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I
RBoeiVHD IN EixoilANOK
|W. L. Clements Library
,/
r
'^
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h
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i
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FA
I
• I
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
0/ the iikikematical CAaroctcTM U9cd in iku Work*
+ signifies yk/itf» or addidoiu
I^HBB 0
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 multiplicand 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
• by .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
0
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
0
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
0
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 0 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 0 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 0 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
0 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 0 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 0 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.
Ch. 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 0 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 0 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 0 \ ^ P^^^ chainSv
Acres 11|0
Or,
Ch.
20=40 ! perches.
160)1760(11 acres.
160
0
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 they 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.
■
-^
Ok
Cit
^
c*
M
»M
1^
2.
s
•
00
C/3
•
00
?
a:
•
CO
C/3
•
o
i
s
01
•
(0
•
^
^
•
Ok
•
o
m
125
o
»l«
^
, ■
>■■
'tjg
Oi
•
b9
O
C0
90
K
Ot
Oo
5 1
—
00
O
g
Z
b
o
•
• *
hr
"k
■
3
> kOl
^«
u
w *»
M
kO
o%
3
00 00
Pt
<o
00
03
•
cf>.
•*!
'
•
Ok
•
10 M
-LU
».
_j_i_
0« (M
00
-^1
f"
u
«
*
w
•
s-
1 '
M
^«
,
D
Ox
b
•
•
Ok
t9
•
b
^
■■■M^a*
■«yiB^
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f *^'
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/
^a
o
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o
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•
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to
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to
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00
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to
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o
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00
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•
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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 0 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. 0 E.
0 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 ■ mtmmi mm ml
eet. Inch. Tenths. Feet. Inch. Tentha.
3
•4
9
I
34
0 ,
8
5
ft
2
0
7
Hence Uie descent is
6
8
&
8
9
38
34
13
13
3 ,
I
8
4 «
5
3
8
8
0
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 0 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 0 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 0 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 0 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 0 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 0 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 0 on Ihe N6niits
would coincide with the first 20 minuter on the-
Arch from 25 degrees. Again, let us suppose the.
0 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 0 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^
0 on the Nonius is at 0 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 0 on the Nonius is at 0 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 0 on the Nonius exactly against 0 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 0 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 0 on the Nonius be to the right of the 0 on the
Archy but if to the left of the 0 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 0 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 0 on the Nonius opposite to 0 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 0 41 Sine 9.0T650
Latitude 26 32 Secant 0.04834
i True amplitude W. 0 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 0 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^
0
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
0
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
0
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
0
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^
0
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
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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
0
547775 547*98
549003 549146
550128 55035'
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381 $80925
381 582063
383 583«99
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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 0 1
556302 5564*3
557507 5576*7
558709 558829
559907 56O026
561 101 56»«*i
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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*
0 I 1
2 13 1
I
|6o2ot)0,602i69
605 f44j603 253 (60336 1 1603469
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1^0745 >
6c2277J602386;602494 601603
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60681 1 606919
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441
442
443
444
445
446
447
448
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61 9 1 98 6193021619406
620240 6 20344J 620448
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638090*638
63908^1639
6400^^41640
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648750 648848
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650696 630793
651666 651762
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631038 631
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644242
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65600s 1
6569601
6y 916
G588TC
659821
660771
661718
66266;
i***i
LOGARITHMS OF NUMBERS.
— i-T
No.
460
461
462
464
0
66z7 5>>
66j70i
,664642
665581
6665J8
667453
668330
I
2
!
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466
467 j6693i7|6694io!6b9503
468 ' '• '"
469
662.^52 66 i94 7
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6656751665769
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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
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.670246I670339 670431
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. 67 1®95|67 2 1 90167228 3 (672375 '672467
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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
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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
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725503; /25585«725667 72S748{725830
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727948 728029 728110I718191J71827
728759 718841 7189121729003; 719084
729570 729651*729732 729813)719893
730378 730459 730540
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733278 733358.733438.7335*8
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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
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560
561
561
563
564
565
566
567
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740441:740511 J7405991740678
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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
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lOGARITHMS OF NUMBERS*
No.
5S0
581
58a
583
584
585
586
588
^•89
5V0
59'
59*
591
594
595
596
597
598
600
601
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604
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606
607
608
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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
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611
611
613
614
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616
617
618
619
620
621
622
623
624
62s
626
627
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629
630
631
632
633
634
635
636
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638
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77*587
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773055
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798029 798098
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800098 800167
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801472 801541
802158 802226
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78802
795*4
80245
80965
81684
82401
83117
83832
84546
85259
85970
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87390
88098
88804
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9C918
91620
92322
93022
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94418
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96505
97198
97890
98582
99272
799961
800648
801335
802021
802705
803389
80407 1
804753
805433
806112
LOGARITHMS OF NUMBERS.'
21
Mo.
0 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,
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663
821514
664
822168
665
822822;
666
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667
824126
668
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669
825426
670
826075
671
826723
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674
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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
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688
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689
838219
690
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691
839478;
692
840106'
693
840733!
694
84'359
695
841985
696
S42609;
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843233
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837273
837904
838534
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839792
840420
841046
841672
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842921
843544
844166
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819939
820595
821251
821906;
822560
823213
823865
824516
825166
825815
816464
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827757
8284O2
829046
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830332
830973
831614
832253
832^2
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836071
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838597
820004
820661
821317
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813930
814581
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815880
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827175
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831678
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831956
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834866
835500
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838030
838660
810070
810727
821382
812037
812691
813344
813996
814646
825296
815945
839217
839855
840482
841 109
84«735
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842983
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844229
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839918
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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
0
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
874482J874540
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
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846213
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848066
848682
849296
8499 1 1
850340 1850401; 850462 ^50524
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845656
846275
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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 1185709 1:857 1511857212 857272
87763418576^18577541857815
856306:856366
856910 856970
8575'3!857574
858116 858176
858718
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859918
860518
8611^6
858778
859378
859978
860578
861176
8617141861773
862310(862370
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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
86^879 J568938J868997 869055|869»'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' >
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880242:880299 880356
87s8i3'87587i
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8769681877026
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87Sn9'«78i77
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875929. 875987:8760451876102
876507 876564 8766aj;87668o
877083^8^7 14 1.'877 198 1877256
863858
864452
865045
865637
866228
866819
867409
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869173
86976c
870345
870930
871515
872098
87^681
873262
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875003
87765918777171877774187733* 877889
878234;878292t878349
0
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878809 878866,878924
87926^ ;«79325 8793*^3 '879440' 879497
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875582
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878407 878464
8789S1 879038
879555 879612
8Sor27 880185
880699. 8S07 56
•
{
LOGARITHMS OF NUMBERS:
No.
760
761
76*
763
764
765
766
768
769
0
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
0
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
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91J927
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LOGARITHMS OF NUMBERS.
No>
88 1
881
883
884
885
886
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mmm
w^^m
tOGAMTHMS OCT NUMBERS.
TJ
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940
941
942
943
944
945
946
947
948
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95 »
95a
953
954
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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 ■
0
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
999783 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 0 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
II
7.505118
7.511649
7.518083
7-5*44*3
7.530672
7.536832
48
12
7.542906
7.548897
7.554806
7.560635
7.566387
7.57*065
47
n
7.577668
7.583201
7.58S664
7.594059
7.599388
7.604652
46
»4
7.609853
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
»5
7.667844
7.672345
.7.676799
7.681208
7.685573
7.689894
43
17
7-694>73
7.698410
7.702606
7.706762
7.710879
7.714957
4»
i8
7.718997
7.722999
7.7*6965
7.730896
773479*
7.738651
4*
19
20
7'74»477
7.746270
7.750031
7.753758
7.775477
7.757454
7.761119
40
II
7-764754
7.768358
7-77'93»
7.778994
7.782482
21
7.785943
7789376
7.79*782
7.7^6162
7.7995*$
7.802843
22
7.806 146
7.809423
7.812677
7.815905
7.819111
7.822292
37
23
7.825451
7.828586
7.831700
7.83479*
7.837860
7.840907
36
■
H
7.843934
7.846939
7.849924
7.852888
7.855833
7.858757
35
»$
f.861662
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
»7
7.895085
7.897758
7.9004*4
7.903054
7.905678
6.908287
3*
28
7.9>o879
7.9 » 3457
7.916019
7.918566
7.921098
7.923616
3*
29
30
7 926119
7 928608
7.931082
7.945641
7.933543
7.935989
7.938422
30
7.940842
7.943448
7.948020
7.95OJ87
7.95*74*
7.966601
29
3'
7.955082
7.957410
7-9597*7
7.962031
7.9643**
28
3*
7.968870 ,
7.97 1 126
7.973370
7.975603
7.977824
7-980034
47
33
7.982233
7.984421
7.986598
7.988764
7-9909*9
7.993064
26
34
7.995*98
7.997322
7.999435
8.001538
8.0036 1 1
8.015981
8.005714
*5
35
8.007787
8.009850
8.01 1903
8.013947
8.018005
*4
36
8.020021
8.022027
8.014023
8.O26011
8.027989
8.029959
»3
37
8.03 19 19
8.033871
8.035814
8.037749
8.039675
8041592
22.
.
38
8043501
8.045401
8.047294
8.049178
8.051054
8052922
21
,39
40
8.054781
8.056633
8.058477
8.060314
8.062142
8.063963
20
8.065776
8.067581
8.069380
8.072955
8.07473*
»9
41
8.076500
8.078261
8.080016
8.081764
8.083504
8.085238
18
4a
8.086965
8.088684
8090398
8.092104
8.093804
8.095497
17
43
8.097183
8.098863
8.100537
8.102204
8.103864
8.105519
16
44
8.107167
8. ro88o9
8.110444
8. II 2074
8. 1 13697
8*153*5
»5
4>
8.116926
8.118532
8.120131
8.121725
8 123313
8.124895
*4
46
8. 12647 1
8.128042
8.129606
8.131166
8.132720
8.134268
13
47
8.135810
8.137348
8.138879
8.1404C6
8.141927
8.143443
12
48
8.144953
8.146458
8.147959
8'M9453
8.150943
8.152428
If
49
50-
8.153907
8.155382
8.156852
8.l^8ji6
8.159776
8.161231
10
8.162681
8.164126
8 165566
8.167002
8.16K433
i.169859
9
5'.
8.171280
8.172697
8.174109
8.I7S5»7
\ 8.1769^
8.178319
8
5»
8. 1797 13
8.l8tT02
8.182488
8.183868
8.185245
8.186617
7
53
8.187985
8.189348
8.190707
8.192062
8.*934*3
8.194760
6
54
8.19610a
8.197440
8.198774
8.200104
8.201430
8.202752
5
55
8.204070
8.205384
8.206694
8.208000
*
8 209302
8.2 1060 1
4
56
8.21 1895
8.2I3I.85
8.214472
8.215755
8.217034
8.218309
3
1
57
8.219581
8.220849
8.222113
8.223374
8.224631
8.225884
2
58
8.227133
8.228380
8.229622
8.230861
8.232096
8.2333**
I
59
8.^34557
8.235782
8.237003
8.238221
8.239436
8.240647
0
eon
50 '
40"
30"
30"
10''
U
-
^-iwi?
DvgrtCJi.
•
I
LOGARITHMIC TANGEOTS.
TflOif^t ODejirree,
■
M
O
0"
10*'
20'
30*'
40"
' 50"
59
t
5.685575
5.986605
6.162696
6.287635
6.384545
J-
6.465726
6.530673
6.588665
6.639817
6.685575
6.726968
58
6.764756
6.7995*8
6.831703
6.^6605
6.861666
6.889695
6.916024
57
6.940847
6.964329
7.007794
7.027998
7.047303
56
7.065786
7.083515
7.100548
7.116939
7-13*733
7.'47973
55
7.162696
7.176937
7.190725
7.204089
7.217054
7.229643
54
7.041878
7-*5J777
7.*65359
7.276640
7.*87635
7.198359
53
7
7.308825
7.319044
7.329028
7-338788
7-348333
7.357673
5*
8
7.366817
7-375772
7.384546
7.393146
7.401579
7.40985*
5»
9
7.4"79P
7-4*5939
7.433764
7.44>45'
• 7-449004
7.4564*8
50
7-4637*7 *
7.470906
7.477968
7.484917
7.49 '7 56
7.498490
49
II
7.505120
7.511651
7.518085
7.524426
7.530675
7.536835
48
la
7.542909
7.548900
7.554808
7.560638
7.566390
7.572068
47
>3
7.577671
7.583204
7.588667
7.594062
7.625097
7-59939«
7.604655
46
«4
7.609857
7.614996
7.620076
7.630060
7.634968
45
«$
7.639820
7.644619
7-649366
7.654061
7.658706
7*663301
44
i6
7.667849
7.672350
7.676804
7.681213
7.685578
7.689900
43
17
7.694179
7698416
7.702612
7.706768
7.710885
7.71496*
4*
iS
7.719003
7.723005
7.7*697*
7.730902
7-734797
7.738658
41
«9
ao
7.74*484
7.746277
7.750037
7.753765
7.75746*
7.76 1 1 27
40
7.764761
7.768365
7.771940
7.775485
7.779002
7.782490
39
21
7.78595«
7.789384
7.79*790
7.796170
7.7995*4
7.802852
38
It
7.806155
7.809432
7.812686
7.815915
7.819120
7.822302
37
aj
7.825460
7.828596
7.831710
7.834801
7.837870
7.840918
36
»4
7.843944
7.846950
7.849935
7.852900
7.855844
7.858769
35
»S
7.861674
7.864560 .
7.867426
7.870274
7.873104
7.875915
34
26
7.878708
7.881483
7.884240
7.886981
7.8897O4
7.892410
.33
»7
7.895099
7.89777 »
7.900428
7.903068
7.905692
7.908301
3*
28
7.910894
7.9" 347 i
7.916034
7.918581
7.9*1 >»3
7.923631
7.938439
31
»9
30
7.926134
7.928623
7.931098
7.933559
7.936006
30
7.940858
7.943*65
7.945657
y.948037
7.950404
7^95*758
29
31
7.95C100
7.968889
7.9574*8
7-959745
7.9*2049
7.96434«
7.966621
28
3*
7.971145
7.973389
7.975622
7.977844
7.980054
27
33
7.982253
7.984441
7.986618
7.988785
7.990940 '
7.993085
26
34
7.995*19
7-997343
7.999456
8.001560
8003653
8.005736
*5
35
8.007809
8.009872
8.011926
8*013970
.8.016004
8.OU029
*4
36
8.020044
8.022051
8.024047
8.026035
8.028014
8.029984
*3
37
8031945
8.033897
8.035840
8,037775
8.039701
8.041618
22
1*
8.043527
8.045428 .
8.047321
8.049205
8.051081
8.052949
21
39
40
8.054809
8.056661
"8^7612"
8.058506
8.060342
8.062171
8.063992
20
8.065806
8.069410
8.071201
8.072985
8.074761
19
4«
8.076531
8.078293
8.080047
8081795
8.083536
8.085270
18
4»
8.086997
8.088717
8.090430
8.092137
8.093837
8.095530
»7
43
«.097*«7
8 098897
8.100571
8.102239
8.103899
8.105554
16
44
8.107202
8.108845
8. 11048 1
8 112110
8.113734
8.115352
«S
45
8.116963
8.118569
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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
0
1
SSo8i4!8SoS7i
88i385|Hgi442
8819551X83012
770
771
77a
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
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793
791
794
793
794
795
796
797
798
799
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883093 883150 883207
883661 883718 883775
8842x9 884185 884342
884795 884852 884909
8853611885418 885474
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887054
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881042
881613
882183
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883264I883321
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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
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1 8904 2 1. 89047 7
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889974 890030
890533 890589
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894316I894371
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800
801
80a
803
804
80$
806
807
808
809
810
811
842
8.3
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815
816
817
818
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8 I 9
■ta
LOttARITHMS OF NUMBERS.
17
840
841
844
>
846
X47
848
849
850
851
85X
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854
856
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860
861
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870
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939968
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940964
94446
94195
94245
94*95
943445
943939
944433
mm
t
LOGARITHMS OF NUMBERS,
0
880
8S1
882
883
884
88s
886
887
888 J9484»3
881} 1948902
944483
944976
945469
945961
946452
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947434
8(p
891
892
893
894
895
896
897
898
89^
900
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902
903
904
905
906
907
908
909
I
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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
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9518721951920
9523o8i952356'952405
952792 952841.952889
9S3*76'9533«5l953373
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949390
949878
95«>36s
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95«337
95«8i3
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944631 944680
945»»4i945»73
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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
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955206 955*55^55303
955688;955736
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958086J958f34
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958564:958612 958659
91Q
911
912
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960946 960994 961041 961089
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942843 '9628901962937
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964260 '964307 964354
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97*989 973035 973081
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973031
8
tOGAMTHMS O^ NUMBERS.
W
TIoT
960
961
96ft
963
964
965
966
967
9€$
969
970
971
971
973
974
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980
9S1
98a
98s
984
985
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987
988
990
991
99a
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980549
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977403
977861
978171 978317
978728 978774
979130
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985910 985965 ,986010
986369 986413 J98645S
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989138
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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
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10.97.148
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10,968763
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9.03141s
10-967575
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10.9700S1
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9.03 '0S9
9.031157
9-99-'4^0
9.997466
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10,001510
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903479'
10,965109
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10.967 74J
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9.0334.1
9.997451
9.035969
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9.049400
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10.947856
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10,946713
10 0017 58
10,94948'
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9.051635
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10,001771
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10.944465
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10,938870
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9.060460
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10.939540
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9.997098
9.064453
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9,991083
9-065556
10,934444
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9.063714
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9.064BD6
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10.931148
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10,00296,
10,935194
20
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10.003583
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34 LOGARITHMIC SINES, TANGENTS, AND SECAN'TS.
mt
M
C
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2
3
4
6
/
8
V
10
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81 JJetjrcejj.
LOGARITHMIC SINES. TANGENtS, AND SECAX^TS. 35
9 Degrees.
M
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3
4
5
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7
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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
0
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 267967
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 9.99194719.288652
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
0
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
10.008552;
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
^! 9346579.9-989014
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-tang.
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
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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
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2382
2412
2443
2474
2504
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2566
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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
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Y
4
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17
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M
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1
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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
9.4527061
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
0 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
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38
37
36
35
34
33
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31
30
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28
27
26
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22
21
20
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18
17
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10.032834
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9-953783
9 953722
9.953660
Sine.
Tang.
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
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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
0.045726
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
JO363377
10.3-63 1 14
10.362852
10.362589
10.362327
10.362065
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10.361542
10.36 1 2S0
10.361019
10.360758
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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^
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24
23
22
21
20
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18
*7
16
*5
»4
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12
II
TO
9
8
7
6
5
4
2
I
o
M
64 Degrees.
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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
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9.65754*
9.949751
9-707790
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10 34*458
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9.657790
9.949688
9.708101
la 19 1 898 10.050312
10.342210
57
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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
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9.659025
9.949364
9.709660
10.190340
10.050636
10.340975
5*
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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
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9.660009
9.949105
9.710904
10.289096
10.050895
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9.949040
9.711115
10.288785
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9.660746
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9.711836
10.188164
10.051090
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9.948X45
9.7 1 1146
10.187854
10.051155
ia339009
44
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9.661136
9.948780
9.711456
10.287544
10.051120
10.338764
43
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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
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9.662703
9.948388
9-714314
10.285686
10.051612
'0.337297
37
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9.662946
9-948343
9.714624
10.185376
10.051677
10.337054
36
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9.663190
9.948257
9-714933
10.285067
10.051743
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9.663433
9.948192
9.715242
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10.051808
10.336567
34
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9 948126
9'7»«55«
10.184449
10.051874
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9.663920
9.948060
9.7^860
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10.051940
10.336080
3*
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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
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9.665133
9947731
9717401
10.181599
10.051169
10.334867
27
34
9.665375
9947665
9.717709
10.181191
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10.334625
26
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9.665617
9.947600
9 718017
10.181983
10.051400
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9.665859
9-947533
9718315
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9.666100
9-547467
9718633
10^181367
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9.666342
9.947401
9-718940
10.181060
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22
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9.666583
9-947335
9-719*48
10.180751
10.051665
'0.3334'7
21
40
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9.666824
9.667065
9.947169
9719555
9-719862
10.280445
10051731
10.052797
10.333176
20
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9.947M3
10. 280 138
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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
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9.667786
9-947004
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10.051996
10.332214
16
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9.711089
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10.053063
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9.668267
9.946871
9-7*1396
10.178604
10.053119
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9.668506
9.946804
9.711701
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10.053196
10.331494
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9.668746
9.946738
9.711009
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10.053261
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9.711611
10.277379
10.053396
10.330775
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9.669464
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10.177073
10.053461
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10.176768
10.0535*9
10.330297
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9-669941
9.946404
97*3538
10.176461
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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
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9-670658
9.946103
9.7*4454
10.175546
10.053797
10.329342
4
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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
0
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
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42
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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
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9.7355^60
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9-9439^7
9.943899
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9.943624
9-943555
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9.943348 9-737171
9.943279 9737471
9-734463
9-734764
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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
5019.683284.9.942^^19.740767
51,9.683514:9.94244819.741066;
5219-683743 9 942378 9.741365'
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54 9.6S4201 9. 9.^2239 g. 741962
55J9.684430 9.9421(59 9.742261
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57J9.6S4SS7 9.942029 9-742858
5819.68511519.941959 9.743156
59'9.685343J9.94i889 9.743454
_6o^9.685£M l9j^4^^ 9.743^52;
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
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LOGARITHMIC SIXES, TANGENTS, AND SECANTS, 55
29 Degrees.
M
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LOGARITHMIC SINES, TANGENTSi AND SECANTS. 57
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9-735719 9923755 9-811964
9-735914 9923673 9-81*241
9.736109 9.923591 9-81*517
Co-sine. Sine.
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38 9.731799
39
40
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52
53
54
55
56
57
58
59
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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
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98604
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0.071580
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Co-sec.
CO'Sec.
0.275790
0.275588
0.275386
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0.274983
0.274781*
0.274580
0.274378
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iIXI6\KITHMie Sm&3, TANf^^TS, A9D SECANTS. 59
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33 Degrees.
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9.814728
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9.825166
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1.82735 1
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60 LOGARITUMIC SINES, TANGENTS, AND SECA)fTS«
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1833
2744
2655
2566
^47 7
Tang. I Co-Ung. t Secant
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
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2299 " " '
2213
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9.848717110151283.10.087790
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1136
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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
9.853268
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
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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
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0.134230
0.133965
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o J33436
0.133171
fo. 132906
0.132642
0.132377
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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
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0.125780
0.125516
0.125253
o. 1 24990
0.124727
o 124464
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0.123937
0.123674
o 123411
0.123149
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0.091042
0/»92i34
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0.092318
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0.092686
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0.093148
0.09^3240
0.093333
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0.093704
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0.093889
0.093982
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0.094448
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0.094821
0.094915
0.095008
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0.095196
0.095289
0.095383
0.095477
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0.095665
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0.095947
0.096041
0 096 1 36
0.096230
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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]
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0.228875
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0.118557
0.128185
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0.117841
ai276€9
a 217497
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0.117153
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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
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9
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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
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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
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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 9.896532
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9.880003
9.880265'
9.880528
9.880790
9.88 IC5 2
9.88I3I4
9-881576;
9.8818^9!
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9.882625
9.882887
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9.883672
9.883934
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9.898981
9.898884
9.898787
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9.8983971
9.898299
9.898202
9.898104
9.898006
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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
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9.891768
9.892028
9.892289
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09275
09014
08753
08493
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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
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0.099567
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02092
02190
02288
02386
02484
02582
02680
02778
02877
02975
03074
03172
03271
03369
03468
52 Uegrees.
Vy0*sec.
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2605
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Secunt.
60
59
58
5'
56
55
54
53
5»
5»
50
49
48
47
46
45
44
43
42
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40
39
38
37
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28
27
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14
13
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10
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8
7
6
5
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:
(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
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9.789665
9-896335
9-89333^
10.106669
10.103665
10.110335
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9.789827
9.896236
989359'
10.106409
10.103764
10.110173
57
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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
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«• xjaaMMTaatKmtt»,rkiimsra,AXD9ecun^
_!•_ Sine. Co-une. 1 Tmg. , Co-tMig., SuetPt.
co.«c -Tl
*0 9.841771
9-85*934
9-9M37
.0.01516]
.0.14306b
.0,158119
6c
1 9-UI9C*
9.85681.
9.9S5090
.0.01+910
10,. 43 1 88
<o.. 580,8
59
i;<).a4aoi3
^1566,0
9-9<534J
IO.Oi46(!
la 143310
10.157967
58
Zl9.t4»'6l
^8565*8
9.9*5i9«
10.014404
.0.1434)1
.0.157837
17
4,9.B4ai»4
9-8l644t
g.9>5M
10.014151
1*14311*.
10.1 (7706
56
!
9-I4HM
9.8563*3
,.986.01
10.013899
.0.. 43677
.0,157576
15
6
f->4>II{
9.S56>oi
9-9M354
.0.0.3646'
10.143799
10.. 57441
54
?
».M»«8i
^8*56078
9.986607
10.013393,
.0..4J911,
10.. 573.1
J3
8
9.84»««5
9.853956
9-9>«86o
.0-013.40
10.. 4404+
I0..57l8(
5*
9
9-S4>946
9.815833
99*7.ti
.o^iiUB
I0.<44i67
io..(7os4
5'
2^J076
9.855711
9-987365
9-987618
.O-0U6J5
10.144189.
;oj^69i4
10.156794
49
p;5S*
9-«5Si88
. 0.011381
*0.i444.i
»-S433lt
9.8ll4«l
9-987871
io.0i>.»9
'O.M+13S
■Oi44«i4
to.i447ii
.0..(6664
+8
»J
9*43466
9.8553+1
9.988.13
.O.OI4I77
10. 156534
vr
14
9->43i95
9.8(5119,
9-988376
I40..6H
10156405
46
*'.
9-»4J7»i
9.8(5096
99886*9
.0.01137.
■o. 144404
A 156175
+5
aa
9-854973
9.98ml
10.011118
.0.145017
10..S6.+5
+4
•;
,.854850
9.989134
10.010866
.o.i4i<50
.0..S60.6
«
It
9.»44"4
9-854717
9-989387
■0.010613
10.145173
,0 1(5884,
+»
>9
9<44H3
4.85460J
9-989640
«aoi036o
'0. '45397
10.1(5757
♦"
10
984437*
5.gj44fc
9 989893
10.010107
«.i4((ia
|°_'556»8
39
11
9.144101
9854316
9.990141
10.0098 j(
'0..4S*4+
.0.1 ((498
9-144631
9-854*13
9-990398
IO.0O96O*
10.145767
.0.15(369
38
»i
(Sffito
9.8(4109
9.9906(1
10.009349
.0.14(89.
iau(i40
v
M
9.I44I89
9-8539S6
9.990903
■0.0090,7
W146014
10.155111
36
»i
9-841018
9. 8(386*
9.9911(6
io-008844
.0.146138
10.154,81
31
16
9I41147
9-853738
9.991409
I&OD859I
■o.i46a6i
■0154853
34
»7
9.S45176
9-853614.
9-991661.
.0.008338
4a 146386
10154714
33
iS
9-(4S401
9-<{J49a
iaoo8o86
.0.1465.0
<oi5459(
J*
>9
9-841133
9-<MJ66
tn'-^Vst
10.007833
.0.146*34
10.54467
P
JO
9. 841661
9.8531+1
9 991410
10.007580
.0..467i»
.0.5+338
30
*9
}i
»■ 845790
9-853-18
9-991671
10.007 1 18
.0. .46881
.0.5+.,^
l»
9.8419 ■ 9
9.85*994
9.991915
10.007075
lo-i+joo*
.0.54081
a
JJ
9.846047
9.851869
9. 993 '78
■0,006811
10.147.3.
■0.5395)
■7
34
9.846175
9-8J1745
9.993430
10.006570
.0.1+71(5
10 1 538 S(
16
)i
9.846304
9.8(i6M
9-992683
10,0063.7
10.147380
10153696
»5
Si
9-846431
9-8sni,t
9.993936
9.4941 t9
10. .47504
.0.(3(11
*4
r
9.846560
9.8(1371
ia.005811
.al«76.«
101534+0
*3
38
9 84«6U
9.81*147
9.9944+<
10.00(5(9
'O- .47751
1015JJI1
39
9.S46I.6
9.851.1*
9.99469+
10.005306
.a.147878
.0.53.84
ai
9.846944
9-8(44,97
9-994947
9.99J.99
10 00S05J
10.004801
10.14S0O3
101SJ056
w
9-847071
9-85 .«7.
.O.1+S..8
10.5191,
'»
4>
9.847 '99
9.8(1747
9.995451
10034548
10.148153
10151S0. I*
43
9-847317
9-8,j6u
9.995701
KU»4a95
10.14*378
10.(1*73 '1
44
9.847454
9.B(i4»T
9-991917
10,004043
laUSTii
.o.i()54«'C
4*
9.Mrs8»
9-851J71
4.996KO
10*03790
iOt51+i8 IJ
4*
9.847709
9.85.146
9.99«46«
IO.OOJ5J7
'0.148754
.O..fil9. >4
♦7
9-I4783*
9.8S...1
9.996715
.OJ303l8(
.0. .48879
10.51.6+ IJ
4i
»»47964
«. 8509,6
9.8(0*70
9.9»*96a
1000303*
10.149004
W..(»J* >«
49
»!*!°'i
9-997">
10.001779
I0..49I3O
101(190, .1
i2
9.^X1 8
9.8(0745
9.997473
10,1491(5
10151781^0
5'
9- 84*345
9.8(0619
9.9977*6
<a. 49381
.0.15.65s »
I»
9.84847 »
9.85049)
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
110313
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
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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
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2 48 45
POINTS
SOUTH 1
Oi
5 37 30
H
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8 26 15
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1
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1
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( 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
30'
Lat {
1.00
r
Dep.
aoi
•
1
iP
Lat.
Lat.
Dep.
f
1.00
1.00
2.00
0.0 r
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3
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4
4.00
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4
4.00
0.05
S
6
5.00
o.oa
5.00
ao4
^ 5
6
5.00
0.07
6.00
0.03
6
600
0.05
6.00
0.08
7
7.00
0.03
7
7.00
0.06
7
7.00
0.09
8
8.00
0.03
8
8.00
0.07
8
8.00
0.10
9
9.00
0.04 \
9
9.00
6.08
9
9.00
o.ia
10
11
laoo
0.04
OOj
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11.
10.00
11.00
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10
11
10.00
013
11.00
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0.05
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12.00
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12
12.00
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58.24
58.95
.5^66
6»37
6iio|8'
6r.7j9
6x5^
63.2<r
63.9^
64.6^
65.34
66:05
6d;76.
67.41.
'% <i 1 1. .
68.18
70.79
7189 173:86
69,60
70.51
7«f02_
7»»7ir
7»f44
73.15
74.57
75*18
75.99
70.70
77.4*
78,f»
78.83
79.54
8q,2f
80,96
81.67
82.38
83.09
83«'8o
84.51
53.51
514.**'
54.9"
55.62
56.32
57*03
57.73^
58.43
59- *4
59*84
60.55
6t.25
61.95
62.66
6J.36
74.63
7i.33
76,03 .
76.74,
77-44
78.15.
78.85
79.55
80.26
80.96
81.67
82.37
83.07
83.78
85,22 184.48
li'
64.07.
6447
65-47 •
66.18
66.88.J
67.59
68.29*
68.99
69.70
70-40
7h8l
72.51
7$22
5^10
50 91
51.62
5t.33
5|'03
5$-74
.54-45
;5$.»5
5f.86
57.28
5798
.58.69
.5940
60.10
60.81
-61.52
.^ii3
62.93
63.64
65.0^
^5.76
-6^47
67.18
67.88
'68.59
•69.30
4raco
70.71
7^4*
72.12
7183
73.54
,74.15
74-95
75-66
76.37
77-07
77-78
78L49
79-90
80,61
81*31
82.02
51.73
83*44
84*15
84.^5
0'
45 DEOKEKS.
TABLE fi.
MBAH BEWUiCTIOIf.
° !
o 15
0 15
0 30
0 40
0 Si
' s
' JC
.■35
1 40
' ■!!
1 so
' Si
r. 30
s 40
1 4i
1 fO
I 5S
i 0
3 S
3 <0
5 »i
3 30
3 JS
3 40
3 -li
I iO
1 55
4 0
3.. '
4 10
4 "J
+ "J
4 30
4 40
4 4!
4 S"
4 iS
33 0
31 10
3° 3J
39 iO
19 6
iS 13
z? 4r
16 10
li 4:
1! i
13 44
13 20
11 4»
11 li
JO 46
10 iS
■9 S'
■9 15
i£ 11
17 48
.7 16
■7 4
15 44
l« a
■i 4S
15 17
IS 9
H36
.4 iO
^"34
11 4c
■1 'S
II sr
M 40
11 8
.0 48
',1 "z
App.
AH
Retr.
App
Krir.
■\T
Hefi.
APP
HrtT.
S =
s s
i >o
( 'S
5 zo
S ^S
S 3°
S 35
5 40
5 S°
s u
e i
« 10
6 M
6 10
6 li
(i JO
* 4!
6 50
6 i!
7 0
7 S
7 M
7 35
7 40
7 4!
7 jO
VI
8 S
g !□
3 1;
li
8 3i
8 4a
8 45
8 so
8 5!
9 'S
g !□
9 15
9 3=
9 3;
9 4"
9 4^
9 5°
9 !4
9 46
9 3S
9 30
9 13
;■',
S 47
;s
f ?
7 57
7' 40
7 15
7 jt^
7 11
T >i
T 6
10 JO
10 JO
lii?
11 4C
It jO
" ''
rl so
IJ 0
13 ">
IJ iO
n JO
13 40
14 30
14 10
5 'i
5 S
^S
4 i'
4 47
4 43
4 J9
4 3'
4 »3
4 16
4 '3
4 S
"4 1
T7-
3 S4
3 !'
3 4R
3 4i
3 43
IS
3 35
1 11
10 10
10 30
10 40
10 so
" J£
11 40
II so
21 ip
22 «o
11 3C
ti 40
21 (O
ij 0
»3 "
13 10
»3 30
2] 4^
»3 SO
14 0
14 10
»4 3°
14 40
14 (C
1 35
*i
1 17
1 id
t 15
i!i
2 16
34 0
34 30
3i 0
36 30
3? 30
38 0
}Li°
19 30
40 0
41 0
41 0
43 0
44 0
% °
47 0
45 0
49 0
5D 0
5. 0
SI 0
iJ =■
54 0
li 0
i7 0
! ii
1 it
■'4
' 't
;1
■ J
' 3
0 iS
0 17
0 ss
°Jj
0 Jl
046
° *4
0 43
0 41
040
D 38
° V
6 33
■6 19
6 ij
6 11
6 iB
6 T;
6 M
t ^
I t
( S8
i 55
fIJ
S 45
i 41
i 39
i J*
S 34
Hi
5 »)■
i ij
IS 'O
l.J Ml|
13 J°
15 40
15 so
16 ro
_^JO
16 40
16 JO
17 ID
17 la
'7.30.
n 401
19 10
19 40
3 iB
3 a6
3 »4
3 11
3 '5
J 11
1 IC
H D
»j 10
JI )0
li 30
15 40
IJ JO
16 0
16 10
16 ID
16 30
1 0
' 5S
j3 0
59 0
ba 0
61 a
61 0
63 0
64 0
° IS
034
0 Jl
0 30
::l
□ 16
0 li
0 ij
0 19
a li
0 16
0 li
0 9
0 «
0 0
3 <
3 J
3 '
I S9
I 57
I 54
» 4S
1 4(.
1 44
1 43
2 40
2 3S
.6 40
16 so
S7 '?
17 30
n 45
iS 0
23 IJ
18 30
iS 45
19 3t
30 30
31 0
31 30
3: 0
31 30
33 °
33 3D
' 53
I 50
-Li*
> 37
' 35
' 33
< 30
1 lE
1 16
73 0
74 0
7i 0
I-:
1 =
III
90 0
fvuLB i.
J-Alil,!'. i.. I 1A.1L.„ 9.
Ux in Alt.
DipcftlK Hoi Linn ]
IJip at (llMer. Ui.tooctl
from tlie Observer.
s
Frrl
Ulp
D:p
■i
d
s
i
H, ,i,.,.t ,1- ibf Ejc 111
"
•
\
4 'T ,
M
»3M*',4i''(^'
6j.
9
4=.^
9
. (6
M
4 1=/
;^ (I?
JO
»
S
» 9
14
4 4«'t
M
40
7
1 1-
fO
8
» 44
iS
i i'l
♦
ss
JO
4
4
S
ft
9
» Si
3 10
3 11
40
4S
6465
1
i
«
5
i
J J»
70
ii
I
81
17
J 5*'
90
K 34
9 6
9 J{
i ■ -,:JU'.' . :
•^ table; w.
The Setni-diiunetcr <>f the Sod-
l>
9<n.lJL
li
ai
1 iM
•6 17
"s/iV
>i*jj
II
•A",
2 :?
.6 ,V
153
H
IS 49
•J.4>
■( 47
»f 47
"i 47
U
)6 S
■6 ,
S '«
»*
si
tfi 4
i ;!'
•5 47
"S47
1148
1 ir
16 11
SIS
* •»
•>
■6 1
16 1
•I »
•J P
■I4»
'I 1'
.5 i»
lif
>6 t?
16 ■>
( i^O
TABLE If.
Apluirtnt Timt qf Trantit qf Pole Star.
Thu tfitble is adapted to leap ^ear,^ pardcularly t808. . lo order to
make it serve for other 3rears,tMUme of transit must be tdieii for the
day following t^at ^ven in tho months of January and February. For
the first year after leap year, one minnte is to be added to the time of
transit ^ven in the table ; twd mitfutes for the secondi and three minutes
far the third after leap year.
Again, to reduce this table to a different iheridian than that to which
it is adapted} viz. Greenwich; if die longitude is between 45® £, and
A5^ W, there is no correction to be applied. If (he longitude is between
45^ and 135" £, one minute ia to be added ; but if it is between 45® and
135® W, one ntiDUteiatobe subtractedt If ike longitude is between
135® £, and 180^f tw4 mimitea are to.be added, but subtracted if the given
longitude is between 135® W, and 180®.
This taUe is useful to find the tim^ when the altitude of the pele star
ought to be observed, to find the latitude by its meridian altitude ; it b
also useful in finding the variation of the compass by the pole stat.
•
1
Jan.
Feb.
Marc
P. M-
April. 1 May.
JuAe.^
July.
Aug. Sept.
Oct.
Nov.
Dec
P ^.
P. M
P.M.
3h56'
P. M.
A. M.
A.M.
8hi7'
A. .W.
A. Ma' A. M.
A. M>
P. M.
6h 9>
2h4'
oh 0'
iohi9'
6hi3'
4h 9'
|2hi3'
ob25'
toh25/
81.22'
1
6 4
3 5»
2 0
0 7
10 15
8 13
6 9-
4 5
2 10
0 21
10 21
8 18
3
6 0
3 4«
« 57
0 3
10 12
8 9
6 5
4 '
2 6
0 18
10 17
8 13
4
S 55
3 44
•
« SI
0 0
A. M.
10 8
« 5
6 I
3 57
» 3
0 14
10 13
8 9
<
5 5«
i4?
I 49
II s6
10 4
8 1
5 S7_
3 53
' 59
0 10
10 8
8 5
6
547
3 3*
« 45
II 52
10 0,
7 57
5 53
3 49
» 55
0 7
.0 5
8 0
7
5 4*
3 3*
I 42
11 49
•9 56
7 53
5 49
3 45
« 5»
0 3
10 I
7 56
«
-
CP.M.
tI2 0
t
8
5 3»
3 28
I 38
«« 45
9 5*
7 49
5 44
3 42
148
It 56
9 57
7 52
9
S 31
3 H
» 34
It 4t
9 48
7 45
5 40
338
» 45
II 52
9 53
7 47
10
5 a9
3 20
3 16
' 3«
II 38
9 45
7 4«
5 36_
5 3*
3 34
r^i
II 48
9 49
7 43
7 38
fi
5 »5
I 27
It 37
9 4*
7 36
J3t
« 37
H 45
9 45
12
c 20
^ ^
I 23
" 33
9 37
^ K
5 28
3 26
I 34
It 4i
9 4*
7 34
n
S 16
3 «
I 20
u ao
9 33
7 a8
5.J14
3 23
I 30
XX 37
9 37
7 so
H
5 "
3 4
1 16
II 23
9 49
7 24
5 20
3 19
I 27
IX 34
9 33
7 »S
1$
5 7
3 0
t 12
fi 19
9 *5
7 20
5 «6
3 15
I 23
11 30
9 *9
7 2«
i6
{ 3
a 57
I 9
II lO
9 21
7 16
5 »»
3 »»
I 19
II 16
9 25
7 16
"7
4 59
s 53
» 5
11 12
9 n
7 Ji
5 8
3 8
I 16
11.22
9 20
7 12
i8
4 55
i 50
1 1
II 8
•9 «3
7 7
5 4
3 4
C 12
11 19
9 16
7 7
'9
4 50
2 46
0.58
'« 4
9 9
7 .3
5 0
3 0
« 9
II 15
9 i2
7 3
20
4 42
% 4*
a 38
2 54
0 50
II I
9 5
6 S9
4*56
2 57
.« 5
II II
9 8
9 4
6 59
10 57
9 I
655*
4 52
2 54
1 1
II 7
6 54
t%
438
2 34
0 47
■0 S3
11 5»
6 $1
448
2 50
0 58
*x 4
9 0
6 so
n
4 31
2 30
0 43
10 50
8 54
6 47
4 44
2 46
0 C4
II. 0
8*56
6 45
H
4 29
2 27.
0 40
10 46
8*50
6 42
4 40
2 43
0 51
10 5f
8 $2
6 41
^s
4 *5
2 23
0 36
10 42
8 46
6 38
4 36
^ 39
0 47
10 52
8 48
6 36
26
4 ti
2 19
0 32
10 |8
8 42
^ 34
4 32
2 3>
0 43
10 48
8 44
6 32
^7
4 »7
2 15
0 29
10 34
8 38
6 20
4 2S
2 32
0 40
10 .j4
8 39
6 27
28
4 '3
2 II
0 25
10 31
834
6 26
4 24
2 28
0 36
10 41
8 35
6 21
29
4 8
2 8
0 21
10 27
8 30
6 22 4 20
2 24
0 33
10 37
8 31
6 19
30
4 4
0 ]8
10 23
8 26
6 17 4 16
2 21
0 29
10 33
8 16
6 14
51
4 0
0 14
8 22
4 L2 i
» >7
10 29 I
6 10
TABLE 19.
irt
JDifference of Altitude qf the Pole Star and the Pole^ at different
distaneea of the Star from the Meridian,
As the pole star is generally known^ that no opportunity*
therefore, may be lost for determining^ the latitude, this table
is inserted) the use of which is as follows : —
Fmd the interval between the time of observation of the
. altitude of the pole sur, and that of its passing the meridian,
and take out the corresponding equation from the table ;
which added to, or subtracted from the true altitude of the
pole star, will give the latitude of the place of observation.
Examples.
I. Let the corrected altitude of the pole star be 46^ 10^ N^
observed 8h. 30' before its passage over the meridian. He-
■ iiir#»H fh#» latitude ?
quired the latitude
True altitude of the pole star
Equation from Uble 12 to 8h. 30
Latitude ...
46« 10/ N.
-f. 1 5
47 15 N.
II. At Ih. 10^ after the pasfage of the pole star over the me*
ridian, its altitude corrected was 58<» 51^ N, Required the
latitude ?
True altitude of the pole star
Equation from Uble 12 to Ih. 10^
Latitude - •
58* 51' N.
1 42
51 9N.
TABLE 12.
Difference of Altitude of Pole Star and Pole.
^gument. Distance of the Star from the Meridian^ in Sidersal Time
Subtract.
Min.
0 Hour.
1 Houv.
9 Hours.
^ Hours. 1 4 Hour*.
5 Hours. 1
0
I" 46;9
1" 43:3
!• 3»>
I" 15:6
0^ 53:4
0** 17:7
60
5
1 46.9
1 4a.7
1 31.4
I 13.9
0 51.4
0 25.4
55
10
I 46. S
I 4a.o
I 30.1
I 12.2
0 49-4
0 23.2
50
ao
I 467
I 41.2
J 28.9
I 10.5
0 47.3
0 20.9
45
40
1 46.5
I 404
I 27.6 z 8.7
1 26.2 I 6.9
0 45.2
0 i8«6
as
I 46.3
1 39.6
0 43.1
0 16.3
35
30
I 46.0
I 38.8
I a4*8
I 5-1
0 40.9
0 14.0
30
35
40
» 45-7
« 45-3
« 37.9
I 23.4
I 3*
0 38.8
0 36.6
0 11.6
20
1 36.9
I 21.9
I 1.3
0 9.3
45
> 44^9
« 35.9
I 34.8
I 20.4
0 59.4
0 34.4
0 7.0
'5
50
I 44.4
* 43-9
I 43.3
I 18.8
0 57.4
0 32.2
0 4-7
10
5$
« 33.7
I 17.2
0 55-4
0 29.9
0 2.3
5
60
I 32.6
f 15.6
0 53.4 0 27.7 1
0 0.0
0
Min.
11 UoursI
10 Hours.
9 Hours.
8 Hours
7 Hourt. '
6 Hours.
Add.
Z
TABLE 13.
'Sun'sDeclinauonror the Years I3ca, 1812, 1816) 1820,
5" ~
' 'J 5
1 tj 0
3 " ii
; 11 4)
F<b.
Mar.. April
May,
June,
N.
11 5
11 ao
»»34
N.
»3 S
*3 3
Aug.
N.
il 1
17 46
"» I'
Sept
'lt^
Xnv.
Dec.
>.
S. i N. ,
N.
'S 7
15 41
16 17
N. : -.
S.
'5 44
[6 ;
16 10
16 ;5
S.
11 3»
11 39
11 49
:S SI
lii
13 13
•3 iS
13 16
13 13
k'<.
.6 18
le 10
7 31 4 36
7 8|4S9
6 45. i «
6 11. S 4S
i 59' 6 7
5 i(
7 H
7 JJ
6 It
IS
4 Si
4 i
J 46
J 13
1 37
1 14
1 so
N.'
s7
0 4t
0 3D
tM
■ '7
1 40
IJO
^i^
6 It 36
■I 11 i6
11 SI 47
■3 »' 37
15 »■ 17
M (i, i 36. 6 30
'S 34, S >3, 6 S3
'f li 4 49; 7 "5
14 (6 4 ih; 7 38
.4 37,4 1- 8 0
16 34
■ 6 ji
■7 7
17 13
17 39
21 40
1146
11 51
11 57
11 19
11 M
•6 9
'5 5'
!iJ4
,S .6
'4 iS
14 40
14 11
11 7
ti 46
II 16
II 6
^4J
9 41
J3 38
'J 57
» 4; 9 49
18 j^
13 6
13 10
23 '4
13 17
13 10
11 7
11 59
II
u
10 t9
10 i
'9 Si
■ 9 «7
19 >3
;i,t
■ S 31
i» lb
- 4 17 19
7 =- 17 45
« 34 IS 33
16 11 6
I, 1:0 i4
igiio 30
10 10 lE
=1 10 s
11,19 !'
13K9 33
141., 14,
=6118 5s
17, 18 40
iGliK ij
19' r8 V
3° i7_n
Jl.'? 36
11 37
" Si
.( 34
10 5r
9 4^
0 S3''D S*
0 I9,ii I J
0 6III 34
/,.|„„
1 19 11 54
1 .6J13 33
: 391 '3 S»
] 3 '4 >i
3 I6fi4 30
19 8
19 11
K> 13
1036
1048
'°J1
ir 9
11 19
11 )9
11 3>
^"7*
13 11
13 14
*3 16
U 17
»J *J
ijrf
ij 17
ij 17
ij 16
!3_i5
13 13
13 11
13 18
2} Ij
8 56i'S 4B
9 "S >9 3
9 40'i9 17
ro J 19 31
10 141945
1 1 49110 36
13 "1 11 1
13 31 11 31
13 iC 11 4
EXPLAN'ATION AND USE OF THIS TABLE.
The Declination of the Sun is an arch of a meridian conlaincd
between its centre and the etjuinoctiali which arch is recUoncd in
(Ie)^reea, minutes, Gcc.
In the first quadrant of the ecUptic. from about the SIstot
March, to the 2 1st of June, the Sun's declination is North, and
increasing ; onii in the thii-d quadrant, between the 22d of Sep-
tcmberand 21st of December, the Sun's decIina?;on is South, and
incrcasi»K ^n the second quadrant of the ecliptic, from about
the 2 1st of June tothc22d of Scinember, the Sun'a declination is
rOorth, and decreasing ; and in the fourth quadrant, between the
2ist of Diircmbcrand the 2lat of March, the Sun's decJinaiion is ■
South, and decreasing ; which will be readily perceived by in-
tipectin^ [)ic tattle.
In ihistabictheSun'sdeclination is given, fromlhefear 1806
to 1623 inclusive, calculated fur the utalMitof noon, each daf, at
TABLE 13.
J 79
Sun's Declination for the Years 1809, 1813, 1817, 1821.
•
t
Jan.
Feb.
Mar jApril^
Mav.
'June.
1 July.
Aup.
Sept
N.
Oct
Nov.
i3>
Dec.
S.
s-
b.
N.
N.
L.^
N. -
s.
b
± 1^
«> f
0 •
0 /
0 /
<» /
1 ^ '
n /
« / fp /
«
/
9 t
« /
1
23 2
17 7
7 37
4 30
'5 A^^ 3
23 9
18 6[8 22
3
7
14 24
21 49
2
22 56
16 50
7 14
4 S3
15 20
22 II
23 S
17 5^ 8 0
»7 3S|7 3«
3
31
'4 43
21 58
3
22 51
'6 33
6 5'
S 16
15 38
22 18
23 0
3
54
15 2
22 7
4
22 45
16 15
6 28
5 39
15 55
22 e6
22 55;
17 19
7 16
4
»7
15 21
22 15
$
6
22 38
»? 57
6 5
5 45
6 2
16 13
22 3Z
22 39
22 50'
22 44
•7 3
6 54
6 31
4 40
5 4
15 40
22 23
22 31
'5 39
6 24
16 30
16 47
IS 58
22 30
7
22 24
15 20
5 '9
6 47
16 4-J 22 45
22 38
16 30
6 9
S
27 16 16
22 37
8
22 16
"5 '
4 5S
7 9
'7 3
22 51122 31
16 13
5 46
5
50 16 33
22 44
9
22 8
14 42
4 32
7 3»
17 19
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1
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20 33
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II S2 0 0
II
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44
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9 49
1 23
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20 4i
23 26 19 56
II 1 1.0 14
11
43
20 31 23 26
25
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9 7
1 46
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20 5^
23 25 19 44
23 23 19 30
10 51}
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12
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25
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20 57
23 25
18 44
8 45
13 28
21 6
10 30
1 10
2; 23
*;
18 29
8 22
2 33
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21 17
23 21 19 17
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12
45
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28
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8 0
2 57
14 6
21 27
23 I9:»9 3
9 48
' 57
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S
21 I9I23 iS
29
n 57
3 20
14 25
21 36
23 16
18 49
9 2?
2 21
r
'3
25
21 29 23 15
|0
■ ii
17 4'
3 43
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23 «3
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9 5
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17 2J
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5
23 l\
the Meridian of Greenwich, or the meridian, at which we begin
to reckon the Longitude. It is to be taken out with the month ut
the top, and the day in the left hand column, at the same time,
noting whether it be North, or South, as expressed at tiie top of
each column. The declination being here given to the nearest
minute, it will^be found sufficiently exact tor the most common
and useful problems, wherein it is concerned.
The sun's declination is necessary to hnd the latitude, wliether
at sea or land, from the meridian altitude observod ; it is also re-
quisite for finding the latitude from two altitudes observed with
the interval of time measured by a watch ; it serves lor comput-
ing the sun's azimuth, having his altitude and the latitude of the
plaoe given, in order to hnd the valuation of the compass ; it is
required, jointly with the latitude of^the place and the sun's
horary angle, to compute his altitude, if neglected to be observed
at-the time of taking the moon's distance from the sun for finding
ihe longit^C} being Useful to facilitate the calculation of the effect
IM
TABLE 13.
Sun's Declinatum for the Years 18I0| 1814, 1818, 1822,
15
us
•
Jan. 1 Fe
b. Mar.j
April 1 May.
June-Uuly. jAng. |Se|»t
Oct
Nov. 1 Dfc-
8.
8
•
S
0 t
N." "
' N.
N
•
N
0 /
N.
N
Q '
b.
S.
:>.
o •
0
«
':rT"
* '" [ • '
• *
1
i5 3
»7
12
7 43 4 a4 14 57
22
I
23 10
\% 9
8 27
% *
14 ao
21 46
2
22 5S|i6
54
7 20
4 47 15 »6i2l
9
23 6
«7 54
8 5
^ *i
14 39
21 56
3
22 53 16
37
6 57
5 10 15 33:22
16.23 »
>7 39
7 43
3 4«
1458
»* 5
4
22 46 16
19
6 34
5 33 15 5»
22
24
22 56
17 23
7 21
4 12
IS 1712* I3|
5
6
22 40
«5 43
6 II
5 48
5 56
16 8
16 26
22
3'
22 5i|i7 7
6 59
6 37
435
i$.35
«5 53
[22 21
»a 33
6 19
22
37
22 46. 16 51
458
22 28
22 261I5
24
5 a4
6 41 16 42
22
44I22 39^ 16 34
6 14
5 21
16 ii;22 36
8
22 f8|J5
6
5 "
7 4' 16 59
22
49^22 33 16 17
5 52
5 44
16 29122 42
9
22 10 14 47
4 3S
7 26 17 15
22
55,22 26
16 0
5 *9
6 7
16 46 22 49
10
...
22 I 14
27
1 '4
7 49 17 3«
23
00 22 19
«5 43
5 7
6 30
17 4 22 54
If
XI 521 14
8
3 5«
8 11
"7 47
23
4
22 II
15 25
4 44
6 53
17 2023 00
12
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48
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23
8
22 3
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1 15
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ai 33, »i
28
3 4
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23
12
21 55 14 50
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7 38
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14
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8
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9 «6|i8 32|
23
16
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3 35
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i6
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21 0
12
47
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23
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27
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23
21
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25:21 8
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24
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23
26
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12 56
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21
so 12
II
2
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N.
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23
23
27
28
20 46
12 37
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N.
S.
0 52
10 13
19 38
23 a?
19 59
10
4«
" 44
20 35
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10 34
19 51
23 *7
2%
«9 45
10
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23
28
20 23
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0 29 10 56
20 5>23 2S
13
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9
57
0 53
12 24
20 30
23
27 20 12
Ji 37
0 6
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20 18 23 27
H
19 17
9 35
I 17
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20 42
23
27 19 59
II 16
0 18
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20 30
23 27
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»9 3
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9
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21 4
23
23
25; 19 47
10 56
0 41
» 5
II 59
20 42
23 as
8
50
'3 23
24
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10 35
12 20
20 54
23 a4
27
18 33
8
28
2 27.
13 43
21 14
23
22
19 20
10 14
I 28
12 40
2» 5
23 aa
28
18 17
8
5
151
14 2
21 24
23
*9
19 7
9 53
■ 52
13 0
21 16
23 19
»9
18 I
3 14
14 21
21 34
23
17
'8 53
9 32
2 «5
13 21
21 27
23 16
30
i7 45
3 37
4 ■
14 39
21 43
»3
13
18 39
18 24
9 10
2 38,13 40
21 37
23 la
lii
17 28
21 52
1
8 49
14 0
23 8
of refraction and parallax upon the distance ; it is also necessary
to calculate the apparent time from an obsered altitude of the sun
at a distance from the meridian, the latitude being given ; or to
compute the time of the sun's setting or rising ; which, tfafough
a less accurate method thmi the former of obtaining the time, ijfiay
yet be U6cful when that cannot be had. For any of these purposes
the sun's declination must be found to the time given nearly, re-
duced to the meridian of Greenwich^ making proportion accord-
ing to its daily increase, or decrease, by the help of table 14, as
in the following examples.
1st Required the Sun's Declination at noon in Mew-Torkj in
J^iongitude 74© 8' West, on the 1st of April, 1811.
Dec. for April 1st, 181 1, at Greenwich, in Tab. 13 =k 4* 18' N.
Equation for Long. Table 14. = 4. 4 50"
. Required Declination =• 4? 22' 50*',^.
TABLS; 13.
IJl
Sun*s Declination for the Years 1811, 1815, 1819, 1823.
•
1
Ha
Feb. 1
Mar.
April
May. (June.
July.
Aug.
Sept
Oct.
Nov. 1
Dec.
8^_
Q '
N.
N.-
N.
N.
N.
N.
S.
S.
• 1
2>.
0 /
• '
9 *
tt '
9 /
«
*
Q '
0 /
» /
I
43 4
12 $9
22 $4
22 48
22 41
17 167 48
4 18
14 53
41 59
43
11
18 13
834
»s«
14 15
2144
ft
16 59.7 4$
4 44
IS II
22 7
43
7
17 58
8 III 3 191
H 34
21 53
3
i6 4» 7 3
S 5
15 29
22 15
43
2
17 44
7 49 3 43|»4 531
22 2
4
16 24;6 40
528
15 47
22 22
22
S«
17 47 7 47, 4 o,«5 *4j
22 fl
J
"6
16 6:6 17
15 47 5 S3
5 50
16 4
22 29
22
5^
17 II
7 5 4 49115 3"|
22 19
22 35
22 27
22 20
6 13
16 21
22 36
22
47
16 55
6 42. 4 52115 49
22 27
7
15 «9
5 30
6 36
16 38
22 42
22
4«
16 38 0 to 5 15
10 7
44 34
S
15 10
5 7
6 58
16 $5
22 48
22
35
«6 22' 5 57 5 38
16 25
42 41
9
22 12
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7 41
17 "
22 53
22
28
16 4 5 35 6 «
16 42
2247
10
11
22 3
14 jj
14 ij
4 20
3 57
7 43
17 27
42 59
22
21
«5 47 5 '4
15 30 4 49'
0 24
16 59 12 53
17 *6 22 58
21 <4
8 5
"7 U
43 3
22
'i
647
21 4C
13 53
3 33
8 27
17 58
43 7
22
5
IS 12
4 47 7 »o
17 31
43 3
13
21 3S
13 33
3 9
8 49
18 14
23 11
21
57
'4 54
4 4 7 34
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43 8
»4
i6
21 2C
13 »3
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9 '«
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23 15
21
49
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3 41! 7 55
'S *
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2^1 14
12 52
2 22
» 59
9 33
18 43
23 18
23 21
21
40
14 17
3 17 8 17
4 54 8 40
18 21
23 16
21 2
12 32
9 54
18 57
21
30
1358
18 36
It
23 19
17
20 52
20 40
20 28
11 11
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10 15
19 II
23 23
21
20
13 40
4 31
9 4
18 51
23 21
i8
II 50
1 II
10 36
19 25
23 25
21
10
13 20
2 81
9 44
19 6
43 H
«9
10
21
II 29
047
10 57
1938
23 26
21
0
13 I
« 45
9 46
19 21
23 25
20 15
II 7
10 46
0 24
s.
N.
0 0
II 18
19 CI
45 47
20
49
12 41
I 21
N.
S.
0 58
10 7
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20 2
II 39
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20
38
12 22
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43 47
21
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19 21
19 6
10 24
0 24
II ^9
20 16
23 28
20
26
12 2
0 35
10 50
20 2>.23 28
43
*4
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i6
10 .2
0 47
1% 19
20 28
23 27
20
M
[1 41
0 II
II 12
20 14 23 28
9 40
I 11
12 39
20 39
23 27
20
2
11 21
0 12
" 11
20 27;23 27
9 «8 I 35
12 59
20 50
23 26
'9
50
II I
0 36
0 59
II 54
20 39:23 26
18 52
8 s6!i s8
13 19
21 I
23 24
>9
37
10 40
12 15
20 5IJ23 24
47
28
18 36
8 33
1 22
«3 38
21 12
23 22
«9
44
10 19
I 23
12 35
21 2123 22
18 21
8 II
4 45
n 57
21 22
23 20
"9
10
9 58
I 46
12 55
21 13:23 20
29
30
18 5
III?
17 34
3 9
14 16
21 32
23 17
18
5<>
9 37
2 9
13 16
21 24; 23 17
3 34
3 55
'4 35
11 41
43 »4
18
43
9 15
4 33J13 3«»;4i 34|43 131
,
11 50
18
28
8 54
1
n 5$
1
•
\n 91
N. B. To find the equations in Table 14,— seek the Sun*s de-
clination to the nearest degree in the top line of the table ; then,
under this declination anfl against the given Lon. in the left hand
column, is found the equation for Lon. and in the same column
with the dec. and against the given time from Noon, in the right
hand column, b found the equation for time ; both which equa-
tions must be added, or subtracted, according to the directions at
the hiead of the Table.
2d Required the Sun's Declination on the Utof May, 1811, at
5 h. 48 min. P. M. in Longitude 72° W.
Dec. May ist, 1811, table n. = U© 53' N.
Equat. for Lon. :=- +3 41'
Equat. for Time = + 4 27
Reduced Dec.
15 1 B N.
182
TABLE 14.
When Sun's dec. increases. When Sun's dee* deereastf •
Add in W. lon. | Add af. noon. | Sub. in W. Ion. I Sub. a£ noon^
Sub. in E. lon. | Sub. be. noon, j Add in E. Ion. } Add be. noon.
Lon.
Sun*s Deri illation.
[ Ti. fr.
noon*
o«
t -^
. 0^0*
40 . 6*'
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1
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O'O^
ohom
3
0 12
1 0 12
0 12
0,1;
0 11
0 1$ 0 11; 0 II
. 0 12
6
0 24
0 24
0 24
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0 23
0 22, 0 22: 0 21
0 24
9
0 3S
0 35
0 35 0 34;
0 34
0 33i 0 32! 0 32
0 36
12
0 47
0 47
0 47
0 46I
0 45
0 44; 0 43t 0 42
0 48
«$
0 59
0 59
0 58
0 57
0 56
0 55
0 54
0 53
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21
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248
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2 57
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60
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75
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78
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5 *2
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84
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87
5 41
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5 31
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5 '8
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90
$ 53
5 52
5 48
5 42
$ 34
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6 4
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96
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5 37
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99
6 28
6 27
6 23
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6 8
6 3
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5 48
6 36
lOX
6 40
6 39
6 35
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6 19
6 14
6 7
5 58
6 48
105
b 52
6 51
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7 d
loB
7 4
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635
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7 15
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114
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11 25
II 14
10 s-
10 47
10 36
10 21
II 48
180
II 4t)i
n 43
»i 37
It 25
II 8
10 58
10 46 ID 32
12 0
.ISiib.in'W lon.[Sub.afno<.i
lAddiiiE. loii. lAJJbe.nuo
Lon
Sun's Ueilinalioii. itlmefVl
]
11°
rj" 1+'
15°
16°
-L'."
JT
~^
Noon.
o"
o'o-
c'c" o'o"
o>o»
O'O'
?^'
ohom
3
a 5
0 9
0 8
6
Q 18
0 iS
□ 16
0 3"
a }O0 lH
0 14
0 24
0 40
0 3f
0 36
0 31
0 3r
0 SI
0 JO
048
0 46
0 J9
^ 35
31
' 9
?1
= !!
0 SI
3 57
3 46
3 54
D 49
I n
^4
1 ]i
1 19
' '9
' 'I
; H
1 M
t 14
0 56
1 36
■ 4S
3°
J^'*!
■ 39
1 31
1 18
1 13
I 1!
33
36
t 53
» 3
■ 49
; S9
ri6
, 56
' S'
' 37
. 46
I 31
' 39
' 3]
77^
1 14
39
1 14
a !
' SS
1 41
4t
2 14
I 34
I Z9
I 14
1 19
! s6
I S
1 49
' 57
1 46
1 48
43
» 44
2 39
1 34
I 13
1 >3
■ 53
SI
a 5S
I 44
I 3S
2 30
1 A
3 14
S4
3 i
' )9
1 J3
i 47
1 19
X 20
11
3 15
3 9
3 3
1 se
z 4S
J 38
3 48
TT
SM
3 '9
1_13
5_5
yss
i_47
2_jb
Lii
3 3!
3 19
3 11
3 "4
3 5
I iS
» 43
1 19
4 '1
69
7i
lis
4 '7
J 39
3 49
3 59
4 9
4 19
i 31
3 4»
3 5'
3 ^3
3 3'
3 41
1 S"
3 '4
3 '3
1 Ji
3 40
3 49
I 3
3 '■
3 36
I J.
I S9
J 7
3 IS
» 36
25!
4 36
4 48
Gi
4 37
4 19
4 9
i 5S
3 44
3 30
8+
4 47
4 3>i
3 s»
3 JS
8?
45
4 49
4 ^7
4 16
4 '
J 46
3 s(
J3.
93
4 i9
5 9
4 49
4 59
4 37
4 46
_4__li
4 34
rrg
3 54
4 '
3 34
3 41
6 □
t
S "9
' ?
4 iS
* 4J
4 IS
] 48
99
S 39
1 =9
5 'I
S 5
4 Jl
4 34
3 55
!"s
5 49
i",
6 10
5 39
S 49
5 59
6 9
i 37
S 47
5 J*
i '4
i 13
S JI
S 4*
11
S "7
S z6
4 S^
i 8
4 »5
4 33
4 4"
448
4 1
4 1'
4 J3
7 24
6 ig
5 5'
5 IS
i 1^
4 JO
7 56
'■7
1'°
6 59
f. IS
(i I
i «
5 »!
5 4
4 is
11;
7 '
6^9
^
m
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S 3-1
S 4»
5 '-
( '9
4 S3
7 "
6 44
i SO
5 0
'3»
'3S
14+
7 3^
7 41
7 i^
8 i
8 n
7 9
7 »3
7 33
7 4»
6 46
6 j6
7 14
7 13
1=1
* 39
5 35
5 43
Vi
6 14
S 7
> )4
5 35
3 48
9 36
I''o
8 '}
JLli
3 ^
S *'
7 3»
6 48
6 21
5 49
— ^
6 57
6 3°
5 57
■0 °
S '0
7 30
7 S
b 37
i 4
» S4
7 39
7 13
6 4i
9 4
8 ji
3 10
7 48
653
6 18
10 jfi
■ 6S
9 '4
9 14
9 3!
;!;
M 41
8 ja
B 19
3 IS
S3S
7 57
5.5
7 19
7 3^
7 46
6 IS
6 31
6 39
.0 48
■74
9 4S
■'1
9 iS
9 3R
9 4I
;;;
9 6
a 13
a;
7 i*
X 3
7 3>
7 40
h 4t'
S 5i
II 24
II 36
• 1 4S
.;_
!£_L
9 SK ■. i'") .5
8 49 1 2117 4t'7 4
12 0
184
TABLE 14.
When Sun*s dec increases. When Sutf« dec decreaM.
Add in W. Ion I Add aft noon. I Sub. in W. Ion. I Sub. aft. aoM
Sub. in E. Ion. | JJ>ttb. bef. nowi. | Add in E. Ion. | Add
bef noon.
Lon.
3
6
9
12
»5
i8
ai
*4
»7
30
33
36
39
4»
4$
48
5«
54
57
60
63
66
69
72
75
78
81
84
87
90
93
96
99
102
ro8
III
"4
"7
t»3
126
129
132
"35
138
141
144
"47
150
l«53
156
159
162
165
168
171
»74
177
180
Sun's Declination.
19*30'
o'o*
o 7
o
o
o
o
o
o
o
I
1
"3
20
»7
34
40
47
54
I
8
I
I
I
t
I
I
I
2
2
2
"4
21
28
IS
41
48
55
2
9
16
20«
6
o
o
o
o
o
o
o
o
o
I
6
12
18
a5
3*
38
44
50
57
4
2 22
2 29
2 36
2r43
2 50
2
3
3
3
3
56
3
10
"7
^4
3
3
3
3
3
4
4
4
4
±
4
4
4
4
30
37
44
5»
58
4
II
18
45
i?
38
45
52
59
6
12
19
26
i3
40
6
6
6
6
6
6
6
6
46
53
o
7
'4
20
27
34
4'
48
10
16
22
»9
36
42
48
54
I
8
14
2C
26
33
40
46
5»
58
5
12
20" 30'
o'o"
o 6
o
o
o
o
o
o
o
o
o
II
17
23
»9
35
41
47
53
59
4
10
16
22
28
33
39
45
5*
59
2
2
2
2
2
2
2
'2
2
2
4
10
16
21
^7
33
39
45
5*
59
3
3
3
3
3
3
3
3
3
3
4
9
"5
21
a7
33
39
46
5»
59
4
4
4
4
4
4
4
4
4
4
4
10
16
22
28
34
40
46
54
58
'5
5
5
5
5
5
S
5
5
5
3
9
"5
21
26
3*
38
44
5"
58
21"
o'o*
5
10
«5
21
*7
3»
38
44
50
55
o
5
ic
16
22
27
3*
38
44
49
I
I
z
2
2
2
2
2
2
54
59
4
10
16
21
26
3*
38
2 44
49
54
59
5
II
16
21
»7
33
39
3
3
3
3
4
4
4
4
4
4
44
49
54
59
5
lO
15
21
»7
33
438
4 43
448
4 54
5 o
6
II
"7
23
29
21030'
o
o
o
o
o
o
o
o
o
o
o
5
9
"4
"9
^4
19
34
39
44
49
o
2
2
2
2
2
2
2
2
2
2
2
2
3
3
3
3
3
3
3
3
3
3
3
3
4
4
4
4
4
4
4
4
4
4
4
4
53
5^
3
8
«3
18
43
28
33
39
43
48
53
58.
3
8
13
18
»3
28
32
37
4*
47
5»
57
2
7
12
"7
22
a7
3*
37
4»
47
5*
57
2
7
II
16
21
26
3«
36
4'
46
5«
56
22
o'V
o 4
8
12
16
21
»5
29
34
39
43
O
o
o
o
47
5"
55
59
4
8
12
16
21
26
30
34
38
4*
47
51
55
59
4
9
13
17
2r
»5
30
34
38
43
48
53
* 57
r I
3 5
3
3.
3
3
3
3
3
9
13
17
21
26
30
35
3
3
3
3
3
4
4
4
4
4
39
43
47
5"
56
o
4
9
14
"9
22*30'
o
o
o
o
o
o
o
o
o
o
o
3
6
10
14
18
21
»4
28
32
36
o
o
o
o
o
o
39
42
46
50
54
57
o
3
7
II
14
17
21
»5
29
32
35
39
43
47
2
2
2
2
2
2
2
50
53
57
I
5
9
12
16
20
«3
2 26
2 29
2 33
2 36
2 40
» 43
2 46
2 50
a 54
2 58
3
3
3
3
3
3
3
3
3
4
8
12
16
»9
22
26
30
^ 34
*
23'
o'o»
o 2
o
o
o
o
o
o
o
o
o
4
7
9
12
«4
17
'9
22
as
o
o
o
o
o
o
o
o
o
o
o'
o
o
o
17
30
3»
34
36
39
4*
44
47
i?
5"
54
56
59
I
4
6
9
II
24
16
19
21
24
26
29
31
34
37
u?
4"
44
46
49
5>
54
56
59
I
4
»3**5'
timefr
Koon
2
2
2
2
2
2
2
2
2
6
9
II
13
»5
>7
20
22
25
28
O 2
O
O
O
O
O
O
o
o
o
4
5
7
9
to
12
14
«5
17
o
o
o
o
o
o
o
o
o
o
19
20
22
»5
»7
29
30
3»
34
o
o
o
o
o
o
o
o
o
o
35
37
39
40
4»
44
45
47
49
50
o
o
o
o
o
5«
54
55
57
59
o
2
4
5
7
9
10
12
H
«5
«7
«9
20
22
*4
a5
27
29
30
3»
34
35
37
39
40
OhOBi
O 12
O 24
O j6
0 48
1 o
I 12
« «4
I 36
I 48
% o
2 12
2 24
2 36
248
#.t
3 a4
3 36
348
4 o
4
4
4
4
5
5
5
5
12
*4
36
4«
o
12
24
36
54«
6 o
6
6
6
6
7
7
7
7
7
8
12
36
4«
o
12
H
36
48
o
8 12
8 24
8 36
8 48
o
12
a4
36
48
o
9
9
9
9
9
10
10 12
10 24
10 36
10 48
11 o
II 12
II 24
II 36
11 48
12 O
r *_
— ^
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