the young surveyor to be negligent in making every possible measurement, since an omission renders it necessary to assume all the notes taken to be correct, the means of testing them no longer existing. (284) Balancing a Survey. The subsequent applications of this method require the survey to be previously Balanced. This operation consists in correcting the Latitudes and Departures of the courses, so that their sums shall be equal, and thus "balance." This is usually done by distributing the differences of the sums among the courses in proportion to their length; saying, As the sum of the lengths of all the courses Is to the whole difference of the Latitudes, So is the length of each course To the correction of its Latitude. A similar proportion corrects the Departures." It is not often necessary to make the exact proportion, as the correction can usually be made, with sufficient accuracy, by noting how much per chain it should be, and correcting accordingly. In the example given below, the differences have purposely been made considerable. The corrected Latitudes and Departures have been here inserted in four additional columns, but in practice they should be written in red ink over the original Latitudes and Departures, and the latter crossed out with red ink. 29.55 10.00 10.10 10.4110.29 10.06 10.06 10.35 10.35 The corrections are made by the following proportions; the nearest whole numbers being taken : *A demonstration of this principle was given by Dr. Bowditch, in No. 4 of "The Analyst." 12 This rule is not always to be strictly followed. If one line of a survey has been measured over very uneven and rough ground, or if its bearing has been taken with an indistinct sight, while the other lines have been measured over level and clear ground, it is probable that most of the error has occurred on that line, and the correction should be chiefly made on its Latitude and Departure. If a slight change of the bearing of a long course will favor the Balancing, it should be so changed, since the compass is much more subject to error than the chain. So, too, if shortening any doubtful line will favor the Balancing, it should be done, since distances are generally measured too long. (285) Application to Platting. Rule three columns; one for Stations; the next for total Latitudes; and the third for total Departures. Fill the last two columns by beginning at any convenient station (the extreme East or West is best) and adding up (algebraically) the Latitudes of the following stations, noticing that the South Latitudes are subtractive. Do the same for the Departures, observing that the Westerly ones are also subtractive. Taking the example given on page 175, Art. (282), and beginning with Station 1, the following will be the results: It will be seen that the work proves itself, by the total Latitudes and Departures for Station 1, again coming out equal to zero. To use this table, draw a meridian through the point taken for Station 1, as in the figure on the following page. Set off, upward from this, along the meridian, the Latitude, 221 links, to A, and from A, to the right perpendicularly, set off the Departure, 155 links. This gives the point 2. Join 1....2. From 1 again, set * This is most easily done with the aid of a right-angled triangle, sliding one of the sides adjacent to the right angle along the blade of the square, to which the other side will then be perpendicular. The advantages of this method are its rapidity, ease and accuracy; the impossibility of any error in platting any one course affecting the following points; and the certainty of the plat "coming together," if the Latitudes and Departures have been "Balanced." CHAPTER VI. CALCULATING THE CONTENT. (286) Methods. WHEN a field has been platted, by whatever method it may have been surveyed, its content can be obtained from its plat by dividing it up into triangles, and measuring on the plat their bases and perpendiculars; or by any of the other means explained in Part I, Chapter IV. But these are only approximate methods; their degree of accuracy depending on the largeness of scale of the plat, and the skill of the draftsman. The invaluable method of Latitudes and Departures gives another means, perfectly accurate, and not requiring the previous preparation of a plat. It is sometimes called the Rectangular, or the Pennsylvania, or Rittenhouse's, method of calculation.* (287) Definitions. Imagine a Meridian line to pass through the extreme East or West corner of a field. According to the definitions established in Chapter V, Art. (278), (and here recapitulated for convenience of reference), the perpendicular distance of each Station from that Meridian, is the Longitude of that Station; additive, or plus, if East; subtractive, or minus, if West. The distance of the middle of any line, such as a side of the field, from the Meridian, is called the Longitude of that side.† The difference of the Longitudes of the two ends of a line is called the Departure of that line. The difference of the Latitudes of the two ends of a line is called the Latitude of the line. *It is, however, substantially the same as Mr. Thomas Burgh's "Method to determine the areas of right lined figures universally," published nearly a century ago. The phrase "Meridian Distance," is generally used for what is here called "Longitude"; but the analogy of" Differences of Longitude" with "Differences of Latitude," usually but anomalously united with the word "Departure," borrowed from Navigation, seems to put beyond all question the propriety of the innovation here introduced. (288) Longitudes. To give more definiteness to the develop ment of this subject, the figure in the margin will be referred to, and may be considered to represent any space enclosed by straight lines. Let NS be the Meridian passing through the extreme Westerly Station of the field ABCDE. From the middle and ends of each side N Fig. 193. C draw perpendiculars to the Meridi an. These perpendiculars will be M K A W Y U (X OD the Longitudes and Departures of the respective sides. The Longitude, FG, of the first course, AB, is evidently equal to half its Depar- F ture HB. The Longitude, JK, of the second course, BC, is equal to JL+LM+MK, or equal to the Longitude of the preceding course, plus half its Departure, plus half the Departure of the course itself. The Longitude, YZ, of some other course, as EA, taken anywhere, is equal to WX-- VX-UV, or equal to the Longitude of the preceding course, minus half its Departure, minus half the Departure of the course itself, i. e. equal to the Algebraic sum of these three parts, remembering that Westerly Departures are negative, and therefore to be subtracted when the directions are to make an Algebraic addition. $ E To avoid fractions, it will be better to double each of the preceding expressions. We shall then have a GENERAL RULE FOR FINDING DOUBLE LONGITUDES. The Double Longitude of the FIRST COURSE is equal to its Departure. The Double Longitude of the SECOND COURSE is equal to the Double Longitude of the first course, plus the Departure of that course, plus the Departure of the second course. The Double Longitude of the THIRD COURSE is equal to the Double Longitude of the second course, plus the Departure of that course, plus the Departure of the course itself. |