E, CF, &c., should be measured to the hundredth part of a foot.

It has not been thought proper to introduce methods of measuring angles with the instruments which have been described, as that has been pretty fully done in the works to which I have already referred. My object in the present performance is to supply what has been omitted. In doing this, I have introduced but little to be found in the authors alluded to. It may not be improper to introduce here the method of verifying the correctness of an angle by the principle of repetition. Place the transit over the angular point B (see the preceding figure,) after having adjusted it for observation, set the vernier to zero, and bring the telescope to bear on the staff at A in one of the lines, including the angle to be measured; clamp the lower plate to the tripod, unclamp the upper plate, and bring the telescope to bear on C a staff in the other line comprehending the required angle; the vernier will point out the measure of the angle A B C. In this position clamp the plates together, unclamp the lower plate from the tripod, and bring

the telescope to bear on A, by revolving the instrument bodily on its axis. Now clamp the lower plate; unclamp the upper plate, and bring the telescope to bear on C; the vernier will show twice the angle A B C. We may repeat this operation at pleasure, the last reading of the vernier being divided by the number of times the angle had been measured, will give a mean result more to be relied on than any single observation, however carefully made.

It is best to repeat the measure of an angle, that the vernier may pass over the entire circumference of the graduated plate, in which case 360° must be added to the last reading of the vernier, previous to dividing by the number of observations.

PROPOSITION 4.

Random Lines.

To run a right line between two given points, when several intermediate stations have to be taken on account of intervening obstacles.

Case 1. When the line can be run from the place of beginning:

C

n

2

m

3

મ

4

m

10

7

B

Having adjusted the instrument at A, the point from which the line is to be run, and directed the sights towards B, as nearly as can be guessed at, this be

ing the point to which the line is to be run from A, direct stakes to be set at

every 20 perches (or otherwise, as may

suit the nature of ground,) in the di

rection of the sights, which let be designated in the order in which they are

3

placed by n1, n3, n3, n, which being continued until we arrive at C, so that

C B may be perpendicular to A C. The line A C in the direction of the sights, must be carefully run by Proposition 2d, Case 1st, Measure C B. Let n1 m1, m3 n3, n3 m3, &c., be drawn parallel to C B. The triangles A B C, A m1 n1, A m3 n2, &c., being similar we have,

AC: An1:: BC: n1 m1

AC: A no: CB: n2 m2, &c.

AC:AF::CB:FE.

That is, as the whole distance measured from A to C, is to any part of the line measured A n1, or A F; so is C B the distance from the termi

mination of the random line, or line run from the point at which it should have terminated, to the distance from (the line measured,) the points, m1 m3, E, &c., in the true line A B.

Having found n1 m2, we have n3 m3, equal to twice n1 m1; also, n3 m3, equal to three times n1m2, &c; for as the stakes are equidistant from each other, the several offsets are some multiple of the first.

1. As an example, suppose it is required to run a line between two farms having a stone at which to commence, and also another to which to run. Having run the random line as directed, and leaving stakes at every 20 perches distance from the beginning of the line, A C was found to be 160 perches, and B C 2.4 perches; it is required to find the distances to be set off from the random line at every stake left in the line, as also the distance to be set off at 96 perches from the place of beginning, As A C: An1 :: B C : n1 m1

As A C: FA:: CB: FE

160: 96 :: 2.4: 1.44 offset at 96 per. These distances being severally laid off from the random line, to the right or left, according as B is to the right or left of C, will give points in the line A B as required.

2. Being requested to run a line between two neighbors, I ran a random line 146 perches, and found I had missed the true corner by 1.23 perches. What perpendicular offsets must be laid off from this random line, that stakes may be set at every 10 perches from the beginning of the line.

Reckoning from the beginning the several distances will be .08; .17; .25; .34; .42; .50; .58; .67; .76; .84; .93; 1.01; 1.09 and 1.18 perches. In general tenths of a foot will be sufficiently exact.

3. Required the perpendicular distances to be set off from a random line 647 perches in length, terminating at the perpendicular distance of 47.3 feet from the point at which the line in question must end, the stakes in the random line being set at every 40 perches.

Result, 1st dist. 2.9; 2d, 5.8; 3d, 8.8; 4th, 11.7; 5th, 14.6; 6th, 17.5; 7th, 20.5; 8th, 23.4; 9th,