facilitate these calculations, which are of so frequent occurrence and of so great use, Traverse Tables have been prepared, originally for navigators, (whence the name Traverse), and subsequently for surveyors. The Traverse Table at the end of this volume gives the Latitude and Departure for any bearing, to each quarter of a degree, and for distances from 1 to 9. To use it, find in it the number of degrees in the bearing, on the left hand side of the page, if it be less than 45°, or on the right hand side if it be more. The numbers on the same line running across the page,† are the Latitudes and Departures for that bearing, and for the respective distances—1, 2, 3, 4, 5, 6, 7, 8, 9,— which are at the top and bottom of the page, and which may represent chains, links, rods, feet, or any other unit. Thus, if the bearing be 15°, and the distance 1, the Latitude would be 0.966 and the Departure 0.259. For the same bearing, but a distance of 8, the Latitude would be 7.727, and the Departure 2.071. Any distance, however great, can have its Latitude and Departure readily obtained from this table; since, for the same bearing, they are directly proportional to the distance, because of the similar triangles which they form. Therefore, to find the Latitude or Departure for 60, multiply that for 6 by 10, which merely moves the decimal point one place to the right; for 500, multiply the numbers found in the Table for 5, by 100, i. e. move the decimal point two places to the right, and so on. Merely moving the decimal point to the right, one, two, or more places, will therefore enable this Table to give the Latitude and Departure for any decimal multiple of the numbers in the Table. For compound numbers, such as 873, it is only necessary to find separately the Latitudes and Departures of 800, of 70, and of 3, and add them together. But this may be done, with scarcely any risk of error, by the following simple rule. The first Traverse Table for Surveyors seems to have been published in 1791, by John Gale. The most extensive table is that of Capt. Boileau, of the British army, being calculated for every minute of bearing, and to five decimal places, for distances from 1 to 10. The Table in this volume was calculated for it, and then compared with the one just mentioned. In using this or any similar Table, lay a ruler across the page, just above or below the line to be followed out. This is a very valuable mechanica. assistance. Write down the Latitude and Departure for the first figure of the given number, as found in the Table, neglecting the decimal point; write under them the Latitude and Departure of the second figure, setting them one place farther to the right; under them write the Latitude and Departure of the third figure, setting them one place farther to the right, and so proceed with all the figures of the given number. Add up these Latitudes and Departures, and cut off the three right hand figures. The remaining figures will be the Latitude and Departure of the given number in links, or chains, or feet, or whatever unit it was given in. For example; let the Latitude and Departure of a course havmg a distance of 873 links, and a bearing of 20°, be required. In the Table find 20°, and then take out the Latitude and Departure for 8, 7 and 3, in turn, placing them as above directed, thus: Taking the nearest whole numbers and rejecting the decimals, we find the desired Latitude and Departure to be 820 and 299.* When a 0 occurs in the given number, the next figure must be set two places to the right, the reason of which will appear from the following example, in which the 0 is treated like any other number. Given a bearing of 35°, and a distance of 3048 links. Here the Latitudes and Departures are 2496 and 1749 links. It is frequently doubtful, in many calculations, when the final decimal is 5, whether to increase the preceding figure by one or not. Thus, 43.5 may be called 43 or 44 with equal correctness. It is better in such cases not to increase the whole number, so as to escape the trouble of changing the original figure, and the increased chance of error If, however, more than one such a case occurs in the same column to be added up, the larger and smaller number should be taken alternately. When the bearing is over 45°, the names of the columns must be read from the bottom of the page, the Latitude of any bearing, as 50°, being the Departure of the complement of this bearing, or 40°, and the Departure of 40° being the Latitude of 50°, &c. The reason of this will be at once seen on inspecting the last figure, (page 170), and imagining the East and West line to become a Meridian. For, if AC be the magnetic meridian, as before, and therefore BAC be the bearing of the course AB, then is AC the Latitude, and CB the Departure of that course. But if AE be the meridian and BAD (the complement of BAC) be the bearing, then is AD (which is equal to CB) the Latitude, and DB, (which is equal to AC), the Departure. As an example of this, let the bearing be 6310, and the distance 3469 links. Proceeding as before, we have The required Latitude and Departure are 1561 and 3098 links. In the few cases occurring in Compass-Surveying, in which the bearing is recorded as somewhere between the fractions of a degree given in the Table, its Latitude and Departure may be found by interpolation. Thus, if the bearing be 103°, take the half sum of the Latitudes and Departures for 101° and 101°. If it be 10° 20', add one-third of the difference between the Lats. and Deps. for 101 and for 1010, to those opposite to 101°; and so in any similar case. The uses of this table are very varied. The principal applications of it, which will now be explained, are to Testing the accuracy of surveys; to Supplying omissions in them; to Platting them, and to Calculating their content.* *The Traverse Table admits of many other minor uses. Thus, it may be used for solving, approximately, any right-angled triangle by mere inspection, the bearing being taken for one of the acute angles; the Latitude being the side adjacent, the Departure the side opposite, and the Distance the hypothenuse. Any two of these being given, the others are given by the Table. The Table will therefore serve to show the allowance to be made in chaining on slopes (see Art. (282) Application to Testing a Survey. It is self-evident, that when the surveyor has gone completely around a field or farm, taking the bearings and distances of each boundary line, till he has got back to the starting point, that he has gone precisely as far South as North, and as far West as East. But the sum of the North Latitudes tells how far North he has gone, and the sum of the South Latitudes how far South he has gone. Hence these two sums will be equal to each other, if the survey has been correctly made. In like manner, the sums of the East and of the West Departures must also be equal to each other. We will apply this principle to testing the accuracy of the survey of which Fig. 175, page 151, is a plat. Prepare seven columns, and head them as below. Find the Latitude and Departure of each course to the nearest link, and write them in their appropriate columns. Add up these columns. Add up these columns. Then will the difference between the sums of the North and South Latitudes, and between the sums of the East and West Departures, indicate the degree of accuracy of the survey. The entire work of the above example is given below. (26)); for, look in the column of bearings for the slope of the ground, i. e. the angle it makes with the horizon, find the given distance, and the Latitude corre sponding will be the desired horizontal measurement, and the difference between it and the Distance will be the allowance to be made In the preceding example the respective sums were found to be exactly equal. This, however, will rarely occur in an extensive survey. If the difference be great, it indicates some mistake, and the survey must be repeated with greater care; but if the difference be small it indicates, not absolute errors, but only inaccuracies, unavoidable in surveys with the compass, and the survey may be accepted. How great a difference in the sums of the columns may be allowed, as not necessitating a new survey, is a dubious point. Some surveyors would admit a difference of 1 link for every 3 chains in the sum of the courses: others only 1 link for every 10 chains. One writer puts the limit at 5 links for each station; another at 25 links in a survey of 100 acres. But every practical surveyor soon learns how near to an equality his instrument and his skill will enable him to come in ordinary cases, and can therefore establish a standard for himself, by which he can judge whether the difference, in any survey of his own, is probably the result of an error, or only of his customary degree of inaccuracy, two things to be very carefully distinguished.* (283) Application to supplying omissions. Any two omissions in the Field-notes can be supplied by a proper use of the method of Latitudes and Departures; as will be explained in Part VII, which treats of "Obstacles to Measurement," under which head this subject most appropriately belongs. But a knowledge of the fact that any two omissions can be supplied, should not lead # A French writer fixes the allowable difference in chaining at 1-400 of level * lines; 1-200 of lines on moderate slopes; 1-100 of lines on steep slopes. |