-notch and a moveable shutter bar; this latter type has the advantage that the aperture size can be varied to suit the lighting conditions, and can also be bi-directional.
2. A short slit or V-notch, as in foresights 1 above. Slits and V-notches limit the eye aperture in one plane only, and when the foresight is long - e.g. a crosshair - both sights must be accurately in parallel. The V-notch has the advantage that the aperture can be varied by simply moving the eye along the notch to suit the light, but is harder for the observer to align accurately, and may not be as free from parallax error as the slit. Both types are bi-directional.

3. A long narrow slit, as in foresight 2 above. As a backsight, a long slit has a wide angle view, combined with the other advantages of a slit. In the Crevel, a long slit at right angles to the measuring scale allows the scale to be viewed from different angles.

4. Short slits are used to view linear or degree scales, with the slit at right angles to the length of the scale - i.e. parallel to the scale calibration marks.



Optical Resolution of Open Sights. The maximum resolution of the human eye - i.e. the smallest target detail or movement the eye can detect - depends on the ambient light level and the colour, contrast and pattern of the target, but is usually taken to be at least 1 arcminute. This represents a tolerance of plus or minus 3 mm. at a target distance of 10 M., a figure more than adequate for small building or civil engineering work; the standard tolerance for a commercial bubble level is +/- 1 mm. per metre, corresponding to a tolerance of +/- 3 arcminutes. Tests of various foresights show that some types lower the maximum resolution which can be obtained, but in no case is it less than 3 arcminutes.

A resolution of 1 arcminute means that the eye is capable of dividing a full 360º circle into 21,600 parts, a subdivision which cannot be matched by any simple measuring scale. For example, it may be just possible, under ideal seeing conditions, to estimate (guess?) scale increments as small as 0.1 mm.. If a scale subdivision of 1 mm. represents 10 arcminutes, the length of a 360º scale would be 2160 mm., with a diameter of 687 mm. - too large to be of practical use. So the limiting factor in simple instruments is not the absence of a telescope, but the inability of a simple low tech. instrument scale to provide small enough minor calibration marks.

Instrument Scales. The Medieval Quadrant, Octant, and Octant, partly solved this problem by providing only a part of a full degree scale, and this approach is adequate for small construction work. But the only degree scales readily available today are on plastic protractors, the largest of which has a radius of 3 inches (75 mm.), with minor calibration marks each 1/2º (30 arcminutes); not only is the radius too small to match a minimum focal length of 160 mm. (See ‘Open Sights’, above) but the minor calibration increment is too large to be of use.

One approach tried was to provide a degree scale with these notes, which could be downloaded, printed, and stuck on the instrument. But the non linearity of some printers, coupled with the size instability of paper, made this impractical.

The solution adopted here is to use linear scales made from ordinary 300 mm. plastic or wooden rulers. Advantages of this approach include :-
1. Linear scales fit in well with the method of remote measurement by similar triangles (See ‘Range Finding’ below).
2. Such rulers are cheap and available the world over.
3. Many well known makes have a calibration accuracy of at least 0.1 %; plastic rulers made under the brand name ‘Marbig’ (among others) consistently meet this criterion.
4. It is likely that anyone without training in geometry will be more familiar with millimetres than degrees.
5. A horizontal linear scale can be mounted on edge, with respect to a horizontal line of sight, allowing direct reading of the scale through the instrument backsight and in most cases eliminating the need for an alidade - this can’t be done with a protractor.

Scale Resolution. A direct comparison between linear and degree scales isn’t possible, but a rough idea can be given..A 300 mm. ruler calibrated in millimetres, mounted 160 mm. - the minimum focal length - from a backsight, has a scale resolution of 1 part in 160. This translates to an increment of 21.5 arcminutes, compared with a protractor scale of 30 arcminutes. The measuring instruments in these notes have a focal length of either 200 or 300 mm.: the scale resolution at 200 mm. is 17.2 arcminutes, and at 250 mm. is 13.75 arcminutes. Tests have shown that it is easy to estimate an increment of 0.5 mm. when reading a millimetre scale from 200 mm. away through a slit backsight, and with practice to make a good estimate down to 0.1 mm., which gives a maximum scale resolution of approximately 1.7 arcminutes. Consideration has been given to fitting a vernier attachment, but there is little justification for what would be a complicated mechanical device.
Comparing the width of the visual field with a part degree scale, gives a figure of 73.74º for a 300 mm. scale at a focal length of 200 mm. compared with a 90º field for a quadrant, a 60º field for a sextant, or a 45º field for an octant..



Re - Thinking the Plumb Line. The plumb line and bob weight is probably older than the beginning of recorded history, and even today remains the simplest way of getting a reliable vertical datum. The ancient Egyptians developed two simple instruments based on the plumb line - the Merkhet and Bay, for aligning the Pyramids, and the Groma, for surveying and restoring boundary lines after each annual Nile flood.
The Groma - five plumb lines hanging from a horizontal wooden or iron cross - was particularly versatile, in that it presented a sight line as a wide angle vertical plane, rather than as a single line; this enabled its use in both flat and undulating country, and kept the use of range poles to a minimum. So versatile was it that the Romans later adopted it for setting out streets in towns and military camps, rejecting more sophisticated instruments developed by the Greeks.
The traditional plumb line, found in every builder’s tool box, is a length of builders’ twine about 1-1.5 mm. diameter, and up to 30 M. long, terminating in an iron bob up to 600 gm. in weight, needed to keep the line taut under all conditions. Using a heavy line and large bob has its problems; small air currents blowing on the line and bob (and tremours if hand held) will set the bob swinging, and it is almost impossible to keep it still; if the line is to be used as part of an instrument, something must be done to curb this.
A plumb line used as part of an instrument will normally be relatively short, and if the line is also the foresight crosshair (the usual case), it should be as thin as possible - nylon fishing line or sewing thread down to 0.1 mm. diameter is strong enough for the purpose. Even cotton sewing thread, although not as strong and rather hairy, can be used. If we assume the wind pressure on a line is roughly proportional to its surface area, the pressure on a 0.1 mm. line will be only 1/100th of that on a 1mm. diameter line, for the same length.
The energy stored in a bob weight is proportional to its mass. If the bob is swinging in still air, the air’s viscosity will exert drag on the bob, slowly absorbing its energy and bringing it to rest. It follows that a lighter bob will come to rest more quickly than a heavy one, given the same line length and initial deflection. Since the bob need only be heavy enough to keep the line taut, for a short length of thin nylon it can be relatively light.

The situation changes markedly if the bob weight is suspended in water, or even in a more viscous liquid such as oil; the viscous drag reduces the bob’s energy, and therefore its swing, much more quickly. This technique was used in Medieval times, and seems to be so self-evident that its use in the old world can almost be taken for granted. The drag is roughly proportional to surface area, so the larger the surface of the bob, the better. An additional advantage of immersing the bob, is that it is completely shielded from air currents.

The requirements of low weight and large surface area are conflicting, but can be satisfied reasonably well by simply using a small open ended and inverted tin can, as the bob weight. One of the advantages of this shape is that viscous drag occurs on both the inside and outside surfaces of the can, like this:-

Tests were carried out on several versions, using different can sizes and line lengths, and comparing them with ordinary plumb lines.

The following table shows the times taken for three bob weights to come to rest, when each was hung from a line of the same length, then pulled the same distance away from the vertical and released.

1. 200 gram cylindrical brass bob weight in “still” air. Did not stop swinging.
(Test stopped after 7 minutes).

2. 200 gram cylindrical brass bob weight in water. No discernible swing after 3 minutes.

3. 200 gram surf fishing sinker with large surface area, in water. No discernible swing after 1.2 minutes.

4. Small tin can, open at bottom, in water (as in sketch above). No discernible movement after 3 swings - less than 5 seconds.

Because the bob is light in weight, the restoring force is small, and several undesirable effects, leading to unwanted deflection of the bob, have been identified. They are:-
a. Surface tension of the water This effect can be removed by either making sure the bob is completely immersed ( i.e. the water level is higher than the top of the bob ) or by putting a little soap or detergent in the water.
b. Deflection from external magnetic fields. This only showed up when a non magnetic water container was used - e.g. a glass jar. Using a tin can for the container shields the bob from external fields. To avoid this, a bob made from an aluminium can was tried, and after modification (see 'Dumpy Plummet') performed satisfactorily.

c. Deflection from electrostatic charge, and buoyancy. Both these effects occured together, when the bob was made from a plastic pill bottle, and can be avoided by not using a plastic bob.

Plumb lines of this type have been used in the Dumpy Plummet and the Calibrated Target.
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Similar Triangle Theory Part of the art of surveying is the ability to measure the horizontal and/or vertical distance between two points (range finding), one or both of which, or the terrain between them, is remote or inaccessible - e.g. the height of a tree or hill, the width of a deep gully, or the distance of a remote point. For building and engineering construction, as distinct from large scale land surveys, this is normally done by placing a sighting instrument such as a theodolite at a convenient third point and resolving the resultant triangle by linear and angle measurements and trigonometric calculations. An alternative method is to resolve a triangle by making linear measurements and similar triangle ratio calculations, which eliminates the need to use trigonometric ratios and only requires a knowledge of simple arithmetic.

In Fig.1, it is required to measure the distance between A, which is a remote point,, and point B, which is close, by locating a point C, from which both A and B can be seen, measuring the distance BC with a chain or tape, measuring the angles B and C with a sighting instrument and calculating AB from the sine rule or similar. The same result can be got by measuring BC as before, finding the direction of C from AB, either as a reading on a linear scale or by aligning an alidade arm (depending on the instrument), then transferring the instrument to point C, and turning it until B is in line with the previous reading (the instrument scale is now parallel to AB ), and calculating the length of AB from the scale length DE {Fig. 2). Like this :-



Each method has advantages and disadvantages, but similar triangle ratios make a surprisingly versatile method capable of solving complex problems with simple arithmetic - amply demonstrated by Old World architecture. Unfortunately, measurements using either method usually result in awkward numerical calculations easiest done with a hand calculator. For the second method, a partial solution is to design instruments so that their dimensions and scale calibration numbers combine to avoid awkward long divisions. All measuring instruments in this series use either a backsight to scale distance of 200 mm., which allows multiplication by 5 and division by 1000, or a distance of 250 mm., allowing multiplication by 4 and a division by 1000.
Consider two points A,B as described above, and locate a point C from which both A and B can be seen. Place at C an instrument such as the Crevel with linear scale ON. Then if ON is parallel with AB, and scale readings are taken at D and E, ABC and DEC are similar triangles having a common apex C.
Then if any one of the lengths AB,BC or AC is known, the lengths of the other two sides of ABC can be found from the scale readings and instrument dimensions - e.g. the length of the sight line from the vertical backsight slit to the scale.

In similar triangles ABC,DEC, the lengths of each pair of sides have a common ratio. Also for any distance XC between AB and point C, the values of XC and xC have the same ratio.The line CF at right angles to AB is a special case of XC, and has the same ratio.

SoBC / EC =AC / DC = AB / DE = XC / xC = FC / GC.....................(1).

The length of DE can be obtained by values OE and OD, read from the scale ON, and the length of EC can be derived in several ways, depending on which instrument is in use.(see below). Then if any one side of ABC is measured with a tape measure etc., the other sides may be found from similar triangle theory. Like this:-

To find the length of AB. AB / BC =DE / EC.
So

This method requires the measuring scale ON to be parallel with AB. For vertical height measurements, the instrument can be oriented by a plumb line or bubble indicator. For horizontal measurements, the instrument must first be placed at B, and its scale aligned with the direction of A by sighting.. This bearing must be preserved when the instrument is relocated at C. (This is no different to the procedure needed for Fig. 1, where the instrument must also be placed first at A to measure its angle.) The procedure for doing this varies with each instrument, and is included in the instrument’s instructions.

This basic triangulation method will suit most situations, but in some circumstances one or other of the following variations may be more convenient. .


Differential Triangulation is a method of finding the distance between two points, when both are inaccessible; e.g. finding the vertical height of a hill or tree. It is not necessary for the angle ABC to be a right angle.

.


To measure the length of AB when the distance BC1 cannot be known.

Let E1 C1= E2 C2 = K1, a constant. Let D1 E1 and D2 E2 be readings on a linear scale, origin O and full scale N.

In triangles A B C1, D1 E1 C1, . and ---------------(1)

In triangles A B C2, D2 E2 C2, , and --------------(2)

Combining (1) and(2) :- B C1 x D1 E1 = B C2 x D2 E2 ---------------(3)

B C2 = B C1 - C1 C2. So:- B C1 x D1 E1 = (B C1- C1 C2) x D2 E2. ---------------(4)


Simplifying (4) gives:- ----------------(5)

And substituting. (5) in. (1) :- ----------------(6)

For both the dumpy plummet and the Crevel, K1 = 200 mm.; D1 E1 and D2 E2 are in millimetres. For the Dumpy Plummet, E1 and E2 = 100, and for the Crevel, E1 and E2 = 150. C1 C2 and AB are in Metres.
Then since C1 C2, D1 E1 and D2 E2 can be measured, AB can be found.


Proportional Triangulation.

This method will find the distance between two points, when one is accessible; e.g. finding the distance across a river or steep gully, or finding the vertical height of a pole etc. when the base is accessible.

To measure the length of A B, when A is inaccessible.

Let E F and F G be readings on a linear scale.

In similar triangles ACD, E G D, the line BD divides A C and E G in the same proportion.

So:- and

Since B C, E F and F G can be measured., A B can be found.

This theory has been applied to the measuring instruments described in these notes - the Dumpy Plummet and the Crevel - to derive working equations suitable for everyday use.



This is the end of what perhaps could be called a short course in simple surveying instrument making; if you have read this far, it seems almost certain you have a definite interest in the subject. The designs and ideas presented here are by no means all original; for many of them we must thank the ancient Egyptians, Greeks, Romans, and Chinese. Others have been contributed by friends, some of whom have worked in the field. I hope that this combination of ancient wisdom, simple mathematics, practical experience, and use of available modern materials, will prove of value to you. And if you find errors in these notes, or have some information or experience which could improve them, don’t keep it to yourself; let me know. The notes can only be improved by feedback.

God bless you in your endeavours to make life a little easier for the less privileged people of the world.