with little sticks or stones; then let him measure with his rod, taking care at each remove to place the rod exactly at the spot where its extremity reached before it was taken up. If there be two persons, it is not necessary to mark off the line by sticks or stones. One person must keep some objects in a line by his eye, and direct the other, who measures, to keep in the direction, by word or a signal with the hand. In measuring with a cord of any kind, if there be but one person to perform the operation, he should begin by placing sticks in the direction to be measured, but backwards from his starting point: he must next make a loop at one end of his cord so as to fix it by a stick in the ground; then going forward and stretching his cord; having his back turned to the place where he is going, he looks along the row of sticks (three are sufficient) and stretches his cord in the exact line. He then fixes a stick for a mark, goes back to pull up the stick which fixed the end of his cord, fixes it again at the new mark and repeats the operation as far as may be necessary. If there be two persons, each walks forward with one end of the cord in his hand, the one in advance sticking up a picket at the end of the stretched cord, which the one who follows takes up. The number of pickets mark, of course, the number of times the length of the cord has been gone over. In this mode, it is the business of the hinder person to see that the other stretch the cord in the direct line, which he does by keeping certain advanced objects, natural or set up by him, in his eye. If no sticks are to be had, stones may supply their place, and if there are no stones, the spot where the cord ends, must be marked by scratching the ground, and each should count to check the other.

TO RAISE A PERPENDICULAR FROM ANY POINT ON A GIVEN LINE.

Measure on the given line A B, two equal distances D and E, on either side of the

given point C; fold a cord in two; marking the middle; fasten the two ends of the cord by pickets at D and E. Stretch the cord tight, and from its middle F, draw the line FC, which will be the perpendicular required.

Care must be taken in fastening the two ends of the cord, after the middle has been found, to take up exactly the same quantity of cord, or the two sides of the triangle, when the cord is stretched, will be of unequal length.

For want of a cord, a long stick or perch may be used, Having measured off, as be

fore, equal distances on either side of the point C; let one person hold the perch by one end at D, while another describes an arc with it, then doing the same thing at E, the intersection F, of the two arcs will be the point from which to draw the line to C.

FROM A GIVEN POINT OUT OF THE LINE, TO LET FALL A PERPENDI

CULAR TO IT.

Fold a cord, whose length must be more than double the nearest distance from the

point to the line, into two equal parts; fix the middle at the given point F. Stretch the two halves till they meet the line A B in D and E, find the middle between D and E, as at C, and draw F C.

TO RAISE A PERPENDICULAR AT THE END OF A LINE.

Having marked the middle of your cord, fix one end at A,

and the other end at such a

distance from this, that when D
the cord is stretched at the
middle it may form nearly an
equilateral triangle; fix a stick
at the apex C of the triangle;
detach the end A and move
with it towards D, then keep-
ing the cord touching C,
stretch it straight in a line

B

with C B, and its extremity D will be the point from which to let fall the required perpendicular.

The operation may be performed by sticks. For this purpose lay the stick or perch with one end at A, and in a direction to form an angle with A B of about 60 degrees; then keeping fast the end C, where a picket must be placed, move the perch round this point, as round a pivot, till the other end coincide with the line A B; plant a picket at B, and slide the perch along towards D, keeping it in a line with CB; in such position, one end of the perch touching Q, the other extremity D will be the point for the perpendicular required.

Another and more expeditious method of raising a perpendicular is by means of the numbers 3, 4, and 5, or any multiples of these. Thus measure off and mark upon your cord 12 equal divisions, say feet, fix the two extremities of the cord at the point where a perpendicular is to be raised as at A. Stretch the cord along the given line towards B, set

B

a picket at the 4th division; turn the cord tight outside of this picket, then seizing the cord at the ninth division, stretch it into a triangle, and fix the picket at C, when CA will be perpendicular to A B.

A cross-staff is very useful for setting off perpendiculars; but we have supposed the traveller to be without instruments; he may, however, have the means of supplying the place of a cross-staff by a very simple contrivance, which will answer well enough for certain purposes. Draw two lines perpendicular to each other on a piece of paper, place this on the cover of a book, keeping it there by five pins or needles, one at each extremity of the lines, and one at their intersection; place the book as horizontally as possible on a stone, or raise a little mound of earth for the purpose, and use this as a cross stick; or stick the pins at one of the angles of the book cover and along its two contiguous outer edges. Two flat sticks may be fastened cross-wise, and stuck firm on the top of another stick, having first marked two lines perpendicular to each other along the upper surface of the flat sticks, on which lines set up pins as before. Perpendicular lines are easily obtained by folding a piece of

paper.

TO DRAW A LINE PARALLEL TO A GIVEN LINE.

From the extremes of the given line, or at any convenient distances along it, raise perpendiculars by any of the means already mentioned; take equal distances along these, or a given distance, if required, and the line joining the two points, marking such distances, will be the parallel line required.

TO MAKE AN ANGLE ON THE GROUND EQUAL TO A GIVEN ANGLE.

Set off any number of equal parts from B to C, and from B to A, and with the same parts measure A C; then from

B

any point b, with the length or radius B A describe an indefinite arc d e; from any point c in this arc as a centre describe with the length A C an arc f g intersecting the arc de in a. Join c b and a b and the angle a b c will be equal to the angle ABC.

TO MEASURE AN ANGLE OF A BUILDING, ETC., HAVING NO PROPER · INSTRUMENT.

If from the inside, measure off equal distances along the two walls, from the angle, and place pickets or stones at the points, then measure across from point to point, that is, the cord of the arc subtended by the angle which the walls make. Now draw a line on paper and take from a line of equal parts (a slip of paper frequently doubled will do for a make-shift scale of equal parts) as many as you have measured feet or paces, &c. and mark that distance on your line; from one end describe an arc with this length as a radius, and intersect this arc by another struck from the

other extremity of the base line with a radius corresponding to your measured chord. Draw a line from the first point of the base to the intersection, and you will have on paper an angle similar to that you measured. Apply a protractor and you will have the value of the angle. At all events it will be thus set down, and may be ultimately measured. If it be more convenient to measure the angle from the outside, draw lines in prolongation of the two walls or faces of the building, and operate with these as just described. This method is equally correct with the former, as opposite angles are equal. This problem may be usefully applied by the traveller, when, being without instruments, he would give as correct a plan as possible of the exterior walls of ruined cities, edifices, &c. On the spot it will be sufficient to draw the plan as correctly as possible by the eye, writing the dimensions by the sides of the lines.

TO ASCERTAIN THE LENGTH OF A LINE AS A B, ACCESSIBLE ONLY AT ITS TWO EXTREMITIES.

Choose a point O, accessible to A and B, draw A O, O B,

prolong A O to D, and make O D equal to A O prolong BO, to C, and make O C equal to B O, and C D, will be equal to A B the required fine. The length of a small lake or marsh may be taken in this way.