not taken on the ground, are likely to be required. If the map be very long, more than one may be needed.

(273) Drawing-Board Protractor. Such a divided circle, as has just been described, or a circular protractor, may be placed on a drawing board near its centre, and so that its 0° and 90° lines are parallel to the sides of the drawing board. Lines are then to be drawn, through the centre and opposite divisions, by a ruler long enough to reach the edges of the drawing board, on which they are to be cut in, and numbered. The drawing board thus becomes, in fact, a double rectangular protractor. A strip of white paper may have previously been pasted on the edges, or a narrow strip of white wood inlaid. When this is to be used for platting, a sheet of paper is put on the board as usual, and lines are drawn by a ruler laid across the 0° points and the 90° points, and the centre of the circle is at once found, and should be marked C. The bearings are then platted as in the last method.

C

B

Fig. 188.

E

(274) With a scale of chords. On the plane scale contained in cases of mathematical drawing instruments will be found a series of divisions numbered from 0 to 90, and marked CH, or C. This is a scale of chords, and gives the lengths of the chords of any arc for a radius equal in length to the chord of 60° on the scale. To lay off an angle with this scale, as for example, to draw a line making at A an angle of 40° with AB, take, in the dividers, the distances from 0 to 60 on the scale of chords; with this for radius and A for centre, describe an indefinite arc CD. Take the distance from 0 to 40 on the same scale, and set it off on the arc as a chord, from C to some point D. Join AD, and prolong it. BAE is the angle required.

09

40

A

60

D

The Sector, represented on page 36, supplies a modification of this method, sometimes more convenient. On each of its legs is a scale marked C, or CH. Open it at pleasure; extend the compass from 60 to 60, one on each leg, and with this radius describe Then extend the compasses from 40 to 40, and the dis

an arc.

tance will be the chord of 40° to that radius. It can be set off as above.

The smallness of the scale renders the method with a scale of chords practically deficient in exactness; but it serves to illustrate the next and best method.

(275) With a Table of chords. At the end of this volume will be found a Table of the lengths of the chords of arcs for every degree and minute of the quadrant, calculated for a radius equal to 1.

To use it, take in the compasses one inch, one foot, or any other convenient distance (the longer the better) divided into tenths and hundredths, by a diagonal scale, or otherwise. With this as radius describe an arc as in the last case. Find in the table of chords the length of the chord of the desired angle. Take it from the scale just used, to the nearest decimal part which the scale will give. Set it off as a chord, as in the last figure, and join the point thus obtained to the starting point. This gives the angle desired.

The superiority of this method to that which employs a protractor, is due to the greater precision with which a straight line can be divided than can a circle.

A slight modification of this method is to take in the compasses 10 equal parts of any convenient length, inches, half inches, quarter inches, or any other at hand, and with this radius describe an arc as before, and set off a chord 10 times as great as the one found in the Table, i. e. imagine the decimal point moved one place to the right.

If the radius be 100 or 1000 equal parts, imagine the decimal point moved two, or three, places to the right.

Whatever radius may be taken or given, the product of that radius into a chord of the Table, will give the chord for that radius. This gives an easy and exact method of getting a right angle; by describing an arc with a radius of 1, and setting off a chord equal to 1.4142.

If the angle to be constructed is more than 90°, construct on the other side of the given point, upon the given line prolonged, an angle equal to what the given angle wants of 180°; i. e. its Supplement, in the language of Trigonometry.

This same Table gives the means of measuring any angle. With the angular point for a centre, and 1, or 10, for a radius, describe an arc. Measure the length of the chord of the arc between the legs of the angle, find this length in the Table, and the angle corresponding to it is the one desired.*

(276) With a Table of natural sines. In the absence of a Table of chords, heretofore rare, a table of natural sines, which can be found anywhere, may be used as a less convenient substitute. Since the chord of any angle equals twice the sine of half the angle, divide the given angle by two; find in the table the natural sine of this half angle; double it, and the product is the chord of the whole angle. This can then be used precisely as was the chord in the preceding article.

An ingenious modification of this method has been much used. Describe an arc from the given point as centre, as in the last two articles, but with a radius of 5 equal parts. Take, from a Table, the length of the natural sine of half the given angle to a radius of 10. Set off this length as a chord on the arc just described, and join the point thus obtained to the given point.†

When the Latitudes

(277) By Latitudes and Departures. and Departures of a survey have been obtained and corrected, (as explained in Chapter V), either to test its accuracy, or to obtain its content, they afford the easiest and best means of platting it. The description of this method will be given in Art. (285).

*

This Table will also serve to find the natural sine, or cosine, of any angle. Multiply the given angle by two; find, in the Table, the chord of this double angle; and half of this chord will be the natural sine required. For, the chord of any angle is equal to twice the sine of half the angle. To find the cosine, proceed as above, with the angle which added to the given angle would make 90°. Another use of this Table is to inscribe regular polygons in a circle by setting off the chords of the arcs which their sides subtend.

Still another use is to divide an arc or angle into any number of equal parts, by setting off the fractional arc or angle.

The reason of this is apparent from the figure. DE is the sine of half the angle BAC, to a radius of 10 equal parts, and BC is the chord directed to be set off, to a radius of 5 cqual parts. BC is equal to DE; for BC 2.BF, by Trigonometry, and DE =2.BF, by similar triangles; hence BC DE.

D

E

Fig. 189.

B

F

·

A

CHAPTER V.

LATITUDES AND DEPARTURES.

(278) Definitions. The LATITUDE of a point is its distance North or South of some "Parallel of Latitude," or line running East or West. The LONGITUDE of a point is its distance East or West of some "Meridian," or line running North and South. In Compass-Surveying, the Magnetic Meridian, i. e. the direction in which the Magnetic Needle points, is the line from which the Longitudes of points are measured, or reckoned.

The distance which one end of a line is due North or South of the other end, is called the Difference of Latitude of the two ends of the line; or its Northing or Southing; or simply its Latitude.

The distance which one end of the line is due East or West of the other, is here called the Difference of Longitude of the two ends of the line; or its Easting or Westing; or its Departure.

Latitudes and Departures are the most usual terms, and will be generally used hereafter, for the sake of brevity.

This subject may be illustrated geographically, by noticing that a traveller in going from New-York to Buffalo in a straight line, would go about 150 miles due north, and 250 miles due west. These distances would be the differences of Latitude and of Longitude between the two places, or his Northing and Westing. Returning from Buffalo to New-York, the same distances would be his Southing and Easting.*

In mathematical language, the operation of finding the Latitude and Longitude of a line from its Bearing and Length, would be called the transformation of Polar Co-ordinates into Rectangular Co-ordinates. It consists in determining, by our Second Principle, the position of a point which had originally been determined by the Third Principle. Thus, in the figure, (which is the same as

* It should be remembered that the following discussions of the Latitudes and Longitudes of the poims of a survey will not always be fully applicable to those of distant places, such as the cities just named, in consequence of the surface of the earth not being a plane.

CS to run due North and South, CS will be the Latitude, and AC the Departure of the line AS.

angle, in which the "Latitude" and the "Departure" are the sides about the right angle. We therefore know, from the principles of trigonometry, that

ACAB. cos. BAC,

BC AB. sin. BAC.

Hence, to find the Latitude of any course, multiply the natural cosine of the bearing by the length of the course; and to find the Departure of any course, multiply the natural sine of the bearing by the length of the course.

If the course be Northerly, the Latitude will be North, and will be marked with the algebraic sign +, plus, or additive; if it be Southerly, the Latitude will be South, and will be marked with the algebraic sign -, minus, or subtractive.

If the course be Easterly, the Departure will be East, and marked +, or additive; if the course be Westerly, the Departure will be West, and marked, or subtractive.