And so on, this proof may be carried through every line of the polygon, until we come to the last line JA, when its Bearing added to the interior angle JAB + or 180° as the case may require will give the original starting Bearing of the line AB.

We have been thus prolix in explaining how the Bearings of the above four lines of the polygon are obtained, as they contain cases in each quadrant of the circle, BC, being a SouthEast Bearing, CD, North-East, DE, North-West, and EF, South-West; the same rule is however applicable to the remaining sides.*

* If the sum of the preceding Bearing and forward angle after deducting 180° amounts to more than 360°, deduct 360° from the total, the remainder will be the Bearing of the next line.

CHAPTER VII.

ON THE PROOF OF THE TRAVERSE, AMOUNT OF ERROR ALLOWED, AND METHOD OF CORRECTION.

We now offer for consideration the following Theorem, viz. : That in every Survey, correctly taken, the sum of the distances gone North from a certain point, will be equal to the sum of the distances returned South to the same point, and that the sum of the distances gone East, will be equal to the sum of the distances returned West.

The truth of the above is self-evident, for the meridians. within the limits of an ordinary Survey having no sensible difference from parallelism, it must necessarily follow, that if a person travel any way soever within such limits, and at length come round to the place where he set out, he must have travelled as far to the North as to the South, and to the East as to the West, though the practical Surveyor will always find it difficult to make his work close with this perfect degree of exactness.

We will however explain this more fully with the assistance of a diagram.

W

i

F

m

Let the line NS run due North and South, and EW due East and West. If we fix on the point A, as a starting point and a person walks from A to b, on the line NS, say 400 yards, and wishes to return to A, he must walk back 400 yards; in going therefore from A to b, and back from 6 to A, he has walked 400 yards North and returned 400 yards South. In the same manner if he fixes on the point b as a starting point, and walks to B on the line EW say 300 yards to get back to b, he must return 300 yards; in walking therefore to B, he has gone 300 yards East, and returned 300 yards West.

Supposing now, he walks from A to B, say 500 yards in the direction of the line AB, he will then have gone North from A, 400 yards, and East from A, 300 yards.

In a continuation of the figure, having walked or measured from A to B, he proceeds on and measures from B to C, in doing so, he goes a certain distance South and a certain distance

East of B to arrive at C, thence he measures to D, going a certain distance North and East of C to arrive at D, from D he measures to E, from E to F, and so on, going North, South, East or West, from the preceding station as the direction of the line may be, until he arrives back at his original starting point, A. In making this tour therefore, he has gone the same distance North as he has returned South, and the same distance East as he has returned West.

Let the vertical and horizontal lines drawn through the several stations A, B, C, &c., represent, the former, a series of meridians or North and South lines or lines of longitude and the latter, a series of East and West lines or lines of latitude; as these lines of latitude and longitude are all respectively perpendicular and parallel to each other, it follows that the angle formed by the intersection of the meridian line of one station, and the latitudinal line of the next station as at b, k, l, m, &c., must be a right angle or 90°.

Now supposing all the lines AB, BC, &c., to have been carefully measured with a Chain, and that having obtained the Bearing of the line AB, by astronomical observation, we have deduced the Bearings of all the other lines by the rule, (page 277); we then have the data in each line, of a side and two angles to find the other two sides.

For instance, in the triangle AbB, we have the side AB, and the two angles bAB, AʊB, (the latter being invariably 90° or a right angle) to find the other two sides Ab and bB the former being the difference of latitude, and the latter the departure of the station B from A. In like manner, in the triangle B/C, we have the side BC, and the two angles CBk, (obtained by deducting NBC from 180°) and BkC (a right angle) to find the other two sides Bk and Ck, the former being again the difference of latitude and the latter the departure of the station C from B, and so on for every line round the figure.

The object of calculating all the sides of these several rightangled triangles on each line, is to obtain the difference of

latitude and departure of each station from the preceding one, which difference being found, the sums of all differences of latitude of lines going North must equal the sums of all differences of latitude of lines going South, or

Ab + Cl + Dm + Gp + Js = Bk + En + Fy + Hq + Ir and the sums of all differences of departure of lines going East, must equal the sums of all differences of departure of lines going West, or

bB+kC + D + rJ + sA = mE + nF + yG + pH + qI and if this is not the result of the above calculations, the Survey has not been truly taken.

We have before stated, that in the measurement of angles, a certain correction is allowed in practice, to obtain the result of the Theorem, which forms the basis of the work, so also in the measurement of Chain lines, a correction is necessary to meet the errors, that notwithstanding the greatest care will occur. In actual practice the columns of latitude and departure will not balance exactly, for inaccuracies must arise from observations and chaining in the field, which no care could obviate. To adjust these differences, previous to defining the meridian distances, the rule is, that should the discrepancy amount to

of a Pole or 5 Links for every station, it will be clear an error has been made in the field measurements, which must

be discovered by a re-survey. When differences, however, are within these limits, the amount of error allowed is 1 Link in 10 Chains, additive or subtractive from the sums of the Northings and Southings to correct the latitude, and from the sums of the Eastings and Westings to correct the departure. This error must be apportioned among each of the distances of the Survey by the following proportions, viz. :

As the sum of all the distances is to the whole error, so is each distance to its correction.

This must be done for the latitudes and also for the departures, and is entered in a column appropriated to each, called the North and South correction, and the East and West correc