For levelling steep inclines it also saves a great number of settings up, as, instead of levelling for, say every 14 feet rise or fall, the gradient of the total distance can be taken and also the distance measured by stadia reading, when the incline is not too great for taking one reading with telescope level, or by gradient reading when this cannot be done, and by adding the staff reading to the distance divided by the gradient, and deducting the height of the instrument the difference of level can at once be ascertained.
Example: Sighting a staff at a gradient which falls conveniently upon it, say 1 in 35 and this reads 8·7 feet. Distance measured, as explained later, say 735 feet, then 735/35 = 21 feet + 8·7 feet = 29·7 feet; deduct the height of instrument, say 4·9 feet, difference of level 24·8 feet.
For measuring long distances beyond the range of the stadia lines or points in the diaphragm, or for measuring distance on inclines, the gradiometer will also be found very useful, as by taking the difference of any two suitable gradients, the base distance is given without calculation for difference of hypo and base.
If the gradient be not very steep or below the height of the staff, the simplest method is to sight the staff with the telescope level and take the staff reading; say this is 12·45 feet, then set the gradient drum to 1 in 100 and again take the staff reading and, say this is 4·30 feet, the difference between these readings = 8·15 feet. Strike out the decimal point which multiplies it by 100 and we have the base distance 815 feet.
For shorter distances a larger base upon the staff may be taken, thus giving greater accuracy; for instance, if the gradient drum after taking the level reading be set to 1 in 50 and the resulting difference divided by 2, any error in taking exact readings is reduced by one half, or 1 in 33-1/3 and divide difference by 3; or 1 in 25 and divide difference by 4: or 1 in 20 and divide difference by 5, etc. Any error of reading would be reduced by one third, one fourth, one fifth, etc.
The difference of readings obtained by either of the following two gradients will also give base measurement without any calculation whatever: 100 and 50 | 63-2/3 and 40 | 60 and 37½ | 50 and 33-1/3 | 33-1/3 and 25 | 25 and 20 | 20 and 16-2/3 | 12½ and 11-1/9 | 11-1/9 and 10.
Example: Suppose the top of staff is below level altogether, turn the drum until the top of staff is sighted in the telescope; say the gradient of this is 27½ go on turning until gradient 25 is indicated and take the staff reading; say this is 12·75, then move the drum until gradient 20 is indicated and take the staff reading: suppose this to be 3·40, then
| 12·75 |
| - 3·40 |
| = 9·35 |
Omit the decimal point and the measurement reads 935 feet, which is the horizontal distance. The two most suitable gradients would of course be used according to the position.
Distances may be set out with equal facility with the gradiometer by the subtense method, by working out a subtense suitable for the distance. This is easily done by dividing the distance required by any two numbers having a difference of the required subtense, the result being two gradients, which, when worked with, will give that subtense at the required distance.