4.0 Introduction 

1. A vertical angle is an angle formed by two connected lines in the vertical plane*, that is, between a low point and two higher points. Since these angles are in the vertical plane, the lines that form them will usually be lines of sight. A vertical angle BAC can be formed, for example, by the line of sight AB from station A on a river bank to a higher waterpump installation, and the line of sight AC from station A to a much higher waterstorage tank. 2. Whenever a line is not horizontal, it has a slope. The slope can be uphill or downhill. Its steepness depends on the difference in height between its points. 3. As you have learned (see Chapter 2), the slope of the ground affects the measurement of distances. Ground slope is also very important in the design of fishfarms, since you can use it to reduce your construction costs. You need to build bottom slopes in canals, to allow the water to move by gravity*; and in ponds, to allow good drainage. And you must build slopes in the dikes for ponds and dams (see the next manual in this series, Constructions for Freshwater Fish Culture). 

4. The slope of a line is called the gradient. It may be defined as:


5. The slope of a line is therefore expressed in various
ways:


Remember that:






Converting percentage of a slope into degrees, or degrees into percentage 

6. Depending on the instrument you are using to measure a slope directly, you may sometimes have to convert the percentage of the slope into degrees, or the degrees into percentage. For help with such a conversion, you should use either Table 4 or the graph given in Figure 3.  
Note: from the table and the graph you can see that:

Examples:
a slope of 17 percent is equal to (10 + 5 + 2) percent, which is equivalent to 5°42'40'' + 2°51'40" + 1°08'40" = 8°101'120'' = 8°103' = 9°43';
a slope of 9º43' is about equal to (9º + 30' + 15'), which is equivalent to 15.84 percent + 0.87 percent + 0.44 percent = 17.15 percent or 17 percent.
Measuring and calculating slopes 

7. There are two groups of methods for determining slopes.

Measuring slope directly





8. To calculate the slope, proceed as follows:

First measure the difference in level


Note: to make your calculations easier:
Remember: you must measure the horizontal distance! 
Then measure the horizontal distance 
Using slope to calculate horizontal distances 

9. In Chapter 2, Sections 2.6 and 2.7, you learned that when measuring a distance AB on sloping ground, you need to correct this measurement in order to find the true horizontal distance AC, but only when the slope exceeds 5 percent (or about 3 degrees). To make these corrections, you may use either the method described below, or the method which will be described in Section 5.0, step 17. To calculate horizontal distances from distances measured over sloping ground, proceed as follows:  
10. Measure the distance AB (in metres) on the ground
between points A and B (see Chapter 2).
11. Measure the average ground slope S in degrees between points A and B (see this chapter, Sections 4.1 to 4.7). Note: if the slope is measured in percent, you will have to convert it into degrees (see Table 4 or Figure 3). 
Measure the groundlevel distance


12. Enter this average slope S (in degrees) in Table 5 to obtain the value of cosine S (cos S). If the slope does not correspond exactly to any of the angle values given in the table, you will have to calculate cos S by using proportional parts (see example in Table 5). 13. Calculate the horizontal distance AC (in metres) using the formula:

Measure the slope


Calculate the horizontal distance

AC = AB cos S

MAIN TABLE

TABLE OF PROPORTIONAL PARTS, P Example To calculate intermediate cosine values using the proportional parts, for cos 7°38' for example, proceed as follows:

Choosing a method to use for measuring slopes 

14. There are several good ways to measure slopes. The method you use will depend on several factors:
Each of the various methods is fully explained and illustrated in the following sections, except for the method to use with levelling devices (see Chapter 5). Table 6 will also help you to compare the various methods and to select the one best suited to your needs. 

Plumbline

Clisimeter

TABLE 6
^{1} *Simple **more difficult ***most difficult.

1. A clinometer is an instrument for measuring slopes or vertical angles. There are various types of clinometers, but they all include a graduated arc similar to a protractor (see Section 3.3, step 11). To use the clinometer, you hold it in your hand and read the slope against this arc. You also usually refer to a freehanging plumbline called the pendulum. There is a line of sight* on the top of the clinometer. You can easily make your own simple clinometer; four models are described in Sections 4.1 to 4.4.
Making the pendulum clinometer, model 1 

2. Get a protractor with a 0º to 90º scale, or
make one yourself as described in Section 3.3.
The protractor should be fairly large (for example, about 20 to 25 cm diameter)
to provide reasonable accuracy.
Note: if you use Figure 2 to make your protractor, you can easily draw a larger 0º to 90º protractor. To do this, get a piece of string and tie a pencil to one end. Measure 20 to 25 cm from the pencil along the string. Hold the string at this point on the centrepoint A of the protractor in Figure 2. With the string stretched tightly, draw an arc with the pencil above the rounded edge of Figure 2. Then add graduations on your new protractor by projecting lines from the graduations in Figure 2. Glue the protractor to a piece of thin wooden board or plywood and cut carefully along its outline. 
Glue the protractor to a wooden backing and cut
it out


Prolong the graduations from the protractor

Draw a larger arc with pencil and string


3. Attach a plumbline (see Section 4.8, step 2) to a small nail driven into the centrepoint A of the protractor. Make the plumbline with a thin piece of string about 35 to 40 cm long and a small weight, such as a heavy nut or a small stone.  4. Glue a 30cm sighting device along the 90º side of the protractor. To make the sighting device, get a soda straw or a narrow tube; or get a thin length of wood and attach two pins along it in a straight line.  
5. Measure the vertical distance from the level of your eyes to the ground, then measure the same vertical distance on a wall and mark it clearly. You will also need to mark this vertical distance clearly on a pole or staff, which you will use for sighting.
Make a mark at your eye level

Mark the reference level on the pole


6. Stand on horizontal ground about 15 paces in front of the mark, and aim at it through the sighting device on your clinometer.  7. Check that the plumbline string indicates 0º. If it does not, adjust the small nail holding the plumbline. When the string indicates 0°, your clinometer is ready to use.  
Check for accuracy by sighting at the mark you have made 

8. Sighting either uphill or downhill with the clinometer, you can measure a slope by moving the protractor around.
9. Take a position with the clinometer. Make sure to stand up straight so you do not change your eye level. Sight at a point. This point should be:
Make sure point A is at the top, 
10. With the clinometer in sighting position, press the plumb.line with your finger against the bottom scale. Be careful not to move the plumbline from its vertical position. Read the scale at the point where the plumbline intersects the degree graduation. This reading is the slope, in degrees.
Note: you can convert your degree measurement into a percentage (see Section 4.0).
Hold the plumbline in place with your finger

1. You can make another type of clinometer from wood or metal. This model also has a plumbline, but its reference scale gives you the slope in percent.
2. Cut a 51 x 51 cm square board from a piece of wood, or build one from strips of wood or metal.
Reinforce the board if necessary

3. Provide a sighting line* along the upper edge of the square.
A block of wood helps you place the nails correctly


4. Provide a centrepoint from which to hang a plumbline. Put a mark on the board 1 cm down from the top edge and 1 cm in from the sighting point which is furthest from your eye. If the board is wooden, drive a nail into this mark; if it is metal, weld a small nail to the mark, or drill a hole through it.  
5. Make a plumbline about 65 cm long, using a piece of thin string and a weight. A plumbbob (a small lead weight) will make the best weight for the plumbline but, if you do not have one, you can use any object which has its weight evenly distributed from a single point. A heavy nut or washer, or a wooden disk with a hole in the centre, will work.  
6. Attach the plumbline to the hole or nail at the centrepoint of the board. 7. Loosely attach a ruler graduated in centimetres along the bottom edge of the board; use large clips, or tie the ruler on with string. Position the ruler so that its zero graduation is directly under the centrepoint. Make sure that the distance between the centrepoint of the plumbline and the zero mark on the bottom edge of the ruler is 50 cm. 
Adjusting your clinometer 

8. Aim the board at a mark which you have aligned at eye level. Standing straight and looking along the board's upper edge, align the two sighting points with this mark. Your sighting line* should now be horizontal and your plumbline should be vertical.  
9. Put your thumb on the plumbline to hold it against the ruler at the bottom of the board, and check to see if the line is at zero. If it is not, adjust the position of the ruler so that the zero graduation and the plumbline fall exactly in line. 10. Check to see that your clinometer is correctly aligned by sighting again. When it is, glue or nail the ruler firmly in place. Your clinometer is now ready to use. 
Using the clinometer to measure a slope 

11. You can measure both uphill and downhill slopes with your clinometer in the following ways: 




12. Place a pole or a staff clearly marked at eye level (see Section 4.1, step 5) on a point you can easily see, usually 15 to 20 m away.  
13. Aim the clinometer at this mark and, when the plumbline
has stopped swinging, press it with your finger to the ruler at the bottom.
Be careful not to move the plumbline from its vertical position. Then,
read the graduation (in centimetres) at this point.
14. Since every centimetre on the ruler equals 2 percent of slope, calculate the slope as a percentage by multiplying the number of centimetres you read on the graduation by 2. Example If you read 2.5 cm on the ruler, the slope is found as: 2.5 cm x 2 = 5% 
1. The third model of clinometer is a little more complicated to make, but it is more accurate. It is also easier to use if you are measuring on ground that is soft enough for you to drive in the supporting staff.
Making the clinometer, model 3 

2. To make the supporting staff, get a straight stick or a piece of wood about 2 m long. Shape one of its ends into a point, so that you can easily drive it into the ground. About 25 cm from the pointed end, mark a line to show how deep you will drive the staff in.  
3. Get three pieces of wood exactly the same, 40 cm long, 4 to 5 cm wide, and about 1 cm thick. Secure them tightly together with nails or screws to form a triangle with three equal sides. 

4. Prepare a ruler graduated in millimetres. Get a piece of wood about 25 cm long, 4 cm wide and 0.25 cm thick. Mark the centre with 0, then mark graduations from this centrepoint up to 100 mm on either side.  
5. Loosely attach this ruler to one of the triangle's sides with string or clips. 
Tie the ruler to the triangle with string


6. On the same side of the triangle, make a sighting device. Drive two nails vertically into the side near each of its ends. Make sure the nails are at equal heights and on the line. 

7. Drill a small hole exactly at the centre of the triangle's
summit, opposite the zero point of the ruler.
8. Attach the triangle near the top of the supporting staff with a nail; make sure that the triangle remains free to swing around this axis. 9. Prepare a plumbline about 40 cm long (see Section 4.2). Attach it to the nail at the centre of the triangle's summit. 
Nail the triangle so it swings freely

Adjusting the clinometer 

10. Drive the supporting staff vertically into horizontal ground until you reach the reference* level you marked above its pointed end. 11. Measure the vertical distance between the ground and the sighting line* of the clinometer exactly. This distance should be about 130 cm. Prepare a pole or staff that shows this height (see Section 4.1, step 5). Note: the height of the sighting line for this clinometer may be different from your eye level. 

12. About 15 paces away, make a mark on a wall set at the same height you just measured. Aim with the sighting line at this mark. 
Sight at the mark


13. Adjust the position of the ruler so that its 0graduation lines up exactly with the plumbline. Check again for sightingline accuracy and adjust the 0graduation if you need to, then glue or nail the ruler firmly in position on the triangle. The clinometer is now ready to use.  14. Exactly measure the distance (in centimetres) between the point at which the plumbline is attached and the point where the sighting line intersects the plumbline. This distance should be about 32 cm, and is the standard distance D of your clinometer. Be sure to measure D precisely.  
Adjust the ruler so the plumbline is at zero

Measure the distance D from the nail to the sighting
line

Using your clinometer to measure a slope 

15. You can measure either uphill or downhill slopes by reading the appropriate one of the two scales. 16. Place a pole or staff clearly marked at the sightingline level (see step 11) on a point B of the slope you are measuring, about 1520 m away. 
17. At point A, drive your clinometer support vertically into the ground, down to the reference level. With the sighting line, aim at the mark on the pole or staff; to do this, slowly swing the triangle around the nail at its top until you sight the marked level.  
Swing the triangle around until you sight the
top
of the pole 

18. When the sighting line is level with this mark, press the plumbline with your finger against the ruler. Be careful not to move the plumbline from its vertical position.  19. Read the graduation N (in millimetres) on the ruler at the point where the plumbline intersects the sighting line.  
Hold the plumbline in place

20. If the standard distance of the clinometer (see step 14) is D (in centimetres), calculate the ground slope S% as:
S% = (10 x N) ÷ D 
Example
If D = 32 cm and you read a graduation of 4.8 cm = 48 mm on your clinometer, the slope is equal to:
(10 x 48) ÷ 32 = 15%
1. The fourth clinometer model is similar in principle to the preceding one, but it has several improvements: it is much smaller in size; it is easier to make; and it provides a direct reading of the slope, so that you do not need to make any calculations. The model 4 clinometer may also be used to measure vertical angles (see this Section, step 17).
Making the clinometer, model 4 

2. Get a small piece of thin wooden board, about 14 x 21 cm. The best material would be plywood. 
3. On this board, glue a sheet of squareruled millimetric paper so that its printed lines are parallel to the sides of the board. 4. Draw a line AB, parallel to the larger edge of the board and about 1.5 cm from it. 

5. Find the centre of line AB and mark it C. From this point lay out perpendicular CD, which should measure 10 cm. You may adapt one of the methods from Section 3.6, or use the lines on the paper to guide you.  6. Through point D, raise perpendicular* EF, which is parallel* to AB.  
7. Taking point D as zero, measure 10 cm to the left and 10 cm to the right of point D, along EF. Divide these two distances into millimetres and mark the main graduations. Once again, the lines on the paper will help you.
Note: instead of drawing the above lines yourself, you can use Figure 4. Make an enlarged a copy of it and cut it out. Glue this figure to the wooden board, with line AB parallel to the board's longer edge.
8. Make a plumbline 17 cm long, using very thin thread (such as a nylon fishing line) and a small weight. Drive in a small nail exactly at point C on the board, and hang the plumbline from it. Slightly below the nail, at K on line CD, drill a hole that a woodscrew will pass through.
9. Make a sighting line* along line AB. To do this, you can drive thin nails in at points A and B. Or, get two metal strips (you can cut them from a tin) and cut small, vshaped notches out of one end of each strip. Then, bend the other end so that the strips can be attached perpendicular to the board. Screw them to points A and B, making sure that the vnotches (your sighting guides) are directly over the two marked points A and B. Align these vnotches with line AB.
Mark sighting line AB with nails...

... or with vnotch sighting guides

10. Get a wooden staff 2 m long to use as the support, and make a point on the bottom end. Loosely attach the clinometer board near the top of this staff with a screw through the hole K you made on line CD in step 8. Tighten the screw so that the board can be turned around. Check that the head of the screw lies slightly below the surface of the board so it will not disturb the plumbline.
Attach the board so it can turn easily

Make sure the plumbline swings freely

11. Clearly mark a reference line* about 25 cm above the pointed end of the supporting staff, showing the depth to which you need to drive it into the ground at each station. Measure the distance between this reference line and the sighting line AB. 12. Then prepare a pole or staff with a reference line and a sighting line at exactly the same height as line AB. This will be your sighting pole. 
Using the clinometer for measuring a slope in percent 

13. You can measure either uphill or downhill slopes by reading the appropriate one of the two scales. 14. Place the sighting pole you made in step 12 on point Y of the slope you are measuring, about 1520 m away. Drive it in vertically up to the reference line. 
15. At point X, drive your clinometer support vertically into the ground up to the reference line. With the sighting line, aim at the mark on the sighting pole. Rotate the board around its screw until you sight the marked level.
Turn the board until you sight the top of the pole

16. Where the plumbline crosses line EF, read the graduation (in millimetres). This gives you the slope in percent.
Note: check carefully to see that the plumbline hangs freely from its support. The board should rotate without disturbing the vertical position of the plumbline.
17. If you must measure a vertical angle in degrees instead of a slope, you may use the model 4 clinometer (as described above). The only difference in this case is that you use the curved scale GH (in Figure 4) rather than the bottom scale.
4.5 How to use the clisimeter 

1. The clisimeter is a simple instrument for measuring horizontal distances, as explained in Section 2.7. It can also be used to measure a slope or a vertical angle, but it can only give a rough estimate of these, accurate to within 10 percent. The lyra clisimeter is a commonly used model. It is made up of a sighting device, an attached ring, and a weight, shaped like a pear, which keeps the clisimeter in a vertical position when hung from its ring. The instrument folds neatly into the weight for transport. 
2. When you look through the sighting device, you see three scales. As described before (see Section 2.7, step 3), the central scale is used to measure horizontal distances. The other two scales are used to measure vertical angles and slopes. You will use the left scale, which is graduated in per thousand (%o) or tenths of percent (%):
The scale inside the clisimeteruse the left scale
to measure slopes

100 on the scale %o = 10% Examples 15 per thousand equals 15 ÷ 10 = 1.5
percent

3. The left scale is graduated from zero in two opposing directions:
You can use the clisimeter by yourself or with an assistant:
4. If you are working alone, you need a pointed stake clearly marked at two levels: the reference level above the pointed bottom, showing the depth to which you will drive the stake into the soil; and the eye level, which is the vertical measurement from the reference level to your eye level. It is best to have the eye level at the top of the stake. (This stake is like the one you learned to make in Section 4.1, step 5.)
Make a sighting pole marked at eye level

Sight at the marked pole

5. If you have an assistant, you can also use a simple rod marked at eye level, but it will be faster to use your assistant instead of this rod. To do this, determine the point on your assistant which is at the same level as your own eyes and sight at that point instead.
Find your eye level on your assistant

Sight at the eye level you have chosen


6. Place the marked stake at point B on the slope
you need to measure, or send your assistant to point B, either with or without
marked rod.
7. Taking a position at point A about 10 to 15 m away, hang the clisimeter vertically from your left forefinger and bring the sighting device up to your left eye. Make sure to stand up straight so you do not change your eye level. 
Sight at the point


8. While looking at the marked level with your right eye,
read the graduation on the left scale of the sighting device. This is the
slope you are measuring, expressed in per thousand. Note: to make reading the graduation easier, move your head slightly from right to left. The graduation will seem to extend out of the instrument into the landscape. Then, read the graduation corresponding to the marked level. 
Read the left scale

9. You will need an assistant for this method. Sight with the graduation on the left scale (which corresponds to the slope) at the marked level (on a rod such as the one described in Section 4.1, step 5, for example) corresponding to the height of your eyes.
10. Ask your assistant to move the marked rod forward or backward until the eye level line is even with the clisimeter graduation.  11. When the rod is properly aligned, ask your assistant to mark the point on the ground with a stake. Move up to this stake and repeat the procedure.  
Move the sighting pole until you see it
at the correct graduation 
Note: if you need greater accuracy, you can hang the clisimeter
at fixed height from a stick. If you do this, remember to adjust the marked
level on the rod to this height.
4.6 How to use the optical clinometer 

1. An optical clinometer is a precise pocket instrument for measuring vertical angles and estimating tree heights. It is commonly used by foresters. It can also be used to measure slopes quickly, with a method similar to that described for the clisimeter (see Section 4.5). 
Clinometer


2. When you look through the sighting device of the clinometer, you can see a crosshair and two scales. The left scale is graduated in degrees and the right scale is graduated in percent. Both scales have a positive (+ ) section for measuring uphill slopes and a negative section () for measuring downhill slopes. 
Sight through hole


3. Keeping both eyes open, sight with one eye through the optical clinometer, moving it until the crosshair lines up with the marked level you wish to measure (such as a rod). With the clinometer lined up in this position read the graduation at the crosshair. 
Read the graduation at the crosshair

4.7 How to use miscellaneous levelling devices 

1. In Chapter 5, various levelling devices will be discussed. These devices can also be used to measure a slope. To set a graded line of slope, see Section 6.9. 
Sight with the clinometer


2. In Section 3.5, you learned about
theodolites and how you can use them to measure horizontal angles. Most
theodolites are designed to measure vertical angles as well.
For this purpose, they are fitted with:

Theodolite

3. Levelling devices help you measure the difference in levels between two points. After you have measured the horizontal distance between these points, you can calculate the slope as explained earlier (see Section 4.0, step 8).
Measuring difference in levels between two points

4.8 How to set out and check verticals 

1. A vertical is a line with a 90º slope. You will often have to set out verticals, especially when you are building walls for a canal or building. You have already used vertical lines, to measure distances over sloping ground for example (see Section 2.6, step 19). 
Most walls are vertical

Setting out and checking verticals with a plumbline 

2. A plumbline is a simple device which forms a vertical line*. The idea of the plumbline is based on the fact that any heavy object will fall vertically, making a 90º angle with the horizontal plane at ground level.  3. In a plumbline, a fairly heavy object, the plumb, is attached to the end of a thin line. When the plumb hangs freely without moving, the line is vertical.  
Gravity makes objects fall vertically

A plumbline

Making your own plumbline 

4. You can make a simple plumbline from:

5. You can make an improved plumbline for measuring buildings in progress and other constructions. Start with a piece of wood or heavy metal about 10 cm square.  
A simple plumbline

An improved plumbline


6. Find the exact centre of the square piece by drawing
two diagonal lines on it. Drill a small hole through the point where they
cross.
7. To make the plumb, get a heavy, solid block of wood (such as red acajou) or metal  the largest side of this block should be 10 cm across or less  if you can, shape the block into a cone. 8. If the block is wooden, drive a small nail into the exact centre of its top surface. If the block is metal, have a small hook welded to this point. 9. Attach the end of a thin line (nylon fishing line is strongest) about 1 m long to this nail or hook on the block and pass the other end through the central hole of the wooden or metal square piece. Fix the line on the other side of this hole either by tying it into a heavy knot or by tying a small piece of wood or metal (such as a nut) on to its end. Note: you can change the dimensions of the plumbline, depending on the materials you have. The line can be longer, if necessary. 

Don't let the end of the line pass through 
Using a plumbline to set out a vertical 

10. Remember that a free plumbline will hang vertically. 11. You can use a simple plumbline to see if a wall is vertical. To do this, hold the top end of your plumbline close to the wall and check to see if the distance between the wall and the top end of the line is equal to the distance between the wall and the centre of the weight at the bottom. This distance will be easier to check if the weight is pointed on the bottom. 
Checking a vertical with a plumbline


12. When using the improved plumbline along a wall:  



Samesize weight touches the wall

Measure a smaller weight

Note: if you need to make the plumbline shorter to measure along walls of different heights, you can pull the line up through the centrehole in the square at the top. Let it back down through the hole to measure higher walls.
Pull up the line to measure shorter verticals

13. Some mason's levels (see Section 6.1) have an additional bubble level for checking verticality. You can use this level when you are building walls, for example. This method is particularly useful when the vertical you are checking is fairly small. Hold the mason's level vertically against the surface you need to check. If the surface is vertical, the bubble will be at the exact centre of the bubble level.