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CHAPTER III.

SURVEYING WITH ANGULAR INSTRUMENTS.

WHAT has been said in reference to chain surveying applies only to small surveys in open country, where the chain lines can be seen from end to end and can be chained through continuously. The same methods apply to any kind of a survey, large or small; but when a survey of large size is made, especially in towns and congested districts, and very large surveys are carried out on trigonometrical lines, the aid of an instrument with which to measure angles must be sought.

The principal instrument for the work is called the Theodolite. It is an expensive part of an engineer's outfit; and although many use cheap forms of implements, under the names of prismatic compasses, miners' dials, etc., for really good work and accuracy there is nothing equal to the transit theodolite. Its use can only be learned by practice, and its easy manipulation and real value are only appreciated by those who have had considerable experience in its employment. Fig. 19 shows a good form of instrument made by Messrs Thornton & Company, King Street, Manchester, and its description is as follows:

it or

ab is the telescope, having a diaphragm at c. This is usually a brass ring, having spiders' webs stretched across it, a glass with fine lines upon platinum iridium points, the upper and lower horizontal wires being what are termed stadia hairs, the use of which will be explained later. aa are the four capstan screws which hold the diaphragm in place. On the other side of the telescope is a milled screw to move the object-glass in or out by means of a rack and pinion and so focus the telescope. The eyepiece, which is also capable of a sliding adjustment, magnifies the image, which is usually inverted. This will puzzle the beginner at first. He soon gets used to it, however. The telescope can be so made as to give an erect image, but this detracts from the power of the lenses. A spirit-level e is fixed on the top of the instrument, fixed by capstan screws. g is the vertical circie fixed to the horizontal axis of the instrument, while hh are verniers which should read zero when the collimation line is horizontal. They are read by microscopes. ll are the supports, and m clips to fix the telescope. Q is the vertical tangent screw, while P clamps the verniers so as to allow of the final adjustment being made by the tangent screws. The telescope should be capable of being moved through a complete circle. In a transit instrument rr are adjusting screws by means of which the horizontal axis of the telescope can be made truly perpendicular to the vertical one. S is the upper or vernier plate rotating on the vertical axis T. s carries the supports ll. On this is a compass to take magnetic bearings, and also two spirit-levels u at right angles to each other, and also

fixed by capstan screws. Then there are the verniers W and microscope V. The edge of the lower plate (which also rotates) is divided into 360°. Theodolites are known as 5-iu., 6-in., etc., according to size of plates. For ordinary work a 5-in. instrument is used, reading to single minutes, minutes or 30 seconds, and minutes = 20 seconds. The two plates called the limbs are capable of being clamped to allow of the use of the slow-motion tangent screws, and the lower plate is clamped to the vertical axis by the

screw X. Y is the upper parallel plate, passing through which are three or four levelling screws LS resting on the lower parallel plate Z. By means of these screws the instrument is levelled up, that is, the two limbs are set truly horizontal to the vertical axis on screws truly vertical. Three levelling screws are better than the four with which the older instruments were provided, because four are liable to stick, and necessitate the use of both hands. Finally, just appearing in the figure is a small hook Z,; the hook should be set in the exact centre of the instrument, that is, in the centre of the vertical axis. It is to the hook that the plumb line is attached when setting the theodolite over a station. In addition to the example given, there are

various forms of theodolite, some having special features of their own, such as Haskold's device for sighting down tunnel shafts.

For further information the reader may refer to any book dealing specially with surveying such as Whitelaw's or Heather's Mathematical Instruments.

The next matter to be explained is the use of what are called the verniers of a theodolite; and as their manipulation is often a source of difficulty to the beginner, it will be explained somewhat fully. There are usually two and sometimes three on the lower plate, and also two at h on the vertical circle (fig. 19). They are to enable single and even or minutes to be read off. The actual construction is shown in figs. 21 and 22.

Fig. 21 shows a vernier such as is usually adopted, reading to 1 minute.
In this case the length of the vernier is equal to 14° 30′ of the graduated

lower limb.

Platinum Iridio Points

FIG. 20.-Diaphragms for Theodolites.

This space is divided into 30 equal parts. Therefore each

division of the vernier is equal to

If the zero of the vernier be set to the zero on the lower plate we find that because there is a space of 29 half degrees divided into 30 parts, the first division of the vernier lags behind the corresponding division on the lower plate by an amount of a half degree, that is, 1 minute, there being 60 minutes in a degree. The second division is therefore behind to the extent of 2 minutes, and so on till the last is 30 minutes behind, and marks 14° 30′ instead of 15 degrees.

=

Therefore, to set off a desired number of degrees and minutes, set the zero of the vernier to the given numbers of degrees required on the lower plate, and then by means of the tangent screw bring the division of the vernier corresponding to the given number of minutes to coincide with the next division which happens to be in advance of it on the lower limb.

To take an example. Assume we require to set off 14° 13', the zero of the vernier should be set to 14°, and the position of division 13 on the vernier should then be noted. Then, having clamped the two plates, bring the division 13 to coincide with the division in advance of it. We then have 14° 13' set off.

The reading of a given angle is of course only the converse process, that is, the zero of the vernier is observed and the number of degrees below it found. This is the number which is nearest it. The division on the vernier which corresponds with one on the lower plate is then sought, and this gives the number of minutes required. Care must be exercised in reading the actual degrees where the divisions do not correspond, because, assuming the reading to be 29° and 30°, it is the 29, or the lesser of the two, which must be read off, because the vernier gives the odd fraction in minutes, which is indeed the purpose for which it is employed.

Naturally the minuteness of the angle which it is possible to read depends on the length of the vernier and also on the graduation of the lower limb. Some verniers are divided into minutes, in which case they have 60 divisions 14° 30' long

of 29, and it is then possible to read to 30 seconds or a minute, the length of the whole vernier being 29° 30′ (fig. 21A). The method of setting off and reading is, however, slightly different.

Assume we require to set off 51° 41' 30". Here the number of minutes is more than 30, so we set off 51° on the lower plate, and bring up the division of the vernier corresponding to 41′ 30′′ minus 30′ = 11′ 30′′, to coincide with the one in advance of it. Where, however, the number of minutes and seconds is less than 30 minutes, no such subtraction is necessary, the angle being set off direct.

There still remains a third kind of vernier-that illustrated in fig. 22. It reads to 20 seconds or of a minute, the lower limb being divided in thirds of a degree. The length of the vernier is 19° 40', and this is divided into 60 parts and read to 20 seconds. The setting off and reading is the same as that for 30 seconds.

Its

The vernier itself is simple enough and gives great accuracy. use, however, requires a little practice to understand. Pupils in engineers' offices and other beginners for whom this work is specially intended should read the vernier in the office during any spare time they have on hand. It is also advisable to read both verniers, so as to counteract errors of graduation, and also as a check, but the compass bearing should likewise always be read as well, as it provides a most valuable check on the results.

The practical use of the theodolite can now be explained. Suppose in fig. 13 that we were able to measure the lines BC and BK direct, but that an obstacle prevented us from measuring CK, and prevented the plotting of the triangle BKC unless the included angle CBK could be obtained. It is here that the theodolite is brought into play. It is, of course, assumed that the reader is acquainted with the common properties of triangles and other plane figures, because without such knowledge theodolite surveys would be impossible. The principal properties of plane triangles most frequently made use of in theodolite surveying are

i. The three angles of any triangle are together equal to 180 degrees. ii. The angles at the base of an isosceles triangle are equal.

iii. An exterior angle of any triangle is equal to the sum of the two interior opposite angles.

iv. All the angles of an equilateral triangle are angles of 60 degrees.

v. In a right-angled triangle the hypotenuse squared equals the sum of the squares of the other two sides.

Reverting to fig. 13, we require to measure the angle at B.

Set the theodolite up as nearly level as possible over the hole left by the ranging pole at B. Get the plumb line and suspend it from the hook Z (fig. 19), steady it, and notice if the bob falls into the hole. This will necessitate a little adjustment of the instrument, then put the plumb bob in your pocket. Now level up by turning the telescope over two levelling screws, and adjust it by the spirit-level lying in the direction across the two screws. Then shift over the other two and do the same. Repeat the process again, and when the bubbles of the levels are in the centre of the run the levelling up is complete, or, in other words, the theodolite will "traverse." Secondly, look at the compass and allow it to settle, then turn the whole instrument until zero on the compass ring graduation is opposite the north end of the needle. Having made sure this is so, clamp the screw X. Then, if this has not already been done, set the plates to zero, and carefully direct the cross hairs on the pole at C. Clamp the plates and complete the adjustment by means of the tangent screw s (fig. 19). Note the compass reading, and see that the instrument has not gone off level. If it has, readjust it. Then unclamp the plate and sight on to K. Clamp again and use the tangent screws. Observe the compass and note down how many degrees the needle has moved through. Then read off the angle by means of the microscopes and verniers. This should then correspond with the compass reading. The next step is to shift the instrument to C, keeping all the plates clamped. Level up and sight on to B, keeping the plates clamped again. Allow the compass to settle. It should read as before, looking in the direction BC. Finally, measure the angles DCX and XCF, and so on, wherever angles are required. They should then be entered in the field book as in fig. 23.

Always take plenty of angles. When coming back to a previous station, take the bearing and note how it agrees with the back bearing. In other