Thomas Jefferson Lee.

A collection of tables and formulæ useful in surveying, geodesy, and ... online

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\ Mean Solar Chronometer No. 2440, by Parlanson fy



Instruments.



t
^

i


Times of ob-
senration by
Mean Solar
Chronometer
JVb. 2440.


True Sidereal
times of ob-
servation.


MBRIDIAN BISTANCB8.


— ^ eoa p.

t


!

In Sidn time


In arc


1

3
3
4
5

6

7


h. m. s.
1 33 02.5
1 34 28
1 35 42.7
1 36 38.2
1 39 07.5
1 41 11.2
1 44 28.2


h. m. s.
20 05 34.1
20 06 59.8
20 08 14.7
20 09 10.4
20 11 40.1
20 13 44.1
20 17 01.7


h.m, 8.
4 58 23.2
4 56 57.5
4 55 42.6
4 54 46.9
4 52 17.2
4 50 13.2
4 46 55.6


O t tf

74 35 48
74 14 22.5
73 55 39
73 41 43.5
73 04 18
72 33 22.5
71 43 54


t If

—24 18.1
-24 54.5
-25 19.8
—25 41.4
-26 34.7
-27 27.1
—28 40.8



Observer, Major J, D. Graham.
Computer, Do,



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LATITUDE.



205



Record and Computation.

Latitude, from observed double altitudes of Polaris.

(Gfnwer's/nn.)

Jwt hovn htfort Us upper meridian ptmage,
ficial horizon of Mercury.
Br^dahmn.



+ a(Astnp)'.
tang^.



+ 1 11.63
4- 1 11.41
+ 1 11.30
-h 1 11.04
+ 1 10.63
4- 1 10.28
+ 1 09.68



(^ 8inp)^
(^eosp.)



ObMT'd don- 1 Trae altitudes
ble alts, of! of Star, as cor-
Polaru out reeled for re-
oftheMeri-i fraction and
dian. I errors of in-

, suument,
! =A.



-0.32
■0.33
-0.33
0.33
0.34
-6.35
-0.37



O / // Of"*



93 01.30 ! 46 31 58.6



93 02.45
93 03.50
93 04.40
93 06.15
93 08.20



46 32 36
46 33 08.6
46 33 33.6
46 34 21
46 35 23.5



93 10.50 46 36 38.5



Latitude de-
duced from
each obser-
vation

= L.



46 08 51.8
46 08 52.6
46 08 59.7
46 09 02.9
46 08 56.6
46 09 06.3
46 09 07



Latitude— deduced from a mean of 7 altitudes of Star )
Polaris' )



A&> 08' 59".4



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206



ASTRONOMT.



Apparent declination of Star 88» S8f ^(K^5.

Apt. N. P. D. of Star = V 31' 29".5 = 5489".5 = A

Refraction (Ther.5T* — Bar. 30.013 inches) .... —55^.4

Index error of Sextant +3' 50"

Errors of excentricity Ac of Sextant -f- 1' 28"

k, m, $.

Apparent AR. of the Star PoUarii (« Una Mmarii) . . 1 03 57.3

Sidereal time at mean noon at this station 11 00 27.1

Sidereal interral from mean noon, of Suu-'s culmination .14 03 30.2
Retardation of mean on Sidereal time — 2 18.2

Mean time of colmination of Star PoUarii 14 01 12

Chron. No. 2440, fast of mean time at time of observation 4 29 24.8

Time by Chronometer of colmination of Star PoUarii . . 6 30 36.8

The redaction of the mean time of observation to sidereal time, in
the preceding example, might hare been omitted by using table of
•tflL tf» ore into sieim Hme, pages 152, Ac Thos — (1st observation)

Mean time of observation 1^33" 02^.5

Mean time colmination of Polaris 6 30 36 .8

Hoar angle, p, in intervals of mean time 4 57 34 .3

Sidereal eqoivalents, in are 4^ =s 60^09'5P.39

57»» = 14 17 20 .45

34> =r 8 31 .40

.3 rr 4 .51

p,inarc = 74** 35' 47".75

Form for computation — (Ist observation)

iMt term. 9d term. 3d term,

log cos p(-h)= 9.4242480 sin p =9.98411
" ^ = 3.7395327 A = 3.73953

Acosp == 3.1637807 Asinp «= 3.72364 ...=3.16378

= 1458".l

Ist term = — 24' 18".l (A sin p)«= 7.44728 ...=7.44728

log cL =4.38454 log )3= 8.89403

A =46^3158.6 tang A =0.02325

— — ^-— — — — ^— — 9.50509

46 07 40 .5 1.85507 3d t'm =l4)".32

2d term = + 1 11 .63 = 71".63

2d term =+1' 11".63

46 08 52 .13
3d term = — .32

LaUtode =46^08'51''.81



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LATITUDE. 207



VIII. Determination of the Latitude by transits over the
prime vertical.

Suppose a Transit instrument so placed, that the transit
axis is on the meridian, or very nearly so, and that the
axis is horizontal, and the coUimation nothing :

1. Call the time T, at which a star, whose declination
is D, passes the middle wire of the instrument on the
eastern side of the meridian, the clock correction to reduce
the observed time to the true £, and the right ascen-
sion of the star AR ; and let T' and £' denote the cor-
responding quantities for the western transit. Then the
two- hour angles, in sidereal time, will be, the eastern
negative,

/ = T + E — AR , <' = T' -f E' — AR .

Let the unknown Latitude of the place be L, and the
Azimuth of the line of collimation, a. The spherical
triangle, formed by great circles connecting the Zenith,
the Pole, and the place of the Star, gives the following
relations :

cos t cos D sin L — sin D cos L



cot a =



cos D sin /
cos t' cos D sin L — sin D cos L



cos D sin /'
Whence,

tang L= tang D^2!4iMi<)

^ ^ COS i (P — t)

If the instrument is very nearly on the prime vertical,
COS i{t+t)=:2 cos 0° = 1, and

tang L = tang D sec. i (i' — /)

for the passage over the middle wire of the instrument.



Digitized by VjOOQIC



f



208



ASTRONOMT.



2. Gall the time of passage of the Star, from a side
wire to the middle wire, f .

Let the distance, in arc, of one of the latersd wires
from the middle wire, measured on a great circle, be
15 /; / being the equatorial interval of the wire, in time.

Then, to reduce the transit over a side wire, to the
centre wire,

, L

^-^ 8in(L + D).sin(L — D)=b V/l^i

The upper sign of the term ± V /, is to be used for
wires crossed by the Star earlier than the middle wire in
the eastern transit, and later in the western transit, and
the lower sign in the opposite cases. An approximate
latitude may be used for L.

3. Should the optical axis not coincide with the middle
wire, substitute / ± c, for / in the above, according as
the error of coUimation c, lies on the same or opposite
sides of ^

4. The preceding formula gives the latitude on the
supposition that, the axis of the instrument is parallel to
the horizon. If the instrument is on the prime vertical,
but the north end of the axis is, for instance, n seconds
too high, the axis is parallel to the horizon of a place
whose latitude is n seconds less than where the instrument
is placed, and the true latitude is, therefore,

L + n

5. But should the instrument not be on the prime ver-
tical, the true latitude becomes

L -|- n sin a



\



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LATITUDE. 209



a being the Azimuth of the centre wire of the telescope,

supposed in collimation.

This may be found from the time elapsed between the

£. and W. transits of the same star, thus :

cot u = tang i (/' — ^) sin D .

_ sin u
sm a = cos v — ?- •
cosL

a is taken between 0° and 90° when the north end of the
transit axis is between the north and west, and between
90° and 180° when the same end is between the north
and east.

If n is called plus when the north end of the axis is
too high, and vice versoy the signs of the corrections are
indicated by those of the quantities resulting from the
formula.

When a is nearly 90°, the correction is exceedingly
small ; so that, when the instrument is placed nearly east
and west, we may proceed in all the computations as if
it were exactly so.

6. The instrument should be set up in the firmest
manner. A change of Azimuth between the east and
west transits of a Star will affect the result much less
than an equal change of level.

It is better, in order to obtain a close result in the
shortest time, to observe several Stars on the same even-
ing, and between the first and last observations to deter-
mine with the level the inclination of the axis several
times, and then to interpolate for transits between the
times of observation of the level. It is of course under-
stood that the changes of inclination must be small, which
will be the case if the instrument is properly placed.
27



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210 ASTRONOMY.

7. In order to point the telescope rightly, the hour an-
gles and zenith distances of the Stars to be observed must
be computed for the time of transit.

When the telescope is on the prime vertical, calling j9
the hour angle, and z, the zenith distance of the Staj,
then

coap = tang D cot L

sin D

cos z = -: — r

Bin L

An aUowance must be made for the time of crossing the
first wire, and for change of zenith distance from the first
to the middle wire.

8. To correct, for errors of Collimation, irregularity in
the pivots, etc., the instrument may be reversed between
the transits over each vertical; i. e., the wires on one side
of the centre wire are observed, the instrument reversed
in its Y's, and the transit over the same wires continued,
but in an inverse order. So that, in each vertical the same
wire is at one time as far north as it is at another south of
the optical axis.

Then let L = the latitude sought,

D = the apparent declination of the Star,
t = the hour angle, illuminated axis northy
= i diflf. of sidereal time of transit over the

same wire, for same position of axis.
/ = hour angle, illuminated axis ^ou^A,

tangP



tangL=



cos i (if + i) . cos i (/'—<)



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LATITUDE.



«11



IX. To determine ike Latitude of a place^ by observing the
difference of the meridional zenith distances of two Stars
on opposite sides of the zenith^ with the zenith and equal
altitude telescope.

Compute an approximate latitude by the formula.

where A and ^' are the polar distances of the south and
north Stars respectively, and (z— z') the quantity meas-
ured by the micrometer. Then,

1. The correction for level is applied by adding the an-
gle which the vertical axis of the instrument makes with
the zenith, when its inclination is southward, or subtract-
ing it when to the northward. This correction is found
by multiplying the value of 1 division of the level scale,
in arc, by one-half the mean change, in level divisions,
which any one end of the bubble undergoes by reversing
the instrument on the meridian; or, if o and «, o' and ef
denote the readings of the object and eye-ends of the bub-
ble, for south and north stars respectively; corrections for
level = i (o' — e!) — J (o — e) X the value of 1 division
of the level scale in arc.

2. The correction for error of meridional position of the
central vertical wire, is found by computing the usual '-re-
duction to the meridian" for each star; then the difference
between the reductions for the northern and southern stars
is taken, and one-half that difference added or subtracted,
according as the reduction for the northern star is greater
or less than that for the southern; or,



correction for position =



m'— »i



^



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212 ASTRONOMY.



191 being the reduction for stars south^ and tn! for stars
north of the zenith.

3. The correction for Refraction is applied similarly to
this last, but with a contrary sign; or,

r—r^
correction for refraction = — ^

(r — r') being small, no note need be taken of the state
of the barometer and thermometer at the time of obser-
vation. It is sufficient to use the actual tabular quantities.

Including all the corrections, the general expression for
Latitude will be

L- g +__.«+ - .b

(mi-m) {r-r')
■•" 2 "*" 2 •

a and 6 being the arc values of 1 division of the micro-
meter and level scale respectively.

4. Should the Star be observed on one side or the other
of the central wire, the reduction to the meridian be-
comes

225
m= -J- sin 1" . sin 2 A . j^'

= [6.4356974] sin 2 A. j5*

p being the hour angle of the Star in seconds of time.
Sine 2 ^ is negative when the Star is south of the equa-
tor or sub-polo.



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LATITUDE. 213



5. To find the value o, of 1 division of the microme-
ter, note the time by chronometer of the transit of Polaris
over the moveable wire placed vertically, and set succes-
sively to, say, every hundredth division of its scale.
Then let x be the angular distance from ihe meridian at
which any reading of the screw was had; />, the hour
angle of the Star at the same instant, and A its polar dis-
tance,

sin 0? = sin ^ sin p .

The value of x is computed for each reading, and the
difierences of these values, divided by the differences of
the corresponding micrometer readings, give values for
the screw.

6. The value by of 1 division of the level scale will be
best found by using, in conjunction with the micrometer,
a distant point as a mark, or the central wire of another
instrument used as a collimator; for the space above or
below the mark, passed over by the horizontal wire of the
micrometer, during the bubble's run over the scale, as the
telescope's elevation is gradually altered, may afterwards
be measured by the micrometer screw.

7. To correct, as much as possible, an erroneous deter-
mination of the value of the micrometer screw, select stars
for observation such, if practicable, that the greatest Z.
D. of a pair will belong as often to the N. Star as to the
S. Star; for if the Z. D. of the N. Star is the greatest,
the observed quantity is subtractive; if least, additive.
For, as a general rule, the error of latitude, arising from
an erroneous value to the micrometer screw, will be the
least when in a set of stars.



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314



A8TROR0MT.



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X. Knowing the time and the latitude of the placey to
find the .Azimuth of the Sun or a Star,

^i,A + S, = ^ip^\^y

A=i(A4-S)=Fi(A-S)

the upper or negative sign is used when x is greater
than ^.

Where

A = the azimuth counted from the northy which must
be subtracted from 180° if counted from the
south.

S = the angle at the star, called the angle of varia-
tion.

>. = the co-latitude of the place.

^ = the north polar distance of the sun or star.

p = the hour angle at the pole.

XI. Without the use of a chronometer j by observing the al-
titude of the sun or star at the same instant with the ob-
servation of the azimuth.

Let Z = the zenith distance, corrected for refraction,
parallax, and semidiameter.

sm Z sin X
2A;=Z +A+X



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/



216



ASTRONOMT.



XII. To find the amplitude of a celestial object at its rising
or setting; by amplitude is meant the complement of the
azimuthy or distance from the east or west points of the
horizon.

This is a particular case of the preceding problem.
When the object appears to be in the horizon, its zenith
distance, instead of being 90^, is, on account of refraction
and parallax, 90"" + k.

Where k = hor. refraction — hor. parallax
= 33' 45" — hor. parallax.

For stars, the hor. par. = and A = 90° 33' 45", for
the sun, k = 33' 45" _ 8" 6 and A = 90° 33'36".4; the
mean refraction and mean hor. par. are here used as these
observations are not susceptible of a great degree of ac-
curacy.

XIII. To find the true meridian by the method of equal alti-

tudes of the Sun.

The instrument remaining stationary, observe the read-
ings of the horizontal limb when the altitude of the Sun's
centre, or of either limb, is the same in the forenoon and
afternoon.

Then, the correction to the mean of these two readings
for the change in the sun's declination in the interval, is

i(D-D')



C=s



where



cos L . sini(^ — *')



D — jy = the change in the sun's declination in the

interval of the observations,
{t — /') = this interval of time, expressed in arc
L = the latitude of the place.



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AZIMUTHS. 217



XIV. To find the azimuth of Polaris ai its greaiest eastern
or western elongation.

Cos p = tang A cot X = cot D tang L = tang L tang A.
Cos L sin A = sin A = cos D, where,

p = the hour angle of the Star, A = the required azimuth,
D= its declination, L = the lat. of the place,

A = its polar distance, x = the co-latitude.

The first equations give the hour angle of the Star at its
greatest elongation; hence the sidereal time of elongation.

The second, the azimuth of the Star at its greatest elon-
gation.

The azimuth at any hour angle is found hy the methods

X and XI, or by the formula

sin 2)
A (in seconds) = — y- -{ a -|-a ' sin 1" cosp tang L }>

The most approved method is to observe a series of
azimuths of Polaris about the elongation, say for not more
than 30 minutes before and after, and to reduce them to
the elongation; to do this, compute from the known lati-
tude, the azimuth of the Star at its greatest elongation
= A, and call the sidereal time from elongation t; the
correction to the azimuth will be,

c = (112.5) f sin Vf tang A
log (112.5) sin 1" = 6.7367274.

The quantities found in the tables for " reduction tc

the meridian "(2 —. — ~-j correspond very nearly to

(112.5) f sin 1", when t does not exceed 15'; so, by en-
tering the table with the time from elongation, and mul
tiplying the tabular quantities by tang A, we obtain the
28



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318



ASTR0190MY.



required correction in seconds of arc. This will be found
a convenient substitute for the more rigorous method.

In these observations, the optical axis of the telescope
of the theodolite must be made to describe a truly vertical
plane.

If the axis of the telescope is not horizontal, the cor-
rection to the azimuth will be



4



[(w •^tDf) — (e-\- ef)] tang 5|c's altitude



where d = the value of one division of the level scale,
w= the inclination of the level to the weai,
e ^ the inclination of the level to the easty

u/ and efy the same values after reversing the level.

XV. Correction for Run in Reading Microscopes.

As it is difficult to adjust the microscopes so that five
revolutions of the micrometer screw shall carry the wire
exactly over one of the five-minute spaces on the limb of
the instrument, (if it be so graduated,) it is preferred to
observe the number of revolutions and the part of a revo-
lution made by the screw while the wire passes over the
space; then

Let m = the mean of first readings^ that is, the readings
obtained by turning the screw in the direction
of increasing numbers from zero of the comb.
m' = the mean of second, or reverse, readings.
Then, (mean) Run = r = »i — m! -\- 300, and

300. m 300 (r + J»' — 300)
true (mean) reading = = ^ — ■ -

= the number of minutes and sec-
onds to be added to the degrees and minutes of the limb.



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L0N6ITUDB.



219



XVI. Lunar distances.



To determine the true distance of the moon from the sun,
or a star; the apparent distance, together with the apparent
altitudes of the moon and the sim, or star, being given.

Let,



d = apparent distance
H = moon's app't altitude
A = sun's app't altitude
P = moon's hor. par. at place
p = sun's hor. parallax
S = moon's hor. semidiam-

eter
s =■ sun's semidiameter
R == refraction for moon's

altitude
A = observed altitude of

moon's limb



df = true distance
H' = moon's true altitude
A' = sun's true altitude
P' = moon's par. in altitude
p' = sun's par. in altitude
S' = moon's augm. semidi-
ameter
D = observed distance
r = refraction for sun's al-
titude
a = observed altitude of
Sim's limb.



P = ^ — jt . E. sin* L; where jt •
zontal parallax.



moon's equatorial hori-



E = the elipticity, log E = 7.6233789; L = the latitude
of place.



S = [9.43637] P

H = A±S'
F = P cos H

H'=H.f.(F— R)



S' = S -(- augmentation

A = ait*

p^ =p cos A

A' =A — (r— ;/)



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390 ASvaovoKT.



For a Star or a Plami.
hf=zh—r c/=D±S' ^

81114(^=008^ iW + H') + C J>co8^ i(A' + HO— C y

The r§dueid distance being thus found, the longitude
mty be deduced from it as follows:

Suppose that at 6^ 05"* 56' mean time, 29th April, 1838,
at a place whose longitude is presumed to be 4*^ 45* OO*
west of Greenwich, the result of observations gave the re-
duced distance between the sun and moon, c? = 71** 06'
35"

Mean time obs'n = 5^ 05°» 56'
Approx. long'de =4 45 00

9 50 56 approx. Greenwich m.
time of observation.
By Naut. Aim. at I3?= 70^ 41' 30'' 70° 41' 30"

(April 29th) XIP = 72° 07' 47" (/ = 71° 05' 35"

1 26 17 24' 06"

Increase of distance in 3*« = 5177".0 8 rf' = 1445"
Then 5177" : 10800-:: 1445" : a? = 0^ 50- 14«.5

Add 9"^



Greenwich mean time deduced = 9*"" 50" 14'.5
Mean time at place =5 05 56.

Longitude, deduced = 4^ 44° 18'.5



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LONCMTUDE*



331



The reduction of this proportion is very much fadli-
tated by the use of ProportUmal Logariihms^ or logs.

of -7=r given in treatises on Navigation, in conjunction

with those in the Nautical Almanac.

The proportion, however, requires a correction for
second differences, when greater accuracy is desired,
arising from the irregularity of the moon's motion.

A closer approximation to the true value of the quan-

tityo^bemg ^ = ^^^3^

In which B = 7 the sum of the second differences,
and A = the middle first difference — B ; thus,



From the Jfautical Almanacy Jlpril 29, 1838.

let difference. 3d difference.

+ V 26' 36"



+ V 26' 17" = A.
+ 1° 26' 59"



— 0' 19"

— 0' 18"



At VI = 69" 14' 54"

IX = 70° 41' 30"

XII = 72° 07' 47"

XV = 73° 33' 46"
X =60«> 14'.5 = O'" 83736 (table page 173.)
B = — 9".2 ; A = A, — B = 5177" + 9".2 = 5186".2
Sd= 1445" ; i B at = — 2".66 ; A + J B a; = 6183".64



whence



X =



10800* X 1445"



5 183". 64
and, longitude deduced = 4'" 44"" 14'. 6



= 60" lO'.e.



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1



222



A8TR0N0MT.



Reduction of the Moon? 8 Equatorial Horizontal Parallax to
the Horizontal Parallax in any Latitude,



m


HORIZONTAL PARALLAX.


•4


54*


56'


58'


60'


62^


o


tt


II


n


II


n





0.0


0.0


0.0


0.0


0.0


8


0.2


0.2


0.2


0.2


0.2


16


0.8


0.8


0.9


0.9


0.9


20


1.3


1.3


1.4


1.4


1.5


24


1.8


1.9


1.9


2.0


2.0


28


2.4


2;5


2.6


2.6


2.7


32


3.0


3.1


3.3


3.4


3.5


36


3.7


3.9


4.0


4.1


4.3


40


4.5


4.6


4.8


5.0


5.1


44


5.2


5.4


5.6


5.8


6.0


48


6.0


6.2


6.4


6.6


6.8


52


6.7


7.0


7.2


7.4


7.6


56


7.4


7.7


8.0


8.2


8.5


60


8.1


8.4


8.7


9.0


9.3


64


8.8


9.1


9.4


9.7


10.0


68


9.3


9.6


10.0


10.3


10.6


72


9.8


10.1


10.4


10.8


11.2


76


10.2


10.6


10.9


11.3


11.7


84


10.7


11.1


11.5


11.9


12.0


90


10.8


11,2


11.6


12.0


12.4



The moon's horizontal parallax, given in the second page of each
month, in the * 'American Nautical Almanac,'* for noon and midnight,
is the equatorial parallax for Greenwich mean noon and midnight; from
thence it is to be deduced for the time and place of observation. The
correction for latitude, on account of the spheroidal figure of the earth,
can be made from the table above. Thus, supposing the hor. equat. par.
to be 58'; the hor. par. in lat 52^ would be 58' — 7".2 = 57' 52".8.



Digiti



zed by Google







LONGITUDE.






223


Augmentation of


ike Moon'


s Semidiameterj


on account of






her apparent altitvde.






t pi


HORIZONTAL SBMIDIAMETBR.














F


14' 30"


15' 0"


15' 30"


16' 0"


16' 30'


17' 0"


o


f/


tt


//


n


II


V





0.00


0.00


0.00


0.00


0.00


0.00


3


0.71


0.75


0.80


0.86


0.92


0.97


6


1.41


1.50


1.60


1.71


1.83


1.94


9


2.11


2.25


2.40


2.56


2.73


2.90


12


2.81


3.00


3.20


3.41


3.63


3.86


15


3.50


3.74


3.99


4.25


4.52


4.80


18


4.17


4.46


4.76


5.07


5.39


5.73


21


4.84


5.18


5.52


5.89


6.26


6.65


24


5.49


5.88


6.27


6.68


7.11


7.54


27


6.13


6.56


7.00


7.46


7.93


8.42


30


6.75


7.23


7.71


8.22


8.74


9.28


33


7.35


7.88


8.40


8.96


9.52


10.12


36


7.93


8.50


9.07


9.67


10.28


10.92


39


8.49


9.10


9.72


10.36


11.02


11.66


42


9.03


9.68


10.34


11.02


11.72


12.44


45


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Online LibraryThomas Jefferson LeeA collection of tables and formulæ useful in surveying, geodesy, and ... → online text (page 16 of 17)