parture: proceed 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 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) The corrections of As, 47: 4.28. .02 difference of lat. as &c. &c. in table 1.

rxAB, CC=r × (AB+ BC), DD"'=r × (AB+ BC+ CD), &c. Then through the points A, B', C". D''', &c. draw the corrected boun dary lines ABCDA, which being determined, the area may be found by divid ing 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. respectively.

As, 47:4:: .22 .02) The corrections of As, 474.22: .02 departure as in ta&c. &c. ble 1.

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 + 5.17 = 9.76 = the meridian distance of the fourth point from the first, and 9.76 + 4.08 = 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 11.149.71 = 1.43 = 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, 3.333.33 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 most westerly point of the survey can be found by beginning at any other point.

After the most easterly, or most westerly point of the survey is discovered, call that point the first station, and proceed to find the meridian distances for the several lines in the order in which they were surveyed; that is, the first dep. will be the first meridian distance, which place in the column of meridian distances opposite the said departure; to the same meridian distance add the said departure, to which sum add the next departure, if it be of the same name with the foregoing departure; but subtract, if it be of a different name, which sum or difference call the ext meridian distance, and set it in the column

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 Northings 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

* Demonstration. Let NS be a meridian passing through the most westerly station, from the points B, C, D, E, F, G, and H; let fall the perpendicu lars Bbb, Dd, Ec, Fe Gf and HI, on the meridians NS.

Now, if from the area of the figure dDEFGHI, the area of the figure dDCBAHI be taken, there remains the area of the survey. The area of the multangular figure dDEFGHI is equal to the sum of the areas of the trapezoids of which it is composed, viz. dƊEc, cEFe, eFGf, and fG HI; but (by Prob. 10.) (dD + cE)× de twice the area of the trapezoid dDEc; and ¿D+cE equal to the sum of the meridian distances of the points D

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 will give twice the area, half of which gives the area of the survey.

The most westerly point of the survey being made the first station, and the several meridian distances being calculated, &c. the foregoing table will stand thus:

=

and E from the first meridian line NS, and dc or ag 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, (dD+cE) dc+ (cE+eF)Xce+(eF+ƒG)×ef+(fG+IH)׃1=twice the area of the figure dDEFG HI= the sum of the south area column. And, in like manner, we demonstrate that (dD+bC) × db + bB × bA + AI× IH=twice the area of the figure dDCBAHI≈ the north area column; therefore (dD +cE) Xde+(cE+eF)xce (eF+ƒG) ×ef + (fG + 1H) ׃l—((dD+ bC) Xdb+bB× bÂ÷Â1×1H)=twice the area of 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.

TABLE II.

St. Courss. DC, N.) S. | EW.c.s. c.w N. S. E. W. M.D. N.Area. S. Area

* The meridian distances, in this column, are the sum of two adjacent meridian distances; but at the most westerly point, the meridian distance is nothing, hence the first Dep. is the first meridian distance, and, in like man• ner, the last Dep. is the last meridian distance.

This is not the first station in the actual survey, but only the most westerly point of the survey as calculated by the foregoing method from the field. notes, which, for convenience sake, I call the first station in making out this Table: