A Specimen of the Pennsylvania Method of CALCULATION; which for its simplicity and ease in finding the Meridian Distances is supposed to be preferable in practice to any thing heretofore published on the subject.

Find, in the first place, by the Traverse Table, the lat. and dep. for the several courses and distances, as already taught; and if the survey be truly taken, the sums of the northings and southings will be equal, and also those of the eastings and westings. Then, in the next place, find the meridian distances, by choosing such a place in the column of eastings or westings as will admit of a continual addition of one, and subtraction of the other; by which means we avoid the inconvenience of changing the denomination of either of the departures.

The learner must not expect that in real practice the columns of lat. and those of dep. will exactly balance when they are at first added up, for little inaccuracies will arise, both from the observations taken in the field and in chaining; which to adjust, previous to finding the meridian distances, we may observe, that if in small surveys the difference amount to two-tenths of a perch for every station, there must have been some error committed in the field; and the best way in this case will be, to rectify it on the ground by a resurvey, or at least as much as will discover the error. But when the differences are not within those limits, the columns of northing, southing, easting, and westing may be corrected as follows:

Add all the distances into one sum, and say, as that sum is to each particular distance, so is the difference between the sums of the columns of northing and southing to the correction of northing or southing belonging to that distance: the corrections thus found are respectively additive when they belong to the column of northing or southing which is the less of the two, and subtractive when they belong to the greater; if the course be due east or west, the correction is always additive to the less of the two columns of northing or southing. The corrections of easting and westing are found exactly in the same

manner.

The following example will sufficiently illustrate the manner of applying the rule.

also the first distance is 70; say then, ference between the columns of northing and southing is .4, In this example, the sum of the distances is 791, and the dif

791 70.4 : .04,

subtracted. to the southing 53.6, from which the correction .04 should be which fourth proportional .04 is the first correction belonging

But as only two of these corrections amount to half a tenth, we must use .1 for each of the corrections .09 and .07, and neglect the correction .04; thus the correct southings become 53.6, 29.0, 135.6.

In like manner from the remaining distances we obtain to

And consequently, by neglecting .04 and .03, and using .1 for each of the two .06 and .07, the northings when corrected are 62.9, 101.2, 54.0, 00.1.

In obtaining these corrections, it is commonly unnecessary to use all the significant figures of the distances: thus, for the ratio of 791 to 70, we may say, as 80 to 7.

The latitudes and departures being thus balanced, proceed to insert the meridian distances by the above method, where we still make use of the same field-notes, only changing chains and links into perches and tenths of a perch. Then by looking along the column of departure, it is easy to observe, that in the columns of eastings opposite station 9 all the eastings may be added, and the westings subtracted, without altering the denomination of either. Therefore, by placing 46.0, the east departure belonging to this station, in the column of meridian distances, and proceeding to add the eastings and subtract the westings, according to the rule already mentioned, we shall find that at station 8 these distances will end in 0, 0, or a cipher, if the additions and subtractions be rightly made. Then multiplying the upper meridian distance of each station by its respective northing or southing, the product will give the north or south area, as in the examples already insisted on, and which is fully exemplified in the annexed specimen. When these products are all made out and placed in their respective columns, their difference will give double the area of the plot, or twice the number of acres contained in the survey. Divide this remainder by 2, and the quotient thence arising by 160 (the number of perches in an acre), then will this last quotient exhibit the number of acres and perches contained in the whole survey; which in this example may be called 110 acres, 103 perches, or 110 acres, 2 roods, 23 perches.

FIELD-NOTES of the two foregoing methods, as practised in Pennsylvania.

Cast up by perches and tenths of a perch.

N. Courses.

Dist. N. S. E. W. M. D. N. Area. S. Area.

Note.-In the foregoing methods the first meridian passes through the map; but as it is more convenient to have it pass through the extreme east or west point of the same, I have given the following example to illustrate this method.

Of computing the area of a survey by having the bearings and distances given, geometrically considered and demonstrated.

Let BCDEFGHA, pl. 14, fig. 11, represent the boundary of a survey of which the following field-notes are given; it is required to find the area.

Find the difference of latitude and departure answering to each course and distance by the Traverse Table or rightangled plane trigonometry, according to the directions already given, and place them under the succeeding columns North or South, East or West, according as they are north or south, east or west; then if the survey does not close, correct the errors by saying,* as the sum of all the distances is to each

*This arithmetical rule was given by Mr. Bowditch in his solution of Mr. Patterson's question of correcting a survey in No. of the Analyst. Also, the editor, Dr. Adrain, has given precisely the same practical rule,