have to refer only state, when single minutes are required, the values of the sines and cosines of angles; thus,

Cot 62° 27'

The reduction of this

vulgar fraction, which will be found equal to 5216767, will show the value of the application of logarithms.

The above approximate result, 521719, is seen to be correct for three places of decimals, but for application to the plotting of long base-lines the more accurate process must be applied.

To determine intermediate points in the curves between F and H, and between G and H (fig. 2), we can apply the formula given in Case 1 for the measurement of offsets from the tangent lines F B and G B, as it will be observed in the diagram which illustrates Case 1, that so long as the tangential angle B A D remains comparatively small, the length calculated for B S may be measured from D at right angles to the tangent line without appreciable error, to determine a point in the curve. Hence, substituting the term "distance" measured along the tangent line for the term "chord" in the equation (Case 1), we obtain for Cases 2 and 3 the formula,

If, therefore, the square of the number of links in the distance from the tangent peg (fig. 2) measured along the tangent line to the point at which the offset is desired to be taken be divided by the length expressed in links of the diameter of the circle, part of the circumference of which forms the required curve, the result will give the approximate number of links in the offset, the length of which is to be measured with the tape or measuring rod in a perpendicular direction to the tangent line. The formula is based upon an assumed length of radius, and the points upon the curve fixed thereby can be marked in the field by laths, pointed at the ends, so as to be easily pressed into the ground. (See pages 210, 211.)

Where great accuracy is not required, the application of

~ NEW

the above formula derived from Case 1 (pages 206, 207) may
be adopted, by the method shown in Case 3, for setting
out offsets at right angles to a tangent line (pages 210, 211).
The first offset nearest the junction of the curve with the
tangent line is thus easily calculated, being equal to the length
measured along the tangent line, divided by twice the radius.
of the curve, and if the remaining offsets be set out at the
same distance apart, measured along the tangent line, the
succeeding offsets, Nos. 2, 3, 4, and 5, will be respectively
equal in proportion to the square of these numbers, or 4,
9, 16, and 25 times the length of the first offset. As the
middle point in the curve is approached, the tangential

NOTE - BY THIS METHOD, A NEW TANCENT LINE MAY BE SET OUT,
WHEN THE OFFSETS MARKED OUT BY THE METHOD SHOWN
IN CASES 3&2, BECOME TOO LONC TO BE APPROXIMATELY
ACCURATE.

angle between the chord and the tangent line increases,
and the amount of error in the length of the offset so
calculated will also be increased. This error can, however,
be kept small by setting out a new tangent line as shown
above in fig. 3.

When the angle formed at B is smaller than that shown in
Case 4, fig. 1 (pages 214, 215), it will probably be found a more
convenient mode for setting out the curve to adopt one of the
following methods :-(1) In the case of a radius not greater
than one chain, to divert one or both or a portion of one of
the lines A B and B C, making them parallel to one another.

K

BH=RX AB-AD

AD

FICURE 1-TO DETERMINE THE LENGTHS OF THE
TANCENT LINES B.F.AND B.C.ALSO THE POSITION OF
THE MIDDLE POINT HIN THE CURVE F.HC.WITHOUT
THE USE OF TRICONOMETRICAL TABLES-MEASURE
FROM THE INTERSECTION PEC B.ANY EQUAL LENGTHS
BAAND BC ALONG THE LINES BK.AND BJ-MEASURE
THE DISTANCE FROM ATO CAND BISECT IT IN D.,THEN
MEASURE THE DISTANCE DB- THE SIDES OF THE
TRIANCLES ADBAND CDBWILL BE EACH PROPORTIONAL
TO THE SIDES OF THE TRIANGLES EF.BOR E.GB-

URACY

NOTE-IN FICURE 2, THE ANCLES AT B
CAN BE TAKEN WITH A BOX SEXTANT,
SUFFICIENTLY ACCURATE FOR ALL
PRACTICAL PURPOSES.

INTERSECTION,

B

PEC

THE CHAIN IS STRETCHED FROM C TOWARDS BIN LINECB C

THE OFFSETS CAN BE
MEASURED WITH A TAPE
FROM POINTS ON THE
CHAIN LINE

E

BETWEE

ANCLINES

M WHICH THE OFFSETS ON THIS
OF BISECT
OF THE LINE
LINE FROM
AIN LINE

NOTE-THE ZERO ON THE CHAIN LINE IS PLACED AT THE TANCENT PEC

ARE MEASU

LINE

BISECTION

LATH

TH

FICURE 2

RADIUS OF CURVE

TANCENT

PEC AT
END OF
CURVE

CASE 3

ABE = EBC

WHEN TRICONOMETRICAL TABLES ARE
AVAILABLE THE FOLLOWING METHOD
MAY BE ADOPTED-

(1) TO DETERMINE THE LENGTH OF THE TANCENT LINES BF
AND B.C.TO BE MEASURED ALONG THE LINES B.A. AND BC
RESPECTIVELY, FROM THE INTERSECTION PEC AT B.

(2) TO DETERMINE THE DISTANCE OF THE MIDDLE POINT H.OF THE CURVE F.H.C.FROM THE INTERSECTION PECB