w particular distance, so is the whole error in departure to the . correction of the corresponding departure, each correction being so applied as to diminish the whole error in departure: pro in his elegant solution of the said question, analytically demonstrated. As the demonstration of this important rule may give great satisfaction to those who have not an opportunity of seeing the Analyst, I have inserted Mr. Bowditch's demonstration of said rule, which is as follows, viz. Demonstration 1. That the error ought to be apportioned among all the bearings and distances. 2. That in those lines in which an alteration of the measured distance would tend considerably to correct the error of the survey, a correction ought to be made; but when such an alteration would not have that tendency, the length of the line ought to remain unaltered. 3. In the same manner, an alteration ought to be made in the observed bearings, if it would tend considerably to correct the error of the survey, otherwise not. 4. In cases where alterations in the bearings and distances will both tend to correct the error it will be proper to alter them both, making greater or less alterations according to the greater or less efficacy in correcting the error of the survey. 5. The alterations made in the observed bearing and length of any one of the boundary lines ought to be such that the combined effect of such alterations may tend wholly to correct the error of the survey. Suppose now that ABCDE (pl. 14, fig. 12) represent the boundary lines of a field, as plotted from the observed bearings and lengths, and that the last point E, instead of falling on the first A, is distant from it by the length AE. The question will then be, what alterations BB, CC", DD”, &c. must be made in the positions of the points B, C, D, &c. so as to obtain the most probable boundaries AB"C"D"A " If AB' be supposed to be the most probable bearing and length of the first boundary line, the point B would be moved through the line BB, and the following points C, D, E would in consequence thereof be moved in equal and parallel directions to C, D, E, and the boundary would become AB'C'D'E'. Again, if by correcting in the most probable manner the error in the observed bearing and length of BC (or BC), the point C be moved to C", the points D' and E' would be moved in equal and parallel directions to D" and E", and the boundary line would become AB"C"D"E". In a similar manner, if by correcting the probable error in the bearing and length of CD (or C"D") the point D" be moved to D", the point E" would be moved in an equal and parallel direction to E", and the boundary would become ABC"D"E”. Lastly, by correcting the probable error in the bearing and length of the line DE (or D"E") the true boundary AB"C"D"A would be obtained. If we suppose the lines BBCC"DD", &c. to be parallel to AE, it would satisfy the second, third, fourth, and fifth of the preceding principles. For the change of position of the points B, C, &c. being in directions parallel to AE, the whole tendency of such change would be to move the point E directly towards A, conformably to the fifth principle; and by inspecting the figure, it will appear that the second, third, and fourth principles would also be satisfied. For, in the first place, it appears that the bearing of the first line AB would be altered considerably, but the length but little. This is agreeable to those principles, because an increase of the distance AB would move the point E in the direction Eb parallel to AB, and an alteration in the bearing would move it in the direction Eb' perpendicular to ceed the same way for the corrections in latitudes. These corrections being applied to their corresponding differences of latitude and departure, that is, add when of the same name and AB. Now the former change would not tend effectually to decrease the distance AE, but the latter would be almost wholly exerted in producing that effect. Again, the length of the line BC would be considerably changed without altering essentially the bearing ; the former alteration would tend greatly to decrease the distance AE, but the latter would not produce so sensible an effect. Similar remarks may be made on the changes in the other bearings and distances, but it does not appear to be necessary to enter more largely on this subject. It remains now to determine the proportion of the lines BB, CC", DD", &c. To do this we shall observe, that in measuring the lengths of any lines the errors would probably be in proportion to their lengths. These supposed errors must, however, be decreased on those lines where the effect in correcting the error of the survey would be small, by the second and fourth principles. , In observing the bearings of all the boundary lines equal errors are liable to be committed; however, it will be proper, by the third and fourth principles to suppose the error greater or less in proportion to the greater or less effect it would produce in correcting the error of the survey. Now the error of an observed bearing being given, as for example GFI (pl. 14, fig. 13), the change of position GI of the end of the line G would be proportional to the length of the line FG (= FI), so that the supposed errors both in the length and in the bearing of any boundary line would produce changes in the position of the end of it proportional to its length. There appears, therefore, a considerable degree of probability in supposing the lines BB, C"C", D'D'", &c. to be respectively proportional to the lengths of the boundary lines AB, BC, CD, &c. The main point to be ascertained before adopting this hypothesis is, whether a due proportion of the error of the survey is thrown on the bearings and lengths of the sides. Now it is plain by this hypothesis that the error in any boundary line is supposed to be wholly in the bearing if the line be perpendicular to AE, and wholly in its length when parallel to AE; and if the length be the same in both cases, the change of position of the end of the line would in both cases be exactly equal. Thus, if FGH be the boundary line, GI the change of position of the point B in the former case, and GH in the latter, we should in this hypothesis have GI-GH. To show the probability of this hypothesis it may be observed, that in measuring the lengths of a line FGH of six or eight chains of fifty links each, an error of one link might easily be committed by the stretching of the chain or the unevenness of the surface. This error would be about so of the whole length. If we, therefore, suppose GI to be so of FG, the angle GFI would be about 10'. Now, with such instruments as are generally made use of by surveyors, it is about as probable that an error of 10 was made in the bearing as that the above error, so part, was made in measuring the length. We shall therefore adopt it as a principle, that the most probable way of apportioning the error of the survey A E is to take BB, C'C', D"D", &c. respectively proportional to the boundary lines AB, BC, CD, &c. Hence the following practical rule for correcting a survey geometrically. Draw the boundary lines ABCDE F. means of the observed bearings and subtract when of different names, then the corrected difference of latitude and departure will be obtained, and the table will stand thus: The errors being corrected thus As 47 : 4 : : .28 : .02 {": corrections of difference of lat. A. #. : . . : #!"; corrections of departure as in The latitudes and departures being thus balanced, it is necessary to calculate the several meridian distances, in order to compute the area of the survey. As beginning at the most easterly or most westerly point of the survey admits of a continual addition of the one and subtraction of the other, the most easterly or most westerly point can be easily discovered from the foregoing table, thus: The first departure corrected is 3.98, which is the meridian distance of the second point of the survey from the first, to which add 0.61 the next dep. corrected, and their sum is 4.59, the meridian distance of the third points of the survey from the first ; and in like manner 4.59–H5.17 =9.76= the meridian distance of the fourth point from the first, and 9.76–H4.08=13.84 = the meridian distance east of the fifth point from the first; after the same manner, continue to add the dep, when east, but subtract when west: the next dep, is west, therefore 13.84 —2.70=11.14 = the meridian distance of the sixth point from the first, and 11.14 — 9.71 =.43= the next. Now the next departure is 4.76, which is west, and 1.43 is the meridian distance of the seventh point from the first, which is east; therefore 4.76 — 1.43=3.33= the meridian distance of the eighth point from the first; as 3.33 is the greatest meridian distance west of the eighth point of the survey from the first, because the next departure is east 3.33; then, 3.33—3.33=0, which closes the survey: consequently, the eighth point of the survey is the most westerly point, and for the same reason as 13.84 is the greatest meridian distance east, which is the meridian distance of the fifth point of the survey. In like manner, the most easterly or w Then through the points A, B, C", D", &c. draw the corrected boundary lines ABCDA, which being determined, the area may be found by dividing the figure into triangles in the usual method. The proportional parts BB, CC", &c. may be found expeditiously by means of a table of difference of latitude and departure, by finding the page where the sum of the lines AB+BC+CD+DE in the distance column corresponds to AE in the departure or difference of latitude column; then find AB, AB+BC, &c. in the distance column, and the corresponding numbers will be equal to BB, CC", DD", &c. respect ively. most westerly point of the survey can be found by beginning at ence call the next meridian distance, and set it in the column of meridian distances opposite the departure last used; and in like manner, continue to add the departure twice when of the same name, but if of a different name subtract twice, and the last meridian distance will be zero, if the additions and subtractions are rightly performed; because the sum of the north ings is equal to the sum of the southings after the survey is corrected, which is evident from theo. 1, and the foregoing table. Then," multiplying the upper meridian distance of each station by the corresponding northing or southing, and place the product in the north or south area, according as the latitude is north or south, the difference of the sum of these products * Demonstration. Let NS be a meridian passing through the most westerly station from the points B, C, D, E, F, G, and H5 let fall the perpendiculars Bb, Cb, Dil, Ec, Fe, Gs, and HI, on the meridian NS. Now, if from the area of the figure dBEFGHI the area of the figure dDCBAHI be taken, there remains the area of the survey. The area of the multangular figure dLEFGHI is equal to the sum of the areas of the trapezoids of which it is composed, viz. dBEc, cEFe, eR'Gf, and fg|HI; but (by prob. 10), (d.D+cE):Kdc= twice the area of the trapezoid dBEc;s and dB+cE equal to the sum of the meridian distances of the points D and E from the first meridian line NS, and dc or dg = the southing of the point E from the point D. In like manner the area of every other trapezoid is found; but these are the south column areas: that is, (d.D+cB)x de-HeB+es')2 ce+(eF-Hfg)xef-H(f6+IH)xf1= twice the area of the figure dBEFGHI = the sum of the south area column. And, in like manner, we demonstrate that (d.D+bC)xdb+bBxb4+AIXIH= twice oi. of * figure dBCBAHI = the north area column; therefore, (dB+cE)xdc+(cE+eF)xce–H (eF+fG)xe G+IH)×f I— [(d.D +bC)xdb+bBxb4+AIx IH] = twice the ..". the survey; consequently, the sum of the south area column — the sum of the north area column = twice the area of the survey. Q. E. D. |