When the paper is filled, put on a new sheet, and begin by fixing on it two points, such as C and D, which were on the former sheet, and from them proceed as before. The sheets can then be afterwards united, so that all the points on both shall be in their true relative positions.

(454) Method of Intersection. This is the most usual and the most rapid method of using the Plane-table. The principle was referred to in Articles (259) and (392). Set up the instrument at any convenient point, as X in the figure, and sight to all Fig. 309. B

the desired points A, B, C, &c., which are visible, and draw indefinite lines in their directions. Measure any line XY, Y being one of the points sighted to, and set off this line on the paper to any scale. Set up at Y, and turn the table till the line XY on the paper lies in the direction of XY, on the ground, as at C in the last method. Sight to all the former points and draw lines in their directions, and the intersections of the two lines of sight to each point will determine them, by the Fourth Method, Art. (8). Points on the other side of the line XY could be determined at the same time. In surveying a field, one side of it may be taken for the base XY. Very acute or obtuse intersections should be avoided. 30° and 150° should be the extreme limits. The impossibility of always doing this, renders this method often deficient in precision. When the paper is filled, put on a new sheet, by fixing on it two known points, as in the preceding method.

(455) Method of Resection. This method (called by the French Recoupement) is a modification of the preceding method of Inter

section. It requires the measurement of only one distance, but all
the points must be accessible.
Lay it off on the paper as ab.
till the line ba on the paper coincides with BA on the ground, as
in the Method of Progression. Then sight to C, and draw an inde-
finite line by the ruler. Set up at C, and turn the line last drawn
so as to point to B. Fix a needle at a on the table, place the
alidade against the needle and turn it till it sights to A. Then the
point in which the edge of the ruler cuts the line drawn from B
will be the point c on the table. Next sight to D, and draw an
indefinite line. Set up at D, and make the line last drawn point
to C. Then fix the needle at a or b, and by the alidade, as at the
last station, get a new line back from either of them, to cut the last
drawn line at a point which will be d. So proceed as far as de-
sired.

Let AB be the measured distance.
Set the table up at B, and turn it

(456) To orient the table.* The operation of orientation consists in placing the table at any point so that its lines shall have the same directions as when it was at previous stations in the same survey.

The French phrase, To orient one's self, meaning to determine one's position, asually with respect to the four quarters of the heavens, of which the Òrient is the leading one, well deserves naturalization in our language.

With a compass, this is very easily effected by turning the table till the needle of the attached compass, or that of the Declinator, placed in a fixed position, points to the same degree as when at the previous station.

Without a compass the table is oriented, when set at one end of a line previously determined, by sighting back on this line, as at C in the Method of Progression, Art. (453).

To orient the table, when at a station unconnected with others, is more difficult. It may be

Fig. 311.

S

effected thus. Let ab on the table represent a line AB on the ground. Set up at A, make ab coincide with AB, and draw a line from a directed towards a steeple, or other conspicuous object, as S. Do the same at B. Draw a line cd, parallel to ab, and intercepted between aS, and bS. Divide ab and cd into the same number of equal parts. The table is then prepared. Now let there be a station, P, p on the table, at which the table is to be oriented. Set the table, so that p is over P, apply the edge of the ruler to p, and turn it till this edge cuts cd in the division corresponding to that in which it cuts ab. Then turn the table till the sights point to S, and the table will be oriented.

(457) To find one's place on the ground. This problem may be otherwise expressed as Interpolating a point in a plat. It is most easily performed by reversing the Method of Intersection. Set up the table over the station,

O in the figure, whose place on the plat already on the table is desired, and orient it, by one of the means described in the last article. Make the edge of the ruler pass through some point, a on the table, and turn it till the sights point to the corresponding

Fig. 312.

B

station, A on the ground. Draw a line by the ruler. The desired

Make the ruler pass through

point is somewhere in this line. another point, b on the table, and make the sights point to B on the ground Draw a second line, and its intersection with the first will be the point desired. Using C in the same way would give a third line to prove the work. This operation may be used as a new method of surveying with the plane-table, since any number of points can have their places fixed in the same manner.

This problem may also be executed on the principle of Trilinear Surveying. Three points being given on the table, lay on it a piece of transparent paper, fix a needle any where on this, and with the alidade sight and draw lines towards each of these three points on the ground. Then use this paper to find the desired point, precisely as directed in the last sentence of Art. (398), page 277.

(458) Inaccessible distances. Many of the problems in Part VII. can be at once solved on the ground by the plane-table, since it is at the same time a Goniometer and a Protractor. Thus, the Problem of Art. (435) may be solved as follows, on the principle of the construction in the last paragraph of that article. Set the table at C. Mark on it a point, c', to represent C, placing c vertically over C. Sight to A, B and D, and draw corresponding lines from c'. Set up at D, mark any point on the line drawn from c' towards D, and call it d'. Let d' be exactly over D, and direct d'e' toward C. Ther. sight to A and B, and draw corresponding lines, and their intersections with the lines before drawn towards A and B will fix points a' and b'. Then on the line joining a and b, given on the paper to represent A and B, ab being equal to AB on any scale, construct a figure, abcd, similar to a'b'c'd', and the line cd thus determined will represent CD on the same scale as AB.

PART IX.

SURVEYING WITHOUT INSTRUMENTS.

(459) THE Principles which were established in Part I, and subsequently applied to surveying with various instruments, may also be employed, with tolerable correctness, for determining and representing the relative positions of larger or smaller portions of the earth's surface without any Instruments but such as can be extemporized.

The prominent objects on the ground, such as houses, trees, the summits of hills, the bends of rivers, the crossings of roads, &c., are regarded as "points" to be "determined." Distances and angles are consequently required. Approximate methods of obtaining these will therefore be first given.

(460) Distances by pacing. Quite an accurate measurement of a line of ground may be made by walking over it at a uniform pace, and counting the steps taken. But the art of walking in a straight line must first be acquired. To do this, fix the eye on two objects in the desired line, such as two trees, or bushes, or stones, or tufts of grass. Walk forward, keeping the nearest of these objects steadily covering the other. Before getting up to the nearest object, choose a new one in line farther ahead, and then proceed as before, and so on. It is better not to attempt to make each of the paces three feet, but to take steps of the natural length, and to ascertain the value of each by walking over a known distance, and dividing it by the number of paces required to traverse it. Every person should thus determine the usual length of his own steps, repeating the experiment sufficiently often. The French Geographical Engineers" accustom themselves to take regular