GEOGRAPHIC TABLES AND FORMULAS

DEPARTMENT OF THE INTERIORFRANKLIN K. LANE, Secretary

UNITED STATES GEOLOGICAL SURVEYGEORGE OTIS SMITH, Director tfj v

Bulletin 650

GEOGRAPHIC TABLES AND FORMULAS

FOURTH EDITION

COMPILED BY '

SAMUEL S. GANNETT

WASHINGTONGOVERNMENT PRINTING OFFICE

1916

ADDITIONAL COPIES

OF THIS PUBLICATION MAY BE PROCURED FROM

THE SUPERINTENDENT OF DOCUMENTS

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AT

25 CENTS PER COPY

CONTENTS.

Rules ior solution of right-angled triangles................................Reduction to center............................................. c ........Graphic reduction to center..............................................Solution of triangles, two sides and included angle being given............Three-point problem ......................................................Given the three sides of a triangle to find the angles.......................Graphic solution of the 'three-point problem................................ 11Micrometer alidades determination of constant and value of division ....... 12Method of fixing a meridian at any time by hour angle...................... 14Table 1. Polaris: azimuth at elongation ................................... 16

2. Polaris: azimuths and altitudes at different hour angles............. 173. Convergence of meridians..........:..................'........... 274. Values in feet of seconds of latitude and longitude................. 295. Offsets, in feet, from secant to parallel............................ 32

Example of use of projection tables............................. 346. Projections for large areas........................................ 367. Projections: scale r^ns, latitudes 0 to 80...................... 488. Projections: scale fl5 ^g 0 , latitudes 0 to 80....................... 589. Projections: scale Tnrcw, latitudes 25 to 50...................... 70

10. Projections: scale ;njwff> latitudes 25 to 65...................... 7611. Projections: scale 24 ooO) latitudes 0 to 25....................... 9012. Projections: scale y^^ny, latitudes 25 to 50...................... 9913. Areas of quadrilaterals; 1 extent, latitudes 0 to 90.............. 10814. Areas of quadrilaterals; 30' extent, latitudes 0 to 90.............. 11015. Areas of quadrilaterals; 15' extent, latitudes 0 to 90............. 11316. Areas of quadrilaterals; 1CK extent, latitudes 0 to 90............. 11917. For conversion of arc into time.,.......:.......................... 12418. For conversion of time into arc................................... 12519. For conversion of mean time into sidereal time.................... 12620. For conversion of sidereal time into mean time.................... 12721. For interconversion of feet and decimals of a mile................. 12822. For conversion of chains into feet................................. 12923. For conversion of feet into chains................................. 12924. For conversion of wheel revolutions into hundredths of a mile...... 13025. Five-place logarithms of natural numbers......................... 14026. Natural sines and cosines ........"................................. 16227. Five-place logarithms of circular functions expressed in arc and time. 17128. Geodetic position computations .................................. 21629. Values of log sec $ (A

CONTENTS.

Table 34. Corrections for curvature and refraction .......................... 33535. For obtaining differences of altitude ............................. 33636. Horizontal distances and elevations from stadia readings ....... .. 35337. For converting metric into United States measures................ 36238. For converting United States measures into metric................ 36339. For interconversion of miles and logarithms of meters............. 364

Convenient equivalents................................................... 376Table 40. Ratios for map scales............................................ 378Base line computations................................................... 379Errors in elevations....................................................... 380Errors in leveling ........................................................ 380Errors in plumbing rods.................................................. 380Comparison of'Centigrade and Fahrenheit degrees........................... 381Constants.................................................. v ............ 382Linear expansion of metals ...............................'................ 382Salary tables ............................................................ 383Index................................................................... 387

ILLUSTRATIONS.

Page.FIGURE 1. Solution of right-angled triangles................................ 5

2. Reduction to center............................................ 63. Solution of triangles; two sides and included angle given......... 84. Three-point problem; computation.............................. 105. Three-point problem; graphic solution.......................... 126. Aspects of Polaris.............................................. 147. Offsets from secant to parallel................................... 338. Construction of polyconic projection............................. 34

GEOGRAPHIC TABLES AND FORMULAS.

Compiled by S. S. GANNETT.

RULES FOB SOLUTION OF BIGHT-ANGLED TBIANGLES.

The "parts" of the figures are H=hy pothenuse, P=perpendicular, B=base,

and the six circular functions of the angle a at the base of the triangle.

Secant

Cosecant

\

\ Tangent \

iSine

CotangentCosine

FIG. 1. Solution of right-angled triangles.

RULE I. The product of two opposite parts = 1, .'. either is the reciprocal of the other.

Example: Tan a X cot or = 1, tan ac= ^

Rule 11. Each part=adjacent part divided by the following -part, .*. each part=the product of the adjacent parts.

Example: Sin a = , ^ sin ^^TJ' B=Hxcos a.

5


REDUCTION TO CENTER.

In fig. 2 letP=place of instrument;C=center of station;Q=measured angle at P between two objects, A and B;y=angle at P between C and the left-hand object, B;r= distance CP;

C'=unknown and required angle at C; D=distance AC;

. (r and D must be reduced to same unit, usually meters.) G distance BC; A=angle at A between P and C; B=angle at B between P and C.

FIG. 2. Keduction to center.

Then0, from the relation between the parts of the triangle,G : r :; sin y : sin B;

hence. _ r sin v sm B = Q-*-

As the angles at A and B are very small, their sines may be regarded as equal to A sin 1" and B sin 1", respectively; hence

and

T / i \ n y B=(m seconds^ sin j,,

sin (Qrfay) r sin y Dsinl" Gsinl"'


In the use of this formula, proper attention should be paid to the signs of sin (Q+y) and sin y; for the first term will be positive only when (Q+y) is less than 180 (the reverse with sin ?/); D being the distance of the right-hand object, the graduation of the instrument running from left to right.

T being relatively small, the lengths of D and G are approximately computed with the angle Q.

The following quantities must be known in addition to the measured angles in order to find the correction for reducing to center:

1. The angle measured at the instrument, P, between the center of the signal or station, C, and the first-observed station to the right of it, A.

2. The distance from the center of the instrument to the center of the station = r.-

* 3. The approximate distances, D, G, etc., from the station occupied to the stations observed. The latter may be computed from the uncorrected angles.

Example: Reduction to center from P to C.Constants: a. c. log sin 1" =5. 31443

log feet to log meters =9. 48402

log constant (for any station) 4. 79845 r= 6.5 feet: log =0.81291

log constant for this station 5. 61136

correction to angle B P A=4".08 +5^.12 =9". 20.

Angle Q-Y (CPA) 23 40'

9. 60365. 39545. 6114

0.61044". 08

Angle Y

(BPC) 3714' or 322 46'

9. 7818

5.31625. 6114

0. 70945". 12

8 GEOGRAPHIC TABLES AND FORMULAS.

GRAPHIC REDUCTION TO CENTER,

Approximate closure errors of triangles may be tested in the field before distances have been computed by scaling from the plot the distances between stations in miles and the perpendicular distance in feet from signal to line joining instrument and distant station.

Then, since 1 foot at a distance of 40 miles subtends an angle of 1" (nearly),

length of perpendicular in feet X 40 .. . , - i r- z - = correction in seconds, number of miles

Example: Station P. Correction for swing on line B P, 30 miles in length from instrument to signal

3.8 feet X 40 K ,, ,= m = 5 - 1'correction for swing on line A P, 25 miles in length,

_2.6 feet x 40 _ . 25 - * -4

and correction to angle B P A = Q to reduce from instrument to signal = 5.1" + 4.2" = 9.3", agreeing closely with the exact computation.

APPROXIMATE SPHERICAL EXCESS IN SECONDS.

This may be obtained -by dividing the area of the triangle in square miles by 75.5.

SOLUTION OP TRIANGLES.

Given two sides and included angle, to solve the triangle:

FIG. 3. Solution ol triangles; two sides and included angle given.

Let x be an auxiliary angle; then

tan X ^TI or log tan x=\og a log J;

tan i (A-B) = tan (-45) tan \ (A+B);i (A+B)-H (A-B) = A;

. i (A+B)-i (A-B) = B; from which remaining parts can be computed.


Example:Given log 0=4.3666779 Given log &=4.2050498

(1) tan i=0.1616281 1=55 25'25". 49

~-

(5) Log tan (i-45)=10 25' 25". 49=9.2647300(6) Log tan 79 22 33 .00=0.7268100

Given C (spherical angle) 21 14' 54". Given J sph. exc. .

C (planeangle) = 21 14 54.00 (?)

180-C=A+B=15S 45 06.00;:.!

J(A+B)= 79 22'3?" JO (4)

(7) 9.9915400= tan (A-B)

(10)

44 26 31 .11

sum=A=12349'04" clifference=B= 34 56 01

11 (8) 89 (9)

Check.

A=123 49' 04". 11 a. c. A= 34 56 01 .89 ' C= 21 14 54 .00

log 'a =4.3666779 log sin A=0. 0804975log sin B=9. 7578744log sin C=9. 5592012

Sum=180 00 00 .00 log c =4. 0063766 log b =4. 2050498

THREE-POINT PROBLEM.

If three points, forming a triangle of which the sides and angles are known or can be computed, be visible from a fourth point, P, it is required to determine the position of P.

Set up the theodolite at P and measure the two angles subtended by any two of the given sides.

This problem is of use in cases where, the regular triangulation hav- ing been completed, additional points are required for the topographic survey, or are needed for special service. The angles should be carefully measured, and in the computations the logarithms should be car- ried to seven places1 of decimals.

Three cases of its application are given, as in others, such as when P falls upon one or another of the sides of the known triangle, or on the prolongation of either, the case resolves itself into the solution of a simpk triangle with one side and the angles given; or the problem is indeterminate, as when P is situated on the circumference of the circle passing through the three known points a contingency which rarely occurs.

In making the computations, first prepare a plat of the various lines on such a scale that the shortest line will be about 2 inches in length, letter the plat the same as above, write by each line (a, &, c) the log of its length, and place in the proper positions the values for the angles A, P', P". The computations for Case III differ only in


the method of combining P', P", and A to find S. Blank forms for this computation are provided.

CASE II. CASE III.Bf 7C

FIG. 4. Three-point problem; computation.

P' .............. 50 06 12P".............. 43 50 38A............... Ill 10' 54

sum.............. 205 07 44i sum.... 1......... 102 33 52

S=180-i sum.......... 77 26 08

, sum................A ................

A sum...............S= (A sum) ....

Computation.

logc.... 3.7975307log sin P'.... 9.8849100

(add) cologfc.... 6.1373320colog sin P".... 0.1594574

log tan Z....' 9.9792301

Z.... 43 37 49.6 Z+45 0 .... 88 37 49.6

log cot (Z+45).... 8. 3785397 (add) log tan S.... 0. 6519386

log tan e.... 9. 0304783 (sign +)

e.... 6 07 21.7S.... 77 26 08.0

(When tan eis +) S+e-AB P....- -----S-e-ACP ....

0

BPA..... 43 ABP..... 83 PAB..... 52

180

50 38 33 29.7 35 52.3

00 00.0

83 33 71 18

APC ACP CAP

29.7 .46.3

(When ta S-a=ABPS+e=A C P

50 06 12 71 18 46.3 58 35 01.7

180 00 00. 0

01 is )

BPC.... PCB.... CPB....

As all the angles and a side in each triangle are now known, the other sides, or the distances from P to the three given points, can be readily computed.

P B ............. 7194.87PA............. 8999. 89PC............. 8107. 98PA............. 8999. 89

P B ............. 7194. 94PA............. 1388. 54PC............. 8107. 91PA............. 1388. 54

The results are verified when both triangles give the same value for the line P A.

P B ............. 5256. 29PA............. 2609. 75PC............. 6203. 63PA.............. 2609. 75

GEOGRAPHIC TABLES AND FORMULAS. 11

GIVEN THE THREE SIDES OF A TRIANGLE TO FIND THEANGLES. ' ' ,.

Let Y be an auxiliary quantity and S the half sum of the three sides; Y== / (S-a) (S-Z>) (S-c)/

V sY Y Y tan i B = Q r tan 4 C = Q a 2 S 6 2 S c

Example:Given a=43. 75C6

6=40. 8954 c= 4.1908

sum=88. 8428 S=44. 4214

S-a= 0.6648 log=9. 82269108-6=3.5260 log=0.5472823S-c=40. 2306 log=l. 6045564

log of product=l. 9745297 log of S =1. 6475922

log of Y2 =0. 3269375=log of quotient, log of Y =0.1634688

log Y =0.1634688 log S-a =9.8226910

log tan A =0. 3407778 (quotient)} A =65 28' 27" A=130 56' 54"

log Y =0.1634688 logS-6 =0.5472823

log tan B =9. 6161865B =2227/ 06// B=4454'12"

log Y =0.1634688 log S c =1. 6045564

log tan \ C =8. 5589124-JO = 2 04' 27" C= 4 08' 54"

180 00'00"

GRAPHIC SOLUTION OF THE THREE-POINT PROBLEM.

1. When new point is within the triangle formed by the three points, point sought is within the triangle of error.

2. When new point is on or near the circle passing through the other points, the location is uncertain.

3. When new point is within either of the three shaded segments of the circle (see diagram below), orient on middle point; then the line from middle point lies between true point and point of intersection of lines from other two points.


4. When new point is without the circle, orient on most distant point; then the point sought is always on the same side of the line from most distant point as the point of intersection of the other two lines.

NOTE. Since a location can be made from any three points, whether correctly plotted or not, therefore always check such locations by means of a fourth point if possible.

FIG. 5. Three-point problem; graphic solution.

MICROMETER ALIDADES DETERMINATION OF CONSTANT AND VALUE OF DIVISION.

R', R" = readings of micrometer screw. R =R' R" = difference of readings.

d = value in seconds of arc of 1 division of micrometer head.A = angle subtended by targets in seconds of arc.C = micrometer constant or ratio.H = distance to targets, supposed at right angles to line of

sight. B = length of base, or distance between targets.

(1) d =

(2) C =

BHR sin 1"

1 HRdsm B


EXAMPLE.

Readings taken on two targets 21.25 feet apart at right angles to the line of sight and at a measured horizontal distance of 2859.5 feet from the point of observation.

R' R" R550.0-88.0 = 462.0540.5-76.5 = 464.0

etc. etc. etc.

Computation of d by formula (1): B =21.25 ft. __: ..log. 1.32736 H= 2859.5ft_ _. colog. 6.54371 sin 1".........colog. 5.31443R = 462.075 div. colog. 7.33528

462.075 mean of 20 readings.

Computation of C by formula (2): B = 21.25ft....colog. 8.67264 H = 2859.5 ft....log! 3.45629 R = 462.075 div..log. 2.66472

C = 62180 .......log. 4.79365d =3".317-------log. 0.52078

For computing distances use this formula:

(V H BC(3; M=^

When the base is not at right angles to the line of sight as at &, or at the same elevation as the point of observation, the factors sin a and cos V must be introduced, a, being the angle between the base and line of sight and V the vertical angle at A.

The full formula for' distances then becomes

(4) H =&C sin a cos V

R

The plotted position of the base & should be prolonged on the field sheet in order to permit the measurement of the angle a with a large paper or other protractor, with greater accuracy.


METHOD OF FIXING A MERIDIAN AT ANY TIME BY HOURANGLE.

[Extracted from United States Land Survey Manual.]

The annexed diagram (fig. 6) will show in their proper relation the various aspects of Polaris in its daily apparent motion around the. north-polar point.

This must be carefully studied, as the illustration of Table 1, for finding at any hour the hour angle and azimuth of Polaris, and the resulting meridian, at times when more direct methods are not available.

Hour angle of Polaris. In fig. 6 the full vertical line represents a portion of the meridian passing through the zenith Z (the point directly overhead), and intersecting the northern horizon at the north point N, from which, for surveying purposes, the azimuths of Polaris

W.elong.5

FIG. 6. Aspects of Polaris.

are reckoned east or west. The meridian is pointed out by the plumb line when it is in the same plane with the eye of the observer and Polaris on the meridian, and a visual representation is also seen in the vertical wire of the transit, when it covers the star on the meridian.

When Polaris crosses the meridian it is said to culminate; above the

GBOGEAPHIO TABLES A"ND FOEMULAS. 15

pole (at.S), the passage is called the upper culmination, in contradis- tinction to the lower culmination (at S').

In the diagram which the surveyor may better understand by hold- ing it up perpendicular to the line of sight when he looks toward the pole Polaris is supposed to be on the meridian, where it Avill be about noon on April 14 of each year. The star appears to revolve around the pole, in the direction of the arrows, once in every 23h 56m.l of mean solar time; it consequently comes to and crosses the meridian, or culminates, nearly four minutes earlier each successive day. The apparent motion of the star being uniform, one quarter of the circle will (omitting fractions) be described in 5h 59m , one half in llh 58m , and three quarters in 17h 5Ym . For the positions S15 S2 , S3 , etc., the angles SPsn SPs2 , SPs3 , etc., are called hour angles of Polaris, for the instant the star is at S1? s2 , or s3", etc., and they are measured by the arcs Ssn Ss2 , Ss3 , etc., expressed (in these instructions) in mean solar (common clock) time, and are alwaj7 s counted from the upper meridian (at S), to the west, around the circle from Oh Om to 23h 56m.l, and may'have any value between the limits named. The hour angles, measured by the arcs Ss15 Ssa , Ss3 , Ss4 , Ss6 , and Ss6 , are approximately lh 8mv 5U 55m , 9h 4m , 14:h 52^, 18h Olm, and 22h 48m , respectively; their extent is also indicated graphically by broken fractional circles about the pole.

Suppose the star observed at the point S3 ; the time it was at S (the time of upper culmination), taken from the time of observation, will leave the arc Ss3 , or the hour angle at the instant of observation; similar relations will obtain when the star is observed in any other position; therefore, in general:

Subtract the time of upper culmination from the correct local mean time of observation; the remainder will be the hour angle of Polaris expressed in time.

The observation may be made at any instant when Polaris is visible, the exact time being carefully noted.

The General Land Office publishes annually the Ephemeris of the Sun and Polaris and Tables of Azimuths of Polaris.

The azimuth of Polaris at any hour angle on any day in the year may be obtained from this table by a very simple computation. This publication replaces tables formerly printed on pages 16 to 25 of Geographic Tables and Formulas.


TABLES.

TABLE 1. Azimuth of Polaris when at elongation for any year between 1916 and 1924.

Latitude.

1011121314

1516171819

2021222324

2526272829

3031323334

3536373839

4041424344

4546474849

50

1916

1 09.609.910.110.410.7

11.011.411.712.112.5

13.013.514.014.515.1

15.716.317.017.718.4

19.220.020.921.822.7

23.724.825.927.028.2

29.530.932.333.835.3

37.038.740.642.544.5

1 46.7

1917

1 09.309.609.810.110.4

10.711.011.411.812.2

12.713.113.614.214.7

15.316.016.617.318.1

18.819.720.521.422.4

23.324.425.326.627.8

29.130.431.933.434.9

36.638.340.142.044.1

1 46.2

1918

1 09.009.209.509.810.0

10.410.711.111.511.9

12.312.813.313.814.4

15.015.616.317.017.7

18.519.320.121.022.0

23.024.025.126.227.5

28.730.031.532.934.5

36.137.839.741.643.6

1 45.7

1919

1 08.708.909.209.409.7

10.010.410.811.111.6

12.012.5

' 13.013.514.1

14.715.315.916.617.4

18.118.919.820.721.6

22.623.624.725.927.1

28.329.631.032.534.1

35.737.439.241.143.1

1 45.3

1920

1 08.408.608.909.109.4

09.710.110.410.711.2

11.712.212.613.213.7

14.314.915.616.317.0

17.818.619.420.321.2

22.223.324.325.526.7

27.929.130.632.133.6

35.337.038.840.742.7

1 44.8

1921

1 08.108.308.608.809.1

09.409.810.110.510.9

11.411.812.312.813^4

14.014.715.215.916.6

17.418.219.119.920.9

21.822.924.025.126.3

27.528.830.231.833.2

34.836.538.340.242.2

1 44.3

1922

1 07.808.008.208.508.8

09.109.409.810.210.6

11.011.512.012.513.0

13.614.214.915.616.3

17.017.918.719.620.5

21.522.523.624.725.8

27.128.429.831.232.8

34.436.137.939.841.7

1 43.8

1923

107.407.707.908.208.5

08.809.1

' 09.509.810.2

10.711.211.612.212.7

13.313.914.615.216.0

16.717.518.319.220.1

21.122.123.224.325.5

26.728.029.430.832.4

34.035.637.439.341.3

1 43.4

1924

1 07.207.407.607.8

.08.2

08.508.8

.09.209.509.9

10.410.811.311.812.4

13.013.614.214.915.6

16:417.218.018.819.8

20.721.722.823.925.1

26.327.629.030.431.9

33.535.237.038.840.8

1 42.9

The above table was computed with the mean declination of Polaris for each year. A more accurate result will be had by applying to the tabular values the following correction, which depend on the difference of the mean and the apparent place of the star. The deduced azimuth will, in general, be correct within 0'.3.

For middle of

February..........

April..............

June..............

Correction.

-0.5

-0.4

-0.3

0.0

+0.1

+0.2

For middle of

July...............

August. ............

September........October............November....... . .

Correction.

/

+0.2

+0.1-0.1-0.4-0.6-0.8

GEOGKAPHIC TABLES AND FORMULAS. 17

TABLE 2. AZIMUTH AND APPARENT ALTITUDE OF POLAHIS AT DIFFERENT HOUR ANGLES.

[From U. S. Coast and Geodetic Survey Report for 1895.]

The accompanying tables are intended for field use, to facilitate placing an instrument in the meridian. They are also suitable for determining the approximate latitude or meridian. They contain the azimuth of Polaris at intervals of fifteen minutes in hour angle for each degree of north latitude from 30 to 60, and the apparent altitude at the same intervals and for each fifth degree of latitude. 68 The tables are computed for the declination of Polaris 88 46', but the rate of change in both azimuth and altitude is given with the argument V increase in declination. 6 The tables are intended to be used in connection with the American Ephemeris, where are given the apparent right ascension and declination of Polaris for each day in the year.The approximate local time will in general be known with sufficientaccuracy from standard time and the approximate longitude of the place. The following example explains the use of the tables and the deriyation of the hour angle of Polaris:

Position, latitude 36 20' N., longitude 5h 20m 30s W. of Greenwicli.h. m. s.

Time of observation, July 10,1895, standard (75th naer.) mean time 8 52 40 p. m. Reduction to local time 20 30

Local mean time 8 32 10Reduction to sidereal time (Table III, Amer. Ephem.) + 1 24Sidereal time mean noon, Greenwich, July 10,1895 7 12 38Correction for longitude, 5" 20m 30s (Table III, Amer. Ephern.) + 0 53

Local sidereal time ' 15 47 05Apparent right ascension of Polaris, July 10,1895 . 1 20 18

Hour angle before upper culmination 9 33 13

"The tables were computed with the following formulas:

_ _ _ __ __ _tan a = co~s ,p tan"8~sin


Declination of table 88 46 Apparent declination, July 10,1895 " 88 44 47

Increase in declination 1 13= 1''2o /

Values from tables (interpolated) azimuth 0 54 12, apparent altitude 35 21.8 Correction for 1'.2 increase in declination +52 1.0

0 55 04 35 20.8 East of north

It is to be remembered that Polaris is east of the meridian for twelve hours before upper culmination, and west of the meridian for twelve hours after. By setting the instrument at the apparent altitude and sweeping near the meridian Polaris can ordinarily be found and the instrument placed in the meridian some time before dark. With transit instruments not provided with horizontal arc, the value of the azimuth adjusting screw may be readily determined and used.

Without the American Ephemeris these tables may be conveniently used for obtaining the approximate meridian or latitude, in connection with Bulletin 14, United States Coast and Geodetic Survey, where are given the approximate mean times of culminations of Polaris, and the mean declinations for various epochs.


TABLE 2. Azimuth and apparent altitude

Hour angle before or after upper culmination.

h. m.0 150 300 45"1 001 15

1 301 452 002 152 30

2 453 003 153 303 454 004 154 304 455 005 155 305 456 006 156 306 457 007 157 307 458 008 158 308 45

9 009 159 309 45

10 0010 1510 3010 4511 0011 1511 3011 45

Elongation: Azimuth . . .

Hour angle.

Azimuth of Polaris computed for declination 88 46'.

Latitude 30.

0 / II

0 05 400 11 180 16 530 22 230 27 48

0 33 050 38 130 43 120 47 580 52 32

0 56 521 00 581 04 471 08 19

. 1 11 331 14 281 17 041 19 191 21 141 22 481 24 001 24 511 25 201 25 271 25 121 24 341 23 361 22 161 20 351 18 34

1 16 131 13 331 10 341 07 171 03 43

0 59 540 55 490 51 310 46 590 42 160 37 230 32 200 27 090 21 510 16 280 11 010 05 31

1 25 27h. m. s.5 57 09

Latitude 31.

0 ' //

0 05 430 11 250 17 040 22 380 28 060 33 260 38 380 43 400 48 290 53 060 57 291 01 371 05 281 09 021 12 18

1 15 15. 1 17 521 20 091 22 051 23 401 24 531 25 441 26 131 26 191 26 041 25 271 24 271 23 061 21 251 19 22

1 16 591 14 171 11 161 07 571 04 22

1 00 300 56 230 52 010 47 270 42 42

0 37 450 32 390 27 250 22 040 16 380 11 080 05 34

1 26 20h. m. s.5 57 02

Latitude 32.

o / n

0 05 470 11 330 17 150 22 530 28 25

0 33 490 39 040 44 090 49 020 53 42

0 58 071 02 181 06 121 09 481 13 06

1 16 051 18 441 21 021 22 591 24 351 25 481 26 401 27 091 27 151 26 591 26 211 25 211 23 591 22 161 20 12

1 17 481 15 041 12 011 08 401 05 02

1 01 070 56 580 52 340 47 570 43 08

0 38 080 32 590 27 420 22 180 16 480 11 140 05 38

1 27 16h. m. s.5 56 55

Latitude 33.

0 1 II

0 05 510 11 410 17 270 23 090 28 45

0 34 130 39 320 44 400 49 360 54 190 58 481 03 011 06 581 10 361 13 56

1 16 571 19 371 21 571 23 551 25 321 26 461 27 381 28 071 28 141 27 571 27 191 26 181 24 551 23 101 21 05

1 18 391 15 531 12 481 09 251 05 44

1 01 470 57 340 53 080 48 280 43 360 38 330 33 200 28 000 22 320. 16 590 11 220 05 42

1 28 14h. m. 8.5 56 48

Latitude 34.

0 ' //

0 05 550 11 490 17 400 23 260 29 06

0 34 380 40 000 45 120 50 120 54 59

0 59 301 03 461 07 461 11 271 14 49

1 17 521 20 341 22 551 24 551 26 321 27 471 28 39-1 29 091 29 151 28 591 28 191 27 171, 25 531 24 081 22 00

1 19 331 16 451 13 371 10 121 06 29

1 02 290 58 130 53 430 49 000 44 05

0 38 590 33 430 28 180 22 470 17 100 11 290 .05 45

1 29 16h. m. s.5 56 40

Latitude 35.

0 / II

0 06 000 11 580 17 530 23 440 29 28

0 35 040 40 300 45 460 50 500 55 401 00 151 04 341 08 361 12 201 15 45

1 18 501 21 341 23 571 25 571 27 361 28 511 29 441 30 141 30 201 30 031 29 231 28 201 26 551 25 081 22 59

1 20 291 17 391 14 291 11 011 07 0 15

1 03 120 58 540 54 21

. 0 49 340 44 350 39 260 34 060 28 380 23 030 17 220 11 370 05 49

1 30 20h. m. s.5 56 33

GEOGRAPHIC TABLES AND EORMULAS.

of Polaris at different hour angles.

21

Azimuth of Polaris computed for declination 88 46'.

Latitude 36.

o / //

0 06 050 12 080 18 070 24 020 29 510 35 310 41 020 46 220 51 290 56 23

1 01 021 05 241 09 291 13 161 16 43

1 19 501 22 361 25 011 27 031 28 42

1 29 591 30 521 .31 211 31 271 31 101 30 301 29 261 27 591 26 111 24 001 21 281 18 361 15 241 11 531 '08 041 03 580 59 370 55 000 50 100 45 080 39 540 34 300 28 590 23 190 17 35

' 0 11 460 05 53

1 31 28 'h. m. s.5 56 25

Latitude 37.

o / //0 06 100 12 180 18 220 24 220 30 150 36 000 41 350 47 000 52 110 57 09

1 01 511 06 171 10 251 14 141 17 44

1 20 541 23 421 26 081 28 121 29 52

1 31 091 32 031 32 331 32 391 32 211 31 401 30 351 29 071 27 171 25 041 22 301 19 361 16 211 12 481 08 561 04 471 00 220 55 420 50 480 45 420 40 240 34 570 29 200 23 370 17 480 11 540 05 580

1 32 40h. m. s.5 56 17

Latitude 38.

O > H

0 06 150 12 280 18 380 24 430 30 410 36 310 42 110 47 39

'0 52 550 57 57

1 02 431 07 121 11 241 15 161 18 49

1 22 011 24 511 27 191 29 241 31 061 32 241 33 181 33 481 33 541 33 361 32 541 31 481 30 181 28 261 26 121 23 361 20 391 17 221 13 451 09 501 05 381 01 090 56 250 51 270 46 170 40 550 35 240 29 430 23 550 18 020 12 040 06 02

1 33 55h. m. s.5 56 09

Latitude 39.

o r H

0 06 200 12 390 18 540 25 040 31 080 37 020 42 470 48 210 53 410 58 47

1 03 371 08 101 12 251 16 211 19 57

1 23 111 26 031 28 331 30 401 32 231 33 421 34 371 35 071 35 131 34 541 34 111 33 041 31 331 29 391 27 231 24 451 21 451 18 251 14 451 10 471 06 311 01 590 57 110 52 090 46 540 41 280 35 520 30 070 24 140 18 160 12 130 06 07

1 35 14h. m. s.5 56 00

Latitude 40.

o / //

0 06 260 12 500 19 110 25 270 31 360 37 360 43 260 49 040 54 290 59 40

1 04 341 09 121 13 301 17 291 21 08

1 24 251 27 201 29 521 32 001 33 441 35 041 35 591 36 301 36 351 36 161 35 321 34 241 32 52I 30 561 28" 381 25 571 22 541 19 311 15 48

' 1 11 471 07 271 02 510 57 590 52 530 47 340 42 030 36 220 30 320 24 350 18 310 12 230 06 12

1 36 36h. TO. 8.

5 55 52

Correction for 1' increase in declination of Polaris.

Latitude 30.

- 5- 9 14-18-23-27-31-35-39-43

-46-50-53-56 58-61-63-64-66-68-69-69-70-70-69-68-67-66-65-64-62-60-57-54-51-48-45 42-38-34-30-26-22-18-13- 9- 4

-698.

+ 2

Latitude 40.

ii- 5-10-16-21-26-31-36-40-45-49

-53' -57-60-63-66-69-72-74-75

. -76_ 77-78-78-78-78-77-76-75-73-72-69-66-64-61-58-54-50-46-42-38-34-29-24-20-15-10 5

-788.

+ 3

Hour angle before or after upper culmi- . nation.

h. TO. :0 15 .0 30 0 45 '1 001 151 301 452 00 .2 152 30

2 453' 00 '

3 15 i3 30 !3 454 004 '15 I4 30 :4 455 005 15 '5 305 45 ,6 006 156 306 457 00 7f5 :7 30 :7 45 '8 008 15 :8 30 !8 45 j9 00 i9 15 '9 30 ;9 45 11000 ;

10 15 3 C3 s PH

Mer

id-

iona

lO

f/lC

0 5 10 15 20 25 30 35 40 45 50 55 60

Mea

n va

lue

of

1"

ofla

titu

de.

-

0 5 10 15 20

-25 30

.

35 40 45 50 55 60

Mea

n va

lue

of 1

" of

lati

tude.

.

63

50.8

5.7

3.6

0.4

7.3

4.2

25

0.0

94

9.9

6.8

3.7

1.5

7.4

44

9.3

1

101.

55

75

26.3

4.2

02

6.0

625

.91

.77

.62

.48

.34

.20

25.0

524

:91

.76

24.6

2

101.

73

61

49

.31

.18

49.0

548.9

2.7

9.6

6.5

4.4

1.2

8.1

548.0

247.8

947.7

6

101.

56

76

24.6

2.4

8.3

4.1

92

4.0

52

3.9

0.7

6.6

2.4

7.3

3.1

82

3.0

422

. 89

101.

73

62

47

.76

.63

' .4

9.3

6.2

3.1

04

6.9

7.8

4.7

1.5

7.4

4.3

146.1

8

, 10

1.57

77

22

.89

.75

.60

.46

.32

.17

22.0

321.8

8.7

4.5

9.4

5.3

12

1.1

6

101.

74

63

46.1

84

6.0

545.9

2.7

9.6

6.5

2.3

9.2

645.1

344.9

9.8

6.7

244.5

9

101.

59

78

21

.16

21

.02

20.8

7.7

3.5

8.4

4.3

0.1

52

0.0

019

.85

.71

.57

19.4

2

101.

75

64

44

.59

.46

.33

.19

44

.06

43

.93

.80

.66

.53

.40

.26

43

.13

42

.99

10

1.6

0

79

19.4

2.2

8.1

318.9

8.8

4.6

9.5

5.4

0.2

618

.11

17.9

7.8

217.6

8

101.

76

- 6

5

42

.99

.86

.73

.59

.46

.32

.19

42

.05

41

.92

.78

.65

.52

41

.38

.

101.6

2

80

17.6

8.5

3.3

8.2

417.0

916

.95

.80

.65

.51

.36

.21

16.0

715.9

2

101.

76

66

41

.38

.25

41

.11

40

.98

.84

.70

.57

.44

.30

.16

40

.03

39

.89

39

.76

101.

63

81

15.9

2. 7

7.6

3.4

9.3

4.1

915

.05

14

.90

.75

.61

.46

.31

14.1

7

101.

77

67

39

.76

.62

.48

.34

.21

39

.07

38

.94

.80

.66

.52

.39

.25

38

.12

101.

64

S2

14.1

714.0

213.8

7.7

3.5

8.4

3.2

913

.14

12.9

9.8

5.7

0.5

512

.'41

101.

77

68

38

.12

37

.98

.84

.70

.57

.43

.29

.15

37.0

13

6.8

8.7

4.6

03

6.4

6

101.

66

83

12.4

1.2

612

.11

11.9

6.8

2.6

7.5

2.3

8.2

311.0

810

.94

.79

10.6

4

101.

78

69

36.4

6.3

3.1

936.0

53

5.9

1.7

7.6

4.5

0.3

6.2

235.0

834.9

434.8

0

101.

67

84

10.6

4.4

9.3

5.2

01

0.0

59.9

0.7

6.6

1.4

6.3

2.1

79

.02

8.8

7

101.

79

70

34

.80

.66

.52

.38

.24

34.1

133.9

7.8

3.6

9.5

5.4

1.2

73

3.1

3

101.

68

85

8.8

7.7

3.5

8.4

3.2

88

.14

7.9

9.8

4.6

9.5

5.4

0.2

57.1

0

101.

79

71

33.1

33

2.9

9.8

5.7

1.5

7.4

3.2

9.1

532

.01

31.8

7.7

3.5

93

1.4

5

101.

69

86

. 7.1

06.9

5.8

0.6

6.5

1.3

6.2

16.0

75.9

2.7

7.6

2.4

85.3

3

101.

79

72

31.4

5.3

1.1

731.0

330.8

9.7

4.6

0.4

6.3

2.1

830.0

429.9

0 '

29.7

6

101.

70

87

5.3

3.1

85

.03

4.8

8.7

4.5

8.4

4.2

9.1

54

.00

3.8

5.7

03

.55

101.

79

73

29.7

6.6

1.4

7.3

3.1

929

. 04

28.9

0.7

6.6

2-.

48

.33

.19

28.0

5

101.

71

88

3.5

5.4

0.2

53.1

12

.96

.81

.66

.52

.37

.22

2.0

71

.92

1.7

8

101.

79

74

28.0

527.9

1.7

7.5

2.4

8.3

4.2

027.0

526.9

1.7

7.6

3.4

826.3

4

101.

72

89

1.7

8.6

3.4

8.3

3.1

91

.04

0.8

9.7

4.5

9.4

5.3

0.1

50.0

0

oo


TABLE 5. Offsets, in feet, from the secant to the parallel.

[Prepared by E. M. Douglas.]

Latitude.

Deyrees.2526272829

3031323334

3536373839

4041424344

4546474849

5051525354

5556575859

6061626364

6566676869

70

o.

Feet.1.61.61.71.81.8

1.92.02.12.22.2

2.32.42.52.62.7

2.82.93.03.13.2

3.33.43.63.73.8

4.04.14.34.44.6

4.7 '4.95.1

- 5.35.5

5.86.06.36.66.9

7.17.57.98.38.7

9.1

mile.

Feet.0.7.7.8.8.8

.9

.9

.91.01.0

1.01.11.11.21.2

1.31.31.41.41.5

1.51.61.61.71.7

1.81.81.92.02.1

2.12.22.32.42.5

2.62.72.83.03.1

3.23.43.63.73.9

4.1

1 mile.

Feet.0.0.0.0.0.0

.0

.0

.0

.0

.0

.0

.0

.0

.0

.0

.0

.0

.0

.0

.0

.0

.0

.0

.0

.0

.0

.0

.0

.0

.0

.0- .0

.0

.0

.0

.0

.0

.0

.0

.0

.0

.0

.0

.0

.0

.0

1 miles.

Feet.0.5.6.6.6.6

.7

.7

.7

.8

.8

.8

.8

.9

.9

.9

1.01.01.01.11.1

1.21.21.31.31.4

1.41.41.51.61.6

1.71.71.81.91.9

2.02.12.22.32.4

2.52.6

2.82.93.0

3.2

2 miles.

Feet.0.9r.o1.0

. 1.01.1

1.11.21.21.31.3

1.41.51.51.61.6

1.71.71.81.81.9

2.02.12.22.22.3

2.42.52.62.72.8

2.93.0

3.13.23.3

3.43.63.83.94.1

4.34.54.85.05.2

5.5

2 miles.

Feet.1.21.21.31.31.4

1.41.51.61.61.7

1.81.81.92.02.0

2.12.22.22.32.4

2.52.62.72.82.9

3.03.13.23.33.4

3.63.73.94.04.2

4.34.54.74.95.15.'3

5.65.96.26.5

6.8

3 miles.

Feet.1.21.31.41.41.5

1.51.61.71.71.8

1.91.92.02.12.2

2.22.32.42.52.6

2.72.82.93.03.1

3.23.33.43.63.7

3.84.04.1'

4.34.4

4.64.85.05.35.5

5.76.06.36.77.0

7.3

COMPUTATION OF OFFSETS FROM SECANTS TO PARALLELS.

The formulas for computing geodetic coordinates and azimuths may be used to compute the offsets or azimuths for a secant line as follows: The distance, s (for metric measures), in the formula will be the distance lc or cd, figure 7, to the intersection of the parallel and secant from the central meridian. The azimuth of the secant at the central meridian c will be 90 or 270. - The latitude to be used in finding the constants A, B, C, etc., may (without appreciable error) be taken as that of the parrallel. As the cosine of 90 is 0 the first term of the latitude computation will be 0, and us the sine of 90


equals 1, the second term will reduce to $2C. The third and fourth terms will in all cases be negligible. Therefore, the seconds of latitude found from the second term when reduced to feet or other unit will be the offset eg. The azimuth of the secant where it intersects the parallel will be 90 plus or minus s (reduced to seconds of longitude) multiplied by the sine of the latitude, or

log s (in seconds) + log sine


b

~

c-

-

TV

g 2.908

CO

>0 N

e2.911

\

'

r

0) N

f

Z.9J5

00 CO ">

h.

8.723

z8-734

y6.745 m,

^ k 2.919 j 8.75S I

.006" ' 4015'

.006"' 10'

*

.006"

' 05'

06 " , 40QO

8O15' 10' 07*72' 05' 9000'

FIG. 8. Construction of polyconic projection. 15'of latitude and longitude; scale 1:48000. Construc- tion lines (to be drawn in pencil) dotted; final projection lines full.

EXAMPLE OF USE OP PROJECTION TABLES.

Let it be required to construct a projection for the area between parallels of 40 0(K arid 40 15' and meridians 90 00/ and 90 15' on a scale of 1:48000 (4,000 feet=l inch). For this scale it is customary to show meridians or parallels at intervals of 5 minutes, though any other desired interval may be adopted.

Through the center of the paper (see diagram, fig. 8) draw two fine pencil lines a-b and c-d exactly perpendicular to each other. The vertical line will be the meridian of 90 07' 30" and the intersection of the horizontal line with the vertical line will be a point on the parallel of 40 07' 30". From the column headed '' Meridional distance," Table 10, page 81, opposite 40 in column "Latitude of parallel," take


the value of a latitude interval of 5', which is 7.588 inches; lay off half of this interval or 3.794 inches, on the central meridian above and below the horizontal line; these distances will give points e and/, on the parallels of 40 10' and 40 05/ , respectively. The distance, 7.588 inches, laid off above and below the latter points will give points g and h for latitudes 40 15' and 40 00'. Through each of these points draw a line parallel to the horizontal line and perpendicular to the vertical line first drawn.

In a similar manner lay off points on the horizontal lines through e, f, g, h, by meas- uring from the central meridian east and west distances obtained from columns headed "Abscissas of developed parallel" in Table 10, page 81, for the appropriate latitudes and for the longitude intervals of 2y and 7$'. Thus, for 40?, the tabular value for 2$' is 2.919 inches, for 5' it is 5.837 inches, and for 7J' it is 8.755 inches, which fix the points i, j, k, and 1. In order to find corner points of the 5-minute area take from the columns headed "Ordinatesof developed parallel" in Table 10, on .the same page, opposite the latitude, the distances for the "Longitude intervals" 2$' and 7$' (the value of 2y for the 1:48000 scale is inappreciable, being less than 0.001 inch); lay these distances off perpendicularly northward from the horizontal lines, giving points x, y, z, etc., the required corners, and through these points draw curves of the parallel concave toward the north and meridional lines joining corresponding points as x, y, zon each parallel. After testing the accuracy of the plotting by comparing the length of the diagonals/ i=f l, h m=h n, etc., the projection may be inked in.

In a similar manner projections may be constructed for other scales or areas. Table 8, for the scale of 1:63360 (1 mile to 1 inch), may be used for any even fraction or multiple of a mile. The distance between parallels being found from column "Meridional distance"; distances not given may be found by simple proportion except for "Ordi- nates of developed parallel," which increase as the square of the distance from the central meridian, or actually as the vers sin of angle of convergence. For scales of any number of thousands of feet to 1 inch use suitable fractions of the distance given for scale 1:12000 (1,000 feet to 1 inch) in Table 12.

For maps of large areas Table 6 gives the actual or full scale distances in meters. These may be divided by the proper scale ratio and the distances so found platted with a metric scale or reduced to feet by the table on page 291. The first part of this table gives the actual lengths of arcs of 1 along the meridian for each degree of latitude as its middle point (see p. 36), the second part gives the lengths of arcs of the parallels for 1 of longitude for each degree of latitude (see p. 37), and the third part gives lengths of the arcs of the parallel for V of longitude. The remainder of Table 6 entitled " Coordinates of curvature " (see pp. 39-47) is for use in making polyconic projections; these are rectangular coordinates of the curves of parallels. The X values are the distances measured east and west from the central meridian, each to be taken for the proper latitude, and the corresponding Y values give the offsets at right angles to the horizontal lines to fix intersection points of the curved parallels with the other meridians. For projections of large extent the meridians differ sen- sibly from straight lines and they as well as the parallels must be drawn as curves.

36 GEOGEAPHIC TABLES AFD FOEMULAS.

TABLE 6. For projection of maps of large areas.

[The ratio of the yard to the meter as stated by Clarke, namely, 1 meter = 1.093623 yards = 39.370432 inches, is that used in the table.]

LENGTHS OF DEGREES OP THE MERIDIAN.

Latitude.

O

012345

6789

10

1112131415

1617181920

2122232425

2627282930

3132333435

3637' 38

3940

4142434445

Meters, a

110, 567. 2110, 567. 6110, 568. 6110, 570. 3110, 572. 7110, 575. 8

110, 579. 5110, 583. 9110, 589. 0110, 594. 7110, 601. 1

110, 608. 1110, 615. 8110, 624. 1110, 633. 0110, 642. 5

110, 652. 6110, 663. 3110, 674. 5110, 686. 3110, 698. 7

110,711.6110, 725. 0110, 738. 8110, 753. 2110, 768. 0

110, 783. 3110,799.0110, 815. 1110, 831. 6110, 848. 5

110, 865. 7110, 883. 2110, 901. 1110, 919. 2110, 937. 6

110, 956. 2110, 975. 1110, 994. 1111, 013. 3111, 032. 7

111, 052. 2111, 071. 7111, 091. 4111, 111. 1111, 130. 9

Statute miles.

68. 70468. 70468. 70568. 70668. 70868. 710

, 68.71268. 71568. 71868. 72168. 725

68. 73068. 73468. 73968. 74468. 751

68.75768. 76468. 77168. 77868. 786

68. 79468. 80268. 81168. 82068. 829

68. 83968. 84868. 85868. 86968. 879

68. 89068. 90168. 91268. 92368. 935

68. 94668. 95868. 96968. 98168. 993

69. 00669. 01869. 03069. 04269. 054

Latitude.

O

454647484950

5152535455

5657585960

6162636465

6667686970

7172737475

7677787980

8182838485

8687888990

Meters, a

111, 130. 9111, 150. 6111, 170. 4111, 190. 1111, 209. 7111, 229. 3

111, 248. 7111,268.0111, 287. 1111,306.0111, 324. 8

111, 343. 3111, 361. 5111, 379. 5111,397.2111, 414. 5

111,431.5111, 448. 2111, 464. 4111, 480. 3111, 495. 7

111, 510. 7111,525.3111, 539. 3111, 552. 9111, 565. 9

111, 578. 4111,590.4111,601.8111,612.7111, 622. 9

111, 632. 6111,641.6111, 650. 0111, 657. 8111, 664. 9

111, 671. 4111, 677. 2111, 682. 4111, 686. 9111, 690. 7

111, 693. 8111, 696. 2111, 697. 9111, 699. 0111, 699. 3

Statute miles.

69. 05469. 06669. 07969. 09169. 10369. 115

69. 12769.13969. 15169. 16369. 175

69. 18669. 19769. 20969. 22069. 230

69. 24169. 25169. 26169. 27169. 281

69. 29069. 29969. 30869. 31669. 324

69. 33269.34069. 34769. 35469. 360

69. 36669. 37269. 37769. 38269. 386

69. 39069. 39469. 39769. 40069. 402

69. 40469. 40569. 40769. 40769. 407

a These quantities express the number of meters and statute miles contained within an arc of which the degree of latitude named is the middle; thus, the quantity 111,032.7, opposite latitude 40, is the number of meters between latitude 39 30' and latitude 40 30'.


TABLE 6. For projection of maps of large areas Continued.

[Extracted from Appendix No. 6, U. S. Coast and Geodetic Survey Report for 1884.]

LENGTHS OP DEGREES OF THE PARALLEL.

Latitude.

0

01234

56789

1011121314

1536171819

2021222324

2526272829

3031323334

3536373839

404142434445

Meters.

111,321 !111, 304111, 253111, 169111,051

110, 900110, 715110, 497HO, 245109, 959

109,641109, 289108, 904108, 486108, 036

107, 553107, 036106, 487105, 906105, 294

104, 649103, 972103, 264102, 524101,754

100, 952100, 11999, 25798, 36497, 441

96, 48895, 50694, 49593, 45592, 387

91, 29090, 16689, 01487, 83586, 629

85, 39684, 13782, 85381,54380, 20878, 849

Statute miles.

69. 17269. 16269. 13069. 07869. 005

68. 91168. 79568. 66068. 50468. 326

68. 12967.91067. 67067. 41067. 131

66. 83066. 51066. 16965. 80865. 427

65. 02664. 60664. 16663. 70663. 228

62. 72962. 21261. 67661. 12260. 548

59. 95659. 34558. 71658. 07157. 407

56. 72556. 02755. 31154. 57953. 829

53. 06352. 28151. 48350. 66949. 84048. 995

Latitude.

o 4546474849

5051525354

55 '56575859

6061626364.

6566676869

707172 '

7374

7576777879

8081828384

858687888990

Meters.

78, 84977, 46676, 05874, 62873, 174

71, 69870, 20068, 68067, 14065, 578

63, 99662, 39560, 77459, 13557, 478

55, 80254,11052, 40050, 67548, 934

47, 17745, 40743, 62241, 82340, 012

38, 18836, 35334, 50632, 64830, 781

28, 90327, 01725, 12323, 22021, 311

19, 39417,47215, 54513, 61211, 675

9,7357,7925,8463,8981,949

0

Statute miles.

48. 99548. 13647. 26146. 37245. 469

44. 55243. 62142. 67641.71940. 749

39. 76638. 77137. 76436. 74535. 716

34. 67433. 62332. 56031. 48830. 406

29. 31528. 21527. 10625. 98824. 862

23. 72922. 58921. 44120. 28719. 127

17. 96016. 78815. 61114. 42813. 242

12. 05110. 8579. 6598.4587.255

6.0494.8423.6322.4221.2110.000

38 GEOGEAPHIC TABLES AND FORMULAS.

TABLE 6. For projection of maps of large areas Continued.

[Extracted from Appendix No. 6, U. S. Coast and Geodetic Survey Report for 1884.]

ARCS OF THE PARALLEL IN METERS.

Latitude.

o /

24 001020304050

25 001020304050

26 001020304050

27 001020304050

28 001020304050

29 001020304050

30 001020304050

31 001020304050

32 001020304050

Value of 1'..

1695. 91693. 71691.51689.31687. 01684. 81682. 51680. 31678. 01675. 71673. 31671. 0

1668. 71666. 31663. 91661. 51659. 11656. 71654. 31651. 81649. 41646. 91644. 41641. 91639. 41636. 91634. 31631. 81629. 21626. 61624. 01621.41618.81616. 11613.51610.81608. 11605.41602.71600. 01597. 31594.5

1591. 81589. 01586. 21583. 41580. 61577. 81574.91572. 11569. 21566. 31563. 41560. 5

Latitude.

o /

33 001020304050

34 001020304050

35 001020304050

36 001020304050

37 001020304050

38 001020304050

39 001020304050

40 001020304050

41 001020304050

Value of 1'.

1557. 61554. 71551. 71548. 71545. 81542. 8

1539. 81536. 81533. 71530. 71527. 61524. 6

1521. 51518. 41515. 31512. 21509. 11505. 91502. 81499. 61496.41493. 21490. 01486. 81483. 61480. 31477. 11473. 81470. 51467. 21463. 91460.61457.31453. 91450.61447.21443. 81440. 41437.01433. 61430. 21426. 7

1423. 31.419.81416.31412. 8"1409. 31405. 81402. 31398. 81395. 21391.61388.11384. 5

Latitude.

o /

42 001020304050

43 001020304050

44 001020304050

45 001020304050

46 001020304050

47 001020304050

48 001020304050

49 001020304050

50 001020304050

Value of 1'.

1380. 91377. 31373. 71370.01366. 41362. 71359. 11355. 41351. 71348. 01344. 31340. 5

1336. 81333. 11329. 31325. 51321.71318.01314. 21310. 31306. 51302. 71298. 81295. 01291.01287. 21283. 31279. 41275. 51271.61267. 61263. 71259. 71255. 81251.81247. 81243. 81239.81235. 81231. 71.227. 71223. 6

1219. 61215. 51211.41207.31203. 21199.11.195. 0

. 1190.81186. 71182.51178.41174. 2


TABLE 6. For projections of maps of large areas Continued.

COORDINATES OF CURVATURE.

Natural scale. Values of X and Y in meters.

Latitude 24.

Longi- tude.

o / 1 002 003 004 00

6 006 007 008 009 00

10 0011 0012 0013 0014 00

15 0016 0017 0018 0019 00

20 0021 0022 0023 0024 00

25 0026 0027 0028 0029 0030 00

X

101,753203, 500

. 305,237406, 959

508, 660610, 336711,981813, 590915,159

1, 016, 6811; 118, 1521,219,5661,320,9191,422,205

1,523,4201,624,5581,725,6141, 826, 5831, 927, 460

2, 028, 2402, 128, 9182,229,4882,329,9462,430,287

2,530,5052,630,5962,730,5542, 830, 3742,930,0523,029,582

Y

3611,4453,2506,778

9,02813, 00117,69523, 10929,245

36, 10243, 67951,97760,99470, 731

81,18692, 360

104, 251116,859130, 184

144,225158, 981174, 451190, 634207, 530

225, 138243,458262, 487282, 225302, 671323, 825

Latitude 25.

Longi- ' tude.

o / 1 002 003 004 00

5 006 007 008 009 00

10 0011 0012 0013 00

.14 00

15 0016 0017 0018 0019 00

20 0021 0022 0023 0024 00

25 0026 0027 0028 0029 0030 00

X

100, 951201, 896302, 831403, 749

504, 645605, 514706, 349807, 146907, 899

1,008,6031,109,2521,209,8411,310,3641,410,815

1,511,1901,611,4831,711,6881,811,8001, 911, 813

2,011,7222, 111, 5222,211,2072, 310, 7712,410,210

2,509,5182, 608, 6892, 707, 7182, 806, 6002, 905, 3293,003,900

Y

3721,4893,3515,957

9,30713, 40118, 23923, 82130, 146

37,21545, 02653, 57862,87372,909

83,68595,202

107, 458120, 453134,186

148,656163, 862179,805196,482213,894

232,038250, 914270, 521290,859311,925333, 718

Latitude 26.

Longi- tude.

o / 1 002 003 004 00

5 006 007 008 009 00

10 0011 0012 0013 0014 00

15 0016 0017 0018 0019 00

20 0021 0022 0023 0024 00

25 0026 0027 0028 0029 0030 00

X

100, 118200, 231300, 332400,416

500,476600,506700, 501800, 456900, 364

1,000,2181,100,0151, 199, 7471,299,4091,398,994

1,498,4981,597,9141,697,2371, 796, 4601,895,578

1,994,5852, 093, 4752, 192, 2432, 290, 8822, 389, 387

2, 487, 7532, 585, 9732,684,0422,781,9532,879,7022, 977, 281

' Y

3831,532

.3,4476,128

9, 57413, 78618, 76324, 50531,011

38,28246, 31655, 11464, 67574,998

86,08297,928

110,534123, 899138,023

152, 905168, 544184, 939202, 089219, 993

238, 650258,061278, 222299, 132320, 788343, 197

40 GEOGEAPHIC TABLES AND FOEMULAS.




Latitude 27.

Longi- tude.

0 /

1 002 003 004 00

5 006 007 008 009 00

10 0011 0012 0013 0014 00

15 0016 0017 0018 0019 00

20 0021 0022 0023 0024 00

25 0026 0027 0028 0029 0080 00

X

99,256198,505297, 742396, 960

496, 154595, 316694,440793, 522892,554

991,5291,090,4421, 189, 2871,288,0571, 386, 746

1,485,3481, 583, 8571, 682, 2671,780,5701, 878, 762

1, 976, 8362, 074, 7862, 172, 062, 270, 2892, 367, 830

2, 465, 2222, 562, 4592, 659, 5352, 756, 4452, 853, 1812,949,739

Y

3931,5733,5396,291

9,82914,15419,26425, 15931,839

39, 30347, 55156, 58366,39876, 995

88, 374100, 534113, 474127, 193141,690

156, 966173,018189,845207, 447225, 823

244, 970264,889285, 577307, 035329, 259352,249

Latitude 28.

Longi- tude.

O 1

1 002 003 004 00

5 006 007 008 009 00

10 0011 0012 0013 0014 00

15 0016 0017 0018 0019 00

20 0021 0022 0023 0024 00

25 0026 0027 0028 0029 0030 00

X

98,363196,719295, 062393, 385

49] , 682589, 945688, 168786, 347884,472

982,5371,080,5371, 1 78, 4641, 276, 3121,374,075

1,471,7451,569,3151,666,7811, 764, 1351,861,371

1,958,4812, 055, 4602, 152, 3022, 248, 9982, 345, 544

2,' 441, 9322, 538, 1562,634,2102, 730, 0872,825,7792,921,284

Y

4031,6123,6276,447

10,07314, 50519, 74125, 78232, 827

40, 27648, 72857, 98368,04078, 899

90, 558103,017116,2/5130, 331145, 185

160, 835177. 280194,518212, 550231, 374

250, 988271,391292, &82314, 559337, 321360, 866

Latitude 29.

Longi- tude.

o / 1 002 003 004 00

5 006 007 008 009 00

10 0011 0012 0013 0014 00

15 0016 00 ,17 0018 0019 00

20 0021 0022 0023 0024 00

25 0026 0027 0028 0029 0030 00

X

97,439194, 872292, 291389, 689

487,059584, 394681,687778, 931876, 120

973,2461,070,3021,167,2821,264,1781,360,983

1,457,6911,554,2951, 650, 7871,747,1611,843,410

1,939,5272, 035, 5052, 131, 3382, 2i7, 0202,322,539

2, 417, 8932,513,0742, 608, 0752, 702, 8902,797,5112,891,931

Y

4121,6493,7106,595

. 10,30514,838

. 20.19426,37433, 376

41, 19949, 84559,31369, 60180,706

92, 631105, 375118, 935133, 311148,602

164,506181,324198, 953217, 392236, 640

256, 695277,558299, 224321, 694344, 964369, 036





Latitude 30.

Longi- tude.

o / 1 002 003 004 00

5 006 007 008 009 00

10 0031 0032 0013 0014 00

15 001C 0017 0018 0019 00

20 0021 0022 0023 0024 00

25 0026 0027 00

. 28 0029 0030 00

X

96, 487192, 967289, 432385,875

482,288578, 665674, 998771, 279867,502

963, 6581,059,7413,155,7441,251,6581,347,477

1,443,1931,538,8003,634,2901,729,6541,824,887

1,919,9822,014,9302, 109, 7252, 204, 3592, 298, 825

2,393,1162, 487, 2242, 581, 1442,674,8672, 768, 3852,861,694

Y

4211,6843,7896,735

10, 52315,15320, 62326,93434,084

42, 07450,90360, 57071,07482,415

94, 591107, 603321,449136, 127151,637

167,977185, 147203, 143221, 966243,616

262,089283, 383305, 498328, 432352, 383376, 749

Latitude 31.

Longi- tude.

0 /

1 002 003 004 00

5 00'6 007 008 009 00

10 0011 0012 0013 0014 00

15 0016 0017 0018 0039 00

20 0021 0022 0023 0024 00

25 0026 0027 0028 0029 0030 00

X

95, 505191,002286,484381, 943

477, 371572, 760668, 103763,392858, 619

953,7771,048,8583,143,8543,238,7581,333,561

1,428,2573,522,8371, 617, 2941,713,6213,805,810

1,899,8521, 993, 7402,087,4682,181,0272, 274, 411

2, 367, 6102,460,6382, 553, 4272, 646, 0292, 738, 4182, 830, 585

Y

4293,7173,8636,867

10, 72915, 450.21,02727, 46134, 751

42,89751,89861, 75372,46284, 024

96, 437309, 701123, 815138, 777154, 586

171,241188, 741207, 085'226, 270246, 295

267,159288, 860313,396334, 765358, 966383, 997

Latitude 32.

Longi- tude.

o / 1 002 003 004 00

5 006 007 008 009 00

10 0011 0012 0013 0014 00

15 0016 0017 0018 0019 00

20 0021 0022 0023 0024 00

25 0026 0027 0028 0029 0030 00

X

94,494388,980283,449377,894

472, 307566, 680663,004755, 272849, 475

943,6051,037,6551,131,6161,225,4801,319,239

1,412,8851,506,4111,599,8081, 693, 0671, 786, 382

3,879,1441,971,9462,064,5792, 157, 0352, 249, 305

2, 341, 3852,433,2642,524,9352, 616, 3902, 707, 6212, 798, 621

Y

4373 , 7483,9336,991 .

10,92215, 72721, 40427,95435, 375

43, 66752, 821)62, 86173, 763.85, 529

98, 164311,664126, 029141,256357, 346

174,296192,305210, 772230, 295250, 672

271, 901293, 981316, 910340, 080365, 307390, 770




Natural scale. Values of X and Y in meters.^

Latitude 33.

Longi- tude.

o / 1 002 003 004 00

5 006 007 008 009 00

10 0011 0012 0013 0014 00

15 0016 0017 0018 0019 00

20 0021 0022 0023 0024 00

25 0026 0027 0028 0029 0030' 00

X

93, 454186,899280, 328373,731

467, 100560, 428653,704746, 922840,072

933, 1461,026,1361,119,0331,211,8291, 304, 515

1,397,0831,489,5261,581,8341,673,9981, 766, Oil

1,857,8661,949,5532,041,0622, 132, 3872,223,521

2,314,4532,405,1752, 495, 6802, 585, 9612, 676, 0072, 765, 812

Y

4441,7773,997,7, 106

11, 10215, 98621,75728, 41435,957

44,38553,69763, 89374,97186, 931

99, 771113,491128,089143, 564159,914

177, 138195,234214, 201234,037254, 740

276; 309298, 741322,034346, 187371, 197397,061

Latitude 34.

Longi- tude.

o / 1 002 003 004 00

5 00-6 007 008 009 00

10 0011 0012 0013 0014 00

15 0016 0017 0018 0019 00

20 0021 00 22 0023 0024 00

25 0026 0027 0028 0029 0030 00

X

92,385184,762277, 121360,454

461,751554,004646, 205738,344830, 413

922,4031, 014, 3051,106,1101,197,8091,289,395

1,380,8581,472,1901,563,3811,654,4231,745,308

1, 836, 0261, 926, 5692,016,9292, 107, 0972, 197, 065

2,286,8232, 376, 3632, 465, 6772,554,7562, 643, 5912, 732, 175

Y

4511,8034,0577,212

11,26816,22522,08228,83936, 494

45,04854,49964,84676, 08988,227

101,258115, 180129, 993145, 696162, 287

179, 763198, 124217, 368237, 493

. 258, 497

230,378 .303, 134326, 763351, 262376, 629402, 863

Latitude 35.

Longi- tude.

o / 1 002 003 004 00

5 006 007 008 009 00

10 0011 0012 0013 0014 00

15 0016 0017 0018 0019 00

20 0021 0022 0023 0024 00

25 0026 0027 0028 0029 OC30 00

X

91,289182,568273, 830365, 064

456,261547,412638,509729,542820, 501

911,3791,002,1651,092,8501,183,4261,273,884

1,364,2141, 454, 4071,544,4541,634,3471,724,076

1,813,6321,903,0061,992,1902, 081, 1742,169,949

2,258,5072, 346, 8382, 434, 9342,522,7872, 610, 3862, 697, 724

Y

4571,8284,1127,310

11,42116, 44522,38129,22936, 987

45, 65655, 23465,7217.7, 11589,415

102, 619116, 728131,738147, 650164, 460 -

182, 168200, 772220, 268240, 657261, 936

284, 102307, 154331,089355, 905381,598408,168




Natural scale. Values of X and Y meters.

Latitude 36.

Longi- tude.

o / 1 002 003 004 00

6 006 007 008 009 00

10 0011 0012 0013 0014 00

15 001C 0017 0018 0019 00

20 0021 0022 0023 0024 00

25 0026 0027 0028 0029 0030 00

X

90, 164180, 319270,455360, 562

450, 631540, 653630, 618720, 517810,340

900,078989, 720

1, 079, 2591,168,6841,257,987

1, 347, 1561,436,1841,525,0611, 613, 7771, 702, 324

1, 790, 6911,878,8701,966,8512,054,6252,142,183

2, 229, 5162, 316, 6132,403,4672,490,0682, 676, 4072, 662, 475

Y

4621,8504, 1627,399

11, 56016, 64522,65229, 58337, 435

46,20955, 90306, 51578,04690,494

103, 856118, 133133, 323149,423166, 433

184,350203, 173222, 899243, 527265,055

287, 479310, 798335, 009360, 111386, 099412, 971

Latitude 37.

Longi- tude.

O 1

1 002 003 004 00

5 006 007 008 009 00

10 0011 0012 0013 0014 00

15 0016 0017 0018 0019 00

20 0021 0022 0023 0024 00

25 0026 0027 0028 00'29 0030 00

X

89, 012178, 015266, 997355, 951

444,865533, 730622, 536711,273799,932

888,503976, 975

1, 065, 3401,153,5871,241,707

1, 329, 6901,417,5261,505,2061, 592, 7211,680,059

1, 767, 2111, 854, 1691,940,9222, 027, 4622,113,777

2, 199, 8602,285,6992,371,2872, 456, 6122,541,6672,626,441

Y

4671,8704,2077,479

11, 68516, 82422,89629,90137,838

46,70656, 50367,22978,88291,462

104, 967119, 395134,745151,015168, 203

186, 307205, 326225, 258246, 099267, 849

290, 503314,061338, 519363, 874390, 125417,267

Latitude 38.

Longi- tude.

o / 1 002 003 00'4 00

5 006 007 008 009 00

10 0011 0012 0013 0014 00

15 0016 0017 0018 0019 00

20 0021 0022 0023 0024 00

25 0026 0027 0028 0029 0030 00

X

87, 833175, 656263,458351,230

438, 962526, 643614,263701,812789, 280

876,657963, 933

1,051,0981,138,1411,225,053

1,311,8231,398,4411,484,8991,571,1851,657,289

1,743,2021,828,9141,914,4151,999,6942,084,743

2, 169, 5512,254,1092,338,4062,422,4332, 506, 1812, 589, 639

Y

4721,8884,2477,549

11,79516, 98323,11230,18338,195

47,14557,03467,86079, 62292, 319

105,949120,511136,002152, 421169,767

188,037207,229227, 341248, 370270, 315

293, 172316, 939341,613367, 192393, 672421,050





Latitude 39.

Longi- tude.

o /1 002 003 004 00

5 006 007 008 009 00

10 0011 0012 0013 0014 00

15 0010 0017 0018 0019 00

20 0021 0022 0023 0024 00

25 0026 0027 0028 0029 0030 00

X

86. 627173, 213259, 8b9346, 403

432,925519,396605, 803692, 138778, 388

864, 545950, 598

1,036,5361,122,3491,208,027

1,293,5591,378,9341,464,1441,549,1771,634,023

1, 718, 6711, 803, 1131, 887, 3371,971,3332, 055, 091

2, 138, 6022,221,8542,304,8382, 387, 5452,469,9632, 552, 084

Y

4761,9034, 2817,611

11,89117, 12123, 30030, 42838,504

47,527, 57,496

68, 40980,26693,064

106, 802121,479137, 093153, 642171, 124

189, 537208, 878229, 146250, 337272, 450

295, 481319,429344, 289370, 059396, 736424, 317

Latitude 40.

Longi- tude.

0 /

1 002 003 004 00

5 006 007 008 009 00

10 0011 0012 0013 0014 00

15 0016 0017 0018 0019 00

20 0021 0022 0023 0024 00

25 0026 0027 0028 0029 0030 00

X

85,394170, 778256, 140341, 470

426, 757511,990597, 158682, 252767,260

852, 171936, 975

1,021,6611,106,2181, 190, 636

1,274,9041, 359, 0121, 442, 9491, 526, 7041,610,267

1,693,6281,776,7751,859,6981,942,3872, 024, 833

2, 107, 0232, 188, 9482, 270, 5972,351,9612, 433, 0292, 513, 790

Y

4791,9164,3117,663

11,97217,23823,46030,63738, 768

47,85267,88868,87580,81193, 695

107,525122, 300138, 017154, 675172, 272

190, 805210, 272230, 671251,998274, 252

297, 430321,528346,543372,473399, 314427,063

Latitude 41.

Longi- tude.

O 1

1 002 003 004 00

5 006 007 008 009 00

10 0011 0012 0013 0014 00

15 0016 0017 0018 0019 00

20 0021 0022 0023 0024 00

25 0026 0027 0028 00'29 0030 00

; x

84,136168, 260252,363336, 432

420,457504, 428588, 332672, 159755,897

839, 537923, 067

1,006,4751, 089, 7521, 172, 886

1,255,8661,338,6811,421,3211, 503, 7751,586,031

1, 668, 0791,749,9091,831,5091, 912.-8691,993,978

2, 074, 8262, 155, 4022,235,6952,315,6952, 395, 3922,474,774

Y

4821,9274,3357,706

12, 03917, 33523, 59130, 80738, 983

48,11858, 20969, 25681,25894,212

108,117122, 971138, 773155, 520173, 210

191,841211, 409231, 914253, 352275,719

299, 014323,233348, 374374, 432401, 404429,287





Latitude 42.

Longi- tude.

0 /

1 002 003 004 00

5 006 007 008 009 00

10 0011 0012 0013 0014 00

15 0016 0017 0018 0019 00

20 0021 0022 0023 0024 00

25 0026 0027 0028 0029 0030 00

X

82,851165,691248, 508331,292

414,030496,712579,325661,861744,305

826,648908, 879990, 985

1,072,9561, 154, 781

1,236,4491,317,9481,399,2671, 480, 3951,561,321

1,642,0351, 722, 5241,802,7791,882,.7881,962,540

2, 042, 0242, 121, 2302,200,1462, 278, 7622,357,0672, 435, 052

Y

4841,9354,3547,739

12, 09217, 41023, 69330, 94139, 152

48,32558,45969, 55381, 60594, 614

108,577123,493139, 360156, 175173, 937

192, 642212, 289232,874254,396276,850

300, 234324,544349, 778375, 932403, 002430, 985

Latitude 43.

Longi- tude.

0 /

1 002 003 004 00

5 006 007 008 009 00

10 0011 0012 0013 0014 00

15 0016 0017 0018 0019 00

20 0021 0022 0023 0024 00

25 0026 0027 0028 0029 0030 00

X

81,541163,071244,578326, 050

407, 476488, 844570, 143651,361732, 486

813,608894,415975, 195

1,055,8371, 136, 329

1, 216, 6611,296,8201,376,7951,456,5751,536,148

1,615,5051,694,6321,773,5191,852,1551,930,528

2,008,628'2, 086, 4432, 163, 9632,241,1762,318,0712, 394, 639

Y

4851,9414,3677,763

12, 12917, 46423,76631,03639,272

48,47458,63969, 76681,85494,901

108, 905123,864139, 777156, 640174,451

193,209212,909233, 551255,129277, 642

301,087325,459350, 750376, 974404,109432, 157

Latitude 44.

Longi- tude.

O 1

1 002 003 004 00

5 006 007 008 00.9 00

10 0011 0012 0013 0014 00

16 0016 0017 0018 0019 00

20 0021 0022 0023 0024 00

25 0026 0027 0028 0029 0030 00

X

80,206160, 401240, 572320, 708

400, 797480, 827560, 786640, 662720,445

800,122879, 681959, 110

1,038,3991,117,535

1, 196, 5071,275,3031, 353, 9111, 432, 3201,510,519

1,588,4961,666,2401,743,7381,820,9301,897,955

1,974,6502, 051, 0552,127,1592, 202, 9502,278,4172, 353, 550

Y

4861,9454,3757,778

12, 15217, 49623,81131,09439, 345

48,56358, 74669,89382,00295,072

109,100124,084140,023156, 913174, 753

193,540213, 270233, 942255,552278, 096

301,572325, 977351, 306377, 555404, 722432,801





Latitude 45.

Longi- tude.

o / 1 002 003 004 00

5 006 007 008 009 00

10 0011 0012 0013 0014 00

15 0016 0017 0018 0019 00

20 0021 0022 0023 00

. 24 00

25 0026 0027 0028 0029 0030 00

X

78,847157, 682236,493315, 269

393, 996472, 663551,258629, 769708,184

786,492864, 679942, 735

1; 020, 6471,098,404

1,175,9941, 253, 4041, 330, 6241,407,6401,484,443

1,561,0191, 637, 3581,713,4471,789,2761,864,831

1, 940, 1032,015,0792, 089, 7492, 164, 1002, 238, 1212, 311, 802

Y

4861,9464,3787,783

12, 16017,50823,82631,11439, 370

48,59458, 78269, 93682,05195, 127

109, 162124, 153140, 099156,996174, 842

193, 635213, 371234, 048255, 663278,211

301,690326,097351,427377, 676404,841432, 918

Latitude 46.

Longi- tude:

0 '

1 002 003 004 00

5 006 007 008 009 00

10 0011 0012 0013 0014 00

15 0015 0017 0018 0019 00

20 0021 0022 0023 0024 00

25 0026 0027 0028 0029 0030 00

X

77, 464154, 915232, 342309,732

387,074464,354541, 562618, 684695, 708

772,623849,416926, 075

1,002,5881, 078, 943

1,155,1281,231,1311,306,9401,382,5431,457,928

1,533,0831,607,9971,682,6571,757,0521,831,170

1,904,9991, 978, 5282,051,7452, 124, 6392, 197, 1972, 269, 410

Y

4861,945

. 4,3767,779

12, 15317,49823,81331,09639, 347

48, 56558, 74769, 89382,00095, 067

109,091124,071140, 003156,887174, 718

193, 494213,212233, 869255, 462277,987

301,441325, 820351,120377, 337404,468432, 507

Latitude 47.

Longi- tude,

0 27 then for 2 41.56 0.44

The sum 1457">32.56+2m

is the required

2 27.

27 . 44-1sidereal

44

ShQmO'time.


TABLE 20. For conversion of sidereal lime into mean lime.

127

s

0

12 3 4 5 6 7 8 0

10

11 12 13 14 15 16 17 18 19

20

2122 23 24 25 26 27 28 29

30

31 32 33 34 35 36 37 38 39

40

41 42 43 44 45 46 47 48 49

60

5152 53 64 55 56 57 58 59

60

m 0

h m s 000

0 6 ,- 6 0 12 12 0 18 19

0 24 25 0 30 31 0 36 37 0 42 44 0 48 50 ' 0 64 56

112

179 1 13 15 1 19 21 1 25 27 1 31 34 1 37 40 1 43 46 1 49 52 1 55 59

225

2 8 11 2 14 17 2 20 24 2 26 30 2 32 36 2 38 42 2 44 49 2 50 55 2 57 1

337

3 9 14 3 15 20 3 21 26 3 27 32 3 33 38 3 39 45 3 45 51 3 51 57 3 58 3

4 4 10

4 10 16 4 16 22 4 22 28 4 28 35 4 34 41 4 40 47 4 46 53 4 53 0 4 59 6

5 5 12

5 11 18 5 17 25 5 23 31 5 29 37 5 35 43 6 41 50 5 47 56 5 54 2 608

6 6 15

m 1

h m s 6 6 15

6 12 21 6 18 27 6 24 33 6 30 40 6 36 46 6 42 52

' 6 48 58 6 55 4 7 1 11

7 7 17

7 13 23 7 19 29 7 25 36 7 31 42 7 37 48 7 43 54 7 50 1 7 66 7 8 2 13

8 8 19

8 14 26 8 20 32 8 26 ;*8 8 32 44 8 38 51 8 44 57 8 51 3 8 57 9 9 3. 16

9 9 22

9 15 28 9 21 34 9 27 41 9 33 47 9 39 53 9 45 59 9 52 5 9 58 12

10 4 18

10 10 24

10 16 30 10 22 37 10 28 43 10 34 49 10 40 55 10 47 2 10 53 8 10 59 14 11 5 20

11 11 27

11 17 33 11 23 39 11 29 45 11 35 62 11 41 58 11 48 4 11 54 10 12 0 17 12 6 23

12 12 29

m2 .

h m s 12 12 29

12 18 35 12 24 42 12 30 48 12 36 54 12 43 0 12 49 7 12 55 13 13 1 19 13 '7 25

13 13 31

13 19 38 13 25 44 13 31 50 13 37 56 13 44 3 13 50 9 13 56 15 14 2 21 14 8 28

14 14 34

14 20 40 14 26 46 14 32 53 14 38 59 14 45 5 14 51 11 14 57 18 16 3 24 15 9 30

15 15 36

15 21 43 15 27 49 15 33 55 15 40 1 15 46 8 15 52 14 15 58 20 16 4 26 16 10 33

16 16 39

16 22 45 16 28 51 16 34 57 16 41 4 16 47 10 16 53 16 16 59 22 17 5 29 17 11 35

17 17 41

17 23 47 17 29 54 17 36 0 17 42 6 17 48 12 17 54 19 18 0 25 18 6 31 18 12 37

18 18 44

m 3

h m s 18 18 44

18 24 60 18 30 56 18 37 2 18 43 9 18 49 15 18 55 21 19 1 27 19 7 34 19 13 40

19 19 46

19 25 52 19 31 59 19 38 5 19 44 11 19 50 17 19 56 23 20 2 30 20 8 36 20 14 42

20 20 48

20 26 55 20 33 1 20 39 7 20 45 13 20 51 20 20 57 26 21 3 32 21 9 38 21 15 45

21 21 51

21 27 57 21 34 3 21 40 10 21 46 16 21 52 22 21 58 28 22 4 35 22 10 41 22 16 47

22 22 53

22 29 0 22 35 6 22 41 12 22 47 18 22 53 24 22 59 31 23 5 37 23 11 43 23 17 49

23 23 56

23 30 2 23 36 8 23 42 14 23 48 21 23 54 27 24 0 33 24 6 39 24 12 46 24 18 52

24 24 58

s 0.00

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

0.10

0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19

0.20

0.21 0.22 0.23 ,0. 24 0.25 0.26. 0.27 0.28 0.29

0.30

0.31 0.32 0.33 0.34 0.35 0.36 0.37 0.38 0.39

0.40

0.41 0.42 0.43 0.44 0.45 0.46 0.47 0.48 0.49

0.50

m s 0 0

0 4 0 7 0 11 0 15 0 18 0 22 0 26 0 29 0 33

0 37

0 40 0 44 0 43 0 51 0 55 0 59 1 2 1 6 1 10

1 13

1 17 1 21 1 24 l'28 1 32 1 35 1 39 1 43 1 46

1 50

1 54 1 57 2 1 2 5 2 8 2 12 2 16 2 19 2 23

2 .26

2 30 2 34 2 37 2 41 2 45 2 48 2 52 2 56 2 59

3 3

s 0.50

0.51 0.52 0.53 0.54 0.55 0.66 0.57 0.58 0.59

0.60

0.61 0.62 0.63 0.64 0.65 0.66 0.67 0.68 '

0.69

0.70

0.71 0.72 0.73 0.74 0.75 '

0.76 0.77 0.78 0.79

0.80

0.81 0.82 0.83 0.84 0.85 0.86 0.87 0.88 0.89

0.90

0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99

1.00

m s 3 3

3 7 3 10 3 14 3 18 3 21 3 25 3 29 3 32 3 36

3 40

3 43 3 47 3 51 3 64 3 58 4 2 4 5 4- 9 4 13

4 16

4 20 4 24 4 27 4 31 4 35 4 38 4 42 4 46 4 49

4 63

4 57 5 0 5 4 5 8 5 11 5 15 5 19 5 22 5 26

5 30

5 33 5 37 5 41 5 44 5 48 5 52 5 55 5 59 6 3

6 6

Example: Given 15h 0 0. The table gives

first for 14h 57m 18' 2 27" then for 2 42 0.44

15 0 0 2 27.44 The difference

15i.0m Os _ 2m 27M4 = 14" 57 32.56is the required mean time.

128 GEOGEAPHIO TABLES AND FORMULAS.

TABLE 21. For inlercohversion of feet and decimals of a mile.

Feet.

53106158211264

317370422475528

581634686739792

84589895010031056

11091162121412671320

Miles.

.01

.02

.03

.04

.05

.06

.07

.08

.09

.10

.11

.12

.13

.14

.15

.16

.17

.18

.19

.20

.21

.22

.23

.24

.25

Feet.

13731426.147815311584

16371690174217951848

19011954200620592112

21652218227023232376

24292482253425872640

Miles.

.26

.27

.28

.29

.30

.31

.32

.33

.34

.35

.36

.37

.38

.39

.40

.41

.42

.43

.44

.45

.46

.47

.48

.49

.50

Feet.

26932746279828512904

29573010306231153168

32213274332633793432

34853538359036433696

37493802385439073960

Miles.

.51

.52

.53

.54

.55

.56

.57

.58

.59

.60

.61

.62

.63

.64

.65

.66

.67

.68

.69

.70

.71

.72

.73

.74

.75

Feet.

40134066411841714224

42774330438244354488

45414594464646994752

48054858491049635016

50695122517452275280

Miles.

.76

.77

.78

.79

.80

.81

.82

.83' .84

.85

.86

.87

.88

.89

.90

.91

.92

.93

.94

.95

.96

.97

.98

.991.00


TABLE 22. For conversion of chains into feet.

129

0102030405060708090

0

Feet. 0660

1,3201,9802,6403,3003,9604,6205,2805,940

1

Feet.66

7261,3862,0462,7063,3664,0264,6865,3466,006

2

Feet.132792

1,4522,1122, 7723,4324,0924,7525,4126,072

3

Feet.198858

1,5182,1782,8383,4984,1584,8185,4786,138

4

Feet.264924

1,5842,2442,9043,5644,2244,8845, 544 '6,204

5

Feet.330990

1,6502,3102,9703,6304,2904,9505,6106,270

6

Feet.396

1,0561,7162,3763,0363,6964,3565,0165,6766,336

7

Feet.462

1,1221,7822,4423,1023,7624,4225,0825,7426,402

8

Feet.528

1,1881,8482,5083,1683,8284,4885,1485,8086,468

9

Feet.594

1,2541,9142,5743,2343,8944,5545,2145,8746,534

The feet for each whole number of chains up to 99 are given; and as links are hundredths they can also be read off (by pointing off two places) and added to the chains. Example: Convert 16 chains 27 links to feet. From the table 16 chains is read as 1,056 feet, and 27 links (read 27 chains and point off two places) as 17.82 feet. The sum, 1,073.83 feet, is obtained by mental addition.

TABLE 23. For conversion of feet into chains.

Feet.

000100200300400500600700800900

1,0001,1001,200

00

Chains.0.001.523.034.556.067.589.0910.6112.1213.6415.1516.6718.18

10

Chains.0.151.673.184.706.217.739.2410.7612.2713.7915.3016.8218.33

20

Chains.0.301.823.334.856.367.889.39

10.9112.4213.9415.4616.9718.48

30

Chains.0.461.973.485.006.538.039.5511.0612.5814.0915.6117.1218.64

40

Chains.0.612.123.645.156.678.189.7011.2112.7314.2415.7617.2718.79

50

Chains.0.762.273.795.306.828.339.8511.3612.8814.4015.9117.4218.94

60

Chains.0.912.423.945.466.978.4810.0011.5213.0314.5516.0617.5819.09

70

Chains.1.062.584.095.617.128.6410.1511.6713.1814.7016.2117.7319.24

80

Chains.1.212.734.245.767.278.7910.3011.8213.3314.8516.3617.8819.39

90

Chains.1.362.884.395.917.428.9410.4611.9713.4915.0016.5218.0319.55

Example: Convert 457 feet into chains 450 feet (line 400, column 50) equals 6.82 chains; 7 feet (line 000, column 70; 1.06 divided by 10) equals 0.106 chain; therefore, 457 feet equals 6.93 chains.

41623 16 9

130 GEOGKAPHIC TABLES AND FOEMULAS.

TABLE 24. Converting wheel revolutions into hundredths of a mile,

[Prepared by J. H. Jennings.]

[Scale divisions outside; revolutions inside.]

CIRCUMFERENCE OF WHEEL, 9.5 FEET.

0

0

10

20

30

40

50

60

70

80

90

1

661

117

172228283

339394

450

506

2

11

67122

178233289

344400

455511

3

1772

128183239

294

350405

461516

4

22

78133189244

300

355411

466522

5

28

83139194

250305361

416472528

6

3389

144200255

311

366422

478

533

7

3994

150

205261

316372

428

483539

8

44

100155211

266322378

433489544

9

50105161216272

328

383439

494

550

10

56111167222

278

333

389444

500555


0

0

10

20

30

40

50

60

70

8090

1

560116171225281

336

391

446501

2

11

66121177231286341

396

451506

3

1672

126182236292

347402

457512

4

22

77132188242297352

407

462517

5

2782137193247303

358413

468523

(i

3388143199253308

363418

473528

7

3893

148204258314

369424

479534

8

44

99154209264319

374429

484539

9

50105159215270325

380435

490544

10

55110165220275

330385440

495550


0

0

10

20

30

40

50

60

70

80

90

1

560

114169

223277

331

386441495

2

11

65120

174228283

337392446500

3

1671

125179

234288

342

397451

506

4

2276

131185239

294

348403457511

5

2781

136190

245299

353408462

517

6

3387

142196

250

305

359414468522

7

3892

147201

256

310

364419473528

8

4498

152206

261

316

370424479

533

9

49103158212

267

321

376429484

539

10

54109163218

272

326

381435490

544


TABLE 24. Converting wheel revolutions into hundredths of a mile Continued.

CIRCUMFERENCE OP WHEEL, 9.8 FEET.

0

0

10

L'O

30

40

50

60

70

80

90

1

559

113167221275329

383437

490

2

11

65119172

226280334

388442

496

3

16

70124178231286339394447

501

4

22

. 75129183237291345400

453

506

5

27

81

135

189

242

296

350

405

458

512

6

- 32

86

140

194

248

302

356

410

464

517

7

3891

145199253307361415469

522

8

43

97151205259313366421

474

528

9

49'l02

156

.211

265

318372

426480

533

10

54108162216270324377431485

539


0

0

10

20

30

40

50

60

70

80

00

1

559112165 219

272325

378432485

2

11

64117

170224277

330384437490

3

16

69122

176229282

336389442496

4

21

75128

181235288341394448501

5

27

80133

186240

293346400453506

6

32

85138192245298352405458512

7

37

91144

197251304357410464517

8

43

96149203256

309362416469522

48101155

208261314368421474528

10

53

107160213267320

373426480533

CIRCUMFERENCE OF WHEEL, 10 FEET.

0

0

10

20

80

40

50

60

70

80

90

1

558

111.164217269322

375428481

2

11

63116169222275327

380433486

3

1669

121174227280

333385438491

4

21

75127

180232285338391444496

5

26

80132185238290343396449502

6

3285

137190243296349401454507

7

3790

143195248301354406459512

8

42

96148201253306359412465517

9

48101153206259311364417470523

10

53106158211264317370422475528


TABLE 24. Converting wheel revolutions into hundredths of a mile Continued.

CIRCUMFERENCE OF WHEEL, mi FEET.

0

0

10

20

30

40

50

60

70

80

90

1

5

58110162

214267

319

371424

476

2

10

63115

167220272

324

376429

481

3

16

68121

173226277329

381434

486

4

21

73126

178231282334

386439492

5

26

79131183236288

340392445

497

6

31

84136188241293

345

397450

5

Documents

GEOGRAPHIC TABLES AND FORMULAS