the plan or plot is said to be made upon a scale of twenty feet to the inch and similarly, whatever be the relation of the numbers that represent corresponding parts.

OF THE DIAGonal Scale.

This scale is thus constructed. Take a line, AD, (Pl. 5, Fig. 3,) equal in length to 4, 1, 3, or 1 inch, or of any other convenient dimension, and describe on it the square ADCB. Divide the sides AD, AB, each into ten equal parts, join A and a, and draw the other nine parallels as in the figure. Produce DA, and lay off the distance AD any convenient number of times, from A to F, from F to G, from G to E, &c., and number the points of division 1, 2, 3, &c. Then divide the line DC into ten equal parts, and through the points of division draw parallels to ED, as in the figure. Now, the small divisions on the line AD are each one tenth (.1) of that line, and therefore .1 of AF, or FG. It follows from the proportion

of similar triangles, that the part of the line .01, intercepted between the lines AB and Aa, is equal to .1 of Ba; the like intercepted part of the line .02, equal to .2 of Ba, the like intercepted part of .03, equal to .3 of Ba, and similarly for the other parallels.

These latter spaces, being tenths of the divisions A1, 12, &c. which are themselves tenths of AD, are hundredths of the line AB or AD.

Naming the line AD the unit of the scale, if it were required to take in the dividers the value of the unit and any number of tenths, let the dividers be placed at F, and extend to the figure between A and D, which designates the tenths. If two or more units were required, the dividers must be placed at that point on the line AE, which is designated by a number equal to the number of units. If units, tenths, and hundredths, are wanted, place the dividers at the intersection of the proper line of units with the line designating the hundredths, and then extend them to the intersection of the line designating the tenths with this line of hundredths: thus, for the number 2.44 the dividers are placed at b, and extended to

If the line AD, instead of being regarded as a unit, be taken to represent ten, then, each of the divisions A1, 12, &c. will represent one, and the parts which were hundredths before, will now be tenths. If the line AD be taken to represent one hundred, the spaces A1, 12, 23, &c. will represent tens, and the tenths in the last case, units.

Diagonal scales are generally cut on two of the brass plates which border the plain table.

OF THE SECTORAL SCALE OF EQUAL PARTS.

134. The sector is an instrument generally made of ivory or brass. It consists of two arms, or sides, which open by turning round a joint at their common extremity. There are several scales laid down on the sector; those, however, which are chiefly used in plotting, are the scale of chords already described, and the scale of equal parts now to be explained.

On each arm of the sector, there is a diagonal line that passes through the point about which the arms turn; these lines are divided into equal parts. On the sectors which belong to the cases of English instruments, the lines are designated by the letter L, and numbered, from the centre of the sector, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, to the other extremity. On the sectors which belong to cases of French instruments, they are designated, "Les parties egales," and numbered, 10, 20, 30, &c. to 200. In the English sectors there are 20 divisions between the lines numbered 1, 2, 3, &c., so that there are 200 on the scale.

The advantage of the sectoral scale of equal parts, is this. When it is proposed to make a plan, of any given number of parts to the inch, or to the part of an inch, take the inch or part of the inch from the scale of inches on the sector; then open the sector, and place one foot of the dividers at the point designated by the number, and extend the sector till the other foot reaches to the corresponding number on the other arm; then lay the sector on the table without varying the angle. Now, regarding the lines on the sector as the sides of a triangle, of which the line measured across is the base, it is

sector, the angle of the sector remaining the same, the bases of the triangles so formed will be proportional to their sides: hence, the distances which are to represent lines on the plan, are to be measured across the sector, and from the numbers which represent the true lengths of the lines.

If a line be so long that the whole of it cannot be taken from the scale, it may be divided, and a part of it taken at a time. If a line be given on the paper, and it is required to ascertain the length of the line on the ground, to which it corresponds, we have only to take the line in the dividers, apply it to the scale, and see to how many parts it is equal. On the edges of the French sectors are scales of inches in English and French measure.

Example.—If a line of sixty-seven feet were to be plotted to a scale of twenty feet to the inch, take one inch from the scale of inches; then place one foot of the dividers at the twentieth division, and open the sector until the dividers will just reach the twentieth division on the other arm; the sector is then set to the proper angle: the required distance to be laid down on the paper, is found, by extending the dividers from the sixty-seventh division on one arm, to the sixty-seventh division on the other.

OF GUNTER'S SCALE.

135. This is a scale of two feet in length, on the faces of which a variety of scales are marked. The face on which the divisions of inches are made, contains, however, all the scales necessary for plotting. They are, the diagonal scale of equal parts, and the scale of chords, already explained.

CHAPTER V.

OF SURVEYING WITH THE COMPASS.

136. THE line about which the earth revolves, is called its axis.

137. Every plane passing through the axis, intersects the surface of the earth in a line, which is called a meridian.

138. Every point of the surface has a meridian line passing through it; since, through such point and the axis a plane may always be passed.

139. All these meridian lines intersect each other at the two points where the axis pierces the surface; but, as the part of the surface surveyed is so inconsiderable that the curvature of the earth is neglected, we may, without any sensible error, regard the meridians as right lines, and parallel to each other.

140. When the compass is placed on its stand, and the needle allowed to settle to a state of rest, the direction it assumes has been named the magnetic meridian (125). Although this line is different from the true meridian, as will be shown hereafter, yet, in the surveys made with the compass, we shall use the term, meridian, as synonymous with magnetic meridian, that is, to designate the direction of the magnetic needle.

141. If the right hand be turned towards the point where the sun rises, the direction pointed by the farthest end of the needle, is called North; the direction shown by the nearest end is called South; and the line thus indicated, is called a north and south line, as well as a meridian.

142. The line perpendicular to the meridian, is called an

East and West line; the East point being on the right hand, and the West on the left.

143. In departing from any point, if the direction taken lies between the north and east lines, it is called north as many degrees east, as is equal to the angle which the line run, makes with the meridian of the point: thus, if it makes an angle of 30°, it is called north, 30 east, and written, N. 30° E. This angle is called the bearing or course, and the line run, the distance. If the course lies between the north and west lines, the bearing is called north west; if between the south and west, south west; if between the south and east, south east; the bearing, as before, being written between the letters which designate the direction of the lines.

144. If, after having passed over a line, the bearing be taken to the back station, this bearing is called the back sight, or, reverse bearing.

145. The perpendicular distance between the east and west lines passing through the extremities of a line, is called the northing, or southing, of that line, according as it was run, north or south: the term difference of latitude designates this, perpendicular distance in either case.

146. The perpendicular distance between the meridians, passing through the extremities of a line, is called the departure of that line, and is east or west, according as the line lies on the east or west side of the meridian passing through the point of beginning.

147. The meridian distance of a point, is its distance from any assumed meridian. The meridian distance of a line, will be designated by the meridian distance of the middle point of that line.

OF THE TRAVERSE TABLE.

148. A table, called a Traverse Table, is used in computing the area of a survey made with the compass. Its use will be here explained.

This table shows the difference of latitude and departure,