A reconstruction of Shortrede's traverse tables (1864)

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Denis Roegel

To cite this version:

Denis Roegel. A reconstruction of Shortrede’s traverse tables (1864). [Research Report] 2013.

�hal-00880847�

Shortrede’s traverse tables

(1864)

Denis Roegel

6 november 2013

This document is part of the LOCOMAT project:

http://locomat.loria.fr

In surveying, a traverse is a sequence of lines which are connected. Each line has a given

length and is related to the previous one by a given angle. The lengths and angles are

measured during a survey. Traverses are either open or closed. A closed traverse ends

where it started (figure 1).

Traverses can be made for many reasons, for instance to determine the positions of

cer-tain markers in a field, to determine the area within a boundary, to prepare construction

work, etc.

483.52 N47 ◦28 000 00E 392.28 S8 ◦27 030 00W 886.04 S56 ◦27 000 00W 452.66 N 26 ◦16 030 00E 279.33 N 39 ◦ 18 0 00 00 W 421.97 S 80◦ 200₃₀00 E A B C D E F N

Figure 1: A closed traverse. (adapted from Moffit & Bouchard’s Surveying, perhaps [6])

The traverse of figure 1 shows the lengths and bearings. The bearings are the angles

from the North or South directions, with the indications of the quadrant. This traverse

should be read from A to B, from B to C, etc., and back from F to A. Then, the first

bearing is 47

◦

28

0

from North, in the East direction, and the length of that line is 483.52.

The second bearing is South-West, and so on.

If the first point of a traverse is at a known location, say (x

A

, y

A

), one possible

calculation is that of the coordinates of all other points of the traverse. The calculation

can be arranged by hand as shown in table 1, and it involves the computation of the sines

and cosines of the bearings, multiplied by the lengths. The vertical variation is called

the “latitude” and the horizontal one is called the “departure.” In that example, the first

latitude and first departure are both positive. Since the traverse is closed, the sum of all

latitudes should equal zero, and so should the sum of all departures. However, because of

measurement errors and roundings, the sums are usually not equal to zero. The closure

in latitude is the sum of the latitudes, and in the previous example, it is equal to +0.44.

The closure in departure is −0.37.

A

N 47

◦

28

0

00

00

E

483.52

0.676019

0.736884

326.87

356.30

B

S 8

◦

27

0

30

00

W

392.28

0.989123

0.147090

388.01

57.50

C

S 56

◦

27

0

00

00

W

886.04

0.552665

0.833404

489.68

738.43

D

N 26

◦

16

0

30

00

E

452.66

0.896680

0.442680

405.89

200.39

E

N 39

◦

18

0

00

00

W

279.33

0.773840

0.633381

216.16

176.92

F

S 80

◦

20

0

30

00

E

421.97

0.167773

0.985826

70.79

415.99

A

2915.80

948.92

948.48

972.68

973.05

−948.48

−973.05

+0.44

−0.37

Table 1: Calculation of latitudes and departures. (from Moffit & Bouchard’s Surveying,

perhaps [6])

2

Traverse tables

The purpose of a traverse table is to facilitate the above calculations by giving the

prod-ucts of λ sin β and λ cos β, for various values of the lengths λ and various values of the

bearings β.

An early traverse table was that of Hamilton published in 1790 [10]. It gave the

traverses for every degree of bearing and for distances 1 to 100, as well as the hundreds

from 200 to 900, to up to four places (figure 3). In 1791, Gale published a table “of the

quantity of northing, southing, easting, and westing, made on any course, to every degree

and fifteenth minute of the quadrant, at any distance from 1 to 100.”

At the end of 1790s, the famed French Tables du cadastre [14] included a traverse

table giving 5 places traverses for each thousand of a quadrant and for distances 0.1 to

10. This table was never published.

In 1836, a 2-places traverse table was published for distances 1 to 100 and angles

from 0.25 degrees to 45 degrees, by steps of 0.25 degrees [1]. Then in 1848, Elias Loomis

published his traverse table [12]. It gave the traverses for distances 1 to 10 and for every

quarter degree of the quadrant to up to 5 places (figure 5). But the most popular traverse

tables seem to have been those of John Theophilus Boileau, which went through numerous

editions [3, 4, 5]. Boileau gave the latitudes and departures to 5 places for every minute

of the quadrant and for distances 1 to 10.

A comparison of all these tables reveals that for some of them, the latitudes increase

when the bearings increase, and for others the latitudes decrease, depending whether the

bearings are measured from the parallels or from the meridians. Shortrede, instead, drops

the terms latitude and departure, and only uses cosine and sine.

Robert Shortrede was born in 1800 and died in 1868 [2].

He studied at Edinburgh

University and in 1822 he entered the Infantry in India. He was appointed to the Deccan

survey and later to the Great Trigonometrical Survey

1

of India where he remained until

1845. He then came home to England on sick-leave. During that time, he also wrote a

paper on the attraction of the Himalayas and its effect on the Great Arc [21].

Figure 2: A map of the Great Trigonometrical Survey from 1870 (source: Wikipedia).

In 1849, Shortrede returned to India, working on the Revenue Survey, and remained

in India until 1856. His regiment then went to Persia and he returned to England in

1

_{The Great Trigonometrical Survey was a project of surveying India which spanned most of the 19th}

century (figure 2). In the 1830s, Shortrede and George Everest were the two surveying officers [7, p. 262].

numbers and sines and tangents in 1844 [20, 19], and 1849 [23, 22]. He also published

traverse tables in 1864 [24], and his last tables were azimut tables published postumously

in 1869 [25]. Some of his tables of logarithms were reprinted after his death [26].

In addition, in 1856, Laundy mentioned that Peter Gray had in his possession a

manuscript of a table of quarter-squares extending to 200000 by Shortrede, and that

Laundy might have taken another route if he had been made aware of this table

ear-lier [11]. We may therefore assume that since Shortrede was at that time in India, he left

some of his papers to Peter Gray, and that the table of quarter-squares was computed in

the 1840s, before Shortrede returned to India.

Some of Shortrede’s tables were edited by others, in particular by Edward Sang in

the case of the traverses.

4

Shortrede’s traverse table (1864)

Shortrede’s traverse tables were computed using common 7 figure tables [24, p. v]. A

notable feature of Shortrede’s table is the special slanted type used, which was especially

designed for that table. The table was also stereotyped and went through the electrotype

process, perhaps the first process of this kind applied to tables, according to Shortrede’s

preface.

Each page gives the latitudes and departures for angles differing by 2

0

, and for

dis-tances from 1 up to 100. There are 50 disdis-tances and 20

0

on each page. The first page

therefore covers the bearings from 0

◦

0

0

to 0

◦

20

0

and distances 1 to 50. The second page

covers the bearings from 0

◦

0

0

to 0

◦

20

0

and distances 51 to 100. The third and fourth page

cover the bearings from 0

◦

20

0

to 0

◦

40

0

and so on. Each page is then divided in two parts,

the departure (cosines, assuming the bearings measured from East or West) on the left,

and the latitudes (sines, assuming again the bearings measures from East or West) on

the right. On each line of a column, there are 11 values, corresponding to steps of 2

0

in

the bearing. The first line of the left column of the first page contains the departures for

the bearings 0

◦

0

0

, 0

◦

2

0

, 0

◦

4

0

, 0

◦

6

0

, 0

◦

8

0

, 0

◦

10

0

, 0

◦

12

0

, 0

◦

14

0

, 0

◦

16

0

, 0

◦

18

0

, and 0

◦

20

0

, hence

11 bearings. The latitudes and departures are given to three places and the integer parts

are only given for the first and last values. When the latitudes or departures are smaller

than 1, the integer part is omitted.

A nokhta (



) is used to show changes in the integer part of the latitudes or departures.

Given the slow variations of the values, a nokhta also always stands for a 0. The position

of a nokhta depends on the direction of variation of the tabulated values. Since the

cosines are decreasing, a nokhta marks the last value of a line with a certain integer part,

or the first value with a new integer part, when the content of the line is read from the

right. For instance, on the first page of tables, for the distance 50, there is nokhta for the

bearing 14

0

, because the departure goes from 49.999 for 16

0

to 50.000 for 14

0

.

On the right column, the values are increasing, and therefore the nokhtas mark the

first value of a line with a certain integer part. For instance, on the second page for the

bearings 40

0

to 60

0

and for the distance 71, a nokhta is given for the bearing 50

0

, because

the latitude is 0.991 for bearing 48

0

, 1.033 for bearing 50

0

and 1.074 for bearing 52

0

.

The last line of a page, right after the lines for distances 50 and 100, is a special line.

This line is an extension of the lines above and adds two digits, so that the base values of

the sines and cosines are also given to 7 places in case they are needed. For instance, the

6th page of the tables ends with a line with the values 323, 254, 181, 105, 025, 942, 856,

766, 673, 577, and 477. Above the first value 323, we have the departure 99.993, and its

last digit corresponds to the first digit of 323, so that the extended value of the departure

is 99.99323. Above the second value 254, we have also the same departure 99.993 and

the last digit again corresponds to the first digit (2) of 254, although theses digits do

not appear identical. The departure 99.993 is in fact rounded, and its value to 7 places

is thus 99.99254. And so on, the last value of the line gives the departure 99.98477. In

some cases, more than one digit needs to be altered in the departure or latitude as given

to 3 places. For instance, on that same line, but on the right side, the extension 987 to

the departure 1.280 gives the new value 1.27987, which is 100 sin 44

0

.

Each half page contains angles in degrees in the four corners. These angles correspond

to the four quadrant. The angles for the first quadrant are given in a larger type. At the

top of the first page, we have therefore 0

◦

in the left corner, and 0

◦

in the right corner.

The value on the right is the number of degrees for the last angle of the line. For this

reason, on the fifth page, we have 0

◦

in the left corner, but 1

◦

in the right corner. On the

seventh page, we start and 1

◦

and end at 1

◦

20

0

and therefore we have 1

◦

and 1

◦

.

From 0

◦

to 45

◦

of the first quadrant, the angles are read at the top and so are the

minutes. When 20

0

appears above 40

0

, it is understood that it is the value at the top

which should be used. From 45

◦

to 90

◦

of the first quadrant, the footer should be used in

place of the header, and of course the sines become cosines and the cosines become sines.

Therefore, on the last page of the table, the angles at the bottom are 45

◦

(in the right

corner) and 45

◦

(in the left corner), because the range is from 45

◦

to 45

◦

20

0

, the number

of minutes being now read in the lower position. Similarly to the fifth page from the

beginning, the fifth page from the end covers the range from 45

◦

40

0

to 46

◦

, and therefore

gives the angles 45

◦

(lower right corner) and 46

◦

(lower left corner). On the first page of

the table, these angles are 89

◦

(right) and 90

◦

(left).

The three other values in the corners correspond to the three other quadrants. The

second quadrant is the one comprising the angles 90

◦

to 180

◦

, the third quadrant comprises

the angles 180

◦

to 270

◦

, and the fourth quadrant comprises the angles 270

◦

to 360

◦

. When

the angle is between 90

◦

and 270

◦

, the cosine should be negative, but the table will only

gives its absolute value. Similarly, when the angle is in the third or fourth quadrants, the

sine is normally negative, but the table will also only give its absolute value.

Consider now the second quadrant, which corresponds to the value given at the lower

right of the four values in the corners. At the top of the pages, this value starts with

180

◦

, it decreases and reaches 135

◦

at the end of the table. Indeed, we basically have

cos(180 − α) = − cos α. Since the values of the angles in this quadrant are decreasing,

the minutes to take into account are the ones on the second lines, so that the first cosine

on the last page, 36.272, corresponds to 135

◦

20

0

in that quadrant. This also explains why

that last page has 135

◦

in both upper corners, because both the first and the last values of

a line correspond to 135

◦

and some minutes. The second quadrant then continues at the

and 135

◦

, because the first value of a line there corresponds to 134

◦

40

0

. When looking up

an angle, there should be no hesitation, since the four quadrants are mutually exclusive,

and one either looks up the sine or cosine part. Each range increases or decreases, and

that will decide which minute figure should be read. The other two quadrants are similar

to the first and second quadrant.

An example should make this clear. When looking up the cosine of 42

◦

50

0

, we know

that we are in the first half of the first quadrant, so the 42

◦

should be sought at the top of

a page. Even if we didn’t know that, we could merely look up 42

◦

in one of the corners.

It will only appear on six pages. When that angle is found, we see readily that the angle

increases, and therefore that those of the minutes which increase should be used, so that

the angle 42

◦

50

0

is readily found, and its cosine is 0.733. Some angles appear in several

parts of the table, but the sines or cosines are then the same. For instance, when looking

up 45

◦

, we find that angle on the last page, both at the top and at the bottom, but we

can see that the two columns headed 0

0

are identical.

Shortrede’s traverse tables are supplemented by several auxiliary tables which we

haven’t reconstructed here. These auxiliary tables give the traverses for the middle of

each degree and for odd distances (1, 3, 5, . . . , 99). Finally, Shortrede has included small

tables of tangents and cotangents for every 10

0

of the quadrant.

Figure

3:

Excerpt

of

Hamilton’s

table

of

tra

v

ers

es

(1790)

9

The following list covers the most important references

2

_{related to Shortrede’s table. Not}

all items of this list are mentioned in the text, and the sources which have not been seen

are marked so. We have added notes about the contents of the articles in certain cases.

[1] Anonymous. A table of logarithms, of logarithmic sines, and a traverse table. New

York: Harper & Brothers, 1836.

[2] Anonymous. Obituary of Major-General Robert Shortrede. Monthly notices of the

Royal Astronomical Society, 29(4):120–121, 1869.

[3] John Theophilus Boileau. A new and complete set of traverse tables, shewing the

differences of latitude and the departures to every minute of the quadrant, and to

five places of decimals, together with a table of the length of each degree of latitude

and corresponding degree of longitude from the equator to the poles. 1839.

[not seen]

[4] John Theophilus Boileau. A new and complete set of traverse tables. London:

William H. Allen & Co., 1872.

[5] John Theophilus Boileau. A new and complete set of traverse tables. New York:

D. Van Nostrand company, 1904.

[10th edition]

[6] Harry Bouchard and Francis H. Moffitt. Surveying. New York: International

Textbook Company, 1962.

[not seen]

[7] Matthew H. Edney. Mapping an empire: The geographical construction of British

India, 1765–1843. Chicago: The University of Chicago Press, 1997.

[not seen]

[8] James Whitbread Lee Glaisher. Report of the committee on mathematical tables.

London: Taylor and Francis, 1873.

[Also published as part of the “Report of the forty-third

meeting of the British Association for the advancement of science,” London: John Murray, 1874.

A review by R. Radau was published in the Bulletin des sciences mathématiques et

astronomiques, volume 11, 1876, pp. 7–27]

[9] James Whitbread Lee Glaisher. Table, mathematical. In Hugh Chisholm, editor,

The Encyclopædia Britannica, 11th edition, volume 26, pages 325–336. Cambridge,

England: at the University Press, 1911.

[10] Robert Hamilton. Mathematical tables. Edinburgh: William Creech, 1790.

[contains

traverse tables]

2

_{Note on the titles of the works: Original titles come with many idiosyncrasies and features (line}

splitting, size, fonts, etc.) which can often not be reproduced in a list of references. It has therefore

seemed pointless to capitalize works according to conventions which not only have no relation with the

original work, but also do not restore the title entirely. In the following list of references, most title

words (except in German) will therefore be left uncapitalized. The names of the authors have also been

homogenized and initials expanded, as much as possible.

The reader should keep in mind that this list is not meant as a facsimile of the original works. The

original style information could no doubt have been added as a note, but we have not done it here.

and subtraction alone. London: Charles and Edwin Layton, 1856.

[reconstructed

in [15]]

[12] Elias Loomis. Tables of logarithms of numbers and of sines and tangents for every

ten seconds of the quadrant, with other useful tables. New York: Harper &

Brothers, 1848.

[13] Denis Roegel. A reconstruction of Edward Sang’s table of logarithms (1871).

Technical report, LORIA, Nancy, 2010.

[This is a reconstruction of [16].]

[14] Denis Roegel. The great logarithmic and trigonometric tables of the French

Cadastre: a preliminary investigation. Technical report, LORIA, Nancy, 2010.

[15] Denis Roegel. A reconstruction of Laundy’s table of quarter-squares (1856).

Technical report, LORIA, Nancy, 2013.

[This is a reconstruction of [11].]

[16] Edward Sang. A new table of seven-place logarithms of all numbers from 20 000 to

200 000. London: Charles and Edwin Layton, 1871.

[Reconstruction by D. Roegel,

2010 [13].]

[17] Robert Shortrede. Scheme of a table for all time. Journal of the Asiatic Society of

Bengal, 10(116):595–612, 1841.

[18] Robert Shortrede. Table of proportional logarithms. Journal of the Asiatic Society

of Bengal, 10(117):713–715, 1841.

[19] Robert Shortrede. Compendious logarithmic tables, etc. Edinburgh: Adam &

Charles Black, 1844.

[20] Robert Shortrede. Logarithmic tables to seven places of decimals. Edinburgh:

Adam & Charles Black, 1844.

[partly edited by Edward Sang]

[21] Robert Shortrede. Observations to determine the latitude of Dera and the

disturbing force of the Himalayas. Monthly notices of the Royal Astronomical

Society, 8(8):189–191, 1848.

[22] Robert Shortrede. Logarithmic tables containing logarithms to numbers from 1 to

120,000, numbers to logarithms from .0 to 1.00000, to seven places of decimals.

Edinburgh: Adam & Charles Black, 1849.

[23] Robert Shortrede. Logarithmic tables to seven places of decimals, containing

logarithmic sines and tangents to every second of the circle, with arguments in

space and time. Edinburgh: Adam & Charles Black, 1849.

[24] Robert Shortrede. Traverse tables to five places for every 2

0

of angle up to 100 of

distance. Edinburgh: William Blackwood and sons, 1864.

[edited by Edward Sang]

[26] Robert Shortrede. Logarithms of sines and tangents for every second. London:

Charles and Edwin Layton, 1873.

360◦

₀

◦ 180◦ ₁₈₀◦

Cos.

359◦

₀

◦ 180◦ ₁₇₉◦ 360◦

₀

◦ 180◦ ₁₈₀◦

Sin.

359◦

₀

◦ 180◦ ₁₇₉◦

0

0 100 500 200 400

0

0 100 500 200 400 1 1 ·000 000 000 000 000 000 000 000 000 000 1 ·000 ·000 001 001 002 002 003 003 004 005 005 ·006 1 2 2 ·000 000 000 000 000 000 000 000 000 000 2 ·000 ·000 001 002 003 005 006 007 008 009 010 ·012 2 3 3 ·000 000 000 000 000 000 000 000 000 000 3 ·000 ·000 002 003 005 007 009 010 012 014 016 ·017 3 4 4 ·000 000 000 000 000 000 000 000 000 000 4 ·000 ·000 002 005 007 009 012 014 016 019 021 ·023 4 5 5 ·000 000 000 000 000 000 000 000 000 000 5 ·000 ·000 003 006 009 012 015 017 020 023 026 ·029 5 6 6 ·000 000 000 000 000 000 000 000 000 000 6 ·000 ·000 003 007 010 014 017 021 024 028 031 ·035 6 7 7 ·000 000 000 000 000 000 000 000 000 000 7 ·000 ·000 004 008 012 016 020 024 029 033 037 ·041 7 8 8 ·000 000 000 000 000 000 000 000 000 000 8 ·000 ·000 005 009 014 019 023 028 033 037 042 ·047 8 9 9 ·000 000 000 000 000 000 000 000 000 000 9 ·000 ·000 005 010 016 021 026 031 037 042 047 ·052 9 10 10 ·000 000 000 000 000 000 000 000 000 000 10 ·000 ·000 006 012 017 023 029 035 041 047 052 ·058 10 1 11 ·000 000 000 000 000 000 000 000 000 000 11 ·000 ·000 006 013 019 026 032 038 045 051 058 ·064 1 2 12 ·000 000 000 000 000 000 000 000 000 000 12 ·000 ·000 007 014 021 028 035 042 049 056 063 ·070 2 3 13 ·000 000 000 000 000 000 000 000 000 000 13 ·000 ·000 008 015 023 030 038 045 053 061 068 ·076 3 4 14 ·000 000 000 000 000 000 000 000 000 000 14 ·000 ·000 008 016 024 033 041 049 057 065 073 ·081 4 15 15 ·000 000 000 000 000 000 000 000 000 000 15 ·000 ·000 009 017 026 035 044 052 061 070 079 ·087 15 6 16 ·000 000 000 000 000 000 000 000 000 000 16 ·000 ·000 009 019 028 037 047 056 065 074 084 ·093 6 7 17 ·000 000 000 000 000 000 000 000 000 000 17 ·000 ·000 010 020 030 040 049 059 069 079 089 ·099 7 8 18 ·000 000 000 000 000 000 000 000 000 000 18 ·000 ·000 010 021 031 042 052 063 073 084 094 ·105 8 9 19 ·000 000 000 000 000 000 000 000 000 000 19 ·000 ·000 011 022 033 044 055 066 077 088 099 ·111 9 20 20 ·000 000 000 000 000 000 000 000 000 000 20 ·000 ·000 012 023 035 047 058 070 081 093 105 ·116 20 1 21 ·000 000 000 000 000 000 000 000 000 000 21 ·000 ·000 012 024 037 049 061 073 086 098 110 ·122 1 2 22 ·000 000 000 000 000 000 000 000 000 000 22 ·000 ·000 013 026 038 051 064 077 090 102 115 ·128 2 3 23 ·000 000 000 000 000 000 000 000 000 000 23 ·000 ·000 013 027 040 054 067 080 094 107 120 ·134 3 4 24 ·000 000 000 000 000 000 000 000 000 000 24 ·000 ·000 014 028 042 056 070 084 098 112 126 ·140 4 25 25 ·000 000 000 000 000 000 000 000 000 000 25 ·000 ·000 015 029 044 058 073 087 102 116 131 ·145 25 6 26 ·000 000 000 000 000 000 000 000 000 000 26 ·000 ·000 015 030 045 061 076 091 106 121 136 ·151 6 7 27 ·000 000 000 000 000 000 000 000 000 000 27 ·000 ·000 016 031 047 063 079 094 110 126 141 ·157 7 8 28 ·000 000 000 000 000 000 000 000 000 000 28 ·000 ·000 016 033 049 065 081 098 114 130 147 ·163 8 9 29 ·000 000 000 000 000 000 000 000 000 000 29 ·000 ·000 017 034 051 067 084 101 118 135 152 ·169 9 30 30 ·000 000 000 000 000 000 000 000 000 00 29 ·999 ·000 017 035 052 070 087 105 122 140 157 ·175 30 1 31 ·000 000 000 000 000 000 000 000 000 00 30 ·999 ·000 018 036 054 072 090 108 126 144 162 ·180 1 2 32 ·000 000 000 000 000 000 000 000 000 00 31 ·999 ·000 019 037 056 074 093 112 130 149 168 ·186 2 3 33 ·000 000 000 000 000 000 000 000 000 00 32 ·999 ·000 019 038 058 077 096 115 134 154 173 ·192 3 4 34 ·000 000 000 000 000 000 000 000 000 00 33 ·999 ·000 020 040 059 079 099 119 138 158 178 ·198 4 35 35 ·000 000 000 000 000 000 000 000 000 00 34 ·999 ·000 020 041 061 081 102 122 143 163 183 ·204 35 6 36 ·000 000 000 000 000 000 000 000 000 00 35 ·999 ·000 021 042 063 084 105 126 147 168 188 ·209 6 7 37 ·000 000 000 000 000 000 000 000 00 999 36 ·999 ·000 022 043 065 086 108 129 151 172 194 ·215 7 8 38 ·000 000 000 000 000 000 000 000 00 999 37 ·999 ·000 022 044 066 088 111 133 155 177 199 ·221 8 9 39 ·000 000 000 000 000 000 000 000 00 999 38 ·999 ·000 023 045 068 091 113 136 159 182 204 ·227 9 40 40 ·000 000 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1 10 ·999 999 999 999 999 999 999 999 999 998 10 ·998 ·128 134 141 147 154 160 166 173 179 186 ·192 1 2 11 ·999 999 999 999 999 999 999 999 998 998 11 ·998 ·140 147 154 161 168 175 182 188 195 202 ·209 2 3 12 ·999 999 999 999 999 999 999 998 998 998 12 ·998 ·151 159 166 174 182 189 197 204 212 219 ·227 3 4 13 ·999 999 999 999 999 999 998 998 998 998 13 ·998 ·163 171 179 187 195 204 212 220 228 236 ·244 4 15 14 ·999 999 999 999 999 998 998 998 998 998 14 ·998 ·175 183 192 201 209 218 227 236 244 253 ·262 15 6 15 ·999 999 999 999 998 998 998 998 998 998 15 ·998 ·186 195 205 214 223 233 242 251 261 270 ·279 6 7 16 ·999 999 999 998 998 998 998 998 998 998 16 ·997 ·198 208 218 227 237 247 257 267 277 287 ·297 7 8 17 ·999 999 999 998 998 998 998 998 998 997 17 ·997 ·209 220 230 241 251 262 272 283 293 304 ·314 8 9 18 ·999 999 998 998 998 998 998 998 997 997 18 ·997 ·221 232 243 254 265 276 287 298 309 321 ·332 9 20 19 ·999 999 998 998 998 998 998 998 997 997 19 ·997 ·233 244 256 268 279 291 303 314 326 337 ·349 20 1 20 ·999 998 998 998 998 998 998 997 997 997 20 ·997 ·244 257 269 281 293 305 318 330 342 354 ·367 1 2 21 ·999 998 998 998 998 998 997 997 997 997 21 ·997 ·256 269 282 294 307 320 333 346 358 371 ·384 2 3 22 ·998 998 998 998 998 998 997 997 997 997 22 ·996 ·268 281 294 308 321 335 348 361 375 388 ·401 3 4 23 ·998 998 998 998 998 997 997 997 997 997 23 ·996 ·279 293 307 321 335 349 363 377 391 405 ·419 4 25 24 ·998 998 998 998 998 997 997 997 997 996 24 ·996 ·291 305 320 335 349 364 378 393 407 422 ·436 25 6 25 ·998 998 998 998 997 997 997 997 997 996 25 ·996 ·303 318 333 348 363 378 393 408 424 439 ·454 6 7 26 ·998 998 998 998 997 997 997 997 996 996 26 ·996 ·314 330 346 361 377 393 408 424 440 456 ·471 7 8 27 ·998 998 998 997 997 997 997 997 996 996 27 ·996 ·326 342 358 375 391 407 424 440 456 472 ·489 8 9 28 ·998 998 998 997 997 997 997 996 996 996 28 ·996 ·337 354 371 388 405 422 439 456 472 489 ·506 9 30 29 ·998 998 998 997 997 997 997 996 996 996 29 ·995 ·349 367 384 401 419 436 454 471 489 506 ·524 30 1 30 ·998 998 997 997 997 997 996 996 996 996 30 ·995 ·361 379 397 415 433 451 469 487 505 523 ·541 1 2 31 ·998 998 997 997 997 997 996 996 996 995 31 ·995 ·372 391 410 428 447 465 484 503 521 540 ·558 2 3 32 ·998 998 997 997 997 997 996 996 996 995 32 ·995 ·384 403 422 442 461 480 499 518 538 557 ·576 3 4 33 ·998 997 997 997 997 996 996 996 995 995 33 ·995 ·396 415 435 455 475 494 514 534 554 574 ·593 4 35 34 ·998 997 997 997 997 996 996 996 995 995 34 ·995 ·407 428 448 468 489 509 529 550 570 590 ·611 35 6 35 ·998 997 997 997 996 996 996 996 995 995 35 ·995 ·419 440 461 482 503 524 545 565 586 607 ·628 6 7 36 ·997 997 997 997 996 996 996 995 995 995 36 ·994 ·431 452 474 495 517 538 560 581 603 624 ·646 7 8 37 ·997 997 997 997 996 996 996 995 995 995 37 ·994 ·442 464 486 508 531 553 575 597 619 641 ·663 8 9 38 ·997 997 997 997 996 996 996 995 995 994 38 ·994 ·454 476 499 522 545 567 590 613 635 658 ·681 9 40 39 ·997 997 997 996 996 996 995 995 995 994 39 ·994 ·465 489 512 535 558 582 605 628 652 675 ·698 40 1 40 ·997 997 997 996 996 996 995 995 995 994 40 ·994 ·477 501 525 549 572 596 620 644 668 692 ·716 1 2 41 ·997 997 997 996 996 996 995 995 994 994 41 ·994 ·489 513 538 562 586 611 635 660 684 709 ·733 2 3 42 ·997 997 996 996 996 995 995 995 994 994 42 ·993 ·500 525 550 575 600 625 650 675 700 725 ·750 3 4 43 ·997 997 996 996 996 995 995 995 994 994 43 ·993 ·512 538 563 589 614 640 666 691 717 742 ·768 4 45 44 ·997 997 996 996 996 995 995 994 994 994 44 ·993 ·524 550 576 602 628 654 681 707 733 759 ·785 45 6 45 ·997 997 996 996 996 995 995 994 994 993 45 ·993 ·535 562 589 616 642 669 696 723 749 776 ·803 6 7 46 ·997 996 996 996 995 995 995 994 994 993 46 ·993 ·547 574 602 629 656 684 711 738 766 793 ·820 7 8 47 ·997 996 996 996 995 995 995 994 994 993 47 ·993 ·558 586 614 642 670 698 726 754 782 810 ·838 8 9 48 ·997 996 996 996 995 995 994 994 993 993 48 ·993 ·570 599 627 656 684 713 741 770 798 827 ·855 9 50 49 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0 51 50 ·997 996 996 995 995 995 994 994 993 993 50 ·992 ·593 623 653 682 712 742 771 801 831 860 ·890 51 2 51 ·996 996 996 995 995 995 994 994 993 993 51 ·992 ·605 635 666 696 726 756 787 817 847 877 ·908 2 3 52 ·996 996 996 995 995 994 994 993 993 992 52 ·992 ·617 648 678 709 740 771 802 832 863 894 ·925 3 4 53 ·996 996 996 995 995 994 994 993 993 992 53 ·992 ·628 660 691 723 754 785 817 848 880 911 ·942 4 55 54 ·996 996 995 995 995 994 994 993 993 992 54 ·992 ·640 672 704 736 768 800 832 864 896 928 ·960 55 6 55 ·996 996 995 995 995 994 994 993 993 992 55 ·991 ·652 684 717 749 782 814 847 880 912 945 ·977 6 7 56 ·996 996 995 995 994 994 993 993 992 992 56 ·991 ·663 696 730 763 796 829 862 895 928 962 ·995 7 8 57 ·996 996 995 995 994 994 993 993 992 992 57 ·991 ·675 709 742 776 810 844 877 911 945 979 1 ·012 8 9 58 ·996 996 995 995 994 994 993 993 992 992 58 ·991 ·686 721 755 789 824 858 892 927 961 995 1 ·030 9 60 59 ·996 996 995 995 994 994 993 993 992 991 59 ·991 ·698 733 768 803 838 873 908 942 97712 1 ·047 60 1 60 ·996 995 995 995 994 994 993 992 992 991 60 ·991 ·710 745 781 816 852 887 923 958 99429 1 ·065 1 2 61 ·996 995 995 994 994 993 993 992 992 991 61 ·991 ·721 757 794 830 866 902 938 97410 046 1 ·082 2 3 62 ·996 995 995 994 994 993 993 992 992 991 62 ·990 ·733 770 806 843 880 916 953 99026 063 1 ·100 3 4 63 ·996 995 995 994 994 993 993 992 992 991 63 ·990 ·745 782 819 856 894 931 96805 042 080 1 ·117 4 65 64 ·996 995 995 994 994 993 993 992 991 991 64 ·990 ·756 794 832 870 908 945 98321 059 097 1 ·134 65 6 65 ·996 995 995 994 994 993 992 992 991 991 65 ·990 ·768 806 845 883 922 960 99837 075 113 1 ·152 6 7 66 ·995 995 995 994 993 993 992 992 991 990 66 ·990 ·780 819 858 896 935 974 13 052 091 130 1 ·169 7 8 67 ·995 995 994 994 993 993 992 992 991 990 67 ·990 ·791 831 870 910 949 989 29 068 108 147 1 ·187 8 9 68 ·995 995 994 994 993 993 992 991 991 990 68 ·989 ·803 843 883 923 963 04 044 084 124 164 1 ·204 9 70 69 ·995 995 994 994 993 993 992 991 991 990 69 ·989 ·814 855 896 937 977 18 059 100 140 181 1 ·222 70 1 70 ·995 995 994 994 993 992 992 991 991 990 70 ·989 ·826 867 909 950 991 33 074 115 157 198 1 ·239 1 2 71 ·995 995 994 994 993 992 992 991 990 990 71 ·989 ·838 880 922 96305 047 089 131 173 215 1 ·257 2 3 72 ·995 995 994 993 993 992 992 991 990 990 72 ·989 ·849 892 934 97719 062 104 147 189 232 1 ·274 3 4 73 ·995 994 994 993 993 992 992 991 990 989 73 ·989 ·861 904 947 99033 076 119 162 205 248 1 ·291 4 75 74 ·995 994 994 993 993 992 991 991 990 989 74 ·989 ·873 916 96004 047 091 134 178 222 265 1 ·309 75 6 75 ·995 994 994 993 993 992 991 991 990 989 75 ·988 ·884 928 97317 061 105 150 194 238 282 1 ·326 6 7 76 ·995 994 994 993 992 992 991 991 990 989 76 ·988 ·896 941 98630 075 120 165 209 254 299 1 ·344 7 8 77 ·995 994 994 993 992 992 991 990 990 989 77 ·988 ·908 953 99844 089 134 180 225 271 316 1 ·361 8 9 78 ·995 994 994 993 992 992 991 990 990 989 78 ·988 ·919 96511 057 103 149 195 241 287 333 1 ·379 9 80 79 ·995 994 993 993 992 992 991 990 989 989 79 ·988 ·931 97724 070 117 164 210 257 303 350 1 ·396 80 1 80 ·995 994 993 993 992 991 991 990 989 988 80 ·988 ·942 99037 084 131 178 225 272 319 367 1 ·414 1 2 81 ·994 994 993 993 992 991 991 990 989 988 81 ·988 ·954 02 049 097 145 193 240 288 336 383 1 ·431 2 3 82 ·994 994 993 993 992 991 991 990 989 988 82 ·987 ·966 14 062 111 159 207 255 304 352 400 1 ·449 3 4 83 ·994 994 993 992 992 991 990 990 989 988 83 ·987 ·977 26 075 124 173 222 271 319 368 417 1 ·466 4 85 84 ·994 994 993 992 992 991 990 990 989 988 84 ·987 ·989 38 088 137 187 236 286 335 385 434 1 ·483 85 6 85 ·994 994 993 992 992 991 990 989 989 988 85 ·987 1 ·001 051 101 151 201 251 301 351 401 451 1 ·501 6 7 86 ·994 994 993 992 992 991 990 989 988 988 86 ·987 1 ·012 063 113 164 215 265 316 367 417 468 1 ·518 7 8 87 ·994 993 993 992 991 991 990 989 988 987 87 ·987 1 ·024 075 126 177 229 280 331 382 433 485 1 ·536 8 9 88 ·994 993 993 992 991 991 990 989 988 987 88 ·986 1 ·036 087 139 191 243 294 346 398 450 501 1 ·553 9 90 89 ·994 993 993 992 991 990 990 989 988 987 89 ·986 1 ·047 100 152 204 257 309 361 414 466 518 1 ·571 90 1 90 ·994 993 993 992 991 990 990 989 988 987 90 ·986 1 ·059 112 165 218 271 323 376 429 482 535 1 ·588 1 2 91 ·994 993 992 992 991 990 989 989 988 987 91 ·986 1 ·070 124 177 231 285 338 392 445 499 552 1 ·606 2 3 92 ·994 993 992 992 991 990 989 989 988 987 92 ·986 1 ·082 136 190 244 298 353 407 461 515 569 1 ·623 3 4 93 ·994 993 992 992 991 990 989 988 988 987 93 ·986 1 ·094 148 203 258 312 367 422 476 531 586 1 ·641 4 95 94 ·994 993 992 991 991 990 989 988 987 986 94 ·986 1 ·105 161 216 271 326 382 437 492 547 603 1 ·658 95 6 95 ·994 993 992 991 991 990 989 988 987 986 95 ·985 1 ·117 173 229 285 340 396 452 508 564 620 1 ·675 6 7 96 ·993 993 992 991 991 990 989 988 987 986 96 ·985 1 ·129 185 241 298 354 411 467 524 580 636 1 ·693 7 8 97 ·993 993 992 991 990 990 989 988 987 986 97 ·985 1 ·140 197 254 311 368 425 482 539 596 653 1 ·710 8 9 98 ·993 993 992 991 990 990 989 988 987 986 98 ·985 1 ·152 209 267 325 382 440 497 555 613 670 1 ·728 9 100 99 ·993 993 992 991 990 989 989 988 987 986 99 ·985 1 ·164 222 280 338 396 454 513 571 629 687 1 ·745 100 323 254 181 105 025 942 856 766 673 577 477 353 170 987 805 622 439 256 073 890 707 524 400 200 500 100

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0 100 500 200 400 1 1 ·000 000 000 000 000 000 000 000 000 000 1 ·000 ·017 018 019 019 020 020 021 022 022 023 ·023 1 2 2 ·000 000 000 000 000 000 000 000 00 999 1 ·999 ·035 036 037 038 040 041 042 043 044 045 ·047 2 3 3 ·00000 999 999 999 999 999 999 999 999 2 ·999 ·052 054 056 058 059 061 063 065 066 068 ·070 3 4 3 ·999 999 999 999 999 999 999 999 999 999 3 ·999 ·070 072 074 077 079 081 084 086 088 091 ·093 4 5 4 ·999 999 999 999 999 999 999 999 999 999 4 ·999 ·087 090 093 096 099 102 105 108 111 113 ·116 5 6 5 ·999 999 999 999 999 999 999 999 999 998 5 ·998 ·105 108 112 115 119 122 126 129 133 136 ·140 6 7 6 ·999 999 999 999 999 999 998 998 998 998 6 ·998 ·122 126 130 134 138 143 147 151 155 159 ·163 7 8 7 ·999 999 999 999 998 998 998 998 998 998 7 ·998 ·140 144 149 154 158 163 168 172 177 181 ·186 8 9 8 ·999 999 998 998 998 998 998 998 998 998 8 ·998 ·157 162 168 173 178 183 188 194 199 204 ·209 9 10 9 ·998 998 998 998 998 998 998 998 998 997 9 ·997 ·175 180 186 192 198 204 209 215 221 227 ·233 10 1 10 ·998 998 998 998 998 998 998 997 997 997 10 ·997 ·192 198 205 211 218 224 230 237 243 250 ·256 1 2 11 ·998 998 998 998 998 998 997 997 997 997 11 ·997 ·209 216 223 230 237 244 251 258 265 272 ·279 2 3 12 ·998 998 998 998 997 997 997 997 997 997 12 ·996 ·227 234 242 250 257 265 272 280 287 295 ·302 3 4 13 ·998 998 998 997 997 997 997 997 997 996 13 ·996 ·244 252 261 269 277 285 293 301 309 318 ·326 4 15 14 ·998 998 997 997 997 997 997 997 996 996 14 ·996 ·262 271 279 288 297 305 314 323 332 340 ·349 15 6 15 ·998 997 997 997 997 997 996 996 996 996 15 ·996 ·279 289 298 307 316 326 335 344 354 363 ·372 6 7 16 ·997 997 997 997 997 996 996 996 996 996 16 ·995 ·297 307 316 326 336 346 356 366 376 386 ·396 7 8 17 ·997 997 997 997 996 996 996 996 996 995 17 ·995 ·314 325 335 346 356 366 377 387 398 408 ·419 8 9 18 ·997 997 997 996 996 996 996 996 995 995 18 ·995 ·332 343 354 365 376 387 398 409 420 431 ·442 9 20 19 ·997 997 997 996 996 996 996 995 995 995 19 ·995 ·349 361 372 384 396 407 419 430 442 454 ·465 20 1 20 ·997 997 996 996 996 996 995 995 995 995 20 ·994 ·367 379 391 403 415 428 440 452 464 476 ·489 1 2 21 ·997 996 996 996 996 995 995 995 995 994 21 ·994 ·384 397 410 422 435 448 461 474 486 499 ·512 2 3 22 ·996 996 996 996 996 995 995 995 994 994 22 ·994 ·401 415 428 442 455 468 482 495 508 522 ·535 3 4 23 ·996 996 996 996 995 995 995 994 994 994 23 ·994 ·419 433 447 461 475 489 503 517 531 544 ·558 4 25 24 ·996 996 996 995 995 995 995 994 994 994 24 ·993 ·436 451 465 480 494 509 524 538 553 567 ·582 25 6 25 ·996 996 995 995 995 995 994 994 994 993 25 ·993 ·454 469 484 499 514 529 545 560 575 590 ·605 6 7 26 ·996 996 995 995 995 994 994 994 993 993 26 ·993 ·471 487 503 518 534 550 565 581 597 613 ·628 7 8 27 ·996 995 995 995 995 994 994 994 993 993 27 ·992 ·489 505 521 538 554 570 586 603 619 635 ·652 8 9 28 ·996 995 995 995 994 994 994 993 993 993 28 ·992 ·506 523 540 557 574 590 607 624 641 658 ·675 9 30 29 ·995 995 995 994 994 994 993 993 993 992 29 ·992 ·524 541 558 576 593 611 628 646 663 681 ·698 30 1 30 ·995 995 995 994 994 994 993 993 992 992 30 ·992 ·541 559 577 595 613 631 649 667 685 703 ·721 1 2 31 ·995 995 994 994 994 993 993 993 992 992 31 ·991 ·558 577 596 614 633 652 670 689 707 726 ·745 2 3 32 ·995 995 994 994 994 993 993 992 992 992 32 ·991 ·576 595 614 634 653 672 691 710 729 749 ·768 3 4 33 ·995 994 994 994 993 993 993 992 992 991 33 ·991 ·593 613 633 653 672 692 712 732 752 771 ·791 4 35 34 ·995 994 994 994 993 993 992 992 991 991 34 ·991 ·611 631 652 672 692 713 733 753 774 794 ·814 35 6 35 ·995 994 994 993 993 993 992 992 991 991 35 ·990 ·628 649 670 691 712 733 754 775 796 817 ·838 6 7 36 ·994 994 994 993 993 992 992 991 991 990 36 ·990 ·646 667 689 710 732 753 775 796 818 839 ·861 7 8 37 ·994 994 993 993 993 992 992 991 991 990 37 ·990 ·663 685 707 730 752 774 796 818 840 862 ·884 8 9 38 ·994 994 993 993 992 992 991 991 990 990 38 ·989 ·681 703 726 749 771 794 817 839 862 885 ·907 9 40 39 ·994 993 993 993 992 992 991 991 990 990 39 ·989 ·698 721 745 768 791 814 838 861 884 907 ·931 40 1 40 ·994 993 993 992 992 992 991 991 990 989 40 ·989 ·716 739 763 787 811 835 859 882 906 930 ·954 1 2 41 ·994 993 993 992 992 991 991 990 990 989 41 ·989 ·733 757 782 806 831 855 880 904 928 953 ·977 2 3 42 ·993 993 993 992 992 991 991 990 989 989 42 ·988 ·750 775 800 825 851 876 901 926 951 976 1 ·001 3 4 43 ·993 993 992 992 991 991 990 990 989 989 43 ·988 ·768 794 819 845 870 896 921 947 973 998 1 ·024 4 45 44 ·993 993 992 992 991 991 990 990 989 988 44 ·988 ·785 812 838 864 890 916 942 969 99521 1 ·047 45 6 45 ·993 993 992 992 991 990 990 989 989 988 45 ·988 ·803 830 856 883 910 937 963 99017 044 1 ·070 6 7 46 ·993 992 992 991 991 990 990 989 989 988 46 ·987 ·820 848 875 902 930 957 98412 039 066 1 ·094 7 8 47 ·993 992 992 991 991 990 989 989 988 988 47 ·987 ·838 866 894 921 949 977 05 033 061 089 1 ·117 8 9 48 ·993 992 992 991 990 990 989 989 988 987 48 ·987 ·855 884 912 941 969 998 26 055 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0 100 500 200 400 51 50 ·992 992 991 991 990 989 989 988 988 987 50 ·986 ·890 920 949 97909 038 068 098 127 157 1 ·187 51 2 51 ·992 992 991 990 990 989 989 988 987 987 51 ·986 ·908 938 968 99829 059 089 119 149 180 1 ·210 2 3 52 ·992 991 991 990 990 989 988 988 987 986 52 ·986 ·925 956 98717 048 079 110 141 172 202 1 ·233 3 4 53 ·992 991 991 990 989 989 988 987 987 986 53 ·985 ·942 97405 037 068 099 131 162 194 225 1 ·257 4 55 54 ·992 991 990 990 989 989 988 987 987 986 54 ·985 ·960 99224 056 088 120 152 184 216 248 1 ·280 55 6 55 ·991 991 990 990 989 988 988 987 986 986 55 ·985 ·977 10 042 075 108 140 173 205 238 270 1 ·303 6 7 56 ·991 991 990 989 989 988 987 987 986 985 56 ·985 ·995 28 061 094 127 161 194 227 260 293 1 ·326 7 8 57 ·991 991 990 989 989 988 987 987 986 985 57 ·984 1 ·012 046 080 113 147 181 215 248 282 316 1 ·350 8 9 58 ·991 990 990 989 988 988 987 986 986 985 58 ·984 1 ·030 064 098 133 167 201 236 270 304 339 1 ·373 9 60 59 ·991 990 990 989 988 988 987 986 985 985 59 ·984 1 ·047 082 117 152 187 222 257 291 326 361 1 ·396 60 1 60 ·991 990 989 989 988 987 987 986 985 984 60 ·983 1 ·065 100 136 171 207 242 277 313 348 384 1 ·419 1 2 61 ·991 990 989 989 988 987 986 986 985 984 61 ·983 1 ·082 118 154 190 226 262 298 334 371 407 1 ·443 2 3 62 ·990 990 989 988 988 987 986 985 985 984 62 ·983 1 ·100 136 173 209 246 283 319 356 393 429 1 ·466 3 4 63 ·990 990 989 988 987 987 986 985 984 984 63 ·983 1 ·117 154 191 229 266 303 340 378 415 452 1 ·489 4 65 64 ·990 989 989 988 987 987 986 985 984 983 64 ·982 1 ·134 172 210 248 286 323 361 399 437 475 1 ·512 65 6 65 ·990 989 989 988 987 986 986 985 984 983 65 ·982 1 ·152 190 229 267 305 344 382 421 459 497 1 ·536 6 7 66 ·990 989 988 988 987 986 985 984 984 983 66 ·982 1 ·169 208 247 286 325 364 403 442 481 520 1 ·559 7 8 67 ·990 989 988 987 987 986 985 984 983 982 67 ·982 1 ·187 226 266 305 345 385 424 464 503 543 1 ·582 8 9 68 ·989 989 988 987 987 986 985 984 983 982 68 ·981 1 ·204 244 284 325 365 405 445 485 525 565 1 ·606 9 70 69 ·989 989 988 987 986 985 985 984 983 982 69 ·981 1 ·222 262 303 344 385 425 466 507 547 588 1 ·629 70 1 70 ·989 988 988 987 986 985 984 984 983 982 70 ·981 1 ·239 280 322 363 404 446 487 528 570 611 1 ·652 1 2 71 ·989 988 988 987 986 985 984 983 982 981 71 ·981 1 ·257 298 340 382 424 466 508 550 592 633 1 ·675 2 3 72 ·989 988 987 987 986 985 984 983 982 981 72 ·980 1 ·274 316 359 401 444 486 529 571 614 656 1 ·699 3 4 73 ·989 988 987 986 986 985 984 983 982 981 73 ·980 1 ·291 335 378 421 464 507 550 593 636 679 1 ·722 4 75 74 ·989 988 987 986 985 984 984 983 982 981 74 ·980 1 ·309 353 396 440 483 527 571 614 658 702 1 ·745 75 6 75 ·988 988 987 986 985 984 983 982 981 980 75 ·979 1 ·326 371 415 459 503 547 592 636 680 724 1 ·768 6 7 76 ·988 987 987 986 985 984 983 982 981 980 76 ·979 1 ·344 389 433 478 523 568 613 657 702 747 1 ·792 7 8 77 ·988 987 986 986 985 984 983 982 981 980 77 ·979 1 ·361 407 452 497 543 588 634 679 724 770 1 ·815 8 9 78 ·988 987 986 985 985 984 983 982 981 980 78 ·979 1 ·379 425 471 517 563 609 654 700 746 792 1 ·838 9 80 79 ·988 987 986 985 984 983 982 981 980 979 79 ·978 1 ·396 443 489 536 582 629 675 722 768 815 1 ·862 80 1 80 ·988 987 986 985 984 983 982 981 980 979 80 ·978 1 ·414 461 508 555 602 649 696 743 791 838 1 ·885 1 2 81 ·988 987 986 985 984 983 982 981 980 979 81 ·978 1 ·431 479 526 574 622 670 717 765 813 860 1 ·908 2 3 82 ·987 987 986 985 984 983 982 981 980 979 82 ·978 1 ·449 497 545 593 642 690 738 786 835 883 1 ·931 3 4 83 ·987 986 985 985 984 983 982 981 979 978 83 ·977 1 ·466 515 564 613 661 710 759 808 857 906 1 ·955 4 85 84 ·987 986 985 984 983 982 981 980 979 978 84 ·977 1 ·483 533 582 632 681 731 780 830 879 928 1 ·978 85 6 85 ·987 986 985 984 983 982 981 980 979 978 85 ·977 1 ·501 551 601 651 701 751 801 851 901 951 2 ·001 6 7 86 ·987 986 985 984 983 982 981 980 979 978 86 ·976 1 ·518 569 620 670 721 771 822 873 923 974 2 ·024 7 8 87 ·987 986 985 984 983 982 981 980 978 977 87 ·976 1 ·536 587 638 689 741 792 843 894 945 996 2 ·048 8 9 88 ·986 986 985 984 983 982 980 979 978 977 88 ·976 1 ·553 605 657 709 760 812 864 916 96719 2 ·071 9 90 89 ·986 985 984 983 982 981 980 979 978 977 89 ·976 1 ·571 623 675 728 780 832 885 937 99042 2 ·094 90 1 90 ·986 985 984 983 982 981 980 979 978 977 90 ·975 1 ·588 641 694 747 800 853 906 95912 065 2 ·117 1 2 91 ·986 985 984 983 982 981 980 979 978 976 91 ·975 1 ·606 659 713 766 820 873 927 98034 087 2 ·141 2 3 92 ·986 985 984 983 982 981 980 978 977 976 92 ·975 1 ·623 677 731 785 839 894 94802 056 110 2 ·164 3 4 93 ·986 985 984 983 982 981 979 978 977 976 93 ·975 1 ·641 695 750 805 859 914 96923 078 133 2 ·187 4 95 94 ·986 985 984 982 981 980 979 978 977 976 94 ·974 1 ·658 713 768 824 879 934 99045 100 155 2 ·211 95 6 95 ·985 984 983 982 981 980 979 978 977 975 95 ·974 1 ·675 731 787 843 899 95510 066 122 178 2 ·234 6 7 96 ·985 984 983 982 981 980 979 978 976 975 96 ·974 1 ·693 749 806 862 919 97531 088 144 201 2 ·257 7 8 97 ·985 984 983 982 981 980 979 977 976 975 97 ·973 1 ·710 767 824 881 938 99552 109 166 223 2 ·280 8 9 98 ·985 984 983 982 981 979 978 977 976 975 98 ·973 1 ·728 785 843 901 958 16 073 131 188 246 2 ·304 9 100 99 ·985 984 983 982 980 979 978 977 976 974 99 ·973 1 ·745 803 862 920 978 36 094 152 211 269 2 ·327 100 477 374 267 157 044 927 807 683 556 426 292 524 341 158 974 791 608 424 241 057 873 690

0

0 100 500 200 400

0

0 100 500 200 400 271◦

₈₉

◦ 269◦ ₉₁◦

Sin.

271◦

₈₈

◦ 268◦ ₉₁◦ 271◦

₈₉

◦ 269◦ ₉₁◦

Cos.

271◦

₈₈

◦ 268◦ ₉₁◦