also the first distance is 70; say then,

791 70.4: .04,

ference between the columns of northing and southing is .4, In this example, the sum of the distances is 791, and the dif

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

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

Cast up by perches and tenths of a perch.

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. 4 of the Analyst. Also, the editor, Dr. Adrain, has given precisely the same practical rule,

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 B'C'), 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 ABC"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 AB'C' D'''Ê"". 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 BB'CC"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 altera