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Full text of "The Theory and Practice of Surveying"

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I 





RBoeiVHD IN EixoilANOK 

|W. L. Clements Library 




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I ■ 

I 



THE 

THEORY AND PRACTICE 

SURVEYING; 

COMTAIinNO 

AU the Instructions requisite for the s)ct^uljnraetice 

of this Art 

ROBERT GIBSON. 

ILLUSTRATED BY COPP£R4^LATB& 



WI«0I£ OOBUCnUD, KSWLT ARBAMOSD, AMD OB&ATLT SNLA^OBW; 

WITH USEFUL SELECTIONS, 



Am A VXW IBT m AGCVBAVB 



MATHEMATICAL TABLES. 



J^ D. P. JlDJUUa, 

TZACHim OV TBS ItATSnCAVIOi^ 



NEW-YORK : 



FUqpSHED BT EVEBT DUYCKmCK, 

Na 108 PBA&L4TBSBT. 
G. Xonj'y prifUor* 

1814. 






Bb it BMMMxamaMDt That oa Hbe tvenf^-eightli day of Harob, in ihe 
tbirty-fiftb year of the lodependeiice of the United States of America, Evert 
Ikijfckinck, of the aaid district^ hath depoated in this office the title of a hookj 
the light whereof he claims as proprietor, in the words foUowing, to wit : 

" The Theory and Practice of Surveying; containing all the Inatnictions re« 
quittte for the skilfol p— stiaa of this Ait By Robert Gibson, mostrated by 
Copper»PIates. Tht whole oorreeted, newly arranged, and gmtly enlarged, 
with tisefol Selections, and a new aet of accurate Mathematical TaUea. By 
D. P. Adams^ Tcdieher vf l^ Matbemalica." 

In conform!^ to the act of the Congren of the United Statea» eatided, *' An 
act Ibr the ODCOwragemeitt of learmng, by aeenring the copies of maps^ charts, 
and books, to the anthers and pr^tfietora of such copies^ during the times 
therein mentioned.'* And also to an act, entitled, ** ^n act, suppletnentary tfi 
an act, entitled^ An ai^t for the enoouragement of learning, fay securing the 
copies of mapS) charts, and books, to (he authors and proprietors of such co- 
pies, during the times therein mentioned, and extending the benefits thereof to 
the arts (if designing engraving, and etching historical and other prints," 

cifARLBs ctmrbN, 

Clerk 9f thf JMstrict pfJ^c0»rTk. 



4 • .. 
» k 

f 



9 






CONTENTS. 



PARTI. 



Page 



i 






« 

r 



Sect 1 Decimal Fvactiont 2 
% lovolutioD and Evolu- 
tion 15 

3. Of Logarithms 23 

4. Elements of Geome- 

try 37 

Mathematical Instra- 
ments 74 

5. TngonomeUy 99 

PART II. • 

Sect 1. The Chain 134 

The Circumferentor 152 
The Theodolite 159 
The Semicircle 164 
Mensuration of An- 
gles by these In- 
struments 169 
The Protractor 171 
2. Mensuration of 

heights ir9 

—-Of Distances 194 

3. Mensuration of Areas 200 

General Method 233 

Pennsvlvania Method 244 

4 Of Od'-sets 250 

5- Method of siiFve?iiig 

by Intersections 257 
6> Changing the soale 

of Maps 263 

r. Method of Dividing 

Land 27} 

9 Maritime Sorveyhig 281 



PARTIL 



Pftge 



Sect 9. LevtUigq: 2S4 

Promiscuous Ques- 
tions 295 

PART in. 

Sect. 1. Introductory Princi- 
ples 298 
2. DesGiiptioD oi In- 
struments 305. 
3.' To find the Latitude 
by the Meridian Al- 
titude of the Sun 316 

pass 318 

LIST OF TABLES 

Ix>garithms of Kumbers 1 

Sines, Tangents, and Secants 20 
Natural Sines « 71 

j Points of the Compass / 82 
Traverse Table 83 



!Mean Refiractkn 
; Sun's Parallax 
IDip of the Horizon 
Dip for Dif. Dist of land 
Semidiameter of the Sun 
Transit of Pole Star 
Difference of Altitude of 

Pole Star and Pole 
Sun's Peclinatioii 
Reduction Table 



} 



174 
175 

iirid. 
176 

ITT 
178 

182 



' , 



EXPLANATION 



0/ the iikikematical CAaroctcTM U9cd in iku Work* 



+ signifies yk/itf» or addidoiu 


I^HBB 


mntct) or subtmction 


Xor. J . 


multit^icatioii. 


•?■ 


dlTision. 


• '^ • • • 


proportion. 


» . . 


cqualitf. 


^ . . 


sqiureToot. 


^ . . 


cube rooti &c'. 


0) • . 


diff. between two ui 



knowtt which Is the grciuer. 
Thus, 

5 + 3, denotes that 3 is to be added to 5. 

6 -* 3| denotes that 2 is to be taken from 6. 

7 X. 3, or 7 . 3, denotes that 7 is to be multiplied bf 3. 
8-^4, denotes that 8 is to be divided bj 4. 

2: 3:: 4:6, shows that, 2 is to 3 as 4 is to 6. 

6 -f 4 s lOy shows that the sum of 6 and 4 is equal to 10. 

j/ 3, or 3i, denotes the square root of the number 3. 

^ 5, or 5^, denotes the cube root of the number 5. ' 

7^y denotes that the number 7 is to be squared. 

8', denotes that the nugiiber 8 is to be cubed. 



THtf 

THEORY AND PRACTICE 



<a 



SURVEYING. 



THE word Surveying, in the Mathematics, 
signifies the art of measuring land, and of 
delineating its boundaries on a map. 

The Surveyor, in the practice of this art, directs 
his attention, at first, to the tracing and measur- 
ing of lines ; secondly, to the position of these 
lines in respect to each other, or the angles form- 
ed by them ; thirdly, to the pltm, or representa- 
tion of the field, or tract, which he surveys ; and 
fourthly, to the calculation of its area, or super- 
ficial content. When this art is employed in 
obsendng and delineating Coasts and Harboui*a^ 
in determining their variation of tiie CJompass^ 
their Latitude, Longitude and soundings, together 
with the bearings of their most remarkable places 
from each other, it is usually denominated Mari- 
time Surveying. This branch of Surveying, how- 
ever, demands no other qualifications tiian those, 
which should be thorou^y acquired by every 
Land-Surveyor, who aspires to the character of 
an accomplished and skimil practitioner. Survey- 
ing, therefore, requires an intimate ao^uaintance 
With the several parts of the Mathennatics, which 
are here inserted as an introduction to thi$ treH«- 
t4se. 

B 



2 BECIiVIAL FRACTIONS! 

PART 1. 

Containing Decimal FracHcns, Invol^ian and Evo^ 
lution, the Nature and Use of Log4mthms3 Geo- 
meti^4ind Plane Trigwomelby. 

SECTION I. 

DECIMAL FRACTIONS, 

If we suppose unity or any one thing to be di- 
vided .into any assigned number of equal parts, this 
^number is called the denominator; and if we chuse 
to take any number of such parts 1^33 thw Ih^ 
whole, this is called the numerator ojF a fraction. 

The numeratoT, lit the vulgar forfo^ ifi always 
written over the denominator, an4 thes6 are sepa- 
rated by a sipall line thus ^ or f ; the first of theee 
is called three-fomlhs, and the latter five-ejghtb^ 
of an inch, yard^ &c. ^r of whatever the whole 
tiling originally cpnsisted : the 4 and the 8 are tiie 
denominsiors, showing into how many equal parts 
the unit is divided > and the three and the five are 
the numerators, showing how many of those {)arts 
are under consideration. 

Fractions are e?cpre3sed in two forms, that ig^ 
either vulgarly or decimally. 

All fractions whose denominators do not consist 
of a cipher, or ciphers, set after unity, are cal- 
led vulgar ; and their denominators are .always 
written under their numerator^. The treatment 
of these, however, would be foreign to our pre- 
sent purpose. But fractions whose denominators 
consist of an unit prefixed to one or more ciphers, 
are called decimal fractions ; the numerators of 
which are written without their denominators, 
and are distinguished from integers by a point 
prefixed : thus j^^, ^^^^^ and -^^^ in the decimal 
form, are expressed by «2 .42 ^72. 



MJCIMkAL FHACTMNS. 3 



o. 



Tlie denomintttoi^ df ^icb^ frsfestSidiis consisting 
always^of an unit, prefixed to as many ciphers as 
there are plfieed ^ %ures in tfee mmierators, it 
follows, that any imraber of ciphers- put after 
those niuneiiat^rs^ will neither increase nor lessen, 
their value : for ^v, r^ ami iWtt ^^ afl' of the 
sanae value, and will stand m the decimal form 
thus ,3 •30- .300^; feut a^ cipher, or ethers prefix- 
ed to those auitaerators^ lessen their value in a ten- 
fold proportion : for -nr, T*inr and tiU , which in 
the decimal ftirro we denote by ,3 .03. and .003, 
are fractions, of wtikh the first is ten times greater 
than the second ; and the second, ten times great- 
er than the third. 

Hence it appears, that as tlie value and deno- 
mination of any figure, or number of figures, in 
conmion arithmetic is enlarged, and becomes ten, 
or an hundred, or a- thousand times greater, by 
plaeing one or tiw^, or ti;u:ee ciphers after it ; so 
in deeunal; arithmetic, the value of any figure, or 
BQfi^ber of figures,, decrea^pe^ and becomes ten, 
OF a hufidred, or a tiiousand times less^ while the 
denomination of it increases, and becomes so ma* 
ny time&greater, by prefixing on6, or two, or three 
cifdptere ta it) : and tiiat any niunber of ciphers^ 
before an integer, or after a decimal fraction, ha$^ 
no effect in changing their values. 



. / 



DECIMAL FRACnONS. 

SCALE OF NOTATION. 

Inteffers. Decimals. 

r- ^ N 

7 3 4 2 18 6 



_^C a 5 C2 g s 

t=!9 ^ 2 5 § sr 



v^* C^ p^ c (^ * 
60 




ADDITION OP DECIMALS. 

Write the numbers under each other according 
to the value or denomipation of their places ; 
which position will bring all the Decimal points 
into a column, or vertical line, by themselves. 
Then, beginning at the right hand column of 
figures, add in the same manner as in whole num- 
bers, and put the decimal point, in the sum di- 
rectly beneath th^ other points. 

EXAMPLES. 

Add 4.7832 3.2543 7.8251 6.03 2.857 ap4 
3,251 together. Place them thus, 

4.7832 
3.2543 
7*8251 
6.03 
2.857 
3.251 



SumfHr28,0Qp6, 



• 
•«i. 



DECIMAL FRACTIONS. 6 

Add 6^2 121.306 .75 2.7 and .0007 to^ 

sether. 121.306 
* .75 

2.7 ^ 

.0007 



Sum «^ 30.9567 

■■■■■ataHiMiHBatf 

What is the sum of 6.57 1.026 .75 146.5 
8.7 526. 3.97 and .0271 ? 

Answer 693.5431. 

What is the sum of 4.51 146.071 .507 .0006 
132. 62.71 .507 7.9 and .10712 ? 
Answer 354.31272. 

SUBTRACTION OP DECIMALS. 

Write Ihe figures of the subtrahend beneath 
those of the minuend according to the denominar 
tion of thehr places, as directed in the rule of 
addition ; then, beginning at the right hand, sub- 
tract as in whole numbers, and place the decimal 
point in the di0erence exactly under Uie other 
two pointa. 

EXAMPliES. 

From 38.765 take 25.3741 
25.3741 



Difference si 3.3909 



From 2.4 take .8472 
.8472 



iffs 1.5528 



mm^i^m^mm 



9. 'DBS£JMAL PRACTI0I9S; 



From 71 .tdi take &483724a 
Difference » 62.966217^ 
From 84 take 82.3412. 
Diff. = 1.65^. 

MULTIPLICATION OF DECIMALS. 

Set the multiplier under the multiplic and with- 
pui any regard to the situation of the decimal 
point ; and haidn^ multiplied asin whole nmnbersy 
cut off as many places for decimals in the product^ 
counting from the right hand- towards the left, as 
there are in both the multiplicand and multiplier : 
but if there be not a sufficient number of places 
' in the product, the defect may be supplied by pre- 
fixing ciphers thereto. 

For the denominator of the product, being an 
unit, prefixed to as many cijphers, as the denomi- 
nators of the multiplier and multiplicand contain 
of ciphers, it follows, that the places of decimals 
in the product, will be as many aa in Uie: numbers 
from whence, it arose. 

Multiply 48.765 by ,003609 
.003609 



•^^^rmmm^f 



438885 , 
292590 
146295 

Product— .17.5992885 

Multiply .121 
• b y .14 

484 
121 



Product =.01694 



Decimal FRAcrtows. 

Midtiply 121.6 by 2.76 

2.76 



7296 
8512 
2432 



Product ==335.616 

. Miiltroly .0089789 by 1085 
rroduct =5= 9.7421065 

Multiply .248723 by .13587 
Product = .03379399401. 

DIVISION OP DECIMALS. 

Divide as in whole numbers ; observing that the 
divisor and quotient together must contain as ma- 
ny decimal places as there are in the dividend. If, 
therefore, the dividend have just as many places 
of decimals as the divisor has, the quotient will 
be a whole number without any decmial figures. 
If there be more places of decimals in the divi- 
dend, than there are in the divisor, point off a^ 
many figures in the quotient for decmials, as the 
decimal places in the dividend exceed those in the 
divisor ; the want of places in ihe quotient being 
supplied by prefixing ciphers. But if there be 
more decimal places m the divisor, than in the di- 
vidend, annex ciphers to the dividend, so that the 
decimal places here may be eaual, in number, to 
those in the divisor ; and then tne quotient willbe 
a whole number, without fractions. 

When there is a remainder, after the division 
has been thus performed, annex ciphers to this 
remainder, and continue the operation till nothing 
remains, or till a sufficient number of decimals 
shall be found in the quotient. 



DECIMAL PRACnONa 

EltAMPT-WS. 

Divide .144 by .12 

.12).144(1^ es quotient 
12 



24 
24 




Divide 63.72413456922 by 2718 
2718)63.72413456922(.02344522979 «» quotient 
5436 



lArtMi 



9364 
8154 

12101 
10872 

12293 
10872 

14214 
13590 



6245 
5436 

8096 
5436 

26609 
24462 

21472 
19026 



24462 
24462 







DECIMAL FRACTIONS. ^ 

There being 1 1 decimal figures in the diridend, 
and none in the divisor, 1 1 figures are to be cut off 
in the quotient ; but as the quotient itself con- 
sists of but 10 figures, prefix to them a cipher to 
complete that number* 

Divide 1.728 by ,012 

jOl 2) 1.728(144= quotient. 
12 



52 
48 



48 
48 




Because the number of decimal figures in the 
divisor and dividend, are alike, the quotient will 
be integers. 

Divide 2 by 3.1416 
3.141 6)2.0000,0(0.6366l84-»qaotient 
1 8849 6 



115040 
94248 

207920 
188496 

194240 
188496 

57440 
31416 



260240 
251228 

9012+ 
C 



10 DECIMAL FRACTIONS, 

In this example there are four decimal fi^uMs 
in the divisor, and none in the dividend ; uiere* 
fore, according to the rule, four ciphers are an- 
nexed to the dividend, which in this condition, is 
yet less than the divisor. A cipher must then be 
put in the quotient, in the place of integers, and 
other ciphers annexed to the dividend ; and the 
division being now performed, the decimal figures 
of the quotient are obtained. 

^ 

Divide 7234.5 by 6.5 Quotient=ni3. 

Divide 476.520 by .423 =1126.5+ 

Divide .45695 by 12.5 =..0365+ 

Divide 2.3 by 96 =.02395+ 

Divide 87446071 by .004387 —=19933000000. 
Divide .624672 by 482 —-=.001296. 

REDtrCTION OP DECIMALS. 

RblE I. 

To reduce a Vulgar FVaeiion to a Ifecimal of the 

same value. 

Having annexed a sufficient number of ciphers, 
as decimals, to the numerator of tiie vulcar frac- 
tion, divide by the denominator ; and me quo« 
tient thence arising, will be the decimal fraction 
required. 

EXAMPLES. 

Reduce ^ to a decimal fraction. 
4)3.00 

.7d==:decimal requii^d. 

For I of one acre, mile, yard, or any things 
is equal to ^ of 3 acres, miles, yards, &c. there- 



DECIMAL FRACTIONS. 11 

fore if 3 be divided by 4, the quotient is the an- 
swer required. 

Reduce I to a decimal fraction. Answer .4 
Reduce 41 - - - - .48 

Reduce t% - - - ^ .11 46789 
Reduce * . - . . .7777+ 

Reduce H - - - - .9130434+ 

Reduce h ky ij h and so on to irV to their cor- 
responding decinial fractions ; and in thi^ opera- 
tion the various modes of interminate decunals 
may be easily observed, 

RULE II. 

To reduce QuantUies of the same^ or of different 
OenamincUions to Decivud Fractions of higher 
denominations^ 

If the given quantity consist of one denomina- 
tion only, write it as the numerator of a vulgar 
fraction ; then consider how many of this make 
one of the higher denoinination, mentioned in the 
question, and write this latter number under the 
rormer, as the denominator of a vulgar fraction. 
When this has been done, divide the numerator 
by the denominator, as directed in the foregoing 
rule, and the quotient resulting will be the deci- 
mal fraction requirM. 

But if the given quantity contain several deno^ 
minations, reduce them to the lowest term for the 
numerator; reduce likewise that quantity, whose 
fraction is sought, to the same denomination for 
the denominator of a vulgv fraction ; then divide 
as before directed. 

' EXAMPLES. 

Reduce 9 ipches to the Decimal of a fqo^^ 



12 DECIMAL FRACTIONS. 

The foot being equal to 12 inches, the vulgs^ 
fraction will be ^ ; then 12)9.00 

.Tdaedecinial frao- 
[tion required. 
Reduce 8 inches to the decimal of a yard. 

8 inches. 



1 y^rd X 3 X 12 = 36 inches. 

36)8.0(.22 + = Answer. 
72 



80 
72 



8 

Reduce 5 furlongs 12 perches to the decunal of 
a mile. 

1 mile 5 furlongs 

8 40 



8 fur. 200 

40 'as vulgar fraction 

- — 320 



320 per. 



320)200.0(.625 = decunal soughjt. 
1920 



800 
640 



1600 
1600 



Reduce 21 minutes 54 seconds to the decimal of 

a degree. Ans. .365 
Reduce .056 of a pole to the decimal of an A<;re, 

Ans. .00035 



DECIMAL FRACTIONS. 13 

)[l6duce 13 cents to the decimal of an Eagle. 

Ans. .013 
Reduce 14 minute to the decunal of a day. 

Ans. .00972+ 
Reduce 3 hours 46 minutes to the decunal of a 

week* Ans. .0224206+ 

RULE III. 

To find the value of Decimal Fractions in terms of 

the lower denofninations. 

Multiply the given decimal by the number of 
the next lower, denomination, which makes an 
integer of the present, and point off as many pla- 
ces at the right hand of the product, for a re- 
' mainder, as there are figures in the given deci- 
mal. Multiply this remainder by the number of 
the next inferior denomination, and point off a re^* 
mainder, as before. Proceed in this manner 
through all the parts of the integer, and the seve- 
ral denominations, standing on the left hand^ are 
the value required. 

EXAMPLES* 

Required the value of .3375 of an acre. 

4 = number of roods 

[in an acre, 

1.3500 

40 = number of percln 
[es in a rood.- 



14.0000 
The value, therefore, is 1 rood 14 percheg. 



U DECIMAL FRACTIONS. 

%hat is <he value of -6875 of a yard! 

3= number of feet in a 

[yard. 

2.0625 

12s: number of inches in 
— ~— [a foot.. 

.7500 

12aB: number of lines in 

[an incb* 



9.0000 
The answer here is 2 feet 9 linesw 

What is the value of .084 of a furlong ? Ans. 3 

per. 1 yd. 2ft. 11 in. 
Wnat is the value of .683 of a degree ? Ans. 40 

m. 58 sec. 48 thirds. 
What is the value of .0053 of a mile ? Ans. 1 

per. 3 yds. 2 ft. 5 in.+ 
What is the value of .036 of a day ? An?- 51' 

50" 34'". 

PROPORTION 

IN DECIMAL FRACTIONS. 

Having reduced all the fractional parts in the 
given quantities to their corresponding decimal^, 
and having stated the three known terms, so that 
the fourth, or required quantity, may be as much 
greater, or less than the third, as the*secrfnd term 
is greater, or less than the first, then multiply the 
second and third ternis together, and divide the 

Eroduct by the first term,' and the quotient will 
e the answer ; — in the same denomination with 
the third term. 

EXAMPLES. 

If 3 acres 3 roods of land can be purchased for 
93 dollars 60 cts. how much will 15 acres 1 rood 
cost at that rate ? 



bBCIMAL FRACTIONS. ' 15 

3 acs. 3 rde. as d.'j^ acres.. 
15 acs. 1 rd. ea 15.25 acre^ 
893 , eOcts. sS93.60 
Then 3.75 : 15.25 : : 93.60 : 

15.25 



46800 
' 18720 
4680O 
9360 

$■ 

3.75) 1427:4000(380.64=«Ailswer. 
1125 



3024 
3000 



2400 
2250 

1500 
1500 



If a clock eain 14 seconds in 5 days 6 hour^ 
how much will it gain in 17 days 15 hours ? Aw. 
47 seconds. 

If 187 dollars 85 cents gain 12 dollars 33 cents 
interest in a year, at what rate per cent is flii» v»> 
terest? An8.6.56-f 



SECTION II. 

» 

INVOLUTION AND EVOLUTION. 

Involution is the method of raising any nuitH 
ber^ coBtsidered sm ^^T««t, to any requved power. 



16 t)ECIMAL PRACTIONS. 

Any number, inribether given^ or assumed at 
pleasure, may be called the root, or first power 
of this number ; and its other powers are tb^ pro- 
ducts, that result from multiplying the number 
by itself, and the last product by the same num- 
ber again ; and so oa to any number of multipU- 
cations. 

The index, or exponent, is the number donot- 
ing the height, or degree of the power, being 
always greater by one, than the number of multi- 
plications employed in producii^ the power It 
is usually written above the root, j^ in the follow* 
ing EXA^iPLE, where the method of involution is 
plainly exhibited. 

Required the fifth power of 8 ) ^the root, or first 
first mujtiply by - - 8 J ~ power. 

then multiply the product 64 = 8* = square, or 

by 8 [second power. 

&c. 512 = 8* = cube, or 
8 [third power. 

4096 = 8«=biquadraW 
8 [or fourth power. 

32768 = 8» = Answer. 



EXAMPLES FOR EXERCISE. 

What is the second power of 3.05 1 Ans. 9.3025 
What is tlie third power of 85.3 ? Answer, 

620650.477 
What is the fourth power of .073 ? Answer^ 

090028398241 
What is the eighth power of .09 ? Answer, 

.00.00:00.0043046721 



tNVQLlTTION. 17 

Note. When two, or more powers are .nmlti- 
plied together, their product is that power, whose 
index is the sum of the indices of the factors, or 
powers multiplied. 

EvotutiOTsr is the method of extracting any re- 
quired root from any given power. 

Any number may be considered as a power of 
some other number ; and the required root of any 
given power is that number, which, being multi- 
plied into itself a particular number of times, pro- 
duces the given power; thus if 81 be the given 
number, or power, it« square, or second root, is 9 ; 
because 9 x 9^=9' =81 ; and 3 is its biquadrate, or 
fourth root, because 3x3x3x3= 3^=* 8 1 . Agaki, 
if 729 be the given power, and its cube foot be re- 
quired, the answer is 9, for 9x 9 x9==: 729 ; and if 
the sixth root of that number be requued, it is 
found to be 3, for 3x3x3x3x3x3=729.^ 

The required power of any given number, or 
root, can always be obtained exactly, by multi- 
plying the number continually into itself ; but 
there are many numbers, from which a proposed 
root can never be completely extracted ; — yet by 
approximating with decimals, these roots may be 
found as exact as necessity requires. The roots 
that are found complete, are denominated rational 
roots, and those, wnieh cannot be found complet- 
ed, or whicjb only approximate, are called surd^ 
or irrational roots. 

Roots are usually represented by these cha- 
racters or exponents ; 

t^, or i which signifies the square root ; thus, 
V9, or94=3 

i t 

^' or^ cube root ; ^ 64, or 64^=4 

X i. 

^, or* biquadrate root ; ^ 16, or 16*=2 &c. 

D 



18 EVOLUTION. 

S 

Likewise 8^ signifies the square root of 8 cub^ 
ed ; and, in general, the fractional indices imply, 
that the gi^en numbers are to be raised to such 
powers as are denoted by their nimierators, and 
that such roots are to be extracted fit>m these 
powers, as are denoted by their denominators. 

RULE 

For extracting the Square Root. 

Separate the given number into periods of two 
figures, by putting a point over the place of units, 
another over the place of hundreds, and so on, 
over every second figure, both toward the left, 
hand in whole numbers, and toward the right hand 
in the Dcfcimal places.— When the number of 
integral places is odd, the first, or left hand pe- 
riod, will consist of one figure only. 

Find iHe greatest square in the first period on 
the left hand, and write its root on the right hand 
of the j^ven number, in the manner of a quotient 
figm'e m division. 

Subtract the square, thus found, from the said 
period, and to the remainder annex the two 
figures of the next following period^ for a divi- 
dend. 

Double the root above mentioned for a divi- 
sor, and find how often it is contained in the said 
dividend, exclusive of its right hand figure, and 
set this quotient both in the place of the quotient 
and in the divisor. — The best way of doubling 
the root, to form each new divisor, is to add the 
last figure always to the last divisor, as it is done 
in the subsequent examples. 

Multiply the whole augmented divisor by this 
last quotient figure, and subtract the product froril 
the said dividend, bringing dowii to it the next 
period of the given nua),ber for a new dividend* 



( 
I 



EVOLUTION. 19 

Repeat the same operation again ; that is, find 
another new diyisor, by doubling all the figures 
now found in the root ; from which, and the last 
dividend, find the next figure of the root as be- 
fore ; and so on through all the periods to Uie 
last 

Note 1. After the figures belonging to the giv- 
en number are all exhausted, the operation may 
be continued in decimals, by annexing any num* 
ber of periods or ciphers to the remainder. 

2. The number of integral places in the root, 
is always equal to the number of periods in the 
integral part of the resolvend. 

3. When vulgar fractions occur in the givep 
power, or number, they may be reduced to deci- 
mals, then the operation will be the same as be-* 
fore dictated. 

EXAMFIiES. N 

Required the ^uare root of 1710864. 



• • • m 



1 
1 



1710864(1308,=AngWer, 
1 



23 
3 



71 

69 



2608 



20864 
20864 



20 EVOLUTION. 

Required tlie gquai-e root of 16007.3104. 



I 
1 



1 6007.3104(1 26.52= Answer. 
1 



22 
2 



60 
44 



246 
6 



1607 
1476 



2525 
5 



13131 
12625 



25302 



50604 
50604 



EXAMPLES FOR EXERCISE, 

Bequired the square root of 298116. Ans. 546, 
Required the square root of 348.17320836. Ans. 

18.6594. . 
Required the square root of 17.3056. Ans. 4.16, 
Required the square root of .000729. Ans. .027, 
Requued the square root of 17f Ans. 4.168333+ 

A GENERAL RULE 

For extracting any Hoot whatever. 

Find by trial a number, which, when involved 
to the power denoted by the index of the i;equirr 
ed root, shall come nearest to the given number, 
whether greater or less ; and let that number bo^ 
called the assumed root, and when thus involved, 
the assumed power. 



EVOLUTION. 21 



Let the giyen power> or number be repre- ) ^ 
sentedby ^ . y * 

the index, or exponent^ in the question by X. 
the assumed power, by A. 

the assumed root, by . Q. 

and the required root by R. 



ThenX+lxA+X— lxG:X+lxG+X— TxA 

That is, as the sum of X+1 times A and X — 
1 times G, 

is to the sum of X+1 times G and X — 1 
times A^ 

so is the assumed root, Q, 

to the required root, R, — nearly ; and the 
operation may oe repeated as many times as we 
chuse, by Rising always the root last found for the 
assumed root, and this, involved according to the 
given index, for the assiuned power.^ 

EXAMPLES. 

1. Required the Cube root of 789. 

"^ *■ This is a very general approxiaiating rule/* stys Dr. Hut- 
ton, *' of which that tor the cube root is a particular case, and 1$ 
the best adapted for practice and for memory, ai any that 1 have 
yet seen. It was first discovered in this form by myself, and the 
iiiTestieation and use of it were giftn at large hi my Tracts->->pa^ 
is &C.'* 



iB2 EVOLtmON. 



X+l 



i»4 and X-l=2. 
And 4X729=2916 4X789=3156 
2x789=1578 2X729=1458 



Then 4494 4614 : : 9 : 9^(H- 

9 



4494)41526(9.24034[Ans. 
40446 



10800 
8988 

18120 
17976 



144QD 
13482 



918 &c. 

In the foregoU^ example the answer is strictly 
correct in its iotegralpart and also in the three first 
decimal places ; but if more decimals were wanted, 
and if their exactness were likewise requisite, the 
present answer might be taken for the assumed 
rooty and the whole operation should be repeated. 

2. Required the biquadrate root of 2.0743. 

Here G=2.0743, ft=1.2,A=L2*=2.0736, X=4, 

X+l=5, and X- 1=3. 
And 5x2.0736=10.3680 5x2.0743=10.3715 
3x2.0743= 6.2229 3x2.0736= 6.2208 



Then 16.5909 16.5923 

[ : : 1.2 : 1.2001+Ans. 



J 



LOGARITHMS. & 

Required the fifth root of 21035.8 Ans. »7.3213+ 
Requkedthe sixth root of 21035.8 Ans. »5.25407 
Required the cube root of 999 Ans. «9.9966-f 
Required the foUrth root of 97.41 Ans. »= 3.141 6 
Required the cube root of .037 Ans. =.33322+ 
Required the cube root of 2 Ans. » 1.2599-f 
Required the seventh root of 21 035.8 Answerer 

[4.1454; 



SECTION III. 

OP LOGARITHMS. 

Logarithms are a series of numbers, so contiiv-* 
ed/ that by them the work of multiplication may 
be performed by addition ; and the operation of 
division may be done by subtraction. Or, — ^Lo- 
garithms are the indices, or series of numbers in 
arithmetical progression, corresponding to another 
series of numlj^rs in geometrical progression. 
Thus,- 



1 



0,1,2)3, 4, 5, 6, &c. Indices or Logarithms. 

1, 2,4, 8, 16, 32, 64, &c. Geometrical progression. 

Or, 

0, 1, 2, 3, 4, 5, 6, &c. Ind. or Log. 

1, 3, 9, 27, 81, 243, 729, &c. Geometrical Series. 

Or, 

(0, 1, 2, 3, 4, 5, 6,&c.LorL: 

{ 1, 10, 100, 1000, 10000, 100000, 1000000, &c. 
Geometrical series, — ^where the same indices serye 
equally for any Geometrical series, or progress 
sion. , ' ' 

Hence it appears that there may be as many 
l[inds of indices, or logarithms, as there can b^ 
taken kinds Df geometncal series. But the Loga- 
rithms most convenient for o^nifn^n w%n are those 



U LOGARITHMS. 

adapted to a geometrical aeries increasing in a 
tenjbld ];Nrogr6ssi<Hi» as in the last of the foregoing 
examples* 

In the geometrical series 1, 10, 100^ 1000, &c. 
if between the terms 1 and 10, the numbers 2, 3^ 
4, 5> 6, 7, 8» 9 were interposed, indices might also 
be adapted to them in an arithmetical progres- 
sion» suited to the terms interposed between 1 
and 10, considered as a geometrical progression. 
Moreover, proper indices may be found to all the 
numbers, that can be interposed between any two 
terms of the Geometrical series. 

But it is evident that all the indices to the num- 
bers under 10, must be less than 1 ; that is, they 
must be fractions. Those to the numbers between 
10 and 100, must fall between 1 and 2 ; that is, 
they are mixed numbers, consisting of 1 and some 
fraction. Likewise the indices to ihe numbers be- 
tween 100 and 1000, will fall between 2 and 3; 
that is, they are mixed numbers, consisting of 2 
and some fraction ; and so of the other indices. 

Hereafter the integral par| only of these indices 
wQl be called the Index ; and the fractional part 
will be called the Logarithm. The computation 
of these fractional parts, is called making Loga- 
rithms ; and the most troublesome part of this 
work is to make the Logarithms of Prime Num- 
berSy or those which cannot be divided by any 
other numbers than themselves and unity. 

RULE 

For Computing the Logarithms of Numbers. 

Let the sum of its proposed number and the next 
less number be called A. Divide 0.8685889638xt 

t The number 0.8685889638^ is the quotient of 2 divided by 
303585093, which is the logarithifr of 10, according to the first 



2. 



, OP LOGARITHMS. 25 

toy A, and reseire the quotient. Divide the re- 
served quotient by the square of A, and reserve 
this quotient. Divide the last reserved quotient 
by the square of- A, reserving the quotient still ; 
and thus proceed as long as division can be made. 
Write the reserved quotients orderly under one 
another, the first being uppermost. Divide these 
quotients respectively by the odd numbers 1, 3, 5, 
7, 9, 11, &c.; that is, divide the first reserved quo- 
tient by 1, the second by 3, the third by 5, the 
fourth by 7, &c. and let these quotients be written 
orderly under one another ; add them together, and 
their sum will be a logarithm. To this logarithm 
add the logarithm of the next less number, and the 
«um will be the logarithm of the number proposed. 

ibrni of Lord Napier, the inventor of logarithms. The manner in 
"Which Napier's logarithm of 10 is fouiid, may be seen in most books 
«f Algebra, but it is htre omitted, because students of Surveying 
are too generaH]^ unacquainted with the principles of that science* 
and the subiect is too extensive for the present treatise. Those, 
however, who have not an opportunity for entering thorou^ly into 
this subject, may with more prt)priety grant the truth of one number, 
and thereby b^nabled to try the correctness of any logarithm in the 
tables, than receive those tables, as truly ooroputed, without, any 
means of examining their accuracy. 



£ 



26 



OF LOGARITHMS* 



EXAMPLE I. 



/ 



Kequired the Logarithm of the number 2. 

4 

Here the next less number is 1, and 2+1 =3^ 
A. and A% or 3*= 9 ; then 

3)0.868588964 

. 9)0.289529654-5- 1 =0.289529854 
9)0X)32l69962-i- 3=0.010723321 
9)0.003574440-J- 5=0.000714888 
9)0.000397160-?- 7=0.000056737 
9)0.0000441 a9-e- 9=0.000004903 
9)0.000004903-5- 1 1 = 0.000000446 
9)0.000000545-r- 1 3=0.000000042 



i«i«IM 



0.000000061 --15=0.000000004 



To this Logarithm 0.301029995 
add theLogarithm of 1 =0.000000000 

Tlieir Sum =0.301029995 =Log. of 2. 

The manner in i^hich the division is here carried 
on, may be readily perceived by dividing, in the 
first place, the given decimal by A, and the suc- 
ceedmg quotients by A* ; then letting these quo- 
tients i*emain in their situation, as seen in tiie ex- 
ample, divide them respectively by the odd num- 
bers, and place the new quotients in a column by 
themselves. By employmg this process, the ope* 
ration ii considerably abbreviated. 



OP LOGARITHMS. 27 

EXAMPLE 2. 

Required the Logarithm of the number 3. 

Here the next less number is 2 ; and 3+25 = A, 
•ndA»=26. 

5)0.868588964 
25)0.173717793+ 1=0.173717793 
25)0.006948712-f- 3=0.002316237 
25)0.000277948'i- 5=0.000055599 
25)0.00001 1118-r 7=0.000001588 • 

25)0.000000445-7- 9=^0.000000049 
0.00000001 8-5-11 =0.000000002 



To this Logarithm 0.1 76091259 
add the Logarithm of 2=0.301029995 

Their Sum =0477! 21 254= Log. of 3. 

Then, because the sum of the logarithms of 
numbers, gives the logarithm of their product; and 
the difference of the logarithms, gives the logarithm 
of the quotient of the numbers : from the two pre- 
ceding logarithms, and the logarithm of 10, which 
k 1, a great many logarithms can be easily made, 
as in the following examples. 

Example 3. Required the Logarithm of 4. 

Since 4=2x2, then to the Logarithm of 

2=0.301029995 
add the Logarithm of 2=0.301029995 

The sura ^Logarithm of 4=0.602059990 



28 OP LOGARITHMS. ^ 

Example 4. Required the Logarithm of 5. 

lO-i-2 beiDg=5, therefore from the Log. of 

10^1.000000000 
subtract the Log. of 2— asoi 029995 

the remainder is the Log. of 5=0.698970005 

Example 5. Required the Logarithm of 6. 

6=s3x2, therefore to the Logarithm of 

3=0.477121254 
add the Logarithm of 2:^=0.301029995 

** their sun «Log. of 6 »0.778l 5 1 249 

Example 6. Required the Lf^arittun of 8. 

9—2', therefore multiply the Logarithm of 

2=0.301029995 
bj 3 

The product ;p=Log. of 8=0.903089985 

I 

Example 7. Required the Logaritlim of 9. 

p s=z3\ therefore the Logarithm of 

3:^0.477121254 
being multiplied, by " 2 



•mam 



the products Log. of 9=0.954242508: 



OF LOGARITHMS. 29 

Example 8. Required the Logarithm of 7. 

Here the next less nmnber is 6, and 7-^6«133s% 
A,and A*«=169. 
13)0.868588964 

^^"■^"■■— •■■■■■■■■• 

169)0.066814536-^1=0.066814536 

m 

169)0.00039d352-r3sO.00O131784 

] 69)O.0000O2339-;-5 =0.000000468 

0.000000014-r 7 =0.000000002 



To this Logarithm=-0.066946790 
add the Log. of 6=0.778151249 

Their sum=0.845098039=Log. of 7. 

oi3 and 4. 
of 7 and 2. 



r^e-in r 



of 12 
of 14 



rim x-^„ J of 15 is equal to the sum J of 3 and 5. 
1 lie ^g'<. of 16 of the Logs. .1 of 4 and 4. 






of 18 
of 20 



of 3 and 6. 
of 4 and 5. 



The Logarithms of the prime numbers, 11, 13, 
17, 19, &c. being computed by the foregoing gene- 
ral Rule, the Logarithms of the intermediate num- 
bers are easily found by composition and division. 
It may, however, be observed, that the operation is 
shorter in the larger prime numbers; for when any 
given number exceeds 400, the firgt quotient, being 
added to the Logarithm of its next lesser number. 
Will give the Logarithm sought, true to 8, or 9 
places ; and therefore it will be very easy to exa- 
mine any suspected Logarithm in the Tables. 

For the arrangement of Logarithms in a Tahlcy 
Ike method of finding the Logarithm of any natural 
mmber^ and of finding the natural nmnber corres- 



3Q OF LOGARITHMS. 

ponding to any given Lcgarithm^ therein : likewise 
for particular rides concerning the Indices, the read- 
er will consult Table 1, with its explanation, at the 
end of this Treatise. 

MULTIPLICATION, 

Two, or more numbers being giveiiy tojind their pro- 
duct by Logarithms. 

RULE. 

Having found the Logarithms of the given num- 
bers in the Table, add them together, and their 
► sum is the Logarithm of the product ; which Lo- 
garithm, being found in the Table, will give a na- 
tural number, that is, the product required. 

Whatever is carried from the dechual part of the 
Logarithm is to be added to the affirmative indices ; 
but subtracted from the negative. Likewise the in- 
dices must be added together, when they are all of 
ti^e same kind, that is, when they are all affirma- 
tive, or all nega,tive ; but when they are of different 
kinds, the difference must be found, which will be 
of the same denomination with the greater. 

Exansple 1. Required the product of 86.25 
multiplied by 6.48 

Log. of 86.25=1.935759 
Log. of 6.48=0.811575 

Product= 558.9= 2.747334 

Example 2. Required the product of 46.75 and 

.3275 

Log. of 46.75= 1.669782 
Log. of .3275=— 1.515211 

Product = 15.31+ = 1.184993 



<^ LOGARITHMS; 3r 

Example 3. Required the product of 3.768, 
2.0i>3 and .007693. 

Log. of 3.768= 0.576111 
Log. of 2.053= 0.312389 
Log. of .007693=— 3:886096 

Product«=.0595ix =—2.774596 

Example 4. Required the product of 27.63, 
1.859, .7258 and 0.3591. 

Log. of 27.63= 1.441381 
Log. of 1.859= 0.269279 
Log. of .7258 =—-1.860817 
Log. of .03591=— 2.565215 

Product nearly= 1.339 = 0.126692 

DIVISION. 

Two numbers being gwen, tojindhow numy times 
me is c<mtainea in the other, by Logarithms. 

RULE. 

From the Logarithm of the Dividend subtract ' 
the Logarithm of the Divisor, and the remainder 
will be the Logarithm, whose corresponding natu- 
ral number wUl be the Quotient required. 

In this operation, the Index of the Divisor must 
be changed from affirmative to negative, or from 
negative to affirmative ; and then the difference 
of the affirmative and negative Indices must be 
taken for the index to the Logarithm of the Quo- 
tient. Likewise when ope has been borrowed in 
the left hand place of the Decimal part of the Lo- 
garithm, add it to the Index of the Divisor, if affir- 
mative ; but subiract it, if negative ; and let the 



32 OF LOGARITHMS. 

Index, thence arising, be changed and worked 
with, as before. 

Example 1. Divide 558.9 by 6.48. 
Log. of 558.9 =2.747334 
Log. of 6.48 =0.811575 

Cluoiient = 86.25 =1.935759 

Example 2. Divide 15.31 by 46.75. 

Log. of 15.31= 1.184975 ... 1 

Log. of 46.75= 1.669782 

ftuotient=.3275=— 1.515193 ^ ' 

Example 3. Divide .05951 by .007693. 
Log. of .05951 =—2.774590 
Log. of .007693=--3.886096 



# 



auotient=7.735 = 0.888494 

Example 4. Divide 4»651 by 22.5. 
Log. of .6651=— 1.822887 
Log. of 22.5 = 1.352183 

€luotient=.02956=— 2.470704 



PROPORTION, 
Or the Rule of Thru in Logarithms. 

RULE. 

Haying stated the three given terms according 
to the rule in common Arithmetic, write them or- 
derly under one another, with the signs of propor- 
tion; then add the Logarithms of the siecond and 
third terms together, and from their sum subtract 



OP LOdARITHMS. 3* 

^e Logarithm of the first term, and the remainder 
will be the Logarithm of the fourth term, or An- 
swer. 

Or, — ^add together the Arithmetical Complement 
of the Logarithm of the first term, and the Loga- 
ritlims of the second and thud terms; the sum, re- 
jecting 10 jfrom the index, will be the Logarithm 
of the fourth temi, or term required. 

N. B. The Arithmetical Complement of a Loga- 
rithm i^«hat it wants of 10,000000, or 20,000000, 
and the eMiest way to find it is to begin at the left 
hand, and subtract every figure from.9, except the 
last, which should be taken from 10; but if the 
index exceed 9, it must be taken from 19, — It is 
frequently used in the rule of Proportion and 
Trigonometrical calculations, to ph^nge Subtracr 
tions into Additions^ 

ISXAMFLEfl. 

M. If a eloekgain .14 seconds in 5 days 18 
kours, how much will it ^in in 1 7 days 1 5 hours ? 
5.75 days :Log.= 0.759668 

17.625 days : : Log.= 1 .246129 
14 Seconds ': Log.= 1.146128 

2.392257 



Anmer=42". 91 « 1.632589 

Or Ihus ; 5.75 days : Arith. Co. Log.= 9.240332 

17.625 : : Log.= 1.246129 

14 Seconds: Log.= 1.146128 



Answers 42". 91 *=1.632$89 



F 



^ OP LOGARITHMS. 

2d. Find a fourth proportional to 9.485, 1.960 
and 34^7.2. 

98.45 : Log.=l,9932l6 

347.2 J S»Log.«= 2.540580 
1.969 :* Log. =0.294246 

2.834826 

Answer=6.944 =0.841610 

3d. What number will have the same proportion 
to .8538 as .3275 has to .0131 

.0131 : Log. =—2.117271 

.3275 : : Log.=— 1.515211 
.8538* : Log.=— 1.931356 

—1.446567' 



An3wer=21.35= 1.329296 

4th. Required a third proportional number t« 
9.642 and 4.821 

9.642 : Log. =0.984167 



» I 



4.821 : : Log =0.683137 

4.821 Log! =0.683137 

1.366274 



Answer=:2.411 =s=0.38210t 
INVOLUTION. 

Tojind any proposed power of a given number hy 

Logarithms. 

Rule. Multiply the Logarithm of the given num- 
ber by the Index of the proposed power, and the 



OF LOGARITHMS. 35 

product will be the Logarithm^ whose natural 
number is the power required. 

When a negative Index is thus multiplied, its 
product is negative, but what was carried from the 
decimal part of the Logarithm must be affirmative; 
consequently the difference is the index of the pro- 
duct, which difference must be considered of the 
^ame kind with the greater, or that which was madtf 
the minuend. 

EXAMPLES^ 

!• What is the second power of 3.874 ? 

Log. of 3.874=0.588160 
Index = 2 



Power required=: 1 5.01 =1.1 76320 

2. Required the third power of the number 2.768. 
Log. of 2.768=0.442166 
Index = 3 



Answer=21.21 =1.326498 

3. Required the second p<mer of the number .2857. 
Log. of .2857=— 1.455910 
Index = 2 



Answer=.08162=— 2.91 1820 

4. Required the third power of the number .7916. 
Log. of .7916=— 1.898506 
Index = 3 



Answer3:.496l =—1.695518 



3» OP LOGARITHMS. 

EVOLUTION. 

Tb extract any proposed Root of a given numbet Jjf 

Liogarithms. 

RULE. 

Find the Logarithm of the given number, and 
divide it by the Index of the proposed root ; the 
quotient is a Logarithm, whose natural number ie 
the root- required. 

When the index of the Logarithm to be divid- 
ed, is negative, and does not exactly contstin the 
divisor without some remainder, increase the index 
by such a number, as will make it exactly divisi- 
ble by the index, carrying the units borrowed as 
«o many tens to the left hand place of the decimal, 
and then divide as in whole numbers. 

EXAMPLEld. 

1 . Required the square root of 847, 
Index 2)2.927883 =Log. of 847. 

1.463941 =Q,uot =Log.of 29.103+=an!i. 

2. Reqiwred the cube root of 847. 

Index 3)2.927883 =Log. of the given number. 

0.979961 =Cluot.=Log. of 9.462=ans. 

* [nearly. 

3. Required the square root of .093. 
index 2)— 2.968483= Log. of .093. 

—1.484241 =Cluot.=LQg.of.304959=ans. 

4. Required the cube root of 12345. 
Index 3)4.091491 = Log. of 12345, 

|,363830=auot.=5Log- of 23.116.=Ai^& 



GEOMETRY. 37 



SECTKHT IV. 
£LEMEJVTS OF 

PLANE GEOMETRY. 



' r. 



« .. 



DEFINITIONS. 
See PtAtB 2. 

t 

1 . Geometry is that science wherein we consider 
. the properties of magnitude. 

2. A point is that which has no parts, being of 
itself indivisible ;aB A. 

\^* 3. A line has length but no breadth ; as AB. fi- 

I ' gures 1 and 2. 

) 4. The extremities of a line are points, as the 

extremities of the line AB are the points^ and B. 
figures 1 and 2. 

5. A right line is the shortest that can be drawn 
"between any two points, as the line AB. fig. 1. but 
if it be not the shortest, it is then called a curve 
line, as AB. fig. 2. 

6. A superficies or surface is considered only as 
havinglength and breadth, without tliickness, as 
ABCD. fig. 3. 

7. The extremities of a superficies are lines. 

8. The inclination of two lines meeting one 
pother (provided they do not make one continued 



38 GEOMETRY. 

line) or the opening between them, is called an 
angle. Thus in fig. 4. the -inclination of the line AB 
to the line J?(7 meeting each other in the point jB, 
or the opening of the two lines BA and BC, is 
called an angle, as ABC. 

Note, When an angle is expressed by three let- 
ters, the middle ofle is that , at the angular 
point. 

9. When the lines that form the angle are 
right ones, it is then called a right-lined angle, as 
ABC, fig. 4. If one of them be right and the 
other curved, it is called a mixed angle, as B. fig. 
5. If both of them be curved, it is cafled a curved- 
lined or spherical angle, as C fig. 6. 

10. If a right line, CD (fig. 7.) fall upon ano- 
ther right line, AB, so as to incline to neither 
side, but make the angles ADC, CDB on each 
side equal to each other, then those angles are 
railed right angles, and the line CD a perpen- 
dicular. 

11. An obtuse angle is that which is wider or 
greater than a right one, as the angle ADE. fig. 
7. and an acute angle is less than a right one, as 
EDB. fig. 7. 

1 2. Acute and obtuse angles in general are call"^ 
ed oblique angles. 

13. If a right line Cj^. (fig. 8.) be fastened at 
the end C, and the other end B, be carried quite 
round, then the space comprehended is called a 
circle ; and the curve line described by the point 
JB, is called the circumference or the periphery of 
the circle; the fixed point C, is called its centre. 



GEOMETRY. 39 

14. The describing line CB. (fig. 8.) is called 
the semidiameter or radius, so is any line from the 
centre to the circumference : whence all radii of 
the same or of equal circles are equal, 

1 5. The diameter of a circle is a right line drawn 
thro' the centre, and terminating in opposite points 
of the circumference > and it divides the circle and 
circumference into two equal parts, called semicir- 
cles; and is double the radius, as ^^ orDE. fig. 8. 

16. The circumference of every circle is sup- 
posed to be divided into 360 equal parts called 
degrees, and each degree into 60 equalparts called 
minutes, and each minute into 60 equal parts call- 
ed seconds, and these into thirds, fourths, &c. these 
parts being greater or less as the radius is.- 

17. A chord is a.right line drawn from one end 
of an arc or arch (that is, any part of the circum- 
ference of a circle) to the other; and is the measure 
of the arc. Thus the right line HG, is the mea- 
sure of the arc HBG. fig, 8. 

18. The segment of a circle is any part there- 
of, which is cut off by a chord : thus the space 
which is comprehended between the chord HG- 
and the arc HBG, or that which is comprehend- 
ed between the said chord HG and the arc 
HDAEG are called segments. Wlience it is plain, 
%8. 

1 . That any chord will divide the circle into two 
segments. 

2. The less the chord is, the more unequal are 
the segments. 



40 GEOMETRY. 

3. When the chord is greatest it becomes a dn 
anieter, and then the segments are equal ; and each 
segment is a semicircle. 

19. A sector of a circle is a part thereof less than 
a semicircle, which is contained between two ra- 

, dii and an arc : thus the space contained between 
the two radii CH, CB, and the arc HB is a sec- 
tor, fig. 8. 

20. The ri^ht sine of an arc, is ^ perpendicular 
line let fall from one end thereof, to a diameter 
draV^ii to the other end : thus HL is the right 
sine of the arc HB. 

» 

The sines on the same diameter increase till 
they come to the centre, and so become the ra- 
dius ; hence it is plain ^at the radius CD is the 
greatest possible sine, and thence is called the 
^hole sine. 

Since the whole sine CD (%. 8.) must be per- 
pendicular to the diameter (by def. 20.) therefore 
prod ucing DC to E,ihe two diameters AB ^ndDE 
cross one another at right angles, and thus the 

Sriphery is divided into four equal pails, as J32?, 
A, AEy and EB ; (by def. 10.) and so BD be- 
comes a quadrant or the fourth part of the peri- 
phery: therefore the radius DC is always the 
sine of a quadrant, or of the fourth part of the 
circle BD. 

Sines are said to be of as many degrees as the 
arc contains parts of 360 : so the radius being 
the sine of a quadrant becomes the sine of 90 de- 
grees, or the fourth part of the circle, which is 360 
degi'ees. 



GEOMETRY. 41 

21. The versed sine of an arc is that part of 
the diameter that lies between Jhe right sine and 
the circumference : thus LB is the versed sine of 
the arc HB\ fig. 8. 

22. The tangent of an arc is a right line touch- 
ing the periphery, being perpendicular to the end 
of the diameter, and is terminated by a line drawn 
from the centre through the otlier end : thus i? JSTis 
Ihe tangeht of the arc HB. fig. 8. 

23. And the line which terminates the tan- 
gent, that is, CKy is called the secant of the arc 
HB. fig. 8. 

24. What an arc wants of a quadrant ia called 
the complement thereof : Thus DH is the com- 
plement of the arc HB. fig. 8. 

25. And what an arc wants of a semicircle is 
called the supplement thereof : thus AH is the 
supplement of the are HB. fig. 8. 

26. The sine, tangent, or secant of the com- 
plement of any arc, is called the co-sine, co-tan- 
gent, or co-secant of the arc itself: thus FH is the 
sine, DI the tangent, and CI the secant of the 
arc DH: or they are the co-sine, co-tangent, or 
co-secant of the arc HB. fig. 8. 

27. The sine of the supplement of an arc, is 
the same with the sine of the arc itself; for draw- 
ing them according to def. 20, there results the 
self-same line ; thus HL is the sine of the arc 
HBy or of its supplement AJDH. fig. 8. 

28. The measure of a right-lined angle, is the 
arc of a circle swept from the angular point, and 

G 



49 GEOMETRY. 

contained between the two lines that fonn ihe 
angle : thus the angle HCB (fig. 8.) is measur- 
ed by the arc HBi and is said to contain so many 
degrees as the arc HB does ; so if the arc HB 
is 60 degrees, the angle HCB is an angle of 60 
degrees. 

Hence angles are greater or less according as the 
arc described about the angular point, and termi- 
nated by the two sides, contains a greater or less 
number of degrees of the whole circle. 

29. The sine, tangent, and secant of an arc, 
is also the sine, tangent, and secant of an angle 
whose measure the arc is : thus because the arc 
HB is the measure of the angle HCB, and since 
HL is the sine, BK the tangent, and CK the 
secant, BL the Tersed sine, HF the co-sine, BI 
the co-tangent, and CI the co-secant, &c. of the 
arc BH; then HL is called the sine, BK the 
tangent, CK the secant, &;c. of the angle HCB^ 
whose measure is the arc HB. fig. 8. 

30. Parallel lines are such as are equi-distant 
from each other, as ABy CD. fig. 9. 

31. A. figure if a space bounded by a line or 
lines. If the liries be right it is called a recti- 
lineal figure, if curved it is called a curvilineal 
figure ; but if they be partly right and partly cur- 
ved lines, it is called a mixed figure. 

32. The most simple rectilineal figure is a trian- 
gle, being composed of three right lines, and i% 
considered in a double capacity ; 1st, with respect 
to its sides ; and 2d, to its angles* 

33. In respect to its sides it is either equilateral, 
having the three sides equal, as u4. fig. 10. 



I 



GEOMETRY. 4i 

34. Or isosceles, having two equal sides, as B". 
fig. II. 

35. Or 8calene> having the three sides unequal, 
as C. fig. 12. 

36. In respect to its angles, it is either iight- 
angled^ having one right angle, as D. fig. 13. 

37. Or obtuse angled, having one obtuse angle, 
a8£. fig. 14. 

38. Or acute angled, having all the angles acute, 
as F. fig. 15. 

39« Acute and obtuse angled triangles are in 
general called oblique angled triangles, in all which 
any side Riay be called the base, and thu other two 
the sides. 

40. The perpendicular height of a triangle is 
a line drawn from the vertex to the base perpen- 
dicularly : thus if the triangle ABC, be propos- 
ed, and BC be made its base^ then if from the 
vertex A the peipendicular AD be drawn to BC, 
the line AD will be the height of the triangle 
ABC, standing on jBCas its base. Fig. 16. 

Hence all triangles between the same parallels 
have the same height, since all the perpendiculars 
are equal firom the nature of parallels. 

41. Any figure of four sides is called a quadri* 
lateral figure. 

42. duadrilateral figures, whose opposite sides 
are parallel, are called parallelograms : thus 



i 

^ 



44 GEOMETRY. 

ABCD is a parallelograip. Fjg. 3« 17^ and AB^ 
fig. 18 and 19. 

* 43. A parallelogram whose sides are all equal 
and angles right, is called a square, as ABCJDi. 
fig. 17. 

44. A parallelogranl whose opposite sides are 
equal and angles right, is called a rectangle, or an 
oblong, as ABCD. fig. 3. ^ 

45. A rhombus is a parallelogram of equal sides, 
and has its angles oblique, as A. fig. 18. and is 
an inclined square. 

46. A rhomboides is a parallelogram whose op- 
posite sides are equal and angles oblique ; as j&. 
fig. 19. arid may be conceived as an inclined rect- 
angle. 

47. Any quadrilateral figure that is not a paral- 
lelogram, is called a trapezium. Plate 7. fig. 3. 

48. Figures which consist of more than four 
sides are called polygons ; if the sides are alLequal 
to each other, they are called regular polygons. 
They sometimes are named from the numln^ of 
their sides, as a five-sided figure is called a penta- 
gon, one of six sides a hexagoa, SCc. but if their 
sides are not equal to each other, then they are 
called irregular polygons^ as an irregular penta-^ 
gon, hexagon, SCc. 

49. Four quantities are said to be in proportion 
when the product oCthe extremes is equal to that 
of the means : thus if A multiplied by 2>, be 
equal to B multiplied by C, then A is said to Ia to. 
JBasCistoZ). 



*'» 



• 



GEOMETRY. 45 

POSTULATES OR PETITIONS. 

1. That a right line may be drawn from any one 
given point to another. 

2. That a right line may be produced or con- 
tinued at pleasure. 

3. That from any centre and with any radius, 
the circumference of a chcle may be described. 

4. It is also required that the equality of lines 
and angles to others given, be granted as possible : 
that it is possible for one right linff to be per- 
pendicular to another, at a given point or distance; 
and that every magnitude has its half, third, fourth, 
&t. part. 

Note, Though these postulates are not always 
quoted, the reader will easily perpeive where, and 
in what sense they are to be understood. 

AXIOMS or self^mdent TRUTHS. 

1. Things that are equal to oije and the sam« 
thing, are equal to each other. 

2. Every whole is greater than its part. . 

3. Every whole is equal to all its parts taken 
together, 

4. If to equal things, equal things be added, the 
whole will be equal. 

5. If from equal things, equal things be deduct- 
ed, the remainder* will be equal. 



i 



46 GEOMETRY. 

6. If to or from unequal things, e^ual things be 
added or taken, the sums or remamders will be 
unequal. 

7. All right angles are equal to one another. 

8. If two right lines not parallel, be produced 
Cowards their nearest distance, they will intersect 
each other. 

9. Things which mutually agi^ee with each other, 
are equal. 

NOTES. 

A theorem is a proposition, wherein something 
is proposed to be demonstrated. 

A problem is a proposition, wherein something 

is to be done or effected. 

# 

A lemma is s6me demonstration, previous and 
necessary, to render what follows the more easy. 

A corollary is a consequent truth, deduced 
from a foregoing demonstration. j 

A scholium, is a remark or observation nmde 
upon something going before. 



THEOREMS. 47 



GEOMETRICAL THEOREMS. 

THEOREM I. 

FL.lyJig.20. 

IF a right Une falU on another^ as AB^ or EB^ doet wn 
CDi U either makes with it two right anglety or two angles 
equal to two right angles. 

1. IfAB be perpendicular to CDj then (by def. 
10.) the angles CBA, and ABD, i^ill be each 9 
rignt angle. 

2. But if EB fall slantwise on CD, then are the 
angles DBE+EBC=DBE+EBA (=DBA)+ 
AmC, or two right angles. Q. E. I>. 

Corollary 1. Whence if any nxunbers of right 
lines were drawn from one point, on the same 
side of a right line ; all the angles made by these 
lines will be equal to two right lines. 

2. , And all the angles which c^n be made about 
a point, will be equad to four right angles. 

THEO.II. 

I • 

Fl.\.Jg.2l. 

If one right line crass another^ (as AC does BD) the oJifiS" 
site angles made by those Unes^ will be equal to each other : 
that isy AEB to CED^ and BEC to AED. 

By theorem 1. BEC + CED = 2 right angles, 
and CED + DEA=^ 2 right angles. 

Therefore (by axiom 1 .) BEC+ CED = CED+ 



48: GEOMETRICAi; 

DEA : lake CED from bofli, and there remains 
BEC=DEA. (by axiom 5) Q. E. D. 

After the same manner CED +AED^2 right 
angles ; and AED + AEB = two right angles ; 
wherefoiJe taking AED from both, there remiains 
CED=AEB.Q.E.D. 

THEO, III. 

Fi- 1./^. 22. 

J/a risht line cr09d two fiaralieh, as GHd9ea AB and CD, 

then^ ^ 

1. Their external angles are equal to each other ^ that isj 
GEB = CFH. 

2. The alternate angles will be egual^ that w, AEF •» EFD 
and BEF «= CFE. 

3. The external angle 'will be equal to the internal 
and opfiosite one on the same sidey that w, GEB = EFD 
mid AEG =, CFE. 

4. And the sum of the internal angles on the same 
side, are equat to two right angles ; that is^ BEF + DFE are 
equal to two right angles^ and AEF -f- CFE are equal to tzi>o 
right angles. 

1. Since AB is parallel to CD, they may be 
considered as one broad line, crossed by another 
line, as GH; (then by the last the».) GEB^CFB, 
and AEG=^HFD. 

2. Also. GEB=AEF, and CF/T^ EFD; but 
GEB= Ci^// (by part 1. of this theo.) therefore 
AEF = EFD. The same way we prove FEBc=z 
EEC. 

3. AEF^EFD; (by the last part of this theo^ 
hut AEF= GEB (by theo. 2.) Therefore GEIi 
^ EFD. The same wa v wo provn AEG^ CFE. 



THEOREMa 4^ 

4. ForwkceOEB = EFDytohoih ^MPEB, 
tteo (by axiom ^.)GEB +FEB^EFD +FEB, 
hut GEB + FEB, are equal to two right angles 
(by theo. 1.) Therefore EFD + FEB are equal 
to two ri^t angles : after the same manner we 
prove that aWF + CFE are equal to two right 
angles. Q.E.D. 

THEO, IV- 

Jn any triangle ABC^ wie of Ue legSf at JBC^ being produced 
iowardt 2>, U mil make the external angle ACD equal to the 
two internal opfumte angles taken together. Viz, to^B and A, 

Through C> let CE be drawn parallel to AB ^ 
then since BD cuts the two parallel lines BA^ 
CE; the angle ECD = JB, (by part 3- of the last 
theo.) and again, since ^Ccuts the same parallels, 
the angle ACE «= A (by part. 2. of Hie last.) 
Therefore ECD + AvE «= ACB ^B -{: A. 
CUED. 

THEO. V. 

Pl. I. Jig. 23. 

In any triangle ABC^ aU the three angles^ taken together^ 
are equal to two right angles^ viz, A + B + ACB =» 2 right 
angles* 

Produce CB to any distance, as D, then (by the 
last) ACD=B+A; to both add^Ci?; then^Cl? 
+ ACB= A + B + ACB; h\x\ACD + ACB =2 
right angles (by theo. 1.) ; therefore the three an- 
gles .4 + jB + ACB = 2 right angles. Q. E. D. 

Cor. 1. Hence if one angle of a' triangle be 
known, the sum of the other two is alsokSown : 
for since the three angles of every triangle con- 
tain two right ones, or 180 degrees, therefere 180 

H 



30 • GEOMEl*BlCAL 

«— the given ftngle wiU be equal to tbe sum of fi)6 
other t\ro ; or 180 — the sum of two giveo anglesi^ 

gives the other one* 

. • • • 

Cor. 2. In every right-angled f ritfngle, the two 
acute angles are =90 degrees, or to one right an- 
gle : therefore 98 — one acute Bugley gives ih«r 
other. 

THEO. VL 

If in anif 4vh ttiangle*, ABC, DKF^ there be two eide$f 
ABs ACin the on^, ^eeveraky equal to DE^ DF in the other f 
end the anffle A contained betvfeen the two Mee in the one^ 
equal to D tn the other i then the remaining' angles qf the one, 
mil be eeveralfy equal to thoae of the other^ viz, B set £ and 
C^=s F: ani the baae of the one BCf trill be equal to £F, thai 
bfthe other. 

If the triansle ABC he supposed to be laid on 
&e triangle 3EFy so as to make the points A 
and B coincide with D and JSJ, which they will do, 
because AB ~ DE (by the hypothesis') ; and since 
the angle A^D, the line -4Cwill fall along I? JF^ 
and inasmuch as they are supposed equaU C will 
fall in F; seeing therefore the three points of one 
l^oincide with those of the other triangle, they are 
manifestly equal to each other ; therefore the nx\r 
glejBr=jBandC=t*\F,andJ3C = JBF. Q.£.p. 



LEMMA. 

If two oideeofa triable a b c be equal to each other^ that iSf 
ac.-^ ebf the angleo which are o/ifioeite to thoee equal riiegj will 
ttUo be equal to each other s viz, a=^b. 

Tor let the triangle a 6 c be divided iiito tw^ 



% 1 



THEOHEMS. 51 

triangles aed^dcb^hy making the angle acds^, 
deb (by postulate 4.) then because ac=b c, and 
cd common, (by the last) the triangle a d c^d cb; 
and therefore the angle a^b. Q. E.D. 

Cor. Hence if from any point in a perpendicular 
which bisects a given line, there be di*awn right 
lines to the extreineties of the given one, they 
with it will form an isosceles triangle. 

THMO. ru. 

Fl. \.Jlg, 25. 

« 

" The mgie B€D at the e^tte ^f a circle ABED^ i» dpuble 
the angle Bi4D at tkt cireutiifipremei et^mdmg ufum the aama 
are B£D. 

t 

Through the point Ay and the centre Ci draw 
the line ACE : then the angle ECD = CAD, + 
CD A ; (by theo, 4.) but since u4C=*C2> being radii 
^of the same circle, it is plain (by the preceding 
lemma) that the angles subtended by them will be 
also equal, and that their sum is douUe to either 
of them, that is, DAC + ADC is double to CAD, 
and therefore ECD is double io CAD; afte^^the 
same martner BCEh double to CAB, wherefore, 
6CE + ECD, or BCD is double to BAC+ CAD 
or to BAD, Q.E.D. 

Cor. I . Hence an angle at the circumference is 
measured b^ half the arc it subtends or stands on. 

Cor. 2. Hence all angles at the circomiference of 
a circle which stands on the saqae chord 3,sAB, are 
equal to each other, for they are all measured by 
half the arc they stand on, viz. by half the arc AB^ 



t 

52 GEOMETRICAL 

Fisr> 26. 

Cor. 3. Hence an angle in a segment greater 
than a semicircle is less than a right angle ; thuft 
ADB is measured by half the arc AB^ but as the 
arc AB is less thaaa semicircle, therefore half the 
arc ABy or the angle ADB is less than b^lf a semi- 
circle, and consequently less than a right apgl^. 

Fig. 27. 

Cot. 4. An angle in a segment less than a semi- 
circle, is greater than a nght angle, for since the 
arc AEC is greater than a semicircle, its half, 
which is the measure of the angle ABC, must be 
greater than half a semicircle, that is, greater than 
a right angle. 

Fig. 2g. 

Cor. 5. An angle in a semicircle is a right angle;, 
for the measure of the angle ABB, is half of a 
semicircle AJSD, and therefore a right angle. 

Tff£0. vin. 

If from the centre C of a circle ABE^ there be let fall the 
perpendicular CD on the chord AB^ it vriM Haect it in the 
point D. 

Let the lines AC and CB be drawn from the 
centre to the extremities of the chord, then since 
04= CB, the angles CAB^ CBA (by the lemma.) 
But the triangles ADC, BDC are right angled 
ones, since the line CD is a perpendicular; and so 
the angle ACD^DCB; (by cor. 2. theo. 5.) then 
have we AC, CD^ and the angle ACD in one tri- 
fingle ; severally equal to CH, CD, and the angle 



THEOREMS. 53 

J8CD in the other : therefore (by thcQ. 6.) A= 
J)J^. Q. E. D. 

Cor. Hence it follows, that any line bisecting 
9 chord at right angles, is a diameter ; for a 
line drawn from the centre perpendicular to a 
chord, bisects that chord at right angles ; there- 
fore, conversely, a line bisecting a chord at right 
angles must pass through the centre, and conse- 
quently be a diameter. 

THEO, T3L 

Pl. \.Jig. 29. 

If from the centra qfa circle ABE there be drawn a fierficii' 
dicuiar CD on the chord AB^ and produced till it meets the cir- 
cle in Fy that line CF^ mil bisect the arc AB in the jfioint F. 

Let the lines ^Fand BF be drawn, then in the 
triangles -4jDF, BDF; AD^BD (by the last ;) 
l>jPis conmion, and the angle ADF—BDFheiug 
both right, for CD or DF is a perpendicular. 
Therefore (by theo. 6.) AF= FB ; but in the 
same circle, equal lines are chords of equal arcs, 
since they measure them (by def. 19.) : whence 
the arc AF^FBy and so AFB is bisected in F, 
by the line CF. 

Cor. . Hence the sine of an arc is half the 
chord of twice that arc. For -^D is the sine of 
the arc AFy (by def. 22.^ AF is half the arc, and 
AD half the chord AB (by theo. 8.) therefore the 
corollary is plain. 

THEO. X. 

Pl. I, Jig. 30. 

In any triangle ABD^ the half of each sidf is the sine of the 
ofifioeite angle. 



54 GEOMETRICAL 

Let the circle ADB be drawn through tfae 
points Ay By 2>; then the angle DAB is measured 
by half the arc BKDy (by cor 1 theo. 7.) viz. the 
chord of BK is the measure of the angle BAD ; 
therefore (by cor. to the last) BE the half of BU 
is the sine otBAD : the same way may be ^woved 
that half of AD is the sine of ABD, and the bs^lf 
of ^JS the sine of ADB. ^ Q. E. JO. 

TffEO.XL 

If a right Hne GBctU two of her right Hnet AB^ CDy 90 a4 
to make the alternate angle* AEfP^ EFD equal to each other^ 
thej^ the lines AB and CD will befmraUel. 

If it be denied that AB is parallel to CD, let 
IJSTbe parallel to it ; then IEF={EFD)^AEF 
(by part 2. theo. 3.) a greater to a less, whiqh is 
absurd, whence IK is not parallel ; and the like 
we can prove of all other lines but AB i there- 
fore ^jB is parallel to CJO, Q.E^D. 

THEO. XU. 

Ft. I. Jig. Z. 

Iftvo equal and parallel line% A By CD, be joined by tmo 
other Hnea ADy BCy those shall be also equal and fiarcUlel* 

Let the diameter or diagonal J37>be drawn, and 
we will h&ve the triangles ABDy CBD : whereof 
AB in one is=to CD in the other^.B2> common to 
both, and the angle ABD~ CDB (by part 2. theo, 
3. ;) therefore (bv theo, 6.) AD=CBy and the an- 
de CBD —AUBy and thence the lines AD and 
JBC are parallel, by the preceding theorem. 

Cor. 1. Hence the quadrilateral figure -4BC2) is 
a parallelogram, and the diagonal BD bisects the 



* ' 



THEOBEMS. 65 

same, inasmuch as the triangle ABD = BCD^ as 
now proved. 

Cor. 2. Hence also the triangle ABD on the 
aanie base AB^ and between the same parallels 
wHh the parallelogram ABCD, is half the paral- 
lelogram. 

Cor. 3. It is hence also plain, that the opposite 
ttdes 6f a parallelogiam are equal ; for it has been 
proved that ABCD being a parallelogram, AB 
wiU be « Ci>and AD ^ BC. 



THMO.XOL 



Pl. hjig. 31. 

AU fiarattelogramt on the same or equal btL%e% and between 
4he tame parallcU* are equal to one another^ that iay if BD s 
OH^ and the lines B If and AF parallel^ then the fiarallelogram 
ABDC ri= BDFK ^ BFHG. 

For AC^BD^EF(^ cor. the last ;) to both 
add CE then AE = CF. In the triangles ABE, 
CDF; AB = CD and AE = CFand the ancle 
BAE'^DCF (by part 3. theo. 3. ;) therefore flie 
triangle ABE^CDFy (by theo. 6.) let the trian- 
gle CKEhe taken from both, and we wilt have 
the trapezium ABKC—KDFE s to each of these 
add the triangle BKD^ then the parallelogram 
ABCD^BDEF i in like manner we may prove 
the parallelogram EFOH^^BDEF, Wherefore 
ABnC=^BDEF-EFHO, Q.E.D. 

Cor. Hence it is plain that triangles on th^ 
same or equal bases, and between the same paral-* 
lels, are equal, seeing (by cor. 2. theo. 12.) they 
are the halves of their respective parallelogram. 



56 GEOMETRICAL 

TJIEO.XIV. 

Pl, l.Jig'. 32. 

In every right-angled triangle^ ABC^ the square of the 
hyfioihenuae or longest nde^ BCj or BCMH^ u egual to the 
€um qf the sguarei made on the other tvfo sides AB and ACy 
that isy ABDE and ACGF. 

Through A draw AKL perpendicular to the 
hypothenuse ^CJoin AH^ AMy DC and BG ; m 
the trianffles, BBC, ABH, BD = BA, being 
sides of flie same square, and also BC—BH, and 
the included angles DBC=ABH, (for DBA^ 
CBH being both right, to both add ABC, then 
DBC = ABH) therefore the triangle BBC = 
ABH(y>y theo. 6.) but the triangle DBCi& half 
of the square ABBE (by cor. 2 theo. 12,) and Ihe 
triangle ABH is half the parallelogram BKLH. 
The same way it may be proved, that the square 
ACGF, is equal to the parallelogram KCLSl So 
ABDE+ACGFihe sum of the squares= BKLH 
+ KCMLy the sum of the two parallelograms or 
square BCMH ; tlierefore the sum of the squai'es 
on AB and ^C is equal to the square on J3C 

a. E.D. 

Cor, 1. Hence the hypothenuse of a right-an- 
gled triangle may be found by having the sides ; 
Sius, the square root of tlie sura of the squares of 
the base and perpendicular, willbe the hypothenuse. 

Cor, 2. Having the hypothenuse and one side 
jgiven to find the other; the square root of the dif- 
ference of the squares of the hypothenuse and giv- 
en side, will' be the required side. 

THEO. XV. 

PjL.l.^g,3S. 

In all circles the chord ^60 degrees is always equal in length 
to the radius* 



THEOREMS. 57 

Thtu in the circle AEBD^ if the arc AEB be an arc of 60 
degreesy and the chord AB be dratvn : then AB = CB = AC. 

In the triangle ABC, the angle ACB is 60 de- 
grees, being measured by the ^rc AEB; therefore 
the sum of the other two angles is 120 degi-ees 
rby Cor. l.theo. 5.) but since ^C=*CJB, the angle 
CAB= CBA (by lemma preceding theo. ?•) con- 
sequently each of them will be 60, the half of 120 
degrees, and the three -angles will be equal to one . 
another, as well as the three sides : wherefore AB 
=BC^AC. Q. E. D. 

Cor. Hence the radius, from whence the lines 
-on any scale are formed, is the chord of 60 degrees 
€m the line of chords. 

THEO. XVI. 

If in two triangle* ABC, abcy all the angles of one be each 
retfiectiveUf equal to all the angles of the other, that ia\ A z=. a, 
B = bj C =s c : then the aides ofifiosite to the equal angles wtU 
ke firofiortionalj viz. 

AB : ab : : AC : ac 

AB : ab : : BC s be 

and AC : ac : : BC : be 

For the triangles being inscribed in two circles, 
it is plain since the angle A= a, the arc BI)C= 
b d c, and consequently the chord BC is to b c, as 
the radius of the circle ABC is to the radius of 
the circle ab cy (for the greater the radius is, the 
greater is the circle described by that radius ; and 
consequently tlie greater any particular arc of that " 
circle is, so the chord, sine, tangent, SCc. of that 
arc will be also greater. Therefore, in general, the 
chord, sine, tangent, SCc. of any arc is proportional 
• to the radius of the circle ;) the same way the chord 

I 



58 GBOMETMCAL 

AB is to tbe chord oJ, in the same proportion* 
So AB :ah:: BC: be ; the same way the rest may 
be preyed to be proportional. 

THEO. XVIL 

Pt. 1. Jig. 35. 

tffrom a point A witkotU a circle DBCE there be drawn two 
itnes ADE^ ABC, each of them cutting- the circle in two/ioints ; 
the product of one whole line into its external part ^ viz. AC 
into A By will be equal to that i^fthe other line into ita external 
part J viz. AE into AD, 

Let the lilies DQ BJEy be drawn in the two tri- 
angles ABE, ADC; the angle AEB^ACD (by 
cor. 2. theo, ?•) the angled is common,and (by cor. 
l.theo. 5.) the angle ADC^ABE; therefore the 
triangles ABE, ADCi are mutually equiangular, 
and consequently (by the last) AC : AE : : AD: 
AB; wherefore AC mivHiplied by ABy will be 
equal to AE multiplied by AD. Q- E. D. 

THEO.XVJXL 

Pl. 2. Jig. 1. 

Triangles ABCy BCDy and paraffelograme ABCF and 
BDEC^ having the same altitude^ have the same proportion be* 
tween themselves as their bases BA and BD, 

Let any aliquot part of AB be taken, whfch 
will also measure JBD: su{^ose that to be Ag, 
which will be contained twice in AB, and three 
times in BDy the parts Agj ^B, Bh, hiy and i D 
being all equal, and let the Imes gC, hCy and i C, 
be drawn : then (by cor. to theo. 1 3.) all the small 
triangles AgCy gCBy BChy 8^c. will be equal to 
each other; and will be as many as the parts into 
which their bases were divided ; therefore it will 
be as the sum of the parts in one base, is to the 



THEOREMS. 59 

sum of those in the other, so will be the sum of the 
small triangles in the first, to the sum of the small 
triangles in the second triangle ; that is, AB : 
BJf : : ABC : BBC. 

Whence also the parallelograms ABCF and 
BBECj being (by cor. 2. theo. 13.) the doubles of 
the triangles, are likewise as their bases. Q. E. B. 

Note. Wherever there are several quantities 
connected with the sign (: :) the conclusion is al- 
ways drawn from the first two and last two propor- 
tionals, 

THEO. XIX. 

TrianglcB ABC^ DJEF^ sanding upon equal bases AB and 
D£f are to each other as their altitudes CQ and FH. 

Let BIhe perpendicular to AB and equal to 
CO, in which let KB =^ FH, and let AZ and AK 
be drawn. 

The triangle AIB^ACB (by cor. to theo. 1 3.) 
and AKB=BEF; but (by theo. 18.) BI: BK: : 
ABI: ABK That is, CG : FH: : ABC : DEF. 
Q. E. D, 

THEO. XX. 
Pt. 2. Jig. 3. 

If a right Hue BE be drawn parallel to one side of a triangle 
ACD^ it mil cut the two other sides proportionally^ viz. AB : 
BC : : AE : ED. . 

Draw CEnndBD; the triangles S^Cand^jBJD 
being on the same base BE and under the same 
paraUel CD^ will be equal (by con to tbeo. 1 3.) 






60 GEOMETRICAL 

therefore (by theo. 18) AB.- BC. : (BE A : BEC 
or BEA : BED) :. AE . ED. k E. D. 

Cor. I. Hence also AC: AB : : AD : AE ■ 
For ^<^' AB (AEC : AEB : : ABD : AEB) 

Cor. 2. It also appears that a right line, which 
divides two sides of a triangle proportionally, must 
be parallel to the remaining side. 

.. ^o»;- 3. Hence also, theo. 16. is manifest j since 
the sides of the triangles ^JB£, ACD, being equi- 
angular, are proportional. ^ ^ 

« 

THEO. XXI. 

Pt. %Jlff. A. 

If two triangle* ABC,ADE^ have an anele BAC in th. .- 
equal to an angte DAE, in the other, SthelSllaioltZl 
equal angles, firofiortional ; that u,, AB : AD--AC^Jn' 
then the triangles mU be mutually equiangular. ' ^ ^"^^^ 

In^^take^rf»^i>, and let rfe be parallel 
to BC, meetmg AC in e. ^ 

Because (by the first cor. to the foregoing theo ^ 
AB: Ad (or AD) : : AC: A.e, and (b| thlh^^ 
thesis, or what is given m the theorem) ^jB • AD • 
AC : AE; therefore Ae = AE seei^fc Ws 
the same Propo/tion to each ; and (by theo. 6.) 
the triangle^rf. = ADE, therefore the an<rle 
Ade= Dmd Aed ^E, but since ed and fiC are 
parallel (by part 3. theo. 3) Ade^ B, and Aed~. 
C, therefore jB=rl> and C=£. Q. E. D. 

TJIJSO. XXIL 
-Px. 2,^^. 5. 
Equiangular triangle, ABC, JDEF, are to one another in 




THEOREMS. 61 

€1 dufiHcate firofiortton of their homologous or like aides ; or o« 
the squares AKy and DM qf their homologous sides. 

Let the perpendiculars CG and FH be drawn 
as well as the diagonals BI and EL. 

The perpendiculars make the triangles ACjS 
and D^Jff equiangular, and therefore similar (by 
iheo. 16.) for because the angle CAG=FDH, and 
the right angle AGC=DHF, the remaining angle 
AC'G=DfH, (by cor. 2. theo. 5.) 

Therefore GC- FH- • (AC- DF- :)AB ' DE, 
or M^hich is the same thing, GC • AB : : FH • DE 
for FH multiplied by AB = AB multiplied by 
FH. 

By theo. 19. ABC • ABI ■ '■' (CG ' AI or AB 
as before : : FH DE or DL : •• ) DFE : DLE 
therefore ABC ^ ABI •■ DFE .• DLE, or AB<J: 
AK •• •• DFE ■• DMy for AK is double the trian- 
gle ABI, and DM double the triangle DEL, by 
cor. 2. theo. 12. Q. E. D. 



THEO. XXIII. 

Pl. %fig, 6. 

Like fiolygons ABC D By ab c dey are in a duplicate firoficr^ 
tion to that of the sides ABy a by which are between t/ie equal 
angles A and B and a and by or us $he squares of the aides 
ABy ab. 

Draw ADy ACy ad, ac. 

By the hypothesis AB - ab ' : BC-bc; and there- 
by also the angle B =6; therefore (by theo. 21.) 
BAC = bac; and ACB= a cb' in like manner 
JEAD^e a rf, and EDA=eda. If therefore from 
the equal angles A, and a, we take the equal ones 



62 GEOMETRICAL 

JEAD +BAC:b:€ a df+ 6 a c the remaining angle 
I)AC= dae, and if from tl|e equal angles D and 
diEDA=eday be taken, weshsJl have ADC=a d 
€ ' and in like manner if from C and c be taken 
JBCAf =^bcay we shall have ACD *=acd; and so 
the respective angles in every triangle, will be 
equal to those in the other. 

By theo. 22. ABC- ahc-- the square of AC to 
the square of acr, and also ADC ' adc : .' the square 
of AVf to the square of a c ; therefore from equa- 
lity of proportions ABC • abc • •• ADC- adc; 
in like manner we may shew that ADC : ad e : 
JEA D* ead' Therefore it will be as one antece- 
dent is to one consequent, so are all {he antecedents 
to all the consequents. That is, ABC is to a ft c as 
the sum of the three triangles in the first polygon, 
is to the sum of those in the last. Or ABu wiU be 
to a 6 c, as polygon to polygon. 

The proportion oiABC U>ahc (by the forego- 
ing theo.) IS as the square of AB is to the square 
of a 6, but the proportion of polygon to polygon, 
is as ABC to a 6 c, as now shown : therefore the 
proportion of polygon to polygon is as the square 
of AB to the square of £W. 

. rjzfio.xxiv. 

Let DHB be a quadrant qf a circle described by the radiug 
CB; HB an arc of it ^ and D Hits comftlement i HL or FC 
the nnCi FHor CL its co^nne^ BK its tangenSy Dlits co^tan^ 
gent i CKits aecant^ and CI its co-secant. Fig, 8. 

1. The co-sine of an arc is to the sine, as the rar 
dius is to the tangent 



THEOREMS. 63 

2. l^e radius is to the tangent of an arc, at 
the co-sine of it is to the sine. 

3. The sine of an arc is to its co-sine, as the ra- 
dius to its co-tangent ; . 

4. Or the radius is to the co-tangent of an arc^ 
as its sine to its co-sine. 

5. The co-tangent of an arc is to the radiusj a 
the radius to the tangent. 

6. The co-sine of an arc is to the radius^ as the 
radius is to the« secant. 

7. The sine of an aire is to the radius, as the tan- 
gent is to the secant. 

The triangles CLH and . CBK, l>eing similar, 
(by tbeo. 16.) ^ 

hCL:LH::CB:BK. 

2. Or, CB : BK: : CL : LH. 

m 

The triangles CFH and CDI, being similar. 

3. CF (or LH) : FH : : CD : DI. 

4. CD : DI : : CF (or LEt) : FH. 

The triangles CDI and CBiT are similar : for 
the angle Cll) =s KCBy being alternate ones (by 

i)art 2. theo. 3.) the lines CB and DI being parall- 
el : the angle CDI— CSiST being both right, . and 
consequently tiie angle DCl^^VKBy wherefore, 

5.DI:CD::CB:BK. 



64 GEOMETRICAL 

And a£;ain, making use of the similar triangle 
CLH ani CBK. 

%.CL:CB::CH:CK. 

7. HL ' CH • BK • CK. 



GEOMETRICAL PROBLEMS. 

PROB, I. 

To make a triangle of three given rfght Hnea BOj LB^ LO^ 
0f which any two must be greater than the third, 

Lajr BL from B to L; from B with the Ime BO, 
describe an arc, and from L with LO describe 
another arc ; from O, the intersecting point of 
those arcs, di*aw BO and OL, and BOL is the 
triangle required. 

This is manifest from the construction. 

PROB. II. 

Px. 2. Jig. 8. 

At a point B in a given right line BCj to make an angle equal 
to a given angle A, 

Draw any right line ED to form a triangle, as 
EAB, take B>=^jy, and upon BFmakethe tri- 
angle BJ^G, whose side ^G=-4J5:, and GF=ED 
(by the last) then also the angle ^ = -4; if we 
suppose one triangle be laid on the other, the sides 



PROBLEMS. 65 - 

wiU mutually agree with each other, and therefore 
be equal ; for if we consider these two triaogles to 
be made of the same three" given " lines^ tbey are 
manifestly one and the same triangle. 

Otherwise! 

Upon the centres A and £, at any distance, let 
two arcs, i>£, FG^ be described; make the arc 
J^=i>£, and through Band G draw the line 
BG^ and it fedone- 

For since the chords EH, GW, are equal, thp 
angles^ and B are also equal, as before (by def.l7.) 



To bitect or divide into tan equal fiartt, any given rigitt* 
lined angte^SAC. 

Tn tlie lines-^S and AC^ icom the poinfiM set 
«ff equal distances AE,==AD, tfien, wilh any dis- 
tance more than the half of DE, <lrscribe two arcs 
to cut each other in some point F; and llie rij^ht- 
line ^F, joining the points -^ and i", wii I bisect 
the given angle BAC. 

For if DF and FE be drawn, the triangles 
ADF, AEF, are equilateral to each other, viz. 
AD=AE, DF=FE, and ^F common, where« 
fQKDAF=EAF, as before. 

PROS. IV. 
Tobiwtaright-Sne. AB. 

With any distance; more thanlialf the line, from 
K 



• *6 GEOMKTRICAfi 

A and By describe two circles CFDyCOH, cutiii^ 
each other in the points V and D ; draw CD in- 
tersecting AB in Ey then AJS—EB. 

For, if ACy AD, JBC, BDy be drawn, the trian- 
gles ACDj BCDy will be mutually equiljiteral, 
and consequently the angle A.CE=BCE : there- 
fore the triangle ACEy BCE, having^C=JBC, 
CE common, and the angle ACE=jBCE; (by 
theo. 6.) the base AE= the base BE. 

Cor. Hence it is manifest, that CD not only bi- 
sects AB, but IB perpendicular to it, (by def. 11.) 

PMOB. V. 

Pl. %Jig. 11. 

On a given point Ay in a right One EFy t9 erect a perpen* 



FVopai the point A lay off on each side, the equal 
dista[£es, AiCy AD ; and from C dnd U, as cen- 
tres, with any internal greater than AC or AD^ 
describe two arcs intersecting each other in J?; 
from .^ to j8 draw the line Aloy and it will be the 
perpendicular required. 

For, let CBy and BD be drawn ; then the trian- 
gles CABy DABy will be mutually equilateral 
and equiangular, so CAB^DAB, a right angle, 
(by def. lo!) 

FROB. VI. 

Px. 2. Jig. 12. 
To raise a perpendicular on the end B of a right line AB, 

From any point D not in the line ABy with the 
&tanc^ from D to J3,let ackclebe described cut- 



PROBLEMS. &J 

iing AS in £ ; draw from E through D the right 
line EDCy cutting the periphery in C, and joia 
CB ; and that is the perpendicidar required. 

EBC being a semicircle, the angle JSJBC will 
be a right angle (by cor. 5. theo. 7.) 

FROB. VTI, 
Px. %.Jis* 13. 

JPtfum a given pAnt A, to let fall a fierfiendicular u/ion a given 

right tne BC, 

From any point D, in the given line, take the 
distance to the given poin^ A, and with it describe 
a circle AOE^ make G^=-4G, join the points A 
and E, by the line AFE, and ^!r will be the per- 
pendicular required. 

Let D^,I>JE;,be drawn ; the angle ADF^ FDE, 
JDA =DEy being radii of the same circle, and DF 
eommon ; therefore (by tbeo. 6.) the angle DFA 
^JDFE, and FA a perpendicular. (By def. 10.) 

FROB. VIII. 

Pl. %Jlg, U. 

Through a given fi^int A^ to draw a right Une AB^fiarallel to o^ 

given right line CD. 

• 

From the point A, to any point F^ in the line 
CX), draw the line AF; with the interval FAj and 
one foot of the compasses in F^ describe the arc 
AE^ and with the like interval and one foot in 
Aj describe the arc BFy making BF=AE ; 
through A and B draw the line AB, and it will 
be parallel to CD. 



68 GEOMETRICAL 

By prob- 2- The angle BAF^AFE, and by 
iheo. 1 1. BA and CD are parallel* 

JPi?0J5. IX. 

(7/iM tt given Une AB to defctipe a equare ABdB. 



»••.. 

* 



Make jBCper^ndicular and equal to AB; and 
from A and C, with the line ABy or BCj let two 
arcs be described, cutting each other in I) ; from 
whence ta A and C, let the lines ADy DC be 
di^wn ; so is ABCD the square requiied. 

For all the sides are equal by construction ; 
therefore the triangles ADb and BACy are mutu- 
ally equUateral and equiangular, and ABCD is an 
equilateral parallelogram, whose angles are right. 
For B being risht^ D is also right, and DaC, 
DC Ay BACy A CBy each half a right angle, (by lem- 
ma preceding theo. 7. and cor. 2. theo. 5,) whence 
DAB and BCD will each be a right angle, and 
(by def. 44.) ABCD is a square. 

SCHOLIUM. 

By the same method a rectahgle or oblong, may 
be described, the sides thereof being given. 

> PROB. X. 

* 

jPx. ^'fig^ 15. 

To divide agraeu right Hne AB^ into any proposed numbtr qf 

equ^lfiartM. 

Draw the indefinite right line APy making any 
9^pgle with AB^ also draw BQ parallel to AP^ i« 



PROBLEMS. 69 

each of which, let there be taken as many equal 

Starts AM^ MNy SCe. J3o, on, SCc. as you would 
ave AB divided into ; then draw Mm^ Nn, SCc. 
iritereecting AB in Ey F, SCc. and it is done. 

For MN and mn being equal and parallel, FN' 
will be parallel to EM; and in the same manner, 
GO ta FN (by theo. 12,) therefore A3I, MN, 
NOy being all equal by construction, it is plain 
(from theo. 10.) that AE, EFy FG, SCc. will like- 
wise be equal. 

PROS. XI. 

Pl. 2. Jig. 16. 

To find a tKrd firofiortimal to (wo given right Unesy A and JS. 

Draw two indefinite blank lines C£, CJD, any- 
wise to make any angle. Lav the line Ay from C 
to F; and the line J3, from C, to G ; and draw the 
line FG; lay again the line Ay from Cto H; and 
through H, draw JEf/ paralM to FG (by prob. 8.) 
30 is Clihe thi^d proportional required. 

For by cor. 1. theo, 20, CG : CH: : CF: CI. 

Or, B'A::A ; CI 

PROfi. XII. 
Pl. %fig, 17. 
Three right Knee A^B^ C^ given to find a fourth fir oportionaL 

Having made an angle DEF anywise, by two 
indefinite blank right lines, JED, JBl^, as before; lay 
the line A% from EioG ; the line By from E to I; 
$ind draw the line 10 ; lay the line C, from E to 



7(1 GEOMETRICAL - 

Hj and (by prob. 8.) draw If JT parallel thereto, 8e 
'vUl £a be the fourth proportional required. 

For, by cor. 1. theo. 20. EG : EI: :EH: EK. 

Ot,A:B::C:EK. 

PROB. XIII. 

Pl, S.Jig. I. 
Ti90 fight Une$j A and B^ given to find a mta% proftortionat. 

Draw an indefinite blank line, as AF^ on which 
lay the line Aj from^ to jB, and the line jB, from 
B to C, on the point J^yVhich is the joining point 
of the lines A and B ; erect a perpendicular BD 
(by prob. 5.) bisect AC in E (by prob. 4.) and de- 
scribe the semicircle ABC; and from the point Z>, 
where the periphery cuts the perpendicular J51>, 
draw the line BDj and that will be the mean pro* 
portional required 

For if the lines ADy DC, be drawn, the angle 
ABC is a right angle (by cor. 5. theo. 7.) being an 
angle in a semicircle. 

The angles ABDy BBC, are right ones (by def. 
10.) the line BB being a perpendicular; wherefore 
the triangles ABBy j)BCy are similar: thus the an- 
^leABB=BBCy being both right, the angle BAC 
IS the complement of BBA to a right angle (by 
cor. 2. theo. 5.) and is therefore equal to BDCy the 
angle ABC being a right angle as before ; conse- 

2uently (by cor. 1. theo. 5?) the angle -4i> JB =3 
WB9 wherefore (by theo. 16.) 

AB : BB : : BB : BC: 
Ot,A:BB::BJD:B. 



PROBLEMS, 7J 

FROB. XIV- 

Pl. 3. Jig. 2. . 

To divide a right line ABj in the fioint E, so that AE ahall have 
the name proportion to EByoa two given lines C and D have. 

Draw an indefinite blank line, AF^ to the ex- 
trenaity of the line ABj to make with it any an- 
gle; lay the line C, from A to C; and 2>, from C 
to D J and join the points B and X>, by the line 
BD ; through C draw CE parallel to BD (by 
prob. 8.) so IS £ the point of division. 

For, by cor. 1. theo. 20. AC: AD : : AE : AB. 
Ox,C:D ::AE:EB. 

PROB. XV. 
PL.3.Jig.S. 

To describe a circle about a triangle ABC^ or fiofHch ia the tanit' 
thing) through any three fiointa^ Af By C^ which are not 
situated in a right tine* 

By prob. 4. Bisect the line AC hj the perpendi- 
cular i)Ey and also CB^ by the perpendicular FO^ 
the point of intersection iff, of these peipendiculars, 
is the centre of the circle requu^d ; from which take 
the distance to any of the three points A^ JB, C^ 
and describe the circle ABC, and it is done. 

For, by cori to theo. 8, The lines DE and FG, 
must each pass through the centre^ therefore, their 
^int of intersection H, must be the centre. 

SCHOLIUM. 

■ 

By this method the centre of a circle may b<i 
found, by having only a degmeut of it given. 



n GEOMETRICAL 



FROB. XVI. 



To make an angle of any number of degreet^ at the point A^ of 
the line AB^ MUfifiote qf 45 degreea. 

From a scale of chords take 60 degrees, for 60* 
is equal to the radius (by cor. theo. 1 5,^ and with 
that distance from ^, as a centre, descrioe a circle 
from the line AB; take 45 degrees, the quantity of 
the given angle, from the same scale of chords, and 
lay it on' that circle from aioh ; through A and b, 
draw the line AhC^ and the angle A wiU be an 
angle of 45 degrees, as required. * 

If the given angle be more than 90% take its half 
(or divide it into any two parts less than 90^ and 
lay them after each other on the arc, which is de- 
!gcribed with the chord of 60 degrees ; through the 
extremity of which, and the centre, let a line be 
drawn, and that will form the angle i^equhed^ with 
the given line. 

PROB. xvir. 



Pi. S.Jig. 5. 
To meaaure a given angle^ ABCL 

If the lines which include the angle, be not as 
long as the chord of 60** on your scale, produce 
them to that or a greater length, and between them 
so produced, with the chord of 60** fromr jB, de- 
scrioe the arc edj which distance ed, measured on 
the same line of chords, gives the .quantity of the 
angle BAC, as required ; this is plam from def. 1 7« 



S 



PROBLEMS. 



73 



PROB. XVIII. 
Pt. 3. Jig. 6, 

To make a triangle BCE equal to a given quadrilateral 

figure ABCD. 

Draw the diagonal ACy and parallel to it (by 
prob. 8.) DjE, meeting -4-B produced in E; then 
draw CjE, and ECB will be the triangle required. 

For the triangles ADCj AEC, being upon the 
same base AC, and under the same parallel EIJ^ 
(by cor. totheo. 13.) will be equal, therefore if 
A.BC be added i% each, then AiBCD =^BEC. 

PROB. XIX. 
Pt,Z\/ig.7, 

To make a triangle DFBty equal to a given five-aided figure 
ABODE. 

Draw DA and DB, and also EH^nd CF, pa- 
rallel to them (by prob. 8.) meeting AB produced 
in H and F; then draw JDHf DF^ and the trianr 
glcf HDF is the one required. 

For the triangle BE A - DHA, and DBC = 
BFB (by cor. to theo. 13j therefore by addinj 
these equations, DEA + J0J3C= DHA-k^ DFI 
if to each of these ABB be added ; then DEA + 
ABB + BBC^ ABCBE = (BHA + ABB + 
BFB.^BHF 

PROB. XX. 

Px. 3.fig» 8. 

To project the Unee of ckordsj dneay tangents and iccantk^ 
With any radius. 

h 



n MATHEMATICAL 

On the line ABy let a semicircle ADB be de^ 
scribed ; let CDF be drawn perpendicular to thur 
line from the centre C ; and the tangent BE per- 
pendicular to the end of the diameter ; let the quad- 
rants, ADy DBy be each divided into 9 equal parts^ 
every one of which will be 10 degrees; iithen from 
the centre C^ lines be drawn through 10» 20, 30,40, 
&c. the divisions of the quadrant SD, and continu- 
ed to BE J we ^all there have the tangents of 10, 
20, 30, 40, &c. and the secants C 10, U 20, C 30, 
&C. ^re transferred to the line CF, by describing 
the urcs 10, 10 : 20, 20 : 30, 30, &c. If from 10^ 
20, 30, &C, the divisions of the quadrant J32>, there 
" be let fall perpendiculars, let these be transferred 
to the radius CjB, and we shall have the sines of 
10, 20, 30, &c. and if from A we describe the arcs 
10, 10 : 20, 20 : 30, 30, &c. from every division of 
the arc AD ; we diall have a line of chords. 
The same way we may have the sine, tangent, SCc. 
to every single degree on the (][uadrant, by ^ubdir 
viding each of the 9 former divisions into 10 equal 
parts* By this method the sines^ tangents, SCc. may 
be drawn to an^ radiqs ; and then, after they are 
transferred to Imes on a rule^ we shall have tlia 
scales of sines, tangents, ^STc. ready for use. 



MATHEMATICAl 

DRAWING mSTRUMENTS, 

The strictness of geometrical demonstration ^dr 
mits of no other instruments, than a rule and a pair 
of compasses. But, in proportion as the practice of 
geometry was extended to the different arts, either 
connected with, or dependent upon it, new ins^tru- 
ments became necessary, some to answer pecyHay 



DRAWING INSTRUMENTS. 16 

purposes, some to facilitate operation, and others 
to promote accuracy. 

As almost every artist, whose operations are 
connected with mathematical designmg, furnishes 
himself with a case of drawing histruments suited 
to his peculiar purposes, they are fitted up in va- 
rious modes, some containing more, others, fewer 
instruments. The smallest collection put into a 
case, consists of a plane scale, a pair of compasses 
with a moveable leg, and two spare points, which 
may be applied occasionally to the compasses ; one 
of these points is to hold ^ink ; the other, a porte 
crayon, for holding a piece of black-lead pencil. 
' . What is called a full pocket case, contains the 
following instruments. 

A pair of large compasses with a moveable point, 
an ink point, a pencil point, and one for dotting; 
either of those points may be inserted in the com- 
passes, instead of the moveable leg. 

A pair of plain compasses somewhat smaller 
than uose with the moveable leg* 

A pair of bow compasses. 

A drawing pen with a protracting pin in the up 
]^r part. 

A sector. 

A plain scale* 

A protractor* 

A parallel rule. 

A pencil and screw-driver.* 

* Large collections are caUed> magazine comcb cf tnstHi* 
menta ; these generally contain 

A pair of six inch compasses witii a moveable leg) an ink 
point, a dotting point, the crainm pcHqt, so contrived as to hold 
a whole pencil, two additional pieces to lengthen occasionally 
one leg of the compasses, and thereby enable them to measure 
greater extents, and describe q^ircles of a larger radius^ 

A pair of hsur compasses. » 

A pair of bow compasses. 

A pair of triangular compasses^ 



76 MATHEMATICAL 

In a case with the best instruments, the protrac-^ 
tor and plain scale are always combined. The in- 
struments in most general use are those of six in- 
ches ; instruments are seldom made longer,but often 
smaller. Those of six inches are, however, to be 
preferred, in general, before any other size ; they 
will effect all that can be performed with the short-* 
est ones, while, at the same time, they are better 
adapted to large work. 

OP DRAWING COMPASSES. 

Compasses are made either of silver or brass, but 
with steel points* The joints should always be 
framed of different substances ; thus, one side,, or 
part> should be of silver or brass, and the other of 

A sector. 

A parallel rule. 

A protractor. 

A pair of proportional compasses, either vith or without 
an adjusting screw. 

A pair of wholes and halves. 

Two drawing pens^ and a pointriL 

A pair of small hair compasses, with a head similar to those 
•f the bow compasses. 

A knife, a fiie^ key, and screw-driveri or the compasses in 
one piece. 

A small set of fine water colours. 

To these some of the following instruments are often added* 

A pair of beam compasses. 

A pair of gunners callipers* 

A pair of elliptical compasses* 

A pair of spiral ditto. 

A pair of perspective compasses. 

A pair of compasses with a micrometer screw. 

A rule for drawing lines, tending to a centre at a great dis- 
tante. 

A protractor and parallel rule. 

One or nvore parallel rules. 

A pentographer, or Peiitagraph. 

A pair of sectoral compasses, formingi at the same time, a 
pair of beam and eidUper compassci. 



DRAWING INSTRUMENTS. 77 

steel. The difference in the texture and pores of 
the two metals causes the parts to adhei'e less to- 
gether, diminishes the wear, and promotes unifor- 
mity in their motion. The truth of the work is as- 
certained by the smoothness and equality of the 
motion at the joint, for all shake and irregularity is 
a certain sigh of imperfection. The points should 
be of steel, so tempered, as neither to be easily 
bent or blunted ; tiot too fine and tapering, and 
yet meetinfic closely when the compasses are shut, 
^ As an inltrument of art, compasses are so well 
Icnown, tliat it would be superfluous to enumerate 
the various uses ; suffice it then to say, that they 
are used to transfer small distances, measure given 
spaces, and describe arches and circles. 

If the arch or circle is to be described obscurely, 
the steel points are best adapted to the purpose ; 
if it is to be in ink or black lead, either the draw- 
ing pen, or crayon points are to be used. 

To ust a pair of compasses. Place the thumb and 
middle finger of the right hand in the opposite hol- 
lows in the shanks of we compasses, then press the 
compasses, and the le^s will open a little way; thia 
being done, push the mnenwost leg, with the third 
finger, elevating, at the same time, the furthermost, 
■witn the nail ofthe middle finger, till the compas- 
ses are sufficiently opened to receive the middle and 
third finger ;. they may then be extended at pleasure, 
by pushmg the fucthermost leg outwards with the 
middle, or pressing it inwards with the four finger. 
In describing circles, or arches, set one foot ofthe 
compasses on the centre, and then roll the head of 
the compasses between the middle and four finger, 
the other point pressing at the same time upon the 
paper. They should be held as upright as possil;>le, 
and care should be taken not to press forcibly upon 
them,but ratlier to let them act by their own weight ; 
the legs should never be so far extended, as to fpnii 



78 MATHEMATICAL 

an obtuse angle with the paper or plane> on which 
they are used* 

The ink and crayon points have a joint just un- 
der that part which fits into the compasses ; by this 
they niay be always -so placed as to be set nearly 
perpendicular to the paper; the end of the shank of 
the best compasses is framed so as to form a strong 
spring, to bind firmly the moveable points, and pre- 
vent them from shaking. This is found to be a. 
more efiectual method man that by a screw. 

Two additional pieces are often applied to these 
compasses ; these, by lengthening the leg, enable 
them to strike larger circles, or measure greater 
extents, than they would otherwise perform, and 
that without the inconveniences attending longer 
compasses. When compasses are furnished with 
this additional piece, the moveable leg has a joint, 
that it may be placed perpendicular to the paper. 

Thehow compasses, are a small pair, usually with 
a point for ink ; they are used to describe small 
arches or circles, which they do much more conve- 
niently than large compasses, not only on account 
of their size, but also from the shape of the head, 
which rolls with great ease between the fingers. 

Of the drawing pen and protracting pin. The 

{)en part of this instrument is used to draw strait 
ines : it consists of two blades with steel points 
fixed to a handle ; the blades are si) bent, that the 
ends of the steel points meet, and yet leave a suflS- 
cient cavity for the ink ; the blades may be opened 
more or less by a screw, and, being properly set, 
will draw a line of any assigned thickness. One 
of the blades is framed with a joint, that the points 
may be separated, and thus cleaned more conve*^ 
niejitly ; a smaU quantity only of ink should be 
put at one time into the drawing pen, and this 
should be placed in the cavity, betweeto the blades, 
by A common p6n,or fsedfer ; the drawing pen acts 



DKAWING INSTRUMENTS. 79 

better, if the pen, by whidi the ink is inserted, be 
made to pass through the blades. To use the 
drawing pen, first fe^d it with ink, then regulate it 
to ihe thickness of the requii-ed line by the screw. 
In drawing lines, incline the pen a small degree^ 
taking care, however, that the edges of both the 
blades toudi the paper, keeping the pen close to 
the rule, and in the same direction during the whole 
operation: the blades should always be wiped 
very clean, before 'the pen is put away. 

These directions are equally applicable to the 
ink point of the compasses, only observing, that 
when an arch or circle is to be described, of more 
than an inch radius, the point should be so bent, 
that the blades of the pen may be nearly perpen* * 
dicular to the paper, and both of them touch it at 
the same time. 

^ 7%c ^otr acting piuy is only a short piece of steel 
wire, with a very fine point, fixed at one end of the 
upper part of the handle of the drawing pen. It is 
used to mark the intersection of lines, or to set off 
divisions from the plotting scale, and protractor. 



OP THE SECTOR. 

Amidst the variety of mathematical instruments 
that have been contrived to facilitate the art of 
drawing, 4heite is none so extensive in its use, or 
of such general application, as the sector. It is an 
universal scale, uniting, as it were, angles and pa- 
rallel lines, the rule and the compass, which arfe 
the only means that geometry makes use of for 
measuring, whether m speculation or practice. 
The real inventor of this valuable instrument is 
unknown ; yet of so much merit has the invention 
appeared, that it was claimed by GaUUoy and dis- 
puted by nations, 



80 MATHEMATICAL 

This instrument derives its name from the tentl^ 
definition of the third book oiEuclidy where be de- 
fines the sector of a circle. It is formed of two equal 
rules called legs ; these legs are moveable about 
the centre of a joint, and will, consequently, by 
their different openings, represent every possible 
variety of plane angles. The distance of the ex- 
tremities of these rules are the subtenses or chords^ 
or the arches they describe. 

Sectors are made of dijSferent sizes, but their 
length is usually denominated from the length of 
the legs when the sector is shut. Tnus a sector 
of six inches, when the legs are close together, 
forms a rule of 12 inches when opened ; and a 
foot sector is two feet long, when opened to its 
greatest extent. In describing the lines usually 
placed on this instrument, I refer to those com- 
monly laid down on the best six- inch brass sectors. 
But as the principles are the same in all, and the 
differences little more than in the number of sub- 
divisions, it is to be presumed that no difficulty 
will occur in the application of what is here said 
to sectors of a larger radius. 

The scales, or lines gi-aduated upon the faces of 
the instrument, and which are to be used as sec- 
toral UneSy proceed from the centre ; and are, 1 . 
Two scales of equal parts, one on each leg, marked 
UN. or L. Each of these scales, from the great ex- 
tensiveness of its use, is called the litfit of lines, 
2. Two lines of chords, marked cho. or c. 3. Two 
lines of secantSy marked sec or s. A line of poly- 
gonsy marked pol. Upon the other face, the sec- 
toral lines are, 1. Two lines of sines marked sin. 
or s. 2. Two lines of tangents, marked tan. 3. 
. Between the lines of tangents and sines, there is 
another line of tangents to a lesser radius, to sup- 
ply Hie defect of the former, and extending from 
45^ to 75^, 



DRAWING INSTRUMENTS. Si ' 

Each pair of these lines (except the linfe of'po* 
lygona) is so adjusted as to make equal angles at 
the centre, and consequently at whatever distance 
the sector be opened, the angles will be icilway s re- 
spectively equal. That is, the distance between 10 
and 10 on the line of lines, will be equal to 60 and 
60 on the line of chords, 90 and 90 on the line of 
sines, and 45 and 45 on the line of tangents. 

Besides the sectoral scales, there are others 
on each face, placed parallel to the outward 
edges, and us6d as those of the comnion plain 
scale. There are on the one face, I. A line of 
inches^ 2. A line of latitudes. 3. A line of hours. 
4. Aline of Inclination of meridians. 5. Aline 
of chorcls. On the other face, three logarithmic 
scales, nanfeiy, one of numbers, one of sines, and 
one of tangents ; these are used when the sector 
is fully opened, the legs forming one line. 

To read and estimate the divisions on the sectoral 
lines. The value of the divisions on most of the 
lines are determined by the figures adjacent to 
them ; these proceed by tens, which constitute the 
divisions of the first order, and are numbered ac- 
cordingljr ; but the value of the divislona on the 
line of lines, that are distinguished by figuix^s, is 
entirely arbitrary, and may represent any value 
that is given to them ; hence the figures 1, 2, 3, 4^ 
kc. may denote either 10, 20, 30, 40 ; or 100, 200, 
300, 400, and so on. 

7%e line of lines is divided into ten equal parts^ 
numbered 1, 2, 3, to 10 ; these may be called divi- 
sions of the first order ; each of these are again 
subdivided into 10 other equal parts, which may 
be called divisions of the second order ; and each 
of these is divided into two equal parts, forming 
divisions of the third order* 

The divisions on all the scales are contained be- 
tween four parallel lines ; those of the first order 

M 



ffii MATHEMATICAL 

extend to the most distant ; those of the thirds ij^ 
the least ; those of the second, to the intermediate 
parallel. 

When the whole line of lines represents 100, thu^ 
diviBions of the first order, or tlu>se to which the 
figures are annexed, represent tens ; those of ii» 
second order, units ; those of the third order, the 
halves of these units. If the whole line represents 
ten, then the divisions of the first order are units ; 
those of the second,tenth8,and the thirds,twentieths« 

In the Hne of tangentSy the divisions to which the 
numbers are affixed, are the degrees expressed by 
those numbers. Every fifth degree is denoted by a 
line somewhat longer than the rest; between every 
number and each fifth degree, there are four divi- 
sions, longer than the intermediate adjacent ones^ 
these ate whole degrees ; the shorter ones, or those 
of the third order, are 30 minutes. 

From the centre ^ to 60 degrees, the line of sines 
is divided like the line of tangents ; from 60 to 70^ 
it is divided only to every degree ; from 70 to 80, 
to every two degrees ; from 80 to 90, the division 
must be estimated by the eye. 

The divisions on the line <f chords are to be es« 
timated in the same manner as the tangents. 

The lesser line <f tangents is graduated every 
two degrees from 45 to 50 ; but from 50 to 60, to 
every degree ; from 60 to the end, to half degrees. 

The line of secants from 4o 10, is to be esti^ 
mated by the eye ; from 20 to 50 it is divided to 
ever)' two degrees ; from 50 to 60, to every degree ; 
and from 60 to the end, to every half degree. 

The solution of questions on the sector is said 
to be simpky when the work is begun and ended on 
the same line ; eompoundy when the operation be- 
gins on one line, and is finished on the other. 

The operation varies also by the manner in which 
the compasses are applied to the sector. If a mea* 



tDRAWING INSTRUMENTS. 89 

tare be taken on any of the sectoral lines, begin-' 
ning at the centre^ it hi called a lateral distance* 
But if the measure be taken from any point in one 
line, to its corresponding point on the line of the 
fame denomination, on the other leg, it is called a 
1ransper$e or paralld distance. 

The diyisions of each sectoral line are bounded 
by three parallel lines ; the innermost of these is 
that on which the points of the compasses are to 
be placed, because this alone is the line which goes 
to the centre, and is alope^ t|ierefore^ the sectoral 
line. 

We shall now proceed to give a few general in^ 
stances of the manner of operating with the sector. 

MuiUpKealion by the line of lines. Make th^ 
lateral oistance of one of ihe factors the pai*aUel 
distance of 10 ; then th^ parallel distance of the 
other factor is ihe product! * 

Example. Multiply 5 by 6, extend the conh> 
passes from the centra of the sector to 5 on the 
primary divisions, and open the sector till this dis- 
tance become the parallel distance from 10 to 10 
on the same diyisions ; then the parallel distance 
from 6 to 6, extended from the centre of the sector, 
ahall reach to 3, which is now to be reckoned 30. 
At the same opening of the sector, the parallel 
distance of 7 shall reach from the centre to 35, 
that of 8 shall reach from the centre to 40, &c. 

Division by the line of lines^ Make the lajtera) 
distance of the dividend the parallel distance of the 
divisor, the parallel distance of 10 is the quotient. 
Thus, to divide 30 by 5, make the lateral dbtance 
of 30, viz. 3 on the primary divisions, the parallel 
distance of 5 of the same divisions ; then the pa-^ 
rallel distance of 10, extended from the centre, 
shall reach to 6. 

Proportion by the line of lines. Make the lateral 
distance of tiie second term the parallel distance 



84 MATHEMATICAL 

of the first term ; the parallel distance of the 
term is the fourth proportional. 

Example. To find a fourth proportional to 8, 4» 
and 6, taxe the lateral distance oi 4, and make it 
the parallel distance of 8 ; then the parallel dis-^ 
tance of 6, extended from the centre, shall reach 
to the fourth proportional 3. ^ 

In the same manner a third proportional is found 
to two numbers. Thus, to find a third proportion- 
al to 8 and 4, the sector remainihg as in the former 
example, the parallel distance of 4, extended from 
the centre, shall reach to the third proportional 2, 
In all these cases, if the number to be made a pa- 
rallel distance be too great for the sector, some ali- 
<]^uot part of it is to be taken, and the ansiwer mul- 
tiplied by^^the number oy which the first number 
was divided. Thus, if it were required to find a 
fourth proportional tcf 4, 8, and 6 ; because the la« 
teral distance of the second term 8 cannot be made 
the parallel distance of the first term 4, take the 
lateral distance of 4, viz. the half of 8, and make it 
the parallel distance of the first term 4 ; then the 
parallel distance of the third term 6, shall reach 
from the centre to 6, viz. the half of 12. Any other 
aliquot part of a number may be used in the sam% 
tvay. In like manner, if the number proposed be 
too small to be made the parallel distance, it may 
be multiplied by some number, and the answer is 
to be divided by the same number. 

To protrtut angles hy the line of Chords. Case 
1. When the given degrees are under 60. 1. With 
any radius on a centre, describe the arch. 2. Make 
the same radius a transverse distance between 60 
and 60 on the line of chords. 3. Take out the 
transverse distance of the given degrees, and lay 
this on the arch, which will mark out the angular 
distance required. 
. Case 2. When the given degrees are more thai\ 



DRAWING INSTRUMENTS. 85 

SO. 1. Open the sector, and describe the arch as 
before. 2. Take j or | of the given degrees, and 
tate the transverse distance of this 7 or ^, and lay 
it off twice, if the degrees were halved, three times 
if the third was used as a transverse distance. 

Ccise 3. When the required angl^ is less than 6 
degrees ; suppose 3. 1. Op^n the sector to the 
given radius, and describe the arch as before. 2. 
Set off the radius. 3. Set off the chord of 57 de- 
grees backwards, which will give^the arc of three 
degrees. 

Given the riidius of a drcle, f suppose e^piai to 
two inches J J required the sine and tangent qf2&^ 30' 
to that radius. 

Solution. Open the sector so that the trans- 
verse distance of 90 and 90 on the sines, or of 45 
and 45 on the tangents, may be equal to the given 
radius, viz. two inches ; then will the transverse dis- 
tance of 38^ 30', taken from the sines, be the 
length of that sine to the given radius ; or if taken 
from the tangents ; will oe the length of that tan- 
gent to the given radius. 

But" if the secant of 2S^ 30' was required ? 

Make the given radius, two inches, a transverse 
distance to and 0, at the beginning of the line of 
secants ; and then take the transverse distance of 
the degrees wanted, viz. 28^ 30'. 

A tangent greater than 45^ fsuppose 60^^ is 
found thus. 

Make the given radius, suppose two inches, a 
transverse distance to 45 and 45 at the beginning 
of the scale of upper tangents ; and then the re- 
quired nxunber 60^00'may be taken from this scale. 

Given the kngth of the sine, tangent^ or secant of 
any degrees ; to find the length of the radius to that 
sinty tangent^ or secant. 

Make the given length a transverse distance to 
its given*degrees on its respective scale : then, 



86 MATHEMATICAL 

In tie sines. The transrerse distance of 90 and 
90 will be the radius sought. 

In the lower tangents. The transverse distance 
of 45 and 45, near the end of the sector* will be 
the radius sought. 

In the upper tangents. The transverse distance 
of 45 and 45, taken towards the centre of the sec-? 
tor on the line of upper tangents, will be the centre 
sought. 

In the secanL -JThe transverse distance of and 
0, or the beginning of the secants, near the centre 
of the sector, will be the radius sought 

Given the radius and any line representing a sine, 
tangent, or secant ; to find the degrees corresponding 
to that line. 

Solution. Set the sector to the given radius, 
according as a sine, or tangent, or secant is con^ 
cemed. ' 

Take the given line between the compasses ; 
apply the two feet transversely to the scale con^ 
cerned, and slide the feet along till thev both rest 
on like divisions on both legs ; then will those di- 
visions shew the degrees and parts corresponding 
to thp given line. 

To find the length of a versed sine to a given num* 
her ofdegreeSy and a given radius. 

Make the transverse distance of 90 and 90 on 
the sines, equal to the given radius. 

Take the transverse distance of the sine com- 
plement of the given degrees. 

If the given degrees are less than 90, the differ^ 
ence between the sine complement and the radius 
gives the versed sine. 

If the given degrees are more than 90, the sum 
of the sine complement and the radius gives the 
versed sine. 
' To open the legs 'of the sector, so that the corres- 



©RAWING INSTRUMENTS. 8^ 

jponding donbk scales of Unes, chords^ sines. Mid 
taa^ents, may make each a right angle. 

On the lines, make the lateral distance 10, a 
diBtance between eight on one leg, and six on the 
ether leg. 

On the sines, niake the lateral distance 90 a trans- 
Terse distance from 45 to 45 ; or from 40 to 50 ; or 
from 30 to 60 ; or from the sine of any degrees to 
their complement 

Or an we sines, make the lateral distance of 45 
^ transTerse distance between 30 and 30. 

■ 

OF THS PLAIN SCALE. 

The divisions laid down on the plain scale are of 
iwo kinds, the one having more immediate relation 
to the circle and its m-operties, the other being 
merely concerned wiui dividing straight lines. 

Though arches of a circle are the most natural 
Measures of an ai^le, yet in many cases right lines 
are substituted, as being more convenient ; for the 
comparison of one right line with another, is more 
natural and easy, than the comparison of a right 
line with a curve ; hence it is usual to measure the 

Suantities of angles not by the arch itself, which is 
escribed on the angulju* point, but by certain lines 
described about that arch. 

The lines laid down on the plain scales for the 
measuring of angles, or the protracting scales, are, 
1. A line of chords marked cho. 2. A line of sines 
marked sin. of tangents marked, tan. of semitan- 
gents marked st. and of secants marked sec this 
last is often upon the same line as the sines, be- 
cause its gradations do not begin till the sines end. 
There are two other scales, namely, the rhumbs, 
marked ru. and longitudes, maiked lon. Scales of 
latitude and hours are someihnes put npoit the 



88 MATHEMATICAt 

piaia scale ; but» as dialling is now but seldom 
studied, they are only made to order. 

The divisions used for measuring straight lined 
are called sccdes of equal parts^ and are of various 
lengths for the convenience of delineating any fi** 
gure of a large or smaller size, according to the 
mjcy or purposes of the draughts^man. They are, 
indeed, nothing more than a measure in miniature 
for laying down upon paper, &c. any known mea^ 
sure, as diains, yards, feet, &c. each part on tlie 
scale answering to one foot, one yard, &c. and the 
plan will be larger or smaller, as the scale contains 
a smaller or a greater number of parts in an inch* 
Hence a variety of scales is useful to lay down 
lines of any required lengthy and of a convenient 

{^ropoilion with respect to the size of the drawing. ' 
f none of the scales happen to suit the purpose^ 
recourse should be had to the line of lines on the 
sector ; for, by the diJSerent openings of that in- 
strunoent, a line of any length may be divided 
into as many equal parts as any person chooses. 

Scales of equal parts are divided into two kind£, 
the one simple, the other diagonally divided. 

Six of the simply divided scales are generally 
placed one above another upon the same rule ; 
they are divided into as many equal parts as the 
length of the rule will admit of ; the numbers 
placed on the right hand, shew how many parts in 
an inch each scale is divided into. The upper 
scale is sometimes shortened for the sake of intro- 
ducing another, called the line of chords. 

The first of the larger, or primary divisions, on 
every scale is subdivided into 10 equal parts, which 
small parts are those which give a name to the scale : 
thus it is called a scale of 20, when 20 of these di- 
visions are equal to one inch. If, therefore, these; 
lesser divisions be taken as units, and each repre- 
sents one league, one mile, one chain, or one yard. 



DRAWING INSTRUMENTS. d» 

,&c. then will the larger divisicms be so many tens } 
but if the subdiyisions are supposed to be tens, the 
la^er divisions will be hunchreds. I 

. To illustrate tbis> suppose it were required to 
setoff from either of the scales of equal parts ff, 36> 
or 360 parts, either miles or leagues. Set one foot 
of your compasses on 3, among the larger or pri* 
mary divisions, and open the other point till it 
falls on the 6th subdivision, reckoning backwards 
or towards the left hand* llien will this extent 
represent, 7^ 36, or 360 miles or leagues, &c. and 
bear the same proportion in the plan as the line, 
measured does to the thing represented. 

To adapt these scales to feet and inches, the 
first primary division is ofteDduodecimally divided 
by an upper line^; therefore, to lay down any num- 
ber of &et and inch^, as for instance, eight feet 
eight inches, extend the compasses from eight of 
the larger to eight of the upper small ones, and 
that distance laid down on the plan will repre-* 
sent eiffht feet eight incites. f 

Of the scale of equal parts diagonally divided. 
The use of this scale is ibe same as those already 
described. But by it a plane may be more accu- 
rately divided than by the fonner ; for any one of 
the larger divisions mAy by this be subdivided into 
100 equal parte ; and, therefore, if the scale con-* 
tains 10 of the larger divisions, any number under 
1000 may be laid down with accuracy. 

The diagonal scale is seldom placed on the same 
side of the rule with tlie other plotting scale. 
The first division of the dmgonal scale, if it be a 
foot long, is generally an inch divided into 100 
equal parts, and at the .c^posite end there is usu- 
ally half an inch divided into an 100 equal parts^ 
If the scale be six inches long, one end has com-* 
monly half an inch, tlie other a quarter of an inch 
subdivided into 100 equal parts. 

N 



gfO ^ MATHEMATICAL 

The nature of this scale will be better undeiy 
stood by considering its construction. For this^ 
purpose : 

First Draw eleTen parallel lines at equal dis« 
tances ; diyide the upper of these lines into such a 
number of equal pails> as the scale to be express- 
ed is intended to contain ; from each of tliese di- 
visions draw perpendicular lines through the 
eleven parallels. 

Secondly. Subdivide the first of these divisions 
into ten equal parts, both in the upper and lower 
lines. V 

Thirdly. Subdivide again each of these subdivi- 
sions, by drawing diagonal lines from the 10th be^ 
low to the 9th above; from the 8th below to the 
7th above ; and so on, till from the first below to the 
above ; by these lines each of the small divisions 
is divided into ten parts, and, consequently, the 
whole first space into 100 equal parts; for, as each 
of the subdivisions is one^^tenth part of the whole 
first spac^ or division, so each parallel above it is 
one-tenth of such subdivision, and, consequently, 
one^hundreth part of the whole first space : and if 
there be ten of the larger divisions, one-thousandth 
part of the whole space* 

If, therefore, the larger divisions be accounted 
as units, the fii-st subdivisions will be tenth parts of 
an unit, and the second, marked by the diagonal 
upon the parallels, bundreth parUi of the unit. 
But, if we suppose the larger divisions to be tens, 
the first subdivisions will be units, and the second 
tenths. If the larger are hundreds, then will the 
first be tens, and the second units* 

The numbers therefore, 576, 57,6, 5,76, are all 
expressible by the same extent of tiie compasses : 
thus setting one foot in the number five of the 
larger divisions, extend the 6tber along the sixth 
parallel to the seventh diagonal* For, if the five 



DRAWING INSTRUMENTS. 9l 

• 

larger divisions be taken for 500, seven of the first 
suMivisions will be 70, which upon the sixth pa- 
rallel, taking in six of the second subdivisions for 
units, makes the whole number 576* Or, if the 
five larger divisions be taken for five tens, or 60, 
fleven of the first subdivisions will be seven units, 
and the six second subdivisions upon the sixth pa« 
rallel, will be six tenths of an unit. Lastly, if the 
five larger divisions be only esteemed as five units^ 
then will the seven first subdivisions be seven 
tenths, and the six second subdivisions be the six 
hundredth parts of an unit. 

Of the Une of chords. This line is used to set 
ofi* an angle from a given point in any right line, 
or to measure the quantity of an angle already 
laid down. 

Thus to draw a line that shall make with ano^ 
ther line an angle, containing a given number of 
degrees, suppose 40 degrees. 

Open your compasses tothe extent of 60 degrees 
upon the line of chords, (which is always equal to 
the radius of the circle of projection,) and setting 
one foot in the angular pomt, with that extent de- 
scribe an arch ; then taking the extent of 40 de- 
grees from the said chord hne, set it ofi* from the 
[iven line on the arch described ; a right line drawn 
•om the given point, through the pomt marked 
upon the arch, will form the required angle. 

The-degrees contained in an angle already laid 
down, arfe foimd nearly in the same manner ; for 
instaftce> to measure an an^le* Prom the centre 
desqibe an arch with the chord of 60 degrees, and 
the length of the arch, contained between the lines 
measured on the line of chords, will give the num-- 
ber of degrees contained in the angle. 

If the number of degrees are more than 90, 
they must be measured upon the chords at twice : 
thus, if 1 20 degrees were to be practised,60 may be 
taken from the chords, and those degrees be laid off 



92 MATHEMATICAL 

twice upon Uie arch. Degrees taken from the 
chords are always to be counted from the begui* 
mng of the scale. 

Of the rhumb lint. This is, in fact, a line of 
chords constructed to a quadrant divided into 
eight parts or points of the compass, in order to 
facilitate the work of the navigator in laying 
down a ship's course. 

Of the line i^langUudes. llie line of longitudes 
is a line divided into sixty unequal parts, and so ap- 
plied to the line of chords, as to shew, by inspection, 
the number of equatorial miles contained in a de- 
gree on any panulel of latitude. The graduated 
line of chords is necessary, in order to shew the 
latitudes ; the line of longitude shews the quantity 
of a degree on each parallel in sixtieth parts of an 
equatorial degree, th^t is, miles. 

2^he lines of tangents; semOangents, and secants, 
serve to find the centres and poles of projected cir- 
cles in the stereographical projection of thesphere. 

The line of situs is principally used for ^e or- 
thographic projection of the sphere. 

The lines o/ilaiitudes and hours are used con« 
jointly, and serve very readily to mark the hour 
lines in the construction of dials ; they are gene- 
rally on the most complete sorts of scales and sec* 
tors ; for the uses of which see treatises on dialling. 

OF THE PROTRACTOR. 

• « 

This IB an instrument used to protract, or lay 
down an angle containing any number of degrees, 
or to find how many degrees are contained in any 
given .angle. There are two kinds put into cases 
of mathematical drawing instruments ; one in the 
form of a semicircle, the other in the form of a pa- 
rallelogram. The circle is undoubtedly the only 
natural measure of angles ; when a straight line is 
therefore u^ed, the divisions thereon are derived 



DRAWING INSTRUMENTS. 93 

from a circle, or its properties, and the straight line 
is made use of for some relative convenience : it 
is thus the parallelogram is often used as a protrac- 
tor, instead of the semicircle, because it is in some 
erases more convenient, and that other scales, &c. 
may be placed upon it* 

2%e semicircular protrnc(ory is divided into 1 80 
equal parts or degrees, which are numbered at 
every tenth degree each way, for the conveniency 
of reckoning either from the right towards the left, 
or from the left towards the right ; or the niore 
easily to lay down an angle from either end of the 
line, beginning at each end with 10, 20,'' &c. and 
proceeding to 180 degrees. The edge is the di- 
ameter of the semicircle, and the mark in the mid- 
dle points out the centre, in a protractor in the 
form of a paraBdogrum : the divisions are as in the 
semicircular one, numbered both vrays ; the blank 
side represents the diameter of a circle. The side 
of the protractor to be applied to the paper is made 
jSat, and that whereon the degrees are marked, is 
chamfered or sloped away to tlie edge, that an 
angle may be n[K>re easily measured, and the di- 
visions set off with greater exactness. 

Application of the protrcuior to vse, 1 • A humi- 
her of degrees being given, to protract, or lay donm 
an angUy whose measure shall be equal thereto. 

Thus, to lay down an angle of 60 degrees from 
the point of a line, apply the diameter of the pro- 
tractor to the line, so that the centre thereof may 
coincide exactly with the extremity ; then with a 
protracting pin make a fine dot against 60 upon the 
limb of the protractor ; now remove the protract 
tor, and draw a line from the extremity through 
that point, and the angle contains the' given numr 
ber of degrees. 

2. To find the number of de^ees contained in a 
given angle. 



• •} 



94 GUNTER'S 

Place the centre of the protractor upon the an- 
gular point, and the fiducial edge, or diameter, ex- 
actly upon the line ; then the degree upon the limb 
that is cut by the line will be the measure of the 
given angle,whichx in the present instance, is found 
to be 60 degrees- 

3* Pram a given point in a line, to erect a perpen- 
dieuiar to that line. 

Apply the ^otractbr to the line, so that the cen- 
tre may coincide with the given point, and the di- 
vision marked 90 may be cut by the line ; theij a 
line drawn against the diameter of the protractor 
will be the perpendicular, required. 

OF PARAXLEL KIJI^0, 

Tarallel lines occur so continually in every spe- 
cies of matfaemati<;al drawing, that it is no .wonder 
80 many instruments have been contrived to deli« 
neate them with more expedition than could be 
effected by the general geometrical methods. For 
this purpose, rtues of various constructions have 
been made ; and particularly recommen(^d by 
theit inventors ; tbeir use however is so apparent 
ns to need no explanation, 

ounter's scale. 

The scale generally used is a ruler of two feet 
in length, having drawn upon it equal parts, chords, 
sines, tangents, secants, &c. These are contained 
on one side of the scale, and the other side contains 
the logarithms of these numbers. Mr. Edmund 
Ounter was the first who applied the logarithms of 
numbers, and of sines and tangents to straight lines 
drawn on a scale or ruler; with t^hich, proportions 
in common numbers, and trigonometry, may be 
solved by the application of a pair of compasi^ea 



SCALE. ' * 95 

« 

only. The method is founded on this property. 
Thai the logarithms of the terms of equal ratios arc 
equidifferent This was called Gunter's Propor- 
tion, and Gunter's Line ; hence the scale is gen- 
erally called the Gunter. 

Of the Lo^arithmical Lines, or Gv/nter's Scale. 

The lo£anthmical lines^ on Gunter's scale> are 
the eight following : 

S^Mhumb, or fine rhumbs, is a line contaiiiing 
the logarithms of the natural sines of every point 
and quarter point of the compass, numbered from 
a brass pin on the right hand towards the left with 
8,7,6,5,4,3,2,1. m 

T^Rhumby or tangent rhumbs, also corresponds 
to the logarithm of £e tangent of every point and 

Suarter point of the compass. This line is nunih 
ered from near the middle of tUb scale with 1. 2. 
3. 4 towards the right hand, and back again with 
the numbers 5, 6, 7 from the right hand toward? 
the left. To take off any number of points below 
four, we must begin at 1, and count towards the. 
right hand ; but to take off any number of points 
alK>ve four, we must begin at four, and count to- 
wards the left hand. 

Numbersj or the line of numbers, is numbered 
from the left hand of the scale towards the rights 
with 1, 2, 3, 4, 5, 6, 7, 8, 9, 1 which stands exactly 
in the middle of the scale i the numbers then go on 
2, 3, 4, 5, 6, 7, 8, 9, 10 which stands at the right 
} hand end of the scale. These two equal parts of 

' the scale are divided equally, the distance between 

V the first or left hand 1, and the fixst 2, 3, 4, &c. is 

I exactly equal to the distance between the middle 

1 and the numbers 2, 3, 4, &c. which follow it. 
The subdivisions of the^e scales are likewise simi- 
lar, viz. they are each one-tenth of the primary di- 
visions, and are distinguished by lines of about 
half the length of the primary divisions. 



% GUNTER'S * 

These subdirisions are again divided into ten 
parts, where room will permit ; and where that }a 
not the case, the units must be estimated, or gues- 
sed at, by the eye, which is easily done by a little 
practice. 

The primary divsions on the second part of the 
scale, are estimated aocprdins to the value set upon 
the unit on the left hand of the scale : If you call 
it one, then the first 1, 2, 3, kc. stand for I, 2, 3, 
kc. the middle 1 is 10, and the 2. 3. 4. &c. folio w-« 
ing stand for 20, 30, 40, &c. and the ten at the right 
hand is 100 : If the first 1 stand for 10, the first 2, 
3, 4, &c. must be counted 20, 30, 40, &c. the mid* 
die 1 will be 100, the second 2, 3, 4, 5^ &c. will 
stand for 200, 300, 400, SOO, &c. and the ten at the 
rijdit hand for 1000. 

If you consider the first 1 as ^ of an unit, the 2, 
3, 4, &c. following will be ^,-;^, ^s^ &c. the middle 
1 wUi rtand for m unit, and the 2, 3, 4, &c. follow-^ 
ing will stand £6r 2, 3, 4, &c. also the division at 
the right-hand end of the scale will stand for I0«. 
Tl)^ intermediate small divisions must be estimated 
according to the value set upon the primary ones. 

Sine- The line of sines is numbered from the 
left hand of the scale towards the right, 1,2, 3, 4, 
5, &c. to 10 ; then 20, 30, 40, &c. to 90^ where it 
terminates just opposite 10 on the line of numbers. 

Versed sine. This line is placed immediately 
under the line of sines, and numbered in a contrary 
direction, viz. from the right hand towards the left 
10, 20,30, 40, 50, to about 169 ; the small divisions 
are here to be estimated according to the number of 
them to a degree. By comparing the line of versed 
sines with the line of sines, it will appear that 
^e versed sines do not belong to the arches with 
which they are marked, but are the half versed sines 
of their supplements. Thus, what is marked the 
versed sine of 90 is only half the versed sine of 90, 



SCALE. 97 

the versed sine of 120^ is half the versed sine df 
60^, and the versed sine iaarked 100* is half the 
versed sine of 80*», &c. 

The versed, sines are numbered in this manner 
to render them more commodious in the solution 
of trigonometrical, and astronomical problems. 

Tangenls. The line of tangents begins at the 
left hand, and is numbered I, 2, 3, &c. to 10, then 
20, 30, 45, where there is a little brass pin just un- 
der 90 in the line of sines ; because the sine of 90* 
is equal to the tangent of 45^ It is numbered 
from dS"" towards the left hand 50, 60, 70, 80, &ۥ 
The tangents of arches above 45* are therefore 
counted backward on the line, and are found at the 
flame points of the line as the tangents of their 
complements. 

Thus, the division at 40 represents both 40 and 
50, the division at 30 serves for 30 and €0, &;c. 

Meridional Parts. This line stands immediate^ 
ly above a J&ie of equal parts, marked Eqml Ft 
vdth whi^ it must alwaysbe compared when used. 
The line of equal parts is marked from the right 
hand to the left with 0, 10, 20^ 30, Sic. ; each of 
these large divisions re^sents 10 degrees of the ^ 
equator, or 600 miles. The first of these divisions' 
is sometimes divided into 40 equal parts, each re- 
presenting 15' minutes or miles. 

The extent from the brass pin on the scale of 
meridional parte to any division on that scale, ap- 
plied to the line of equal parts, will give (in de* 
grees) the meridional parts answering to the lati- 
tude of that division. Or the extent from any dfi- 
nAsicn to another, on the line of meridional parts, 
fipplied tb the line of equal parts, will give the 
meridional difierence of latitude between the two 
places denoted by the divisions. These degrees 
are reduced to leagues by multiplying by 20, or 
to miles by multiplying by 60. 

O 



dd GUNTER'S, &c. 

The use of the hghtithmcdl lints &n Gunler^s Scale. 

By these line^ atid a pair of compasses^ all the 
problems of Trigonometry, &c- may be solved. 

These problems ate all solved by proportion ; 
Now in natural numbers, the quotient of the first 
teiin bv the second is equal to the quotient of the 
third by thfe fourth: Iherefor^^ logarithmically 
ispeaking the dlfl^renc!^ between the first and se-- 
cohd term is equal to the difierence between the 
third and fdurtH, consequently on the lines on the 
scale, the distance between the first and second 
term will be equal to the distance between the 
third and fourth. And for a similar reason, be- 
cause four proportional quantities are alternately 
proportional, ihib distance between the first and 
third tenns, will be equal to the distance between 
the second and fourtli. Hence the following 

GefieralMule. 

The eitent of the compasses from the first 
term to the second, will reach, in this same di- 
rection, from th6 third to the fourth term. Or, 
the extent of the compasses from the first term to 
the thii^, will reach, in the same direction, from 
the second to the fourth. 

By the same direction in the foregoing rule, is 
meant that if the second term lie on the right 
hand of the fil^st, the fourth will lie on the right 
hand of the third, and the contrary. This is true, 
except the two first or two last terms of the pro- 
portion are on the line of tangents, and neither of 
Ihcm under 45** ; in this case the exter^t on the 
tangents is to be made in a contrary direction : 
Forbad the tangents above 45'' been laid down in 
their proper direction, they would have extended 
beyond the length of the scale towards the right 
hand ; tliey are therefore as it were folded back up- 



TRIGONOMETRY. W 

on the tangents below 45^, and consequently lie in 
a direction contraiy to their proper and natural 
order. 

If the two lagt terms of a proportion l)e on tbq 
line of tangepts^ anc} one of them greater and iim 
other less than 46^ ; the extent from the 6i^t term . 
to the second will rBaf;h froqi the third beyond Uie 
si^le. To remedy this inconvenju^cey ?pply the 
extent between the two first terms from 45* bjaGk* 
ward upon the line of tmigei^s^ and ki^p th^ left 
fafand poii^ of tjbe compasses where it faUs ; ibring 
the right hand point from 45'' to the third term of 
the proportion > this extent now in the pompasses 
applied from 45^ backward will reach to the fourth 
term> or the tangent required. For, had the line 
of t^n^ents heen continued forward beyond 4d^» 
the dft?)si6n9 would have fallen above 45"* forward ^ 
in the sam^ loaqner ^s they f^U under 45** bac^^ 
ward, 



SECTION T. 

TRIGONOMETRY, 

The word Trigonometry signifies the measuring 
nf trioMgles. But, under this n^me is generally 
comprehended the art of determining tbe pos^ioBS 
and dimensions of the several unkoown parts of 
extension, by means of some parts, which are al^ 
ready known. If we conceive the difierent points, 
which may be represented in any ^pace, to be 
joined together by right lines, there are jthree things 
ofiei^d Kf* our consideration ; 1. tbe length of 
these lines ; 2. the angles which they form with one 
anoth^ ; 3. the angles formed hy the planes, in 
which these lines jtre drawn, or are supposed to bo 
traced. On the comparison of these tbtee objects. 






100 TRIGONOMETRY. 

depends the solution of all questions, tbat can be 
proposed concerning the measure of extension, 
and its parts ; and tte art of determining all these 
things trom the knowledge of some of them, is 
reduced to the solution of these two general 
questions. 

1. Knowing three of the six parts, the sides and 
angles — ^whicn constitute a rectilineal triangle ; to 
find the other tliree. 

2. Knowing three of the six parts, which com?* 
pose a spherical triangle ; that is a triangle formed 
on the surface of a sphere by three anSies of cir- 
cles, which have theu* centre in the centre of the 
same sphere — ^to find the other three. 

The first question is the object of what is called 
Plane Trigonometry, because the six parts, con- 
ddered here, are in the same plane : it is also de- 
nominated Rectilineal Trigonometry. The second 
question belongs to Spherical Tiri^onometry, 
wherein the six parts are considered m different 
planes. But the only object here is to explaip 
the solutions of the former question : viz. 

Plaot Trigonometry. 

Plane Trigonometry is that branch of geometry, 
which teaches how to determine, or calculate th^e 
of the six parts of a rectilineal triangle by having 
the other three parts given or known. It is usually 
divided into Ri^t angled and Oblicjue angled Tri- 
gonometry, according as it is applied to the men-« 
duration of Right or Oblique angled Triangles. 

In every triangle, or case in trigonometry, three 
ibf the parts must be given, and one of these parts, 
at leas^ must be a side ; bdbause, with the same 
angles, the sides may be greater or less in any 
propojkioR, 



I? 



I'RIGONOMETRY. 101 

Right Angled PLAxnc Trigonometry. 

1. In every right*aiigled plane triangle ^BC, if 
the hypothenuse A Che made the radius, and with 
it a circle, or an arc of one, be described from 
each end ; it is plain (from def. 20.) that BC is 
th^ sine of the angle A, and AB is the sine of the 
angle C ; that is, the legs are the sines of their op- 
posite angles. 

If one leg AB be made the radius, and with it, 
' on the point ^1, an arc be described ; then BC is 
the tangent, and AC is the secant of the angle A^ 
by de£ 22 and 25. 

3. If BC be made the radius, and an arc be des^ 
cribed with it on the point C; then is AB the tan^ 
gent,and^Cis the secant of the angle C, as before. 

Because the sine, tangent, or secant of any giv- 
en arc, in one circle, iis to the ^ine, tangent, or se- 
cant of a like arc (or to one of the like number of 
<legree3) in another circle ; as the radius of the one 
is to the radius of the other ; therefore the sine, 
tangent, or secant of any arc is proportional to the 
Wie, tangent, or secant of a like arc, as the radius 
of the given arc is to 10.000000, the radius from 
whence the logarithmic sines, tangents, and se- 
cants, in most tables, are calculated, that is ; 

If AC be made the radius, the sines of the an- 
gle A and C, dejscribed by the radius AC, will be 
proportional to the sines of the like arcs, or angles 
)R the circle, that the tables now mentioled were 



5 



103 TRIGONOMETRY^ 

calculated for. So if BCwas required, having the 
aagles and AB given, it will be, 

As8.C'AB::SU^'BC. 

That iS| as the sine of the angle C in the tables, 
, is to the length of AB ; (or sine of the an^le C, 
in a circle whose radius is AC; J so is the sine of 
the ar^le A in the tables, to the length oiBC. (or 
fiine of the same angle, id the circle, whose radius 
hAC.) 

In like manner the tangents and secants rejpre^ 

seiEited by making either leg the radius, will be 

' proportional to the tangents and secants of a like 

arc, as the radius of the given arc is to 10.000000, 

the radius of the tables aforesaid. 

Hence it is plam, th;.t if the name of each side 
of the triangle be placed thereon, a proportion 
will arise to answer the same end as before : thus 
if AC^ be made the racUus, let liie word radius 
be written thereon ; and as BC and ABy are the 
eines of their opposite angles ; upon Hie first let 
&Aj or msB of the aoffle A, sod on the other left 
8.C^ orsiae of the an^ C, foe wr^ieou Theo« 

When a side is nequired, it may Im <ribtained by 
this proportion, viz. 

As the naioe of the side given 
is to the side gi^en, 

Sols the name of the side requined 
to ihi^ side required. 

Thus, if the an^es A and C, and the hypo^ 
thenuse -4Cwere given, to find the sides ; the pro- 
portion will be 

-%. 1. 

l.RiAC''S.A'Ba 



f 

I 



TBIGONOMETRY- 1Q3 

That i&9 as radius is to AC, so is tJie sine of the 
angle A, to BC. And, 
2. li:AC''S.C'AB. 
That is» as radius is to AC, so is the sine of the 
angle C to AB. 

^ When an angle is required^ we use this proper* 
tion,viz. 
As the side that is made the radius^ 
is to radius, 
So is the other given side, 
to its name. 
Thus, if the legs were given to find the angle JF^ 
and if AB be made the radius, it will be 



-«(r-2. 



AB.R'.BCTA. 

That is, as AB^ is to radius, so is jBC, to the tan- 
gent of the angle A: 

After the same manner, the sides or angles of 
ml] right angled plane triangles tnaj be found^ from 
their proper data. 

We here^ in plate 4, give all the proportion 
requisite for the solution of the six cases in rightr 
angled trigonometry ; making every side possible 
the radius. 

In the following triangles this mark — in an 
angle denotes it to be known> or the quantity of 
degrees it contains to be given ; and this mark^ 
on a side, denotes its length to be given in feet, 
^ards» perches, or miles, &c. and this mark% either 
in an angle or on a side, denotes the angle or side 
to be required. 

From these proportions it may be observed ; 
that to find a side, when the angles and one side 
are given, any side may be made the radius ; and 






104 TKlGOrrOMETRY. 

to find an angle> one of the given sides must b^ 
made the ramus. So that in the Ist, 2d, and 
3d cases, any side as well required as given may 
be made the radius, and in the first statin^s of 
tile 4th, 5th, and 6th cases, a given side only is 
made the radius. 

RIGHT ANGLED TRIANGLES. 

* 

CASE L 

The anglea and h}ffiothenu»e given j to find the base and fiei^r 

fiendicular. 

Pl. 5. \Fig. 4. 

In the right angled triangle ABC, suppose the 
angle A = 46^ 3u . and ccmsequently the angle C 
= 43*. 3Gf' . (by cor. 2. theo. 5.) ; and AC 250 parts, 
(as feet, yards, miles, &c.) required the sides AB 
and BC 

1st. BY CONSTRUCTION. 

Make an angle of 46^ 30', in blank lines, (by 
prob. 16. geom.; as CAB; lay 250, which is the 
dven hypothenuse, from a scale of equal parts, 
from AioC; from C, let fall the perpendicular 
(JBC, by prob. 7. geom») and that will constitute 
the triangle ABu. Measure the lines BC, and 
AB, from the same scale of equal parts that AC 
was taken from ; and you have the answer. 

2d. BT CALCULATION. 

1. Making AC the radiuSy the required sides are 
found by these propositions, as in plate 4, case L 

R.'AC-'S^.'BC. ' 

R : AC ' • S.C : AB. 



TBIGONOMETRT. 103 

That is, a£ radius, =90" 10.000000 

is to -4C =250, 2.397940 

So is the sine of As46*. 30' 9.860562 



to BC, =18L 4 2.258502 



As radius, «=90* 10.000000 

is to ACy • -=250 2.397949 

So is the sine of C»:43». 30^ 9.837812 



to AB, ?=172* 1 2.235752 



» 

If from the Bum of the second ai^ third logs, 
that of the first be taken» the numbWwill be the 
log. of the fourth ; the number answeruig to which 
wm be the thing required ; but when the first log. 
IB radius, or 10.000000».reject the firpt figure of the 
sum of the other two logs, (whicji) is the same thing 
a3 to subtract 10.000000;) and that will )t>e the 
1<^. of the thing required* 



2. Making AB the rqdius. 

Secant A : AC: : R : AB. . 
Secant A : AC : : T.A : BC. 



That is, As the secant of ^«=46« 30' 10.162186 

13 to AC, »250 2.397940 

So is the radius =»90« IOjOOOOOO 



12.397940 



io AB, c=172. 1 2.235762 



106 



TRIGONOMETRY. 



As the secant of A 


=46»30' 


10.16218a 


is to vie, 


= 250 


2.397940 


So is the tangent of A 


=46»3flr 


10.022750 



12.420690 



toBC, 



=181.34. 2.258502 



3. Making BC the radius. 

Sec. C : AC : : R : BC. 

Sec. C: AC:: T.C : AB. 

That is, as the secant of C=43* 30'' 

is to ACy = 250 

So i^dius = 90* 



10.139438 

2.397940 

10.000000 



toBC, 
As the secant of C 

is to ACt 
So is the tangent of C 



*=181.34 
=43* 30' 
= 250 
==43»30^ 



12.397940 

2.258502 

10.139438 

2.397940 

9.977250 

12.375190 



to AB, = 172. 1 2.235752 

Or, having found one side, the other may be ob- 
tained by cor. 2. theo. 14. sect. 4. 

« 

3d. By Ounter^s scak* 

The first and third terms in the foregoing pro- 
portionSj being of a like nature, and those of the 
second and fourth being also like to each other ; 
and the proportions being direct ones» it follows ; 
that if the third term be greater or- less than the 
first, the fourth term will be also greater or less 



•TRIGONOMETRT. 107 

than the second ; therefore the extent in your 
compasses, from the first to the thhrd tenn> will 
reach from the second to the fourth. 

Thus, to extend the first of the foregoing pro- 
portions ; 

1. Extend from 90^ to 46*» 30', on the line of 
sines ; that distance will reach from 250 on the 
line of numbers, to 181, for BC. 

% Extend from 90*^ to 43<» 30', on the line of 
fiines ; that distance will reach from 250 on the 
line of numbers, to 172, for AB. 

If the first extent be from a greater to a less 
number ; when you apply one point of the com- 
passes to the second term, the other must be turn- 
ed to a less ; and the Contrary^ 

By def. 20. sect. 4, The sine of 90** is equal to 
the radius ; and the tangent of 45** is also equal 
to the radius ; because if one angle of a right 
angled triangle be 45% the other will be also 45''; 
and thence (by the lemma preceding theo. 7. 
sect. 4.) the tangent of 45** is equal to the radius : 
for this reason the line of numbers of 10.000000, 
the sine of 90% and tangentrof 45" being all equal, 
terminate at the same end of the scale. 

The two first statings of this case, answers the 
question without a secant : the like will be also 
made evident in all the follo.wing cases. 

4th. Solution by Natural Sines. 
From the foregoing analogies, or statements, it 



108 TRlGdNOMETRIr. ^ 

is obvious that if the hypoth^Use be multiplied 
by the natural sine of either of the acute angles, 
the product will be the length df the i^ide opposite 
* to that angle ; and multiplied by the natural co- 
sine of the same mgle, the product will be tbe 
length of the other side, or that which is conti^ 
guous to the angle. Thus : 
the given dug. =47* 3d'. 
Nat Sine —.725374 Nat. Cos. =.688355 
Hyp.= 250 260 



ifm 



36268700 34417^50 

1450748 * 1376710 



Perpend. =181.343500 Base= 172.088750 

CASE IL 

The base and angles given f tojindthe /terfiendicuiar and ky* 

pothenuae^ 

Pt. 5. Jig. 5. 

In the triangle ABC there is the angle A 42*^ 
2(/, and of course the angle C 47* 40^ (by cor. 2, 
theo. 5^ and the side AS 190, given^ 4<rflnd JJC 
and AC^ 

1st By Construction. * 

Make the angle CAB (by prob. 16. sect. 4.) in 
blank lines, as fiefore. From a scale of equal parts 
4ay 190 from -4 to jB : on the point jB, erect a per- 
pendicular BC f by prob. 5. vsect 4.) the point 
where this cuts the other blank line oi the angle, 
will be C : so is the triangle ABC constructed ; 
let AC and BC be measured from the same scale 
of equal parts that AB was taken from, and the 

snswers are founds 



TtilGbKonttitttlr. 



109 



2d. By CalcukMon. 

1. Makir^ AC tfu radius. 

8.C : AB : : R : AC. 
S.C : AB : : 8iA : BC. 

That is, as the sine of C =47" 40f 

is to AB, '^ 190 

So is radius = 90* 



to^C 

As the sine of C 

is to AB, 
So is the sine of A 



=47« 40^ 
= 190 
=42»20^ 



toBC, 



=sl73. 1 



9.868785 

2.278754 

10.000000 

12.278754 



2.409969 

9.868785 
2.278754 
9.828301 

12.1070d§ 
2.238270 



2. Making AB the radius. 

i? : AB : : T.A : BC. 
B : AB : : Sec. A : AC. 

That is, zs radius ss 90* 

IS to AB, « 190 

So is the tangent of ^«:42> 20^ 

tojBC, ==173. 1 

As radius »90 

is to AB, =190 

So is the secant of u4=42» 20' 



10.000000 
2.278754 
9.959516 . 

2.238270 
10.000000 

2.278754 
10.131215 



to.4C, 



257 



2.409969 



110 TRIGONOMETRY. 

3. Making BC the radius. 

T. C : AB : : Sec. C : AC. 
T. C : AB : : R : BC. 
That is, as the tangent of C— 47* 4Cf 10.040484 

bto^^, = 190 2.278754 

So islhe Secant of C=47»40f 10.171699 



12.450453 



to^C, «= 257 2.409969 

As the tangent of C^Xl" 40 10.040484 

isto^JS, = 190 . 2.278754 

So is the radius ^^SO" 10.000000 



12.278754 



to BC = 173. 1 2.238270 

Or, having found one of the required sides, the 
•tber may be obtained, by one, or the other of the 
cors. to uieo. 14. sect. 4. 

3d, By Ounter^s Scale: 

1. When AC\b made the radius. 

Extend from 47' 40^, to 90** on the line of sines : 
that distance will reach from 190 to 257, on the 
line of numbers, for AC. 

2. When AB is made the radius, the first stating 
is thus performed : 

Extend from 45* on the tangents (for the tan- 
gent of 45** is equal to the radius, or to the sine of 
90"* as before) to 42** 20' ; that extent will reach • 
JQ^om 190, on the line of numbers, to 173, for jBC% 



TRIGONOMETRY. ill 

3. When BC is made the radius, the second stat-> 
ing is thus performed : 

Extend from 47"* 40' on the line of tangents, to 
45**, or radius ; that extent will reach from 190 to 
173, on the line of numbers, for BC; for the tan- 
gent of 47"* 40^, is more than the radius, therefore 
the fourth number must be less than the second^ 
as before. ' 

The two first statmgs of this case, answer the 
question without a secant. 

4<A. Solution by Natural Sines. 

ABy.R. ABi^SotA 

= ACj and = BC. 

SoiC. Sot a 

Nat. S of* C, side ABxR. 
Thus .739239) 190.000000 (257.02 Src.=AC. 

147.8478 



4215220 
3696195 



5190250 
5174673 

1557700 
1478478 

and, 
.673443= Nat. S. of A. 
190= side AB. 



60609870 



in TRIGONOME'raiY. 

N9t.IS.ofC. 673443 



.739239) 127.954170 (173.09=jBC. 
739239 



^" 



5403027 
5274673 

2283540 
2217717 

■ ■.LI ' 

6502300 
6653151 



CASE III. 



Hu angle't and fierpeiidkular given ; to Jimt the batt and 

hyfiothenute. 

ft. 5. Jig. 6. 

In the triangle ABC^ Ihei:^ is the angle A 40*, 
and consequently the angle C 50% with BC 170, 
given : to find AC and AB. ^ - ^ 

1st. By Construction. 

Make an angle CAB of 40* in blank lines ; (by 
prob. 16. sect. 4.^ with BC 170, from a line of equal 
parts draw the Imea jBiP parallel to AB (by prob. 
8. sect. 4.) the lower line of the angle, and from 
the point where it cuts tihe other line in C, *let 
fall a peq>endicular BC (by prob. 7. sect 4.) and 
the triangle is constructed : the measures of AC 
and ABj from the same scale that BC was taken, 
will answer the question. 



». 



TRIGONOMETRY. ll3 

What has been said in the two foregoing cases^ 
is sufficient to render the operations in this, both 
by calculation, Gunter's scale, and Natural sines, 
so obvious, that it is needless to insert them ; hew- 
ever, for the sake of the learner, we give for 

Answers ; AC 264. 5, and AB 202l 6. ' 

CASE IV. 

The b(ueand hyfiothtnuac given ; to find the angles and fief - 

pendUtUar. 

Ph. 5. fig. 7. 

In tlie triangle ABCy there is given, AB 300 
land AC 500 : the angles A and v, and the per^ 
petidicular BCj are required. 

isL By Construction, a ,. 

From a scale of equal parts lay 300 from A to 
By on B erect an indefinite blank perpendicular 
line, with AC 500, from the same scale, and one 
foot of the compass, in^, cross the perpendicular 
line in C ; and the triangle is .constructed 

By prob. 17. sect, 4, measure the angle A, and 
let be be measured from the same scale of equal 
parts that ^Cand AB were taken from ; and the 
answers arc obtained. 



2d. By Calculation. 

1 . MaJcing AC the radius- 

AC: R::AB: S.C 
R: AC::8.A.BC. 



114 



TRIGONOMETRY. 



That is, as AC 
is to radius. 
So is AB 


= 500 
= 90* 
= 300 

of C;=36« 52* 
5. 90»— 36' 52' = 

90» 

500 

= 53» 08' 

= 400 


2.698970 

10.000000 

2.477121 




12.477121 


to the sine 

By con 2. theo. i 
angle A. 
As radius = 
is to ACy = 
Sbisthesineof^ 


9.778151 
53*08' the 

10.000000 
2.698970 
9.903108 


to BC, 


2.60207a 



2. Making AB the radius. ^ 

AB : R : : AC : sec. A. 
B. : AB : :.T.A : BC. 



That is,98 AB 
is to radius 
So 18-40 



300 
90» 
500 



2477121 

10.000000 

2.698970 

12.698970 



to the secant of A,=^ 53". 08' 10.221849 



As radius — 

is to AB, = 

So is the tangent of A 

to BC, = 



90" 
300 
53*. 08' 

400 



10.000000 

2.477121 

10.124990 

2.6021 1 1 



Or BC may be found from cor. 2. theo. 14. 
sect. 4. 



/ 



TRIGONOMETRY. 115 



3d. By Chmkr^s Scale. 

1. Making AC the radius. 

Extend from 500 to 300, on the line of numbers ; 
that extent will reach from 90^> on the line of sines, 
to 36*. 52^ for the angle C. 

» 

Again, extend from 90* to 53°. 08', on the line 
of sines, that extent will reach from 500 to 400, 
on the line of numbers, (or BC. 

2. Making ^C the radius, the second stating is 
thus performed. 

Extend from rj^dius, or the tangent of 45", to 
53*. 08', that extent will reach from 300 to 400, 
for^a 

4/A. SohUion by Natural Sines. 

R>iAB, ACxSotA. 

— = Soi C; and = BC, 

AC R 

Thus, AC AB 

6,00) 300.0000,00 
.600000 =Nat. sine 36' 52f. 

and, 

Nat. sine of ^ = 53* 8^ = .800034 
^C =r 500 



400.017000 =.5e. 



116 TRIGONOMETRT. 



CASE r. 



I%e Jitrfiendieular and hytiothentue gwefit tojind the anglea and 

base. 



Pi. 5. Jig. 8. 

In the triangle ABC there is BC 306, and AG 
370 given; to find the angles A and C; and the 
base AB. 



1st By ConstructioTL 

Draw a blank line from any point, in which, at 
I By erect a perpendicular, on which lay BC 306, 
from a scale of equal parts : from the same scaled 
with AC 370, in the compasses, cross the first 
drawn blank line in A, and the triangle ABC, is 
constructed. 

Measure the angle A (by prob. 17. sect. 4.) ; and 
also AB, from the same scale of equal parts the 
other sides were taken from, and the answers are 
now found. 

The operations by calculation, the square root, 
Gunter*s scale, and Natural sines, are here omit- 
ted, as they have been heretofore fully explained : 
the statings, or proportions, must also be obvious, 
from what has already been said. 

Answelrs ; The ansjle A 55"* 48' ; therefore the 
angle C 34* 12^, and AB 208. 



TRIGONOMETRY. U"? 

CASE ri. 

The dose and fierfiendkiUar given i to find the angles and 

AyfiotAemue, 

Fl. 5. Jig. 9. 

In the triangle ABC, there is AB 225, and jBC 
272» given ; to find the angles A and C, and the 
bypothenuse AC. 

< 

Ist By Construction^ 

' Draw a blank line, on which lay AB 225, from 
a scale of equal parts ; at B, erect a perpendicu- 
lar; on which lay BC, 272, from the same scale : 
Join A and C, and the triangle is constructed. 
' As before, let the angle Ay and the bypothenuse 
AC be measured ; in order to find th^ answers. 

2d. By Calculation. 
L Making AB the radius. 

AB :R::BC: T. A. 

R. : AB : ; sec. A ' AC. 

2. Making BC the radius. 
BC:R::AB: T. C. 

R.BC:Sec.C:AC. 

By calculation ; the answers from the foregoing; 
proportions are easily obtained, as before. 

But because AC, by either of the said propor- 
tions is found by means of a secant ; and smce 
|here U no line of secants on Gunter's scale ; after 



118 TRIGONOMETRY. 

baying found the angles as before> let us suppose 
* Ai} the radius, and then 

\.8.A:BC: :R,:AC. 
or 2. & C •• AB ' '• a .• AC. 

t 

These proportions may be easily resolved, ei- 
ther by calculation, or Gunter's scale, as before ; 
and thus the hypothenuse ^C may be found with- 
out a secant. 

From the two given sides, the hypothenuse 
may be easily obtained, from cor. 1. theo. 14. 

sect. 4. 

« 

Thus the square of AB = 50625 
Add the square of BC = 73984 

124609 (353 =» AC 
9 ^ 



65)346 
325 

703)2109 
2109 



From what t^as been said on logaritiuns, it is 
plain, 

1. That half the logarithm of the sum of thop 
squares of the two sides, will be the logarithm of 
the hypotlienuse. Thus, 

The sum of squares, as before, is 124609 ; its 
log. is 5.095549, the half of which is 2.547774.; 



TRIGONOMETRY. 



119 



and the correfsponding number to this, in the ta- 
bles, will be 353, for AC. 

2. And that half of die logarithm of the difler- 
ence of the squares of -40 and ABy or of ^C and 
BC, will be the^ logarithm of JBC, or of. AB. 

The following examples are inserted for the ex- 
ercise of the learner. 

1 nv^„ \ the angle C 64" 40' S AB ^„. , 
1. Given, j »^^ 33^^ j ^5,^ required. 



2.GiTen, > 



the ai^le C 47» 20' 
AB 17 



SAC .J 
< D/-Jrequired. 



3. Given, I *'-'"«'|?^,^ H?«,«ired. 



. Given, f 



^B2 

AC ^ 



/the angles ., 
land^C 'eqmred. 



3 Given l- ^^^'^ ( *^® *°8*® Vnuired 

J.uiven,j- ^021.6 land^J3 "?<l""^ect. 

c. n- \ ^8 2871.64 /the angles _^ . , 

6. Given, j ^c 3176.2 landic ^q""^^* 



Tbe answers are omitted, that the learner may 
^eisolTe tliem for himself by the foregoing me- 
thods ; by which means he will find and see more 
distinctly their mutual agreements : and become 
more expert, and Ijetter acquainted with the sub- 
ject. 



(120) 



OBLIQUE AlfGLED 



PLANE TRIGONOMETRY. 



B 



EFORE we proceed to the solution of the 
four cases of Oblique angled triangles, it is neces^ 
sary to premise the following theorems. 

THEO. L 

Pl. 5. Jig. 10. 

Jrt any plfLWe triangle ABC^ the side^ are ftroJtortioruU to the 
mnea qf their o/ifioMite angles ; that w, ^. C : AB : : 8, J : 
jBCjandS. C : AB ; : S. B : AC; also S. B : AC : : S. A : 
BC. 

By theo. 10. sect. 4. the half of each side is the 
sine of its opposite angle ; but the sines of those 
angles, in tabular parts, are proportional to the 
sines of the same in any other measure; and there- 
fore the sines of the angles will be as the halves 
of their opposite sides ; ahd since the halves are as 
the wholes, it follows, that the sines of their an» 
gles are as their opposite sides; that is, & C:: 
AB :: S. A : BC, SCc. Q. E. D. 

THEO. 11. 

JTig' 11- 

hi any fUane triangle ABC^ the sum of the two given aides AB andt 
BC^ including a givcii angle ABC, is to their dWerence^ aa the 
tangent (fhalf the aunt of the two unknown ongTea A and Cia t^ 
the tangent (jif half their difference. 

Produce AB^ and make JfB= BC, and joinlfC; 
«)et fall the perpendicular BEy aind tliat will bisect 



I^RIGONOMETRY* liJl 

the Angle HBC (hy theo. 9. sect 4.) through B 
draw SD parallel. ioAC,. and make HF ^DC^ 
and join BF; take BI^BA, and draw IQ paral- 
lel to BD or AC. ' 

It is then plain that AH will be the fium, and 
J37 the difference of the sides AB and BC ; and 
nince JIB=BC, and BE perpendicular to HC, 
therefore HE^ EC (by theo. 8. sect. 4.) ; and since 
BA=BI, and BD and IG parallel to ACy therefore 
GD^DC^FHy and consequently irG=FD,and 
iHG=|F2> or ED. Again, EBC being half 
HBUy will be also half the sunoi of the angles^ and 
C (by theo. 4. sect. 4^ also, since JETB, HFy and 
the included an^le H, are severally* equal to BC, 
CDi and the inauded angle BCD : therefore (by 
theo. 6. sect. 4.) HBF=DBC=BCA (hy part 2. 
theo. 3. sect. 4.; and since HBD^A (oy part. 3. 
theo. 3. sect. 4.) and HBF^ BCA : therefore BFD 
is the difference, and EBDy half the difference of 
the angles A and C : then making BE the radius, 
it is plain, that EC will be the tangent of half the 
sum, and ED the tangent of half the difference of 
the two unknown angles A and C : now lO being 
parallel to AC; AH: IH: : CH: GH. (by cor. 1. 
theo. 20. sect. 4.) But the wholes are as their 
halves, that is, AH: IH: : CE : ED, that is as the 
sum of the two sides AB and JBC, is to their differ- 
ence ; so is the tangent of half the sum of the two 
Unknown angles A and C, to the tangent of half 
their ditference. Q« E. D^ 



R 



1 22 TRIGONOMETRY. 



THEO. m. 



Fig. 12. 



In any right lined plane triangle ABD ; the bate AD mU he to the 
9umqfthe other Mes^ Ao^ BD^ aa the difference ^ thoae mde9 
is to the difference (if the segments of the o(ue^ made by the per-- 
pendktdcar BE; vis. the d^crence between AE and £D* 



Produce JBD, till BO=AB the lesser leg; and 
on jB as a centre, with the distance BG or BA, 
describe a circle AGHF; which will c\xi BDy and 
AD in the points JEf and F; then it is plain, that 
GD will be the siiin, and HD the difierence of 
the sides AB and BD; also since AE^EF (by 
theo. 8. sect 4.) therefore, FD is the difference of 
AE ED, the segments of the base ; but (by theo. 
17. sect. 4.)^1D : GD : : HD : FD ; that is, the 
base is to the sum of the other sides, as the differ- 
ence of those sides is to the difference of the seg* 
ments of the base. Q. E. D* 



THEO. IV. 



Fig. 15. 

If to hay the nan qf two guantUieet be added half their d^er^oce; 
the sum vrill be the greateat of them $ and if from half the eum be 
aubtracted half their difference ; the remainder tvill be the leaat qf 
them. 

Let the two quantities be represented by AB 
and BC : (making one continued line ;) whereof 
AB is the greatest, and BC the least ; bisect the 
^hole line AC in E ; smd make AD^BC; then 



TRIGONOMETRY. 123 

it is plain, that AC is the sum, and DB the differ- 
ence of the two quantities ; and AE or ECy their 
half sum, and DE or EB their half difTerence. 
Wow if to AE we add EB, we shall have AB the 
greatest quantity ; and if from EC we take EB^ 
we shall have BC the least quantity. Q. E. 2>. 

Cor. Hence, if from the greatest of two quanti- 
ties, we take half the difference of them, the re- 
mainder will be half ttieir sum ; or if to half their 
difference be added the least quantity, their sum 
will be half the sum of the two quantities. 



OBLiaUE ANGLED TRIANGLES. 



CASE L 



TWO Me9% and an angle ofifiomte to one .of them given ; to Jmd 

the other angles and me. 



Fl. S.Jig. 11. 

In the triangle ABC^ there ia given AB 240, tht angle A 46"* 30^^ 
and BC 200 ; tojind the angle C| bemg acute, the angle B, and 
tfie Me AC. 

1st By Construction. 

Draw a blank line, on which set AB 240, from 
a scale of equal parts ; at the point Ay of the line 
AB, make an angle of 46* 30 , by an indefinite 
blank line ; with BC 200, from a like scale of equal 
parts that AB was taken, and one foot in jB, des* 
cribe the arc DC to cut the last blank line in the 
points D and C. Now if the an^le C had been re- 
quired obtuse, lines from D to B, and to -4, would 
constitute the triangle; but ab it is required acute. 



1S4 TRIGONOMETRY- 

draw the lines from O to J3 and to A^ and the tri^ 
single ABC is constructed. From a line of chorda 
let the angles B and C be measm-ed ; and AC 
from the same scale of equal parts that AB and 
jBC were taken ; and you will have the answers 
required, 

2df. By Cakv^im. 

This ifi performed by theo. 1. of this sect- 
thus; 

AbBC c= 200 / 2-301030 

is to the sine of .^ = 46^ 30V 9.860562 

Sois^JB 8 240 2-380211 



12-240773 
to the sfaie of C, « 60*. 31' 9.939743 

180* — the sum of the angles A and C, vill^ire 
the an^le B, hy cor. 1. theo. 5, sect 4. 
A 46». 30^ 
C 60. 31 



180»— 10T». l'=72*. 59'=B. 

As the sine of ^ — 46*. 30^ 9.860562 

istoiSC, = 200 2.301030 

So is the sine of B a 72<'.d9^ 9.980555 



12.281585 



— — /* 



to AC, => 263. 7 2.421023 

3d. By Omier's Scak, 

Extend from 200 to 240, on the line of numbers ; 
tint distimce wiH iieach from 46" 30' on the liiie 
tii sine% to €©• 31' forth^e angle C<, 



TRIjGONOMETRY. 126 

Extend from 46* 30', to 72* 59', on the line of 
siiies ; that distance wUl reach from 20Q to 263.T 
on the luie of numbers, for AC. 

Note. The method by Natural Smes will be ob- 
vious from the foregoing analogies. 

•»• • - . ' 

CASE II. 

Tiv# angUa and a Me given ; tojind the other eidcM. 

PL.5./ig. 15. 

In the trUmgle ABCy there ie the imgle A 46« 30' AB 230i 
emd the angle 3 37* 30', given tojind AC and BC. 

1st. By Constructiofi. 

Draw a blank line, upon which set AB 230, 
from a scale of equal parts ; at the point ^ of the 
line ABy make an angle of 46* 30', by a blank 
line ; and at the point B of the line AB make an 
angle of 37* 30', by another blank line : the inter- 
action of those lines gives the pomt C, then the 
triangle ABC ia constructed. Measure AC and 
BC from the same scale of equal parts that AB 
was taken ; and you have the answer required. 

2dk By Cakvlalian. 

By («or. 1. theo. 5.'sect. 4.) 180" — the sum of the 
angles A and B—C. 
A 46" 30' 
B 37. 30 



J 80»— 84\ 00'=96« 00^ = C. 



126 



TRIGONOMETRY. 



By de£ 27. sect 4. The sine of 96''3=:the sine 
of 84^ which is the supplement thereof; therefore 
instead of the sine of 96% look in the tables for 
the sine of 84% 



j^theo. 1. of this sect. 


• 




1 


As the sine of C = 
is to ABf = 
So is the sine of ^ ^ 


96* 00* 

230 

46*30' 

• 

167.8 


. 9.997614 
2.361728 
9.860562 


1 




12.222290 




to BC, «= 


2.224676 


• 


As the sine of C =« 
is to -4 B, 5= 
So i^ the sine of J3 =» 

• 


96* 00' 

230 

37* 30' 


9.997614 
2.361728 
9.784447 


1 




12.146176 





i(iACy 



140.8 



2.148561 



3d By Gunter^s Seak, 



Extend from 84* (which is the supplement of 
96"") to 46^ 30^ on the sines ; that distance will 
reach from 230 to 168, on the line of numbers, for 

Ba 



Extend from 84'* to 37*. 30^, on the sines ; that 
extent will reach from 230 to 141, on the line of 
numbers, for AC. 



TRIGONOMETRY. ^27 



CASE III 



« fyBMeaanda contained angle gtven ^ tQ JStuf the ^ther gngkf 

and Me * 



Pl. 5. ^g. 16. 

lA the triangle ABC, there ie AB 240» the angle A S6^ W andAp 
180^ given ; to find the anglee € and B, and the Me MC. 



XsL By Crnistructiotts 

Draw a blank line, on which from a scale of 
equal parts, lay AB 240 ; at the point A of the 
line AB, make an angle of 36'' 40% by a blank 
line ; on which from A, lay AC 180, from the 
same «cale of eq|ual parts ; measure the angles C 
and B, and the side JBC, as before ; and you have 
Hie answers required. 

2d. By Caknlatian^ 

By cor. 1. theo. 6. sect. 4. 180« — the angle A 
36\ 40' r= 143*. 20' the sum of the angles G and 
B : therefore half of Uy. 20', will be half the 
9um of the two required angles, C and B. 

By theo. 2. of this sect. 

As the sum of the two sides AB and AC =» 420 
is to their' difference, « 60 

So is the tap^ent of half the sum of ) _ 710 4rv* 
the two unknown angles C and B ) ~ 
to the tangent of half their difference = 23* 20' 



128 TRIGONOMETRY. 



By theo. 4. 



To half the sum of the angles C and B=^7V iOf 
Add half their difference as now found = 23 20. 



The sum is the greatest angle, or ang. C=95 00 

Subtract, an4 you have the le^st angle, or J3=s4820 

• . 

The angle C and B being found ; BC Is had, as 
before, by theo, 1. of this sect. Thus, 

a.B.AC::8:A:BC. 
48* 20: : 180 : : 36- 40 : 143. 9. 

Sd. By QunUr's Scale. 

Because the two first terms are of the same kind, 
extend from 420 to 60 on the line of numbers ; 
lay that extent from 45"* on the line of tangents, 
and keeping the left leg of your compasses fix- 
ed, move the right leg to 7 P. 40'; that distance 
laid from 45*" on the same line wiU reach to 23*. 
30', the half difierence of the requinBd angles. 
Whence the angles are obtained, as before. 

The second proportion may be easily extended^, 
from what has been already said. 

CASE IK 

Ft. S.Jig. 17. 

The three Mes gruen^ to find the angles, 

if} the triangle JBC, there is given, AB 64, AC A7y BC 34 ; the 

angles A^ By C, are required. 



•" TRIGONOMETRY. 129 

1st. By Cfmstrmtiim. 

The construction of this triangle must be mani- 
fest, from prob. 1. sect. 4. 

2df. By Calculation. 

From the point C, let fall the perpendicular CD 
on the base AB ; and it will divide the triangle 
into two right angled ones, ADC and CBB ; as 
well as the base AB^ into the two segments, AD 
and DB. 

AC 47 
BC 34 

Sum 81 

Difference 13 y 



By theo. 3. of this sect. 

As the base or the longest side, AB 64 

is to the sum of the other sides, ^Cand BC, 81 

So is the difference of those sides 13 

to the difference of the segments of ) laAA 

the base AD DB. ^ ^^'^^ 

By theo. 4. of this sect. 

To half the base, or to half the sura ) ko 

of the segments AD and DB. . ) 
Add half their difference, now foundi 8.23 

Their sum will be the greatest segment AD 40.23 



^ 



130 TRIGONOMETRY. 

Subtract, and their difference will be ) 03 72 
the least segment DBy \ 

In the right angled triangle ^l^C^ there is ^C47» 
and AD 40. 23» given, to find the angle A. 

This is resolved by case 4. of right angled plane 
trigonometry, thus, 

AD : i? : : AC : Sec. A 
40. 23 : 90^ : : 47 : 3P 08' 



Or it may be had by finding the angle ACD, 
the complement of the angle A j without a secants 
thus. 



AC.R: :AD:S.ACD. 

4|:90V :40 23 : 58^52' 

90 — 58« 52'= 31«. 08', the angle A. 

Then by theo. 1 . of this sect. 

BC:8.A: :AC:S.B. 
34 : 31* 08' : : 47 : 45" 37. 



By cor. 1. theo. 5. sect 4. 180* — the sum of ^ 
and B=C. 



A 3P. 08' 
£45. 3T 



180»— 76. 45=103«. 15', the angle C 



TRIGONOMETRY. 131 

3d. By GwKter's Scale. 

The first proportion is extended on the line of 
numbers ; and it is no matter whether you extend 
from the first to the third, or to the second term, 
since they are all of the same kind : If you extend 
to the second, that distance applied to the third, 
will give the fourth ; but if you extend froffi the 
first to the third, that extent will reach from the 
second to the fourth. 

The methods of extending the other prQportions 
have been already fully treated of 

jdn examfile in each caae of obBque angled trianglfs, 

AC 290^ A 
1. Given, ^ C69 •30' >B requu^ed. 

AB 350 SBC 



C C . 24^ Ha ^AB 
r. Given, < JB 128*. 30 V required. 
(AC 32463 5C 

« . 

AC 6 "^ A 

3. Given, ^C 124».30'VB required. 

BC 4. 5 Sab 



AB A6)A 
4. Given, < AC 92> B required. 

BC 52) C 



/ 



13» TRIGONOMETRY. 



Additional Exercises with their Answers. 



aUESTIONS FOR EXERCISE- 

1. Given the Hypothenuse 108 and ibe Angle 
opposite the Perpendicular 25'' 36 ; i^uired the 
pd^ and Perpendicular* 

Answer. The Base is 97.4, and the Perpendi- 
cular 46.66. 



2* Given the Base 96 and its opposite Angle 7P 
45' ; required the Perpendicular and the Hypo- 
thenuse. 

Answer. The Perpendicular is 31.66 and the 
Hypci|;henu8e 101.1 • 

3. Given the Perpendicular 360 and its opposite 
Angle 58* 20' ; required the Base and the Hypo- 
thenuse. 

Answer. The Base is 222, and the Hypothec 
nuse 423. 

4. Given the 5ase 720 and the Hypothenuse 
980 ; required the Angles and the Perpendicular. 

Answer. The Angles are 47^/. and 42M3', 
and the Perpendicular 664.8 

5. Given the Perpendicular 110.3 and the Hy- 
pothenuse 176.5; required the Angles and the 
Base. 

Answer. The Angles are 38U1' and 51M9', 
and the Base 1 37.8. 

G. Given the Base 360 and the Perpendicular 
480] required the Angles and the Hypothenuse. 



TRIGONOMETRY. 133 

Answer. The Angles are 53* tf and SG** 52^, 
and the Hypotfaenuse 600. 

7. Given one Side 1 29, an adjacent Angle 56* 
30 , and the opposite Angle 81' 36' : required the 
third Angle and the remaining Sides. 

Answer. The third Angle is iP 54', and the 
remaining Sides are 108.7 arid 87.08. 

8. Given one Side 96.5, another Side 59.7, and 
the Angle opposite the latter Side 3P30' : requir- 
ed the remaining Angles and the third Side. 

Answer. This Question is ambiguous ; the given 
Side opposite the given Angle being less than the 
other given Side (see Rule I. ;) hence, if the Angle 
opposite the Side 96^5 be acute, it will be 57* 38', 
the remaining Angle 90* 52^, and the third Side 
114.2 ; but if the Angle opposite the Side 96.5 be 
obtuse, it will be 122*" 22?, the remaining Angle 
26* 8', and the third Side 50.32. 

« 

9. Given one Side 1 10, anottier Side 102, and 
the contained Angle 11 3*" 36 : required tlie remain- 
ing Angles and the third Side. 

Answer. The remaining Angles are 34"* 37' and 
3P 4r, and the third Side is 177.5. 

10. Given the three Sides respectively, 120.6, 
125.5, and 146.*^ : required the Angles. 

Answer. The Angles are 5P 53^, 54^ 58', and 
73*9'. 

The student, who has advanced thus far in this 
work with diligence and active curiosily, is now 
prepaid to study, with ease and pleasure, tlte fol- 
lowing part; which comprehends all the necessary 
directions for the practice of Surveying. 



(134) 



PART ir. 



Or the Practical Surveyor's Ouide. 



SECT. I. 



Containing afiarticular Detcrifttion nfthe »everat ItulntmenU 
a»ed in Survetfing, mth their retfiective V*et, 



THE CHAIN, 

JL HE stationary distance, or nierings of gi'ound, 
are measured either by Gunter's chain of four 
poles or perches, which consists of 100 links ; 
(and this is the most natural division) or by one 
of 50 links, which contains two poles or perches : 
but because the length of a perch differs in many 
places, therefore the length of chains and their 
respective links wiU differ also. 

The English stattUe-perch is 5i yards, the two- 
pole chain is 11 yards, and the four-pole one is 22 
yards ; hence the length of a link in a statute- 
chain is 7«92 inches. 

There are other perches used in difierent parts 
of England, as the perch of woodJ^md mtasure, 
which is 6 yards; that of church-land measure^ 
which is 7 yards, and the forest measure perch, 
which is 8 yards. 



OPTHECHAJfiS. , 1^ 

m 

For the more ready reckoning the links of a 
fouF'pole chain, there is a large ring, or sometimes 
a round piece of brass, fixed at every 10 links ; and 
at 50 links, or in the middle, there are two large 
rings. In such chains as have a brass piece 
jit every 10 links, there is the figure 1 on the first 

i>]ece, 2 on the second, 3 on third, SCc. to 9. By 
eading therefore that end of the chain forward 
which has the least number next to it, he who car- 
ries the hinder end may easily determine any nun>* 
ber of links : thus, if he has the brass piece number 
8, next to him, and six links more m a distance, 
that distance is 86 links. After the same manner 
10 may be counted for every large ring of a chain 
which has not brass pieces on it ; and the number 
of links is thus readily determined. 

The two-pole chain has a large ring at every 10 
links, tnd in its middle, or at 25 links, there are 2 
l^rge rings ; so that any number of links may be 
Ae more readily counted ofjf^ as before. 

The surveyer should be careful to have his chaia 
measured before he proceeds on business, for the 
rin^s are apt to open by frequently using it, and 
its lenglh is thereoy increased, so that no one can 
be too circumspect in this point. 

In measuring a stationary distance, there is all 
object fixed in the extreme point of the line to be 
measured ; this is a direction for the hinder chain- 
man to govern the foremost one by, in order that 
the distance may be measured in a right line ; for 
if the hinder chainman causes the other to cover 
llie object, it is plain the foremost is then in a right 
line towards it. For this reason it is necessarj' to 
have a person that can be relied on, at the hinder 



136 OF THE CHAIN. 

end of the chain, in <H*der to keep the foiK^iost 
roan in a right line ; and a surveyor who has no 
such person, should chain himself. The inaccura* 
cies of most surreys arise from bad chaining, that 
is, from straying out of the right line, as well a&from 
other oimssions of the hinder chainmaii : no per-, 
son, therefore, should be admitted at the hinder 
end o[ the chain, of whose abilities in this respect, 
the surveyor was not previously convinced ; since 
the success of the survey, in a gi^est measurei de-- 
pends on his care and skill. 

In setting out to measure any stationary distance, 
the foreman of the chain canies with him 10 iron 
pegs pointed, each about ten inches lon^ ; and 
when he has stretched the chain to its full length, 
he at the extremity thereof sticks one of those pegs 
perpendicularly in the ground ; and leaving it 
there, he draws on the chain till the hinder man 
checks him when he arrives at that peg : the chain, 
being again sti^tched, the fore man sticks down 
another peg, and the hind man takes up the former; 
and thus they proceed at every chain's length con- 
tained in the Ime to be measured, counting the sur- 
plus links contained between the last peg, and the 
object at tlie termination of the line, as before : so 
that, the number of pegs taken up bv the hinder 
chainman, expresses the number oi chains ; to 
which, if the odd links be annexed, the distance 
line required in chains and links is obtained, which 
must be registered in the field book, as will hereaf- 
ter be shewn. 



If the distance exceeds 10, 20, 30, SCc. chains, 
when the leader's pegs are all exhausted, the hind- 
er chainman, at the extremity of the 10 chains, 
delivers him all the pegs ; from whence they pro- 



Of tut OHAm* m 

ceed tp measure as before^ till the leader's pegs arq 
again exhausted, and the hinder chainman at th^ 
extrepiity of these 10 chaind again, delivers him the 
pegs ; from whence ihey proceed to measure the 
whote distance line in the like manner ; then it is 
plain, tlmt the number of pegs the hinder chainman 
bas, being added to 10, it he had delivered all the 
pegs once to the leader, or to 20 if twice, or to 30 
if wrice, SCc^ wU) give the number of chains in thai 
distance ; to which if the surplus links be added^ 
ihe length of the stationary distance is kaown in 
chains and links* 

It is customary, and indeed necessary, to have 
red, or other coloured cloth, fixed to the top of each 
peg, that the binder man at the chain may tlie more 
readily find them ; otherwise, in chaining through 
corn, high grass, briars, rushes, SCc. it would be ex* 
tremely difficult to find the pegs which the leader 
puts down : by this means no lime is lost, which 
otherwise n^ust be, if no cloths are fixed to th^ 
pegs» as before. 

It will be necessary here to observe, that all 
slant, or inclined surfaces, as sides of hills, ar^ 
measured horizontally, and not on the plane or 
furface of the hill, and is thus effected. 

• 

liet ABChe a hill, the hindmost chainman is td 
hold the end of the chain perpendicularly over the 
point A (which he carithebettereffectwithaplum- 
met and line, tlian by letting a stone df op, which 
is most usual) as d is oyer A, while the leader puts 
down his peg at e : the eye can direct the horizon- 
tal position near enough, but if greater adeuraay 

T 



138 01? THE CHAIN. 

Were required, a quadrant applied to the ehain^ 
Would settle that. In the same manner the rest 
may be chained up and down ; but in going down, 
it is plain the leader of tlie chain must hold up the 
end thereof, and the plummet thence suspended, 
will mark the point where he is to stick nis peg. 
The figure is sufficient to render the whole evident ; 
and to shew that the sum of the chains will be the 
horizontal measure of the base of the hill; for 
de=Aojfg=op, hi=pqy SCc. therefore dexfgy^M, 
SCc.^=:Aoxapxpq9 SCc. = AC^ the base of tne hilL 
If a whole chain cannot be carried horizontally, 
half a chain, or less, may, and the sum of these 
half chains, or links, wUl give the base, as before* 

If the inclined side of the hill be the plane sur- 
face, the angle of the hill's inclination may be ta- 
ken, and the slant height may be measured on the 
surface; and thence ^y case 1. of right-angled 
trigonometry^ the horizontal line answering to the 
top, may be lound ; and if we have the angle of 
inclination given on the other side, with those al- 
ready given ; we can find the horizontal distance 
across the hill^ by caie 2. of oblique trigonomeiry. 

All inclined surfaces are considered as horizon- 
tal ones ; for all trees which grow upon any inclined 
surface, do not grow perpendicular thereto, but to 
the plane of the horizon : thus if Ad^ ef^ gh, SCc. 
were trees on the side of a hill, they grow per- 
pendicular to the horizontal base ACy and not to 
the surface AB: hence the base wHl be capable to 
contain as many trees as are on the surface of the 
hill, which is manifest from the continuation of 
them thereto. And this is the reason that the area 
of the base of a hill, is considered to be equal in 
value to thQ hill itself. 



OF THE CHAIN. 13^ 

Beindes, the irregularities of the surfaces of hills 
iri gieneral are such, that they would be found im- 
possible to be determined by the most able mathe* 
maticians. Certain regular curve surfaces have been 
investigated with no small pains, by the most emi- 
nent ; therefore an attempt to determine in general 
the infinity of irregular surfaces which oflTer them- 
selves to our view, to any degree of certainty, 
would be idle and ridiculous, and for this reason 
also, the horizontal area is only attempted. 

Again, if the circumjacent lands of a hill be 
planned or mapped, it is evident we shall have a 
plan of the hill^ base in the middle : but were it 
possible to put the hill's surface in lieu thereof, it 
would extend itself into the circumjacent lands, 
and render the whole an heap of confusion : so 
that if the* surfaces of hills could be determinedj 
no more than the base could be mapped 



'S 



Roads are usually measured by a wheel for that 
purpose, '^alled the Perambulator, to which there 
is fixed a machine, at the end whereof there is a 
spring, which is struck by a peg in the wheel, once 
in every rotation ; by this means the number of 
rotations is known ; if such a wheel were 3 feet 4 
inches in diameter, one rotation would be lOi feet^ 
which is half a plantation perch ; and because 320 
perches make a mile, therefore 640 rotations will 
be a mile also ; and the machinery is so contrived, 
that by means of a hand, which is carried round 
by the work, it points out tlie miles, quarters, and 
perches, or sometimes the miles, furlongs, and 
perches. 

Or roads may be measured by a chain more ac- 
curately ; for 80 four-pole, 160 two-pole chains, or 
320 perches, make ft n;iile as before : and if ro^ds 



14tt O? THE GHAIN- 

are measured by a statute<K;hain, H will ^ve you 
tile miles English, but if by a plantation chain, the 
miles will be Irish. Hence an English mile con* 
tains 1^760^ and an Irish mile SQ40 yards ; and be- 
cause 14 naif yards is an Irish, and 1 1 half yards k 
$n English perch^ therefore 1 1 Irish perches^ ot 
Irish miles, are equal to 14 English ones. 

Since some surveys are ta^en by a fbur-pole, and 
others by a two-pole chain ; and as ground fbir 
houses is measured by feet, we will shew how to re- 
duce one io the other, in the following problems* 

PnOB. I. 

« 

To rfdttce tw^pfiU cfynm an4 Unt9 tofour»^oU oncf. 

If the number of chains be even, the half o£ 
^em will be the four^ole ones, to which annex 
the given linkfi, thus, 

1« In 16, 37 i>ftwQrpo)ecfafun3>bpw2iKwy four- 
fold opes ? 

Apswer 8^ 37. 

But if the number of chains be o^d, take the 
half of them for chains, and add 50 to the links» 
and they will be four-pole chains apd links, thus* 

2. In 17. 42 of two-pole chams, how many 
fpur-pole ones ? 

Answers. 92» 



«^ THE CHAIN. Hi 

PBOB. B. 

Double the chains, tp which annex the Vix^, if 
they be less than 50 ; but if they exceed 5<^, dou- 
ble the chains, add one to them, and take 50 from 
the links, and the remainder will be the links, thus, 

1. Iij8. 37 of fdur-pote chains, how many 
2. twQ-pole ones ? 

16. 37 



2. In 8, 82 of four-pole chains, how nmny 
2. 50 two-pole ones ? 

17. 32 Answer, 



T0 rtdueefcur^polt ehaina and Unkg, tQ fierchea, attd deeinigls 

qfa^ertk 

The linksof a four-^pole chain are decimal parts 
of it, each link before the hundreth part of a 
diain ; therefore if the chain and links be multiplied 
by 4, (for 4.perches are a chain) the product will he 
the perches and decimal pwts of a perch* Thus, 

Ch* JLr. 

How many jperchee in 13* 64 o^ femr-pole 

Answer 54, 56 perches* 



142 OP THE CHAIN. 



PROB. 

ToftdttceiiifO'iioleckaimandlini^ to fier'eh€9 and decimah ^ 

a perch. 

They may be reduced to four-pole ones (hy 
prob. 1.^ and thence to perches and decimals (by 
the last,; or. 

If the links be multiplied by 4, carrying one to 
the chains, when the links are, or exceed 25 ; and 
the chains by 2, adding one, if occasion be i the 
product will be perches, and decimals of a perch* 

Thus, 

> 

Ch. L. 
1. In 17. 21 of two-pole chains, how many 
2, 4 perches. 

Answer, 34. 84 perchear. 



Ch. L. 

^. In 15. 38 of two-pole chains, how mnay 

2. 4 perches. 
• * " • • 

Answer, 3f . 52 perches. 



PROB. V. 

To reduce fierche$f anddecimaU of afierch^ tofour^tiole chaitu 

and Unka. 

m 

Divide by 4, so as to have two decimal places 
in ihe quotient, and that will be four^pole chains 
and links. Thus, 



OP THE CHAIN. 143 

Iq 31. 52 perches, how many four-pole chains 
^nd links ? 

Ch. L, 

4)31.52(7. 88 Answer. 

35 



32 



PROB. VI. 



To reduce fierchee and decimal* of dperchj to two-fioU chain* 

and Itnk; ' 



The perches mav be reduced to four-pole chains 
(hy the last) and uova thence to two-pole chains 
(hy prob. 2.) or. 

Divide the whole number by 2, the quotient will 
be chains ; to the remainder annex the given de- 
cimals, and divide by 4^ the last quotient will be 
the lirJ^s. Thus, 

In 31.52 perches, how many two-pole chains and 
links? 

Ch. L. 
2)31.52(15. 38 Answer. 

11 



4)152(38 
32 

■ H »«iiii<i » 



> 



m of THE CHAIN, 

pitod. vii. 

To reduce cbaina and 6'izf «> to /het and decimal fiarte qf a 

foot* 

If they be two-pole chains, reduce them to foup- 
pole ones : (by prob. L) these beins multiplied by 
the feet in a four-pole chain> Will give the feet and 
decimals of a foot. Thus» 

Ch. L. ' 

In 17« 21 of t>vt>^pole diaiiis^ how many feet ? 

Ch* L* 

8. 1^1 of fottr-pote chains. 
66 feet = 1 chain. 



■«airta«taMriMi 



$226. Feet Inches 

522A Answer 574. lOi. 



F«et 574.86 
12 



>■■■*! 



Inches 10.32 
'T* 4 • 



<i. i 

1.28 



PBOB. riii 

I 

To reduce feet and inches to chaine ^nd links* 

Reduce the inches to the decimal of a foot> and 
annex that to the feet ; that divided by the feet in 
afoui^Ale chain, will give the fottr-pble chains and 




OP THE CHAIN. J45 

liid^s in, the quotient : these may be reduced to 
two-pole chains and links^ if required^ by prob« 2t 
Thus, 

Feet. Inches. 
In 217. 9 how many two-pole chains? 
12)9.00.(75thed6Ginialof 9 inches. . 

60 



6iB)217.75(3. 29 of four-pole ich^ins^ or 

197 

■ ■ Cw. Xf. 
655 6. 29 



61 



«i*i 



How to lake a 8urv^ by th$ Chain only. 



PROS. L 



To nirvey a fUeee qf ground^ by giving round U^ and the me* 
thod if taking the angie$ of the Jietdy by the chain only. 

FL*6.Jig.6. 

Let ABCDEFG be a pieJe.of ground to be sur- 
veyed : beginning at the point ^, let one chain be 
laid in a direct line from A^ towards Cr, where let 
a peg be left, as at c ; and again, the like distance 
from ^ in a direct line towards B\ where another 
peg is'also to be left, as at d: let the distance from 
«I to c be measured, and placed in the field-book, in 



146 OF THE CHAIN. 

the 8ec6nd coIuhib under the denoaiinatioo of aa^ 
glesy in a line with station No. 1 ; and in the same 
Une, under the title of distances, in the third column , 
let ihe measure of the line AB in chains and links 
be inserted. Bein^ now arrired at J3, let one chain 
be laid in a dnrect line from JEf towards A, where let 
a peg be left, as at /, and again, the like distance 
from J3 in a direct line towards C, where let also 
another peg be left, as ate ; the distance from c to 
/ is to be inserted in the field-book in the second 
column, under angles, in a line with station No. 2 ; 
and in liie same bne, under the title of distances 
in the third column, let the measure of the line 
BC, in chains and links, be inserted : after the 
same manner we may proceed from C to !>, and 
thence to E; but because the angle at i5, viz. FED, 
is an external angle, after having laid one chain 
from E to A, and to g, the distance from g to hit 
measured, and inserted in the column of angles, in 
a line with station No. 5. and on the side of the 
field-book against that station, we make an asterisk, 
thus *, or any other mark, to signify that to be an 
external angle, or one measured out of the ground. 
Proceed we then as before, from JB to F^ to O, and 
thence to Ay measuring the angles and distances, 
and placing them as before, in the field-book, oppo- 
site to their respective stations ; so will the neld- 
book be completed in manner following. 

N. B. After this manner the angles for inac- 
cessible distances not^y be taken, and the method 
of constructing or laying them down, as well as 
the construction of the map, from the following 
field-notes, must be obvious from the method of 
taking them. 

The form of the field-book, with the titie. . 



OP THE CHAIN. 



147 



A field-book oi part of the land of Grai^e, in the 
parish of Portmarnock, barony of CooTock, and 
county of Dublin ; being part of the estate of 
L. P. Esq. let to C. B. &rmer. Surveyed Janu- 
ary 30, 1782. 

Taken by a four-pole chain. 



*^» 



Remarks. 



Mr. J . D'a part ol Grange 

Mr. L. P's part of Portmar- 

nock strand 

Widow J. G's part of Grange 




Distan. 
Ch. L. 

" 17.66 
18.50 
28.00 
20.00 
14.83 
19.41 
24.53 



CiloM at Um first ■tation. 



Explanation of the remarks. 

Mr. J. D's part of Grange bounds, or is adjacent 
to the surveyed land from ihe first to the third sta- 
tion ; Mr. L. P's part of Portmarnock bounds it 
from the third to the fourth station ; the strand 
then is the boundary from thence to the sixth, and 
from the sixth to the first station, the widow J. G's 
part of Gi*ange is the boundary. 

m 

It is absolutely necessary to insert the persons* 
names, and town-lands, strands, rivers, bogs, ri- 
vulets, SCc. which bound or circumscribe the land 
which is surveyed, for these must be expressed in 
the map^ 



In a survey of a town-land, or estate, it is suffi- 
cient to mention only the circumjacent tewn-land?^ 



148 OF THE CHAIN. 

without the occupiers' names : but when a part 
only of a town-land is surveyed, then it is neces- 
sary to insert the person or persons' names, who 
hold any particular parcel or parcels^ of such town- 
land, as bound the parts surveyed. 

When an angle is very obtuse, as most in our 

Present figure are, viz. the angles at Ay B, Cy 
?, and G : it will be best to lay a cbam from the 
angular point, as at Ay on each of the containing 
sides to c and to d ; and any where nearly in the 
middle of the angle, as at e : measuring the distan* 
ces ce and ed; and these may foe placed for tiie 
angle in the fi^ld-bpok. Thus, 



No. Sta. Angle. 

Ch. L. Ch. L. 
1.03) 



17.65 



Pot when an anglb is very obtuse, the chord 
line, as erf, will be nearly equal to the radii Ac 
and Ad; so if the arc ced be swtept, and lhe"*chord 
line tdhe laid on it, Jt will be difficult to determine 
exactly that point in the arc where ed cuts it : but 
if the angle be taken in two parts, as re, the arc, 
and the angle thence, may pe truly determined 
and constructed. 



After the same manner any piece of ground 
may be surveyed by a two-pole chain, ' 



OP THE CHAIN. 



149 



PROS. II. 



To take a turvey of a fdeee of ground fitm any fioint vritUn 
itt from nhenet alt the aiglet can be teen t iy the ekmn 
only, 

"L> 6. Jig, 6. 

Let a mark be fixed at any point in the ground, 
as at Ht from whence all the angles can be seen ; 
let the measures of the lines Ai, HB, HC, SCc. 
be taken to every angle of the field from the point 
H; and let those be placed opposite to No. 1, 2, 
3, 4, ^c. in the second column of the radii : the 
measures of the respective lines of the mearing, 
vix. AB, BC, CD, bS, SCc. being placed in the 
third column of distances, will complete the field- 
book. Thus, 



Remarks. 



No 



1 
2 
3 
4 
5 
6 
7 



Ch.L. 



Distan. 
ICb. L. 



20.00 
21.72 
21.74 
25.34 
17.20 
29.62 
21.20 



17.65 
18.50 
28.00 
20.00 
14.83 
19.41 
24.53 



CloK at tbe fint «Utia|i. 

If any line of the field be inaccessible, as sup- 
pose CD to be, then by way of proof that the 
distance CD is true, let the measure c^ tbe angle 
CHD be taken by tbe line oo, with tbe chain : if 
tliis angle corrcspnndsvutbits containing sides, the 
length of the line DOw truly obtained, and the 
whole work is truly taken. 



15* OP THE CHAIN. 

Note, That in setting off an angle, it is necessa- 
ry to use the largest scale of equal parts, vis. that 
of the inch, which is diagonally divided into 100 

I^arts, in order that the angle should be accurately 
aid down ; or if two inches were thus divided for 
angles, it would be the nibre exact ; for it is by 
no means necessary that the angles should be l^id 
from the said scale with the stationary distances. 

PBOB. Ill 



To take a survey by the chain only^ v>hen all the anglcB cannot 

be seen from one fioint vnthin. 

Fl. 6.J!g. 7. 



Let the ground to be surveyed be represented 
by 1,2, 3, 4, SCc. Since all the angles cannot be 
seen from one point, let us assume 3 points, as Aj 
JB, C, fr^prf Whence they may be seen ; at each of 
which let a mark be put, and the respective sides 
of the triangle be measured and set down in the 
field-book ; let the distance from ^ to 1, and from 
B to 1, be measured, and these will determine the 
point 1 ; let the other Ihies which flow from A, J5, 
C as well as the circuit of the ground, be then 
measured as the figure directs ; and thence the 
map may be easily constructed. 

There are other methods which may be used ; 
]as dividing the ground into triangles, and mea- 
suring the 3 sides of each ; or by measuring the 
base and perpendicular of each triangle. But 
this we shall speak of hereafter. 



# 



OP THE CHAIN. 151 

PROS. IV. 

Ho9 to tnke any inaccetMle dUianee by the chain only, 

Pl. 8. Jig. 8. 

Suppose AB to be the breadth of a river, or any 
other inaccessible distance, which may be required. 

Let a ctaff or any other object be set at J3, draw 
yourself backward to any convenient distance C, 
so that B may cover A : from B, lay oflf any other 
distance by ue river's side to £, and complete the 
parallelogram EBCD : stand at D, and cause a 
mark to be set at jF, in the , direction of A ; mea- 
sure the distance in links from E to F, and FB 
will be also given. Wherefore EF: ED : : FB : 
AB. Since it is plain (from part 1. theo. 3. sect. 
4. and theo. 2. sect. 4,) the triangles EFDBFA 
are mutually equiangulftr. 

If part of the chain be drawn from B to C, and 
the other part from BioE ; and if the ends at E 
and Cbe Kept fast, it will be easy to turn the chain 
over to JD, so as to complete a parallelogram ; by 
reckoning off the same number of links you had in 
SC, from i5 to J), and pulling each part straight. 



( 152) 



THE 



CIRCUMFERENTOR. 



X HIS instrument is composed of a brass circu- 
lar box, about five or six inches in diameter; wiUi- 
in which is a brass ring, divided 6n the top into 
360 degrees, and numbered W, 20, 30, SCc. to 360 : 
in the centre of the box is fixed a steel pin finely 
pointed, called a centre-pin, on which is placed a 
needle touched hy a loadstone, which alwajs re- 
tains the same situation ; that is, it always points 
to the North and South points of the horizon 
nearly, when the instrument is horizontal, and the 
jieedle at regt. 

The box is covered with a glass lid, in a brass 
rim, to prevent the needle being disturbed br 
wind or rain, at the time of surveying : there is 
also a brass lid or cover, which is laid oter the' 
former to preserve the glass in carrying the in- 
strument* 

r 

This box is fixed by screws, to a brass index, or 
ruler, of about 14 orl5 inches in length, to the 
ends whereof are fixed brass sights, which are 
screwed to the index, and stand perpendicular 
thereto : in each sight is a large and a small aper- 
ture, or slit, one over the other ; biit these are 
changed, that is, if the large aperture be uppermost 
in the one sight, it will be lowest in the other, and 



THE cmcUMPERENTOR^ I53 

so of the small ones : therefore the small aperture 
in one is opposite to the lar^e one in the other ; in 
the midd}e of which last^ there is placed a horse 
hair, or fine silk thread. 

The instnunent is then fixed on a ball and sock- 
et ; by the help of which and a screw, you can rea- 
dily fix it horizontally in any given direction ; the 
socket being fixed on the head of a three4egged 
stafi^ whose legs, when extendedj support the in- 
strument whilst it is used. 



To take JIM notet by the CircumfirefitoK 
Pi. 6. Jig. 6. 

Let your instrument be fixed at any angle as A^ 
your first station ; and let a person stand at the 
next angle B, or cause a stafi", with a white sheet, 
to be set there perpendicularly for an object to take 
your view to : then having placed your instrument 
horizontally Twhich is easuy done by turning the 
box so that tne ends of the needle may be equi- 
distant from its bottom^ and it traverses qr plays 
freely) turn the.flower-de4uce, or north part of the 
box, to your eye, and looking through the small 
aperture, turn ibe index about, till you cut the per- 
son Or object in the next angle J3, with the horse 
hair, or thread of the opposite sight ; the degrees 
then cut by the south end of the ne^e, will give 
the number to be placed in the second column of 
yourBeld-book in a line with station No. 1, andex- 
presses the number of degrees the stationary line is 
from the nbrtb, counting quite round with the sun. 

Most needles are pointed at the south end, and 
have a small rinff at the north : such needles are 

X ' 



154 THE CItlCUMPERENTOR. 

better tJian those which are pointed at each end, 
because the surveyor cannot mistake by counting 
to a wrong end ; which error may be frequently 
tomiyitted, in using a twt>-pointed needle. 

Two-pointecJ needles have sometimes a ring, but 
more usually a cross towards the north end : and 
the south end is generally bearded towards its ex- 
tremity, and sometimes not, but its arm is a naked 
right bne from the cap at the centre. 

Having taken the degrees or beaiing of the first 
stationary line ABy let the line be measured, and 
the lengtii thereof in chains and links be inserted 
in the third column of your field-book, under the 
title of distances, opposite to station No. 1. 

It is customary, and even necessary, to cause a 
sod to be dug up at each station, or place where 
you fix the instrument : to the end, that if any 
error should arise in the field-book, it may be the 
more readUy adjusted and corrected, by trying 
over the former bearings and stationary distances^ 

Having done with your first station, set the in- 
strument over the hole or spot where your object 
stood, as at B, for your second station, and send 
bim forward to the next angle of the field, as at 
C; and having placed the instrument in an hori- 
zontal direction, with the sights directed to the 
object at C, and the north of the box next your 
eye, count your degrees to the south end of the 
needle, which register in your field-book, in the 
second column opposite to station No. 2 ; then 
measure the stationary distance J5C, which insert 
in the third column, and thus proceed from angle 
to angle^ sending your object before you, till you 



THE CmCUMFERENTOR. 15& 

■ t 

I 

return to the place where you began^ and you 
wijl have the field-book complete ; observing al- 
ways to signify the parties names who hold the con- 
tiguous lands, and the names of the town-lands, 
rivers, roads, swamps, lakes, SCc that bound the 
iand you survey, as before ; and this is the man-' 
Mr of taking field-notes by what is called fore- 
sights. * 

But the generality of mearsmen frequently set 
ihemselvesm disadvantageous places, so as often to 
occasion two or more stations to be made, where 
one may do, which creates much trouble and loss 
of time ; we will therefore shew how this may b^ 
remedied, by taking back-sights, thus : let your 
object stand at the point where yoi^beginyaur sur- 
vey, as at A ; leaving him there, proceed to your 
next angle JB, where fix your instrument so, that 
you may have the longest viewposdble towards C, 
llaving set the instrument in an horizontal position, 
turn the south part of the box next your eye, and 
liaving cut your object at Ay reckon the degrees 
to th^ south point of the needle, which wfll be 
the same as if they were takep from the object to 
the instrument, the direction of the index being the 
same. Let the degree be inserted in the field- 
book, and the stationary distance be measured and 
annexed thereto, in its proper column ; and thus 
proceed from statipp to station, leaving your ob- 
ject in the last point you left, tilFyou retHm to the 
first station ^t ' 

By this method your stations are laid out to the 
l)est advantage, and two men may do the business 
of three, for one of those who chain, may be your 
object ; but in fore-sights, you must have an olb^ 
ject before you^ besides.two ^ainm^n* 



156 THE CIRCUMFERENTOR. 

It was said before^ that 4 surveyor should have a 
person with him to carry the hinder end of the 
chain, on whom he ccui depend : this person should 
be expert and ready at taking oflP^ets, as well as ex- 
act in giving a faithful return of the length of every 
stationary line. One who has such a person, and 
who uses backnsights, will be able to go over near 
double the ground he could at the same time, by 
taking fore-sights, because of overseeing the chain* 
ing ; for shomd he take back-sights, he must be 
olniged, after taking his degree, to go back to the 
foregoing station, to oversee the chaining, and by 
this means to walk three times over every line, 
which is Q labour not to be borne. 

Or a back and a fore-sight may be taken at ontf 
station, thus ; with the south of the box to your 
eye, observe from JB the object A, and set down 
'the degree in your field-book, cut by the south end 
of the needle. Again from B observe an object 
at C, with the north of the box to your eye, and 
pet down the degree cut by the south point of the 
needle, so h^ve you the bearings of the lines AB 
und BC; you may then set up your instrument at 
D, from whence teke a back-sight to C, and a fore* 
sight to E: thus the bearings may be taken quite 
round, and the stationary distances being annexed 
to them, will complete the field-book. 

But in this last method, c^ire must be taken to 
see that the sights have not the least cast on either 
side ; if they have, it will destroy all : and yet 
with the same sights you may take a survey by 
fore-sights, or by back-sights only, with as great 
truth as if the sights were ever so erect, provided 
the same cast continues without any alteration ; 
but, upon the whole, back-sights only will be found 
the readiest method. 



THE ClRCtTMFERENTOR. 157 

If your needle be pointed at each end, in taking 
forensights, you may turn the north part of the hot 
to your eye, and count your degrees to the south 
part of the needle, as before ; or you may turn 
the south of the box to your eye, and count your 
degrees to the north end of the needle. 

But in back-sights you may turn the norih of 
the box fo your eve, and count your degrees to 
the north point of the needle ; or you may turn 
the south of the box to your eye, and count your 
degrees to the south end of the needle. 

The brass rkig in the box is divided on the side 
into 360 d^rees, thus ; from the north to the easA 
into do, from the north to the west into 90, from 
the SQUth to the east into 90, and from the sotith 
to the west into 90 degrees ; so the degrees are 
numbered from the nomi to the east or west, ami 
from the south to the east or west. 

The manner of using this part of the instrument 
is this ; having directed your sights to the object, 
whether fore or back, as before, observe the two 
cardinal points of your compass, the point of the 
needle lies between, (the nortn, south, east and west 
being caUed |jbe four cardinal points, and are grav- 
ed on the bottom of the box) putting down tliose 
points, together by their initial letters, and there- 
to annexing the number of degrees, counting from 
the north or south, as before, thus ; if the point of 
your needle lies between the nortii and east, jQorlii 
and west, south and east, or south and west points 
in the bottom of the box, then put down NEy 
NWy SE, or SW, annexing thereto the number of 
degrees cut by the needle on the side of the 
ring, counting fromrthe north or south as before, 



158 tlffi CmCTTMFERENTOIt. 

But if the needle point exactly to the norths 
iouthy east» or weBt> ^ou are then to write down 
N, 8, E, or Wy without annexing any degree. - 

This is the manner of taking field notes, where* 
by the content of ground may be universally de- 
termined by calculation ; and they are said to be 
taken by the quartered compass^ or by the four, 
nineties. 



Tbjind the number (ifdegrecM contained in any given angle. 

Set up your instrument at the anmilar pointy 
and thence direct the sights along each le^ of the 
angle, and note down their respective beanngs, as 
before ; the difference of these bearings, if less 
than 180|^will be the quantity of degrees contain- 
ed in the given angle ; but if more, take it from 
360, and the remainder will be the degrees contain- 
ed in the given angle. 



(159) 



{THE 



THEODOLITE. 

X HIS instrument is acircle, commonly of bra^s, 
of ten or twelve inches in diameter, ivhose limb is 
divided into 360 degrees, and those again are sub- 
divided into smaller parts, as the magnitude of it 
will admit ; sometimes by equal divisions, and 
sometimes by diagonals, drawn from one concen* 
trie circle of the Imib to another. 

In the middle is fixed a circumferentor, with a 
needle ; but this is of litUip oi; no use, except in 
finding a meridian line, or the proper situation of 
the land. 

Over the brass circle is a pair of sights, fixed to 
a moveable index, which* turns on the centre of 
the instrument, and upon which the circumferen- 
tor'-box is placed. 

This instrument will either give the angles of 
the field, or the bearing of every stationary dis- 
tance line, from the meridian ; as th« circumferen- 
tor and quartered oompass do. 

To take theanglei qf the field, 

Pl, e.fig. 6. 

Lay the ends of your index to 360*, and 180*; 
turn tiie whole Qbout with the 360 frgm you ; direci^ 



160 THE THEODOLITE. 

llie sights from A to G, and screw the instrument 
fast; direct them firom Ay to cut the object at B ; 
Hie degree then cut by that end of the index which 
is opposite you, will be the quantity of the angle 
GABy to place in your field-book ; to which an- 
nex the measure of the line ABy in chains and 
links ; set up your instrument at By unscrew it, 
and lay the ends of your index to 360 and 180; 
turn the whole about with the 360 from you, or 
1 80 next you, till you cut the object at A ; screw 
the instrument fast, and direct yoursightia to the 
object at C, and the degree then cut by that end 
of the index which is opposite to you, will be the 
quantity of the angle AjSC. Thus proceed from 
station to station, still laying the index to 360, 
turning it from you, and observing the object at 
the foregoing station, screwingthe instrument fast, 
and observing the object at the following station, 
and counting the degrees to, the opposite end of 
the index, will give you the quantity of each res- 
pective angle. 

JLEMMA. • 



M the angles qf any fiolygQUy are equal to twice a» many 
right angles as there are sides less by four, Thusy all the an* 
gles A^ By Cy Dy Ey Fy Gy oTc equal to tvfice as many right an» 
gles as there are sides in thejigurey less by /our. 

Pl. 6, Jig. 6. 

Let the polygon be disposed into triangles, by 
lines drawn from any assigned point jBT within it, 
as by the lines HAy HBy HC, 8Cc. It is evident 
tlien (by theo. 2. sect. 4. part !•) that the three 
angles of each triangle are equal to two right ; and 
consequently, that the angles in all the triangles 
are twice as many right ones as there are sides : 



I 



♦ 



THE THEODOLITE. 161 

but all the. angles about the point Hj are equal to 
four right (by cor. 2. theo. 1. sect. 4.) ; therefore 
the remaining angles are equal to twice as many 
right ones as there are sides in the figure, abating 
four. dE.JD. 



SCHOLIUM. 

Hence we may know if the angles of a surrey 
be truly taken ; for if their sum be equal to twice 
ias many right angles, as there are stations, abat- 
ing four right angles, you may conclude that the 
angles were truly taken, otherwise not. 

If you take the bearing of any line with the cir- 
cumferetitor, that bearing will be the number of 
degrees the line is from the north ; consequently 
the north must be a like number of degrees from 
the line, and thus the north, and of course th^ 
south, as well as the east and west, or the situation 
of the land, is obtained. 

m 

To take the bearing qfeach reapective line from the meridi^ 
an; or tofier/orm the office qfthe circum/erentOTj or quartered 
comfiaaa by the theodolite. 

« 

Set your instrument at the first station, and lay 
the index to 360* and 180% with the flower-de4uce 
of the box next 360 ; unscrew the instrument, and 
turn the whole about, till the north and south 
points of the needle cut the north and south points 
in the box ; then screw it fast, and the instrument 
is north and south, if there be no variation in the 
needle ; but if there be, and its quantity known, 
it may be easily allowed. 

The circumferentor-box may then be taken off! 

-JL 



16^ THE TH80D0LITE- 

Direct the sights to the object at the second sta- 
tion, and the degree cut by the opposite end of 
the index will be the bearing of that lin^ from the 
north, and the same that the cireumferentor would 
give. 

After having measured the stationary distance, 
set up your instrument at the second station ;. un- 
screw it, and set either end of the index to the de- 
gree of the last line, and turning the whole abo^t 
with that degree towards you, direct your sights 
to an object at the foregoing station, and screw the 
instrument fast ; it will then be parallel to its for- 
mer situation, and consequently north and south ; 
direct then your sights to an object at the follow- 
ing station, and the degree cut by the opposite end 
ojfthe index, will be the bearing of that line. 

In like manner you miy proceed thipugh the 
whole. 

If the brass circle be divided into four nineties, 
from 360 and 180^ and the letters N, S, E, W, be 
applied to them; the bearings may be obtained by 
putting down the letters the far (u* opposite end of 
the index lies between, and sumexing thereto the 
degrees from the N* or 8; and this is the same aa 
the quartered compass. 

If you keep the compass box on, to see the mu- 
tual agreement of tide two instruments ; after having 
fixed the theodolite north and south, as before ; 
turn the index about with the north end or flower*- 
de-luce next your eye, and count the degree to tlie 
opposite, or south end of the index, and this will 
correspond with the degree cut by the so\ith end 
of the needle. 




THE THEODOLITE. 163 

At the second, or next station, unscrew the in- 
strument, and set the south of the index to the de- 
je of the last station ; turn the whole about, with 
south of the index to you, and cut the object at 
thfe foregoing station ; screw the instrument fast, 
and with the north of the index to you, cut the 
object at the next following station, the degree then 
cut by the south of the index, will correspond with 
the degree cut by the south end of the needle, and 
go through the whole. 

Some theodolites have a standing pair of sights 
fixed at 360 and 180, besides those on the movea-* 
ble index ; if you would use both, look through 
the standing sights, with the 180 next you, to an 
qbject at the foregoing station : screw the instru- 
ment fast> and direct the upper sights on the rnove^ 
able index, to the object at the following station, 
and the degree cut by the opposite end of the in- 
dex,* will give you the quantity of the angle of the 
field. 

Two pair of sights can be of no use in finding the 
angles &om the meridian; and inasmuch as one pair 
is sufficient to find the angles of the field, the se- 
cond can be of no use : besides, they obstruct the 
free motion of the moveable iiulex, and therefore 
are rather an incumbrance than of any real use. 
Some will have it, tbat they are useful with the 
others, for setting ofi* a right an^le, in taking an 
off-set : and surely this is as easily performed by 
the one pair on the moveable index : thus, if you 
lay the mdex to 360 and 180, and cut the object 
either in the last or following station, screw the in- 
strument fast, and turn the index to 90 and 270, 
and then it will be at right angles with the line. So 
tjiat the 9mall sights, at those of the circle, can bo 



164 THE SEMICIRCLE. 

of DO additional use to the instrument, and there- 
fore should be laid aside as useless. 

This instrument miy be used in windy and rainy 
weather, as well as in mountainous and hilly 
grounds ; for it does not reqiure an horizontal po^ 
sition to find the bearing, or angle, as the needle 
doth ; and therefore is preferred to any instrument 
that is governed by the needle. 



THE SEMICIRCLE. 



T 



HIS instrument, as its name imports, is a half 
circle, divided from its diameter into 180 degrees, 
^nd from thence a^in, that is, from 0, to 360 de- 
grees : it is generally made of brass, and is from 
8 to 18 inches diameter. 

On the centre there is a ndoTeable index with 
sights, on which is placed a circmnferentor-box, 
as in the theodolite. 

This instrument may be used as the theodolite 
in all respects ; but with this difference, when you 
are to reckon the degree tb that end of the index 
which is off the semicircle, you may find it at 
the other end, reckoning the degree from 180 for-^ 
^vards, 



(165) 



IHE 



PLANE TABLE. 



A 



PLANE TABLE is an oblong of oak, or 
other wood, about 15 inches lonj^, and 12 broad ; 
they are generally composed oi 3 boards, which 
are easily taken asunder, or put together, for the 
convenience of carriage. 

There is a box frame, with 6 joints in it, to take 
off and put on -as occasion serves ; it keeps the 
table together, and is likewise of use to keep down 
a sheet of paper which is put thereon. 

[' The outside of the frame is divided into inches 

and tenths, which serve for ruling parallels or 
squares on the paper, or for shifting it, when occa- 
sion serves. 



The inside of th6 frame is divided into 360 de- 
grees, which, though unet^ual on it, yet are the de^ 
grees of a circle produced from its centre, or cen- 
tre of the table, where there is a small hole. 

The degrees are subdivided as small as their 
distance will admit ; at every tenth degree are two 
numbers, one the number of degrees, the other 
its complement to 360. 

There is another centre hole about i of the 
table's breadth from one edge, and 13 in the (nid- 



166 THE PLANE TABLE. 

die between the two ends. To this centre hole 
on the other side of the firame, there are the divi- 
sions of a semicircle, or 180 degrees ; and these 
again are subdivided into halves, or quarters, as 
the size of the instrument will admit. 

. That side of the frame on which the 360 de- 
grees are, supplies the place of a theodolite, the 
other, that of a semicircle* 

TheiSB is a<;ircumferentor-bo:x of wood, with a 
paper chart at the bottom, applied to one side of 
the table bj^a do ve-^tail joint, fastened by a screw. 
This box fbesides its rendering the plane table ca- 
pable of answering the end of a circumferentor) 
L very useful forplacmgthe instrument in the 
same position every remove. 

There is a brass tuler or index, of about two 
inches broad, with a sharp or fiducial edge, at each 
end of which is a sight ; on the ruler are scclles 
of equal parts, with and without* diagonals, and a 
scale of •dior(ki ; the whole is fixed on a ball and 
sockets and set on a tb:ee*legged staff. 

To take the angle$ qf a Jleld by the toble. 

Havingplaeedthe instrument at the first station, 
turn it about till the north end of the needle be 
over the meridian, or flower-de-luce of the box^and 
there screw it fast. Assign any convenient point, 
to which aj^ly the edge of the index, so as through 
the sights you may see the object in the last sta- 
tion, and by the edge of the index from the point 
draw a line. Again, turn about the index with its 
edge to the same pointy and through the sights ob- 



- » 



THE PLANE TABLE. ^67 

serve the object in the second station, andfi:om the 
point, by the edge of the index, draw another line; 
so is the angle laid down ; on that last line set off 
the distance to the second station, in chains and 
links ; apply your instrument to the second sta- 
tion, taking the angle as before ; and after the like 
manner proceed till the whole is finished. 

This method may be used in good weather, if 
the needle be well touched and play freely ; but 
if it be in windy weather, or the needle out of or- 
der, it is better, afier having taken the first angle 
as before, and having removed your instrument to 
the second station, and placed the needle over the 
meridian line as before, to lay the index on the last 
drawn line, and look backward through the si^ts ; 
if you then see the object in the first station, the 
table is fixed rights and the needle is true ; if not, 
turn the table about, the index lying on the last 
line, till through the sights you see the object in the 
first station : and then screw it fast, and keeping 
the ed^e of the index to the second station, direct 
your sights to the next ; draw a line by the edge 
- of the mdex, and lay off the next line ; and pro- 
ceed through the whole without using the needle, 
as you do with the theodolite. 

If the sheet of paper on the table be not large 
enough to contain the map of the ground you 
survey, you inust put on a clean sheet, when the 
other is full ; and this is called shifting of paper> 
and is tlius performed. ' 

Pl. 6, Jig. 8. 

Let ABCD represent the sheet of paper on the 
plane table, upon which the plot JB, Fy 6?, H, /, 



168 THE PLANE TABLE. 

Ky Lj M, is to be drawn ; let the first station be 
E; proceed as before from thence to F^ and to G; 
then proceeding to Hy you find there is not room 
on your paper for the line GH; however draw as 
much of the line GHy as the paper can hold, or 
draw it to the paper's edge. Move your instru- 
ment back to the first station Ey and proceed the 
contrary way to M, and to L s but in going from 
thence to JST, you again find your sheet will dot 
bold it ; however, draw as much of the line LK 
on the sheet as it can hold. 

Take that sheet ofi* the table, first observing 
the distance oo of the lines GH and LKy by the 
edge of the table ; take off that sheet, and mark 
it with No. 1, to signify it to be the first taken off. 
Having then put 7)n another sheet, lay that dis- 
tance 00 on the contrary end of the table, and so 
proceed as before, with the residue of the survey, 
firom to -fir, to JST, and thence to o ; so is your 
survey complete. 

In the like manner you may proceed to take off^ 
and put on, as many sheets as are convenient ; and 
these may afterwards be joined together with, 
mouth glue, or fine white wafer, very thin. 

If the index be fixed to the first centre, using 
the 360 side, it will then serve as a theodolite, 
and when to the second centre, using the 1 80 side, 
it will serve as a semicircle ; by either of which 
you may survey in rainy weather, when you can^^ 
not have paper on the table. 



(16B) 



To MEAsttRfi Angles op Altitude by thje CtR* 
cuMFERExrroR^ Theodolite, Semicircle, 

OR Plai^ Table. 

1. To take an angle of altitude^ 6y the eircum/erentdr, 

JjET the glass lid be taken off, and let the 
instrument be turned on one side, with the stem 
of the ball into the notch of the socket, so that the 
circle may be perpendicular to the plane of the 
horizon ; let the instrument be placed in this situa- 
tion before the object, so that the top thereof may 
be seen through the pights ; let a plummet be sus- 
pended from the centre pin, and the object being 
then observed, the complement of the number of 
degrees, comprehended between the thread of the 
plummet, and that part of the instrument which is 
next y oiu- eye, wilf give the angle of altitude re- 
quired. 

2. If an angle of altitude is to'be taken by the 
tiieodoiite, or semicircle, let a thread be run through 
a hole at the centre, and a plummet be suspended 
by it ; turn the instrument on one side, by the help 
of the ball and notch in the socket for that purpose^ 
so that the thread may cut 90, having 360 degrees 
neirt vou ; screw it fast in that position, and through 
the sights cut the top of the objects ; and the de- 
grees then cut by the end of tlie index next you, 
are the degrees of elevation requu-ed. An angle 
of depression is taken the contrary way, 

Z 



170 OP ANGLES OP ELEVATION, SCc 

3. By the plane table an angle of altitude is ta-- 
ken in the like manner, by suspending a plummet 
from the centre thereof, having turned the table on 
one side, and fixed the index to the centre by a 
Bcrew, so as to move freely, let the thread cut 90^ 
look through the sights as before, and you have 
the angle of eleyationj wd on the contrary that 
of depression* 



(171) 



«HE 



i'ROTRACTOR; 



Ti 



HE protractor is a semicircle annexed to a 
8cale> and is made of brass, ivory, or horn ; its di- 
ameter i^ generally about five or six inchea 

The semicircle contains l&ree concentric semi^ 
circles at such distances from each other, that th^ 
spaces between tiaem may contain figures. 

The outward circle is numbered from the right 
to the left hand, with 10, 20, 30, SCc. to 180 de- 
grees ; the middlemost the same way, ftom 180 to 
360 degrees ; and the innermost from the upper 
edge of the scale both ways, from 10, 20, 30, SCc, 
to 90 degrees. 

It is easy to conceive that the protractor, though 
a semicircle, may be made to supply the place of a 
whole circle ; for if a line be drawn, and the cen- 
tre-hole of the protractor be laid on any point in 
that line, the upper edge of the scale corresponding 
with that line, the divisions on The edge of the se- 
micircle will run from to 180, from right to left: 
again, if it be turned the other way, or downwards, 
keeping the centre-hole thereof on the aforesaid 
point in the line> then the^ divisioiis will run from 



^ 



172 



THE PROTBACTOR- 



180 to 360, and so completes an entire circle 
with the former semicircle. 

The use of the protractor is to lay off angles^ 
^nd to delineate or draw a map, or plan of any 
ground from the field notes ; aiKi is performed in 
the following manner, 



T^ protract a Jield^boot, when the oHslf wre taken from the 

mendtan* 



fjL. 6. Jig. 9, 



I 



On your paper rule lines parallel to each other, 
at an inch asunder (being most usual\ or at any 
other convenient distance ; on the left end of the 
pavallels put N. for north, and on the right 8. for 
south ; put E. at the top for east, and W. at the 
bottom of your paper for west. 

Then let the following field-book be that which 
is to be protracted, the bearings being taken from 
the meridian, whether by a circumferentor, theo- 
dolite, or semicircle, and measured with ^ two^ 
pole chain. 



C h. L. 

55-20 
12.36 
29.20 
55.20 
40.00 
76.00 
87.02 

(;io8e at the first ttnUom "'\ 



JVo. 


Bearing. 


1 


283i 


2 


3481 


3 


317 


4 


266 


5 


193 


6 


124 


7 


63* 



THE PROTRACTOR. 1713 

Pitch upon any convenient point on your paper 
for your first station, as at I, on which lay the cen- 
tre-hole of your protractor, with a protracting 
pin ; then if the degrees be less than 180, turA the 
arc of your protractor downwards, or towards the 
west ; but if more than 180, upwards, or towards 
the easti. 

Or if the right hand be made the north, ^nd 
the left the south, the west will be then up, and 
the east down. 

In this case, if the degree be less than 180, turn 
the arc of your protractor upwards, or towards 
^e west; and if more> downwards, or towards 
the east. 

By the foregoing field-book, the first beanng is 
283i, turn the arc of your proti-actor upwards^ 
keeping the pin in the centre-hole, move the pro- 
tractor so that the parallel lines may cut opposite 
divisions, either on the ends of the scale, or od 
the degrees, and then it is parallel. This must b^ 
always first done, before you lay off your degrees. 

Then by the edge of the semicircle, keeping the 
protractor steady, with the pin prick the first bear- 
ing 2831, and from the centre point, through that 
point or prick, draw a blank line with the pin, on 
which from a scale of equal parts, or from the 
scale's edge of the protractor, lay off the distance 
55C. 20L* so is that station protracted. 

At the end of the first station, or at 2^ which 
is the beginning of the second, with the pin place 
the centre of thfe protractor, turning the arc up, 
1»eeau8e the bearing of the second, station is more 



174 THE PROTRACTOR- 

than 180^ vis. 348i Place your protractor pa- 
rallel as befere, and by the edge of the seflfiicirele^ 
^vith thepiB prick at that degree, through which 
and the end of the foregoing station^ draw a blank 
line, and on it set the distance of that station. 

In the like mariner proceed through the whole, 
only obserre to turn the arc of your protractor 
<l[>wn, when the degrees are less than 180; 

If you lay off the stationary distances by the 
edge of the protractor, it is necessary to observe, 
that if your map is to be laid down by a scale of 
40 perches to an inch, every division *on the pro- 
traetor's edge will be one two^Ie chain ; i a dir. 
vision will he 25 links, and ^ of a division will be 
I2i links. 

If your map is to be laid down by a scsde of 20 
perches 4;o an inch, two divisions w3i be one two^ 
pole chain ; one division will be 25 links ; t a 
division 12i tinks^ and i of a division will be 
eilinks^ 

In general, if 25 links be multiplied by the num- 
ber of perches to an inch, the map is to be laid 
down by, and the product be divided* by 20 (or 
which is the same thing, if you cut off one and take 
the half), you mil have the value of one divifeion 
on the protractor's edge, in links and parts* 



Examples, 



1. How many links in a division, if a map be 
ad. down by a scale of 8 perches »to an inefa ? 



THE PROTRACTOR. 175 



25 
8 



2|0)20|0 

10 links. Answer* 

2. How many links in a division, if a map be 
laid down by a scale of 10 perches to an inch ? 

25 
10 



2|0)25|0 



12.5 or 12Hinks. Answer. 



And so of any other. 

To protract a/Uld^bookf taken by the anglea of the field, ' 

Note. We here suppose tlie land surveyed i*^ 
Icept on the right hand ^s you survey. 

Draw a blank: line with a ruler of a len^h greater 
than the diameter of the protractor ; pitch upon 
any convenient point therein, to which apply the 
centre-hole of your protractor with your pin, turn- 
mg the arc upwards if the angle be less than 180, 
and downwards if more ; and observe to keq) the 
upper edge of the scale, or 180 and degrees upon^ 
the line : then prick off the number of degrees con- 
taiiied in the given angle, and draw a line firom the 
first point through the point at the degrees ; upon 
which lay the stationary distance. Let this line be 
lengthened forwards and backwards, keeping yoiu* 
fnrst station to th^ right, and second to the left ; 



176 THE PROTRACTOR. 

and lay the centre of your protractor over the se* 
cond station^ with your pin, turning the arc up- 
wards, if the angle be less than 180, and down- 
wards, if more ; and keeping the 180 and degrees 
on the line, prick off the number of degrees 
contained in the given angle, and through that 
point and the last station draw a line, on which 
lay the stationary distance ; and in like manner 
proceed through the whole. 

In all protractions, if the end of the last station 
falls exactly in the point you began at, the field- 
work and protraction are truly taken, and perform- 
ed ; if not, an error mu&t have been committed in 
one of them : in such case make a second pro- 
traction ; if this agrees with the former, and neither 
meet nor close, the fault is in the field-work, and 
not in the protraction ; and then a re-survey must 
be taken. 

REMARKS. 

The accuracy of geometrical and trigonometri- 
cal mensuration, depends in a great degree on the 
exactness and perfefction of the instruments made 
use of ; if these are defective in construction, or 
difficult in use, the surveyor will either be subject 
to error, or embarrassed with continual obstacles. 
If the adjustments, by which they are to be ren- 
dered fit for observation, be troublesome and in*- 
convenient, they will be taken upon trust, and the 
instrument will be used without examination, and 
thus subject the surveyor to errors, that he can 
neither account for, nor correct. 

In the present state of science, it may be laid 
down as a maxim, that every instrunient snould be 



WST OP IHSTRDMENTa |77 

«o contrivedf that the observer may easily examine 
and rectify the principal parts ; for however care- 
ful the instrument-maker may be, however perfect 
the execution thereof, it is not possible that any 
instrument should long remain accurately fixed 
in the position in which it came out of the maker's 
hand, and therefore the principal parts i>^uld be 
moveable, to be rectified occasionally by the ob- 
gerver. 

AK ErnmSRATlOJSf OF mSTRCBIKIfTS USSFtTL TO 

A stTRvinroii ; 

Fewer or more of which will be wanted, accord- 
ing to the eitent of his work, and the accuracy 
required. 

A case of good pocket instruments. 

A pair of beam compasses. 

A set of feather-edged plotting scales. 

Three or fotir parallel rules. 

A pair of prc^rtional conqMiBses. 

A pair of trianigulair ditto* 

A pantagraph. 

A cross staff. 

A cireumfeirentof • 

An Hadley^s sextant. 

An artificial hofizottr 

A theodolite. 

A surveying compass. 

Measuring chains, and measuring tapes. 

King's surveyinj^ quadrant 

A perambulator, or niteasuring wheel. 

A spirit le vd; with telescope. 

Station staves; i»ed with the level 

A protraeter^i with qr without a nonius. 

To b€ added for county and marine aurveying ; 

Ah astronomical quadrant,or circular instrument. 

A. a 



178 LIST OP INSTRUMENTS, 

A good r^ncHag and reflecting tdiescopcr 
A copying glass. 

For marine purveying ; 

A station pointer. 

An azimudi compass. 

One or two boat compasses. 

Besides these, a number of measuring rods^iroo 
pins, or arrows, &c. will be found y,ery conyenienty 
and two or three offset staves, which are straight 
pieces of wood, six feet seven inches long, and 
about an inch and a quarter square ; they should 
be accurately divided into ten equal parts, each of 
which w ill be equal to one link. These are used 
for measuring octets, and to examine and adjust 
the chain. 

Five! or six staves of aboutfive feet in length, and 
one inch and an half in diameter, the upper part 
painted white, the lower end shod with iron, to be 
struck into the ground as marks. 

Twenty or more iron arrows, ten of which are 
always wanted to use with the chain, to count the 
number of links, and preserve the direction of the 
chain, so that the distance measwed may be reaUy 
in a sU^ghtline. 

The pocket measuring tapes, in leather boxes, are 
often very convenient and usefuL They are made 
to the dilSerent lengths of one^ two, three, four 
poles, or sixty-six feet and 100 feet ; divided, on 
one side, into feet and inches, and im the other 
into links of the chain. Instead of the latter, are 
sometimes placed the centesimals of a yard, or 
three feet into 100 equal parts. 



(179) 



SECTIOK IL 

r 

MENSURATION 

« 

OF HEIGHTS AND DISTANCES^ 

\$U Of Heights. 
Pl* s.JIs. 18. 

X. HE Snstrament of least expence for taking 
heights, is a quadrant, divided into ninety equal 
parts or degrees ; and those may be subdivided 
into halves, quarters, or eighths, according to the 
radius, or size of the instrument : its construction 
will be evident by the scljeme thereof. 

Prom the centre of the c^uadrant let a plummet 
be suspended by a horse hair : or a fine silk thread 
of such a len^h that it may vibrate freely, near 
Hie edge of its arc : by looking along the edge 
ACj to the top of the object whose height is re- 
.quired ; and holding it perpendicular, so that the 
plummet may neither swing from it, nor lie on it; 
the degree then cut by the hair, or thread, will be 
the angle of altitude required. 

If the quadrant be fixed upon a ball and socket 
on the three-^legged staff, and if the stem from the 
4>all be turned into the notch of the socket, so as 
to biing the instrument into a peipendicular posi- 
tioo, the. angl^ of altitude by this means, can be 
acquired with much greater certainty. 

An angle of altitude may be also taken by any 
of the instruments used in surveying i; a^ has beeo 



180 OF HEH^HTS. 

particularly shown in treating of their description 
anduse& 

Most quadrants hare a pair of sights fixed on 
the edge AC^ with small eircular holes in them ; 
whidi are useful in taking the sun's altitude, re- 
quisite to be known in many astronomical cases ; 
wis is effected by letting the sun's ray, which pas- 
ses through the upper s^ht, fall upon the hole in 
the lower one ; and the degree then cut by the 
thread, will be the angle of uie sun's altitude ; but 
those sights are useless for our presept purpose^ 
for lopkiog along the quadrant's ed^ to the top 
i»f the object will be sumcient, as beiore* 

PROS. I. 



njtni the k^gki ff a ^tr/iendkttlm^ oiffeet tu $ne iitttion^ wMck i» 



A steeple. 

{The angle of altitude, 53 degrees* 
lllstance from the observer to the foot 
of the steeple, or the base, 85 feet. 
Height of the instrument, or of the ob« 
senrer^ 5 feet. 

Bequired, the height oi ibt steeple* 

The figure i$ constructed and wrought^ in all 
respects, as case 1)* of right-angled trigoncNEnetry ; 
only there must be a line drawn parsulel to, and 
beneath AB of 5 feet for the obserrer's height, to 
represent the plane upon M^ikb the ol^ct staoMb; 



«<• 



OF HEIGHTS. 181 

to whkh tiie perpendicular must be continued, 
and that will be the hei^ of the object. 

Thus, AB is the base, A the angle of altitude, 
BCihe height of the steeple from the instrument, 
or from tile observer's eye, if he were at the foot 
fk it ; JDCthe height of the steeple kbove the ho- 
rizontal surface. 



Various stbtb^ fiir BCy as in case 2. of light* 
an^ed f^ane tri^nometry. 



90* 
53=A, 



1. JR C; AB : : 8. A : BC 
37" 85 53* 112.8. 



2. B,:AB::T.A:BC. 
90* 85 53* 112.8. 

3. T.C:AB::ll.:Ba . 
37* 85 90" 112.8 

ToBC 112.8 

Add DB 5. the height of the observer. 

Tfadr sum is 117. 8 or 118 feet, the height of 
the steeple required. 



M2 Of HEIGHTS- 



PROB. 11. 



Pl. 5./f . 30. 



njhd the hdghi ff a fierfiend&cuUar object^ onan hdrizontaifiiant ; 
by having the length qf the shadow given. 

Provide a rod, or stafl^ whose length is given, 
let that be det perpendicular, by flie help of a 
quadrant, thus ; apply the side of the quadrant 
AC, to the rod, or staff ; and when the thread cuts 
90^. it is then perpendicular ; the same may be 
done by a carpenter's or mason's plumb. 

Having thus set the rod or staff perpendicu- 
lar ; measure the length of its shadow, when the 
sun shines, as well as we length of the shadow of 
the object, whose height is required ; and you 
have the proper requisites given* Thus, 

ah, ihe length of th« shadow of the sta^ 15 feet. 

I he, the length of the staff, 10 feet 

AB, the length of the shadow of the steeple, or 
object, 135 feet. 

Required BC, the height of the object 

• 

The triangles abc, ABC, are similar, thus ; 
the angle h^B, being both right ; the lines ac, 
AC are parallel, being rays, or a ray of the sun ; 
whence Ae angle a~A (by part 3. theo. 3. sect 
4.) and consequently e=C. The triangles being 
therefore mutually equiangular^ are similar (by 
theo. 16. sect 4) it will be. 



OP HEIGHTS I8|t 

nb.'hc: : AB : BC. 

15 10 135 90. the steeple's height, required. 

The foregoing method is most to be depended 
on ; however^ this is mentioned for variety's sake.^ 



PROB. ni. 



JPx. $.Jig. 21. 

Tq uU:€ the iUtiiude of aiufpendkular odjeet, at the foot qfa hSl^ 

jTom the hUTa die. 

Turn the ceotre A of the quadrant, next your 
eye, and look along the side Ac^ or 90 side, to the 
top and bottom of the object ; and noting down 
the angles, measure the distance from the place of 
observation to the foot of the object, Thus^ 

Angle to the foot of the object, 55^ 
p. „ f or 55*. 15' 
wven, ^ ^^j^ to the top of it, 3H or 3P. IS' 

Distance to the foot of it, 250 feet. 
Requiredy the height of the object. 

By Gmstruction^ 

Draw an indefinite blank line ADy at any point 
in which A make the angles EAB of 55*. 15^ and 
EACoiZV. 15'; lay 250 from^ to B; from JB, 
draw the perpendicular BE (by prob. 7 of geome* 
try TcroBsing AC in C; so will BC be the height 
ot tne object required. 

« 

In the triangle ABC there is given> 



ji84 OF HEIGHTS. 

ABE the complement of EAB to 90% wluch 
is 34*. 45'. 

. CAB the difference of the ^ven angle 34*.0(/. 
The side ^B, 250. Required, BC. 

Thb is perfonned as. case 2. of oblique angular 
trigonometry. Thus, 

180 —the sum of ABE 34*. 45', and CAB 24*. 
Oar^4CB 121M5'. Then, 

S. ACB : AB : : S. CAB : BC. 

121*. 15' S50 24". OO' U9, the height reqoir- 

TO. 

PBOB. IT. 

To takethe $UStttk qfafier/iendicular objgct, on the tqfiofahUl^ 
at one §tatkm / Hfhen the toh and bottom qfU can be oecn Jrom 
t%e Jbot qf the ML 

. As in proU t. take an aogl^ to- the top, and 
another to the bottom of the object ; and measure 
from the place of observation to the foot of the 
object^ and you hare all the given requisites. 
Thus, 

V 

A Totver on a hiU. 

C Angle to the bottom, 48'. SeT. 
Given, 7 Angle to the t<^, 67". OO'. 

( -Disttothe foot of theobject, 136 feet. 
Required, the hei^f <^ the olyect. 



.•N, 



OF EEBIGHTS. IBO 

• * 

JBy Canstructiim. . 

Make the angle BAB^AS!' 3(y, and lay 136 
feet from ^ to ^ ; from B^ let fall the perpendi- 
cular BB ; and that will be the height of ibe hill ; 
produce BB upwards by a blank line : again, at 
A, make the angle jD^C=67* 00' by a blank line, 
and from C where that crosses the perpendicular 
produced^ dtaw the line CB, and that w31 be the 
beight of the object required. 

Let^Cbe drawn^ 

In the triangle ABQ there is given> 

The angle ACB the complement of BAC^ 
23*, 00'* 

CAB the difibrence between the two given wor 

And the side AB 136. To find BC, 

SC:: AB .' •' 8. CAB : BC. 
23* 136 18*.30'110^. 

If BB were wanted, it is easily obtained, by 
the first cas^ of right-angled plane trigonometry* 

PROB. r 



Tc fake en imfcurible ficrfiendtaUar alfUudtf on- a horit^tOii 

fiiane. 

TllilB is done at two statioQs, thuu : 



m OF HEIGHTS. 

Let DChe a tower whieh cannot be approached 
by means of a moat or ditch, nearer than B ; at 
o^ take an angle of altitude, to C: measure any 
conrenient dmanee backward to A^ which note 
Hown; at A^ take another angle to C; so hare 
you the giren requisites, thus : 

i First angle, 56*. 00'. 
GiTen^ { Stationary distance, 87 feet 
( Second angle, 37*. 00'. 

The height of the tower CDy is required^ 

By Cmstruetiofu 

tJp6n aft indefinite blank line, lay off the rti» 
tionary distance 87, from ^ to £ ; firom i3, set off 
yoUr first ; and firom A^ yoiu^ second angle ; fix)m 
C, the point of intersection of the lines which f<Hin 
these angles, let fall the perpendicular CD j and 
that will be the height of the object required. 

The external angle CBJO, of the triangle ABC,^ 
is equal to the two internal opposite ones. A, and 
ACB (by theo. 4.^ sect 4.) : wherefore if one of 
the internal opposite angles be taken from the ex« 
temal aDgle» the remainder will be the other m- 
teraal opposite one> thus ; 

CBDSS^'-A yi^^ACB W. 

Therefore in the triangle ABC; we have the 
angles A, and AGB, with the side AB given t» 
«nd JBC. 

S.ACB:AB::S.A:Ba 
W 87 37^ 169.4 



OP HEIGHTS. 187 

Having found BC, we have in the triangle BCD 
ihe angle CBD 55% conee^ueutly BCD 35", aod 
BC 169.4 ; to find DC 

This is performed by the first case of right-an- 
gled trigonometry^ three several ways ; thus : 

hB: BC: : S. CBD : DC. 
90* 169.4 55* 138.& 

The height required. 

% SecCBD : BC: : t. CBD : DC. 
55' 1694 5^ 138.8. 
The height required. 



3. Sec. BCD : BC : : R : CD. 
35* 169.4 90« 138.8. 
The height required. 

If BD, the breadth of the moat, were requir- 
ed ; it may also be found, by three different stat- 
ings, as in the first case of right-angled plan^ trig* 
onometry. 

PROB. ri 

i*A. S.Jlg. 24. 

LetBC, a may^-pole, whose height is lOO feet, be 
broken at D ; the upper part of which, DC^ fall9 
upon an horizontal plane, so ijpAi its extremity, C, 
is 34 feet from the bottom or foot of the pole, 

Requi):tsd^ the segments BD and DC 

By Consfruction. 

Lay 34 feet from Aio B; on B exect the per- 
pendicular JBCof lOQ feet ; and draw AC > bisect 



188 OF HEIGHTS. 

AC (by prob. 4. geom.) with the perpendicular 
line, JS-Fy and from 2>, where it cuts tne perpen- 
dicular BC^ draw AD^ which will be the upper 
segment ; and DB will be the lower. 

By cor. to lemma, preceding theo. 7. geom. 
AD=DC; and fby the lemma) the angle 
C^CAD. ■ , ^ 

In the triangle ABC, find C as in case 6, of rights 
angled trjgonometry, thus ; ^ 



1, BC: R::AB : T. C=GAD. 

100 90* 34 18* 4/ 



By theo. 4. geom. The external angle ABD = 
37* 34', or to twice the angle C, i. e. to C and 
GAD. 



Then in the triangle ABD, there is ABD 37* 
34', therefore alsp its complement DAB 52P 36^, 
and AB 34, given, to find AD and BD. 

By the second case of right-angled trigone^ 
Hictry. 

a 9. ADB : AB : : R : AD or DC. 
3V 34' 34 90* 55.77. 



•. 



100—55.77=44.23 required. 

l^se may be had jErom other stations, as in the 
second cfts^ afoireiaid. 



OF HEIGHTS, m 



PROB. riL 



Pl. $.Jig:. 25. 



To take the altitude tf a fierpendkular object en a hillfjromafilane 

beneath it. 

This is dom at two stations, thus ; 

Let the height DC, of a wind-mill on' a hill be 
required. 

From any part of the plane whence the foot of 
the object can be seen^ let angles be taken to the 
foot and top ; measure thence any convenient dis- 
tance towards the object, and at the end thereof^ 
take another angle to the top : and you have the 
proper requisites, thus ; 

First station- Angle to the foot JD^B 21* (X/. 

Angle to the top CAB 35*»0(y, 
Stationary distapce^i? 104 feot. 

Second station. Angle to the top dS"" 30. 

DC required. 
By Construction. 

On an indefinite blank line, lay the stationary 
distance AB 104 feet ; from A, set off the second, 
and from JB, the third given angle ; and from the 
intersecting point C of the line formed by them, 
let fall the perpendicular CE/ from A set off the 
-first angle^ and the line formed by it will deter- • 
mine the point D« Thus have we the height of 
the hill^ as well as that of the wind-mill 



i90 OP HEIGHTS. 

The angle CBE — ^^^CJB^asihthe last prob. 
In the triangle ABC^ find AC thus ; 

S. ACB : AB : : 8. ACB (or sup. of CBE) : AC 
IS*. 30' : 104 : : 13P.30' : 333.6 

The angl^ CAE—DAE^CAn. 

The 9ngle ACD^AEDxEAD, bj theo. 4. 

In the triangle CAD^ find CD thus, 

S. ADC: AC: : S. CAD : DC 

Ill^ : 333.6 : : 14 : 8646 required. 

CE, BEf or DEy may be found by other various 
statings^ as set fortii in the first and second cases 
of right-angled trigcHiometryt 

PROB. rm. 

Tojnd the length tfan Meet, that •tanda obUguely 011 the tep tf 

ahiUtjhmafiltmebtneath. 

Let CD be a tree whosle length is required. 

This is done at two stations* 

Make a station at By from whence take an ai^le 
to the footy and another to the top of the tree; 
ineasure any convenient distance backward to A^ 
from whence also let an angle be taken to the foot> 
and another to the top \ aixl you have the v&fir 
sites given, Thus^ 



OF HEIGHTa Idt 

First station. Angle to the foot JE;.BD«:3e*. SOT. 

Angle to the top EBC'^W. 3Xf. 
Stationary distance AB « 104 feet. 

Second station. Angle to the foot EAD^2i\ 30^. 

Angle to the top £^C=32*. OO*. 

Xet DC and JDE be reqiured. 

The geometrical constructions of this and the^ 
next problem are omitted ; as what has been al- 
ready said, and the figures, are looked upon as mS' 
ficient helps. 

EBC—A^ACBy or 44*. 3(y— 32».= 12». S0», 
as before. 

In the triangle ABCy find BC. Thus, 



1. 8.ACB.'AB::8.A: 

12*. 30* 104 32» 254.7. 

MBD^EAD^ADBiOt 36*.3()'-24*. 30^— 12* 00^^ 
In the triangle ADB» find DBt thus ; 

2. 8. ADB : AB : : 8. DAB : DB, 
12- 00^ 104 24". 30*. 207,4 

CBE^DBE^CBDfitU: 30'— 36- 30^«*8*0flr 

In the triangle CBD there is given, CB 254.7, 
DB 207.4, and the angle CBD 8* 00^; to find DC. 

This is performed as case 3. of oblique angled 
trigonometry, thus ; 



Ite Ol' HEIGHTS. 

3. BC X BD : BC-- BD : : T. ofl BBC-¥ BCD / 
462.1 47.3 86«.00'« 

T.odBDC—BCD, 

55». 40'. 
86». 0(ir+55». 40'=.141». 40'= Bi>C. 
86\00r -55«. 40' = 30''. ^'^^BCD. 

4. & BCD : BD : : S. CBD : DC. 

30*. 2ff 207.4 8*. 00" 57.15 length of 
the free. 

To find DE in the triangle DBE. 

Say R. : BD : : S. DEE : DE, 

90". 207.4 36\ 30' 123.4 height of tbe 
hUL 



PROS. IX 

To find th$ height (/m htaceenAle object CD. onahiUBC. fiom 

grmmd that ia not harizontaU 

Pu ^.Jig. 1. 

From any two points, as O and A, whose dish 
lance GA, is measured, and therefore given ; let 
the angles HGD, BAD, BAC, and EAG, be ta- 
ken ; because GH is parallel to EA (by part 2. 
theo. 3. geomO the angle HGA^EAG; therefore 
EAGy^ HGJJ=AGD: and (by cor. 1. iioBf^. I. 
geom)180— the sumof £^6?andJ5^jD=(?-4D/ 
and, (by cor. 1. theo. 5. ceom.( 180 — the smn of 
the angles AGD and GaD^GDA : thus we hav* 
the angles of the triangle AGD, and tiie side AG 
given ; thence (by case 2. of obi. ang. trig.) AD 
may be easily found. The angle DAB — CAB 
=DAC\ and 90*— BAD^ADC; and ISO^^the 
sum of 2?^C and ADC^ACD .• so have we th« 



OF HEIGHTS. 193 

several angles of the triangle ACH given, and tba 
side AD ; wbeoce (by case 2, of obi. trig.') CI} 
majr be easily found. We may also fina -4C, 
^hich with the angle BACj will give CB the 
height of the hill. 

The solutions of the several problems in heigl^ 
and distances, by Gunter's scale, are omitted ; be* 
iDause every particular stating has been already 
afaewn by % in trigonometiy. 



Cc 



I 



(m) 



2d. OF DISTAIfCES. 



X HE principal iDstnimente used In suireying , 
will give the angles or bearingB of lines ; which hss 
been particular^ ehewn, when we treated of tbenii 

PBOB. L 

Let A and B be two houses on one side of a 
riTer, whose distance asunder is 293 perches: 
there is a tower at C on tiie other side of tne river^ 
that makes an angle at A^ with the line AM of 
dS"" 20' \ and another at B^ with tbe line BA of 
66"" 20' ; required the di^nce of the tower from 
each house, mz. AC and BC. 

This is performed as case % of oblique angled 
trigonometry, thus ; 

1. 8. C: AB : : 8. A: BC. 
Wr 20' 293 63^ 2tf 270.5. 

2.8.C:AB::S.B:AC. 
60*20' 293 66*20^ 30a<8. 

pBOB.n: 

Let B and C, be two housesivhose direct dis^ 
lance asunder, JSC, is inaceessible : however it i^ 



»♦ 



OP DISTANCES. 195 

known that a house at A is 252 perches from B^ 
and 230 from C; ^ndthat the angle BAC^ is found 
to be 70*. What is the distance BC^ between the 
two houses ? 

« • 

This is performed as case 3. of oblique angled 
trigonometry, thus; ^ 

1. AB^AC : AB^AC : : T, of * C + B , 
482 22 55*. W 

T. of ♦ C— JB 

3*44' 

55^3*. 44'== 58*. 44'=C55*— 3«. 44'«51% 16 
^B. 

%S.C:AB::8.A:Ba 
58*. 44' 252 70» 277. 



PEOB. lU. 



Suppose ABC a triangular piece of ground^ 
which by an old survey we find to be thus ; 
AB 260» AC 160, BC 150 perches, the mearing 
lines AC and J9C, are destroyed or plowed down, 
and the line AB^ only remailung. What angles 
must be set off at A and B^ to run new mearings 
by exactly where the old ones were ? 

This is performed as in case 4. of oblique an- 
gled trigonometry, thus ; 

1. AB : AC+BC : : AC-^BC: AD^DB. 
260 310 10 11.92 



196 OF DISTANCES. 

190+ 5.96=]35.96»^1>. 
1 30— 3.9fi» 124.04 =I>J3. 

2. AD : n : : AC : Sec. A, 
136 90':: 160 31^47'. 

3. BC:8.A::AC:8,B. 
150 31^ 4,r 160 34% 10, 

F ROB, TV, 

Tl. ^.Jlg. 4. 

liOt 27 and C, be two trees in a boe, to which 
you can have no nearer access than ^A and B ; 
there is riven, BAB 100», CAB 36«. 30*. C'J?^ 
12I*. BBA 49<>, and t&e line AB 113 perches. 
Required, the distances of the ti^es i>C. 

180»--the sumof 1>B^ and BAB^ADB^2\\ 
180*— the sumof CAB and CBA^ACB^^, 30, 

In the triangle ABD^ find JDJ5, thus ; 

1. S, ABB :AB .• .- 8. DAB : DB, 

3l« lis : : 100» 2I6. 

And in the triai^le ABtt $nd BC, thus ; 

i 8. ACB : AB ': : 8 CAS : BC. 
22«30' 113 36»30' 175.6. 

In the triangle DBC, you have DBC=ABC^ 
ABD^iT!^; Iike*ri«th6 si^iBD, B€, as befom 
found, given to find DC. 



3. BD+BC: BD—BC: : T.oU DCB-^CDBr 
391.6 40.4 54* 



OF DISTANCES: 199 

T. of 4 DCB-^CDB. 

«• 05'. 

54* + 8« 05' =62» 05' =DCB, 
54»— 8''05'=4&« 55'=CDjB. 

4. S. CDB • jBC- • & DEC- DC. 
45» 55* 175.6 72f» 232.6. 

L£MMA. 

Pi. 6./ir. 12. 

Jffrom a /koto C,ofa trtangte ABC, irucrtbed in a circle, there ie 
aperpendiindar CD, Utfau vhan the oMioaUe Me AB ; that fur* 
ftetuaeiUaria to <me tf the Met, kichiaing the angk, a* the other 
Me, inebidtng the angk, it to the diameter tfthe circle, L e. DC: 
4C:;CB ; C£. 

Let the diameter CE be drawn, aAd join JSJ3 ; it 
is plain the angle CEB= CAP (by cor. 2. theo. 
7. geom.) and CBtlh aright angle (by cor. 5. theo« 
?• geom.) and«^2>C : whence ECB=ACD. 
The triangles CEBt, CAD, fere therefore mutually 
eqiliatigmar, and (hf theo. 16. geom.) DC- AC: : 
CB:CEyQvDC:CB::AC:Ct:, Q. E. D. 

* 

PBOB. r. 



Pl. 6. Jig. 5. 

Iiet three gentlemen's seati, A^ B, C, be situate 
in a triangular form : there is given, AB2.5 miles, 
AC % 3, and BC 2. It is requu^d to build a church 
at Et that shall be equi-distant from the seati^ A, 
B, C. What distance njtist it be from each seat, 
and by what angle may the place of it be found ? 



198 OP DISTANCES; 

By GmstrueHoB. 

By prob. 15. s^om. Find the centre of a circle 
that will pass ttut>ugh the points^ A^ B^Cr and 
that will be the place of the church ; the measure 
of which, to any of these points, is the answer for 
the distance : draw a line from any of the three 
points to the centre, and the angle it makes with 
either of the sides that contain the angle it was , 
drawn to ; that angle laid off by the direction of 
an instrument, on the ground, and the distance 
before found, being ranged thereon, will give the 
place of the church reqiured. 



By Calculaiion. 

1. AB : AC+BC' - AC-^BC: AD^DB, 
2.5 4.3 .3 .916. 

1.25+.259^1MB^AD, 

By cor. 2. theo. 14. gecnn. The square root of 
the difference of tlu^sqwues of the nypotheirase 
AC, and given leg AD, will give DC, 

That is, 5.29— 2.274064» 3.015936. 

Its square root is 1 .736 =s CD* 

Then by the preceding lemma, 

2. CD • AC : : CB: the diameter. 
1.736 2.3 2 2.65. 

the half of which, viz. 1.325 is the semi-diameter, 
or distance of the church from each seat, that is, 
AEy CEf BE. 



OF DISTANCED 



19& 



From the centre J?, Iet«fa)l a perpendicular 
upon any of the sides as EFj and it will biwct ia 
M : (by theo. 8. geom.) 

Wherefore AF=> CF=^i ^0=1.15. 

In the right angled triangle AFE^ you have AP 
1.15, and AE the radius 1 .325 given^ to find FAEy 
thus; 

3. AF.'fR. : .' AE : Sec. FAE. 
1.15 90* 1.325 29* 47'. 

Wherefore directing an instrument to make an 
angle of 29* 47', with thie line AC ; and measur- 
ing 1.325 or. that line of direction, will giTe the 
place of the church, or the centre of a circle that 
will pass through A, B, and C. 

The above angles F^JET, may be had without a 
secant, as before, thus ; 

AE : R .-.' AF: & AEF. 

1.325 9(f .115 60^. 13^. 

It»cop)pIemexit 29*. 47'^ will give FAEy as be^ 
fore. 

The questions that may be proposed on this 
head, being innumerable, we have chosen to gite 
only a few of the most usefuK 



\ ' . 






( 200 ) 

« 

5BCTIQN PI. 

Mensuration of Areas, or the various me- 

THODS OF calculating THE SUPERFICIAL 
CONTENT OF ANY FIELD* 

« 

DEFINITION. 

X HE area or content of any plane surface, id 
perches, is the number of square perches which 
that surface contains. 

Pl, 7. Jig. L 

Let A BCD represent a rectai^ular parallelo* 
gram, or oblong : let the side aS, or x)C, con* 
tain 8 equal parts ; and the side A Dp or BCp 
three of such parts ; let the line AB be loored i^ 
the direction of -41>, tillithas come to JEFj where 
AEy or BF (tiie distance of it from yts first Bitua- 
tion) may be equal to one of tb^ equal p^rts. Her^ 
it is evident, thsit the generated oblong ABEF\ 
will contain as many squares as the dide AB con* 
tains ecj^ual parts, which are 6 ; each s(}uace har« 
ing for its side one of the equal parts^ mto which 
ABj or ADj is divided. Again, let AB move 
on till it comes to GH^ so as CrEy or HFj may be 
equal to AEy or BF; then it is plain that the ob- 
long AOHBj will contain twice as msiny squares, 
as-tne side AB contains equal parts. After the 
same manner it willappear, that the oblong ^jDC!B 
will contain three times as many squares as the 
side AB contains equal parts; and in general, that 
every rectangular parallelogram, whether square 
or oblong, contains as many squares as the pro- 
duct of the number of equal parts in the base, 
multiplied into the number of the same equal parts 
in the height, contains units, each square having 
for its side one of the equal parts. 



• * 



T^Jind the CkmtdU of Oromd. 201 

Hence arises the solution of the following prob- 
lems. 

PROB. L 

Tojind the content qf a equate fiiete (if ground. 

i. Multiply the base in perches, into the per- 
pendicular in perches, the product will be the con- 
tent in perches ; and because 160 perches make 
an acre, it must thence follofr, that 

Any area, or content in perches, being divided 
by 160, will ^ive the content in acres ; the remain- 
ing perches, if more than 40, being divided hv 40, 
will give the roods, and the last remainder, i^any, 
will be perches. " , 

Or thus : 

2. Square the side in four-pole diains^ ^nd 
links, and the product will be square four-pole 
chains and links : divide this by 10, or cut off one 
more than the decimals, which are five in all, from 
the right towards the left : the figures on the left 
are acres ; because 10 square four-pole chains 
make an acre, and the remaining figures on the 
right, ai'e decimal parts of an acre. Multiply the 
five figures to the ri^ht by 4, cutting 5 figures 
from the product, and if any figure be to the left 
of them, it is a rood, or roods ; multiply the last 
cut off figures by 40, cutting off five, or (which is 
the same thing) by 4, cutting off four ; and the re- 
maining figures to the left, if any, are perches. 

1. The first part is plain, from considering that 
a piece of ground in a square form, whose side is 
a perch, must contain a perch of ground ; and that 
40 such perches make a rood, and four roods an 

Dd 



202 lS»Jlnd the QmUnt ^ Graimd. 

acre ; or which is the same thing, that 160 squam 
perches make an acre, as before. 

2. A square four-pole chain (that is, a piece of 
ground four poles or perches every way) must 
contain 160 square perches; and 160 perches make 
an acre, therefore 10 times 16 perches, or 10 square 
four-pole chains, make an acre. 

* Note. The chains given, or required, in any of 
the following problen^s, are supposed to be two- 
pole chains, that chain being most commonly used ; 
but they must be reduced to four-pole chains or 
perches for calculation, because the links will not 
operate with them as decimals. 



EXAMPLis. 

Pt' i.j%. 17. 

Ck.L. 

liet A BCD be a square field, whose side is 1 4 29^ 
required the content in acres. 

Ck. L. 

By problem 4. section 1. part 2. 14. 29 are equal tm 

29.16 perches 
29.16 



17496 
2916 

26244 

5832 


A. R. P. 

5. 1. 10. content 


160)850.3056( 
40)50(1 rood. 



10 perches. 



I 



Tojhd the Content of Ground. 203 

Or thus : 

CA. L. Ch. L. 

14. 29 are equal to 7. 29 of four-pole chains^ by 
profo. 1. sect 1. pt. 2.. 7. 29 

6561 
1458 
5103 

A.R. P. 

Acres 5|3144I cont. as before 5. 1. 10 

4 



Rood 1 125764 

40 



Perches 10130560 



It is required to lay down a map of this piece 
of ground, by a scale of twenty perches to an 
inch. 



. Take 29; 16 the perches of the given side, from 
the small diagonal on the common surveying scale, 
where 20 small, or ttro of the large divisions, are 
an inch : make a square whose side is that length 
(by prob. 9. geom.) and it is done. 



PROB. IL 



To jpnd the Me qf a s^fuare, whoae content U given. 

Extract the square root of the ^iven content in 
perchedj and ^ou have the side m perches, and 
oonsequeittly in chains* 



^ 



2M IhJM ti0 CoUe$U <f GnmneL 



EX4Mrt9. 



It 19 required to lay out a square piece of ground 
which shall contain 12 A. 3R. 16P. Required the 
number of chains in each side of the square ; and 
to lay down a map of it, by a scale of 40 perches 
to ap inch. 

A. R. P. 

12. 3. 16. 
4 



51 
40 



Ch. L. 



2056(45.34+ perches = 22. 33^ by prob. 6, 

« - 

85)456 [sect 1. pt. 2. 

903)3100 
9064)39100 &0. 

Tq draw the ms^p. 

From a scale where 4 of the large, or 40 of the 
small divisions are an inch, take 45.34, the perches 
of the side, of which fnake a square. 



PMOB. Ill 

To find the content of an obiong fiiece of ground. 

Multiply the length by the breadth, for the 
content. 



T»fnd the Omimi of Grmmd. 205 

EXABIFLE. 

Fl. \.Jlg.2. 

t 

Let ABCOhe an oblong piece of ground, whose 
length AB is UC. 2dZ. and breadth SC. 371/. Re- 
quired the content in acres, and also to lay down 
a map of it, by a scale of 20 perches to an inch. 

Ch.L. Perches. 



15732 
3496 

A. R. P. 



160)506.9200(3. 0. 27. content. 
26 perches, or near 27. 



Or thus : 
4 pole ch. 
Ch. L. Ch. L. 
14.25 = 7.25 
8 



\m _ /^m \ By prob. 1. sect. 1. pt. 2. 



5075 
2175 
2900 

Acres 3| 16825 

4 



r ..f 



Rood 167300 

4 



Perches 26|9200 



206 To find ti^ Content of Grtmii. 

To draw the map. 

Make an oblong (by schol. to prob. 9, geom.) 
whose lenn^, from a scale of 20 to an inch, may 
be 29percnes, and breadth, 17.48. perches. 

PROB. IV. ^ 

The content qf cm cblong pkce qf ground* and one Me given^ to 

Jind the other. 

Divide the content in perches, by the given side 
m perches, the quotient is the side required in 
perches ; and thence it may be easily reduced to 
chains. 

EXABIPLE. 

There is a ditch 14 CK. 25 L. long, by the side 
of which it is required to lay out an oblong piece 
of ground, which shall contain 3 A. OR. 37P : what 
breadth must be laid off at each end of the ditrJi to 
enclose the 3 A. OR. 37P? 

A. R. P. . 

3. 0. 27, 

4 

12 
40 

Perch. Of. L. 



29)507(17.48 = 8. 37. breadth. 
217 
N 140 
240 
8 



To find the Content of Ground. 207 

Th« map is constructed like the last. 

PROS. r. 

To find the content of a piece of ground^ in form of an obUgiBe 0ir 
guiar parallelogram i or q;' a rhombust or rhomboides. 

Multiply the base into the perpendicular height 
The reason is plain from theo. 1 3. geom. 

«p 

Example, ^/j^- 



Pl, 7. fig. 2. 

Let A BCD be a piece of ground in form of a 
rhombus, whose base ^ B is 22 chains, and perpen- 
dicular DEy or FC^ 20 chains. Required the con- 
tent. 



Ch. Ch. 
22= 
20 



— 10 \ ^ P^^^ chainSv 



Acres 11|0 



Or, 



Ch. 

20=40 ! perches. 



160)1760(11 acres. 
160 




08 To[find the QmUnt €f Otaund. 

« 

The conrerse of this is done hy prob. 4. and the 
map is drawn, by laying off the peipendicideir on 
that part of Ibe base from whence it was taken ; 
joining tM^ extremity thereof to that of the base 
by a righ^ine^ and thence completing the paral- 
lelogram. 

PBOB. VL 

Tofini the content <ifa trianguktr fikce q/* ground. 

Multiply the base by half the perpendicular, or 
the perpendipilar by half the base ; or take half 
the product of the base into the perpendicular. 

The reason of this is plain, from cor. 2. theo. 
12. geom. 

EXAMPLE. 

Pl. l.Jlg. 16. 

Let^i9Cbe a triangular piece of ground, whose 
longest side or base JBC, is 24 C. 38Z. and perpen- 
dicular AD, let fall from the opposite angle, is 1 3 
C 28L, Required the content. 
Ch. L. Ch, L, 
l.Base24. 38= 12. 38 ) . ^ i . . 

f perp. 3. 39 j * ^^^ ''^'''^' 

11142 
3714 
3714 



Acres 4|I9682 

4 



Rood 178728 



40 



Perches 3 lj49 120 
A. R. P, 

Content 4. -0. 31. 



Ch.L. Ck,L. 

Perp. 13.28 ae 6.78 ) ibur>pole chains by 
vperp. 6.39 » 3^9 y |«ob. 1. sect. 1. pt. 2. 

Or 2dl7. Perp. 6.78 of four-pole cnaiiu. 



ibaae 6.19 



6102 
678 
4068 

A. R P. 



4119682 = 4. 0. 31. ' 

Or 3dly. Base 12.38 four-pole chains. 
Perp. 6.78 

9904 
8666 
7428 



>*. * 



83.9364 

A. R. P. 

Its! « 4]1^682 = 4. 0. 31. 

Or the base and perpendicular may be reduced 
to perches ; and the content may l>e thence ob- 
tained) thus : 



£e 



210 3*0 JM the Conteni of <7roim<9 

C%. L. Pereha. 
Peip. 13.28 = 27.12 J 

Half %i>eip. UM J »y P"^****' *' ««*• *-P*-2' 

Perches, Ch. Lti 
1. Base 49.52 »» 24.38 
Iperp. 13.56 



29712 
24760 
14856 
4952 

160)671.4912(4*. 0. 31. 
31 

Perches. 
2. Perp. 27 12 
Half base 24.76 



16272 
18984 
10848 
5424 



A. R. P. 

671.4912 » 4. O. 31. 



But, square perches may be reduced to acres, 
&c. rathor more ^ommodiously, by diyiding by 40 
and 4> than by 160; thus, 

4|0)67|1. ' 

4)16. 31 

A. 4. 0. 31 



Tojind the Content of Ground, SU 

Perches. 
3. Base 49.52 
Perp. 27.12 

9904 
4952 
34664 
9904 



1342.9824 

■ A. R. P. 

671.4912 = 4. 0. 31. 



The map may be readily drawn, having the dis- 
tance from either end of the base, io the perpen- 
dievlar given ; as may be evident from^the figure* 



PJROB. rii. 



ne content of a trkmgvltr fdece of ground^ and the base given, (o 

find the perfiendictUar, 



Divide the content in perches, by half the base 
in perches ; and the quotient will give you the per- 
pendicular, in perches and so in chains. 

Examples. 

Pl. \.fg. 16. 

Let BC be a ditch, whose length is 24C. AOL. 
by which it is required to lay out a triangular 
piece of ground, whose content shall be 4A, IR. 
lOP, Required the perpendicular, 



212 ToJMtheCanUta<fOr«tmd. 

C%. JL. Perches. 
Base 24.40 » 49^ 
Half the base >» 24^ 

A. R. P. 

4. 1. 10. 
4 



17 
40 



Perches. 



24.8)690(27.28 



1940 



2040 



560 



64 



Perches. Ch. L. 
Answer perp. 27.28. « 13.45. 

This perpendicular being laid on any part of th^ 
base, and lines run from its extremity to the ends 
of the base, will lay out the trianele Tby cor. to 
(faeo. 13. geom.) so that the perpendicular may be 
set on that part of the base which is most conve- 
nient and agreeable to the parties concerned. 



Ti>fi»A the Caatmt rf Chmmd. 213 
LEMMA. 

Ifjrom ha(f the turn qf the sides <if any plane trkmgle ABC^ each 
particuiar side he taken ; and if the haif sum^ ana the three re- 
mamaers be. multifUied contmtuUlv into tach other ^ the square roof 
qf tins product vfiU be the area rf the^ triangle, 

Bi§ect any two of the angles, as A and B^ with 
the lines Abj BD meeting in D j draw the per- 
pendiculars DE, DF, DG. 

The triangle AFD is equiangular to AED ; 
for the angle FAB—EABhy construction, and 
AFD=.^EDy being each a right angle, and of 
consequence ADF=ADE ; wherefore AD • 
JDE : : AD : DE : and since AD bears the same 
proportion to DF, that it doth to DE, DF^DE, 
and the triangle AFD^AED. The same way 
DE=DO, and the triangle DEB=DGB, and 
FDi=:DE=DG ; therefore D will be the cen- 
tre of a circle that will pass through E, Fj G. 

In the same way if .4 and C were bisected, the 
same point 2> would be had ; therefore a line from 
Dio C will bisect C, and thus the triangles DFC, 
DGC will be also equal. 

Produce OA to H, till AH=EB or GBj sd 
will HC be equal to half the sum of the sides, vis. 
to ^AB, + i AC + iBC; for FC, FA, EB, are 
severally equal to CG, AE, BG ; and all these 
together are equal to the sum of the sides of the 
triangle ; therefore FC f FA + EB or Cff, are 
equal to half the sum of the sides* 

FC= CH—ABSor AF=AE, and HA=EB; 
therefore HF=AB; md AF=CH--'BC; for CF 



SI 4 To find the Content of Ground. 

=.CG, and AH=GB ; therefore BC^HA^FC, 
and AH = CU—AH. 

Continue jDC, till it meets a perpendicular 
drawn upon H in K; and from K draw the per- 
pendicular Kly and join AK. 

Because thd angles AHKdLndAIKareiwo right 
ones, the angles SlA and K together, are equal 
to two right ; since the angles of the two trian-^ 

fles contain four right : in the same way FDE + 
^AE=(2 right angles=) FAE+IAH; let FAE 
be taken from both, then FDE=lAHy and of 
course FAE ^ K ; the quadrilateral figures 
AFDEy and KHAlj are therefore similar, and 
have the sides about the equal angles propoition* 
al ; and it is plain the triangles tiFD and CHK 
are also proportional : hence^ # 

FD'HA::FA: HK 
FD:FC ::HK:HC 



Wherefore by multiplying the extreme, and 
means in both, it will be the square of FD x HK 
X HC^FCx FAX HAxHK ; let HAT be taken 
from both, and multiply each side by CH ; then 
the square of CH x by the square of FD^FC^ 
FAkHAxCH. 

It is plain, by the foregoing problem, that 1 AB 
X JDJB, +i BC X DO + i ACxFjD = the area of 
the triangle ; or that half the sum of the sides, viz. 
CH>< l^jD=the triangle ; wherefore the square of 
Cif X by the so uare of FD= I^X K4 X jEL4 X C/T, 
that is, the half sum multiplied continually into 
the differences between the half sum and each side, 
will be the square of the area of the triangle, and 
its root the area. Q. E. 2>. 



To find the Content of OrounJL fZld 
Hence time following problem will be evident. 

PROS. riiL 

Tht tktceMt$ qfatUan$ Mangle ghfen tojhtd i&e are({, 

BULE. 

From half the sum of the three sides subtract 
#ach side severally ; take the logarithms of half 
the sum and three remainders, and half their total 
will be the logturithm of the area : or, take the 
square root of the continued product of the half 
mm and three remainders for the area. 

Examples. 
1. Jnthe triangk ABC, are 



Ciren, \ ^C= 12.28 J ^"'■"P^if.K '"*""! 

s 9.00) 



Sum 31.92 



Half sum 15.96 Log. 1.203033 

5.32 — 0.725912 

Remaindera ^ 3.68 ~ 0.565846 

6.96 — 0,842609 



2)3.337402 



Aqswer, Sqr. Ch. 46.63 Log. 1.668701 

or, 4.663 Acres. 



Or, 15.96 X 5.32 >i 3.68 x 6.96 » 2174.71113216 ; 



SI6 '^fifui the Ckmieni ^ GhrminA 

the square root of whkh is 46^639 for tke area as 
before. 

2. What quantity of land is contained in a tri- 
angle, the 3 sides of which are, 80, 120 and 160 
perclMSs refi|>ectiyely ? Answer»29A. 7P. 



PROB. IX. 



Two Met of a/UmtC'triangle and their included angle given, tm 

find ike area* 



Rule. 

To the log. sine of the siven angle (or of its sup*^ 
plement to 18(y*, if obtuse; add the logarithms of 
the containing sides ; the sum, less radius, will be 
the logarithm of the double area. 

Examples. 

Suppose two sides, 4^^ JC, of a triangular lot 
jtBCf form an angle of 30 degrees, and measure 
one 64 perches, and the other 40.5, what must the 
content be ? 

Given angle 30*. sine 9.698970 

r^r^i^i^tr^^ aiA^a i 64. log. 1.806180 
Containing Mdes } ^^^ ,^| ^^^^^^ 

2)1296. log. 3.112605 
160)648(4A. 8P. answer. 
8 



To find the ContMt of Ground. 217 

2: Required the area of a triangle/ two sides of 
which are 49.2 and 40.8 perches^ and their con- 
tained angle 144t degrees? Answer, 3A. 2R. 22P. 

3. What quantity of ground is inclosed in an 
equilateral triangle, each side of which is 1 00 pereh- 
es^eithor angle being 60 degrees? Answer,27A. lOP. 

Demonstration qf this problem. 

Pl, tl. Jig. 5. 

Let ^H be perpendicular to Ah and equal to 
ACy and HE^ jFCO, paralel to AB ; then making 
AH{^ AC) radius, AF{^ CD) will be the sine 
of CAD, and the parallelograms AB EH (the pro- 
duct of the ^iven sides,) and ABGF the double 
area of the triangle) having the same base AB, ara 
in proportioti as their heights AH, ALE; that is, 
as radius to the sine of the given angle ; which pro- 
portion gives the operation as in the rule above. 



PROB. 



Tojind the area tfa trafiezM^ viz, a Jigute bounded by four right 
Hn^e, two ffvfhkh are parallel, buttmeqmU 



R«LE. 

Multiply the sum of the parallel, sides by their 
[>endicuU " ' ^ .^ ^ %r.x. m . 

the area. 



perpendicular distance, and take half the product 
for th 



NoTS. On this 10th problem are founded most of the cal- 
culations of differences by latitude and departure, and those 
hj offsets', following in this treatise* 

Ff 



218 Tojind Ike Cwteall tf Qrwaf. 

Examples. 

1. Required the area of a trapezoid, of which 
the parallel sides are, respectively, 30 and 49 
perches, and their perpendicular distance 6L6? 



30+49 



^^\^\ Multiply. 



2)4866.4 
Answer, 2433*2 =15A. 33.2P. . 

2. In the trapezoid ABCD the parallel side* 
are, :4D, 20 perches, JBC, 32, and their perpendi- 
cular distance, AB^ 26 ; required the content ? 



/ 



Answer, 4A. 36P. 



P ROB. XL 

To find the Content of a trapezium. 

Rule. 

Multiply the diagonal, or line joining the re- 
motest opposite angles, by the sum of the two per- 
pendiculars falling from the other angles to that 
diagonal, and half the product will be the area. 

Example, 

Pl. 7. Jig. 3, 

Let ABCD be a field in form of a trapezium^ 
the diagonal AC 6 1.4 perches, the peipendicular 
Bh 13.6 and Dd 27.2, required the content ? 



I 
I 



1P»Jind the CoHieM cf ChinM. • 21t 



Diagonal = 64.4 ) ^^^^x^y^ 
13,64+27.2=40.8 1 "'«»"PV 



2)2627.52f 

160)131376(8A. 33iP. Answer 
1280 



331 perches. 



Note. The method of multiplying together the 
half sum.s of the opposite sides of a trapezium for 
the content is erroneous, and the more so the more 
oblique its angles are. 

To draw the map set oS Ah 28 perches, and Ad 
34.4, ancl there make the perpendiculars to their 
proper lengths, and jqjn their extremities to those 
of the diagonal. 

PROB. XII. 

To find the ana tf « cbrcU^ ar^tn eiSfim, 

Rule. 

Multiply the square of the circle's diameter, or 
the product of the longest and shortest diameters 
of the ellipsis by .7854 for the area. Or, subtract 
0.104909 from the double logarithmof the circle's 
diameter, or from the sum of the logarithms of 
those elliptic diameters, and the remainder will be 
the logarithm of the area. 

» 

Note. In any circle, the 

Diam. muUL < i « o i ^ i cq S produces the Cir. 
Circum. dir. j wy-^-^^Joy, ^ ^^^^^ ^^^ ^j^^ 



fi20 To find the CkmienS qf GrmmiL 

t 
Examples. 

1. How many acres are in a circle of a mile 
diameter ? 

1 Mile3=320 per. log. 2.505150 

2.505150 



5.010300 
0.104909 



4|0)8042|5. log. 4.905391 

4)2010.25 

Answer, 502A. 2R. 25P. 

2. A gentleman, knowing that the area of a cir- 
cle is greater than that of any other figure of equal 
perimeter, walls in a circular deer park of 100 
perches diameter, in which he makes an elliptical 
nsh pond 10 perches long by 5 wide ; required the 
length of his wall, content of ^ park, and area of 
his pond? 

Answer, the wall 314.16 perches inclosing 49^. 
14P. of which 394 perches, of i of an acre nearly, 
is appropriated to the pond« 

\ 

\ fBOB, XilL 

The area of a circle given^ tojlndits diameter. 

Rule, 

To the logarithm of the area add 0.104909, and 
half the sum will be the logarithm of the diameter. 
Or, divide the area by .7854, and the square-root 
of the quotient will be the diamete^. 



Tojhd the Content of Ground. asi 

Examples. 

A horse in the midst of a meadow suppose. 
Made fast to a stake by a line from his nose. 
How long must this line be, that feeding all 

round. 
Permits him to graze just an acre of ground ? 



Area in perches 160 log. 2.204120 

0.104909 



2)2.309029 

2) 

Diametei; 14.2733 log, 1.154514 



I ^ 



Aii8wel-> 7.13665 per. = IHF. 9 In. 

PROS. XIV. 

Mlowancefor roade. 

It is customary to deduct 6 acres out of 106 for 
road^ ; the land before the deduction is made may 
be termed the gross^ and that remaining after such 
deduction, the neat. 

Rule. 



The gross div. ^ u^ j /^r; J quotes the neat. 
Theneatmul. J ^7*-"^^ ( prod, the gross. 



Examples. 

1 . How much land nmst I inclose to haye 850 A 
2R. 20P. neat ? 



SK Tajki the QmtaiU of &rMm£ 



40 
4 



20. 
2.5 
— — Acres. A. R. P. 



850.625X1.06»901.662d=901.2.26. ibb ans. 

2. How much neat land is there m a tract of 
901 A. 211. 26P. gross? 

40126. 
4| 2.65 

Acres. A. R. T. 

1.06)901.6625(850.625== 850. 2. 20. the answ. 
848 



&c 



Note. These two operations proTe each ether. 



PROB. 



To Jind the area of a piect <^ ground be it ever <o irregtUar by di^ 

indmg it into trkmglea and tra/iezia, 

I^L. 7. Jig. 4. 

We here admit the surrey to be taken and prch 
tracted ; by having therefore the map, and know- 
ing the scale by which it was laid down, the con- 
tent may be thus obtained. 

Dispose the given map into triangles, by fine 
pencilled lines,, such as are here represented in the 
scheme, and number the triangles with 1> 2, 3, 4, 
&c. Your map being thus prepared, rule a table 
with four columns ; the first oi which is for the 
number of the triangle, the second for the base of 
* it, the third for the jierpendicular, and the fourth 
for the content in perches. 



Tojlni the Content of Ground. 223 

Then proceed to measure the base of number 1, 
from the scslle of perches the map was laid down, 
and place that in the second column of the table^ 
under the word base ; and from the angle opposite 
to the base, open your compasses so, as when one 
foot is in the angular point, the other being mov- 
ed backwards and foi^wards, m^y just touch the 
base line, and neither go the least above or be- 
neath it; that distance in the compasses measured 
from the same scale, is the length of that perpen- 
dicular, which place in the third column, under the 
word perpendicular. 

If the perpendiculars of two triangles fall on one 
and the same base, it is unnecessary to put down 
the base twice, but insert the second perpendicu- 
lar opposite to the number of the triangles in the 
table, and join it with the other perpendicular by 
a brace, as No. I & 2, 4 & 5, 6 & 7, 9 & 10, SCc. 

Proceed after this manner, till you have mea- 
sured all the triangles ; and then by prob. 6. find 
the content in perches of each respective triangle, 
which severally place in the table opposite to the 
number of the triangle, in the fourtli colunm, un- 
der the word content. 

But where two perpendiculars are joined to- 
gether in the table, by a brace having both one 
and the same base ; find the content of each (be- 
ing a trapezium) in perches, by prob. 11. which 
place opposite the middle of those perpendicu- 
lars, in the fourth column, under the word con- 
tent 

Having thus obtained the content of each re- 
spective triangle' and trapezium, which the map 
r:ontains, add them all tosrether, and their sum will 



r • 



224 



To find the CoatetU of Ch'ound. 



be the content of the map in perches ; which be- 
in^ divided by^l60, gives the content in acres. 
Tnus, for 



EXAUFI^S. 



No. 



1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 



Base. 



24.8 

28.2 
39.8 

494 

387 
40.0 

42.8 

26.2 
24.0 



i:*erpend, 



17.0 
16.3 
16.0 
19.6 
16.2 
29.0 
15.0 
6.7 
17.0 
13.0 
10.2 
12.3 
17.9 
11.6 
10.0 



I 



Content. 



412.92 

225.6 

712.42 

1086.8 
129.64 
600. 

481.5 

234.49 

259.2 



Content in perches 4142.57 



This being divided by 160, will give 25A. 3R. 
22P. the content of ihe map. 

Let your map be laid down by the largest scale 
your paper wUl admit, for then the bases and per- 
pendiculars can be measured with greater accura- 
cy than when laid down by a smaller scale, and if 
possible measure from scales divided diagonally. 

If the bases and perpendiculars were measured 
by four-pole chains, the content of every triangle 



To/itc? the Ckmtent of Otottlid. 2i5 

and trStpeKiuiQ, may be had as before, in problems 
6, and 11. and consequently the whole content of 
the map. 

If any part of yom- map has short or crooked 
bounds, as those represented in plate 7. jfig. 5* 
then by the straight edge of a transparent jbom, 
draw a fine pencilled line as AB to balance th« 
parts taken and left out, as also another, BC • 
these parts when small, may be balanced very 
nearly by the eye, or they may be more accurate- 
ly balanced by method the third. Join the points 
A and C by a line, so will the content of the tri- 
angle ABCf be equal to that contained between 
the line AC, and the crooked boundary from A to 
JB, and to C: by this method the number of triangles 
will be greatly lessened, and the content become 
mote certain ; for the fewer operations you have, 
the less object will you be to err : and ^ an error 
be committed, the sooner it may be discovered. 

The lines of the map should be drawn small, and 
Heat, as well as the bases ; the compasses neatly 
pointed, and scale accurately divided ; without 
an which you may err greatly. The multiplica- 
tions should be run over twice at least, a^ aho the 
addition of the column content. 

From what has been said, it will be easy to sur- 
vey a field, by reducing it into triangles, and mea- 
suring the bases and perpendiculars by the chain. 
To ascertain the content only, it is not material to 
know at what part of the base the perpendicular 
was taken : since it has been shewn (in cor. to 
theo. 13. geom.) that triangles on the satne base, 
and between the sanae parallels, are equal ; but if 
vcu would draw a map from the bases and perpen- 



226 



To find the dmtent of Ground. 



diculars, it is evident that you must know at wbat 
part of the base the perpendicular was taken^ in 
order to set it off in its due position ; and hence 
the map is easily constructed. 



PROS. jtr/. 



To determine the area qfa/tiece of ground, having! the m^ given, 
by reducing it to one triangle equal thereto^ am thence Jindmg 
it4 content' 

9 

PL.8.Jlg.5. 

LeiABCDEFGHhez map of ground, 
which you would reduce to one tnangle equal 
thereto. 

Produce any line of the map, as AH, both ways,, 
lay theed^e of a parallel ruler, from A to C, baring 
B above it ; hold the other side of the ruler, or 
that next you, fast ; open till the same edge touches 
B, and by it, with a protracting pin, mark the 
point b, on the produced line, lay the edce of the 
ruler from b to J5, having C above it, nold the 
other side fast, open till the same edge touches C, 
and by it mark the point c, on the produced line. 
A line drawn from cio JO will take in as much^s 
it leaves out of the map. 



Again lay the edge of the ruler from Hio F, 
having G above it, keep the other side fast, open 
till the same edge touches (?, and by it mark the 
point gy on the produced line ; lay the edge of 
the ruler from g to E^ having JPabove it, keep the 
other side fast, open till the same edge touches JF^ 
and by it mark the point /, on the produced line. 
Lay the edge of the ruler from/ to i>, having E 



To find the Content of Ground., ^27 

%boye it, keep the other side fast, open till the same 
edge touches E^ and hy it mark the point e, on the 
produced line. A line draivn from D to e, will 
take in as much as it leaves out Thus have )rou 
the triangle cD e^ equal to the irregular polygon 
ABCDEFGIL 

If when the ruler's edge be applied to the points 
A and Cy the point B falls under the ruler, hold 
that side next the said points fast, and draw back 
the other to any convenient distance ; tlien hold 
this last side fast, and draw back the former edge 
to jB, and by it mark &, on the produced line ; and 
thuij a parallel may be drawn to any point under 
the ruler, as well as if it were above it. It is best 
to keep the point of your protracting pin in the 
last point in the extended line, till you lay the 
edge of t^e ruler from it to the next station, or 
you may mistake one point for another. 

This may also be performed with a scale, o^; ru- 
ler, which has a thin sloped edge, called a fiducial 
edge ; and a fine pointed pair of compasses. Thus, 

Lay that edge on the points A and C, take the- 
distance from tne point jd to the edge of the scale, 
so that it may only touch it, in the same manner as 
you take the perpendicular of a triangle; carry 
that distance down by the edge of the scale paral- 
lel to it, to h ; and there describe an arc on the 
point h ; and if it just touches the ruler's edge, 
the point h is in the irue place of the extended 
line. Lay tlien the fiducial edge of the scale from 
h to D, and take a distance from C, that will just 
touch the edge of the scale ; carry that distance 
along the edge, tUl the,.point which was in C, cuts 
the produced line in c ; keep that foot in r, and 



'22fi To find the Cantmt of- dround. 

describe an arc, and if it jiuit touches the ruler's 
edge, the point c is in the true place of the extend- 
ed line. Draw a line from c to i>, and it will take 
in and leave out equally : in like manner the other 
side of the figure may be balanced by the line e 
D. 

Let the point of your compasses be kept to the 
last point of the extended line, till you lay your 
scale from it to the next station, to prevent mis- 
takes from the number of {)oints. 

That the tri?in?le c D e, is equal to the ri^t- 
lined figure ABCDEFGH, will be evident from 
profi^ems 18. 19. geom. for thereby, if aline were 
drawn from h to C, it will give and take equally, 
and then the fiffure hCDEFGH, will be equal 
to the map. Thus the figure is lessened by one 
side, and by the text balance line will lessen it by 
two, and so on, and will give and take equally* 
In the same manner an equality will arise on the 
other side. 

The area of the triangle is easily obtained, as 
before, and thus you have the area of the map. 

It is best to extend one of the shortest lines of 
the polygon, because if a very long line be pro- 
duced, the triangle will have one angle very ol>-, 
tuse, and consequently the other two very acute ; 
ji^'^ in which case it will not be easy to determine ex- 
actly the length of the longest side, or the points 
where the balancing lines cut the extended one. 

This method will be found very useful and rea- 
dy in small enclosures, as well as very exact ; i^. 
may be also used in large ones, but great care must 
be taken of the points on the extended line, which 
will be crowdef);, as well as of not nii.ssinga station. 




To find the CoiUeta of Chround. 22^ 
PROB. XVII. 

A mafi wUhUs area being gtven^ and ita 4cale otmtted to be eUher 
drawn or mentioned ; to find the acaie* 

%JAST up the map by any scale whatsoever, and 
t will be . 

As the area found • 

Is to the square of the scale by which you cast up, 
: : The given area of the map 
To the square ot the scale by which it was laid 
down. 

The square root of which wiU give the scale. 

Example. 

A map whose area b 126 A. 3R. 16P. being 
given ; and tlie scale omitted to be either dntwn 
or mentioned ; to find the scale. 

Suppose this map was cast up by a scale of 20 
perches to an inch, and the content thereby pro- 
duced be 31A. 2R. 34P. 

As Ihe area found, 31 A. 2R. 34P.=5074P. 
Is to the square of the scale by which it was cast 

up, that is to 20x20=±;400, 
: : The given area of the rtiap 126A. 3R. 16P. 
«20296P. 



To the square of the scale by which it was laid 
down. 



5074 : 400 : : 20296 : 1600 the square of the 
required scale. 



■ » 



230 To Jind the Content of Orowni. 

Root 
1600(40 
16 






8(00 

I 

Answer. The map was laid down by a scale of 
40 perches |p an incb. 

PROB. xriii. 

How iojtnd the true content of a survey^ th<mgh it he taken by a 

chtdn that ia too long cr too ahort. 

Let the map be constructed, and its area found 
as if the chain were of the true length. And it 
will be. 

As the square of the true chain 

Is to the content of the map, 

: : The square of the chain you surveyed by 

To the true content of the map. 

Example. 

If a survey be taken with a chain whidh is 3 
inchies too long ; or with one whose length is 42 
feet finches, and the map thereof be found to con- 
tain 920 A. 2R. 20P. Required the true content. 

As the square of 42F. Oln.=the square of 504 

inches=254016. 
Is to the content of the map 920 A. IR. 20P.= 

147260P. 
: : The square of 42F. 3ln. =the square of 507 

inches =257049. 
To the true content 



3V> find the Omtent of Chromut. 231 

P. P. 

250416 : 147260 : : 257049 : 149019 

A. R. P. 
160(149019(931. 1. 19 Answer. 



■■"J 



501 



■Va 



219 



40)59(1R. 
19P. 



S* 



(232 ) 



Method of determining* the Areas op right* 
LINED Figures universally, or by calculation. 



DEFINITIONS, 



'• JMLeRIDIANS are north and eouth lines, 
i^hich are supposed to pass through every statioB 
of the survey. 

2, The diflTei-ence of latitude, or the northing or 
southing of any stationary line, is the distance that 
one end of the line is north or south from the 
other end ; or it is the distance which is intercepted 
on the meridian, between the beginning of the 
stationary line and a peipendicular drawn from 
the other end to that meridian. Thus, if N. S. be 
a meridian line passing through the point A of the 
line ABf then is Ab the diflference of latitude or 
^louthing of that iincr 

3. The departure of any stationary line, is the 
neai'est distance from one end of the line to a me- 
ridian passing through the other end. Thus Bb it 
the departure or easting of the line AB ,• but if 
CB be a meridian, and the measure of the station- 
ary distance be taken from JB to -4 ; then is BO 
the difference of latitude, or northing, and ACibB 
departure or westing of the line BA. 



COMPUTATION, &a 233 

4. Tfiat meridian which pas^s through the first 
station, is sometimes C0lled the first meridian ; and 
sometunes it is a meridian passing on the east or 
west side of the map, at the distance of the breadth 
thereof, from east to west, set off* from the first 
station. 

5. The meridian distance of any station is the 
distance thereof from the first meridian, whether 
it be supposed to pass through the first station, or 
on the east or west side of the map. 



THEO. L 

In every survey which is truly taken, the sum of 
the northings will be equal to that of the south- 
ings ; and Uie sum of the eastings equal to that of 
the westings. 

Let a, 6, c, c,yj gy A, represent a plot ot parcel 
of land. Let a be the fii^ station, h the second, 
c the third, SCc. Let NS be a meridian line, then 
will all lines parallel thereto, which pass through 
the several stations, be meridians also ; as ad^ o$, 
cd, SCc. and the lines 5o, cs, de, SCc. perpendicular 
to those, will be the east or west lines> or depar- 
tures. 

The northings, ti+go+hq=mzao+bs+cd+fr ihci 
southings : for let the figure be completed ; then it 
is plain that go+hq+rk=^w+ks+cd, and ei — 
rk=fr. If to the former part of tliis first equation 
ei — r* be added, and/r to the latter, iheu go-hhq 
+ei=sao'^bs+cd+fr ; that is, the sum of the north- 
ings Is equsil to that of the southings* 

Hh 



^.. 



234 COMPUTATION 

The eastings cs+qa^^-^-de+if+rg+oh^tbe west* 
ings. For aq+yo ^azj ^^de+i/^Hrg+oht and fio«= 
cs — yo. If to the former part of this first equa- 
tion, cs — yo be added, ana bo to the latter, then 
€S+aq^ob+deHf+rg'H>h ; that is, the sum of the 
eastings is equal to that of the westings. Q. jE. i>. 



SCHOLIUM. 



This theorem is of use to prove whether the ^ 
lield-work be truly taken, or not ; for if the sum 
of the northings be equal to that of the soutit* 
ings, and the sum of the eastings to that of the . 
westings, the field-work is right, otherwise not. 

Since the proof and certainty of a survey de- 
pend on this truth, it will be necessary to shew 
how the difierence of latitude and departure for 
any stationary line, whose course and distance are 
given, may be obtained by the table, usually call- 
ed the Traverse Table. 



To find the difference of Latitude and departure, 

by the Traverse Table. 

This table is so contrived, that by finding there- 
in the given course, and a distance not exceeding 
120 miles, chains, perches, or feet, tlie difierence 
of latitude and ^departure is had by inspection: 
the course is to be found at the top of the table 
when under 45 degrees ; but at the oottom of the 
table when above 45 degrees. Each column sign* 
^d wi(h a course consists of two parts, one for the 



OP AREAS. 23d 

difference of latitude, marked Lat. the oilier for 
the departure, marked Uep* which names are both 
at the top and bottom of these columns. The 
distance i^ to be found in the column marked Diat 
next the left hand margin of the page. 

Example. 

In (be use of this table, a few observations only 
are accessary. 

I.. If a station consist of any number of even 
chains or perches (which are almost the only mea-^ 
sures used in surveying) the latitude and depar- 
'ture are found at sight under the bearing or course, 
if less than 45 degrees; or over it if more, and in 
a linct with the distance. 

f 

2. If a station consist of any number of chains 
and perches, and decimals of a chain or perch, un- 
der the distance 10, the lat and d^. will be found 
as above, either over or under the bearing ; the 
decimal point or separatrix being removed one 
figure to the left, which leaves a figure to the 
right to spare. 

If the distance be any number of chains or 
perches, and the decimals of a chain or perch, the 
fat. and dep. must be taken out at two or more 
' operations, by taking out the lat. and depi. for the 
covins or perches in the first ^lace ; and then for 
the decunal parts. 

To save the repeated trouble of additions, a ju- 
dic^ious surveyor will always limit his stations to 
whole chains, or perches and lengths, which can 
commonly be done at every station, save the last- 



236 COMPUTATION 

1. In order to illustrate the foregoing observa- 
tions, let us suppose a course or bearing, to be & 
35^ 1 5' E. and tbe distance 79 four-pole chains. 
Under 35*. Ij5', or 35i de^ees ; and opposite 79, 
we find 64. 52 for the latitude, and 45. 59 the de« 
parture, which signify that the end of that station 
differ in latitude from the beginning 64. 52 chains, 
and in departure 45, 59 chains. 

Note. We are to understand the same things if 
the distance is given in perches or any other mea- 
sures, the method of proceeding being exactly the 
$ame in every case* 

Agiain, let the bearing be 545 degrees and dis- 
tance as before ; then over said degrees we find 
the same numbers, only with this diiOference, that 
the lat. before fecund, will now be the dep. and the 
dep. the lat. because 54i is the complement of 35£ 
degrees to 90, vis. lat 45. 59. dep. 64. 52. 

2. Suppose the game course, but the distance 
7 chains 90 links, or as many perches. Here we 
find the same nnmbers, but the decimal point must 
be removed one figure to the left. 

Thus, tinder 35i, and in a line with 79 or 7.9, 
are 

Lat. €. 45 
Pep. 4. 56 

tlie 5 in the dep. being increased by 1, because the. 
p is rejected ; bijt over 54* we get 

Lat. 4. 56 
Dep. 6. 45 



OF AREAS. 23T 

3. Let tbe course be as before, but the distance 
7.789 then opposite 

7. 70 Lat. 6. 29 Dep. 4. 43 
9 7 6 



7. 79 6. 36 4. 49 



Or opposite 

7. 00 Lat 5. 72 Dep. 4. 03 
.79 .64 .46 



7. 79 6. 36 4. 49 



THEO. Ih 



When the first meridian passes through the map. 

If the cast meridian dhtancea in the middle <^ each line be 
multifiUed into the particular Bouthingy and the west meridian 
distances into the fiartieular northings the sum q/* these firO' 
ducts mill be the area of the mafi. 

Pl. iO.flg. 1. 

Let the figure abkm be a map, the lines, ab hk 
to the southward, and km ma to the northward, 
NS the first meridian line passing through the first 
station a* 

The meridian! zdy^ao 1 ^a. \(im 
Distances east J tu^oxt^bj/Jj ^ ^^^J ow 

•The meridian ") efxgx 1 _* Ixp 
pistances west/ hh^gafmyJ} ""^^^^Jg/ 



28« COMPUTATION 

These four areas am-i-ow+xp+gl will be .the 
area of the whole figure cmswiprk, which is equal 
to the area of the map abkm. Complete the 
^gure. 

The parallelograms am and ow, are made of the 
east meridian distances dg and tu, multiplied into 
the southings ao and ox. The parallelograms xp 
and gl are composed of the west meridian dis- 
tances </*and hhy multiplied into the northings *j:^ 
0[kdga (my J but these four parallelogranis are 
equal to the area of the map ; for if from them be 
taken the four triangles marked Z^ and in the 
place of those be substituted the four triangles 
marked O, which are equal to the former ; then it 
is plain the area of the map will be equad to the 
four parallelograms. Q. E. D. 

THEO. TIL 

'1/ the meridian'dia$ance wktn eu9t^ he muUifilied into the 90uth' 
ing9^ and the meridtan diatanee when veat be mulfifiHed into 
the northmga^ the aum q/* theae leaa By the meridian distance 
when we9ty multijfUied into the aouthing%^ ie the area ^ the 
survey. 

Pl. 10. Jig. 2. 

Let ai che the map. 

The figure bein^ completed, the rectangle i^is 
made of the meridian distance eg when east» mul- 
tiplied into the southing an ; the rectangle vk is 
made of the meridian dMance xtv, multi,^ied into 
the northings cz or ya. These two rectangles, or 
parallelograms, af+yk, make the area of the figure 
^nyikdj vcom which tsJcing the rectangle otf^ made 
of the meridian distance to when west, into the 
southings oh or 6m, the remainder is the area of 
the figure dfoKkd, which is equal to the area of 
the map. 

Let bou= F, urih^Ly ric^Oy wrc= Z=,akfv^ 
Ky and efh^B, ade^A. I say, that T+Z+B= 
K+IA-A. 



OF AREAS. 239 

T:=^L+0, add Z to both, then T+2^L+0+ 
Z; but Z+0« JT, put JSTinstead of Z^tO; then F+ 
Z^L+K, add to both sides the equal triangles B 
and X then F+Z+jB=jL+A+^. If therefore B+ 
y+Z be taken from air, and in lieu thereof we 
put L+K+Aj we shall have the figure dfokikd^ 
ahc, but that figure is made up of the meridian 
distance when east, multiplied into the southing, 
and the meridian distance, when west, multiplied 
into the northing less by the meridian distance^ 
when west, multiplied into the southing. Q. E. D. 

COROLLARY, 

Since the meridian distance (when west) multi- 
plied into the southing, is to be subtracted, by the 
same reasoning the meridian distance when east, 
multiplied into the northing, must be also sub- 
tracted. 

SCHOLIUM. 

From the two preceding theorems we learn how 
to find the area of the map, when the first meri- 
dian passes through it ; that is, when one part of 
the map lies on the east and the other on the west 
side 01 that meridian. Thus^ 

Rule. 

The merid.l east fmultiplied fsouthings") 
Dist. when J west 1 into the Inorthings J 
their sum is the area of the map. 

But, 

The merid. feast 1 multiplied f northings > 
Dbt. when 1 west J into the 1 southings J 

the sum of these products taken from the formeS' 

gives the area of tne map. 



240 COMPUTATION 

These theorems are true,* when the surveyor 
keeps the land he surveys, on his right hand, 
which we suppose through the whole to be done ; 
but if he goes the contrary way, call the south- 
ings northings, and the northings southings, and 
the same rule will hold good. 

General RuU for finding the Meridian distaiiees. 

1. The meridian distance and departure, both 
east, or both west, their sum is the meridian dis- 
tance of the same name. 

2. The meridian diitance and departure of dit 
ferent names ; that is, one east and the other west> 
theu' difference is the meridian distance of the 
same name with the greater. 

Thus in the first method of finding the area, ai^ 
in the following field-book. 

Tlie first departure is put opposite the north- 
* ing or southing of the first station, and is tlie first 
meridian distance of the same name. Thus if the 
first departure be east, the first meridian distance 
will be the same as the departure, and east also ; 
and if west, it will be the same way. 

The first meridian distance 6.61 E. 

The next departure 6.61 E, 

• The second meridian distance 1 3.22 E. 

The next departure 1 .80 £^ 

The third meridian distance 1.5.02 E. 



OF AA£A8. 241 

At fltatioB 5, ^ meridiaQ dktance 5.78 £» 
The next departure 7.76 W* 



The next meridian distance 1,98 W. 



**wi*i 



At gtation 11, the meridian distance 0.12 W« 
The next departure 5.84 £. 

The next meridian diatonce 5.72 E« 



ik 



M 



tn the 5th dtid tiih stations^ the ttieridian di&* 
tance being less than the departures^ and of a con** 
trary name, the map will cross the first meridian^ 
und will pass as in the 5th line» from the east to 
the west line of the meridian; and in the^ 11th 
line it will a^ain cross from the east to the west 
aide, which will evidently appeari if the field-work 
be protracted, and the meridian line passing through 
the first dtation, be drawn through the map# 

The field-book cast up by the first method, will 
be evident frx>m the two foregoing theorems, and 
therefore requires no fiirtfaer explanation ; but to 
fmdihe areOi kSf ^ second method, take this 

RuiiBtf 

When fh^ meridian distances are east, put the 
products ttf north and south areas in their proper, 
columns ; but when west, in their contrary co^ 
lumns ; that is, in the column of south area, when 
the difierence of latitude is north ; and in nortli 
when south : the reason of which is plain, from the 
two last theorems. The difference of these two 
ci>Iumns will be the area of the map« 

li 



34^ 



ilM-Book, MtOind t 



Lat. and Merid 
halfDep Dist 



-KO.^ 



No. 

St. 

1 
2 



Bearings.! C. L. 



^ 



NE 75 



ikrika^ 



i3.ro 



N£20i 



East 



SW33J 



>*■*• 



10.30 



N 3.5^ 
E 6.61 



N 9.67 15.03 



E 1^0 



16.20 1 






35.30 



SW76 



North 



SW84 



16.00 



6.61 £ 
13.23 £ 



16.82 



aOO 24.92 £ 
E 8.10 33.02 £ 



S 89.44 23 28 



W 9.74 



S 3.87 



3.54 £ 



Area. 



Oedaet. 



d 



S 685.: 



1632 



5.76 £ 



" 



w rat 1.98 w 



8 



9 



10 



11 



12 



13 



NW5SJ 



NE36i 



9.00 



11.60 



11.60 



19^20 



N 9.00 
0,00 



S 1.21 
W 5.77 



1.98 W 
1.98 W 



23.3994 
144.9430 



22.3686 



i^." »■ 



7.75 W 
13.52 W 



N 6.94 18.16 W 
W 4^ 22.80 W 



1 7.8300 



N 15.38 



£ 5.7411.S2W 



NE32I 14.00 






SE76| 



SW 15 



SW 16| 



12.00 



10.85 



10.12 



N 12.93 
£ 2.68 



S 2.75 
E 5.84 



S 10.48 
W 1.40 



S 9.69 
|W 1.46| a 



If .06 W 



126.0304 

I 



262.3828 



«k»« 



8.64 W 
5.96 W 



0.12 W 
5 72 E 



111.7152 



9.3775 



0.3300 






4.32 
2.92 



1.46 Ej 
0.00 1 **' 



45.2786 



1474 



1285.1012 
178.0499 



«i^i^ 



MH^MIlftl 



Content in Chains, 1107.05 1 3 



*«^«*i 



178.0499 



taMlM*MM«te 



mmt^btmm 



J 



T%e foregoing Field-Book, Method 11. 243 



iJSf U needlesa here to insert the celurtma of bearing or disgance^, 
in chainsy t hey beinj^ the same aa before* 



mmmm' 



No. 
St. 



1 



7 



10 



11 



13 



13 



Lat. and 
half Dep. 



N 
E 



^.54 
6.61 



sr 

E 



Mei'id. 
Dist. 



N. Area. 



islj E ^''''' 



9.65 
1.80 



0.00 
8.10 



S 
W 



39.44 
9.74 



S 
W 



3.87 
7.76 



N 



9.00 
0.00 



1.31 



15.03 £ 
16.82 £ 

34i93~E 
33.03 £ 



144.9430 



33.38 E 
ld.S4 E 



5.78 E 
1.98W 



1.98W 
1.98W 

7.75 W 



W 5.77|13.53W 



N 
W 



6.94 
4.64 



N 15.38 
I 5.74 



N 

£ 



1393 
3.68 



I8.16W 
S3.60W 



17.06W 
n .33 W 

"8.64^ 
5.96W 



S 
£ 

S' 

W 1.40 



3.75! 0.1 3 W 
5.84: 5.73 E 



10.48 '4.33 



s 
w 



9.69 
1.46 



E 
3.93 £ 



1.46 E 
0.00 



■•»■ 



9.3775 



0.3300 



178.0499 



S. Area. 






685.3633 



33.3686 



17.8300 



136.0303 



363.3838 



111.7153 



45.3736 



14.1474 



Area in chainsy as before. 



r 



384.1013 
178 0499 



1107.0513 
I >■■■■,■■ ^ 



244 COMPUTATION 

CmiMifUCthn *of the Mapfnm eUher ikeUi^tkeU IhUei 

Pl. 10. Jtg, 3. 

Draw the line NS for a north and south line, 
which call the first meridian ; in this, line assume 
any point, as 1 , for the first station* Set the north* 
ing of that stationary line, which is 3.54, from 1 
to 2, on the said meridian line. Upon the point 2 
raise a perpendicular to the eastward, the ineri« 
dian distance being easterly, and upon It set 13.22, 
the second number in the column of meridian dis« 
tance from 2 to 2, and draw the line 1 2, for the 
first distance line : from 2 upon the first meridian, 
set the northing of the second stationary line, that 
is, 9.65 to 3, and on the point 3 erect a perpendi* 
cular eastward, upon wnich let the meridian dis- 
tance of the second station 16.82^ from 3 to 3, and 
draw the line 2 3, for the distance line of the se« 
cond station. And since the third station has nei- 
ther northing nor. southing, set the meridian dis- 
tance of it 33.02, fromS to 4, for the distance line 
of the third station* To the foinlh station there 
is 29.44, southing, which set from 3 to 5 ; upon 
the point 5, erect the peipendicular 5 /}; on which 
lay 13,54, and draw the line 4 to 5. 

In the like manner proceed to set the northings 
and southings on the nrst meridian, ftnd the meri- 
dian distances upon the perpendiculars raised to 
the east or west ; the extremities of which con* 
nected by right lines, will complete the map. 

A Sfiecinteu qf the Penntylvama Methtd qf CALCULATION ^ 
vfhkh^Jbr ii^ SbnfihcUy and J&tue^ m£namg the Meridian DU^ 
tuneeat it mfifioeed to he ptrferahU m fractieeto any TThiMg here^ 
tajbre fiuhMihed on the Subject. 

X IND hi the first place, by the Traverse Table, 
the lat. and dep. for the several courses and dis<i 

taqcesj 2(s silready taught; and if th^ survey be 



OFAREAa 245 

tndy taken, the sums of the tiortbiB|s and south-* 
ings will be equal, and also those of the eastings 
and westings. Then, in tl^ next place, findtbe 
meridian distances, by choosing such a»place in the 
column of eastings or westings, as will admit of a 
continual addition of one, and subtraction of the 
other; by which means we avoid the inconyenience 
of changing the denomination of either of the de- 
partures. 

The learner must not expect that in real prac-- 
tice the columns of lat. and those of dep. will ex- 
actly balance when they are at first added up, for 
little inaccuracies will arise, both from the obser- 
Tations taken in the &eld^ and in chaining ; which 
to acyust, previous to finding the meridian dis- 
tances, we may obserye> That if, l(i small sur- 
veys, the difference amount to two-tenths of a 
perch for every statioi^ there must have been some 
error committed in the field ; and the best way in 
this case, will be to rectify it on the ground by a 
re*survey, or at least as much as will discover the 
error; iBut when the differences are within those 
limits, the columns of northing, southing, easting, 
and westing, may be corrected as follows : 

Add all the distances into one sum, and sky, as 
that siim is to each particular distance, so is the 
difference between the sums of the columns of 
northing and southing to the correction of northing 
^ or southmg belonging to that distance : the correc- 
tions thus found are respectively additive, when 
they belong to the column of northing or southing, 
which is the less of this two, and subtractive when 
they belong to the greater ; if the course be due 
east or west, the correction is always additive to 
the less of the two columns of northing or souths 
ing. The corrections of easting and westing are 
found exactly in the same manner. 



A 



346 COMPUTATION 

This rule Was investigated two different ways, 
by N. Bowditcht Author of the Practical Navi-' 
gator, and R. Adrain, Prof. Math, and N. Phil. 
Columbia Col. N. Tork, as may be seen in the 
Analyst No. lY, published m 1808. 

The following example will sufficiently illus* 
irate the manner of applying the rule. 

In this example the sum of the distances is 79 1, 
and the difference between the columns of north- 
ing and southing, is .4, also the first distance Is 
70; say then, , 

791 : 70 : : A : .04 

which fourth proportional .04 is the first correc* 
tion belon^^ing to the southing 53.6, fi;om which 
the correction .04 should be subtracted. 

In this manner the several corrections of the 
southings 

53.61 .04^ 

29.1 > are found to be .09 > respectively. 
135.73 .07) 

But as only two of these corrections amount to 
half a tenth, we must use .1 for each of the cor< 
rections .09 and .07, and neglect tlie correction 
^04 ; thus the correct southings become 

53. 

29, 

135, 

In like manner from the remaining distances 
we obtain to 

62.9 ) .04 

the northings 101.1 f the additive corrections .06 

54.0 ( .03 

00.0 ) ,07 




OP AREAS. 



247 



And consequently, by neglecting .04, and .03^ 
and' using .1 for each oi the two .06 and .07, tb9 
northings 

62.9 
when corrected are 101.2 

54.0 
00.1 

In obtaining these corrections, it is commonly 
unnecessary to use all the significant figures of 
the distances : thus, for ti^e ratio of 791 to 70, we 
may gay, as 80 to 7. 





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09 



248 COMPUTATION 

The latitudes and departures being tfaus liaklie* 
ed, proceed to insert the meridian distances 'by 
the above method, where we still make use of the 
same field notes, only changing chains and links 
into perches and tenths of a perch. Then by look^ 
ing sdong the column of departure, it is easy to 
observe, that in the columns of easting, opposite 
station 9, all the easting may be added, and the 
westings subtracted, without altering the denomi- 
nation of either. Therefore by placing 46.0, the 
east departure belonging to this station in the co« 
lunm of meridian distances, and proceeding to add 
the eastings and subtract the westings, according 
to the rule already mentioned, we shall find that 
at station 8, these distances will end in 0^ 0, or a 
cypher, if the additions and subtractions be right- 
ly made« Then multiplying the upper meridian 
distance of each station by its respective northing 
or southing, the product will nve the norUi or 
south area, as in the #xamples already insisted on, 
and which is fully exemplified in the annexed spe- 
cimen. When these products are all made out, 
and placed in Iheir respective columns, their dif- 
ference will give double the area of the plot, or 
twice the number of acres contained in the survey. 
Bividethisremainderby 2, and the quotient thence 
arising by 160 (the number of perches in an acre), 
then will this last quotient exhibk the number of 
acres and perches contained in the whole survey ; 
which in this example may be called 110 acres, 
103 perches, or 110 acres, 2 quarters, 23 perches^ 



OP AREAS. 



249 



l^IELD'S^OTESf qf the tvio foregoing MetAod$, aa PfactUcd in Fcnric^^ 

aylvarda. 



Cast up by perches and tenths of a perch. 



ST 


ooarscM. 


DUL 


«. 


r 


52.9 


"IE" 


nacT 

'^85.3 
288.2 


N. Area. 1 8. Ar«ai.{ 




1 

s 
d 

4 
5 


N 75.00 E 

• 


54.8 

648 
141.S 
640 
36.0 
46.4 


142 
38.6 


3341.26 
11680.36 






N 20.30 E 




14^ 




302.6 
317.0 




. 


East. 




iir.7 


64.8 




381.8 
446.6 

36&7 
29a8 

166.0 






' 


S 33.30 W 

8 76.00 Vi 

North- 




46.0 
21.4 
46.7 


77.9 




43395.99 


1 




15.5 


62.1 




354i.85 
590.45 




« 


36.0 


4.9 


166.6 
166.6 

1*0.5 
744 


5977.60 




7 
8 

ft 

U 
W 

ts 


S 84.00 W 




46.1 


1034.'. 6 




N 53.15 W 


46.4 


87.8 


37.2 


37.^ 

oa(» 

46.0 
92.0 

113.4 

154.8 






N 36.45 £ 


76.8 
56.0 


61,5* 

5i.r 


'" ^ 1 




2829.00 
5862.78 






N t2s&3e £ 




a. 2 






S 76.45 e' 


48.0 


11.0 


181.5 

2i^8..£ 

217.0 
205.8 

194 V 

18^:4 




1996.50 




S 15,00 W! 43A 




^41.9 






9092.30 




<5 16.45 V 40.5 


iKi9.8 


38.8 
229.8 


J46.2 


11.7 


• 


rs31.08 

06I51T7 
K«745.16 

!viQ6.01 
177031 fcO** 


- 


■ I 


z^A.l 




30745.16 


. 








Area) 


2 
bperehet. 


1 

1 






















r 

■1 






( 250 ) 



SECTION IV. 



OF OFF-SETS. 

JlN taking; surveys it is unnecessary and unugudl 
to make a station at every an^^ular point, because 
the field-work can be taken with much greater ex* 
pedition, by u$ing ofi^sets and intersections, and 
with equal certainty ; especially where creeks> kc. 
bound tlie survey. 

Off- sets are perpendicular lines drawn or mea-* 
sured from the angular points of the land, that lie 
on the right or left hand to the stationary distance^ 
thus, 

Pl. II. Jiff. 24 

Let the black lines represent the boundaries €^ 
a farm or township : and let 1 be the first station ; 
then if you have a good view to 2,'omit the an^- 
lar points between 1 and 2, and take the bearing 
and length of the stationary line I, 2, and insert 
them in your field-book : but in chaining from 1 to 
2, stop at d opposite the angular point a, and in 
your field-book insert the distance from 1 to rf, 
which admit to be 4 C. 25L. as well aslhe measure 
of the off-set orf, which admit to be IC. 12L. thus: 
by the side of your field-book in a line with the 
first station, say at 4C. 25L. L. IC. 12L. that is, 
at 4C. 25L. there is an off-set to ihe left hand of 
IC. 12L. 



OP OFP-SETS- 251 

This done, proceed on your distance line to e 
opposite to the angle by and measure ehy supposing 
then 1 « to be 7C. 40L. and eh 3C. 40L. say (still 
in SI line with the first station in your field-book) 
« at 70. 40L. L. 3C. 40L." That is, at 7C. 40L. 
there i% an ofF-set to the left of 3C. 40Ij. proceed 
then with your distance line iof opposite to the 
angle c, and measure /c ; suppose then 1/ to be 
13C. and fc 1 C. 25L. say in the same line as before^ 
at 13C- L. IC. 25L. Then proceed from/ to 2, 
ar 1 you will have the measure of the entire sta- 
tionary line 1,2, which insert in its proper column 
by the bearing. 

In taking off-sets, it is necessary to have a perch 
chain, or a ^ta(ff of half a perch, divided into links 
for measuring them ; for by these means the chain 
in the stationary line is undisturbed, and'the num- 
ber of chains and links in that line from whence, 
or to which, the off-sets are taken, may be readily 
known. 

Having arrived at the second station, if you find 
your view will carry you to 3, take the bearing 
from 2 to 3, and in measuring the distance line^ 
stop at I opposite g ; admit 2/ to be 4C. lOL. and 
the offset ^ IC. 20L. then in a line with the se- 
cond station in your field-book, say at 4C. lOL. R. 
JlC. 20L. that is, the off-set is a right hand one of 
IC. 20L. Again at m, which suppose to be lOC. 
25L. from 2 ; take the ofi^set mh of IC. 15L. and 
in a line with the second station, say at lOC. 2SL« 
B. IC. 15L. In the same line when you come to 
the boundary at i, insert the distance 2t, 13C. lOL. 
thus, at I3C. lOL. 0; that i% at 13C. lOL. there is 
no off-set. At », which is 15C. from 2, take the 
off-set nk 45L. and still opposite to the second sta- 
tion say at l^C. L. 45. L. 



252 OF OFF-SETS': 

Lei the line, 3, 6, represent the boundary, which 
by means of water, briers, or any other impedi- 
ment, cannot be measured. In this case make one 
or more stations within or without the land, where 
the distances mav be measured, arid draw a line 
from the beginning of the first to the end of the 
last distance, thus; make stations at 3, 4, and 5, 
taking the bearings, and measuring the distances 
.as usual, which insert in your field-book, and draw 
a mark like one side of a parenthesis, from the third 
to the fifth station, to shew thai a line drawn from 
the third station to the farthest end of the fifth sta- 
tionary line will express the boundary. Thus, 



No. Sta. Deg. Ch. L: 
^3 172i 5.45 

4 200 ia25 
4 250 3.36 



Suppose the point/) of the boundary to be inac^ 
cessible, by means of the lines 6p or p7, being 
c^verflowed, or that of a quarry, furze, &c. might 
prevent your taking their lengths: in this case 
take the bearing of the line 6, 7, which insert op-, 
posite to the sixth station in your field-book with 
the other tearing ; then direct the index to the 
point p, and insert its bearings on the left side of 
the field-book, opposite to the sixth station, annex- 
ing thereto the words InL for hmmdary ^ and hav- 
ing measured and inserted the distance 6; 7, set the 
index in the direction of the line 7/?, and insert its 
bearing on the left of the seventh station of the 
field-book, annexing thereto the words Tnt for 
boundary : the crossing or intersection of these two 
bearings will determine the point /?, and of course 
the boundary 6p7 is also determined. 

If your view will then reach in the first station^ 



OP OFF-SETa 



25a 



taike its bearing, stationary line, and off-sets, as be- 
fore, and you have the field-book completed. Thus, 

The FieldrBook. 



Remarks and intersect. 



318 Inu to a tower 



33lj.Int. to ditto 



1 358 



3 297} 



I55j Int. fpr bound. 
374 Int. for ditto. 






3 

4 
5 
6 



C.L. 



OFF-SETS. 



22.13 At4C.25L.L. IC. 
12L. at 7C. 40L. 
L,. 3C. 40L. at 
13C. L. IC. 35L. 

22.12 At 4C. lOL. R. IC. 
20L.at 10C.35L. 
R. IC. 5lL. at 
isC. loL. 0. at 
15C.L*45L. 



172J 

200 

256 

125 

105j 



I 



5.45 
13.25 

3.36 
15.15 
15.10 



AtlC. 20L.L. 3C. 
20L. at 7C. 45L. 
L. 2C. 32L. at 
lie. 25L. 0. at 
12C.25L.R.36L 



Cloie at the fint statioo. 

If you would lay down a tower, house, or anj 
other remarkable object in its proper place ; from 
any two stations take bearings to the object, and 
their intersection will determine the place where 
Tou are to inseil it, in the mannes that the tower 
is set out in the figure, from the intersection taken 
at the first and second stations of the above field- 
book. 

A protraction of this will render all plain, on 
which lay off all your off-sets and intersections^, 
and proceed to find the content by any of the me- 
thods in section the 4th. 



2^1 



aV W-SETS. 



3^ foregoing ^fidd^ooJc may be othermae Jtepti 

t/lUSt 






No 



Remarks mi intersection. «^ Deg. 



3l8lnt toa tower 



238 J Int. for ditto. 



155} Int for bound. 



374 In« for boundary. 



1 368 



297J 



3 
4 




L. han. 
Off- set 
Ch. L. 


Dist. 
Ch.L. 


R. ban. 
Ofl-^et 
Cti.L. 


1.13 


4.25 




3.40 


' 7.40 




1.25 


13.00 






23.12 




1 

i 


4.10 


1.30 


• 


10.25 


1.15 




11.10 


• 


0.45 


15.00 
21.21 






5.45 






13.25 






3.36 






15.15 




2.20 


1.20 




2.32 


7.45 
11.25 


V 




12.25 


0.36 


•• 


15.20 








V 1 



Haw to tost up off-sets by the pen. 

1, 2—\f=2f—le=fey le^ld^ed. 

Then !(/ X ida^lday by prob. 6, page 183, and 
ic4xda+fc=b€fc^ and 2/ x hfc^cfd; the sura 



• 



OP OPF-SETa 29S 

of all which will be labcil ; the area contaioed 
between the stationary lini^ 1, 2, and the bounda- 
ry^ 1 abc 2« 

In the same manner you may find the area of 
2ihg2, of iA:3i, as well as what is without and with* 
inside of the stationary line 7, h 

If therefore the left hand off-sets exceed the 
right hand ones, it is plain, the excels must be ad- 
ded to the area within the stationary lines, but if 
the right hand off-sets exceed the left hand ones, 
the difference must be deducted from the said area; 
if the ground be kept on ttie right hand, as we have 
all along supposed ; or in words, thus ; 



To Jind the contents of off-sets* 



1. From the distance line, take the distance to 
the preceding offset, and from that the distance of 
the one preceding it, &c. in four-pole chains ; so 
will you have tlie respective distances from offset 
to off-set, but in a retrograde order. 

2. Multiply the last of these remainders by i 
the first offset, the next by i the sum of the first 
and second, the next by half the sum of the se- 
cond and third, the next by half the sum of the 
third and fourth, &c. The sum of these will be 
ttie area produced by the off^ts. 

Thus, in the foregoing field-book, the first sta- 
tionary line is 22C. 12L. or llC. 12L, of four-pole 
c^iii^. See the figure. 



I 



266 OF OFF-SETS. 

Ch. L. Ch. L. eh. L. 

From 11.12=1,2 6.50=1/ 3.90=le 

Take 6.50=1/ 3.90= le 2.25= li 

a^B^HBH^Ma^'W' ^MM^BMMaMi^MaB «^MMiHaa^^iM»aw 

4.62=^ 2.60=c^ 1.65=ed 



Ch. L. 

lrf=2.25X32L. half the first djar-set= .7200 

ed^ 1.65X1C. 26L.i the sum of the 1st and 2d 2.0790 

e^=2.60xiC. 32L.) the sum of 2d and 3d=3.4320 

2^= 4.62X37L. half the last off-set = 1 .7094 

Content of left oflP^ets on the first dist. " 

. in square four-pole chains 7.9404 

In like mannerthe rest are performed. 
The sum of the left hand ofiT-sets will be 14.0856 
A^d the sum of the right hand ones 3.6825 

Excess of left hand oif-seti in squ. 4 pole C« 10.4031 

Acres L04031 



.16124 
4 



Perches 6.4496 



Excess of left hand off-sets above the right hand 
ones, 1 A. OR. 6P. to be added to the area within 
the stationary lines^. 



•c 



(257 ) 



SECTION V, 



To find the area of a piece tf Ground by vtteraecthm only, when 
ail the an^lea o/thejield can be teen from any two Stattont on the 
QltsMe <^ the ground* 



Pl, 12.j5^. I. 

IjET ABCDEFG be a field, H and / two 
places on the outside of it, from whence an ^ob- 
ject at every angle of the field may be seen- 

Take the bearing and distance between H and 
/, set that at the head of your field-bool(:, as in the 
annexed one. Fix your instrument at H, from 
whence take the bearings of the several angular 
{K>int8 Ay By C, Dy &c. as they are here represent- 
ed by the lines HAy HBy HCy HDy &c. Again 
fix your instrument at /> and take bearings to the 
same angular points, represented by the lines I A, 
IBl Wy IDy &c. and let the first bearii^s be en- 
tered in the second coliunn, and the second bear- 
ings in the third column, of your Q^ld-book ; then 
it is plain that the points of intersection, made 
from the bearings in the second and third columns 
of every Ihie, will be the angular points of the 
field, <N* the points Ay By C, Z>, &c. which points 
being joined by right lines, will give the plan 
ABCDEFGHA required. . 

^ hi ' 



258 



BY INTERSECTIONS. 



Bear. 180 Dis. 28G. of the Sta. H and I. 



No. 
A 


Bear. 


Bear. 
331t 


2611 


B 


265J 


31 7i 


C 


248 


307^ 


D 


238i 


289 


E 


2151 


262J 


F 


2084^ 


286i 


G 


220 


300 



The same'may be done from any two stations 
within-side of the land, from whence all the angles 
of the field can be seen. 



This method will be found useful in case the 
stationary distances from any cause prove inec- 
cesiible, or should it be required to be done by one* 

J^arty, when the other in whose possession it is, re- 
uses to admit you to go on the land. 



Tojindtht chuent 6fajield by calcuIoHon^ vfhkk was takfn ^ 

intersection^ 



In the triangle AIHy the angles AHIy AIH, 
and the base Jal being known, the perpendicular 
Aa, and the segments of the base Ho, Alm^y be 
obtained by trigonometry : and i» the same man- 
ner all the other perperdiculars Bb^ fk, Dd^ Ety 
tfy Gg^ and the several segments at &, c, dy e,/, and 
g : if therefore the several perpendiculars be sup- 
posed to be drawn into the scheme (which are here 
omitted to prevent confusion arising from a 
multiplicity of lines) it is plain that if from 
bBCDEeb, there be taken bBAGFeb, the re- 
maindea; will be the map ABCDEFGA. 



BY INTERSECTIONS. 259 

As before half the. sum of JBft, and Cc muHipli- 
ed by hc^ Will be the area of the trapezium bBCcj 
after the same manner, half the sum of Cc, and 
JDrf, multiplied by cd, will give the area of the tra- 
pezium cCDd; and again, half the suniof jDd, and 
i?e Multiplied by dk, gives the area of the trape- 
zium dDEe ; and the sum of these three trapezia 
will be the area of the figure bBCDeb. 

Again, in the same manner, half the sum of Bb 
and Aa multiplied by oA, will give the area of the 
trapezium BoAa ; and half the sum of aAy and 
gOyJby ag, gives the trapezium aAOg ;. to these 
add the trapezia g6?i^, nnd fFEe, which are found 
in the like maimer^ and you will have the figure 
bBAGFEeb, and this taken from bBCJDeb, will 
leave the map ABCDEFQA. Q. E. F. 



It will be sufficient to protract this kind of work, 
and from the map to determine the area as well 
as in plate 10. fig. 3. to find the areas o£the pieces, 
3, 4, 5, 6, 3, and 6, 7, 7, 6, from geometrical con* 
structions. 



Hbwjo determine the station where a fault has been committed in 
a Jteld book^ vfithout the trouble of going round the whole grotmd . 
a second time* { 



Prom every fourth or fifth station, if Ihey be not 
rery long onen, or oftener if they are, let an inter- 
section he taken to any object, as to any particular 
part of a castle, house, or cock of hay, &c. or if all 
these be wanting, to a long staff with a white sheet 
or napkin set thereon, to render the object more 
conspicuous, and let this be placed on the summit 
of the land, and let the respective intersection^ iso 



\ 



260 BY tNTBBSlCTJONS. 

taken be imierted on the left band side of the field- 
book» oppoftite to the stations from wfai^ce tbey 
were respectively taken. 

In your protraction as you proceed, let every 
intersection be laid off from the respective stations 
from whence thev were taken^ and let these lines 
be continued ; if they all converge or meet in one 
point, we (hence conclude all is right, or so far as 
they do converge ; but if we find a line of inter- 
section to divei^e or fly off from the rest^ we 
may be mn^ that either a mistake has happened 
between the station the foregoing intersection waa 
taken at, and the station fr*om Vhence the intersec- 
tion line direrges, or there must be an error in the 
intersection ; but to be assured in whicb of these 
the fault is, protract on to the next intersection, 
and having set it off, if it converges with the resty 
though the . foregoing one did not, we may con- 
clude the fault was committed in taking the last 
intersection but one, and none in any station, ahd 
that so fkr i^ true as is protracted ; but if this as 
well as the foregoing intersection diverge or Hy 
ttom the point of concourse or convergmg p6int 
of th^ rest, the error must have its rise jronor some 
station or stations, at or after that, from^ whence 
the last converging intersection line was' taken : 
so that by going to that station on the ground, 
and proceeding on to that where the ne^^tg, or from 
whence the follov^ing diverging intersection* was 
taken, we can readily and with little trouble set 
all to rights. 

■ > 

But in most tracts of Jfand, ome object cannot 
be seen from every station, or from perhaps one 
^ fourth of them ; in tliis case we are under we ne- 
cessity to move the pole after we begin to loae 
sight of it, to some other pa{i of the laad> where 



if 



BY INTERSECTIONS. 26| 

it may be seen from as many more stations as pos- 
sible ; which is easily done by viewing the bound- 
ary before it be sui-veyed : the pole then being 
fixed in an adTsAtageous place, Ibe first intersec- 
tion to it is best to be made from the same st ation 
from whence the last one was taken, and then as 
often as may be thought convenient, as before ; 
in like manner the wliole may be done by the re- 
moval of the pole. • 

When we here speak of stations, we do Botme^n 
suck as are usually taken at every particular angle 
of the field): for it is to be apprehended, that every 
skilful surveyor, particularly such who use eall;ai- 
lation, wiU take the longest distances possible, 
not only to lessen the number of stations, for the 
case of either protraction or caleul ation, but with 
greater certainty to account for the land passed 
By, on the right hand or on the left, which is tai- 
ken by off-sets : and surely it will be allowed that 
any measure taken on the ground, and the con- 
tent thence arithmetically computed, will be much 
more accurate than that which is obtained from 
any geometrical projection. 

Prom what has been said it is plain, that from 
this method any fault committed in a survey can 
be readily determined, and therefore must be 
much preferable to the present method of taking 
diagonals^ or the bearings and lengths of lines a- 
cross land, to accomplish that end ; which ' last 
method is too frequently used by surveyors to ap- 
proximate or arrive near the content, which will 
ever remain uncertain, let these diagonals be ever 
so many, till the station or stations wherein the 
error or errors were committed, be found ; and 
the fault or faults .be coi^ciieid. 






262 BY INTERSECTIOrfS. 

I * 

Where one dia^^onal is taken, it may perhaps 
close or meet with one part of the survey and not 
with the other ; in this case, if the surveyor would 
discover his error, he must survey that part of the 
land which did not close, and this may be half or 
more, of the whole. And should the diagonal 
close with neither^part, but be too long, or too 
short, or should it rail on either side of the assign* 
ed point it was to close with, he ought to go over 
the whole, and make a new survey of it in order 
to discover his error, 

A number of diagonals are frequently taken, 
the sum of the lengths of which very often e:|- 
ceeds the circuit of the ground, and after s^l Hney 
are but approximations, and the content remains 
uncertain as before ; therefore he who returns, a 
map, made up by the assistance of diagooal^ 
where there remains a misclosure in any one jpart, 
runs the risque of being detected in an error, and 
must suffer uneasiness in his mind, as he caqnot 
be certain of the return he makes. 

The frequent misclosures which are botched 
up by diagonals, occasion the many and frequent 
scandalous broils and animosities between survey- 
ors, which tend to the loss of character of the one 
or the other, and indeed often to the disrepute of 
both, as well as to that of the science they pro- 
fess. 

But these may be easily remedied by intersec- 
tions, and the bearing or line to be adjusted where 
the fault was committed, and till this be found, 
nothing can be certain. 






( .263 ) 



SECTION VL 



To ENLARGE OR DIMINISH IlIAPS. 



To enlarge or dtminkh a mafit or to reduce a mafi/rom one scale to 
another ; aUo the numner ^umtmg separate mafis f^ lands vfhich 
join eagh other, into one Mdfi of any assigned size. 

JL/A Y the map you would enlarge, over the pa^ 
per on which you would enlarge it, and with a 
fine protracting pin, prick through every angular 

5oiftt of yeurmap, join these points on your paper 
laying the map you copy before you) by pencil- 
Jed or popped lines, and you have the copy of the 
map you are to enlarge ; in this manner any pro- 
tractiom may be cogied on paper, velluin> or 
parchment, for a fair map. 

If you would enlarge a map to a scale which 
is double, or treble, or quadruple to that of the 
map to be enlarged, the paper you must provide 
for its enlargement must be two, or three, or four 
times as long and broad as the map ; for whicli 
purpose in large things you will find it necessary 
to join several sheets of paper, and to cement them 
with white wafer or paste, but the former is best. 

Then pitch upon any point in your copied map 
for a centre ; from whence if distances be taken 
to its extreme points, and thence if those distance? 
be set in a right line with (but from) the centre. 



264 * To enlarge w dimifUsh Ma^s^ 

and these last points fall within your paper^ the 
map may be increased on it to a scale as lar^e 
again as its own ; and if the like diistanqes be agam 
set outwards in rio^bt lines from the centre, and if 
these last points fi^l within your paper, it wili conr 
tain a map increased to a scale three times as large 
as its own, &c. 



Ft. l^Jtg' ^ 

Let the pricked or popped lines represent the 
copy of a down or old survey, laid down by a 
scale of 80 perches to an inch, and let it be re^ 
quired to enlarge it to one laid down by 40 tn an 
inch. 

Pitch upon your centre as 0, from whence thro' 
a lay the fiducial edge of a thin ruler, with a fine 
pointed pair of compasses, take the distant^ fix)m 
a to the centre O, and lay it by the ruler's edge 
from a to ^;' in the like nvinner take the distdtice 
from the next station b to the centre Q, and lay it 
over in a right line from 6 to JB, and join the points 
A and Bhy the right line AB ; in the like man* 
ner set over the distance from every station to the 
centi'e, from that station outwards, and you will 
have every point to enlarge to ; the joining of 
these constantly as you go on by right lines, will 
give you the enlarged map required. 

In taking the distance from every station io the 
centre, set one foot of the compasses in the sta- 
tion, and the other very lightly over the centre- 
point, so lightly as scarcely to touch it, otheH^ise 
the centre-point will become so wide, that it may 
occasion several errors in the enlarged map : for 



Jb eUlafge or diminish Maps: ' 265 

if you err from the exact centre but a little, that 
error will become double, or treble, or quadruple^ 
as you enlarge to a scale that is double, or treble, 
or quadruple of the ^iven one ; therefore great 
accuracy is required m enlarging a map. 

When you have doite with a station, give a dash 
with a pen or pencil to it, such as at the station a 
and b ; by this means you cannot be disappointed 
in missing a station, or in laying your ruler over 
one station twice. 



Froml^hat has been ^aid it is plain, that, if a 
map is to be enlarged to one whose scale is double 
the given one, that the distances from the respec- 
tive stations to the centre, being set over by the 
ruler's edge, will give the points for the enlarged 
one. And thus may a map be enlarged from a 
scale of 1 60 to one of 80, from one of 80 to fine of 
40, from one of 20 to one of 10 perches to an inch, 
&c. For to enlarge to a scale that is double, the 
ilumbet of perches to an inch for the enlarged map 
must be half of those to an inch for that to be en- 

« 

larged : to enlarge to a scale that is treble the giv- 
en one, the number of perches to an inch for the 
ertlarged map, will be one third of those for the 
other ; if to a scale that is quadruple the given 
one, the number of perches to an inch fdr the en- 
larged map, will be one fourth of those for the 
other, &c. therefore if you would enlarge a map 
which is laid down by a scale of 120 perches to 
an inch, to one of 40 perches to an inch, the dis- 
tance from the several stations to the centre, be- 
ing set twice beyond the said stations, will mark 
out the several points required, for these points 
will be three times further from the centre than 
the: stationary points of the map are. 

Mm 



2^ }^ eniarge or dimni^ MflpL 

In the same manner, if you would enlarf^e ai 
map from a scale of 160, to one of 40 perches to 
an mcb, the distance from the several stations to 
the centre, being set three times beyond said 
stations, will lay out the points for your enlarged 
map, for these points will be four times further 
from tjie centre than are the stations of the map^ 

When a map is enlarged to another, whose scale 
is double, or treble, or quadruple, &c* of the given 
one, every line, as well as the length and breadth 
of the enlarged map, will be double, or treble, or 
quadruple, &c. those of the given one, for it must 
be easy to conceive that those maps are like : but 
the area, if the scale be double, will be four times; 
if treble, nine times ; if quadruple, sixteen times 
that of the given figure ; that is, it will contain 
four, nine^ or sixteen times'as itiany square inches 
^s the 'given one (for it has been shewn that like 
polygons aie in a duplicate proportion with the 
homologous sides)« Yet these figures being cast 
up by their respective scales will produce the 
same content. 

Thus much is sufficient for enlarging maps, and 
from h^nce, diminishing of them wfll he obvious ; 
for one fourth, one third, or half the distances 
from the several stations to the centre, will mark 
out points, which if joined, wiU compose a map 
similar to the given one, whose scale will be four 
timesji three times, or twice as small as the given 
one. 

■ 

Thus, if we would reduce a map from 40 to 80^ 
from 20 to 40, from 10 to 20 perches to an inch, 
&c, half the distance of the stations from the 
centre wjll give the points requisite for drawing the 



lb enlarge or dimmish M(^B. 26T 

map ; if we would reduce from 40 to 120» from 
20 to 60, from 10 to 30 perches lo an inch, &c, 
one third of the distances to the centre, will give 
the points for the map ; and if we would reduce 
from 40 to 160, from 20 to 80, from 10 to 40 
perches to an inch, &c. one fourth of the dis* 
tances to the centre, will give the points for the 
map. 

By the methods here laid down I have reduced 
a map from a scale of 40 to one of 20 perches to 
an inch, which contained upwards of 1 200 acre8> 
and consisted of f 224 separate divisions, without 
the least confusion from iJie lines ; for none can 
}irise if the methods here laid down be strictly ob^ 
served. 

I have a]so from the same methods reduced a 
large book of maps, each of which was an entire 
skin of parchment, and the whole contained up* 
wards of 46000 acres, to a pocket volume ; and 
afterwards connected all these maps into one map, 
which was contained in one sitin of parchment : 
therefore upon the whole I do recommend these 
methods for reducing maps to be much more ac- 
curate than any of the methods commonly used, 
such as squaring of paper, ijsing a parallelogitim, 
proportionable compasses, or any other method 1 
ever met with, though the figures to be reduced 
were ever so numerous, irregular, or complicated. 



To unite ueparott maji$ of iantU mMehJoin each ot^erg into one nutfi 

qfany dssignea aife. 



If there be several large mapd coHtained in a 
book> each of wbichi suppose to take up a skin 



268 ' To enlarge or diminish Map^* 

of parchment, or a sheet of the largest paper; 
which maps of lands join each other ; and it be re-r 
quired to reduce them to so smaU a scale, tlmt 
all of them when joined together may be contain-- 
ed in one skin, half a skin, or any assigned sized 
piece of parchment, or paper. 



Having pricked off and copied the several maps 
on any kind*of paper, unite them by cutting with 
ecissors along the edge of one boundary which is 
adjoining the other, but not cutting by the edge 
of both, and throw aside the parts cut off; then 
lay these together on a large table, or on the 
floor, and wnere the boundaries agree, they will 
fit in with each other as indentures do ; and after 
this manner they are easily connected : measure 
then the length and breadth of the entire connect- 
ed niaps, and the length and breadth of the parch- 
ment or paper you are confined to ; if the former 
be three, four, or five times greater (that is, longer 
and broader) than the latter, reduce each copied 
map severally to a scale that is three, or four, or 
five times less, as before ; and the same parts of 
the boundaries you cut by in the large maps, by 
the same you must^ also cut in small ones, and 
unite the small as the large ones were united ; ce- 
menting them together with white wafer : thus 
will your map be reduced to the assigned size, 
which copy over fair, on the par<:hment, or paper 
you were confined to. 

But it is not always that a person is confined to 
a given area of parchment, or paper ; in such 
cases, if there are many large maps to be united 
into one, reduce each of them severally to a scale 
of 160 perches to an inch, and unite those by the 
poijtiguity or boundaries^ as before : or if you have 



To enlarge or diminish Maps. 289 

a few, it will be sufficient to reduce them to a scale 
of 120, SCc. But having the maps given, and the 
ficale by which they are laid down, your reason 
will be sufficient to direct you to know what scak 
they should be reduced to. 



Directions concerning mrveysin general. 

If you have a large quantity of ground to sur- 
vey, which consists of many fields or holdings, 
and that it be required to map and give the re- 
spective contents of the same, it is best to make 
a survey of the whole first, and to be satisfied 
that it is truly taken, as well as to find its con- 
tent ; and as yout go round the land, to make a 
note on the side of your field-book at every statior^ 
where the boundary of any particular field or 
ijolding intersects or meets the surround ; then 
proceed from any one of those stations, and in 
your field-book say, "proceed from such a station,^ 
and when you have gone round that field or divi- 
sion, ^insert the station you close at, and so through 
the whole : a little practice can only render this 
sufficiently familiar, and the method of protraction 
must be evident from the field-notes. When the 
whole is protracted, and you are satisfied of the 
closes of the particular divisions, cast up each se- 
verally, and if the sum of their contents be equal 
to the content of the whole first found, you may 
safely conclude that all is right. 

The protraction being thus finished and cast up, 
transfer it on clean paper, vellum, or parchment, 
^s before ; be careiul to draw yom* lines with a 
line pen, write on it the names of the circumjacent 
lands, and set No. 1, 2, 3, 4, ^c. in every parti- 



270 7b enlarge or diminish Map9. 

cular field or division; let every tenant's particular 
holding be distinguished by a difierent coloured 
paint foing run finely along the boundaries ; let 
all the road9, rivulets, rivers, bridges, bogs, pond8> 
houses, casfles, churches, beacons (or whatever 
else may be remarkable on the ground) be dis- 
tinguished on the map. Write the title of the 
map in a neat compartment either drawn, or done 
from a good copper-plate graving, with the gen- 
tieman's arms. Prick ofi* one of your parallels 
with the map, and on it make a mariner's com- 
pass, and draw a flower-de-luce to the north, and 
this will represent the magnetical north ; after 
which set on the variation, which eipress in fi- 
gures, and through the centre of the eompass, let 
a true meridian line be drawn of about 3 inches 
long, by which write True Meridian. Let a scale 
be drawn, or it is sufficient to express the number 
of perqhes to an inch» the map was laid down by. 
I>raw a reference table of three, pr, if occasion be, 
of four or more colunms ; in the^ first insert the 
number of the field or holding : in the next its 
name, and by whom occupiea : in the third the 

2uantity of acres, roods, and perches it contains : 
'you have unprofitable land, as bog or mountain, 
let the quantity be inserted in the fourth colunitt ; 
and, if it be required, you may make another co^ 
lumn for statute measure, and then the mz^^ is 
completed. 



iHvisian of Land. S271 



■<• 



SECTION VIL 

THE METHOD OF DITIDING LAND^ OR OF TAKING 
OFF OR INCLOSING ANY GIVEN QUANTITY. 

ExAM?IiE I. 

N 

Pl. 12. Jiff. 1. 

Let ABCDj SCc. be a map of ground, contain- 
ing 1 1 acres, it is required to cut off a piece as 
DEFGIDy that shall contain 6 acres. 

Join any two opposite stations as D and Cr, with 
the line DGy (which you may nearly judge to be 
the partition line) and find the area of the part 
DEFG, which suppose may want 3IL 20P. of the 
quantity you would cut off: measure the line DGj 
which suppose to be 70 perches. Divide 3R. 20P. 
or 1 40P. by 25, the * of DG, and the quotient 4 
will be a perpendicular for a triaj^le wnose base 
is 70, and the area 140P. Let Mm be drawn pa- 
rallel to DGj,2A ihe distance of tlie perpendicular 
4, and from /, where it cuts the boundary, dr^w a 
line to Z>, and that line !>/, will be the division 
line ; or a, line from G to H will have the same 
effect ; all which must be evident from what has 
been already said. 

But if hills, trees &c, obstruct the view of the 
points D and I from each other, it wUl be neces- 
sary in order to lOin a partition line, to know its 
bearing ; and it may be proper on some occasions, 
4o have its length ; both tjiese may be easily cal- 
culated from the common field-notes only, as in 
the following example, without the trouble of any 
other measurement on the ground, or any depen- 
dance on^ the map and scale. 



I 



272 



DiviHon of LatuL 



Example ii. 



Pl. IX Jig, 3. 



Let ABCBEFGHlA be a tract of land, to be 
divided into two equal parts, by a right line from 
the comer /to the opposite boundary CD; rfequir- 
ed the bearing and length of the partition line IN^ 
by calculation, from the following field-notes, viZi 



Field-Notes aod Area. 



Boon. 



AB 
BC 
CD 
DE 
EF 
FG 

an 

HI 
lA 



Bearing. 



*»i 



N.. 19^ o'E. 
S. 77. E. 

E. 

OW. 

30E. 



S. 27. 
S. 52. 
S. 



15. 

West. 

N. 36. OW. 

North. 
N. 63. 15W. 



Perch. 



108. 

91. 
115. 

58. 

76. 
70.9 

47. 
64.3 

59. 



152 A. 



IR. 25.9P. 



Operation. 



mmmtimt 



lABCl 



lAiN. 62«*jW. 
AB N. 19 E. 
BCiS* 77 E. 
CI 



Per.f N. . S. 



59 

188 

9i 



«-^a 



27.5 
102.1 



^0.5 
I 109 A 



E. 



35.2 
88.7 



W. 



52.2 



71.7 



E 



PP 



Ar"ea;8722.3p"erches[l 29.6 {129.6 |123.9 {123.9 P 



152A. IR. 25.9P.= 24385.9 perch. 
haIf,tobedividedoja;=:J2192*9 > ., ,. 
the part lABCI = 8722.3 j ®"^^' 



Triangle ICNI «= 3470.6 perches* 



Division, of Land, 



273 




IN. E, 

S. 27. E. 



Per. 



115 



Area, 6522.1 per. 



TT 



109.1 



■5: 






102.5 
6.6 



E. W. 



71 7 

52«2 



123.9 



109.1 ;109.t 122.9 1123.9 



Si 

-1 



o 



Then, ( . ICDI : CD : : ICNI : CN \ Th. 
as I 6522.1 : 115 : : 3470.6 : 61.19 ] Sec, 
trhich determines the point N in CD. 



18 
1 



iCni. 


Per. 

61.2 


""1*. 


s. 


JC#. 


"w- 


IC as before 
CN S. 27 E. 
NI 


109.1 




17.7 
27.8 




54.6 






54.6 




99.5 









As dif. lat 54.6 
Radius S. 90 deg. 
: Depart. 99.5 
Tang. Bear, ei"! 5' 



AaS. Bear 6l*15' 

Depart. 99.5 

: Radius S- 90 deg. 

Distance 113.49 



A „=™^r S ^^ runs N. 61 • 15' E. ) , ,% - 
Answer, ^j^^^ g gj ^^ ^ 113.5 per. 

In the part lABCIy the difference between the 
northings and the southings of the three lines, lA^ 
AB and BC (109.1) is the difference of latitude, 
and that of their eastings and westings (71.7) the 
departure of the line Ci, which is placed thereto, 
so as to balance the columns ; see theo. 1. sect. 5. 
hence the content is obtained, as already taught, 
without the bearing or length of the line CL 

For the triangle ICDIj the diff. lat. and dep. of 
IC are taken from the preceding table, which in 
going from / to C will be northing and easting : 
those of C^ are found by the bearing and distance, 
and of DI by balancing the colufnQi> a» before 
fi^r CL 

Nn 



274 JDimian qf Land. 

The tKflcrence of latitude ^34.6) and departure 
(99.5) of ibe line NI, in the third iable, are itmnd 
by balatieing those of IC and CN^ and a^ they 
are the base and perpendicular of a right aisled 
triangle, of if^iich the line ifl ia-the bypotibemise, 
and the angle opposite to the deptitoite, tbe bear- 
ing, we have the answer by two trigonometrical 
sta tings, aa above ; and tkms may any tract be ac- 
curately divided, or aiiy proposed quantity readi- 
ly cut off or inclosed* 

No>v the itudent or practitioner may calculate 
the content of the part ABCNIA (the bearinj 
and distance^ or the diff. lat. and dep. of €N an< 
of NI beitig known) and if it be fcnmd equal to 
the intended quantity, it proves the truth of the 
'#peration. 



}£XA2>U>LS HI. 



!♦• 13.^^. 3. 



It is proj)osedto cutoff 38A. 16P|. to the south 
end of this tract, by a line running from £ due 
West 40 perches to a well at O, aiicl from thence 
a right line to a point M in the boundary HI; the 
place of My and the bearing and length of the line 
OM are required ; the field-notep being as in es- 
au^ple 2d 



tewer, J ^^^ ^ 7g.^ ^ 3g JJ3 \ perch^^l 



Division of ^and. 27^ 

In this example we find, 

The anea of OEPGHO = 5270.5 

Coiwequently of HOMH =^ «26.a 
. Bif. lat. of the Ibe HO^HV = 35.2 
Departure of ditto =mQV ~ 38.2 ■ 

As HI happens to be a meridian) the area of 
jBOMH divided by half Or(I9.1) quotes HM 
(43.23) without finding the area of HOIH, as we 
did of ICDI in example 2d. and HM—HV^ 
VM= 8.03 = dif. lat. of OiM, which with its dep, 
VO = 38.2. gives the bearing and distance as t>e« 
fore. • 

Example it. 

Pl. 12. Jig. 4. 

A trapezoidal field ABCDj bounded as under 
specified, is to be divided into two equal parts by . 
a right line £F parallel to AB or CD ; required 
AF or BF? 



Bou. 1 Bearing. 


Per. 


AB 
BC 
CD 
DA 


SottUi. 
N. 80 W- 

N. 39* W. 
S. 80 £. 


30* 

6a 

45.5 
89^4 


13 A. 3R, 7P, 1 



In the triangle CBG are given BC and all the 
angles (known by the bearings) to find BG, and 
tbenee the area by prob. 9. sect. 4. which+half the 
area of ^^C2>=area ofEFG ; then as the area 
of CBG to that of JBFG, so isthesquareof BG to 
the square of JPTf?, and FGSG^BK 



276 



Operation at lar^c. 



Angle G 39' 3Cfy log. 8, Ck>. Ar. 0.19649") 
Side jBC 60 pen log. 1.7781.5 Vadd 

Angle C 40* 30', sine 9.81254 > 



i*««iii7W«*«< 



Side BG 61 '. 26 per. 
Side £C 60 per. 
Angle J3 100° 0', sine 

2)3619.8, log. 



J. 78718") 
1.77815 Vadd 
9.993363 

3.55868 



As CJ8G = 1809.9 Co, Ar. 6;74235^ 

1103.5 = BCEF I 

To EFG = 291 3.4, log. 3.46440 V 

So8qr.B(?6l.26,log. | l^s^IsJ 



add 



To sqr. FG 77.72 
Ans,S|5'= 16.46 per. 



(2)3.78111 
1.89055 



By the application of this ipetbod a tract of land 
ma^ be divided accurately, in any proportion^ by 
a line running in any assigned direction. 

Note. When the practitioner would wish to be 
very accurate, it will be much better to work by 
four-pole chsdns and links thaa by perches and 
tenths ; one tenth of a perch square being equal 
to 6t square links. 



. ( 277 ) 

EXABJnPLE \. 

T%efoUomng FiddrNoUs (front A. Burns) art 
of a^ece ^ land, which i$ proposedy as an exam- 
pit J to be divided into three e^piatparts by two fj^ht- 
iints mwmng from the sixth and seventh stations ; 
andprovedy by caknlating tht content of the middle 
part. 



Si.| Bearing. kP.C 



2 N.E. 26i 



13.44 



3 

4 



S.£. 71118-96 



S.£. 26 j 13.44 



SW. 7418 96 



S.E. 



45 



8.47 



7 S.E. 63JU3.44 



8 



9 



10 



11 



13 



N.E. 45 



S.E. 26i 



8.47 



13.44 



S. W . 45 



S.W. 63J 13.44 



NW. 76 

r^N.w. 



8.47 



124.73 



36fS0.09 



A. R» P. 
Area 167 1. 24. 



278 Division of Land. 

Example vi. 

m 

Pl. 8. /gr. 5. 

The plet ABCDEFGHA is proposed to be di- 
Tided, geometrically^ in the proportion of 2 to 3, 
by a right line from a given point in any bounda- 
ry or angle tliereof, suppose the point JO. 

Reduce the plot to the triangle cDe, as already 
taught ; divide the base ce in the point Ny so that 
^iV be to Nc in the ratio of two or three, by prob. 
14. page 53 ; draw DNy and it is c^ne. 

Example viy. 

Pl. 12.^. 3. 
ExamJiU ^d m<ty likevfUe be fiexfi^rm^d gepmetrkdltii • 

Produce CH both ways for ^ base, and reduce 
the whole to a triangle, i)fiakingj| the vertical point i* 
then bisect the base in iV, and draw IN* But^ . - 

Notwithstanding this geoinetvical method b de- 
monstrably true in theory, it is not as safe, on 
practical occasions requiring accuracy, as the cal- 
culation, even when perforiDed with tbe greatest 
care ; for which reason we will not enlarge on it 
here. 

Example viil 

r 

Sufiftost 864 turea to be laid ota inform of a right-at^led fiaraUel- 
ogram^ of which the Mes 9haU be m proportion a9 5 to3i regutr- 
ed their dimendom ? 

For the greater side, multiply the area by the 
greater number of the given proportion, and divide 



Divisian of Land. 279 

l)y the less, or, for the less side, multiply by the 
less number, and divide by the greater; -the square 
root of the quotient Will be the side required : 
thus. 



864A-=138240P 1.38240 

5 3 



3)691200 5)414720 

Answ, V 230400«480. V 82944=288, 



EXAMPLS IX. 

If it be required to lay out any quantity of 
^ound, suppose 47 A. 2R. 16P.]nform of a paral- 
lelogram, of which the length is to exceed the 
breadth by a given difference, for instance 80 
perches, then add tlie square of half this diflerence 
to the area; and take the square-root of the sum; 
to whifcb add half the difference for the greater 
side, and subtract it therefrom for the less ; thus, 

/ 

. 2)80 47A. 2R. 16P.=» 7616 perches. 
1600 



40 



40 t^ 9216^.96 



1600 half difi*. add and subt.— 40 

C the length » 136 

Ans. < — 

( the breadth « 56 

Any proposed qiumiity of ^ound may be laid 
•ni or inclosed in the form 



280 Division oj Land. 

Square - - by prob. 2d. 

of a ) Parallelogram, 1 side giv. by pro. 4th. 

Triangle of a givea base, by pro- 7th. ( . 
Circle - - by prob. 13th. ) ^' 

It is sometimes most convenient, when land is 
to be laid out adjacent to a creek, river, or other 
crooked boundary, to measure off^ts to llie am- 
gles or bending thereof, from a right line or lines 
taken near such boundary, and to deduct the 
area of these ofF-sets from the given quantity, and 
then to lay off the remainder from the right-line 
ifr lines, in the desired form. 

In laying out new lands, attention must be paid 
to the allowance for roads, as exemplified in prob- 
14th. 

« 

r Example x. 

It is required to divide off 30 acres, to thp south 
east end of the tract, of which the field-notes are 

fiven in example 4th, by a right-line to run N. 20* 
J. See example 4th. ' ■ 



v . 



» 



»■ 



» • 



it. 




n ^^' 



< aft! ) 



SECTION VIII. 



Of SURVfitlNfi HARBOURS, SHOALS, SANSS, &C. 



Pt, 13. Jig. 1. 



T 



H£R£ are three methods whereby this may 
be performed ; for the observations may be made 
either on the water or on the land. Those made 
Qn the water are of two kinds, one by the log-line 
and compass (as in plane sailing measuring) the 
coiirse and distance round the sand ; and then to 
be plotted as a large wood, or any inclpsure* taken 
by the circumferentor. 

This method I omit' for two reasons ; first, be- 
cause it is to be deduced from the writers of navi- 
gation : and, secondly, because the distances thus 
measured are liable to the errors of currents, 
which generally attend shoals or sands near the 
shore. ^ 

The second method, where there are nfi dis- 
tances to be measured <in the water, though still 
tiiere is one inconvenience, common also to the 
former, because the bearings or observations are 
to be taken on that unstable element (an error 
scarce mentioned by practical artists) I shall 
brieiSy hint at; and so rather choose a third, which 
is liable to neither of these imperfections. 

Oo 



dBB Ctf Surveying h&rbMtSydio(jiSi$amd$;Sl^i:i 

Let a boat be manned >out irHb a signal flag^ a 
fog and line, lead and line, and to observe tiieb^sKiv 
ings of any land mark, a compass witii £igbts« 

Take two or more objects or places, as A, B, O, 
on the sboi^, from whence the boat may be seen 
on the several parts of this shoal^ and 'determine 
their relative position by bearing and distance 
either before or after the other necessary obser^ 
vatiohs are made. 

One of the boat's crew is to sound till be finds 
himself on the edge of the sand, by the d^^Ui of 
waler? and then to come to an anchor ; which be 
is to signify to two persons on the shore, at B and 
C, by his signal. And then from those known 
feind-mat'ks, 2? and C, the observers are to take 
the bearings of the boat, and to register their oh^ 
dervations ; which, when dene, they are to sigmfy 
to the* crew by waving a flag, or by some o^er 
signal. 

And in the mean time, to prevent mistakes, let 
the crew take the bearings of each of these Unitf- 
ijfiarks : then weigh anchor^ which suppose at 2). 

. Then by sounding, proceed to £, aiuj make Uke 
observations. And so at E, F, 6, kc^ till . you 
have surrounded your sand. 

And if in this process^ you are about to lose 
the sight of one of your land-marics, 8U]^ose G, let 
your assistant at c', or B, who^t that time win 
also be about to lose the sight of the boat, by sis^ 
nals (before agieed on) remove to some other ob<^ 
Ject before-hand agreed on, suppose io jff, or JTw 
and then to proceed ?W5 beforei 



Of SiuwqftMg hmrbo9ir$, Aocd$y $ands, Sl't. 88$ 

'^ hnitfy^ if the sand runs so fai* out at S3a, that 
tfae>6l96ct 'Cannot be seen by the boat, nor the 
boat *%< the observer on shore ; there may be 
rockets fired by the boat's crew, and also by the 
observers on the shore in the liight, whereby those 
bearings jpay be taken almost at as great a clis- 
tjemce as tlie light can be seen. For supposing 
thi^y rise but a quarter of a mile above the appa- 
rent horizon^ its stay will be about 9 seconds, ^nd 
its distance for this quarter of a mile will be visi- 
ble about 44 miles. 



But rockets rise much higher, and then the dish 
tances are much greater, whereby they are visible. 

Or two boats may lay at anchor instead of the 
iilod mark^s and then you may work as before* 

• • * 

Now, since the land-marks B and C are fixed, 
their position may be laid down in the draught, as 
iq common surveying, by plotting the distance be- 
tween B and C. And then by plotting the line 
£tD, and the line DC, according to their position, 
their common intersection Will give the point 2). 
And in like nianner E, F, 67, &o. may be plot- 
ted ; and so the shoals completed. And this ttom 
ih^ bearings taken at B and C. 

II thb be a standing lake, environed by bogs, 
or other ijiy>ediments, the observations at i), Ey 
JP, &c. by taking their, opposites, may suffice to 
plot the same from the land-mark. A, B, Q, &c. 
as well as tJauose taken on the land : or, indeed, by 
tte f^urge and distance, as in navijgftion, if the 
water be smooth and without a current. 



Ih sea flboalsi it is convsi9B|€9it'k> note at fi^eh f br 
aervation ibe depth 6f t)ie water fovD4 by the^lea^^ 
and ti^e drift and setting of the^ cwrent by thcf Io0 
and compass, while the boa^ is at anchors whicb 
may be done with ea^e and eKpedition enoi^^ 
For/ while the boat rides at am anchor, her dtero 
points out the setting of the current, and ihe.kig 
and glass will measure its drHi. 

. And these ought to be noted on the drangtit^ 
which may be thus : 

• 

The currents may be shewn, by drawinga dart 
pointitfg out its settii^, and its drfft b^ the sfoman 
Qi^ital letters, the depth of Ifae water by tbe^small 
figures, and rocks by little crosses, && ' 






SGCTION IX. . 

LEVELLINC, 

Fi. IX Jig. 2. 

• i 

JjIEVKLLING is the art of ascenaining the 
perpendicular ascent or descent of one place (or 
more) aboye or below the horizontal level of ano* 
ther, for various intentions ; and of marking out 
courses for conveyance of water, &c, 

* 

•The true levd is a curve conforming tof the sur- 
face of the earth ;»as ASG, 

The cmpafptf levd is a tangent to ibid curre : 



OP UEVELLlJUef, 



SKk 



^, orsllowante for the earth's cnr- 
Tattite, isr flie difference between the apparent 
livfel aM tb^ true, as BD. The quantity erf this 
0b^re<ttion ntoy be tciwnrnp by haTm]^, hi the right- 
angled^^angle CABy the two fegs, JK!=ihe'sBr 
IHidiameter of the earth (=* 1*267500 percheii) and 
^D^^the distance of the object,, to find the hypo-i 
thenuse CDy from which taking CB : (=CA) the 
remainder will be the correction BD ; but it iQay 
be obtained Mare practlcalty thinrr 

Square the , distance in 

S four-pole chains, and dirvide by lOO, 1 
or in perches, and divide by- 12100, > 
or in^ miles, and multiply by 8> j 
ifor the correction in inches. 



£xam!ple. 

Required the correction for 20 foilr-pole chains 
^80 percheii=i mile. 

800)20 x2d=4e0(.5 
1 2S00)80 X 80 =r:640i(.$ 
i=.25, and .25x25^8=* .5 
that is> .5, or i idcb^ the correction required. 

But, to sate the trouble of calculation, we ii^ 
$ert the following table ol cop¥ttiom« . " * 1 



>< « • 



^*v 



^8B 



OP LEVELLING: 



A TViAfe of CorrecUoki: 
Tha distances in four^pole cimins. 



Dlstar. 


CorrecjDisiar. 


;Corrf. 


Chatii> 


iotheik 


Chaio>. 


ilQbt:^ 


I 


,0013 


37 


«.«! 


3 


J,Q05 


38 


0^8 , 


3 


3,01125 


39 


1,05 


4 


J,03 


30 


1,12 


5 


•),03 


31 


,19. 


6 


J,04 


33 


1,37 


7 


J,06 


33 


1,35 


8 


S08 . 


34 


1.44 


9 


XIO 


85 


1,53 


10 


Js\2 


36 


1,63 


11 


vM5 


37 


:.7l 


13 


0»18 


38 


1,80 


13 


0,31 


39 


1,91 


14 


0,34 


40 


3,00 


15 


>.38 


45 


2,38 


16 


j,32 


50 


3^13 


17 


0yS6 


SS 


S,7« . 


18 


3,40 


. 60 


f 4,50 


19 


0,45 


65 


5,31 


30 


'.50 


70 


6,13 • 


31 


.>«5 


75 


7/)3 


32 


'>,60 


80 


8,00 


23 


',67 


85 


9,03 


34 


0.73 


90 


10,13 1 


35 


J,78 


95 


11,38 1 


36 |i,84 1 


100 1 13,50 1 



. ■ 



till 



I • 



The first thing necessary in le;yelling> is the ad- 
justiDgof the level> which may be performed seve- 
ral ways ; The following is very easy and practical. 

Choose some grouna which is not above 4 or 5 
feet out of the level, for the distance of 8 or 10 
chains len^, and suppose it be AB (fig. 3.) and 
find the middle between A and B, which suppose 
to be C; plant the instrument at C: du*ect the 
tube to a station-stafi*, held up at A, and elevate or 



depress the tuber till tbe bubbfe is exactly in the 
middle of r the divisions ; then by signals direct 
your assist^ at A^ to rise or depress the vane^ 
sliding on the- station staff, till the horizontal hair 
in the glasd cuts the middle of that yane : then 
see how mapy feet> inches, and parts, are cut by 
thQ upper p^rt of tlie vane^ which suppose to be 3 
&et 4 uichej^ and 6 tenths. 

t 

In like manner direct to the other staff at B^ 
and suppose the upper edge of that vane to cut at 
the height of 6 feet, 5 inches and two tenths, then 
will these two ranes be on a level. 

From 6 feet 5.2 inches subtract 3 feet 4,6 inchep^ 
and reserve the remainder 3 feet 0.6 inches. 

• 

Now, remove the instrument as dose to the 
higher station-staff as you can ; so that the middle 
of the telesdope may almost touch it. Then bring 
the telescope as n<ar to a level as the Judgment 
of the eye i?ill direct 

Measure irom the ground, the height of the top 
of the telesoope ; and also of the bottom, in feet^ 
inches, and parts; suppose theiii to be 4 feet, 10.5 
inches, and 5 feet 0.3 inches; then half the sum 
of the heights 4 feet 11.4 inches is the height of 
the centre of the glass ; and to this add half the 
breadth of the vane, which suppose to be 1 iBch 
and 5 tenths, and to tlie sum 5 feet 0.9 inches, add^ 
the preteding remainder 3 feet 0.6 inches ; then 
$et' the person at B move his vane, till the upper 
edge cut 6 feet 1.5 inches, the sum of the preced-. 
ingnnmbcr^; 



9» ' 



•/-„ ^ 



aw jop gMvmJumi 

< Kow, fiA elevate Qr<depc^ss .Ae bnt cr Urn Irafe* 
iAe, an !ibe halt «iit liie mideOe <sf Ifae vaoe ai.i4» 
and' ttt^ttie'same tidw <te bubble irtaoda ai tbe ni^ 
dl« of^^ divkioiM ; and theo-wiiLtlie itetanowafe 
ibe <iuly tidjuEfted. .1 . n\ 

. , • - ' • 

If you have a mind to be more accurate, W^Jfesil^ 
the operation ; but when you place the instrumeiit' 
at C, turn the tube at right angles to the line AB,' 
andiheve set It level ; then {>rQceed with a repe-^ 
tition of the work. Only observe to cross-lever iJt 
kk thU adjustment^ tod in all future juses \M3at- 
soever. * 

* ■ 

6r the lej^el may be adjusted thus : As bef^re^ 
first plant the instrument in the middle between. 
A and B (fig. 4.) and observe the heights on t^ 
^atMa-ilKiias, .woidi fOf^m ia be afi. a^xer ; .afod 
oQQsequantly tltmrdiiEi^xmcef as be£^ce>:i0^3 i^ 
OJft joiobea Nmr neasure fhim C toKwrds Ibe 
bSgbast'groQfid J(^ Mme distaiotce tha^ jipqmafl . al-- 
moat to A j suppose 4. chaius to />$ and i)JSi wMl ^ 
be 1^chains> and JDA atie chsju : Then p)^t ^ 
iMtrament ^ IK dkebt. the teleaeope ^tp ^, . a^ 
siMitig the ?bubbte4o tbe imiddle of ilfaeid^visao^ 
diMot your assktant io mme the i$ine» till 4t^ 
hair cats the aiiddle of U; ^uid noie dawn. the ^^sely, 
incfeM^ and parts iHit by the upper e4ge ^ tb€» 
rme ; which mppose to be 3 feet 8 A inches :. To 
tbiiiadd^he difference 3 feet '(KG incfaesr «t9d«ttif^ 
sum 6 feet 9 inobes reserve; u- 



*i . 



Now direct the telescope to the stafi*at B, level 
it^ and djrect your assistant to move the vane* lift 
tl^ hair cuts the middle thereof ; and then^ ir the 
upper edge of the vane cuts the foregoing surti 6 
feet 9 incfies, the hair and bubble are ti-nly adjust- 



C^ LEVELLING. 28a 

ed.* ' Bst if not; wj^ As BD leas AD^ is to the 
diffeieQos bettr eea the numbers cut by the upp€^r 
efc^ •£ '• the vane, and the aumber 6 feet 9 inches; 
do IB 1h6 4isteiice> AD to a number^ which added 
to that cut by the raxie, when less than -.6. feet d» 
and subtracted from the number cut by the vanCj 
whonit is greater than 6 feet 9, will give a num- 
ber to which let the assistant fix the rane ; then so 
eieyate or depress the hair or the bubble, till the 
hair, cuts the middle of the vane at B, and the bub- 
ble stands in the middle of the divisions ; for then 
Ihp. level wUl be adjusted. The operation may^ 
be again repeated, and at every station cross le- 
relled, which will confirm tlie former adjustment.^ 

Or it will be still better to set the station staves 
equally distant from the instrument (suppose about 
16 or 20 perches each) at an angle of about 60% or 
m as t6 torm nearly an equilateral triangle there^ 
with, and level the 2 vanes {A and B figi 5.) as be* 
|&f^,'whicb will be then both hi the !^me horizon'- 
tal level; whether the instrument be right acQusted 
oi* hot, because one will be^as much above or be* 
low the tfue level oi the instrument, as the other, 
being ^ki the same distance from it ; then remove 
Ute inurnment as neafr as may bi to cme of thern^ 
Mppeise A^ and rieiis^ or lower the vane A to the 
exact lev^l'of the visual ray in the instrument, no- 
fhig frreciBelvbow much it is moved, and have the 
other TUie B moFe just a» much, in order to bring 
them again to a level, allowing /or the correction 
of the apparent level if it be a sensibte Quantity ; 
then adjust the instrument to the level ot the vane 

T9a<]^ustthe rafter level (plate 13. fig. 6.) which 
inay be 10, 12, or 14 feet in the span AB ; set it 
on a plank or hard groimd nearlv level, and mark 



290 OF LEVELLING- 

f 

where the plumb line cuts the beam , mn^ sOpp^ae 
at Cy then invert the position by setting the. foot A^ 
in the place of B, and B in that of A% umikiBSi 
where the line now cuts, as at e s the middle pointy 
between c and e will be the true leveUiiu; ijoark-. * 

To continue a level course with this iqstrur 
iiaent> set the foot A to the starting jdaca^ aod!^ 
move B upward or downward toward 1) or ^>. tjll, 
tlie point B be determined and marked for a level^ 
with A J then carry the instrument forward in the 
direction of C till tJbe foot A rests at B9 whence 
the point C is levelled as before, &C. Sights amy 
be placed at r and 5, and the instrument adjurted. 
to themi as before, by reversing them in the di^ 
rection of some distant object 

After the instrument is duly adjusted^ you m^/ 
proceed to use it. Let the example be, this an,-*. 
nexed (fig, 7.) where A every where repres^nt^ 
the level, and B the station staves ; and suppose 
the route be made from a to e y fir^ plant tne in^ 
fitrument between the staves a and 6 ; at .i4 direct 
tlie level to aJS, In-ing the bubble to the middle of 
the divisions, and instruct your assistant so to place 
the vane, that tlie hair in the telescope 'cute the 
middle of the vane, then in a book divide into two 
columns, the one entitled JSorA; 5t^i/5, the other 
Fore sights^ enter the feet, inches, and parts cut by 
the upper edge of the vane at aJB, in the column 
entitled Back sights. 

• 

Then look toward toe other staff 6 B^ bring the 
l)ubble to the middle of the divisions* and d^^ect 
your assistant to place the vane so, that the hair 
cuts the middle of the vane ; then enter the feet,, 
inches, and parts cut by the upper edge of the 
vane, in the column of Fwe sights. 



CfF LEVELUNa 



291 



' Kbw/ pliht the instrument at A^, still keeping 
the; staff fii eiactly in the same place, and cairr 
<}ie stai^ifB forwards to the place cB ; . now look 
back to the staff hB^ and enter the numbers cut , by 
Ifae vane there under the title Back sigkts ; then 
look farw?irds to rB, and enter, the observation 
under the title JFhre sights. Do the like when the 
mstrument is planted at A\ A\ &c. Always taking 
cafe to keep the staff in the same place when you 
looked at it for a ¥ort sight; till you have also 
taken with it a Back iight^ 

• : Hliviftg finished your level, add up tlie colttmn 
ef Back sights tMo one swiHy and the column of 
Fart sights also into one sum ; and the difference^ 
between these sums is the ascent or descent re^' 
quired. And if the sum of the Fore sights be 
glreat^r ttein the sum of the Back sights^ then e is 
fower* than a;^ but if the sum of the Fwe sights 
he less than the snm of the Back sights, e is higher 
than a. For example, let the numbers be as in the 
folio ving table- 



t • 



'i 



i4i 



Backsights. i Foresights. 

f-'M^f - J, - 1 [ - • J J - J " " ^'" - - - I— »g^ " ' " I H I ■ m tmm i mm m l 

eet. Inch. Tenths. Feet. Inch. Tentha. 



3 

•4 

9 
I 



34 



, 



8 



5 
ft 
2 

7 



Hence Uie descent is 



6 
8 
& 

8 
9 



38 

34 



13 
13 



3 , 



I 
8 






4 « 



5 
3 

8 
8 




3 



a 

8 



■•■^ 



2d2 OF UCVZLLHTS. 

ObservaUav. 

1« Aiki if the distances thus taken we 6hort> the 

curTature of the earth may be rejected. For, if 

ihe di^Dce from the instrument be dvery where 

' about 100 yards/ all the curvatures in a mile*b 

worlc will be less than half an inch, 

% If the distance from the instnmient to the 
hindennost staff, be eyery where equal to the 
distance from the instrument to the corresponding 
staff; the curyature of the earth, and the munite 
errors of the instrument^ will both be destroyed. 
Hence it will be much best to set the iistrmneiit 
as equally distant from both staves as itiay be. 

3. Ifthe distances of the instrument from the 
staves, be very unequal and very long, tlie pur- 
vatui^s must be accounted for, and the (Ust^^c^s 
Jn order thereto, tnust be measured- 

4. Therefore it appears, that the best method to 
take a level is to measure the sevei'al distances 
from the instrument to the back and forward sta- 
tion staves ; and enter them in the .field-book, ac- 
cording to the titles of their several columns, as 
in the following example ; and correct the heights 
from the table of allowances, which may be done 
at home when you are about to sum up the 
height?* 



ss^.ws^nsx 



s»s 



- 


BiucMnffd9# 






ForwBras* 


JDifttan. Height Correcud* 




Diauiu 


iici|(iit 


Coii'ccied. 


LmM* Inches. 


inches. 


Links. 


Inches. 


Inches* 


.370 


3^35 


S,24 


418 


4^86 


4i34 


4^0 


6,10 


6,08 




S23 


7,18 


7,1? 


766 


5,38 


5,31 




289 


6,75 


6,6r 


584 


7,25 


7,21 




530 


^,53 


9,50 


326 


8,15 


8,14 




485 


11,25 


11,22 


' *t$B 


)0»S5 


10,20 




376 


8,65 


8,(3 . 


«o 


6)02 


6,29 




730 


10,?4 


IO32B 


3658 


46,47 


3i,46 


S7M . 


3146 


» ' 


« 




. .1 . 


46,47 


4M)* 




,. 




r 

« 


I 


1M4 



So that the fall in 68 chains is aboijft II inches* 
^atid J of an hicfa. 

Lastly, Though hitherto we have considered the 
level with one telescoi)e only, the same observa- 
tions wsiy be applied to a level with a double te- 
^ lescope ; and I would advise those who use the 
^ double telescope, at every station to turn that end 
of the telescope forward, which before was the 
'coiitrary way. 



ui 9Mrt giturai metJkod tfirvtfBng^ adbfttid to tke^utweying ^ 

roads atudbUiy grou^ i§ e^kidiuui in thtfo^tmi^g taoaw^dt^ tn 
ivhich the meastires are given in Unka. 



Examples. 



Pl. lZ.J!g.B. 



Required the bearing and distance of the place 
B from Af and its perpendicular ascent or descent^ 
above or below the horizontal level of A. 



294 



OPLKVELLIWG. 



^iCourstor 



1 

3 

4 



Bearing 



NE79«>I5 
NETS Oi 
NESO 30 
SE85 13 
SE70 Oi 



Elev. or 
Depres. 



lOMT 
Disu 



D 2 i 4f 

E 14 00 
O 11 30 
E 19 15 



Diflt 



738 

684 

9761 

930 

620 



705 
635 
947 
911 
585 



Parpen. Dih 
Ascent {Lai. 
or dene. 



3189 
^253.4 
336.1 
1«5.4 
204.0 



3948 3783 3l7.6 
1 I Desc. ' 



I 



.- A 



I3]j 

1641 

603' 

.754 

200. 



623 



692 

613 

90ft. 
549 

349t 

E. 



Ab Dif. Lat 622 
Is to radius S. 20*, 

So is Dep. 3492 
To T. Bear. 79* 54'. 



As S. Bean 79*45' 
Is to Dep. 3492» 

So is radius S. 90* 
To Dist 3547. 



As 100 links : 66 feet : : 217.6 links : 14aj&. 
feet, the descent B below tbe level of A. 

Hence, B bears N. 79* 54' E. from A^ 
Nearesthoriz- dist. 3547 links. 1 
Sum of obi. dist. 3948 links. > answer. . 
Sum of horiz. dist. 3783 links, j 
Perp. desc. 217.6L. ==1,43.6.F. J 

With the angular elevation or depression in the 
third column, and the oblique distance in the fourth 
(as course and distance) are found the borixontal 
distance in the fifth, and ihe perpendicular ascent 
or descent on the sixth, for each station (as differ- 
ence of latitude and departure :) then, with the 
bearing and horizontal distance we get the dif- 
ference of latitude and departure in the two last . 
columns. 

The ascents 'and descents in the sixth column 
are distinguished by the letters E and D in the 
third, signiTying elevation or depression ; and be- 
ing added separately, the difference of their rams 



PROMiaqUOUS aUESTIONS. 295 

IB set a^HlRrbottoiii of the column with tiie nam^ 
.of the greater, and shews the perpendicular de- 
scent of jB below the horizontal level of A.. 

In like manner the northings and southings in 
ihe j^eventh column are distinguished by tfa^ let- 
itrsciV and S in the second, &c. 



PROMISCUOUS dUESTIONSu 

The perambulator, or surveying wheel, is po 
cbiitrived as to turn just twice m the length of a 
pt 'le or I6i feet ; what then is the diameter ? 

Answ. 2.626 feet. 

2. Two sides of a triangle are respectively 20 
and 40 perches ; required the third, so that the 
Qontent may be just an acre ? 

Answ. either 23.099 or 58.876 perches^ 

3. I want the length of a line by which my 
gardener nmy strike out a round orangei-y that 
shall eontain just half an acre of ground. 

Answ. 274 yardsr 

4- What proportion does the arpent of France^ 
which contains 100 square poles of 18 feet each, 
bear to the American acre, containing 160 square 
poles of 16.5 feet each, considering that the 
length of (he French foot is to the American a« 
16 to 15? 

Answ. as 512 to 605. 



296 ^llOxMlScUdtrS aCESTIQNS: 

5. The ellipse in Grosvener square iQeasuiie» 
840 links the longest way, and 612 the shortegt^ 
within the rails : now the wall being 14 inches 
thick, it is required to find what quantity of 
ground it incloses, and how much it stands upon. 

Answ. it incloses 4 A. 6P. and ^ands on 17601 
square feet. 

6. Required the dimensions of ah elliptical acre 
with the greatest and least diameters in the propor- 
tion of 3 to 2 ? 

Answ. 17.479 by 11.653 perches*. 

7. The paving of a triatigular court at ISdL per 
foot, came to 100/. The longest of the three sides 
was 88 feet : what then, was the sum of the ot6er 
two equal sides ? 

Answ. 106,85 feet? 

• 8. In 110 acres qf statute measure, in which the 
pole is 164 feet, how many Cheshire acres, where 
the customary pole is 6 yards, and how many of 
Ireland, where the pole in use is 7 yards ? 

Answ. 92A. IR. 28P. Cheshke ; 67p, SR. 25P. 
Irish. 

9. The three sides of a triangle containing* 6A. 
IR. 12P. are in the ratio of the three numbers, 9, 
8, 6, respectively ; required the sides ? 

Answ. 59.029, 52.47, and 39.353. 

10. In a pentangular field, beginning with the 
douth side, and measuring round towards the east> 
the first or south side is 2735 links, the second 
3115, the third 2370> the fourth 2925, and the fifUi 
2220 ; also the diagonal from the first angle to the 
third is 3800 links, and that from the third to the 
fifth 4010 ; required the area of tiie fi,eld ? 

Aiisw. U7A. 2R. 28P. 



I'ROMiSCtrOtrs aUESTiONS. 297 

■ 

* 11. Required the dimensions of an oblong gai- 
den contafaiing three acres, and bounded by 104 
• ^rches of pale fence ? 

Answ. 40, perches by 12, 

12. How. many acres are contained in a sqiiam 
meadow, the diagonal of which is 20 perches more 
than eithel of its sides ? 

• — Answ. 4 A. 2K. IIP. 

.3, If a man six feet high travel round the earth, 
much greater will be the circumfE^rence desr 
clibed by the top of his head than by his feet ? 

Answ. 37.69 feet 
N. B. The required difi^rence is equal to th^ 
circumference of a circle 6 feet radius, let the 
magnitude of the earth be what it may. 

14. Required the dimensions of a parallelogram 
containing 200 acres, which is^40 perches longer 
than wide ? * 

Ans^v. 200 perches by 160. 

1 5. What difference as there between a lot 28 
perches long 6y 20 bsoad^ and two others, each T>f 
half the dimensions .? 

Answ. 1 A. 3R. 



: I 



y^q 



• 



(^> 



PART III 



C$n4a9mf the Attr^nwrncal methods qfjlndm^ the Latitudef Vchfih 
4ition qf'the conrfiass, k3^c.. vfth a d^MCriptign.ofthc <f^<t»nie|B»- 
uaed vi these qfieratums. 



SECTION I. 



IKTRODUCTORT FRINCIPLEff. 

m3 ay aod nkfat arise frcao the circiMiBOlalkMi 
of the Earth. That imaginary line about wliich 
the rotation is performed, ie called the Axis^ and 
its extremeties are called Poles. Thai towards 
the most remote parts of 'Europe is called the 
North Poky and its opposite the South Pqk. The 
Earth^s Axis being produced wil] point out the 
Celestial Poles. 

The Equator is a great circle on the Eattbi 
every point of which is equally distant from th6 
Poles ; it divides the Eartn into two equal parU^ 
called Hemispheres ; tliat having the ISorth Pole 
in its centre is called the Northern Hemisplure — • 
and the other, the Southern Hemisphere. The 
plane of this circle being produced to the fixed 
stars, will point out the celestial Equator or Ecjui- 
noctial. The Equator, as well as all other great 
circles of the sphere, is divided into 360 equal 
parts, called degrees ; each degree is divided ^to 
60 equal parts, called minutes s ^nd the sej^aj^ei^* 
fnal division is continued. 



INTRODtrCTOflY PRINCIPLES. 299 

NoTiu The ancients having no instruments by 
ivhich they could make observations with any to- 
lerable degree of accuracy, supposed the length of 
the year, or annual motion of the earth, to be com- 
pleted in 360 days : and hence arose the division 
of the circumference of a circle into the same numr 
het of equal parts, which they called degrees. 

The Meridian of any place, is a semi-circle pas- 
sing through that place, and terminating at the 
Poles of the Equator. The other half of this cir- 
cle is called the opposite Meridian. < 

The Latitude of any place, is that portion of 
the Meridian of that place, which is contained be- 
tween the Equator and the given place ; and is ei- 
Ihi&t North or Souihy according as the given place 
is in Notlbeni or Southern Hemisphere, and there- 
fore eaimot exceM 90*. 

Th6 Parallel of Latitude of any place, is a cfr- 
de ' pa^in^ through that place, parallel to the 
Equator. 

. The Difference of Latitude between any two 
places, 13 an arch of a meridian intercepted bet 
tween the corresponding parallels of latitude of 
those places. Hence, if the places lie between 
the Equator and the same Pole, their difference 
of latitude is . found by subtracting the less lati- 
tude, from the greater : but if they are on oppo- 
site sides of tlie Equator, the difference of lati- 
ttlde IS equal to the sum of the latitudes of both 
places. 

' 'The Pii'st Meridian is an imaginary semicircle, 
pjissin^ through any remarkable place, arid is" 
therefore arbitrary. Thus, the British esteem that* 



SOO INTRODUCTORY PRINCIPLEJS. 

to be the First Meridian which passes through Ih^ 

, JEloyal Observatory at Greenwich ; and the French 

reckon for their First Meridian, that which passes 

.. through the Royal Observatory at Paris. — ^Fqr- 

, ^ueriy nmnv French geographers reckoned the 

jpaeridian of the island of Ferro to be their First 

^JVIeridian ; and others, that which was exactly 20 

degrees to the west of the Paris Observatory. The 

Germans, again, considered the meridian of tlie 

Peak of Teneriffe to be the First Meridian. By 

Ahh inode of reckoning, Europe, Asia, and Africd^ 

«re in east longitude ; and JNorth and Soutli A- 

itieiica^ in west longitude. At present, the first 

ineridian of any country is generally esteemed to 

be that which passes through the principal Obser*- 

t^atory, or chief city of that country. 

The Longitude of any place is that portion of 
tjie Equator which is contained between the first 
meridian, and the meridian of that place : and is,, 
usually reckoned either east or westy according aa 
the given place is on the east or west side of tlie^ 
first meridian ; and, therefore, cannot exceed 100**. 

• * 

t 

^ The Difference of Longitude between any two 
places is the intercepted arch of the Equator. be^ 
Iween the meridians of those places^ and catmot 
exceed 180'. 

There are three different Horizons, the appa- 
rent, tlie sensible, and the true. The apparent or: 
visible Horizon is the utmost apparent view of the 
sea or land. The sensible is a plane passing tlu-ough 
the eye of an observer, perpendicular to a plumb • 
line har/^ing freely; And the true or rational Ho-, 
xh^^ti is a plane pa-sing Ihronirh the centre of thcj. 
parth, parallel to the sensible Horizon. . .^ • • 



. INTRODUCTORY PRINCIPLES. 301 

Altitudes observed at sea, are ineasured freoi 
the visible Horizon. At land, when an astronom- 
ical quadrant is used, or when observations are 
taken with a Hadley's quadrant by the method of 
reflection, the altitude is measured from the sen- 
sible Horizon ; and in either case, the altitude 
must be reduced to the true Horizon. 

The Zenith of any given place is the point im-' 
mediately above that place, and is, therefore, tlje 
elevated pole of the Horizon : The Nadir is th^ 
other pole, or point diametrically opposite. 

m 

A Vertical is a great circle passing through the 
Zenith and Nadir; and, therefore, intersecting 
the Horizon at right angles. « 

The AH itude of any celestial body in that por- 
tion of a Vertical, which is contained between its 
centre and the true Horizon. The Meridian Alti- 
tude h the distance of the object from the true Hori- 
zon, when on the Meridian of the place of obser- 
ration. When the observed Altitude is correct- 
ed for the depression of the Horizon, and the er- 
rors arising from the instrument, it is called, l^e 
apparent Altitude ; and when reduced to the truer, 
Horizon, by applying the parallax in Altitude, it 
iscajled the Inte Altitude. Altitudes are express- 
ed in degrees, and parts of a degree. 

The Zenith Distance of any object is its dis- 
tance from the Zenith, or the complement of its 
Altitucje. 

The declination of any object is that portion of 
its meridian which is contained between the equi- 
noctial and the centre of the object ; and is either 
north or south, according as the star is between 
the equinoctial nnd thr north or ^^onth pole. 



392 INltlODtJCTOflY tTtmClPL^S. 

The Ecliptic is that great tiircle, in which the 
btintiiil tevolutian of the Earth round the Snin h 
J>erfoWied. It is «o named, because Eclipses can- 
not happen but when the moon is in or near that 
circle. The inclination of the Ecliptic ttnd JEqui- 
tioctial is at present about 23* 28' ; and by com- 

{)aring ancient with modem 6bset*vations, Ihe ob^ 
iquity of the Ecliptic is found to be diminishing 
.-i.^rti|c]i diminution, in the present cefltUry, is 
about half a second yearly, 

I 

The Ecliptic, like all other great circles of the. 
8][^ere, is divided into 36{>' ; and is fur&erdivided 
into twelve equal parts, called Signs :, each Sign, 
therefore, contains 30*. The names and cfaarac-^ 
ierB of these Signs are as follows : t 

Aries, T Cancer, ® Libra, ^ Capricomus^ >f 
Taurus, 6 Leo, Q Scorpio, H Aquarius, ^ 
CeiiuAit ^ Virgo, ^KSagittarius,^ Pisces, X 

♦ . 1 

Since the Eoliptic and Equinoctial are great 
efrcles, they, tlierefore, bisect each other }ntiv6 
jyt>ints, which ftre called the Equinoelial Points. 
The Sun is in one of tliese points in Mjirch, and 
in the other in September ; hence, the first i« calK 
ed the VemcUy and the other the Autumnal Equi^- 
nox — ^and that sign which begins at the Vernal 
Equinox is called Aries. Those points of the 
Ecliptic, which are equidistant from the equinoc- 
tial points, are called the Solstitial Points ; the - 
first the summer, and the second the winUr solstice. 
That great circle which passes through the equi- 
noctial points and the poles of the earth, is caSled 
the Equinoctial Colure: and the great circle which 
passeslhrough the solstitial points imd ib^ poleeof 
the earth, is called the Solstitial Colnri^- ^^ - , . ^»^ 



IWTftOOUGTORY PRINGEPLES; 

. lYben tb^ Sua «D^rs ArieSii k bis tJbe I^^uir 
qoctia}; ^od^ therefore^ has no d^cliB^tioGi.. Frox^ 
IheQcc^ it moves forward mthejp^cliptk^.accoi^lp^ 
tp the order of the sigjin, and advaxices towajrdjs tJE^ 
north po^e, by a kiud of retarded molion, till it eo^ 
lers Cancer, and is then most dlntaiait fron^ th^ 
Equinoctial ; and moving forward in the Eclipt icy 
tjbe Sun appajrently recedes from the noilh p^I^ 
wkh>ao accelerated motion till it enters Ltt)rai an4 
l(eing again in the E^j^uinoctial, has no declin^^ipa^A' 
tjbe Sun moving through the sign« Lifoca, Scorpio^ 
and Sagittarius, enters Capricorn ; and then it^ 
gputh declinatjon is greatest^ and is, theiiefore^ 
WfioA distaiit iroiothe north- pole; and moving fbih 
ward Uuougli the signs. Capricorov Aq-narius, afi4 
Pisces, again enters Aries : Hence, a period of 
the seasons is completod^ and this period ifi called 
a, SoljBLi: YeajT. 



4» • 



The rfgns' Aries j Taums, Genunt, Cancer, Ledj, 
5itid Yirgo, are called Northern Signs^ becauec^ 
t|iey: are contained in that part of the Ecliptio: 
whkh is between the Etq^uinoctial and North Pole 7, 
audi therefore, while the Sun is in these signs, its. 
^ai^lisation 19 north : the other six sagn&aji^ ca^^d. 
Sj^hfrn SignSn The signs in the first and. foui:t||; 
quarters of th^^ Ecliptic are^called Ascending Signs c 
because^ while the Sun is in these signs, it ap^ 
proaches the north pole — and, therefore^ in th^ 
northern, temperate, and frigid zones, the Sun'a 
niwidiaii. altitude daily increases; or, which is th^ 
a9nie„ the Hun ascends to a* greater height above« 
the horizon every day. The signs in the secQo4, 
and tbicd ^quarters of the Ecliptic are called JDc- 
steviding JSigns^ 

ff 

The 'tropica are circles parallel to the Equi- 
R^ciia], whMa; dagtanoe therefrom^ is equal to the 



304 INTRODUCTORY PRINCIPLES. 

obliquity of the Ecliptic. The Northern Tropic 
touches the Ecliptic at the beginning of Cancer, 
and is, therefore, called the Tropic of Cancer ; and 
the Southern Tropic touches the Ecliptic at the 
beginning of Capricorn, and is hence called the 
Tropic of Capricorn. 

Circles about the poles of the Equinoctial, and 
passing through the poles of the Ecliptic, are call- 
ed Polar Circles ; the distance, therefore, of each 
Polar Circle from its respective Pole, is equal to 
the inclination of the Ecliptic and EguinoctiaL. 
That Circle which circumscribes the North Pole^ 
is called the ArtiCy or North Polar Circle ; and 
that towards the South Pole, the AiUartic^ or Swiih ^ 
Polar Circle. 

That semicircle which passes through a star, or . 
any given point of the heavens, and the Polet; of, 
tJba £clq[>tic, ia called a Circle of JLatitude. ^ ^ 

The Reduced Place of a Star is that jpoiiiV of. 
the Ecliptic, which is intersected by the circle of. 
l^ttide passing through that star. . ^ . . 

, The Xatitude of a Star is that portion of the 
circle of latitude contained between the Star and 
itg reduced place — and is eithef north or souths ac- 
cording as the Star is between the Ecliptic and 
the north or south pole thereof. 

The £ongitude of a Star is that pbrti6n of the 
Ecliptic, contained between the Vernal Equiopx 
^Jk^i the reduced place of the ^tar. 



'» 



(305) 



,1 . ' » 



SECTION II. 



' »' « 



Descr^tion of the LutrumentB reqiMle m A^irunQtiticut ^ 

Obaervatione' ' -^ 



TH£ QUADRANT. 



t 



T Is generally allowed that we are indebted tcr 
John lladley, Esq. for the invention, or at least ; 
for the first public account of that adoiirahle voir f 
strmnent, conunonly called Hadley's Q,i^tdrant» r. 
Who in the year 1731, first communicated its prin- 
iiiples to the Royal Society^ which were by thorn 
published soon after in their Philosophical Tran&r , . 
actions ; before this period, the Cross Stafii* ande."* 
avis's Quadrant were the only instruments used 
r measuring altitudes at sea, both very imper- 
fect, and liable to considerable error in rough tfea^ 
ther; the superior excellence hower^r of Hadr i 
ley's . duadrant, soon obtained its general use 
among ^amen, and the many improvements thh 
instrument has received from ingenious men at 
various times, has rendered it so correct, that it is 
now applied, i^ith the greater success, to the im- 
p<irtant purposes of ascertaining both the latitude 
and Ifor^itude at sea> or land. 

The Octant or Frame, is generally 'made of 
ebony, or other hard wood, an^ consists of ah arch 
firmly attached to two radii, or bars, which are 
strengthened and bound by the two braces, in or-" 
der to prevent it from warping- 

Rr 



m TH£ aU ADH ANT. 

The Arch, or Limb, althou^b only the eigttfc 
pail of a circle, is on account of the double reflec- 
tion, divided into 90 degrees, numbeiedO, 10, 20^ 
30, &ic. from Uie right towards the left ; these are 
subdivided into 3 parts, containing each 20 mi- 
nntes, which are again subdivided into single mi- 
nutes, by means of a scale at the end of the Index. 
The arch extending from towards the right hand 
is called the arch o/ excess, 

llie Index is a flat brass bar, that turns on the 
centre of the instrument ; at the iower end of the 
Judex there is an oblong opening : to one side of 
this opening a Nonius scale is fixed to subdivide 
the divisions of the arch ; at the bottom or end of 
tlie index, there is, a piece of brass which bends 
under the arch, carrying a spring to make the 
Nonius scale lie close to the divisions ; if is also 
furnished with a screw to fix the Index in any de- 
sired position. 

Some instruments have an adjusting or tangent- 
screw, fitted to the Index^ that it may be moved 
I more slowly, and with greater regularity and ac- 
curacy than hy the hand ; it is proper, howerer,r 
to" observe, that the Index must be previously fix- 
'cd near its right position by the above mentioned 
screw, before the adjusting screw is pift in motion*^ 

The Nonius is a scale fixed to the end of the 
Index for the purpose, as before obserted, of di- 
viding the subdivisions on the Arch into Minutes ; 
tt sometimes contains a space of 7 degrees, or 21 
subdivisions of Ihe limb, and is divided into 20 
equal parts; hence each division on the lyonius 
will be one-twentieth part greater, that is, one xm- 
j^ute longer than th« divisiofis on the Arch ; con- 



TliE atJADRANT. 305 

Mtjuently, if the fir^t divisk>n of tte Nonius mark- 
ed 0, be set {)red9ely op|>o6ite to any degi^e, the 
rfeiative position of the Nonius and the Arch must 
he altered one minute before the next division ou 
the Nonius will eoincidi with the next diviBion on 
the A|-ch, the second dirfsion will require a chanp^e 
.of 2 minutes^ the third of 3 minutes, and so on, till 
ihe 20th stroke 6n the Nonhis arrires at the next 
20 minutes on the Arch ; the on the Nonius will 
then have moved exactly 20 minutes from the di- 
vision whence it set out, and the intermadiate di^ 
visions of each minute^ have been regularly, point- 
#ed out by the divisions of tlie Noniuii. 



The divisions of the Nonius scale are in the 
.above case reckoned from tlie middle towaitls the 
right, wd from the left towards the middle ; there- 
Core th^ first 10 minutes ai'e contained on the right 
of the 0, and the other 10 on the left. But this 
method of reckoning the divisions being found in^ 
convenientt they are more generally counted, be- 
ginning from the right-hand towards the left; and 
then 20 divisions on the Noniuis ^re equal to 19 
on the limb, consequently one division on tlk> 
Arch will exceed one on the Nonius by one-twen- 
tieth part, that is, one minute. 



The on the Nonius, points out the entii-e dcr 
grees and odd twenty mbiutes subtended by the 
objects obsei-vcd ; and if it coincides with a divi- 
sion on the Arch, points out the required angle : 
thus, suppose the on the Nonius stands at 2o de- 
grees, then 25 degrees will be the measui^ of the 
• angles observed; if it coincides with the next dir 
vision on the left hand, 25 degrees 20 minutes is 
tbe apgle ; if with the second division bey Qnd 25 



308 THE, QUADRANT. 

degrees^ then the angle will be 25 degrees 40 mi- 
nuti^s ; and so on in every instance where the on 
the Nonius coincides with a division on the Arch;- 
but if it does not coincide, then look for a divi* 
Bion oh »the Nonius that^stands directly opposite 
to one on the Arch, andlhat division on the No- 
nius gives the odd minutes to be added to that on 
the Aitrh nearest the right-hand of the on the 
Nonius ; for example, suppose the Index division 
does not coincide with 25 degrees, but that the 
next division to it on the Nonius is the first coin* 
cident division, then is the required Angle 25 de- 
grees 1 minute ; if it had been the second division^ 
the Angle would have been 25 degrees 2 niiimtes», 
and so on to 20 minutes, when the on Ihe N6niits 
would coincide with the first 20 minuter on the- 
Arch from 25 degrees. Again, let us suppose the. 
oh the Nonius to stand between 50 degrees-and 
50 degrees 20 minutes, and that the 1 5tb division 
on the Nonius coincides with a division on the 
Arch, then is the angle 50 degrees 15 minutes. 
Further, let the on the Nonius stand between. 
45 degrees 20 ipinutesand 45 degrees 40 minutes,, 
and at the same time, the 14th division on the No-: 
nius stands directly opposite to a division on the 
Arch, then will the Angle be 45 degrees 34 mi- 
nutes. 

The Index Glass is a plane speculum, or mirror 
of glass qiiicksilvered, set in a brass frame, and so 
placed that the face of it is perpendicular to the . 
plane of the insti*ument, and imipediately over the 
pentre of motion of the Indexi This muTor being 
fixed to the Index moves along with it, and has its 
direction changed by the motion thereof. 

This glass is designed to reflect the image of the 
gun, or any other object, upon either of the two . 
horizon glasses, from whence it is reflected to tbe^ 



•THE aUADRANT. 30» 

eye of the obserrer. The brass frame, .with the 
glass, is fixed to the Index by the screw ; the other 
screw serves to place it in a perpendicular posi? 
tion, if by any accident it has been put out of or* 
der. 

The Horizon Glasses are two small speculuma 
on the radius of the Octant ; the surface of the 
vippef one is parallel to the Index glass when th^ 
on the Nonius is at on the Arch ;. these mir- 
jrors receive the rays of the object reflected from 
the Index glass, and transmit them to the observer. 
The fore Horizon glass is only silvered on its lower 
half, the Upper half being transparent, in order that 
the direct object may be seen through it. The 
back Horizon glass is silvered at both ends ; in 
the middle there is a transparent slit, through 
which the Horizon raav be seen. Each of tliese 
glasses is set in a brass if ame, to which there is an 
axis ; this axis passes through the wood work, and 
is fitted to a lever on the under side of Hhe quad- 
rant, by which tlie glass may be turned a few de- 
grees on its axis, in order to set it *parallel to the 
Index glass. 

To set the glasses perpendicular to the plane of 
the quadrant, there are two sunk screws, one be- 
fore and one behind each glass : these screws pass 
through the plate on which the frame is fixed mto 
another plate, so tha,tby loosening one and tighten- 
ing the other of these screws, the direction of the 
frame, with its mirror, may be altered, arid thus 
hp set perpendicular to the plane of the mstrument. 

The Dark Glasses, or Shades, are ysed to pre- 
vent the bright rays of the Sun, or the glare of the 
Mot>n, from hurting the eye at the time of obser- 
vation; there are generally three of them, two red, 
aad one green. They areeacb set in a brass frame 



310 THE QUADRANT. 

which turn on a centre, so that they may be used 
separately or together, as the brightness oi the 
object nuiy require. Thp green ^lasg may be used 
also alone, if the Sun be very famt; U is likewise 
itBed in taking observations of the Moon ; when 
these glasses are used for the fore observation^ 
they are set immediately before the fore Horizon 
glass, but in front of the other Horizon glass, 
when a back observation is made. 

The Sight Vanes are pieces of brasa, standing 
perpendicular to the plane of the instniment r that 
one which is opposite the fore horizon, is called 
the fore Sight Vane^ the other the back Sf^ht Vane. 
There ai-e two holes in the fore Sight Vane, the 
lower of wljich, and the upper edge of the silvered 

Eart of the fore Horizon glass, are equidistant 
om the plane of the instrument, and the other is 
opposite to the middle of the transparent part of 
tliat fflass ; the back Sight Vane has only one hole, 
which is exactly opposite to the middle of the 
transparent slit in the Horizon glass to which it be- 
longs : but as the back observations are liable tl> 
many inconveniences and errors, we shall not give 
any directions for tlieir practice. 

ADJUSTMENTS. 

The several pai'ts of the Quadrant being liable 
to be out of order from a variety of accidental cir- 
cumstances, it is necessary to examine and adjust 
thenn so that the instrument may be put into a 
proper state^ previous to taking observations* 

An instrument properly adjusted, must have the 
Index glass and Horizon glasses perpendicular ta 
the plane of the Quadrant ; the plane of the fore 
Horizon glass parallel, and that of the back Hpri- 



THE atTADfeANT. 311 

zotk glass perpendicular to the plane of the Index 
gl^s!$, ^heii the on the Nonius is at on the 
Arch ; hence the Q^iadrant reauires five adjusir 
meirts, the first three of which oeing once iRajde» 
are not so liable as the last two to be out of order ; 
howisver tliev shoulti all be occasionally exauunr 
ed \h case of an accident. 



h 7\f 9et the PImu of the lnde± (Uas$ ficrfiendicular to tf»t ^ 

the Instrument, 

Place the Index near to the middle 6f the Arch, 
and holding the Cluadrant in a horizontal position, 
with the Index glass close to the eye, look ob- 
liquely down the glass, in such a manner that vou 
may see the Arch m the Quadrant by direct vie w^ 
and by reflection at the same time ; if they join 
in one direct line, and the Arch sieen by reflectiota 
^orms an ex^ct plane, or strait line, with the Arch 
seen by direct view, the glass is perpendicular to 
the plane of the Qtuadrant ; if not, it must be re- 
stored to its right position by loosening the screw, 
4xr tightening it^ or vice versa, by a contrary 
operation. * 

n- T9 set the Fore Horizon Glass fiafaUeUto the Index Glass^the 

Index being at 0^ 

* 

Set the on the Nonius exactly against on 
the Arch, and fix it there by the screw at the un- 
der side. Then, holding the Quadrant vertically, 
with the Arch lowermost, look through the Sight 
Vane, at the edge of the sea, or any other well 
defined and distant object- Now, if the Horizon 
in the silvered part exactly meets, and forms one 
coritinued line with that seen through the unsil- 
^vered part, the Horizon glass is parallel to the In- 
dex glai??. But if the noriTOns do not ^^oincid^y 



312 THE atJAOftANT, 

ihen looseB the button-sciiew in the iiiidclle <tf <litf 
lever, on the umler side of the Quadtaiit, lamd 
move the Horizon glass on its axis, by tuftiMg ^e 
nut at the end of the adjusting lever, till ybU have 
made them perfectly coincide ; then fix the lev^r 
firmly in this situation by tightening the btittbti- 
screw. This adjustment ought to be repeated bfe- 
fore and after eVery observaticm. Some obser- 
Ters adopt the following method, which is Called 
finding the Index error. Let the Horizon glass 
remain fixed, and move the Index till* the iknage 
and object coincide ; then observe whether'© on 
the Nonius agi-ees with on the Arch, if it does 
not, the number of minutes by which they differ Is 
to be added to the observed altitude or ^hgle, if 
the on the Nonius be to the right of the on the 
Archy but if to the left of the on the limb^ ^ it is 
to be subtracted. ' . 

It has already been observed, that that part of 
the Arch beyond 0, towards the right hand, is cali- 
ed the Arch of excess : the Nonius, when ihe *t) 
on it is at that part, must be read the contrary- 
way, or which is the same thing, you may read off 
the minates in the usual way, and then their oom- 
plement to 20 minutes will be the red numnerj. to 
be added to the degrees and minutes pointed cHtt 
by the on the Nonius, 

« 

IIL To set the Fore Horizon Glata fierfiendicuiar to (he Plane of 

the Qtuulranf. 

Haying previously made the above adjustrpentt 
incline the Quadrant on one side as much as possi- 
ble, provided the Horizon continues to be seen 
in both parts of the glass ; if when the instrument 
is thus inclined, the edge of the sea seen through 
the lower hole of the Sight Vane continues to-fonn 



THE QUADRANT. 318 

>- 1^« imlbroken liiie> the Homon> glass is par^ 

i fi^jCtiy adjusted ; but if the reflected Hqrizon be 

. ^emifated from that seen by direct vision, tlxe spe- 

^.ciihimis Oot perpendicular to the plane of IJtfe 

: Q,u^drant : then if the limb of the Quadrant in in^ 

cli|ie<i towarda the Horizon, with the face of iko 

iastrUiOient upwards, and the reflected nea appeal's 

higher than the real sea, you must slacken . the 

. s^rew before the Horizon glass, and tighten that 

, .which 3S /behind it; but if the reflected sea ap-' 

geais lower, the contrary naust be performed, 
are niust be always taken in this adjustment to 
loosen one screw before the other is screwed up, 
and to leave the adjusting screws tight, or so as to 
draw with a moderate force against each other. 

This adjustment may be also made by the Sun, 
MoOn, or a Star; in this case the Quadrant is to be 
i^held in ^ vertical position ; if the image seen by 
' reflection appears to the right or left of the object 
,jieen directly, then the glass must be acyueted as 
.t)efore by the two screws. 

It Will be necessary, after having made this ad- 
' justment, to examme if the Horizon glass still con- 
tinues to be parallel to the Index glasj$, as some- 
times by turning the sunk screws the plane of the 
Horizon glass will have its position altered. 

t 

USE OF HADLEY'S aUADRANT. 

, . The use of the Quadrant is to ascertain the An- 
<gle suhfteiyled by two distant objects at thfi eye qf 
vihe pbeerver ; but principally to observe f h^ al- 
titude of a cele^al object above the Horizpa : 
this l^ pomted our by the Index when one of the 

8s 



314 tllE QUADRANf. 

objects seen by reflection is made to Goincide with 
the other, seen through tlie transparent partof tb(^ 
Horizon glass. 

To takc.an Mtitude of the Sun^ Moon, or a Star, by a Fore 

Observation. 

Having previously adjusted the instnuDen^ 
place the on the Nonius opposite to on the 
Axch, and l-urn down one or more of the screen^ 
according to the brightness of the Sun ; then ap- 
ply the eye to the upper hole in the fore Sight 
Vane, if the Sun's image be very bright, other- 
tvige to the lotv er, and holding the Qxiadrant ver- 
tically, look directly towards the Sun so a& to let 
it be behind the silvered part of the Horizon glass, 
then the coloured' Stints image will appear on the 
speculum ; move the Index forward till the Spin's 
image, whicH will appear to descend, just touchea 
the Horizon with its lower or upper limb ; if the 
upper hole be looked through, the Sun's image 
must be made to appear in the middle of theitrans- 

{>arent part of the Horizon, but if it be the lower 
lole, hold the Quadrant so that the Sun's iinage 
may be bisected by the line joining the silvered 
anq transparent parts of the Horizon glass. 

The Sun's lunb ought to touch that part of the 
Horizon immediately under the Sun,, but aslhis 
point cannot be exactly ascertained, it will be 
therefore necessary for the observer to give the 
Quadrant a slow motion from side to side, turn- 
ing at the same time upon hk heel, by wjiich mo- 
tion the Sun will appear to sweep the Horizon, 
and must be made just to touch it at the lowest 
part of the Arch; the degrees and • minutes thea 
pointed out by the Index on the Limb of the 
Quadrant will be the observed altitude of that- 
limb whioh ig brought in contact with the Hqrizoar 



THE aUADRANT. 315 

When the meridian or greatest altitude is re- 
quired, the observation should be commenced a 
short time before the object comes to the meri- 
dian ; being brought down to the Horizon, it will ^ 
appear for a few minutes to rise slowly ; when it " 
is again to be made lo coincide with the Horizon 
by moving the index forward ; this must be re- 
peated until the object begins to descend, whert 
Che Index is to be secured, and the observation to 
he read off, ' ^ 

From tlus dbscriptioii of the Quadrant and its use, the manner of 
adjusuDg and Rising die Sextant will be readily apprehended. Ouv 
limits wiQ not allow a particular descnption ot this excellent in- 
•truioeDt^ 

The Artijidal Horizon. , 

In many cases it happens that altitudes are to be 
taken on land by the Q^uadrant or Sextant; whicb^ 
for want of a natural horizon, can i»nly be obtain- 
ed by an nrtificinl one. There have been a vari- 
ety of these sorts of instruments made, but the 
kind now described is allowed to be the only one 
that can be depended upon. It cbnsists of a wood 
or metal framed roof, containing two true parallel 
classes of about 5 by 21 inches, Yixed not too tij]^t 
m the frames of the roof. This serves to shelter 
from the aha wooden trough filled with quicksil- 
ver. Jn making an observation by it with the 
Quadrant, or Sextant, the reflected image of the 
sun, moon, or other object, is brought to coincide 
with the same object reflected from the glasses of 
the Quadrant or Sextant : half the HiigJe shown 
upon the limb is the altitude above the horizon or 
level required. It is necessary in a set of obser- 
vations that the roof be always placed the same 
way. When done with, the roof folds up flat^ 
ways, and, with the quicksilver in a bottle, kc\ i^ 
pejcked into a portable flat ca.^, 



(316) 



SECTION m. 



' To find the Latitude by the Meridkm Altitude tf the Stm, 

The Latitude of a place is its distance from the eauator* eijther 
Korth or South ; and is measured by an arch of a Meridian cdn^ 
tained between the Zenith and the equinoctial. Hence; if thedi** 
tanc(^ of nny heavenly body from the Zenith, when on t!^ Meridian*, 
and its declination, or the number of degrees and minutes it is to the 
North watxU or Southward of the equinoctial, be given, the Latitude 
may thence be tound. 

The Altitude of the Sun, observed by a Quadrant, or Sextant, re- 
quires fonr corrections in order to obtain the true iQdtnde ; these ate^ 
the Semidiameter, Dip, Refraction* and Parallax. 

By the Semidiameter of the Sun is meant the angle subtended by 
the distance from its centre to its apparent Circumference. The 
quai titv of this angle is given for every sixthday in the year in table 10. 

The Dip of the Horizon is a vertical angle contamed betweexi a 
Horizontal plane passing through the eye of an observer, atid ft line 
drawn from his eye to the visible Horizon. This Dip isfouxid m 
Table 8, when the visible liorizon is formed by the apparent junction 
of the water and sky ; but in Table 9, when land intervenes. In tHfe 
case, the line that separates the land and water is used as the Hon- 
zojK and its distance from the observer must be duly estimated.. 

The Refraction of any celestial body is the difference between its 
apparent place, and that wherein it would be seen, if the space be* 
tween the observer and object, was either a void, or of a uniforni 
density. l'a!>Ie 6 contains this Refraction. 

That part of the heavens, in which an object appears, when view- 
ed from the surface of the earth, is called its apparent .place; -and 
tile point, wherein it would be seen, at the same instant, if viewed 
from the centre of the earth, is called its true place ; thedfifitrcncc 
between the true and apparent places, is called ihePanUl^KX* The 
Sun's Parallas^ in Altitude is found in Table 7. 



RULE 

For finding the Latitude from the Sun^i MeriMan 

Altitude. 

Having observed with the Quadrant, or Sextant, 
^he altitude of the Sun's lower limb above the vi- 
fiible horizon, — or the line of separation of the 
land fx'om the water, when that horizon is:obstruct- 
ed by land — add thereto the semidiameter, taken 
from table 10 at the given day of the month, or ^he 
pne nearest to it, and from this suni subtract the 






IHE LATITUDE. 



319 



X)ip, from table 8 or 9, corresponding to the height 
of the observer's eye above the surface of the 
water ; .and this resixlt will be the apparent alti- 
tude of the Sun's centre. Then take* the refrac- 
tion from table 6, and the parallax from table t» 
corresponding to this altitude, and the diiference 
of Uhese quantities, called the correction, being 
subtracted from the apparent altitude, the remain-* 
der will be the Sun's true altitude ; the comple- 
ment of which will be ite zenith distance, north or 
soiith, according as the Sun bears south or nortb^ 
at the time of obseiTation. 

When the observation has been made by bring- 
ing the Sun's image in the Quadrant, or Sextant^ 
to a just coincidence with its image in an artificial 
horizon, half the angle shown on the instrument Ir 
the Sun's apparent altitude, which must be cop^ 
rected by the corresponding refraction and paral- 
lax only, in order to obtain the true altitude. 

Take the Sun's declination from table 13, an- 
swering to the given year, month, and day, observ- 
ing whether it be north or south, and reduce it, as 
there directed, by the help of table 14, to the lon- 
gitude of the place of observation. Then the sum, 
or difierence of the zenith distance, and declina- 
tion, according as they are of the same, or of a 
contra^ denofloination, will be the latitude of the 
place of observation, of the same name with the 
greater of those two quantities. 



VAHTATION OF 



Examples. -^'■ 

Irt. March lotfi, 1811. hl-mje- 2il. MarlPrh. ISJi.inbut.W 



I . 

M. r Ml r^l^ 
a«i>^<tivneti3- 
np— ublcA 

Ap. AIL 
Currcctloa 


. = 19- jij no' ^ 
= +16 05 
B -<13 19 

= 50 o; 49 

B — U 


f.lr]t 

«a> 50" 40" 
p Ap, AtL 

TniE a: 

Zraltli ! < ' 

tatitiidet 


^.?=4»' iV •W 
= -43" 


Troe Alt. 


50 mw 


. ,T. 


Zcnllh niU. 
iUriuced Dec. 


= 59 .57 .«N 
- 4 1% ■-■US. 

;5 i; .jn. 


-=9S « JT'K. 



a^At at 19 

JUL, =j'&> 33' OO" & t-OTTocUon ^ ^ ~*^ ., 



SECTION IV. 

VARIATION OF THE COan\V-S. 

TIk varialirtn of the compas.-i U Uie 'itviation^ 
tlrtj points of Uie inariuei's ctunpuss from the cop*- 



THEGOMl^ASS; 319 

responding points of tbe horizon, and is termed 
east or west variation^ according as the magnetic 
jieedle, or north point of the eompa^vs, is inclined 
to the eastward or westward of the true north point 
of the horizon. 

The true amplitude of any celestial object is an arch of the hori* 
aon contained between. the true east or west points thereof, and the 
centre of the object at the time of its rising or setting ; or it is the 
degrees and minutes, the object rises or sets to the northward or 
southward of the true east or west points of the horizon. 

Tlie magnetic amplitude, is an arch contained between the east 
or w est points of the compass and the centre of the object at rising 
or setting ;. or it is the bearing of the obJ4:ct, by compass* when in 
the honzoQ. . 

The triie azimuth of an object is an arch of the horizon contained 
between the true meridian and the azimuth circle passing thi'ough 
the centre of the object* 

Tlie magnetic azimuth, is an arch contained between the magnetic 
meridian and the azimuth circle passing through the centre of the 
object; or it is the ^>earingof theobject» by compass, at any time 
when it is above the horizon. 

The true amplitude, or azimuth, is found by calculation, and the 
magnetic amj^itode, oraeimoth, by an azimuth compass* 

THE AZIMUTH COMPASS. 

From the accounts of the compasses, heretofore 
given in the descriptiou of surveying instruments, 
it is presumed that the nature and properties of thp 
azimuth compass will be readily conceived by a 
contemplative inspection ; the directions for its 
tises are as follow : 

To observe the Sun's am/iHtude. 

Turn tlic compass-box until the vane containmg Xht magnifying 
glass is directed towards the sun : and when the bright speck, w^ 
riys of the sun collected by the magnifying glass, falls upon the slit 
VI the other vane, stop the card by means of the nonius, and^read off 
the amplitude- 

Without using the magnifying- glass, the sight maybe directed 
through the dark glass towards Uie sun ; and in this case, the card 
is to be stoppod iirheu the tun is bisected by the thread !n the other 
ipine. 

•The observation should be made when the sun's lower limb ap- 
pears sonicwhat more than his semidiameter above the horizon, 
btetvse his centce is ireally then in the horizon, although it ia ftp" 



^20 VARIATION OF ^ 

pftrentlf elevated oo accoiiiit of the refraction of the atoMipliere : 
this Is i>artictilarly to be lu^ced in high latitudes. 

To obaerue the Sun*a Jtzmuth- 

Raise the magnifving-glass to the upper part of the vane, and 
•Rov^ theboK. asbeuHv directed, until the bright speck fall oo the 
other vanet or on the Ime in the horizontal bar ; the card is then to 
be stopped^ and the divisions being read off, will be the son's mag^ 
lietic azimuth. 

If the card vibrate cxmsiderabljr at the time of observatkn, it wiS 
be better to observe the extreme vibrations, and take their meaa as 
tiie magnetic azimuth- When the magnetic azimuth is observed* 
tlie altitude of the object must be taken, in order to obtain the true 
azimuth. 

It wHl conduce muph to accuracy if several azimuths be observed, 
vridi the corresponding altitudes, and the mean of the whole taken 
for the observation- 

To find the variation of the Compass by an amplitude. 

Rule — 1. To the log. secant of the latitude, 
rejecting; the index, add tlie log. sine of the sun'f 
declination, corrected for the time and place of 
observation; their sum will be the lofg, sine of the 
true amplitude, to be reckoned from the east in 
the mommg, or the west in the afternoon, towards 
the north or south, according to the declioaiJon. 

2. Then if the true and magnetic amplitudes, 
be b(tlh north or both south, their difference k the 
Tariation ; but if one be north and the other south, 
their sum is the variation ; and to know whether 
it be easterly or westerly, suppose the observer 
looking towards that point of the compass repre- 
senting the magnetic amplitude : then if the true 
amplitude be to the right hand c^ the magnetic 
amplitude, the variation is east, but if to the left 
band, it is west 



THE COMf>AS^. m 

EXAMPLE J. 

luly 3, 1812, m latitude 9o SS' S. the Son was obtenfed to rise B^ 
3S^ 4^ N : required the vanation of the compasa 

Latitude 9o 36' S. - Secant 0.0061$ 

' ■ DccUnation 22 59 N- - Sine 9.59158' 

True amplitude R 23 20 N- * Sine 9*5^71 
Mag.aaiplitade£.12 42 N* 

> Variatkm * 10 38 west, beoanse the true amplitude i9 

. totlielefitaf themagoetsc* 

EXAMPLE IT* 

September 24^ 1812, in latitude 2<^ 32^ N. and longitude 7"^ W. 
the Suo*s centre was observed to set W. Cp 1$' S» about Gx- P- M. 
ji^quired the variation of the compass* 

Sun's declination (y> S(f & ^ 

CcH-r. for long, reo W. + 5 

Corr. for time 6h. P. M, -f 6 

Reduced declination 41 Sine 9.0T650 

Latitude 26 32 Secant 0.04834 

i True amplitude W. 46 S. Sne )ftl.2464 

Mag. acnpMtttde W. 6 15.8* ' 

VariBtion 5 29 east, bectaae the true 
amplitude is to the right hand of the magnetic 

To Jin/d the Variation of the Comftatt bf an Ar^uth 

♦ • . • . ' 

Rule. L-^-^Reduce the Sun's decliDation to the 
' tSne and place of obseiration^ and confute the 
true 'altitude of the Sun's centre. « 

2. Siibtract' the Sun's declination from 90*^ 
wfaeb the latitude and declinationf are ef Ihe same 
Bame> or add it to 90*j when tliey are of conftrory 
Nintmes^and ihe sum, orremainder, will be the 
* Sun's polar distance. : ^ ^ 

3« Add together the Sun's polar distaneer the 
latitude of tber place, and the altitude of the Sun; 
take the difference between half their sum and the 
polar distance, and note the remainder. 
4. Then add together 
the log. secant of the altitude ) rejecting their 
the log, secant of the latitude \ indicesr 
the log. CO. sine of the half sum, 
and ihe log. co. sine of the remainder. 

Tt 



322 



VARIATION OF 



5. Half the sum of these four logarithms will be 
the sine of an arch, which doubled, will be the 
Sun's tjrue azimuth; to be reckoned fix>m the soutl^ 
in north latitude, and from the tiorth in south lati- 
tude : towards the east in the morning, and to- 
wards the west in the afternoon. 

6. Then if the true and observed azimuths be 
'both on tiie east, or both on tJiie west side of the 
meridian, their difference is tlie variation : but if 
one^ be on the east, and the other on the we^ side 
of the meridian, their sum is the variation ; and to 
know if it be east or west, suppose the ohsierver 
looking towards that point of the compass repre- 
senting the magnetic azimuth ; then if the ti^e 
azimuth be to the right of the magnetic, the vari- 
ation is east, but if the tjfue be to the left of the 
raagnetica the variation Is west 

EXAMPLE. 

November 2, 1812, in latitude 2*» 32' N, and 
longitude 75* W. the altitude of the Sun'tf lower 
limb was observed to be Id*" 36^ about 4b. lOm. 
P. M. bis magnetic azimuth at that time being S» 
58^ 32^ W. and the height of the eye 18 feet; re- 
quired the Variation of the compass. 

Sun's de. Nov. 2, at d. 14» 48' S. Ot». alt Sun's lower lisd^ ISo W 



Cerr. for long- 75^ W« 
Co- for ti. 4h. 10m* af- a 


14 55 

90 00 


Semidiameter 16" > 

Refraction 

True altitude 

Secant 0.01663 
Sesant 004463 

- Ca sine 9.46345 

Camne9.9292» 

19.45399 

- ^ne 972699 

I because the true azhmith 


+ 12 


Reduced decGisaUoa 


15 48 

S 


Polar distance 

Altitude 

Laititode 


104 55 
15 45 

25 32 


15 45 


Sum 

Half 
Kemainder 


146 12 

r3 6 

31 49 

32 14 

2 


1 


True azimuth S^ 
Mag. azimuth S- 

Variation 
right of the magnetic' 


64 28 W, 
58 32 W. 

5 56 east 


* 

is to the 



THB COMPASS. 323 



TPa^nvm tmt merUian Une to a mafiy itavhg the variation end 

mognttkai meridian given' 

Qn any magne^cal Qaeridian or paralld, upon which the map ib 
protracted, set off an angle €rom the north towardatheeaHk equal to 
Sie degrees or quantity of variation, if it be westerly, or from the 
north towards the itest* if it he easterly, and the line which consti- 
tutes such an ai^le wiiii the maguetical meridian, will k^ a true 
meridian line. 

For if the variatoi be westerly, the ma|;netica] meridian will be 
the quantity of variation of the west side of the true meridian^ but 
if easterly, on the east side; therefore the true meridian must be a 
like quantity on the east side of the magnetical one, when the vari- 
ation is westerly, and on the west side when it is easterly* 

To iay out a true meridian Sne by the drcumferentoT' 

If the variation be westerly, turn the box about till the north of the 
needle points as many degrees from tiie flower-de-luce towards the 
east of the box, or till the south of the needle points the like number 
of degrees ^tota the south towards the west, as are the number of 
degrees contained in the variation, and the index witt'be then due 
north and south : therefore if a line be struck out in the direction 
thereof, it will'be a true meridian line* 

If the variation wat easterly, let the north of the needle point as 
many degrees from the flower-de-luce towards the west of the boit,* 
or let the south of the needle point at maby degrees towards the 
east, as are the number of degrees contained in Uie variation, and 
then the north and south of the box will ddincide with the north and 
9oa|h pdnts of the horizon, and consequently a line being laid out by 
the direction of the index, will be a true meridian line* 

This will be found to be very useful in setting an horizontal dial, 
for if you lay the edge of the index by the base of the stile of the 
dial, and keep thean^ar point of the stile toward the south of the 
box, and allow the variation as before, the dial will then be due north 
and soiltlH aftd in i{s proper situation, provided the plane upon 
which it is fixed be duly horizontal, and the sun be soutn at noon^ . 
but in places -where it is north at noon, the angular poipt of the hi- 
dex must be turned to thie north. 

Horn mope may be traced by the hetfi nf a true mendkm Sne* 

If all maps had a true meridian line laid oat upon them, it would 
be easy by producinf^ it, and drawing parallels, to make out fieJd* 
notes ; and by knowmg the variation, and allowing it upon every 
bearing, and having the distances, you would have notes sufficient 
for a trace. But a true meridian line is seldom to be met with, there- 
fore we are obliged to have recourse to the foregoing method. It is 
therefore advised toJUiy out a true meridian line upon every map. 

Tojtnd thed^ertnce between the firtaent variation^ and that at 
a time when a tract was formerly surveyed, in order to trace or run 
out the original Unes- 

If theoM variation be specified in the map or writing and the pre- 
sent be known, by calculation or otherwise, then the difference is im- 



324 VARIATION, Sec. 

mediately leen fay inspection ; but as it more f nequcsitly hapfieiifl.* 
that neiUier is certainly knowa, and as die variation of aifierent in- 
strumcuts is not always alike at the same time, the following prac- 
tical method will be found to answer every purpose. 

Go to any part of the premises wh^re any two adjacent caneri 
are known ; and, if one can be seen from the other, take their bear- 
ing ; which, compared with that of the same Hne in the former sop- 
vey, shows the difference. But if trets, hills* &c- obstruct the view 
of the object, run the line according to the given bearing, and 6b- 
serve the nearest distance between the line so mn and the conier, 
then. 

As the length of the whole line 

Is to 5r.3 degrees,* 

So is the said distance 

To the difference of variatkn required- 

EXAMPLE. 

Suppose it be required to run a line which some years agoboi^ 
KE. 45*", distance 80 perches, and in running this line by the riven 
bearing, the aorner is found 20 links to the left hand ; what sSlow- 
ance must be made on each bearing to trace the old lines* and what 
16 the present bearing of this partioilar line by the conpaas ^ 

p. Deg. L. 

As 80 : 5r .3 :; 20- 
25 20 



2|000 1146.0Cd*. 34' 
60 

2)681760.0 

Answer, S4 minutes; or a little better than half a fle^Ke to tha 
left hand, is the allowance required, and the line in questioii beats 
N. 440 25 • K 

J^ott. The diffierent variatidhs do not affect the area m tiie calcn- 
lation, as they are similar in every part of tlkfr survey- 

•sr-S Is the radiua of a circle (nearly) in auch parts aathedr- 
cumference coDtaim 36a 



FI^U 



TABLE L 



OLOGARITHMS OF NUMBERS. 



£»t.ftiVATioir. 



JL4OOARITHMS arc a scries of numbers so contrive'd,that the sum 
of the Logarithms of any two numbers, is the logarithm of the product 
of these numbers. Hence it is inferred, that if a rank, or series of 
numbers in arichmetieal progression, be adapted to a scries of numbers 
in geometrical progression, any term in the arithmetical progression 
will be the logarithm of the corresponding term in the geometrical 
progression. • 

This table contsdns the common logarithms of all the natural num- 
bers from to 10000, calculated to six decimal places ; such, off ac- 
count of their superior accuracy, being preferable to those, that are 
computed only to five places of decimals. 

In this form, the logarithm of 1 is 0, of 10, 1 ; of 100, 2 ; of 1000, 3 
Sec. Whence the logarithm of any term between I and 10, being 
greater than 0, but less than 1, is a proper fraction, and is expressed 
decimally. The logarithm of each term between 10 and 100, is 1, with 
a decimal fraction annexed 5 the logarithm of each term between 100 
and 1000 i3 2, with a decimal annexed, and so on. The integral part of 
the logarithm is called the Index, and the other the decimal part.— 
Except in the first hundred logarithms of thi» Table, the Indexes are 
not printed, being so readily supplied by the operator from this gene- 
ral rule; the Index <^ a Logarithm u alvfO^M one /«# than the number 
^Jigurea contained in its corresfionding- natural number^'^xcluaive pf 
fractionB^ when there are any in thatnumber. 

Hie Index of the logarithm of a number, consisting in whole, or m 
parts, of integers, is affirmative ; but when the value of a number is 
less than unity, or 1, the index is negative, and is usually marked by the^ 
sign, — , placed, either before, or above the index. If the first signi- 
ficant figure of the decimal fraction be adjacent to the decimal pomt, 
the index is 1,— or its aritl^metical complement 9 ; if there is one 
cipher between the decimal point and the first significant figure in the 
decimal, the index is — 2, or its arith. comp. 8 ; if two ciphers, >the in- 
dex is «-t 3, or 7, and so on ; but the arithmetical complements, 9, a, 
7 t(c« are rather more conveniently used in trigonometrical calculations. 

A 



LOGARITHMS OP NUMBERS. 

The decimtl parts of the logarithms of numbers, consbtiag of the 
same figures, are the samey whether the number be integral fractionalt 
or misled : thus, 



of thp n^tur^ 
number 



'23450 

23450) 

234.50 

U3.450 

2.3450 

2.3450 

•02345 

.002345 



the Log.< 



4.370143 
3.370)43 
2.370143 
1.370143 
O.370143 
1.370143 
2.370143 
>370143 



er<'8. 



370143 
370143 
370143 



M. B. The arithmetical complement of the logarithm of anjr number, 
b found by subtracting the given logarithm from that of the radius, or 
)>y subtracting each of its figures from 9, except the last, or right-hand 
figure, which is to be taken from 10. The arithmeUcal complement 
pf an index is found by subtracting it from 10. ' 



PROBLEM I, 

« 

Tojind the logarithm qf any given numherm 

RULKS. 

t 

1. If the number is under 100, its logarithm is found in the first page 
of the table, immediately opposite' thereto. 

Thus the Log. of 53, is 1.724276. 

S. If the number consists of three figures, find it in the first colomu 
pf the following part of the table, opposite to which, lAid under 0, is iu 
logarithm. 

Thus the Log, of 384 U 2.58433 l-^pre^xing the index 2, because 
the natural number contains 3 figures. 

Again the log. of 65.7 is 1.817565— prefixing the index 1, because 
there are two figures only in the integral part of the given number. 

3. If the given number contains four figures, the three first are to be 
found, as before, in the side column, and under the fourth at the top of 
the table is the logarithm required. 

Thus tl^e log. of 8735 is 3.941263-^for against 873, the three first 
figures , found in the left side column, and under 5, the fourth figure 
found at the top, stands the decimal part of the logarithm, vir .941263, 
to which prefixing the ifide^, 3, because there are four figures in the 
natural number, the proper logarithm is obtained. 

Again the logarithm of 37.68 is 1.5761 1 1 — Here the decimal part of 
the logarithpd is found, as before, for the four figures ; but the index 
is 1, because there are two integral places only in the natural number. 

4. If the given number exceeds four figures, find the dilFerence be- 
tween the logarithms answering to the first four figures of the given 
number, and the next following logarithm ; multiply this difierence by 
the remaining figures in the given number, point off as many figures 
|o t^e right-hand as there are in the multiplier^ and the remainder, addr 



LOGAllITHMS OF NUMBERS. 

t " • 

t . 

ed to the log^arithm) answering to the first four figuresi will be the re^ 
quired logarithm^ nearlf. • 

Thus ; to find the logarithm of 738582 ; 
the log. of the first four figureSi riz. 7385 .868350 
the next greater logarithm • = 868409 

Dif. » 59 

to be multiplied by the remaining figures = 82 



rf*i 



118 
472 



48|38 

Jthen to .868350 
add 48 



tde sum 5.868398^ with the ptoper index prefixed^ is the required 
logarithm. * . 

5. The logarithm of a vulgar-^fraction is found hj subtracting the 
logarithm of the denominator from that of the numerator ; and tiiiat of 
a mixed quantitf is found by reducing it to an improper fractioni and 
proceeding as before. 

Thus to find the Logarithm of | ; 
from the log. of 7 =s 0.845098 

subtract the log. of 8 3= 0.903090 

Remainder «b 9.943008 « the required log. 



PROBLEM 11. 

« 

Vofivd t^ number answering to any ^enlogarithrfti 

RuLtts« 

I 

1. Find the next less logarithm to that given in the tolumn marked 
o at the topi and continue the sight along that horizontal lihe^ and a 
logarithm the same as that given^ or yery near it^ will * be found ; theii 
the three first figures of the corresponding natural number will be found 
opposite thereto in the side column^ and the fourth figure immediately 
above it,,at the top of the page. If the index of the given logarithm is 
3, the four figures thus found are integers; if the ir^dex is 2, the three 
first figures are integers, and the fourth is^i decimal^ and so on* 

Thus the log. 3.132580 gives xht Nat. Numb. 1357 

2.132580 gives 135.7 

1.132580 gives 13.57 

0.13258D gives 1.357 

9.132580 gives . ".1357&cJ 

2. If the given logarithm cannot be eiactly found in the tablci and if 
inore than K)ur figures be wanted in the correspondil^g natural. num<* 
blBr ; then find the difference between th6 giten i^nd the ni^xt lens loga-; 



LpGAEITHMS OF NUMBERS. 

riduiiSy to which ABpex |i» nmaf Giph«r«.s|^ere ajre figures required 
above four in the natund number ; which-divide by the difference be- 
tween the nextlesS) and next greater logarithms), and the quotient an- 
nexed to the four figures fi)rmeriy founds will give the requii>ed natural' 
number. 

Thus to find the nstuml nnmber of the log. 4.S3899i ; 
the next less log. is .82898S whkh gires 6V3S ; 
the next greater log. is 629046 



^mm^ 



Dif. ^ 64 
next less log. = 828982 
girenlog. =828991 

Dif. with one o annexed = 90 
then 64) 90 (1.4 
.64 




therefore 1.4 being .annexed to 6735} the required aataral- mimber» 
6735 i Af is now obtained. 



■1 



TABLE I. 



X«»AHlTBIfS OV NUMBBK». 



No. 


Log. 


No. 


Log. 


No. 


Log. 


No. 


Log. 


No. 


Lo . 


I 


0.000000 


2L 


1.322219 


.4i 


1.6 11784 


61 


1.785350 


81 


1.908485 


2 


0.301030 


22 


«.34S4»3 


4* 


1.623249 


62 


1.79*39* 


82 


1.9U814 


3 


0.4771*1 


13 


.1.361728 


43 


1.633468 


63 


I.79934* 


83 


1.9 "9078 


4 


0. 60^060 


24 


1.38021 1 


44 


'•643453 


64 


1.806180 


84 


. 1^9*4*79 


5 


0.698970 


«5 


"397940 


45 


1.653213 

'1.6627 $8 


65 

66 


1.812913 


85 


. i.9;»94«9 
1*934498 


6 


o-77«i5* 


26 


«-4»4973 


46 


1.819544^ 


86 


7 


aS4s09S. 


»7 


1.431364 


4? 


1.672098 


47 


1.826075 


87 


M395i9 


8 


0.903090- 


.»« 


1447158 


48 


1.68 1241 


68 


1.832509 


88 


1.944483 


9 


0.954243 


19 


1.462398 


49 


1.690196 


69 


1.838849 


89 


1.949390 


lO 


1.000000 


39 
31 


1.4771*1 


?o 


1.698970 


70 


1.845098 


90 
9' 


1.954*43 


11 


1.041393 


i.49«3*» 


5» 


i.707570 


71 


1.851258 


1.959041 


11 


1.079 181 


3» 


i«505i5o 
1.518514 


5* 


1. 716003 


7» 


185733* 


9» 


1.963788 


»3 


1.113943 


33 


53 


1.724276 


73 


1.8633*3 


93 


1.968483 


14 


1.146128 


34 


»-53i479 


54 


««73»394 


74 


1.869232 


94 


1.9731*8 


^5 


1.176091 


^ 


1.544068 


SS 


1.740363 


75 


1.875061^ 


95 


1.9777*4 


i6 


1.204720 


36 


1.556302 


.56 


j.7481^8 


76 


t.880814 


96- 


1.98127 1 


17 i.*30449 


37 


1.568202 


57 


1.755875 
1.7634*8 


77 


1.886491 


^l 


1.986772 


tS , 1.155273 


38 


1.579784 


.58 


78 


1.892095 


98 


1.991226 


19 » I.i7«754 39 1 


1.591065 


59 


1.770852 


79 


1.897627 


99 


1-995635 


so i.^ojojo 40 1 


f. 60 2060 


60 


1.778151 I 80 


1.90309b 


100 


1.000000 



LOGARITHMS OF NUMBERS, 



No. 



too 

101 
lOl 

103 

104 
105 
106 
107 
108 
ib9 



no 
III 
iia 

"3 
114 

X16 
117 
118 
1T9 



120 
III 
lis 
121 

134 

126 

127 
128 
129 



004321 0047 5 1 005180 
oo86oo!oo92o6 009451 
012837:013259 013680I014100 
017033017451:017868 '018284 
021189 021603 0220161022428 
025J06.025715 026 1 34' 026 5 33 
0293844029789 030195 1030600 
033424 033826 034227^034628 
037416 037825 038223 1 038620 



041393 

045323 
049218 

053078 

056905 

06069I 



06445]^ 064832,065206 



068186 
071882 

07554^ 







J. 



1 



-J, 



I 



0000001000434000868 



001301 
005609 
009876 



041787 042i82;042575 
04^714 046105I046495 
049606*049993 1 0503 80 
«^53463i653846 054230 
057286 057666 1058046 
061075,061452 0618^9 



079181 
082785 
086360 



08990^ 090258 



093422 
096910 
100370 
103804 
107210 
11059O 

"3943 

I 1727 I 

120574 
I 23^52 
1 27 105 

130334 

«33539 
136721 

139879 

l'4SO'5 

146128 
149219 
152288 

«$533* 
158362 

161368 

1^353 
167317 

170262 

173186 



17^1 
178977 

181844 
184691 
187521 
19O332 
1931255 
195900 

19S657 
201397 



0685571068928 
0722501072617 
07 59 12- 076276 



079543 
083144 

086716 



093772 
097257 
100715 
104146 

"07549 
II 09 26 



079904 
083503 
087071 
O90611 
094122 
097604 
01059 
04487 
07888 
1 1 262 



1 14277 
117603 

120903 
I 24178 
127429 
130655 

»3385« 

137037 

140194 
»433*7 



146438 
149527 

> 5*594 
155640 

158664 
161667 
164650 
167613 

»70555 

"73478 



176381 
179264 
182129 
184975 
187803 
I90612 

'934031 
196176 

19S932 
201670 



1 



065580 
069298 

072985 
076640 



001734 
006038 

010300 

014520 

018700 

022841 

026942 

031004 

035029 

039017 

042069 
046885 
050766 
054613 
058426 
062206 
065953 
069668 

07335a 
077004 



1461 1 

"7934 
21231 

24504 

27752 
30977 
34«77 
37354 
40508 

_4S*39 
^46748 

49835 
•52900 

5594? 
58965 

61967 

♦4947 
67908 

70848 

73769 



76670 

7955* 
82415 

85259 

88084 

9089 i 

93681 

9*45* 
99206 

201943 



080266 

083861 

087426 

O90963 

094471 

097951 

01403 

04828 

0S227 

2^598 

14944 
18265 
21560 
24830 
28076 
31298 
34496 
37670 
40822 

43 W 



080626 
^0842 19 
0&7781 
091315 
,094820 
098297 
01747 
05169 
08565 
11934 



080987 081347 
084576084934 
0881361088490 
091667 09201? 



47058 
50142 
53205 
56246 
59266 
62266 
65244 
682OJ 

7 1 14' 
74060 

76959 

79R39 
82700 

85542 

88366 

91171 

93959 
96729 

99481 

10221 6 



15278 
18595 
21888 
25156 
28399 
3T619 

348 »4 
379*7 
41136 
44«63 

47367 

50449 
53510 

56549 
59567 
62564 

65541 
68497 
7»434 
7435« 



77248 
80126 
82985 
85825 
88647 
91451 

94*37 
97005 

99755 
202488 



.l_«_l. 



002166 

006466 

010724 

014940 

019116- 

023252 

0273 50 ' 

031408 

035430 

OJ9414; 



002598. 003029 

0068941007321 
011147:011570 

015360015779 
Oi9532|Oi9947 
0236641024075 
027757 J028 164 
031812)032216 
03583o[oi6229 
039811.040207 



8 



043362i043'75j 
047275:047664 
0511521051538 

0549961055378 
0588051O59181 

0625821062958 

0663 26 1 066699 

07003 8 j 070407 

073718J074085 

077368.077731 



095169 



O98644 098990 



02090 
05510 
08903 

12270 



15610 
18926 
22216 
25481 

28722 

3«939 
35»33 
38303 
41450 

44574 



47676 
50756 

53»*5 
56852 

59868 

62^63 

65838 

68792 

71726 

74641 



77536 
80413 
83270 
86108 
88928 
91730 

945 »4 
97281 

200029 

202761 



095518 



01434 
05851 
09241 
12605 



«5943 
19256 

22543 
25806 

29045 

322 

3545> 
3861)^ 

41763 
44885 



044148 
048053 
051924 

055760 
059563 

063J33 

067071 

070776 
074451 
078094 



081707 
085291 
088845 
092370 
095866 

09*535 

02777 
00191 

09578 

12940 



60 *3 



47985 
51063 

54119 

57154 
6016S 

63161 
66134 
69086 
72019 
74931 



77825 
80699 

83554 
86391 

89209 

91010 

94792 

97556 
20030; 
203033 



16276 
19586 
22871 
26131 
29368 
2580 
35768 
38934 
42076 
45196 



48294 
51370 

544*4 

5745 V 
60468 

63460 

66430 

69380 

72311 

75222 



78113 
80986 
83839 
86674 
89490 
92289 
95069 

9783* 

200577 
203305 



003460 

007748 
OIJ993 
016197 
020361 
024486 
028571 
032619 
036629 
040662 



044540 
048442 

05*309 
O56142 
059942 
063709 

067443 

071145 
074816 

078457 



081067 

085647 

089198 

092721 

09621 5 

099681 

03119 

06531 

09916 

13275 



16608 
19915 
23198 
26456 
2969O 
329CO 
36086 

39249 
42389 

45507 



48603 
51676 

54728 

57759 
60769 

63757 
66726 
69674 
72603 
75S«» 



78401 
81272 
84123 
86956 
89771 
92567 

95346 
98107 

200850 

203577 



8 



003891 
00817^ 
01241; 
oi66ij 

020775 
024896 
02897^ 
033021 
03702S 
04099S 

044931 
04883c 
05269^ 
056524 
06032c 
064083 
067814 
071514 
075182 
078815 

082426 
086004 
08955a 
093071 
09656a 

00026 

0346a 

0637c 

I025J 
13605 



1694c 

20245 
23525 
26781 

300 IS 

332IS 

36403 
39564 

42703 

45818 



48911 

5»9« 
55031 
58061 
6io6t 

6405^ 
6702s 
69968 
72895 
7580a 



7868Q 

8i55« 
84407 

87439 
90051 

9284^ 

9562J 

98381 

201124 

203 84I 



LOGARITHMS OF NUMBEKS. 



No. 

■ 60 



1041 10 


1 
104391 


2 
104661 


3 
«H93l 


4 
105104 


5 


6 


' 


B 


9 




10547 S 


»0574S 


106016 


1^186 


i06isfr 




i6> 


106816 


13709s 


10736s 


M7634 


107 90J 


10B171 


108441 


108710 


108978 


109147 




iti( 


J0951S 


109783 


110051 


110118 


11OS86 


1.0853 


111.10 


111388 


11.654 


111911 




161 


..»rSS 


111454 


111710 


..1986 


iijlSi 


li35'8 


1.3783 


114049 


1143.4 


i'4ITS 




i<4 


1.4844 


115109 


i'S373 


1.5633 


115901 


116166 


116430 


116694 


1169S7 


117111 




u] 


1:7484 


l'7?47 


118010 


i'8i7 3 


118535 


1.8798 


11,060 


1I9J11 


1I95&4 


.i,84< 




166 


1101 OS 


110370 


iloSji 


110891 


""ij 


111414 


111675 


1.1936 


111196 


iii4{t 




167 


M1716 


111976 


i.3»36 


113496 


"37SS 


114015 


114174 


114533 


M479HWSOJ1J 




.«S 


11S309 


115568 


»I816 


116084 


116141 


116600 


116858 


11711s 


»1737> 


1176JO 




>ti 


117887 
130449 


11B144 


11I4OO 
130960 


118657 


«)!9'3 


U9170 


119416 


119681 


.19938 


13093 




130704 


111115 


13.470 


i3'7»4 


131979 


131133 


131488 


Ji=-4: 




'7' 


131996 


133150 


»33504 


»M7S7 


134011 


114164 


134S»7 


1J4770 








'7» 


»3SSi8 


135781 


»3603J 


136185 


136537 


1367*9 


137041 


137191 


137544|ir7^,; 




»7J 


1J8046 


»38i97 


13B548 


138799 


139049 


139199 


139550 


139800 


■4005^1 i-!^3CO 




"74 


140S49 


140799 


141048 


M'197 


141546 


141 795 


141044 


141193 


1^:;,^ 24:-9: 




'7S 


14303 S 


1,3186 


»435J4 


143781 


144030 


144177 


144514 


144771 


i;,.i^ i4;i6t 




176 


MSS>3 


1457 S 9 


146006 


146151 


146499 


14674s 


146991 


147136 


i.^4Si 147-18 






»47973 


148..9 


148464 


148709 


148954 


149198 


149443 


149687 






178 


IJ0410 


150664 


150908 


151.51 


15139s 


15.631 


15.88. 


131115 






179 


'51853 

15(173 


153096 
»555i+ 


»557!i 


153580 
155996 


153811 


154064 


154306 


154548 


i>----yii03' 




IBC 


156136 


15647; 


S5671B 


156948 






IS7679 


157918 


158158 


158398 


158637 


158877 


159116 


15935! 


^■■wl^'^ji' 




r8i 


160071 


160310 


160548 


160787 


161015 


16.163 


16150. 


16.738 








iSj 


161451 


i6i6as 


161915 


rt3i6J 


163399 


161636 


163 87 J 


164109 


if+;4? 


26458. 




1S4 


i648iS 


165054 


165190 


165515 


16576. 


165996 


166111 


166467 




266937 




'4 


167171 


16,V 


i6?64i 


16787J 


16S110 


168344 


168578 


168811 


K.g046 


169179 




1S6 


16951] 


169746 


169930 


170113 


170446 


170679 


1709.1 


171144 




17.609 




187 


171841 


171074 


171306 


171538 


171770 


173001 


171133 


173464 


1- ;h6 


1739*7 




IBS 


174158 


174389 


274610 


174850 


175081 


17531. 


175(41 


175771 




17613; 




.89 
190 


176461 
S7S7S4 


17669, 


1769»1 


177151 
»794I9 


177380 


177609 


177838 


17806; 


»7Sli:6 


178515 




178981 


?79iio 


179667 


17989s 


280.13 


18035. 


180578 


1S0806 




191 


iS.oH 


i8ii«i 


i8i4SS 


181715 


181941 


181,69 


18139s 


181611 


1K1S49 


ISJOTJ 




191 


1B3301 


183517 


»8J753 


183979 


18410s 


184431 


184656 


184881 


185.07 


18533: 




19} 


'8SSS7 


185781 


186007 


i86lji 


1S6456 


ig6eK. 


,86905 


187 <10 


1873(4 


187(78 




194 


i87!loi 




188149 


188473 


188696 


18S910 


*S9'43 


189366 


1H95S9 


iSuS.: 




■9< 


190035 


19015; 


1J0480 


190701 


190915 


191147 


191369 


191591 


I9.B13 






196 


191156 


191478 


191699 


191910 


193.41 


193363 


193583 


193804 


19401^ 






197 


194466 


194687 


194907 


195117 


195347 


195567 


195787 


196007 


196:16 






.'g 


19666s 


196884 


197104 


197113 


19754*1197761 


197979 


198.9a 


1^84, 6 






199 


198853 


19907^ 


199^9 


199507 


199715 199941 


30016. 


J0037S 


3O0i95 






100 


301030 


301147 


301464 


301681 


30.898 


301. .4 


301331 


30*547 


301764 


701980 




391 


30J196 


3034" 


301618 


303844 


304059 


304175 


304491 


J04706 


30491 ! 


305.3* 






30535' 


305566 


305781 


305996 


306111 


306415 


306639 


306854 


107068 


107181 




tat 


307496 


307710 


307514 


308137 


308351 


308564 


30B778 


108991 


309»4 


309417 




104 


309630 


JO984J 


310056 


31016S 


310481 


310693 


31O9O6 


l.iiiS 


3113JO 


311541 




10( 


j"r54 


311966 


311177 


311389 


3.1600 


]ii8ii 


3.3013 


313*34 


J1J44S 


111656 




m6 


313867 


314078 


3141S9 


114499 




314910 


3.5.30 


315340 


3'S!S0 


,iS76t> 




107 


315970 


3i6iBp 


316390 


3 '6599 


3.6309 


317018 


317117 


317436 


31764s 


317854 




loB 


3.B063 


3.S171 


S.8481 


J.86S9 


3.8898 


3.9106 


J'93'4 


319511 


319730 


3>993< 




109 




310354 
JI1416 


310561 
311653 


310769 


310977 


311.84 


hi??'. 

313458 


31.598 


31.805 


311011 

314077 






311*19 


311839 


313046 313151 


313665 


31387. 






3i*a8i 


3144SS 


314694 


J14899 


315.051315310 


315516 


1*5711 


JI5916 


316.3. 






316316 


316541 


316745 


316950 


3i7'SS!3i73S9 


317561 


117767 


3*797* 


318.76 




iij 


J183S0 


118583 


318787 


318991 


319.94(319398 


319601 


3198=5 


3jooo8;3iOi.i 






330414 


330617 


330819 


331011 


33 1115 133 1417 


331610 


33.831 


11103413 1 iij6 




atj 


331438 


33164O 


331841 


333044 


333»46|333447 


333649 


333850j3!4Oi'.J34i5J 






334454 


J3465? 


334856 


315056 


335*57 i33J4S8 


335658 


3»859'336059'336i6o 




117 


336460 


336660 


336860 


137060 


337160337459 


337659 


337Bs8,338058'33ais7 




iiB 


138456 


J386s6 


33S85S 


339054 


339153 133945' 


319650 


33984913400471340146 






340444 


340641 


340841 


341039 


341137 3*143! 


34>63i 


341830,341018:341115 









1 


2 


3 


4 1 5 


6 


7 1 8 ; 9 





LOGARITHMS OF NUMBERS. 



No. 
%%l 

212 
223 
224 
225 
216 
227 
228 
229 

130 

232 
m 

236 

«7 
238 

240 
241 
24a 

*43 
244 

246 

247 
248 

^49 

251 

253 
254 

256 

257 

260^ 

a6i 

262 

263 

364 

265 

266 

267 

268 

269 

270 
271 
272 

273 
274 

«75 
276 

277 

278 

279 



34*4*3 

34439» 

346353 

348305 

350248 
352182 

354108 

356026 

357935 

361728 
363612 
3654^8 
367356 
369216 
371068 
373912 
374748 
376577 
378398 



380211 
382017 
383815 
385606 
387390 
389166 
390935 
392697 

39445* 
396199 



397940 

399674 
401400 

403120 

404834 

406540 

408240 

409933 
41 1620 

413300 



1 



342020 

344589 

346549 
348500 



35044*350636 



35*375 
354301 
356217 

3581*5 
360025 

361917 
363800 
365675 
367542 
369401 

371*53 
373096 
37493* 

376759 
378580 



380392 

38*197 
383995 

385785 
387568 

389343 
39*"* 

39*873 

394627 

396374 



398114 
399847 

401573 
403292 

405005 
406710 
408410 
'410 102 
41 1788 

4*3467 



4.14973 
416640 

41 830 1 

4199^6 

421604 

423246 

424882 

42651 1 

428135 

429752 

A I 



415140 
416807 
418467 
420121 
421768 
423410 



342817 

344785 

346744 
348694 



35*568 

354493 
356408 
358316 
360215 



36210c 
363988 
365862 
367729 

369587 
37«437 
373*80 

375««5 
37694* 
378761 



380573 

382377 

384J174 
385964 

387746 
389520 

39«*88 
393048 
394802 

396548 



398*87 

400020 

401745 
403464 

405*75 
40688 1 

408579 
410271 

411956 
413635 



42504^425208 



426674 
428297 
4,29914 



431364, 

43*969' 

434569 

436163 

43775« 

439333 

440909 

44*480! 
444045' 
445604 





4»5307 

4*6973 
4*8633 

420286 

4**933 

4*3573 



4315*5 
433**9 
4347*8 

4363** 

439909 

43949» _. - 
441066 441224 



426836 
428459 

430075 
43 1687 
433*90 
434888 
436481 
438067 
439648 



442636 

444201 
445760 

I 



44*793 
4443^7 
4459*5 



343014 
344981 

346939 
348889 

350829 

352761 

354685 

356599 
358506 
360404 



362294 
364176 
366049 
467915 
369772 
371622 

373464 
375*98 
3771*4 
378943 



380754 
382557 

384353 
^6142 

387923 
389697 
39*464 
393**4 
394977 
3967** 



398461 
400192 
401917 

403635 
405346 

407051 

408749 
410440 

412124 

413802 



43*846 
433450 
435048 
436640 
438226 
439806 
441381 

44*950 

4445*3 
446071 

• 3 



4 



343*'* 
345*78 

347*35 
349083 
351023 

35*954 
354876 

356790 
358696 

360593 



362482 

364363 
360236 

368101 

369958 

37*806 

373647 
375481 
377306 

379**4 



380934 
382737 

384533 
386321 

388101 

389875 
391641 

393400 

395*5* 
396896 



4*5474 

4*7139 
418798 

420451 

422097 

4*3737 

4*537* 

4*6999 
428621 

43023^1430398 



398634 
400365 
402089 
403807 

4055*7 
407221 
408918 
4 10608 
412292 
413970 

415641 
417306 
418964 
420616 
42226 1 
423901 

4*5534 
427161 

428782 



432007 
433610 

435*07 
436798 

438384 
439964 
44*538 
443106 
444669 
446226 



343409 
345374 
347330 
349278 
351216 

353*47 
355068 

356981 

358886 

360783 



362671 

36455* 
366423 

368287 
370143 
37*99* 
37383* 
375664 
377488 
379306 



381115 
382917 
384712 
386499 
388279 
390051 
391817 

393575 
395326 

397070 



398808 

400538 
402261 
403978 
405688 

40739* 
409087 

410777 
412460 

4'4*37 



415808 

4*747* 
419129 
420781 
422426 
424O64 

4*5697 
4*73*4 
428944 
430559 



432167 
433770 
435366 

436957 
43854* 
440122 
441605 
443*63 
444825 
446382 



343606 
345570 
3475*5 
34947* 
35*4*o 
353339 
355*60 

357*7* 
359076 
360972 



362859 

364739 
366610 

368473 
370328 

37**75 
374015 
375846 
377670 
^79487 



381296 

383097 
384891 

386677 
388456 

390228 
39*993 

39375* 
395501 

397*45 



398981 
400711 

402433 
404149 

405858 

407561 

409257 

410946 

412628 

4*4305 



4*5974 
417638 
419295 

4*0945 
4*2590 

424228 

425860 

427486 

429106 

4307*0 



432328 
433930 
4555*6 
437**6 
438700 
440279 
441852 

4434*9 
444981 
446537 



343802 
345766 
347720 
349666 
35*603 

35353* 
35545* 
357363 
359266 

361161 



363048 
364926 
366796 
368659 

370513 
37*360 

374*98 
376029 

37785* 
379668 



381476 

383*77 

385070 

386856 
388634 
390405 
39*169 

3939*6 
395676 
3974*8 



399^54 
400883 

402605 

4043*0 



8 



343999 
34596* 

347.9*5 
349860 

35*796 
3537*4 
355643 
357554 
359456 
361350 



344196 

346 1 571 
348110 

350054 
35*989 
3539*6 
355834 
357744 
359646 
361539 



363236 

36^113 
366983 
368844 

370698 

37*544 
37438* 
376212 
378034 

379849 



381656 
383456 
385*49 

387034 
388811 

390582 

39*345 
394*01 

395850 
39759* 



3634*4 
365301 

367169 

369030 

370881 

37*7*8 

374565 

376394 
378216 

380030 



381837 
383636 
385428 
387212 
3^8989 

390759 
392521 

394*77 
396025 

397766 



3993*7 
401056 

40*777 
404492 



406029 406199 
407731 407900 
409426 409595 
411114 411283 
412796 412964 



41447* 



416141 
4*7804 
419460 
421110 

4**754 



4*4393^4*4555 



426023 



429268 
430881 



43*488 



435685 

437*75 
438859 
440437 
442009 



414639 



399501 
401228 

40*949 
404663 

406370 

408070 

1409764 
I411451 

41313* 
414806 



416308 
417970 
419625 

4*1*75 
4**918 



427648 427811 



426186426349 



429429 
43*042 



^3*649 



434090 434*49 



435844 
437433 
439017 
440594 
442166 



443576J44373* 

445*371 445*93 
446692 446848 



8 



416474 

4*8135 
419791 

4**439 
423082 

424718 



4*7973 
429591 
43**03 



432809 

414409 
436003 

43759* 

439175 
440752 

44*3*3 
443888 

445448 
447003 



LOGARITHMS OF mJMBERS. 



15^ 





1 


2 


3 


4 


5 


' f ' 


"s 


9 


lio 


«4"S» 


4473 '3 


447468 


447613 


447778 


447933 


44808844814* 
4496331449787 


44B397 


4485** 


■Si 


44S706 


44ii!6i 


449015 


449.70 


449314 


449478 


44994' 


450091 


181 


4!0i49 


4 5040 J 


450557 


45071. 


450865 


45.0.8 


451171 


4513*6 


45-479 


4(1*33 


il] 


4ii'8* 


451940 


4(1093 


45»i47 


45*400 


4S1J53 


45*706 


451859 


4530.1 


«3»*i 




»!4 


4533'* 


4SJ471 


4J3614 


4537 77 


4S3930 


454081 


454*3! 


454187 


454S40 


45469* 




185 


4!4>45 


454997 


45!'49 


4iSS0> 


455454 


455*06 


45S7S8 


4I5910 


4i»06t 


*(*i'4 




1S6 


456366 


456518 


456670 


456 81' 


45697] 


4571*5 


457176 


4(74*8 


457579 


457T3t> 




i87 


4i7SI» 


458033 


«B.84 


«8336 


458487 


45863! 


458789 


458940 


4J9O91 


4(»*4« 




lit 


4(9391 


4!9I4} 


459*94 


459845 


459995 


460.46 460196 


460447 


460597 


4*0747 




^9 
490 


4tol,« 


461048 


46.198 


46.348 


461498 


+5.649 


46^9 
463196 


46.948 


46109B 
463594 


461*4! 
4*3744 




46.39« 


4**548 


4*1*97 


461847 


4*1997 


463.46 


463445 




tgi 


463S9] 


4*4041 


16419' 


4*43+0 


464489 


464639:464787 


4649 J6 


465085 


465*34 




ifi 


♦6i3lj 


4*5531 


4*5680 


465819 


465977 


466.161466174 


4664*3 


4«*S7' 


466719 




19! 


466868 


46?0.6 


467164, 


46?3'» 


467460 


467608:467756 


4*7904 


4*8051 


468100 




»M 


46834? 


46S495 


i6864l 


468790 


468938 


469085 


469*33 


4693*0 


4695*7 


469*75 




*9S 


469>M 


469969 


470116 


470163 




470557 


470704 


4708JI 


47099* 


47t«4J 




>96 


47.191 


47M38 


47'i*S 


471731 


47'B73 


47101; 


47117' 


47*317 


47*464 


47rtiO 




297 


47>7S6 


471903 


473049 


473'9S 


473341 


473487 


473633 


473779 


47J9*( 


474070 




19B 


474116 


474361 


474508 


474653 


474799 


474944 


475090 


475*35 


475381 


47J5»' 




099 


47S67I 


475816 


475961 


476107 


47615* 


476397 


+7654* 


4766B7 


4768J, 47697*1 




300 




477166 


47741 ' 


477 Si' 


477700 


477844 


477989 


478133 


47**78, 47«4M 




JOI 


478J66 


47K71' 


478*55 


47899< 


479 '43 


479*87 


47943' 


479575 


479719 


479H6J 




30* 


480007 


4K0.(i 


4*0194 


480438 


480581 


+80715 


480869 


48.0.1 


48.156 


481x99 




J03 


4S1+43 


481586 


48.719 


48.871 




481.59 


481301 


t'***' 


481588 


48*7 3 « 




J04 


481874 


4J30'6 


483159 


483301 


4B344S 


483587 


483730 


483871 


48401s 


484' 57 




JOJ 


♦84300 


484441 


484584 


4847*7 


484869 


4850.. 


485153 


485*95 


485437 


48(571 




306 


481711 


485863 


486005 


486.47 


486189 


486+30 


486571 


486714 


486I15 


486997 




307 


487138 


4B7»*3 


48741' 


4*7563 


487704 


487845 


48,986 4881*7 


4<8rt9 


^!*s'; 




308 


4S8{{i 


488691 


488833 


488973 


489..4 


489*55 


4S9396 489537 


489677 


489818 




309 


4S9;h 


490099 


490139 


490380 


490510 


49066. 


490S01 -42094. 


49108. 
49148' 


49.311 




310 


441361 


491501 


49.641 


491781 


491911 


491061 


491101 J491341 


4916" 




JII 


491760 


491900 


493040 


49J'79 


493319 


493458 


493597 


493737 


493876 


4940j^ 




3i» 


494'!( 


.; 94194 


494433 


494571 


494711 


494850 


494989 


4951*8 


495167 


♦9540* 




311 


49, '544 


4956S3 


4958" 


495960 


496099 


496137 


496376 


49*5 '4 


49**53 


446791 




314 


496930 


497068 


497106 


497344 


497481 


49761. 


497759 


497897 


498035 


498'7J 




3>S 


49«J" 


498448 


498586 


498714 


498S61 


498999 


499 '37 


499*75 


4994.1 


49955« 




3.6 


499687 


49,814 


499961 


500099 


500136 


500374 


5005.. 


50064* 


5007*5 


(OO9M 




317 


S01OS9 


501196 


59'") 


50.470 


501607 


50.744 


50.880 


501017 


50*154 


501190 




3.! 


501417 


50.564 


501700 


501B37 


50*973 


503.09 


503145 


503381 


503518 


503*54 




3 '9 


503 791 


503917 


504063 


504 199 


504335 


50447. 
505818 


504607 


504743 
506099 


504878 
506 >34 


JOJ014 




J?0 


505150 


505186 


50541 ' 


505557 


505691 


505963 


(0*371 




3i[ 


506505 


52*640 


50677s 


S06911 


507046 


S07IE. 


50731* 


(074S' 


507586 


S07711 




JXZ 


507856 


50799' 


508.15 


508160 


508395 


508530 


508664 


508799 


50*933 


SO906S 




313 


509101 


509337 


50947' 


509606 


509740 


509874 


510008 


510.43 


J 10177 


S"H't 




3H 


5 '0545 


510679 


5^8.3 


5 .0947 


S..0R1 


5. .1.5 


!"348 


51148* 


51.6.6 


S.17IC 




3»i 


5..S83 


(110. 7 


SH150 


.1*84 


51*417 


51*551 


SI1684 


5118.8 


JI19SI 


5.3084 




J16 


513118 


JiJJS' 


513484 


5,36.7 


513750 


5.3883 


514016 


514149 


514181 






3»7 


J'4548 


S 146 Bo 


5.48.3 


514946 


5' 5079 


515*11 


S'5344 


('(476 


(.(to. 


5'574» 




3 18 


5'iS74 


516006 


516139 


5.6171 


5.6403 


S'6!35 


516668 


5 16800 


5.6,31 


i.706- 




J»9 


5 17 196 


517318 


SJ2460 
S'8777 


517591 


517714 


S>7i5l 


517987 
519303 


S.8..9 


5.8.5. 

(19565 


5,83*1 




3 JO 


('»!.4 


5.8645 


5.S909 


519040 


5.9171 


("9434 


5-9*97 




J3I 


J. 9818 


$19959 


510090 


51011. 


510351 


5*0483 


5106.4 


!*D74( 


5*0876 


SI.O07 




33» 


SII13! 


SII169 


(11400 


511530 


51166. 


51.791 


51191* 


5**053 


511183 


(.13 'H 




33J 


S"444 


S11S75 


(11705 


511835 


511966 


513096 


5131*6 


5*3356 


J.3486 


5.3*" 




334 


513746 


513876 


5^4006 


514.36 


514166 


5*43*6 


5*45*6 


514656 


514785 


5149 >i 




33i 


515045 


515174 


515304 


iiJ434 


iii!63 


5*5693 


515811 


5*59(1 


5160S1 


J16..0 




IJ« 


51*! Jy 


516468 


516598 


5167 17 


516856 


516985 


517114 


5*7*43 


517371 


J17501 




337 


517630 


^517759 


5.7888 


5 180. 6 


S1B.45 


5i8i74|s*»40*l5i8!3' 


5*8660 


;i87n 




33» 


5189.7 


51904s 


519.74 


519301 


519430 


519??9 519687, S19«1S 


(19943 


53007* 




339 


530100 


51£JiB 


510456 5J0584 


5307.11 S3084o'!]O968 531O9S I3i*=3 


53'iSi 







_.'__ 


2l3!4!5l6 7:8 


9 





tDGARITHMS OF NUMBERS 






340 

34« 

34» 

343 

344 

34$ 
346 

347 
348 

349 



350 

3S> 

3$» 

353 

354 

355 
356 

357 
358 

359 

560 
361 

363 
364 
365 
366 
367 
368 

3^9 



370 
37« 

373 

374 

375 
376 

377 
378 
379 







547775 547*98 
549003 549146 

550128 55035' 
55145'^ 55*571 
552668 551790 
553883 554004 
555094 555*'5 



53 "479 

53*7J4 
534026 

535*94 
536558 
537819 
539076 
540329 

541579 
542825 



544068 
545307 

546543 



I 



531607 
53*882 

534153 

5354*1 
536685 

537945 
539201 

540455 
541704 

542950 



544"9> 
54543] 
546666 



57749* 
5786^9 



380 579784 

381 $80925 
381 582063 

383 583«99 

384. 584331 . 

38s 585461 585574 

386 586587 586700 

387 587711 587823 

388 588832 588944 

389 58995Q 59OQ61 

390 591065 591176 

391 591177 591*88 
391 593*86 593397 

393 594393 594503 

394 595496 595606 

395 596597 596707 

396 597695 597805 

397 598790 598900 

398 599883 59999* 

399 600973 601082 

I 1 



556302 5564*3 
557507 5576*7 
558709 558829 
559907 56O026 
561 101 56»«*i 
561293 56*41* 
56J481 563600 
564666 564784 
565848 565966 
567026 567144 
568102 568319 
569374 569491 
57*^543 570660 
571709 57*815 
571871 571988 

574031 574'47 
575»88 575303 

57634* 57645V 
577607 

578y4 

579898 
581039 

58*177 
58331a 
584444 



i^i" 



8 



53*734 

533009 

534*80 

535547 
5368 1 1 

538071 

5393*7 
540580 

541829 
543074 



544316 

545554 
546789 

548021 

549*49 
550473 
551694 

551911 
554126 



531862 531990 

533136 533*63 
534407 534534 
535674 535800 
536937 537063 
538197 5383** 
53945* 539578 
540705 540830 

54*953 54*078 
54 3'99 5433*3 

544440 544564 
545678; 545802 
5469*3! 547036 
5481441548266 

549371-549494 
550595:550717 
55|8i6|55»938 



553033 



555336 i55457 



556544 

557748 
558948 
560146 

561340 
562531 
563718 

564903 
566084 

567162 



568436 
569608 

57077^ 

57194* 
573104 

574*63 
575419 
576571 

5777*1 
576868 



580011 

581153 
581191 

5831416 

584557 
585686 

586811 

587935 
589056 

590173 



591*87 
59*399 
593508 

594613 
595717 
596817 

597914 
599009 
600101 
601190 



554*47 554368 



553154 



555578 



556664 
557868 
559068 
560265 
561459 
56165O 

563837 
565O11 

566201 
567379 



5^8554 

5697*5 
570893 

572058 

573**0 

574379 

575534 
5766S7 
577836 

578983 



580116 
581167 
581404 

583539 
584670 

585799 
586915 

588047 

589167 

590*84 



556785 
557988 
559188 
560385 
561578 
562769 

563955 

565139 
566320 

567497 



568671 
569842 

571010 

57*174 
573336 

574494 
575650 
576802 

57795* 
579097 



591399 
592510 

593618 

5947*4 
5958*7 

5969*7 
598014 

599119 
6002 JO 

601299 



580240 
581381 
582518 

58365* 

584783 
585911 

587037 
588160 

589*79 
590396 



591510 
592621 

5937*9 
594834 
595937 

597037 

598134 
599218 

600319 
601408 



531117 

533391 
534661 

5359*7 
537189 

538448 

539703 

540955 
541103 

543447 



544688 

5459*5 
547159 
548389 
549616 

550840 

55*059 
553*76 

554489 
555699 



556905 
558108 
559308 
560504 
561698 
562887 

564074 
565*57 
566437 
567614 



568788 

469959 
571116 

572291 

57345* 
574610 

575765 
576917 
578066 
579*1* 



580355 

581495 
582631 

583765 
584896 

586024 
587149 
588272 
589391 

590507 



591611 
592732 
593840 

594945 
596047 

597146 
598*43 
599337 
600428 
60 1 5 1 7 



53**45! 
533518' 

534787 
536053 

5373*5 

538574 

5398*9 
541080 

54*3*7 
543571 



541105 

54*45* 
543696 



544811 
546049 

547*8*1 
54851* 

549739 
550961 

55*181 

553398 
554610 

5558*0 



557026 
558218 

5594*8 
560614 
561817 
563006 
564191 

565376 
566555 

56773* 



56S905 
570076 

571*43 

57*407 
573568 
574726 
575880 

57703* 
578181 
579326 



580469 
58160S 

58*745 

583879 
585009 

586137 
587262 

588384 
589503 
590619 



53*37* 53*500 53*627 
533899 

. ,_, I 535167 

536179 536306 53643* 

537567 537693 

538699 538825 53895 

539954 540079 540*04 
54*330 541454 
^42576 54*701 
5438*0 543944 

545060 
546296 

5475*9 
548758 
549984 

551206 

55*4*5 
553640 

55485* 
556061 



544936 
546172 

547405 

548635 
549861 

551084 
55*303 

553519 

554731 
555940 



557146 
558348 
559548 

560743 
561936 

5631*5 
564311 

565494 
566673 

567^49 

5690*3 
570193 
571359 
5725*3 
573684 
57484* 
575996 
577*47 
578295 

579441 



557*67 
558469 
559667 
560863 
562055 
563244 

5644*9 
565612 

566791 

567967 



569140 
570309 
571476 
572639 
573800 

574957 
576111 
577262 
578410 

579555 



59173* 
59*843 
593950 
595055 
596157 
597*56 

598353 
599446 

600537 

601625 



580583 580697 5K08U 
5817*1 581836 581950 
582858 582971 583085 
583991 584105 584I18 
585112 585*35 585348 
586250 586362 586475 
587374 587486 587599 
588496 588608 588720 
589615 589716 589838 
590730 590841 590953 



591843 

59*954 
594061 

595165 
596267 

597366 

598462 



545183 
546419 

54765* 
548881 

550106 

5513*^ 
55*546 
553762 

554973 
556182 

557387 
558589 
559787 
560982 

561174 
563362 

564548 
565730 
566909 
568084 

569257 
5704*6 

57159* 

57*755 
5739»5 
575072 
576226 

577377 

5785*5 
579669 



59*955 
593064 

594*7* 
595176 

596377 
597476 

59857* 



5995561599665 

600646 600755 
601734 601843 

T"! 8 



592066 

593*75 
594282 

595386 

596487 

597585 
598681 

599774 
600864 

601951 



tmmm'tm 



i^* 



B 



LOGARITHMS OF MUMBEItS. 



I 



No. 



400 
401 
402 

403 
404 
405 

406 
407 
40 s 

_409_ 

410 
411 
412 

4»3 
414 

4»5 
416 

4>7 

418 
419 



4X} 
421 
422 
423 
424 
425 
426 
427 
428 
429 



430 

431 

43 a 

433 

434 

455 
436 

437 

438 

439 



V 



450 

451 
452 

453 

454 

455 
4S6 

457 
458 

450* 



I 1 



2 13 1 



I 



|6o2ot)0,602i69 
605 f44j603 253 (60336 1 1603469 
604.'.26: 604334*604442; 6045 50 
605^05 60,-413:6055211.605628 

'606381 

1^0745 > 



6c2277J602386;602494 601603 
603577 603686 
604658*604766 
605736 605844 
60681 1 606919 



606489 6'.^ 596' 606704 
607562 607 669! 60777 7' 

6o8526|6.>8633;6o874o!6g8847; 

[609594 6097C1 .609140816099141 

J610660J6107671610873J610979! 

(6 f 1723161 18291611936,612042 

6127841612^49016129961613101 



8' 



602711 602819 602928. 
603794 6o3902j6040io' 
604874 6049821665089! 
60595 1 606059 
607026 607133 



.440- 
441 

442 

443 

444 

445 
446 

447 
448 
449 



613842 
614897 

615950 

617COO 

^18048 
619093 
620136 
62iif 6 
622214 



607884«^0799 1 .608098 608205 
608954 609061 '609167 609274 
6i0O2r6ior28!6i0234 610341 
611086 6iii92!6ii298 61 1405 
612148 6122541612360 61x466 



'6061661 

607241' 

.6083121 

609381' 

|6i0447* 
161151 I 

;6i2572 



6i3947;6i4053 614159 
61 50031615108 615213 
6i6o55!6i6i6o 616265 
6J7i05i6i72io!6i73i5 
6181531618257 618362 
61 9 1 98 6193021619406 
620240 6 20344J 620448 



613207 6133141613419,613525 
614264 Ci4370|6i4475 614581 
615319 6154241615529 615634 
616370.6164751616580 616685 
617420 6i75a4i6i7629i6i7734 



621280 
6223 i« 



621384 621488 
622421 622525 



623353 
624385 

6^54'5, 
626443 

627468 



623249 
624282 
625312 
626340 
627366 
62S389 62^491 

629410 '' 

630428 
631444 

632457 



633468 

*34477 

635484 
636488 

637450 

6384S9 
639486 

64048 1 

641474 

642464 

■^43455 
644439 

645422 
6^16404 
6473S3 
648360 

649335 
650307 

651278 

65:1246 



6234561623559 
6244881624591 
625518,625621 
626546 626648 
627571 627673 
628593 62i^95 
62951 11629613:629715 

63"530(6so63 11630733 



03 '545 
632558 



653213 
654176 

6,-5138 

6 :609o| 



633569 

634578 

635584 
636588 

637590 

638589 

639586 

640581 

641573 



6316471631748 
632660 632761 

633771 
634779 
635785 
636789 



642563.642662 



633670 

634679 

635685 

636688 

637690.637790 

6386891638789 

639686J639785 

640680 640779 

641672 641771 



642761 



6437^9 
644734 



64355* 643650 

644537 644635 

645520j6456i9;6457r7 

64650216466001646698 

6474811647 579 '647676 

648458,6485551648653 

649432,649530^649627 

6504051650502 650599 

65'375'65i472!65i569 

652343i65244o;G52536 



653309:653405 653502 
654273,654369,654465 
655234. 65533i}65542? 



656194 656290 



657056j657i5r657247 
6^8cII 658107I658202 



65^065 
6599 16 

660865 
66 1 8 13 







656386 

657343 
65S298 



659060 659x55:659250 
66001 1 '660106 660201 



660960 661055 



66 1907; 662002 662^96 



I I 2 



6611 50 






618466 618571J 

619511 619615 

620552 620656 

621592621695 

622628 622732.622835*622939 



623663 623766 
624694*624798 
625724'625827 
62675ii626855 

627775J627J^78 
628797628900 
6298171629919 
630834I630936 
631849 631951 
632862.632963 



618675I618780 
619719*619823 
620760(620864 
621799I621903 



6136 .0 
614686 

615740 
616790 
617839 
618884 
619928 
62096S 
622007 
623042 



60 ](p3 6 
604118 
605197 
606174 
607348 
608419 
609488 
610554 
611617 
612678 



;623869(623972 
6249011625004 
625929*626032 
626956,627058 
62798o'628o82 



63387*!653973 
634880*63498 1 

6358861635986 

636889636989 

637890 63799>^ 
638888,638988 

639885*639984 
640879 640978 

641870I641970 

642860 642959 



634074J634 
635081J635 
6360861636 
637089J637 

638090*638 
63908^1639 
6400^^41640 
641077^641 
642069 642 
645058 643 



643847643946 
644832 644931 

6458i5'6459i3 
646796 646894 

647774 647872 

648750 648848 

649724 649821 

650696 630793 

651666 651762 

6526^3 652730 



653598 6536^^5 
654562 654658 

6555-3 655619 
656481 656577 

657438 657534 
658393 65S488 
659346 659441' 
66^296 660391 
661245661339 
662191 662205 

"~4 ~ 



629CO2J629 
6300211630 
631038 631 
632052(632 
633064-633 



04 
23 
39 
53 
65 



624O76 
625107 
626135 
627^61 
628184 
629206 
630224 
631241 

632*55 
633266 



75 
82 

87 
89 
90 
88 

83 
76 
68 

56 



644044,644143 
6450291645127 
646011 646109 
64699 1 1647089 
647969648067 
648945 649043 
649919 6500T6 
650S90.6509S7 
651859 651956 
652826 652923 



653791I653888 
6547541654850 
655714 655810 
6566731656760 
657629;657725 
658584:658679 
659536 659631 



634276 
635283 
636187 
637289 
63S289 
639287 
640283 
641276 
642267 

643^55 

644242 
645226 
646208 
647187 
648165 
649140 
650113 
651084 
652053 
653019 



660486 
661434 
662380 



660581 
661529 
662474 



653984 
654946 

655906 

^56864 

657820 

658774 
650726 

660676 

661623 

662569 



613756 

61479* 
615845 

616895 

617943 
618989 

62OC32 

621072 

61211a 

623146 



624179 
6^5109 
626238 
627263! 
628287 
62930S 
63035 

63134a 
632356 

633367 



634376 

635383 
6363SS 

637390 

638389 
6395?? 

640382 

64«375 
642366 

643354 



8 



644340 

645314 
646306 
647285 
648262 

64923: 

650210 

651181 

6521 5«> 
653116 



§54080 
655042 
65600s 1 
6569601 

6y 916 
G588TC 
659821 
660771 
661718 
66266; 



i***i 



LOGARITHMS OF NUMBERS. 



— i-T 



No. 

460 
461 
462 

464 







66z7 5>> 

66j70i 
,664642 
665581 
6665J8 
667453 
668330 



I 



2 



! 



3 



466 

467 j6693i7|6694io!6b9503 

468 ' '• '" 

469 



662.^52 66 i94 7 
663795. 66j«89 
664736J6648JO 
6656751665769 

6666i2J6667^S 
667546'667640'667733!'>67S26 



668479J668572;668665 



470 

47' 
47 a 
47 3 
474 

475 
476 

477 
478 
479 



6630411663135 6b3230; 

P63985 664078 664172' 

664924; 66 50 18. 6651 I2| 

665862,6659561 6t)6o 50 j 

666791^:666892 666986 

667920 

668852 

669782 



669596 
670524 



671451J671543 



.670246I670339 670431 
67ii73l67i265!67i35< 

. 67 1®95|67 2 1 90167228 3 (672375 '672467 

: 6730111673113)6732051673297 673390 

673942}674034;674i26ib74i**^ 
67486i;674953;67 5045 675136 



66875S 
669689 
67061H670710 



480 
481 
482 

483 

4«4 

4H 
436 

487 
488 
489 



490 
491 
492 

493 

494 

495 
496 

497 
498 

500 
501 
502 

503 

504 

505 
566 

507 

508 

J09_ 

510 

511 
S12 

514 

515 
516 

S«7 
518 

5«9 



67?775ii675><70 
,6766941676785 
677607J677698 
6785181678609 
679428)679519 



68o335|68o426 6^0517 



675962 
676876 
677789 
678700 
679610 



6812411681332 
6821451682235 
683047 683137 
683947 684037 
6818451684935 
68574i,68583( 
6X6636.686726 
6875^91687618 
688420I688509 
689309 6^9398 



690196690285 
6910811691170 
691965)692053 
692847J692935 

6937271693815 
6946051694693 

695482*695569 

696356J696444 

697229;6973'7 
698188 



681422 
682326 

683227 
684117 
685025 
685921 
686815 
687707 
6S8598 
689486 



690373 
691258 



698100 



698970 699057 
6998381699924, 

7007041700796 
701568,701654 
701430I702S17 

703291 I703377 
7041 50. 704236 
705008' 705094 
705864.705949 
706718 706803 



707570 707655 
708421; 708506 

709170 7093S5 
710117 7JO202 

710963 711048 
7 1 1807 711892 
712650 7 '2734 
713490 7'3S74 
7»4330 7«44»4 
715167 715251 



693023 
69390J 
694781 
695657 
696531 
697404 
698275 



676053 

6769O8 
677881 
678791 
679700 
680607 



681513 
682416 

683317 

684217 
6851 14 
680010 
6869O4 

687796 
688687 
689575 



690462 
691347 



6743 »o 
675228 

676145 

677059 

67797-i 
678882 
679791 
68069R 



6633241663418 

664266 

665206 

666143 

667079 



681003 
682506 
683407 
684307 
685204 
686100 
686994 
687886 
6S8776 
68Q664 



692142 692230! 



693111 



690550 
691435 
692318 



699144 
70001 1 
700877 
701741 
702603 

703463 
704322 

705179 
706035 

706888 



693991 
694868 
695744 
696618 
697491, 
698362 



693199 



694078 
694956 
695832 
696706 
697578 
798448 



67^636 

672560 

673482 

674402 

675320 

676236 

677151 

678063 

67JJ973 
679882 

6807 Jiy 



706974 



699231 699317 699404 
700098 700184 700271 
700963 701050 701136 
701827 7OI913 701999 
702689 702775 702861 
703549 703635 703721 
704408 704494 704579 
705265 7053501705436 
706 1 20! 706206 1 'O6 29 1 I 



681693 
682596 
683497 

6S4396 
685294 
686189 
6S7083 
687975 

6^J8865 
689753 

690639 

691524 

692406 

693287 

694166 

695044 

695919 

696703 

697665 

698535 



668013 
668945 
669875 
670802 
671728 

672652 

673574 
674494 

675412 

676328 

677242 

67S154 

679064 

679973 
680S79 



8 



663607 

6-. "15 48 



963512 
664360166445-^ 

065299 1665 ?iy Jo 
666237 J6663J '!<■•'■' I J 
067173166-/C >.'? -^^ J 59 
66Sio6l66:;it,'.,jb)^^93 
669038*66913 j'!09i24 
669967 1670060*! ()? J 1 53 
67o895|670^88|6-' ioSd 

67 1821167 I9C3|^)72005 

672744;a* 28.<() 
673666I67375S 
674586I674677 

675503*675595 
6764191676511 



681784 
682686 
683587 
6?<4486 
685383 
686279 
687172 
6S8064 

688953 
689841 

690727 
691612 



677333 
978245 

679155 
6-^0063 

680970 



681874 
082777 

683677 
6S4576 

685473 
686368 

687261 

688153 

689042 

689930 



67 74^4 



672VJ29 
673850 
674""69 
675687 
676602 
677516 



6783361678427 
679246 679JJ7 
6801 54 6.S024J 
6 >io6oi68r 1 5 1 



681964 

681867 

6i'-5767 

684666 

685563 

6R6457 

»j8735I 

688242 

689131 

6900 19 



690816 
691700 



692494 692583 



6933-5 
694254 

69)»3i 

696007 

696880 

69775^ 
698622, 



699491 
700358 
701222 
7020S6 



693463 
694342 
695219 
696094 
696908 
697839 
698709 



6820 






682957 

683857 

6847.^6 

685652 

686547 

687440 

6S8331 

689220 

690107 

n.;0905 1690993 
6917891691877 
6926711692759 



699578 
700444 
701309 
702172 



/02947 703033 
703807I703893 
704665! 704-' 5 1 
7055221705607 
706376:706462 
:07059|707i44.7072-9!7073i5 



69355* 

694430 

675307 
696182 
697055 
697926 



693639 
694517 

^tJ5>H 
696269 

61^7142 

6980 1 3 



6987961698883 

— ^ I ■ I II m' 

699664! 69vj 7 5 I 
700531 70o6[ 7 



701395 
702258 

703 1 19 



701482 
702344 
703205 

65 



7039791704^65 
704837 j 7049 22 
705693I705778 
706547,706632 

7074O0I707485 



707911 7079961 
708764I708846J 
7 09609 1 709694: 



707740 70-'826 

708591 708676 

709440 709524 

710287J710371 

711132 7«i2i6j7ii30i 

711976 7120601712144 

7i29Oij7i2986[7'307Oj 
713742I713826 7*39«o 



712818 
713658 







( 



1 



2 



708o8i!7o8i66 

70893 1 17090 15 

709779' 70^jS 63 

7i0456l7'054oJ7io625 710710 

7ii385J7ii47o|7M554 
7i2229;7i23i3, 71239: 



7o825i'7o83-;6 
709 too' 709 185 



709948 

710794 
7M638 

712481 



710033 
710879 
71172-; 
712566 
713406 



7»3i54,7»3238l7i3322 . ... 

7«5994;7'4078 7i4«62|7r42/:,0 
7i4497|7i458i|7i4665:7»4749l7»48-32 7'49'6!7i5Goo 7150S4 
7»5i35 7i54'8 7 » 5502,7 i5586i7'-s669 7' 57>3'7»58> 6 ^ 7i S92Q 

7 8 19 






w* 



rw 



LOGARITHMS OF NUMBKR&, 



521 

522 

5»3 

524 

526 

5»7 
528 

530 

$31 

53» 
533 
$34 
535 
536 
537 
53« 
539 
540 

541 

54a 

543 

544 

545 
546 

547 
548 

549 



2 



8 



I 



716003 716087 7»6i70;7i6254,7i6337 
716S38 716921 :7i7004j7i70Ji8 717171 
7.1767 1;7 17754 7»7«37!7i79aO|7 18003 
718502 718585 718668.718751 718834 

7»933i.7i94H 71949/ 7»9S8o<7»9663 
72Q159 720242 720325 720407,720490 
720986 721068 721151 721233 721316 
72181 1 721893 721975,722058:722140 
722634 722716 722798 722881 722963 
7234S6 723538 723620 723702 .723784 

724Z76;724358 7244407245" 724603 
725095 725176 725258 725340 725422 
725912 725993 '72607 5' 726 1 56.726238 
726727 726809 726890 726972:727053 
727541:727623 727704 7277851727866 
7283541728435 728516 72859VI728678 
729165729246 7a9327:7294o8;729489 
729974'730O55 730136,7302171730298 



716421 7*6504 716588 7i667i|7i6754 

717254 717338 7»742«[7»75«>4 717587 
7180S6 718169 7182531718336,718419 
718917:719000 7190831719165 719248 

719745 719*28 7i99«i 7^9994 710077 
720573 720655 7*07381710821 710903 
721398 721481 72i563;72i646 721728 

722212 722305 7223871722469 7»1552 

713045 723*27 723209 713191 7*3374 
713866 723948 724030 



7141 11 724'94 



714685,724767 7248497*4931 725013 
725503; /25585«725667 72S748{725830 
726320,726401 726483 726564! 716646 
717134 717216 727197 727379I727460 
727948 728029 728110I718191J71827 
728759 718841 7189121729003; 719084 
729570 729651*729732 729813)719893 



730378 730459 730540 
7307821730863 730944 73'024i73ii05J73«»86 731266 73«347 



73'589 73*669 73«750 73*830, 73i9ii;73*99» 732072 732151 



•I' 



7323944732474 732555 732635731715 



733*97 

7339'J9 
734800 

|735599 

•736396 

737*93 

737987 
738781 
739572 



733278 733358.733438.7335*8 

734079734*597342401734320 

734S80 734960 735040:735*20 735200 735*79 735359 



731796 732876 732956 

733598 733679 733759 
734400 734480 734560 



730611 '730702 
7314181731508 
732133 '732323 

733037;733»*7 
73383917339*9 



550 
55* 
552 
553 
554 
555 
556 
557 
558 

iil 
560 
561 
561 

563 
564 

565 
566 

567 
568 
569 



740363 
74**52 

74*939 
7427^5 

7435*0 

744293 

745075 

745855 
746634 

7474J2 



735679 735759 735838 7359*8 
736476 736556 736635 736715 
737272 737352 73743* 7375** 
738067,738146 738215 738305 
738860 738939 7390*8:739097 
73965**739730 739810I739889 



740441:740511 J7405991740678 
74ii30'74i309:74«388 74*467 

74io;8j742096i742i75 
742804:7428821741961 

743588J743667I743745 
74437* |744449 744528 

745*53 745a3*i745309 



745933 
7467*2 

747489 



746011746089 
746790:746868 
747567747645 



748!88|748266i748343 748421 

748963 749040.7491 18 749«95 
749736j7498*4i74989i;749968 

750508.750586.750663 750740 

75*»79;75*356f75*433. 75*5*0 
7510481751125 752102.751279 

752816 752893, 7529701753047 
753583J753^6o;753736 753813 

754348.754425^75450*1754578 

755M2I755189755265J75534* 



570 755875 

571 756636 

572 757396 

573 758*55 

574 7589*2 

575 759668 

576 760422 760498 

577 761176I761251 

578 7619281762003 
5 79^ 1 762679 1 762754 

I t 1 



75595*;756027 756103 
756712,756788,756864 



757472:757548 
758230*758306 
758988 759063 



759743 



759819 
760573 

761326 

762078 

762829 



757624 
758382 

759*39 
759894 

760649 

76140a 

762153 

762904 



742254 

743039 
743823 
744606 

745387 
746*67 

746945 

747722 



735998 736078.736157 
736795 736874736954 
737590737670.737749 



738384:738463 738543 73*611 



739*77739256 739335 
739968 740047.740126 



734640 

73S439 
736237 

737034 
737829 



740757 740836 7409 1 5 
74*546. 74i624'74i703 
742332'7424»*i742489 
743**8 743* 96} 743*75 



7394*4 
740205 



7347*0 
7355*9 
7363*7 

737**3 
737908 
738701 

739493 
740184 



748498 
749272 
75004c 
750817 

75*587 
752356 
753*23 
753889 

754654 

7554*7 



756180 
756940 
757700 

758458 
759214 

759970 
760724 
761477 
762228 
762978 

4 



743902 

744684 
745465 
746245 

747023 
747800 



740994 
74*781 
742568 

743353 



74398017440581744*36 
744762|744840!7449'9 



748576 

749350 
750123 
750894 
75*664 

75*433 
753*00 

753966 

754730 

755494 

756256 

757016 

757775 

758533 
759290 

760045 

760799 

761552 

762303 

763053 

5 



7455431745621 



746^23 

747101 

747878 



748653 

749427 
750200 

75097* 

75*74* 
752509 

753277 

75404a 
754807 

75^570 



756332 

757092 

75785* 
758609 

759366 

760121 

760875 

761627 

762378 

763128 



74640.1 
747*79 
747955 



74873* 
749504 
750277 
75*048 
751818 
752586 

753353 

754*19 
754883 



745699 

746479 

747256 

748033 



748808 

749582 

750354 
75**25 

75*895 
75*663 

753430 

754*95 
754960 



7556461755722 



756408:756484 
757168I757144 

757927I758003 
758685I758761 



75944' 
760196 

760950 

761702 

762453 
765203 



7595*7 

760172 
761015 

761778 
761529 

763178 



8 



74*075 
741860 

74*647 
74343* 
744**5 
744997 
745777 
746556 

747334 
748110 



748885 

749659 

7P43* 
751102 

75*97* 

75*740 
753506 

754»7* 
755036 
755799 



756560 

7573*0 

758079 
758836 

75#59^ 
760347 
761101 

761853 
762604 

763353 



•Wl 



lOGARITHMS OF NUMBERS* 



No. 

5S0 
581 
58a 

583 
584 

585 
586 

588 
^•89 

5V0 

59' 

59* 

591 

594 

595 
596 

597 
598 

600 
601 

6o2 
60s 
604 
60s 

606 

607 

608 

609 



MM 



J. 



1 



|7634»8 763SO3 76^578 7t)36s3 

1764176 764251 764326 764400 

•764923 764998 765072 765147 

765669 765743 765818 765892 

766413 (766487 766562 766636 

767 1 56 1767*30 767304 7673791 

767898J767972 765046I768120 

■768638.768712,768786 7688601 

7693771769451 7695*5(769599 

770ti5{77oi89 770263,770336 



3 I 4 



6io 
611 
611 

613 
614 
61$ 
616 

617 
618 
619 



620 
621 
622 
623 
624 
62s 
626 
627 
62$ 
629 



630 

631 

632 

633 

634 

635 
636 

637 
638 

639 



770852 

77*587 
77232* 

773055 
773786 

774$ «7 
775*4^ 
775974 
776701 

777427 



778151 

778874 
779596 

780317 
781037 

781755 
782473 
783189 

783904 
784617 



785330 
786041 

786751 
787460 
788168 
78887$ 

78958" 
790*85 
790988 
791691 



770926 770999 
771661 771734 
77*395!77*468 
773 »*8 773*01 
77S860j 773933 

7745 90; 774663 

77$3«9 77539* 
776047 776120 

776774 776846 
777499*777572 



7782i4;878296 

778947i7790»9 
779669:779741 

7803891780461 

781109(781181 

781827I781899 

782544:782616 

78326O1783332 

783975 •784046 
784689I78476Q 



792392 

79309* 
793790 
794488 
79$«85 
795880 

79^574 

797268 
797960 

79865* 



79934" 
800029 

800717 
801404 
802089 

802774 
803457 
804139 
904821 
805501 



785401178547* 
786112:786183 
786822' 786893 
787 $3 "1 787602 
788239788310 
788946 789016 
789651 78972s 
790356 7904*6 

79«059'79"*9 
791761 791831 



792462 792532 
793162 793231 
793860 793930 
7945587946*7 
795*54 7953*4 
795949 796019 
796644'7967f3 

797?37 797406 
798029 798098 
798720 798780 

799409.799478 
800098 800167 
800786 800854 
801472 801541 
802158 802226 
802842 802910 
803525 803594 
804208 804276 
804889 804957 
805569 805637 



771073 
77»8o8 

77*54* 
773*74 
774006 

774734 

775465 
776193 
776919 

777644 

778368 

779091 

779813 
780533 

781253 

781971 

782688 

783403 
7841 18 

784831 



785543 
786254 

786964 

787673 
788381 

789087 

789792 

79O496 

791199 
79«90i 



792602 

793301 
794000 

794697 

795393 
7960S8 

796782 

797475 
798167 

798858 



8 



637*7 

64475 
65221 

65966 

766710 

67453 
68194! 

68934} 

69673 

70410 



71146 

7i«8i 

72615 

73348 

74079 
74809 

7$S3« 
70265 

7699* 

77717 



7844" 
79163 

79885 
80605 

813*4 
82O42 

8*759 
83475} 
84189 
84902 



85615 
86325 
87035 
87744 
88451 
89157 
89863 

90567 
91269 

91971 



92672 

9337" 
94070 

94767 
95463 
96158 
96852 

97545 
98236 

989*7 



JM^ 



899616 

800305 

'800992 

801678 

802363 

•803047 

803730 

804412 

!8cf5093 

:• 805773 ! 

3 I 4 I' 



799547 
800236 

800923 
801609 
S02295 
802979 
803662 
804344 
X05025 
805705 



63802 
64550 
65296 
66041 
66785 

675*7 
68268 
69008 
69746 

70484 



63877: 
64624! 

65370; 

66 II 5 . 

66859 

67601; 

68342 

69082' 

69820' 

70557 



71220 

7«95i 

72688 

734*1 
74152 

74882 

75610 

76338 
7 7064 1 

77789'. 



78513 
79236 

79957 
80677 
81396 
82114 

82831 
83546 
84261 

84974 



85686 
86396 
87106 
87815 
S8522 
89228 

89933 
90637 

91340 
92041 1 



9*74*i 

9344' ( 

94^39; 
94836; 

9553*, 
96227 j 

96921! 

97614; 
98305' 
98996' 



63952 
64699 

65445 

66190 

66933 

67675 

68416; 

69156; 

69894' 

7063 1 ' 



71293 

72028 
72762 

73494 

74**5 

74955 
75683 

76411 

77>37. 
77862 



7»5«S 

79308 

80029 

80749 

8i46ii 

82186: 

82902 

/83618! 

y%433*' 
|S245! 

85757 ' 
86467 

87177 
87885 
88593 
89299 
90004 
90707 
91410 
92111 



92812 

935" 
94209 

94906 

95602 

96297 

96990 

97683 

98374 
99065 

799685 799:754 
800373,800442 

801060*801129 
801747*801815 
802432 802500 
803116 803184 
803798*803867 
804480 804548 
805161 805229 
805841 805908 



71367: 
7*102! 

7*835 
73567 
74*98 
75028, 

75756 
76483 
77*09 

77934 



78658 
79380 
80101 
80821 
81540 
82258 
82974 
83689 
84403 
85116 



85828 
86538 
87248 
87956 
88663 
89369 

90074 
90778 
91480 
92181 



764027 
764774 
765520 
766264 
767007 

767749' 
768490 

769230 

769968; 

77070$ ! 

771440; 
772175 
772908 
773640 

774371 
775*00 
775829 

776556 

777282 
778006 

778730 

77945* 
780173 

780893 

781612 

782329 

783046 

7^3761 

784475 
785187 



764101 
64848 

65594 
66338 

67082 

67823 

6S564 

69303 

70042 
70778 

7* 5 '4 
72248 

72981 

73713 
74444 
75173 
75902 
76629 

77354 
78079 



785899 
786609 

787319 
788027 

788734 
7894401 
790144I 
790848 

791550 
792252 



92882 

93581 

94*79 
94976 

95671 

96366 

97060 

9775* 
98443 

?9r34 

998*3 
800511 

801198 

801884 

802568J 

B03252: 

I803935 

804616, 

805297 

S05976 



792952 
793651 

794349 

795045 

79574' 
796436 

797129 

797821 

798512 

799203 



799892 
800580 
801266 
801952 
802637 
803321 
804003 
804685 
805365 
806044 



7 I 7 



78802 

795*4 
80245 

80965 

81684 

82401 

83117 

83832 

84546 

85259 

85970 
86&80 
87390 
88098 
88804 
89510 
90ti5 
9C918 
91620 
92322 



93022 

937*1 

94418 

95»i5 
95811 
96505 

97198 
97890 
98582 
99272 



799961 
800648 

801335 

802021 

802705 
803389 
80407 1 

804753 

805433 
806112 



LOGARITHMS OF NUMBERS.' 



21 



Mo. 


1 


640 


i;o6iSj, 


641 


806858 


641 


807535 


643 


80821 1 


644 


^08386 


645 


809560 


646 


810233 


647 


810*^04 


64S 


811575 


649 


812245 


650 


812913 


651 


813581 


652 


814248 


653 


814913 


654 


815578 


655 


816241. 


656 


8 1 6904' 


657 


817565 


658 


818226 


659 


818885 


660 


819544 


661 


820201 


662, 


820858 


663 


821514 


664 


822168 


665 


822822; 


666 


«234?4. 


667 


824126 


668 


824776- 


669 


825426 


670 


826075 


671 


826723 


67a 


827369 


673 


82S015. 


674 


828660 


675 


829304 


666 


829947. 


677 


830589 


678 


831230 


679 


831870' 


680 


832509 


681 


833147; 


682 


833784! 


683 


834421'. 


684 


835056, 


685 


835691 


686 


836324; 


687 


836957; 


688 


837588 


689 


838219 


690 


838849. 


691 


839478; 


692 


840106' 


693 


840733! 


694 


84'359 


695 


841985 


696 


S42609; 


697 


843233 


698 


843855, 


699 


844477 



1 



806248: 
806926: 
807603; 
808279. 

808953' 

8^1,627' 

8 

810971 
1642 
2312 



806316J8Q63S4 

r.0706 r 

8o773^'' 

1808414! 

8oyoSS 

80976 2j 



806994 
S07670 

808346 

809021 
8o<;6q4 



806451 

807129 
8o-'So6 
808481 
8091 56 

809S29 



806519. 

: 807197! 
j 807873' 

; 808549; 



8 



■ tl0 



806587 
807264] 

8^79411 , / 

So86i6|8o8684!8o875 
;So*^223' 809 290: 8093 58'! 8094 
,0098961809^64'" 



806655 806723,806790 

80733 2; 807400 1 807467 
808008; 808076 i8o8f4j 



2y80i 
3648' 

43'4 

49^0; 

5644' 
6308- 

6970I 
7631 
S292 
89511 



819610 
820267; 
820924! 
821579 

822233; 
822887- 

823539. 
824191; 

824841* 

825491' 

826140 

826787I 

827434 

828080 

828724 

829368 

83001 1 

830653 
831294 

83'934 



0367 
.03X 
1709 

2378 

3047 

3714 
4381 

5046 

%7ll 

6374 
7036 

7698 

8358 
9017 



•8 

'8 

Is 

8 

18 
18 



, 







832573! 
8332111 

833848' 

834484; 

835120 

835754, 
836387; 
837020! 
837652' 
838282' 

S389121 

8i954fj 
840169' 

840796 

841422 

842047, 

842672 

843295; 

843918! 

844539 ; 
I -I 



819675 
820333 
820989 
8 2 1 644 
822299 
822952 
823605 
8242^6 
824906 
825556 

826204 
826852 

827498 
828144 
828789 
829432 
830075 
S30717 
831358 
831998 

832637 

833275 
833912 

834548 
835*83 
835817 
836451 

8370^3 
837715 
838345 



0434= 
1 1 Ob; 

1776. 
244 5 1 



3i>4 

3781 

4447 
5»73 

5777 
6440! 



7102 8 



7764 
8424 

9083 



819741 
1820399 
1821055 
1821710 
1822364 
1823018 
823670 
824321 

!82497i 
1825621 



,826269 
J826917 
1827563 
'828209 
;828853 
[829497 

1830139 

•830781 

1831422 

832062 



0501 

"73 

1843 
2512 



3181 
3848 

45'4 
5'79 
5843 
6506 

7169 
7830 
8490 

_9»49 

819807 
820464 
821120 
821775 
822430 
823083 

823735 
8243K6 

825036 

825686 



:8 

'8 
8 
8 

I 

18 

8 
8 
8 
8 
'8 
'8 



838.975 
839604 

840232 

840859 

841485 

842110 

842734 
843357 
843980 
844601 



831700 

833338 

833975 
83461 1 

835247 
835881 

836514 

837146^ 

837778I 
838408; 



826334 
826981 

827628 
818273 
8289(8 
829561 
830204 
830845 
831486 
832126 



0;69l 
1240- 
1910 
2579; 

3247, 

39'4l 

4581! 

5246 

5910 

6573 

7235 
7896! 

8556 

9215 



rl 



839038! 

839667] 
} 840 294 1 
,•8409211 

;84i547! 
8421721 
842796' 
843420I 
844042! 
8446641 



832764 
833402 

834039 

834675 
835310 

835944 
836577 
837210 
837841 

838471 

839101 
839729 

840357 
840984 

S41610 

842135 
842859 

843482 

844 1 04 
844726 



9Ji73 
820530 

821186 

821841 

822495 

813148, 
823800; 
8 2445 1 
825101. 

8257 5' 

826399 
827046 
827S92 
828338 
828982 
S29625 
830268 
830909 
831550 
832189 



0636' 
1307' 
1977 
2646; 

3981; 



808818 
809491 



4647 



53»2| 
5976J 

66391 

730» 
7962 

8621 

9281 



0031 8100981810165 
0703 8107701810837 
1374 8114411811508 
2044]Si2jii;8i2i78 
8i27i3' 8i278o ! 8ii8 4y 

" 3381 0134^8^813514 
4048 AJ41 1418141$! 
4714 8147801814647 



5378l8i5445l8«55«* 
8161091816175 



6042 
6705 

7367 
8028 
8688 
9346 



832828 
833466 
834103 

834739 
835373- 
836007 

836641 

837273 
837904 

838534 



3 I 4 



839164 
839792 
840420 
841046 
841672 
842297 
842921 

843544 
844166 

844788 
~5 



819939 

820595 

821251 

821906; 

822560 

823213 

823865 

824516 

825166 

825815 



816464 
827111 

827757 
8284O2 

829046 

829690 

830332 
830973 
831614 

832253 



832^2 
833550 
834166 
834802 

835437 
836071 

836704 

837336 

837967 
838597 



820004 
820661 
821317 
821972 
822626 
823279 
813930 
814581 
825131 
815880 

826528 
827175 
S17811 
828467 
829111 
819754 
830396 

831037 
831678 

832317 

831956 

833593 
834230 
834866 
835500 

836134 
836767 

837399 
838030 

838660 



810070 
810727 
821382 
812037 
812691 
813344 
813996 
814646 
825296 
815945 



839217 

839855 
840482 

841 109 

84«735 
841360 

842983 

843606* 

844229 

S44850 



839289 
839918 
840545 
841172 
841797 
842421 
843046 
843669 
844291 
844912 



816771 

817433 
818094 
818754 
819412 



8*6593 
827240 
827886 
828531 
829175 
829818 
X 30460 



816J38 
817499 
818160 

819478 



820136 
820792 
81144S 
821133 
811756 
813409 
814061 
824711 
825361 
826010 



826658 
817305 
817951 
818595 
819239 
829S82 
830525 



831101 831166 
831741831806 

831381 '831445 



833OIOJ8J3083 
8336571833721 
834293:834357 



834919 

835564 
836197 

836830 

837462 

838093 

838723 



834993 
83502; 

836261 

836894 

837525 
838156 



839352 
839981 



839415 
840043 



84o6o8*8406; 



841234 
841860 

842484 

843108 

843731 



841297 
8419:2 

842547 

843170 

843793 



844353 8444' 5 



844974 



845036 



8 



^mm 



JLOGARITHMS OF NUMBERS. 



700 
701 
70a 
703 

704 

705 
706 
707 
708 

J7Q9 

7J0" 

711 

711 

7»3 

7 '4 

7»S 
716 

7»7 
718 

7»9 







72a 

7ii 
722 

723 

724 
725 

726 

727 
728 

730 

731 
73» 
733 
734 
73 J 
736 
737 
738 
739 
740 

74« 
742 

743 

744 

745 
746 

747 
748 
749 



845098 
845718 

846337 
846955 

847573 
848189 

848805 

849419 



845160 845222 
845780 845842 
846399*846461 
8470I7J847079 
847634 847696 



848251 
848866 
84948 1 



8560331850095 

850646I 250707 



851258:851320 
85i870'35i93i 
852480I852541 
853090:853150 

8536981853759 

854306 854367 

8549>3 854974 
85S5»9'855S8o 

856124,856185 

856729:856789 



357332 



857393 



848312 
848928 
849542 
850156 
850769 

85138^1 
851992 

852602 

853211 

8,-3820 

854427 
855034 
855640 
856245 
856850 



8579354857995 
8585371858597 
859138:859198 

859739I959799 
8603 *{ 8; 860398 



860937 
861534 
862131 
862728 



863323 
863917 
8645 1 1 
865 1O4 
865696 
866287 
866878 
867467 
868056 
868644 



860996 
861594 
862191 
862787 



863382 
863977 
864570 
865163 
865755 
866346 

866937 
867526 
868115 
868703 



750 

75« 

75* 
753 
754 
755 
756 

757 
758 
7S9 



869232 
869818 

8 7 0404 
870989 

871573 
872156 



869290 

869877 
870462 

871047 

871631 

872215 



872739,872797 

87332^»}873379 
873902I873960 

8744 82J 8745 40 

87 596 I; 

875640 
876218 

876795 

877371 
877947 
878522 

8790961 



875119 
875698 

876276 

876853 



857453 
858056 

858657 

^59258 
859858 
860458 
861056 
861654 

862251 

862847 



863442 
864036 

864630 

865222 

865814 

866405 

866996 

867585 

868174 
868762 



869349 
869935 

870521 
871106 
871690 

872273 

872855 

873437 
874018) 

874598 



3 i 4 



1. 



845284.845346 
845904, 845966 
846523 846584 

847 141 j 847202 
847758'8478i9 

8483741848435 
8489891849051 

849604; 849665 

8502171850279 

850830' 8 50891 850952J85 1014^85 1075J85 II 36 



8454081845470 845532 
846028*846090 84615 1 
846646 '8467081846770 
847264(847326'847388 
84788 1 1847943 '848004 
848497 J8485 59 1848620 
849112J849174 849235 
849726 ;849788;849849 



845594 
846213 
846832^ 
847449 
848066 
848682 
849296 
8499 1 1 



850340 1850401; 850462 ^50524 

1: 



845656 
846275 
846894 
847511 
848127 
848743 
S49358 
849972 
850585 
851197 



851442I851503 85(564'85i625. 8516861851747 851808 
852053^8521 14 83iai75;852236 852297I852358 852419 
852663.852724 853785;85.2846i852907 852968 853029 
853272 853333 853394|853455:8535i6;853576 853637 
853881:853941 8 54C02 854063 854124.854185.854245 
854488*854549 854610I854670 854731^854792 8^4852 
85^095:855156 855 2i6'855277, 855337. 85539»;855459 
855701,855761(855822 855882 855943J856003 856064 
856427 {856487.'856548!8566o8 .56668 
J5703 1 1 8570 9 1: 857 151 185 7212 857272 

87763418576^18577541857815 



856306:856366 
856910 856970 



8575'3!857574 
858116 858176 



858718 
859318 
859918 
860518 
8611^6 



858778 

859378 
859978 
860578 
861176 



8617141861773 
862310(862370 
862906:862966 

863501186356 



8582361858297. 858357:858417 
858838. 858S98 858958:859018 
859438 :859499'859559;8596 19 
860038 86009^ 860158(860218 
860637 860697.8607571860817 
861236 861295 861355*86.1415 



861833,86(893:861952*862012 862072 



864096 1 864 1 55 
864689I864748 
865282J865341 



865874 
866465 



865933 
866524 



867055 867114 



862430,862489 8625491862608 
863025 863085 863i44'863204 




867644 



868233I868292 
8688»i 



869408 
869994 



870053 
870579 870638 



863620 863680 863739 863798 

864214 864274 864333,86439* 
8648o8i864867{864926, 864985 

86540^1865459. 8655 18.865578 
865992'86605i;866i 10 866169 
866583. 866642'86670i(86676o 
8671731867232,867291.8673^0 

86r762{86782i;80'/88o;867939 
86835o'868409:868468j868527 

86^8 79 J568938J868997 8690 55|86 9»'4 
869466J 869 5*25 1869584 869642J869701 



867703 



875177 
875756 
876333 
876910 



870 1 1 1 1870 170^870228:870287 
87o696}870755i87o8 13,870872 
871164 871223 87 128 1187 1339.8713981871456 
871748 871806 87i865'87i923.87i98ij872040 
872331 872389 872448:872506 872564,872622 
872913}872972.873030J873088 ;873i46;873'04 

873495i873553|87j6ii:873669"873727:873785 
87407^ 874134 874192.874250 874308 874366 

874656-8747 14- 874772:874830 874887 874945 



875235:^75293,87535 «'875409'875466 8755M 



877429'877486 

878004)878062 

878579I878637 

879*53*8792' > 
379669*879726 879784 

880242:880299 880356 



87s8i3'87587i 
876391I8V6449 

8769681877026 

877544I877602 

87Sn9'«78i77 
R78694 878751 



875929. 875987:8760451876102 
876507 876564 8766aj;87668o 

877083^8^7 14 1.'877 198 1877256 



863858 
864452 
865045 
865637 
866228 
866819 
867409 
S67998 
S68586 
869173 

86976c 

870345 
870930 

871515 

872098 

87^681 

873262 

873844 
874424 

875003 



87765918777171877774187733* 877889 



878234;878292t878349 







112' 



878809 878866,878924 
87926^ ;«79325 8793*^3 '879440' 879497 
8*»984i ;879898i879956 88ooi3'8'^oo70 
880413 88o47»'?Rd52S 880585 880642 



875582 
876160 

8767^7 
8773*4 






878407 878464 
8789S1 879038 
879555 879612 
8Sor27 880185 

880699. 8S07 56 



• 



{ 



LOGARITHMS OF NUMBERS: 



No. 



760 
761 
76* 
763 
764 
765 
766 

768 
769 










8SJ8I4 
S8I385 

881955 



I 



880S7I 
H8I442 

88^012 



8825241882581 
883093 883150 



770 
771 
772 

773 

774 

775 
776 

777 

778 

78a 
781 
781 

783 

784 

78s 
786 

787 

7«8 

790 
791 
79» 
793 
794 
795 
796 

797 
798 
799 



883661 
S84229 

884795 



883718 
884185 
884852 



2 



885361 885418 
885926 88 5983 

88649 1 1 886547 

887054i887fii 



88o9x 

8814^1^ 

882OV9 

8826JC 

883207 

883775 
SS4342 

884909 
•885474 



«l-0985 
S'S.556 
SS2126 



881042 
881613 
882183 



888179 888236 

8887411888797 

•889302I889358 

•889862'8899i8 

[890421 890477 

P90980 891035 



8826951882752 
883264:883321 
853832 883888 



884399 
884965 



884455 
885022 



88553if885587 



886039.886096:886152 



881099 
881670 
882240 
882809 

883377 

883945 
884512 

885078 

885644 
886209 



886604 ^66o|8X67i6| 886773 



881156 

881727 
882297 
882866 

883434 
884002 

884569 

885135 

885700 

886265 



887167 8S7223.887280I887336 
8876171887674 887730 887786.887842' 887898 
" " 888292 888348i888404'88846o 
888853 8889091888965*889021 
889414 889470I889526' 889582 
889974 890030 890086 890141 



890533.890589 
891091 891 147 



89153-' 891593 891649891705^91760,891816 



•892095,^92150 
J892651JS92707 
1893207:893262 
I893762;8938i7 

1 8943 1 6! 89437 1 



890644 890700 
89 1203; 89 1 259 



894870 
895423 

895975 
896526 

897077 



897627 
898176 
898725 
899273 
899820 
900367 
900913 
901458 
902003 
902547 



892206 892262 892317 892373 
S92762 89281818928731892929 
8s33'8 893373 893419.893484 
893873 8939^8 893984.894039 
S94427 894482I894538. 894593 
S94980 895036 89509i!895i46 

8^5533 895588;895643;895699 

89608 5, 896 1401896195 '896251 

896636.896692 896747 896802 

897297 897351 



800 
801 
802 
803 
804 
80$ 
806 
807 
808 
• 809 



810 
811 

8<2 

8,3 
814 
815 
816 
817 
818 
819 



903090 
903632 
904174 
904715 
905256 
905796 
906335 
906873 

90741* 
907948 



897682 
898231 
898780 
899328 
899475 
900422 
900968 

9015*3 

902057 
902601 



8 



881BI3 881270 
881784 881841 
882354 88141 1 
882923 882980 
883491 883548 
884059 884115 
884625 884681 
885 192!885248 885305 
885757I885813 885870 



886829 
887392 

887955' 
8885161888573 

8890771889134 



886311 

886885 

87449 

88801 1 



889638 
890197 
890756 
891314 



8^9694 

890253 
890812 

891370 



89i872!89i9i8 

892429! 892484 
89298 5! 893040 
8935401893595 
894O94I894150 



80492 

895478 

89603c 

896581 

897131I897187JX97242 



8977371897791 



903*44 
903687 

904228 

904770 
905310 
905850 
906389 
906927 
907465 
908002 



897847 897902 
89828618983411 898 ^96' 89845 I 
89883 5'898890| 898944' 898999 
8993831899437 899492;899547 



899930J899985 
9OO476 900531 



901022 
9O1567 
902 1 1 2 
902655 



908485 908539 
909021 909O 7 4 
909556 909609 



9 1 0090 



910144 



910624 910678 
9iii58|9ii2ii 

9ri690|9ii743 
912222 912275 

9127531912806 
913284:913337 







1 



903198 

903741 
904183 

904824 

905364 

905904 
906443 

906981 

907519 

908056 

908592 
909 1 28 
909663 
910197 
910731 
91 1264 
911797 
9«i3i8 
912859 

9*3390 



901077 
9O1622 
902 I 66 
901710 



900039 9000 9 4 



894648 
895201 

895754 
896306 

896857 
897407 



897657 
898506 
899054 
899602 
900149 



894704 

895157 
895809 
896361 
896917 
897462 

898072 



886378 

886942 

887505 
888068 
888619 
889190 

889750 
890309 
890868 
891426 

39*983 



89154D 
893096 
893651 
894205 

894759 
895312 

895864 

896416 

896967 

897517 



898067 



891595 
893151 
893706I 
894161I 

894814 
895367 
895920 
896471 
S970U 
897571 



898561 

899109 
899656 
9002O3 



9005861900640 900695 9007 49 



901 131 ,'901 186 

90i676;90i73« 
902211-90127 



90 1 240 
901785 
902329 



903253 
9O3795 

904337 
904878 
905418 
9059^8 
906497 
9070f5 

907573 
908 1 09 

908646 
909181 
909716 
910251 
910784 

9'"3»7 
911850 
911381 
911913 

9*3443 



^02764 9028181902873 
903307:903361 

9038491903903 
904391,904445 

90493 1 '904986 

905472,905526 

9060111906065 

906550 906604 

907089. 907 142 

907626,907680 

908 1 63 908217 



908699 908753 
909235 909288 
909770:909823 
910304 910358 
910838 9 1089 1 
911371I911424 
91 1903 911956 
9124351911488 
9ii966!9i30i9 
913496 913549 



903416 
903958 

904499 
905040 

9O5580 

906 1 1 9 

906658 

9O7196 

907734 
908270 

908807 
9^9342 
909877 
91041 1 
910944 

9"477 
9 I 1009 
912541 
913072 
913601 



90 1 29 5 
901840 
902384 

902927 



903470 
904012 

904553 
905094 

905634 
906173 
9067 1 1 
907250 

907787 
908324 



908860 

909395 
909930 
910464 
910998 
911530 
9 1 2063 
912594 

9*3"5 
913655 



898615 
899164 

899711 
900158 
900804 
901349 
90 I 894 
902438 
90298 1 

903514 

9d4o66 

904607 

905148 

90568^ 

906227 

906766 

907304 

907841 

908378 



908914 
909449 
909984 
910518 
911051 
914584 
911116 
911647 
9*3*78 
913708 



8981x2 
898670 
899118 
899766 
900312 
900858 
901404 
90 1 948 
90 1492 
903036 

903578 
904120 
904661 
9052O2 
905742 
906281 
906820 
907358 

90789! 
908431 



908967 
909501 
910037 

9*357* 
911104 

9*«637 
911169 
911700 

91313* 
913761 



8 



LOttARITHMS OF NUMBE!IS. 



. If 




9Xi8i.4J9i3S67 



I 3 



914874 
915400 

91J927 
9164J4 
916980 

9*7505 

918030 

2i!ii! 
91^078 

919601 

920123 

920645 

921 166 

921686 

922206 

922725 



91J920 

9»4J96j9»4449 
91492$ 914977 

9«545J.9i5505 
91(980916031 

916507 916559 
9«7033 917085 
917558 917610 



918083 
918607 



919130 
919653 
920175 
920697 
921218 

9*1738 
922258 

922777 
923244 9aja96 
923762I923814 

924279 



918135 
918659 



9*3973 
914502 

915030 

9»555* 
916085 

916612 

917138 

917663 

918188 

918712 



4 



IIP*-- 

75 



7' r 'ff 



■M 



919183 919235 
9X9705919758 



914026 

914555 
915083 

9 1 56 11 
916138 
9^6664 
917190 

917715 
918240 

918764 



.!. 



. . . , 9«433i 
924796I924I4S 

925312 

925828 

92634* 



925364 
925879 

9*6394 
9268 57 j 926908 

927370(9274** 
927883,927935 

928396:928447 

928908 928959 

929419.929470 
929930 92998 1 
930440 93049' 
930949 93 1000 

93«458 931509 
931966 93*017 



930228 
92P749 
921270 
921790 
922310 
922829 
923348 
923865 

9*4383 
924899 

9*54*5 

925931 
926445 
926959 

9*7473 
927986 
928498 
929010 



920280 
920S01 
921322 
921842 
922362 
92)881 
923399 
923917 



914679 914131 914184 
914608 914660 914713 
915136 915189 915241 
915664 915716 915769 
916191 916243 916296 
916717 1916770 916822 
91^243.917295917348 
917768917820,917873 
9i,8292;9i8345j9i8397 



914237I914290 
914766:914819 
915294I915347 
9 15822 '9 15874 
9'6349'9i64Qx 
916875 916927 



919287 
919810 
9*0332 
920853 
921374 
921894 
922414 

9**933 
9*345" 
9*3969 



929521 
930032 
930541 
931051 

93 ' 560 
932068 

932575 

933082 

9335^8 
934094 



■ 



93*474 93*5*4 
93*981 93303* 
933487 933538 

933993 934044 

934498 934549 934599 
935003 935054 93 5 »04 
935507 935558 935608 
936011 936061 936111 
936514936564 936614 
937016 937066 9371 16 
9375*89375631937618 
93X019 938069*938119 
938520 938570J938620 
939020 939<^70 939'20 

9395»9 939569j9396i9 
940018 940068 940 II 8 

740516 940 566 19406 1 6 

941014 9410641941114 

941511 941561J941611 

942008 9423;8|942i07 

942504942554942603 
943000 9430491943099 

94i494 943544'9435V3 
9 439^9 9440 381 94; .,^ 8 

. 1 1 2^ 



924434 
924951 

925467 
925982 
926497 
927011 

9«75*4 
928037 

928549 

929061 

929572 
930083 
930592 
931102 
931610 
932118 
932626 

933133 
933639 
934145 



924486 
925002 
925518 
926034 
926548 
927062 
927576 
928088 
928601 
929112 



934650 

935<54 
935658 
936162 
936664 

937167 
937668 
938169 
938670 
939170 



929623 
930134 
930643 

93'i53 
931661 

932169 

93*677 
933183 

933690 

934»95 



9|^8j6{9 1 8869; 918921 

9193921919444 
9199141919967 
920436,920489 
920958 9210:0 
921478 921530 
921998 922050 
922518 922570 
923037 923088 



919340 
919S62 

9*0384 
920906 

921426 

921946 

922466 

922985 

9*3503 
924C21 



924538 

9*5054 
925570 

926085 

926600 

927114 

927627 
928140 
928652 
9*9163 



939669 
940168 
940666 
941163 
941660 
942157 

94*61,3 
943 '48 

943643 



934700 

935*05 
935709 
936212 

936715 

937217 
937718 
938219 
938720 

939**0 



939719 
940218 
940716 
941213 
941710 
942206 
942702 
943198 
943692 



944'37 944'86 



929674 
930185 
930694 
931203 

93171* 
93**20 

93*7*7 
933*34 
93374^ 
934*46 

93475 » 

935*55 

935759 
936262 

936765 
937267 

937769 
938269 

938770 
939270 



917400 

9J79*5 
918450 

918973 



9*3555 

9240 72 

9*4589 
925106 

925621 

9*6137 
926651 

927165 
9*7678 
928191 
928703 
9292^4 

9*97*5 
930236 

930745 
93**54 
93*763 



923607 
924124 



924641 
925157 

925673 
926188 
926702 
927216 

9*T730 
928242 
9*8754 
929266 



919,496 
920019 

9*054* 
921062 

921582 

922102 

922622 

923140 

923658 

924176 



9*7453 
9*7978 
9i850&| 
9190261 



929776 
930287 
930796 

93*305 
931814 

932*71 93*3*1 



93*778 

933*85 

93379' 
934296 



939769 
9402C7 
940765 
94**6; 
941760 
942256 

94*75* 
943247 
94374* 
944236 

5 



934X01 
935306 
935^09 

936313 
936815 

937317 
937819 

938319 
938820 

9393*9 



939819 

940317 

940815 

94*313 

94i8o<^ 

942306 
942801 

943*97 

94379* 
944285 



924693 
925209 
925724 
9*6239 
926754 
927268 
927781 
9*8293 
928805 

9293«7 



919549 
920071 

92059« 
921114 
921634 

9***54 
922674 

9*319* 
923710 

924228 



c 



mm 



I 6 



93*829 

933335 
933841 

9343 47 

934852 

93535^ 
935860 

936363 
93^865 

937367 

9378^9 
938370 

938870 
9393 69 
939868 
940367 
940865 
941362 
941859 

94*355, 
942851 

943346 
943841 

944335 



929827 
930338 
950847 
93*356 
931864 

9J237* 

93*879 
933386 

'933892 
934397 



9*4744 
925260 
925776 
926291 
926805 

9»73*sr 
9*783* 

9*8345" 
928856 

9 29 3 6$ 



934902 
93S4«6 
9359*0 

936413 
936916 

9374«8 

9379*9 
9384*0 

938920 
9394*9 



929878 

93038ft 
930898 

93140? 

93*9*$ 

932423 

932930 

933437. 

93J943. 
934448 



939918 

940417 
940915 

9414** 
941909 

942405 

942900 

943396 
943890 

944384 



8 



934953 
935457 
935960 
936463 
936966 

937468 
937969 
93847^3 
93897^ 
93946^4 



9399^^ 
940465 

940964 

94*4^2 

941958 

942454 
942950 

943445 
943939 
944433 



«Nfl 



\n 



LOGARITHMS OF NUMBERS. 



No> 

88 1 
881 
883 
884 
885 
886 



' 







I 



.1, 



2 



944483 
944976 
945469 
94^961 
946452 

^46943 

947434 
947914 

!94»4«3 
< 948902 

949390 
949878 

950365 
950851 

9$»337 
95«8i3 



94453* 94458* 
945025.945074 
945518945567 

9460101946059 
946501,946550 
946992:947041 

9474831947532 
9479731948021 

948462(948511 

9489511948999 



944631 

945124 

945616 

946108 

946600 

947690 

947581 

94^070 

948<;6o 

949048 



949439*949488 
9499261949975 

950413 950462 

9509Oo!95O949 

95>3«6'95i435 

9518721951920 
952308j952356'952405 
95279!' 952841.952889 

953*76 9533«5i953373!9534»> 
953760:5^3808,953856 953905 



944680 

945»73 
94^65 

946157 
946649 

947139 
947630 

948119 

948608 

949097 



9447*9 
945*22 

945715 
946207 

946698 

947189 

947679 

948168 

948657 
949146 



944779 
945*72 
945764 
946*56 
946747 

947*38 
9477*8 
948*17 
948706 
949195 



949 S 36 
950024 
9505 1 1 
950997 

95X483 
951969 

95*453 



954242|954»9i|95J339 

9547*5'954773l9548*? 
955*06 955*55955303 



955688J955736 
956168 956216 
956649J 956697 

9571*81957176 

957607 957655 
958086J958f34 



955784 
956^164 

956745 
957**4 

957703 
958181 



(958564; 95861*9 58659 



949585)949634 
95007319501*1 
950560 950608 
951046 951095 



95»53* 
952017 

95*502 



952938^952986 
953470 
953953 



954387 
954869 

95535' 

95583* 
956312 

956792 
957*7* 

9S775> 
9582*9 

958707 



954435 
954918 

955399 
955880 

956361 

956840 
9573*0 
957799 
958*77 



951580 
95*066 
95*550 

95J034 
9535»8 
954001 



9544^4 
954966 

955447 

9559*8 

956409 
956888 

957368 
957847 

9583*5 



944828 

945 J^» 
945«>3 
946305 

946796 

947*87 

947777 



948755 
949*441 



949683 
950170 

950657 

95«>43 
951629 
952114 

95*599 
953083 

953566 
954'"'49 



8 



944*77 
945S70 
945862 

946354 
946845 

947336 
9478*6 



948*66 948315 



94973 » 
950*19 

9S0705 

95119* 

951677 

95**63 

95*647 

953»3i 
953615 
9 54098! 9 54 146 



948804 

949*9* 

949780 
950*67 

950754 
951*40 

95t7»6 

95**11 

95*696 

953180 

953663 



9449*7 

9454«9| 
9459*11 

946403 

946894 

94*^385 
947875 

948364 
948853 

9493^ 

949829 

950516 

950803 

951289 

95»f74 
95*259 

95*744 
953*18 

9537" 
954194 



958755 958803 



95453* 
955014 

955495 
9559'6 

956457 
956936 

957416 

9578^4 

958373 
958850 



I959041 959089 959137 959184 
9595i8|959566;9596i4 959661 
959995; 960O42:96oo9Oi96oi 38 
96047 1 1 960518 960566 960613 
960946 960994 96 1041 961089 



9614*1 
961895 



96i469'96i5i6'96i563 



9619431961990 
962369i9624i7!96*464 
94*84319628901962937 
963315 9'53363* 



9*0 
9*1 
922 
9*3 
9*4 

9*5 
926 

9*7 
928 

_9*9, 
930" 
931 
93* 
933 
934 
935 
936 
937 
938 

939 



963410 

963788.963835 963882 
96426o'964307 964354 
964731,964778(964825 
965*02 965*49:965296 
96567* 965719 9^5766 
966142 9661891966236 
966611:9666581966705 
967080:9671*7 967173 

967548967595196764* 
968016 968062 968109 



962038 
96*511 
962985 

9^3457 



959*32 959*809593*8 
95970J9 959757 959804 
960185 960233 960*81 
960661 •9607091960756 
961 136,961 r84!96f23i 
96161119616581961706 



954580 954628 
95506* 955110 

955.543 95559* 



956024 
956505 



95^7* 



954677 

955«58 
955640 

956110 



956984 95703* 



957464 

95794* 
9584*0 

958898 



956553I95660I 



9575" 

957990 
958468 

958 j46 958994 



957080 

957559 
958038 
958516 



95937519594-13 



95985*1959900 959947 



96208 5 {962 1 3 21962 1 80* 9622*7 •96**75 



9625591962606 
96303*1963079 
963504196^552 



9603*8.960376 
9608041960851 



959471 



960413 
960899 



96i*79'96t3*6'96i374 



961753I961801 



96*653; 96*70 1 !96*74B 
9631*6 

1963599 



9639*9*1963977 ;964024i964O7i 



96440 1 1 964448 964495 
96487*1964919 '964966 
965343!965390'965437 



968483 968530.968576 
968950 968996*969043 
969416^969462 969509 
96988*; 9699281969975 

970347 970393(97044^ 
97081* 9708581 970904 

971276 97«3**'97i369 

971740 971786 971832 

972203 97**49 97**95 
97^666 97*7'*, 97*758 







1 



965813 
966283 
96675* 
967220 
967688 
968156 



96454* 

965013 
965484 



96586019659071965954 
9663*9 ,966376 {966423 
966798 966845 '966^9* 



967267.967314 
967735-967782 
968203 968249 I 



967361 
967829 
968296 



9^3«74 963«» 
963646 ' 963693 

964118 964165 

964590)964637 
965060J965108 

965531(96557* 
966001 966048 



966470 
966939 
967408 



966517 
966986 

967454 



967875'96792* 



970011 
970486 
970951 

97'4'5 
971879 

97*34* 
972804 



97*388 
972851 



97*434:97*f8o 
972897 972943 

6 



968343I968389 

968903 
969369 
969742l969788<969835r 



968623 968670:968716 968763 9688101968856 
969090 969136:969183 19692*9!969*76|9693*3 
969556 969602 '969649 J969695' 

9700681970114 970161 

970^33 '970579 9706*6 

970997^97 10441971090 

97i46i;97i508 971554 
971925 971971.972018 



961848 
96*32 

96*795 
963268 

963741 

96421a 
964684 

965155 
965615 

966095 
966564 
967033 
967501 
967969 
968436 



970*07:970*54 9703< 
970672 970719 970765] 

971137 97'»*3 971**< 
971600,97164719716931 



972064!97*iio{97ai 56 



97*5*7;97»573 
97*989 1973035 

7 I 8 



97*619] 
97308J 



■nv"i««i 



mmm 



w^^m 



tOGAMTHMS OCT NUMBERS. 



TJ 



O. 



940 
941 
942 

943 
944 
945 
946 

947 
948 

9$o 

95 » 

95a 

953 

954 

955 
9 $6 

957 
958 

959 



960 
961 
96A 

963 
964 

965 
966 

967 
968 
969 



9731*8 

973590 
974051 

9745»a 
97497* 
97543* 
975891 

976350 
9768CS 

977266 



9777*4 
978180 

978637 

979093 
979548 

980003 

980458 

98091a 

981365 

981819 

982271 
98*7*3 

983175 
983626 

984077 

9845*7 

984977 
985426 

985875 
986324 



970 

97" 

97* 

973 

974 

975 
976 

977 
978 

980 
981 
98a 

983 

984 
985 

986 

987 
988 

990 
991 

99* 

993 

994 

995 
996 

997 
998 

V9 



98677* 
987219 

987666 
988113 
988559 
989005 

989450 
989895 

990339 

990783 



1 

973 « 74 
973636 
974097 

974558 
975018 

975478 

975937 
976396 

976854 
9773'* 

977769 
978226 
978683 
979138 

979594 
980049 

9J'0503 

980957 

981411 

981S64 



973**0 
973682 
974 «43 



973266 
973728 
974189 



974604:974650 

975064:975110 

9755*4 975570 
975983.976029 
9764421976487 
9769001976946 

977358l97?403 



977815 
978272 



977861 
978317 



978728 978774 
979184 979230 



982316 
982769 
983220 
983671 



979639 
980094 

980549 

981003 

981456'; 



979685 
980140 
980594 
981048 
981501 



984122I984167 
984572.984.617 



98 19091 98 1954 

982362 
982814 

983*65 
983716 



982407 
982859 
983310 
983762 

984*1* 
984662 

985112 

985561 



986816 
987264 
987711 
988157 
988603 
989049 



98686] 
987309 
987756 
988202 
988648 
989094 



9850221985067 

9854711985516 

985920 985965 986010 
986369!9864f 3 1986458 

986906 

987353 
987800 

988247 

988693 

989138 

989583 
990028 

99047* 
99O916 



9894941989539 
98993 9(989983 
990383*9904*8 
990827*990871 



9912701991315 

99i7i3i99'75'' 
992156199**00 
992598 99*642 



993039 
993480 



9939*1 993965 
99436*1994405 



994801 
995*40 



991226 
991669 
992111 

99*553 
992995 

993436 

993877 

19943 17 

994757 

9951 96 

995635 

996074 
9^6512 
996949 
997386 

9978*3 
998*59 998303 

998695I998739 
999130:999174 
999565' 999609 



993083 
993524 



994845 
995284 



995679^9957*3 
9961 17 996161 

996555 996599 
996993 997037 
997430 997474 



997867 997910 



99MS9 
991802 



973313 
973774 
974*^5 
974696 
975156 
975616 
976075 

976533 
976991 

977449 



973359 
973820 

974*81 

97474* 
975202 

97s66i 

976121 

976579 
977037 
977495 



6 



973405 
973866 

9743*7 
974788 
975248 
97570- 
976166 
976625 
977083 

977541 



978363J978409 
978819 978865 



9779061977952 977998 
978454 
978911 
979366 
979821 
980276 
980730 
981184 
981637 
982090 



979*75 
979730 
980185 
980640 
98 I 093 
981547 
982000 

982452 
982904 
983356 

983807 
984257 

984707 

985157 
985606 

986055 
986503 



986951 

987398 
987845 
988291 
908737 

989183 
989628 
990072 
990516 
99O960 



991403 
991846 



99224^1 992288 



992680 

9931*7 
993568 

994009 



99*730 

993172 

993613 
994053 



9793*1 

979776 

98023 X 
980685 

981139 
981592 

982O45 



982497 
982949 

983401 
983852 
984302 

98475* 
985202 

985651 

986100 

986548 



978043 
978500 
978956 

9794'* 
979867 

980322 

980776 

981229 

981683 

982135 



982543 
982994 
983446 
983897 

9^*4347 
984797 

985247 
985696 
986144 
986593 



986995 987040 987085 

987443 987487 987531 

987890,987934987979 
988336 988381I988425 

988782 988826 

989227 989272 



989672 989717 
990117 990161 
990561 990605 



991004 



991049 



991448199149* 
991890:991934 

99*333,99*377 
992774992818 
993216,^93260 
993657I993701 
9940971994141 
9945371994581 

994977 9950*1 
99i4i6j99546o 



994449*994493 
994889 994933 
995j28!995i7» 

995767:995811 995854^995898 

996205j996249|996293i996336 
996643J996687 996730I996774 
997080 ■ 







1 



1 



997517 997561 
. 997954 997998 
9983461998390 998434 
998782 9988261998S69 

999*18 999*611999305 
9996521929696I999739 

2 ' i 3 r 

mmmmmmmmmmm 



997i*4i 9971681997212 
997605 !997«48 



998041 . 998085^ 9981 z8 



998477 9985211998564 
998913*998956 999000 
999348:999392 99943 S 
99 9783 9998*6 999870 

5 • 



973451 

973913 

974373 
974834 

975294 

9''5753 
976212 
976671 
977129 
977586 



8 



982588 
983040 
98341; 1 
983942 
984392 

98484* 
985292 

985741 

986189 

986637 



988871 
989316 
989761 
990206 
990650 

991093 



991536 

991979 
992421 

992863 

993304 

993745 
994185 

994625 

995064 

995504 



99594* 
996380 

996818 

997255 
997692 



973497 

973959 
9744*0 
97488c 

975340 

975799 
976258 

976717 

977175 

97763* 



978089 
978546 
979002 

979457 
979912 

980367 

98082 1 

9^1*75 
981728 

982181 



982633 

983085 

983536 
983987 

984437 
984887 

985337 
985786 
986234 
986682 



987130 

987577 
988024 
988470 
988915 
989361 
989806 
990250 



990694 9907 3^ 



991137 



991580 
992O23 
992465 

99*907 
993348 

993789 
994229 

994669 
995108 

995547 



995986 

9964*4 
996862 

997299 

997736 



998608 

999043 
999478 



97813 

97859 

97904 

97950 

97995 
98041 

98086 

981321 

98177 
98222 



98267 

98313' 
98358 

98403 
98448 
98493 
98538 

98583^ 
986271 

98672 



98717- 
98762: 
98806I 
98851. 
98896< 
98940 
98985( 
99029^ 



99118: 

991621 
99206; 
992 50< 
99295 

99339: 
99383: 
994*7: 
99471: 
99515^ 
99559 



99603< 
99646I 

9969O; 

99734: 
997 7 7< 



99817a 9982 1( 



99865: 
99908; 

9995* 



6' I 7 18 



99913 99995' 



r 



■Mil 



9 



t ABLE Sr. 



I^ogarithwic SincMj Tangeni§^ mid Seeuniti 

"^is table conuhiB the logarithmicy or, ai they are tometinies call- 
adt the artificial sinea, tangents, and secants, to each degree and mui-» 
vte of the qiuidrant, with their complements or co-sines, co-tangents, 
and co-secants, to six places of figures besides this index. 

Tq find the Imogorithmic Sinty Co-Siney ^e. f^ any /(Tmrnber tf D(tgne^ 

and A£nutee, 

If Ac givehdiegreesbe under 45, thef a^ to be taken from the top^ 
and the minutes f om the left side column, opposite to which in that 
column with the name of the logarithm at the top, will be found tibe 
required logarithm* But if the degrees be more than 45, they will be 
found at the bottom of the page, and the minutes in the right side 
4olumn ; likewise the name of the logarithm is to be taken from the 
bottom of the page. 

When the given degrees exceed 9C, they are to be subtracted from 
lao degrees, and the logarithm of the remainder taken out as before. 
Or the logarithmic sine, tangent, 8cc. of degrees more than 90, is ihe 
logarithmic co-sinci co-tangent, &c. of their excess above 90 itegrots. 



BXAXFLfiS. 



J^equired tIbe log. sine of 

co-sine of 



36 32 
6i 18 
54 17 
42 50 
19 27 



tangent of 

co-tang, of 

secant of 

co-secant of 70 33 

sine of 108 36 

or sine of 71 %4 

or co-sine of 18 36 



}• 



logarith 

9.774729 
9.681443 
10 143263 
IO.OS2877 
10.025519 
10.025519 

9.976702 



fojlnd the Degrtf tend ABnute^ nearest corree/ionding to a given Logd,^ 

rithndc Sine, Co'-ainey ^c. 

• 

Look in the column marked at the top or bottom with the name of 
ibegiYen logarithm, and when the nearest to it is found, the corres- 
pond'mg degrees and minutes will be those required, observing that 
when the name is at the top of the column, the degrees are to be tak- 
en from the top and the minutes from the left side column, but if the 
Dame is at the bottom, the corresponding degrees Will bo there like- 
fjppe> mA tho iwimtoa iatiie right tide ooliinaf* 



tOGAWTHMlC SiNES, TANGENTS, AND SECANTS. 3.^ 

XXAMPLES. 

I 

The degrees and minutes corresponding to the 

log. sine 9.265390 are lOo 37' 
co-sine 9 :>28461 70 16 

tangent 9.70156 26 42 

secant ]ra254U 56 9 

* ■ 

The logarithmic sines^ dec. taken o»t to degrees and minutes onlf 
are in general sufficiently accurate, but in some of the more rigid as* 
troDomical calculations, it is frequently necessary to tbke them out to 
the nearest second ; 'when this is the caee they arc to be found in the 
fallowing mamier : 

7b faid tht $ine, tOH^cnt, CV. qf on arch exfirtesed in degrees j ndnuiea 

mid 4econd4, 

m 

RULX. 

T\r\i the sine, tangent, iic, answering to the giTen degree and 
Aiinute, and also that answering tathe next greater minute ; multiply the 
difference between them by the given numJberof seconds^ and divide the 
product by 60 ; theni the quotient added to the sine, tangent, Sec. of 
the given degree and minute, or subtracted from the co'Sine, co-tan- 
gent, &c. will give the quantity required, nearly. 

If .the arch be less than three degrees, it will be necessary to use the 
following rule ^— 

To the. anthmetieai complement of the given degrees and minutes 
reduced to seconds, add the logarithm of the given degrees, minutes^ 
and seconds, reduced to seconds, and the log.-sme, tangent, &c. of the 
g^iven degrees and minutes, the sum, rejecting 10 from the index, will 
be the log. -sine) tangent, 8ec. of the proposed number of degrees^ 
minutes, and seconds. 



Tojind the degreea^ minutes^ and aeconday anawering to a given logarithmic 

wiCy tangent^ isfc. 

Rule. 

]Pind the degrees minutes and seconds answering to the next less Ioga» 
rithmic sine, tangent, &c. which subtract from that given ; multiply tho 
femainder by 60, and divide the product by the difference between the- 
next less and next greater logarit^ons, and the quotient will be the se- 
conds to be. annexed to the degrees and minutes before found. 

If the given logarithm is that of the sine or tangent of a small arch-^ 
then, to the arithmetical complement of the next less logarithm in the 
tables, add the given logarithm, and the logarithm of the degrees and 
minutes, in seconds, answering to the next less logarithm, the sum, re- 
jecting radius, wiU he the logarithiA of the number ef a^conds ici the 
required arch* 



9 


< 


LOCABTTHmC SINES) 










4 


Sine Degree. 




« 






M 
O 


0" 


10" 


20". 1 30" 


40" 


50^ 




► 




5-685575 


5.986605 ■ 6.162696 


6.287635 


6.384545 


59 


I 


6.46 3 7 2( 


6.5306jr3 


6.588665 


6.639817 


6.685575 


6.726967 


58 




% 


6.764756 


6799518 


6.831703 


6.861666 


6.889695 


6.916024: 


1 57 




3 


6.940847 


' 6.964328 


6.986605 


7.007794 


7.027997 


7.047303 


I56 




4 


7.065786 


7.08351c 

7.176936. 


7.100^548 


7.116938 


7'n^73i 


M47973 


55 




5 


7.162696 


7.190725 


7.204089 


7.217054 


7.229643 


54 




6 


7.241877 


7.253776 


7.265358 


7.276639 


7.287635 


7.*9835« 


53 




7 


7.308824 


7.319043 


7.3190*7 


7.338787 


7.34833* 


7.35767* 


5* 




8 


7.J66816 


7-375770 


7.384544 


7.39314s 


7.401578 


7.4«>f850 


5« 




9 

lO 


7.417968 


7.415937 


7.433762 


7.44'449 


7,449002 


7-if56426 


50 

49 




7-4637*5 


7.470904 


7.477966 


7.484915 


7.49*754 


7.498487 




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7.511649 


7.518083 


7-5*44*3 


7.530672 


7.536832 


48 




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7.548897 


7.554806 


7.560635 


7.566387 


7.57*065 


47 




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7.583201 


7.58S664 


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7.599388 


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46 




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7.614993 


7.620072 


7.625093 


7.630056 


7.634963 


45 




i6 


7.639816 


7-644615 


7.649361 


7.654056 


7.658701 


7.663297 


44 




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7.672345 


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7.681208 


7.685573 


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43 




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773479* 


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21 


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7789376 


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7.7^6162 


7.7995*$ 


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22 


7.806 146 


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7.819111 


7.822292 


37 




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7.828586 


7.831700 


7.83479* 


7.837860 


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36 


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35 




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7.864548 


7.867414 


7.870262 


7.873092 


7.875902 


34 




24 


7.878695 


7.881470 


7.884228 


7.886968 


7.889690 


7.892396 


33 




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7.895085 


7.897758 


7.9004*4 


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7.905678 


6.908287 


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7.9 » 3457 


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7 928608 


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8.001538 


8.0036 1 1 
8.015981 


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35 


8.007787 


8.009850 


8.01 1903 


8.013947 


8.018005 


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36 


8.020021 


8.022027 


8.014023 


8.O26011 


8.027989 


8.029959 


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8.033871 


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8.037749 


8.039675 


8041592 


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8.056633 


8.058477 


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8.062142 


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8.072955 


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8.080016 


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8.088684 


8090398 


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8.093804 


8.095497 


17 




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8.097183 


8.098863 


8.100537 


8.102204 


8.103864 


8.105519 


16 




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8.110444 


8. II 2074 


8. 1 13697 


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8.120131 


8.121725 


8 123313 


8.124895 


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46 


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8.129606 


8.131166 


8.132720 


8.134268 


13 




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8.1404C6 


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12 




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If 




49 
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8 165566 


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20' 


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6.162696 


6.287635 


6.384545 


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6.588665 


6.639817 


6.685575 


6.726968 


58 




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6.831703 
6.^6605 


6.861666 


6.889695 


6.916024 


57 




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7.027998 


7.047303 


56 




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7-13*733 


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7.217054 


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7.884240 


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7.8897O4 


7.892410 


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22 


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21 


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8.092137 


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8.192115 


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8.237068 


8.238286 


8.239501 


8.240713 





?r-i 


j 60" 


50" 


40" 


30" 


20" 


10" 


M 



Hqpxct.^ 



S4 



LOGAMTIIMIC SINESI. 









Sine 1 Degree 








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0" 


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8.243060 


20'" 


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8.245459 


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57 




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8.264190 


8.265334 


8.266475 


8 267613 


8.268749 


56 




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8.271010 


8 272137 


8.273260 


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8.276614 


8.277726 


8.278835 


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8.281045 


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54 




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8.285431 


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8.287608 


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53 




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8.290852 


8 291928 


8 293002 


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48 




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8.409803 


8.410621 


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31 




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28 




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8.436029 26 




34 


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8.440632 25 


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35 


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18 


43 


8.476498 


9.999805 


8.476693 


11. 523307 


10.000195 


11.523502 


•7 


44 


8.480693 


9.999801 


8.480892 


11.519108 


10.000199 


II. 519307 


16 


4$ 


8 484848 


9.999797 


8.485050 


1 1.5 14950 


10.000203 


11.515152 


«5 


46 


8 488963 


9.999794 


8.489170 


11.510830 


1 0.000206 


11.511037 


»4 


47 


8.493040 


9.999790 


8.403250 


11.506750 


10000210 


11.506960 


»3 


48 


8.497078 


9999786 


8.497293 


II. 50*707 


10.000214 


11.502922 


12 


49 


8.501080 


9.999782 


8.501298 


1 1.498702 


10.000218 


1 1.498920 


11 


$0 


8.505045 


9.999778 


8.505267 


11.494733 


(0.000222 


'1.494955 


10 
9 


8.508974 


9.999774 


8.5O9200 


11.490800 


10.000226 


11. 49 1026 


$a 


8.512867 


9.999769 


8.513098 


1 1.486902 


10.00023 1 


11.487133 


8 


$3 


8.516726 


9.999765 


8.516961 


11.483039 


10.000235 


11.483274 


7 


54 


8.520551 


9.999761 


8.520790 


11 479210 


10.000239 


11.470449 


6 


S5 


8.524343 


9-999757 


8.524586 


11.475414 


10.000243 


11.475657 


5 


$6 


8.528102 


9.999753 


8.528349 


1 1.47 165 1 


10000247 


11.471898 


4 


$7 


8.531828 


9999748 


8.532080 


11.467920 


10.000252 


11.468172 


3 


$» 


8-5355*3 


9.999744 


8.535779 


11464221 


10.000256 


11.464477 


a 


S9 8.539186I 


9.999740 


8.539447 


11.460553 


10.000260 


11.460814 


I 


60 


8.542819 


9-999735 


8.543084 


11.456916 


10.000265 


11.457181 





H 


Co-sine. 


Sine. Co-Ung.' Tang. ' Co-sec, 


Secant. 



^ De^re^i 



LOGARITHMS OF NUMBERS! 



No. 



760 
761 
761 
763 
764 

765 
766 

7^7 
768 
769 







1 



SSo8i4!8SoS7i 
88i385|Hgi442 
8819551X83012 



770 
771 
77a 

773 
774 
775 
776 

777 
778 
779 
780 
781 
782 

783 
784 

785 
786 

787 
7«8 

Jit 
793 
791 
794 
793 
794 
795 
796 
797 
798 
799 



8825241882581 S826JC 
883093 883150 883207 
883661 883718 883775 
8842x9 884185 884342 
884795 884852 884909 
8853611885418 885474 
8859261885983 

886491 



2 



88092 

88i4v^ 
8820O9 



887054 
887617 



886547 
8871 1 1 

88767^ 



888179 888130 



.<50985 
^^1556 
882126' 



4 



881042 

881613 

882183 

882695188275a 

883264I883321 

8S3832 883888 

884399 884455 

884965 885022 

88553, '885587 

886039:8860961886152 

886604 58666o| 8X6716 



881099 881156 88iai3 



881670 
88224O 
882809 
883377 

883945 
884512 

885078 
885644 
886209 



886773 
887336 



881727 881784 



887167 887223:887280 

887730 887786:8878421887898 
888292 8883481888404^888460 
88874118887971 888853 888909I888965! 889021 
889302I889358 889414 889470(889526' 889582 



;889862;8S99i8 
1 8904 2 1. 89047 7 
•?9098o 891035 
1891537 891593 



889974 890030 
890533 890589 
891091 891 147 



890086:890141 
890644:890700 
891203; 89 1 259 



•892095, ?92i50 
892!35 1 {892707 
893207.893262 
893762I893817 
894316I894371 



894870 
895423 

895975 
896526 

897077 



800 
801 
80a 
803 
804 
80$ 
806 
807 
808 
809 



810 
811 

842 

8.3 

814 

815 

816 

817 

818 

819 



897627 
898176 
898725 

899273 

899820 

900367 
900913 

901458 

902003 
90^547 

903090 
903632 
904174 
904715 
905256 
905796 

906335 
906873 

907411 

907948 



8Q4925 
895478 
89603c 
896581 

89713* 



891649 891705^91760:891816 



896085 896140 
896636896692 

8971871^97242 



897682 
898231 
898780 
899328 
899475 
900422 
900968 

9015*3 
902057 
9O2601 



892206 892262 892317 892373 
8927628928 1 81892873, 892929 
8si33»8!893373 893429i893484 
893873893928^893984; 894039 
894427 894482I894538 894593 
894980 895036 895091 895146 

8^5533^955881895643:895699 



882297 
882866 

883434 
884002 

884569 

885135 

885700 

886265 

886829 
88739a 
887955 



882354 
882923 

883491 

884059 

884625 

885192 

885757 
886321 



8 



886885 

•87449 
88801 1 

888516J888573 

889077I889134 

889638)8^9694 



890197 
890756 
891314 



890253 
890812 
891370 



891872:891928 



892429 892484 
89298 5 1 893040 
8935401893595 
894094(894150 



8977371897792 
898286)898341 

898835''898890 

8993831899437 
8999301899985 
90O476'90053i 



896195 896251 
896747 896802 
897297 89735* 



908485 
909021 
909556 
910090 



903144 
903687 
904228 
904770 
905310 
905850 
906389 
906927 
907465 
908002 



9OIO22 
9O1567 
902112 
902655 



9O1077 
9O1622 
902 I 66 
902710 



908539 
909074 
909609 
910144 



903198 

90374* 
904283 

904824 

905364 
905904 

906443 

906981 

907519 

908056 



903253 
903795 

904337 
904878 



897847 897902 
8983961898451 
8989441898999 
899492 899547 



900039 



9OO094 



894648 
895201 

895754 
896306 

896857 

897407 



900586(900640 
90113 1 1901186 
9Oi676[90i73i 
902221-902275 
^02764 

903307 903361 
9038491903903 

90439 »i 90444 5 
90493 1 '904986 



897657 
898506 
899054 
899602 
900 I 49 
900695 
90 1 240 
901785 
9023 29 
9028181902873 



881270 
881841 
882411 
882980 
883548 
8841 1 5 
884682 
885248 
885813 
886378 

88694a 
887505 
88S068 
888629 
889190 

889750 
890309 
890868 
891426 
39*983 



894704 

895257 
895809 
896361 
896912 
897462 

898012 
898561 
899109 
899656 
900 2O3 
900749 
901295 
901840 
902384 
902927 



892540 
893096 
893651 
894205 
894759 
895312 
895864 
896416 
896967 
897517 



881328 
881898 
882468 

883037 
883605 
884172 

884739 
885305 

885870 
886434 



88699& 
8S7561 
888123 
888685 
889246 
889806 
890365 
890924 
891482 
892039 



910624 910678 
9111581911211 



11690 911743 

12222 



9 

9 

912753 

9^3284 





912275 
9 L2806 
9»3337l 

"1 



908592 
909 1 28 
909663 
910197 
910731 
91 1264 
911797 
912328 
912859 

9*3390 



9054i8[905472;905526 



9059^8 
906497 

9070^5 

907573 
908109 

908646 
909181 
909716 
910251 
910784 
911317 
911850 
912381 
912913 

9*3443 



9060 12 90606 5 
906550:906604 
907089.907142 
907626,907680 
9081.63 908217 



908699 908753 
909235.909288 
9097701909823 
910304*910358 
910838 910891 
9113711911424 
911903I911956 
9 12435 '912488 
9i2966!9i30i9 
9*3496 9*3549 



903416 

903958 
904499 

905040 
905580 
906 1 1 9 
906658 
9O7196 

907734 
908270 



903470 
904012 

904553 
905094 
905634 

906173 
906712 
907250 
907787 
908324 



3~|~r"i' 



908807 
969342 
909877 
9 104 I I 
910944 

91*477 
912009 
912541 

913072 
913602 



908860 
909395 
909930 
910464 
910998 

9**530 
912063 
912594 

9*3*25 
9*3655 



898067 

898615 

899164 

89971 

900258 

900804 

901349 

901894 

902438 

90298 1 

903524 
90^4066 
904607 
905148 
905688 
906a a7 
906766 
907304 
907841 
908378 



892595 

89315* 

893706 

894261 
894814 

895367 
895920 
896471 
^97012 
897572 



908914 
909449 
909984 
910518 
91 1051 
91*584 
912116 
912647 
9*3*78 
9*3708 



898122 
898670 
899918 
899766 
900312 
900858 
901404 
90 1948 
90 2492 
903036 

903578 
904120 
904661 
905 2O2 
905742 
906281 
906810 

907358 
907895 

908431 



908967 
909502 
910037 
91057* 
911104 
911637 
912169 
912700 

9»323« 
9*3761 



8 I 9 



■ta 



LOttARITHMS OF NUMBERS. 



17 




840 
841 

844 



> 

846 

X47 
848 
849 



850 
851 
85X 

«53 
854 

856 

8s8 

860 
861 
86z 
86s 
864 
865 
866 

867 
868 
8^9 



870 
871 
872 

873 

875 
876 

87.7 
878 

J?79 



913814 

914343 
91487a 
915400 

9159*7 
916454 

916980 

917505 

918030 

9*8555 



913867 ;9i39ao 

914396,914449 

914925 914977 

9«5453!9«$505, 
9 1 5980 19 16033 

9165O7 916559 

917033 917085 

917558 917610 

918083 918135 



9 1 8607 



91^078 
9 1960 1 
92012) 
920645 
921 166 
921686 
922206 
922725 

9*3*44 
923762 

9242791924331 



919130 

919653 
920175 
920697 
921218 

9*1738 
922258 

922777 
923296 
923814 



924796 
925312 
925828 
926342 



92484S 
925364 
925879 
926394 



926857J926908 

9*7370j9*74*» 
927883,9*793$ 
928396,9*8447 



918659 



928908 



929419 



928959 



929470 



929930929981 

930440 j 93049" 
930949193 1000 

93«4$8j93'$09 
931966 93*017 
93*474 93*5*4 
93*981,93303* 
933487 
933993 



934498 
935003 
9355071935558 



933538 
934044 

934549 
935054 



919183 
919705 
920228 

9*0749 
921270 
921790 

922310 
922829 

9*3348 
923865 

924383 
924899 

9*54*5 

9*593 > 
926445 

926959 

9*7473 
927986 

928498 

929010 

929521 
930032 
930541 
93 105 1 
93 1 560 
932068 

93*575 

933082 

933588 
934094 



934599 
935104 

935608 

93611 1 
936614 
937116 



936011 936061 
9365i4;936564 
937016 937066 
9375i8,937S&8!9376i8 
93801 9 93 S069 938 1 1 9 
938520 93857O1938620 
939020 939Q70;939'*o 



939519939569:9396*9 
940018 940068I940118 

740516 940566I940616 

941014 941064I941114 

941511 941561J941611 

942008 9423;8|942i07 

94*504 94*5 5494*603 

943000 943049 ;943099 
94i494 943 544^943 5V3 
943989 944038 



1 



94-.^ 



9*3973 
914502 

915030 

9*5558 
916085 
916612 
917138 
917663 
918188 
91871* 
919235 

9*9758 

920280 
920S01 
921322 
921842 
922362 
929881 

9*3399 
923917 



9*4434 
9*495* 
925467 
925982 
926497 
927011 

9*75*4 
928037 

9*8549 
929061 

929572 
930083 
930592 
93*102 
931610 
932118 
932626 

933133 
933639 

934145 



924486 
925002 
925518 
926034 
9*6548 
927062 
927576 
928088 
928601 
929112 



934650 

935*54 
935658 
936162 
936664 
937167 
937668 
938169 
938670 
939170 



939669 
940168 
940666 
941163 
941660 
942157 

94*6 1)3 
943148 

943643 
944'37 



914026 

9*4555 
91508J 

915611 

916138 

91.6664 

917190 

9*77*5 
918240 

918764 

919287 
919810 
920332 
9*0853 
921374 
921894 
922414 

9**933 
923451 

923969 



8 



^T" 



929623 

930134 
930643 

93**53 
93*661 

932169 

93*677 

933183 
933690 

934195 



934700 
935205 
935709 
936212 

936715 

937217 

937718 
938219 

9387*0 
939220 



939719 
940218 

940716 

94***3 
941710 

942206 
942702 
943198 
943692 
944186 



914079 914*31.914*84 
914608 914660 914713 
915136 915189 915241 
915664915716 915769 
916191 916241 916296 
916717:916770 916822 

91,7243.917295 917348 
9*7768 9i7820i9i7873 

91.829219*834519*8397 
918816 ) 9188 69: 9189 21 

91939^19*9444 
919914,919967 

920436,920489 



9*4*37.9*4*90 
914766:914819 

91529419*5347 
915822^915874 

9*6349:9*6401 

916875 916927 



919340 
919862 

929384 
920906 

921426 

921946 

922466 

922985 

9*3503 
924C21 



9*4538 

9*5054 
925570 

926085 

9266CO 

9*7**4 
927627 
928140 
928652 
9*9163 



9*4589 
925106 

925621 
926137 
926651 
927165 
927678 
928191 
928703 
9*9*14 



9*9674 
930185 

930694 

931203 

931712 

932220 

93*7*7 

933*34 
933740 
934246 



93475* 

935*55 

935759 
936262 

936765 

937267 

937769 
938269 

938770 

939*70 



939769 
940 zC 7 
940765 
941263 
941760 
942256 

94*75* 
943*47 
94374* 
944*36 



920958 

921478 

921998 

922518 

9*3037 

9*3555 
924072 



924641 
9*5*57 

9*5673 
926188 

926^02 

927216 

927730 

928242 

928754 

929266 



9*97*S 
930236 

930745 
93**54 
93*763 

932271 

93*778 
933*85 

933791 
934*96 



934801 
935306 
935809 

936815 

937317 
937819 

9383*9 
938820 

939319 



939819 
940317 

940815 

9413*3 

94180V 
942306 
942801 
943*97 

94379* 
9442X5 



9240ZO 
921530 
922050 

922570 
923088 
923607 
924124 



9*4693 
925209 
925724 
926239 
926754 

9*7*68 
927781 
928293 
928805 
9*93*7 



9*9776 
930287 
930796 
931305 

93*8*4 
93*3*1 
93*8*9 
933335 
933841 
934347 



93485* 

93535^ 

935860 

936363 
936865 

937367 
937869 

938370 
938870 
939369 



939868 

940367 
940S65 
941362 
941859 

94*355, 
942851 

943346 
943841 
94433 5 



917400 

9*79*5 
918450 

9*8973 



9*7453 
917978 
918502 
9 1 9026 



919496 
920019 
920541 
921062 
9**582 
922102 
922622 

9*3*40 
9*3658 
924176 



929827 
930338 
950S47 

931356 
931864 

9i*37* 

93*879 
933386 

93389* 
93439" 



93490* 
935406 
935910 

936413 
936916 

937418 

9379*9 
9384*0 

938920 
939419 



9399*8 
940417 
940915 
941412 

941909 
94*405 
942900 

943396 
943890 
944384 



8 



919549 
920071 

9^059« 
921114 

9**634 
922154J 

922674 

923 1 92 

923710 

924228 



9*4744 
925260 

925776 
926291 
926805 

9273 i$f 

927832 

9*834^ 
928856 

929368: 



929878 

93038* 
93089I 

931407 

9319*1 

932423 

932930 
933437. 
933941 
93444* 



934953 

935457 
935960 

936461 
936966 
937468 

937969 
938470 

93897^ 
93946'^ 



939968 

94046:^ 
940964 
94446 

94195 

94245 

94*95 

943445 

943939 

944433 



mm 



t 



LOGARITHMS OF NUMBERS, 







880 

8S1 

882 

883 

884 

88s 

886 

887 

888 J9484»3 

881} 1948902 



944483 
944976 
945469 
945961 
946452 
^46943 

947434 



8(p 
891 
892 
893 
894 
895 
896 
897 
898 
89^ 



900 
qor 
902 
903 
904 
905 
906 
907 
908 
909 



I 



I 



944532 944581 

945025:945074 
9455i8'94556t 

9460 10; 94605 9 
946501:946550 
9469921947041 
947483194753* 



947924] 947973;94 8021 
948462*948511 
9489511948999 



949439' 949488 
949926 949975 
9^04131950462 
9509Oo'95O949 
95i386'95i435 
9518721951920 
9523o8i952356'952405 
952792 952841.952889 

9S3*76'9533«5l953373 
95376oJ9538o8;953856 



949390 
949878 

95«>36s 
950851 

95«337 
95«8i3 



3 

944631 944680 

945»»4i945»73 
945616194^665 
946 1081 946 1 57 
946600I 946649 
947690 947 '39 



947581 
948070 
948c 60 
949048 



947630 
9481 19 



9447*9 
945222 

945715 
946207 

946698 

947189 

947679 

948168 



948608 j 9486 57 
949097 l949«46 



954H*j954»9'J954339 
9547a5'954773l9548a? 
955206 955*55^55303 



955688;955736 
956i68'9562i6 
956649(956697 

957»28|957i76 
9576071957655 
958086J958f34 



955784 
95^-64 

956745 
957224 

957703 

.958181 

958564:958612 958659 



91Q 
911 
912 

9«3 
914 

916 

917 
91S 
919 



949 S 36 
950024 
9505 1 1 
95O997 

95*483 
951969 

95*453 
95*938 

9534** 

953905 



954387 
954869 

95535" 

95583* 

95631* 
956792 

957272 

9S775* 
958229 

958707 



959041 959089 959137 959184 
9595«8|959566|9596i4 959661 
959995i960O42,96oo9Oi96oi38 
96047 1 '9605 18:960566 960613 
960946 960994 961041 961089 
961421 96i469'96i5i6-96i563 



961895 9619431961990 
962369' 96241 7 1962464 
942843 '9628901962937 
963315 9 63363 963410 

963788,963835 963882 
964260 '964307 964354 
96473 ".964778I964825 
965202 965249:965296 
965672 965719 965766 
966142 966189.1966236 
96661 1.966658 196670 5 
967080967 127 967173 

967 54«:967595. 96764* 



962038 
96251 1 
962985 

963457 



949585.1949634 
9500731950121 

950560(950608 
951046 951095 



95»5i* 
952017 
952502 
952986 
953470 
953953 



954435 
954918 

955399 
955880 
956361 
956840 

9573*0 
957799 
958*77 
958755 958803 



951580 
952066 

95*550 
953034 
953518 
95400 1 
954484 
954966 

955447 
955928 
956409 
956888 
957368 
957847 

9583*5 



944779 
945272 

945764 
946256 
946747 

947*38 
947728 
948217 
948706 
949195 



949683 
950170 
950657 

951143 
951629 

95*114 
95*599 
953083 
953566 

954*49 



95453* 
955014 
955495 
955976 

956457 
956936 
957416 

9578^4 

958373 
958850 



959*3* 959*801959328 
959705 959757(959804 
960185 960233*960281 
960661 '9607091960756 
961 1361961 1841961231 
96i6ii'96i658l96i7o6 
962085 |962i32i962i8o 



944828 
945311 

945«i3 



946305 946354 



946796 
947*87 

947777 
948266 

948755 
949*44 



8 



944877 

«45S70 
945862 



946845 

947336 
947626 

948315 
948804 

?49*?» 

949780 
950267 

950754 
951240 

951726 
952211 
952696 
953180 
95J663 



949731 
950219 

9S0705 
95119* 
951677 
952163 

952647 
953131 
953615 
9540981954146 



954580 V54628 954677 
955062955110955158 

955.543195559*955640 
9 56024] 95607ft 956120 



-^ 



944917 

945419I 

9459'n 

946403 

946894 
94*^385 

947875 
948364 

948853 

9493jy 

949829 
950316 
950803 
95**89 
95ir4 
95**59 
95*744 
953**8 
9537" 
1954194 



956505 
956984 

957464 
957942 

958420 

958898 



956553 
957032 

9S7S" 

957990 
958468 

958946 



956601 
957080 

957559 
958038 

958516 
958994 



962559)962606 962653 '962701 :96s748 



9630321963079 963 126 
963504'96?55*l963599 



963929J 963977. 964024; 
9644011964448,964495 
964872, 964919*964966 
965343 



930 
931 
93* 
933 
934 
935 
936 

937 
938 
939 .97^666 



968483 9685301968576 
968950 968996*969043 
969416,969462 969509 
969882.9699281969975 
970347:970393,970440 
970811 970858)970904 

97i276'97i3**'97i369 

971740 971786 971832 

1972203 972249 972195 

97i7i*,97*758 



964071 
964542 
965013 



- 9653909654^7 -965484 

9658i3j965860}9659O7 1965954 



963174 963131 
963646*963693 



966329 966376-966423 
966798 966845 '966892 

967267:9673141967361 
967735 !967782|967829 
968203 i968249;968296 



966283 
966752 

967220 
967688 
9681 56 

968623 968670:^68716 -968763 
9690901 969 1 36 ;969 183 1969229 
969556 969602 '969649 J96969 5 
970021 970068 1970x14970 161 
9704861970^3 1970579 970626 



959375<9594-t3l9S947i 
959852»959900 959947 

960328 960376 960423 
960804)96085 1 960899 

96f279'96i3i6'96i374 
9617531961801 961848 
9622171961275 962322 

96279 J 
963268 

963741 

96421 
964684 

965155 
965625 

966095 

966564 

967033 

967501 

967969 

968436 



964118 964165 

9645901964637 
9650601965108 
9655311965578 
966001 966048 
9664701966517 
966939 966986 

967408 967454 
967875'96792a 

968343I968389 



1 



i**p 



970951 

97*415 
971879 
972342 
972804 



970997,9710441071090 
97i46i'97i5o8|97i554 



9719*5 
971388 

9728<;i 



m^mm 



971971 ;9'7 1018 
97*434:97*^80 

972897 (972943 

5 I 6 



968810:968856 
96927619693*3 
9697411969788 

9702071970154 
970672 970719 

971137 97T«83 
971600:971647 



968903I 
969369I 
96983 5I 
970300I 
970765I 

97 1**9 
971693 



972064l97liio!97ii56 
97152719715731972619 
97*989 973035 973081 



;97t573 
973031 

8 



tOGAMTHMS O^ NUMBERS. 



W 



TIoT 



960 
961 
96ft 
963 
964 
965 
966 

967 
9€$ 
969 



970 

971 

971 

973 
974 
975 
976 

977 
978 

980 
9S1 
98a 
98s 
984 
985 
986 

987 
988 

990 
991 

99a 

993 

994 

99$ 
996 

997 
998 



973118 

973590 
974051 

9745" 
97497* 

97543* 
975891 

976350 

9768CS 

977266 



9777H 977769 



9781H0 

978637 

979093 
979548 

980003 

980458 

98091a 

981365 

981819 



98137 1 
981733 

9*3175 
983616 

984077 
9845*7 
984977 
985416 

985875 
986314 



98677* 
987119 
987666 
988113 
988^59 
989005 
989450 
989895 

990339 
990783 



991116 
991669 
9911 II 

99*553 
992995 

993436 

993877 

19943 "7 

994757 
995196 



995635 
996074 
9^6512 

996949 
997386 



1 

973«74 
973636 

974097 
974558 
975O18 

975478 

975937 
976396 

976854 



973110 
973681 

974^43 



973166 
973718 
974189 



9773'* 977358 



978116 

978683 

979138 

979594 
980O49 

9P0503 

980957 
981411 

981864 



974604 '974650 

975064I975110 

9755*4 975570 
975983,976019 
9764411976487 
9769001976946 



977815 



979*84 
979639 
980094 
980549 
981003 
981456 



977403 



977861 



978171 978317 
978728 978774 



979130 
979685 
980140 

980594 
981048 
98 1 501 



9819091 981954 

982362 

981814 

983*65 



982407 
981859 

9833 «o 
983761 

984*'* 
984661 
985111 
985561 



981316 

982769 

983110 

983671 983716 

984111J984167 

984571:984617 

9850211985067 

98547i|9855J6 

985910 985965 ,986010 

986369 986413 J98645S 

986906 

987353 
987800 

988147 

988693 

989138 

989583 
990018 

990471 

99O916 



986816986861 
987164987309 



987711 

988157 
988603 

989049 

989494 



987756 
9881OZ 
988648 

989094 
989539 



9899391989983 
99038319904*8 

990827^990871 



9912701991315 
99'7«3)99'757 
992i56;99**oo 

99*598 99*64* 



993039 
993480 
993921 



993083 

9935*4 
993965 



99436^994405 



994801 
995140 



995679 



996555 
996993 
997430 



9978i3'997867 



998*59 
998695 



994845 
995284 



995723 



996117 996161 



996599 

997037 

997474 
997910 



973313 
973774 
974*35 
974696 
975156 
975616 
976075 

976533 
976991 

977449 



973359 
973820 

974*81 

97474* 
975101 

975661 
976111 

976579 
977037 
977495 



977906J977952 
9783631978409 



978819 

979*75 
979730 
980185 
980640 
981093 
981547 
982000 

982452 
982904 

983356 
983 H07 
984157 

984707 
985157 

985606 

986055 

986503 



986951 

987398 

987845 
988191 

908737 

989183 
989618 

990071 
990516 
99O960 



99»359 
99180* 

99**44 
992686 

993 '*7 
993568 

994009 



994889 



991403 
991846 
991188 
991730 
993171 
993613 
994053 



99444^ 994493 



994933 



995jr28!99537* 



995767:995811 



978865 
979321 
979776 
980131 
980685 
981139 
98159a 
982O45 



982497 
982949 
983401 

98385* 
984301 
984751 
985101 
985651 
986100 
986548 



973405 
973866 

9743*7 
974788 
975148 
97570: 
976166 
976625 
977083 

97754* 



97345" 
973913 
974373 
974834 
975*94 
9 



8 



973497 

973959 
9744*0 
97488c 

975340 



977998 

978454 
978911 

979366 

9798*1 
980176 
980730 
981184 
981637 
982090 



982543 

98*994 
983446 

983897 

9^4347 
984797 

985247 
985696 
986144 
9*6593 



986995 

987443 
987890 

988336 

988782 

989227 



987040 
987487 

987934 
9S8381 

988816 

989171 



989671 989717 
990117 990161 



990561 



990605 



991004 ^ 991049 



991448)99149* 
991890991934 

99*333,99*377 

9917741991818 

993116^993160 

9936571993701 
9940971994141 

994537{99458i 
994977 :9950a I 
99^416 (995460 



9983031998346 



998739 



999130:999174 
999565' 999609 







nr 



998782 
999218 
9996521 

2" I 



996105 
996643 
997080 
997517 

997954 
998390 



995854:995898 



996a49)996293;996336 



996687 



9988161998869 
999261 999305 
999696 999739 



3 



996730I996774 



997i24[997i68|997iii 

997561 

997998 

998434 



997605 '997648 



998041. 99808^ 998128 



998477 998511 
'998913*998956 

999348999392 
999783 9998*6 



5 * 



•li 



5753 975799 
976211 976258 



976671 

977»*9 
977586 



978043 
978500 

978956 

9794'* 
979867 

9^0322 

980776 

981119 

9S1683 

981135 



981588 
983040 

983491 
983942 

98439* 

98484* 
985192 

985741 

986189 

986637 



987085 

98753* 

987979 
988415 

988871 
989316 
989761 
990106 
990650 
99'093 

991536 
991970 
992421 
992863 
993304 

993745 
994185 

994625 

995064 

995504 



99594* 
996380 

996818 

997255 

99769a 



998564 
999000 

999435 
999870 



976717 
977*75 
97763* 



978089 
978546 
979002 

979457 
979912 
980367 
980821 
9S1175 
981728 
982181 



982633 

.983085 
983536 

983987 
984437 
984887 

985337 
985786 
986134 
986682 



987130 

987577 
988024 

988470 

988915 

989361 

989806 

99015O 

990694 
99"3 7 

991580 
992O23 
992465 

99*907 
993348 

993789 
994229 

994669 
995108 

995547 



995986 

9964*4 
996861 
997199 

997736 
998171 
998608 

999043 
999478 
<y999!3 



973543 
974005 

974466 
974916 
975386 

975845 
976304 

976761 

977110 
977678 



978135 
978591 

979047 
979503 
979958 
980411 
980867 
981320 

981773 
982226 



982678 
983130 

983581 
984031 
984481 
984931 
985381 
985830 
986179 
986717 



987174 
987612 
988068 
988514 
988960 

989405 

989850 

990294 
990738 
991181 



991625 
992067 
991509 
991951 

99339* 
993833 
994*73 

9947*3 
99515a 

99559* 



99603O I 
996468 
996905 
997343 

997779 
998216 

99865a 
999087 
999522 
999957 



r I 8 



Q 



Mir 



S3 lOGARITHMlC SINES, TASGENTS, AXD SECANTS. 



M sine. 


Co-sine. T.,iiK. , Co-IaiiR. , Sccuit , Co-MC. , x 




o 19.019135 


9.997614 9.011610 


10.978380 


10-001386 


.0.980765 


60 






9.01043 J 


9.997601 9.011834 


■0,77'66 


.0.001399 


,0.979S6S 


59 




« 


9.01 itji 


9.997588 9.014044 


10,975956 


10.001411 


10.978368 


i8 




J 


9.o»i8j5 


9-997574 9-Oiiii' 


■0,974749 


.0,0014*6 


10.977.75 


i' 




4 




9.997561 


9,016455 


io-97Ji4i 


10.001439 


10.975984 


i* 




i 


9.015203 


9-997547 


9,017655 


10.97134s 




10.974797 


Si 




6 !9.0i6}86 


9-997 534 


9.01B851 


10.97.148 


10.001466 


,0.973614 


54 




T 9.D.7i67 


9.997510 


9 030046 


10 969954 


10.001480 


10.971433 


5J 




8 I9.01B744 


9!l97i07 


9.031137 


10,968763 


10.001493 


.o.,71h6 


i' 




i 9.01S9I8 


9,997493 


9.03141s 


10-967575 


10,001507 


10.9700S1 


S' 






9.03 '0S9 
9.031157 


9-99-'4^0 
9.997466 


9-°i36_09 


,0,966391 


10,001510 


,09689.1 


S° 




903479' 


10,965109 


.0.001534 


10.967 74J 


■»! 






9.0334.1 


9.997451 


9.035969 


10.964031 


.0,00154^ 


10.966579 


4« 




'J 


9.034581 




9037144 


.0,961856 




10,96 54' 8 


47 






9.035741 


9,997415 


9,038316 


10,961684 




.0.964159 


46 




'5 


9.OJ6S96 


9.99741 ' 


9.0394s; 


10,960515 


10.001589 


10,963.04 


45 






9.038048 


9,997397 


9.040651 


■0.959349 


10,001603 


.0961951 


44 




•7 


9,039197 


9,997383 


9.04'Si! 


10,958187 


10.001617 


10,960803 


4J 




18 


9.040341 


9,997369 


9.041973 


.0,957017 




,0,95965! 


4» 




19 


904'485 


9- 9973 S 5 


9.044130 


10.955870 


.0,001645 


10,958515 


4' 




iT- 


9041615 
9 043761 


9-99734' 


9,045184 


10.954716 


10.001659 


'0.9i7J75 


40^ 




9,997317 


9.O464J4 


10,953566 


10.001673 


10.956.38 


39 




I. 


9-044895 


9 99731J 


9047581 


10.951418 


10.00168; 


10.95510} 


38 




»j 




9,997199 


9048:17 


10,95.173 


10.001701 


10 953974 


37 




»4 


9.047154 


9,997185 


9.049869 


10.95013. 


io.OOJ7ii 


10.951846 


36 




»( 


9.048179 


9.997171 


9.051008 


to.948991 


10,001719 


io.95>7i' 


a 




16 


9.049400 


9-997157 


9.051144 


10.947856 


10,001743 


.0,950600 


34 






9,050519 




9,053^77 


10,946713 


10 0017 58 


10,94948' 


33 




tS 


9.051635 


9,997118 


9054407 


io,945?93 


10,001771 


,0,948365 


J* 




»9 


9.051749 


9,907114 


90555J! 


10.944465 


10-001786 


10,947151 


3' 




32. 


9053859 


9-997 '99 


9^56659 


10.943341 


10.001801 


.0,946.4. 


30 




1' 


9.OH966 


9,997185 


9.057781 


10,941119 


".0,0018,5 


10,945034, 


"2 




3* 


9.056011 


9-997170 


9,058900 


10,941100 


,o,ooiS3C 


,0.943919 


18 




33 


9.057171 


9,997156 


9,060016 


10.939984 


10.001844 


,0.941818 






34 


9.05S171 


9.99714' 


9,061130 


10,938870 


10-001859 


10,94.719 


X6 




IS 


9.059367 




9,061140 


10.937760 


10,001873 


.0,940633 


15 




36 


9.060460 


9-997 Ml 


9-063348 


'0,936651 


.0.001888 


10.939540 


14 




37 


9.061551 


9.997098 


9.064453 


'0-935547 


10.001901 


10,938449 


»3 




J> 


^.06163 9 


9,991083 


9-065556 


10,934444 




10,93736. 








9.063714 


9.997068 


9066655 


'o,93334S 


100019JI 


.0.936176 








9.064BD6 


9.997043 


9.067752 


10.931148 


.0002947 
10,00296, 


10,935194 


20 




*i~ 


9.065885 


9.997039 


9.068846 


10,931154 


10,934115 


't 




41 


9.066961 


9.997014 


9,069938 


10.930061 


,0,001976 


10933038 


iS 




43 


9.06B036 


9997009 


9,071017 


.0.918973 


1D.00J991 


.0.931964 






t4 


9.069 lOV 


9 996994 


9,072113 


10.917B87 


10.003006 


10.930893 


16 




t! 


9.0-70176 


9,996979 


9,073197 


10,916803 


.0,00301, 


'0919814 


»s 




|6 


9.07114= 


9,996964 


9.074178 


10.915711 


[O.DO3036 


10-918758 


14 




47 


9,071306 


9 996949 


9,075356 


10,914644 


10.00305, 


10,917694 


IJ 




48 


9.073366 


9,996934 


9,076431 


.0,913568 


IO,OOJ065 


10.916634 






49 


9.074414 


9.996919 


9.077505 


10.911495 


10.003081 


10.915576 






50 ,9.075480 


9,99''904 


9.078576 


.0,911414 


.0,003096 


10914110 






S> |9'076i33 


9.996S89 


9.079644 


10.910350 


10,003111 


.0.913467 


"T 




(t 9.077583 


9.996874 


9,0807 10 


10,9.9190 


1 0,003 ' 16 


10.9114.7 


< 




SJ 19.078631 


9,99685a 


9,081773 


io,9iBii7 


10.003,41 


10,911369 


7 




54 9 079676 


9.996843 


9.0S1BJ3 


10.917167 


10.003157 


10,910314 


6 




55 19.08^719 

56 ;9-oS'759 


9.996818 


9,083891 


10.916109 


.0.003171 


10.919181 


S 




9,996811 


9.084947 


10.9.5053 


10.003 IS8 


109.8141 


4 




i7 .9081797 


9,996797 


9.086000 


10.914000 




10,917103 


3 




58 i9.083Sj2 


9,99678^ 


9,087050 


10,911950 


10.QO]l.a 


,0.916168 






Sg 9.084864 


9,996766 


9.088098 


10.9.1901 


10,003134 


10.915136 






60 9.08(894 


9.996751 


9.089144 


10.910856 


.0,003149 


.0.9.4106 






" (;i>-Miie. Sine. Cn-Unfr 1 Tnnj?. 


Co-BCC. 


Secant, M 





«3 Degreei, 



XOGARITtiEMIC SINES, TANGENTS, AND SECANTS. 



on 



7 Degrees. 



: Sine. ; Conine. 

o 9.085894)9.99675 1 
i<9.o86922!9. 996735 
:t!9.o87947J 9. 996720 
3. 9.08897019. 996704 



Tang;. , Co-tang-. , Secrmt 1 (Jo-sic. 



4 

5 
6 

7 
8 

9 
10 



9 

9 
9 

9.089990)9.996688 9 

9 



II 
12 

13 
14 

16 

X7 
18 

20 

21 

2Z 

aj 
^4 

»S 

26 

a7 
28 

29 

30 



9.091008 
9.092024 
9.095037 
9.094047 
9.095056 
9.096262 



9.097065 
9.098066 
9.099065 
00062 
01056 
02048 

03037 
04025 
05010 

0599* 



9- 
9- 
9- 
9- 
9- 



3i 
3* 
33 

34 
3$ 
36 

37 

38 

39 
40 



4« 
4*1 9 
43 9 
9- 
9. 
9- 
9- 
9- 
9- 
9- 



44 

45 
46 

47 
48 

49 

JO 

51 

5« 

S3 

54 

55 

56 

57 

58|9 

59! 9 

60*9 



9.996673 

9 996657 
9.996641 
9.996625 
9.996610 

9*99 659 4 
9.996578 
9.996562 
9.996546 

9.996530 
9.996514 
9.996498 
9.996482 
9.996465 
9.096449 

9-996433 



06973 9 9964*7 
07951 9' 996400 

08927 9.996384 
09901 9.996368 
10873*9.996351 
11842I9.9963359 
1 2809 9. 996318(9 
U774 9-996302 
9.996185 
9.996269 

9.996252 
9.996235 
9.996219 
9.996202 



»4737 
15698 



16656 

17613 

18567 

»95'9 
^469:9*996185 

21417 9.996168 

22362 9-996151 

23306.9.996134 

24248 9.996117 

25187 9.996100 



26125 9.996083 
27060 9.996066 

i7993 9- 996049 
28925 9.996032 
29854 9.996015 

30781 9 995998 
31706 9.99 ;98o 

316309 995963 
3355* 9995946 
344709 9959*8 



35387 9.99S9«» 
36303.9.995894 
37216 9.9958-76 

38128 9.995859 
39037 9.995841 
39944 9.995823 , 
40850 9. 99 5806 1.9 
417549.99578819 
42655 9-99J77i|9 
43555 9.90575^ 



9 

9 
6 

9 

9 
9 
9 
9 
9 
9 
9 
9 
9 

9 
9 
9 
9 
9 
9 
9 
9 



089144 
090187 
O91228 
09^266 
093302 
094336 

095367 
096395 
O97422 
O98446 
099468 

00487 
01504 
02519 

0353* 
04542 

05550 
06556 

07 5 59 
08560 

09J£9 

10556 
11551 

>i543 

"3533 
14521 

15507 
1 649 1 

1747* 
18452 

»9429 

20404 

a»'377 
22348 

233 « 7 

24 284 
25249 
262 II 

27 17^ 
28130 
29087 

3^041 
JO994 

31944 
32893 

33839 
34784 
35726 
36667 
37605 
38541 



39476 

40409 
41340 
42269 
43196 
44121 

45044 
45966 
4688 s 
47803 



0.910856 

0.909813; 

O.908772. 

0.9077341 

o 906698 
0.905664 

0904633 

0.903605; 

0.902578* 

0.901554 

0.900532 



0.899513 

0.898496 

0897481 

0.^96468 

0.895458 

0.894450 

0.893444 

0.892441 
0.891440 
0.89044I 



Q. 889444 
O.88H449 

0.887457 
0.886467 

0.885479 
0.884493 
0.883509 
O.8S2528 

0.88 1 548 
0.880571 



0.879596 
0.878623 
0.877652 
0.876683 
0.S75716 

0.874751 
0.873789 
0.872828 
3.871870 
0870913 

0.869959 
C.869006 
0.868056 
0.867107 
0.866 1 61 
0.865216 
0.864274 
0.863333 
0.862395 
0.861458 



10.003249 
10.003265 
10.003280 
10.003296 
10.003312 
10.003317 
10.003343 
10.003359 
10.003575 
10.003590 
10.003406 



10.005422 
10.005458 
10.005454 
10003470 
10.005486 
10.005502 
10.005518 
10.005535 
10.005551 
10.005567 



M Co-sinc bin. 



i Oulanp. 



10.003583 
10.003600 
10.005616 
10.005652 
10.005649 
10.005665 
10.005682 
10.005698 
10.005715 

10.005751 

10.005748 
10.003765 
10.005781 
10.005798 
10 005815 

IO.OQ3832 

10.003849 

10.003866! 

10.005885 

10003900 

10.005917 
10005934 
10.005951 
10.003968 
10.005985 
10.004002 
1O.OO4O20 
10.004037 
10.004054 
10.004071 

10.004089 

10.004 106 

10.004124 

10.004141 

IO.OO4I 59 

10.004171 

10.004194 

10.004212 

10.004229 

10.004247 



0.860524 
0.859591 
O.S5860O 
0.857751 

o 856S04 

0.855879 
0.854956 
0.854054 
0.855115 
0.851197 

Tanj^. ' C<>-J5cc. » Secant. ' m 



M 



O.9I41OS 60 

0.913078 59 

0.912055 58 

O.9IIO5OJ 57 

0.910010; 56 

0.9089921 55 

O.9O7976I 54 

0.906963" 53 



0.905953 
O.9O4944 
0903938 

0.902935 
0.901954 
0.900955 
0.899958 
0898944 
0.897952 
0.896965 
0895975 
O.89499O 

o 894008 

0.895027 
O.892O49 

0.891075 
0.890099 
0.889127 
0.888158 
0.88719: 
0.886226 
0.885263 
0.884502 



0.885544 
0.882587 
0.881453 

o.t:8o48i 
0.X79551 
0878585 
0877638 
0.876694 
0.875752 

0» 748i 5 
0.873875 
0.872940 
0.872007 
0.871075 
0.870146 
o 869219 
0.868294 
0.S67570 
0.866449 
0.865530 

0.864613 
0.863697 
0.8627S4 
0.861872 
0.860965 
0.^(10056 
0.859150 
0.858246 

0.85734? 
0.856445 



51 
51 

ii 

49 

48 

47 
46 

45 
44 
43 
4» 

4« 

40 



39 
38 
37 
36 

35 
34 
33 
31 
31 

il 
29 
28 

17 
26 

15 
24 

13 

22 

21 . 
20 



«9 
18 

17 
16 

15 

13 

12 

If 
10 



9 
8 

7 
6 

5 

4 

3 
1. 
I 

o 



E 



^^^m 



34 LOGARITHMIC SINES, TANGENTS, AND SECAN'TS. 



mt 



M 



C 
I 

2 

3 

4 

6 

/ 
8 

V 

10 



II 

12 
»i 

>4 

'5 
16 

17 

i8 
«9 

20 



21 
22 

23 
14 

»5 

26 

27 
28 
29 
.10 



3i 
32 
33 
34 
35 
36 
37 
38 

39 

40 



bine. 



Co-siue. I Tang. 



9 
9 
9- 
9 
9 
9 
9 
9 
9 



4'j9- 

4219. 

43 9- 

441 9- 

45,9- 

46.9- 

47|9- 

4819- 
49 9. 

50 2: 

51 9- 
52'9' 
53 9- 
54'9- 
559. 
569. 



57 
58 
59 

M 



(J( 



43555 
44453 

45349 
46243 

47136 
48026 

48915 
49802 

50686 

51569 

52451 



53330 
54208 
550S3 

55957 
56830 

57700 

58569 

59435 
60301 

6 II 64 



62025 
62885 

63743 
64600 

65454 
66307 

67159 

68008 

68856 
69702 



7054' 
71389 

72230 

73070 

73908 

74744 
75578 
76411 

77242 

78072 

78900 
79726 
80551 
81374 
82196 
83016 

83834 
84651 

85466 

86280 

87092 

87903 

8S712 

89519I 

90325: 

91 130. 

91933 

92734 
93S34 
94*? 32 



8 Degrees. 
Co-tang. 



9-995753 9- 

9-995735 9- 

9 9957*7 9- 
9.995699 9. 

9.995681 9. 

9.995664.9. 

9.995646 9. 

9 9956*8|9. 

9.99561019. 

9-99559';9- 
9-995573 . 9- 

9995555|9- 
9995537|9- 
9-9955 •9l9• 
9•99550l,9• 
9.995482 9. 
9.995464 9. 
9.995446 9. 

9-9954i7|9- 
9.995409)9. 

9995390 9- 



9.99537* 9- 

9 995353 9- 

9 995334 9- 
9.995316 9. 

9995i97|9- 
9.99527819. 

9.995260J9- 
9-995H«|9' 
9.995221 9- 
9.99 5203 *9^ 

9.995»''4|9 
9-995«65'9- 
9.995146. 9- 
9.995127 9- 
9,995 io8;9- 
9.995089J9. 

9.995070.9- 
9-99SO.SI 9- 
9.995032 9 
9 995013 9- 



9994993! 9 
9.994974 9 
9-994955!9' 
9-994935|9 
9.994916I9 
9.994896*9' 
9-994877J9- 

9994857 9- 
9.99483819. 

9.9948i8'9- 



9.994798! 9- 
9 994779'9- 
9-994759 9- 

9-994739 9- 
9.994720 9- 
9.994700 9. 
9.994680 9. 
9.994660 9. 
9,994640 9. 
9.994620 9. 



47803 
4871k 
49632 

50544 
5»454 
5*363 
53169 

54«74| 
55077 
55978 
56877 



57775 
58671 

59565 
60457 

6»347 
62236 
63123 

64008 
6489Z 

65774 



66654 
67532 
68409 
69284 

70157 
71029 
71899 
72767 

73634 
74499 



75362 
76224 

77084 

7794* 

78799 

7965s 
80508 

81360 

82211 

83059 



83907 
84752 

85597 
86439 

87280 

88120 

8895H 

89794 
90629 
91462; 



92294 
93124; 

93953! 
94780 

95606 
96430 
97253 

9S074 
98S94 

99713 



0852197 
0.851282 
a850368 
0.849456 
0.848546 
0.847637 
0.846731 
0.845826 
0.844923 
0.844022 
a843i2} 



0.842225 
0.841329 
0.840435 

0.839543 
0.838653 

0.837764 

0836877 

0.835992 

0.835108 

0.834226 

0.833346 
0.832468 
0.83 1 59 1 
0.830716 
0.829843 
0.828971 
0.828101 
o 827233 
0.826366 
0.8 7.5 50 1 



o 824638 
0.823776 
0.822916 

0.821201 
0.820345 
0.8 1 9490 
0.818640 
0.8i77i.9 
0,816941 



O.816093 
0.815248 
O.S1440; 
0.813561 
0.812720 
0.811880 
0.8 J 1042 
0.810206 
0.809371 
0.808538 

0.807706 

0.806876! 

0.806047 1 

O.80522O1 

0.804394' 

0803 570 

0.802747 

0.801926 

0.801 106 

O.K00287 



SI lie. 



Sine. Cotfing". Tjin^. 



Secant. 


Co-scc. 
10.856445 


M 

60 


10.004247 


10.004265 


10.855547 


59 


10.004283 


io.854651 


58 


10.004301 


10.853757 


57 


10.004319 


10.852864 


56 


10.004336 


10.851974 


55 


10.004354 


10.851085 


54 


10.004372 


10 850198 


53 


10.C04390 


10.849314 


5» 


1 0.004 4O9 


10.848431 


5> 


10.004427 


10^47 549 
10846670 


50 


10.004445 


49 


10.004463 


10 845792 


48 


10.004481 


10.844917 


47 


10.004499 


10.844043 


40 


10004518 


10.843170 


45 


10.004536 


10.842300 


44 


10.004554 


10.841431 


43 


10.004573 


1O.S40565 


4» 


10.004591 


10.839699 


4' 


10.004610 


10.838836 


40 
39 


10.004628 


^0837975 


10.004647 


10.837115 


38 


10.00J666 


10.836257 


37 


10.004684 


10.835400 


36 


10.004703 


10.834546 


35 


10 004722 


10.833693 


34 


10.004740 


!0 832841 


31 


10.004759 


10.831992 


32 


10.004778 


10.831144 


31 


10.004797 


10.830298 
10.829453 


30 
29 


10.004816 


10.004835 


10828611 


18 


10004854 


10 817770 


27 


10.004^^73 


10.8^6930 


26 


10.004892 


10. 8 2609 2 


*5 


10.00491 1 


10,825256 


24 


10.004930 


10.824422 


13 


10004949 


io,8235g9 


22 


10.004968 


10 822758 


21 


10.004987 


10.821928 
10.821100 


20 
»9 


10.005007 


10005026 


10.820274 


18 


10.005045 


10.819449 


17 


10.005065 


10.818626 


16 


10.005084 


10.817804 


15 


10.005104 


10*816984 


»4 


10.005123 


10.816166 


13 


10.005143 


10.815349 


12 


10.005162 


10.814534 


1 I 


10.005182 


10.813720 
10.812908 


iO 

9 


10.005202 


10.005221 


10.812097 


8 


10005241 


10.811288 


7 


10.005261 


10.810481 


6 


10.005281 


10.809675 


5 


10.005300 


10.808870 


4 


10.005320 


10.808067 


3 


10.005340 


10.807266 


a 


10.005360 10.806466 


I 


10.005380 10.00566^; 





Co fcec 


Secant. 


?t 



81 JJetjrcejj. 



LOGARITHMIC SINES. TANGENtS, AND SECAX^TS. 35 



9 Degrees. 



M 

o 

I 
a 
3 

4 

5 

6 

7 
8 

9 

lO 

II 

12 

'3 

«4 

'5 
i6 

17 
i8 

'9 

20 

21 
22 

ij 

H 
25 

26 

27 
28 { 
29 

30 



Sine. 

9.19433* 
9. 195 1 29 



Co-sine. 



9.994610 
, _ ,,9-994600 
9.195925 ;9.99458o ^. — -,-rj 
9. 1967 19I9.994560 9.202159 



rang. 



Co-tang-. 

10.800287 
10.799471 
10.79865$ 

10 797841 

9. I975UJ9.99454O 59.20297 I 10.797029 

9. 198^02 9.9945i9'9.203782|i0.7962i8 



9.199713 
9. 200^29 
9.20134s 



Secant. 1 Co-sec 



9.199091:9.994499 

9.199879:9.994479 
9.200666'9.994459 

9.20145119994438 
9.202234,9.994418 



9.203017,9.994398 



9.204592 10.795408 
9.205400 10.794600 



9.206207 
9.207013 
9.207817 



9.208619 



^0-793793 
10.792987 

10.792183 



10.005380 
10.005400 
10.005420 

10005440110.803281 
1 0.00 5 460 1 10.802489 



io.ito$668 

10.8048711 
10.804075 



10.791381 



9.203797:9.994J77|9-20942O 10.790580 
9.204577 9.994357 '9.210220 10 789780 
9.205354 9.994336:9.211018)10.788982 
9.206131:9.994316^9.211815110.788185 



10.005481 
10.005501 
10.005521 
10.005541 
10.005562 



10.801698 
10.8C0909 
10.800121 
10.799334 
10.798549 



10.005582! 10.797766 






M 

"Eo 

59 
58 
57 
56 
55 
54 
53 
s» 

5' 

50 



9.206906,9.994295 

9.207679'9.994«74 
9 2o8452;9-994*54 
9.209222 9.994233 

9.209992I 9 994 212 

9.21076019.994191 
9.21 1526 9.994171 



9.21 26 If 
9.213405 
9.214198 
9 214989 
9.215780 



3' 
3* 

33 
34 
35 
36 

37 
38 



9.212291 
9.213055 
9.213818 
9.214579 
9.215338 
J9.216097 
9.216854 
9.217609 



9.218363 
9.219116 
9 219868 
9.220618 

9-"«367 
9.2221 15 

9.222861 

9.223606 



39 |9 »H349 
40 

41 

4a 

43 

44 

45 
46 

47 
48 
49 
50 



9.225 092 

9^5833 
9.226573 
9.2273II 
9.228048 
9.228784 
9.229518 
9.230252 
9.230984 
9.231715 
9.232444 



9.994150 
9.994129 
9.994108 
9.994087 
9.994066 
9.99404s 
9.994024 

9 994003 

9.993982 

9.993960 

9-993939 
9.993918 

9-993897 
9.993^75 
9-993854 
9.993832 
9.9938 1 1 

9-993789 



9 216568 
9.217356 
9.218142 
9.218926 
9.219710 
9.220492 
9.221272 
9 222052 
9.222830 
9.223607 

9.224382 
9.225156 
9.225929 



10.787389 
10.786595 
10.785802 
10.785011 
10.784220 

10.783432 
10.782644 
10781858 
10.781074 
10.780290 
10.779508 
ia778728 
10.777948 
10.777170 
10.776393 



10.775618 

10.774844 

10.774071 

9.226700I 10.773300 



10.005602! 10.796983 
10 005623I 10.796203 
10.005643 j 10.795423 
10.005664I 10.794646 
10.005684, I0.793869' 
10.005705; 10.793094 
10.005726' 10.792321 
1OOO5746 IJ.79154S 
10.005767 10.790778 
10.0OS7SS 10.790008 



49 



10.005S09 
iO.005829 
10005850 
10.005871 
10.005892 
10.005913 
10.005934 
10.005955 
10.005976 
10.005997 



9.993768 

9993746 
9.993725 

9-993703 
9.99368 r 
9.993660 
9.993638 
9-993616 

9-993594 
9-99357* 



9.227471 
9.228239 
9.229007 
9.229773 
9.230539 
9.231302 



51 9-»33«7« 9993550 

52 9233899 9-9935*8 

53 9-234625 9-993506 

|54 9-235349 9-993484 

55 9.236073 9993462 

56 |9.236795i9-993440 

57 !9-2375>5j9-9934»8 

58 ;9-238i35l9-993396 

59 9-238953;9-993374 

60 9.239670 . 9-99335 « 

Co sine. ' Sine. 



9.232065 
9.232826 
9.233586 

9-234345 
9-235»03 

9235859 
9.236614 

9.237368 

9.238120 

9.238872 



9.239622 
9.240371 
9.241118 
9.241865 
9.242610 

9-243354 
9.244097 
9.244839 

9-245579 
9.246319 



■tavVM 



Co-tang 



10.772529 
10.771761 
10.770993 
10.770227 
10.769461 
10.768698 



10.767935 
10.767174 
10.766414 
10.765655 
10.764897 
10.764141 
10.763386 
10.762632 
10.761880 
10.761128 



10006018 



47 
46 

45 

44 

43 
42 

41 

40 



10.789240 
10.788474 
10.787709 
10.786945 
10.786182 
10.785421 
10.784662 
10.783903 
10.783146 
10.782391 



10.781637 



10.006040 10. 780884J 
10 006061 1 10.7801 32 



10.006082 
10.006103 
10.006125 
10006146 
10.006168 
10.006189 
10.006211 



10.760378 
10.759629 
10.758882 
10.758135 

10.757390 
10.756646 

.•0.755903 
10.755161 

10.754421 
10.753681 



Taag. 



10.006232 
to.006254 
10.00627 5 
10.006297 
10.006319 
10.006340 
10.006362 
10.006384 
10.006406 
10006428 



10.C06450 
10.006472 
10.006494 
10.006516 
10.006538 
10.006560 
iO.006582 
10.006604 
10.006626 
10.006649 



Co-sec. 



10.779382 
10.778633 
10.777885 

10 777139 

•0.776394 
10.775651 

10.774908 



10.774167 
10.773427 
10.772689 
10.771952 
10.771216 
10770482 
10.769748 
10.769016 
10.768285 
10.767556 



10.766828 
10.766101 

10.765375 
10.764651 

10.763927 

10.763205 

10.762485 

10.761765 

10.761047 

10.760330 



decant 



39 
38 

37 
36 

35 
34 
33 

32 

31 

J?- 
29 

28 
27 
26 

25 
24 
23 
22 
21 
20 



19 
j8 

17 
16 

H 

13 
12 

If 

10 

"T 

8 

7 
6 

5 

4 
3 

2 
I 




u 



60 Uegrc^,. 



^MNP«l 



bo LOGAUITH.MIC SINES, TANCENTS, AXD SECANTS. 



10 T)egreefl. 



M ■ Sir.t:. <!o-hine. 



o 
I 

2 

3 

4 

S 
6 

7 
8 

9 

10 

1 1 

12 

»4 

i6 
17 
ig 

20 

Zl 
22 

23 

a? 

26 

2? 

2g 

50 



5' 

t53 

54 
55 
56 
57 
58 

59 

6o_ 

M 



Taiic 



9.246519 
9.247057 



Co-tan j«^. J 



9.239670 9.91} 35 5' 
9.2403^6,9.993329 

[9.^41101 9-99330719 -47794; 
'9.241814.9.9^528519.248550, 
19.242520 9 993262:9.249264, 
;9.243237;9.o93240i9.249998i 
19.243947 19 99521 7. 9.2507301 
^9.244656.9,993:95 9.251461 
l9.24s363.9-993«72 9.252191 
,9.2.^6069 9.993149 9.252920J 
9-^4677$ • 9-993 '27 9.2S364B 

|9.2474"^^ 9-995 »04 9-254574 
19 24818119.993081 9.2551CO 

;^,248SS3;9.99305q|9.255824 

;9.249583|9 993Ci6'9.256547 

.9.250282j9.9930i3;9 257269 

I9. 250980 I9-992990I 9.257990 



3» 

32 
3i 
34 
35 
36 

.;7 
38 

39 

40 



19-251677 

:9-252373 

9.253067 

*9. 253761 

'9 2j-44S3|'9 992i!7S|9-26i578 
9.992852:9.262292 
9.992829I9.263005 



9.255144 

J9255834 

'9-256523 
i9 257211 

19.257898 

19.258585 

19.259268 

9.259951 

9 260633 



4» 

42 

43 

44 

4i 
46 

47 
48 

49 

50 



9-261314 
9.261994 
9.262673 

9-2633 5 » 
9.264O27 

9.264703 

9-265377 
9.266051 
9.266723 
9 267395 



9.268065 
9.268734 
9.269402 
9.270069 
9.270735 



9.992967 I9.25871O 

9-992944J9-259429 
9.992921 19.260146 

9.992898,9.260863 



9.992806 
9.992783 

9992759 
9.992756 

9.992715 

9.992690 

9.992666 



9 992645 
9.992619 
9.992596 
9.992572 

9 992549 
9.992525 

9.992501 

9.992478 

9.992454 

9.992430 

9.992406 
9.992582 
9.992558 

9-992335 
9.992511 

9.992287 
9992265 



9.271400 
9.272064 
9.272726I9.992239 

9.27558819.992214 
9.274049 9*992190 



9.265717 
9.264428 
9.265138 

9-265847 
9.266555 

9.267261 

9 267 967 

9.26S671 

9-269575 
9.270077 
9.270779 

9-271479 
9.272178 

9.272876 

9-273573 
9.274269 

9.274964 



9275658 
9.276551 
9.277045 

9-277734 
9.278424 

9.2791 13 

9.279801 

9.280488 

9.281174 

9.281858 



9.282542 



[9.274708 9 992166 
!9-275367"9-992i42 9.285225 

9.276025.9.9921 i8'9.2859C7 

9.276681 9.992095 9.284588 
;9-277337 9-992069.9.285268 
.9-27799« 9 992044I9.285947 

9.278645 9.99202OJ9.286624 
.9.279297 9-991996 9-287301 
I9.279948 9-99»97i, 9-287977 

9.280599 9.991 94719.288652 

OO'Sine. Sine. 'Co^tang.^ Tang. 

79 De^^rees. 



0.755681 

0.752943 
0.752206 

0.751470' 

0.750736I 

0.750002! 

o 749270: 

0.748559 
0.747809! 

0.747080; 

0.746352; 



0.745626 



I 



0.744900} 
0.7441761 
o 743453! 

0.74273 1 1 

0.742010 
0.74I290 

0.740571 
0.739854 

0-739»37 



0.738422 
0.757708 
0.736995 
0.756283 
0.735572 
0.734862 

o.734»S3 

0.733445 
0.732739 

0.732055 



0.73*329 
0.750625 
0.729923 
0.729221 
0.728521 
0.7 27 82 1' 
0.727124 

0.726427 
0.725751 

0.725036 



0.724342 
0.725649 
0.722957 
0.722266 
0.721576 
0.720887 
0.720199 
0.719512 
0.718826 
0.718142 



0.717458 
0.716775 
0.716095 
0.715412 
0.714732 
0.7 14053 j 

o.7'3376i 
0.712699! 
0.712023' 
0.711348 



Secant. | 


Co-sec 1 


« f 


10.006649,10.760330 60 1 


i0Oo6b7t 10759614 


59 


10.006693 


10.758899 


58 


10.006715 


10.758186 


57 


10.006738 


•0.757474 


56 


10.006760 


10 756763 


55 


10.006783 


10.756053 


54 


10006805 


»0-755344 


53 


10.006828 


«0.754637 


52 


iaoo685i 


» 0.7 5393 < 


5» 


10.006873 


10.752522 


50 
49 


10.006896 


10.006919 


10.751819 


48 


10.006941 


10.751 117 


47 


10.006964 


10.750417 


46 


10.006987 


10.749718 


45 


10.007010 


10.749020 


44 


10.007035 


10.748323 


43 


10.007056 


10.747627 


42 


10.007079 


10.746933 


4» 


10.007 102 


ro.746239 


40 
39 


10.007125 


10-745547 


10.007148 


10.744856 


38 


10.007171 


10.744166 


37 


10.007194 


'0.743477 


36 


10.007217 


10.742789 


35 


10.007241 


10.742102 


34 


10.007264 


10.741417 


ii 


10.007287 


10.740732 


32 


JO.007311 


10.740049 


31 


10007554 


> 0-7 393^7 


30 
29 


10.007557 


10.738686 


10.007 5 8 1' 


ia738oo6 


28 


10.007404 


«o.737327 


27 


10.007428 


10*736649 


26 


10.007451 


^0-735973 


25 


10.007475 


10.735297 


M 


10.007499 


10.734623 


23 


10.007522 


«>-7^3949 


22 


10.007546 


10-733277 


21 


10.007570 


10.732605 


20 

'9 


10.007594 


J0.73I935 


10.007618 


10.731266 


iS 


iaoo7642 


10.730598 


17 


10.007665 


10.729931 


16 


10.007689 


10.729265 


>5 


10.007713 


10.728600 


14 


10.007737 


10.727936 


U 


10.007761 


10.727274 


12 


10.007786 


10.7266:2 


21 


10.007810 


10.725951 


10 
9 


10.007834 10.725292 


10007858 10.724633 


8 


10.007882 10.723975 


7 


10.007907' 10.723319 


6 


10.00793 J ] 10.722663 


5 


10.007956 10.722009 


4 


10.007980 10.721355 


3 


1 0.008004: 10.720703 


ft 


10.008029; 10.720052 


I 


10.008053 


10.719401 




M 


Co.«ec. ' 


Sijecant 



LOGAHITHMIC SINES, TANGENTS, AND SECANTS. 37 



11 Dejfpces. 



Sine. Co-gine. Tan^ 



o 
I 

a 

3 

41 

5 
6 

7 
8 

9 

10 

Ji 

12 

»5 

r6 

17 
r8 

«9 

20 



21 

22 
23 
24 

as 

26 

27 

28 

29 

JO 

32 

33 
34 
35 
3& 
37 
38 
39 

41 
42 
43 
44 
45 
4* 
47 
48 

49 
50 



S» 
5a 
53 
54 
55 
56 
57 

58 
59 
60 

•I 



9.280599 
9.281248 
9.281897 

9.282544 
9.283 19O 

9283836 

9.284480! 
9-2851241 
9.2S5766I 
9.286408; 

9. 2 8? 048^ 



9.2876881 
9.288326-1 
9.288964 
9.289600 
9.290236 
9.290870 
9.291504 

9 292137 
9.292768 

9-^93399 



9 
9 

9 

9 

9 

9 

9 

9- 

9 

9 

9 

9 

9 

9 

9 

9 

9- 

9 

9 

9 



9.99 

9.991 

9.991 

9.991 

9.99 

9.991 

9.991 

9.99! 

9.991 

9.99] 



9.288652 
9.289326 
9.389999 
9.290671 
9.291342 
9.292013 
9.292682 

9.293350 
9.294017 
9.294684 

9-^9 5349 

9.296013 

3.296677 

9a97339 
9.298001 

9.298662 

9.299322 

9.299980 

9.300638 

9.301295 

9.301951 



Co-tang 



294029 
294658 
295286 

a959i3 
296539 

297164 

297788 

2984 I 2 

299034 

299655 

00276 
00895 
01514 
02132 
02748 

03364 
03979 

04593 
05207 

05819 



06430 
07041 
07650 
08259 
08867 

09474 
0080 

0685 

1289 

'893 

*495 
3097 
3698 

4*97 
4897 

5495 
6092 

66«9 

7284 

7879 



9.990986 
9.990960 
9.990934 



9^990908 
9.990882 
9.990855 
9.990829 
9.990803 
9.99C777 
9.990750 
9.990724 
9.990697 
9.990671 



9.990645 
9.990618 
9.990591 
9.990565 
9.990538 
9.990511 
9.990485 
9.990458 
9.990431 
9.99O404 



9,302607 
9.303261 

9-3039'4 
9.304567 

9.305218 

9.305869 

9.306519 

9.307168 

9.307815 

9.308463 

9.309 109 

9'309754 

9.310398 

9.311042^ 

9.311685 

9-3«3»7 
9.312967 

9.313608 

9.314247 

9.314885 



M Co-sine, i Sine. 



0.7 1 1348 
0.710674 
0.710001 
0.709329 
0.708658 

0.707987 
0.7073*8 

0.706650 
0.705983 
0.705316 
0.70465 1 



0.703987 
0.703323 
0.702661 
0.701999 
0.701338 
0.700678 
0.700020 
0.699362 
0.698705 
0.698049 



0.697393 
0.696739 

0.6960S6 
0.695433 
0.694782 
0.69413 1 
0.693481 
0.692832 
0.692185 
0.691537 



0.690891 
0.690246 
0.689602 
0.688958 
0688315 
0.687673 
0:687033 
0.686392 
0.685753 
0.685115 



Secant 

ro.008053 

10.008078 

10.008103 

10.008127: 

10.008 1 52I 

10.008177; 

10.008201 1 

10.008226: 

10.008251' 

10.008276 

^o.oo830I 



Co-sec. 



10.008326' 

io.008351] 

10.0083761 

1 0.00840 1 j 

10.008426' 

10.008451' 

10.008476 

10.008502 

10.008527 

10.008 552; 

iO.008578 
10.008603 
10.008628 
10.008654 
10.OC8679 
10.008705 
10.008730 
ro.008756 
10008782 
10.008807 



10.719401 
10.718752 
10.718103 
10.717456 
10.716810 
10.716164 
10.715520 
10.714876 
16.714234 
10.713592 
10.712952 



M 



10.008833 

faoo8859 
10.00888) 
10.008910 
10.008936 
10.008962 
10.008988 
10.0090 14 
10.009040 
1 0.009066 



0.679408 1 10.009305 
0.678778' 10.009329 



■«^" 



9.315523 10.684477 10.009O92 

9,316159 f 0.683841 10.009 1 18 

9.316795 10.683205 10.009 145 

9.317430 10.682570 10.009171 

9.318064 10.681936 10.009197 

9.318697 10.681303 10.009223 

9.319329 10.680671 10.00925c 

9.3 1996 1 10.680039 10.009276 
9.320592 
9.321222 

9.321851 
9.322479 
9.323106 

9323733 
9.324358 
9324983 
9.325607 
9.32623 1 1 
9.326853 1 

9-327475 ' 

Co-tang. Tang. Co-sec. 



10.712312 
10.711674 
10.711036 
10.710400 
10.709764 
10.709130 
10.708496 
10.707863 
10.707232 
10.706601' 

10.705971 
10.705342 
10.704714 
10.70408- 
10.703461 
10.702836 
10.702212 
fo.701588 
10.700966 
10.700345 



ia699724 
10.699105 
10.698486 
10.697868 
10.697252 
10.696636 
10.69602 E 
10.695407 
10.694793 
10.694181 



10 693570 
10.692959 
10.692350 
10.691741 
10.691133 
10.690526 
10.689920 
10.689315 
10.688711 
10.688107 



0.678149 10.009355 
0.677521 ' 10.009382! 
0.676894I 10.0094091 
0.676267110.0094351 
0.675642' 10.009462I 
0.675017* 10.0094891 
0.674393 10.0095 1 5| 
0.673769 10.009542. 
0.673 147I 10.009569' 
0.672525 10.009596 



106^7505 
10.6S6903 
10.686302 
10.685703 
10.685103 
10.684505 
10.68^908 
lO 683311 
XO.6B2716 
10.682121 



Secant. 



60 

59 
58 
57 
56 
55 
54 
53 
52 

5' 

50 



47 
46 

45 
44 
43 
42 

41 
40 

39 
38 
37 
36 

35 
34 

33 

32 

31 

30 



29 
28 

26 

25 

24 
23 

22 

21 

20 



9 
8 

7 
6 

5 
4 

3 

2 

I 

o 



9 
8 

7 
6 

5 

4 

3 

2 

I 

o 



M 



••ki 



78 Degrees- 



:38 LOGABITHMIC SIXES, TANGENTS AND SECANTS. 



12 Decrees. 



M I Sine. Co-fiine Tang. , Co-Ungr. Srcant. Ck)-h*c. 



__0!9-3»7879 9 9 90404 9-3a7474; 10.672526 
J 9.3l?47,; 9 q 903 7 8 9.328095; 10.67 19O5 
2 9.319066 9.9903^1 '9.3287 15I ic 671285 
319.319658 9.990324 9.329334,10.670666 
4!9.320249 9.990297 9.32^953,10.670047 
5 9.320840 9.990270 9.330570, 10 669430 
6|9.32i430 9.990243 9.331187. 10.668813 
719.322019 9. 9902 15 9.331803: 10.668197 
819.322607,9.990188 9.332418* 10.667582 
91^.32319419.990 1 6 1. 9.3 33033, 10.666967 
10 ; 9 3^3780 9.990134 - 9J33646 10.666354 

1119.324366 9.990107.9-334*59' *o-66574' 
12 9.32495019.990079 9.334871:10.665129 

»3i9-3*S534J9-99005*;9-33548a 10.664518 
9.990025I9.336093, 10.663907 



I4;9.32fiii7 
15 9.326700 
i6!9.32728i 
1719-3*7862 
189 328442 
19! 9.3 29021 

20* 9.329599 

a»:9-330»7^ 
22!9.330753 
»3;9-33>349 
a4!9-3S»903 
iv9-33*478 

26J 9.33305 1 
a7|9333624 

?.8, 9.334195 
2919.334767 
30'9-335337 



9-9^9997i9-336702 
9.989970:9.337311 
9.989942 9.3379*9 

99899 > 5 

9.989887 

9.989860 



10.663298- 

10.662689' 

10.662081. 

9-33^5*7j>o.66i473 

9 339133; 10.660867! 

9-3397^9' 10.660261J 

9.340344 10.6596561 
9.340948; 10.659052 
93415^2110658448 
9.342155 10.657845 
9 3427 ;7| 10.657243; 
9-343358 10.65664a: 



3i|9- 335906 
32J9.336475 

33'9-337043 

U 9-337610 

35 9-33^'76 
30 9338742 

37 9-339307 

38 9-339871 

39 9-340434 

40 9-340996 



10.656042! 

10.655442: 
10.654843! 
10.654245I 



4* 
42 

43 
44 
45 
46 
47 
48 



9.341558 
9.342119 
9.342679 

9343239 
9-343797 



9.989271 
9.989243 
9.989214 
9.989186 
9.989157 



9-343958 

9- .3445 58 

9 34S>57 

9 34575^ 

9.346353 10,653647 

9.346949,10653051 

9-347 545 "0.652455 

9.348141 

9-348735 

9-349329 
9.349922 

9350514 
9.3 5 1 106 

9.351697 



9.352287 
9.352876 

9353465 



10.651859 

10.651265 

10.650671 

10.650078; 

10.649486 

10 648894 

10.648303 



10.647713; 
10.6471241 
10.646535 



9-354053 '0.645947 
9 354640 10.645360 



9-34435519-989128 9-355227| 10.644773 
9.344912J9.989100 9.355813110644187 
9-345469;9-98907i|9-356398|'0.6436o 



49J 9. 3460241 9.989042 9.3569821 10.643018 



^! 9 346579 . 9-989014 
9-347i34'9-988985 
9.34768719.988956 



5» 
5* 

53 
54 



9.348240 



9-357566)10.642434 



9.348792 

55 9-349343 

5619-349893 

57 9.350443 

58 9.350992 



9358140 

. . . 9-35873» 
9.988927)9.359313 



10.641851 
10 641269 
10.640687 
10 64007 



9.988898:9.359893, 
9.988X69.9.360474110.639526 
9.988840 9.361055^ 10.638947 
9 988811:9.361632; 10.638368 
9-98878219.36221010.637790 

59 9.3515^0:9.988753 9.362787110.637213 

60 9.352088 9.988724 9.363364' 10.636636 



M 



Co -sine ' Sine. Co-tanp;'.! 'ianjj^. * Co-sec. 



0.009596 
0.009622 
C.009649 
0.009676 
o 009703 
0.009730 

OOOS757 
0.009 785 

0.009812, 

O.OO9839 

0.009866; 



O.009893J 

0.009921 1 

0.009948^ 

0.009975 

0.010003 

0.010030 

0.0100581 

0.0100851 

O.O101131 

O.O1014O1 

0.0101681 

0.010196 

0.010223 

0.0 102 5 1 

0.010279 

0010307 

0.010335 

0.010363 

O.OIO39O 

OOIO418 

0.0 1 044 7 
0.010475 
O.OIO5O3 
0.01053 I 
0.010559 
0.010587 
0.010615 
0.0*0644 
0010672 
0.010700 

0.0 10729 
0.010757 
0.010786 
0.010814 
0.010843 
0.010872 
0.010900 
0.010929 
0.0 109 58 
0.010986 



0.01 1015 
0.0 1 1044' 
0.011073 

0.0III02 

O.OIII3I 
o.ot J160 
O.OII189 
0.01 1218 
0.0:1247 
0.01 1276 



0.682I2I 
0.681527 
0.680934 
0.680342 
0.679751 
0.679160 
0.678570 
0.677981 

0.677393 

0.676806 
0.676220 



M 

60" 

59 

58 f 

57 

56 

SS 

54 
SS 
5* 
5» 

50 



0.675634 
0.675050 
0.674466 
0.673883 
0.673300 
0.672719 
0.672138 
0.671558 
o 670979 
0.670401 

0.669824 
0.669247 
0.668671 
0.668097 
0.667522 
0.666949 
0.666376 
o 665805 
0.665233 
0.664663 

0.664094 
0663525 
a662957 
0.662390 
0.66 1 S24 
0.661258 
0.660693 
0.660129 
0.659566 
0.659004 



0.658442 
0.657881 
0.657321 
0.656761 
0.656203 
0.655645 
0.655088 
0.654531 
0.653976 
0.6534H 

0.652866 
0.652313 
0.651760 
0.651208 
0.650657 
0.650107 
0.6.19557 
0.649008 
0.648460 
0.647912 



49 
48 
47 
46 

45 
44 
43 
4* 

4> 

40 



39 
38 

37 
36 
35 
34 
3S 
3» 

3» 
30 



29 
28 

27 
26 

15 

23 
la 
21 
20 



19 
18 

n 
16 

>5 

14 

«3 

12 

II 

to 



9 
8 

7 
6 

5 

4 

3 

3 

I 
o 



Secant. ' >i 



77 Dctjrees. 



ijOGABTTHMIC SINES, TANGENTS, AND SECANTS. 39 



13 Decrees. 



J 



M f Sine. Co-sine. 



!9 3S30S8 
9-35a63s 
9.353181 
9 3S37»6 

9.354*71 
9.354815 

9-3SS35^ 
9-35$90i 
9356443 
9.356984 

'9'3>75M 

11 ;9.3 58064 

12 I9.3 58603 

13 9-359I4* 
9.359678 

9.360215 

9.36075* 
9.361287 

9.361822 

9.362356 

9.362889 



o 
I 

2 

3 

4 

5 
6 

7 
8 

9 
10 



14 

«5 

16 

»7 
18 

19 

20 



I. 



21 
21 

24 

*5 

26 

27 
28 

*9 

30 



3' 
3* 
33 
34 
35 
36 
37 
38 
39 

11 

4« 
42 

43 
44 
45 

46 

47 
48 
49 
50 

5» 

5* 

53 
54 
55 
56 
57 
58 

59 
60 



9.363422 

9363954 
9.364485 

9365016 

9.365546 

9.366075 

9.366604 

9.367 13 1 

9.367659 

9.368185 



9.3687 1 1 
9.369236 
9369761 
9.370285 
9 370808 

9-371330 
9.371852 

9-37*373 
9.372894 

9-3734<4 



9-373933 
9.374452 
9 374970 

9-375487 
9.376003 

93765 «9 

9-377035 
9-377549 
9.378063 

9378577 



9.379089 
9.379601 

9.3801 13 
9.380624 

9.38 M 34 
9.381643 
9.382152 
9.382661 

i9-383»68 
9383675 



9.988724 
9.988695 
9 988666 
9 988636 
9.988607 

9-988578 
9.988548 

9.988519 
9.988489 
9.988460 
9.988430 



9.988401 
9.988371 
9.988342 
9.988312 
9.9S8282 
9.988252 
9.988223 
9.988193 
9988163 
9988 T33 

9.988103 
9.988073 
9.988043 
9.988013 
9.987983 

9.987953 
9.987922 

9.987892 

9.987862 

9.987832 



9.987801 
9.987771 
9.987740 
g.987710 
9.987679 
9.087649 
9.98761X 
9.987588 

9-987557 
9.987526 



9.987496 
9.987465 

9.987434 

9.987403 

9-98737* 

9-98734« 
9.9873 »o 

9.987279 
9-987*48 
9.987*'7 

9.987186 

9.987155 
9.987124 

9.987092 

9.987061 

9.987030 

9.986998 

9.986967 

9.986936 

9.9S6904 



Tang:. 



.363364 
.363940 
5-3645 15 
Q.365090 
•J. 36 s 664 
^.366237 
9.366810 
9.367382 

9-367953 
9.368524 
9.369094 

9.369663 

9.370232 

9-370799 
9.371367 

9-37'933 
9.372499 

9.373064 
9-373629 

9.374«93 
9374756 



9'3753«9 
9.375881 

9.376442 

9-377003 

9377563 
9.378122 

9.378681 

9-379*39 
9-379797 
9380354 

9.380910 
9.381466 
9.382020 

9.38*575 
9.383129 

9 3836<?2 

9.384234 

9.384786 

9-385337 
9385888 



Co-ta ng. 

0.636636 
0-636060 
0.635485 
0.634910 
0.634336 
0.633763 
0.633190 
0.632618 
0.632047 
0.631476 
o. 630906 

0.630337 
0.629768 
0.629201 
0.628633 
0.628067 
0.627501 
0.626936 
0.626371 
0.625807 
0.625244 



9.386438 
9386987 
9.387536 
9 388084 
9.388631 
9.389178 

9-3897*4 
9.390270 

9.390815 

9.391360 



M Co-sine. Sine. 



9.39 » 903 
9.39*447 
9-39*989 
9.39353« 
9394073 
9.394614 

9.395 « 54 

,9-395694 

;9 396233 

9^396771 

Co .tang. 



Secant. Co-sec. 



0.624681 
0.624119 
0.623558 
0.622997 
0.622437 
0.621878 
0.621319 
0.620761 
0.620205 
0.619646 



0.619O90 
0.618554 
0.617980 
0.617425 
0616871 
0.616318 
0.615766 
0.615214 
0.614663 
0.614112 



0.613562 
0.613013 
0.612464 
o.6i 1916 
0.61 1369 
0.610822 
0.610276 
0.609730 
0.609185 
0.608640 



0.608097 

0.607553 
0.607011 

0.606469 
0.605927 
0.605386 
0.604846 
0.604306 
0.603767 
1.603229} 



0.0 
0.0 
0.0 
0.0 
00 
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 
0.0 
0.0 
0.0 



0.0 
0.0 
0.0 
0.0 
0.0 
0.0 
0.0 

0.0 

0.0 

0.0 



00 

0.0 
0.0 
0.0 
0.0 

ao 
00 
0.0 
ao 
0.0 



0.0 
0.0 
0.0 
0.0 
0.0 
0.0 
0.0 
0.0 
0.0 
0.0 



276 
305 

334 
364 

393 
422 

45* 
481 

5«» 

540 

570 



599 
629 

658 

688 

718 

748 

777 
807 

837 
867 

8"97 

9*7 

957 
987 

2017 

2047 
2078 
2108 
2138 
2168 



2199 
2229 
226a 
2 290 
2321 

*35> 
2382 

2412 

2443 

2474 



2504 

*535 
2566 

*597 
2628 

2659 

2690 

2721 

*75* 
2783 



2814 
2845 
2876 
2908 

*939 
2970 
3002 

3033 
3064 
3096 



Tang. I Co -sec. 



0.647912 
0.647365 
0.646819 
0.646274 
0.645729 
0.645185 
o 64464a 
0.644099 

0.643557 
0.643016 
0.642476 



0.641936 
0.641397 
0.640859 
0.640322 
0.639785 
0.639248 
0.638713 
0.638178 
0.637644 
0.637 1 II 



M 



60 

59 
•58 
57 
56 

55 
54 
53 
5* 
5» 
50 



0.636578 



49 
48 

47 
46 

45 
44 
43 
4* 

4» 

40 



39 



0.636046; 38 

o.6355»5 37 
0.634984 36 

0.634454I 35 

0.633925! 34 

0.633396; II 

0.632869 32 

0.632341 • 31 

0.631815I 30 



0.6312891 29 

0.630764 28 

0.630239 27 

0.629715 , 26 

0.629192 25 

0.628670 24 

0.628148 23 

0.627627 22 

0627106 21 

0.626586 20 



0.626067 
0.625548 
0.625030 
0624513 
0.623997 
0.623481 
0.622965 
0.622451 
0.621937 
0.621423 



0.6 209 1 1 
06 203 99 
0.619887 
0.619376 
0.618866 
0.618357 
0.617848 
0.617339 
0.616832 
0.616325 



Secant. 



»9 
18 

17 
16 

»5 

14 

>3 
12 

If 

10 



9 
8 

7 
6 

5 

4 

3 

2 

I 

o 



M 



76 Degrees-. 





KUqEW. 


■ 




T 


Sit"' 




...,„_, „™, 


-- 






I 

1 


• 






S3 






If 


Y 






4 






'J 












17 

m 


■:■ 












:;,■ 
















If* 
3« 


i^''- " 






,„ 






14 


^ ■ 




'-■ 


;; 






i6 


^:t. 




■ 








S 


'ii-- 






■ 


^ 






43 
44 








: 


.',; 




1 ■ ■ 


4" 


l 












t 


S2_ 
I' 


.y. 








': 




■ 


J4 


m! 








.; 




^1 ' 


I'' 


?-^ 






\ 






la 


I '.\ 






\ 




^ 


> 


"(■ , - , L 


.... > „ ,. . ■ , ,.,„ 


1 ... , ■ ..-,r..,"' 


M 


^fl 


■ 


■H^H 


■ 


■ 


1 


■ 



LOGAIUTHMXC SIXES, TANGENTS, AND SBCAKTS, 41 



15 Degrees. 



M 



o 
I 

2 

3 

4 

5 
6 

7 
8 

9 
10 

II 

12 

'3 

«4 

>5 
i6 

"7 
i8 

19 

20 



Sine. ,€o^ine. 



9.41299619.984944 



9.41J467 

9-4U938 
9.414408 

9.414878 



9.9S4910 
9.984876 
9.984S42 
9 984808 



9.4«S547l9-984774 



9.41C81SJ9.984740 9-43>075 



9.410283 
9.41675 1 

9.417217 

9.417684 



9.418150 
9418615 
9.419079 
9419544 

9.420007 
9.420470 



9.420933 5-9^4363 



94*1395 
9.421857 
9.422318 



21J9.422778 

22|9.423*3« 
23:9.423697 

24 9.424156 

25 9-4*4615 

26 9.4*5^73 
a7,9-4*5S30 

28 9-4»5987 

29 94*6443 
30-9 426899 



31 
3* 

33 
34 
3S 

36 

37 
38 

39 

40 



9-4*7354 
9.427809 

9.428263 



9.4287179983770 



4X 

42 

43 

44 

4S 
46 

47 
48 

49 
JO 

5» 
5* 
53 
54 
55 
56 
57 
58 

59 
60 



9-43 '879 9-9835*3 



M 



9.429170 
9.429623 

9430075 
9.430527 

9430978 



9.984706 
9.98467^ 
9.984638 
9.984603 



9.984569 

9.9^4535 
9.984500 

9.984466 

9-98443* 
9-984397 



9.984328 
9.984294 
9.984259 

9.984224 
9.984190 
9.984155 
9.984120 
9.984085 
9.984050 

9984015 
9.983984 

9.983946 

9.983911 



9.983875 
9.983840 
998380.5 



9-983735 
9.983700 

9.983664 

9.983629 

9-983594 



9.431429I9.983558 



9-43*3*9 
9.432778 

9-433**6 



9 983487 
9.983452 

9.983416 



9-433675i9-98338i 

9434112J9.983345 
9.434S61) 99^3309 
9.4350169983*73 
9.43546219-983*38 
9.435908 9.983202 

9.983166 



9436353 
9.436798 

9-437*4* 



9983 » 30 

9.983094 
9.437686 9-983058 

9.438i29;9-983022 
9.438572.9.982986 
9.4390i4Js>.982950 

9-439456 19-9829 '4 
9 439897 ;9-982878 

9.440338 9.982842 9.457496 

^ ■ » ■ ■ ■ . I !■ I I I ■ - I II 

"Co-iine.i Sine. Co>tan}|^. 



Tang. 



9.428052 

9.428557 
9 429062 
9.429566 
9.430070 
9*430573 



9-43 '577 
9.432079 
9.432580 
9.433080 



9.433580 
9.434080 

9-434579 
9.435078 

9-43557^ 
9-4l6073 
9.436570 
9.437067 

9437563 
9.438059 



9.438554 

9 -43 9048 

9-439543 
9.440036 

9.440529 

9.441022 

9.441514 

9.442006 

9.442497 

9.442988 



9.443479 
9-443968 
9.444458 

9-444947 

9-445435 

9-445923 
9.44641 1 

9.446898 

9.447384 

9.447870 



Co-tang. 



0.571948 

0.571443 
10.570935 

0.570434 
0.569930 
10.569427 
0.568925 
o 568423 
0.567921 
0.567420 
0.566920 

0.566420 
0.565920 
o 565421 
0.564922 

0.564424 

0.563927 

10.563430 

10.562933 

0.562437 

0.56I94I 



9.448356 

9.448841 

9.449326 

9-4498 «o 
9.450294 

•450777} 
9.451260 

9-45»743 
9.452225 

9 .452706 1 

9-453*87 
9.453668 
9.454148 
9.454628 

455'07 
9.455586 
9.456064 

945654* 
9.457019 



o. 56 1 446 
o 560952 
0.560457 

0559964 
0.559471 

0-558978 
0.558486 
0.557994 

0.557503 
0.557012 



0.556521 
0.556032 

0.55554* 
0.555053 

0.554565 

0.554077 

0.553589 

0.553102 

0.552616 

0552130 



0.551644 
0.551159 

0.550674 
0.550190 
0.549706 
0.549223 
0.548740 
o 548257 

0.547775 
0.547294 



Secant. 



0.0 1 543 1 
0.015465 
0.015500 
0.015534 
ix>.oi55-68 
0.015603 
0.015637 
0.015^2 
0.015706 
0.015741 



0.546813! 
0.54633*1 
0.54585* 
0-54537* 

0.544893' 
0.544414 

0.543936I 

0.54345* 
0.542981 

0.542504 
TlnJ"" 



0.015056 
0.015090 
0.O15124 
0.015 158 
0.015192 
0015226 
0.015260 
0015294 
0.015328 
0.015362 
0.015397 



0.015776 
0.01 58 10 
0.015845 
0.015880 
00J5915 
p.01^950 
0015985 
0.01^019 
0.016054 
0.016089 



0.016125 
0.016160 
0.016195 
0.016230 
0.0162^5 
0.016300 
0.016336 
0.01637 1 
0.016406 
0.016442 



0.016477 
0.016513 
0.016548 
aoi6s84 
0.016619 
0.016655 
0.01669! 
0.016727 
0.016762 
0.016798 



Co-sec. , M 



0.016834 

0016870 

0.016906 

0.01^942 

0.016978 

0.017014 

0.017050 

0.017086; 

0.017 122; 

0.017 1 58: 

Co^icc. I 



0.587004 
Q.586533 
0.586062 
0.585592 
0.585122 

0.584653 
0.584185 

0.583717 
0.583249 

0.582783 
0.582316 



0.581850 
0.581385 
0.580921 
0.580456 
0.579993 

0.579530 
0,579067 
0.578605 
0.578143 
0.577682 



0.577222 
576762 
0.576303 
0.575844 

0.575385 
0.574927 

0.574470 
0.574013 

0.573557 

0.57^101 



0.572646 
0.572I9I 

0.571737 
0.571283 

0.570830 
0.570377 

0.569925 

0.569473 

0.569022 
0.568571 



0.568I2I 
o 567671 
0.567222 

0.566774 

0.566325 
0.565878 
0.565431 

0.564984 
0,564538 
0.564091 



0.563647 

0.563202 
0.562758 
0.562314 
0.561871 
0.561428 
0.560986 
0.560544 
0.560103 
0.559662 



74 Degrees^' 






60 

59 
58 
57 
56 
55 
54 
53 
5* 
5' 
50 



49 
48 

47 
46 

45 
44 

43 

43 

4" 

40 



3^ 
38 
37 
36 
35 
34 
33 
3* 
31 
30 



*9 
28 

27 
26 

*S 
*4 

*3 
22 
21 

20 






'9 
18 

17 
16 

"5 

>4 

13 
12 

II 

10 



9 
8 

7 
6 

5 

4 
3 

2 

f 

o 



•3 UWABrrilMIC SCtt% Ti 






" 




"] 


5: 




i 


V : , 


54 








5J 




8 




f 




9 ■> .. 


S' 

J2. 




T'}>; 


49 
4» 




a;: 

l8;l-i, 


4* 
4f 




I9'" !■ 






»4 u t 


14 
1) 




i« '. ^ 


11 




199-^ ■-■: 


]■ 






JO 






«9 




j*94i*ii)4998iM» 


9.4;ii3i,IOi>-.i 






3J 9.4S4* '99.981615 


9.47199s '05: ; 






34 94SS044 9-981487 


9.47J4O7,i0.5!f.;, 






J(».*5S469 9.9S'i49 


9.473919 io.5:f.. - 


»1 




j6 94SSK9J9 98«(" 


9.«T4J8i (O.S = ^l 


«4 




)7 9.4i6j'6 99»''tl4 


9.47484I,'0■S^^■^ 1 


>J 




jB 9.456739 !'98>436 


9-47JJOJ 'o.i^r ■ 






J9 9.45"ti!l.9»'J99 


9.475;6j .o.i..:- 






4P94P(S*9.98'>«' 


9 47t»i lo.iJ--- 






41 94i8«)&9.98ijti;9.4'66(;j lajii, , 


"^ 




41 9^(8417 s-sKoXilg^rTm 'OS!;' 






43 9-4sKB^Xo,.,SiM? y-1-^:'"^^ ■'-• ii- 






4494. 'i-^^ .,..":■-.., ., -:■■■ 


<6 




46 9 4^-- -....■.., 


'4 




4K vt''-'. .'■ ■■ .i''->' . i ■ ■' ■-! 


IJ 




49 9.40iSf<4 ijySicjiy 9.45<.-.;45 '^-■'">< 






io 9.461'*! 9.9809I1 9 4S0SI1. '.»(..,... 


to 




«> 9,461199 9.9*09^19 481*57, i&.iis- - 


2 




ii 9461^16 9.98090J 9.4817 rj' !.:■,>-. ■-, 
« 9-46j'3»9 9SoS66 9.4«iir^ 








(49.4614489.98082754X1.1 






}( 9-4618649.980789 9 4*! 






i6 9.454311, ^.ijKoi 50 9.4R.; , 






r 9.4r4inji uoKOT.j o.jlj.. 






i8 9..,u,, 


" 




fioiJr '^ 












H !.:■ 







tOGABITHMIC SIKE3, tAKGBNTS, Am 8ECANTa 43 

17l)«ffree5. 



M 

"cT 
I 

2 
I 

4 

5 

6 

7 
9 

lO 

II 

li 

H 

1$ 

i6 

17 
iS 

20 

21 
22 

is 

»4 

26 

a? 
;S 

2^ 
30 



Sine. 



^-47.6536 9 '979579*9-49^^57 



31 
3> 

13 
34 

i^ 

37 
38 
39 

il 

41 
4> 
43 
44 
45 
46 

47 
4? 
49 
50 

51 

53 
54 
55 

56 
57 
58 
59 
60 



9-465935 
9,466348 
9.466^61 



Co-sinc i Tan g. > Cortan y. 
9.980596 ;9,4iS339 



9 9805S8;9.48579» 
9 980519 9.4S6242' 
9.467 1 7 jj9 9804S0. 9.486693 
9467585 9.980442 9.487143 
9.4679^ 9-9^0403 9.487593 
9.46^407 9-980364I9.438043 
9[.46&8i7 9'980325'9.438492 
9,469227 9 980286 9.488941 j 

9469637 9.980247 9.489390; 
9.470046 9.980208:9.4898381 

9.980 1 69 '9 490286 
9,980 130! 9.490733 
9.980091:9.4911801 
9 980052.9.49 1 627^ 



9470455 
9470863 
9 47>a7i 

9-471679 
9.4720816 
9.472492 
9.47289^ 

9473304 
9-4737 '0 
9.4741 15 



9 4745 » 9 
9 4749*3 
9-475327 
9-475730 
9-476133 



9.98001219.49*073; 
9-979973l9 49*5>9' 

9-979934;9-49*965| 
9-979895;9-4934»o, 
9.979855:9493854; 
9 9798i6 i ^.494299 

9979776,9494743 



947^938 

9.477J40 

9-47774« 
9478141 

9478542 

947^94* 

947934* 

9 47974* 
9.4801^0 

9.4X0539 
9.480937 



9481731 
9.482128 



9.482525 
9.48*921 
9.-^^83316 
9483712 
9.484107 
9.484501 
9.484895 
9.485289 
9485i&82 
9.48607 5 



9-979737 

9979697 

9.979658 
9979648 



9.495186 
9.49^630 

9.496073 
9.496515 



9^979.539 9 497399} 
9 979499»9-49784»i 

9 979459 9-498»8a} 
9.979420 9498722} 

9-499 > 63; 



9.9793.80 

9 979340 
9-979300 
9.979260 
9.979220 
9-979»8o 



9.4996OJ 
9 5OQO42 
9 50O481 
9.5009^ 
9501359 



, _. 9-979»40[9-50i797 
948i334|9.979i00j9 502x35 

9.979059.9-50*672 



9.9790 i9;9 50H09 ; 
9 978979I9 503546: 



0.5146^1 
o« 5 1 4269 
0511758 

0.513307 
0.512857 

0.512407 
0.51 1957 
o 511508 
0.5II059 

0.5 106 10 

0.510162 

O.5097I4 

lte.509267 
o 508820 

o 508373 
o 5079*7 

0.507481 
0.507035 

0.506590 
0.506146 
0505701 

0.505257 

0.504814 

0.504370 
05039*7 
0.503485 
0^503043 

o 502601 
a502i59 
o 501718 
0.501278 

o 500837 

0.500397 

0499958 
0.499519 

0.499080 

0.498641 
0.498203 

0.497765 
0.4973*8 

Q. 4968 9 1 



9978939 
9.978898 

9.978858 
997&817 
9-978777 
9-978737 
9.978696 
9.978^55 
9.978615 



9 486467 
9 486860 



9978574 
9978533 



9487*5»i9-978493 
9487643:9978452 
9.488034.9.97841 1 



9.48SAZ4 
9488814 
9489204 

9489593 
9 489982 



CO'Aine 

mmmmmmmm 



9 50398* 

9. 5044^8 : 

9504*54; 
9505289 

9 5057*4: 
9.506159 

9.506593' 
9.507027. 
9.50 7460 . 

9 507893 
9.508326, 

9-5^8759. 

9»S09J9^ 
9.509622; 

9.510054 



9.978288 
9 978247 



'Mne. 



9.978370 _ 
9.9783299 510485 



9.51O916 
9.5"346 



9.978206 9. 5 1 1''76 



0.496454 
0496018 
0.495582 

0.495*46 
0.4947 1 1 
0.494*76 
0.493841 
0.493407 
49*973 
o 49*540 



0.492107 

0.491674 
0.491241 
0490809 
0.490378 
0.489946 
0.489 CI 5 
LO.489084 
0.4^654 
0.488224 

Co-UnR.'T'ang. 



Secant. 



019404 

o 1 9442 

.019481 

019520 

.019558 

019597 
.019636 

•019675 
.019714 

-019753 
.019792 



Uo-sec. 



0.01983 1 
0.O19870 
0.0199O9 
0.019948 
0.019988 
0.020027 
0.020066 
0.020105 
0.020145 
0.020184 



0.020224 
0.020263 
0.620303 
p. 020342 
0.020382 
0.02042 1 
0-620461 
0.020501 
0.020541 
0.020580 



0.020620 
0. 020660 
0.020700 
0.020740 
0.020780 
0.0208 20 j 
o.oao86o' 
0.0209061 
0.020941 j 
0.02098 1 1 



0.021021 
0.C21061 
0. 02 1 102 

0.021 142I 

0.021183 

0.021223' 

0.021263! 

0.021304 

0.021345; 

O.O21385I 



0.021426 

0.021467' 

0.021507' 

0.021548 

0.021589' 

0.021630 

0.02167 1 

0.021712 

0.021753 

0.621794 



o. 
o 
o. 
o. 
o. 
o. 
o. 
o. 

0. 

o. 
o. 



o. 
o 
o 
o. 

o. 
o 

0. 

o. 
o. 
o. 



o. 
o. 
o. 
o. 
o. 

0. 

o 
o. 
a 
o. 



0. 

o. 
o. 
o. 

0. 


o. 

0. 

o. 

o. 



o. 
o. 
o. 

0. 

o. 
o. 
o. 
o. 
o. 
o. 



o. 

o. 
o 

0. 

o. 
o 

o. 
o. 

0. 
0. 



34065 

3365* 
33239 

3*827 

3*4«5 
32004 

3»593 
31183 

30773 

30363 
29954 

*9545 
*9>37 
28739 
28321. 
27914 
27508. 

27102J 
26696 
26290, 

15?£5, 
25481 

25077* 

*4673- 
24270 
23867 

23464. 
23062 
22660 . 
22259 
218^ 

17458 
21058 
2065 8 j 
20259' 
19860^ 
19461! 
1*063; 
18666; 
18269 I 
17872) 






«7475 

17079 
16684 

16288 

'5893 
'5499 
1 5 105 

147 IX 
14318 
13925 



»3533 
'3*40 

1*749 
1*357 
119661 

"576; 

11186 

10796 

10407; 
10018' 



T^ Ulegrees. 



Co-sfc 



Secant 



60 

si 

58 
57 
56 
55 

54 
51 
5* 
5' 

tl 

47 
46 
45 
44 
43 
42 

4*1 

40 

]l 

37 
36 

35 
34 
53 
3* 
3' 
30 

29 
28 

*7 
26 

*5 
*4 
*3 
22 

21 

20 t 

18 

17 
16 

«5^ 

«4 

'3 
12. 

II 

10 

8 

7 
6 

5 

4; 

3' 

2: 

I 

o 



m! 






'/ 



im 



W, *AxoiWTs. \im« 



~ — ~ — ' - -- ■-- ■ - ■ 


- -. . ,. — 1 ,, 


T^r 




■" !*■;'". 






u 
u 


17- 


S.4- 


V' 




4 4v. 




i2. 


q^-.. 


. ._ !. ,■, _^ 




9..|..-- 


:* 


It 


^..i.,.. 


J4 




if 


Jl 


'4 








,.iO,,,.. 






j7" 


^SO.Sj* 


9.9-1..,. , . . I . 


^') 




S.JOltJI 


»,97^-.' . ■! 






SiOifiOT 








9-S'»9>4 


9.97'->-;. 




Si 


9.JOJJ60 


99:"' . 




36 


? sojTJi 


9.9- f. 




r 


9.S04110 


9.9- r,' ■ 




i* 






19 




JO_ 












*' 


9 !■:■• 




41 






■1) 


9)' 




44 


9. (.:■'. . 


!' 


46 


9.sc.-.r . 

9.5:,-! . 


;j 


48 


9. !■■■■■■ 
9-i-J; 






<|,;ioJ. . . . 




i6 


9.i..i4C.4.L,;,>.L.o,, ;;, . 


■ 1 


(S 


9i"iP7 *y^i?i' 'JSi'-i^ 




IV 


9.S'li7i t-^7V» 9 !ifii' ■ 




60 


9 !iilS4iii.<)Tch70 9V!''.r- 




u ' (Jn-ainr. Sine Co-twiir ...... 





£0'GA.rtlTHMrC Sliims, TANGENTS, AIH) SECANTS. 4S 









19 De^reies. 








O 


Sine 


Co-sine. 


Tanpr. 
9.536972 


Co-tanj^ 


Secant. 1 Co-sec. | m 


9. $12642 


9.975670 


10.463028 


10.024330 ro.4873.:8 


15" 


• 1 


9; 5 13009 


9.975627 


9-537382 


10 462618I 10024373 10.486991 


59 


2 


9.5^3375 


9.975583 


9-537792 


10.462208J 10.024417 10.486625 


58 


3 


9S»374i 


'9.975539 


9.538202 


ro.46 1 798J 10 024461 ; 10.486259 


57 


4 


9.514107 9-975496 


9.538611. 


10.4613891 10.0245041 10.485893 


56 


5 


9-5«4472 


9.975452 


9.539020 


10.460^80! iao24548| 10.485528^ 


55 


6 


9.5 '4837 


9.975408- 


9-539+29 


JO.46057 1 ; 100245.92 10.485163 


54 


7 


9.515202 


9-975365 


9 539837 


10 460163. 10.024635' 10.484798 


53 


8 


9.515566 


9-97532i;9-540245 


•04597 55| 10.024679 10.484434 


52 


9 


9-5«5930 


9.975277 9540653 


10.459347:10.024723 10.484070 


51 


lo 
II 


9.516294 


9-975253 9 54'06i 


'0-45^939 


10024767 10.483706 


50 . 
49 


9-516657 


9-975*89 


9.54T468 


10,458532 


10.02481 1. 10.4J83343 


12 


9.517020 


9-975145 


9 541875 


10 458125 


iao24855 10.482980 


48 


"3 


9.517382 


9.975*01 


9.542281 


10.457719 


10.024899110.482618 


47 


'4 


9-5»7745 


9975057 


9.542688 


10.457312 10.024943110.482255 
10.456906 10.024987I 10.481893 


t 46 


M 


9.518107 


9.975013 


9-543094 


45 


i6 


9.519468 


9.974969 


9-543499 


ro.456501 


10.025031 10.481532 


44 


17 


9.518829 


9.974925 


9-543905 


10.456095 


10.025075 


10-484171 


.43 


l8 


9.519190 


9.974880 


9.544310 


10.455690 


10025120 


10.480810 


42 


'9 


9.51955' 


9.974836 


9-544715 


10.455285 


10.025164 


10 480449 


41 


20 
21 


9.519911 


9.974792 
9-974748 


9.545119 
9-545524 


10.454881 
10.454476 


10.025208 
10.025252 


10.480089 


40 
39 


9.520271 


10.479729 


22 


9.520631 


9.974703 


9545928 


10.454072 


10.025297 10.479369 


38 


.^3 


9.520990 


9.974659 


9-546331 


10.453669 


10.025341 10.479010 


37 


*4 


9.521349 


9.974614 


9-546735 


10.453265 


10.025386 


10.478651 


36 


«5 


9.521707 


9-974570 


9.547138 


10.452S62 


10.025430 


io.478293 


35 


26 


9.522066 


9-974525 


9-547540 


10.452460 


10.025475 


10.477934 


34 


27 


9.522414 


9.974461 


9-547943 


10.453057 


10.025519, 


10.477576 


33 


28 


9.522781 


9.974436 


9.548345 


10451655 


10025564 


10.477219 


32 


29 


9.523138 


9-97439* 


9-548747 


10.451253 


10.025609 


10476862 


31 


30 
31 


9523495 
9 523852 


9.974347 


9.549149 


10.450851 


10.025653 


10.476565 
10.476148 


30 
29 


9974302 


9.549550 


10.450450 


10.025698 


32 


9.524208 


9-974257 


9.549951 


10.450049 


10.025743 


10.475792 


28 


33 


9.5^4564 


9.974212 


9.55035a- 


10.449648 


10.025788 


10.475436 


27 


54 


9.524920 


9.974»67 


9.550752 


10.449248 


10.025833 


ic.4750«o 


26 


35 


9.525275 


9^974 '22 


9.5511^2 


10.448848 


10.025878 


10.474725 


25 


36 


9.525630 


9-974077 


9-551552 


10.448448 


10.025923 


10.474370 


24 


37 


9.525984 


9.9740 ?2 


9.551952 


i 0.448048 


10.025968 


10.474016 


23 


38 


9.526339 


9.973987 


9-552351 


10.447649 


10.026013 


10.473661 


22 


39 


9.526693 


9-973942 


9.552750 


10.447250 


10.026058 


10.473307 


21 


40 
41 


9.527046 
9.527400 


9973897 


9.553149 


10.446851 


10.026103 
10.026148 


10.472954 


20 

19k 


9.973852 


9.553548 


10.446452 


10.472600 


42 


9-5»7753 


9.973807 


9 553946 


10.446054 


10.0^6193 


10.472247 


18 


43 


9.52810^ 


9.973761 


9-554344 


10.445656 


10.026239 


10.471895 


17 


44 


9.528458 


9.973716 


9-554741 


10.445259 


10.026284 


10.471542 


16 


45 


9.528810 


9.973671 


9-555«39 


10.444861 


iO.026329 


10.47 11 90 


'5 


46 


9.529161 


9.973625 


9555536 


10.444464 


10.026375 


10.470839 


14 


47 


9-5295 »3 


9973580 


9-555933 


10.444067 


10.026420 


10.470487 


13 


48 


9.529864 


9.973535 


9-556329 


10-443671 


10.026465 


10.470136 


12 


49 


9.530215 


9.973489 


9.556725 


10.443275 


10.026511 


10.46978^ 


II 


50 
51 


9530565 


9-973444 
9 973398 


9.557121 


10.442879 


10.026556 


10.469435 


10 


9-530915 


9-557517 


10.442483 


10.026602 


10.469085 


9 


5* 


9.531265 


9.973352 


9.557913 


I0.4420ii7 


10.026648 


10.468735 


8 


53 


9.531614 


9973307 


9.558308 


10.441692 


10.026693 


10.468386 


7 


54 


9-531963 


9.973261 


9.558702 


10.4.^1298 


10.026739 


10.468037 


6 


55 


9.532312 


9-973215 


9,559097 


10.440903 


10026785 


10.467688 


s : 


56 


9.532661 


9.973169 


9-55949' 


1 0.440 5O9 


10.026831 


10.467339 


4 


>7 


9.533009 


9-973124 


9.559885 


10.440115 


10.026876 


10.466991 


3 


58 


9-533357 


9.973078 


9.560279 


10.439721 


10.026922 


10.466643 


2 


S9 


9533704 


9.973032 


9.560673 


10.439327 


10.026968 


10.466296 


I 


f>0 


9.534052 


9.972986 


9.561066 


10.438934 


10.027014 


10.465948 




K i 


Co-si nc 


Sine. Co-tang'.' Tang. ' Co-sec. * Secant. | 



70 Decrees. 



46 L0i:jimTinncSM*.TAWGtTr3,AimsBMS 

20 l>esr-F*. 



Smc , Co*ine Tapg 
9tJ4JM|9-9T>MO »)'"*■ 



L,l.7ll.)« 


;,->"-7l>M 


9.971151 


9.S6»39S 




j.ifi»4K6 


,.97IOill 


fl-SftSHTI 




9-i6«'.'.i 


9.9719^1 


9.i69^/ 


9.971917 


9.(7Co- 


9.97 1I70 


fl-i'-'4- 



,540149 

.i^osjo 

5*09) 

9-S4r9iJ 
9!4J»<»3 

9.S4»6j.' 
J41971 
9S43JK 
r-S*]649 
l-i419<T 
9-i441» 5 
9.)44A6j 
9.J45O00 
<4i3J« 
9.54(*74 
I). 54601 1 
¥-546 J4' 
9.(46683 
9.(4TOIia 



9 Hv^i' h'9r-JSIi' 9 
9.5493*0 9>»T»St4 9 
<;-(4969j 9.9?38»7 9 



9-(S»!49 



H7a490 
9.97e44» 
9-970394I 



9.55JOIO 9.9»oj45j 



9n3J4* 

19.551670 
J9.5I4W 
i 9-<)4H'9 



,fTO»97 
-9"0*49i 
g.97M00 

Sine." 



9.581907 11 

|9.5«iiS6 11 

i9.]g26e5 II 

9-58J04I 'I 

9.;B]4ii II 

hsSjioo II 

|9_£*^ .J 

Co-Uiig 



iri 



tOGAHrniMIC SINES, TANGENTS, AND 8SCAKT8. 4*7 



21 Degrees* 



o 
I 

% 

3 

4 

5 
6 

9 

o 



3 
4 

5 

6 

7 
3 

9 
TO 

21 

a2 

«3 

»4 

as 
26 

*7 

28 

*9 
i2 



Sine. 



9 

9» 

9- 

9- 

9- 

9- 

9- 

9* 

9- 

9- 

9- 



9- 

9- 

9 

9- 

9- 

9 

9- 

9 

9- 

9- 



3*j 
33| 
34 

as 

36 

37: 

39 

-40. 

4« 
4* 

43 

44j 

47' 
-48. 

'49! 



9- 

9- 

9- 

9. 

9- 

9- 

9* 

9. 

9 

9- 



5 

531 

54' 

5$: 
56: 

57 

58 

59 
60 



543 »9 9-97015* 
54658 9.970*03 

54987 9-9700$ 1 
553* $[9.97000* 
9.969957 



$5643 
5597' 
56299 
56626 9.969811 



5^953 

57280 
57636 



5793* 
58x58 

$8583 
58909 

59234 
59558 
59883 
60207 

60531 
60^5$ 



561501 
61824 
62146 
62468 
62790 
63112 

63433 

63755 
64075 



Co-Bin(^. 



9.969909 9,586062 
9.969860(9. 586439 
9.586815 
9969762 9.587190 
9.9697149.587566, 
9.969665 9.587941 



9.969616 
9.969567 
9.969518 
9.969469 
9.969420 
9.969370 
9.969321 
9.96927a 
9.969223 
9969173 



61178 9.969124 



9.969075 
9 969025 
9.968976 
9.968926 
9.968877 
9.968827 
9.96S777 
9.968728 
9.968678 

9.968628 



Tang. 



9.584177 

9-584555 
9.58493 » 

9-585309 
9 585686 



9-588316 
9.588691 
9.589066 
9.589440 
9.589814 
9.590188 
9.590561 

9-590935 
9.591308 

9.591681 



Co-tang. 

io»4i5823 
10.415445 
10.415068 
10414691 
10.414314 
10.413938 
10.41 3 561 
10.4131^5 
10.412810 
10.412434 
10.412059 



10.411684 
10.411309 
10.410934 
10.410560 
10.410186 
10.4^09812 
10.409438 
10.409065 
10.408692 
10.408319 



9.592054 
9.592426 
9.592798 

9-59317 i 
9.593542 

9 5939«4 
9.594285 

9.594656 
9.595027 
9-59539? 
9.595768 
9. 5^6138 
9 596508 
9.596878 

9.597*47 
9.597616 

9.597985 



64396,,, 

6471619.968578 

65036 9.968528 

65356)9-968479 
65676!9.968429 

65995!9-968379 
663i4'9.968329 

66632-9.968278 

566951 '9.96822819.598722 

6726919.968 1 78, 9.599091 

6758719.968128 9.599459 
67904^9 968078 9.599827 
68222,9.968027 9.600194 

68539:9-967977 9600562 
68856 9.967927 9.600929 
69172 9.967876,9.601096 
69488 9 96782619.601662 
69804 9 96777 5 J9. 602029 
70120 9.967725 19.602395 
70435 9-967674'9.6o276i 



10.407946 

10.407574 
10.407202 
10.406829 
100^06458 
10.406086 
10.405715 
10.405344 

10.404973 
10.404602 



10404232 

1 10.403862 

10.403492 

10.403122 

10.402753 
1OL402384 
10.402015 
9.598354I 10.401646 
10.401278 
10^00909 



70751^9.967624 9.603127 



9- 
9. 
9- 
9. 
9- 
9- 
9- 
9- 
9- 

9- 

Co Bine. Sine. 



71066 9.967573 
7i38o;9.96752z 

7i695'9.96747i 
71009 9.967421 
7*3*3 9967370 
72636 9.9<>73t9 
72950 9.967268 
73263*6.967217 
73575 9.967«66 



9.603493 
9.603858 
9.604223 
9.604588 
9 604953 
9.605317 
9.605682 
9.606046 
9.60641c 



10.400541 
10.400173 
10.399806 

10.399438 
10.39907 1 
10.398704 
10.398338 

<o.39797i 
10.397605 

10.397*39 



Secant. 



10.029848 
10.029^97 
10.02994$ 
10.029994 
io.030043 
100)30091 
10.030140 
10.030189 
10.030298 
10 030286 
10.030335 



1 0.0303 84 
10.030433 
i 0.03048 2 
10.030531 
10.030580 
i 0.030630 
10.030679 
iao30728 
10.030777 
10030827 



10.030876 
10.030925 
10.030975 
10.031024 
10.031074 
iao3xi23 
10.031173 
10.031^23 
10.031272 
10.031322 



10.031372 
10.03 1422 
10.031472 
10.031521 
10.03 1 57 1 
10.03 162 1 
10.031671 
10.031722 
10.031772 



Co-sec. 



1 0.44 5 67 1 
10.4^$ 342 
10.445013 
10.444685 
10.444357 
10.444029 
10.443701 

10.443374 
10.443047 
10- 4427 20 

io.4423941 



10.442068 
10.441742 
10.441417 
10.441091 
10.440766 
10.440442 
10.440117 
10.439793 
10.439469 
10.439145 



ro.438822 
10.438499 
ia438i76 

10.437854 

"0-43753* 
10.437210 

iO.436888 

ia4365e7 

10.436245 

10.4359* 5 



10.435604 
10.435284 

10.434964 
10.434644 

10.4343*4 
10.434005 

10.433686 

10.433368 

10.413049 



10031822*10.432731 



10.03 1872110.432413 
1C.03 1922 10.432096 
I0.03i973lia43i778 
10.03202;! ia43 1 46 1 
10.032073 1 '0.431144 
10.0321241 10.430828 
10.032174' 10.4^0512 
ro.032225! 10.430196 
10.032275110.429880 
10.032326110.42956^ 



Co-tanp 



10.396873 
10.396507 
10.396142 

«o.395?77 
10.395412 

10.395047 
10.394683 
10.394318 

10393954 
10.393590 



Tanj-. 



10.032376: 10.429249 
10.032427 1 10.428934 
10.032478! 10./128620 
10.032529; 10.428305 
1O.032579 10.427991 
10.032630' 10.427677 
1003268 1 1 10.427364 
10.032732110.427050 

10.032783 '; 10.42673; 
10.0328 h!>0.426425 



60 

59 
58 
$7 
56 

55 

54 
53 
5* 
5' 



49 

4« 

46 

45 
44 
43 
4* 
41 
40 

39 
38 

37 

36 
35 
34 
IZ 
3* 
31 
30 J 



29 

28 

*7 I 

26 

*5 

24 

*3 
22 
21 

20 



«9 
18 

17 
16 

15 

14 

»3 
12 

II 

10 






9 

8 

7 
6 

5 

4 

3 

2 

1 

o 



Co -sec. ' becanl • m 



68 ^Idegrcca* 



4S LOGARlTHMie SIN£S, TAXSKXTS, AND S£eANT8. 

. 22 Def^ees. 



M Sine. 



o 
I 

2 

3 

4 

5 
6 

7 
8 

9 

10 



ji 

12 

'3 

'4 

'5 
i6 

17 
j8 

»9 

20 



9-573575 
9.573888 

9.574200 

9S74?»2 

9.574824 

9S75'36 

9 575447 

9.57575iJ 
9.576069 

9576379 
9.576689 



9-576999 

9577309 
^.577618 

9-577927 
9.578236 

9578545 
9578853 

9.579162 
9.579470 
9-57977 7 
9.580085 
9.580392 
9.580699 
9.581005 
9.581312 
9 581618 
9.581924 
9.582229 

9-58^535 
9.582840 



21 
22 

23 

24 

^5 
26 

27 
28 

29 

JO __ 

.^i 9-583i45 

32 9-5^3449 

33 9583754, 

34 9 584058 9 965406 

3.5 9 58436if9.965353 
36|9 584665j9-9^530i 
37,9 584968 9965248 
38,9585272 9-965195 
;9:9-585574,9-965i43 



Co-sine. Tang. Co-tang 

9 967166 9.606410 
9.967115 9.606773 



9.967064 
9.967013 
9.966961 
9.966910 
9.966859 
9.966808 
9.966756 
9.966705 
9.966653 



9.966602 
9.966550 

9 966499 
9.966447 

9 966395 
9.966544 

9.966292 

9.966240 

9966188 

9 966136 



9.966085 
9.966033 
9965981 
9.965929 
9.965876 
9.965824 



9.607137 
9 607500 
9.607863 
9.608225 
9.608588 
9.608950 
9 609312 
9.609674 
9.610036 

9610397 
9.610759 
9.61 1 120 
9.61 1480 
9.61 1841 
9.612201 
9-612561 
9.612921 
9.613281 
9.613641 



9.614000 

9614359 
9.614718 

9615077 

9615435 
9615793 



9.965772 9.616151 
9.965720 9616509 
9.965668 9616867 



9.965 6^5 

9.965563 
9.965511 

9965458 



9.617224 



10.393590 
10,393227 
40.3928^3 
10.392500 
10.392137 
10.391775 
10.391412 
10 391050 
10.390688 
10.390326 
10.389964 

10.389603 
10.389241 
io.3888"8o 
10.3X8520 
10.388159 
10.387799 
10.387439 
10.387079 
10.386719 
fo.386359 



10.386COO 
10 385641 
10.385282 
10.384923 
10.384565 
10.384207 
10.383849 
10.383491 

10.383133 
10.382776 



Secant. 



10.032834 
10.032885 
iO.032936 
10.032987 
10.033039 
10 03 ^090 
IO.P3314I 
10.033192 
10.033244 
iao33295 
10.053347 



1^033398 
10.033450 
10.033 50J 
10033553 
10.033605 
10.033656 
10.033708 
10.033760 
10033812 
10.033864 

10.033915 
10.033967 
10.034019 
10.034071 
10.034124 
10.034176 
10.034228 
J 0.034280 
J0.034332 
10.034385 



9.617582 10.382418! 10.034437 
9.617939 10.382061^10.034489 
9.618295 10.381705 10.034542 
9.618652I 10 381348 10 034594 
9.619008! 10.380992110.034647 
9.619364J 10.380636110.034699 
9.619721. 10.380279I 10.034752 
9.620076 10.379924:10.034805 
9.620432; 10.379568110.034857 
40.9 585877J9.965090 9.620787 10.379213110.034910 



Co -sec. 



42(9 $8648219-964984 
4.?'9-586783j9-96493i 

44'9.587085!9.964879 
45'9-587386!9.964826 

4619.587688, 9964773 



4i|9vc86i79;9.965037 9.621142 10.378858I 10.034963 



9.621497 1 10.3 78 503 1 10.03 50 16 
9.621852 10.378 148 10.035069 

9.622207 10.377793 ; 10.035121 

9.6225611 10.377439' 10.035174 

9.62291 5 1 10.377085! 10.035227 



47 J9. 587989, 9.964720 19. 623269: 10.37673 J '10.035280I 
48*9.588289 9.964666 9.623623 ' 10.376377 10 035334I 



49:9.588590 9.964613 
50I9. 588890 9.964560 

9.589190,9964507 

9 589489,9-964454 
9.589789 9.964400 

9.59C0S8 9-964347 
9.590387'9.964294 
9.590686 9.964240 
9.590984-9.964187 

9 59»282 9-964^3 



51 
52 

53 
54 
55 
56 
57 
58 
59 
60 9 591878 9 964026 



9623976 10.376024 10.035387 
9.624330 10.375670 10035440 

9.624683 10 375317J10.035493 

9.625036! 10.374964,10.035546 

9.625388 10.37461 2 jio.03 5 600 

10.374259 10035653 

10.373907110035706 
10.373 55 5 •«o-03 5760 
10.373203.10.035813 



9.625741 
9 626093 
9.626445 
9.626797 
9.627149 



9.591580 9 964080I9.627501 

9.627852 



M 



(Josiiie. ' i^ine. 



Cot:tn«»' 



10.372851-10.035867 
10.372499 10.035920 
10-372 1 48 , 10.0 35974 

Tiiv.ir I Co-sec 



.426425 
.426112 
.425800 
.425488 
0.425176 
0.424864 
0.414553 
0*424242 

0-423931 
0.423621 
0.423311 



0.423001 
0.422691 
0.42238a 
0.422073 
0.421764 
0.421455 
0.421 147 
0.420838 
0.420530 
0.420223 



59 
58 
57 
56 
55 
54 
53 
5» 
5« 
50 



0.419915 
0.419608 

0.419^1 

0.418995 

0.418688 

0.418382 

0.418076 

0.417771 

0.417465 

0.417160 



49 
48 

47 
46 

45 

44 

43 
42 

4« 

40 



39 
38 
37 
36 
35 
34 
33 
32 
31 
30 



0.416855 
0.416551 
0.416246 
0.41594a 
0.415639 

0.415335 

0.415032' 

0.414728! 
0.4144261 
0^414123; 

0.413821 > 

O.413518I 

0.413117 

0.410915 

0.412614^ 

o.4ia3iaj 

0.412011! 

0.411711 

0.41 1 4 ID 

0.411110 



0.4108101 
0.44051 1 1 
a4ioaiij 
0.40991 al 
0.40961 31 
0.409314, 
0.4090 I 6| 
0.408718; 



0.408420 

0.408 I 22i 

Secant. I 



29 
28 
27 

26 

25 
24 

»3 

21 
20 

»9 

18 

17 
16 

>5 
14 
13 
12 

IX 

10 

9 
. 8 

7 
6 

5 

4 
3 

. a , 
I 
o 

M 



67 Degrcest 



fM»l4BITHMlp 9imS» tANOSNTS, AND SECAHTS. 49 



23 Degrees. 






o 
1 

a. 

3 

4 

5 
6 

; 

8 



line 



9.592176 

9-59^473 

9«S9»770 



9.593067 9.963SI I 



9-595«37 
9.59543a 
9.595727 
9.596021 

9,596315 
9.596609 

9.596903 



9.59S953 
9.599244 

9-59953^ 
9.599S27 

9.600118 



9' $93363 

593659 

9-593955 

9 '594*5' 

9-594547 
9.594842 



Co^ 



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9.964026 

9.96397a 
9.963919 
9.963865 



9-963434 
9.963379 

9-9633»5 
9.96327 1 

9.963217 

9.963163 

9.963108 



9.5971969.963054 
9.5974909-96*999 



9-597783 

9. 59807 J 
9.598368 



9.962945 



9.962890 

9.962836 

9.598660)9.962781 



9.962727 
9*962672 
9.9626 1 7 
9.962562 
9.962508 



9.600409 9.962453 
9.600700 9.962398 



9 

10 

ti 

12 

13 
14 

»5 
16 

17 
18 

*9 
^o 

21 

12 

23 

24 

»5 

26 

27 
28 

»9 

J2 
3« 
3* 
33 
34 
35 
36 

37 
38 
39 

41 9.603882 
42(9.604170 

43 
44 
45 
46 
47 
48 
49 

J2 

51 9.606751 9-96**iS 

529.607036,9.961179 

53 

^4 

5$ 

56, 

57 9.60846119.960899 

5« 
$9 
60 



9.600990 
9*601280 
9.601570 
9.601860 
9.602150 
9.602439 
9.602728 
9.603017 
9 603305 
9.603594 



9-604457 

9.604745 
9*605032 

9 605319 
9.605606 

9.605892 

9*606179 



9-963757 
9.963704 

9.963650 

9.943596 

9.963542 

9.963488 



9.962343 
9.962288 
9.962233 
9.962178 
9.962123 
9.962067 
9.962012 

9.961957 
9 961902 

9.96 1 846 



9.961791 

9.961735 
9.961680 

9.961624 

9.961569 

9.961513 

9.961458 

9,961402 

9.961346 



9.606465I9 961290 



9.6073** 9-96 »»*3 

9.60760719.961067 

9.6o7892',9*96iOii 
9.6o8i77l9'960955 



M 



9.608745,9.960843 
9,609029^9.960786 

9.60»3.i3j9-960730 

Co-fiine. 1 iSint. 



9.631704 
9.632053 
9632401 

963*750 
9.633098 

9-633447 
9.633795 
9.634143 

9-^34490 
9.634838 



9.635185 
9 6JS53* 

9.635879 
9.636226 

9.636572 

9,636919 

9.637265 

9^37611 

9 637956 
9.638302 



9.638647 
9.638992 

9-639337 
9.639682 
9.640027 

9.64037 J 
9.640716 

9.641060 
9.641404 
9.641747 



9.642091 

9-64H34 
9.642777 
9.643120 

9.643463 
9.643800 
9.644148 
9.644490 
9.644832 
9645174 



9645516 
9.645857 
9.646199 
9.646540 
9.646881 
9.647222 
9.647562 
9.647903 
9.648243 
9.648 583 

Cotaoff 



10.371797 
10.37*446 



10.036028 
10.036081 



10.371095 10.036135 



Tang I Co^Ung. , Stfcafit. 

9.627852110.37*148 10.035974 

9.628203 

9.628554 

9 628905 

9,629255 

9.629606 

9.629956 
9.630306 
9.630656 
9.631005 

9-631355 



10.370745 

10.370394 
10.370044 

10.369694 



Go-see. 



10. 036189 
10.036243 
10.036296 
10.036350 



10.369344' 10.036404 
10.368995 f 0.036458 
10.368645 10.036512 

10,368296110 036566 

10.367947 ; 10.036621 
10.367599110.036675 
10.367250.10.036729 

10.366902 i0%036783 
10.366553 j 10.036837 
10.366205110.036892 
10.365857 j 10.036946 
10.365510 10.037001 
10.365162 10.037055 



to.364815 
10.364468 



10.037110 
10.037164 



10 364121 10.037219 

«o.363774'io.037273 
10.363428110.037328 
10.363081 j 10.037383 
10.362735110.037438 



10.362389 
10 362044 
10361698 

10.361353 
10.361008 



10.037492 
10.037547 
10.037602 



10.037657 

XO.037712 

10.360663^0.037767 



10.360318 

iO.359973 
10359629 

10 359284 



10.03)822 

10.037877 
10.037933 

10.037988 



10.358940 10.038043 
10.358596110.038098 
10.358253 j 10^038 IJ4 



10.408122 

10.407824 
10.407527 
10.407230 
10.406933 
10.406637 
10.406341 
10.406045 
10.405749 
10.405453 
10.405158 



10.404863 
10.404568 
10.404273 
10.403979 
10.403685 
10.403391 
10.403097 
10.402804 
10.402510 
10.402217 



10.40 1 925 
10.401632 
10.401340 
10.401048 
10.400756 
10.400464 
10.400173 
10.399882 
10.399591 
fo.399300 



10.357909' 10.038209 
10 357566! 10.038265 
10.357223] 10.038320 
10 356880 10.038376 
10.356537 10.038431 
10-356194 10.038487 
10.355852 10.038542 
10.355510 10.038598 
10.355168 10.038654 
10.354826 10.038710 

io.354484jio.Oi8765 
io.354i43|io. 038821 
JO.353801J10.038877 
JO. 1^3460} 10.038933] 
10.353119.10.038989. 

10.35*778,1003904$, 
10.352438 10.039 10 1 
10.352097 10.039157: 
»0.35i7S7 10.039214 
10.351417 10.039270' 

Tanjy. i Co-sec. 



10.399010 
10.398720 
ro. 3 98430 
10.398140 
10,397850 
10.39756^ 
10.397272 
10.396983 
10.396695 
10. 3 96406 

16396118 
10.395830 

'0. 39 5 543 
10.395255 

10.394968 

10.394681 

10.394394 
10.394108 

10.393821 
«0393535 



10.393249 

10.392964 
10.392678 

10.39*393 
to.392108 

10.391823 

10.391539 
10.391255 

10.390971 

10.390687 



Secant. 



•*r 



60 

59 
58 
57 
56 
55 
54 
53 
5* 
51 
50 

49 
48 

47 
46 

45 
44 
43 
42 

41 
40 






39 
38 
37 
36 

35 

34 
33 
3* 
31 
30 



29 

28 

27 
26 

*5 
*4 
*3 
22 
21 
20 



*9 

18 

'7 
16 

»5 

'4 

12 
II 

lO 



9 
8 

7 
6 

5 

4 

J 

2 

1 

o 



M 



66 Degfrces. 



•^f 



■6 iiilj |9 -. 
9 6«Ji»> 9 ■' 



XiQOARITHIliC SIKES, TANGENTS, AND SECANTS. 51 



"25 Degrees. 



Sine. 



Go-ftine, 



o 
1 

2 

3 

4 

S 
6 

7 
8 

9 
JO 

II 
'4 
13 

«4 

«5 
i6 

17 

i8 
«9 

20 



"9.625948 

.9.626219 

• 9.626490 

9.626760 

9.627030 

9.627300 

J9.62757O 

19.627840 

9.628109 

9.628378 

9 628647 



19.957276 
9.957217 

9-957158 

9957099 
9.957040 

9956981 

9.956921 

9.956862 

9.956803 

9.956744 
9.956684 



9.628916 
9.629185 
9.629453 
9.629721 
9.629989 
9^630257 
9 630524 
9.630792 
9.631059 
9.631326 



21 
22 
23 

24 

»5 

26 

*7 
28 

29 

30 



31 
3a 
33 
34 
35 
36 

i7 

38 

39 

40 



4« 
42 

43 
44 

55 

46 

47 
48 

49 
11 

^i 

53 

54 
55 
56 

57 

58 

59 
60 



9-631593 
9.631859 

9632125 

9.632392 

9.632658 

9.632923 

9.633189 

9.633454 

96337 »9 
9.633984 



9.634249 
9.634514 
9.634778 
9.635042 
9.635306 

963 5 570 
9.635834 
9636097 
9.636360 
9.636623 



9.636886 

9.637148 
9.63741 1 

9637673 

9-63793.5 
9.638197 

9.638458 

9.638720 

9.63898.1 

9.639242 

9.639503 
9.639764 
9.640024 
9.640284 
9.640544 
9.640804 
9.641064 
9.641324 
9.641583 
9.641842 



M ' Co-sine. 



9956625 
9.956566 
9.956506 
9.956447 
9.956387 

9.9563x7 
9.956268 

9.956208 
9.956148 
9.956089 



9.956029 
9.955969 
9.955909 
9.955849 
9.955789 
9.955729 
9.955669 
9.955609 

9955548 
99S54&8 



9.955428 
9.955368 

9.955307 
9.955247 
9.955186 
9.955126 
9.955065 
9.955005 
9.954944 
9.954883 



9 9548*3 
9.954762 

9.954701 

9.954640 

9-954579 
9.954518 

9-954457 
9.954396 

9-954335 
9.954274 



9-953783 

9 953722 
9.953660 



Sine. 



Tan g. 

9.668673 
9.669002 
9.669332 
9.669661 
9.669991 
9.670320 
9.670649 
9.670977 
9.671306 
9671634 
9.671963 



9.672291 
9.672619 
9.672947 
9.673274 
9.673602 

9 6739*9 
9.674257 

9,674584 

9.674910 

9-675237 



9.675564 
9.675890 
9.676217 

9.476543 
9.676K69 
9.677194 
9.677520 
9677^46 
9.678171 
9.678496 



9.678821 
9.679146 
9.679471 

9.679795 
9.680120 

9.680444 
9.680768 
9.681092 
9.681416 
9.681740 



9-9542J3 
9.954152 

9.954090 
9.954029 
9.953968 
9.955906 
995384519-687219 



9.682063 
9.682387 
9.682710 
9.683033 
9.683356 
683679 
9.684001 
9684324 
9.684646 
9.684968 

9.685290 
9 685612 
9.685934 
9.686255 
9.686577 
9.686898 



9.687540 
9.68?8&i 
9.688182 



CO'tanjif. 



Co-tang; 



o.S3tS^7 
0.330998 
0.330668 

0.33^339 
0.330009 

0.329680 

0.32935* 
0.329023 

0.328694 
0.328366 
0.328037 



0.327709 
0.327381 
0.327053 
0.326726 
0.326398 
O.326071. 

0.325743 
0.325416 

0.325090 
0.324763 



0.324436 
0.3241 10 
0.323783 
0.3^3457 

0.32313^ 
0.322806 
0.322480 
0.322154 
0.321829 

0.321504 

— ■ .^— 

0.321179 
0,320854 
0.320529 
0.320205 
0.319880 
0.319556 
0.319232 
0.318908 
0.318584 
0.318260 



0-3«7937 
0.317613 
O.31729O 

0.316967 
0.316644 
0.316321 
0.315999 
0.315676 
0.315354 
0.315032 



o.3«47»o 
0.314388 

0.314066 

0.3*3745 
0.3*3423 
0.313102 
0.312781 
0.312460 
0.312139 
0.31 1818 



Secant. 



Tanjr. 



0.042724 
0.042783 
0.042842 
0.042901 
0.042960 
0.043019 
0.043079 
0.043138 
0.043197 
0.043256 
0.043316 

0.043375 

0.043434 
0,043494 

0043553 

0.043613 

0.04367J 



Co-5ec. 

.10.374052 

»o.37378i 

10.373510' 

10.373240I 

10.3729701 

10.372700I 

10.372430 

10.372:60 

10.371891 

10.371622 

*o.37*353 



M 



<.^ 



0043792 
0.043852 
0.043911 



10.371084 
10.370815 

*o.370547 
10.370279 

10.370011 

*o 369743 
0.0437321*0.369476 
to.369208 
10.36894] 
10.368674 

10.368407 
10.368141 
10.367875 
10.367608 

10-367342 
10.367077 
10.366811 
10.366546 
10.366281 
10.366016 



0.043971 
0.04403 1 
0044091 
0.044 1 5 * 
0.0442 1 1 
004427 1 
0.044331 
0.044391 
0.044452 
0.044512 



0.044572 
0.044632 
0.044693 
0.044753 
0.044814 
0.044874 
0.044935 
0.044995 
0.045056 
0.045117 



0.045177 
0.045238 
OP45299 
0.045360 
0.045421 
0.0454S2 
0.045543 
0.045604 
0.045665 
0.04 5726 

0^45787 
0.045848 
0.045910 
0.045971 
0.046032 
0.046094 
0.046 1 5 5 
0.046217 
0.046278 
0.046340 



10.365751 
10.365486 
20.365222 
10.364958 
fO.364694 
10.364430 
10.364166 
10.363903 
10.363640 

J O363377 

10.3-63 1 14 
10.362852 
10.362589 
10.362327 
10.362065 
.10.361803 
10.361542 
10.36 1 2S0 
10.361019 
10.360758 

— ■ ■ ■ ■ ■ M^ 

10.360497 
10.360236 
10.359976 
13.359716 
10.359456 

ta359i96 
10.358936 
10-358676 
10.358417 
10.358158 



(Jo-.sec 



Secant. 



60 

59 
58 
57 
56 
55 
54 
53 
52 

51 

50 



49 
48 
47 
46 
45 
44 
43 
4* 

41 

40 

38 
37 
36 
35 
34 
33 
3* 
31 
30 

29 

2^ 
27 
1^ 

a$ 

24 
23 
22 
21 

20 



*9 
18 

*7 
16 

*5 

»4 

*3 
12 

II 

TO 



9 

8 

7 
6 

5 

4 

2 
I 
o 

M 



64 Degrees. 



^^M^F 1^ ^mRMmic srxEs, TAvcE]n«'AiWMfMI>V^^^^| 




C6 Ofgi*e» 


■ 




H . Sine. (:<rsii>e. 1 1 .n({ , Ca-I.np , Src*n(. 


to-Kt 


H 


T 




c l,.44i!4i 


99S]66o.96?3iti is-]iili« tc^4«j40 


.aUlijI 


to 


1 




. l9 64*>0> 


9-951 i99'9.***i'» i»-J"49« •0044401 


•0117899 


1! 






s :9.64»j60 


9-9iJ!3T'9 684g»j to j. 1 .it io.P44*«i 


><yi^7«w 






J ;9.64»(.i8 




ra.K-jai 


5' 






4 q.M»*IT 


9 9iM' 




J* 






1 9643'J5 


9 9' 




55 






6 ,9.64J!93 


^■•>'.:- 




54 






7 lS.»43*ii' 






53 






^:^■.■ ■'■■■ 




i» 


■ 




9 Jy '■-, 




<» 


■ 








49 
4t 
4» 
4* 
«( 
44 


1 




17 ,9f'4«>iS 












18 ,9M474 


9 9'.- 










19 19646719 


^-'JS-- 










,0 ■,446984 


9- !)■--. 


- I'j 


^ 






11 9'647»40 


9 9iijv' ^."V-, 




;* 






il 9647494 


g.jSiiM 9.6.ys 




1" 






zi ■9-64'749 


9 9i"3'9.*9>-r 








14 ,9648004 


9.0<il68 9.«<nx; 


■ 1- 






Ij 9C)82;H|.,.0(ii^'-'I.|6n.... 
16 '. 'M^ ■ 


■-' if 








:;;::;; 


•0 






Jl »6497'*< 


9 9S'T»» 


9-69'"'53 


■ L.. 10.94; .a«4Sl71 


iijioiTi 








Jl 96500J4 


9.951665 


9.698J69 


10.30161. .0.«48jJ5 


.0.34996* 


Ik 






J! riioay 


9.9s '6o» 


9.698685 


.{>.30fli5 tao«g)98 


W. 3*97 11 








1* 9 6iOiM 


9-9J'iM 


9.699001 


io.jO09eq ic-:iiS4f>. 


'- i4M6i 


.6 






Jf ■» 640791 


9951476 


9.699316 


.o.joch.-.^i 




's 






j6 9 6s'044 


(.951411 


96996)^ 


lO.SOOjf.- 










,7 9.5(1197 


9-9! '!*9 


9.699947 






'> 






38 19.6SIH9 


9-9inH6 


9.700163 


10.19,^-; 




a 






39 IsSS'Soo 


995"" 


9700578 


10.19.)., :■ 






^ 




4° I9*^i'0i» 


9 9inj, 


9700893 


to. 19^1 c 






A 




41 .9651304 


9.9110,6 


9701108 


.O.19K-., 




'i 


■ 




J^^ 9651555 


9.9SXOJ1 


9-70.513 


iO.»9S.t 




li 


■ 




4J 96;rf06 


9.95096S 


9. 70. 8)7 


.O.i98.i>. ■ .., . 






■ 




44 9.653017 


9950905 


9.76»'i» 


.0.1978411 


'O-'^-liOfi 


<~-H<'<>*i 




■ 




45 9.6s3Jt>li 


9.95084' 


9.70^66 


iD.l97i)4 


10.049' 59 


.o,H6<.9i 


li 


■ 




46 9-6iJ5j8 


9.950778 


9.701780 


10. 197 110 


1O.O4911: 


10.14^(41 




■ 




47 9.«IJ»o8 


9.950714 


9.703091 


10.19691.5 


'O.O^QjSt 




'J 


^M 




+g ,9-6S4059 


9-950650 


9.7OJ409 


10.196^.^.'--: --- 






H 




49 9«»309 


9.950586 


9.703713 


.0,.96.-' 










50 |9:65*iSB 


9:9505 »» 


9.704036 


'0-*9i''' - 










i. ,654808 


9950458 


9.704350 


.o.mi'> 




9 






i» .9.65 1058 19-950)94 


9.70466' 






t 






5J r9.6iS]07 9.9IOJP 


9.7C4" ■ - 










54 :9.*IiH6l9-9SO«i6 


9.70^!., 










(j 9*i5Soj.9.950«)i 


9-70 i'-' 








i6 9.6,6014 


9.950138 


9.70.;.;- 








J7 >.6J«1M 


9.950074 


9, 70', 2. 








(8 96S«J5' 


9.9S»10 










59 9'>S6'99 


9-94994J 


li.Tt^:-; . 








60 '9-657047 


9-94»ll8< 


9 T07 ' 








..Conine- Smc, iu,-u,>. , , ,,:..^, 


^^^^^ 


^ 




•■i UcgitM. ^ 




■ 


■ 



U)GABITBMIC SINES^ 7ANGENT8, AND SECAKTS. 53 



27 Decrees. 



M 


Sioe. , Co-«ine. 


Tang. 


Co-tang 1 Secant. | Co-sec. 


M 

60 


9.657047 


9.949881 


9.707166 


10.191834110.050119 


10,342953 


I 


9.657195 


9*949816 


9.707478 


10.191521! 10.050184 


10.342705 


59 


% 


9.65754* 


9.949751 


9-707790 


10.191210! 10.050148 


10 34*458 


58 


3 


9.657790 


9.949688 


9.708101 


la 19 1 898 10.050312 


10.342210 


57 


4 


9.658037 


9.949623 


9.708414 


10.2915861 10.050377 


10.341963 


56 


5 


9.658284 


9-949558 


9.708716 


10.291274; 10.050442 


10.341716 


55 


6 


9658531 


9-949494 


9.709037 


10.290963: 10.050506 


10.341469 


54 


T 


9.658778 


9.949429 


9.709349 


10.290651 


10.050571 


10.341222 


53 


S 


9.659025 


9.949364 


9.709660 


10.190340 


10.050636 


10.340975 


5* 


9 


9.659271 


9.949300 


9.709971 


10.190029 


10.050700 


10.340729 


5' 


10 

11 


9.659517 
9659763 


9949*35 


9.710282 


10.189718 


10.050765 


10.340483 

10 340237 


50 
49 


9.949170 


9.710593 


10.189407 


10.050830 


la 


9.660009 


9.949105 


9.710904 


10.289096 


10.050895 


•0-33999 » 


48 


"3 


9.660255 


9.949040 


9.711115 


10.288785 


10.050960 


«o 339745 


47 


"4 


9.660501 


9948975 


9.711525 


10.188475 


10.051015 


10.339490 


46 


'$ 


9.660746 


9948910 


9.711836 


10.188164 


10.051090 


10.339254 


45 


i6 


9.66099 1 


9.948X45 


9.7 1 1146 


10.187854 


10.051155 


ia339009 


44 


«7 


9.661136 


9.948780 


9.711456 


10.287544 


10.051120 


10.338764 


43 


i8 


9.J661481 


9.948715 


9.711766 


10.187134 


10.051285 


10.338519 


42 


'9 


9.661726 


9.948650 


9.713076 


10.186924 


10.051350 


10.338274 


4' 


20 
11 


9.661970 


9.948584 


9.713386 


10.286614 


10.051416 


10.338030 


40 
39 


9.662214 


9.948519 


9.713696 


10.286304 


10.051481 


10.337786 


11 


9.662459 


9.948454 


9.714005 


10.285995 


10.051546 


>o.33754» 


38 


»3 


9.662703 


9.948388 


9-714314 


10.285686 


10.051612 


'0.337297 


37 


14 


9.662946 


9-948343 


9.714624 


10.185376 


10.051677 


10.337054 


36 


*$ 


9.663190 


9.948257 


9-714933 


10.285067 


10.051743 


ia3368io 


35 


26 


9.663433 


9.948192 


9.715242 


io.284758 


10.051808 


10.336567 


34 


*7 


9 663677 


9 948126 


9'7»«55« 


10.184449 


10.051874 


«0-3363*3 


33 


28 


9.663920 


9.948060 


9.7^860 


ia284i40 


10.051940 


10.336080 


3* 


*9 


9664163 


9-947995 


9.716168 


10.183831 


10.052005 


10.335837 


31 


33 


9.664406 


9.947929 


9.716477 


10.283513 
10283115 


10.052071 


'0.335594 


30 
29 


9.664648 


9947863 


9.716785 


10.051137 


'0.33535* 


3« 


9.664891 


9947797 


9-7 17093 


10.282907 


10 051203 


10.335109 


28 


33 


9.665133 


9947731 


9717401 


10.181599 


10.051169 


10.334867 


27 


34 


9.665375 


9947665 


9.717709 


10.181191 


"0.051335 


10.334625 


26 


3$ 


9.665617 


9.947600 


9 718017 


10.181983 


10.051400 


'O.334383 


*5 


36 


9.665859 


9-947533 


9718315 


10.281675 


iao5i467 


'0.334'4' 


*4 


37 


9.666100 


9-547467 


9718633 


10^181367 


10.051533 


10.333900 


*3 


38 


9.666342 


9.947401 


9-718940 


10.181060 


fo.051599 


'O.333658 


22 


39 


9.666583 


9-947335 


9-719*48 


10.180751 


10.051665 


'0.3334'7 


21 


40 
41 


9.666824 
9.667065 


9.947169 


9719555 

9-719862 


10.280445 


10051731 
10.052797 


10.333176 


20 

'9 


9.947M3 


10. 280 138 


»0.332935 


4» 


9 667305 


9.947136 


9.720169 


10.279831 


10.052864 


10.332695 


18 


43 


9.667546 


9.947070 


9.710476 


10.279514 


10.052930 


'O.332454 


17 


44 


9.667786 


9-947004 


97*0783 


■0*179117 


10.051996 


10.332214 


16 


45 


9.668017 


9946937 


9.711089 


iai7i9ii 


10.053063 


'0.33'973 


'5 


46 


9.668267 


9.946871 


9-7*1396 


10.178604 


10.053119 


'O.331733 


'4 


47 


9.668506 


9.946804 


9.711701 


10.178198 


10.053196 


10.331494 


'3 


48 


9.668746 


9.946738 


9.711009 


10.17799' 


10.053261 


10.331254 


12 


49 


9.668986 


9.946671 


9.7*1315 


J0.177685 


10053329 


10.331014 


1 1 


50_ 
5> 


9.669225 


9.946604 


9.711611 


10.277379 


10.053396 


10.330775 


10 
9 


9.669464 


9946538 


9-7**9*7 


10.177073 


10.053461 


10.330536 


5» 


9.669703 


9. 94647 i 


9-7*3*3* 


10.176768 


10.0535*9 


10.330297 


8 


$3 


9-669941 


9.946404 


97*3538 


10.176461 


10053596 


10.330058 


7 


54 


9.670181 


9.946337 


9.7*3844 


to.176156 


10.053663 


10.329819 


6 


55 


9.670419 


9946270 


0.724149 


10.1758^1 


10.05373c 


10.329581 


S 


5^ 


9-670658 


9.946103 


9.7*4454 


10.175546 


10.053797 


10.329342 


4 


57 


9.670896 


9.946136 


9-7*4759 


10.175141 


10.053864 


10.329104 


3 


58 


9671134 


9.946069 


9.715065 


10.174935 


10.053931 


10.328866 


1 


59 


9.671371 


9.946002 


9-725369 


10.174631 


10.053998 


10.328628 


I 


60 


9.671609 


9-945935 


9.725674 


10.174316 


10.054065 


10.328391 




M 


f M 


Co-sine. 


Sine. 


Co-tang. 


Tang. * Co-icc. ' Secant ' 



$2 Degrees. 



54 LOGARITH>HC SIKLS, TAXGENTS, AKD SECANTS. 



28 Degrees. 



M 



3 

4 

5 
6 

7 
8 

9 

20 



21 
22 

26 



SSine. 

9.671609 
9.671847 
9.672084 
9.672321 
9.672558 
9.672795 



Co-sine 



994$9i5 
9.945S68 

9.945800 

9-945733 
9.945666 
9.945598 
9.67303219.945531 
9.673268 9945464 



9.673505 
9.673741 

9-67j»)77 



9-94539^ 
9.945328 

9-945^^' 

9.674213 9.945193 

9.67444«|9-945«25 
9.6746S4I9 945058 

9.674919 9.944990 

9.944922 

9.944854 
9.944-^^6 

9.944718 

9.944650 

9- 944 v^ 2 



9<>75»5S 

9 67539^ 
96756^4 

9.675859 
9 676094 
9^6318 

9 676562 
9.676796 
9 677030 



9.944514 
9.944446 
99443 



9.677264J9.944309 
9 67749> 9.944241 



9.677731 

27 9.677964 

28 9.678197 

^9i9-67S433 
30I9. 678663 



3' 

52 
33 
34 
35 
36 
37 
3« 
39 

d2 
41 
42 
43 
44 
45! 



9678895 
9,679128 
9 67936c 

9679592 
9.679824 

9.680056 
9.680288 
9680519 
9.680750 
9.680982 

9.681213 
9.681443 
9.681674 
9.681905 
9.682135 
46:9 682365 
4719.682595 
48:9.682825 
4919.683055 



9.944172 
9.944104 



Tan,^. , Co -tang-. 

o 274326 
0.274021 
0273716 

0.273412 
0.273108 

0.272803 

0.272499 

0.272195 

0.271891 

0.271588 

0.271284 



9.725674 
9.725979 
9.726284 
9.726588 
9.726892 
9.727197 
9.727501 
9.727805 
9.7281O9 
9.72S412 
9.728716 

9.729020 

9-729323 
9.7Z9626 

9.729929 

9-730233 

9-730535 
9.730838 

9-73»i4» 
9-73 '444 
9-"'3'746 



9-''32048 

9-73235' 
9732653 

9-732955 
9-733257 
9733558 
9.7355^60 



9-944036 9-734'62 



9-9439^7 
9.943899 



9-943830 
9-943761 

9-943693 
9.943624 

9-943555 
9.943486 
9.943417J9-736871 

9.943348 9-737171 
9.943279 9737471 



9-734463 
9-734764 



9 73S066 

9-735367 
9-735668 

9-735969! 
97362691 

9-7365701 



9.943210 



9-737771 



9.943i4r 9-73^071 
9.943072 9-73837 1 
9.943003 9-738671 



■ 



9.942934|9'73897i 

9.942864J9-739271 
9-942795|9-739570 
9.942726 9.739870 



9.942656 
9.94258/' 



9.740169 
9.740468 



501 9.683284 .9.942^^19.740767 

51,9.683514:9.94244819.741066; 

5219-683743 9 942378 9.741365' 



I 



53;9-683972 9.942308 9-741664 
54 9.6S4201 9. 9.^2239 g. 741962 
55J9.684430 9.9421(59 9.742261 
56/9.684558 9.942099 9.742559 
57J9.6S4SS7 9.942029 9-742858 
5819.68511519.941959 9.743156 
59'9.685343J9.94i889 9.743454 
_6o^9.685£M l9j^4^^ 9.7 43^52 ; 

M '(;o-siiit:. ' Sine. C()-l;in:<. ■ 



0,270980 

0.270677 

0.270374 
0.270071 
0.269767 
0.269465 
o 269162 
0.268859 
0.268556 
0.268254 



0.267952 
0.267649 

0.267347 

0.267045 
0.266743 
0.266442 
0.266140 

0.265838 

0.265537 
0.265236 



0264934 
0.264633 
0.264332 
0.264031 
0.263731 
o 263430 
0.263129 
0.262829 
o. 262529 
0.262229 



Secant. \ Co-sec 



0.26 1929 
0.261629 
0.261329 
0.261029 
0.260729 
0.260430 
0.260130 
0.259831 
0.259532 
0259233 



0.258934 
0.258635 
0.258336 
0.25803S 
0.257739 
0.257441 
0.257142 
0.256844 
0.256546 
0.25624S 



liiny-. 



0.054065 
0.054132 
0.054200 
0.054267 

0.0^4334 
0.O54402 

0.0 5 4469 
0.054536 
0.C54604 
0.054672 
0054739 
0.054^107 
0.054875 
0.054942 
0.C55010 
0055078 
0.055146 
0.055214 
0.055282 
0.055350 
0.055418 



0.055486 
0055554 
0.055623 
0.055691 

0.055759 
0055828 

0.055896 

0.055964 

0.056033 

0.056101 



0.056170 

0.056239 

0.056307 

0.056376 

0.056445 

0.056514 

0.056583 

00566521 

0.056721 

0.056790 



0.056859 
0.056928 
0.056997 
0.057066 
0.057136 
0.057205 
0.057274 
0.057344 
0.057413 
0.057483 



C.057552 
0.057622 
0.057692 
0.057761 
0.057831 
0.057901 
0.057971 
0.O5804 I 
0.0581 1 • 
0.058181 

Co- sec. 



0.328391 
0.328153 
0.327916 
0.327679 
0.327442 
0.327205 
0.326968 
0.32673a 
0.326495 
0.326259 
0.326023 



0.325787 
0325552 
0.325316 
0.325081 
0.32484s 
0.324610 

0.324576 
0324141 
0.323906 

o 323672 



032343^ 

0.3252C-4 
0.312970 

0.322736 
0.322502 
0.322269 
o. 522036 
0,321803 
0.321570 

0.3*1337 



0.321105 
0.32087 a 
0.320640 
0.320408 
0.320176 

0.319944 
J0.3I97I2 
o 3 1948 1 
0.319250 
0.319018 



0.318787 
0.318557 
0.318326 
0.318095 
0.317865 
0.317635 

0.317405 

0.31717s 
0.316945 

0.3 1 67 1 6 



0.316486 
0.316257 
0.316028 

0.315799 
0.315570 
0.315342 
0315113 

0.314885' 
0.31465:^1 

0.31442 9J 



60 

59 
58 
57 
56 
55 
54 
53 
52 
5« 
50 



49 
48 
47 
46 

45 
44 
43 
42 

41 
40 



59 
38 
37 
56 

35 
34 
33 
3* 
3» 
30 



29 
18 

27 
26 

25 

24 
23 

22 

21 

20 



19 
18 

«7 
16 

>$ 
14 
13 

12 

11 
10 

9 
8 

7 
6 

5 

4 
3 

2 

1 

o 



, 



J-i- ^« '.Li. 



LOGARITHMIC SIXES, TANGENTS, AND SECANTS, 55 



29 Degrees. 



M 



Sine. 

9.685571 
9.685799 
9.6860Z7 
9.6S6254 
9.686482 
9.686709 
9.6869:16 
9.6X7 1 6j 



(>i-stne. 



9.9418:9 
9.941749 

9941679 
9.041609 

9.941539 
9.941469 
9.941398 
9.941328 



Tanjr 



9.687^89:9.941258 
9 687616 9.941187 
9.687843 1 9.941 117 

9. 688069.19, 941046 
9.940975 
9.94090^ 
9.940834 
9.940763 
9.940693 
9.940622 

9.94055 » 
9.940480 
9.940409 

9.940338 
9.940267 

9.940196 
9.940125 
9.940054 

9-9399^^ 

}9.9399«i 

9.930840 



9.688295 
9.688521 
9.688747 
'9.688972 
•9.689198 
9.689423 
9 689648 

9.689873 
I9.690099 

^9.690323 



9-74375^ 
9.744050 

9.744348 
9.744645 

9-744943 
9.745240 

9-745535i 

9.745^35 
9.74613a 
9.746429 
9^746726 

9.747023 

3-747319 
9.747616 

9.7479»3 
9.748209 

0,748505 

9.748801 



19.690548 
;9.690772 
I9.690996 
'9.691220 

9691444 

9.691668 
!9.69i892 

9.69211 519.939768 

9.692339 * 993 9697 

9.692562I9 939625 

9.692785;9.939554 

9 693008 9.939482 

9.693231 9.9394«o 

9-693453 9-939339 
9.693676 9.939267 

J9.693898 9-939«95 
19.6^4130 9.939123 
19.694342 9939052 
.9.694564 9.938980 

J9.694786 9.938908 9 
J9.695007J9.938836J9 

9 
9 
9 
9 
9 
9 
9 
9 



;9.695229|9. 938763 
'9 695450:9-93^^691! 
I9.695671J9.938619 
•9.6,5892,9.938547 

9.696113(9.938475! 

9.69633419.938402! 

9.696554;9.93833o! 
. ?-<>9677 5 ! 9;9 38258 

9.696995*9.938185 
9.697215J9.938113 
9 69743 5 '9-9380401 
9-697654;9.937967| 
9-697874I5.937895: 
:9.698094 9.937822 



M 



(9.698313 
19.698532 

: 9.6987 55 
9. 698970 

'('f)-biiU". 



9-937749! 
9-937676; 

9.937604 
9937S3I; 



749097 

749393 
749689 

749985 
750281 
750576 
750872 

751167 

751462 

7S«757 
752052 

752347 
752642 



752937 
75323* 
753526 
753820 

754»I5 

754409 

754703 
754997 
755*9* 
755585 

755878 
756172 
756465 

756759 
757052 

75734s 
75763^ 

75793* 
758224 

7585«7 



Sine. 



758810 
9.759102 

759395 
759687 

759979 
760272 

760564 

760S56 

761148 

761439 

Co-tauM". 



Co-tanp 

256248 
0.255950 
0.255652 

0.255355! 
0.255057' 

0.254760; 

o 25446Z 

0.254165 

0.2^3868 

0.253571 

0.253274 

0.252977 
0.252681 
0.252384 
0.252087 
0.251791 
0.251495 
0.251 199 

Q. 250903 
0.250607 
0.25031 1 



0.250015 
0.249719 
0.249424 
0.249128 
0.248833 

0-248538 
0.248243 

0.247948 

0.247653 

0.247358 



0.247063 
0.246769 

o. 246474 

0.246180 

o 145885 

0.245591 

0.245297 

0.245003 
0.244709 
0.244415 



0.244122 
0.24382S 

0.243535 
0.243241 

0.242948 
0.242655 
0.242362 
0.242069 
0.241776 
0.241483 



0.241 190 
0.240898 
0.240 60 5 
0.240313 
0.240021 
0.239728 
0.239436 
0.239144 
0.23885Z 
0,238561 



Tang-. 



Secant. 

0.058181 

0.058251 

0.058321 

0.058391 

0.058461 

0.058531 

0.058602 

0.05S672.; 

0,058742. 

0.058813^ 

0.058S83I 



(vO-HCC. 



C.05.8954 

0.059025. 

O.O59CO5 

0.059166 

0.059237 

0.059307 

0.059378 

0.059449 

0.059520 

0.059591 



0.059662 
0.059733 
O.O598O4 
0.059875 
0.059946 

0.0600 1 8 
0.060089 
0.060160 
006023 a. 
0.060303 



0.060375 
o 060446 
0.060518 
0.060590 
0.060661 
0.0O0733 
0.060805 
0.060877 
0.060948 
0.061020 



0.06 1 092 
0.06 i 164 
0.061237 
0.06 1 309 
0.06138 1 
0.061453 
0.061525 
0.061598 
0.061670 
0.061742 



0.061815 
0.061887 
0.061960 
0.062033 
0.062105 
0.062178 
C.062251 
0.062324 
0.062396 
0.062469 



Co>sec- 



0.3 

0.3 
0.3 

0.3 

0.3 

0-3 

<5-3 

0.3 

0-3 
0.3 

0-3 



0.3 
0.3 
0.3 
0.3 

0-3 
0-3 
0.3 

0.3 
P-3 



4429 
4201 

3973 
3746 

35*8 
3291 

3064 

2837 
261 c 
2384 

2157 



M 



60 

59 
58 
57 
56 
55 
54 
53 
5^ 
51 
50 



*93ii 

»705, 

1479 
125s 

1028 

0802 

0577 

0352 
0127 



0.309 90 2 

0.309677 
0.309452 
O.309228 
0.309004 
0.308780 
0.308556 
0.308332 
0.308108 
0.307885 
0.307661 

0.307438 
0.307215 
0.306992 
0.306769 
0306547 
0.306324 
0.306102 
0.305880 
0.305658 
0.305436 



49 

48 

47 
46 

45 
44 
43 
42 

4» 

40 



39 
3» 
37 
36 

35 

34 
33 
32 

31 

29 
28 

27 
26 

25 

24 

23 
22 

21 

20 



0.305214 
0.304993 
0.304771 
0.304550 
0.304329 
0.304108" 
0.503887 
0.303666 
0-303446 
0.303215 



0.303005 
0.302785 
0.302565 
0.302346 
0^02126 
0.30 1 906 
0.301687 
0.301468 
0.30124:) 
0.301030 

Secant, i M 



9 
8 

7 
6 

5 

4 
3 

2 

I 

o 

9 
8 

7 
6 

5 

4 

2 
I 

o 



0<) Drgrccs. 



LOGARITHMIC SINES, TANGENTSi AND SECANTS. 57 



31 Degrees. 



M 

1 


Sine. 


Co.slne.| Tang. Co-tang. 


Sebant. 


Co-dec. 


1 »'.i 


o 


9.71 1839 


9.9J3066 


9.778774 10.221226 


10.066934 10.28816:! 60 1 


n .9,712050 


9.932990 


9.77906O; 10.220940 


10.067010 10.287950] 59 1 


. a 19.71226Q 


9.932914 


9-779346! ic 220654 


10.067086 i 0.287 740 


S8 


3 


9.712469 


9.932838 


9.779632 16.220368 


10.067162 10.287531 


1 

57 


4 


9.712679 


9.932762 


9.779918 10.220082 


10.067238 10.287321 


56 


$ 


9.712889 


9.932685 


9.780203 10.219797 


10.067315 10.287111 


55 


6 


9.713098 


9 932609 


9.780489110.219511 


ioo673Qi'io.286902 


54 


7 


9.713308 


9932533 


9.780775 10.219225 


10.667467 10.286692 


53 


8 


9-7i35»7 


9.-9324S7 


9.781060 10.218940 


10067543 to.286483 


52 


♦ 

9 


9.713726 


9,932380 


9.781346 10.218654 


10.067620; 10.286274 


5« 


>'0 


9-7«3935 


9-93*304 
9.932228 


9.781631 10.218369 


10067696 


10.286065 


50 

49 


ki 


9.714144 


9.781916 


10.218084 


10.067772 


10.285856 


>* 


9-7«435» 


9.932151 


9.782201 


10.217799 


10.067849 


10.285648 48 1 


.'3 


9.714561 


9-932075 


9.782486 


10217514 


10.067925 


10.285439 47 1 


H 


9.714769 


9.931998 


9.782771 


10.217229 


10.068002 


10.285231 


46 


«5 


9.714978 


9.931921 


9.783056 


10.216944 


10.068079 


10.28)022 


45 


^6 


9.715186 


9.931845 


9.78334* 


10.216659 


10.068155 


10.284814 


44 


«7 


9-7«5394 


9.931768 


9.783626 


[O.216374 


10.068232 


10.284606 


43 


18 9.715602 


9.931691 


9.783910 


10.216090 


10.068309 


10.284398 


42 


19 9-715809 


9-93*614 


9.784195 


10.215805 


10.068386 


10.284191 


4» 




9.716017 
9.716224 


9931537 


9.784479 

9784764 
9.785048 


10.215521 


10.068463 


10.283983 


40 
39 


9931460 


10.215236 


10.068540 


10 283776 


%2 


9.716432 


9.93 «J 83 


10.21495a 


10.068617 


10 283 5 W 


38 


a3 


9.716639 


9.931306 


9^785332 


10 214668 


10.068694 


10.283361 


37 


*4 


9.716846 


9 93 » 229 


9.785616 


10.214384 


10.068771 


10.283154 


36 


*5 


9-717053 


9.931152 


9.785900 


10.21410C 


10.068848 


10.282947 


35 


a6 


9.717259 


9.931075 


9.786184 


10.213816 


10.068925 


10.282741 


34 


47 


9.7 1 74661 


9.930998 


9.786468 


10.213532 


10 069002 


10 282534 


23 


28 


9717673 


9 930921 


9.786752 


10 213248 


10.069079 


10.282327 


32 


»9 


9.717879 


9.930843 


9.787036 


10.212964 


10.069157 


10.282121 


3' 


30 
3» 


9.718085 
9.7 1 829 1 


9.930766 


9787319 


10.212681 


10.069234 


10.281915 
10.281709 


30 
29 


9 930688 


9.787603 


10.212397 


10069312 


3» 


9.718497 


9.930611 


9.787886 


10 212114 


10.069389 


to. 28 1503 


28 


33 


9.718703 


9-930533 


9.788170 


10.211830 


10.069467 


10.281297 


27 


34 


9.718909 


9.930456 


9788453 


10.211547 


10.069544 


io.<28i09i 


26 


35 


9.719114 


9.930378 


9.788736 


10.21 1264 


10.069622 


10.280886 


25 


36 


9.719320 


9.930300 


9.789019 


10.210981 


10.069700 


10.280680 


24 


37 


9.719525 


9.930223 


9.789302 


10.210698 


10.069777 


10.280475 


23 


38 


9-7«973^ 


9-930145 


9.789585 


10.210415 


10069855 


10.280270 


22 


39 


9-719935 


9.930067 


9.789868 


10.210132 


10 069933 


10.286065 


21 


40 
4" 


9.720140 


9.929989 


9.7901 5 1 


10.209849 
10.209567 


10.070011 
10.070089 


10.279860 


20 
19 


9.720345 


9.929911 


9790433 


10.279655 


'4» 


9.720549 


9.929833 


9.790716 


10.209284 


10.070167 


10279451 


j8 


43 


9.720754 


9.929755 


9.790999 


10.209001 


10.070245 


10. I7 9 246 


17 


44 


9.720958 


9.929677 


9.791281 


10. 2087 '9 


10.070323 


10.279042 


16 


4S 


9.721 102 


9.929599 


9-79 "563 


10.208437 


10.070401 


10.278838 


«5 


46 


9 721366 


9.929521 


9.791846 


10.208154 


10.070479 


to.178634 


»4 


47 


9.721570 


9.929442 


9.792128 


10.207872 


10.070558 


10.278430 


»3 


48 


9.7*1774 


9.929364 


9.792410 


10.207590 


10.070636 


10.278226 


12 


49 


9.721978 


9.929286 


9.792692 


10.207308 


10.070714 


10.278022 


11 


50^ 
51 


9.722181 


9 929207 


9792974 
9.793256 


10.207026 


10.070793 


10.277819 


10 
9 


9.72238c 


9929129 


10.206744 


10.070871 


10.277615 


$» 


9.722588 


9 929050 


9793538 


10.206462 


10.070950 


10.277412 


8 


S3 


9.722791 


9.928972 


9.793819 


I0i'206i8i 


10.071028 


10.277209 


7 


54 


9.722994 


9 928893 


9.794101 


10 205899 


10.071 (07 


10.277006 


6 


55 


97231^7 


9.928815 


9794383 


10.205617 


10.071185 


10.276803 


5 


56 


9.723400 


9,928736 


9-794664 


10.205336 


10.071264 


10.276600 


4 


57 


9.723603 


9.928657 


9-794945 


10.205055 


10.07134} 


10.276397 


3 


58 


9.723805 


9-928578 


9.795227 


10.204773 


10.071422 


10.276195 


2 


59 


9724007 


9.928499 


9795508 


10.204492 


10.071501 


10.275993 


I 


t 60 

M 


9.724210 


9.92842a 


9.795789 
Co'tang. 


i 0.2042 1 1 


10.071580 
Co-iec. 


10.275790 




M 


Co-sine. 


bine. 


Tang. 


Secant. 








5S 


Degreeji., 






« 



59- LOGARITHMIC SINES, TAKGtNTS, AKD SECANTS; 



32 I>egree94 



M 

O 
1 

2 

4 

5 
6 

7 
8 

9 

lO 



11 

12 



9.714210 9.92842o'9.795789; 



Sine. , CcKsine., Tang. , Co-tanp. 

0.20421 1 
0.203930 
0.203649 
0.203368 
0.203087 
O.202S06 
0.202525 
0.202245 
0.201964 
0.201684 
o. 20 1 404 



9.72441a 
9.724614 
9.724816 

9.725017 
9.725219 
9.7254*0 
9.725622 
9.725823 
9.726024 
9.726225 

9.726426 



9.928342 
9.928263 
9.928183 
9.928104 



9.796070! 

9-7963^>l 
9.796632 

9.7969J3 



9.928025 ;9.797«94 

9.927946,9.797475 

9.927867;9-79775S 
9.927787 19-798°36 
9.9277o8'9.7983i6 
9.927629 9.798596 



9.927549.9.798877; 
9.72662619.927470 9-799»57l 

13 9-726827;9-9a7390'9 799437, 



>4 

15 
16 



.72702719.9273*0 9.7 
.72722819. 927231 9.7 



17 
18 

'9 



9 
9 

9.7274^^:9-927 > 5 
9.727628I9. 92707 



7997171 
99997 



9.800277J 

9.800557' 
9.800836. 
9.801116, 
9.801396I 

9.801675 

9.801955' 

9.802234 

9.802513 

9.802792 

9.803072 



41 
42 

43 

44 

45 
46 

47 
48 

49 
5^ 



9.727828J9. 92699 

9.728027.9.92691 

20! 9.728227 I 9^^683 

21,9.728427,9.92675 

22 9.72S!626 9.92667 

23 9 72882519 9^659 

24;9.7*9024i9-92^5i 
2519.72922319.92643 

2619.729422^9 92635 

27 9.72962 1 '9.926270 9.803351 

?.8 1 9. 7 2984019.926 1 9019.803630 

2919.73001819.926110 9.803908 

30I 9.73021 7 9.926 029 9.80 4187 , 

3119.730415 9.925949 9 804466 
3219.730613,9.925868 9.804745' 

33'9-7308ii|9.92S788;9.805023! 

34J9.73f009i9-92S707;9-805302: 
9.925626J9.805580,' 
9.925545!9.8o5859' 

9-925465',9-8<^U7. 
99*5384 9-^06415; 

9-7iJ996|9 9253%3^9-8o6693; 
9-73*'93 9-925*** 9-806971 
9.732390 9-9*5»4i 9 807249 
9.73*587 9-925060 9.807527 
9.732784 9 92497^ 9.807805 
9.732980 9.924897 9-808083 
9.733177 99*4816 9.808361 
9-733373 9-9*4735 9-808638 
9-733569 9.924654 9.808916 
9 733765 9-9*457* 9-809193 
9-733961 9.9*4491 9809471 

9.809748 

9.810025 
9.810302 
9.810580 
9-734939 9-924083 9.810857 
9-735135 9.924001 9.811134 
9735330 9-923919 9-811410 
9-735525 9923837 9811687 
9-735719 9923755 9-811964 
9-735914 9923673 9-81*241 
9.736 109 9.923 591 9-81*517 

Co-sine. Sine. 



3519.731*06 
36I9.731404 
37! 9.73 1602 

38 9.731799 

39 

40 



5» 

52 
53 
54 
55 
56 
57 
58 

59 
60 

M 



9-73415719-924409 



9 734353 9-924328 
9-734549I9924246 
9-734744 9-924164 



Co-tang. 



0.201 123 

0.200843 

0.200563 

0.200283 

0.200003 

99723 

99443 
99164 

98884 

98604 

98325 

98045 
97766 

97487 
97208 

96928 

96649 

96370 

96092 

95813 



Secant. 



o. 
o. 
o. 
o. 
o. 

o. 
o 
o. 
o. 
o. 
o. 
o. 
o. 
o. 
o. 



o. 
o. 
o 
o. 
o. 
o 
o 

0. 

o. 
o. 

o. 
o. 
o. 
o. 

o. 
c. 
o. 
o. 
o. 
o. 



o. 
o. 
o. 
o. 
o. 
o. 
o. 
o. 
o. 

0«: 



9S534 
95*55 
94977 
94698 
94420 
94141 
93863 
93585 
93307 
93029 

92751 

92473 
92195 

91917 

91639 
91362 

91084 

90807 

90529 
90252 



89975 
89698 

89420 

89143 
88866 

88590 

88313 
88036 

87759 
87483 

Tanj;. 



0.071580 
0.071658 

0.071737 
0.071817 
0.071896 

0.071975 
0.072054 

0.072133 

0.07x213 

0.072292 

0.072371 



0.072451 
0.072530 
0.072610 
0.072690 
0.072769 
0.072849 
0.072929 
0.073009 
0.073089 
0.073169 



o 073249 
0.073329 
0.0-3409 

0.073489 

0.073569 

0.073649 
0.073730 

0.073810 

0.073890 

0.073971 



0.074051 
0.074132 
0074212 
0.074293 

0074374 

0.074455 
0.074535 
0.074616 
0.074697 

0.074778 



0.074859 
0.074940 
0.075021 
0.075103 
0.075184 
0.075265 
0.075346 
0.075428 
0.075509 

0-07559' 

0.075672 
0.075754 
0.075836 
0.075917 
0.075999 
0.076081 
0076163 
0.076245 
0.076327 
0.076409 

Co-sec. 



CO'Sec. 



0.275790 

0.275588 

0.275386 

0.275184 

0.274983 

0.274781* 

0.274580 

0.274378 

0.274177 

0.273976 

0.27377S 



1' 



0-173574 

0.273374 

0273173 

0.27*973 

0.272772 

0^72572 
0.272372 
0.272472 
0.271973 

0.271773 



0.271573 
0.271374. 
0.271 175! 
0.270976J 

o 270777! 

0.270578; 

0.270379} 

0.270180 

o 269982 

0.269783, 

0.2695851 
0.269387; 
0.2691891 
O 26899 i{ 

0.268794J 
0.2685961 

0.26S398. 

0.268201 
0.268004 
0.267807 



0.267610 
0.267413 

0.267216 
0.267020 
0.266823 
0.266627 
0.266431 
0.266235 
0.266039 
0.265843 



0.265647 
0.265451 
0.265256 
0.265061 
0.264865 
o 264670 

0.264475 

0.264281 
0.264086 
0.263891 

Secant. 



2f 



^7 Degrees. 



iIXI6\KITHMie Sm&3, TANf^^TS, A9D SECANTS. 59 



\.«m 



33 Degrees. 



Sine. Co-ftine. 



9.736109 

1 '9.736303 

2 .9.736498, 

3 9.736692 

4 ,9.736886 

5 '9.7370801 

6 (9-73ra74' 

7 |9-737467| 

8 19-737661 

9 I9.73785S1 

10 J 9. 7 38048 1 

11 19.738241 
1% '9.738434 
13 19738627 
«4 [9 7i'88ao 

15 97390 '3 

16 9.739206 

17 9-739398 

18 9-739590 

19 9739783 
ao 9739975 



9.92359' 
9.923509 

9.923427 

99*3345 
9.9»3i63 

9.9231.81 

9.92JP98 

9.923016 

9.922933 

9.922851 

9.922768 



21 

22 

aj 

»4 

»5 
26 

27 

28 

»9 

,11 

3i 
32 

33 
H 
35 
36 

37 
38 

39 

40 



41 
42 

43 

;44 

45 
46 

47 
48 

49 

11 

5« 

5* 

53 

54 

55 

56 

57 

58 

59 
6j 



9.740167 

9740359 
9.740^50 

9.74074* 

9740934 

9-74« »«5 
9-74>3"6 

9.74" 508 

9-741699 

9.741889 



9.742080 
9.742271 
9.742462 
9.742652 
9.742842 
9-743033 
9-743213 

9.7434 » 3 
9.743602 

9-74179^ 



9.743982 

9.744171 
9744361 

9.744550 

9-744739 
9.744928 

9.745117 

9.745306 

9.745494 
9.745683 



9.74587* 
9.746060 
9.746248 
9.746436 
9.746624 
9.746812 
9.746999 
9.747187 
9747374 



M (^o-biiie 



9.922686 
9.922603 
9.922520 
9.922438 

9-9"355 
9.922272 

9.922189 

9.922106 

9.922023 

9.921 940 

9.921857 

9.921774 
9.921691 

9.921607 

9.921524 

9.921441 

9-9a»357 
9^1274 

9.92 II 90 

9.921 107 



9.921023 
9.920939 
9.920856 
9.920772 
9r^92o688 
9.920604 
9.920520 
9.920436 
,9.920352 
9.920268 



^.920184 
9.920099 
9.920015 
9.919931 
9919846 
9.919762 
9.919677 

9-9I9593 
9.919508 

9.9194H 

9-9«9339 
9.919254 

9.919169 
9.919085 
9.919000 
9.918915 
9.918830 
9.918745 
9.9 1 86 5 If 
9-9«8s74 
S^ine. 



Tanjf. 



9.812517 
9.812794 
9.8 13070 

9-813347 
9.813623 

9.8x3899 

9.814115 

9.814452 

9.814728 

9.8150Q4 

9-8 'SV9 

9-815555 
9'8i583i 

9.816107 

9.816382 

9.816658 

9.816933 
9.817209 

9.817484 

9.817759 

9.818035 



9.818310 
9.818585 
9.818860 
9.819135 
9.819410 
9.819684 

9819959 

9.820234 

9. 820508 
9.820783 



9.821057 
9.821332 
9.821606 
9.82^1880 
9.822154 
9 822429 
9.822703 
9.822977 
9.823150 
9823524 



9.823798 
9 824072 
9.824345 
9.824619 
9.824893 
9.825166 

9.825439 
9.825713 

9.825986 

9.826259 

9.826532 

9.826805 
9.827078 
1.82735 1 
t.827624 
1.827897 
1.828 1 70 
1.828^42 

1828715 
1.828987 

Co-taiig'. 



Co-tang'. Secant. 



87483 
87206 

86930 

S6653 

86377 
86101 

85825 
85548 
85272 
84996 
0.1.84721 



o. 
o 
o. 
o. 
o. 
a 
o. 
o. 
o. 
o. 



0. 
o. 
0. 
P.. 
p. 
o. 
o. 
0. 
o. 
o. 



o. 
o. 
o. 
o. 

0. 

0. 

o 
o- 
o- 
o. 

o! 

o. 

0. 

o. 
o. 
o. 
o. 
o. 

0. 

o. 



o. 

o. 
o. 
o. 
o. 
o. 
o. 
o. 
o. 
o. 



o. 
o. 
o. 
o. 

Q. 
o. 
o. 
o. 
o. 
o. 



84445 

84169 

83893 

83618 

8334^ 
83C67 

82791 
82516 

82241 

81965 



78943 
78668 

78394 

78120 

77846 

77571 

77*97 

77023 

76750 

76476 



76202 

75928 
75655 

75381 

75«07 

74834 
7456.1 
74*87 

740 »4 

7374' 



81690 

81415 

8U40 

80865 

80590 

80316 

80041 . 

79766 

7949* 

79217 



734<i8 

73'95 
72922 

72649 

7*376 
72103 
71830 

7'558; 
71*85! 

710131 



0.076409 
0.076491 
0.076573 
0.076655 
0076737 
0.076819 
0.076902 
0.076984 
0.077067 

0.077149 

0.077232 



0.077314 

0.077397 
0.077480 
0.077562 

0.077645 
0.077728 
0.0778 1.1 
0.077894 

0.077977 
0.078060 



0.078143 
0.078226 
0.078309 
0.078393 
0.078476 
0.078559 
0.078643 
0.078726 
0.078810 
0.078893 



Go-sec 



0.078977 
0.079061 

0.079144 

0.079228! 

0.079312! 

0.079396 

0.079480 

0.079564 

0.079648 

0.079732 



0.079816 

0.07 9 90 1 

0.079985 

0.080069 

0.080154! 

0.080238! 

0080323 

0.080407 

0.080492 

0.080576 



Ta-ff. I 



o.oZoCbi\ 

0.080746; 

0.080831; 

0.08C915' 

0.081000' 

0.08 1 08 5 j 

0.08 1 1 70' 

0.081255 

0.08 1 34 1 

0.081426 

Co-stc. 

wmtmmmmimm 



0.263891 
0.263697 
0.263502 
0.263308 
0.263114 
0.262920 

o. 262726 

0.262,533 
0.262339 
0.262145 
0.261952 



0.261759 
0.261566 
0.261373 
0.261 180 
0.260987 
0.260794 
0.260602 
0.260410 
0.260217 
0.260025 



0.259833 
0.259641 
0.259450 
0.259258 
0.259066 
0.258875 
0.258(584 
0.258492 
0.258301 
O.2581II 



0.257920 
0.257729 

0.257538 

0.257348 
0.257158 
0.256967 
0.256-77 
0.256587 
0.256398 
0.256208 



0.256018 
0.255829 
0.255639 
0255450 
0.255261 
0.255072 
0.254883 
0.254694 
0.254506 
0.254317 



0.254129 
0.253940 

0.25375* 
0.253564 

10.253376 

0.253 '88 
0.253001 
0.252813 
0.252626 
0.252438 

S> c <tni 



60 

5 • 
5^ 
5- 
56 

55 
54 
53 
5* 
5' 
50 



49 
48 

47 
46 

45 

44 

43 
42 

4' J 

40 



39 

38 

37 •• 

36 

33 

34 

33 

3* 

3' 

30 : 



29 
28 

27 

lb 

*5 
24 

*3 
22 
21 
20 



»9 

18 

17 
16 

'5 

'4 

»3 
12 

II 

id 



9 
8 

7 
6 

5 

4 

3 

2 

1 

o 

M 






^m 



56 i)egr?e9. 



60 LOGARITUMIC SINES, TANGENTS, AND SECA)fTS« 



34 Uegreei. 



w , 


Sine. \ 


06-81 ne. 


Tang. 


Co-tanfi^. 1 


Secant ^ 


Co-8t:C.' 


60 





9.747562 


9.9;8574 


9.82«9«7 


10.171013 


10.081426 


10.252438 


I 


9-747749 9-9 '8489! 


9.829260 


10.170740, 


10.081511 


10.252251 


59 


a 


9-747936 


9.918404 


9.829532 


10.170468 


10.081596 


10.252064 


58 


3 19-748 »»3 


9.918318 


9.829805 


10 170195 


10.081682 


ia25i877 


57 


4 19-7483 10 


9.918233 


9 830077 


16.169923 


10081767 


10.251690 


56 


5 19.748497 


9.918147 


9.830349 


10.169651 


10.081853 


10.251503 


55 


6 9.74«6«3 


9.918062 


9.830621 


10.169379 


10.081938 


10.151317 


54 


7 


9.748870 


9917976 


9.83^893 iai69i07 


10.082024 


10.251130 


53 


8 


9.749056 


9917891 


9.83(165 10.168835 


10.08 21 09 


10.250944 


5« 


9 


9749243 


9.917805 


9.831437 


10.168563 


10082195 


10.250757 


5" 


10 
fi 


9 7494*9 


9.917719 


9.831709 


10.168291 


10.082281 


10.250571 
10.250385 


50 
49 


9.749615 


9-9 '7634 
9.917548 


9.83 198 1 


10.168019 


10.082366 


12 


9.749801 


9.832253 


10.167747 


10.082452 


XO.250199 


48 


13 


9.749987 


9.917462 


9.832525 


10.167475 


10.082538 


10.350013 


47 


>4 


9.750172 


9.9«7376 


9.832796 


iai67204 


10.082624 


10.249828 


46 


>$ 


9 750358 


9.917290 


9.833068 


10.166932 


10.082710 


10.249642 


45 


16 


9-750543 


99172OJ 
9.917115 


9-833339 


10.166661 


10.082796 


I0.H94S7 


44 


17 


9.750729 


9.83361 1 


10.166389 


to.082882 


10.249271 


43 


18 


9.750914 


9.917032 


9.833882 


10.16611S 


10.082968 


10.2490S6 


42 


'9 


9.751099 


9.916946 


9.834154 


10.165846 


10083054 


10.248901 


4* 


20 

21 


9.751284 


9916859 


9.834425 
9.834696 


10.165575 
10.165304 


10.083141 
10.083227 


10.248716 


40 
39 


9.751469 


9.9>6773 


10248531 


22 


9.75*654 


9.916687 


9.834967 


10.165033 


10.083313 


10.248346 


38 


?3 


9-75*839 


9.916600 


9.835238 


10.164762 


10.083400 


10.248161 


37 


H 


9.752023 


9.916514 


9835509 


10.164491 


10.0^3486 


10.247977 


36 


25 


9.752208 


9.916427 


9.835780 


10.164220 


10.083573 


I0.24779» 


35 


26 


9.752392 


9.916341 


9 836051 


10.163949 


10.083659 


10.247608 


34 


27 


9.752576 


9.916254 


9.836322 


10.163678 


10.083746 


10.247424 


33 


28 


9.752760 


9.916167 


9.836593 


10.163407 


10.083833 


10.247240 


3» 


29 


9.752944 


9.9 1608 1 


9.836864 


10.163136 


10.083919 


10.247056 


3" 


30_ 
31 


9-753ia8 


9.915994 
9.915907 


9:837*34 
9-837405 


10.162866 


10.084006 
10.084093 


10.246872 


30 
29 


9-7533»2 


10.162595 


10.246688 


3* 


9-753495 


9.915820 


9837675 


10.162325 


10.084180 


10.246505 


38 


33 


9.753679 


99*5733 


9837946 


10.162054 


10.084267 


10.246321 


27 


34 


9.753862 


9.915646 


9.838216 


10. 16:784 


10.084354 


10.246138 


26 


35 


9.754046 


9.915559 


9.838487 


10.161513 


10.084441 


10,245954 


^5 


36 


9.754229 


9.9*5472 


9.838757 


10.161243 


10.084528 


10.245771 


*4 


37 


9.754412 


9.915385 


9.839027 


10.160973 


10.0846 1 5 


10.245588 


^3 


38 


9754595 


9.915297 


9.839297 


10.160703 


10.084703 


10.245405 


22 


39 


9754778 


9.915210 


9.839568 


10.160432 


10.084790 


10.245222 


21 


40 
41 


9.754960 


9.915123 


9.839838 


10 160162 


10.084877 
10.084965 


10.245040 
10.244857 


2Q 

«9 


9755»43 


9.9*5035 


9.840T08 


10 159892 


4* 


9-755326 


9.914948 


9.840378 


10.159622 


1C.085052 


la 244674 


18 


43 


9755508 


9.914860 


9.840647 


»o.*59353 


10.085140 


^0.244492 


17 


44 


9.755690 


99*4773 


9.840917 


10.159083 


10.085227 


10.2443101 16 


45 


9-755872 


9.914685 


9.841187 


10.158813 


10.085315 


10.2441 28J 15 


46 


9.756054 


9.914598 


9.841457 


10.158543 


10.0^(5402 


10243946' 14 


47 


9.756236 


9 9*45*0 


9.841726 


10.158274 


10.085490 


•0.2437641 »3 


48 


9.7564 « 8 


9.914422 


9. 84 1 996 


10.158004 


10.085576 


10.243582- 12 


49 


9 756600 


99*4334 


9.842266 


«o.* 57734 


10.085666 


10.243400! "' 


50 
5' 


9.756782 
9.756963 


9-9^*4246 
9.9*415^ 


9.842535 


10.157465 
10.157195 


10.085754 


1O.2432181 


10 1 


9.842805 


100858^^2 


10.243037, 


9 


5* 


9.757 «44 


9.914070 


9.843074 


10.156926 


10.085930 10.242856 8 
iao86oi8 10.242674 7 


53 


9-757326 


9.913982 


9.843343 


10.156657 


54 


9-75750: 


9.913894 


9.843612 


10.156388 


10.086106 10.242493 6 


55 


9.757688 


9.913806 


9.843882 


10.156118 


10.086194 10.24.2312: 5 


56 


9.757869 


9.9137*8 


9 844 15 1 10.155849 


10.086282110.2421311 4 


57 


9.758050 


9.913630 


9.844420 »o. 15 5580 


10.Q86370 10.241950; 3 


58 


9.758230 


99*354* 


9. 844689 j 10.1553 11 


10.0864591 10.241770: 2 


59 ,9.7584" 


6.9*345319-844958 10 1550421 


10.086547)10.241589 ' 1 


60 9-75859ij9-9i3365l9'845ai7 


»0.»5477^ 


io.oii6635 10.241409' 
Co-scc Secant. u 


' M . Co- sine.' Sine. Co-tang* 


TanfT. » 



55 Pefrecs. 



•^ 



LOGAKITHMtC SINES, TANGENTS, AND SECANTS. 6L 



55 



Degrees. 



M 



o 
I 

a 

3 

4 

5 

6 

7 
8 

9 

10 



Sine. 



975»59i 
9.758772 

9.75895a 
9-759 »32 
9.750311 

9.759492 

90J9j72^9-9 

9-759»52J9-9 
9.760031 9.9 

9.760111 9.9 



Cocaine. 



9.9 
9.9 
9.9 
9.9 
9.9 
9.9 



9.760390 



II 
iz 

J3 

'4 

"5 
16 

17 
18 

»9 

20 

21 

22 

*i 

24 

as 
26 

*7 
28 

*9 

JO 

3» 
31 

34 
35 
36 

37 
38 
39 

-12 
41 

4* 

43 

44 

45 
46 

47 
48 

49 

JO 

5» 

5* 

S3 

54 

55 

56 

Sf 

58 

59 
60 



9.9 



9.9 



9.760569 

9.76074819.9 

9.7609I7I9.9 

9.761 106 

9.761285 

9.761464 

9.761642 

9.761821 

9.761999 

9-762177 



9.762356 
9.762534 
9.762711 
9.762889 
9.763C67 
9.763245 
9.763422 
9.763600 

9-763777 
9763954 
9.764131 
9.764308 
9.764485 
9.764662 
9.764838 
9.765015 
9.765^91 
9.765367 

9765544 



9-9 

9-9 
9.9 

99 
9.9 

9-9 
9.9 



9.9 
9.9 
9.9 
9.9 

9-9 
9.9 

9-9 

9.9 

9.9 
9.9 

9 
9.9 

9.9 

9.9 

199 
9.9 

9.9 



3365 
3276 

3187 

3099 
3010 

2922 
1833 

2744 
2655 

2566 

^47 7 



T ang. I Co-Ung. t S ecant 



9 845**7, 
9. 845496; 10 
9.845764, 

9.846033 
0.8463021 

9.846570' 
9.846839 



0.154236 10.086813 
•■53967 10.08690 1 
.153698' 10.086990 
1. 1 5 3430 1 10.087078 
^ . *, J. 153 161 ! 10.087167 
9.847107 10.151893 10.087256 
9-847376) 10. 1 52624. io.o«7345 

9 847644.10.152356:10.087434. 
9.847913 10152087. 100875231 

~ - - • • — I 



238819.848181! 10.151819 10.087612 
2299 " " ' 

2213 

2121! 

2031! 



y. w.^u ivailt^. I^ioik^. AW.WO^U t J, 

9.848449110.151551110.087701^ 
9.848717110151283.10.087790 



1942! 

1853 

1763 

1674 
1584 



'495 

1405 

t3«5 
1226 

1136 
1046 

0956 
0866 
0776 
0686 

0596 

0506 
P4«5 

0325 

0*35 

0144 
0054 



9-848986| 

9.849*54; 
9.849522 

9.849790 

9.850058 

9.850325 

9.850593 



9.909963 
9.909873 
9.76572019.9097821 

9.76589619.909691 
9.766072 9.909601 
9.766247 9^909510 
9.766423 9.909419 
9.766598 9.909328 
9.766774 9.909237 
9.766949 9.909146 
9.767124 9.909055 
9.767300 9.908964 
9-767475 9-908873 



9.767649 



9.908781 



9 76782419.908690 

9 76799919-908599 
9.768173 9.908507 

9.768348 19.9084 16 

9. 7685Z2!9. 908324 

9.768697I9.908233 

9.76^871 9.908141 

9.769O45I9.908049 

9 'r69ii9'9.907958 

u ' Co-sine. Sine. 



9.850861 
9.851119 
9.851396 
9.851664 
9.851931 
9.852199 
9.852466 

9-852733 
9-853001 
9.853 268 

9'853TF5 
9.853802 

9.854069 

9854336 
9.854603 
9 854870 

9-855*37 
9.85 54*4 

9.855671 
9.855938 



9.856204 
9.856471 
9.856737 
9.857004 
9.857270 

9-857537 
9.857803 

9.858069 

9.858336 

9.858602 



9.858868 
9.859134 
9.859400 
9. 859666 
9.859932 
9.860198 
9.860464 
9.860730 
9.860995 
9.861261 



Co-tang-. 



-»54)73 
154504 



0.151014 
o. 1 50746 
0.150478 
0.150210 
o. 149942 
0149675 
0.149407 



0.149 1 39 
0.148871 
0.148604 
0148336 
0.148069 
0.147801 

0.147534 
0.147267 

0.146999 

0.146732 

0.146465! 
0.146 1 98 
o 14593 1 
0.145664 

o»45397 
0.145130 
0.144863 
0.144596 
0.144329 
0.144062 



0.143796 
0.143519 
0.143263 
o. 142996 
0.142730 
0.142463 
0.142197 
0141931 
0.141664 
0.141398 



0.141132 
0.140866 
o. 140600 

0.140334 

0.140068 
0.139802 

0.139536 
0.139270 
0139005 



'ian 



54 Degrees. 



Sl 



1.0866 j 5 
086724 



0.087879 
0.087969 
0.088058 
0.088147 
0.088237 
0.088326 
0.088416 



0.08850s 
0.088595 
0.088685 
0.088774 

o.o88«64 
0.088954 
0.089044 
0.089134 
o 089224 
0.089314 



0.089404 
0.089494 
0.089585 
0.089675 
0.089765 
0.089856 
0.089946 
0.090037 
0.090127 
0.0902x8 



0.090309 
0.090399 
0.090490 
0.090581 
0.090672 
0.090763 
0.090854 
0.090945 
0.091036 
0.091127 



0.091219 
0.09 13 10 
0.091401 
0.091493 
0.091584 
0.091676 
0,091767 
0.091859 
0.091951 



0.138739 JO.092042 



CO'S^c. 



0.241409 

0.241228 

0.241048 

o. 240868 

0.240688 

0.240508 

0.240328 

0.240 148 

0.239969 

0.239789; 51 

0.239610 50 



0.239431 
0.239252 
0,239073 
0.238894 
0.238715 
0.238536 
0.238358 
0.238179 
0.238001 
0.237823 



0.237644 
0.237466 
0.237288 
0.237 11 1 
0236933 
0.236755 
0236578 
0.236400 
0.236123 
0.236046 



0.234104 
0.233928 

0.233753 
0.233577 

Q. 23 3402 

0.233226 

0233051 

0.232876 

0.232700 

0.232525 



0.232351 
0232176 
0.232001 
0231827 
0.231652 
0.231478 
0.231303 
0231129 

0.230955 
0.230781 



('o-seo ' Secant 



0.235869 
0235692 
0235515 

0.235338 
0.235162, 
0.234985* 24 
0^234809' 23 
0.234633 22 

0.234456; 2t 
0.234280 20 



»9 
18 

17 
16 

«5 

H 
13 

12 
II 
10 

I 

7 
6 

5 

4 
3 

2 

I 

o 



v2 LOGARITHMIC SIXES, TANGENTS, AND SBCAKTS. 



St* l>ejr*«cs. 



M 



Sine. C'>-sijv-. TaniT' Co-tanj?" i Secant. 



o 9.769219 9-9079)K 9.861Z61 
r 9.769393 9.907866 9.861527 
2'9.'>69566 9.907774 .9.861792 
3 9.769740*9.^07682 9 862058 
419.769913 9.907590 9.862323 
5:9.770087,9.907498 9.862589 
6*9.770260.9.907406 9.S62854 

7!9-7704^3,9-.9073«4 ^863119) 

8 9.7706061^.907122 9.863385- 

9 9.770779 9.907119 9.863650 
10 g.77095« 9.907017 986391 5 



I! 9.771^25 9.4106945 9.864180 
1 2-9.77 1298 9.906852 9 864445 

13 9-77 »470 9.906760 9.8^4710 

14 9 771643 9.906667 9.864975 
1(1977181$ 9.90657519.865240 

1619.771987 9.90648 2 i<> 865505 

i7i9-77a»S9 9.906389I9.865770 
«8J9.77^33>j9-906«96j9-866035 
19' 9.772563' 9.90620419.866300 
20 9.772675!9.906i 1 1 {9.866564 

9.772847l9.9o6oi8t9.866829 
9.773018 Q.905925 9-^67094 



21 
22 

14 

»? 

26 

*7 
28 

29 

JO 

31 
3* 
33 

34 
J-) 
36 

37 
38 

39 

40 



9-773190.9905831 

9-77336«'9-90?'39 

9.773533:9-905645 
9.773704.9.905552 

9.773875j9-905459 
9.774046I9.9Q5366 

9.774*«7!9'905*7a 



41 
42 

43 
44 
45 



9-774388 

9-774558 
9-7747*9 
9.774899 
9.775070 

9.775240 
97754»0 
9.775580 

9-775750 
9-77 59*0 
9.77609O 



9.905179 



9.776259 



9.905085 
9.904992 
9.90489S 
9.904<>04 
9.9047 1 1 
9.904^17 

9.9045*3 
9.904429 

9-904335 
9.904241 



9.8673(8 
9.867623 
9.867687 
9.868152 
9.868416 
9.868680 

9.868945 
9.869109 



9.904147 



9.776429I9. 904053 



9.776598' 
9-776768 

9-776937 
461^.777106 

47 9-777175 
48,9.777444 

49'9.777«13 

5o;9.77778i 

5«:9-777950 
52 9.778119 
53.9.778287 

54:9-778455 
55I9. 778624 

56'9.778792 

57I9.778960 

589. 779128 

^9 9.^^79295 

6 9 779463 



9903959 
9.903864 

9.903770 

9.903676 

9.903581 

9.903487 

9.903392 

9 903^98 



9.903203 
9.903108 
9.903014 
9.9021)19 
9.902824 

9.902''2Q 

9 902n34 



9.869473 
9.869737 
9.870001 
9.870265 
9.870529 
9.870793 
9.871057 
9.871321 
9.871585 
9.871849 



9.872112 

9.872376 
9.872640 

9.872903 

9-873 '67 

9.873430 

9.873694 

9-873957 
9.874220 

9.874484 



0.874747 
9.875010 
9.875273 

987 ^■'> 36 
• 5SC0 
^.8-6063 
9.876326 



M Co-sme. ' 



9,90253919.8765891 
9 90244419.87685 It 

9.902349 0.8771 14' 



0.138739 
ai38473 
a 138208 

0.13794a 

0.137677 

©.1374" 

ai37i46 
a 1 36881 
0.136615 
0.1 3613 50 
0.136085 



0.135820 

0.135555 
0.135290 

0.135025 

0.134760 

0.134495 
0.134230 

0.133965 
0.133700 
o J33436 
0.133171 
fo. 132906 
0.132642 

0.132377 

o. 1321 13 
0.131848 
0.13 1 584 
0.13 1 320 
0.131055 
0.130791 

0.130527 
0.130263 
0.129999 

0.129735 

0.129471 
o 129207 
o 128943 

0.T28679I 
O.L284I5 
o. 1 28 1 5 1 [ 

a 1278881 

0.1A7624 

0.127360; 

0.127097 

0.1268331 

0.126570' 

0.1263061 

o.ia6o43; 

0.125780 

0.125516 



0.125253 
o. 1 24990 
0.124727 
o 124464 
0.124200 

0.123937 
0.123674 
o 123411 
0.123149 

O.I22J<'<6 



0.091042 

0/»92i34 
0092226 
0.092318 
0092410 
0.092502 
0.002594 
0.092686 
0.092778 
0.092871 
0.092963 



0.093055 
0.093148 
0.09^3240 

0.093333 
0093425 
0093518 
D.O93611 
0.093704 
O.09J796 
0.093889 

0.093982 
0.094075 
0.094 1 68 
0.094261 
0.094355 
0.094448 
o.09454r 
0.094634 
0.094728 
0.094821 



0.094915 
0.095008 
o 095 102 
0.095196 
0.095289 
0.095383 

0.095477 
0.095571 

0.095665 

0.095759 

0.095853 
0.095947 
0.096041 
096 1 36 
0.096230 
0096324 

0.096419 
ao965i3 
0.096608 
0.096702 



0>-sec. 



0.096797 
0.096892 
0.096986 
0,097081 
0.097176 
0.097271 
0.097366 
0.09746 1 
0.097556 
0.097651 



0.230781 
0.130607 

0.X30434' 
0.230260, 
0.230087, 
0.119913 • 
0.2197401 
0.229567] 

ai29394 

ar2922it 

0.219Q4S 



0.228875 
asiSjoi 
aii853t> 
0.118557 
0.128185 
0.128013 
0.117841 
ai276€9 
a 217497 
[y.ii73 *5 



0.117153 
0.126981 
0.116810 
0.126639 
0.116467 
0.216196 
0.116115 
0.115954 
0.125783 
0.215612 



0.115441 
0.215271 
0.115101 
0.114930 
0.224760 
0.124590 
0.224420 
0.224250 
0.1140S0 
0.1239 ip 



0.213741 
0.113571 
o 123401 
0.223231 
0.223063 
0.222894 
0.222725 
0.111556 
0.121387 
0.222119 



Sine. 

mmmmmm 



■Co-Uiii^ ' lang • Co-.s c 



ai2i050 
a22i88i 
0.121713 
0211545 
a22i376 

0.22I2OS 
0.221040 

o 2^871 

0.220705 
0.220537 ; 
Secant 



M 



^S Degrees. 



lX>OAItITHMI0 SINBS, TANG£}!7TS, AKD SECANTS. 6^ 



3T Degrees. 



• 



M 



O 
1 

2 

J 

4 

5 
6 

7 

%\ 

9 
lO 



Sine. 



Co-sine. 



9-77946i 
9.779611 
9.779798 
9.779966 
9.780133 
9.780300 

9.780467 
9.780634 
9.780801 
9.780968 

9.7S1134 



It 9.781301 

12 9.781468 

13 9'78i634 
14 

15 

16 

»7 
18 

*9 
20 



9.781800 
9.781966 
9.78213% 
9.782298 
9.782464 
9.78x630 
9.781796 



21I9.782961 

22«, 9-783 «*7 

23 9-783*9» 

24 9-783458 

25 9.783623 

26 9-783788 

27 9-7^3953 , .,,.,, 
9.784118 9.899660. 

9.784282 9.899564: 



28 
29 

JO 

31 
3» 
33 

34 
35 
36 
37 
38 
39 

i2 

41 
42 

43 
44 
4? 
46 

47 
4S 

49 

50 



9-784447 9-899467 



9.902349 
9.902253 
9.902158 
9.902063 
9.901967 
9.9O1872 
9.901776 
9. 90 168 I 
9.901585 
9.901490 
9.901394 



Tang. 



9.877114 

9-877377 
9.877640 

9 877903 
9.878165 

9.878428 

9.878691 

9.878953 

9 879216 

9.879478 

9-87974« 



9.9O1298 

9.901202 

9.901 106 

9.901010 

9.9OO914 

9.90081 81 

9 900722 

9900626 

9.9005291 

9.900433 ; 

9.900337 
9.9OO240 
9. 900 1 44 

9.900047 
9.899951 
9.899854 
9.899757 



Co- tang*. 



9.784612 9.899370 
9.784776 9.899273 



9.784941 
9.785105 
9.785269 

9785433 

9-785597 
9.785761 

9.785925 

9.786089 



9.786252 
9.786416 
9.786579 
9.786742 
9.786906 
9.787069 
9.787232 

9-787395 
9787557 
9.787720 



51 9.787883 9.897418 

52 9,788054 9.897320 

53 9.788208 9.897222 

54 9.7883709.897123 

55 9.788532 9.897025 

56 9.788694 9.896926 

57 9.788856 9.896828 

58 9.789018 9.896729 

59 9.789180 9.896631 
'60 9 789342 9.89653 2 

Co^sine. ' S ne. 



9.880003 
9.880265' 
9.880528 
9.880790 

9.88 IC5 2 

9.88I3I4 

9-881576; 
9.8818^9! 

9.882101* 
9^88i36jt 

9.882625 

9.882887 

9883 14» 
9 883410 

9.883672 
9.883934 

9.884196- 

9.884457 
9884719 

9.884980 



9. 899 r 76 
9.899078 
9.898981 
9.898884 

9.898787 
9.898689 
9.898592 
9 .S98494 ; 

9.8983971 
9.898299 

9.898202 

9.898104 

9.898006 

9.897908 

9.897810 

9.897712 

9.897614 

9.897516 



9.885242 
9.885503 
9.885765 
9.886026 
9.886288 
9.886549 
9-886810 
9-887072 
9-887333 
9.887594 



9.887855 
9.888116 

9.888377 
9.888639 

9.888900 

9.889160 

9.889421 

9.889682 

9.889943 

9.890204 

9.890465 
9.89O725 
9.89O986 
9^191247 

9 891507 
9.891768 
9.892028 
9.892289 
9.892549 
9.892810 



Co -tang 



o. 
o. 

0, 

o. 
o. 
o. 
o. 

o. 

O- 

o. 
o. 

o. 

o. 
o. 
o. 
o. 

0. 

o. 
o. 

o. 
o. 



o. 
o. 
o. 
o. 
o. 
o. 
o. 
o. 
o. 
o. 



a 
o. 
o. 
o 
o. 
o. 
o. 
o. 
o. 

0. 



o. 
o. 
o. 
o. 
o. 
o. 
o. 
o. 
o. 
o. 



o 
o. 
o. 
o. 

0. 

o. 

0» 

o. 
o. 
o. 



22«86 

22623 
22360 

2 209 7 

31835 

21572 
21309 

21047 
20784 
20522 
20259 



9997 

9735 

9472 

9210 

894H 

8686 
8424 

8161 
7899 
7637 



737S 

7113 

6852 
6590 
6328 
6066 1 
5804 

5543 

5281 
5020 



4758 

4497 

42135 

3974 

37«« 

345i 
3190 

2928 

2667 

2406 



2145 
1884 
1623 
1361 
IIOO 

0840 

0579 

0318 

0057 
09796 



09535 
09275 

09014 

08753 
08493 

08232 

0797* 
07711 

07451 

07190 



Tang. 



Secant. 

0.097651 

0.097747 

0.097842 

0.097937 

0.098033 

0098 128 

0.O98224 

0.O98319I 

0.09&415 

0.098510 

0.098606 

0.098702 
0.098798 
0.098894 
0.09X990 
0.099086 
0.099 1 82 
0.099278 

0.099374 
0.099471: 

0.099567 



Co-stj< 



0.099663 
o.099?6oj 
0.099856] 
0,0999531 
0.1 00049 1 



o. 



0.100243 
0.100340 
00436 

00533 



o. 
o. 



o. 
o. 
o. 
o. 
o 
o. 
o. 
o. 
o. 
o. 



o. 
o. 
o. 
o. 
o. 
o. 
o. 
o. 
o. 
o. 



o. 
o 
o. 
o 
o- 
o. 
o. 
o. 
o. 
o. 



001 



46J 



00630 1 

00727 
00824 
00922 
01019 
OI1I6 
01213 
01311 
01408 
01506 



1-01603 
01701 
01798 
01896 

01994 

02092 
02190 
02288 
02386 

02484 



02582 
02680 

02778 
02877 

02975 
03074 

03172 

03271 

03369 
03468 



52 Uegrees. 



Vy0*sec. 

umattmmmm 



O 

o 

K> 

O 

o 
o 
o 
o 
o 
o 



220537 
220369 f 

220202 
220034 
9867 
9700 

9533 
9366 

9199 

9032 

8366 



8699 
8532 
8366 
8200 
8034 
7868 
7702 

7536 

7370 
7204 



M 



7039 

6873 
6708 j 

6542! 

6377| 

6212) 
60471 
5882, 

57i8i 
5553' 



5388' 
5224, 
5059 

4895; 
473 «! 

4567 
4403 
4*39 
4075 
39IJ 

374*8} 

3584 

3411 

3*58 

3094; 

5^93 « 

2768' 

2605 

2443 
2280 

21 17 

»955 
1792 

1630 

1468 
1306 
1144 
0982 
0828 
0650 



Secunt. 



60 

59 
58 
5' 
56 
55 
54 
53 
5» 
5» 
50 



49 
48 

47 
46 

45 

44 

43 
42 

4" 

40 



39 
38 

37 
36 
35 
34 
33 
3a 
3» 

Jl 
29 
28 
27 
26 

45 
*4 
*3 
22 

2f 

20 

«9 
18 

»7 
16 

M 

14 

13 
12 

II 

10 

T 

8 

7 
6 

5 

4 

3 

2 
I 
o 



M 



: 






(M LOGARITHMIC SINES, TANGENTS, AND SECANTS; 



38 Degrees. 



M 

O 


Sine. Co-sine., Tan^. | C(Kang. | Secant. \ 


Co-scc. 


M 


9.789342 


9.896532 


9.891810 


10.107190 


10.103468 


iaiio658 


60 


I 


9.789504 


9896433 


9.893070 


10.106930 


10.103567 


lai 10496 


59 


% 


9.789665 


9-896335 


9-89333^ 


10.106669 


10.103665 


10.110335 


5« 


3 


9.789827 


9.896236 


989359' 


10.106409 


10.103764 


10.110173 


57 


4 


9.789988 


9.896137 


9.893851 


10.106149 


10.103863 


fO.lIOOllI 


56 


5 


9790*49 


9.896038 


9.894111 


10.105889 


10.103961 


10.1O9851 


55 


6 


9.790310 


9.895939 


9.89437 « 


10.105619 


10. 10406 1 


10.109690 


54 


7 


9.790471 


9.89584O 


9 894631 


10.105368 


I a 104160 


10.109519 


53 


S 


9.790632 


9.895741 9.894892 


10.105108 


10.104159 


10.1O9368 


5* 


9 


9-79P793 


9.895641 9.895152 


I a 104848 


10.104359 


iai09207 


5« 


lO 

II 


9.790954 


9.895542 9895412 


10.104588 


10.104458 


10.109046 


50 
49 


9.791115 


9-895443 9-89567* 


10.104318 


10.104557 


10.108885 


11 


9-79»»7S 


9-895343 


9.895932 


10. 104068 


10.104657 


10.208715 


48 


n 


9.79*436 


9.895244 


9.896192 


10.103808 


10.104756 


16.208564 


47 


«4 


9.791596 


9.895145 


9.896451 


10.103548 


10.104855 


10.208404 


46 


«5 


9-79«757 


9-895045 


9.896712 


10.103288 


IO.I049S5 


iaio8i43 


45 


16 


9-79'9«7 


9.894945 


9.896971 


10.103019 


10.105055 


10.108083 


44 


«7 


9.792077 


9.89484619.897231 


10.102769 


10.105154 


10.207913 


43 


i8 


9.791137 


9.894746 9.897491 


10.101509 


10.105254 


10.107763 


4» 


'9 


9.792397 


9.894646i9.89775> 


10.101249 


la 1053 54 


10 107603 


4" 


lO 

11 


9-792557 


9-894546J9.898010 


10.101990 


10.105454 
10.105554 


10.207443 


40 
39 


9.792716 


9.894446 9 89S270 


10.101730 


10.107184 


11 


9.792876 


9.894346 


9.898530 


10.101470 


10.105654 


10.107114 


38 


IS 


9.793035 


91.894246 


9.898789 


10.101211 


10.105754 


10.106965 


37 


»4 


9-793«95 


9.894146 9.899049 


10.100951 


10.105854 


10.206805 


36 


*$ 


9-793354 


9.894046 


9.899308 


10.100692 


10.105954 


I a 106646 


35 


i6 


9-7935 »4 


9.893946 


9.899568 


f 0.100432 


la 106054 


la 106486 


34 


^7 


9-793675 


9.893846 


9.899817 


10.100173 


10.106154 


10.106327 


a 


i8 


9.793832 


9893745 


9.900086 


10.099914 


10.106255 


10.106168 


32 


19 


9-79399" 


9-893645 


9.900346 


10099654 


10.106355 


10.106009 


3« 


30_ 
31 


9;7?4«50 
9794308 


9.893544 


9.900605 


10099395 


10.106456 


10. 205850 


30 


9.893444 


9.900864 


10.099136 


10.106556 


.10.205692 


19 


31 


9.794467 


9893343 


9.901 114 


10.098876 


10.106657 


10.105533 


2i 


33 


9.794626 


9.893243 


9.90138J 


16.098617 


10,106757 


10.105374 


*7 


34 


9.794784 


9.893142 


9.9O1641 


10.098358 


10106858 


10.105116 


16 


35 


9-79494* 


9.893041 


9.9O1901 


10.098099 


10.106959 


10.105058 


25 


36 


9.795101 


9.89294019.902160 


10.097840 


10.107060 


I a 104899 


24 


37 


9.795259 


9.892839 


9.902419 


10.097581 


10.107161 


10.104741 


13 


38 


9-79S4«7 


9.892739 


9 902679 


10.097321 


10.107161 


10.204583 


11 


39 


9795575 


9.892638 


9.902938 


10.097062 


10.107362 


10104415 


It 


40 
4» 


9-795733 
9.795891 


9892536 


9.903197 


10.096803 


10. 107464 


10.204167 


10 
>9 


9.892435 


9903455 


10.096545 


10.107565 


10.104K09 


4* 


9.796049 


9.892334 


9.903714 


10.096286 


10.107666 


10.103951 


t8 


43 


9.796206 


9.892133 


9.903973 


ro 096027 


10.107767 


10.103794 


«7 


44 


9.796364 


9.892132 


9.904131 


10.095768 


10.107868 


10.103636 


16 


45 


9-796521 


9.892030 


9.994491 


10.095509 


10.107970 


10.203479 


•5 


46. 


9.796679 


9.891929 


9.904750 


10.095250 


10.10S071 


1C.103311 


14 


4? 


9796836 


9 891827 


9.905008 


10.094992 


fo.108173 


10.103164 


«3 


48 


9-796993 


9.891716 


9.905167 


10.094733 


10.108274 


10.103007 


11 


49 


9-797150 


9.891614 


9.905526 


10.094474 


10.108376 


I a 2018 50 


II 


12. 
5' 


9797307 
9797464 


9.891513 
9.891421 


9.905784 


10.094216 
10.093957 


10.108477 


10.101693 


10 


9.906043 


10.10S579 


10.101536 


9 


5* 


9.797621 


9.891319 


9.906302 


10.093698 


10.108681 


10.101379 


8 


53 


9-797777 


9.S91117 


9.906560 


10.093440 


10.108783 


10.201113 


7 


54 


9-797934 


9.891115 


9.906819 


10 093 181 


10.108885 


10.102066 


6 


55 


9.798091 


9.8910J3 


9.907077 


10.092923 


10 108987 


10.101909 


5 


56 


9.798247 


9.8909 1 1 


9.907336 


10.092664 


10.109089 


10.101753 


4 


•57 


9.798403 


9.890809 


9-907594 


10.091406 


10.109191 


10.101597 


3 


58 


9-798560 


9.890707 


9.907851 


10.092148 


10.109293 


10.101440 


1 


59 


9.798716 


9.890605 


9.9081 11 


10.091889 


10.109395 


10.101184 


I 


60 


9.798872 


9.890503 


9.908369 


iao9i63i 


10.109497 


10 201128 





M Co-sine. 


Sine. 


Co-tan{2^. Tang • Co-sec. ' 


Secant m | 








51 J 


l^grces.' 









' JUMABITHMIC StNES, TANGB!fT9, ANB SECANTS. 6fi 



T 


Dine. 


Co-wie 


Tang, Co-ting. 


SecBnt_ 


Co.«ec, 


-^\ 


^jsiTI 


^90503 


9.908369 


.0.0,1631 


lo 109497 10.10.118 6d 




9.79^18 


9,890400 


9.,086iS 




O91J71 


1O.1096DO .0.100^7, 59 




9.799184 


9.890.93 


,908886 




091114 


10.109701 10100B.6 58 




9-799J39 


9.890195 


9.909144 




090856 


10 .09805 id,,oq66i 57 




<l 79949 J 




9.90940, 




090598 


.0.109907 10.WO5O5, 56 




9.799'>' 




9909660 




090340 


10.110010 iOK>0349 5; 


6 


».7»<0« 




9 9099 18 




090081 


iDi.Dii,io.wo.9+! 54 




9.79996* 


988978; 


9910177 




08,8,3 


.o.i.oi.s 'o-ioooja 53 


a 


9.800117 


9 889aSi 


9.910435 




089565 


.ai.03i8,.o.. 99883 5. 




9.800171 


9.889579 


9.9 1069 J 




CX9307 


10..10411 10.. 99718; 51 




9.800417 


9.889477 


9^095. 




08,049 


10,1105,3 '0'99S73 50 


77 


9.800^81 


9-»»9314 


9.9.11D9 


To 


■^7";^ 


10.1106,6 iai994iB 4, 




9.800737 


9.B8917' 


9911467 




088533 


10.110719 '0.199.6J; 48 


ij 


9800891 


9.889.68 


9,9117.4 




088176 


1011083, 10,19910s 




'4 


9801047 


98890*4 


9.9"98. 




0880.8 


10.110936 10.19B953 


46 




9.SOI30I 


9 888961 


9.911140 




087760 


.O.IMO39 10,1,87,9 


45 


',1 


9.»<J.Ji6 


9888158 


9,91.498 




08750, 


10.1.114.10-198644 




<7- 


9.80,5.. 


9888755 


9.91.75* 




087,44 


iO.I1114i;10 198489 




it 


9.SO-66S 


9 8886 51 


9.913014 




081^986 


10,111349,10.1,8335 






9-8018.9 


9.888548 


9.913171 




086719 


10.1M45. .0,i,8iBi 






9801973 




9.9135.9 




086471 


.D.M1556 .0.1,80.7 






9.801.18 




99 '3787 




086.13 


10.111659 10,1, 7R71 


IT 




9.8oii8» 


9 ' 


9.914044 




0859;* 


10.111763. 10.1977'B 


38 




980143& 


9^i.'- 4 


991430. 




085698 


lo 111866 


10.1,7564 


37 




9.801(89 


9 ■■. 


,914560 




085440 


10.111970 


.0.197411 


36 




9.Bo«74j 




9 9'48'7 




085.83 


10.111074 


10,197,57 


.15 


iS 


9.8MB97 


9 '■ ' 


9-9' 5075 




0849,5 


lalliis! 


10,197103 


34 




9 8OJ05Q 


9 < 


99 '533. 




C84668 




10.196,50 


33 




9.S01104 




9-9'SS90 




0844.0 


io,.ijB6 


10 .,57,6 


31 




9-*033i7 




9.9.5847 




084153 


10.1 11490 


10,1,6643 


i' 


7" 


9.80,511 


I _i 


9'9'6_^04 


10 


083896 


10,111594 


I0.r,648!» 


30 


9.»OJI(.4 


9.88730. 


9.9.636, 




083638 


.0.1 1.698 


10.196336 


i» 


9.80,8.7 


!,.887I98 


9 9>**>9 




083381 


10.11 .801 


10 .96.83 






9.801970 


9.B87O9, 


9.9.6877 




083.13 


10.11.907 


10.196030 






9. 8041 1] 


9.8)6989 


9.9'7.ft 




081866 


10.II30I. 


10.195877 






9.804.7* 


9.S8688; 


9-9 '739' 




081609 


1D.I1J1I5 


.0.1957.4 




i6 


9.804418 


9.K86780 


9 9>7648 




08,351 


IO.1.J110 


10.. 95571 




3' 


,.t04j8. 


9' 886676 


9.9 0905 




o8m9s 


10-W3314 


10.195419 






9.804734 


9.88*571 


99'8.63 




081837 


10.1.3419 


10.195166 




19 


9.8048I6 


9.8864** 


9.91 84W 




081580 


ID.I.3S34 


iai9Sii4 






9.805039 


9.8S6](. 


9.918677 




0813^ 


.0-1.3538 


.0,19496. 


V 


«<~ 


9.805191 


9.886,57 


9.91 89 J4 


10 


08.966 


10.. 3743 


.0 194B0, 


*» 


9 80SJ4J 


;.886„> 


9.919191 




080809 


.0,..3848 


.0 194*57 




*1 


9.S0549S 


9 886047 


9.9.9448 




08055. 


10.113953 


la 194505 




44 


9 80 J 647 


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9-997979 




KI.149S07 


iC.5.S»« 8 


13 


9.848199 


f. 85036! 


9.998131, '0 001 769 


10.149631 


1015.401 7 


54 


♦.«.;«-if 9.850141 


9.99B48+ 10,00.516110.149758 


.015. 1J4 6 


Si 


9.8^>-;,; .,,850116 


9-99e737!.o.O0ii6j! 10 149884 


lO.Sli+S 5 


J6 


9.8.1 ■:.,-„ .1.849990 


9-998989 10.0010. .JIO-MOOIO 


101 (lot. 4 


17 


9.84.,--... .j.Bt9864 


9. 999 141 .0.0007581 laijo.je 


101(08,4 3 


I» 


9.84,.-. ^.849738 


9-999495, '0-OOOS05, .ai50i6i 


io.(Oj68 1 


J9 9-«4;:. 919-^9611 


9-999747:>O.O00iiJ,. 0.1(0389 


.0.506+. ^ 


60 9-84.- -i ').S49485 


lO.OOOOOO 10.000000! .0.1 (Of 15 








ir'CQ-:,r.,<=- • Sine. 1 Co-luij. TWR. 1 (krt«. 


SeCHlt. M 



TABLE. III. 

JVatural JSmet. 

In this table tbe natoral aineB arc €xbxbited taeteiy degree and 
Qiinute of the quadrant, and arranged sa that the degrees correftpond* 
ing to the sines are to be taken from the top* of the page with their 
minutes in the left side columns, and the degrees, answering to the co- 
sines from the bottom with their miniites in the right side columns* 

The natural sine or co-sine of anjr number of degrees, &c» more 
than 90^ is the same as the natural sine or co-sine of its supplementt 
Ibund by subtraethig them Crom l&Qo ; or the natural sine or co^sine 
of an arch greater than 90^ is the natural co*sine or sine of its excess 
abore 90*. 

To find the mOmral Siae or iCo^umt qf a givtn Mumttr qf Degrec^^ 

3£nute8^ and Seeonda : 

OTf tojind thidegrecM, Minnteaf and Secandsf corre^onding to a givers 

natural Sine or Co^sine. 

These are to be fbund as directed for the logarithmic sines^ 8cc. ex* 
cept that the differences to loo'^ are to be taken from the bottom of 
that column containing the given degx^ees in the former ca8e> or the 
nearest naturid sine or co-sine in the Utter* 

XXAJiPXSI. 

Required the natural Sine orS2<>21'45^or its;Supplement U7^ 38' 15^. 

The natural sine of 32© 3 1*^ is 535090 

The difference at the bottom of the column containing the na« ^ 
tural sine of the given degrees and minutes n 409, thb i , 
multiplied by 45, pointing off two figures in the product, I "^ 



184 



IS 

Sum.is the natural sine re<^ired » . . • 535374 

.Required the natural Co-sine of 7 w 40' 35", or IQ8o 19' 35''. 

The natural co-sine of 7Y<» 40' is 314545 

The difference 460, multiplied;by 85, pointing off twofigures, is — 115 

Remaiader is the natural co-sine required * ^ * 31 4430 

XXAMPLB XII, 

Required the Degrees, Minutes, and Seconds, answering to the natu- 

ral &ine 495994. 
The natural sine next less to that given is 495964, answering to 
39e 44' ; the difference between this natural sine and the given one is 
30, to which two cyphers being added, and that divided by 432, the 
difference at the bottom of the column, gives the quotient 7* to be an- 
nexed to 290 44'. Hence 29' 44' 7", or its snppletnent 150« 15' 53^ 
are the degrees) 8cc. required.. 



72 tfAntUH. SINEid. 

\ 

EXilMPLE IT. 

Required the degrees^ Minutes^ and Seconds, answering to the natu^ 

ral Cocaine 368805« 

The natural Co-sin^ next greater to that gir^ft b 366936^ to which 
answers 68o 31'; the difference between this natural sine and the given 
one is 131, to which two cyphers being added^ and that dirided by 
451, the difference fonnd at the bottom of the cohimn, gives the quo* 
tient 39 '. Hence 68o 21' S9"»or iuaupplenaenti Ui<» 38' 3i"are the 
degreeSf See. required^ 

Tojind the natural verged Sine qfa given J/umier o/ Degree; A^ttuie^f 

and Seconder 

If the given arch be less than 90^9 find its natural Ohsine, which 
subtract from lOOOOOO, and the remainder will be the natural versed 
sine required. But if the given arch exceed 90^| find the natural co- 
sine of its supplement, which add to lOOOOOO, and the sum will be 
the natural versed sine required. 

EXAMPlB t. 

Required the natural tersed Sine of 30^ 39^ 
The natural co-sine of 20<^ 39' is 935752, which subtracted from 
lOOOOOOi leaves 064248| the natural versed sine of 20^ 39'< 

EXAMPLE tu 

Required the natural versed Sine of 146<> 38^ 40/^^ 
The natural co-sine of 2S^ 21' so" (the supplement of l46o 38' AO") 
is 835274» which added to LOOOOOO^ the sum 1835^274 is the naiunl 
versed sine required. 

Tojind the Degreea^ t3^e, corre9ponding to a given natural vetoed Sine. 

Take the difference between the given natural versed sine and 
1000000, and the remainder will be a natural co-sine ; the degrees, 
&c. corresponding to which, will be those required, if the given natu- 
ral versed sme be less than 1000000^ but if otherwise, it will be their 
supplement. 

Example i. 

• • 

Required the Decrees; &c. answering to the natural versed ^ine 098965. 
The above subtracted from 1000000, leaves 901035> which taken as 
a natural co-sine^ corresponds to 25^ 42' 20'' • 

EXAMPLE II. 

Required the Degrees, Scc.answeringto the natural versed Sine 1 160172. 
Here lOOoOOO subtracted from the above, leaves 160172, which ta- 
ken out as a natural co-sine, corresponds to 80® 46' 59''; therefore its 
suppleinent 999 13' 1" are the degrees^ fcc. required. 



ftiVT^UAL SlNSa 






o 
I 

2 

^ 

4 

5 
6 

7 
8 

9 

JO 



0* 



11 
»3 

i6 

I? 
i8 

•9 

20 

21 
22 

23 
^4 
*S 

a? 
c8 

ig 

30 

3» 
32 
33 
34 
35 
36 
37 
38 

39 

40 

4« 

4» 
43 

44 

45 
46 

47 
48 

49 
12 
5» 
5» 
53 
S4 
55 
56 
57 
58 
59 
60 

M 



000000 

000291 
ooo;8z 

D00873 
OOII64 
001454 

001745 

002036 
002327 
002618 
002909 



003200 
003491 

003782 

004072 
004363 
004654 

004945 
005236 

005527 
005818 

006I09 
006399 
006690 
0069SI 
007272 
007563 
007854 
008145 
008436 
008727 



OI74S* 

017743 
018034 

018325 

018616 

018907 

019197 

019488 

019779 
020070 

020361 



3» 



Q34899I052336 
035 1 90 052626 



035481 

03577* 
036062 

0363 53 
036644 
036934 
037225 
037516 
037806 



020652 
020942 
021233 
021524 
021815 

QZ2106 
022397 
022687 
O2297R 
023269 



03809? 
038388 
038678 
038969 
039260 
039550 
039841 
040132 
040422 
040713 



023560 
023851 
024141 
024432 
024723 
025014 
025305 
025595 
0258X6 
026177 



OO9OI7 
009308 
009599 
OO989O 
010181 
010472 
010763 
01 1054 
011344 
01 1635 



01 1926 
Ol21!7 
012508 
012799 
OI3O9O 
013380 

or367i 
013962 
014253 
014544 



052917 

053*07 
05349« 

053788 
054079 
05436y 
054660 
054950 
055241 



05553* 
055822 

056112 

056402 
05^693 
056983 
057274 
057564 
057854 
J58145 



026468 
026759 
027049 
027340 
027631 
027922 
028212 
028503 
028794 
029085 



041004 
04 I 294 
041585 
041876 
042166 
042457 
042748 
043038 
043329 
043619 

043910 
044201 
044491 
O44782 
045072 

0453^3 
045654 

045944 

046235 

046525 



058435 
058726 
059016 
059306 

059597 
059887 

060177 
060468 
060758 
061049 



4« 



.1 



069756 0871(6 
070047 j 087446 

0703371087735 
0706271088025 

07O917 088315 

071207I088605 



07M97 
071788 
072078 
072368 
072658 

072948 

073*3^ 
073528 
073818 
074108 

074399 
074689 

074979 
075269 

075559 



075849 
076139 
076429 
076719 

077009 

077*99 

077589 
077S79 
078169 
078459 



061339 
061629 



061920079329 
062210 079619 



062500 
o627i9i 
063081 
063371 
063661 
063952 



014835 
01 5 1 26 
01 5416 
015707 
015998 
016289 
016580 
016871 
017162 

01745* 



89' 



029375 046816 



C29666 
029957 
030248 
030539 
030829 
03 11 10 
03i4i<i 
034702 
051992 



0322*3 

03*574 
032864 

033155 

033446 

033737 



0343 '8 
03 4 609 
034899 

88' 



047106 

947397 
047688 

047978 
048269 
048559 
048850 
049 J 40 
04943 J 



049721 
050012 
050302 



064242 
064532 
064823 
065113 
065403 
065693 
065984 
066274 
066564 
066854 



088894 
089184 
089474 

089763 
0900^3 



090343 
090633 
09O922 
09 1 2 1 2 
091502 
O91791 
O9208 I 
092371 
09 2660 
09 29 50 



07^*749 
079039 



079909 
080199 
080489 
080779 
081069 
081359 



093239 

0935*9 
093819 

094108 

094398 

O94687 

094977 

095267 

095556 

095846 

096135 
096425 
096714 
097004 
097293 
097583 
097872 
098(62 
098451 
O98741 






a' 



>««»«• 



104528:121869:139173 1564346; 

104818' 122158 139461 156722 5* 

105107! 122447 139749 157009 V 



105396 
105686 
105975 
106264 
•06553 
106843 

10713* 
10742 1 



107710 
107999 
108289 
108578 
108867 
109156 
109445 
109734 
110023 
1103 13 

110602 
11C891 



111469 
111758 
112047 
1123^6 
1 12625 
112914 
113*03 



122735 
123024 
123313 
1 2360 1 

123890 

124179 
124467 

124756 



125045 

1*5333 
125622 

125910 

1 26 1 90 
126488 
126776 
127065 

»*7353 
127642 



127930 
128219 



140037 1572^^ 
140325' I 57 584 

1406 1 3 1578" 1 , 5 
140901.158158! 5 



■» 

^'1 



I4ii89;i58445|5 

141477:158732 5 
141765:15902015 



I42053n 59307 

142341 I59S94 
142629 1598^1 
1429 171 1 60 1 68 
143205; 160455 

14349 ?!«6o:^.n 
143780! 161030 



144068 

144356 
144644 

«4493* 



145220 

145507 



11 1180 i28507 145795 



081649 
081939 
082228 
082518 
082808 
083098 
083388 
083678 
083968 
084258 



067145 

067435 
067725 



0505931068015 
0508831068306 
0511741068596 



11349* 
113781 

1 14070 

"4359 
114648 

114937 
115226 

'155>5 
1 15804 

116093 



099030 
099320 
099609 
099899 
100188 
100477 
100767 
101056 
101346 
101635 



084547 
084837 
085127 
085417 
085707 
085997 



034027: 05i464;068886[o86286 



05 175 5 {0691 76! 086 576 

05 20451069466! 086866 

J0523361069756I08-156 

1 87^1 ~Ho^\ ~8 J*" 



5* 

4' 
4 
4 
4' 
4 
4 
4 
4 
4 



2879C 1460H3 

129O84 14637 I 

1*9373 146659 

129661 146946 

129949 «47*34 

130238 147522 

130526 147809 

130815 148097 



131103 

13139> 
.'31680 

131968 



148385 
148672 
148960 
149248 



132256 149535 



«3*545i 149823 



i 10382 
116671 
116960 
117249 

117537 

117826 
1181J5 

1 1 8404 
1 18693 
118982 



i3*833ji50iii 
I33i2i!i50398 
133410'j 150686 

135698 150973 



161317 
161604 
161891 

162178141 

162465 3< 
162752 3! 
163031, 3' 
•63326 V 
•63613 J 
t6390oJ3- 
i64i8''|3. 

i64474'3 
16476 1 ji 

i6504F'i< 

2< 

li 

2* 



133986 

*34*74 
134563 
134851 
135139 
1354*7 
135716 
136004 
136292 



15 1 261 
151548 



165334 
165621 

165908 

i66iv)5 2( 

166482 

166769 

167056 

16734* 
167629 
167916 



i65i2J3 
16X4S9 
168776 



1518361169063 
1521231169350 
15241 il 169636 



2( 
II 

r 

r 
1 



101924 
102214 
102503 

102793 
103082 

103371 
103661 

1*^950 
104239 
10452S 



6^' 



119270 

119559 
119848 

120137 
1 20426 
120714 
121003 
121292 
121581 
121869 



136580 
136868 
137156 

137445 
•37733 

138021 

138309 

138597 
1388X5 

139173 



«J 



>o 



152698 
152986 

153-73 

153561 

153848 

154136 

» 544*3 
154710 

I 5499H 

155285 

15557* 



169923 

170209 
170496! I 
170785; « 

17JO69 

171556 
1-1643 
171929 
172236 

1-2^2 
I727i>9 



i55iJ(>0 173075 



I 56147 
•56434 



81' 



173362 

173648 



Natural Co-sines. 




J»|Ttlfft4JU «J«C»3b 



M 

O 
I 

2 

3 

4 

5 
6 

7 
8 

9 

lO 

1 1 

12 

>3 

14 

'5 
[6 

17 
(8 

»9 
ti 

12 

^3 
14 

i6 

t7 
i8 
'9 

;o 

II 

12 

3 

4 

5 
6 

7 

8 

9 

o 

I 

2 

3 

4 

5 
5 

7 

i 

} 
) 



10' 



173648 

•"3935 
174221 

174508 

174794 
175080 

175367 

»7^939 
176^26 

176512 



ir 



176798 
17*085 

i7737» 

177657 

'77944 
178230 

178516 

178802 

179088 

179375 



192807 

•93093 
«93p8 

195664 



12' 



19080 • 
191095 
191380 
(91666 
191951 

192^37 
19^5221209619 



179661 

«79947 
180233 
180519 
18080, 
181O9I 

»8i377 
i8i663 

181950 ^,--_, 
182236 199368 

182522 
182808 



193949 
194234 
194520 
194S05 
195090 
•95376 
195661 
195946 
196231 
•965'7 

196802 
197087 
1^7372 

197657 
197942 
198228 

«985»3 
198798 
199083 



209903 
210187 



1.-> 
J 



>^o 



2J79I 2 224951 
208196 225234 
208481 225518 
208765 225801 
2O9O5C 22608s 
209334 226368 
226&5I 
226935 
217218 
210472*^227501 
210756 227784 



; 



21 1040 
211325 
21 1609 
211893 
212178 
212462 
212746 
213030 
213315 

i'3>99 



183094 

'83379 
•83665 

183951 

184*37 
184523 

184809 

185095 



185381 
185667 
185952 



'99653 
199938 
200223 
2 10508 
ZOO793 
201078 
201363 
201648 
201933 
202218 



213883 
214167 
214451 

»»4735 
215019 

215303 

215588 

215872 
216156 
2 1 6440 

216724 

217008 



228068 
228351 
228634 
228917 
229200 
229484 

229767 
23CO50 

230616 




241922 
242204 
2424>:>6 
242769 
243051 

M3331 
243615 
243897 
244179 
244461 

244743 



230899 
231182 

231465 
231748 
232031 

232314 
232597 
232850 
233163 

23H4S 

233728 
23401 1 



245025 

245307 
245589 
245871 
246153 

246435 
246717 
246999 

247281 
247563 



2^^819 
259100 
259381 
259662 
259943 
260224 
260505 
260785 
261066 
261547 



275637 
275917 

276197 
276476 
276756 

277035 

277315 
277594 



2v2372 
292650 
292928 
293206 
293484 
293762 
294040 
294318 



^ 



59 
58 

56 
55 

53 

261628 278432!l95i52r3ii782 32 8317 .50 



277874!»94596 



18^ 



309017 
309294 
309570 
309847 
P0123 
310400 
310676 



l9o 



32SH3 
32611S 

32639s 

32666K 

326943 



r60 



327*18 



3*0953. 3*7493 
311*29 3x770* 



278 1 53 29487413 1 i$o6 328042 



247845 
248126 
248408 
248690 
248972 
249253 

M9535 

249817 

250098 
250380 



26 1 908 
262189 
262470 
262751 
263031 
263312 
263592 

263873 
264154 

264434 



202502 
202787 
203072 



^17292 234294 
217575*234577 
217859 234859 



186238 203357 



186524 
186810 
1 87096 
•87381 
187667 
187953 



203642 
203927 
204211 

204496 
204781 
205065 



188238 205350 



188524 
(88810 
189095 



205635 
205920 
206204 



218143 
218427 
218711 
218995 
219279 

219562 
219846 
220130 
220414 
220697 
220981 
221265 
221548 
221832 
2221 16 



235142 

235425 
235708 

235990 

2^6273 



189^81 206489 
189667 206773 
189952 207058 
190238 2C7343 



190523 

190809 



79' 



207627 
207912 



78" 



222399 
222683 
222967 
223250 

223534 
223817 

224101 

224384 

224668 

224951 



236556 
236838 
237121 

237403 
237686 

237968 

238251 

238533 
2388161 
239098 



250662 

250943 
251225 

251506 
251788 
252069 
252351 
252632 
252914 

253195 



278712 
278991 
279270 
279550 
279829 
280108 
2S0388 
280667 
280946 
,281x25 

281504 
281783 
282062 
282341 
282620 
282900 

283179 
283457 



770 



239381 
239663 
239946 
240228 
240510 

240793 
241075 

241357 
241640 

241922 



253477 
253758 
254039 

254321 
254602 
254883 
255165 
255446 

255727 
256008 



264715 
264095 
265276 

265556 
265837 
266(17 

266397 
166678 
266958 283736 
267238 284015 

267519 
267799 
268079 
268359 
268640 
268920 
2692CO 
269480 
269760 
270040 



i9543o;3i2059 32S592I49 

295708i3«i33$l32»867,4i 

295986I31261 11329 »4'.47 
296264 '312888 3 294i6'46| 
196542 3131^1329691 '4; 

313440 329565^44 



270320 
270600 
270880 
271160 

271440 
271720 
272000 
271280 
272560 
272840 



296819 
197097 

297375 
297653 

2979 30 

298208 

298486 

298763 

29904 I 

299318 

299596 

299873 

3':>oi5i 

300428 

300706 

300983 
301161 
301538 



313440 
313716 

313992 
314269 

31454s 



314821 
1315097 

315373 

315649 

315925 
316201 

316477 
3i6-'53 
317019 
3 • 7305 



441 

33014043 

3305 I4f42 

330789141 
331063(40 



331338*39 

331612 38 

331887 
332161 

33243s 

332710 

332984 

333533 

333^7 



317580 

317856 

3i8i?2 

3018151318408 



284294 

284573 
284852 

285131 

285410 301093 3(8684 
285688 302370 318959 
285967 302647 3^923$ 
286246 301914 3 1951 1 
286545 303102 319786 
286803 303479 320062 

2870821303756(320337 
287361 [304033 310613 
2876391304310 320889 
2879181304587 311164 
288196I304864 321439 
288475J305141 321715 

288753I305418 321990 
2890321305695 322266 
2893to,'305972 321541 
28958913062493 228 16 



334081 

33435$ 

334629 

334903 
335>78 
335452 
335726 
336000 
336274 
336547 



256289 
25657 r 
256852 
257133 

257414 
257695 

2579:6 

258257I275078 

258538 

258819 



7AO 



76* 



70' 



273120 28986713065261323092 
273400 290145 306803 323367 
273679 390424I307080 

273959 290702J307357 
274239 2909811307633 
274519 291259 307910 
27479^ 291537 308187 
275078 291815:508464 
275358 292094-308740 



275637 



74" 



::!■ 



323367 
323641 

323917 

324193 
324468 

324743 
325018 

325293 



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730 I ^2"* 71° 



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TJIBIE jr, 



THE 



A IJ G L E S 



Which every Point and Quarter Point of the Compass 

makes with the Meridian. 



KORTH 


POINTS 

9i 


O 1 /' 

2 48 45 


POINTS 


SOUTH 1 














Oi 


5 37 30 


H 










OJ 


8 26 15 


Oi 






N. b. £. 


N- b. W. 


1 


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1 


li 


14 3 45 


11 










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16 52 30 










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19 41 15 


4 




, 


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N. M. W. 


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25 18 45 


2 


O* i>« E* 


S. S.W. 


« 




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2d 7 39 


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H 






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3 


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S. ^r - biD« 






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36 33 45 


3i 










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31 










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47 48 45 


4 
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♦ 


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41 

4 






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E. N. E. 


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6 


E« S« E. 


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6i 


70 18 45 


«i 










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7 


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7 


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9 




74 


81 33 45 










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84 22 30 










7^ 


87 11 15 


7i 






Ease 


West. 


8 


90 


8 East 


West 



( 83 ) 



TABLE V. 



A TRAVERSE TABLE, 

7b every Degree and Quarter Degree of the Comfia99 or fforizon, 

EXPLANATIOK. 

This Table is calculated foi" the easy and expeditious solution of 
the several cases of Right-angled Plane Trigonometry. It is general* 
ty esteemed a useful and requisite assistant to the Surveyor, the 
Navigator, and to every on^ who has any concern with trigonometry 
in the exercise of his profession. The manner of using it must be very 
obvious to all, who are acquainted with the principles of that excellent 
branch of geometry ; but to those, who have only a superficial know-< 
ledge of the subject, the following description and examples will be ' 
necessary. 

In this Table, one of the acute angles—whether given, or required*^ 
If less than 45<>, is found, to the nearest 15' at the top of the page ; but 
if more than 45^, it must be sought at the bottom, where the numbers ^ 
are found in a retrograde order. And whether the angle under consid- ' 
eration, be at the top, or bottom, the Hypothenuse, if less than 130, is 
always in 9LDi9tance column ; against which, in a column marked Lati-^ 
iudey h found the side contiguous to the angle ; and in a column^ 
marked Defiarturey the side opposite the angle. 

When the given numbers exceed the limits of the table, any aliquot 
parts, such as a half, one third, &c. may be taken ; and those found 
Cbrresponding are to be doubled, trebled &c. that is^ multiplied by the 
alone figure^ that the given number is divided by. 

1. Let the Hypothenuse of a right angled tirkuigle=96 and one of 
the acute anglesc=33o 45' ; required the sides. 

Under 33o 45' at the top of the table, and against 96 in a Distance 
column, are found 79.^4 in a Latitude column for the side contiguous 
to the given angle, and'53.34 in a Departure column for the side oppo- 
site the given angle. 

3. Let the sides of a right imgled trianglei besB69.33 and 66.03 } 
required the angles and Hypothenuse. 

By inspecting this table, till these two sides are found agunst each 
cither in adjoining coiumns of Latitude and Departure) the angle op-> 
posite the longest side is found to be 530 30% the other, 36<» 30^ an4 
the Hypothenuse, 111.^ 

In this manner all the cases of Right-angled Plane Trlgonometi^' 
oan be readily solved ; but for more p^miettliir directions} \^%^ Q^ t(»|9 
subject should be Consulteili 



9i 



4 nSOBEfiS. 





15' 


1 

Dep. 

0.00 


^1 

1 


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Lat { 
1.00 


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aoi 


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1 


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4 


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S 

6 


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^ 5 

6 


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6 


600 


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7 


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7 


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7 


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8 


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22 


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30 
31 


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3' 


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31 00 


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3* 


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31.00 


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3« 


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33 


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33 


3300 


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33 


3300 


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34 


34.00 


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34 


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a30 


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3400 


0.44 


35 
36 


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3S 

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37 


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38.00 


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38 


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38 


38.00 


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39 


39.00 


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39 


39.00 


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39 


39-00 


051 


40 
4« 


40.00 


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40 
4J 


40.00 


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4700 


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48 


48.00 


0.21 


48 


48.QO 


0.42 


48 


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0.46 


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54 


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60.00 


0.52 


60 


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Dep. 


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30 


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41.99 
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11.00 

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14.00 
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20.00 

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22.99 
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26.99 
27.99 
28.99 
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0.79 
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26.99 
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30.99 

31.99 
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35.98 
36.98 

37-98 
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40.98 
41.98 
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60.91 


60.90 


3.46 


60.87 


61 


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3.14 


61. 90 


3.5« 


61.88 


3.79 


61.87 


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62.91 


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61.90 


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3.86 


67.87 


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68.91 


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68.87 


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68.85 


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70 
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69.90 

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70.89 


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71.88, 


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71 87 


4.40 


71.85 


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72.86 


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72.84 


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73.86 


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74.90 


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95.82 


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102.8 


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1028 


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103.8 


5.90 


103.8 


6.35 


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106; 


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104.8 


5 95 

6.01 


104.8 
105.8 


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105.8 


647 


105.8 


693 


107, 


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106.8 


6.07 


106.8 


6.53 


106.8 


7.00 


108 


107.9 


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1078 


6.12 


107.8 


6.59 


107.8 


7.06 


109 . 


108.9 


S-70 


108.8 


6.18 


108.8 


6,65 


10S.8 


7.1 3 


110 

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109.8 


5 76 
5-8 1 


139.8 


6 24 
6.19 


109.8 


6.72 


'IQ9.8 
110.8 


7.19 


tio.8 


110.8 


110.8 


6.78 


7.26 


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II 1.8 


5.86 


tii.8 


6.35 


111.8^ 


6.84 


111.8 


7.33 


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112^8 


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114 


113.8 J 


5.97 


113-8 


646 


113.8 


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116 


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117.8 


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119 


118.8 1 6.13 


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118.8 


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110 


119*8 ,' 6.18 


119.8 


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119.8 


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8.98 


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40.90 


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45,89 


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3,69 


46.84 


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49.86 


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51.86 


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51,84 


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51.81 


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51,84 


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56.0 


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Lat 


Dep. 


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6<1X5 ; 4 z6 


60,83 4.52 


60.81 


4.79 


60.79 


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62 


61.85 


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61.79 


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63 


62.85 


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62.81 


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62.78 


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65.84 


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63.82 


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4.53 


64.82 


482 


64.80 
65.80 


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64.7£ 
65-77 


5.38 

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67 


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66.82 


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66.77 


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67.81 


5.04 


67.79 


534 


67.77 


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69 68 83 


481 


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68.79 


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68.76 


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70 


69.83 
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488 


69.81 


5.19 


69.78 


5-49 


69.76 


5.80 
5.88 


71 


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70.80 


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70.78 


5-57 


7a76 


7% 7T 82 


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71.80 


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7178 


565 


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5.96 


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72.80 


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72.78 


573 


7275 


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73.80 


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73 77 


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73.75 


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74.79 


5.56 


74-77 


588 


74.74 


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76 


5-30 


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5.63 


75.77 


5.96 
6.04 


75-74 


6.29 


77 t 7681 


5 37 


76.79 


5-7' 


76.76 


76.74 


6.38 


78 1 77.81 


5-44 


77.79 


5.78 


77.76 


612 


77.73 


6.46 


. 79 ! 78.81 


55" 


78.78 


5.85 


7876 


6.20 


78.73 


6 54 


80 


7981 


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79.78 
80.78 


5-93 


79-75 


628 


79.73 


662 


81 


80.80 


5.65 


6.00 


80.75 


6.36 


80.72 


6.7* 


82 


81.80 


5.7» 


81.77 


6.08 


81.75 


6.43 


81.72 


6.79 


83 


82.80 


5.79 


82.77 


6.15 


82.74 


6.51 


82.71 


687 


84 


83.80 


5.86 


83.77 


6.23 


83.74 


6.59 


83.71 


6.96 


85 
86 


84.79 


5 93 

6.00 


84.77. 


630 


84.74 


6.67 


84.71 


7.04 


85.79 


85.76 


•6.37 


85.73 


6.75 


85.70 


7.12 


87 


86.79 


6.07 


86.76 


6.45 


86.73 


683 


86.70 


7.20 


88 


8779 


6wi4 


87.76 


6.52 , 


8773 


6.90 


8770 


7.29 


89 


88.78 


6.21 


88.76 


6.60 


88.73 


6.98 


88.69 


7.37 


90 
9" 


89.78 


628 
6.35 


89.75 


6.67 


8972 


7.06 
T"«4 


89.64 
90.69 


7.4f 
7-54 


90.78 


90.75 


6.74 


90.72 


9» 


9178 


642 


9>.75 


6.82 


91.72 


7.22 


, 91.68 


7.62 


93 


92.77 


6.49 


9».74 


6.89 


92.71 


7.30 


92. 68 


7-70 


94 


93.77 


6.56 


93-74 


6.97 


9371 


7.38 


93.68 


7.78 


95 
96 


94-77 


663 


9474 


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94.71 
95.70 


7.45 
7.53 


94.67 


7.87^ 
7.95 


95-77 


6.70 


95.74 


95.67 


97 


96.76 


677 


96.73 


7.19 


96.70 


7.61 


96.67 


8.oi 


98 


97.76 


6.84 


9773 


7-26 


97 70 


7.69 


97.66 


8.12 


99 


98.76 


6.91 


98.73 


7.34 


98.69 


7.77 


98.66 


8 20 


lOO 

101 


99.76 


698 


99.73 


7.4' 


99.69 


7.92 


99.66 
100.6 


8.28 

8.ir 


100.8 


7.05 


100.7 


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101.6 


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102.7 


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102.6 


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7.71 


103.7 


8.16 


103.6 


8.61 


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104.6 


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8.55 


108.6 


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76.51 


8.72 


76.47 


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77.54 


8.49 


77.50 


8.83 


77.46 


9.17 


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78.57 


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78.53 


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78.49 


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78.45 


9.29 


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81.55 


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81.51 


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81.47 


9.28 


81.43 


9.64 


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8.68 


82.51 


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82.47 


9.40 


82.42 


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88.43 


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88.38 


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9.62 


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91.36 


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74.22 -1076 


74.18 


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75.21 10.91 


75.17 j".»1 


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76.25 


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76.20 j 11.05 


76.15 I11.3S 


76.10 11 71 




78 


77-44 


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77.»4 


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78.23 


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87.09 


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86.98 


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88.02 


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96.00 


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96.99 


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96.92 


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97.98 


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97.91 


14.63 


97.85 


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99.03 


13.92 


98.97 


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98.90 


14.78 


98.84 


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99.82 


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102.8 


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14.61 


103.9 


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103.8 


'$.97f 




105.0 


M'75 


104.9 


15.21 


104.8 


15.67 


104.8 


16. i7 




107 


106.0 


14.89 


105.9 


»5-35 


105.8 


15.82 


■05.8 


16.29- 




108 


106.9 


1 5.03 


106.9 


15.50 


106.8 


15.96 


106.7 


16.43I 




109 


107.9 


I $.17 


107.9 


15.64 


107.8 


i6.fi 


107.7 


i65» 




tio 
III 


108.9 


«S-4S 


108.9 


15.78 


108.8 


16.26 
16.41 


108.7 


'6.7jr 
16.89M 




109.9 


109.9 


1593 


109.8 


109.7 




111 


110.9 


>f59 


110.8 


16.07 


IIO.8 


16.55 


iia7 


i7o4r 




113 


If 1.9 


'5-73 


I11.8 


l6.2l 


II 1.8 


16.70 


111.7 


17- "# 




114 


II 1.9 


1587 


II 2.8 


16.36 


1I2«7 


16.85 


112.7 


'7-34r 




116 


1 1 3.9 


16.00 


II 3.8 
114.8 


16.50 


II3-7 


1700 


113.7 


iif^ii 




114.9 


16.14 


16.65 


114.7 


17.15 


114.7 


17-63 




»'7 


115.9. 


16.28 


115.8 


16.79 


115.7 


17.29 


115.6 


17-83 




118 


IJ6.9 


16.42 


116.8 


»^93 


116.7 


'744 


116.6 


'7-9ffl 




119 


"Z'! 


16.56 


U7.8 17.08 j 


117.7 


>7-59 


n7.6 i8.ia 




120 


118.8 


16.70 


118.8 17.22 1 


118.7 


»7,74 


ia8.6 [i8.2j[ 


, 


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9^ 


Dep; 


Lat ] 


De p. 


Lat. 

r 


Dep 


Lat 1 


Dep. 1 Lat. 


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15' 










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1 

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IS' 1 
L21. 1 Dep. 


30' 1 


45' 


Lit. 


Dep. 


Lat. 


Dep. 


Lai. Dep. 


1 


0.99 


0.16 


a99 > aj6 


0.99 


0.17 


0.99 , 0.17 


2 


1.98 


0.31 


1.97 ! 0.32 


»-97 


0-33 


1.97 


0^34 1 


3 


2.96 


CX47 


2.96 


0.48 


2. ,6 


0.50 


2.96 


0.51 


4 


3-95 


0.63 


3.95 


0.64 


3.95 


0.66 


3-94 


a68 


S 


4-94 


0.78 


4 94 


a8o 


• 4 93 


-•83 


4-93 
5.91 


a85 


6 


5-93 


0.94 


5.92 


0.96 


5.94 


0.99 


1.02 


7 


6.91 


I.IO 


6.91 


1.13 


6.90 


1.16 


6.90 


1.19 


8 


7.90 


1.25 


7.90 


1.29 


7.89 


1.32 


7.88 


1-35 


9 


8.89 


1.41 


8.88 


"•45 


8.88 


«'49 


8.87 


1.52 


10 

II 


9.88 


1.56 


9.87 
10.86 


1.6 1 

».77 


9.86 


1.65 
182 


9.86 


1-69 


to.86 


1.72 


10.85 


10.84 


1.86 


12 


11.85 


1.88 


11.84 


«-93 


11.84 


1.98 


11.83 


2«0} 


«3 


J 2.84 


2/)3 


12.83 


2/>9 


12.82 


2.15 


12.81 


2.20 


>4 


13.83 


2,19 


13.82 


2.25 


13.81 


431 


13.80 


4*37 


'5 
i6 


14.82 


»35 


14.80 


2.41 


»4.79 


2.48 
2.64 


15.77 


4.54 


IS 80 


2.50 


<5.79^ 


a-57 


15.78 


2.71 


17 


1679 


2.66 


16.78 


4.73 


16.77 


2.81 


16.75 


2.88 


i8 


17.78 


2.82 


17.77 


2.89 


17.75 


4.97 


17.74 


3.05 


«9 


18.77 


2.97 


18.75 


3.05 


18.74 


3.«4 


18.73 


3.44 


ftO 


19 75 

20.74 


3.«3 


1921 


3.21 


19.73 


330 
3.47 


19.71 

20.70 


3.39 


3.29 


40.73 


3.38 


2071 


356 


22 


4173 


3.44 


21.71 


3-54 


21.70 


363 


21.68 


3-73 


23 


22.72 


3.60 


22.70 


3.70 


22.68 


3.80 


22.67 


3.90 


»4 


23.70 


3.75 


43.69 


3.K6 


2367 


3.96 


23.65 


4.06 


*5 


24.69 


4.07 


24.67 


4.02 
4.18 


24.66 


4.13 
4.29 


24.64 
25.62 


4.43 


26 


25.68 


25.66 


2564 


440 


27 


26.67 


4.22 


26.65 


4*34 


26.63 


4.46 


26.61 


4.57 


28 


27.66 


4.38 


47.64 


4.50 
456 


27.62 


4.62 


27.60 


4.74 


»9 


28.64 


4.54 


28.62 


28.60 


4.79 


28.58 


4.91 


30 


29.63 


4.69 


29.61 


484 


29.59 


4.95 


49.57 


5.08 
5.45 


30.62 


4.85 


3060 


4.98 


30.57 


5.12 


30.5s 


3» 


31.61 


5.01 


31.58 


5.14 


31.56 


5.28 


31.54 


54« 


33 


3a.59 

33.58 


5.16 


34:57 


5.30 


3*-55 


5-45 


3454 


5.59 


34 


5-34 


33.56 


5.47 


33 53 


5.61 


3351 


5.76 


35 
36 


34.57 


5.48 


3454 


5.63 


34.54 


5.78 


34.49 


5 93 
6.10 


3556 


5-63 


35-53 


5-79 


35-5' 


5-94 


35.48 


37 


36.54 


5-79 


36.52 


595 


36.49 
37.48 


6.11 


36.47 


627 


38 


37.53 


5-94 

O.IQ 


37.51 


6. II 


6.27 


3745 


6.44 


39 


38.5* 


38.49 


6.27 


38-^7 


6.44 


38.44 


6.60 


40 
4' 


39. 5 « 


6.26 


39.48 


643 


39-45 
4044 


6.60 


39.44 


6.77 


40.50 


6.4« 


40.47 


6.59 


6.77 


40,41 


6.94 


4» 


41.48 


6.57 


4«-45 


6.75 


41.42 


6.93 


4".39 


7.11 


43 


44.47 


6.73 


44.44 


6.91 


42.41 


7.10 


44.38 


7.48 


44 


43*46 


6.88 


4343 


7-^7 


43.40 


7.26 


43.36 


7-45 


45 
46 


44.45 


7.04 


44.41 


7-41 


44.38 


743 
7-59 


44.35 


7.62 


45.43 


7.20 


45.40 


7.39 


45-37 


45.34 


7-79 


47 


46.42 


7.35 


46.39 

47.38 


7.55 


46.36 


7.76 


46.34 


7-96 
8.13 


48 


47.41 


7-5» 


7.72 


47.34 


7.92 


47.31 


49 


48 40 


7.66 


48-36 


7.88 


48.33 


8.09 


48.29 


8.30 


50 


4938 


7.82 


49.35 


8.04 


49-3 « 


8.25 


49-»8 


8.47 


50.37 


7.98 


50.34 8.20 


50.30 


k.42 


5a 26 


864 


S« 


5«.36 


8.13 


51.32 


8.36 


51.29 


8.58 


51.25 


8.8 1 


S3 


54-35 


8.29 


54.31 


8.52 


54.47 


5-75 


52,23 


8.98 


54 


53.34 


8.45 


53.30 


8.68 


53.46 


8.91 


53.42 


9.14 


55 


54.34 


8.60 


54.28 


8.84 


54.*5 


9.08 


54 41 


9-31 


56 


55-31 


8.76 


55.47 


9.00 


5543 


9.24 


9.48 


57 


56.30 


8.92 


56.46 


9.16 


56.22 


9.41 


56.18 


9.65 


S« 


57.49 


9.07 


57.*5 


9.32 


57.20 


9*57 


57- '« , 


9.82 


59 


58.27 


9.23 


58.23 


9.4S 


5819 


9.74 


58.15 


9-99 


60 


59.26^ 
Jiep. 


9 39 


59.42 
Dep. 


9.64 


59.18 1 


9.90 


5$-13 J 


10.16 


Lat. 


Lat. 


Dep^ 


Lat 


Dep 


Lat. 


i 





f 


45/ 1 


30' 1 


- IS 


f 

t 



^^ 






^ 


M^a^KM i»a^KJO« 






A< 




. "' 1 


. 15' 


30' 


45' 


Xut 


Dep. 


Lat. 


9.81 


Ut 


Dep. 


Lat 


Dep 


60.25 


9.54 


6021 


60.16 


10.07 


60. 1 2 


>0.33 


62 


61.24 


9.70 


61.19 


9-97 


61.15 


10.23 


61.10 


1050 


63 


62*22 


9*86 


62.18 


10.13 


62.14 


1040 


62.09 


10.67 


64 


63.21 


laoi 


63.17 


10 29 


63.12 


10.56 


63.08 


10.84 


65 

66 


64.20 


10.17 


64.15 


1045 


64.11 


10.73 
10.89 


64.06 


11.01 


!r? 


10.32 


05.14 


10.6 f 


6$.09 


65.05 


11.18 


67 


66.18 


10.48 


66.13 


iO.77 


66.08 


11.06 


66.03 


11.^5 


68 


67.16 


10.64 


67.12 


10.93 


67.07 


11.22 


67.02 


11.52 


69 


68.1$ 


10.79 


68.10 


11.09 


6805 


11.39 


6800 


11.69 


70 
7i 


69,14 


»o.9$ 1 
11.11 


69.09 

70.08 


11.25 


6904 


11.55 


68.99 


11.85 


70.13 


11.41 


70.03 


11.72 


69.97 


12.02 


7» 


71.11 


11.26 


71.06 


"•47 


71.01 


11.88 


70.96 


12.19 


73 


72.10 


11.42 


72.05 


".73 


7t.oo 


12.05 


7*95 


r2.36 


74 


7309 


1158 


73.04 


11.90 


72.99 


12.21 


7^93 


12.53 


75 
76 


74.08 
75.06 


11.73 


74.02 


1206 


74.96 


1S.38 


7391 


12.70 


11.89 


75.01 


12.22 


12.54 


74.90 


1287 


77 


76.05 


i2«05 


76.00 


12.38 


7594 


12.71 


7589 


13.04 


78 


77.04 


12.20 


76.99 


12.54 


76.93 


12.87 


76.87 


13.21 


79 


78.03 


12.36 


77.97 


12.70 


77.91 


13.04 


77.86 


U'3^ 


80 
81 


79-02 


12.51 


78.96 


12.86 


78.90 


13.20 
>3.37 


78.84 


«3.55 


80.00 


12.67 


79 95 


13.02 


79.89 


79.83 


13.71 


82 


80.99 


ia.83 


80.93 


13.18 


80.88 


"353 


8a82 


13.89 


«3 


81.98 


12.98 


81.92 


«3 34 


81.86 


13.70 


81.80 


14.06 


84 


82.97 


t3.i4 


82.91 


13.50 


82.85 


13.86 


82.79 


14.23 


«5 
86 


83.95 


'3.30 


8389 
84.88 


13.66 


83.83 


«4.03 
14.19 


83.77 


'439 


84.94 


»3.4S 


13.82 


84.82 


84.76 


14.56 


87 


85.93 


13.61 


85.87 


13.98 


8581 


1436 


85.74 


M.73 


88 


8692 


»3-77 


86.86 


14.15 


8679 


14.52 


86.73 


14.90 


89 


8790 


13.9a 


87.84 


I4.3> 


87.78 


14.69 


87.71 


15.07 


90 
9»' 


88.89 


14.08 


88.83 


•4 47 


88.77 


14.85 


88.70 


15.H 


89.88 


14.24 


89.82 


14.63 


89.75 


15.02 


89.69 


15.41 


91 


90.87 


«4.39 


90.80 


«4 79 


90.74 


15.18 


90.67 


15.58 


93 


91.86 


M$S 


9*79 


14.95 


9«.7l 


«535 


91.66 


«5-75 


94 


92.84 


14.70 


9278 


15.11 


92.71 


15.51 


92.64 


15.92 


95 
96 


93.83 


14.86 
1502 


93.76 


1527 


9370 


15.68 
15I7 


93.63 


16.09 


94.82 


94.75 


'5-43 


94.68 


94.61 


16.26 


97 


95.81 


»5i7 


95-74 


"5-59 


95.67 


16.01 


95.60 


16.43 


98 


96.79 


J5-33 


96,73 


«5-75 


96.66 


16.17 


96.58 


16.60 


99 


97.78 


«S-49 
15.64 


97-71 


15.91 


97.64 


16.34 


97-57 
98.56 


16.77 


100 
lOl 


9877 


98.70 


16.07 


9863 


16.50 


16.94 


99.76 


1580 


99.69 


16 24 


99.61 


16.67 


99-54 


17.10 


102 


ioa7 


15.96 


100.7 


16.40 


100.6 


16.83 


100.5 


17.27 


'V 


101.7 


16.11 


101.7 


16.56 


101.6 


17.00 


101.5 


"744 


104 


102.7 


16.27 


102.6 


16.72 


102.6 


17.17 


102 5 


17.61 


105 
106 


103.7 


16.43 


103.6 


16.88 


103.6 


'733 
17.50 


103.5 
104.5 


n-7t 


104.7 


16.58 


104.6 


104.5 


17.9s 


107 


105.7 


16.74 


105.6 


17 20 


105.5 


17.66 


105.5 


18.1a 


108 


106.7 


16.90 


106.6 


17.36 


106.5 


17.83 


106.4 


i8;K 


109 


107.7 


17.05 


107.6 


17.5a 


107.5 


17.99 


107.4 


18.46 


110 
III 


108.6 
109.6 


17.11 


10S.6 


17.68 


108.5 


18.16 
18.32 


108.4 


tH 


17.36 


109.6 


17.84 


109.5 


109.4 


IIS 


110.6 


17.0 
I7.i8 


iia5 


18.00 


110.5 


18.49 


110.4 


«8.97 


"3 


11 1.6 


111.5 


18.16 


III. 5 


18.65 


II 1.4 


19.M 


«»4 


112-6 


17.83 


112.5 


18.32 


112.4 


18.82 


1124 


1931 


•M 


113-6 


17.99 


113.5 ••849 


ii34 


16.98 


113.3 




116 , 


. iM 


18.15 


1145 {"8.65 


114.4 


19.15"" 


"4.3 


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117 


115.6 


18.30 


115.5 j 18.81 


115.4 


19.31 


115.3 


19.V 


118 


116.5 


18.46 


116.5 !«8.97 


116.4 


19.48 


116.3 


■9.4 


119 


117.5 


18.6s 


117.5 ''9»3 


117.4 


19.64 


117.3 1 20. 11 


ISO 


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•8-77 


118.4 
Hep. 


19.29 


118.4 '9-8o 


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1.96 0.37 




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3-93 0-73 


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6.88 i.iB 


6.88 


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7.87 1.46 


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8.t6 


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8.86 1 1 60 


8.8j I 64 


8.84 


1.68 


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I.Tfl 


9.84 i_^78 


9.83 


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9.8. 


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,o".l, 


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10.8 1 '. 1-96 


10.81 


1.00 


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».<»r 




ii.li 


».o8 


...gl 1 I..4 


ii.80 


*.I9 


11.79 


1.14. 




11.80 


*.ib 


11.79 1 ^i' 


IJ.78 


1.37 


14.T7 


tt 


14 


"3-79 


i+J 


13-78 1 »-49 


iJ-77 


»-i5 


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1.60 


14.7S »-67 


.14-''S 


HL 


'4-74 


1.30 


16 


TTTT 


i.7r 


15-74 , »-*S 


'S73 


t.91 


15-71 


1.9S 




16.74 


*-9{ 


16.73 


3-oj 


■6,71 


3.10 


1670 


3- '7 


tS 


17-73 


3-'3 


17.71 


3.K> 


17.70 


3.18 


17-68 


3.36 


"9 
10 


ig.71 


3. JO 


1870 


J.jS 


■8.68 


3.46 


18.67 


354 


19.7£_ 


347 


1,68 


Jjl 


19.67 


3-64 


.9^5 


3-73 






















10.6S 


3*5 


10.66 


3-74 




3.83 


10 63 


3-91 
4-10 




11.67 


381 


i..6i 


J.91 


...6j 


4.0I 






M.65 


3-99 


1163 


4.09 


li.«l 


4-'9 


iito 


4.19 


M 


U64 


4-17 


1J.6» 


4-»7 


13.60 


4-17 


«3.S8 


4.** 


16 


14.6» 
IS.61 


4.;i 


14.60 4-45 
»ii9 1 4*3 


»4.S« 


4-74 


14-56 


^ 


1S.S6 


»S.S4 


*7 


*6-i9 


Ik 


»6.p , 4-&0 


16.SS 


4.91 


»6.i3 


S-<H 


iS 


JT.(7 


486 


17. ss 4-98 


»7 SI 


5.10 


ij.ji 


S-" 




rfi6 


i.04 


18, 54 1 S.'6 


^.5. 


S.18 


H.49 


J-*i 


J° 


19- S4 


j.ii 


zq i>. JJ± 


19.50 




>9^7 


S-60 




30-(J 


S.38 


3051 5-S» 


J04S 


Tfi? 


30.46 


S78' 


JI.5. 


5-56 


31 49 1 S.69 


3146 


S,83 


31.44 


t?6 


3J 


3».iO 


S73 


31.47 . iS7 


3».45 


6.01 


3 ••4* 




34 


33.48 


5,90 


33.46 1 6-05 


3343 


6.10 


33.40 


«34 


3{ 


3447 


6.aE 


J±i±'-i:iL 


344' 


638 


34-39 


±si 


]6 


3;-4j 


61s 


3! 43 6.41 


35.40 


6.56 


35.37 


erf 


37 


36-44 


S.43 


jM. 6.S8 


36.38 


6.74 


36-35 




JS 


37-4» 


6.60 


37.39 6 '6 


37.36 


6.91 


37.33 




39 


38.41 


6-,7 


Jg.38 6 94 


3S-J5 


7.11 


J831 




4° 


_29^.J9 


6-9> 


39.36 ' 7-'i 




JJ± 


39-30 






40,3s 


7.11 


40.15 , 7.30 


1577 


7 47 


40.18 






41.3S 


7.19 


4'. 33 


7-47 


4'.3i> 


7.6i 


41.16 


7.aj 




41-3; 


7-47 




765 


41.1S 


7-84 


4i.»S 






43-33 


7-|4 


43.10 


l^i 


43.»6 


toi 


43. w 






44-3* 




_44:i8_ 


801 


44-'J 


!.M 




H.;9 


"46" 


4)-30 


7-99 


45.17 


X,;: 


45-»3 


s.ir 


45-'9 


».i8 


46-19 


t 16 


46.1s 


1.36 


46,11 


8.57 


46.18 




4S 




1.34 


47.13 


1.54 


47.10 


!'" 


47,1* 


895 




48.16 


8.5. 


48." 


8.71 


48,18 


S.93 


48.4 


9-14 


S° 


49-»4 


8.68 


49- '0 


g.90 


49,16 


Ji'-L 


49 '» 


_±i± 




SO. 13 


8.86 


50.19 


9.08 


40.15 


9.19 


so- II 


9.51 




Si.xi 


f-oj 


51.17 


9-^S 


51.13 


9-4S 


i 1 .09 


9,-70 


S3 


51,19 


9.10 


SiiJ 


9.43 


Ji.ii 


9.66 




9*9 


S3.-! 


9-3* 


53-'4 


9.61 


J3..0 


9.84 


-lios 




ii 


S4-'6 


_?ji. 


54 ■» 


9.79 


SViS 




i-i-^i 


ID. 16 


S6 


55. 'S 


9.71 


S5.11 


9.96 


55.06 




55.01 


10.47 


56.13 


9,90 


56-09 


10.14 


56.0s 


10.39 


,6^ 




Is 


S7.I1 


10.07 


j7-o7 


10.31 


S7-03 


10.57 


5698 


10.81 


$9 


S8..0 




58.06 


10. (0 


58.0. 


10.7S 


57.96 




60 


59-09 


.0.4» 


59.04 


ID.6S 


59.00 


'±SL 


S*l.95 


" '9 


1 


Dep. 




LaL 


Ifcp. 


r-nC" 


lirp. 


L.I- 


Det,- 




45"'"" 


3 ' 


1 . t; 






— - 


79 


DEGB 


EES. 






.,. 



10 B£GREES. 



1( 




*i 



0» 



■MP 



70 . 

7«| 

73 ; 

74} 

75 






86 

87 
88 

89 
90 



91 
9* 
93 
94 
95 



96 

97 
98 

99 
100 



101 
lOt 

103 

104 
105 



La. 


JUep. 1 


00.07 


1059 


61 06 


10.77 


62.04 


1094 


63.03 


II 1 1 


6401 


11.29 


65.00 


11.46 


65.98 


11.63 


66.97 


11.81 


67.9s 


11 98 


68.94 


I2.t6 


69^.92 


ii32 


7^.91 


12 50 


71 8i* 


12.68 


72.88 


12.85 


•73.86 


13.02 


74.85 


13.20 


7S'^3 


«3-37 


76.82 


*i'54 


77.80 


'372 


78.78 


M.89 


79-77 


i^.^jy 


8075 


^4.24 


81.74 


14.41 


82.72 


>4S>^ 


83.71 


14 76 


84.69 


"4 93 


85.6i( 


15.U 


86.66 


15.28 


87.65 


»5-45 


88.63 


iv63 


89.62 


15.80 


90.60 


15.98 


9' 59 


.6.15 


94.57 


16.32 


93 56 


16.50 


94. S4 


16.67 


*^5'53 


16.84 


96.51 


17.02 


98.48 


17.19 


17.36 


99.47 


«7.54 


100.4 


17.71 


101.4 


1789 


102.4 


18.06 


«03 4 


18.23 


104.4 


1841 


105.4 


18.58 


106.4 


i«.7S 


107.3 


1893 


1083 


19.10 


'09 3 


19.28 


110.3 


»9.45 


111-3 


19.62 


U2.3 


19.8c 


iiii 


1997 


114.2 


20.14 


1152 


20.32 


116.2 


20.49 


117.2 


20.66 


/1S.2 


20. 84 



De 



I' 



li.tr. 



>f'P 



lu>it>. 



a' 



15 


', 1 


3C 


l^at. 


Dr-p 1 


La 


60.03 


10.85 


59.98 


61 01 


11.03 


60(^6 


61. .19 


11 li 


61.95 


62.9b 


11.39 


62.93 


63.96 


'• 57 


63 9J^ 
64.89 


64.95 


11.74 


65.93 


11.92 


65.88 


66.91 


12.10 


66.S6 


67.90 


12.28 


6784 


68 88 


I 2.46 


68.83 


69.^7 


12 63 


69.81 


70.85 


12.8> 


70.79 


71.84 


12.99 


7178 


72.82 


13.17 


72.76 


73.80 


nzi 


73 74 
74.73 


74.79 


•3 5^ 


75 77 


1370 


75-71 


76.76 


• j.Sd 


76.69 


77.74 


14.06 


77.68 


78.7a 


14.24 

H.4I 


7866 


79-71 


79.64 


80.69 


»4.59 


80.63 


81.68 


«477 


81.61 


82.66 


'4-95 


82.59 


8364 


IS. 13 
15.30 


83.58 


84.63 


8456 


85.61 


'5.48 


85.54 


86.60 


15.66 


8B.53 


87.58 


15.84 


875< 


88.56 


16 01 


88:49 


89>> 


16 19 


8948 


90. >3 


16.37 


90.46 


91.52 


16.55 


91.44 


92.50 


16.73 


92.43 


93.4^ 


16 90 


93 4» 


94.47 


17.08 


94.39 


95-4> 


i7si6 


9538 


9644 


17.44 


96.3^6 


97-4» 


17.62 


97-34 


98.40 


»7 79 


98..U 


99-39 


17.97 


99.31 


A00.4 


18.15 


100.3 


toi.4 


18.33 


)oi.3 


102.3 


18.51 


10.:. 3 


103.3 


18.68 


103.2 


104.3 


18.86 


104.2 


105 3 


19.04 


105.2 


106.3 


19 22 


106 2 


107.3 


19.40 


107.2 


108.2 


»9-57 


lO.l 


1O9.2 


«9-7S 


109.1 


110.2 


19.93 


IIO.I 


11 1.2 


20 11 


Ill.l 


112.2 


20.29 


112 1 


113.2 


20.46 


11 3- 1 


114.1 


20. 6 4 


114.I 


115.1 


20 82 


1150 


116.1 


2;. 00 


Il^.^ 


117-1 


21.18 


117.0 


118.1 j 


.21.35 


118.0 



■ .Ti 



13.85 

»403 
1421 
14.40 

'14*76 



; 84.49 

; 85.47 
8646 

!• 87 44 
|_88.42 

I 89.40 
90.39 
9137 

9235 



12.31 
12.50 
12.68 
12.87 
13.06 



Drp. 



• 94-3 » 
9530 
96.28 
97.26 
98.25 


17.91 
18.09 
18.28 

18.47 
18.63 


* 99.23 
100.2 
101.2 
102.2 
103.2 


19.84 
19.03 
19.21 
19.40 
19.59 


I0-«1 

• 105.1 
j 106. f 
' 107. 1 
f 108.1 


19-77 

19 06 

20; 14 
20.32 

20 52 


109.1 
I to.o 

II 1.0 
II 2.0 

in 


20.70 
20 89 

2r.oS 
21.26 
21.45 


114.0 
1149 
115.9 
116.9 

1179 


21.60 
21.82 

22.01 
2 2.20 
22.38 


( 1>M'- 


1 Lat. 



1:.' 



79 OKUUluK^}. 
O 



a 
s- 
r 

1 


0/ 1 


15/ 


30' ^ 


f 5^^ 


; 


Lat 


Ik^p. 


Lat. 


Dep. 


Lat 


Dep. 


Lat 


Dep J 


1 
1 


0.19 


0.98 


0.10 


0.98 


0.10 


0.98 


o.no 


% 


1.96 


0.38 


1.96 


0.39 


1.96 


0.40 


f .96 ' 


0.4*1 


J 


1,94 


0.57 


a.94 


059 J 


a.94 


0.60 


«-94 


o.«a| 


4 


3 93 


0.76 


3.9 a 


0.78 


39» 


0.80 


19^ 


o;8a f 


5 

6 


5.89 


0.95 


4.90 
5.88 


0.98 


4.90 


1030 


4«90 . 


1.0s 
i.^a 




1.14 


1.17 


5.88 


1 10 


5.87 


7 


6.87 


"•34 


6.87 


"37 


6.86 


1.40 


6.^5 


1.43 




8 


7.S5 


"•53 


7.85 


1.56 


7.84 


"59 


7.83 


i.«3 




9 


883 


1.7a 


8.83 


1.76 


8.82 


'79 


&4)i 


■.8jk 
a.04| 


10 

11 


9.81 
10.80 


1.91 


9.81 


'•95 


9.80 


1.99 


9.79 


1.10 


i#.79 


a.15 


10.78 


a. 19 


10.77 


a.»4 




ft 


11.78 


a. 19 


11.77 


a.34 


II. 76 


a. 39 


11.75 


M4 




n 


11.76 


2.48 


11.75 


»-54 


ia.74 


».59 


ia.73 


a.65 




«4 


1374 


a.67 


«3-73 


a.73 


13.71 


a.79 


«3-7f 


a.85 




»5 


14.71 


2.86 


14.71 


a.93 


14.70 


a.99 


.4/9 


3.05 




i6 


15-71 


3.05 


15.69 


3.'» 


15.68 


3 '9 


i).66 


3.*6. 




"7 


16.69 


3-a4 


16.67 


33* 


16.66 


3-39 


16.64' 


3 4<J 




i8 


17.67 


3-43 


17.65 


3.51 


17.64 


3.59 


17.62 


3.«7 




■«J 


18.65 


3^3 


1863 


37" 


i8.6a 


3-79 


18.60 


3.»» 


( 


20 


fq.6l 


3.8a 


■9.6a 


3.90 


19.60 


3.99 


19.58 


4^7 




SI 


10.61 


4.01 


20.60 


4.10 


ao.58 


4.19 


ao.56 


4.a8 




92 


21 60 


4.20 


11 58 


4.19 


ai.56 


4.39 


11.54 


44» 




»3 


12.58 


4-J9 


aa.56 


4.49 


1154 


4.59 


aa.5a 


4.6S 


1 


»4 


«3 56 


4.58 


a3.54 


4.68 


13.52 


478 


1J.50 


4.89 




i6 


»4 54 


4-77 
4.96 


14.51 
15.50 


4.88 

5.07 


14.50 


4.98 


24.48 
25.46 


5.09 
529 




15.54 


«5-4« 


5.18 ' 


*7 


16 50 


5"5 


2648 


5-*7 


16.46 


5.38 


a6u»3 


5.50 


, 


aS 


»7^ 


5-34 


17.46 


5 46 


2744 


5.58 


17.4' 


.5-70 


1 


»9 


1S.47 


$•$3 


18.44 


5.66 


iB^i 


5.78 


a8.39 


5-9' 






19.45 

30-43 


5 7» 


19.41 
3040 


5.85 


19.40 


5.98 


29-37 


6 II 


, 


5.9a 


6.05 


30.38 


6.18 


30.35 


631 




J2 


31^1 


6.11 


31.39 


6.14 


3 '.36 


6.38 


3'.33 


6.52 


' 


13 


3X.39 


6.30 


3a-37 


6.44 


32 34 


6.58 


32 3' 


6.72 


1 


34 


1338 


6.49 


33.35 


6.63 


3332 


6.78 


33.»9 


6.91 




35 


34 3<> 


6.68 


14-3^3 


683 


i^-io 


6.98 


34. »7 


7.'3 




36 


35-34 


6.87 


35-31 


7.0a 


3528 


7.»8 


3S»5 


7*33 




37 


36.32 


706 


36.19 


r.22 


36.16 


7-38 


36.aii 


7.53 




3« 


37.30 


7.2$ 


37.27 


7.41 


37-24 


758 


37.26 


7.74 




39 


38.18 


".44 


3825 


7.61 


38.2a 


7.78 


3814; 


7.94 




40 
4' 


39'«7 


7.63 


3913 


7.80 


39.29 
40.18 


7.97 
8.'i7 


39. "6 


8.15 




40.15 


7.81 


40.11 


8.00 


40. f4 


8-35 


4* 


41.23 


8.01 


41.19 


8.19 


41.16 


8.37 


4Kia 


8.55 




43 


42.11 


8.10 


41.17 


8.39 


42.14 


857 


41.10 


8.76 




44 


43. « 9 


8.40 < 


43. « 5 


8.58 


43- '2 


!-7 


43-08 , 


8.96 




45 
46 


44.17 


8.59 
8.7S 


44. '4 
45.1a 


8.78 
8.97 


44-10 


8.97 


44.06^ 


9.16 
9.37 




45.if 


4508 


9.17 


45^4 




47 


46.14 


8.97 


46. 10 


9-«7 


46.06 


9-37 


46.0a 


9-57 




4» 


47. • a 


9.16 


47.08 


9.16 


47.04 


9.57 


4699 


9-77 




49 


48.10 


9-35 


48.06 


9.56 


4801 


9.77 


47.97 


9.98 




50 
Si 


49.08 


Q-54 
J9-73 


49.04 

50.0a 


9-75 
9^95 


49.00 
49.98 


9-97 


48.95 


10.18 


1 


50.06 


10.17 


49-93 


»»-39 




5* 


J 1.04 


9.92 


51.00 


10.14 


50.96 


'O.37 


50,91 


10. 59 


J 


53 


5^03 


10. II 


51.98 


10.34 


51.94 


10.57 


51.89 


10.79 


1 


54 


53.01 


10.3* 


52.96 


10.53 


51.92 


10.77 


52.87 


11.00 


1 


55 


53-99 
54-97 


10.49 


53.94 
54.91 


'O.73 
»0.93 


53.90 

54.88 


1097 


J3.85 


ii.ao 


■ 


56 


10.09 


II 16 


54.83 


IT.dO 

11.61 


> 


57 


5595 


10.88 


5590 


11.12 


5586 


11.36 


55.81 




S» 


56.93 


11.07 


56.89 


11.32 


56.84 


11.56 


5678 


1 1^1 


1 ■ 


59 


57.9* 


11.16 


57-87 


11.51 ' 57.8a 1 


11.76 


5776 


ra.oi 


1 


60' 

A' 
'0 


58.90 


11.45 


58.85 
Df^p. 


11.71 


58^ 


11.96 


58.74 
Dep. 


10. aa 
Lat. 


■ 
• 


1)ep 


Lat 


Dcp. Lat. 


i 


0' 


4.W 1 


30/ 


15' 1 


^ 








TB 


D£GK 


££J^, 








1 









UDEGI 


VESS. 






lOS 


ff 


""' ' V'^ 


li' 


SV 


*J' J 


Lai 1 Urp 


Lit 


pep 


L«l. 


u^^ 


IM. 


Dep. 


— 


J9.JS l'l*4 


S9H 


11.90 


5 9.7 8 




59-71 


11.41 


61 


fc.g6 l.l.gj 


60. Si 




60.76 




60.70 


11.61 


61 


Si.a* 


no*. 


li" 


iiag 


61.74 


11 S6 


61.68 


11. Sj 


tx 


6U1. 


11. »i 




H.49 


61.71 


i».76 


61,66 


'3-OJ 


4 


6j.J. 




6j'" 


ii.6<( 


61.70 


11-96 


63.64 


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102 


99-77 


21 21 


99.6S 


21.64 


99 58 


22.08 


99.49 


21.51 


103 


100.7 


21.41 


100.7 


21.85 


100.6 


22.29 


100.5 


22.73 


104 


101.7 


21.62 


101.6 


2207 


101. 5 


22.51 


10 1.4 


aa.95 


12L 
106 


101.7 


21.83 


102.6 
103.6 


22.28 


102 % 


22.73 


ioa.4 


23 '7 


103.7 


22.04 


2249 


'03 5 


V.94 


io^4. 


»3.39 


107 


104.7 


22 25 


104.6 


22»70 


1045 


23.16 


104.4 


23.61 


loH 


105.6 


22.45 


105.5 


22.92 


105.4 


2338 


1053 


23.84 


X09 


106.6 


2266 


106.5 


23-13 


106.4 


^3 59 


106.3 


24.06 


no 
III 


107-6 


22.87 


107.5 


23.34 


107 4 
108.4 


2381 


'O7.3 

108.3 


24.28 


108.6 


23.08 


108.5 


23.55 


24.0a 


24.50 


112 


109.6 


23.29 


109.4 i *3 76 


109.3 


24.24 


109.1 


24.72 


113 


1 10.5 


2349 


110.4 .23.98 


110.3 


24.46 


110.1 


24.94 


114 


ifi5 


2370 


111.4 24.19 


If 1.3 


2467 


iii.i 


25.16 


11$ 

Z16 


112.5 
i'3.5 


239" 


1124 24.40 


112.3 


24.89 


112.2 


25.38 


24 12 


113.4 [2461 


««33 


25.11 


1 13. 1 


25.60 


117 


1T4.4 


2433 


1 14.3 24.81 


114.2 


2^2 


114.1 


25.81 


118 


"M4 »4^i3 


115.3 125.04 


115.2 


25-54 


.1151 


26.04 


119 


1 16.4 2474 


ri6.3 125.25 


116.2 


25.76. 


^ I16.I 


26.26 


120 


117.4 


M95 
Lat 


117.3 2^.46 


117.2 


2597 


1 17.0 


2668 


•J 
2 


'45 


Lat 


Df-p. 


])ep. 


LaL 


\i 





/ 


/ 


30 


t 


15' 



77 l^GHZt:;i. 



•PW 



tM 






UmOYUSSlL 




• 


0' 




1>/ 1 30' ] 


i5' \ 


Lat 


Dep 


Lat. 1 


Dcp. 


Lat. 
0.^7 


iier>. 


Lat 


Uep. 


1 


0.97 


0.13 


Ofc97 


0.13 


023 


0.97 


OH 


% 


r.^5 


0.45 


1.95^ 


0.46 


1.94 


0.47 


, «-94 


0.48 


1 


%,^% 


0.67 


a.91 


0.69 


4.94 


•.70 


4^91 


071^ 


4 


j.9a 1 0.90 


3^«9 


0912 


3.89 


0.93 


389 


9.95 




4.R7 


1 1.12 


4.87 


1.15 


4.86 


l.^7 
1.40 


4.86 


.'•*> 


^35 


f-84 


1.38 


5.83 


5.83 


1.41 


7 


6.K1 


>-57 


6.81 


160 


6.81 


1 63 


6.80 


1.66 


; % 


780 


t.8o 


7.79 


f.83 


778 


i.a? 


. 7.77 


190 


i 9 


»^77 


2.02 


ft.76 


4.06 


»75 


r 4.10 


8*74 


2.14 


to 
^ II 


9.74 


2.25 


973 


2.29 
2.5Z 


9-71 


4.57 


9*71 
10.68 


2 38 


10.71 


2.47 


tail 


10.70 


4.61 


I*- 


11^ 


2.70 


11.68 


4.75 


11.67 


4.80 


11.66 


4.85 


"^ 


\%JtPf 


2.94 


ri.^ 


a.98 


I2«64 


3 05 


12^ 


3.09 


M 


%lM^ 


3 »5 


■3.63 


3.2r 


I3»6< 


3.47 


13.60- 


333 


»5 


14.^ 


3 37 


14.60 


3 44 


*4-W 
15.56 


3-74 


i4*S7 


3i7_ 


r6 


<5$9 


3.<»o 


*$$7 


367 


*5*54 


3.»o 


»7 


l6ir< 


3.82 


16.55 


3.90 


1653 


8.97 


16*51 


404 


■S 


■7.54 


405 


17.51 


4.»3 


17.50 


4.ao 


17.48 


448 


■9 


18.51 


4*7 


•8.49 


4 35 


28*48 


4-44 


18^ 


4 5» 


20 
41 


»9-49 


f50 


19.47 


458 


»9-45 
20.42 


^67 


«9-43 


4.7S 


20.46 


4-7« 


20.44 


4.8 1 


4-90 


40.40 


4-99 


S» 


ai.44 


4-95 


a 1.41 


5«4 


41.39 


5-H 


41.37 


5*3 


»3 


:l2.4i 


5»7 


22.39 


5.»7 


22.36 


f-37 


44.34 


$-47 


H 


a3.38 


5.40 


»3-36 


5.50 


•3*34 


5.60 


43,31 


i-To 


»? 


24 36 


56a 


»4*33 
25.31 


^73 


^4-3' 


5.84 

407 


44.48 


y^ 


36 


*5-33 


5.85 


5.96 


45.28 


25»45 


6.18 


a? 


a6.ji 


6.07 


26.28 


6.19 


20.45 


6.30 


26.23 


^.4* 


j8 


27.48 


6.30 


27.2^ 


6.42 


4M3 


*-54 


27.40 


6.66 


S9 


48;s6 


652 


28.23 


6.65 


48.20 


6-77 


28,17 


685 


30 


29^23 


6.97 


29.20 


6.88 


29.17 


7.00 


29.14 


7*3 
7-37 


JI 


30.21 


30.17 


7.^r 


30.14 


7.H 


30.11 


32 


31.18 


7.20 


31.15 


733 


3I>I4 


7 47 


31*08 


7.61 


53 


32.15 


7.4* 


32.12 


7.56 


32.09 


770 


34t05 




34 


33*3 


7.65 


3309 


7.79 


33.06 


7-94 


33*03 


35 
3^ 


34.10 
35^8 


7.87 


34-07 


8.02 


34-03 


«.»7 


34.00 


i34 


8.10 


35-04 


8.25 


35.01 


8.4O 


34*97 


8.56 


37 


36.05 


8.32 


3602 


8.48 


35.98 


8.64 


35.94 


8.7^ 


3« 


37X^3 


8.55 


36.99 


8.71 


36.95 


887 


36.91 


9-^3 


39 


38.00 


8.77 


37.9^ 


8.94 


37-9* 


9.10 


J7.88 


9.27 


40 
41 


38.97 


9.00 
9.22 


38-94 
39-9« 


9.17 


38.89 


9-34 


38-85 


g.51 


39*95 


9.40 


39.87 


9-57 


39-i^3 


^n 


4» 


40.92 


9-45 


40.88 


963 


40.84 


9-80 


40.80 


9.9* 


43 


41.9O 


9.67 


4i.i»6 


9.86 


4181 


10.04 


4^-77 


10.44 


44 


4a-87 


9.90 


41.8J 


10.08 


4>78 


10.27 


42.74 


10.46 


45 
46 


4385 
4482 


IO<l2 


43.80 


10.31 


43.76 

44-73 


10.51 


43-71 


JO 70 


4478 


10.54 


10.74 


4468 


•0.93 


47 


45.80 


10.57 


45-75 


10.77 


45.70 


10.97 


45.65 


1 1. 17 


48 


46.77 


10.80 


467a 


11.00 


46.67 


II 21 


40.62 


1 1.41 


49 


47.74 


if.oa 


47.70 


If. 43 


47.65 


11.44 


47-60 111.65 1 


50 


48.71 


11.15 


^.67 


11.46 


48.62 


1 1.67 


48.57 1 


11.88 


49.69 J 1 1.47 


4964 


li 69 


49-59 i»«-9* 


4'^*54 ■ 


l2.fX 


5* 


50.67 .11.70 


50.62 


11.92 


50.56 < 12.14 


50.51 12.36 


53 


51.64 :ii 9> 


5».59 


12 15 


51.54.12.37 


5148 12.60 


S4 


52.62 '12.15 


ii.56 


12.38 


52.51 12.6J 


52.45 12.84 


SS 
56 


5359 "»-37 


53 54 


12.61 


53.48 12.84 


55.42 13.07 


54.56 12.60 


54.51 12.84 


5445 13.07 ! 


5440 ♦J.3« 


57 


55.54 21282, 


55.48 1 13.06 


55-43 ^il^ 55.37 "355 


58 


56.51 ;i3.o? 


56.46 (13.49 


5640 .13.54 5^-34 »3-79 


59 


57 49 "3.»7 


57.43 '3 5* 


57.37 113-77 ; 5731 «4-0» 


60 


58.47 'I3v50 


58.40 '13.75 


58.34 ; 14.01 


58.28 1426 1 




Dep Lat. 


Ucp Lai. * «)t. . 


Lat. 


Dtp. 


Lat. 1 





4S' 30 


1 


f^-^« 









UWOBERik 






lU 


T <»' 


- ..' 1 -v 


' is' ■' 


? -^ ~ 


Lit 


0^ 


l.ut 


1..,. 


L»l. 


iirPj. 


L»i- , Ocji.; 










ti 


i9*4 


i3-7» 


i«-j8 


.Jj8 


S9 3' 


14..4 


i-^iS 


14,50 


Cs 


6a4i 


'19S 


6035 


(4.11 


00.19 


'4-47 


60-11 


"4-74 


*i 


6.» 


(4-17 


61. ]i 


■4-^4 


61 lb 


14,71 


61 19 


'4-97 


«4 


6m« 


14.4a 


6i.iO 


14.67 


61.JJ 


>4B4 


fer.i? 


15.11 


ii. 


6J.JJ 


.4.6> 


6j.„ 


IX"— 


(.8.10 


'.*iL 


63.14 


lid.'. 


66 


64.!! 


■4.8s 


e+M- 


• i'J 


64. t8 


15.41 


64.11 


15.69' 


*? 


65 1! 


15.07 


65.11 


<i-j6 


65.15 


15.64 


6i-o8 


>J93 


ts 


6S.«S 


IS. Jo 


56..., 


'i-i9 


66.i> 


,5.87 


6605 


■6,16 


*9 


67.>J 


.j-5» 


^;: 


• iil 


6,. 09 


16.11 


67»» 


16.40 


,TO 


68.1» 


'5-7S 


16.04 


6S.07 


i6.i4 


67-99 


16.64 




















T 


65,18 


'S.97 


5,-11 


16.17 


fcv -4 


16.57 


68.97 


1688 


7* 


TO-. 5 


16.M 


70-s« 


.6.S* 




iG.Bi 


69.94 




7J 


71.U 


■ 6.4. 


7.^ 


'6 73 


7..K 




TQ-91 


'7-JS 


74 




.6«, 


7a oj 


16,6 


71.9^ 


.7 18 


71.88 


'7.59 


7i 


73.08 


1687 


73.00 


12JL 


7t.W 


175" 


71.85 


'7-83 


:s 


74-Oi 


17.10 


7J.98 


17.41 


73.»o 


17-7* 


73-8a 


.ToT 




7 5rfiJ 


173» 


74 9i 


17.6s 


74-87 


.7-*a 


74-79 


.8.30 




76.00 


<7-(J 


7M» 


.7.81 


75-84 


i8.ii 


75-76 


'8.54 




76.9« 


"7 77 


76.90 


18.11 


r6.sii 


1844 


76.74 


18,7s 




-:!;a 


ii.oo 


77.87 


<»J4 


77.79 


iS.SS 




ll-^L 


TT 


71(.»» 


18.11 


'78.84 


.8.J7 


78.76 


.8.9- 


TaTsT 


'<)-»S 




r».so 


>M; 


79-*« 


1B.79 


79 7J 


■ 9-I4 


79-65 


'949 




fcj-g? 


18.67 


IO.T9 


1901 


80.71 


.9.3B 


80.6; 


'9.73 




81,85 


.RV> 


8..7i 


'9*5 


81.68 


19.61 


11.59 


'997 




8«.Si 


'9.11 


81,74 


•»ll 


81.65 


■ 9.84 


A*-il 
























83,(0 119.J5 


8J.7I 


19-71 


83.61 


«o.08 


81.44 


10.44 




84-77 is-ij 


l4.M 


«994 


84.60 


110.31 


»4-i. 


10.68 




84.74 t9-«0 


85.C6 


10.17 


85.57 


10. S4 


as.«8 


10.(1 




W,7« lK).Oi 


86.^5 


10-40 


86.54 


iio.?8 


86^5 


11.15 




87.69 


10 IS 


87.60 


io,6j_ 


«7 5. 


il.OI 


87-41 


"■i2- 


~y^ 


TK:6r 


W-tJ 


88.58 


ao.t6 


88.4, 


11.14 


88-39 


11.*] 




89.64 




*9-H 


ai.09 


8^.46 


»i.48 


&9J6 


1..17 




90.6. 


io.»i 


9a. S« 


ir.ji 


9043 


-ti.7i 


90-33 




0.1 


9<» 


ll.IJ 


9'- JO 


11. ,4 


91.40 


i'-94 


♦"■Ji 


11-34 


V 


9i-54 


1I.J7 


91.47 


n.7T_ 


9=.j8 




9118 


!i.e 


11.60 


9J-44 


i;oo 


9335 


11.41 


93-15 


It 8) 




94-i< 


11.1) 


94.41 


11. tj 


9+.31 


11.64 


94-11 


IJ.06 


•>8 


9!-49 


M.O[ 


9W9 


91.46 


9i.*9 


11.88 


95-19 


23.19 




96.46 




96.J6 


11.69 


9t>xi 


ij.ii 


B6.16 


13.53 


103 


97-44 


ll-JO 


97.J4 


?1IL 


97.14 


1334 


97-13 




lor 


9S.4« 


11,71 


98.3. 


ij.ij 


9«.l. 


13.58 


98. 1. 


iJoT" 




99-J9 


"■94 


«-»* 


xjja 


99.1 8 


138' 


99-oK 


14.14 




100.4 


ij,i7 


10O.J 


13.6. 




14-0* 




14 4" 




101.3 


13-40 




13-84 




141* 




14-71 




■01.3 


ij.61 




*t£L 




145^ 




14^96 


7^ 


lOVj 


13,84 


1031 


14 JO 


lOJ.I 


14.75 


"wlo" 


15.17 




■04J 


14-07 


104-1 


14.51 


104.0 


M-98 


103.9 


1343 




M5» 


14.19 


1051 


14-75 


105.0 


15.1' 


104.* 


15-67 


1^9 


it>6.> 


14 J» 


106,1 


t4.9S 


1060 


11-45 


105,9 


15.91 




107.1 










1568 


106 8 


i6.T{ 






















■OS.X 


14,97 


108,0 


15.44 




15.91 


107.! 


16-3T 




1091 


15.19 


Jt>9,0 


15.67 


.OS., 


16.15 


.088 


=6 61 


'U 




15.41 




lJ-90 


'09 9 


16 J8 


■098 


16.86 






15.64 




16-1 J 


110.9 ib6i 


1m' 


17 10 






1(87 




16.36 


-li'l'ii«L 




17-33^ 


IIG 


l'J.O 


16.0s 


1.1.9 16 S9 


■ ILK ,J7 08 


i<i,7 


17.57 


'17 


114.0 


16.31 


1.3.9 '6.81 


"3K 1J.3, 


1.J.6 


17.81 




lli-O 


Mi.S4 


114.9 »7-05 


"47 17.55 


..4.6 


18.05 


"' 


116.0 


ifi.77 


.15.8 1717 


"5.7 I1T.7B 


115.6 iSiS 


m6., 


16,99 


.16.8 17.50 


i.6.r '1*0. 


.16.6 ,185, 


i 


IMp. 


Lit 


4 


iir 


Ilfp 1 L,U. 


!».„■ 


ur 


0' 


■ 




1 


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7fi 


lUtCvk 


tX3i% 









5 


a 


Dcp! 


16/ 


3C 


Dep. 


45 


j 




Lat. 


Lat. 1 Dep. 


Lat. 


Lilt* : Dtp. 




I 


0.97 


0.24 


0.97 1 0*5 


0.97 


0.25 


0.97 1 0.25 


1 


s 


1-94 


0.48 


i.94 


0.49 


1.94 


050 


f.93 


0.51 


1 


3 


1.9 1 


0.73 


2.91 


074 


2«90 


0.75 


2.90 


0.76 


1 


4 


3.88 


0.97 


3.88 


0.98 


3.87 


1.00 


3.87 


1.02 




5 
6 


4.85 


1.21 


4.85 
5.82 


1.23 

I 48 


4.84 


1.25 


4.84 


1.17 




5.82 


»-4S 


5.81 


1.30 


5.80 


1 S3 




7 


6.79 


1.69 


6.78 


1.72 


6.78 


••75 


677 


1.78 




8 


7.76 


1.94 


7.75 


1.97 


7.75 


2.00 


7.74 


2.04 




9 


8.73 


2.18 


8.72 


A. 21 


8.71 


2.25 


8.70 


1.29 




lO 

11 


9,70 
10.67 


2^2 
T66" 


969 


2.46 


9.68 


% 50 


9.67 


_*'5S 


1 


10.66 


2.71 


10.65 


*-75 


10.64 


1.80 




la 


11.64 


2.90 


11.63 


2.95 


11.62 


3.00 


11.60 


3.06 




*3 


l>6i 


3.'5 


12.60 


3.20 


12.59 


3*5 


12.57 


3.31 




14 


13.5« 


3-39 


13.57 


3-45 


•355 


351 


«3-54 


3-56 




»5 
16 


»4<55 


3.63 
3.87 


14.54 
15.51 


3-69 
3.94 


J4-S2 
"5.49 


3.76 


14.51 


3.82 




4.01 


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18.44 


4.60 


1842 


4.68 


18.39 


4,76 


18.37 


484 




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20.35 


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19.36 
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5.60 




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5.56 


22.29 


566 


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5'-76 


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23.29 


5.81 


23.26 


5.91 


23.24 


6.01 


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7.01 


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28.04 


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35.86 


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42.60 


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11.08 


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19.60 


78.51 


19.94 


78.42 20.28 


78.33 '20 I 


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79.56 


19.84 


7948 


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79.39 


20.53 


79.30 1 20. Ji 


83 


80.53 


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80.45 


20.43 


80.36 


20.78 


80.26 21. 1 


84 


81.50 


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81.32 


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82.29 


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82.20 


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86.26 


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28.87 
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29.94 


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30.84 


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31.88 


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31.84 


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32.84 


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35.65 


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36.71 


9.84 


36.66 


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36.61 


10.16 


36-57 


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37.67 


10.09 


37.63 


10.16 


37.58 


10.42 


37 54 


10.59 


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38.64 


10.35 


38.59 


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3855 


10.69 


3850 


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39.60 


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39-56 


10.78 


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10.96 


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44.38 


11.10 


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45.40 


11.16 


45.35 


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45.29 


11.56 


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11.76 


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46.36 


11.41 


46.31 


11.63 


46.15 


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46.10 


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47.35 


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47.*7 


11.89 


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13.09 


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13.15 
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48.18 


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49.20 


49.15 


13.63 


49.09 


13.84 


$» 


50.23 


13.46 


50.17 


2368 


50.11 


13.90 


50.05 


14" 


53 


51.19 


13.71 


51.13 


1394 


51.07 


14.16 


51.01 


■4 39 


54 


51.16 


13.98 


$1.10 


14.20 


52.04 


»443 


5 •'97 


1466 


55 

56 


53.'3 


14.24 


53.06 


H47 


53.00 


14.70 


52.94 


«493 


54.09 


14.49 


5403 


'4-73 


53.96 


«497 


53.90 


15.20 


57 


55.06 


"4-75 


54-99 


14.99 


54.93 


15.13 


54.86 


15.47 


58 


56.01 


15.01 


55.96 


15.16 


55.89 


15.50 


55.81 


*5.74 


59 


57-99 


15.17 


56.91 


15.51 


56.85 


'5-77 


56.78 


16.02 


60 

M 

.5^ 


57.96 
Dep. 




M-53 


57.89 


iS-78 


57.81 


1603 


57-75 
Uep. 


16.19 


Lit 


l>ep. 


Lat. 


Dep. 


Lat 


Lat. ^ 


> 


45 


1' 


3( 


J' 


15' 1 



74 DEGKEES. 



1$ TfRGRERS. 



Ldit 



68.58 
70.51 

72.44 



74.38 
75.34 

7^r5l 

77.17 



91.73 
93*69 

94.66 

95.63 
9659 



97^6 

9^S» 

99-49 
100.5 

101.4 



12.316 

Z1.78 

*3-03 
23.19 



15' 



Lat 



68.50 
69.46 

70.43 

7»-39 
71.36 



73. 3 » ;>9-99 
74.19 ;20.25 

7515 2051 



76.11 
77.18 



78.1$ 
79-11 

80.08 
81.04 
82.01 



,10.78 
11 04 



11.31 
11.57 
.11.83 
11.09 
11.36 



81.97 
83.94 
84.90 
85.87 
86.83 



11.61 
11.88 

13.15 

*3«4i 

13«67 



87.80 
88.76 

89-73 
90.69 

9« :«5_ 

91.61 115*25 



93.58 

94*55 
95*5' 
96.48 



\^S 5« 
15.78 
16.04 
16. 30 



97 44 
98.41 

99.37 
100.3 

101. 3 

102.3 
103.1 
104.1 
105.1 
106. 1 

107. 1 
108. 1 
109.0 
1 10.0 
llf.O 



26.57 
16.S3 

17.09 
17.36 

17.61 



17.88 

28.14 
i8.4r 
18.67 
18.93 



1 



115-9 .31.06 
Pep. {TZu 



19.20 
19.46 
19.71 
19.99 
30.15 



30.51 
30.77 
31.04 

31.30 
31.56 

Lit." 



Lat I Dcp. 



5878 

59*75 
60.71 

61.67 

61.64 



63.60 
64.56 

65.53 
66.49 . 

6745 '8.?t 
6841 !ii>.97 
69.38 19.14 

70.35 
7I.3» 
72.17 



73.24 
74.10 

75.16 
76.13 
77.09 



78.05 
79.01 
7998 
So 95 
81.91 



81.87 
83.84 
84.80 
85.76 
8673 



87.69 
88.65 
89.61 
90.58 
91.54 



22..;8 
13.15 

*3.5* 
13.78 

14.05 



14.3a 
14.59 
14.85 
15.11 

»5.39. 



9».5« 

93 47 

94 44 
95.40 

0.36 

97.33 
98.29 

9915 

100.1 



26.99 
17.16 

»7.53 

3t7.79 
101.1 ii8.o6 



15.65 
15.91 
16.19 
26.46 
16.71 



102. 1 

103. 1 
104. 1 
105.0 
106.0 



107.0 
107.9 
108.9 
109.9 
itoA 



28.33 
18.59 
18.86 

29- 13 

29.40, 

19.66 

29.93 
30.20 

30.47 

30^72 

31 00 

31.17 

3«.53 
31.80 
31.07 



Lut 



30' 



45i 



Lat. 



58.71 
59.67 
60.63 
61.60 
61.56 



63.51 
64.48 

65.45 
66.41 

67.37 



6833 
69.30 

70.16 

71.11 It 

71.18 



73 '5 
74,11 

75.07 

76.03 

77.00 



77 v6 
78.91 
79.88 
8085 
81.81 

"8177" 

83.73 
^84.70 
85.66 
8661 



87.58 

8855 
8951 

90.47 
9'-43 



106.8 

107.8 
108.8 
109.7 
1 10.7 



II 



9140 
93.36 

94 3* 
95.18 

9615 
97.11 
98.17 

99.13 
100. 1 
101. 1 

101.0 
103.0 
103.9 

104.9 
105.9 



15' 



74 D£GBBKS. 



116 



16 BEGBEES. 



c 

I 


0' 

Lat 
0.96 


Dep. 


\t 


\» 


50' 


45 


» 


Lat. 


Dep. 


Lftt. Dep 


Lat. 


Dep. 


0.28 


0.96 


0.28 


0.96 0.18 


096 


0.29 


2 


1.92 


0.5s 


1.92 


0.56 


1.91 


0-57 


1,92 


0.58 


3 


2.88 


0.83 


2.88 


084 


2.8S 


0.85 


2.87 


0.86 


4 


3.85 


1. 10 


3.84 


1.12 


3.84 


1. 14 


3-83 


1.15 


5 
6 


4.S1 


1.38 


4 *o 


1.40 


4-79 


1.42 


_4i79 

"5-75" 


1.44 


5-77 


1.65 


5.76 


1.68 


5-75 


1.70 


'-73 


7 


6.73 


»-93 


6.72 


1.96 


6.71 


1.99 


6.70 


2.02 


8 


7.69 


2.21 


7.68 


2.24 


7.67 


2.27 


7.66 


2.31 


9 


8.6s 


2.48 


8.64 


2.52 


8.63 


2. $6 


8.62 


2.59 


10 

II 


9.61 


2.76 


9.60 


2.80 


9-59 


2.84 


9.58 


2.88 


10. $7 


3.03 


10.56 


3.08 


10.55 


3.12 


10,53 


3.17 


12 


II 54 


Z'l>^ 


11.52 


336 


11. 51 


3 41 


11.49 


3-46 


»3 


12.50 


3.58 


12.48 


3.64 


12.46 


3.69 


12.45 


3.75 


»4 


13.46 


3.86 


1344 


3-9* 


13.42 


3.98 


13.41 


4-03 


i6 


14.42 


4- '3 
4.41 


^40 
«5-36 


4.20 


14.38 
«5-34 


4.26 


14.36 


4.32 


1538 


4.48 


4.54 


15.32 


4*6 1 


»7 


"634 


4.69 


16-32 


4.76 


16.30 


4.83 


16 2S 


4.90 


iS 


17.30 


4,96 


17.2S 


5.04 


17.26 


511 


17.24 


S-«9 


'9 


18,26 


5.24 


18.24 


532 


l8.22 


5.40 


18.19 


5.48 


20 


19.23 


J;5» 
5-79 


19 20 


5.60 


1918 

20.14 


5.6!i 


«9»5 


5.76 


20.19 


20.16 


5.88 


5.96 


20.11 


6.05 


22 


21 15 


6.06 


If. 12 


6.16 


21.09 


6.25 


21.07 


634 


23 


az.11 


6.34 


23.08 


644 


22.05 


6. 5 J 


22.02 


6.63 


»4 


23.07 


6.62 


23.04 


6.* 


23.01 


6.82 


2298 


6.92 


2^ 

26 


1403 


6.89 


2400 


7.00 


23-97 


7.10 

7.38" 


23.94 


7.20 
7-49 


• 24-99 


7- 17 


24.96 


7.28 


2493 


24.90 


27 


»5-95 


7-44 


25.92 


7.56 


25-89 


7.67 


25.85 


7.78 


2}{ 


26.92 


7.71 


26.88 


7.84 


26.85 


7-95 


26.81 


8.07 


29 ^ 27.88 


7-99 


27.84 


8.12 


27.81 


8.24 


*7.77 


8.36 


30 


28.84 


8.27 


28.^0 


8.39 


28.76 


8.52 
8.80 


2873 


8.65 
8.9i 


31 ! 


29.80 


8.54 


29.76 


8.67 


29.72 


29.68 


31 j 30.76 


8.82 


30.7 a 


8.9s 


30-68 


9.09 


30.64 


9.22 


33 1 3i-7a 


9.10 


31.68 


9.23 


31.64 


9-37 


31.60 


9.51 


34 » 32.68 


9-37 


32.64 


9.51 


32.60 


9.66 


3256 


9.80 


3S 
36 


3364 


965 
9.9a 


33.60- 


9-79 


33 56 


9.94 


33.52 
34.47 


10.09 


34.61 


3456 


10.07 


3452 


10.22 


fO.38 


37 


35-57 


10.20 


35-52 


10.35 


35-48 


10.51 


35.43 


10.66 


3» 


36.53 


10.47 


36.48 


1063 


36.44 


10.79 


36.39 


10.95 


39 


37^9 


10.75 


37.44 


10.91 


37.39 


11.08 


37.35 


11.24 


40 
4' 


38.45 


1103 


38.40 


11.19. 


38.35 


11.36* 


38.30 


11.53 


39-4» 


11.30 


39-36 


11.47 


3931 


11.64 


39.26 


11.82 


4a 


40.37 


11. 58 


40.32 


11.75 


40.27 


"93 


40.22 


12. 10 


43 


41-33 


11. 85 


41.28 


13.03 


41.23 


12.21 


41.18 


12.39 


44 


4*30 


14 13 


42.24 


12.31 


42.19 


12.50 


42.13 


12.68 


45 
46 


43.26 


12.40 


43 20 


12.59 
12.87 


4315 


12.78 


43.09 

44.05 


12.97 


44-2* 


12.68 


44.16 


44.11 


13.06 


13.26 


47 


45.18 


12.96 


45-12 


'15.15 


45.06 


«3.35 


45.01 


»3-55 


48 


46.14 


nn 


46-08 


».>43 


46.02 


13.63 


45.96 


13.83 


49 f 47.10 


i3.5» 


47-04 


13-71 


46.98 


13-92 


46.92 


1 4. 12 


$0 48 06 


13.78 


48 00 


13.99 


47 94 
48.90 


14.20 


47.88 


14.41 


5' 


49.02 


14.06 


48.96 


14-27 


14.48 


48 84 


14.70 


5a 


49-99 


14-33 


49-92 


14.55 


49.86 


»4'77 


49-79 


14.99 


«3 


50.95 


14.61 


50.88 


14.83 


50.82 


15.05 


50.75 


15.27 


54 


51.91 


14.88 


51.84 


15.11 


51.78 


1534 


51.71 


15.56 


56 


5287 
53.83 


15.16 


5f.8o 
5376 


'5 39 


5274 


15.62 


52.67 
53.62 


15.85 


>5-44 


15-67 


53.69 


15.90 


16.14 


57 


5479 


M-7I 


54-72 


'5-95 


54.05 


16 19 


5458 


16.43 


5» 


5575 


15.99 


55-68 


16.23 


5561 


16.47 


55-54 


16.72 


j9 56-71 


i6.z6 


56.64 


16.51 


56.57 


16.76 


56.50 


1700 


60 1 57.68 


16.54 1 


57.60 
l>ep. 


16.79 


57-53 


17.04 


57.45 


17.29 

Lat. 


• 


Dep 


Lat. 


Lai. 


Dep 


T^t 


Dep. 

15 





/ 


45 


/ 


30' 





M 



n DKCiixiEES. 









AQ 


Ajr.urnric«3. 






2J 


1 s 


0' 


15' 


30' 1 


45' 


6c 


Lat 
58.64 


Dep 


Lat. 

58.56 


Dep. 


lit 


Dep. 

17.32 


Lat 


Dftri, 


16.81 


17.07 


58.49 


$8.41 


17.58 


62 


50.60 '17.09 


59.5* 


17.35 


59-45 


17.61 


59-37 


17.87 


63 


60.56 1 17.37 


60.48 


17-63 


60.41 


17.89 


60,33 


18.16 


64 


61.52 1 17.64 


61.44 


17.91 


61.36 


18.18 


61.28 


18.44 


6s 
66 


62 48 ! 17.92 


62.40 


18.19 


62.32 


18.46 

i8.7r 


62.24 


18.73 


63.44 18.19 


63.36 


1847 


63.28 


63.20 


19.02 


67 


64.40 


18.47 


64.32 


18.75 


6424 


19.03 


64.16 


«9.3i 


68 


65-37 


18.74 


65.28 


19.03 


65.20 


"9-3« 


65.11 


19.60 


69 


66.33 ! 19.02 


66.24 


i93i 


66.16 


19.60 


66.07 


19.89 


70 
7i 


67. 29 19.29 


67.20 
68.16 


»9.59 


67 12 ; 19.88 


67.99 


20.17 


6^.25 19.57 


19.87 


68.0S 


20.17 


20.46 


7* 


69.21 J1985 


•69.12 


20 15 


69.04 


20.45 


68.95 


2073 


73 


70.17 j20.I2 


70.08 


20.43 


69.99 


20.73 


69.90 


21.04 


74 


71.13 '20.40 


71.04 


20.71 


70.95 


21. 02 


70.86 


21.33 


75 
76 


72.09 


2067 
,20.95 


72.00 
72^6 


20.99 


7191 


21.30 


71.82 
72.78 


21.61 


73.06 


21.27 


72.87 


21.59 


21.90 


77 


7402 :2i 22 


73.9* 


21.55 


7383 


21.87 


73-73 


22.19 


7S 


74.98 ^21 50 


74.88 


21.83 


74.79 


22.15 


74-69 


22.48 


79 


75.94 121.78 


7584 


22.11 


7575 


22.44 


75.65 


22.77 


80 
81 


76.90 2205 
77.86 121.33 


76.80 


22.39 


76,71 


22.72 


76.61 


23.06 


77.76 


2267 


77.66 ,23.01 


.77.56 


^3.34 


82 


78.82 J22.60 


78.7* 


22.95 


78 62 23.29 


•78.52 


23.63 


83 


79.78 22.88 


79.68 


^323 


79.58 .23.57 


79.48 


23.92 


84 


80.75 Us- 15 


80.64 


23.51 


80.54 ,23.86 


80.44 


24.21 


85 
86 


81.71 
• 82.67 


1*3.43 


81.60 


24,07 


81.50 .24.14 


81.39 


24.50 


|i3-70 


82.56 


82.46 24.43 


82.35 


24.78 


87 


83.63 '23.98 


8352 


a4.35 


83.42 2471 


83.31 


25.07 


88 


84 59 ^24.26 


84.48 


2463 


84-38 ^4.99 


84.27 


25.36 


89 


8555 '2453 


85.44 


24.90 


85.34 25.28 


85.22 


25.65 


90 
9' 


86.51 


24.81 
2508 


86.40 


25.18 
^5.46 


86.29 25.56 


86.18 


25.94 


87.47 


87.36 


87.25 25.85 


8714 


26.23 


9% 


8844 25.36 


.88.32 


a5.74 


88.21 2613 


88.10 


26.51 


93 


89.40 125.63 


89.28 


2602 


89.17 =26.41 


89.05 


2680 


94 


9036 25.91 


90.24 


26.30 


90.13 126.70 


90.01 


27.09 


95 
96 


91.32 '26.19 


91.20 
92.16 


26.58 


91.09 26.98 
92.05 27.27 


90.97 


27.38 


92.28 26.46 


26.86 


91.93 


2767 


97 


93.24 26.74 


93.12 


27-14 


9301 27.55 


9288 


27.96 


98 


94 20 27.01 


94.08 


27.42 


93.96 27.83 


93.84 


28.24 


99 


95.16 27.29 


9504 


27.70 


C4.92 28.12 


9480 


28.53 


100 
LOI 


96.13 127.56 


96.01 


27.98 


95.88 


28.40 

28. V 9 


95-7^ 


28.82 


9709 t27.84 


9097 


28.26 


96.84 


96.71 


29.11 


102 


98.05 


28.12 


9793 


28.54 


97.80 28.97 


97.67 


29.40 


103 


99.01 


28.39 


98.89 


28.82 


98.76 I9.25 


98.63 


29.68 


f04 


99-97 


28.67 


99.85 


29.10 


99 72 29.54 


99.59 


29.97 


105 

106 


100.9 


28.94 


100.8 


29.38 


100.7 
101.6 


2982 


100.5 


30.26 


101.9 


29.22 


C01.8 


29.66 


30.11 


101.5 


30.55 


t07 


102.9 


29.49 


102.7 


29.94 


102.6 30.39 


102.4 


30.84 


108 


103.8 


29.77 


103.7 


30.22 


103.6 30.67 


103.4 


31.13 


109 


104:: '30.04 


1046 


30.50 


104.5 30.96 


1044 


31.41 


110 
III 


105.7 13032 


105 6 


30.78 
31.06 


1055 3i.24 


105.3 


31.70 


106.7 


30.60 


106.6 


1064 3«.53 


106.3 


31.99 


III 


107.7 


3087 


107.5 


3' 34 


107.4 3 1*8 J 


107.2 


3228 


113 


108.6 


31.15 


108.5 


31.62 


108.3 32.09 


108. 2 


32.57 


114 


109.6 


31.42 


109.4 3«-90 1 


109.3 32.38 


109.2 


32.85 


"$ 


110. 5 131.70 


1 10.4 


32.18 
3M6 


110.3 3**66 


IIO.i 


33.14 


ii6 


1 1 1.5 131.97 


111.4 


1 1 1.2 32.95 


111. I 


33.43 


117 


II2.5 '32 25 


112.3 32.74 


112 2 33*23 


112,0 


33-7» 


118 


"3 4 .S*-53 


113.3 3»-02 


113.1 33.61 


1130 


34.01 


ti9 


1144 '32.80 


II4.2 3330 


1141 33.80 


1 14.0 


34.30 


140 

i 


"5-4 J 33.08 
Dip. 1 Lat. , 


«'5-» ,3.3.58 


115.1 


34.08 


1149 


34.58 
L^t. 


Dep Lut. Dep. 


Lat 

1' 


IXtp. 





, 


4i 


y 1 


30 


1 



73 dkgb|:ks. 



118 



J7 DCBKISISS. 



I 


0/ 


IJ/ 


oO' 


' 45' "^J 


Ut. J 
0.96 


Hep. 


Lit. 


Dep. 

0.30 


Uit. 


J Dep. 
i 0.30 


Lat. 


Dep- 


0.29 


0.96 


0.95 


0.95 


0,JlO 


a 


i.gf 


0.58 


f.91 


0.59 


1.9' 


0.6c 


1.90 


0.61 


1 


».87 


a88 


2.87 


0.89 


2.86 


0.90 


2.86 


0.91 


4 


3.83 


1-17 


3.8a 


1.19 


3.81 


1.20 


3.8' 


1.22 


5 

6 


5-74 


1.46 


4.78 
5.73 


J48_ 
1.78 


4-77 


150 


4.76 


1.52 


«-7S 


5.7* 


1.80 


57' 


1.83 


7 


6.69 


1.05 


6.69 


2a>8 


6.68 


2.10 


6.67 


2.13 


8 


7*5 


a.34 


7.64 


*-37 


7.63 


2.41 


7.62 


M4 


* 9 


861 


a.63 


860 


2.67 


8.58 


2.71 


8.57 


2.74 


lO 


9.5« 


^9* 


9-55 
10.5,1 


2.97 


9-54 


3.01 
3-3 » 


9.5* 


3.05 


11 


la^ft 


3.22 


3.26 


1049 


10.48 


335 


11 


11^ 


3-5 » 


1146 


3.56 


n.44 


3.61 


11.43 


3.66 


H 


i*-43 


3.80 


12-4^ 


3.86 


1240 


3-9' 


12.38 


3.96 


14 


13-39 


4.09 


1337 


4'5 


'335 


4.21 


'3.33 ' 


4.27 


'5 
i6 


»4.34 


4^39 ^ 


'433 


4-45 


'4-3' 
"T5S6 


4,51 

4.81 


14.29 


4.57 
4.88' 


15.30 


4.68 


15.28 


4-74 


15.24 


"7 


16.26 


4-97 , 


16.24 


504 


l6.2i 


5.11 


16.19 


5.18 


i8 


17.21 


5.26^ 


17.19 


5-34 


17.17 


5-4' 


17-14 


S49 


>9 


18.17 


5.56 


J8 T5 


563 


18.12 


$.7» 


18.10 


5.79 


40 
21 


'9''3 

20.08 


5.85 


19.10 

20.06 


593 
6.23 


19.07 


6.01 


19.05 


6 10 


6.14 


2003 


6.31 


2(^.00 


6.40 


IS 


21*04 


6-43 - 


2101 


6.52 


20.98 


6.62 


io.95 


6.7' 


«3 


2200 


6.72 


21.97 


6.82 


21.94 


6.92 


Z1.9I 


7.01 


H 


22.95 


702 


22.92 


7.12 


2289 


722 


22.86 


;.3« 




»S-9« 


7.31 


23.88 


7-4' 


23.84 
24.80 


7-5* 


23.81 

24.76 \ 


7.62 


2486 


7.60 


24,83 


7-7' 


7.82 


7*93 


»7 


2582 


7.89 


»«.79 


8.01 


*5-75 


8.12 


25.71 


8.23 


28 


26.78 


8.19 


2674 


8.30 


26.70 


8.42 


26.67 


^l^ 


29 


47.73 


8.48 


27.70 


8.60 


27.66 


8.7a 


27.62 


8.84 


30 
31 


28.69 
29.65 


8.77 


28.65 


8.90 
9.19 


28.61 


9.02 


28.57 


9.'5 


9.06 


29.61 


a9-57 


9-3* 


29.52 
30.48 


9-45 


3a 


30.60 


9.36 


3056 


9-49 


30.52 


9.^62 


9.76 


33 


3i.5<» 


9.65 


31.52 


9-79 


3 '.47 


9.92 


3«-4S 


10.06 


34 


3».5« 


9-94 


3M7 


10.08 


3*43 


10.22 


32.38 


1^37 


35 

36, 


3347 


10.23 


33-43 


10.38 
10.68 


33.38 


10.52 


33-33 


10.67 


34-43 


10.53 


34.38 


34.33 


1083 


34.29 


iO.98 


37 


3538 


10.82 


35-34 


10.97 


35*9 


11.13 


35*4 


11.28 


38 


36.34 


II. II 


36.29 


11.27 


36.24 


11.43 


36.19 


11.58 


39 


37.30 


11.40 


37.a5 


11.57 


37-10 


"-73 


37-'4 


1LJ9 


40 
4« 


38-»5 


11.69 

~s7 


3820 


11.86 
12.16 


38 '5 


12.03 


38.10 


i».i9 
12 50 


39'* » 


39'6 


39.10 


■»33 


39.05 


4* 


40.16 


12. 28 


40.11 


1245 


40.06 


1263 


40.00 


11.80 


43 


4112 


1*57 


41.07 


12.75 


41.01 


12.93 


40.95 


13.11 


44 


4«.o8 


12.86 


41^02 


13.05 


41.96 


'3.at3 


41.91 


'34* 


4J 
46 


4303 


13.16 


4*98 


13.34 


42.92 


'353 


42.86 


13-72 
14.02 


43-99 


'3.45 


43.93 


13.64 


43.87 


'3.83 


43.81 


47 


449S 


*3.74 


44^9 


'394 


44.82 


14.IS 


44.76 


'4.3j 


48 


45.90 


14.03 


4584 


'4-*3 


45.78 


'4.43 


45«7* 


14.63 


49 


4686 


"4-33 


46.80 


'453 


46.73 


'4-73 


4fs67 


'4-94 


50 
5' 


47-5^ 
48.77 


1462 


47.75 
48.71 


'4.83 


47-69 


15.04 


47.62 


'S-24 


14.91 


15.12 


48.64 


'5-34 


48.57 


'5-55 


$» 


4973 


15.20 


49.66 


15.42 


49-59 


15.64 


49.52 


15.85 


S3 


5068 


15.50 


50.62 15.72 


50.55 


'5-94 


50.48 


16.16 


S4 


51.64 


'579 


51.57 1601 


51.50 j6 24 


5'43 


16.46 


5$ 

56 


5260 


16.08 
16.37 


5».53 • 
5348 


16.31 


52.45 '6.54 


52.38 


i6.77_ 
17.07 


53.55 


16.61 


534' 


16.84 


53-33 


57 


54-5' 


16,67 


54.44 .1690 


54-36 


17.J4 


54- »9 


^Hl 


58 


55.47 


16.96 


55 39 i«7.ao 


55 3* 


'7-44 


55.24 


17.68 


f* 


56.42 


17.25 


56.35 


17.50 


56.27 


'7-74 


56.19 


«7-99 


I 60 


57.38 


«7.54 


57-30 


'779 


57.22 


18.04 


57.14 18.29 


Dist. 


Dep 


Lat. 


Dep. 


Li^t 


Dep. 


Lat. 


Dep. Lat. 





/ 


45/ 1 30/ J 


15' 








72 


D£GB] 


BBS. 









iT 


0' 


15' 


W 1 


45/ 




1 ^ 


Lat. 


Dep. 


Lat 


|Dep. 


Lat. 


Dep. 


Lat. Dep. 






6i 


58.33 


17.83 


58.26 


18.09 


58.18 


1834 


58.10 18.60 







6a 


59*9 


18.13 


59.21 


18.39 


59- U 


18.64 


59.05 


18.90 






63 


60.25 


18.42 


60*17 


1868 


60.08 


18.94 


60.00 


I9»2t 






64 


6l-20 


18.71 


61.12 


18.98 


61.04 


19^25 


6095 


19*5" 






66 


62.16 
63.12 


19.00 
19.30 


62.08 


19.28 


61.99 


»9.55 


61.91 


19.8a 






63.05 


*9.57 


62.95 


19.85 


62.86 


20.12 


■ 




67 


64.07 


»9-59 


63.99 


19.87 


63.90 


20.15 


63.81 


20.43 






68 


6 s 03 


19.8S 


64.94 


20. 16 


64.85 


20.45 


64.76 


20.73 






69 


65.99 


2a 17 


65.90 


20.46 


65.81 


20.75 


65.72 


21.04 






70 
7» 


66.94 


to 47 


66.85 


20.76 


66.76 


21.05 


66 67 


21.34 






67.90 


20.7 6 


67,81 


21.05 


67.71 


4I.35 


67.62 


21.65 






7» 


68.85 


21.05 


68.76 


ai35 


68.67 


21.65 


68.57 


21.95 






73 


6981 


21.34 


69.7a 


21.65 


69.62 


21.95 


69.5a 


22.26 






74 


70.77 


21.64 


70.67 


21.94 


70.58 


22.25 


70.48 


22.56 






75 
76 


7^7* 


21.93 


71.63 


22-24 
22.57' 


7>.53 


22.55 


7:1.43 


2286 


. 




72 6S 


22.22 


7258 


72.48 


22.85 


72.38 


23.17 






77 


73.64 


22.51 


73.54 


22.83 


73.44 


23.15 


73.33 


23.47 






78 


74.59 


22.81 


74-49 


23.13 


74-39 


23.46 


74.29 


2378 






79 


75-55 


23.16 


75-45 


»3-43 


75.34 


23.76 


7524 


24.08 






80 
81 


76.50 


43.39 
23.68 


76.40 


23.72 
24.02 


76.30 


24.06 


76.19 


24.39 






77.46 1 


77.36 


77.*5 


24.36 


77.14 


24.69 






82 


78.42 23.97 


78.31 


24.32 


78.20 


24.66 


78.10 


25.00 






.83 


79-37 *4.a7 


7927 


24.61 


79 «6 


24.96 


79.05 


25.30 


' 




84 


80.33 24.56 


80.22 


24.91 


80.11 


25.26 


80.00 


25.61 


: 




85 
86" 


81.29 


25.14 


81.18 


25-2f 


81.07 
8202 


25.56 


80.9s 


25.91 


? 




82.24 


82,13 


25.50 


25*86 


81.91 


26.22 






87 


83 20 


25.44 


83.09 


25.80 


82.97 


26.16 


82.86 


26.5a 


■ 




88 


84.15 


a5-73 


84,04 


26.10 


8393 


26.46 


83^1 


26.83 


■ 




89 


85.11 


26.02 


85.00 


26.39 


84.88 


26.76 


84.76 


27- «3 


1 




90 
9» 


86.07 


26.31 


85.95 


26.69 


85.83 


27.06 


85.72 


27.44 
27.74 






87.02 


26.61 


86.91 


26:99 


86.79 


27.36 


86.67 


. 




92 


87.98 


26.90 


87.86 


27.28 


8774 


27.66 


87.62 


28.05 


< 




93 


88.94 


27.19 


88.82 


2758 


88.70 


27.97 


88.57 


28.35 


t 




|94 


89.89 


27.48 


89.77 


27.87 


89.65 


28.27 


89.53 


28.66 






9S 
96 


90.85 


27.78 


90.73 
91.68 


28.17 
28.47 


90.60 


28.57 


90.48 


28.96 


'• 




91.81 


28.07 


91.56 


28.87 


91.43 


29.27 






97 


91.76 


28.36 


92.64 


28.76 


92.51 


29.17 


92.38 


29.57 






98 


93.7* 


28.65 


93.59 


29.06 


93.46 


*9^47 


93.33 


29.88 






99 


94.67 


18.94 


94.55 


19-36 


94.42 


a9.77 


94.29 


30.18 






100 

101 


95.63 


29.24 


95.50 


29.65 


95.37 


30.07 
30.37 


95-24 


30.49 






96.59 


a9-53 


96.46 


29.95 


96.33 


• 96.19 


30.79 






!ioi 


97.54 


29.82 


97.4« 


30.25 


97.28 


30.67 


97.14 


31.10 






103 


98.50 


30.11 


98.37 


30.54 


98.23 


30.97 


98.10 


31.40 






104 


99.46 


30.41 


99-3» 


30.84 


99.19 


31.27 


99.05 


31.71 






106 


100.4 


30.70 


100.3 


3«.i4 


too. I 


31.87 


lOC.O 


32.01 
32.32 






10 1.4 


30.99 


10K2 


3«43 


lOM 


lOl.O 






107 


102.3 


31.28 


102.2 


31-73 


102.0 


32.18 


101.9 


32.62 






108 


103.3 


31.58 


103.1 


3*03 


103.0 


32.48 


102.9 


32.93 






109 


104.2 


31.87 


104.1 


31.32 


104.0 


32.78 


103.8 


33-23 






,110 
III 


105.2 


32 16 


105 1 


3^62 


104.9 


33.08 


104.8 


33.54 






106. 1 


3M5 


106.0 


32.92 


i05s9 


33.38 


105.7 


33.84 






jiia 


107.1 


3».73 


107.0 


33.21 


106.8 


33-68 


106.7 


34.14 






113 


108.1 


33.04 


107.9 


33-5« 


107.8 


33.98 


107 6 


34.45 






114 


1090 


33-33 


108.9 


33-81 


108.7 


34.28 


108.6 


34.75 






ILL 
1116 


1100 


33.62 


109.8 


34.10 


1097 


34.58 


110.5 


35-36 






110.9 


33'9» 


iiaS 


34.40 


1 10.6 


34.88 






117 

A 


111.9 


34.a« 


111.7 


34.70 


Iff. 6 


35.18 


111.4 


35-67 






118 


112.8 


34-50 


112.7 


34.99 


112.5 


3548 


112.4 


H'H 






119 


113.8 


34.79 


113.6 


35.a9 


113.5 


35-78 


113.3 


36.28 






120 

m 


1 14.8 


35.08 


114.6 1 35- 59 


114.4 


36.08 


114.3 


3658^ 

Lat. 






]>ep. 


Lat 


Dep. 1 Lat. 


Dep. 


Lat. 


Dep. 






0' 


45' 


31 


0/ 


15 


1 





72 ]H&ail£E3. 



120 






18 DEGREES. 










0' 


Dcp. 


• 15' 


3C 


\' 


' ' 45- " 


55. 

• 


Lat. 


Lat. 


Dep. 


Lat, 
0.95 


Dep. 


Lat. 

0.95 


Dep. 


I 


0.9s .0.31 


0.93 


0.31 


0.32 


O.J2 


% 


I 90 


0.62 


1.90 


063 


1.90 


063 


1.89 


0.64 


3 


2.85 


0-93 


2,85 


0.94 


2.85 


0-95 


2.84 


0.96 


4 


3.80 


1.24 


3.80 


1.25 


3.79 


1.27 


3-79 


1.29 


^ 


4.76 
5-71 


JIL 

1.85 


4-75 
5.70 


> 57 


_4.74 
5.69 


i.59 


4-73 


1.61 


6 


1.88 


1.90 


5.68 , 


«93 


7 


6.66 


a.15 


6.65 


2.(9 


6.64 


2.22 


663 . 


2.25 


8 


7.61 


a.47 


7.60 


2.51 


7.59 


a 54 


7.58 


^S7 


9 


8.56 


2.78 


«.55 


2.82 


8.53 


2.86 


8.52 


2.89 


10 

11 


95' 


309 


950 


3.«3 

3.44 


9.48 

10.43 


3.»7 


9-47 


3.21 


10.46 


340 


10.45 


3 49 


10.42 


3.54 


12 


11.41 


371 


11.40 


3.76 


11.38 


3.81 


11.36 


3.86 


13 


1236 


4.02 


ia.3S 


4.07 


1233 


413 


12.31 


4.18 


14 


i3.3« 


4-33 


13.30 


4-38. 


13.28 


4.44 


13.26 


4.50 


M 


14.27 
15 22 


4.64 


14.25 
15.20 


4-70 


14.22 


476 


14.20 


4^2 


16 


4.94 


5.01 


15.17 


5.08 


15.15 


5.14 


»7 


16.17 


5*5 


16.14 


532 


16 12 


5 39 


16.10 


$.46 


18 


17 12 


5.56 


17.09 


5.64 


1707 


5.71 


17.04 


5-79 


>9 


18.07 


5.87 


18 04 


595 


18.02 


6.03 


"799 


6.11 


20 
21 


19.02 
19.97 


6.18 


18.99 
1994 


6.26 
6 58 


18.97 
1991 


635 


18.94 


6.43 


649 


6.66 


19.89 


6.75 


22 


20.92 


6.80 


20.89 


6.S9 


20.86 


6.98 


20.83 


7.07 


*3 


21 87 


7.11 


21 84 


7. 20 


21 81 


7-30 


21.78 


7-39 


24 


2283 


7.42 


22.79 


7.52 


22.76 


7.62 


2273 


7-7 » 


*5 
26 


23.7^ 
a4-73 


7-73 1 


a3-74 


7.i^3 


2371 
2466 


7.93 


23.67 
24.62 


8.04 


8.03 


24.69 


8.14 


8.25 


836 


27 


25.68 


8.34 


2564 


8.46 


25.60 


8.57 


25.57 


8.68 


28 


26.63 


8.65 


26.59 


8.77 


26.55 


8.88 


26.51 


9.00 


29 


27.58 


8.96 


a7.54 


9.08 


27.50 


9.20 


27.46 


932 


30 


* 28.53 


9.27 


28.49 


939 


2845 
29.40 


9.52 


28.41 


9.64 


3» 


29.48 


9.58 


29.44 


9.71 


9.84 


29.35 


9.96 


3a 


30.43 


9.89 


30.39 


10.02 


3035 


to. 1 5 


3030 


10 29 


33 


3».38 


10.20 


3«.34 


10.33 


31.29 


1047 


31.25 


10.61 


34 


3^34 


10.51 


32.29 


10.65 


32.24 


10.79 


32.20 


10.93 


3? 
36 


33-^9 


10.82 
II. 12 


33.»4 


10.96 


33-«9 


11.11 


3314 
34.09 


11.25 


34-24 


34- »9 


11.27 


34.14 


11.42 


11.57 


37 


35»9 


"•43 


35. H 


11.59 


35.09 


11.74 


35-04 


11.89 


3« 


36.14 


11.74 


36.09 


11.90 


36.04 


12.06 


35.98 


12.21 


39 


37.09 


1205 


37.04 


12.21 


36.98 


12.37 


3693 


12.54 


40 
4« 


38.04 


12.36 


37-99 


12.53 
12.84 


3793 


(2.69 
ij 01 


37.88 


12.86 


38.99 


12.67 


3894 


3888 


38.82 


13.18 


4a 


39.94 


12.98 


3989 


»3 »5 


3983 


'3.33 


39-77 


13.50 


43 


40.90 


13.29 


40.84 


«3.47 


40.78 " 


13.64 


40.72 


1382 


44 


41 8j 


13.60 


41.79 


13.78 


4».73 


13.96 


41. 66 


14.14 


45 
46 


42.80 
43.75 


13.91 


4274 


i4.09_ 
14.41 


42.67 
43.62 


1428 


42.61 


14.46 


14.21 


4369 


14.60 


43.56 


"4.79 


47 


44.70 


14.5a 


44.64 


14.72 


44-57 


14.91 


44.51 


15.11 


48 


45 65 


14.83 


45-59 


15.03 


45.52 


15.23 


45^45 


«5-43 


49 


46.60 


15.14 


46.54 


15 35 


46-47 


1555 


46.40 


«5-y5 


50 


47.55 


'545 

15.76 


47 49 


15.66 
'5-97 


47-42 
48.36 


15.87 
16.18 


47.35 
48.29 


16.07 


48.50 


48.43 


16.39 


52 


49-45 


16.07 


49-38 


16.28 


49-3 » 


16.50 


49.24 


16.71 


53 


50.41 


16.38 


50.33 


16.60 


50 26 


16.82 


5019 


17.04 


54 


5>.36 


16 69 


5I.2JJ 


16 91 


51.21 


»7 13 


51.13 


i7.3<> 


55 
56 


5^-3 1 


17.00 


52.23 


17.22 
»7.54 


52.16 
53. >« 


'745 


52.08 
53-03 


17.68 


53.26 


>7.3i 


53.18 


17.77 


18.00 


57 


54.21 


17.61 


54 » 3 


17.85 


54.05 


18.09 


53.98 


18.32 


58 


55.16 


17.92 


55.08 


18.16 


55.00 


18.40 


54.92 : 


18.64 


59 


56.11 


18.23 


5603 


18.48 


5S95 


18.72 


55.87 : 


18.96 


60 

• 


5706 
Dep. 


18.54 


56.98 
Dep. 1 


18.79 


56.90 


19.04 


56.82 ; 


19.29 


Lau 


Lat. 


Dep. 

30 


Lat. 


Dep. 


Lat. 


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0/ - 1 


45' 


15 


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71 l)£pf^£Sv 









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0/ 


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50' 


45' 


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L&t. 1 Uep. 


Lat. 1 Dep. 


Lat. 1 Dep. 


Liit 1 De 

57.76 ;I9.6 


6i 


58.01 18.85 


57 93 


19.10 


57.85 


'19-36 


61 


S8.97 


19.16 


58.88 


19.42 


: 58.80 


1967 


58.71 


19.9 


63 


59.9a 


19.47 


59.83 


1973 


1 59.74 


«9-99 


59.66 


20 2 


64 


6a87 


19.78 


60.78 


20.04 


' 60.69 


20.31 


60.60 J20.5 


6j 

66 


61.82 


20.09 


61.73 
62.68 


20.36 

2067 


61.64 


20.62 


61.55 


120.8 

1 


62.77 


20.40 


62.59 


20.94 


62 50 


21.2 


67 


63.7* 


20.70 


63.63 


20.98 


i 63.54 


21.26 


63.44 


21.5 


6S 


64.67 


21.01 


64.58 


21.30 


' 6449 


21.58 


64.39 


21.8 


69 


65.62 


21.32 


6553 


21 61 


i 6543 


21.89 


6534 


22.1 


70 
7» 


6657 


21.63 


66.48 


21.92 


: 66.38 
67.33 


22.21 


66.29 


22.51 


ty.^ 


21.94 


67.43 


22 23 


22.53 


67.23 


22.8 


7a 


68.48 


22.25 


68.38 


22.55 


68.28 


22.85 


68.18 


23 I 


73 


6943 


2256 


69.33 


22.86 


69.23 


23. J 6 


.69 13 ia3.4 


74 


70.38 


22.87 


70.28 , 


43.17 


•70.18 


2348 


70.07 23.7. 


75 
76 


71.33 
72.28 


23.18 
23.47* 


71 23 

72.18 


23.49 


71.12 


23.80 


71.02 24.1 


»3-8o 


72.07 


24.12 


71.97 ,24.4 


77 


73-«3 


»3 79 


73.13 


24.11 


73.02 


*4 43 


7%9« !a4.7 


78 


74.18 


24.10 


74.08 


H.43 


73.97 


24.75 


73.86 ,-25.0 


79 


75.'3 


24.41 


. 75-03 


24-74 


74-9* 


25.07 


; 7481 


25.3. 


80 
81 


76.08 


24.72 


7598 


25.05 


75.87 
76.81 


25.38 


75-75 


25.7 


77<^4 


25.03 


76.93 


»537 


25.70 


76.70 


26.0. 


82 


77 99 


»5 34 


77.88 


25.68 


77.76 


26 0» 


.7765 


26.31 


83 


78.94 


25.65 


78.83 


»5.99 


78.71 


26.34 


' 78.60 


266! 


84 


79.89 


25.96 


79-77 


26.31 


79.66 


26.65 


79-54 


27.C< 


86 


80.84 


26.27 


80.72 


2662 
26.93 


80.61 


2697 


80.49 


273 


81.79 


26.58 


81.67 


81.56 


27.29 


8144 


27.6. 


S7 ^ 


8274 


26.88 


82.62 


27.25 


82 $0 


27.61 


82.38 J27.9 


88 


83,69 


27.19 


83.57 


27-56 


83.45 


27.9a 


, 83.33 p8.2. 


89 


84.64 


a7.50. 


8452 


27.87 


84.40 


28.24 


84.28 !28 6 


90 
9' 


85.6P 


27.81 


85.47 
86.42 


28 18 


85.35 ^ 56 


85.22 


2S.9; 


8655 128 12 


28.50 


86.30 


28.87 


86.17 


29.2' 


9* 


87 50 


28.4J 


87.37 


2881 


!z*5 


29.19 


87.12 


29.5: 


91 


884$ 


28.74 


88.32 


29.12 


88 19 


29- 5 i 


88.06 


29.81 


94. 


89-40 


29.05 


89,27 


2944 


89.14 


29.83 


8901 


30.2: 


9? 
96 


90.35 


29.36 


90.22 


*9-75. 


90.09 


30.14 


, 8996 


30.5. 


9130 


29.67 


t 9».»7 


30.06 


91.04 ,'30.46 


90.91 


30. 8< 


97 


92.25 


29.97 


92.12 


30.38 


91.99 


30.78 


91.85 


31. li 


9« 


93.ao 


3028 


93.07 


30.69 


94.94 


31.10 


9 2. So 


3».J< 
31.82 


99 


94 »S 


SO 59 


94.02 


31.00 


93.88 


3i~4« 


93.75 


100 


95.11 


30.90 


94-97 


3».32 


9483 


3' 73 
32.05 


^4 69_ 
95.64 


3.2. 14 


101 


96.06 


31. ai 


95.92 


31.63 


95-78 


32,4( 


102 


97.01 


3»p 

31.83 


96.87 


31-94 


96.73 


32.36 


96.59 


32.7i 


103 


97.96 


97.82 


33.26 


97.68 


3268 


97.53 


ii »i 


104 


98.91 


32.14 


98.77 


32.«7 


98.63 


33.00 


98.48 


33.44 


105 

f06 


99.86 


3 MS 


99-72 


32.88 


99 57 
100.5 


33.32 


99-43 


33.7ii 


100.8 


i^.'jd 


100.7 


3320 


33.63 


»0P-4. 


34.03 


107 


101.8 


3306 


101.6 


33.51 


10 1. 5 


33.95 


101. 3 


34.3S 


108 


102.7 


33.37 


102.6 


33-82 


102.4 


34*7 


102.3 


34.7a 


109 


103.7 


33-68 


103.5 


3413 


1034 


34.59 


103.2 


J5.<^ 
35 3{ 


IfO 

III 


104.6 


33-99 


104.5 


34.45 


1043 


34-90 


104. 2 


105.6 


34-30 


105.4 


34.76 


105.S 


35.22 


105. 1 


35^ 


II& 


106.5 


34.61 


106.4 


35.07 


106.2 


3^'lf 


106. 1 


36.| 


113 


107.5 


34.9» 


107.3 


35-39 


107.2 


35.86 


107.0 


36.|| 


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108.4 


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108.3 


35.70 


108. i 


3*-«7 


108.6 


36.1^ 


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116 


109.4 


35-54 


109.2 


36.01 


109 I 


36.49 


1089 
1Q9.8 


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110.3 


35-85 


110.2 


36.33 


1 10.0 


36.81 


37.1 


117 


iit.3 


36.16 


ill.i 


36.64 


1 11.0 


37.12 


11Q.8 


37 1 


118 


112.2 


36.46 


1IZ.1 


36.95 


III.9 


37-44 


HI. 7 


37-1 


119 


113.2 


3677 


113.O 


37.27 


112.0 

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3776 


112.7 


38 3| 


• 

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114.1 
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37.08 


M40 3758 


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113.6 


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11.31 


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11.27 


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4.34 


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4.61 


13.10 


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34.93 
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38.71 


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85.38 


3«.50 


85.24 


31.87 


85.10 


9* 


86.45 


3«47 


86.31 


31K4 


86.17 


32.12 


8603 


32.59 


91 


87.39 


31.81 


87.25 


32.19 


87. ft 


32.57 


8697 


32.95 


94 


88.33 


32.15 


88.19 


32.54 


88.05 


32.92 


87.90 


3.30 


9J 
96 


89.17 


y.49 


89.13 


32.88 


88.98 


33.27 


8884 


3366 


90.21 


3».83 


9ao7 


33.23 


89.91 


33.61 


8977 


340» 


97 


91.15 


33 '8 


91.00 


33.57 


90.86 


33-97 


90.71 


34 37 


98 


92.09 


335* 


9 '.94 


33.9> 


91.79 


3432 


9164 


34-72 


99 


9303 


33.86 


92.88 


34271 


92.73 


3467 


9158 


35.07 


100 


93.97 


34.10 


93.82 


34.61 


9367 


35.02 


93.51 


35.43 


lot 


94.91 


34 S4 


94.76 


3496 


9460 


35.37 


94 45 


35.78 


lOS 


95.85 


34.89 


95.70 


3530 


95.54 


3572 
36.07 


95.38 


36.14 


lOJ 


96.79 


3i*3 


96.63 


35.65 


96.48 


9632 


3649 


104 


97.73 


35 57 


9757 


36.00 


97.4« 


36.41 


97.25 


36.85 


105 
106 


98.67 
99.61 


35 9« 


98.51 


36.34 


98.35 
9929 


36.77 


98.19 


37.20 


36-*5 


99.45 


36.69 


37.12 


99.12 


37-55 


107 


100.5 


36.60 


1004 


37.03'. 


100.2 


3747 


loai 


37.9 « 


loS 


iot.5 


36.94 


101.3 


37.38 


101.2 


37.82 


101.0 


38.26 


109 


102.4 


37.*8 


102.3 


3773 


101.1 


38.17 


101 9 


38.62 


no 
tit 


103.4 


37.62 


103.2 


38.07 


103.0 


38.52 


ioa.9 


38.97 


104.3 


3796 


104.1 


38.42 


1040 


38.87 


103.8 


39-33 


IIS 


105.2 


38.31 


105.1 


38.77 


104.9 


39.12 


104.7 


3968 


«»I 


to6.2 


38.65 


106.0 


39. u 


105.8 


39.57 


105.7 


40.03 


"4 


I07. 1 


38.99 


107.0 


39.46 


106.8 


399* 


106.6 


40.39 


«»5 
ti6 


108. 1 


39.33 , 
39.67 


107.9 
108.8 


39.80 


107.7 


4017 


107.5 


40.74 


109.0 


40,15 


108.7 


40.6a 


108.5 


41.10 


«'7 


109.9 


4002 


109.8 


40.50 


109.6 


40.97 


109.4 


4».45 


irS 


110.9 403^ 


If 0.7 


40.84 


1 10.5 


4t.32 


iia3 


41.81 


119 


111.8 40.70 


111.6 {41.19 


III. 5 


41.67 


111.3 


42.16 


tto 


112.8 41.04 


112.6 41.53 


1124 41.02 


iia.a 


42.52 


i 


Bep. . Vkt, 


IJep. 


Lat. 


Dep. Lat. 


Dep. Lat. | 


(/ 


45 


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30' 


15' 


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69 hegmes. 



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L»t 


Dep. 


IJ' 


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Lai 


Uep. 


Lit. 


n*p. 


Ljt. [ Uep. 




3 


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I.S7 
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o.j6 
i*o3 


0.9.1 
i^Bo 


0.3* 


0.93 
1.K6. 


0,73 


i:g6 0^74 
3,79 I-" 










3-75 i-tS 


3.7» 


"47 


3-71 : ..48 




; 


+67 


I-Tfl 


4.6^ 1 Ki 


4.65 


l.S] 


4-64 1 '-85 




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~r 


iSt 


1.17 


Si* 


i,ao 


5.57 i.»i 






6)4 




6. J I 


1.54 


6-S' 




6.50 


»-i9 




E 


7-47 


1.K7 


7.46 










1.96 




9 


S.40 


jn 


8,39 


J.16 


8^37 


330 


8.J6 


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9-t9. 


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m:'o 


3.S4 
4.30 


\V,l 


3 'J9 
4-ii 


10. Z3 


4.03 
4.40 


II-. 5 


4.08 
4-4S 




I] 


11.14 


4.66 








4-76 




4.8. 




>4 








5.07 


'3.03 


S '3 




i-'9 




_^ 


'4-00 


i-7 3' 
6,0, 


_^,8 
I (,84 


(44 

IT. 


■3.96 

■4-39 


T8S" 
6.13 


1393 
14.86 
'i-79 


5.56 

593 

*30 




iS 


16^83 


6.41 
6.81 


16.78 


IT, 


■6.7 5 
ir.68 


6.60 
696 


1671 

,7.6s 


S.67 
7.04 






iB^ftT 




.8^^ 


.ZIL 


18.6 1 




.g-iS 


-■41 




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ig.61 


TiT 


19 S7 


"761 


i-)'i* 


T^o" 


■ 9.5" 


7-78 






ID. 54 


7.SH 


W.iO 


797 


13.47 


8.06 


J0.43 


81; 








a. 14 


11-44 


8-3-t 


11-40 


Su 


11.36 


8.51 






it.41 




11.37 


S.70 


11.33 


i.8o 


1119 


1.8, 




»5 


'i J4_ 


_L'!_ 


H.!0 


406 


'3-^6 


JJL 


H-ii 


^16 




16 


14.17 


9.31 

9.68 


14,13 
15.16 


9-41 
9-79 


14.19 
ij 11 


9 S3 

9.90 


■H 


9.63 




iK 


16.14 


I0.OJ 


16.10 


10-1 j 












19 




10.39 


V.03 


.0.51 


16:98 


10.63 


16.94 


107! 




Ji. 


is'oi 


10. 7 s 


17 9^ 


10.87 


17 9' 




17.86 








1S94 


11.11 


.S.)(9 


11 14 


-^87 


rr^r 


18.79 


m!49" 






19.87 


11 47 


19 K; I11.60 


1977 


11 73 


19.71 


11,86 






i^.Bi 


il.Sj 




30 70 


1109 


30.65 


111 J 




J4 


n74 


It.. 8 


3.:69 i.l'u 


3' 63 


,1.46- 


31.58 


.».6^ 






]z.6» 




,H6i 'll.Sq 


3' 56 


il,S] 


}^%< 


1197 




~J6" 


3361 


rTTsJ" 


3J.i4 i'J.Ot 


liS" 


13.19 


JJ.4+ 


'3-34 






34 54 


i3>6 


34-48 13.11 




13.56 


34-37 


1J.7. 




I'i 


Ji-4« 


.3.61 


3)4' 'J77 


3S-3'' 


i!93 


]i.'9 


14.08 






36.41 


13.9K 


36 3( j.4.14 


36.19 


14.19 


36.11 


'4 45 






37.34 




;7.iS ;(4 JO 


37.11 




37 '5 


148a 




~ 


~S.xS 


■4^r 


38.1. J1486 


38.5 


'5 03 


33 08 


IS.19 






3911 


'S-°5 


39.14 i,Ml 


,9.08 


'S39 


39.01 


15.56 




11 


40.14 
4..0B 


15.41 


40.08 15.58 
41.01 '15.95 


40.01 
40.94 


15.76 
16 >3 


39 94 
40. 8 7 


'5 93 

16.30 




4! 




16^13 


4-.94 1631 


41.87 


1649 


41.80 


'6 68 




46 
4- 
48 


44-"bi 


16.4S 
.6. 84 

17.56 


41 8 7 16.67 
4J.80 17,03 
44.74 17.40 
4,67 17-76 


"41 Bo 
43 73 
44.66 
45 S9 


16.86 

'7.1J 
17-59 
1796 


4173 
4365 

44.58 

4i ii 


17.0s 
17.41 




J0_ 


47.61 


TsTiT 


46.60 .3.11 
47-SJ '8-48 


46,51 ^33 
4-45 >8.69 


46.44 


.8.53 
.8.90 




47 37 






43. i( 


18.64 


.4^46 18,85 


4K.j8 119.06 


4839 


19,17 






49.48 


18.99 


49.40 ly.!. 


49.31 119.41 


49 '3 


19.64 






{041 


'9-3i 


50 3J 19.57 


50.14 !l9.79 


SO.16 








('■35 


19.71 


51,16 19.9.1 


51.17 1016 


51.08 


10.3! 




















56 


SX.1S 


10.07 


51.19 1030 


51.10 10.51 


51.01 


ia75 






ij.ii 


IQ.43 


SJ.ii 10.66 


5J.03 ,W.B9 


51.94 












5406 11.0, 


53.96 |ii.i6 


S3«T 


*'-4? 




S9 


"'08 


ti.H 


I4.99 --38 


J4.K9 111.61 


54- 80 


*1.86 




(>o 


56.01 


11. JO^ 


55.91 ,11 7! 


55,83 ,11 99 


Sv71 


11. IJ 




















Q 


l)en 


l.at. 


Utp. 1 LbL 


l).p. 1 Lut. 


^^ 


Ul. 









45' 30' 


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■""^ 


' 




68 


D£U1( 


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30' 


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a 


L:.! 


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lj.1 |l>p. 


Lat 1 1)P|>. 




i).p. 


~6i~ 


56-95 


ir,86 


S6.«i 111,. 


5676 


".36 


"6:66 


n.6o 




f7.liS 






5769 




S7.59 


11.97 


H 


!S,8. 


ii.jS 


!8;7i 11 S] 


,861 


13.09 


58,51 


13-35 


^* 




11.94 


59-6 S >3 »o 


59 SS 


13.46 




13,71 




6i.6B 


1319 


6358 Ii3i* 


6048 


13.81 


60.37 


1409 


'bT 


61.61 


'365 


6151 1391 


61.41 


14.19 


61.30 


14.46 


6t 


6.5J 


1+.01 


6144 i4'8 


61,34 


14,56 


fii.13 


14.83 


68 


63.4< 


*4.]7 


6338 His 


63,17 


14.91 


63.16 


15.10 


6, 


fi+4» 1m.73 


64.31 li.Oi 


64.10 


15-19 


64,C9 


M.(7 




6(.J5 i») 02_ 


6<.i4 25.37 


65,13 


iv66 




^5 94 




















66.1S '13.44 




>i73 


6&.06 


i6.ai 


*S.9S 


16, ]i 




671, l^i.So 


67. to 




66.99 


16.39 


66,K7 


16. b8 


»3 


6S.., !i6.i6 


6K0+ 


;&.46 


67.91 




67,80 


17-05 


74 


6y,0S >6,ii 




1681 


68.1)5 




68.-3 


17-41 




7i>.0i li6.3B 


6990 


17 .8 


69.78 


17^49 


69.66 


17-79 


" 


70.,i »7.»4 
71.89 »7.S» 


7083 


i7iS 
179' 


?i.64 


17.^5 


70. i9 
71-51 


18". 16 
iB.Sj 


78 


7»E» '795 


71. TO 


i8,»7 


71-57 


18,59 


7145 


18,90 




71.7! '*■'' 


73.63 


i!.6i 


'I'iO 


1895 


73.3« 


19->7 




7469 iL*!!. 


74.S6_ 


19.00 


74-43 


19.31 


l.t}° 


1964 


















K> 


-5.51 


19,03 




1936 


7i-36 


19.69 


7!.i3 


30,01 




'6-!! 


i!9,J9 


76.41 


19.71 


7S.19 


32.05 


76.16 


30,39 


8j 


77.49 


19.74 


77.36 


30.0S 






77.09 


10.76 


H 


78.4* 


3O.I0 


78,19 (30-44 


7&.15 


30,79 


78.01 


JT.13 




'9'3! 


30.46 


79. 11 jjo.Si 


79-09 




78.95 


31.50 


















86 


*8s:^ 


3081 


80[! Ji.iT 


80.01 


.I'ii 


79.sa 


31,87 


»7 


8. It 


Jl.lS 


S1.08 3 '53 


So 95 


3I-S9 


80.1 1 


31.14 


88 


8i.iff 


Jl !4 


31.01 31 89 




31.15 


81.74 


31.51 


Kq 


83.09 


31.89 


81,9,- 31.1S 


siisi 


31.61 


81.66 


11.98 




84 01 


3^»J_ 


83.88 3t.6i 


Bl-74 


31.99 


«3-!9 


13J5_ 


9' 


84.96 


J 1.61 


84.81 319* 


8467 


33-35 


S451 


liJi 


91 


8J.89 


31.97 


8;. 74 . 33.34 


85.60 


33-71 




,fo, 


9J 


86.81 


33.3J 


ss.6a 33.71 


86.53 


34.08 


86! 38 


34.46 


94 


87.76 


J3*9 


87.61 


34.07 


8746 


34.4s 




14-83 


5i 


88.69 


3±2t 


88.S4 


y_*l 


88.39 


1±^1 


I8.14 


3i^iO 


96 


89.62 


34.40 


89.47 


34-79 


~i;~ 


3S-.8 


89.T7 


35 57 


'I 


90.56 


34'76 


90.40 


35.16 


90,1! 


35!! 


90.09 


3i94 


98 


9 '49 


35.J1 


9>.34 


35. 51 


9..18 


3591 


91.01 


36-31 


99 


9141 


3S-48 


91.17 


35.S8 


91. i[ 


36,18 


9'9S 


36,69 




Sije 


35.84 


9J.10 


36.14 


'1>^ 


3665 


91,88 


37.06_ 






















94-19 


3610 


94-13 


36.61 


93-97 




93.S. 


3743 ■ 




9(.»3 


36. SI 


9!.06 


36.97 


94,90 


37-3R 


9+74 


17.80 


10) 


96.16 


3691 


96,00 


37-33 


9i.83 


37."! 


95-67 


38.1J 


.04 


97.09 


37 17 


9693 


37.69 


96.76 


38.11 


96.60 


1S.54 




9803 


3763 


9786 


j8o6 


97*9 


38,48 


97.51 


38.9' 


i^ 


98.96 


3799 


98.79 


38,41 


98.6i 


38.85 


98.4! 


3918 




99.89 


3B.3S 




38.78 


99. 5! 


19-11 


99-38 


19-6 { 


108 


10Q.8 


38-70 


100.7 


39-14 


100.5 


39.58 


100,] 


43.01 




loi.S 


39.06 


101.6 


39!> 


101-4 


39-95 




40,39 






3941 


101 ( 


39ji_ 




40-31 




4076 


111 


.0J.6 


19-78 


103.5 


40.13 


103,] 


40 68 


103.1 


41-11 




.04.6 


40 '♦ 


1044 


40.59 


104.1 


41.05 


104 


41.50 


IIJ 


roj-i 


40.50 


10! J 


40.96 


1051 


41 4. 


105,0 


41.87 


114 


■06.4 


40,8 s 


lOfi.l 


41.31 




4'.7B 


105,9 


41.14 


"( 






107.1 


4^68 


107. c 


li-'i. 






116 


"TisT 


4~-S7" 


108. ■ 


41.04 


107.9 


41.51 




^",7 




10».* 


4 ".93 


109.0 




108.9 


41.8S 


loi:. 


41.36 


Ilg 




41.19 






109.8 


41-»5 


,09.6 


43.73 


"9 




41-6! 


110.9 


43.13 


110.7 




110.5 








41.oo^ 


111.8 


4149 




livs 






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l>ep. 


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128 






23 DfiGRfiBS. 








I 


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16' 


30' 1 


45' 


Lat. 


Dep. 

0.37 


Lat 

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Dep 


Lat. 


l)<p. 


Lat 1 Dep 


0.93 


0.38 


a92 


0.38 


0.92 ! 0.39 


2 


1.85 


0.75 


^ 1.85 


0.76 


1.85 


0.77 


1.84 ; 0.77 1 


3 


2.78 


1.12 


2.78 


1.14 


2.77 


1.15 


M7 i-»6 1 


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3-71 


1.50 


3.70 


1.51 


3.70 


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3.69 ! «^5 1 


5 
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5.56 


1.87 
225 


463 


1.89 
2.27 


462 


1.91 


4.61 


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230 


5-53 


a.32 


7 


6.49 


2.62 


6.48 


2.65 


6.47 


2.68 


6.46 


a.71 


8 


7.42 


3.00 


7.40 


3.03 


739 


3.06 


7.38 


3.09 


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8.34 


3-37 


8.33 


34i 


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S.30 


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9-27 


3-7 5_ 
4.12 


9,26 


3.79 


9.24 


383 


9.2a 

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3-87 
425 


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10. u 


4.17 


10.16 


4.21 


12 


11.13 


4.50 


11.11 


4-54 


ii.09 


4-59 


11.07 


4.64 


13 


12.05 


4.87 


12.03 


4.92 


12.01 


4.97 


11.99 


503 


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12.98 


5.24 


12.96 


530 


12.93 


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12.91 


5.4> 


i6 


13.91 


5.62 


13.88 


S.68 


13.86 


5-74 


13.83 


5.80 
6.19 


14-83 


599 


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6.06 


14.78 


6.1a 


14.76 


17 


1576 


6.37 


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644 


15.71 


6.{i 


15.68 


6. $7 


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16 69 


6,74 


1666 


6.82 
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»7.59 ' 


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7.27 


17.52 


7.35 


20 
21 


«8.54 


7.49 


18.51 


7-57 


18.48 


7.65 


18.44 


773 
8.12 


>9 47 


7.87 


19.44 


7.95 


19.40 


8.04 


»9'37 


22 


2040 


8.24 


20 .6 


8.33 


20.33 


5i* 


aa29 


8.5 L 


23 


21 33 


8.^2 


21.29 


8.71 


21.25 


8.80 


21.21 


8.89 


*4 


22.25 


8.99 


22.21 


9.09 


22.17 


9.18 


%%i$ 


9.28 i 


26 


23.18 


9-37 


23.14 
2406 


947 


23 10 

24,02 


9-57 


23.06 
23.98 


9.67 


24.1 1 


9-74 


9.84 


9-95 


10.05 


27 


25.03 


io.ix 


24.99 


10.22 


24.94 


I0.3J 


S4.90 


10.44 


28 


25.96 


10.49 


25.92 


10.60 


25.87 


10.72 


25.82 


10.83 


20 


26.89 


10.86 


26.84 


1098 


26.79 


11.10 


26.74 


11.21 


30 
3»- 


27.82 


11 24 


27.77 


11.36 


27.72 


11.48 


27.67 


11.60 
11.99 


28.74 


11.61 


28.69 


11.74 


28.64 


11.86 


2859 


32* 


29.67 


II 99 


29.62 


12.12 


29.56 


12.25 


29.51 


ia.37 


33 


30.60 


1236 


3054 


12.50 
r2.87 


30.49 


12.63 


30^43 


12.76 


34 


3i-52 


'2.74 


3«.47 


31.41 


13.01 


3t-3$ 


ia.15 


35 


32.45 


13.11 


32. ?9 


11.25 


32.34 


•3.39 


3228 


«3-53 
13.92 


36 


33.38 


«349 


33.32 


1303 


33.26 


13.78 


33.20 


37 


34-31 


13.86 


34.25 


14.01 


34.18 


14.16 


34.12 


14.31 


38 


35»3 


14.H 


35.17 


14-39 


35-«i 


14-54 


35.04 


14-69 


39 


36.16 


14.61 


36.10 


14.77 


36.03 


1492 


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15.08 


40 
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37.09 


14.98 
15.36 


37.02 


15.15 


36.96 


15.31 


36.89 


15-47 
15.86 


38.01 


37.95 


15.52 


37.88 


15.69 


37.81 


42 


38.94 


«5-73 


38.87 


15.90 


38.80 


1607 


38.73 


16.24 


43 


39-87 


16.11 


39.80 


16.28 


39*73 


16.46 


39-6; 


16.63 


44 


40.80 


1648 


40.72 


16.66 


4065 


16.84 


40.58 


17 02 


45 
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41.72 


16.^6 


4145 


17.04 \ 41-57 


17.22 
7760 


41.50 


1740 


42.65 


17.23 


42.57 


17.42 


42.50 


42.42 


17-79 


47 


43.58 


17.61 


4350 


17.80 


43.42 


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43-34 


18.18 


4«- 


44-50 


17.98 


44.43 


18.18 


44-35 


18.37 


44.27 


18.56 


49 


45-43 


18.36 


45*5 


18.55 


45.27 


18.75 


45.19 


18.95 


50 
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4636 


18.73 


46.28 


18.9s 


; 46.19 


19.13 


46.11 

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•934 
1972 


47.29 


19.10 


47.20 


19-31 
19.69 


47 12 


19.52 


52 


48.21 


19.48 


48. J 3 


48.04 


19.90 


47.95 


20 if 


53 


49.14 


19.85 


4905 


20.07 


48-97 


20.28 


48.88 


20.5a 


54 


50.07 


2023 


49.98 


2045 


49.89 


20.66 


49.80 


20.88 


55 
56 


51.00 


20.60 


50.90 


2083 
21.20 


50.81 


21.05 


50.72 


21.27 


51.92 


51.83 


5«-74 


21.43 


51.64 


21.66 


57 


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21.35 


52.76 


21.58 


52.66 , 


21.81 


52.57 


22A4 


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53.78 


21.73 


53.68 


21.96 


53.59 


22.20 


53-49 


22.43 


59 


54-70 


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54.61 


22.34 


54-5« 


22.58 


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22.82 


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22.48 


55.^53 

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22.72 


55-43 


22.96 


55-33 


23.20 


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22.85 


56.46 


2).10 


56.25 


23-59 


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57.38' 


1348 


57.28 23.73 


57.18 


23.98 


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58.41 


23.60 


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60.27 


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59.13 24.49 


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24.75 


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14.35 


60.16 
61.09 


2461 
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59.94 
6087 


25.14 


61.19 


24.72 


25.51 


67 


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62.01 


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61.90 25.64 


61.79 


25.91 


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63.05 


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62.94 


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62.82 26.02 


6J.71 


26.30 


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25.85 


63.86 


26.13 


63.75 26.41 


63.63 


26.68 


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26.22 


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65.71 


26.51 


64.67 26.79 


64.55 


2707 


65.83 


26.60 


16.88 


65.60 27.17- 


65.48 


27.46 


74 


66.76 


26.97 


66.64 


27.16 


66.52 27.55 


66.40 


27.84 


71 


67.68 


a7-35 


67.56 


2764 


67.44 *7.9sr 


67.32 


28.23 


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68.61 


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68.49 


28.02 


68.37 28.32 


68.24 


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75 
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6954 


28.10 


69.4A 


2840 


69.29 


28.70 
19.08 


69.17 


29.00 


70^47 


28.47 


70.34 


28.78 


70.21 * 


70.09 


29.39 


77 


7«.39 


2884 


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29.16 


71.14 29.47 


71.01 


29.78 


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29.22 


7219 


2953 


72.06 29.85 


71.93 


30.16 


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29.91 


72.99 3023 


72,85 


30.55 


80 
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30.67" 


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73.78 
74.70 


30.94 


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3034 


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76.03 


3072 


75-89 


31.05 


75.76 ;3».38 


75.62 


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31.09 


76.82 


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76.54 


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31.47 


77.75 


31.81 


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79.60 


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80.23 


33.64 


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81.30 133.68 


81.15 


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82.52 


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82.08 


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83.30 


34.08 


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83.92 


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34.46 
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85.15 


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84.84 


35.58 


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85.76 


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89.94 


89.78' 


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90.86 


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90.70 


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38.27 


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93.48 


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38.21 


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96.83 


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48.68 


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103.7 


42.41 


103.5 


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23.71 


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24.62 


11.09 


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11.20 


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11.30 


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11.39 


a5-53 


11.50 


25.48 


it.6i 


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11.72 


19 


26.49 1 1 1.80 


26.44 


it.91 


26.3t 


It 03 


26.34 


12.14 


30 


27.41 11220 

28.32 ; 12.61 


»7.35 


12.32 
12.73 


27.30 
28.21 


12.44 


27.24 


12.56 


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28.26 


12.86 


28.15 


12.98 


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29.23 13.02 


29.18 


13.14 


29.12 


13.27 


29.06 


13.40 


33 


30.15 W3.42 


30.09 


'355 


3003 


13.68 


29.97 


1382 


34 


31.06 13.83 


31.00 


13.96 


30.94 


14.10 


30.88 


i4.»3 


35 

36 


31.97 ,i4a4 


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14.38 


31*85 


14.51 


41.78 


1465 


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14.64 


32.82. 


«4-79 


32.76 


14.93 


32.69 


15.07 


37 


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33.67 


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33-60 


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34-71 


15.46 


3465 


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34-58 


15.76 


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16.02 


35.49 


16.17 


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16.33 


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36.47 


16.43 


36.40 


16.59 
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36.33 


16.75 
17.17 


ii6 68 


37.38 


1684 


37.31 


37.*3 


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38.37 17.08 


38.29 


17.25 


38.22 


17.4* 


38.14 


17.58 


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39.28 17.49 


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17.66 


39.'3; 


17.83 


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18.00 


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40.12 


18.07 


40.04 


18.2 c 


3996 


18.42 


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18.30 


41.03 


18.48 

1889 


4095 


18.66 
19.08 


40.87 


18.84 


4».02 


18.71 


41.94 


4 1.86 


4«77 


19.26 


47 


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19.12 


4285 


19.30 


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19.49 


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19.68 


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43.85 


19.52 


43.76 


1971 


43.68 


19.91 


43.59 


20.10 


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44-76 


19.93 


44.68 


20.13 


44.59 


20.32 


44.50 


20.51 


50 


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45-59 


20.54 


45'50 


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45-41 


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46.59 


20.74 


46.50 


20.95 


46.41 


21 IS 


46.32 


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47.50 


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47-4» 


21.36 


47.3* .•*«-56 


47.22 


21.77 


53 


48.42 


21.56 


48.31 


21,77 


48.23 1 21.98 


48.13 


22.19 


54 


49-33 


21.96 


49.24 


22.18 


49-14 |"|9 


49.04 (2>.6l 1 


55 



50.25 


a*.37 
22.78 


50. IS 


22.59 


50.05 
50.96 . 


22.81 ' 


49.95 
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51.16 


51.06 


23.00 


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52.07 


23.18 


5'r97 a3.4' 


51.87 -23.64 


51.76 2386 


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n.59 


52.82 23.82 


5**78 


24.05 1 


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59 


53.90 


24.00 


53.79 24.23 


^3.69 


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53.5? *4.70 


60 
1 


54.81 
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24.40 


54-71 1 


24.64 
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54.60 
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54.49 


25,12 


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55.61 15.05 


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55.40 


15-54 


62 


56 64 25.12 


56.53k 


25.46 


56.41 


15.71 


5650 


15.96 


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57.55 »5«^ 


57-44 


15.88 


57.33 


16.13 


57.11 


16.38 


64 


58,47 |a6^3 


58.35 


26.19 


5814 


26.54 


58.11 


16.79 


65 
66 


59.38 


16.44 


59.16 
60.18 


26.70 


59-" 5 


16.96 


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59.94 


17.11 


60.19 


16.84 


27.11 


60.06 


17.37 


17.63 


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61.21 


17.15 


61.09 


17.5* 


60.97 


17.78 


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18.05 


68 


61.11 


17^66 


61.00 


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61.88 


18.10 


61.75 


18.47 


69 


63.03 


18.06 


61.91 


a8.34 


61.79 


28.61 


61.66 


18.89 


70 
71 


63-95 


18.47 


63.81 


28.75 


63.70 


2903 


63.57 


1931 


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28.88 


64.74 


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64.61 


19-44 


64.48 


19.71 


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65.78 


19.19 


65.65 


a9-57 


65,51 


29.86 


65.39 


30^4 


73 


66.69 


19.69 


66.56 


29.98 


66.43 


30.17 


66.29 


30.56 


74 


67.60 


30. 10 


67.47 


30.39 


67.34 


30.69 


67.10 


30.98 


75 
76 


68.51 


30.51 


68.38 


30.80 


68.15 
69.16 


31.10 


68.11 


31.40 


«9 43 


30.91 


69.19 


31.11 


3«-5» 


69.01 


31.81 


77 


70-34 


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31-63 


70-07 


31.93 


69.93 


31.14 


78 


71.16 


3«.73 


71.12 


31.04 


70.98 


3*35 


70.84 


31.66 


79 


71.17 


31.13 


71.03 


3M5 


71-89 


31.76 


7».74 


33.07 


80 
81 


73-08 


31.54 


7194 


31.86 


71.80 


33.18 


71.65 


33.49 


74-00 


3*95 


73.85 


33.»7 


73.71 


33-59 


73^56 


33.91 


89 


74.91 


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74-76 


33.68 


74.61 


34.00 


74-47 


3433 


83 


75.8a 


3376 


7568 


34.09 


75.53* 


34-4* 


75.38 


34.75 


84 


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34.17 


76.59 


34.'50 ' 


76.44 


34.83 


76.18 


35-17 


85 
86 


77.65 


34-57 


77.50. 


34.91 


J7.35 
78.16 


35-*5 


77.19 


35-59 


78.J6 


34-98 


78.41 


353* 


35-66 


78.10 


36.00 


87 


79-48 


35.39 


79.3* 


35.7J 


79.17 


36.08 


79.01 


36.41 


88 


8039 


35-79 


80.14. 


36.14 


80.08 


36.49 


79.91 


36.84 


89 


81.31 


36.10 


81.15 


36.55. 


8099 


36.91 


8081 


3716 


90 
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81.11 


36.61 


81.06 


36.96 


81.90 


37-3* 


81.73 


37.68 


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37.01 


81.97 


37.38 


81.81 


37.74 


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84^05 


37.4* 


83.88 


37-79 


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83.55 


38.51 


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84.96 


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84.79 


38.10 


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84.46 


38.94 


94 


85.87 


38.13 


85.71 


38.61 


85.54 


38.98 


85.37 


39-35 


95 
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86.79 


38.64 


86.61 


39.01 


86.45 


3940 
39.81 


86.17 


39-77 


87.70 


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87.53 


39.43 


87.36 


87.18 


40.19 


97 


88.61 , 


39-45 


88.44 


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88.17 


40.13 


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40.61 


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89-35 


40.25 


89.18 


40.64 


89.00 


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90.44 


40.17 


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90.09 


41.05 


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41.45 


100 
101 


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40.67 


91.18 


41.07 


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4».47 


90.81 


41.87 


91.17 


41.08 


92.09 


4>.48 


91.91 


41.88 


91.71 


41.18 


101 


93.18 


4«'49 


93.00 


41.89 


9x81 


41.30 


92.63 


41.70 


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94.10 


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93.54 


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95.01 


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94.81 


41.71 


94.64 


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105 
106 


95.9a 


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96.65 


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95-55 


43.54 
43.96 


95.35 


43.96 


96.84 


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43.54 


96.46 


96.26 


44.38 


107 


97.75 


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97.56 


43-95 


97.37 


44.37 


97.17 


44.80 


108 


98.66 


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98.47 


44.36 


98.18 


44-79 


98.08 


45.11 


109 


99.58 


44.33 


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44.77 


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45.ao 


98.99 


45-63 


110 

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100.5 


4t;74 


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4518 


100.1 


45.61 


99.90 


46.05 


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101.1 


45-59 


101.0 


46.03 


100.8 


46.47 


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102.3 


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101.1 


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101.9 


46.45 


101.7 


46.89 


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45.96 

46-37 


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101.6 


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103.9 


46.81 


103.7 


47.18 


105,5 


47-73 


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104.9 


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47-69 


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48.15 


106.0 


47.18 


105.8 


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105.6 


48.10 


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48.56 


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48.98 


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107.8 


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107.6 


41.46 


107.4- 


48.91 


107.1- 


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48.88 


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53^45 


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53^4.^^5.63 


'60 

i 


54*38 


25.36 


54-»7 


*$I9 


54ii6 


25.83 


54<}4 .26.07 


Dep. 


Lat. 


bep 


iat. 


lal. 


Dep. } 


Lai. 1 


0* 




45' f 30/ 1 


U 


F-| 


"' 






64 


vmnA 


&ES. 









35I»GBS£S. 



U$ 



:* Lat I Dep* Lat. Dcp. 



5$.a8 

56.19 
57.10 
sS.oo 

59 «» 
60.72 

6 1.6 J 
6a,54 
6344 
64.35 
65.15 
66.16 
67.07 
6797 



6^M 

69.79 
70.69 
71.60 

7«.5«> 



73-4* 

74-Ja 

76.U 

77'04 



77.94 

7a.8s 

79.76 

8a66 
81.57 



«a.47 

83.38 
84.99 

85.19 

86.10 



87.01 

87-9' 
89.84, 

89.7a 
9a63 

91.54 

9«-44 

93-5$ 
94.26 

95.16 



96.07 

9^.97 
97.88 

98.79 
99-*9 



105.1 
106.0 
1 06.9 
107.0 
108.8 



Dep. 



.0' 



4.V 



55-»7 

56.08 
56.98 

57»»9 
58.79 



30.01 
3043 

Zi.vj 
31. 7P 



59.^9 
60.60 
61.50 
62.41 
6331 



32 i» 

5a-S4 
3^96 

3^.39 
3381 



34-»3 
3465 
3508 

35«P 
35'9» 



36.3 J 
3*77 
37 «9 
3761 

38.04 



3^46 

3910 

39-71 
4015 



44-80 

45M 
4564 
4607 

46.49 



46.91 

47-33 
4776 
48.18* 
48.60 



49.02 

49-45 
4987 
50.29 
50-71 



UU. 



77.78 
78.69 

79-59 

8a 50 
81.40 



26.Q2 

26.45 
2687. 

27.30 

27.73 
28.15 
28.5^ 

29.01 

^9 43 
49^6 






45.21 
45.64 
46.07 
4650 

^469* 

I47 37 

.47.78 
i48.»0 

U8.63 
49.06 



55.06 
5*V6 
5686^ 

5^77 
58.67 






Lat Dep Lat Dep. 



59^57 

60.47 
6i.|8 
62.28 

64.08 
64.99 
65.89 
66.79 

67 6? 



26.26 
26.69 
27.12 

^7-55 
27.98 



68.60 
69.50 

7«.4* 
7»y> 
72.11 



73*« 

74-01 
74-91 
7 1-8 a 
76.72 



77.62 
78.5a 
79.43 

81.23 



82.14 
83:04 
83.94 
84.84 

85.75 



86.65 

8>55 
88.45 

8936 

90.26 



91.16 
92.06 
9J.97 
93.87 
94-77 



JO4.9 4948 
105 8 149.91 

106.7 50.34 
107.6 50.76 
108.5 .5'->9 



95.67 
96.58 

97.48 
98.38 
99 28 

Tool 
101. 1 

1 02.0 

J 02.9 
103.8 

104.7 
105.6 



28.41 

28.84 
2927 

29.71 
30.14 



30.57 

31.00 

31.43 
31.86 

32.72 

33-^5 
33.58 

34.01 

3444 



34.87 
3530 

3S-73 
36.16 

36.59. 

3702 

37.45 
37.89 

J8.3* 

38.75 



39.18 
39.61. 
4004 

4P47 
40.90 



4«-33 
41.76 
42.19 

42 62 
43-05 



43.48 
43.9 « 
44.34 
4477 
45.20^ 

45.63 
46.06 

46.50 

4^.93 
4736 



106.5 50.80 



VjA. 



108.3 ■ 
TTfp. 



47.79 
48.22 

48.65 

49.08 

49-5' 

49.94 
SO37 



5«-»3 
51.66 

Xat 



54-94 
55-84 

56.74 

57.64 
58.55 

59.45^ 
60.35 

61.25 

62.15, 

^3.05 

63.95 

64:85 

6^5.7i 
66.65 

^7.55 

68.45' 
69.35 
7a>5 
71.16 
7a.o6 



7296 
73.86 
74-76 
75*66 
76.56 



77.46 
78.36 
79-it6 
80.16 
81.06 



81.96 
^2.86 
83.7(5 
84.67 
85.57 



86.47 

8737 
88.27 
89.17 
90.07 



37-36 

3M0. 



45' 
94 OEOttEKS. 



SO* 



90.97 
91.87 
92.77 
93.67 

94^57 

95-47 
96.37 

97.»8 
98.^8 
99.0& 

99.98", 
100.9 

101.8 
102.7 

j|^03.6 

»04.5 
105,4 

106.3 

107.2 

J108.1 

Dep. 



41.71 

42.14 

42.58 

43.01 

4M4_ 

43.88 

44- 1« 

44.75 
45.*8 

4605 

4649 
46.92 

47.35 
47-79 
48.12 
4866 
,49.09 
.49-53 
4996 

50,40 
50.83 
51.26 
51.70 

51. U 

Lai 
15» 



136 






26 DEGREES. 








I 

2 

3 

.4 

5 

6 

7 
8 

9 

10 

II 

12 

13 

H 

>5 

16 

17 
i8 

19 

20 

21 
22 , 

«3 
»5 

26 

.27 

28 

»9 

30 

31 
3« 

33 

34 
35 

36 

37 
38 
39 

4« 

42 
43 
44 
45 

46 
47 
48 

49 

50 

51 
52 
53 
54 
55 

56 

57 

58 

59 
60 


0' 


15/ 1 


30/ 45' 1 


Lat. 1 


l)ep 


ut. 1 


Dep. 


Lat 1 Dep. \ Lat. | 


Dep. 


0.90 < 
r.8o 
2.70 
J.60 

4.49 

5 39 
6.29 

7-»9 
8.09 

8.99 


0.44 
0.88 ' 

1.32 ■ 

«75 
2.19 


0-90 

'79 
2.69 

3.59 

4.48 

5-38 
6.28 

7.>8 
S.07 
8.97 


0.44 
a88 

1.33 

1-77 

2.21 

2.65 
3.10 

8-54 
3.98 

4-42 


0.89 

1.79 
2.68 

3.58 

4-4'^ 


0.45 
0.89 

«.|4 

1.78 

2,23 


0.89 ' 

1*79 
2.68 

5-57 
4.46 


. 0.45 
0.90 

1-35 
1.80 
2.25 

2.70 

3.15 
3.60 

4.05 
4.50 

4.95 
5.40 

5.85 
6.30 

6.75 


2:63 

3.07 
3-51 
3-95 
4.38 


5-37 
6.26 
7.16 
8.05 
8.95 


2<68 
3.12 

357 
4.02 

4.46 
4-91 

5.80 
6.25 

6.69 

7.14 
7.59 

8.03 

8.48 

8.92 

9-37 

9.82 

to. 26 

10.71 
11.15 


5.36 
6.2$ 

7.»4 
8.04 

, 8-93 
9.82 
10.72 
11.61 
12.50 
13.39 


9.8^ 

10.79 
It. 68 

12.58 
13.48 


4-82 
5.26 
5-70 
6.14 
6.58 


9.87 
1076 

11.66 

>|2.56 

13-45 


4.87 
531 
5.75 
6.19 
6.63 


9.84 

10.74 

11.63 

12.53 
13.42 


14-38 
15.28 
16.18 
17.08 
17.98 

18.87 

19.77 
20.67 

21.57 

22.47 


7.01 

7.45 

7.89 

8.33 

8.77 


i*-35 
15.25 

16.14 

17^4 
'7-94 

1973 
2a63 

21 5a 

22.42 


7.08 

7.5* 
7.96 
8.40 
8.8j^ 

9.29 

9.73 
10.17 

10.61 

11.06 

ti.50 
11,94 
ia.38 
12.83 

13.27 


14.32 
15.21 
16.11 

17.00 
17.90 

18.79. 

19.69 

20.58 

. 21.48 

22.37 " 


14.29 
i$.l8 
16.07 
16.97 
17.86 

18.75 
19-65 

2a 54 

21.43 
22.32 


7.20 
7.65 
8.10 

8.55 
9.00 

9-45 

9.90 

10 15 

10.80 

ri.25 


9.21 

9.64 

10.08 

■0 52 

10.96 


23.37 

24.27 

25.17 
26.07 
26.96 


11.40 
11.84 

12.27 
ia.71 

13.15 


23.32 
24.22 
25.11 

'26.04 
26.91 


23.27 

24.16 
25.06 

«5 95 

26.85 


11.60 
12.05 

12.49, 
12.94 

>3-39 


23.22 
24.11 

25.00 
25.90 
26.79 


11.70 
12.15 
12.60 
1305 
13.50 

>3-9S 

14.40 

'4.85 
15.30 

15.75 


27.86 
28.76 
29.66 

30.56 
31.46 


13.59 

I4."03 

14.47 
14.90 

15-34 


27.80 

28.70 
29.60 

30.49 
31.39 


13.71 

14-15 
14.60 

15-04 
15.48 


27.74 
28.64 

29.53 

30.43 

_3«*32 

31.11 

33.11 
34.0 c 

34.90 

3580 

36.69 

37.59 
38.48 
39.38 
40.27 

41.17 
41.06 • 

42.96 

43.85 

44.75^ 


'3.83 
14.28 
14.72 
15.17 
15-62 

16.06" 

16.51 

16.96 

17.40 
•7.85 


27.-68 
28.58 

^9-47 
30.36 
31.25 


32.36 
33.26 

34-15 

3505 
35-95 


15.78 
16.22 
16.66 
17.10 

17.53 

17-97 
18.41 
18.85 
1929 

»9.73 


3229 

33.18 
34^08 

34.98 
35-87 


15.92 
16.36 
16.81 

17.25 
»7-69 
18.13 
18.58 
19.02 

19.46 
19.90 

20.35 

ZO.79 
21.23 

21.67 

22. If 


32.15 

33«»4 

35-93 
34.83 
35.72 


1610 
1665 
17.10 

17.55 
1B.00 

18.45 
18.90 

«9.35 

19.80 

20.25 


36.85 

37.75 
38.65 

3955 

4045 
41.34 
41.24 

43-14 

44.04 
44-94 


36.77 
37.67 

38.57 
39.46 
40.36 

41.26 
42.15 

43.05 

43.95 

-^*± 

45.74 
46.64 

47.53 
48.43 

4933 


18.19 
18.74 
19.19 
19.63 
1008 


36.61 

37.51 
38.40 

39.29 
4a 18 


ao.17 

20.60 

21.04. 

2148 

21.92 


10.53 
20.97 
21.42 
21.86 
12.31 


41.08 

41.97 
42.86 

43.76 

4465 


10.70 
21.15 
11.60 

12.05 
12.50 


45-84 
46.74 
47-64 
4853 
4943 


22. )6 
22.80 
23.23 

23-67 
24.11 

24.55 
24.99 
2543 
25.86 
26.30 


22.56 
23.09 

2344 
23.88 

.24.33 


45-64 

46.54 

47.43 

48.33 
49.22 


22 76 
23.10 

23.65 
24,09 

24.54 


45-54 
46^3 

47.33 
48.22 
49.11 


22.95 

234! 

13.86 

24.31 

24.76 


50.33 
51.23 
52.13 
53.03 
5J-93 


.5a22 
51.12 
5x02 
52.92 
53.81 


24.77 
15.21 

a^.65 
26.10 

26.54 


10.12 
51.01 
51.91 
52.80 
53.70 


24.99 

25-43 
15.88 

26.33 
26.77 


5aoi 

50.90 

?i.79 
52.69 

53-58 


15.21 
15.66 
26.11 
16.56 

17.01 


Dep. 


Lat 


llep. 


Lat. 


Dep. Lat. 


Dep 


y. 


Q/ 1 45 


1 


30' 15 


1 t^mm 






es 


DEQIU 


gES. 




* 





""■ I 



0/ 



Lat. 

S4.«J 

$M* 

5 8.4* 









Ii6,$« 3*44 



6M< 



7730 
y8.«a 
79.0^ 

taitt 

I1.79 

Is 69 

I4.49 

^5-39 



S6.18 
87.18 



88x>8 4t-^ 



88.98 



J?:^±lliii 



Dcp^ 

a6.74 

a7.i8 
%7.6z 
29 06 

»]8.49 

19.37 
29.81 

30.«5 
3069 



31.14 

31.0 

3».00 



3188 



33-3* 
33.75 

34.*3 
35-^7 



35$" 

35^5 
36.38 
36.ii 

37H 



3r.;o 
3I14 

3»,58 
3|.«l 

3J89 

4^33 
4t.U 



4i.o8 

4*. 5* 



4j|.40 



44 »8 

44-7 « 

45- « 5 

4J.5r9 
^.03 

95-»7 46.47 



90.78 
91.69 
91.5*8 
93.47 
94 37- J 



96,17 
97.07 
97.97 
98.87 



4^9« 

47*34 
4778 
48.23 

^.77 48.66 

iod.7 l49.»0 

49-54 

49-97 

1 504.' 



iot.6 
102.5 

' 034 

1043 

IDl.t 
106. 1 

107.0 
107.9 



5O.8J 
51.29 

$1.73 
$1.17 
51.60 



Dep. I bat 



0' 



Lat. 

54-71 
55.61 

56.50 

•57-40 
58-30 



59- «9 
60.09 

60.99 

6».8» 

61.79 



3i4d 
3i-84 
l3»-»9 
3*71 
33.»7 



■ 34'06 
34-1© 

34-94 

35-38 



3583 
36.»7 
3^7.1 

3>.»5 

J7?9 



77-*l 
•78.03 

78*9* 
29.82 

80*7 2 

8i.6r 

82.51 

<3.4i 
84.31 
85.10 

86: 10 

87.00 
;37.89 

^8.79 
8969 



90.58 
'9»-4* 

93^7 
•94-*7 
95.07 4688 
95.97 47-3* 



96^86 
97.76 
98.66 



104.0 
104.9 
105.8 

1067 
107.6 

Dep. 



Dep. 



38.04- 

3M8 

38.9* 



4^.69 

41.13 

4t5« 
41.02 

41.46 

41.90 

43.H 
43.79 



44.67 
45.11 
45.^ 
4600 
46.44 



k7-7? 
48. ai 

99-55 49-09 
100.4 .|49-54 

49'9* 
504a 

5086 

5i.3« 

51.75 
52:19 

52.63 

53-07 



27.1a 
47.66 
18.11 
18.56 
19.00 



6l;65 



76^46 

i7Ji6 



,1.44 

ti33 

84.} 2* 

85)01 



85.91 
86.81 

87.70 

8860 

_8949 

90.39 
9i.%8 

92.18 

93.07 

94.86 

95-76 
96.65 

97-55 
98.44 



Lat 



45/ 



Jjep. 



Lat 



11.68 
3111 

J*-«7 
IJ.oa 

3346 



67.87 
68.76 

6965 

70-55 
7«-44 



7*^31 
73.»4 
74.1* 

75 o« 

75.90 



36.46 

369" 
37-36 
37.S1 
38.26 



40.60 
410$ 
41.50 

4i.94 
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4»-83 
43*8 

4i-7J 

44- *7 
44.61 



45.07 

4! 5» 
45.96 

4640 
46.85 



47.30 

47 74 
48.19 

48.64 

49.08 



76.80 
7769 
78-58 

79.48. 

_iL 
81.16 

^2il5 

83-Pf 

83^94 
84.812 

86;62 

87»5^. 

88.40 

8930. 



90.19 
91.08 
91^ 
9i.»f 
93,76 



9466 

95.55 
96.44 

97.33 
98.13 



2.--L 12* ^'^ 
99.11 

100.0 

100.9 
ior.8 
10I7 



103.6 

1045 
105 4 
106.3 
107.2 



Dep. 

17.46 

17;9I 

18.36 
18.81 

19.26 

■ ill fc 

19.71 

30.16 
J0.6I 
31.06 
il.51 



31.96 
32.41 

32.86 
33.76 



34-n 
34.66 

?5-l* 
3556 
36.01 



38.71 
3916 
39.61 
40.06 
4051 



40.96 
41.41 
41.86 

41. ft 
4^-76 

43.11 
43.66 
441! 

44-56 

45.46 

459^ 
4636 

46.81 

47.»6 



47.7» 
48 16 

48.61 

49.06 



;2.2i 



54.01 



30' 



Dep. - 1 Lilt 

■ II -■ I 

15' 



63 UtQBAf^. 

8 



13a 




« 


'27 USAitUJS»9* 










0/ 1 


15/ 


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Lat. 


l>ep 


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0.89' 


045 


0.89 


10.4* 


a89 


0.46 


Q-ii 


tW|7 


9 


t.7% 


0.i|l 


1.78 


0.92 


1.77 


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^77 


093 


3 


ft.67 


1.^6 


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1.37 


1W66 


«J# 


».65 


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4 


3.56 


i.Ea 


3.56 


i»3 


3-55 


1.85- 


3-54 


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5 
6 


4-4« 


2a7 


445 


a.29 


4-44 


»3t 


_t44 
5-3* 


4-33 


S.35 

tf.a4 


a.72 


5-31 


5-3* 


4.77 


«.79 


7 


3.18 


ijia 


3.21 


6.21 . 


3-^3 


<^i9 


3-46 


8 


r.»3 


3-^3 


7.11 


3W 


7.10 


3H 


>08 


3.7* 


9 


8.0a 


4.09 


1.00 


4.*» 


7-98 


4«« 


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.4-19 


10 

ti 


8.91 


J.54 
"4-99 


889 


458 


8L87' 


4.^4 


M5 


4^66 


9.80 


9-78 


$•04 


9-76 


$.08 


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5-45 


«9-f7 


5 4* 


«o.H 


5«4 


10.6* 


$•$• 


^3 


11.58 


5.90 


11.56 


5-9f 


tr.SS 


6.00 ' 


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«4 


ia.47 


6.36 


ia.45 


6.41 


12.4* 


6.46 


H^39 


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i6 


«3-37 


6.8 1 

7.a6 


_»3jJ4 

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687 
733 


»3-3< 


693 


I3.t7 


6.9I 

7.45 


I4.a6 


14:19 


7-J9 


«4-t6 


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15 ij 


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15.11 


y.78 


15.08 ^ 


7.?5 


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1604 


8.17 


1600 


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831; 


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i«K03 


8.63 


16.89 


8.70 


16.S5 


8.77' 


16I1 


k.85 


20 ' i7.8a 


9.08 


17.78 


916 


■7.74 


••44 
9-70 


17.70 

18.58 


9-8* 

».78 


9 53 


18.67 


9.6a 


18^3 


ta 


19.60 


9-99 


»9'56 


10.07 


i9.fl 


iai6 


»>47 


to. 84 


»3 


ao.49 


«0.44 


20.45 


'0.|3 


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10.6a 


taj5 


tO.71 


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a I 38 


to. 90 


ai.34 


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a 1.29 


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'91.24 


11.17 


as ( aa.a8 
a6 ' ai.i7 


liJ^ 


22.21 


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22.1S 


•J-w 


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Mn64 
18.11 


11.80 


23.1-1^ 


ir.90 


23.06 


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