If the line AD, instead of being regarded as a unit, be taken to represent ten, then, each of the divisions Al, 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 Al, 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 plain that, if any other line be similarly measured across the 1 sector, the angle of the sector remaining the same, the basey 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 trtie lengths of the lines. It a line be so long that the whole of it cannot be taken inom the scale, it may be divided, and a part of it taken at a tine'. It 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 stak: and see to how many parts it is equal. On the eyes of the French sectors are scales of inches in English and teach ur'asurt. Edecept - If a line of sixty-seven feet were to be plotted vaak o'twenty Rty to the inch, take one inch from the venhe's; then place one foot of the dividers at the truire di ani penthe ector until the dividers will man ar SMRMCRTÀ divisia on the other arm; the sector steet pyeranie: the required distance to be laid A ver the queer. As reund, by extending the dividers from * a oa e arm, to the sixty-seventh of Mr. I COUNTER'S SCALE. 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 Jast and West line ; the East point being on the right hand, and the West on the left. 143. In de parting from any point, if the direction tak en 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, 100, the one hundred being regarded as links, chains, rods, or any other dimension. If (Pl. 6, Fig. 4) FG were a line measured, NFS the meridian, and SFG the bearing, or course; then FG' would be the southing, or difference of latitude, and GGʻ the departure east. It is evident that the distance run, the difference of latitude, and the departure, are, respectively, the hypothenuse, the base, and the perpendicular, of a right angled triangle, of which, the course or bearing is the angle at the base. If there be two bearings, which are complements of each other, or, of which the sum is 90°, the departure corresponding to the one, will be the latitude corresponding to the other. For, if GʻG were a meridian, and G'GF the bearing, instead of GʻFG, then, GG would be the difference of latitude, and FG' the departure. In the table headed Traverse Table,' the figures at the top and bottom show the bearings, to degrees and parts of a degree; and the columns on the left and right of each page, the dig tances to which the latitudes and departures correspond. If the bearing be less than 45', the angle will be found at the top of the page, if greater at the bottom; then, if the dis tance be less than 50, it will be found in the columns ("distances”) of the left hand page; if greater than 50, in the columns of the right hand page. The table is calculated only to quarter degrees, for the bearings cannot be accurately ascet, tained to smaller fractions of a degree. 149. For the same bearing, and lines of different lengths, it is evident, that the latitudes and departures will be proportional to the distances. Therefore, when the distance is greater than 100, it may be divided by any number which will give a quotient less than 100; then, the latitude and departure of the quotient being multiplied by the divisor, the products are the latitude and departure of the whole course. It is also plain, that, for the same bearing, the latitude and departure of the sum of two |