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 cypher, 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 quarters, 23 perches.

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.

EXAMPLE.

Sides of the land. Bearings. Length in Chains.

Find the difference of latitude and departure answering to each course and distance, by the

traverse table or right angled plane Trigonometry, according to the directions already given, and place them under the succeeding columns N. or South, E. or West, according as they are N. or S. E. or West; then if the survey does not close, correct the errors, by saying,* as the sum of all

*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 Doctor Adrain, has given precisely the same practical rule, in his elegant solution to 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, other. wise 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 ABC"DA? 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 boun dary would become ABCDE. 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 ABCDE". In a similar manner, if by correcting the probable error in the bearing and length of CD (or CD'), the point D" be moved to D", the point " would be moved in an equal and parallel direction to E", and the boundary would become ABC DE Lastly, by correcting the probable error in the bearing and length of the line DE(or D), the true boundary AB CD"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 in specting 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

the distances is to each 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 de

would move the point E in the direction Eb parallel to AB, and an altera tion in the bearing would move it in the direction Eb' perpendicular to 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 CFI, (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, CC, 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 of the whole length. If we therefore suppose GI to be 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 part was made in measuring the length. We shall therefore adopt it as a principle, that the most probable way of apportion ing the error of the survey AE, is to take BB, CC, D"D""', &c, respec tively proportional to the boundary lines AB. BC, CD, &c.

Hence the following practical rule for correcting a survey Geometrically, Draw the boundary lines ABCDE by means of the observed bearings and lengths, and find the error of the survey AE, and let the quotient of AE, divided by the sum of all the lines AB, BC, CD, DE, be represented by r, through the angular points B, C, D. &c. draw the lines BB, CC, &c. parallel to AE, and in the same direction that A bears from E. Take BB'