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MATHEMATICAL ,j^ 



AND 

ASTRONOMICAL TABLES, 

FOB THB USB OF 

STUDENTS OP MATHEMATICS, 

PRACTICAL ASTRONOMERS, SURVEYORS, ENGINEERS, 

AND NAVIGATORS ; 

WITH 

AN INTRODUCTION, 

coKTAiiriiro 
THE EXPLANATION AND USE OP THE TABLES, 

ILLU8TKATED BY 

NUMEROUS PROBLEMS AND EXAMPLES. 



BY WILLIAM GALBRAITH, M.A^ 

TBACHBR OF MATHEMATICS IN BDINBUBOH. 



EDINBURGH : 

rUBLUBXD BY 

OLIVER & BOYD, TWllEDDALE-COURT ; 

OEO. B. WHITTAKEB, AND J. W. NO&IE & CO.j 

LONDON. 

1887. 



>:A;>r'r \ : .yi /M 




OLITBK * MTD, nillTnll. 



TO 



Sir GEORGE CLERK, of PENmrcuiCK, 

BART., M.P., P.R.S., 

ONE OF THE LORDS COMMISSIONERS OF THE ADMIRALTY, 

Sir, 

The following Work, which you have allowed me 
the honour of inscribing to you, is intended to promote the pur- 
poses of useful instruction, and the advancement of practical 
science ; and it is therefore confined to subjects having a direct 
utility in the business of life. 

Though I am aware that no patronage can materially influ- 
ence the success of a Work of this nature, which must depend 
upon its merits alone ; yet I have bee^ solicitous to inscribe it 
to you, in the hope, . that practical men, in search of useful 
knowledge, may be induced to consult a Book sanctioned by 
a name intimately connected with many recent scientific im- 
provements; and I confidently trust, that a reference to the 
volume itself will prove that your obliging permission has not 
been undeservedly bestowed. 



I have the honour 'to be, 

SirfJ'-*; . 
With the utmost respect. 

Your most obedient servant, 

WILLIAM (JALBRAITH. 

Edimbuhoh, iVbv., 1826. . ^^r 



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PREFACE. 



7li£ i^mdiciitioii of the mathematical aci^ioes to practical pur- 
^p&m ha9H>f lat<9 mi^d^ ^peat advaiioes in accuracy and precirioiL 
The perfection also w]^di astroiiomical and geodetical operations 
iyaye yf4!che4f 9f^ the extreme delicacy of constructioii to which 
^l^truR^epte have been harried, require correspondent improYe* 
mf3|^t8>in tl^ .^lethods of computaticm and reduction ; and, there* 
fftrtty convenient (aided of moderate expense must be of great 
¥fAfl^ tP. ^hoj^ fiH^^iged either in the details of practice, or die 
t^u^ui^ii^ pf ^E^^tEttctiw- 

. ' Tl^M ai^e two dt^Bsea of tables chiefly in use ; one dther 
Uxgfi sfj^^tpeskfiy^f or attached to expensive works, and which 
th$9f^&lie can with dsfficiilly be procured by the generality of 
p^^|j|p/s^$; th^ other so limited and defective as to be totally 
un^l fjpr constant jceferendeu It has been my study to hold a 
imiM]^ 0oUr^ between these two extremes. By paaking sudi 
<fl^ti$)i^ U> the miwi tables as to render their apjdieadon mpre 
^Wf 9 3¥4tl¥Hit ^eatly ibc|reaffltng their bulk ; by selecting the most 
jfflft^fKQaihacg^r coUectioJQs; hy supplying some new ^hles, and 
f^m^yi^gAe Igxwti^ rules, several very kWrioos processes 
have been rendered more simple and precise, while the requisite 
^u^i^rt^yibr. the nicest purposes has been strictly preserved. 

In most of <>ur initiatary works for popular instruction, die 
processes and examples are unfortunately conducted in sudi a 
m^ftnncr tvs to b^ compa^tively of little advantage in actual prac< 
tice, and, consequently, what has been learned in youth, must, 
in a great degree, be foi^tten in manhood, while new methods 
are then to be acquired. 



I 

^^^" rao 



PBEFACE. 



copic 



To remedy tliis inconvenience, I have selected some of the 
raosi: approved modes of treating the problems frequently re- 
quired by Astronomers, Navigators, and Engineers, from the 
works of persona celebrated for their successful application of 
the exact sciences to the niceties of modem practice. 

I have therefore taken many of the Astronomical Rules and 
Examples from the works of Maskelyne, Pond, and Brinkley ; 
and such as relate to other topics from those of Captains Kater, 
Hall, Sabine, and Parry. To Captain Hall I am under great 
obligations, not only for access to his original papers, but also 
for his friendly advice relative to the application of these me- 
thods to practice. 

To Mr Ivory I am indebted for his very accurate Table of As- 
tronomical Refractions, which I have endeavoured to improve by 
expanding and adding proportional parts to the subsidiary tables, 
thereby facilitating its practical application. 

Besides labouring to improve many of the ordinary Tables, I 
have added several which are new, chiefly for the purpose of 
simplifying some operations and rendering others more accurate. 

The explanations will, it is hoped, be found full and explicit, 
especially towards the beginning. The explanation of some tables 
which follow others, analogous in structure or arguments, is 
sometimes less full, as it is presumed those previously given are 
well understood. For example, the note to Table XXV., at the 
bottom of page 91, can hardly be intelligible to a mere practical 
man who has little mathematical knowledge ; but as the method 
of taking out the quantities from Table V., in whatever quadrant 
of the circle, or division of 24 hours, they are situated, is so fully 
explained before, it was thought unnecessary to repeat the same 
minutia! a second time. Still, however, there may be some parts 
which require to be expanded, in order to be more readily uii- 
derstood, as well as others which might, perhaps with propriety, 
be abridged. 

The Introduction is divided into three parts, followed by a 
copious explanation of the general tables, which may be called 
A- fourth. 

n the first I have shortly described the nature, and investi- 

>d the more simple series for the computation of Logarithms; 

generally, however, only given tho more important 

word.s at length, without iiivct T^ntinT), *o ,ti to Ik* rcTlHv 



I 



FH£FAC£. TO 

prehended by persons who hav^ acquired a knowledge of the 
elemenfayry princ^les of mathematics* In fact, the demonstim* 
tioas can only be understood by .those who hare obtained a toler<* 
Me knowledge of the elements of geometry and algebra, and, 
since the generality of books containing these comprehend also the 
usual investigations in trigonometry, it was thought advisable to 
omit them. = If, for example, a student should purchase Le» 
gmdre^s Elements of Geometry in qrder to study that science, he 
will find it to contain also very elegant investigations of almosi 
aU the useful properties in Plane and Spherical Trigonometry. 
On this account, I have only given the demonstrations of those 
propositions less commonly inserted in the usual treatises. 

On the Barometric Measurement of Altitudes, I have given 
four different methods* The third is in a great degree new, and 
by die original subsidiary tables, calculated expressly for this 
purpose, it will be found easy and accurate. 

The second part contains Spherical Trigonometry, with a 
great variety of its most useful applications. As the rules and ex- 
amples are eith^ new or selected from the best writers on the 
subject, it is hoped this section will prove interesting to students 
of Astronomy and Navigation, since it contains a number of the 
usual methods and examples practised by the most distinguished 
men of science of the day. 

The third part contains a variety of Rules and Formulae for 
the use of Surveyors, Engineers, Navigators, and practical Astro- 
nomers. Those for geodetical purposes are selected chiefly for 
their general utility, and comprehend a sufficient number for 
usual practice,— *an idea which was suggested to me by some of 
my more advanced pupils who have been employed in govern- 
ment surveys. They were first collected in the form of notes 
and transcribed into their albums, to be used when they were en- 
gaged in geodetical^ accurate military or marine surveying; and 
as they may prove generally useful to that class of Students, I 
have arranged them in as natural an order as possible. 

The ingenuity and skill of Captain Kater having devised the 
most beautiful simplifications of the problem of determining the 
figure of the earth by means of the pendulum, and brought the 
experiment within the reach of our more active and intelligent 
military and naval officers, I have added the necessary rules and 
formuls»> for that purpose, in order to initiate, as far as possible. 



vm fR£FAGE. 

odf Cadets and Midshipmen in these ifiteresting redearcbes ; as 
sneh higher objects ofpursuit^not onljr iBTigomte their faculties, 
but -kisp&re them with enthusiasm for the attldnment of profes-' 
sjonal reiiowD. 
. The fourth part contains the necessary Explanation of the 
Tdblte* 

I have thus endeavoured to collect) into as small a space as 
posable, the greatest quantity of useful matter naturally con-; 
nectcd with the subjects treated in the work ; but uritb what 
success I must allow tkcf public to determine. 

WILLIAM GALBRAITH. 

Edivbueoh, Nooember^ 1826. 



. i i 



d \ 



CONTENTS 

or 

INTRODUCTION. 



^AtT I. PROPERTIES of LOGARITHMS 1 

Omitxuctiflii of Logarithms 4 

TrigonometriaJ lonei, eaDad Sinet^ &c. $ 

Midttples and Powen of Arcs • • 19 

PLANE TRieONOMETRY 15 

Its Applicadoii to Sailings in Navigation,. •••. S4 

Ito AppUcatioQ to the Mensuration of Heists and Distances S6 

Its Application to the Determination of the Lines and Angles of Re- 
gular Fortresses, and a Tahle of their Measttxes. 43, 44 

Barometric Measurement of Altitudes « « 44 

II. SPHERICAL TRIGONOMETRY, &c. 

Definitions, Principles, and General Properties < • 66 

Solution of Spherical Triangles, witli their Stereographic Projection OS 

Napier's Rule of the Circular Parts 64 

Maskd/ne^s Rules for determining the Latitude and Longitude, 
from the Right Ascension, Declination, and the Obliquity of 

theEdiptic, &c « «« • 68 

Sohition of Oblique-Angled Spherical Triangles 73 

On finding the Latitude by Obserration , 81 

On Finding the Longitude by Observation. 

1. By Lunars 89 

2. By Chronometers «. .....^ 104 

Equation to Equal Altitudes 107 

3. By Occultations 114 

4. By the ]k|oon*s Transit. 129 

Of the Trannt Instrument. ;... 130 

To take a Transit 131 

Method of Tabulating a Transit « 132 

To bring a Transit Instrument into the Meridian 133 

To determine the Error and Rate of a Clock or Chronometer by the 

Transit Instrument • , 130 

IIL MENSURATION, SURVEYING, AND FOAMUL^, &c. 

MenSuratiOa of Surfaces « 139 

MensuTBtion of Solids... 142 




T III. Land Surveying. 
Levelling. 

HULES and FORMULA, 

The best Form of Trianglra 

Ti> reduce Angles to the Centre of the Sution, 

To compule the S|>herical Eiusa, 

To reduce > Measured Base at any Height to the Levd of the Sea. 

To determine the Uoriionlal Refruction by Rule or Formuls 

To find the Angle made by a given line with the Meridian 

To determine tlie EUipticil; of (he Earth by tliB MeasuKmenl of 



To determine a Degree of Latitude 

To determinaa Degree of Longitude. 

To deteimine an Oblique Degree „ 

Specific Oraiity » i 

To determine the Specific Gravity of Air, Dry, laturotcd with 
Jloisturc, and according to llic actual Slate of the Atmoaphere... 

To detetmiDe the Specific Gravide« in inKUo 

. , To deteimhie ttie Effect! of the Buayaaey of the Atmoaphere on 

. the Pendulum 

Convction of Pendulums vibrating in Circular Am 

Correction of VibratioD for Buayanty 

for ExpannioD _i..,<..i< 

tor Ue^t abuve the Sck...... 

Determination of the Length of ihe Pendulum it diStlent Points 

on liie Eorlh'i Surface 

DetemiiDaliDnuf ttie Figure of ^ Earth bf ihe Pendatum 

Camparisan of the Eiiglifih and French Penduluma 

Velocity of Sound 160, 

Velocityof Ihe Dlscha^e of Wuer.pipu, Riven, and Canala.. 161, 

Fall in a River mused by Obstniodon in the Stream 

Tonnage of ehipi , 

Sueiifjth of Timber 166, 167, 



EXPLANATION OF THE TABLES, &c. | 

I, Miles of Longitude al any Latitude I I 

It. Li^nriihms of Number! 1 1 

Logarithmic Ariihmetic 8 

III. Angles which every Point and QoaiUr Poinl of the 

CotnpBua makes aith the Meridian 7 IT 

IV, Logorithmie Sines, Ae. in every Poinl and QuaiMr 

Point of ilie Cmnp«s.w 7 17 

V. logarithmic bines, Tangents, &c. to Degrees 7 18 

VI. Naltitaf Sines; Tangents; Seiaiil*, and Versines, to 

every Degree of the Quadnw II 63 

V'll. Alerldional t>artBloevery Degfwof iheQuadrant.... H *H 

VIIJ. Tt.»er«e Table H M ^b 



Fycb Putk 



Sij ^., — ., . .. .^ — . ■ 1'v \- I. ■: .1 •/ 

tL Table IX. Diuinal Logarithms •• «• < 

X. Proportioiud Loganthini 13 68 

^.'v v.. XI. D^oftheHoriion ,.,«.i 13 84 

XII. Dip*tdifiereatI)MUMWC%...^ ^..^.w 13 84 

\*-i XIIL Correction of the San'sAUkiids aft 8m ^ 14 84 

> XIV. CoRoetionofa^iWtAliitodft^..^— ..^.^.^^ 14 84 

- XV*' SvB « osmioiaiiMlefy «€»•••••«•«•••«•••«••»«•••••«•«•••• 14 88 

XVI. Soa'f P«rallasin Aliitade,..^..« ••• 14 88 

XVII. Mean Rcftaetiom hy My h/mj^^i ^ 14 



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XVIIL ) 



16 


90 


16 


90 


16 


91 


16 


91 


17 


91 


18 


98 


19 


93 


21 


94 


24 


95 


28 


96 


28 


96 


29 


97 


29 


97 



XIX. VSoUidiary to XVII«.^ .^ 14 89 

XX.J 4 

XXI. Augmentatioii of the MiMft*t fittnadiaBcter in Alti- 
tutte^ and ^.U.r.*«^.w..«««.v«»vi.*««.«..««i4*.. ••••••••• 

^^^ XXII. ReduotioB of the Moon'a Earallaxen the Spheroid... 

XXIII. IiA^aritbma of the Earthfk .^adii on the Spbtreid. 

^tii XXIVtf RedoMkm of the Lati(ude...«..w 

<*(*^ ^ XXV. For deteminlng die Latitude hy the Fele Star.., 

XXVIL AttgttleMM|on^of die Meon^ bemidiaweter hy the 

' '^^ ... NoBageumal .•••.•^^^..•••••♦•♦•w.....* 

<'<^i XJCirii l^uiMtoa of SeooBd Diffec^^ 

<^^i X^VIII. ReducdoD to the M«adiaii.,r. ^^^*..,^^.... 

^^^ XXIX. JUdoetien to elAer Sohitee ^. 

• X^X. T#di«nge MeaaMar hito Sideieal Time^.^ 

Xidtf, To diangeSidtieal. into Mean SolarXloM........ 

XXXII* To oonvert Mean Time into Paru at tht Squalor 

XXXilL Lengthg of Citcalar Arci.... - 

XXXIV. To XLVIII. For compudng the CorrectioM of the 

Fixed Stan 29 98 

XLIX. Mean OUIquity of the EcHptio ^ 32 108 

L. And LI. Correctiona of the Obliquity 32 103 

LII. And LI II. Solar and Lunar Nutatioas of the Xqui- 

^- noKeainTirae -.«.... 32 103 

LIV. Right Ascensions and Declinations of Stars for 182&.. 32 104 

LV. Decimal Numbers for each Day in the Year 32 104 

LVI. Sun's R.A. for 1828 33 105 

LVII. Sun's Dedinadon for 1828 33 106 

LVin. Equation of Time for 1828 33 107 

LIX. Correction of Longitude by Chronometers 34 107 

LX. Latitudes and Longitudes of Places 35 108 

LXI. To convert Space into Time 36 109 

LXII. To convert Timeinto Space • 36 109 

LXIII. Usefid Numbers in Calculadon 36 110 

LXIV, An4 LXV. To find die Time and Height of High 

Water 36 111 

. ~i .^, T LXVI. And LXVII. Tables of Equadon of Third and 

^f<i Fourth Differences 36 118 

LXVIIL TaU^ to^iiid die Ladtude by die Pole 9tar 37 118 



r 



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f 

fT 

HISCEIdEiANfiOUS TABLES IN THE INTRODUCTION. 



XI t .... 

TABiiSri;..^' I. SigM f»f,Trigooo|D^eal Liues^....- ^» 12 

Vi i II. MuldpUsand Powersof Arcs 12, 13, 14, 16 

•'\: Z ... JIL.SIeitSHirM of Forts ,M...M^*»..»f r***i ^ 

iy« Dqn^iioQ of Mercury in Ojass Tubes./...../. 48 

ci) i; V« Elaadc FoRce of Aquepue Vapour (Pi#>n) 48 

Vi) i.' .:<;i^^i;M,j4p0B|it|l««oft^^ttlk^f^a^ 

fe-c^ ' * tures.^.. ..••*i«i«;v*M««»^«t^iV***V)4*A 49 

7 



«• 



m CONTENTS. 

Face. 
TablS VIL I/ogtfhhmg of the EEkd of Latitude on Barometnc 

Ahitades 48, fiO 

VIII. Gorreetum of the Oblique Semidiameter in Lunars by 

Dr Young 100, 101 

IX. Equation of Second Difoence for Three Hours or for 

00' and 100" 102 

X; Correction of Apparent Time depending upon the Equa- 
tion of Second DiJfoence and the Vi^tion of the 
Distance between the Moon and the Sun, or a fixed 

Star, in Three Hours. .-. 102 

XI. Of the Dedmal Fractions of a Day 113 

XII. Decimal Parts of an Hour 113 

XIII. To convert Dechnals of Time into Degrees at the rate 

of fifteen Degrees to an Hour US 

XIV. Variation of the Sun's R.A. and D. in one Second for 

each Month in the Year 138 

XV. Areas of Circular Segments ,i 141 

XVr Polygons 142 

XVII. Regular Bodies 143 

XVIII. Table A. for Correcting the Number of Oscillations 

for the Are of Vibration 166 

XIX. Tables of Specific GraTity 163, 164, 166 

XX. Expansions of Solids and Liquids 166 

XXf . Table for computing the Strength of Timber ....^ 166 

XXII. Table for Correcting Lunars for Spheroidal Figure of 

theEardi. Explanation of Tables 43 

XXIII* Table for finding the Latitude by the Pole Star. 43 



INTRODUCTION. 



PART I. 

OF LOGARITHMIC AND TBIOONOMBTBIGAL TABLES. 

Section I. 

Of the Properties of Loganthms, 

1. Logarithms are a series of numbers^ originally invented by Baron 
Napier, for the purpose of facilitating arithmetieal calculations* 
This end is attained by their enabling us to perform the operations 
of multiplication by addition, of division by subtraction, of involu- 
tion by multiplication, and of the extraction of roots by division.* 

2. It is evident that any two series of numbers, the one being in 
arithmetical and the other in ffeometrical progression, possess these 
properties, thus, for example. Jet the 

At. series be 1 2 3 4 6 ) - 

Geo. series 1 10 100 1000 10,000 100,000/"^ 
Now, if we add any two numbers in the arithmetical series, such 
as 2 and 3, which are equal to 5, and multiply the corresponding 
numbers under them, 100 and 1()00, we have 100,000, the number 
immediately under 5, which was obtained by the addition of 2 to 3. 
Hence, then, it is clear that, if tables of this kind, sufficiently ex- 
tensive, were formed, by a reference to them,* the operation of multi- 
plication could be performed by means of addition. 

In like manner, we perform division by subtraction, for, if from 
5 we take 3, the remainder is 2, under which we get 100, that is 
100,000, the number under 5, divided by 1000, that under 3, gives 
100 as a quotient. 

Roots are readily determined in a similar way ; thus, 4, in the 
arithmetical series divided by 2 gives 2, under which, in the geome- 
trical series, is 100, that is, \h& second, or square root of 10,(X)0 the 
number under 4, is 100, the number under 2, and so on. 

Napier called the first series the logarithms of the corresponding 
numbers in the second. 

3. Since the two series may be assumed at pleasure, we may have 
as many difierent systems of logarithms as we choose. 

4. The series in art. 2 being adapted to the common denary scale of 
arithmetic, is, on the whole, the most convenient for general pur- 
poses, though other systems have, in j^rticular cases, their peculiar 
advantages. 

On considering these series, it appears that the logarithm of 1 is 



* The identity of this process with that porfonned upon the exponents of quantities 
in the corresponding operations of algel»&, will be obvious to those who have acquiTe4 
the rudiments of that branch of matnematics. 



2 INTRODUCTION. 

0, and that of 10 is 1, and hence the logarithms of all numbers be- 
tween I and 10 are greater than and less than 1, that is, they are 
fractions. In the same manner, between 10 and 100 they are greater 
than 1 and leas than 2, that is, they are 1 with some fraction an- 
nexed, and BO on. The whole numbers or integers in the logarithmic 
series are hence easily obtained, being always a unit less than the 
number of figures in the integral part of the corresponding natural 
number. On this account it is customary, in the common printed 
tables, to put down only the fractional part in the form of a decimal, 
the computer supplying the whole number or integer under the 
name of index. 

5. In order to generalize, let ua assume the two following series : 

r', V, K, r^, &c. . . (1) 

p, y' . f. y". &c. . (2) 

in which r is some given number greater or less than unity, and x, 
X', x", x'", &c. any variable quantities chosen in such a manner that 
I /W=3', r''=i/', r^=«", r'"^^'", &c., then the several exponents, 
.*, x'. X", x"-, &c. of the series (1) are called the logarithms of the 
jcmresponding terms in the series (2). 

. Thus if ^, i/, y, y'", &c, be a series of numbers such that r'-^y, 
T''=:y', T''=.y", f'"=.y"', &c., then J^log, y, j;'=;log. y, ii:"=:log. y", 
*"'=log.^"', &g. 

6. For the purpose of adapting the series (I) to the series of na- 
tural numbers I, 2, 3, &c. the given number r must be greater than 
unity, the first index m must be equal to 0, and the several indices 
*', x", s'", &c. must continually increase. For, since by the prin- 
fjples of algebra, x''=V, whatever r may be, this series will increase 
&oni 1 to infinity; and by properly adjusting the values of j-', .t", 
a"', &c. it is evident that the several quantities r*, r*", r*'", &c. 
inay be made to coincide with the numbers 2, 3, 4, &c. For ex- 
ample, let r=;JO; then, since I0°=1, find IO'=I0, the indices of 10, 
yhich would give 10", 1(K', 10"", &c. equal to the numbers 2, 3, 
4, &c., must be fractions between and 1. If we take the number 3 
■jre have 10i:=3.16 nearly, from which we infer that a fraction (j') 
Mimewhat less than \ or 0-5, being made the index of [/) 10, would 
«ivelO''=3. This fraction is found by calculation to be -47712; 
nence 10'""==3; therefore, when r=10, the logarithm of 3 is 
^7712. 

. In like manner, if we assume tlie number 5, whose logarithm is to 
be found in place of that of 3, we have 1 0'^4.64 whence a fraction, 
j;'"'' somewhat greater than J, or .660 being made the index or 
exponent of 10, would give 10^" —5. This fraction more accu- 
rately computed is found to be .69897> t^iat is, when rT=10 the loga- 
rithm of 5 is .69897- 

7. From this it appears, that the value of the logarithm of any 
given number depends upon the value of the number r, and that by 
assuming it equal to different numbers, as many different systems of 
logarithms may be formed as we please. 

In every system, however, since r°=I, the logarithm of 1 must 
be 0. This constant quantity r from the powers of which the na- 
tural numbers are formed, ia culled the radii or hoK of the system 
tu which it belongs. 

8. In the general equation r*=,y, (art. S.), lit iis make 
and observe tlie cnrrcsipondent variation™ of i/. 



nake ,r v«|»^h 



LOGAHITHMIC TABLES. 



If r is greater than 1^ on making sz=0, we have ^=1 ; when xsl 
then tf=:r or the logarithm of the base is=l ; in proportion as « in« 
creases from to infinity^ y will increase from 1 towards r, and after- 
wards to infinity^ so that if we suppose a: to pass through all the in- 
termediate values^ in following tne law of continuity, y will increase 
also in the same manner, though much more rapidly. 

If we put for X, negative values, we shall have 5f=r"', or 

^=— . Here we see, in like manner, that the more x increases the 

more y or -j decreases, so that in proportion as x augments, nega- 

tivelyjr takes all possible values less than 1 as far as^O, in which case 
X becomes infinite. This was the proposition which Napier made to 
Briffgs on their celebrated meeting at Edinburgh, when conversing 
on me propriety of changing the logarithmic scale. 

If r is less than 1 we shall make ^=-r> ^ being greater than 1 and 

we have y'=^fz or ^=6^ , according as jr is positive or negative. We 

fall here upon the same case, with this differeuce> that x is positive 
when 5^ is less than 1, and negative when y is greater than 1. This 
proposal Briggs made to Napier, but immediatly abandoned it ob 
N»pier suggesting ^at mentioned above, which was finally lido^ited. 

l£ r=l, we have ^=1 whatever x may be. 

We may then say generally, that provided r is not unity, th^re 
din always be found a value for or, which renders f equal to any 

given number y. The constant use that is made of the properties of 
ie equation yzizr* requires the denominations of its parts to be fix- 
ed in order to avoid circumlocution. Hence as before remarked, x 
is called the logarithm of the number v, the invariable number r in 
called the base and, finally, the logaritnm of a number, the power to 
i^hich the base must be raised in order to produce that numoer. 

With regard to the base r it is arbitrary, and when we write 
a:=log. y to show that x is the logarithm of the number y or that 
yzzir'y tne base r is alway imderstood, because when once chosen it 
IS supposed to remain fix^. If it should be changed the new base 
ought to be indicated. 

9. From these prihcijples are derived several properties. 

1°. In every system of logarithms, the logarithm of 1 is and that 
of the base r is I. 

S^. If the base r is greater than 1, the logarithms of numbers 
^eater than 1 are jpositive, the others are negative. "Hie Cohtrarjr 
takes place if r is less than 1. 

3°. The composition of a table of logarithihs consists in determin- 
ing all the values of x whetl y is made successively equal to 1, 2„ 3^ 
&c. in the equation y =7^. 

If we suppose re=^ on making 

a;=0, ^i 2j, 3g, &c. , .. . n^ 
We find .y=l, /», ^% ^ , &c. . . . ^** 

The logarithms therefore increase in progression by differences, 
while the numbers Increase in progression by the product or quo- 
tient, according as /tt is an integer or a fraction. 

The ratios Hre tibie arbitrary numbers ^ and /». We may, therefore, 
regard the systems of values of x and y which satisfy the equation 



INTRODUCTION. 

. ys^r", as classed in these two propresaionB, which coincides with 
I what has been already said in art. (2.) 

10. We shall now demonaCrate algebraically the various properties 
of logarithms. 

Let N and n be any two numbers belonging to the series (I) ; and 
for example, let N=r' and n=r^, then N ii-=r' xr^=:r'*' , but, by 
art. 5, the logarithm of r^^' is x-^x'—\og. 7-"+ log. r''=log. N+log. 

N. 

In like manner, if n, n', n" be any set of numbers in the series (I) 
it might be shown that the logarithm of n x n' X n", &c.^log. 
B-|-log. n' + log. n", &c., from which we infer that the logarithm of 
the product of any number of factors is equal to the sum of their 
logarithms. 

N r' 

11. Again — =— ^; but the logarithm of r'~*=ix — *'; therefore, 

N 
the logarithm of — ^x — j''=.log. r* — log. r^^log. N — log, a; hence 

I it appears, that the logarithm of the quotient of any two numbers is 
equal to the difference of their logarithms; and that the logarithm 

of a fraction I — 1 is equal to tlie logarithm of its numerator minus, 

k the logarithm of its denominator. ' 

If N be less than «, tben log. N — log. n is negative; therefore, 
' die logarithms of all proper fractions are negative. 

12. Let N=r' be raised to the wi'* power, then N"'=r"'; but the 
I logarithm of /'" is=mj;, hence the logarithm of N''=:m:c=rB log. r* 

I srmlog. N; for the same reason, since^N=Nm=r™, the logarithm 



f^N: 



._j_ 1og.K 



I mf' power of ai 
n, and that of 

by... 



; from which we infer, that the logarithm of the 

' number is found by multiplying its logarithm by 
le m"' root of any number, by dividing its logarithm 

Section II. 
Of Ike Conslruclion of Tables 0/ Logarithms. 
13. Let r" express generally any term of the aeries, (1), and let N 
I be the corresponding number, then r'^N. Hence to find the !oga- 
l zithm of N iamerely to solve the equation r'^N where x is the un- 
1 known quantity. In order to accomplish this purpose let r=il + b 
[ and N=l + n, then estruct the t/'* root of each side of this equatira, 

I and we obtain (l + 6)-=(l + n)--, which by expansion gives 



>+^<»H}(i-.)(s)+i(i->)(H(i;)+- 

Now suppose J to be indefinitely great with respect to x and I, 
h in reference to —1, — 2, &c., so that 



1 



I then will — and — vs 

—1 and —1 will each become equal to 




LOGABITHMIC TABL£S. 6 

equal to -^2, &c. &c.> hence rejecting 1 from each side of the eqiut* 
tion we have 

- (ft-i b^ + i 6'— i b'+Scc.)=l (n— i «« + J n»— i n*+&c.) 

y y 

hence x, the log. (l + «)=j_^ j.^j J'-TiH&'c. 

but ii=:N — I and 6=r — 1^ therefore^ by substitution^ the above ex- 
pression becomes 

t (N— 1) — i (N-l)« + ,' (N— l)'-i (N-l)« + &c. 

( r-1) -i (.r-l)« +i ( r-l)'-i ( r-l/+&c. 
^^ ^ (r-l)_i (r-i (r-l)* + ;{ (r-l)'-i (r-ly+*«-= 

This quantity M^ which evidently depends upon the base r, is 
called the modulus of the particular system of logarithms to which it 
belongs. As it is obvious the series n — Jn*+J »' — J »*+} «*— 
&c will not converge when n is any whole number greater than uni- 
ty^ before proceeding to the calculation of the logarithms of any 
particular system^ it will be proper to show the manner in which the 
value of X in the last article may be expressed in a converging series. 
This may be effected by means of the following process in which 
M is substituted for the quantity 

(r-1) -i {r-iy + \ ir-\Y^ (r_iy + &c. ; ^"*' 

Log. (1 + n) =M (w— i n« + 5 w'-4 »*+ y— n*— &c.) . . (3)» 

In the above for n put — n, and then 

Log. (1— n) =M (—•»— i r*— J n»— J n*— ^ n*— &c.) . . (4) 

Subtract (4) from (3), then log. (1 + n) — log. (1— ii) = log. 

i±^=2M(ii + Jii« + iii* + ^ii» + &c.) ..... (6) 

Let N= , i then 1%-=,-^^- — , hence 
1 — n N + » 

Log. N=2 m{ (§=-}) +i (§^5)' + J (^;=-;)' + &c.} . (6) 

Again let ii=^-jj---j, then jH^— 2N^' hence by substitution in 
formula (K\ 

T^ 1 1 1 ^ / 

Log. N-log. (N_l) =2 M (^-^ + g^^ _ +5 (2^1). 

+&c); .ndlog.N=2M(jjLj+3-^^-lj^, + _^^ +&e.) 
+ log. (N— 1) (7) 

T ...1 -e l + » N+1 , 1 , 

^ *^(2NqFl+r(OqT)»+5(N+i) »+**=•)+'««• N. • (8) 



• By m^s of this fonnuls the logarithm of a quantity exceeding unity by a very 
•nuul fraction may be readily found. ^ j ^ j 



I 

1 
I 

I 



6 IiNTltODUCTIOK. 

■ Since the \og. of 1;=0, this last series which converges yery 
rapidly, will give the lo/^arithms of all the natural numbers, wiUi 
facility in succesaion. To these thearms might have been added 
others still more convenient, but they are sufficient for ordinary 

cases. 

15. Before proceeding to compute a table of logarithms, some 
value must be assigned to M' Since the value of r is arbilrary, let 

:^ or M shall 

s (R) we hive 



-1)'- 



it be so assumed that . — i\~~i"7 T\i — i—r~ 

be equal to 1, that adopted by Napier. Taking 
Log. 1 = (art. 6.) 

2 = 2^g + 3.+jg; + &c. to 8 terms) . =0.6931472 

3 = 2(g^+3p+^B + &c.)+log. 2 . . =1.0086123 
4=21og. 2(art. 12) . . =1.3862944 
^ = ^(s''"3^'+5i"''^'') + '°S^ =1.6094371) 
6 = Iog. 2+log. 3(art. 10) . . =1.7917595 

8 = 3 log. 2 (art. 12) . . . = 2.0794415 

9 = 2 log. 3 (art. 12) . ... =2.1972246 
10 = log.2+by 3(art. 10) ... =2.3025851 

3tc. 

In this manner the Napierean logarithms of oil the natural num- 
bers may be found. As their accuracy, however, depends upon 
those immediately preceding, being derived Huccessively from each 
other, it would be necessary to check the computations in the actual 
cotiatruction of a table of logarithms by some independent formula, 
such as (6), though this in large numbers would be rather inconve- 
nient from its slow convergency. 

16. To find the value of r, the base, in this system recourse must 
be had to the series (3) art, (14). If log. (1— n) or log. N be put 
=/andM=l, we have /=« — ^ti+jH^— |fl'-(-, Bcc. ; reverting this 

geries, nndl + B, ox N=l+l+i'P+^^P+i^jf^ l^ , &c. Now let 



number whose logarithm is I, that is, the base 



I fc=], then thi 

I'fcl + l + i+^+g^^-J-, &c. =2.7182818. To prevent confii- 

m, however, we shall alwavs designate the base or radix of this 
Jnritem by R, retaining r for tnat of the common logarithms. Hence 
R=2.718,28I,82846. 

[ These are also called hyperbolic logarithms from their application 
p^ the quadrature of the hyperbola; but this designation is improper, 
I «B any system may be similarly employed. 

I — 17. When we have the logarithm of a number N for any particular 
1 value of r, the base, we can readily obtain the log.irithm of tlie same 
I tnimber in every other Htstem. Since, art (5), wbeti the bask is t 
\ we have r'=N, we shall likewise have K*=N when the bast is R, 
Kin which j; is different from X, therefore, R''= r'. 



LOOABITHMIC TABLES. 7 

Now taking the logarithms reladvely to the system whose base is r, 
then 

but Lr»:=zx by h3rpothe8i8, and Z.R^ =X IJR, art (12), whence X LR:=zx, 

or X=-Q. But if R is the base, X will be the logarithm of N in the 
L R 

system having that base, and designating this by L.N to distinguish 

it §*om the other, we shall have L.N=j^ (12) 

consequently we obtain the logarithm of N in the second system, by 
dividing its logarithm taken in the first system by the logarithm of 
the base of the second system. Again from formula (12) we get 

L.Nx^.R=/,N . . (13) 

Hence in every system the logarithm of any number is the product 
of its Napierean logarithm by tne logarithm of R, called the modulus. 

Also since =A-^=/. R, there exists between L N and L.N a constant 

jLi.N 

ratio represented by LR 

/.N 
Since we have by formula (12) L.N=^, as N=10, then art (15) 

8.3025851 = ^, or M = 2302535! = ^-^^^^^^^^^ and 2M = 

0.8685889638 . (14) 

18. It is now easy to construct a table of common Warithms 
whose base r=10, for by formula (13) we have /.N=s^R x£.N, but 
I. R=:M=: 0.4342944849; consequently /.N=:0.4342974819xL.N. 
It therefore only is necessary to substitute this value for M in any 
of the series formerly give for the computation of the Napierean lo- 
garithms to obtain the common ; thus, if in series (8) for 2 M we 
substitute its value 0.86858896 we shall have 

log. CS+l)=0.m58896(^\ri-^^^^^+^^^+ &c.) 

+ log. N, and making N successively 1, 2, 3, &c. 

Log.l= ... 0.0000000 

2=:-86858896(|+^+gij+,&c.) =0^10000 

3=-86858896(^+ ^+ ^+, &c.) + log. 2 =0.4771213 

4 = 2 log. 2. .... =0.6020600 

11 1 

5=-86858896(^+g^+g^ + ,&c.)+log.4 =0.6989700 
6=log.2+log.3 =0.7781513 

7 =-86858896(^ + 3~gp + 6(l3p+' ^^- )+^^«- ^' =0-8450980 

8 = 3 log. 2 .... =0.9030900 

9 = 2 log. 3 . =0.9542425 
10= .... 1.0000000 

19. After Lord Napier had computed his first tables of logarithms 
it occurred to him that it would be proper to change the radix 
R=2.718^18tor=10,atthe same time making the logarithms of in- 
tegers positive, and those of fractions negative, (art 8.), as more con- 
formable to the denary scale notation, and more convenient in practice. 
It appears that Mr Henry Briggs had also conceived the idea ot* 



INTRODUCTION. 

cha.nging the radix, and had computed logaritlims on a plan .some- 
,' what leas commodioua, by making the logarithms of integers nega- 
tive, and those of fractions positive, which, upon a personal com- 
munication with Lord Napier, he rejected, and finally adopted Ids 
lordship's views. He soon afterwards published the first thousand 
Warithms of this kind under the title of Losnrhhmonim Ckil'ms 
Frima. 

Skction in. 
Of the Trlgonotaetrical Lines, called Sints, Tangents, S(C. J 

j 90. The Egyptians and Chaldeans began to study astronomy at ■ 
'v early period. As the determination of the relations and distances 
the heavenly bodies involve the mensuration of lines and angles, 
■ It was necessary to invent some method of ascertaining the value of 
these quantities, at least in an approximate manner, before any use- 
ful results could be obtained. Some of the more elementary pro- 
positions in geometry must have been discovered in the most remote 
antiquity, and the inventive genius of the Greeks filled up the gene- 
ral outline. The properties of geometrical figures thus acquired, 
would, without doubt, be applied to the mensuration of several mag. 
lutudes, and the distances of various points in space- About six 
hundred years before the Christian era, 'I'hales measured the heights 
of the pyramids in Egypt by means of their shadows^ a method 
which depends upon the proportionality of the sides of similar tri- 
angles. This simple property forms the basis of modern trigonome- 
try. If, for example, a pole or gnomon be set perpendicular to the 
horizontal plane, it will, in a clear day, when the sun is not vertical, 
cast a shadow to a given distance, while any other high object, such 
as a steeple near, it will do the same. If straight lines be conceived 
to be drawn from the top of these objects to the extremity of each 
of their shadows, it is evident that, unless they are very distant, by 
this means triangles nearly similar will be formed, whose sides are 
proportional ; that is, as the shadow of the gnomon is to its height 
so is the shadow of the object to its height. Now, suppose the length 
of the shadow of the gnomon to be made the radius with which an 
arc of a circle is described commencing at the bottom of the gnomon, 
and, as will be afterwards explained, measuring the angle between the 
horizontal line and the line from the extremity of the shadow to the top 
of the gnomon, that gnomon will, by the principles of geometry be a 
tangent to the circle. Whence the former proportion becomes as the 
radius is to the tangent of the angle of elevation, so is the length of 
the shadow of the object to its height. It would thus require the 
length of the shadow of the pole or gnomon to be measured each 
time any height was determined. This, however, might be avoided 
by having the measure of a set of triangles whose sideD, to an assum- 
ed radius, and a corresponding series of angles, are previously deter- 
mined by computation. By this means, in such cases, it is only ne- 
cessary to measure the angle of elevation of the object, at a given 
point, and its distance from it, and comparing it with one of those 
computed triangles equiangular to it, to detennine, in a manner similar 
to the former, the height of the object. It is obvious that the same 
principles may be applied to objects situated in any plane, whether 



vertical, horizontjd, or oblique. 

.Several series of triangles of the kind n 



y mentioned have b 



TRIGONOMETRICAL TABLI<:8. 9 

actually computed and anangad in tablea under the designation of 
trigonfllnietrical tables. 

These werd not accomplished at <moe^ but were the improvements 
of successive ages. Hipparchus^ about 150 years before the Chjristian 
era, supposed similar triangles to be inscribed in circles, and employ- 
ed in nis computation the chords subtending the arcs measuring 
them in sexagesimal parts of the radius, linearly 300 years after- 
wards, Ptolomy, in his H§ym^ 2v»r«(ic, recomputed the chords, 
but in his Analemma employs the Jkalf chords instead of the chords 
approaching very nearly to the use of sines, afterwards introduced 
by the Arabians. 

Some notions of the tangents, secants, and versed sines, were, 
towards the beginning of the tentii century, entertained by the more 
learned Arabians. About the beginning of the fifteenth century the 
sciences began to be cultivated in Europe, where the greatest pro- 
gross has been made. At that period MuUer invented Uie tangents, 
and shortly after Maurolycus produced his table of secants. These 
were all in natural numoers to a ^ven radius now generally taken 
at nnity, and, therefore, their appbcation waa in manv cases trouble- 
some. To remove this inconvenience as far as possible, Napier in- 
vented his logarithms, which have brought them perhaps to the last 
degree of perfection. 

Hipjparchus, who has been, followed by most of the modems, em- 
ploye the circle to measure angles. He supposed the whole prcum- 
rerence to be divided into 360 equal parts eadi called a degree. The 
degree was divided into 60 equal parts called minutes, and the mi- 
nute into 60 equal parts called seconds, and the sexagesimal division 
was continued, though now the fractions of seconds are more com- 
monly expressed in decimals, which are more convenient for calcula- 
tion.* 

Whence the semicircle contains 180 degrees and the quadrant 90. 
As Jour right angles can be constituted about a point, 90 degrees 
must be the measure of a right angle. For the purposes of abbrevia- 
tion a degree is marked with a small circle, a minute with one accent, 
a second with two accents, &c. Thus ^T IT 44''.806, denotes 57 
degrees, 17 minutes, 44 seconds, and .806 the decimal, whose value 
is 806 thousandths of a second. This, being an arc whose length is 
equal to the radius as will be afterwards explained, is also expressed 
in degrees and decimal parts of a degree, thus 57°*2957795, a mode of 
using it, which in some cases has its advantages. 

The number of these parts, in either case, contained in the arc 
between the lines constituting the angle, of which arc the angular 
point is the centre, indicates the measure of that angle accordingly. 

Hence, if to any number expressed in sexagesimal degrees onC' 
ninth of itself be added, the sum will be the same number expressed 
in the centesimal degrees ; and if from any number expressed in 
centesimal degrees one- tenth of itself be subtracted, the remainders 
will be the same number expressed in sexagesimal degrees. 



* The French have lately adopted the centesimal division, which, in many cases, is 
preferable to the sexagesimal. The whole circle is divided into 400 degrees, each de- 
gree into ISO minutes, and the centesimal division is continued. Hence the semicircle 
contains 200 desrees, the quadrant 100, and the ratio of the centesimal to the sexage- 
simal is as 9 to 10. 

To convert lexagerimal degrees into centesimal add ( of the arc to itself. 

The converse is effected by subtracting i^g of tlic arc from iuclf. 




10 JKTBODUCTION. 

DEFmiTlONE. 

21. If two straight liaea intersect one another in the centre of a dr- 
f3e, th» arc of Uie circumference intercepted between theis is called 
the meamire of tiie contained angle, whatever be ^ 

the radius of the circlej since the arcs are pro- 
portional to then- radii. Thus, the arc AB or A'B', 
18 the measure of the angle ACB, and is expreased 
in degrees, &c- 

23. The comptement of an arc is its diflerence ft'om 
a qnadrant, its aupplemenl, its diiFerence from a 
semicircle, and its exptemeni, its defect from the 
whole circumference. Thus if AB be any arc, then BD is the com- 
plement, BE the supplement, and BDEFA the explement. 

The same thing holds with regard to the angles of which the (ires 
are the measures, that is, if ACB be any angle, BCD its difference 
from a right angle is called the complement, BCE the supplement 
to two right angles, and BCA, measured by the arc BDEFA, the 

■ eXplement or difference from four right angles. 

■ ■ 33. The tine of an arc, or of an angle of which the arc is the mea- 
- sure, is a perpendicular let fall from one of its extremities npon a ra- 
' dins or diameter passing through the other. 

24. The nersed sine or versine of an arc is that part of the diameter 
■ ■ intercepted between its sine and the circumference. 

S5. The tangent of an arc is a perpendicular to the extremity of the 
" radius at one end of the arc, and limited by a straight line drawn 
^m the centre passing through the other. 

26. The fccon( of an arc is fte straight line drawn from the cen- 
tre to the extremity of the tangent. 

27. It is usual to express the fine, tangent, and secant of the com- 
plement of an arc by the abbreviated terms cosine, cotangent, and 
cosecant. 

28. Let ACDE be a circle of which the diameters 
AD and CE are at right angles to one another. 

Take any arc AB, produce the radius OB, and 
draw BG, AK perpendicular to AO or AD, and 
HB, CI perpendicular to CE ; then BG is the D^^ 
tine, BH or GO the cosine, AG the lersine, OH the 
coversine, DG the suversine, and HE the xiicovrr- 
Wneof the arc AB. Also of that arc AK is the (nn- i^ - 

gent, CI the cotangent, OK the secant, and OI the coteranl. 

29. Since the diameter which bisects an arc, also bisects the chord 
of that arc at right angles, therefore, the sine of an arc is equal to half 
the chord of twice the arc. Thus BG=J BF=half the chord of the 
arc BAF, the double of the arc AB. 

30. In the right-angled triangle OGB, BG^-t-OG'^OBS that is, 
the squares of the sine and cosine are together equal to the square 
of the radius. 

- 31. The triangie GOB being similar toOAK, OG:GB:: OA:AK, 
or the cosine of an arc is to the sine as radius is to the tangent. 
_ .32. Also the triangles 0GB, OAK being similar, as before, 
"OO : OB : : OA : OK, the radius is a mean proportional between the 
.cosine and the secant. 

33. Since DG : GB : : GB : GA, it follows that the sine is s mean 
proportional between the versine and suvcreine. 

34. Again, AD : AB : : AB : AG, or the chord of an arc is a mean 
propa^on&l between the diaipetcr and versine. 




TRIoqSQHSIAiqM' TABLE& 



11 

CVm-.— Since AB«=AD. AG, thai, becaate AD u constuit, AB* 
win« AO, W (4 AB)* ct AO; thiifc k, the Moare of the nne Tarici 
£r«f«I».)n the TnaoQ, or inmvdjrw the cottne, c»f twic« the vc. 
:39- nwtnao^QAiruulIc6iraihn9at,tiiM«ftre'AIC:AO:: 
OCiQli eonwqdenaj the nuHna ii a mMtt preportisnal betwaan 
the tangent and cotangent of an arc. 

36. m the application of algebra to geometry, where the trigQii»< 
metrical lines are employed, it is neceasair to trace their cbangea in 
the several quadrants of the circle, since it is obvious that the tame 
lines treated of ftb«ve, may be applied to each. ^. 
In die first quadrant AC, if the sine BO and cosine 1^ 
GO be supposed potiHve, then ^e sine B'G' on tha 
hame side of ihe diameter AA', attd in the same di- 
rection, still remains potilive ; but the cosine OO' . 
having changed its position with respect to the -'^f] 
centre O, or diameter CC, becomes hwoImw. 
Zb tiie. third qnadrant, the cosine AO^ aod tine 
&B", having bolfc clunged their poaidoDt, are 
both negaiivt. In the fbinth quadrant, the oaaiiM 1 
"^ vini^vesnmedita 6ri^»aljMmHm, OO Is now f o m fwe , white the 
te dV", remabring aa in the third qnadrant, is » —• . 




BNita and secanta depcni&lg njpoit uie 
ugBa determined aetkirdiTigly. . . 

Fnmi article 30, to 86 ind iiu:ltMive,BbdBgraAus, Ac* veoi 

l.dn. =(E"— COS.")* 7t«. =— 



9^ COS. 



= (R«— *in.«)i 



a tan. =:(sec.«-^R«^ 

4. COL =(co8ec«— R«)* 

5. sec =; (Ro+tan.")* 
ft cosec. s=(B«+oot.«)* 



11. versine =^ 



I.I 



R+coa. 
13. cQverf. =^^ 
If radius be supposed unity, then 
= (1— cos.")* 7- tan. = — " 

COS. 



1ft cotec iz J_ 



a.m. 


= (l-»n..)! 


3. On. 


= («c.<-l)t 


4. cot 


=(«»«.._l)i 


Caec. 


=y+uii.") 


ft oooee. 


= (l+cot.V 



n 



JNTROBOCTION, 



' 37- Now, since (7) tan. ^ ~, ihenitfoUowsfromtheprinciples 
.cf elgebra, that when the signs of the sine and cosine are like, the 
i»ign of the tangent h posilive, and when anlite, the sign of the tan- 
gent is negative. In like manner, the signs of the cotangent, secant, 
. u)d cosecant may be determined from formulas (8), (9), and 10). 



Table of the Sign. 



I utfuidrants. . Sine. Cdru 

Mr 5 9 1+ + 

a 6 10 J + — 

' 3 711 i- - 

f 8 12j&c. I— + 



of Tiigfiometrical Lines. 
Tangent. Count^l. Secant. 



T! 



Of Ike Maltiples and Pofvers of Arcs. 
38. In moat treatisea on geometry, such as Leslie's, Legendre's, &c. 
•the elementary propositions containing the principles of trigonometry 
are also given. It is therefore unnecessary to repeat them here, as it 
* only puts the student to the expense of purchasing the same things in 
"(wo or three different works. We shall only give a few of the re- 
' Aults most generally useful, referring to those works on geometry 
\ and trigonometry where the requisite information may be obtained." 



Ifaandia 
;tinity, then 



; two given arcs 



of a circle of which the radiui 
i. 6-l-si 



i.(«+6)= 
,f COS. Ca + 6)=cos. a Cos. 6— sin. a sm. b 

sin. (n— S)=sin. a cos. 6— sin. b cos. a 
cos. Co-i)=cos. a COS. a + sin. b sin. a 
If we divide these equationn, the one by the other ii 
Y<U, 0) by (2), and (3) by (4), then 

sin. a cos li-\-a\- '■ —■ " 
tan. («+&)= : "-'"^°'-"+'' 



.. C<,-6)= 



S.6— 8 



1.6 



'Dividing the two terms of the second numbers by cos. < 
■ubstituting tan. a and tan. b for their values in terms 






tan. (a + b)- 
tan. (a— 6)= 



in.fi 



~1 — tan. a tan. 4 
1 — tan. b 

expressions which give the tangent of the s 
s of the tangents o" " 



(1) 



(5) 
(6) 



(7) 



(8) 
f the difTerence 



If we make a^b in the preceding formula, they giv 



(9) 
(10) 

(11) 



inching litt pnclicc {tTiIif 



TaiOONOM£TRICAL TABLES. 13 

expressions which give the sine, cosine, and tangent of twice the 
arc in terms of the sine^ oonney and tangent of the simple arc. 

39. Returning to equations (1), (2), &c. we have by addition and 
subtraction 

sin. (a+M+sin. (a-^) = 2sin. a COS. A • (13] 



cos. (a+6+cos. (o— 6) = 2 COS. a cos. b . (13j 



sin. (a+^V-^^ (^ — ^) = ^ 8^* ^ c^* A • (14] 

COS. (a— 6)=:cos. (a +6} =2 sin. a sin. b . (15^ 

Let (a+6)=:ti, and (o---^) =v> then bv addition and subtraction 
a=:^ {u+v), 6=^ («—«)> consequently the precedhig formuli^ 
become 



sin. 11+ sin. i; =: 2 sin. X (ii-)-^) <^^ ^ (*^-~^) • (1 

sin. u — sin. « =: 2 cos. i (u — v) cos. i {u+v) (1 

COS. 11 + cos. V = 2 cos. ^ (« + v) COS. i (u — v) • ?ll 

COS. t; — COS. II = 2 sin. j (fi + v) sin. ^ (u — v) . (19^ 

expressions which serve to transform the sum or the difference of 
the sine or cosine into the product, and thus to unite the two terms 
into one. 

If we divide formula (16) by formula (17) they give 
sin tf+sin. v tan, j (u+v ) 

sin. u — sin. v tan. ^ {u — v) ' ' / 

If we multiply these equations member by member, observing to 

substitute sin. 2 a=2 sin. a cos. a, formula (9), then 

sin.'u — sin.*!; = sin. (w+») cos. (v+f) • (91) 

COS.* V — C08.*« = sin. («-f v) cos. (u+v) . (31) 

Since sin. 2 a=.2 sin. a cos. a, and cos. 2 a=cos.* a — sin.* a. 

The second of these equations may be put under the two following 

forms: 

cos. 2 0=1 — 2 sin.* a, and cos. 2 a=:2 cos.* a — 1 

. 1— cos. 2a , ^ 1 + cos. 2a 
whence sm. * a = s f ^^o. cos. * a = g . (39) 

These expressions are used when, for the squares of the sine and co- 
sine, the first power of the cosine of the double arc is substituted. 
40. Let 2a=:u, then a=:^u formula (22), these formulae become 

1 — cos. u 1 + cos. u 
sm.«iii= 2 ' co8.*4tt= 2 — • (23) 

and dividing each corresponding number successively, they giyv 

1— cos. u 
tan.* ii.=jqp^^^ . . • (M) 

1— tan.* I II 
"'^ ^^«' " =l + tan.*iii • . . (25) 

If b in formulae (1), (2) be made 2 a, 3 a, &c. we may obtain mul* 
tiple arcs thus : 

sin. 3a=sin. a cos. 2a+sin. 2 a cos. a 
COS. 3 a=cos. a cos. 2 a — sin. a sin. 2 a 
Substituting for sin. 2 a and cos. 2 a, their values, they become 

sin. 3 a=3 sin. a cos.*a — sin.^ a , 

cos. 3 a= — 3 COS. a sin. a + cos.^ a 
These may be put under the form • 

sm. 3 a=cos.* a(3 tan. a — ^tan.^ a) 
COS. 3 a=:cos.' a(l — 3 tan.* a) 




.»« 



, ^NTnouucnoN. 



In genertil « being any integer, 

Bin Ka = cos." n -j n tan. a ^ ^ g tan.? a+ 

..(»_l). (»-ai.(i.- 3).(n-4) 

^7 1.2.3.4.5 

>' (, n(n-l) 

«""'''"="""n'^TT2"'"'°+ 1.2.374 

The coefficients of the different terms are those of the n" power of 



a...&c.| 



I 



-the binomial, whenci 
ing form : 






3 may be collected under the foUow- 



■teii.*a,icc. 






{cns.aV~lsin.4-^^{-.«-^-lsin..}", 



:«os. nfl = 4{c<,8.o+^-l gin. a }"+ 4 {. 
I 'These formulfe, by development, will giv 
>rtnii are thus easily verified! 

41. It may be shown* that if a: represent any 



,s. a— V-1 sin- fl } (31) 
the two foregoing series. 



»s.«=l — jj + 1:234 — 1.2.3.4.5.6 +'^'^ 
In these expresaiona the arc x is supposed to be divided by the 
CrimdiuB, which is here taken for the unit of length, and consequently 

if we wish to restore it we must write — in place of x and 

' .instead of sin. x in the two members of these equations. 

These formulse might be carried much farther than can be !ntro~ 
1 'fluced into this place. Most of them may be seen by consulting the 
J fcookB already referred to, but above all the anali/sis ivjinilorum of 
iJiuler. 

Tables of Multiples and Powers (^ Arcs. 




' Woodhouu') Trif[ntioinct>T. ihird cdidon, pi^e 34S Gregnr;, pogr IS und E0. ] 



TBIOONOMKnUCAL TABLES. U 

5. 6. 

sin^ «=8m. « COS. a=oof.a 

2 niL'«s:l — ooft.2a 2 cot.' a =1+ cot. 3 a 

4 ain.' a=3sm.a — 8in.3 a 4co8.'it=3co9. a+co8.3a 

8 dnA a =3 — 4 cos. 2 a4-co«.4a 8 cos.^ a =3+4 cot.Sa+co6.4a>ftc. 

42^ Having given a short abstract of the more useful formulsi re- 
lative to multiples and powers of arcs^ we shall now proceed toshetr 
the method of constructing the tables of sines^ tangents^ &c. 

When the radius of a circle is unity^ the semicircumference is 
a 1415926536 nearly. Now there are 180° or 10800" in a semicirde^ 
consequently^ if the former be divided by the latter^ the result will 
be 0.0008908882, the measure of an arc of one minute, which, as 
the arc is so small, may be considered its sine. 

Now, art. 35. 2, cos = (1 — sin.^)i consequently cos. 1' ^ 
0.9999999577. If these values are substituted in formuks, (32^, and 
(^), Bit 41 the sines and cosines may be obtained through the 
whole quadrant. 

Thus let the axe a=l', and, therefore, sin. jr=0.0002908882. Let 

fl=5^ then i21?:1^^5^=0.08726646 the length of « or *, and 

x=+ 0.06726646 



^ T2S = ~ 0.00011076 
+ h£i:5 = +0.00000004 



therefore, x — T23'^ l 2 34.5 * &c. =0.0871 5574= the natural sine 

of 5"*, the logarithm of which is &740206, the log. sine the same are. 
This method is easy when the arc is small, as the series then converges 
very rapidly, but it is rather laborious when the arc is large, in 
which case recourse must be had to other methods depending upon 
the properties of multiple arcs,. as may be seen in most of our trea* 
tises on trigonometry. 

As the sines are computed, the cosines of the same arcs may be 
found firom art 41, formula (33), oc from art. 35, formula (2), tibe tan- 
gents and cotangents, from formula (7) and (8), and the secants and 
cosecants from {9) and 10). 

Section IV. 

Of the application of Tahhs of Sines, Tangents, Secants, ^c. to 
plane Trigonometry. 

CASE I. 

43. In any plane triangle it is shewn in our usual treatises, that 
the sides are proportional to the sines of their opposite angles, or 

The sine of any one angle. 
Is to the sine of another angle i 
As the side opposite to the firsts 
Is to the side opposite to the second. 
These terms may be taken alternately, inversely, &c. 

44. Whe^ one of -the angles is a . right angle, then the pre- 
ceding rule may either be applied, or a modification of it derived 
from the properties which are peculiar to right-angled triangles. 



M INTKODUCTION. 

In right-angled triangleaj it is usual to call that aide subtending 
the right angle the hypoitmst, and the other sides which contain the 
right angle the legs, or the one the base and the other the ptrpenHi- 



Then if 
the radius, the 
98, as follows:- 



of the aides of any triangle ABC, be assumed equal to 
. of the other aides must be determined by art. 





Kadius 




Tangent C 



B of tlie sides being thus known when three of the parta 
a triangle including a side are given, the rest may be found by 
■ the following rules: — 

I, — Tojind a side. 
As the name of the given side. 
Is to the name of the required side ; 
So is the given aide. 
To the required side. 

ll.—ToJiitdan angle. 
As the side made radius. 
Is to the other given xide. 
So is radius. 
To the name of this side. 
f-a Any side may be made radius to find a side, but one of the given 
odes must be made radius to find an angle. 

In the solution of plane triangles, it must be recolleeted that all the 
angles in any triangle are together equal to two right angles, or 
180°. Whence if two of the angles are given, the other ma^ be 
found by subtracting their sum from 180° ; when one angle is given 
the sum of the other two may be found by subtracting it from 180° ; 
and if one be right or 90°, the eum of the other two is also 90°, 
the one is the complement of the other. 

CASK II. 

45. In a plane triangle when the tw. 
are given. 

I. As the sum of the given sides. 
Is to their difference ; 

So is the tangent of half the sum of the oppoahe angli 
To the tangent of half their difference. 
Half the difference added to half the sum of those angles gives the 
greater, and subtracted from half the sum gives the less. 

All the angles being now known, the third side may be found by 
the rules in case T. 

Or, after having found half the sum and half the difference of the 
angles, the remaming side may be found without determining the 
actual angles, as proposed by Thnc.kcrin 1743, and recommended by 

Profeasor Wallace, in the Ed" ' ' " ' ■" 

the following 






mtained angl«^^^| 
.e angle, ^ 



Edinburgh Philosophical Transactions, itt'J 



PLANE TMOONOMETRY. IT 

II. Aa the sine of half the difference cyf the opposite an^let» 
Is to the sine of half their sum^ 
So is the difference of the containing sides ; 
To the remaining side; or, 
III. As the cosine of half the difference of the opposite angles. 
Is to the cosine of half their sum ; 
So is the sum of the containing sides 
To the remaining side. 
These two methods may be used as a verification to each other> 
and will be found somewnat more easy in practice than the first 
method, as several of the quantities may be taken out from the tri- 
gonometrical tables at the same time. 

Should the sides come out in logarithms from some previous openu 
tion, then Giuiss' table for finding the logarithm of the sum ana dif- 
ference of numbers from their logarithms, without first determining 
the natural numbers themselves, would be some advantage, thoum 
it was not thought sufficient to warrant an insertion of it among the 
tables. 

The following method of resolving this problem is oonvenient, par* 
ticularly when the logarithms of the sides are given. 

IV. From the logarithm of the greater of the two given sides, having 
its index increased by 10, subtract the logarithm of the less side, the 
remainder will be the logarithm tangent of an arc, from which, 45^ 
being subtracted, there will be obtained a remainder. To the lo- 
garithm tangent of this remainder add the log. tangent of half the sum 
of the (^posite angles, the sum, rejecting 10 in the index, will be the 
log. tangentof half their difference, from which the angles thonselves 
may be found. 

GA8B III. 

43, In any plane triangle, when the three sides are given, 
I. As the base 

Is to the sum of the sides ; 

So is the difference of the sides 

To the difference of the segments of the base made by a perpendi- 
cular upon it, or upon it produced from the (^posite anffle. 

It may perhaps be convenient to call the longest side Uke base« 
in order that the perpendicular may fall within the triangle. 

When the three sides of a triangle are given, the difference of the 
segments of the base may thus be found. Then half the difference 
added to half the sum, that is, to half the base, will give the greater 
segment adjacent to the greater side ; and half the difference taken 
from half the sum will give the less. From these the angles may 
be found by Rule II. § (44). 

II. In a plane triangle, as the rectangle under any two sides, is to 
the rectangle under the excesses of the semiperimeter above those 
sides ; so is the square of the radius to the square of the sine of half 
their contained angle, as shown in Leslie's Geometry. In practice, 
this rule, when logarithms are employed, may be stated as follows : 

To the arithmetical complements of the logarithms of. the two 
sides containing the required angle, add the logarithms of the dif- 
ferences between those sides and half the sum of the three side^ 
then half the sum of these four logarithms will be the log. sini cnT 
half the required angle. ,^ ^ 

III. To the arithmetical complements of the sides containing the 
required angle, add the logarithm of half the sum of the t\vt%«i %\^<^%^ 



18 



INTEODUCTION. 



and the logarithm of the difference between this half sirni and the 
side opposite the required angle; half the sum of these four loga- 
rithms will be the log. cosine of half the required angle. 

IV. To the arithmetical complement of the logarithm of half the 
sum of the three sides, add the arithmetical complement of the dif- 
ference between half the sum of the three sides and the side oppo- 
»ite the required angle, and the logarithms of the differences between 
tiiat half sum and the sides containing the required angle ; half the 
nun of those four logarithms will be the log. tangent of half the re- 
qnired angle. 

It may be remarked that these three last rules will, in general, be 
die most commodious in practice, though, in particular cases, each 
may have its peculiar advantage when great accuracy is required. 

When the required angle does not exceed 90°, Rule II. may be 

\ wed, when it does. Rule 111. may be employed; and in either case 

Bule IV. will give correct solutions. These observations depend 

upon the variation of the trigonometrical lines in certain parts <rfthe 

■*■- "1- as, for example, near 90°, the sines vary very slowly, so that 



tffrcle. 



tile true value of an arc cannot be obtained by our ordinary tables, 
while the tangents always vary by such perceptible quantities 

I leave no doubt of the real value of the required "" 

I may be easily verified by examining any of o 
lax or seven places of decimals. 

Of the Construction iif Trianglet. 

47- Previous to the numerical solution of any triangle, it is gene, 
rally first constructed geometrically. This is accomplished by means of 
what are termed mathematical instruments, consisting of scales, com- 
passes, &c. contained in a. case, at various prices, to suit the conve- 
nience of purchasers. Printed descriptions of these, as well as of 
many others, are to be found in Jones' edition of Adams' Geometri- 
cal and Graphical Essays. 

In the construction of plane triangles the sides are taken from a 
•cale of equal parts, and the angles are laid down by a scale of chords, 
or more conveniently by a protractor. 

EXAMPLBS. 



48. 1. Given the angles and hypotenuse of a right-angled triangle, 
to find the base and perpendicular. 

Let the hypotenuse AC of the right-angled triangle ABC be 288, 
and the angle A 39° 22' ; it is required to find the 
aides AB and BC. 

Cottstmctim. — In the indefinite straight line AB 
take any point A, and by a protractw or scale of 
diorda, make the angle A equal to 39° 2^; from 
■ny convenient scale of equal parts take AC equal ^ 
to 288, and from C draw CB, perpendicular to AB ; 
tfien ABC will be the triangle reijuired. In order to simplify and 
preserve uniformity, the angles may, in general, be denoted by the 
^ capital letters A, B, C, and the opposite sides by the small lett«rs 
a, b, c. The sides a and c being measured by the same scale from 

which b W— •-!'— — =11 ■>" '' > •" '— ion 1 I nnn » ^Wji 




PLilNE TBIOONOMETRY. 19 

CakuloHoH 

1. By natural numbers^ § (43). 

To find a. 

Aa am. B : sin. A :: b : a, or azn — -; — tr 

sin. B 

1 : (WB4281 : : 388 : ^^^^ ^ ^^ =188,(173=^ 

To find c. 
And sin. B : sin. C^ or cos. A : : 6 : c 

1 .0.773103:: 288 :<^-^73103x 288 

2. By logarithms. 

To find a. 

As sin. Bj or radius 10.000000 

Is to sin. A ao^' 22' 9.802282 

So is 6 288 2.459392 



To a 182.673 2.261674 

To find c. 

As radius 10.000000 

Is to COS. A dO"" 22' 9.888237 

So is 6288 2.459392 



toe 222.653 2.347629 

The solutions may be varied by assuming- any of the sides for 
dius^ according to art. (44), and verified by Ounter's scales. 

2. Oiven the angles and one side, to find the hypotenuse and the 
other side. 

Let the side AB be 758, and the angle C 39° 26^; to find the angle 
A, and the sides BC and AC. 

Ans.—BC is 921.7, and AC 1193.36, and the angle A 50^ 34'. 

Construction. — From a scale of equal parts make AB equal to 758, 
the angle A 50° 34', the complement A C, and draw BC at nght 
angles to AB ; produce AC and BC till they meet in C; dien ABC is 
the triangle required, and a and h measured on the same scale from 
which c was taken will be found to be about 922 and 1193 respeo* 
tively, [ 

3. Oiven the hypotenuse and one side, to find the angles and other 
side. 

Let the hypotenuse AC be 544, and the base 464 ; to find the angles 
A, a and c, and the side BC. 

i^m^The angle A is 31° 28', though C is 58" 32' and BC 284. 

Construction* — ^Make AB equal to ^4 from a scale of equal parti, 
and from B draw BC perpenoicular to AB, then from the centre A 
at the distance AC equal to 544 describe an are intersecting BC in 
C, join AC, and the triangle is constructed. The angle A being 
measured by a protractor or scale of chords, will be found to he 31° 
SB', consequenUy C is 58° 32', and the side BC 284 from the same 
scale by wnich me other sides were laid down. 

4. Given the base and perpendicular, to find the angles and hypo- 
tenuse. 



so 



INTRODUCTION. 



liet the base AB be 558, and tlie perpendicular BC 456 ; required 
the angles A and C and the hypotenuse AC. 

An,.— A 39° 15' 21", and 50° 44' 39", and AC 720.622. 

Conslruclion. — Make AB equal to 558, and draw BC perpendicular 
to AB and equal to 456, join AC, and the triangle is constructed. 1 
The angle A will measure 39^°, and the hypotenuse will be aboai | 
721 nearly on the scale of equal parts. The other side may be found 
by Euclid I. and 47, or Leslie's Geometry II. 10, and 13. ' 

5. Given tlie angles and one side of an oblique-angled plane trian- 
gle, to find the other sides. 

In the triangle ABC, are given the side AC, 532, the angle A 
38" 40', C 92° 46', and consequently the angle B 48= 34' ; to find thir I 
sides AB and BC. ' 

^fw.— AB 708-76, BC 443.34. 

Construction — Draw the indefinite AB, at A 
make the angle BAC equal to 38= 40", and from a 
Hcale of equal parts make AC 532, at C draw CB 
making the angle ACB equal to 92" 46', it will cut j^ 
AB in B forming the triangle ABC which was re- ■" 
quired. 

6: Given two sides, and an angle opposite one of them, to find the 
other angles and the third side. 

In the triangle ABC are given the side AB 274, AC 306, and the 
angle B 78° 13' ; required the angles A and C, and the third side 
BC. 

^H*.— The angle C is 61" 14', the angle A 48° 33", and the side 
BC 203.22. 

Construction. — Make AB equal to 274, the angle B equal to 78* 
13', and with an extent equal to AC, 306, intersect the line BC in 
C ; ABC is the triangle required. 

If in this triangle the side B be greater than C, there may be 
two triangles formed, constituting what is called the ambiguous case, 
that is, it admits of two solutions, either of which answers the condi- 



dons required, unless from some known circumslances 
ust be adopted in preference to the other. 
Thus in the oblique-angled triangle ABC there are 
" 195, and the angle A 32= 40'. 
[na.— The angle B is 61° 50" or 118° 20', the angli 
or 29°, and the side AB is 360.246 or 175.15. 

Constraclion. — Make AB equal to 318 from any 
convenient scale of equal parts, the angle A equal 
^ 32° 40', and with the centre B and distance equal 
" BC 195 describe an are cutting AC in C or C ; 
ABC will be the triangle required. 

CASE II. 

49. Given two sides and the 
:les and the third side. 



f them 



giv. 




intained angle, to find the otberJ 



Ifn the triangle ABC let the side AB be 920 and AC 500, and the I 
mtained angle A 36° 52' ; required the angles B and C, and tlie | 
lird side BC. 
'i Am.— a is 30° 58' 50", C 113" 0' 10", and BC is 600.31. 



PLANE tRIOONOM£T£Y. 



df 



ContiructioH. — ^Make AB equal to 920, at the 
point A make the angle BAG equal 39* 52^, and 
AC equal to 500 ; join EC ; ABC is the triangle 
required. 

By Calculation^ art. 45, 1. 
As AB+BC 1420 
IstoAB— BC 420 
So is tan^ (B+C) 71^ 34' lO' . 

To tan. i (B— C) 41 35 10 

C 113 9 10 

B 2958 50 
As sin. B 29° 58' 50" 
Is to sin. A 36 52 
So is AB 500 

To BC 600.31 

Or by art. 45, II. and III. 

As sm. i (B— C) 41<> 35' 10" 

Is to sin. i (B + C) 71 34 

So is AB— BC 420 

ToBC 600.31 

As cos. i (B— C) 41° 35' 10" 

Is tocos. I (B+C) 71 34 
SoisAB+BC 1420 




B 

3.152288 

2.623249 

10.477162 

0.948123 



9.698714 
9.778119 
2.698970 

2.788375 

9.822001 
9.977125 
2.623249 

2.778373 

9.873877 
9.499963 

3.152388 



ToBC 600.30 2.778374 

The advantage of these two last methods consists in its being unne- 
cessary to find the values of the angles C and B to determine fiC, and 
that several of the quantities are found among the tables at the same 
opening of the book^ and if computed both ways they are a check 
upon each other. 

CASE III. 

50. Given the three sides of a triangle^ to find the angles. 

In the triangle ABC, there are given AB 800^ AC ^, and BC 
562 ; to find the angles. 

Construction, — ^Draw the line AB equal to 800 from a scale of equal 
parts, then from the same scale take an extent 
equal to AC 320, and with the centre A and j^ 
distance 320 describe an arc, in like manner, . ^ 
with the centre B and distance BC 162, intersect j^ 
the former arc in C ; ABC is the triangle required. 

In the solution of this question, if me an^es Aor B are first to be 
determined, then rules II. or IV. § 46, wm be found most conve- 
nient and accurate ; but if C be wanted first, then if great accuracy 
is required it would be improper to use rule II., but rule III. or 
IV. snould be employed, so as to giv^ the angle with aQ the requisite 
accuracy in nice operations. 




SB 



INTBODUCl'lON 


By 


' Calculation. 


AB 800 

AC 320 «r. CO. 

BC 562 art. CO. 


ROLS II. 

i 

• 


Sum ItittS 




Half 841 
Ist diff. 521 log. 
2d diff. 279 log. 


• 
• 


Sum 


• • 


Half 64" r 54".4 i 

2 


sin. 


C 128 3 48 .8 




Bulb III. 
AB 800 
AC 320 ar. co. 
BC 562 ar. co. 


Sum 1682 




Half 841 log. 
Diff 41 log. 


• 
• 


Sum 


• • 


Half 64° 1' 54".9 

3 


COS. 


C 128 3 48 .8 




AR 800 
AC 9S0 
BC 562 . 


RULB IV. 


Sum 1KH2 




Half 841 ar. co. 
Ist diff. 41 ar. CO. 
2d diff. 521 log. 
3d diff. 279 log. 


• 
• 

• 
• 


Sum 


• • 


Half 64" 1' 54".7 ( 

2 


tan. 


C 128 3 49.4 





7.404d6O 
7-250264 



2.716838 
2.445604 

19.907556 
9.953778 



7494850 
7250264 



2.924797 
1.612784 

19.282694 
9.641347 



7075204 
8.387316 
3.716883 
3.445604 

20.624862 

10.312431 



From these solutions it appears that the first and second differ about 
V from each other^ while the second and last only differ 0'^4. 



PLANE TBIGONOMETRY. * 

Had the angle C been nearer 180^^ the first and second solutions 
might perhapslbave differed more considerably, while the second and 
third would have agreed more nearly. Hence it is clear that the 
proper rules, when great nicety is required, must be chosen accord** 
mg to the nature of the angle. 

Examples fob Exeboisx. 

51. 1. What angle will one foot subtend at the distance of Mj 
miles ? Ans.-^\JS. 

2. The hypotenuse of a richt-angled triangle beinff 6473 foet, and 
the acute angle adjacent to Uie base, 29® 50^ 5ff\ what are the base 
and perpendicular? 

Ans,— The base 4746.064, and the perpendicular, 2723.53a 

3. If the base of a plane triangle be 384, and the other two tides 
288 and 192, what is the length of the perpendicular upon the base, 
and the length of the segments of the base made by a line bisecting 
the vertical angle ? 

Ant.— Ferp. 139.4274, segments 230.4 and 163.6. 

4. There are three towns. A, B, C, so situated that the bearinff of B 
and C from A forms an angle double that of A and C from B, and 
that of A and B from C double that of A and C from B, or 
the angle opposite b is double t)f that opposite c, and the circuit 
round all the three is just one hundred miles ; what are their rela- 
lative distances from each other in succession ? 

^ra^.— 19.8073, 35.6861, and 44.5066 miles. 

5. In the right-angled triangle right-angled at B, ffiven the base 
AB 70, and the sum of the h3rpotenuse and perpendicular AC and 
BC 200, to find the hypotenuse and perpendicular, and the remain- 
ing angles ? 

^n*.— The angle ACB is 37? 16', AAC 51° 24', and AC 112^2, 
and BC 87.6a 

6. In an oblique-angled triangle ABC let the side BC be 532, the 
angle BAC IIO" 30^, and the sum of the sides AB, AC 637; requir- 
ed the angles C and B, and the sides AB and AC ? 

Ans.— The angle C is 45<' 5', B 24^' 25', and the side AB 402.3 and 
AC 234.7. 

7. In the oblique-angled triangle ABC, let the side BC be 260, 
the anjde BAC 96° 50^, also the £fference between the sides AB and 
AC lud; required the angles ACB and ABC, together with the 
sides AB and AC ? 

Ans.— ACB is 57° 55', ABC 25° 15', and AB 2ia4, and AC 107.4. 

a Given the base 214, the vertical angle 49° lO', and the sum of 
the other two sides 459 ; to find the sides and remaining angles ?, 

Ans. — ^Tlie acute angle is 33° 44' 48", the obtuse aiigle is 91° 
59' 12", the side opposite the acute angle is 17^75, and the side op- 
posite ^e obtuse angle is 282.245. 

9. Given one of the sides 252, the opposite angle 20° 46', and the 
excess of the base above the remaining side 86; to find the remaining 
angles and sides* 

Ans. — The vertical angle is 94° 22' 28", the remaining angle is 
66° 51' 32", the base is 507.08, and the other side 421.08. 

10, Given the base 1514, the vertical angle 75° 24' 60", and the 
perpendicular 972.41 ; required the remaining sides and angles. 

iii«.~The sides are 1298 and 1172, and the angles are 56^ 4f ^'^ 
and 48^ 31' 5" respectively. 







24 INTRODUCTION. 

52. Tlie various sailings in navigatiun are outy the applications ol" 

trigonometry in particular circumstances. i 

The course is the angle formed between the meridian and tha 

Sointon which the ship sails, the distance ia the hypotenuse, and the 
ifference of latitude and departure, the legs of a right-angled 
triangle. 

Thus let AB represent the meridian ; then if a , 
ship sails north-easterly, the line AC is drawn to D 
the right-hand, making an angle BAG equal to 
the course, and AC represents the distance, AB, 
the difference of latitude, and BC the departure. 
If she sails north-westerly, then'BAD is supposed, 
to be the angle of course shown by the compass, and 
is generally in points and quarter points, AD the 
distance, — AB the difference of latitude and BD 
the departure. Again, if the ship sail south easter- 
ly, AF is the distance, AE the difFerent latitude, EF the do- 
parture, and FAE the course. If, however, AE' be the meridiaa 
difference of latitude, E'F' is the difference of longitude, E'AP* 
is the course, and AF is still the distance. Hence the course and 
distance between two places can be found, by this method, when 
their latitudes and longitudes are known. This is commonly called 
Metcator's sailinff. 

Parallel, middle latitude, and oblique sailings, may readily be ex* 
plained on similar principles, though these can only be completely 
discussed in regular treatises on navigation. 

See Mackay's, Norie'a, Riddle's, Inman's, or Robertson's Na- 



\ 


/ 


/ 




?^ 


k\„ 


/ \ 


d i:' l< 



Examples. 



1. A ship from latitude 47" 30' N, sails S. W. by S. 98 miles ; what 
latitude is she in, and what departure has she made ? 

Am. — Difference of latitude 81.48, departure 54.45 miles, and the 
latitude came to 46° 9' N. 

2. A ship from latitude 48° 32' N. sails between north and weat 
till her departure is 54 miles, and then finds herself in latitude 49* 
fi4' N. ; what course did she steer, and wliat distance did she run ? j 

^nj.— Course 32" 22' N. W., and distance 9ai8miles. , 

3. Coasting along shore I saw a cape bearing N. E. by N. AfW 
standing N. w. 20 miles the same cape bore E. N. E. Required the 
distance of the ship at each station. 

jinj.— From the first station 33.26, and from the second 35,31 

4. Required the course and distance from Ciiithness point in Scot- 
land, in latitude 58° 46' N. longitude 3° 17' W., to New York in 
North America, in latitude 41" 5' N. and longitude 74" 15' W. 

y*nj.— Course 68' 32' or W. S. W. nearly, and distance 2899.8 

5. A ship from latitude 60° 24' N. and longitude 43" W. saili 
between South and West till she is in latitude 56° 30' N., and hu 
made 226 miles of departure ; required her course, distance, and 
longitude ? 

Ans. — Course S. E. nearly, distance 325.4 miles, and the longitud* 
of the ship 35° 47' W. 

6. fleqiured the course and distance between the Isle of May, in lati- 



PLAN£ TRIGONOMETRY. 

tude 56° 12^ N. loi^itade 2^ 33' W., and Heligoland in latitude 64* 
12^ N. longitude 7** 53' B ? 

An8.— Coune 8. 71** 27' B. and Dirt. 377 miles. 

7. Aship from the Isle of May sailed on the following true courses ; 
required her situation ? 



Courses. 


Dist 


Diff. Lat. 


Dtputun. 


S.E. 


40 


N. 


S. 


£. 


W. 




28.3 


2a3 




S. S. £. 


50 




46.2 


19.1 


1 


N.E. 


20 


14.1 




14.1 


1 


S. B. b» S* 


60 




49.9 


33.3 




R. S. B. 


200 




76.5 


184.8 




W. b. s. 


15 




2.9 




14.7 


N. N. W. 


20 


18.5 






7-7 


N. E. b. N. 


76 


63.2 




42.2 




B.S.B.f B. 


60 




146 


5&2 




S.7UE. 


378 


95.8 


218.4 


380.0 


22.4 


DiflT. of Lat 


• 




95.8 


22.4 




122.6 


357.6 


1 

1 

1 


( 

4 


2° 3'S. 


■ 


Lat left 
<Lat in 


• 
• 


56 12 N. 




54 


9N. 



Hence the ship is about 3 miles south of Heligoland light 

Sbgtion V. 

Application qf Plane Trigonometry to the Mensuration tf Heights and 

Distances. 

53. One of the most important applications of plane trigonometry 
is the mensuration of heights and distances. The data are some of 
the sides and angles of a triangle. The sides are measured by rods^ 
lines^ tapes^ or cmins> constructed according to the degree of accu- 
racy required ; and the angles are measured by some angular inrtru- 
ment, such as the quadrant^ sextant^ reflecting circle^ repeating circle, 
or theodolite. The repeating theodolite is perhaps, in general, the 
mort convenient of all for taking the necessary anffles, and the chain, 
properly constructed, the best for measuring the side called the base, 
though, to military engineers, the small pocket circular box-sextant, 
or semicircle, as improved by Sir Howard Douglas, will be found 
highly useful, when accompanied by the box-measuring tape. One 
of Schmalcalder's surveying compasses will also be found very commo- 
dious in military and nautical surveying. A complete description* of 
these instruments would far exceed our limits, and their use is best 



* Those who wish for written descriptions may consult Jones* edition of Adam's 
Geometrical and Graphical Essays, alieady mentioned, Biot's Tiaktf d*AttttMnie 
Fhvfique, DeUmbre's Astronomie, Base du Systeme Metrique, Woodhouie'f, Viiiec^ 
ana Pearson's Treatises of Aationomy. 



K INTRODUCTION. 

learnt under the supeiiiitendeiice of a niiister. In general, it liiay 
be remarked, that an allowance must be made Ibr the height of dife 
eye above the liorizontal plane ; and when the /lahv abnve-mentioneil 
is indined to the horizon, it must be reduced to it according to the 
given inclination, though in nice operations the base is selected bo u 
to be, if not exactly, at least nearly level. Then, from a little atten- 
tion, by driving in stakes at moderate distances, and levelling their 
tops, on which deals properly prepared are laid, an exftct horison- 
tal line may be obtained. This truly level line is to be most careful- 
ly measured, allowance being made for the contraction or expansion 
of the materials of which the chain is composed according to the state 
of the thermometer ; in nice operations reduced to the level of the 
sea; and such other precautiunsas the nature of the case may require 
must be observed, in order to insure the greatest possible accuracy ; 
many examples of which may be seen in the Trigonometrical Sur- 
vey of the British islands under the direction of the Board of Ord- 
nance.* A number of the more useful problems connected with 
trigonometrical surveying may be seen in the third volume of Hut- 
ton's Course of Mathematics by Dr O. Gregory, in Baron Zach's 
Work on the Attraction of Mountains, in the Base du Syateme de 
Metrique Decimal, and in Piussant's Geodesie. 

ExAUPLK I. 

To determine the distance of n tower, inaccessible by reason of an 

intervening river, I measured, on a horizontal plane, the base Al), 

fiOO yards, and at each end took the angle included between the 

other end and the tower, which were 60° 58' a 

«nd75*' 10" respectively : What is the distance 

rf the tower from each end of the base? 

In the annexed figure, 

AB ^ 500 

CAB = 50° 56- 

CBA = 75° W, and consequently 
Angle_C = 180"— (A + B)=53°54' 



Hence, sin. C 53" 54'' 

la to AB 500 

So is sin. A 50° 56' 

To BC 480.46 

So is sin. B 75^ UY 



9.907400 




To AC 590.2 

The perpendicular 

found thus : 

As radius 

Is to AC 598 2 

So is sin. A 50° 56- 



2.776844 
nearest distance Cd may, if required, be easily 



I 



10,000000 
2-776844 
9.890093 



To Cd 464.45 

i?cmort».— These distances i 

tn instrument to 



2.666937 
light have been determined withc^ 
angles. Thus, suppose that, in the 






' There vro Micnl mclhods of approiimatint; lo tlic liciKhii of objecU by mams 
oFmimm, ihadnwn, itaHi, nomclrical •qusm, and tiunlec'ii quadnaUf but m ihvy 
ue seldom Died where atucn accutncr Ih rcquirt'd. ihejr *n oaiinei here. 



PLAN£ TBiqONOMETRY. ftj 

GontiDiuMioii o£ the liase AB^ and the Unet CA» CB^ the four dittanoft, 
^, A£> BF, B&» were taken all equal to 100 feet, and OS nMaautu 
ed 86, and F6 122 feet, the respective chords, to a radius of iflD 
feet, of the exterior angles DAE, FBO, which are eqqal to their ver- 
tical interior angles GAB, CBA. Now, since half tibe chord is the 



sine of half the angle, we have >|^',j:^*49=sin. ^ A^S^** SO', and 
A=6a» 56\ In like manner, sin. ^B— 61=37° 35', and B=76«» lO', 
which results agree with the form^. 

Note 1. — ^The number 100 was chosen for the sake of sjmplicifv; but 
any other convenient number may be adopted, taking care to divide 
half the measure of the chord by it. ^ 

Nate 2. — The same thing may be accomi^shed when the nd^ of 
^le triangles bear any proportion to eacn other, by finding^^om 
tb^m the'^Qgles D AE, FBG. Also the supplements £AB, jSBQoi 
the original angles may be found in the same manner, or otherwise 
by joining AG and BE. 

Example II. 

Wanting to know the breadth of a river, I measured 100 yards in 
a straight line by the side of it ; and at each end of this line I found 
the angles subtended bythe other end, and a tree close by the oppo* 
site side, to be 53° and 79^ 12^ ; what is its perpencUcidar breadtn ^ 

Jn^.— 105.89. 

Example III. 

In order to find the distance between two trees A and B, whidi 
cpuld not be directly measured on account of a pool of water whidi 
occupied the intermediate space, I measured the distance of each 
from a third object C, which were 588 and 672 yards respectively, 
and then at C took {be angle ACB between the two trees 65^ w. 
Required their distance. 

180^ 0' 
Angle C 55 40 

A+B 124 20 

i(A+B 62 10 
s BC+AC 1260 3.100371 

Is to BC— AG 84 1.024273 

So is tan. i (A+B)62<' lO' 0" 10.277379 

To tan. i (A— B) 7 U 53 

Angle A . . 69 21 53 

Angle B . . 54 58 7 

As sin. A 69^21' 53' - . 9.971203 

Is to BC 672 2.827369 

8oissm.C 55 40 9.916859 




To AB 592.96 2.773025 ♦ 

Example IV. 
In the trigonometrical survey of Britain, Oolond Mudge found, 
from computations depending on fbrmer operations, that the loga- 
rithm of the number expressing the distance between Cheviot an^ 
Cross Fell in feet was 5.4654017> and between Cheviot and Wisji 
Hill 6.267^78; and the angle contained by these, corrected for 

^ '?* - ■ b ! — ! — _! . : i — — — 

* In some of the examples the computatioiu hi tnoportioii axe peff^mpo^ by ^compet- 
ing the tines of the aoglea with the Md^, a method wmethnes move easy to begjuoMeci. 



I, aiid 
finding 




% IKTIlODDCriON. 

- fcphetical excess, was 53" 30" 18". Required the other anglf 
the distance between Wisp Hill and Crass Fell, without first 
'ihe value of the given sides in natural numbers. 
,<«*.— The angle at Wisp Hill is 87" 14' 4". 

Crosa Fell 39 15 46 
The distance of Wisp Hill from Cross Pell 235018.6 feet. 

ExAMPLi: V. 
In order to determine the height of a tower, I mea- 
nired in a direct line AB 366 feet on a horizontal 
plane. I then took the angle Cab 37° 30', the height 
Ao of my instrument being 5 feet. Required BO the 
height of the tower. 
Arts. iC =280.84. 
Add Ao 5.00. 



Height BC =285.84. 

EXAMPLB VI. 

Walking along the side of a river, I observed an obelisk on the op- 
posite side, which on account of the river was inaccessible, but 
whose height I wanted to ascertain. For this pur- 
pose I took at B the angle CBD 50" 39' at A the angle 
CAB 33° 30', which was distant from B 368 feet. 
Required the height of the obelisk and the dis- 
tance of the station D from its base. 

Solulion.—BecaMse the angle CBD=CAB+ ACB, 

CBD— CAB=ACB=50°39'— 33°30'=17°9',hence A B 
Mn. C : AB : : ain. A : BC j and in tlie right-angled triangle DBG are 
now given BC and the angle CBD, to find DC and BD, 52] and 
427-2 feet respectively. 

Example VII. 

A solution of this problem, more easy and commodious in practice, 
may be obtained thus : — 

Let CD represent any object whose height is to be determined j 
at the points A and B observe the angles of elevation, ami measure 
tile distance AB, the points A,B,C, and D being in the same plane. 
See preceding figure. 

For in the triangles ABC, CBD, 
Bin. ACB : AB : : sin. A : BC, 

and R : BC : : sin. CBD : CD, from which we have sin. ACB ■ AB 
X BC : : sin. A x sin. CBD : BC x CD or sin. ACB x BC x CD= 
sin. A X sin. CBD x AB x BC ; radius being unity. 



Hence CD= 



in. Ax sin. CBDxAB 



I. ACB, 



; or, making the terms homo- 



geneous, and substituting cosec. for — ~, 

R' X CD = sin. A X sin. CBD x cosec. ACB x AB. 

That is, to the sines of the observed angles of elevation, add the 
cosecant of the difference of tlieae angles, and the logarithm of the 
■measured distance; the sum, rejecting SOfroua tlie index, will bo the 

I ^height of the object. 

1 Let the angles of elevation be 55^ 54', and 33^ 20" respectivelv, 

[ and the distance between the stations 100 feet. Required ib» 

' height of the object. 




PIJ^K£ TRIGONOMETAY 99 

A 1 ^1 ,. „ f 55** 64' sine M18062 
Angles of elevation ^ 33 go sine 9.739975 

Difference 22 34 cosec. 10.415942 

Distance . 100 feet 2.000000 



Height 118.5 2.073979 

Height of the eye 5.5 

Height of object. 124.0 feet 

Example VHI. 

In order to determine the distance of two inaccessible objects lying 
in a direct line from the bottom of a tower 90 feet high^ on the Urp 
of which I took the angles of depression of the two objects; that ci 
the most remote being 24"^ 48', and that of the nearest 58° 30^. Re- 
quired their distance from the tower^ and from each other. 

^11^.-139.842 feet. 

EXAMPLB IX. 

Wanting to know the distance between two boats lyinff at anchor 
in a strai^t line from a liffht-house, which is 110 feet high, on the 
top of which I took the angle of depression of the farthest^ and found it 
to be 18'' 26'^ and that of the nearest 56'' 44'. What was their distance f 

Ans.— 129.5286 feet. 

Example X. 

From the top of a hill I observed two mile-stones on a horixontal 
road^ which ran straight from its bottom, and took their respective 
angles of depression below the horizontal plane passing through the 
place of my eye ; that of the nearer mile-stone was 96^ 1^, and 
that of the more distant 15'' 26". Required the height of the hill. 

Ans, — ^780.17 yards. 

Example XI. 

In order to find the height of an obelisk standing on the top of a 
regularly sloping hUl, I measured from its bottom a distance of 40 
feet, and then found the anffle formed by the inclined plane^ and a 
line fi-om the top of the obebsk to centre of the instrument, to be 41**; 
and, after measuring downward in the same direction 60 feet farther, 
the angle formed as before was only 23° 45^ What was the height of 
the obelisk and the angle of the inclined plane with the horizon ? 

Jm— Height 57.623 feet. Inclination 2P 54^. 

Example XII. 

Wishing to know the height of a tower standing on the top of a 
regularly sloping hill, to the bottom of which I could not approach on 
account of a ditch around it, at the outside of which I tooK the angle 
formed by the inclined plane, and a line from the centre of the m- 
strument to the top of the obelisk, and found it 41 ; but after mea- 
suring downivard m the same sloping direction 54 feet farther, I 
found the angle formed in like manner to be 239 45'. What was the 
height of the obelisk itself, and that of its top above the last place 
of observation, supposing the angle formed by the inclined plane and 
ttt^ hoHWn'tq ftfe 2> 54'| ? 

Ans.^^l.^ feet the height of the obelisk, and 83.51 above the last 
plilce of observation. 



30 



INTBODOCTION. 



Example XIII, 

Being on a horizontal plane, and wanting to know the height of a 
tower on the top of an inaccessible hill, I took the angle of eteyatian 
of the top of the hill 40°, and of the top of the tower 51°; then mea- 
suring in a direct line 100 feet farther from the hill, I took in the 
same vertical plane the angle of elevation of the tower 33° 45. Re* 
quired the height of the tower ? 

^n«._46.666 feet. 

Example XIV. 

In order to know the height of a castle standing on a hill, I took 
the angle of elevation of the top of the castle above the horizontal 
plane 59>, and of the top of the hill 25°; but could not, as in last 
exampie, measure a sufficient distance directly from the castle. I 
therefore measured in an oblique direction 52 yards, making with 
the castle an angle-of 72° 10", at the farther end of which the angle, 
in the same manner, was 04° 30'. What was the height of the cas- 
Ue? 

^ni.— 34.464 feet. 

Example XV. 

Wanting to ascertain the height of a tower standing npon a hill, the 
height of the hill, and the horizontal distance from the nearest place 
of observation, on account of the nature of the ground I proceeded 
as follows :_ 

At A I took the angle 
GCKS'SS', andGCE2°44': 
then having set up a staiFAC 
equa) in height to the centre 
of the theodolite, I measured 
1810 feet up the sloping 
ground AB in a direct linewith 
the tower, keeping the points 
K, E. C. B, in tlie same ver- 
tiwl plane. At R I took the 
angle FDC=I5AI = 1<' 54', 
and EDF=1° 32' Required the height of the tower, the lu-ight of 
tlie hill, and tlie horizontal distance from the first place of observa. 

1. In the triangle DCE, are given the side DC=18]0 feet, the 
angle ECD 175° 22-, EDC 3' 26', and DEC 1° IS"; to find CE=5175.89 
feet.* 

2. In the triangle CKE, the angle K=86'' 22', CEK=92° 44', 
KCE=0° 54' and CE=:5175.89; hence EKri8J.463 feet. 

3. In the triangle CGE, the angle GCE=2= 44', and CE=5] 75.89; 
hence CG=AH=5170 feet; and GE^246.826. 

4. In the triangleABI,AB=1810, the angle BAI^l" 54'; hence 
AI=1809 feet, and BI=60.011 feet. 

If EKj the height of the tower, were only wanted, it may be found 




(Mu cmrputatiDD, tu log. sboulil be nwrved fruin the 6nt tc 



PLAN! TBIGONOMETKY. 



SI 



Sin. jytSb : DC : { sin. CDB : C£=DC sin. CDE. cosec. DEC, 
dn. K : bB (=:t)C. sin. CDS. cosec. DEC): : sin. KCE : K£, arid 
R^K£=:DC. sin. CDE. sin. KCE, sec GCK. cosec. DEC. 

By logarithms, 
sin. CDB 3'' 26^ 
sin. KCE QP 54' 
sec. GCK 3* 38^ 



cosec. DEC P 12' 
log. DC 1810 

EK 81.403 



8777333 

ai961(» 

10.000874 

14.6789S3 

3.267079 

1.910061 



Example XVI. 

At the top of a castle which stood a hill near the sea-shore, the angle 
of depression of a shin's hull at anchor was 4^ 53' ; at the hoCtom of 
the castle the angle ot depression was 4** S^. Required the horimntai 
distance of the vessel, and the height of the hill on which the castle 
stands ahove the level of the sea, the castle itself being 64 feet high. 

Am, — 4373.75, and 306.4 feet respectively. 

Example XVII. 

^rom a window in the lower part of a house, nearly cm a level 
with the bottom of a steeple, I took the angle of elevation of the top 
of the steeple 40°; and from another window 18 feet directly above 
the former, the same angle of elevation was 37^ SO'. Required the 
height and distance of the steeple. 

Ans.—210M, and 250.79 feet respectively. 

Example XVIII. 

Suppose A and C to be two sta- 
tions on sloping ground, O an ob- 
ject on the top of a hill, and the 
angles OCA, OAC, measured with 
a sextant, to be 79^ 29" and 63"" 1 1' 
respectively ; also let the angle of 
elevation of AO above the horizon- 
tal plane be 6° 36', and that of CO 5"" 22' ; what are the horisontal 
distances and height of the object, AC being 410 yards ? 

In the triangle AOC are given all the angles, and the side AC ; to 
find AO and CO. A^ain, in the triangle AGO right-angled at G, 
are given the ancle OAG and the side AO ; to find AG=6604) and 
OG=76.4. Lastly, in the triangle COB, right-angled at B, are known 
CO and the angle OCB ; to find CB 6OO.7, and OB 56.4, and OG— . 
011=76.4—56.4^=20 yards nearly =;HG=CP, the difference of the 
heights of the stations, supposing AP ta. be horizontaL Now in the 
right-angled triangle APu a rc given A C a nd CP> to find AP=: 

^AC-hCP) (AC— CP)}* =V430 X 300 = vR7700t= 409.6 yards. 
Hence the sides of the horizontal triangle APG are ^ven, to find the 
^bgles, which may be determined by cSise ill. Plane Trigonometry, to 
be AGP=37** 31^29", GAP=63° lO' and GPA=79o 9' 31" 

The present may serve as an example of reducing hypotenusal 
lines to their horizontal measure, and of determining the height of an 
object above each place of observation In qmt ccmmion cases. 

EXAMPLB XIX. 

The height of the mountain called the Peak of Teneriffe was found, 

7 




baroiKStricftLlj', by the methods ilescribeit in Gregory's Mechanics, 
V6L h book S.tob e 12,350 «et, w,g.34 English miles, and the angle 
of depreBUtUIQAt'horizon, tVomi tlK mean ofia^.great number of 
observations, 1° 58' 12" ; it is required to determine the diameter of 
the earth, supposing it to be a perfect sphere. 
vl»M.— 79ia«tttjfcs. 

Let C be the .centre of the earth, the circle BTG 

» vertical stctioQ -passing through the centre, AI5 

i the height pf tk? feak, AT the tangential line 

I drawn from its top to the visible horizon, and AD 

I SJline perpetidtculaj' to a plumb-line hanging free- 

f : also, let BE, a tangent to the earth's surface at 

J, meet the other tangent AT in E. Then, in tlie 

triangle ABE, right-angled at B, there are given 

ftAE thecoOTplement of DAT, the angle of depres- 

^6Ti=:W V 48", and AB=2.34, hence R : AB: : tan. 

k-. BE ;: sec. A : AE. But since the triangles , " 

DBE, CTE, arc right-angled at B and T, have the side CBi^'pE 

I fcftil CE common, they are (Leslie's Geom. I, 22, or Hinton'e Geo^ 

L Bleo. 34, COT. 2) equal, and therefore BE=ET; hence, AE'+JiB 

I T^E + pT— AT. In the triangle ATC, right-angled at T, weftwl 

1 «i'AT;;'t*n.A : TC, the radius of the earth. The operation, jiKug 

trfofrted occupies but small compass, which may stiTl be ffrtiii^ 
orteited. For since tan. A -f sec. A— tan. (A-i-^ corap. A) we'sjiBfi 
iocofpwating theproportions from which AE, BE, ^d C^'irJ 
duced, have . eik)\.1 

" ' R" CT=iABtan.(A-l-^comp. A}taii. A; ' 

I or, log. CT=log. AB+ log. tan. CA + ^comp. A)-Hlog. t^.j^T^j^ 
[ Hf^ fndex. 

P 58' 




* Depression 

Half f Oy I! 

Camp. de]]res9.\di) 1 41t tan. 



Sum m t>4 tan. 

Height of Peak 2.34 miles, log. 

Earth's semid. 39.'i6.t! 



Diameter 7913.6 

Distance J36.1 . •2.1338.'595 '"• " 

I If AT were required, ve have only.totiik^ radiu»r("*) fi"m 'he sura 
fof the two last lines, and the remaindel', 21338fe.'», is the log. of 
f 136.1, the distance sought* 

Nole I. — This method cf determining the earth's radius, though 
L elegant in theory, b useless in praclicci at least where any thing 
I VQore than an approximation is wanted, 'by the great irregularity of 
the horizontal refractions. 

Nvte 2. — When the diameter of the e&rlh is known, andlieight of 
'^e abject given, the distance of the vlrible horiston nifty^e easily 
found) (m,Sjtm Hl^a6>AB-.AG— AT^ 



FUUn TUOOROMSTEY. 

By Iflgirittiiu- 

AB SA4 log. . 

BO TftldJS 



ABH^BChsAO 7916.M log. 



4997719 
As before 136.1 mUet, log. 9.183869* 

Kate 3. — ^The depression of the horizoii, or the dip, at k is cdbd 
9k see, is the angle DAT eontained betwoiD the trot and vislhle 
liori jkm. For if an obsenrer^ whose eye is aitatfied at A on the dedc 
of a vessd, takes the altitude of a eelestial object with Hadkr't 
quadrant or sextant^ by bringing that object to the surfSu)e of no 
water at T^ instead of^^tiie true horijott AD, the alttade it ovi- 
dentiy too great by the angle DAT=sTCA. TUs may be oabJrted 
by tlie usual formuUs of trigonomeiry fiur that purpose ; bat •■ it 
will, at any probable altitude, be a snudl quantity, thoee whidtglve 
the cosine or secant of its value are not sufiaeotly coereot ; fhr 
which reason we riiall give the following niediod>«» 

(BCM.AB)xAB«AT«, (Euc lU. 36.), henoe BOxAB^AB^^ATS 
or llBCx AB+AB*=sAT«, and AT* being, at any probable eleva» 
tion, but a small quantity in comparison orAC, it m^be stWy nt* 
glected ; therefore sJ{W^C X AB)=AT. But CT( :JBC) : fi : : AT 

[^(SBC X AB)] : tan. C=tan. DAT a " ^<^^'^ ta J^S^^. 
Now since -^^ is a constant quantiQr, and BC bebg taken In |«M* 

fil at 8066 miles ==20887680 feet, hence the log. of ^blUBlH 

snd tan. DAT=4(19.08114+log. AB). ftnee, hi the present ^m^ 
the arc may be substituted for its tangent, the rtdint, tnerefbre, be- 
comes 5r 17' 44''.8=206264''.8; and we have log. DAT in eeoonde 
ss|(3-6O099 + log. AB in feet). 

liie dip is affi^ted by terrestrial refraction, whidiis very vnritble^ 
and by cuflbrent authors it is estimated at diievent quanlicieai Ikf 
Madudyne estimated it at one-tenth of the whole; M • Delaa^bie, on** 
deventn, and Col. Mudge, one«twdfUi. See Dr Hntloiils OmxHt 
ToL in. page ISa 

• 

Ex. — Beouired the dip, the height of the eye being 40 &et, and 
estimating ue tonrestrial refraction at -^^ 

Constant log. 8.6D090 

Height c^eye 40 fbet 1.60906 

S.2U0S 



408^.6 kg. a.6060B 
Refrac tab. ^ S3 .6 

Dipt 370=:6' 10^'. 



'^mtmmmmtmm^mmmmm^ 



• SeeabodMmeasdby^XiaiiilaWieMiaslirt 
t ne4ipiaqifl«asstosqetltodn«3«aesaBtsr«el«ii^tlalbetiMHiv 



1 






■'34 ' IMTRODUCTION. 

Note 4.—Since ABxBG+AB'=AT', therefore 
AB(AO + AB)=AT..,„dAB = jg^. 
Now,if ABis the unknown quantity, and being small in comparison 
of BG, it may be found approximately by making, first, AB' =. -^^ 
nearly, substituting this value of AB' for AB in formula (1-), and 

:^''=BGfa (2.) 

-wJiich will be sulHciently correct for moat piirpoBea. If not, 1^. 
.operation may be repeated till it is ao. . " 

This is useful in determining the height of an object consider^ 
:dfBtsnt. ., ,^ 

- Now, the mean diameter of the earth is about 7912 miles, or 
1775360 feet =GB, of which the logarithm is 7-620920, and its 
.vHhmetical complement is 2.379080; therefore to twice the log. 
of AT, in feet add the constant log. 2-37i)080, the sum, rejecting 
tens in the index, will give AB', which will be sufficiently correct iF 
AT does not exceed 1000 feet. If more distant, the operation muat 
be repeated. This correction must always be added to heights de- 
termined geometrically u« the usual instruments give their eleva- 
tion only above the tangent AT. ,Jj 

ElTAMPT.E "XX. 

."Given the angles of elevation of any distant object, taken at 
pla«ee in a horizontal straight line, which doea not pass through 
point directly below the object ; and the respective distances betwe^ 
the stations : to find the height of the object, and its distance from 
either station. 

Let AEC be the horizontal plane ; FE the perpendicular height of 
tbe object F abOi^e that plane ; A, B, C, the tiiree places of observa- 
tion ; FAE, FBE, FCE, the respective „ 
angles of elevation, and AB, BC, the 
given distances. Then, since the Iri- 
aiiglea AEF, BEF. CEF, are all right- af 
angled It E„the distances AE, BE, CE, 
wiU loauifestlybe as the cotangents of the 
angles of elevation atA, B, and.Ciand 
we must determine the point E, so that 
these lines uiay have that ratio. 

Coiislrjielioti 
To effect this geonjetiically, we must tak^ ^1, of AC produci 
equal to BC, BNeqiMltoAB; and make 

MG : BM {=BC) : : cot A : cot. B, and 

BN (=A^.- NO : : cot. B : cot. C. 
With the lines MN, MG, NO, construct the triangle MNG. 
join BO. Draw AE -aoidiat the angle SAJiinay be eqtml to MOBJ 
this line will meet BO produced in E, <he point in the hi ' 
plane falling perpendicularly under K. 

DemoMtlratiiM. 



' BE i BA <=:BN^ : : BM : BG. ■: " 

Therefore the triangles BEC, BON are s imtlirt *'«*««(( ubnti^ 

7 




PLAVSi T&iGOiiOM£TRY. 35 

BE : £C :: BN : NG : : cot B : cot. C Whence it ia obvious that 
AG> B£, C£> are. respectively as cot A» cot B, cot-C. 

. Calculatum. 

In the triangle MGN are given all the aidea, to find the GMN, 
equal to the angle AEB. Then^ in the triangle MGB, are given two 
sides^ and the contained angle; to find the angle MGB« equal to the 
ansle EAB. Hence^ in the triangle AEB are known the aide ABj 
and all the. angles; to find AE and BE. And then EFr^AE • tan. 
A==BE.tan. B. 

Analytically. 

Let AB=ir^ BC=j ; also let the cotangents of the angles F AE« 
PBE, FCE, be denoted by the letters a, h, c, respectively. 

Then, putting EF=j^ we have^ to radius 1^ 1 : a:: x : aje^AE, 
1 : bi: X : bx=iBE, 1 : c :: x z cxz=zC^ ; and on AC from E, letting 
fall the perpendicular ED, we have (Euc. II. 12) «« x^^b^ j:^+ 

r« + 2r.BD; hence BDss, ''^ ^^^^ J*— r« i„ Ukg manner CD= 



■ ^ =BD— BC=BD^^: whenceBD= ^^^' T* 

2s 2s 



inereiore c% " == ^ Hence 

2s 2r 

r^^±r^ and .=/ ^CH-0 



Otherwise thus ; 

If AB and CBbe conceived to be bisected itfi M'. and N'/and BD a 
perpendicular upon AC, which are however (»nitt>ed to svold' eom- 
plexity in the figure ; then, (Leslie's Geometry, II, 21.) AE*— 3£* 
=AB X 2M'D, and CE""— BE»=BC x 2N'D ; therefore, AE< X BC 
— BE«xBC=ABxBC+2M'D, and C£« X AB— BE" X ABisAB 
X BC X 2N'D. Ad^bng equals to equals, and AEf> x B0-H::B'<' X AB 
—AC X BE«=AB X BC X AC; consequently AE« X BC+CE" X AB 
=AC xBE« X AC X AB X BC. 

If AB=:BC, then AE'> + CE<'=2AB«+2BE*', the line EB being 
drawn from the vertex E of the triangle ACE, to anvi^aint-B in the 
base. Put AB=:D, BC=zd, EF=a?, and then expretsmg algebraioaUy 
the foregoing theorem. f r- • 

The -equation thence resulting is« 

dx» cot «A+Dar2 cot «C=(D+rf)«" cot «B+(D-(-d) Dd. 
Hence^ transposing all the unknown terms to one side of the equa- 
tion, dividing by the sum of the coefficients, and extracting the 

square root, weshaUhayex== y^^,^^j/^ff^^j^^^^,y 

Thus EF becoming known, the distances AE, BE, CE, are found 
by multiplying the cotangents of A, B, and C,: respectively; by EF. 
Cor^*-When Dzzid^ or I>+.ds=SD;=2d, the ^pression beoosies • 
x=rf-f.V[(icot.«A+icot.«C— cot«B), which is pretty well suit- 
ed to logarithmic computation. The rule -may, in that case, be thus 
expressed. — Double the logarithm cotangents of the angles of eleva- 
tion of the extreme stations, find the natural numbers answering 
thereto^ and take half their sum; from which subtract the natural 
number answering to twice the logarithm contangent of the middle 
angle of elevation : then half the log. of this remainder subtracted 



fiBBtWiAK.ofUwJaMmMd Jl«ame> Iwtwwn tha to! incl'WlBlia, 
or th0 MCond «tid lUrl^MUhia, wiB b* ths lagj of dM hdgl* > «r At 

l.Lst ABaieo fmt, BG 73 fcM; angle POEstSO^ 3S'« die 
angle FBBsiO* SS*, uid the angle TAE~d90' 4»: TMpdnrid ibe 

£^IM AS, BB, CB, md EF, th« htAaht oi t)i« objMt ■ - -l 

j4w^A£=£lW^ ftct, BEsliaU ftet, CEs^Hsl ^M, Md 

BP=»t84fc<«. - -'( 

fi. Lm die difee Kngtei of rievadon be 90^ SO*, 31° 94', '«*d Mr, 

rad the two equal meainred distances 84 feet ; required the baigbt 

of the (rf)j«ct Wm^— 63u064 fteb 

BxawfmXXI. ■ i 

'. C^Tinthftanglesflf elaratMiiatwhif^anolijcetisaecn.AKan'tiiftc 
igivwpointa in a hoHaoat*! plane j tD fioditfpoiitisoaadalti&itdKi 
y;4;f^A<B,wadCt>etlw«hra9 . - 

'poinU of obBervation, and D ^! _p.. 

t4b(^^toia.i>f tbeferpendicu- 
iJfir.fToni.tiK £1*49 ,n>j«ct b> 
-ttief^njipVtalpUne. I^imi-. 

dent tliafc, ,tiie J^ctcif iwtal dift- 

flivtfn AI^ QP, and CD are 
rjjntpof^of ^ t4: tJbe .flotapgenti 

of the vertical anelee at tbe 
^glatipps A. B, C ; U* tbeae c 



^langents be respectiv^r di 
:j]»Ud bv theL, IC,andfi[. 



\1 



AB intsmaUy 




: cictemally at the poiiitajfiwid 
' P in the ratio of Ii ta Af ; and 
; the Unea DE and DF jofaiing 
- bx the vertex D mnt bisect 
.'Internally and exUtraii^y tfaa 

an^le, whence EDFiiaright 

angle., and contained in a se- 
,4picirak ; wharefim OQ EF dfr> 
< foribe amnidrde. In the aame manner, ttvide CB intahully<«tid 
^•ext^nuUr at O and H* ia the ratio of M to N; and in SH daHoOw 
..'AWDMiurde. The point D oamnMn t« both Bflndoirds n«at4»ccur 

. .In their int^vection. — -'y,^. 

From thia construction tlia tligOScftiMtifcal Ckleaktkn' ivtfc^Dy 

deduced. Vat L+M : U : : AB : BE and I^— M: M:: AB ;BF; 

•«rheDcaDE:= ^ =*^i-or fc»dius KE is found. In like 

miwiei N+AI : M : : CB4 BO, and 'n^Mi • CBi BK*'c^se. 
■''%lltilflyDIte59+M. ■ Ktfetriaftfelfe IBK, the ddea.".W;|fmd 
-"«K,-^thth(!bi|tidttded1nikle', ~:ABC, Ure^lveh: wd/tB^Hifore, 



• 8«e Loltt'i OMRiabr, fourth cdltkn, pagt 975- To avoid exi 
aodi, tbapdntH, wUdiriM^dbolBcooiinuattonaf Bi, inthoi 
ontinuaden of BK, U vdJ u the Uqn joloing DE and OF, ii oi 



vuamynxamunmrBY. 



9r 



the triangle BDK the whole an^e B&D and its containing ri J ea aH 
:^en>u'ui4 the^Cftre^ the bafl»lU>» or the horlMfttrftfhtama fitom 
Sip Ata4ioB B, aad oonseqUmtly ibi aUitode^ is dHetmntJL ^ '■ 

It is obviouii, that the oppoitte aemioiicl^s wiU Ukfewiaft, hw thdr 

;iillAeraQCi|iop> give« on th* otnrr Bide« «' second poaition I^ &r that 

point. In practice^ howerer, this ambiguity could be teiflv remov- 

^ . It mj be remarked too^ thattbe point D may lall dtWr whh- 

IQ or without the triangle^ - 

If tbef object bo seen at the same elevation from aU the-Aive 
pointSj the arcs of the drdes will evidei^iy become tangents^ whidi 
Disect at right angles the sides of the triangle ABC. The projection 
'-D^liifl^M^ol^ectcHithe boriacratal pkne, wfllthenb^ die tmtt^ of 
thM^vele MeumscriMng that iinangle; aiWI« thevefore^dieradlntf^ 
distance AD maj be found bv prop. \%, book VI. Leslie'V OtdHmrj, 
as shown in the noteil, page. My* ' -^^-'^'t 

If the three points jof observation should lie in the'saklMJ'MMj||lit 
line^ the centres of ^the determining circles trttl occur IM' thtfrl^ieiir 
its extension ; 9nd henc6 the process ^ (rieulatiMflHnM^^gfMidy 
abridged, and will coincide with the fbregoltlj^'tMfeloMtioi^ '^^ ^^•"^' 

£«amp2e.w^Let the angle of elevation ofHhe llb}ecl ^t Aj ^ W 4ft^ 
that «t B 58^ 15', and that at C ^e^'^lS^ ; also the rfO^ ABM' VMrfl^ 
AC 38, and BC 50. Required ats heWit ? .v ; mT to 

Hmice;L =: cot 50« 45^, M =f cot » W, sttd N si^oot - W^'^B'. 
FroDi th# given sides the vigle ACB s= 9^ 36' Wi 9AC»t4fl^^ 
28^% andBAC 10fi» 16' 28". Also L =0.8170M8, If' !i£ 08188180, 
N=0.9407061; therefore, BE = 10.343, And BF =:^4098/li^ce 
KB =4^.635^ and BK rs 82.2925. In Uke mimner, BOf =fClK8f6> 
BH = 98J23, hence DI= 57.9845, and IB e: 38.1885. T¥^ th^ 
the ai^lo 1KB = 77'' 11' 24", and KIB fiS'' 39' 14^; andiheHdelK 
= 23.677- Jb^owfrom the three sides ID, IK, andKD,' thi^ aaglo 
IKD =;: 107^ K^ 28". To this, by applying the ang1«^ 1KB by lid. 
dition and subtraction we obtain the angle BKBK £= 184^ SI 80^, 
^^and BKD -= AKD = 29° 59' 2". 

' From the sides BK and KD, and the contained angle BKD, are 
: ibupd the «mle KBD ^ 102^18' 39", and KDB =: 47* 44" M", 
.fixmi which BD 2l 21.8086; and the hdfffat of tfie cueist Vi'MftaOa. 
^i :; Shoaldthe point IV be the ibo% of ^ perpendicular, -ftfr ifngls 
KBD' = 2» 29' and KD'B = P 52^ 50", and Biyrr 74.878; ^%Sat 
/!«fac>haght abbve!iy lirffl^ba !« y^tfds. 

'jA\{ r.I .brrrot ?r jy ^ij^Ckherwise ihns: " '' ■"-..''■.•'■ 

Given the angles of elevation of the object from three points in 
c-tiw saiiiia pb|&e foiiaodiig atfrifaigle,' <if whid^ th^ ndes vr^ kild^m to 
. find the nosition c^ t^ ?^J!^.. referred pei^p^dictda^^^ dial^pl^ 
aha it^'&HStuw'abdVVf it " 



00 J yiu'^.v. 



!,.l>-' 






•* -- 



«. t 



t . 



■ ■ .1 . .■«.' t 



:"'■. J- 



I i 






INTHODUtTION. 

may {'all either within or -without, the tri- 
angle. ,la. t^oth casea^ let A, B, and C be the 
points of obBefvatlon, and a, J, and y the 
aiigles of elevation at Uiese points reapective- 
ly. Join A, Bj and C, wkI on AB produced, 
if iteceasaiy, muke AE equal to AC, and AD 
toAB, joii) ED, und upon it construct the 
triangle EDF so that cotangent ,3 : contan- 
gent ,3 ; : AE ; EF, and cotangent ^ : co- 
tangent y ;: AD : DF. Join AF, and from '^ 1' t-'i 
B draw BG, making the angle ABG ennal to the angle ATE; Hid 
join CG. The point G in which the straight lines BG and AF MU 
lersect each other will be the point at which a perpendicular iet 
fall &oni the object would meet the plane, thus ascertaining't'"*" 




tilde, may he fount 
i Demonslration.- 



It is obvio 



B that the straight lines drawn f^vw' 



^^HB.cot 



each of the points of obserration to the point at which n perMoAlv 
ciilV let fall ftom the ubject meets the plane, ought to be in pi'flpMM' 
tion to the eotangente of thfr angles of elevation at these points r«u 
spectively. The proposition therefore resolves itself into this; TW , 
hndapoint in a plane from which straight lines drawn to three 
given points in the same plane shall have to each other a given ra- 
tio which follows from tlie construction just given. 

Solttlitm. — In the triangles ABG, AFE, the angles at Band F' are 
B^aaJ by donstruction, and the angles BAG is common to both; 
"' two triangles are therefore similar. And AG : BG : : AE : EF 
■" ^. Again AG: AE.: 

^^. AF or AG : AC : : AD : AF ; and as the angle at A 

lii'An ttj the two triangles AGC, and ADF ; these triangles 

cwflsequently AG: CG :: AD : FD^: *bt. « : cat y. wfienoe P) 
ABxcot-y. 



Hence EF — - 



of the ( 



cnt-K 

The triangles ADE, ABC having the sides AD, AE 
equal to the sid^s AB, AC of the other, and the angle at 
to both-, are eqnai!, and the side ED is equal to the side BC. There- 
fijrein the triangle ADE, the three aides are given, and those of the 
triangle FDE are already found; whence the angles AED and 
FED, and consequently the angle AEF may he obtained ; and from 
the angle AEF, with the sides AE snd EF, the angle AFE or 
ABG, which is equal to it, may be determined. Then in the tri- 
angle aBG, having the two angles at A and B, and the side AB 
the distance, BG may be found, consequently, with it and the angle 
fi, the height of the object becomes known. 

Example I^t the siile AB be 80 feet, BC=111), and AC =1 

also the angle at A or a^rSO", that at B or ^=60°, at C or y=^ 
required the height of the object. 

From these EP=!I6.329, DF=66.758;'ihe angle AED=.14^ 
EDA=:87'6'23", EAD=87= 6' 23", EAD=58'^ o' 37", OEF=:34- 
(i' 57", AFE, or ABG=7l>" 37' 8", FAE or AGB=4I»° ^8" 16", BG 
=53.673 ; and the height 96.392 feet. 



PIJLNI TBlOOmnSCTRY. 




From a conyenient station V, there could be teen three cftijectf A*, 
' B^ and G, whose distance from each other were AB=::8 tafies, AC=i6 
miles^ BGi=:4 miles; I took the horizontal angles APG=:33'> 45', 
'BPCesaa** aC. it is hence required to determine the respective 
distances of my station from each object. Here it will b* necessary, 
as illastrative and preparatory to the computation^ to describe the 
mknner of 

Construction, 

Praw the given triangle ABC fjrom any ccnveoiBiift scsle. From 
the point A £raw a line AD to make with AB 
an anale equal to 22^ 30^^ and from B a line BD 
•tQ.malM an angle BDA equal to 33'' 4&\ Let a 
circle be described to pass through their inter- ^. 
section D, and through the points A and B. 
Through C and D draw a straight line to meet 
th^iCirclift again in P^ which liaithe point ne- 
quired. For drawing PA, PB, the angle APD 
is evidently equal to ABD, since it stands on 
the same arc AD ; and, for a like reason, BPD =sB AD, . Bo that 
P ia the point where the angles have the assigned value. 

'■' Computation, 

In the triangle ABC, all the sides are given ; to find the angles. 
In the triangle ABD, all the angles are known, and the, aide AB; to 
find one of the other sides AD. Take BAD from BAC, the remaind* 
er, DAC is the angle included between two known sides AD, AC ; 
•from which the angles ADC and ACD may be found. • ;The angle 
CAP = 180^— (APC + ACD). Also, BCP=BCA— AQD; and 
I*BC=ABC+PBA=ABC+ sup. ADC. Hence, the three required 
distances are found by these proportions. 

As sin. APC : AC : : PAC : PC, and : : sin. PCA : PA ; and, 
lastly, as sin. BPC : BC : : sin. BCP : BP. The operation at length 
is as under : 

By Rule IL, Case iii., we have 

Sm. i BAC.= /^^=ViV=i— 25=sin. 14** 2^ 39",' and ' 

BAC == 28»'57' 18". ..!,",■.. 

Sin. i ABC s= jyiiss^VhrmaMfJrzMn. 23" 17' l'% and 

Sin. i ACB = /P4«^f *iVlft--79O6094-8in. 63» 14' 19" j, tad 
ACB = 104" 28' 3r.„ ' ^ . , „,, , . . , 



DAB= a2» aC CABza^WiQ" 180" C 0" 

DBA 33 45 BtABssSS ;lpO ■. J)AC?= 6 2718 

Sum 56 15 DACsr 6 27 18 ■: AI><Jl+ACp5?173 32 42 

180 KADC-|-AC]>)= 86 46 21 



ADB li23 45 



fkl I l> 



NTEODUCTION. 



sin. ADB 123= 46' i 
oAB 8 miles 

iisin. ABD33M6' 



AD log. 

AC 6miles, l«g. + 10 



0.0801536 
O.J>O3O0OO 
9.7447390 

0.727dd26 
10.J781013 



Remainder 3 18 7 tan- 
, J(ADC+ACD)=8fl 46 21 tan. 



4(ADB— ACD) 45 39 17 tan. 



8.7611283 
11.2487967 



ACD = 41 7 4 
ACD 41- 7' 4" sin. 9.819678 
APC 33 46 ar. cosin. 0.2562610 


OMSSfot 


Sum 74 62 4 
180 


i 


PAC 105 7 56 sin. 

AC 6 miles log. 0.7781513 


0.98467M 
0.778151* 


PA 7.10199 miles 0.8513801 


. . . . ;i 


PC 10.42525 miles 

ACB=104= 28' 39" 

ACD= 41 7 4 BCP+BPC= 


i.oi8oeetf 
180° art! 

85 51 3S? 


BCP= 63 21 35 PBC= 
As Bin. BPC 22= 30* 0" ar. cd. 
I«toBC 4 miles 
8q in sin. BCP 63" 21' 35" 


94 8*' 
0.4171603 . 
0.ti020600 -; 
8.9512594 ^ 



To PB 934285 miles 0.6704797 

The computation of problems of this kind, however, may be a litti 
shortened by means of the following 

General Invesligalion.* 
PutAC=a. BC=6, APC=P, BPC=P', ACD=C, and let there b 
Uken for unknown quantities PAC=:r, PBC=:b. The triangles PACi 
and PBC give " " 

Sin. APC : sin. CAP : : AC : CP, and 
Sin. BPC : sin. CBP : : BC : CP; that is. 

Sin. P ! sin. * : : a : — "' . ^ -zrCP, and 

Sin. P- : rin. y : : 6 ; ^t^HJ!-CP. 

Hence, —. — ^ =-: — ^ ; which may be reduced to a sin. P' 41^ J 
«— 4 sin. P sin. y=0. 

■ Sec LMTDii TriffgasoMtdt, swi OMgofy'i Trijonnw Wf . 



PLAN^H»IS88»WF»Y. 4^ 




se q tfen tiy j a sin. P' sin. x — h sin. P (sm. K cos, a;— cos. R. sin. a:;=:0. 

d^;i£^ by sin. x, there results, a sin. F-^ sin. iHt^R ^^^^ 
UZV^^^m 0.^ ^ ^ci ,»;V5m a OA sin. a: 

t--H^^,MVe have S??l^ = cot ^ «y°^y «y JI»^R , 

Sin. a; suy P siki. ftlu^ 

This expression separated into two pa rto j- wh ave 

€B£Ue^;j: ^ a sin. P^ .^^(». Bl T. k^m? i><;T[&,?l 

Xd^itU;;.!!^^^^- 6 sm. P sin. R t^iHSaji .^(U.^ f OG^i* 

.. .. ., cos. R / a sin. P' . -\ ^ . 

a sin. P* — '•"■'— ^. 
cot. a: = cot, R (t^- — p »T» t+ *) ; or, lastly, <!' )A 

cdi/a(ai|iln. F cosec. P co8^vEaMii.^ll+i|jitt^Rrft 01^ t;C D«1A 

Hence, x being thus determined, we get y from tl|^ e^^aj^n jffflfit 
R — a; ; and CP from either of the expressions given 4l>oMf . \yi] { 
We shall now apply the foregoing formula to the flqlTitifMy flf the 

ACB = 104 28 39 found by computation 



ci^ c 



R=199 16 21 ^ .. I, ;:. . 'im 



cot. ar=Y sin. :P' c6(ie<J. P, cosec. R^U R4«coi R; dry^ '^ *■ • '' 

cot. X = C0t.5^^f^^=^-5+l) iild USili^^lo^fay^ '^ *''^ 

we have t^T^O'/'Af = 3 log. a477l9l3i i^'y oT 




R vhose cos, is neg. * W9^ ifr « «r. ca. C. 0.0250452 . , . „ 

>i.i^^ii^«adT_.^$iO»48Bc*^'^.' 0;a892STi .;..;;:•** 



+ 1.00000 






cot.R +. . l?d!Jlf' Sft? .10.4509594^ 



' i *\ 



cot. ar — 105 8 10 9.4321587 

As sin. 33° 45' 0" ar. co. ' 6.265M0 

Istosm. arl05 8 JO t ^ 9;6g4a6aO ^ ^j. 

So ii?^ '"^ ^ "^ ^ '-^^'^ ^^ -^ - -^^'--^ ^ •>: }flfe778ISI3^^.- '-"'; "'^ 

To,£C 104251 1.018(K83 

Whence the msmhWRSf^ mi. HPv:awwvA > < •• • -. ' , 



^ INTBOUUCTION. 

In using these t'ormutee great attention must be paJd to the signs 
of die quantities. 

Example XXV. 

Suppose the objects A,B,G, are seen from D, and have their 

distances AB 7,i miles, EC 12 miles, and AC 8 miles, the angle 

&DA 25°, and CDA 19° ; it is required to determine the distances 

DA, DB, DC. 

^ni— DA 10.0286, DC 16.7857, DB 14.E)095 miles. 

Example XXVI. 
Suppose the objects A, B, C, are seen from D, and have their 
distances AB 8 miles, BC 12, and AC 7}; the angle BDC being 
17° 47' 19". Required the distances DA, DC, and DB. 
Ans.—DB 12, IX; 22.85, and DA 20 mUes. 

Example XXVII. 
If, AB be 8, AC 7.2, and BC 12 miles, and the angle ADB 107" 
66' 13". Required the distances DA, DC, and DB. 
^»M.— DB 6, DA 4,892, and DC 7 miles. 
Example XXVIII. 
Let the objects A, B, C, be in a straight line ; and their distances 
AC 3.626, AB 12, and BC 8.374, the angle ADC being 19°,, ' 
BDC 25°. Required the distances DA, DC, and DB. 
.rfiM.— DA 9.4711, DC 10.861, and DB 16.8485. 

Example XXIX. 

Let the objects A, B, C, as seen from D, be within the triangle 

and let the distance AB be 6 miles. BC 12, and AC 9, the angle BDC 

being 123° 45', and ADC 132= 22'. Required the distances DA, 

DC, and DB. 

^«*.— DA 1.372, DB 5.523, DC 8.018. 

Example XXX. 
A ship from Bombay in latitude 18° 57' N, sailed S. W. by S. 
224 miles. Required Uie latitude come to, and the departure. 
Ans. — The difference of latitude is 186.2, and the departure 124.4 
Latitude of Bombay 18° 57' N. 

Diff. of lat. ISe miles = 3 6 S. .^ 



aces 

4 



Latitude come to 



15 51 N. 
Example XXXI. 



Having occasion to travel dirough the counties of Kent and Sur- 
tey, I perceived the fort built by Lady James, on Shooter's hill, 
vnich bore from me N. N. E. ; and afler going 20 miles in a 
W. N. W. direction, 1 perceived the fort again, which now bore 
N. E. by E. Required my distance from it at each station. 

Ans. — 29.93 miles, and 36 miles. 

Example XXXII. 

From a ship at sea, I observed a point of land to bear E. by 
8., and aA«r safltDg 12 miles N. E., it bore S. E, by E. Required 
the distance of the last place of observation from the point oJ land. 

Ant.—2Q mikes. 



i 



PLANE TRIGONOMETRY. 48 

Example XXXUI. 

SaiUng N. N. W. at the rate isi 6 knoU an hour, at Sb. p. m. I dit- 
covered two light-houses, the northernmost of which bore N. N. E. 
and the other E. by N., and at lOh. dOm» the northernmost light 
bore S.N. E., and the other £. 8. E. The bearing and distance of 
the liffhts from each other are required. 

CMtdation. — ^In the triangle ACD are given the side AC eoual to 

15 mUe«3 t^e angle ADC 3 points, the interval between E oy N. 
and E. S. E. and Uie angle CAD 4 points, the distance between 
6. S. E. the oppodte point to N.N. WC, andE.S.E.; tofindCD = 
19<09. Again, in the triangle ABC are given AC as before equal to 

16 miles, the ansle ABC equal to 4 points, the interval between 
N.N.E. and E.N.E. and the angle ACB also 4 points, the interval be* 
tween the N. N. W. and N. N. £. points ; hence the angle CAB is a 
right angle ,- consequently, we get BC =: 21.21. 

Lastly, in the triangle BCD are given the sides CB, CD, equal to 
21.21 and 19.00 respectively, and we included angle BCD 5 points, 
die interval between N. N. E. and E. by N. ; to find the angles CDB 
= 67' SCy, CBD =66« 16' =6 points, CBE = BCN =2 points, and 
the distance BD = 19.09. 

Example XXXIV. 

The side AB of a pentagon being 180 toises, the face o£ the bas- 
tion AC 60, the normal or perpendicular KJL 30 ; it is required to 
find, by trigonometrical calculation, all the other lines and angles of 
the fortification, supposing the line of defence AH to be equal to a 
line drawn from A to D. 

Solution,>^-'H.ere -5- ^ -^ =:90:=AK. 

Hence, in the right-angled triangle 

AKL, AK (90) : R : : Hi (90):tan. 

LAK=18»20'. Because AB is the side iN. 

300** 
of a regular pentagon, we have > = 

72« = AOB, and ^ = 36<» = AOK, 

whence 90"— 36'=l64° = EAK, and 64<'— 1»> 20^ =: SS^ S4' =: BAO, 
which being doubled is 71" 8", t^ salient angle PAC or DBB. Join 
BG, then will ABC be a triaaivin which u-e given AB, AC, and 
their contained angle BAC ; to ted ABC=:0» 4&. Now sin. ABC {^ 
4ff) : AC (60) : : sin. BAC (18' 26") : BC =: 133.62, equal to the line 
of defence AH or BG. In tM triangle BCG, ABGK— ABCs=l8» 
20^~»' 48' == IP 38^=: CBG. Because BC = BO, we have 

^»^-^^^^^:=iy = 84oir = CGB. 

Again, because AB and EF areparaliel, and AH, BO equal ; we 
have the angles BAH, ABO, AHB, and EOF all equal, that is, each 
equal to 18° 26'. 

In the triangle CGH, we have the angle COB + BOH =84° 11' 
+ IS^ 2^ = ife^ 37 = CGH ; 180°— (CGH+ CHG) = 180°— (102° 
37'+ 18° 260 = 680 57' = the angle HCG ; and the side CH = AH 
—AC = 1^.62— 60 =83.52 =CH. Then sin. CGH (102° 37'): 
CH (83.62) : : sin. CHG (18° 26') : the flank CG or DH = 27062 : : 
sin. HCG (68* 670 : the curtain GH = 73.323. 



A 




K B 




t 




> 


^cf/^ 




E 






/ s 


T\ 



t 



44 INTRODUCTION. 

TABLE OF THE MEASUHES OF THE PRINCIPAL LINES AND 

ANGLES IN REGULAR PORTRESSES, FROM FOUR TO 

TWELVE SIDES INCLUSIVE. 



Names of Sides and Ai]r1«. 


Nin,« of Polygons. 


SquKE 


PHlUg 


He^ 


HepU. |0«*B. 


Nunag 


D«.g 


Ui>.h»; 


Dudec 












































































































































































































1S.7 




S7.1 






51.1 




7.4 
























iiSSIKEa" ■■ 


3?! 


'Wtl 


'wii 




ffi,! 


Z,". 


'^f.i; 






















































,M56!,«.6 










Br™m.<,riw!,>Tc,i«> 


1. 1 ,6 


.7 ■' 18 1 19 1 M 1 SI 


Si 


W 



APPENDIX. 



J 



[f to the 



BAROMETRIC MBASUREUKMT OF ALTITUDES. 

Having given a pretty fall view of the method of measuring the 
heiglits of objects geometrically, we shall here subjoin that of deter- 
■nining them by the barometer, thermometer, and hygrometer. 

That the observatioaa may be carefully and properly made, the 

persons who undertake them should be provided with two portable 

barometers of the best construction, fllled with mercury of the same 

roectfic gravity, on which, by means of a vernier properly adapted t 

toe scale, the height of the mercurial columns may he read o 

jMHhh part of an inch; each barometer being fitted up wit! 

tached thermometer, set in the wooden &ame in the same mi 

the barometer tube is. The ball of each thermometer would be best 

' If nearly of the same diameter as the barometer tube. Besides 

I these, they must also be provided with two other thermometers de- 

I tached from the barometers. Of these barometers, one, with its at- 

I tached and detached thermometers, is to be placed in the shade at 

L tile top of the eminence, while the other remains below. Let them 

' continue in their places at least a sufficient time for the detached 

I thermometer to acquire the temperature of the air, that is to say, till 

\ the contained fluid is stationary. Then the observer on the emi- 

ice must note down the height of the mercurial column in the ba- 

l<rometer, at well as tlie temperatures enhibitcd by the attached and 

detached thermometers ; and, at the Maine lime, the other observer 

St make like observations upon the instrumenti^ below. 




PLANE TRIGONOMETRY. 

tMs numner^ three or four sets of observations be taken^ at each 
tion^ after short intervals of time^ and the mean of the results ftir^ 
nished by these sets respectively be taken^ the probabili^ of error in 
the [true altitude deduced bj the following rules wiU be much 
diminished. When our third method of computation is adopted^ two 
of Darnell's hygrcmieters must be employed to determine the dew 
points at each station. If the observations be repeated on several 
successive days^ the position of the instruments ought to be changed 
at each station alternately^ at the same time comparing each pair of 
instruments to determine their index error should there be any. It 
is also advisable to make the observations in serene weather^ be- 
tween 11 and 12 o'clock. For it has been found that the com- 
puted heights are too small, when the observations have been made 
near sunnse or sunset^ or when the wind blows fresh from the 
south ; and that^ on the contrary^ the computed results are too greai, 
when the observations are made about three o'clock in a hot sum- 
mer day^ or during a brisk wind from the north or east.* 

I. Dr Robison's Method. 

In this method no tables are required ; it will be sufficiently exact 
for most purposes^ and is not difficult to remember. It was deduced 
from the following considerations : 

1. The height through which we must rise in order to produce 
any fall of the mercury in the barometer is inversely proportional to 
the density of the air^ that is^ to the height of the mercury in the 
barometer. 

2. When the barometer stands at 30 inches^ and the air and quick- 
silver are at the temperature of 32^ of Fahrenheit's thermometer^ we 
must rise through 87 feet to produce a depression ^^ of an inch. 

3. But if the air be of a dilierent temperature^ this 87 feet must be 
increased or diminished by about 0.21 of a foot for every degree ^ 
difference of the temperature from 32^. 

4. Every degree of difference of the temperatures of the mercury 
at the two stations makes a change of 2.833 feet in the elevation. 

Hence the following rules : 

I. Take the difference of the barometric heights in tenths c^ an 
inch ; call this D. 

II. Multiply the difference d between 32^ and the mean tempera- 
ture of the air by *21^ and take^the sum or difference of this product 
and 87 feet. This is the height through which we must rise to cause 
the barometer to fall from 30 inches to 29.9 ; and may be called h, 

SODA 
Thus is the approximated elevation very nearly. 

191 

IV. Multiply the difference ) of the mercurial temperatures by 
2.833 feet^ and add this product to the approximated elevation if the 
uTOer barometer has been the warmest ; otherwise subtract it; then 
will the resulting sum or difference be the corrected elevation. 

Or^ this rule may be expressed by the following formula^ where d 
is the difference between 32° and the mean temperature of the air, D 
is the difference of barometric heights in tenths of an inch, m is the 

• One person may perform the whole operation with one set of instruments, by mak- 
ing the observations two or three times alternately at the top and bottom, and taking a 
mean of the results at each station. 



II the mercurial tem- 



46 INTRODUCTION. 

mean barometric height, 3 the difference betw 
perature, and E ia tlie correct elevation. 

E^30(87±0.21,Q^,^,g33 

For an example, suppose that the mercury in the barometer at the 
lower station was 29.4 inches, its temperature 50" of Fahrenheit's 
thermometer, and the temperature of the air 45' ; the height of the 
mercury at the upper station 25.19 inches, its temperature 46, and 
the temperature of the air 39°. 

Here D = 294 — 251.9 = 42.1 
k = 87+(10x-21)= 89.1 
m = ^(29.4+25,19)= 27.295 

zi- — = apprtMimate elevation = 

Correction for temp. 



X 2.833 = 



4123.24 
U33 



n feet 



4111.91 



% 



II. Dr HvtlonS Method. 

1. Observe the height of the barometer at the bottom of any height 
OT depth intended to be measured, with the temperature of the quick- 
rilver by means of a thermometer attached to the barometer, and also 
die temperature of the air in the shade by a detached thermometer. 

2. Let the same thing be done also at the top of the said height or 
depth, and, at the same time, or as near the same time as may be. And 
let those altitudes of the barometer be reduced to the sr.me tempera- 
ture, if it be thought necessary, by correcting either the one or other ; 
that is, augment me height of the mercury in the colder temperature, 
or diminish that in the warmer, by its gnn„th part for every degree 
irf difference of the two. The altitudes so corrected being denoted 
by M and m. 

3. Take the difference of the common logarithms of the two heights 
of the barometer, corrected as above, if necessary, cutting off 3 figures 
next the right hand for decimals, when the log. tables go to 7 figures, 
or only 2, when they go to 6, and so on ; or, in general, remove the 
decimal point 4 places more towards the right hand, those on the left 
are fathoms in whole numbers. 

4. Correct the number last found for the difference of temperature 
of the air as follows : Take half the sum of the two temperatures for 
the mean one ; and for every degree which this differs from the tem- 
perature 31°, take so many times the iJjth part of the fathoms above 
found, and add them if the mean temperature be above 31", but sub- 
tract them if the mean temperature be below 31° ; and the b 

'- lUfference will be the true altitude in fathoms ; or, being a 
by C it will be the altitude in feet. 

Same example. 
TAermomelers. 



Detached 
45 



Attached 
50 



Barometers. 




PLAN£ TJUGONOMKTBY. 4H 



As 9600: 4:: 29.4: .0123 
Mean 42 corr. .0123 



Stand 31 M= 29.3877 log. 4681666 

Diff. 11 m = 25.19 log. 4012282 

485 : 11 :: 609.374 : 16.924 
Corr. 16.924 



The altitude sought 686.298 fathoms. 
Liet the state of the barometers and thermometers be as follows 
to find the altitude. 



Barometers. 
Lower 29.68 
Upper 25.28 
Altitude 719.897 fathoms. 



Thermometers. 
Detached. I Attached. 

57 67 

42 I 43 

M(Uhod m. 

The foregoing methods have been found from experience to give 
results tolerably correct in ordinary circumstances^ thoudi they de- 
viate considerably from the truth in peculiar cases. To obviate 
this> as far as possible^ we have given another method^ which^ it is 
boped^ will prove very accurate. 

In this case let B be the height of the English barometer at the 
lower station, b thut at the upper, t, the temperature by Fahrenheit 
at the lower, and f that at the upper, L the latitude or the place of 
observation, y* the elastic force of vapour at die lower, voAf^ that at 
the upper, and H the height of the oneplace above the oUier in fieet, then 



{ 



3f 
2 



H= 60000 ^(1.376) ^^ (1+0.00268 cos. 2 L) x 

(^+ B+6{i+ox)ooi(/-^)i)^'«- (iin-o.doOl(^)]~^/0} "^^^ 

tti' 

2 



The fiictors (1.375) ^^ and 1+0.00268 cos. 2L, may be re- 
duced into tables; and, if given in logarithms> they will be very 
leadBly applied. If the eentignde thermometer be used, then 

H =5 60845.6 \ (14>75)**(l+0i»268 cos. 2 L) = 

/I+^- i±il. -^loir./'-T ^~^ \ l^B^ 

V B+5((+aoooi8(i-^)} ^ \*{i+o.oooi8«-r)}-tf ; f ^ ^ 

In which case also B^ b,f, andjT' may be given in reference to the 
French standard metre.* 

Hie log. of the constant 60000 feet may be employed with ad- 
vantage, being 4.778151. 

If Laplace's constant 18393 metres, or 60345.6 feet, be taken, the 
constant logarithm would be 4.780646, and the &ctor 1+0.00268 
cos. S L must be used^t 

* 8ee l^kt'ii TnXtk de Phygique, Tome I. p. 531. 
Lapltet's oowiMft is fei^impn the moie tocunte, it may be used in both caseg. 



48 



INTKODUCTION. 



BAROMETRIC TABLES. 
TABLE I. 

TABLE OF THE DEPRESSION OF MEBCUBY IN GLASS TUBB8. 







DepmtloiMtaT 






Ivory. 


LapUce. 


Young. 


In. 


In. 


In. 


In. 


0.05 


0.29494 




0.2964 


aio 


0.14028 


ai3Q40 


0.1424 


0.15 


0.08028 


0X>8&38 


00880 


0.20 


0.05811 


0.05798 


0.0589 


0.25 


0.04075 


0.04117 


0.0404 


0.30 


0.02916 


aosoes 


0.0280 


0.35 


0i)2110 


0.03165 


0.0196 


0.40 


0X)1534 


0.01591 


0.0139 


0.45 


OMin 


0.01174 


0.0100 


0.50 


oxmss 


0.00868 


0.0074 


0.60 


0.00443 


0.00462 


0.0045 


0.70 


0J0O22S 


0.00244 




0.80 


0.00119 


0.00128 





This table is to be used only when two barometers, difiering c^i- 
siderably in thdr internal diameters^ are employed. 

The expansion of the resume of merciny for 1^ Fahr. = O.OOOiB^ 
more correctly than 0.0001, though the dinerence in the nicest hard- 
metric observations is almost insensible. 

TABLE II. 

MB DALTON's table OF THB BLASTIO FOBOB OF AQUB0U8 VAPOVB-- 

Barometer 30 Inches, 



Ttmp. 
Fahr. 


Fotce. 


Temp. 


FoKce. 


Temp. 


Fosoe. 


Temp. 


1 Force. |Teni|».l Warm* 1 


Inches of 


Fahr 


Inches of 


Fahr 


Inches of 


Fahr 


Inches of 


Fahr 


Incbea of 


Mercury. 


A wm^Mm • 


Mercury. 


Jb MAJA* 


Mercury. 


A CH-'A* 


Mercury. 


A Wil lA » 


Mercury. 


Of 


0.064 


20 


0.129 


40 


0.263 


60 


0.524 


80 


1.000 


1 


0.066 


21 


0.134 


41 


0.273 


61 


0.542 


81 


1.040 


2 


0.068 


22 


0.139 


42 


0.283 


62 


0.560 


82 


1.070 


3 


0.071 


23 


0.144 


43 


0.294 


63 


0.578 


83 


1.100 


4 


0.074 


24 


0.150 


44 


0.305 


64 


0.597 


84 


1.140 


5 


0.076 


25 


0.156 


45 


0.316 


65 


0.616 


85 


1.170 


6 


0.079 


26 


0.162 


46 


0.328 


66 


0.635 


86 


litlO 


7 


0.082 


27 


0.168 


47 


0.339 


67 


0.655 


87 


1.240 


8 


0.085 


28 


0.174 


48 


0.351 


68 


0.676 


88 


1.S80 


9 


0.087 


29 


O.IBO 


49 


0.363 


69 


0.698 


89 


1.S20 


10 


0.090 


30 


0.186 


50 


0375 


70 


0.721 


90 


1.860 


11 


0.093 


31 


0.193 


51 


0.388 


71 


0.745 


91 


1.4tOO 


12 


0.096 


32 


0.200 


52 


0.401 


72 


0.770 


92 


1.440 


13 


0.100 


33 


0.207 


53 


0.415 


73 


0.796 


93 


1.480 


14 


0.104 


34 


0.214 


54 


0.429 


74 


0.823 


94 


1.630 


15 


0.108 


35 


0.221 


55 


0.443 


75 


0.851 


95 


1.680 


16 


0.112 


36 


0.229 


56 


0.468 


76 


0.880 


96 


1.630 


17 


0.116 


37 


0.237 


57 


0.474 


77 


0.910 


97 


1.680 


18 


0.120 


38 


0.245 


58 


O.'^O 


78 


0.940 


98 


1.740 


19 0.124 


39 


0.254 


59 


0.507 


79 


0.071 


99 


1.800 



FLANK IWOOHOHETRY. 
TABpE IIL 

LOOARlTHXa OF THE BULK OF QAB, 

a the farmnlt ^ X log. 0.1(83097, in -wioth x 
at Aegreea above 32° Fahrcnlieit. 



c;w 


■ i«. Bdt 


T«np 


l,.B. 


T™v.| L^.U. jT.mp 


L=|.B. ' 


0' 


r.97M13 


35" 


1.994022 


60» 


0013830 


7.5^ 


0.033030 


1 




976181 


26 


.995390 


61 


0.014599 


76 


0.033807 


3 




976960 


27 


.996158 


52 


0015367 


77 


0.034667 


3 




977718 


28 


.996927 


53 


0.016135 


78 


003.5344 


4 




978486 


29 


.997698 


54 


0.016904 


79 


0.036112 


5 




979255 


30 


.998463 


55 


0.017672 


80 


0.036881 


6 




980023 


31 


.999232 


56 


0.018410 


81 


0.037643 


7 




980791 


32 


0.000000 


57 


0.019209 


82 


0.038418 


8 




981560 


33 


0.000768 


68 


0.019977 


83 


0.039IS6 


9 




982328 


34 


0.001,137 


69 


0/120746 


84 


a039964 


10 




983086 


36 


0002306 


60 


0.031514 


86 


0040723 


11 




983865 


36 


0.003073 


61 


0.022282 


86 


0.041401 


12 




984633 


37 


0.003842 


62 


0.023050 


87 


0.042369 


13 




93';401 


38 


0.004610 


63 


a023819 


88 


ao4saaa 


14 




986170 


39 


0.005378 


64 


0.024687 


89 


aoi378a 


15 




986938 


40 


0.006147 


66 


0.025356 


90 


0.044664 


16 




987706 


41 


0.006916 


66 


0.026124 


91 


0.046333 


17 




988476 


42 


0.0O7683 


67 


0.026892 


03 


0.046101 


18 




989243 


43 


0.008462 


68 


0.037661 


93 


0.046860 


19 




989911 


44 


0.00922O 


69 


0.028429 


94 


0.047638 


20 




990780 


46 


0.009989 


70 


0.039197 


05 


0.048406 


ai 




991548 


46 


0.010757 


71 


0.029966 


96 


0.048174 


g 




992317 


a 


0.011626 


72 


0.030734 


97 


0.049943 




993085 


0.013294 


73 


0031902 


98 


0-06O711 


24 




99386S 


49 


0.013062 


74 


6032271 


90 


0.061480. 


P. 1 


'. .1 .2 -3 -4 -6 •« -7 « •» 


totenthi 77 153 238 307 384 461 638 61fi 601 


TABLE IV. 


UMABITHHIG VA1.DH8 OF 1 +0.00368 OX. 3L. 


l«b 


i««. 


"., Let. 


Ul. , Las. 


Lit. 


Log. 


y 


0.001162 


13- 


0.001MS 


26- 


aooo?i6 


f 


sitt 


1 


0.001162 


14 


0«)1027 


27 


0.000684 


3 


0.001160 


16 


0.001007 


28 


O.000661 


41 




» 


woim 


16 


aoQoeee 


89 


0.1)00617 
0.000683 


$ 


aoooi32 


4 


0.001161 


17 


0.000964 


30 


« 


OoOOOBl 


5 


0.001146 


18 


0.000941 


31 


0.000646 


44 


0.000041 


6 


0.00U38 


19 


0.000916 


32 


0.000610 


45 


0.000000 


J 


0MU12» 


20 


0000801 


S3 


0.000473 


46 


9.999^9 


8 


0.001)18 


21 


0.000884 


34 


0.000434 


47 


9.0W9ia 


9 


0.001106 


22 


0.000836 


35 


0.000398 


48 


9.999878 


10 


0.001093 


23 


0.000803 


36 


0.000366 


49 


9.999838 


11 


0.001078 


24 


O.0OO778 


37 


O.00O321 


60 


9.099798 


13 0.001063 135 


0000747 


38 


0.000281 


61 


8.990768 



50 


mxRODUCTION. 


—\ 


^ 


i 




TABLE IV.— Continued. 






52- 


1^- 


fL 


Lgg. 


Xal. 


Log. 


I^L 


Lou. 


9.999719 


62- 


9.999349 


72- 


9.999059 


m' 


9.998882 




6a 


9.999679 


63 


9.999316 


73 


9.999036 


m 


9.998871 




M 


9.999640 


64 


9.999284 


74 


9.999014 


K+ 


9.998862 




bb 


9.999602 


6.1 


9.999263 


75 


9.998993 


!« 


9.C(E)8855 




b6 


9.999566 


66 


9.099222 


76 


9.998973 


!i(i 


9.998849 




bl 


9.999627 


67 


9.999192 


77 


9.998955 


«7 


9.998844 




M 


9.999490 


6K 


9.999164 


7« 


9.998938 


»! 


9.998840 




M 


9.999454 


69 


9.999136 


79 


9.098922 


m 


9,998838 




W) 


9.999418 


76 


9.999109 


JHt 


9.998907 


yo 


9.998838 




61 


9.999383 


Ji. 


9999084 


81 


9.998894 


1 



Example I. 

To determine the height of Arthur's Seat above the sea at Leith 

by the following observations, the height by levelling being 802.66 



■ Bar. 

Leilh Pier 29.567 
Arthur's Seat 28-704 

Fah. ther. 64^0 
50.5 

Sum 104 .5 Constan 

Half 52.25 log. B 
B=29.567, B-i/-=29.5 
h =28.714, 6— 1/-'=28.7 


Att. ther. Dct. Iher. 

56.25 64''.0 
51.75 50.5 

3.5 • 

28.704x0.0001 X 
fi=28.704+0.0 

t log. of 60000 feet 

37-0.062=29.505 io 
4-0.059=28.65510 

1.01256 log. 


Dew ptunt. 
50°.0 / =0.375 
48 .5 /'=0.357 

/+/'=0.732 
3.5=0.010 nearly, and 
10 :=28.714 

4.778151 

0.015367 
J. 1.469895 158 
. 1.457201 % 


Difference 

,,/+/' ,, 0.732 _ 
^+B + 6-'+58.281- 

1 


.0126941 


og.2.]0346S 
138 

0.006181 

206 

as 












2.902731 


H — 802.66 


japlace's 


constant is 




Defect =3,34 feet 

The operation, when ' 
lows: 


used, wou 


u 

d beasfot- 



■ The I and f in the dEnominaton of the fnu:liin» in tha formuU ihoold have b«ta 
1- txA r', Ihe tempentuica t^ the Bitschcd, 10 diatinguiih llicm (rani <ho«e of iho d*. 
tachcd ihennomelets. 



PLAN£ TAIGONOMETRY. 



£1 



Laplace's constant log. in feet 
1 + 0.00268 COS. 2 L for 56'' 
Mean temperature 52.25 log. B 



Difference of logs of corrected altitudes, log. 



1.01256 log. 



Hr=803.12 
H'=802.66 



4.780646 
9.909566 
OMSaffJ 

153 

38 

3.103462 

138 
0.005181 

206 
26 

2.9047^ 
70 

12 



Excess = 0.46 foot^ or 5^ inches 

EXAMPI/B II. 

Hequired the height of the Peak of Snowden above Caernarvon 
quay from the following set of observations ? 

Bar. Att. Ther. X>et. Ther. 



Caernarvon Quay 29.984 . 56.5 
Snowden Peak 26.2?! 42.75 . 

Constant loearithm 
Correction for latitude 53° 4' 
55°.25+43° ^^,.^^^ j^^ g 

B—l/ =29.920 log. 1.475962 
b'— if '=26.262 1.419328 

Difference, 0.056634 log. 



55.25 
43.00 



H = 3561.2 feet 
H'= 3555.4 



Dew Poinu 

. 60^.25 
41.00 

4.780646 
9.99967» 

0.013062 

77 

15 

4 

31 

2.753047 

0.004751 

247 
33 

a651592 



Excess 5.8 

Example III. 
- Captain Sabine found the height of a hill at Spitzbergen^ deter- 
mined geometrically, to be 1644 feet; required its neightDarometri- 
cally from the following set of observations ? 

Observed height of the barometer at the bottom. 

In. In. 

Barometer, (diam. of tube 0.30) 29.6735 Attached ther. 39°.75 
Reduction to 32° F. . —0.0200 
Capacity . — 0.0561 

Capillai^ action (Young) +0.0280 
Index . +0.1960 



Detached 34 .90 

Dew point. 34 .00 
By Darnell's hygro- 
meter. 



!■ . I 1* 



True height 



+ 0.1479 
29.8214 



W INTRODUCTION. 

Obser*«d height of the batometer, &c. at the top, 

Bai-offl^ter, (tliam. of tube 0.15) 28.0075 AtUcbed ther. 36".4 

Reduction to 32° P. —0.0105 Detached . 35 .4 

Capacity . . —0.0443 Dew point Dl. 35 .5 
C^OlMJ action (Young) +0.0880 



+ 0.0330 
Tfue height 28.0405 

Constant logarithm 
Correction for latitude about 80" 
B —i/=2e.8214— 0.0357=29-7857 log. 1.474008 
6 —J/'=28.0405— 0.0375=28.0030 log. 1.447204 



Diffeiwtue 0.026806 log. 
Mean temperature g ^'^+^^- .5— 35.2 log. B 
i+ /tr_ 0ai4i^225_ 0.439 „ 



4.780(>46 
9.998907 



0.003039 
247 



Geo. H = 1644 

Dkference — 9 feet 

By another set «f observations. 

Barometer, at bottom 29.8304, at tap 

Attached therraometer 39°.4 

Detached 35.4 . . 34.2 

Dewpoiht 35.4 . . 34.2 

Constant logarithm 

Ctwrection for latitude 80° . - . 

B-Hi/ =29-8304— 0.0374=29.7930 log. 1,474115 

«—j^"'=28.0624— 0.0360=28.0264 log. 1.447569 

Difference 0.026550 log. 

Mean temperature of the air g ^ 34.8 lo^. B 



./+/'- 



4.780646 
9.998707 



2.424065 

0.001537 

615 



" =1,00762 log- 



J)lff. — 25.7 
,By the firet set or't 
By the secon*! 



■itperiinents H 



1 1635 feet 
1618 



3.209007 



A 



PLANE TRIGONOMETRY. IS 



Captain Sabine thinks there is some error in the aeeoad set of 
perim^ts^ Brising fhmi the larcumstatice, duit Mr Fottcr> his asrfslaiit, 
was obliged to hold die imtramenti to prevent theifa|^tatioii hf tlib 
wind. 

It is proj^ to remark^ that Captain Sabine finds 1644^ for the 
first and 1090.66 fbt the second set of bbsenrations, as stated !n the 
Philosophical Transactions of the Boyal Society of London^ but the 
particular formula he used is not mention^ The vmA for- 
mulas given by Roy^ Shuckburgh^ and Laplace may give the 
height more near the geometrioal method in certain cases^ todi as in a 
mean state of the atmosphere, than that which we have given, 
though there is no doubt but that the drcmnatuioes which have ia* 
duced us to give a new method, inv(dving considerations not usual- 
ly attended to in such measurefneatSy art more conformable to the 
laws of nature, and will In time become more accurate as those 
branches of physical science on which they depend are rendered 
more perfect 

The dew point is supposed to be found hf Daniell't hygroneier. 
If 4hat instrument is not at hand, the dew poutt may be fbiuid Vy two 
good thermomceters, oixie of which has its ball eovered with inn«ff^ynffd 
tisbue-paper, as pr(^>osed by Mr Anderscm, Rector of the Academy of 
^erth) who also gives a fbrmula for the barometric measurement of 
altitudes, in which in scnne of the corrections I have been antici- 
pated. 

Let F, the elastic force of vapour by Dalton's table be thus re- 
duced toj^ticcordiDg to the difference between the naked and covered 

thermometers, then/i=F— ^'^^^P =F— 0.00(»2>/ Xp, in which 

it 18 the difference between the temperatures of the thermometers^ and 
p the barometric pressure. 

Now let ^ be we elastic force at the dew point, then 
^ / _ F-X).00(»ap>< ^,,.,^ ... 

*^""1+0.002084X/— O"" l+0.0021(t-:?)""~'*^ • ^^' 

Seir)9 j^> the temperature of the dew^-point is ankaonvL but may bfe 
determined, first approximately firom the numerator of the fonoula, 
and then substituted in the denominator, and a second approximation 
obt^ed, which will generally be sufficiently correct 

To exemplify this, let the thermometer with the dry baU show 00^ F, 
and that covered with moistened tissue paper 51^ 

t'-^orit. , . . . Si 

Now if thelbafometef be at 30.4 inches we have from liie numera- 
tor of formula <l)/==0.6a4--O.00092x 84 x30.4=^.524--«^ 
'0;S86. nils j^ corresponds, "by the table ca Daiton to 42? 4K8rly, 
-Which l>eing substituted for V in the denominator of the formula 

286 0.286 

^^ *= rvsismsB^^ 15378 = ^-^^ "*** ^^^ «*^^« 

<'=4P.3, the dew point. This is^<»hapis one of tfa6 best methods 
of determining the point of deposition, as tlie instnimentB are not, 
like tlie hygrometers of Deluc andSaussure, liable to be deteriorated 
by time, and besides, may still ansinsr oCherputpotes which none of 
the usual hygrometers can. 

Cor.— -From the same principles, may be derived a formula ix> 
determine the weight of moisture in 100 cubic inches of air or 



64 INTRODUCTION. 

W = iiQ^l^ff_^) ^ *e freezing point. When (p=.-2756 and 

<'=4]^ we get ftom the expression W=:0.1837 grains when the air 
is completely saturated with humidity. But when the temperatures 

.re 60» «.d 4P the W= i^O-O^^^l) =^'^'^^ «'^ ^ ^^ 
cubic inches. Perhaps this method may be conveniently compared 
with Mr DanieH's^ to show their relative accuracy and consistency. 
* It may be added^ that Mr Dalton states from experiments at mo- 
derate heights^ that an elevation of 240 feet ^ives a depression of 1° 
temperature Fah. and an elevation of 390 feet gives a depression of 1°F. 
of tne dew point. Hence^ if t be the temperature and u the dew point 

^, AH .^^ AH 

Method IV. 

For ordinary heights^ such as those usnallv met with m Britain^ 
the following method^ requiring no tables^ which is somewhat simpler 
and more easily recollected than Dr Robison's^ it subjoined. 

Let B be the barcmietric altitude at the lower situation, and d that 
at the upper corrected for the difference of temperature in the usual 
manner, the atmosphere being in its mean state with regard to 
aqueous vapour^ &c. 

Then H^1310a^5±*^?:=*i / l+0.00!m(-^^~3Sf) y in feet. 

Bar. in. Att. Thcr. Det. Ther. 

JBor.— Leith Pier 29.567 55°^ 54° 

Arthur's Seat 28.704 51 f 50 ^ 

2a704 X 0.0001 X 3.5=0.010, and 28.704+ 0.010=2a714=6 

l + (^^i^*--32<')x0.00245=l + 20ix0.00245=i.04961, hence 

^=^3100X^5^X1.04961 = 805feet 

Height by lerelling 803 

Difference .... 2 feet. 

EXAMPLBB FOR EXBRCISE. 

1. If the base of an oblique-angled plane triangle be 40, and the 
other two sides 20 and 30, what is the length of the perpendicular? 

^«*.— 14.52309. 

2. If the base of a plane triangle be 40, and the other two sides 20 
and 90, what are the segments of the base made by a line bisecting 
the vertical angle ? Ans. — ^24 and 10. 

3. The hypotenuse of a right-angled triangle is 19630040, and one 
o£ the legs 19630000 ; required the two acute angles ? 

. i<iM.— 6' 56".4, and 89° 53' 3".6. 

4. If the sides of a plane triangle be in proportion to each other as 
'di4»^ numbers \, J, and ^ ; what are the angles ? 

Ans:—l 17° 1 6' 46", 36° 20' 10", and 26° 23' 4". 

5. At the Observatory on the top of the Calton-hill, 350 feet above 
the sea at Leith, the angle of depression of the horizon marked by 



PLANE TRIGONOMETRY. 



55 



the sea down the frith of Forth was 18' 12" by observation. Now 
supposing the effect of refraction to be one-twelfth part of the whole> 
this must be increased by one-eleventh of itself^ or the true depres- 
sion would be 19' 51^28. Required the earth's diameter ? 
Ans. — ^7946 miles. 

6. Suppose the height of Melville's Monument^ in St Andrew's 
Square^ Edinbursh^ to be 60 feet^ and that the figure placed upon 
die top of it is 12 feet high, at what distance from the monument may 
the statue be viewed under an angle of .3% and what is the greatest 
angle under which it can be seen ? 

Ans. — It will be seen^ under an angle of 3^ at the distance of 
208.23, or 20.75 feet^ and the greatest angle nnder which it can be 
seen from a point in the horizontal plane is 5^ 13'. 

7- It is required to find the distances from the Edystone light- 
house to Plymouth, Start Pointy and the Lizard respectively from the 
following data : 

1^ Plymouth to Lizard 60 \ 

The distances from •< Lizard to Start Point 70 > miles. 

( Start Point to Plymouth 20 1 . 
Plymouth \ (North 

Lizard >- bears from Edystone -J W. S. W. 

Start Point j 

i Lizard 
Ans, — From Edystone to < Plymouth 

^Start 
Thermometers, 
Attach. I Detach. 
38 31 

41 35 



8. Barometers. 
Lower 29.45 
Upper 26.82 



{% by N. 

53.04i 

14.33% miles. 

17-36I 
Required the AUiikde* " 
-^11*.— -409.61 fathoms, 

or 2458 feet, by Hut- 

ton's method. 



BXAMPLB8 BY THB FRENCH AUSASUUBS. 



Attacned"* 
Thennomeler. 



Thormoioetar. 



. Point. 



Obseryer 
Humbert. 



Quindiu, 
Pac. Oc. 



Chimb. 
Pac. Oc 



. Helgiitof 
Barometer. 



0-.50981820°.0cent. 
.76294425 .3 



0-.377275 10^.0 cent. 
.762000.25 .3 



18°.75 cent 
25.30 



— l°.6cent 
+25.3 



16^0 cent 
BO .0 



0°.0 cent 
20.0 



Latitude. 



5° ON. ^, 
H=3543" 



P45'N. 
H=::692aH 



Calculation of the last Example by Method III. 
Constant 18393 metres log. 

(1.375) 100 = ii^ X 0.138303 = 

Latitude l"* 45' log. . . 

B— i/=0.759114 log. 1.880307 
6— J/' =0.377471 log 1.576884 



4.264653 

0.016389 
0.001161 



Difference 
f±J 
B+ 



0.303423 log. 



1 , f+f'-i I 0-022373 _ ^Q^ 
^+ B+ i-^ + 1.136585 ^ ^'^^^^^^ 



1.482048 
0.008467 



H=5925 4 metres . 3.7727I8 

Or 19441 English feet> the height of Chimborazo above the level of 
the Pacific Ocean. 



SPHERICAL TRIGONOMETRY. 



De/lnilions, Prhidples, and General Projierties. 

fl. Spherical Trigattomelry is that branch of mathematics by which we 
B enabled, in all cotes, where three of the six parts of a triangle 
•med by arcs of great circles in the surface of a sphere are given, 
I to compute or determine the other three. 

I 2. In plane trigonometry tlie knowledge of the three angles is not 
I flufficient for ascertaining the sides; for in that case the relalions 
I only of the three sides can be obtained, and not their value ; whwe- 
I W] IQ tpkerical trigonometry, when the sides are circular arcs, whose 
' T^ue depende on meir proportion to the whole circle, that is, on the 
I number of degrees they contain, the sides may always be determined 

when the three angles ore known. Among other remarkable dif- 
I fcrences between plane and spherical triangles are, 
I {!,) That in the former, two known angles always determine the 

Aiird J while in the latter they never do. 

(2.) The surface of a plane triangle cannot be determined from a 

knowledge of the angles alone j while that of a upherical triangle al- 

ways can. 

3. A tpbere or ghibe is a round body formed by the revolution 
ot a, senucircle about its diameter, which remainfi fixed. 

4. The oenlre of the sphere is the same with that of the revolv- 
ing semicircle. 

5. The axis of the sphere is the straight line about which the 
Banidrcle revolves. 

Proposition I. 

6. If a sphere be cut by a plane, the section wijl be a circle. 

I^ the sphere AEBF be cut by the plane ADB ; tlien will the 

section ADB be a circle. Draw tile chord, or ^ 

E ^ diameter of the section AB, perpendicular to 
^ the section ADB, and through the centre C 

draw the axis of the sphere ECGP, which will 

(Euc. III. 3.) bisect the chord AB in the point 

(J. Also, join CA, CB; and draw CD, GD, 

to any point D in the perimeter of the section 

ADB. 

Then, because CG is perpendicular to tlic plane 

ADB, it must be perpendicular both to GA and 

GD. Hence CGA and CGD are two right-angled trianglCB, having 




SPHERICAL TRIGONOMETRY. 57 



the perpendicular CG common, and the hypotenuse CA equal to 
the hypotenuse CD, being both radii of the same sphere ; there- 
fore their third sides 6A, GD, are also equal. In like manner, it 
may be shown, that any other line drawn from G to the circumfer- 
ence of the section ADB, is equal to GA, or GB ; and consequently 
that section is a circle. 

Cor.— If a sphere be cut by a plane through the centre, the section 
is a circle, having the same centre with the sphere, and equal to the 
circle by the revolution of the half of which the sphere was described. 
For all the straight lines drawn from the centre to the surface of the 
sphere are equal to the radius of the generating semicircle. There- 
fore the common section of the spherical surface, and of a plane 
passing through its centre, is a line lying in one plane, having all its 
points equally distant from the centre of the sphere, and is conse- 
quently the circumference of a circle, having for its centre the centre 
of the sphere, and for its radius, the radius of the sphere, that is, of 
the semicircle by which the sphere is described. It is therefore 
equal to the circle of which that semicircle is a part. 

7- Any circle formed from the section of a sphere^ by a plane 
through its centre^ is called a great circle of the sphere. 

Cor.— -An great circles of uie sphere are equal ; and any two of 
them bisect each other. 

They are all equal, because they have all the same radii, as has 
just been shewn, and any two of them bisect one another ; for^ as 
they have the same centre, their common section is a diameter of 
both, and therefore bisects both.* 

8. The pole of a ffreat circle of the sphere is a point in the surface 
of the sphere equidistant from every part of the circumference of 
that circle. 

9. A spherical angle is an angle on the surface of a sphere con<* 
tained by the arcs of two great circles which intersect each other, 
and is the same as the inc£nation of the planes of, or tangents at the 
point of intersection to, these great circles. 

10. A spherical triangle is a figure on the surface of a sphere form- 
ed by the intersection of three arcs of great circles, each of which is. 
less than a semicircle. 

11. A right-angled spherical trj^ele has one right-angle; the 
sides about the right-angle are c^led legs, and that opposite the 
right-angle is called the hypotenuse. 

12. A quadrantal ^herical triangle has one side equal to a qua- 
drant, or 90®. 

13. An obUque'&ngled spherical triangle has none of its angles 
right. 

14. Spherical triangles are also called equilateral, isosceles^ or soa^ 
lene, according as they have three sides equal> two sides equal, or all 
the three sides unequal. 

15. Two arcs> or angles, when compared together, are said to be 
aUke, or of the same affection, when bom are less, or both are greater 
than 90*'. But when one is less, and the other greater than 90^, iJbey 
are said to be unlike, at of different affections or characters. 

16. Every spherical triangle has three sides and three angles; 



* Hence the intenections of ilfe dxcumfeieDces of two great circles aie two ^gcft&ioik- 
diametrically oppoeite to each other. 




58 INTEODUCTION- 

•nd if any three of these ux parte be given, the other three may be 
found. , . , 

17. A iune it a part of the rarfcce of » sphere contuned by the 
•emicircumferences of two irreat circle*. 

la A twiaU circle of the sphere ia that who« pl»ne doa not paa 
through the centre of the spnere. . . , 

Ifl. The small circles of the sphere do not fall under the conuder- 
ation of splierical trigonometry, but such only a* have the Hina 
centre with the apbere itMl£ And hence it is that sphmcal tngo- 
nometry ia <^ so much um in practical astronomy, the apparat 
heavens assuming the shape of a concave sphere whoae centoc 1* the 
uioe as the centre of the earth. . . , 

20. The tides of a sijierical triangle are all arcs of great orcks, 
which, by their intersection on the surface of a sphere, oonatrtate 
that triangle. 

21. If ABDO be a great drcle of the 
qdiere whose centre is C and PCF a dia- 
meter of the sphere perpendicular to its 
I^e, the pointa P, P'^are the poles of that 
circle. And if the small cirde a&cd be 
perpendicular to PP*, we call P. P* the 
poles of that small circle also. 

52. The great circles PAP', PG?", pas* 
ing through the poles P, P' of the great 
drcle ABDC, are called aeeondariet to that 
circle. 

Proposition II. 

53. If two arcs of circles meet each other they make twa anglM^ 
vhich are together equal to two right-angles. 

iLet the arc AB meet the arc CD in the point 
B ; Uien will the two angles ABC, ABD be equal 
to two right-angles. For, auppoae the arc BiG 
to be perpendicular to CD, then the angles 
EBC, EBD are right-angles. 

And since the angle EBD is equal to the angles C^ 
EDA, ABD, the three angles. EBC. EBA, ABD, ■" 

are equal to the two right-angles. 

But the two angles, EBC, EBA, are equal to the angle ABC; 
whence the two angles, ABC, ABD, ate uso equal to two right* 
angles. 

Proposition III. 

84. If two arcs of a circle intersect each other, die vertJCid, or 
opposite angles, will be equal. 

Let the two arcs, AB, CD, intersect each other in 
B, then will the angle AEC be equal to DEB, and 
AED to CEB. 

For since 'Ok arc AH meets the arc CD, the 
angles ABCj AED are together equal' to two 
right-angles, {Prop. II.) ' ■' i 

And because the arc DE meets the arc AB. the 
an^es DEB, DEA are also equal to two right-angles. 

Taking away fi-om eacfa the C(»ntnon''angle'AED, and tlie re> 





SPHERICAL TRIGONOMETRY. 5Q 

Vnaining mgle, AEC will be eqaal to DEB. In the game manner 
it may be proved that the angle A£D is equal to CEB. 

Cor.*— Hence if any number of arcs of circles intersect each other, 
all the angles formed about the pdnt of intersection are together 
equal to four right-wangles. 

Prii^osxtion IV. 

25. The arc of a great circle^ between the pole and the drcum« 
ference of another great circle^ is a quadrant. 

Let ABC be a great circle^ and P its P^e; if PC, an arc of a 
great circle^ pass uirough P and meet Abc in C, the arc PC is a 
quadrant. 

Let the circle^ of which PC is ah arc, meet ABC again in A, and 
let AC be the common section of the planes 
of these great circles^ which will pass through 
E, the centre of the sphere: Join PA, PC. 

Because AP=PC» (def.), and equal straight 
lines in the same circle, cut off equal arcs, 
the arc AP =r the arc PC ; but APC is a 
semicircle, therefore the arcs AP, PC, are 
each of them quadrants. 

Co^. 1. If P£ be drawn, the angle AEP is a rigfat^mgle; and 
P£, being at right'-angles to every nne ft meets with in the plane of 
the circle ABC, is at r^ht- angles to that plane. Thei^fere the 
straight line drawn from the pole of any great circle to the centre 
of the sphere is at right-wangles to the plane of that circle ; and, conr 
versely, a straight line drawn from the centre of the sphere perpen- 
dicular to the ^ane of any great circle, meets the surface of the 
sphere in the pole of that circle. 

Cor, 2. The circle APC has two poles, as has been shewn in 
art. 21., one on each side of its plane, whidi are the extremities of a 
diameter of the sphere perpendicular to the plane APC-; and no 
other points but these can be poles of the circle APC. 

Proposition V. 

26. If the pole of a great circle be the same with the intersection 
t>f other two circles, the arc of the first circle intercepted between 
the other two, is the measure of the spherical angle wnich the same 
two circles make with one another. 

Let the great circles AP, BP, on the surface of the sphere of 
which the centre is O, intersect each other in 
P, and let AB be an arc of another great 
circle of the pole as P, AB is the measure of 
the spherical angle APB. 

Join PO, AO, BO ; ance P is the pole of 
AB, PA, PB are quadrants, and the angles 
POA, POB are right; therefore the angle 
AOB is the inclination of the planes of thecirdes 
PA, PB, and is equal to the spherical angle 
APB; but the arc AB* measures the angle 
AOB, therefore it also measures the spherioJi angle APB. 

Cor. If two arcs of great circles, PA, PC, which intersect each 
other in P, be each of them quadrants, P will be the pole of the 




INTIlOimCTlON, 



gpe^t circle whidi passes through A 
«C3. For since the arcs PA and PB 



id B, the extremities of those 
. . , idratits, the angltis 

I'OA, POB are right-angles, and PO ia therefore perpendicular to 
|iia plane AOB, that is, to the plane of the great drcle which passec 
ttiraiigh A and B. The point A, therefore, is the pole of the great 
drcle which passes through A and B, 

Proposition VI. 

27- An angle made by any two great circles of the sphere is 
e^iual to the angle of inclination of the planes of these circles. 

Let BAE be a spherical angle made by two great circles CBA. 
CEA ; then will this aiigle be equal to the angle ^ 

of inclination of the planes of those circles. For, 
take the arcs AB, AE, each equal to 90% or a ■ 
oiiadi'ant, and through the points B, E draw 
||je arc of the great circle BE, and from D, the 
centre of the sphere, draw DB, DE. _„ _ _ 

^,| Then, because AB, AF are quadrants, 

A and C are the poles of the circle of which BE is a part, ajid the 
jines DB, DE are each perpendicular to the common section AC ; 
jmnsequently BDE is the angle of inclination of the planes CBA, 
C^A. But since DB, DE are equal, being radii of the same sphere, 
the angle BD£!, which is measured by the arc BE, is equal to the 
angle BAE, which is measured by the same arc. 

And if FH be drawn in the plane CBA, and FG in the plane 
^EA, each perpendicular to the common section AC, the angle 
HFG, which is equal to the angle BDE, will also be equal to the 
angle BAE. 

Cor. The angle BAE made by two great circles of the sphere 
BA, EA, is equal to the angle n A m, formed by two tangents ilrawn 
ftom the angular point A, one in each plane, these tangents being 
■each peifpendiciilar to the diameter AC. 

PSOPOBITION VII. 

28. The distance of the poles of any two great circles of the 
sphere is equal to the angle of inclination of the planes of thosc 
'circles. 

'' Let AEB, CED be two great circles, and P, P' their poles ; then 
■will the arc PP' be equal to the angle of their 
inclination AOC or BOD. 

For, since P is the pole of the circle AEB, 
and P' of CED, the arc PA will be equal to 
■ PC, being each quadrants, or 90°; and if PC, A^ 
which is common to each, be taken away, rfie 
remaining arc, PP', which is the distance of 
two poles, ia equal to CA, the measure of 
the angle of inclination AOC. 

' PHOPOBtTION VIII. 

29. The circumference of a secondary is at right angles to the cir- 
dunterence o! its great circle at the point of intersection. 

" T^e (Hrcction of the circumference of a gicnt circle at any point 




SPHEBICAL TBIOONOMETRY. 61 

being the same as the diameter of ita tangtent at that pointy theanffle 
OBT, (figarejprop. V^, is a right-angie, BT beine a tangent to fip 
at the point d. FOB is also a rigbt-angle> and the arc FB is in the 
plane POB^ therefore the direction of tiie circumference PB at B 
must be parallel to PO. But PO is perpendicular to the circle 
ABC ; therefore the circle PBP' is at B perpendicular to the circle 
ABC ; hence the arc PB at B is at right-angles to AB at B. For 
the same reason PAB is also a right-angle. 

Cor, 1. — If a great circle, PBP', be perpendicular to ABC, and 
BP^ BP' be taken each eqtial to a quadrant, or 90^, P, P are the 
poles of the circle ABC. 

Cor. 2. — If any two great circles, PAP', PEP, be perpendicular to 
the circle ABC, they meet at the poles P, P of that cirde. 

PfioapofiiTioN IX. 

30. In an isosceles spherical triangle the angles at the base are equal. 
Let ABE (figure prop. VI.) be a spherieal triangle, having the side 

AB equal to the side AE, the spherical angles AB£, ABB are equaL* 
Cor, 1. — Hence, if two of the angles of a triangle be equal, the 

sides opposite to lliem are likewise equal. 

Cor, 2. — ^A perpendicular drawn from the rertez of bm isoacelea 

spherical triangle to the base, bisects both the base and the vertical 

angles except when the two sides are quadrants ; in which case there 

are an indefinite number of peipen^cahnrs. 

Proposition X. . 

31. If the three sides of one sjpherical triangle be equal to the 
three sides of another, each to each, the angles whidi are opposite 
the equal sides are equal. 

Pboposition XI. 

32. If two sides and the included angle of one spherieal triangle 
be equal to two sides and the included angle in anotherj these two 
tr ingles are equal. 

Propobitioic XII. 

33. If from the angles of a spherical triangle, as poles^ there be 
described on the surface of the sphere three arcs of great drdes, 
which, by their intersections, form another spherical triangle, each 
side of this new triangle will be the supplement of the measure of 
the angle which is at its pole, and the measure of each of its angles 
the supplement to that side of the primitive triangle to which it is 
opposite. 

Proposition XIII. 

34. If the three angles of one spherical triangle be equal to the 
three angles of another, each to eacn, the sides which are oj^HDsite 
to the equal angles are equal. 

Proposition XIV. 

35. If a side and two adjacent angles of one spherical triangle be 
equal to a side and two adjacent angles of another, each to each, 
their remaining sides and angles will be equal. 

* The demoQSttstiQiia, whidi may be seen in ^syMr*! » lieMndre*s Geometrf , arc 
omitted, u they wonld •we^ this work too much, hut may perhaps appear in a mote 
complete tteatise on trigtMioaietry that has been long meditated. 



n '"' iNTnODUCTION- 

Pro POSITION XV. 

36. Tiie sum of any two sides of a Epherical triangle is greaUt 
than the third side ; and the difference of any two sides is less tliaO 
the (bird side. 

Cor. — The shortest distance between any two points on the sur- 
face of a sphere is the arc which passes through these points. 

Pbopobition XVI. 

37- The greater si Je of any aphetical triangle is opposite to the 
greater angle, snd the less aide to the less angle. 

And, in a similar manner, it may be shown that the less side is op- 
posite to the less angles and the less angle to the less aide. 

Fbopqsixion XVII. 

38. The sum of the three sides of any splicrical triangle is less 
than the circumference of » circle, or 300** ; and the difference rf 

any two sides is less than 180". 

Phoposition XVIII. 

39. The sum of the three angles of every spherical triangle is 
greater than two right-angles, or 180", and less than sis, or 5W°. 

Cer. — ^The sum of any two angles of a spherical triangle ia great- 
er than the supplement of the third angle. 

For the angles A+B + C, being greater tlian two right-angles, ot 
than ACB-h ACG, if ACB or C be taken away, the sum of the j-e- 
maining angles A+ B, will be greater than ACG. 

Proposition XIX. 

40. If the sum of any two sides of a spherical triangle be equal 
to, greater, or less than a semicircle, the sum of tlieir opposite 
angles will, accordingly, be equal to, greater, or less than two 
ri^t-angles; and conversely ■ 

And, in a similar manner, it may be shown, that if the sum of the 
two angles B and C lie equal to, greater, or leas than 180', the sum 
of the opposite sides AB and AC, will also be equal to, greater, or 
less than ISO". 

Cor. 1. — If each side of a spherical triangle be equal to, greater, 
or less than liKP, each of the angles will, accordingly, bo right, 
obtuse, or acute ; and conversely. 

Cor. 2. — Half the sum of any two sides of a spherical triangle ia 
of the same kind as half the sum of their opposite angles. 

PRoroaiTiON XX. 

41. In any right-angled or quadrantal spherical triangle, die legs 
or sides are of the same kind or affection as their opposite angles, 
and conversely. 

The same will also hold if the triangle be quadrantal ; for its sides 
and angles being the supplements of the angles and legs of the polar 
triangle, which in this case is right-angled, the stmuarity will be 
the same as before. 

Proposition XXI. 

42. In any right-angled spherical triangle the hypotenuse is less 
or greater than 90°, according as the two Teg«, or the two ftngle«, or 

a leg and its adjacent angle, are alike or unlike. 



SFH£BICAL TiUGONOMETEY. 03 



Section II. 

Solution of Spherical Trian^es. 

Having given a view of the general principles and properties of 
spherical triangles, the solution of the various problems in spherical 
trigonometry ought necessarily to follow. These problems may be 
resolved either by geometrical construction or by arithmetical calcu- 
lation. There are various methods of construction, but the most 
simple^ and generally employed^ is the stereographies in which all 
the circles of the sphere are represented by straight lines or circles. 

Of ike Siereogrofhic Prqfectkm qfihe Sphere. 

Dbfinitionb. 

I. To project an object^ as it is commonly called^ is to rq>reaent 
every point of that object upon the same plane, as it appears to the 
eye in a certain position* 

II. That plane upon which the, object is projected is called the 
plane of projection^ and the point where the eye is situated^ the /iro- 
jecttns point, 

III. The stereograpkic projection of the sphere is that in which a 
great circle is assumed as the plane of projection, and one of its poles 
as the projecting point. 

IV. The great circle, upon the plane of which the projection is 
made^ is called the primitive, 

y. By the semitangent of aiiy arc is meant the tangent of half 
that arc. 

VI. The line of measures of any circle of the sphere is thatdiamew 
ter of the primitive, produced indefinitely, which is perpendicular to 
the line of common section of the circle and the primitive. 

VII. The projection, or representation of any point in the sphere,, 
is the point m which the straight line drawn n'om it to the project^' 
ing point intersects the plane of projection. 

Thjsorem J. . 

Every great circle of the sphere, which passes through the pro- 
jecting point, is projected in a straight line, passing through tiie 
centre of the primitive ; and every arc of it, reckoned from the other 
pole of the primitive, is projectea into its s^nitangent.* 

Con l.-«-^very small drda/whidi passes through the projectiiig 
poiitt, is projected into that straight line which is its cntanum leetioii: 
with the primitive. 

Cor. 2. — Every straight line in the plane of the .jprimidvei ktfid 
produced indefinitely, is the prqjectioa of some circle on the sphere 
passing through the projecting poiat. • 

Cor, 3. — The stereographic projection of any point on the.sorfaoe 

■ ',;••' ■■' / r . ..J- . ■ \- .'"'■*. ■"•■•■ 

* For the Jnvesd^ptfWofd^praptrtlesiif this mtbod.rf^B^^ 
or Keith's Tiestises of Trigai»ikietry» aiid'W«K*a MatlMnsaab . ^v . 



M INTRODUCTION. 

of the sphere, is distant Irom the centre of the primitive by Ute ■&• I 
mitangent of the distance of that point from the pole opposite th« 
projecting point. 

Theorem II. 

Every circle of the sphere, which does not pass through the pro- 
jecting point, is projected into a circle. 

Cor. 1. — The centres and poles of all circles parallel to the primi- 
tive, have their projections in its centre. 

Cor. 2. The centre and poles of every circle, inclined to the pri- 
mitive, have their projections in the line of measures. 

Cor. 3. — All projected circles cut the primitive in two points di- 
ametrically opposite. 

Theorem III. 

The centre of the projection of a great circle is distant from the 
centre of the primitive by the tangent of the inclination of the great 
circle to the primitive, and its radius is the secant of the same. 

Thbobem IV. 
The centre of projection of a small circle, perpendicular to the 
primitive, is distant from the centre of the primitive by the secant of 
the distance of the circle from its nearest pole, and the radius of pro- 
jection is the tangent of the same. 

Theobem V. 
The projections of the poles of any circle inclined to the primi- 
tive, are in the line of measures distant from the centre of the primi- 
tive by the tangent and cotangent of half its inclination. 

Theorem VI. 
Any two circles upon the sphere, passing through the poles of tw» 
great circles, intercept equal arcs upon them. 

Theorem VII. 
If, from either pole of a projected great circle, two straight liiiea 
be drawn to meet the primitive and the projection, they will inter., 
cept corresponding arcs of these circles. 



ffr!^ 



Solution of Right-AngUd Spherical Triangles. 



"^e solution of right-angled spherical triangles may be accom^- 
plished by formula? investigated expressly for thnt purpose. We are" 
indebted to Napier, however, for a comprehensive rule of great ad-' 
vantage to the memory, by reducing all the theorems employed in the 
solution of right-angled triangles to two. This is called ttie rule o/ /A*! 
circular parts, and is perhaps one of the happiest examples of arti-"' 
Hcial memory that is known. 

Definitions. ' 

I. If in a right-angled spherical triangle the right-angle be set 
aside, and the five remaining- parts of the triangle alone be consider- 
ed, consisting of the three sitles, and the two oblique angles, then, 
rtie two sides containing the right-angle, and the complements <rf 



iSFHEfilCAL TRIOONOHETKY. AS 

the other three, n«mtAj, of the two angles, and of the hypoteniue,' 
are called the circular parit. 

II. When, of the five circular parts, any one is taken for the 
middle part, then, of the remaining four, the two which are imme- 
diately adjacent to it on the right and left are called ad;acmaar^/ 
and the other two, each of which is aeparated from the middle part 
by an adjacent part, are called opponte part*. 

This arrangement being made, the tolutioD is obtained hj the fol- 
lowing 

Theobbh. ■ ■■; 

In any right-angled spherical triangle, the rectangle under the ra^ 
dius, and the sine of the middle part, is equal to the rectangle under 
the tangents of the adjacent parts ; or to the rectangle under the co< 
siNBsof the oppoaiTE parts. . 

This tbeoran, or rule, may be easily remembered, by remarking, 
that the first vowels in sine, langenl, cotine, are respectively the suae 
as the first in wiiddle, adjacent, opposite, 

or, R X sin, micl=rect. tan adj. = rect, cos. op.* 

It is usual to convert the equation under consideration into an 
analogy having the unknown i^uantitr for the last term, though, to 
diose acquainted with algebra, it would be more convenient to make 
it alone the first term ofan equation, and the remaining terms, com- 
bined properly according to the rules of algebra, the last 

Fhoblbh T. 

Given three of the six ^arts, as, for examplsj the hvpotentue and 
one of the angles of a nght^angled spherical triangle, to find the 
sides and the remaining angle. 

On the first of May 1826, the sun's longitude was 1" 10° 32' 12", 
and the obliquity of the ecliptic 23° 27' 40" ; required the right as- 
cension and decuoation i\ 

Ans.—-R. A. 2" 32" 27'.3 ; dec. 14» W 47" N. ■ ■ 

Construction. — With the chord of 60° describe r 

the primitive circle EPQP' on the plane of the 
solstitial colure, and draw the diameters £Q 
and PP' at right-angles to one another, then ^ 
will EQ represent the equator, and PP* the Ef- 
polar axis. Lay off from the same line of chords 
E e=23° 27' 40", the obliquity of the ecliptic, 
and draw the diameter e / representing the 
ecliptic, at right-angles to which draw pp', 
and p,p' are the poiea of the ecliptic From .-,_ . 

tangents, (Theorem l.), lay off the sun's longitude 1* 10° 89' 12", 
or 40° 32'.2 on the ecliptic, from A to C, then C will be the placa 
of the sun, and » cm a paralld of declination. Through tlie pointa 
PCP* draw a circle of right ascension, cutting the equatw £Q 




■ should eiihecof the oblique ui{;1ei, oi 
initFail of tbe word in the fermula, iwe Ihat ( 
rim nsd ciwme, for nuiiM read aine, and so Ou. 

+ Foi tbe eipUnattoii of these terms the usual tiutlia on utioiHin; inajr bl oou. 
■alted. To thoie aeqounted with the uae of tlie globes, cgmct l^U lAtfts to ttlcw 
pniblani pULT basMdilr otXaliKd. Itinaj bewldHi, thstibsKn'aloogitad^Hd 1^ 
obUqui^ of the ecliptic, are computed from astnmonUcal lablci. 



06 INTBODUCTION. 

at right-angles in B, then will AB be the right ascension^ BC ^e 
declination^ and BCA the remaining angle or anffle of position^ as 
it is sometimes called^ which^ in astronomy^ is seldom of much use. 

Calculation. — In the triangle ABC there are given AC:=40^ 
92" 12"', and the angle BAC=23'' 2?' 40"^ to find BC, the distance 
of the sun from the equator £Q, or the declination, as it is usually 
called. Now, since in spherical trigonometry the sines of the sides 
are proportional to the stnes of their opposite angles. 
Therefore, 

Assme ABC or radius 10.000000 

IstosmeBAC 23° 27' 40" 9.600021 

So is sine AC 40 32 12 9.812870 



To sine BC 14 59 47 9.412891 

To find AB we may employ the method of the circular parts. 
In the triangle ABC are given AC and the angle BAC, to find AB 
the right ascension. Now, since the side CA, the angle CAB, and 
the side AB are all connected, that which stands in die middle or 
the angle A is called the middle part, and the sides AC and AB ad- 
jacent to it on each side are callea the adjacent parts.* 

Consequently R X cos. A = cot AC x tan. AB ; and resolving 
this into an analogy, as is frequently done in this country, we have, 
Ascot. AC 40^32' 12" 10.067939 

Is to radius 10.000000 

So is cos. A 23 27 40 9.962526 



Totan. AB. 2»*32"27'.3 9.894587 

or, since cot. : R : : R : tan., or tan.= to radius unity (§ 35, page II.) 

COv* 

As radius 10.000000 

Is to ten. AC 40°32M2" . 9.982061 

So is cos. A 23 27 40 9.962526 



Totan. AB 2**32"27'.3 9894587 

the same as hefore. 

To those acquainted with algebra, it is better, after the manner of 
foreign mathematicians, still to retain the form of an equation thus : 
A T> B X COS. A - A ^ ^ ^. * . 

ten. An = — ^ ac ~~ ~ ^^^' ^ **^* ' radius being re- 

S resented by unity; in which case ten must be rejected in the in- 
ex. 

To log. COS. A 23° 27' 40" . . 9.962526 

Add kg. tan. AC 40 32 12 9.932061 

Sum tan. AB 2^32°27.*3 9*894587 

To find the angle ACB, since the parts under consideration are 
still all connected, AC standing in tne middle is assumed as the 
middle part, and the angles A and C are the adjacent parts, whence 



* ft may be remarked, that if the parts are all connected, that which stands in the 
middle is (»lled the midme part, and the other two are called the a4jacent parts. If 
4W0 only are connected, and one stands by itself, then this is called the middle part, and 
the otb^ two are called the opposite parts. 



SPHERICAL TRIGONOMETRY. 57 

R X COS. AC = cot Ax cot. C, and cot. C = — ^— t-=co8.AC xtan. 

cot. A 

A^ hence 

To log. COS. AG iO^" 32" 12^' 0.880606 

Add log. tan. A 23 27 40 9.037496 

Sum = cot. C 71 44 42 .2 . . . 9.618304 

Or the comp. 1^8 15 17 •8, is called properly the angle of peti- 
tion, sometimes useful in computing the parallaxes in solar edipaet 
and occultations of the fixed stars and planets hj the moon. 

By assuming different parts of the triangle ABC for the middle 
part, may be resolved the following 

Examples for Exercise. 

1. On the first of June, 1827> at noon on the meridian of Green- 
wich, the sun's longitude will be 2* 10"* 9' 45'^ the obliquity of the 
ecliptic 23'' 27' 36'^ ; required the right ascension and declination ? 

Ans,—R. A. 4»» 34" 7'.6 ; Dec. 21« 59^ 34" N. 

2. August 12th, 1827, the obliauity of the ecliptic being 23*» 27' 
36", the sun's right ascension wUl be 9** 25" 29*.3 ; required his 
longitude and dedination ? 

^n*.— Longitude 4* 18° 66' 28", Dec. 16° 9' 32" 8. 

3. On the 10th November, 1828, on the meridian of Greenwich, 
the sun's right ascension will be 15** 2™ 32*.7> and declination 17® 14' 
12" S. ; required the sun's longitude and the obliquity of the 
ecliptic ? 

Ans. Longitude T 18^ 6' 7", and obliquity of the ediptic 23* 
27' 34". 

4. On the 2d of March, 1828, when the sun's declination was 
7° 5' 18" S., and obliquity of the ecliptic 23° 27' 35" ; required his 
longitude and right ascension ? 

In*.— Longitude 11' IV 56' 34" ; R. A. 22'* 53™ 24'. 

Problem II. 

When the celestial object is not upon the ecliptic, as the moon^ or 
the planets, and some of the fixed stars, the right ascension and de- 
clination are found by the solution of two right^ngled triangles. 

1. On the 17th of January, 1826, at noon, on the meridian of 
Greenwich, the moon's longitude was 1* 11° 5' 14", and her latitude 
2^ 34' 3" N. ; required her right ascension and declination, the 
obliquity of the ecliptic being 23** 27' 40" ? To resolve this example 
it is necessary to employ two right-angled spherical triangles. 

In the foregoing figure, the longitude of the moon or any star S, 
is AD, the latitude DS, the obliquity of the ecliptic BAG, wie right 
ascension AB and declination BS. Now, supposing a line drawn 
from A to S, there would be formed the right-angled spherical trian- 
gle ADS, right-angled at D, of which AD and JDS are given to find 
the angle DAS and the side AS. If the position 8 of the star is 
without the ecliptic, then to the obliquity of the ecliptic BAC, add 
the angle DAS, the sum will be the angle BAS ; but if S is within 
the ecliptic, that is between it and the equator, subtract the angle 
DAS from the obliquity BAC, and the remainder will be the angle 
BAS. Since the side AS, and the angle BAS, are now known, AB 
the right ascension, and BS the declination, may b^^lbund. 

Calculation. — By the rule of the circular part% first AD and jp& 



INTBODUCTION. 

I ife given to finil AS, and aini^e the last is separated from the two 
I first by the oblique angles, it will be the middle part, and AD and 
t DS are the opposite parts ; therefore, R x cos. AS ^ cos. DS x cos. 
AD, or COB. Ah = COS. DS X cos. AD to radius unity. 

To log. COS. DS 2" 34' 3" . . . 9.999564 

Add log. COS. AD 41 5 14 ... . 9.877204 



-■iog.cos.AS 41 9 11 . . . 9.876768 

* Again, to find DAS, since the right angle does not separate the 
parts, DA standing in the middle is called the middle part, and the 
Bide DS and the angle DAS are the adjacent parts ' " ■ 

I t>A = tan. DS X cot. DAS, and, therefore, cot. DAS 
•in. DA X cot. DS, consequently 
■ To log. cot, DS 2" 34' 3" . . 

' Add log. Bine DA 41 5 14 




Sum=log. cot. DAS 3 54 14 
To this add Ob. Ec. 23 2? 40 



9 817634l^^J 
1 1.16Q956 ^^1 

BS. I 



Sum = angle BAS 27 21 54 

Hence AS and BAS are now known, to find AB and BS. 
First to find AB. In this case the parts are connected ; therefore 
BAS is the middle part, and AB and AS are the adjacent parts, 

< cos. BAS = tan. AB X cot. AS, or tan. AR = ^^jo-. and 



tan. AB = cos. BAS x tan. AS, hence 
, Tolog. coa. BAS27°21'fi4" 
Add log. tan. AS 41 9 11 

Sum = log. tan. AB 37° 49' 5 " 
Or in time R. A. 2" SI" 16-.3 
- To find BS, the angle BAS and side AS a 
disjoined, whence Rxsin. BS = sin. AS> 
I sines of the sides arc proportional to the sin 

e ABS or radius 
eAS 41' 9' 1" 



:. AS ' 



i connected, 
iin. BAS, < 
B of their opposite 



So is sine BAS 2? 21 54 

TosineDecBSi; 
The foregoing method i 



5N. 



I iediptic, provided proper 




is general and applicable to 
attention be paid to the f 
^ ielestiai object witti respect to the ecliptic and equator. Asthispro- 
[ llem and its converse is of frequent occurrence in practical astruno' 
I Shy, rules and formula;, and even tables, have been formed for the 
L jurpose of facilitating the computations. The following rules 
r '%kv the ^"''^ ''^~ i^T....!.-^ fii I — f.^...wj w«... ».^».....»;^...4. i 



I purpose. 



late Dr Maskelyne, will be found very 



for this 



FnosLiCM II- 



Given the right nscension, the declination, and the obliquity of rtfc^~J 
ecliptic, to find the longitude and latitude. ^"^ 



SPHERICAL TRIGONOMETRY. ^9 

Let RA denote the right ascension, O the obliquity of the ecliptic, 
and D the declination^ 

Tan. D-r-sin. RA = tan. A, North or South as the declination i8.% 

Call O in the first six signs of RA South or S. and in the last six 
signs North or N. 

Then A+O = B^ regard being had to the algebraic signs, 

A being less than 45°, and using logarithms. 

Sec. A + cos. B + tan. RA=tanl Ion. of the same kind as RA, 
unless B be more than 90°, when the quantity found of the same 
kind as RA must be taken from twelve signs. 

A being more than 45°. 

Tan. A+cosec A+cos. B+tan. RA = tan. Ion. of the same kind 
as RA, unless B be more than 90^, when the quantity found of thtt 
same kind as RA must be taken from twelve signs. 

Lon. being nearer III. and IX. signs than O and VI. signs. 

Sin. lon. + tan. B ^ tan. lat. of the same name as B. 

Lon. nearer O and VI. signs, than III. and IX. signs. 

Tan. Lon. + COS. lon. -f tan. B =tan. lat. of the same name as B. 

Example. 
On Monday the 12th of June, 1826, the moon's R A at noon, 
was found by observation to be 10^ 39" 31' and her declination 2** 
61' 58" N. ; required her longitude and latitude ? 
D= 2° 51' 58" N. tan. 8.099533 
RA=10^ 30" 31* sine 9.536560 tan. 9.563908 



A 8° 16' 50" N. tan. 9.162973 sec. 0.004551 
O 23 27 40 S. 



B 15 10 50 S. cos. . 9.984575 tan. 9.433497 

Lon. 160 20 17 tan. . 9.553034 sine 9.526946 



Lat. 5 12 59 S tan. 8.960443 

Problem III. 

Given the longitude and latitude of a celestial object, and the ob- 
liquity of the eOnptic ; to find the right ascension and declination. 

Tan. Lat. — sine Lon.=:tan. A, North or South as the latitude is. 

Call O North in the six first signs, and South in the six last signs. 

A + O = B, as before. 

A beinff less than 45% sec. A + cos. B -f tan. lon. =z Tan. RA of the 
same kind as the longitude, unless B be more than 90^, when the 
quantity found of the same kind as the longitude must be subtracted 
i&om twelve signs. 

A being more than 45^, tan. A -f cosecant A+cos. B+tan. Ion. 
= tan. RA of the same kind as the longitude, unless B be more than 
, 90°, when the quantity foimd of the same kind as the longitude must 
be subtracted from twelve signs. 

If RA be nearer III. signs and IX. signs, than O and VI. signs, 
sine RA+tan. B = tan. Dec. of the same name as B. 

And RA being nearer O and VI. signs, than HI. and IX. signs, 
tan. RA+cos. RA+tan. B = tatL Dec. of the same name as B.* 



_^i?*'***;""l5tJ5S2 ^ff**?**^ ^^'ifeP^P^ ^V^^ except in peculiar circumstanccg, 

^wPr Alir«inR^aftHl^t«ltir U A* Phil. Trans, for 1816, pm 1884 ^WrK^wl 
want of n^Ri^ cannot ba gxTCB Bm 



INTRODUCTION. 



On the 1st of January, 1820, the n.^ — g.^u..^ „> i„c .jwi ru- 

malhaut was 1 1' 1" 19' si", the mean latitude 21" 6' 45" S. ; required 



the riRht ascension and decl 

ing 23" 27' 46" ? 
Lat. 21° 6' 45" S. tan. !).586721 
Lon. 331 19 34 



longitude of the Star Po- 

ade21''6'45"S.; requ' ' 

the obliquity of the ecliptic 



A= ; 



9-737901 
R. tan. 9.905639 sec. 0.108420 



' 46 S. 



j B= 62 17 12 S. cosine 
BA=341 65 14 Ungent 

Dec. 30 34 21 S. 



9.667498 tan. 10.279585 
9.5J3819 sine 9.491831 



tan. 9.771416 
Examples for Exercise. 

1. The mean longitude of • Arietis, on the 1st January), 1820, was 
1" 5° 8' 48", and mean latitude O'* 57' 34" N. when the obliquity of 
the ecliptic was 23= 27' 46" ; what was the right ascension and dedi- 
nation ? 

Ans.—R. A. 1" 57" 3* ; Dec. 22° 36' 24" N. 

2. Required the right ascension and declinntion of Pollux, when 
the longitude was 3" 20° 43' 58", the latitude 6° 40" 1?" N. the ob- 
liquity of the ecliptic being 23° 27' 46" f 

Ans.—R. A. 7" 34'" 17.5' ; declination 28" 27' 8" N. 

3. The mean longitude of SpicaVirgiiiis is 6" 21° 19' 50", latitude 
go 2' 24" S. and the obliquity of the ecliptic 23° 27' 46"; required 
the right ascension and declination ? 

Ani.—R. A. 13'' 15" 43.5' ; declination 10° 13' 4" S. 

4. The mean right ascension of « Aquila? is 19" 42"', and declina- 
I tion 8° 24' 4" N. the obliquity of the ecliptic being 23° 27' 46' ; re- 

ulred the longitude and latitude ? 

^n*.— Longitude 9" 29" 14' 14", Latitude 29" 18' 36 " N. 

5. Required die longitude and latitude of « Pegasi, of which the 
' tight ascension is 22" 55" 48', declination 14° 14' 21", the obliqiuty 
1 of the ecliptic being 23° 27' 46" ? 

,iw.— Longitude 11' 20° 68' 47", Latitude 19= 24' 36" N. 

PRDBI^EM IV. 

Given the latitude of the place, and the eun's declination, to find 

liis altitude and azimutli at d o'clock. 

1. At Edinburgh, in latitude 55° 57' 20"' N. on the 2l8t of June, 

1^6, the sun's declination was 23" 2?' 36" K ; required his altitude 
[ §ntl asirauth Bt 6 o'clock in the morning or evening, his declination 
\ seing supposed to remain the same. 
I Const r»clio7i. — Describe the primitive UPON on the plane of the 

meridian. Let HO represent the horizon, ZN 
I tte prime vertical at right angles to the former. 
J idake OP, from a scale of chords equal to the la. 
I HituJe of the plate. North in the present instance; 
1 draw PP', the six o'clock hour circle in this case, i 
L and at right angles to it draw the equator EQ ; ' 

'describe the small circle nm at the distance of 

23° 27' 36" from the equator, representing the 

/>ara11el of declination, and it will cut the six " 

o'clock boar circle PP' in F, the sun's plnce at tlie given time. 




SPHERICAL TRIGONOMETRY. 71 

Through Z, F, and N, describe the azimuth circle ZFN cutting the 
horizon in D, then FD is the altitude, FZ the zenith distance, and 
the angle FZP, or its measure, the arc DO, is the azimuth ; conse- 
quently, the thinffs ffiven and required fall in either of the triangles 
FZP, or FDA, which are supplemental to each other. For, since OP 
is the latitude, PZ is the colatitude, AF is the declination ; conse* 
quently, FP is the polar distance, DF being the altitude, FZ must be 
tne zenith distance. 

Calculation. — ^In the right-angled spherical triangle FPZ, right* 
angled at P, FP and PZ are given, to find the angle FZP and FZ ; or 
in the triangle ADF, right-angled at D, there are given the angle 
FAD, equal to the latitude of me place, and AF, the sun's declina- 
tion, to nnd DF, the altitude, and the side AD the azimuth. 

By the rule of the circular parts FP, PZ, and PZF, are all con- 
nected, therefore PZ is the middle part, and PZF and PF are the 
adjacent parts, where 

Rxsine ZP =tan. PF xcos. PZF, or 

R X cos. lat. =: cos dec x cos. azimuth, therefore 

cos. azimuth = 3 — = cos. lat. X tan. dec. 

COS. dec. 

To log. COS. lat. 66° 67' 20'' . 9.748061 

Add log. tan. dec. 23 27 S6 . 9.637472 

Sum = log. cos. az. 76 20 88 . 9.385633 

Again, to find FZ the coaltitude, the same things being given, 

R X cos. FZ = cos. ZP X COS. FP, or sine alt = sine lat X sine dec. 

To log. sme lat BB"* bT 20" . 9.918347 

Add log. sine dec. 23 27 36 . . 9.600002 

Sum =: log. sine alt 19 15 40 9.518349 

Problem V. 

Given the latitude of the place, and the sun's declination, to find 
the altitude and hour when the sun is due East or West 

£XAMPLK. 

At Edinburgh, on the 2l8t June, 1826, what was the sun's alti- 
tude and hour when due East or West, the declination bdnir 23'' 
27' 36'' N. 

In the last figure, let ZAN meet the parallel n m in K, and sup- 
pose a circle tohe drawn through the points PKP, forming the tri- 
angle ZEIP, right-angled at Z, then ZK is tiie coaltitude, and ZPK 
the hour from noon ; hence 

R X cos. PK =: COS. ZP X cos. zK, or 

-_-. COS. PK _.__ —^ 

COS. ZK = ^tjj = COS. PK X sec PO, or 

COS. £iir 

sine alt. = sine dec. x sec. lat 

Dec. 23° 27' 36" sine 9.600002 

Lat bb 67 20 sec. 0.081653 



Alt 28 42 55 sine 9.681655 
R X cos. ZPK 3= tan. ZP X cos. PK, or 
COS. T = COS. lat X tan. dec. 



^ 



IXTBOWTCnOX. 

1-t. oS* 57 aO* ooe. ».£3a7U 
Dk. S3 37 36 ud. 9.637*72 

Timkt 'T' 51-48' cos. 9.467186 
», that «, « 7* ff" 1? A. M., and 4* 51" 4 



! pDtpose oF determining time accurately 
when an altitude instrament i^ used. A^ the change of altitude, on 
mUA the accnncy of the determination of the time depends, is 
miic:kest vben ibe object is on the prime vertical, the most proper 
ttnie for obeerrii^ an altitude for Uut purpose is, therefore, when 
the object ia due East or West, as anr si 
has then the lent possible effect on tte 
■n this case in a great degree avoided, or a 
particularly that arising from any small e 
at the time of obseiration. To facilitatt 
' responding to the latitude and declinatiot 
name with the latitude), have been giv£ 
nomy, such as those of Slendoza Rios, Slackay, and Las, When the 
latitude and declination are of different names, the altitude must be 
as near die horieon aa is consistent with accuracy, go far as depends 
upon the uncertainty of the horiaontal refraction. Altitudes under 
5° should not be used when great accuracy is required. 
Pboblkm VI. 

Given the latitude of tiie place and the sun's declination, reqitired 
his amplitude and ascensional difference.* 

At Edinburgh, on Uie 21st of June, 1826, from the data given, on 
what point, anil at what time, did the sun rise and set ? 

In the triangle ABC, in the last figure, there are given the ai 
BAC, equal to the colatitade, and BC the sun's decUnation ; to 
AC and AB. 

R X one BC = sine AC X sine BAC, or 

»ne AC = ."""^^^ = sine BC x coaee. BAC. 



imall error in the observation 
ite. Otiier errors are also 
•ast considerably lessened, 
r in the estimated latitude 
s application, tables, cor- 
n (which must be of the same 
n books on Nautical Astro- 



; BAC" 

BC, or dec. 23^ 27' 3fi" N. sine 9.600002 
Latitude, 55 57 20 sec. 10.251939 



AC, 4.i 19 33 sine 9.851 941 

CO, 44 40 27, in which case AC is the ampli- 

tude reckoned from the East or West, to the North and South, ac- 
cording to the name of the declination, and CO is that reckoned from 
the meridian, or Jrom the North or South, according to the name of 
lite declination. 
- Again, in the same triangle AB is the ascensional difference, and 



^ , 



[ R X sine AB = cot. BAC X tan. BC, oi 
Lat 55" 57' 20" tangent, 
Dec. 23 27 3ti tangent. 



r AB = tan. lat. x tan. dec 
10.170286 
9.037472 



A. D. 2"' 39" 52- i 



= time of setting. 

= time of rising, the Utitude and 



I 



' B]F flie —imtiiinl dlftercnca la taeam the ijnie b^iiic i 
M m wm. Uf tliii i>tobUiD, i]ietch<R, the K'ngiht of tli 
-' MMl^JwfMiMiM of Uw uiadiui'iUKupaiis. 



SPHERICAL TEIOONOMETRY. 78 

dinatian being of the same naine, or if instead of sine we read cotine, 
then we would get the time of rising if the latitude and declination 
are of the same name« and the time of setting if of different names. 
This^ however, is only the approKimate' time, as no allowance is made 
far the effects of a change of decHiiation, the horiaontal refraction 
and parallax in the case of the sun and pkmets. For these eee 
Mackay on the longitude, or thej may be found by the fbllowing 
rule. J^irst, let the approximate time be found. . To this time let 
the declination of the object be reduced. With it find the as- 
c^isional difference ag formerly. Now, find ttke sum and dif- 
ference of the natural cosine of the reduced declination and natu- 
ral sine of the latitude, which maybe carried to foar places of fignras 
only, these being sufficieBtly accurate for this purpose, and take half 
the sum of the logarithms of these quantities, to which add the con- 
stant logarithm 7-1761, and the proportional logarithm of the differ- 
ence between the horizontal parallax and the sum of the horiaontal 
refipaction and dip of the horizon, the sum, rejecting 10 in the indeoi, 
will be the proportional logarithm of the correction which is to be 
subtracted from the time of rising, or added to the time of setting, if 
the horizontal parallax is iess than the sum of horizontal refVaction 
and dip,, otherwise the correetion must be added in the first eatse, and 
subtracted in the second. 

ECXAMPLB. 

Required the time of rising and setting of the sun on the jst of 
April, 1826, in latitude 33^ ^ N., and longitude 16^ 20' W. the 
height of the eye, aboye the sea, being 38 feet. 

Dec. 4"" 28^ N. COS. 990^ 

Lat. 33 42 N. sine 6M3 



Sam 15517 log. 4.1906 

Diff. 4421 log. 3.6455 

Dip to 28 feet — 5' 16" TS© 

Hor. refrac. ■■■ 31 17 * ' * 

Parallax + 9 34)181 

■'» const, log. 7-1761 

— 39 24 P.L. 0.6ofie- 



— ff-WP.L. . . W540 
The correction to be subtracted from the time of rising, or added 
to the time of setting. As the moon's horizontal parallax ia in gene- 
ral greater than the effects of dip and refraction, the correction th«s 
obtained would have been appbed with a contrary sign. This me- 
thod of determining time may sometimes be of use when a better 
cannot be obtakied, and in the case- of the sun or moon, a mean of 
the times of appearance of the upper and lower limb may be takm.* 

Sdlfiium qf Oblique^Anf^ed S^erieai Trtangh*, 
The" different cases of oblique-Mif^ed spherical triangWs may h« 
solved by the fellawing theorema:-*< 



* To find the rising and setting of a star or planet, the transit over the meridian must 
be first coniputed as rollows :— From R. A. of the star sabtract that of the sun for noon, 
the remainaer is the approximate time of transit Aeduoe the R. A. of both to tMs time 
and the given longitude, and subtract as befo)«, and the remainder will be the true time 
9i tnmsit, whieh, ]»TOpeilT am^ied to the semidiiunal aic, will give, when corrected for 
dip, &c., the true time of rinng or aetdng. 



INTRODUCTION. 



" -portionul to the sines of the 
isngles opposite to them,* 
Or, sin. AB : sin. AC : : ain. 



4^ ^::>? 



] 



Theobeh II. 

- In oblique-angled spherical triiingles a perpendicular arc being 
drawn from any of the angles upon the opposite side, the cosineB of 
the angles at the base are proportional to tne sines of the segments 

lofthe vertical angle, or cos. B : cos. C : : sin. BAD : sin. CAD. 



Thboi 



M III. 



e things remaining, the cosines of the sides are propor- 
ie cosines of the segments of the base, ur cos. AB : cos. 



' The s 

tional to the cosines of the segments o 
_ AC:: COS. BD:coa. CD. 
^1 Theorem IV. 

li The same construction remaining, the sines of the segments of the 
base are reciprocally proportional to the tangents of the angles at 
the base, or sin. BD : ■- "" '— " '- " 



in. CD : 



: tan. B. 



The same cuu» 
the vertical angli 
the side! 



TtiKonEii V. 
remaining, tlie cosines of the segments ( 
:s am rtTciprocally proportional to the t^igents c 
BAD : COB. CAD : : tan. AC : tan. AB. 

Tbeoreu VI. 



If, from an angle of a spherical triangle, there be drawn a perpen. 
dicular to the opposite side or base, the tangent of half the sum of 
the segments of the base is to the tangent ofhalf the sum of the two 
sides of the triangle, as the tangent of half the diiTerence of those 
sides to the tangent of half the difference of the segments of the base. 
orUn. A(BD+CD):tan. J (AB + BC) : : tan.^ (ABtf) AC) : tan, J 
(BDtf^CD). 

When the three sides or the three angles are not the given parts 
of the triangle, to have sufhcientifa/a fur the solution of the proUem, 
the perpendicular must be so drawn, that two of the given things in 
the obbque-angled triangle may be known in one of the resulting 
right-angled triangles. 

Theorem VII. 

If a perpendicular be drawn from an angle of a spherical triangle, 
to the opposite side or base, the sine of the sum of the angles at the 
base is to the sine of their difference, as the tangent of half the base 
is to the tangent of half the difference of its segments: And the Vine 
of the sum of the two sides is to tlie sine of their difference, as the 
cotangait of half tile angle contained by the sides is to the tangent 



ricic LXXVI., smf (he foUowing in unlcr. 



SPHERICAL TRIGONOMETRY. 76 



of half the difference of the angles which the same sides make with 
the perpoidicular/ or sin. (B+ C) : sin. (B ca G) : : tan. ^ BG : tan. 4 
(BD (» CD). And sin. ( AB + AC) : sin. ( AB ud AC) : : cot. i A : ton. } 
(BAD OP CAD). 

Theobbm VIII. 

The sine of half the sum of any two angles of a spherical trianffle^ 
is to the sine of half their difference^ as the tangent of half the side 
adjacent to these angles^ is to the tangent of half the difference of 
the sides opposite to them. And the cosine of half the sum of the 
same angles^ is to ti^e cosine of half their difference^ as the tangent of 
half the side adjacent to them^ is to the taneent of half the sum of 
the sides opposite^ or sin. 4 (A-f B) : sin. 4 (A <i) B) : : tan. i AB : tan. 
4(BCcir> AC). And cos. i (A+B) : cos. i (A (/)B) : : tan. ^ AB : tan. t^ 
(BC (jn AC). 

CoroUaty. — The sine of half the sum of any two sides of a sphe- 
rical triangle^ is to the sine of half their difference, as the cotan^ 
gent of half the angle contained between them, is to the tangent 
of half the difference of the angles opposite to them : And the cosine 
of half the sum of these sides is to tne cosine of half their difference. 



as the cotangent of half the angle contained between them, is to the 
tangent of half the sum of the angles opposite to them,t or sin. 4 
(AB + AC) : sin. ^ (ABc/jBC) : : cot i A : tan. i (B^dC) cos. } 
(AB X AC) : cos. i (AB uo BC) : cot i A : ten. i (B+ C). 

Theobem IX. 

It will be sometimes more easy in practice to compute an angle 
from the three given sides by the following formulas and rules, than 
by any of those already given : thus, suppose A, B, C^ are the angles 
as before, and a, b, c, the sides opposite ; then 
Sin. iA= / gin. ( i (g+& + c)-^} . sin. { j (a+&+c)— 6 } .^. 

* ^ sin. b sin. c 

Co8.iA= / "i"- i («+&+0 8in. { i (a+6+c)~a ) „ 

* V sm. b sin. c ^ ' 
Ton 1 A — / gJP' { i {a+b+cyU ] . sin. [ ^ {a+b+cy^ ] .^x 
Tan. 4 A-^ sin. { i (a+b+c)^ ] .sin. { i (a + 4 + c) ] ^"^^ 

Mules in Words, 

I. From half the sum of the three sides subtract each of the two 
sides which contain the required angle. Then to the oosecanto of 
the sides which contain the required angle add the sines of the two 
remainders ; half the sum of these foregoing logarithms will be the 
sine of half the required angle. 

il. Find the difference between half the sum of the three sides, 
and the side opposite the required angle. Then to the cosecanto of 
thjB two contaimng sides add the sines of the half sum and difference ; 
half the sum of these four logarithms will be the cosine of half the 
required angle. 

III. To the cosecant of half the sum of the three sides add the 




where 

Logarithmfc Tables, Erroneous rules uid impossible triangle , , ^. 

Ue, be avoided.— See the French Edition of Cagiioli*8 Trigonometry, § 1088, 1108 and 
1109. 
t Legendre, § LXXXIII. 



76 INTBODDCTION. 

Cflsecant ofbalf that sum diminished by the side t^i^ionte the nqnlr- 
ed angle, sikI the since of the same half sum diminished by eadi of 
t}ie sides containing tlie required uiiglei half the sura of these fbur 
logarithms will be the tangent of half the required angle. See re- 
marks annexed to CaHe iii., Plane Trigonometry. 
Tbeobeh X. 
Given two aides and the contained angle, to find the side opposite 
\ ^at angle. 

To twice the sine of half the contained angle, add the sines of the 
(wo containing sides, and &om half the sum of these three logar- 
llluus subtract the sine of half the difference of the sides ; the re- 
mainder will be thi! tangent of an arc, the sine of which being sub- 
tracted from the half sum of the three logarithms already found, 
leaves the sine of lialf the required side. 
Tbeosbu XI. 
The two sides and contained nn^cbeing given, the dUrd^de may 
be found in the following manner. 

To twice the sine of half the contained angle add the sines of the 
two containing sides ; half the sum of these three logarithms, after re- 
jecting 20 in the index, will be the cosine of an arc Also find half 
the difference of the two containing sides- 

To the sine of the sum of these two last arcs add the sine of tLeir 
difference ; half the sum of these two logarithms will be the cosine of 
half the required side. 

It may be remarked, that when the side is not greater than 90°, 

{theorem X. may be used; when it is greater than 90°, theorem XL 

may be employed when great accuracy is required. 

Theohbh XII. 

The three angles of a spherical triangle being given, to find the 

ndes. 

From half the sum of the three angles subtract each of the angles 
next the required side, then to the cosecants of the adjacent angles 
^d the cosines of the two remainders ; half the sum of these four 
logarithras will be the cosine of half the required side. 

Theohbh XIII. 

The same things bdng given ; from half the sum of the three 
angles subtract the angle opposite the required side, then to the 
cosecants of the adjacent angles add the cosine of half the sum and 
the cosine of the difference ; half the sum of these four logarithms 
will be the cosine of half the required side. 

Either of these theorems may be employed, which will give the 
more accurate result. 

Having stated the theorems on which the solutions in oblique-an- 
gled spherical triangles depend, it is necessary to illustrntc them by 
examples which will chiefly consist of those applicable to the usual 
eases that occur in practical astronomy and navigation. 

Problem I- 

Given the latitude of the place, the sun's altitude and declination, 
to find the time and the azimuth. 

At the observatory of Edinburgh, on tlie Calton-hill, in latitude 
fi5' 57' 21" N., on the third of June, 1826, the following observa- 



SPHEBICjML. ^U0QHOM£TBY. 



n 



tions of the ton's hnMr iinb were ttkcn in the mondng ; re^nlnd 
the tiine and azimuth, the barometer beng at 29.56 in., and the 
theaiio»iet«r at 64'' F. ? 

Tnt» by Watch. AUUudes. 

7*1" 20- . 26^61'aO'' 

2 18 . 26 59 30 

3 25 . . . 27 7 15 

4 30 . 27 15 40 
6 27 . 27 23 45 



L 



17 



Meana. 7 3 24 
O obte^rved ZJ}, 
Z. D. 62° 52^.5 log. i I 
Thermometer 64"" F. log. 
Barometer 29.56 
Thermometer 64.0 F. 



35 37 30 

27 7 30 Lomtr limb. 

62 52 30 

2.03692 
9J6751 
9.99358 



r = 106".5 
Z. dist. 
Refiraction 



1' 46".5 log. 
= 62«52'30" 
+ 1 46 .5 



2X)174i 



TraeZ.D. 62 54 16 .5 of the low^r limb. 
Semidiameter •— 15 47 .5 



True Z. D. 62 38 29 of the centre. 
Approximate time, June 2d, 19^ 4" 

Longitude in time add -(- 12 West 



Estimated Ghreenwich time 
Daily variation of dec. 

Prop, part to 17* 18" 
Dec., June 2d, 



19 16 D. L. 0.09503 
T 42' P. L. 1.36878 



+ 6 11 P.L. 1.46381 
22« 9 38 N. 



Reduced declination 22 15 49 N. 

Polar distance 67 44 11 

1. Now in the figure, (page 70), l^ere are given OP the latitude, 
and coateequentl J Z P die colatitude, PK the polar distance, and 
ZK the i£nith distance, the place of the sun being K near' the 
prime vertical, as being most advantageous to determine the time 
with aecnracy, or the three sides of the triangle KPL ; to find the 
angle ZPK the time, and the angle PZK me azimuth from the 
southern meridian PEP. This, wercfore, is solved by means of 
theorem IX. 



78 



INTSaDVCTION. - ' 



k// 



Now the Utitade being BB"" 5T 21^ the eoladtude is 34° IK 99' 
Z. D. 164 38 29 

CoUdtude 34 2 39 cosec. 0,251942 

Polar dist. 6? 44 11 cosec. 0.083647 



Sum 



164 35 19 



Half 

First rem. 
Second rem. 


83 12 39 
48 10 
14 28 28 

a'27-2'.. 
2 


sine 
sine 

• 
• 

* .sine 


Time from noon 3d 


4 
12 


54 


42 




App. time> A. M. 

Tmie by watch 

• 

Watch slow 


7 

7 


5 
3 


18 
24 

64 


for s 


Again app. time 
Equation of time 


7 


5 
2 


18 
23 




Mean time 
Time by watch 


7 
7 


2 
3 


55 
24 





9.872208 
a897850 

19.555647 
9.777^4 



for apparent time 



Watch fast 29 for mean time. 

2. To find the azimuth or the angle KZF, the point K being ihat 
in which the circles n m and ZIN cut each other, there are given the 
three sides of the triangle KPZ. 

KP, or polar disL 67° 44' 11" 

PZ, or colatitude 34 2 39 cosec. 0.251942 

ZK, orZ. dist. 62 38 29 cosec. 0.051515 



Sum 

Half 
Difference 



164 25 19 



82 12 39 
14 28 28 


sine 
sine 


9.995974 
9.397850 

19.697281 


45 7 41 
2 


COS. 

E. 

sin. or . . 


9.848640 


N. 90 15 22 

44 52 19 

2 





S. 89 44 38 E. or reckoned from the 
South in north latitude^ or from the North in south latitude. 

This problem is very useful in navigation, for the purpose of find- 
ing the variation of the compass, which is the difference between the 
true and observed amplitude or azimuth. 



STKEUCiiL JUBOSIOUETRY. 99 

To determine thii, let the obtenrer be suppoaed to look direcfly 
from the centre of the card towards the point representing the true 
azimuth ; then if the observed asimuth is to the l^ of the true aai- 
muth^ .the variation is easterly, but if to the right it is westerly to the 
amount of the difference between them. 

Thus let the true azimuth be 8. 89^ 44' 38^' E. 

Observed . 65 24 38 



Variation 24 30 West 

Or about 2} points westerly. 

These results for time and variation have been deduced strictly 
from the solution of the spherical triangle formed by the data^ but 
they may be found more readily by rules derived from it, as may be 
seen in various books on navigation and nautical astronomy. 

When tables which have proporticmal parts annexed to them are 
used^ the following method may be advantagecmsly employed 
for determining the time. 

Rule. — When the latitude of the place and the declination are of 
the same name, let their difference, but, if o£ contrary names^ let their 
sum, be taken. Under this difference or sum place the zenith dis- 
tance, and let the half sum and half difference of these be taken ; 
then add together the secant of the latitude, the secant of the de- 
clination, the sine of the half sum, and the sine of the half difference ; 
half the sum of these four logarithms will be the sine of half the 
hour angle or time from noon, f^om which the apparent and mean 
time may be obtained as formerly. 

Latitude 56° 67' 21" N. secant 
Declination 22 15 49 N. secant 



Difference 33 41 32 
Zenith dist. 62 38 29 



Sum 96 20 1 half 48» lO' 0^'' 

Difference 28 56 57 half 14 28 28 { 



2» 


35*20' 
1. 


05 P. 


P. 


2 


27 


21 


.05 
2 




4 
24 


64 


42 


.10 





• 

• 


«6 

0.251877 
0.033606 
4S 


sine 
une 


1 

9.872208 

0.307821 

233 




10.566662 


aine 


9.777826 
681 




45 
43 



June 2d, 19 5 17.90 P.M., 

In the above computation the several proportional parts are set 
down and summed all together, which renders the operation some- 
what more easy when our tables are employed. 

Several variations may be made on the six things here proposed, 
that may -serve' as- a useful exercise, which, by a reference to the 
the<»rems and rules already given, wUI be ea^y performed. 



/ »T« l«<Hn'BCIfttTC'nON. 

II. 



RB6^e latitude of the yh.ce and ihe sun's declination ; to And 
ESae when tvilight beffuis and etuU. 

At Vlmt time wiU twilight b^in und end at London, in latitude 
51° 32' N,, on the second erf May, 1827j the sun'* declination being 
15" 14' N. ? 

In figure, (page 70)j suppose a parallel h m to the equator EQ to 
be drawn at tiia distance «f 15° 14' above it, while another paEallel 
to the horizon HO ia drawn at the distance of 18= below it, these 
two would cut one another somewhere between c and jit in S, film- 
ing the triangle ZPS, in which ZP, PS, and ZS, are given to find 
the angle ZPS, the angle between the meridian PEP and another 
meridian passing through the sun at the time he is 18° degrees below 
the horizon, Mb situation when twilight begins and ends. 
, Z sPK zenith distance 108° 0' 

V t or polar distance 74 46 cosecant 0.015534 



PZ or colatitude 


38 28 

221 14 

no 37 
2 37 


cosecant 


0.200168 


Sum 






Half 


9.971256 


Difference 


Bjne 


3.659425 
1^852433 




noon 


4-58" 6' 
2 


cosine 

. the evening 
11 the morning. 




9.426216 
443 


Time from 
Or at 


!) 56 12 ii 
2 4 48 ii 


I27 






PifOBLEM 111. 





Given the right ascensions and declinations, or the longitudes and 
latitudes <rf two celestial objects ; ta find thrif angular distnnce. 

In this problem there are given two sides and flie contained angle 
to find its opposite side. The contained angle is the difference be- 
tween their right ascensions or longitudes, and the containing sides 
are the complements of the declinations or latitudes. If the sun be 
one of the objects, as his latitude is very small, he Tnay be supposed 
to be always in the ecliptic ; then the triangle so formed wSl be 
right angied if the longitudes and latitudes are used, and the com- 

Sutation becomes more simple. By means of this problem the lunar 
istances in the nautical almanac are computed. 
On the Ist of June, 1828, required the distapco bet ween the moon 
and ■ Pegasi, at noon, on the meridian of Greenwich, tlie moon's risbt 
ascension being 295= 23' 46", and declination 16' 11' 45" S., the 
star's right ascension being 22* SS" 13"-85. or 344" 3^ 28", and north 
polar distance 75° 43' 2", or declination 1*" W 58" N. 

344" 3* 28"— 295" 23' 46"t=48" 39' 42" the angle at the pole. 
Instead, however, of following the operation derived from the s]>beri> 
cal triangle, a more simple practical rule may be derivvd from i( ac- 



<;^ing to thetrrem IX. 
Tot • "-- -'--'"- 



twice the sine of half the contained angle add the ctntsea of 
tlie nopon And star's declination*, and ta)(« half theaum oF tb«se 



SPHSBIGAL 'mOOIIOMETRY. 



«l 



thfe* loMritlimfl. iVon din ludf HMM MdbtiMst tht tiiie of half Ae 
^trm of ttie dedinations if they are dicomtrary namet, or tliat of half 
Itieir ^dii«ttnoe if of the aattie name^ the icmdnder will be Ao tan- 
gent oST an arc, the tine of which being inbtractad frcm half the 
««B of the tbree logarithraa already found wiQ gi^ the aina of half 
the required distance. 

Diff.offt.A. 4»'d9'4Q'' 



Half 


24 19 51 


■inexS 

S.OM. 

^■•008i 

■ 

sine 


b=ia9BoeM 

1 


Moon's declination 
Star's dedination 


16 11 45 
14 1« 56 

■ # 


ft9aa4i8 
ftfleeaai 

• 




aoiiMsei 


Sum 


30 88 4S 




U«.509a9l(«) 
9.419717 


Half 


15 14 8H 


Arc 


'56 31 18 


tan. 


10.179674 


Same arc 




sine 


9.Miafl6 (() 


• . ■ . « 

Half distance 


28 27 29 
2 


sine 


9.078076 (a— «) 


True distance 


56 64 58 


• 
• 



Examples for ExercUe. 

1. Required the distance between the moon and sun on July Sd, 
1896, at noon on the vieridian of Greenwich, the kmgitudo of the 
sun being 3^ IQP 28' 44', the longitude of the moon 11" l?"" 59" 3&', 
and lathudfi 2° 51' 40" N. ? 

Afu.—Ui^ 27' 19" east of her. 

3. Required the distance between the moon and sua fnHfaajSOth 
Janoary, 1828, at noon, the sun's longitude being 9^ 29^ iHK>■J9'^ 
that of the moon 11* 17** 54' 42", and latitude 3* 24' 28" ? 

SL Required the distance between the moon and m Aqmihm, at 
boon on the 10th of May^ 1838»'the right ascenrion cxf thefnDda.be- 
ing 6P G8' 43^', the declmatkm 4<» 44' ^' N., the ngbt 4Maa«io& of 
m AqiliUe ib^time, being 19^ 42^ Sfi'i62, and north twiar diataiM SV* 
34' 41"? ' i 1 

>liij.=70^ 64^ fil" wiest of horV 

4i Required the (Satance between the moon and ALdeifaanna^ at 
nadnight on the 16th of December, the moon's R. A, being 9S^ 3h 
W, £b dedination 11° 18' ir'N., the R. A. of Aldebanai hsiitg 4|. 
as- »S7s and N. P. D. 73^ SC 37".4 ? 

iliw.— 33° 21' 10". - 

P^OBLEni IV. 

On finding the latitude by obsCTYation. 

The most simple practical. metliodjoiTfinding the.laititudc^ is firom 
(he meridian altitude of a celestial bckly. whose declinati[onls Iknown. 

ffliould the object be the sun, moony ttf sonae of tho planet^ the 
•Wlade or sanim.dfalaMe of the lo#er or uiiper ll]Dib, or bad|> are 



obsemd, and by tlie applicfttjon of several correctiona that of tht 
centre is obuined. 

When reflecting instnimenta, such as the sextant, repeating circle, 
&C. with an artificial horizon, are employed, the arc read off must, 
from, the principles of optics, be halved before the other corrections 
are implied,* 

A meridian altitude of the sun, moon, or a planet taken at land, 
must be corrected for refraction, parallax, and semidiameter, and at 
sea for the dip of the horiron.t 

Having found the true altitude, take ita complement to 90°, 
which gives the zenith distance, denominated north or south, accord- 
ing as Uie observer is north or south of the object. 

Now. if the zenith distance and declination are of the same name, 
their mm is the latitude ; if of contrary names, their difference is the 
latitude i^the same name with the greater. 

Ex. \. — Edinburgh Observatory, March 28th, 1825, with an arti- 
ficial horizon and one of Troughton's best sextants, the vemicc' of 
which showed 10", Captain Pringle Stokes, R. N. found the mert<^an 
altitude ftf the sun's lower limb to be 73° 32' 15", the index error b^ng 
+ 2'26", the barometer standing at 29-66 inches, and Fahrenheit's 
thermometer 56°; what was the latitude, employing the refractions 
in the tahle in the nautical almanac ? 
, ^sen(e4^1%de . . 73'32'15^ _ 

»:«: LT IS 



Sum 



iT-Bi iiri .. ■ 
H«lf 

aefraction to 29.66 and 56° : 
. Parallax 
Bnaidiameter 

True altitude 

Zenith distance 
Declination 

■( Latitude . 55 57 28 

Ex. 2. — To determine from the observations of Captain Basil Hall, 
B, N., taken June 4th and 6th, 1832, the latitude of San Bins, that 
by estimation being about 21" 32^' N., and longitude lOo" 15' W. = 
7^ J- in time. 

To compute the sun's declination, June 4th, 1822. 




Longitude in time 7" 1™ 
Daily variation 6' 56" 
Prop, part to 7" l" 2' 1".6 

Eq. to sec. diff.— 23" and 7" + 2.4 


D.L. 
P.L. 
P. L. 




1^ 


Correct prop, part 2 4 .0 
DecHnsdon at noon G. 22" 24 4] .0 


>i 






Sun's true dec 22 26 45 .0 N, 


'¥ 


t T*W«i XIIL .nd XIV. hB»t bctn conipuicd, ci 
(imbininK tbe whole m one. 


,„»,„■., 


iW. 


puTxntBtjll 



■WifttiimOMETBT. I 

and the tbermometer 86° Fdirenheit, to mend. ut> tljtSf* Sdfy>t 

Z.D,: ;::••" vvorioi.i ft07M 

Bar. 29.76 " . . .' 9.9963 

Tiler. 06° 9.9964 

' - • • f - ■ • - ■ - ■ I . * ■ , ■ 

r 'J".l' • •.'^' " ■ (MB8B 

,PaxallwO"-a (table 16) 

Faceof the citcle w^st. ' ' 

p ..r J 8t Vernier " aO^W' 0" 

Reading8|2^ SO 10 

Obs. merid. alt sun's /.7. ■ 88' 50* 5 ' 

Sun's asmidiaHieter +15 



Beftaction . -1 ' "1.1 • » 

Parallii^ . + ^<h2" • ' 

■ ■ ■ j'll. I. '.il V 



True alt. sun's centre 89 'O'SI^, 



tiijt.i 



flft -r.. "^ 



*■ .^ 



Zenith dist ^ 54 ' 9.7 fl. 

Decliiiation 29 M W-O^'N. 



i^Ji 



-■ "1 



Ladtttde with face west 21 32 36.3 N. 

To compute the sun's declination, June 6th> 1822. 

Longitude in time f 1°*D.L. 0.5S408 

6 Q^'P.L. 1.406M 

■ ■■■■■— A 

ftop. part to T» 1", r 48" P.L 2J00048 

Eq. to sec. diff.— 24" and 7» + 2.6 

Correct prop, part + 1 50.5 
Dec. at noon 6. 22 38 10.0 



.5 



lime dec. at S. B. 22 40 0.5 N. 



1 ■ 



,v'> 



thit-t^ni|oiiieter 85'' Fak.^ the meridian Z.D. bemg l"" 23^.5. nearly. 

Z^'iypy V i : iPi^Ji I«. r 0.1896 Parallax (K.2i 

Ther. 85° log. 9.96M 

Bar. 29®"''"' "■'' -aio). .ruMin- ■.••;• i r.9.9071 

MiBi^^a6° : ■: i 0.9085 ; ^ 

Hr;ttjl?'.44 \ . - . 01576 

Face of the circle east. 

Readinirsi ^^^ ^^^^'^ " • ^"^ ^ ^' 

*^««»ings^2^ vernier 25 

Obs. zenith dist. suns's /. /. : - jl' S^.-^4o 

Siin's semidiameter . • — ]5 47*0 

Refraction . ' . , -^ ■ ,, ,. gWg 

Parallax-;-:- . . .._^ 4- - 0.2.. 



^)ll'.<MINTEODOCTiOK. ^ 



1 



Mean latitude by sum 21 32 28.75 

Wben the latitude is determined by an astronomicat circle, 
servation is not supposed to be complete, till the observer has. re- 
versed the circle, by this means combining two sets of observations, 
■with the ftce of graduated limb of the instrum«it alternately, as in 
this example, towards the east and west. 

San Bias, S»th May, 1822, tlie barometer being at S19.78 inches, 
Fahres^eit's tliermometer 83", the chronometer too fast for mean 
time 4" 4'" 45", Polaris on the meridian below the pole by chronome- 
ter at'l" S" 41- and its true apparent N. P. D. 1" 38' 28" .46. 



- 


FhoooflnHT* 




nnitfrnnj 


S.,1U«|.. u. 


ol».|.b..«ul 


Am.-d» 




: 


WMti 


I fi 5 

1 -S 51 
1,-8 41 
1 14 3 
1 16 Jl 
1 18 35 


2 36 
60 


5 22 
730 

6 54 

6 


a3''.27 

1 .36 

0.00 

56 .55 

110.44 

199 .41 


70 3 34 5 
3 34.0 
3350 
19 66 19.0 
56 18.0 
55 20.5 


^ 19 56 25.5 

J- 66 36.0 

1 56 25.0 

56 19,0 

56 lao 






19 56 23.33 




« 


374 .03 






62.34 





To compute the correction of altitude o 
of the star from the meridian. 

A 21° 32' 30" coaine 

: } 28 21 30 cosine 

Alt 19 56 23 secant 

*... 62"^ log. . 

i Cot— 1.77 log. . . 

The correction for part II. is in this case insensible. 

To compute the refraction. 
Z. D. 70° 3'.f 



8.467118 
0.026814 
1.794767 



Ther. 
Bar. 
Ther. 



83 



9.78 



F. 



r 147".02 
OrS'27"«2 
Observed altitude 
Refraction 
Correctioti 

True altitude 



2.20325 
9.97112 
9.99445 
9.9i)857 

2.16739 

19' 56' 22'',33 

— 3 27 .02 

- I ^7 



R-|tWni_;,.*iil> S3 53 .M 



log. 
log. 



SPHKMOAL iTttlOaifQMETBY. 8B 

True idtitede below the ^ole IS 6a Mu^N. 

Poltt- ditOmde .... laS^fla^N. 



Latitude firom Polaris SI 38 29 .00 N. 

from Sun . 91 32 28 75 



Mean 91 '38 26 ^ 

Captain Hall ma]^ at 8138 93.07 



"*^i 



Difference . . — ^ . 1 .JO 

Which appears to be oceaaioned by ne^leetng the anplicMaon et* 
the equation of second difference in redaang the Bun's aediBstion to 
the place of observation. 

. It aeema unneoesBary to extend our tvmarks farther with regard 
to ihete obaervationg, more especially if the examples in theimplBna- 
tion of the table XXVIII. be eonsulted. If the obiervMiiAna ne 
taken at tea with a reiecting instnunent, on the principles of Had- 
ley's quadrant^ a correction must be made for the dip. in addition to 
these already giV«ii. Thris may be taken from taole Xf; ; or the 
true altitude may be still more readily found from table XIII.'or 
JQV. sufficiently correct for all the usual purposes at sea. 
^ Ex^ 1. May 1st, 1B86, in longitude 64'' 25' W., fHie iDbterved n»e. 
ridian altitude of ^e sun's ^ £ was 48" 34' 30^, the senlth ^belig 
north of -die sun, and the height of Uie eye 14 f^; whatSras the la- 
titude? ; 
. May let at ship, time 0^ 0°" Dec.'3st 15<'4^10''(N. 
i Lcmg. in time 4 18 P.P. *f 3 14 ^ 

Or. time. May Ist 4 18 R. D. 15 7 S3 [n. 

Observed Altitude . 48»34'.5J 

- Cor. to 48^^ 14 feet, and May -t 11.5^ 

True alt . 4a 46j0 

Z.D. 41 14.0 N. 

Declination 16 7*6 N. 

■ ■' n» 



Latitude . 60 81 .6 N. 

It is unnecessary to push the calculations nearer than tenths of a 
mimstB, aa any observation taken at sea is, from the indistitietneas of 
the horiaon and the uncertainty of the horizontal refitustion, ukiless 
a dip sectiim be used, liable to an error of at least one minute. 

Examples for Exercise. 
•■■'Ir. €W the 1st of September, IB24, in longitude 54° W., the meri- 
dMn ^lOtlitude of the sun's lower limb was ^44' 15'' S., tbe'height 
of the eye being 24 feet ; what was the latitude ? . 
. ^^jfrJj^lS' 30^.9 N. 

• 8r"OH 4e 1st of January, 1826, the meridian altitude of the star 
AlrtfeWrs was 60° 41' 8., the height of the eye being 24 feet;,yrhat 
was the latitude ? 
-^iii:;-^9°*29'.8. 

'8. On the 14th September, 1827, in longitude 103" 1?' Ey ,Iet the 
meridian altitude of the moon's lower limb be 51° 4' N., and the 
'the^e 20 feet ; required the latitude ? , , . a 



^■"TT-furaaomjcTioN. i 



ri-t. 



if the 

& N.. 




4. On the 29th September, 1827, m longitude 20° ^y W. 
observed meridiiui altitude of the moon's upper limb be 83 
and the height of the eye 16 feet ; required the latitude ? 
^n*^21° 25'.7 S. 
As the meridian altitude may, by the inter- y_ 

position of clouds, or other causes, be lost at 
aea when a knowledge of the latitude is neces- 
sary for the safety of the ship, recourse must 
be had to other methods, particularly to that of h| 
double altitudes, and the time between them, 
^g being the most practicable.* This me- 
thod requires solutions in three spherical trian- 
Sles. In the triangle ZPS there are given -" 

'S the sun's polar distance at the time of the first observation, 
PS' that at the second, and the angle S'PS measured by th« 
the elapsed time ; to find the side S'S and the angle PS'S.t Again 
ip, the triangle ZS'S there are given the zenith distance ZS at the 
time of the first observation, ZS' that at the second, and the side S'S 
already foond to determine the angle ZS'S. But PS'S being al- 
leady computed, ZS'P may be obtained. Whence there are in the 
triangle ZST, the sides ZS', and PS', and the contained angle ZS'P; 
if) find the side ZP the colatitude. This is the regular method by 
^>hericBl trigonometry ; but if the polar distance PS be supposed la 
remain the same, that at the middle time, between the observations, 
or, as Professor Lax seems to think preferable, the same as st the 
time of the greater altitude, and, by combining the solutions of the 
several triangles in one, the operation becomes more simple. In 
order to render this method still more easy to practical seamen, 
Pouwes proposed an approsimate method by introducing the lati>- 
tude by accoimt, which, wlien properly restricted according to the 
rules of Maskelyne or the tables of Lu, will generally give the de- 
sired result sufficiently correct for nautical purposes, Mid the com- 
putations may be very readily performed by the tables of Lyni 



tables a 



■ used, Mr Iv( 



is the best. 



When the ci ^ 

particularly in the form that Mr Riddle has given it, which we shall 
adopt here. 

Find the sun's declination for the time of the greater altitude, and 
the true altitudes, reducing the less if necessary for the ship's run 
to what it would have been had it been taken at the same place with 
the greater. This is accomplished by observing the suns bearing 
by compass, at the time of taking the less altitude, and, finding the 
angle contained between that and the ship's course by compass, cur- 
recled for leeway if she makes any, in the interval between the ob- 
servations. With this angle as a course enUx a traverse tabl^, and 
the difference of latitude, answering ta the distance run during ^e 
lapsed time, will be the reduction of altitude. 

If the less altitude be observed in the forenoon, the reduction of 
altitude must be added to it, if the angle between the ship's course 
and the sun's 'bearing be less than eight points ; bnt if that angle be 

E eater than eight points, the reduction is to be subtracted from tlie 
IS altitude. If the less altitude be observed in the alternoon, the 



" ' <A ihe •uihorrtr of 

•Inubic alUtadts arc enl o 

i- A drde U suppiHed 



r ft VDTV dutiagutibed Mclieal luiif^tar, I u 
of siicfi Itrponrace M U gaanlUy mnpotai, 
i 10 pasi tlirounh PS' 1" similar (ci PCI". 



SPH£RI(eM.nnOC9XOiIETRY. 8^ 

i«duttMMi ii'io 'Ite flUbttllcted from 'it, If th« «m|» betw«^ Uie 
shfp'sr c^iifne and die ton's bearinr ift' Ims thm tif^t potnts ; bat If 
greater^ the reduction it to be adcled to the less ftltitnde. With the 
corrected altitudes^ the elapsed timey and the declination^ the la. 
titude at the time of the obsenration of the matest altitude will be 
found, which may be reduced to noon by means of Ae dead 
reckoning. 

1. Take half the interval between the observations, and call it 
the Mf elapsed time. 

H: To the sine of the half dapsed time add the sine of die sun's 
polar distance, the sum, rejecting always ten in the index, will be 
arc first. 

3. To the secant of arc first add the cosine of the polar distance, 
the sum will be the cMne of arc second, whidi will be of the same 
aliectioii or cfasoracter as the polar distance. 

4» To thd cokeoani of arc mrst, add the came of half the sum t^ 
tbe true' altitudes, and the sine of half their diilerenoe, the sum will 
herXht^sineo^ arc third. 

-5. Add together the secant of arc first, the sine of half the turn of 
the true altitudes, die cosine of half their difference, and die sticemt 
of flr4! third, die sum will be the cosine of arc Jburtk, 

6.: The difference of arc second and arc fourth is arciyUi, when the 
aenidi and the elevated pole are on the same side of the great ciMlt, 
pasting through the places of the sun at the times of observmtion, 
otherwise their sum is arcJUih* 

7. To die cosine of arc tmrd add the coxine of are fifths and the 
Slim wiU be the tiite of the latitude. 

Ex, l.--.On the 6th oi June, 1828, in latitude 58° N., and longitude 
40^ W., by account, at 10^ 53^ 20^ A. M. per watch, the altitude of the 
sun's lower limb was 52'' 20^, and at 1^ 17°* 8", the altitude of the 
same limb was 52^ Mt, and die bearing per compass 8. W. by W. 
The ship's course during die elapsed time was S., the wind E.S.E., 
and hourly rate of sailing 8 knots, and the ship making 1 j pts of 
lee-way. Required the true latitude at the time o£ observation of 
die greatest altitude, die height of die eye being 16 feet ? 

Ship's apparent course S. or 0^ 

Lee-way 1^ 

' Ship's ijrue course S. by W.^ W. = 1^ pts 8. W. 

Su^s beaHng at ad obs. S. W. byW. =5 ptsS.W. 

Cbntldn^ angle. . Si 

Interval 1>etw^n the observations = 2^ 23^ 48* = 2^.4 
Distance run =± i=2^.4 X 8 == 19.2 miles. 

l^dw'tbcburseSl points, and distance 19^.2, die difference of lati- 
tude is 14^.84, and since die least altitude was observed in the 
afternbcm^ and the angle between the ship's course and sun's bear- 
ii^ is less than eight points, this reduction is suhtractwe. 



■ ■■' ■. 

■ i ■ ■ 



* Should there be any doubt whether the zenith and elevated pole are on the same 
•id« «f die great drde, jMUtinf through the places of the sun, the latitodeinay 
puted mi bom soppotitionB, which, being comj^aied with that by account, the true lati- 

are 



tads iHaSL teffuinl,.be nadUyjUmMmed wiUrlktle addirioaainoobte, for It fe only 
fourth and it»oMBB,tiiatfMBisqiiiTCalt»mtiOa. ' - ■"•■• ^'^' 



•i^'t 



SPHRRIditL TBIOOMOMETRY. W 

iiHIHttiieiWMtVMN^W^iW. DiunagthedspMltHiteltadup 
!ir#fi.4iaiUpg 3. W. bj W. at the rate of 4 knots fir fcour^ aad the 
K^igtit of die <^aerver'8. eye was Sftleet Required the latitude sit 
the time of taking the fint altitude ? 

P]gu>BLi|;v VI. 
Onjiiiding ikfi ItfMgifiuie. 

I. BY L&NAR8. 

Sipce the rotation of the earth aliout its axis is performed ia « day, 
die sun appears to pass over 900" in 24 hours^ and, conseqii^enfly, 
over 16^ m one hour ; therefore, it is obvious, ttiat the d^Betenct of 
tkne between any two places will give the difFi^i^ of l(»igjlM4^ be- 
tween those places. 

A varietr of methods have been propoi^d for. deierminix^ .(bf 
Imgitiide of a place, but almost all or them depend upon one gnie- 
rd pHneiple, tae comparison of the relative tin^iea upafsr twct.4^fb:-> 
ent meridians ; so tha^, if the time on two different mericKanV be 
known, the difference of these times turn^^ ir^to degrees, at the^te 
of 15* to an hour, will give tlie difference of longitude between tltew* 
meridians. 

At the iun apparently moves froxa the ecist towards th^ W^st, 
it is evident, that all places lying to the eastward of atiy meridian 
will have noon^ or any other hour, sooner^ or if westward^ ^^^^^4 ^.V 
the precise time the sun takes to pass from the iperidian pf t&i pnV 
place to that of the other. Hence, if the time on the meridian of 
Greenwicb> the place fVom which pur longitude i^ Reckoned, an^ 
that of any other pl^ce at the same instant be kn^wp^ the IpAgitode 
of the latter place fjron^ Greenwich is also k^own, by tumii^ file 
difference of time i|itp degrees, at the rate of 15° to ap hour. 

Among the heavenly bodies which frequently present themselves 
for observation, there is none whose apparent velocity is so rapid 
with regard to the sun, planets, and fixed stars near the ecliptic, as 
f^hat of the mopn ; the uumal motion of that object being at a mean 
i«te about I^ 1Y\ ' Hence, her distance from these bodies is co]>-> 
tinually changing in proportion to the time, and an error of 2'^ in 
the distance between the moon and any of these bodies will produce 
mt error of about I' only of longitude. Of all the various modes, 
then, which have bfsen proposed to determine the longitude at sea, 
it is probable the method by lunar observations will continue to be 
the moft practicable. It appears also from the numerous observa- 
tioQn latehr made by several of our most distinguished navigators, 
that a series of lunars taken at land with good instruments, will, 
.when great nicety in the reqjiiisite observations and calculations is 
jMtndad to» give the longitede with sfaiguiar aceuracy. 
■ ; Tbt iDatrumenta generally employed are a good ehronometer iqt 
cgnnectiv^ obeervations tal&en at different times widi one ancyther, 
.tJip^ good quadrants for obtaining the altitudes, and a sextant or re* 
.^frtTTg cirae for taking the distance. These instmments are all 
deacriblBd in our usual treatises on navigation and nautical aatro- 
luxny. 

,If tl|a«uii4ir,«Ur.beetm^BaiBei«Dt diatanoe, from the asorldiaB at 
OiMinM of tikiof^thi distanee^ the tme «ltitBde of either ^ Atm 
4|i|is^ iHH J9«i^ 1|i>:0opitmt# ^ ■ppcmit time sft tlie ahip^ asiddA 
eompannd with the Oreenwidi time, derived from the lunar distance. 



9» 



, ftr * (IftKPBtwsicriON. 



will give, the longitude. The Bame thing majr be obt«iiietl from th* 
moon's altitude, but leas readily,. as her right ftsceiision and declina- 
tion must be very accurately computed by applying the equation of 
second difference. 

This method will be rendered familiar by the following examples.* 
Ea:. I.— September 24, 1827, in latitude W SC south, and longi- 
tude by account 120= west, at 8" 18° 3Cr A. M., the following obaer- 
vadons were made to obtain the true lon^tude ; the height of the 
eyes of the observers being 30 feet above the surface of the «ea, the 
angular inatruments being perfectly adjusted when the English ba- 
rometer stood at 29.4 inches, and Fahrenheit's thermometer at 60°. 

The mean of five distances between the moon and sun's nearest 
limbs -was 44° 33' 45", tlic altitude of the sun's lower limb 22° 4' 15", 
and the altitude of the moon's upper limb ff* 6' 0". 
Time at ship 23" 20^ 18"' 30- To this time by e 

Longitude in time 8 the sun's aemidiameter if 



Est. Green, time 24 4 18 30 
Obs. dist. n. /. 44° 33' 45" 
Sun's semidia. + 15 59 



the n 

augmentation 
hoT, parallax 
reduction to lal. 49° ; 



Moon's semidia. + 
J, Attgrnentation + 

App. cent dist. 45 
Alt sun's/./. 22° 4'!5' 


16 2 
2 

5 48 
lop; 


reiluced parallax 

alt. moon's h. /. 6^ 6' 

2.15567 fST41 
9.99104 
9.99123 


58 42 


». ». 67 55 43 


'2-30*^ 








Thermometer 60 .0 F 

r=187".25 

Or 2' 17".26 


9.99957 

9.98184 

2.13751 .•' = 482",3 
or = B' 2".3 


9.98184 
2.6a<108 







— 0.]04x{60— 50)=104xl0 = — I .04 
+ 0.15 X (30— 29.4) =15 X 6= +0.09 



True refraction for the n 
Alt. sun's I. I. ^2^ 4' 15" Alt 
Dip to 30 feet— 5 27 

31 58 48 
Semidiameter + 15 59 



Semi diameter auptn. 
App. altitude ' 



ilions are readily and rcry accuMltly performed, nccordillR 

StMoneUj from tlie Ublu conalned in Ibl* vort Therr 
tea, w^ u thote of Mendou Kin, I^s, L]riin,^it)l 
Thamaan, whidl, for Bcnenl prmcticc at m, by kbillug rameihitiK of rigorouB ucOib- 
cj, rendfT (he c«l™l«fifiO" more liiniilo. Home of ihem, however, m r»lh« Inlki 
•ntl expontlir 



SPHERie!At* tiMteltfH<*lJf ETEY. 

Parallax + 3 

' ' • ' • — Parallax in alt. 

SuWBT.rit » 12 33 

" •» : .'• i Modn's true Alt. 

SM.^Mu'. ; 58 42 

Paf; iii ah. •. 58' 22 






• / 



. 1 • 



5 36 28 
+ 58 22 

6 34 50 
Secant O.00&47 
P.L. 0.4868S 



P.L. i».4^0 



The reduction c^the apparent to the true dUtance is effectie4;faf 
the sQlu^pn .of t^vo 9pbencal trianglm. First the angle at the MniKp 
is^biuia £et^ fonofid by the qiparent senijlji djatiiBcigt 

a|id apparent distance. Next the true distance is computed from die 
angle at the zenith uid^the.true zenilji ^xBtancee^ aQ44ie9A-^W>lMt 
bj^rC<»pbined in the foIlqwJLMr ^ 



wi{yx^ .manner. 



^ >' 



A^pprdkt. 45'' 5' 48'' 
A^alt.© 22 14 47 
App. alt ) 5 44 29 



secant 
secant 



O.'083S93 

o.msri88 



Sum 




ehce 



73 5 4 

36 32 32 
8 33 16 



r • 



True alt 22 12 33 
True alt ) 6 34 50 



CQsine 
cosine 


9904^ 
9JW51«l 

i)i4)665ffi 
9.997129 


cosine 
cosine 



Sum 
Half 


28 47 33 
14 23 42 
27 1 65 


• • 

cosine 

sine 
sine 

sine 

■ i , i .0?42' 9" .^ 
1 37 15 


P.L. 
P.L. 


19.899515 


Arc 


9J49707 


Sum 
Difference 


41 26 37 
12 38 13 

> 

22 21 54 
2 


9.820638 
9.339993 

• 


\^ -K" 


iaj60«u 


RP > 


9.580316 


t if 
TriuLilist. .. .. 

6 


44 43 30 

44 1 21 

45 38 36 


0i63(Me 

026738 



Time bast 3* 1M8" 1* 
Pneced. time 3 



PL. 



0.36310 












•••'i9n!tf\ji^ bn.f' 



fWflWfitofci'ioM^.^' 



(Imil 11" 



42" 24' 13" 
3" 44 1 31 
« -4S 38 m 
9 47 15 f 



(1" llg" r)x4 = 5'' IS"^', fo wKchand second difference 7" we 
'gei (teinntMe XXVU.) 1" of motion, t%at'at a mean rttfe gives'2 
^^conda of time. 

This, from the explanation of the table, because the first differ- 
"fetd^s are all increasing, must be subtracted from the approximate 
BistatiL'e, and consequently added to the approximate time. 



To the approximate time 4*^ 18" 1' 
^dd cor. from sec. diff. + 2 * 



Greenwich 24^4 11! 3 



'^<:«n2piItatiort of the time d^fived from the figure iu.p^ 70, X 
I Tfieorem IX., page 75 trfler the examples in ( 



22^ 12' U 
89 40 33 
48 50 




Uf Diff) 

1 37 16 
1 37 22 



UdDfff: 



.+'2U 



xampled in page JQ. 




7 59 40.0=119^55' 0"West. 

= 120 446"W.iin ,-- 

— -^-i ' i i> ' ■ ' ^nautical tables. 



> Ex. a.— On September 12th, 1823, in latitude So" 30' N., Idngi- 
:ftUile by account 24° 30' W. at 5" 34" P. M. by watch, the altiUlde 



* The Huatior. of mcoihI diflerance bappona lo be smBll in this exunple. It nuf 
■mount 10 6 secorHU of ditlBDCe, 13 Kcondi of time, or 3' of longitude in sonic cues. 
The coTTection of lecoDd dlfleicnce l« tokea fVoni (he uiUal ubk, and il* eSeCta cMi- 
maled Bceoiding it) the moon'*! meui motion. It ii perTDnned more comclly. however, 
bj m«uu of Tables Sd and 4th Lmmedialelj'foIloBijigthUaitlele, which have been com- 
piiled by the suihor expressly for Ihis purpose. 



SPH£BlfiiM< IWKMVOMKTHY. 

c^xtto sun's low^ Innb waa7^ ^'» tlmt of the moon's lower limb 
wiu 35^ 35', the distance of fheir nearest Ui«bt 95* 19' 6&%fkehmh 
meter being 30.28inches, alfd tbe thermcmifter 7^-4 Fslu^nbeit, the 
height of the eye being 25 tfet; vrhat wagHtbe longitude ? 
Time per watch 6''34"' 

Longitude 24° W. in time + i 36 

<^fl^^6reeu^ t ^^^/^ '^ 

MOMTS'semidiameteratnomi I'if SSf'^ jparallax.. . )^'.3C'' 
^^QfllfBfpetim &e Greenwich time ,-| . ^ ^ , opjrea forT TpJ^^j.^ 



llf.t.Ji A- 



14 i5tJ \ . Equatorial Mr. .\^ffrSf^ 
Augmentation + ,.^ T t?!®^^- *^^^^ ■ ,^ 

I'vtfe te^tnidiameter 14 58 ^ i*d. hor. par. ' WPM 

Alt of sun's LI T^T • ■ 1tfoott*s-36° 36' . v 'i 

2. IT. - 88 S» loir.4^L61S18 Z.D. '54 26 )M.Aii)feP90 

P-P.3r 860 , , PJR'B' 133 

Hi^tfmoAieter 73 4 log. 9.d8Q20 9.08Q20 

Barometer 30 28 log. 0.00080 laOOfltO 

^ ''itiMbbmeter 72 4 log. 9.99902 '-9AM2 



r 398M 2.5989av TB'^B log. I^Mtt 

Ot 6' 38".l ot I' 18".« 

UUMUx + 22.4 ±=— 13 alt mooflfrs A ?. 35'' 35' secaitt 0^077 

♦'t^*^«Mc+-28 =z+ 0.0 red. hcMT. par, 64''24" P. L. O.HQISJ 

'xlf ' 6 36.8 paerallax in alt 44 14 P. L. 0.80044 

; Alt sun's /. /. 7° 37' ^' alt of moon's t /. 36^35' 0" 

Dip to 25 ft. — 4 £8 dip. te 25'feet v^ ' J-Lv' 41 98 

88 30 « 

7 32 2 sfemic&uaeter ; < : <)«'> M K 



Semidiameter + 15 56 am>. altittede . . 35 45 

' re&action . .»**m.4 ..1)^8 

Agp.ilt 7 47 68 pariOlax . :>^: ,^- 14 
Ra-atefion _ 6 S7 



I .» ■ -• • » 



Parallaat - v:(: ' J9 ,; moon's true alt tSfS 37x16 

Dbaeirved distence . . flsKlJJ'AB"^ 

^aSWfi's i^n^Siatfikdi' C ' \ . v :- -+ dS 58 

^^^Sfctoh'ii semJdiaMeter ^ ., ■ • < v. H,' : Iin66 



App. cttitr^ dist. 95 '50^58 



App. distr 
Sun's.app- alt. 
Aloon'aapp- all- 

' Sum 

Half 

Difference 
I Sun's true alt. 
I Moon's true ult. 

Sum 

Half 
Arc 

Sum 
difference 



fEalfiUet. 



•( T 1 MMaraBOIMM-lUON.;' 

95^30' 52' 
7 47 58 seoaiu. 

35 45 1 secaiu 



26 8 57 

7 41 301 
36 27 SBf 



22 4 43 
56 14 30 



78 19 33 
34 10 7 



47 5-2 13 



aO04(13fi 

(l.09{]672 



9.540:^77 
9.953107 
9-996075 
9.9053^ 

19489339 



9.744770 

9.990922 
9.749430 

19-740372 



9.870186 



True dist. 
True distance 
Dist-atd" 



U5° 44' 26" 
95 8 17 
96 30 13 





'3fi' 




(I' 


9" 


I 


21 


56 


r 


19" 


25- 


6 







■ iWe.pastT 
jTiioe , of first distance 

' Apprfiximate app. time at Green. 7 19 25 
'■ To find the corection for second difference. 

liisi. at 3^ 93° 46- 14 " ,.„«, „„ 

6 95 8 17 j If ^% - r 

9 96 30 13 I ll 2q - 7 

12 97 52 2 1 -JJ *« 

To the approximate time (T 20"') X 4, or 5'' 20", and the « 

second difference — 7". the equation from Table XXVII. is 0".9 •r 

about 1", which, since the second difference is negative, ought to be 

added to the proportional part of the distance computed by even 

proportion for the approximate time, and contttuiuently it must b« 

subtracted from the approximate time, or in general this correction 

for the time must be applied with a conlrari/ ilgn to that which is 

employed when correcting an arc, or with the same sign as that olj 

the second difference. . i - ■ '■"" 

Now l°2r58".-l"::3'': 2'oftimenearly. ■ n- i^ iV 

Or ttus operation may be performed by proportional logarithma, 




SPH£IUMlAli VRIOaKOMETKY. 



From approximate apparent time 
Sttb^Kct equation now found 

True apparent time at Greenwich 






>**««■ 



"^.( . i 



7 19 23 
To find the apparent time at the place of observation. 



The reduced declination i» found ag in the explanation of Table 

g;:i^:xxvii., then 
|iti^de 26° SC 0" N- , secant . ., 
Declination 4 17 7 N. secant 



ao489a0 

0.001214 



l* 



iKfference 22 12 53 
Zenith dist 82 18 30 



Suiri^ 



104 31 23 half 52 15 42 
Qi^nce 60 5 37 half 30 2 48 



ame 
nne 



.. i 






2M7 



m A* 

2 



sine 



9.898075 
9.699689 

19.647080 

9.823540 



; I 



App. time 
Greenwich time 



5 34 8 
7 19 23 



tpngitude in time 1 45 15 W. = 26* 18' 45" West. 

' Or about two miles less than Mr E. Riddle makes it in His trea* 
tiae on navigation^ a very useflil work, combining theory^ witli^pViiCj 
tice, a method too much neglected in the present plan of nauif^kl 
instruction. 

Ex, 3.— On the 14th of June, 1827, in latitude 28° 3P 10" N., 
and longitude; 144" W. by account, at abbut ^St^ 82"*,' the distance 
between the sun and moon was observed to be 97'' 22' 4Qf'; w^efi, 
the altitude of the sun's lower limb was 44*" 36^ 40". the altiiiide <6lt 
the moon's upper limb was 35° 38' 20^', the height of the eve being 
20 feet ; required the longitude, thie barometer being at S^.68 inches, 
aiid Fahrenheit's thermometer at 68°. 



Eatimaled time, June 14th, 
liongitude 144° W. iti ti^e 



■..»:. 



90^32" 
+ 9 36 



.'^ 



Af^O!8ilnate'Gi:«enw4dK time,'^une 15th, 5 

1V» tfaiiitinle ^[KKm's 'setE^diaiiieter is 15' 37^' hor. par. 
Adg»eiitatioii:to36^ak. ,< ',+ 9 red. to lat. 28^' 

pi n-ji/U '■ ' .^z , .. > ■'■■ * 

G>iTect semidiameter 
Alt. of sun's /./. 44° 36'.7 



'15 46 cor. hor. par. 
moon's 2^./ 35° 38^.3 



8 

5718^ 
— 2 



67 la 

. 'A 



"M '*' 



Zenith dist. 45 23 .3 lo.« 1.77198 Z.D. 

tli^hteometer '68 9.98401 

tWclAMer 29 .7 9.99563 

l^lietmometer 68 9.99922 

lotii^U ■ 

r56"-3 1.75084, r 1' 17".4 



54 2l.71o.ll, 
"ft 




.. '/V J-^ ri 



1.88873 



Wlh'vMMdiiineter lH' 46" 
PRraHax^-in kit. d 



Alt. 8un-8 I. I. W 36 40" 
Dip. to iX) fbet — 4 36 
Senu(}isnieter + 15 4ti 



moon's ah. 36° 3f*' 
hor. par. 37' 16" 



Apparent central distance 
Now to compute the C' 
]p{r young's method, there are given. 
4 = 97° which by Uble I. gives A = 
a =45 




lii^^ri 


app. alt, 

refraction 

parallax 

true ah. 

limbs 


35 18 8 
- 1 18 
+ 46 32 


^ue sh. 44 47 10 
Observed distance of nearest 
Sun's aemi diameter 
Sfoon'a semidiameter 


36 3 22 
97" 22' 40' 
+ 15 4fl 
+ 15 46 



97 54 12 
nidiametera, by 



8 14 

, As these give in Table 11. 1" for the sim and 1 " for the moon, «r 
^ m all, it is necessary to subtract them from the apparent or evrnl 
tfue distance when they are so small. 
Apparent distance 97° 54' 12 '' 

Mn's app. altitude 44 48 secant 0.14aW>4 



Son 

BJf 

a*jn'i true «lt 
Jloon'fl true -altitude 


178 20 

89 10 
8 54 2 
44 47 10 
36 3 22 


Sun. 


80 50 32 


r 


40 25 Ifi 
82 30 



8.240647 
B.90473H 
9.851100 

9.yo764(i 

IR.231386 



iikni 



SPH£Ri04I« THQOMaHETRY. 



• • F 

Half&t. ^35'3Si'' sine 

•■-.■■ • - ' ■ ". . 'V- ; '. '1 ' ; " 

J;!';' vir <• I t ill. 



♦ t? r 



Cb^ Ibr oblique aemidiAi 



«r 11 7 



Hme^dlst. 
DSAtat 



>-<'a^>*> . . . , f •: 



97 U 5 
6^A7^ 18 62 
9^95 47 



.HI 



9^5078 



, -I 



1 3152 px. Woan 



9° 7' 47" ?^ ^g}{ 



6. • . • It. 1 » 



1 .r 



— 83" 

— 23 



Mfin 
— 23" 



. .Ill 



•t 6 16 1& 

Di«t«t 3^08° 61' 7" 1.02.1V/ 
; > .6 ■SU 18 52 } If J^ 

,^ To 15"* and 23'' the equation of second, ^J^^^Bj^c^aci^ is J!^, wnic& 
for a variation of 1° 32' nearly, gives 2* of tfme to be 'sabtracteJ; 
whence the true time is 6^ 15"" 13' of the 15th of June^ orj^ Ib'^Vf 
after the noon of the 14th. 



?rue altitude 
olar distance 
Latitude 



To compute the time. 
440 47/ i(y/ 

66 41 10 
28 31 20 



cosecant 
secant 



oiKMwga 
OHifiauB 






139 59 40 

09 £9 50 
25 12 40 



.aosine. 
"sine . 




• % 

■ ' • 

Time from noon 

^■•. • -J: ■^ ■ 

' ^ ri|[e ]4th •;" 

^j^liejejit GreeniM^ich 



1»40"86'.3 sine 
2 




13 21 
24 


10.6 


%0 38 
30 15 

• ft: . 


49.4 

13.0 



9.534111 
9.029d6| 

19JSSQBO0 

ii-Mii ■■■■■ ix 

199 




9 36 23 .6 = 144 O' W. 



lii' 



._ilfipihe 29th of March, 1826, in latitude 56° 12' S., and longi- 
tuide by account 97° W. at about f" 20" P. M., the observed distance. 
^HHPaeorthe moon's nearest limb and the star Fomalhaut, was, ftan^ 
a mean of five sets of observations, OP 56' 30" ; the observed altitude 
f ^^lte .moon's lower limb was 32"^ 4'; H^o observed altitude of the 
UpHhlJS'; the barometer being iS4!'inilbcl8, the thermometer 42^1^ 
awLuflLb^ght of the eye 20 feet : what'was the true longittmi F ' 



^ fBlit- iNTRODUCTIONi-fHie 

Est. time 7* 
Jjong- in time 6 * 

Est. G. time 13 30 reduced hor. par. 58 6 

moon's semidiameter 15 53 

augmentation t« 32° + 9 

augm. semidiametcc 16 1 

''itov to correct the oblique semidiameter by Dr Young's 

'■TWni Tables I. and II. we have 



■InH 




5 in Table I. 



170 give Cor. : 



■WH. 



•'.<M»aerTed distance 
HcKHi's aug. Bemidiameter 

App. central distance 
rjytofstar 6" Iff ah. of ir 

sLd. 83 44 log. t aeSUO, Z. D. 57 36 log. » 

Thermometer 42° F. 0.00730 

Barometer 29.2 in 9.98826 

Thermometer 42" 0.06034 

9.99590 9.99590 

r' = 486".4 2.68700 f =: 1' 32" log. 

Or =8' 6.4 ^ 

>*r^.l X — 8 = + 
■>i9-'.Ux -8=+ 



: in table 11. 
61" 5& 30" 
+ 16 1 



33 13 31 
1.96844 



.^•^W*' 



).»>. =8 7 5 

Alt. of Htar 6° 16' 0" 

JDip. to 20 feet — 4 36 



True alt. of star 6 



Aloon's ait. 32° 4' secant 
Hor. par. 68' 6" P.L. 

P.L. 



Par. 



alt. 49' 14" 
alt. of moon's /. /. 
dip. to 20 feet 



app. alt centre 
refraction 
par. in alt., 



32° 4* »■ 

— 4 26 

31 59^ 

+ J^ I 

32 15 ^ 

— 1 3J 

+ «J* 



SPHEBiCHtoJIiOWWtETBY. 

Star's ap^. alt. 6^ il ^M : v- \ WM^ bfn 
Moon's app» alt. 32 16 36 teomt 

100 »M ., ;- - .-n .-.,.■ 



1 , 



%i 



Sum 

Half 

Diff. 

•StanlatRipalt. 

Moon's t. alt. . 

Sum 

Half 
Arc 

Sum 

Dif: 




60 Id 60: 

11 63 41 

e 3«r 

83 3 17 

19 33 29 
38 6 49 

57 39 11 
13 32 37 



'U 



cosiM ; 



r K \^i; 



9.806007 
0MOOOO 

lUNtaSOB 

lft7910iS 



coniie 

sine 
sin* 



■Mil 



9^896967 



?>■; 



* 

V. 
f 



31 13 64 sine 
2 



^iVdl^dflt. 
Dist. at 



>1^i^^ I III! 



«2 ae la 

12^03 1^41 
I£ (n 41 46 



«»4*'a6» 
i.sa 60 



»\« 



Preceding hdto 



12 



y^jpxjj«t* *W». time 



9J9O706 

>' MOS40O 
^> ■» 

19.^9166 

rniiii 

u-.: j Jj ii.'JiI'i - ■*!'*«' 

P.L. 030021 



^. 



13 80 at Greenwich. 



P26'38" 
1 28 66 
1 28 n 



— 46 



-^ 4Sr.5or^44''nearlj 

lO 



;i^Aw td |i^6jdiiafie;.timfc l" 9^ and second ifi£RBlr$nee^i-4¥^yihe 
Mation of lieond difffobm^i is Sr£, to HiMeh tnd fariMtibii f* 29' 



equatio] 



^.Yiiair^y in 3 iMir4. (heL^^^ edaatiim in tinuris mfoout 11* tk> be sub- 
^T^m^r WBenc^jffrott l^^'d^ air eqoadM^ 



^,gi^tn«:4pp«rent 
fp^ m. 69 27 40 



I 





1^99* 49* at O^penwit^h. 
UMi apparent Ane at shi|^.' 
26^ 



fJJ 






56 W'^'O^''"" 



cosecant 
secant 







121*'«- If*^ t^' 
54 48 7 



cosine 

sine 



Star's merid. dist. £. 



4^22-46^* 
2 

^1 I II hiifcl I 

8 4fr 31 



sine 



' 0;9B4t94 

*o 1 ^ 1... 1 ,. 

9.087^2 
9.912309 

19.919348 

9.969674 
54 

20 



100 



^...WTWWCTIftN, ..,.,., 



Stor's merid. dittence E. 8^ 45*" 31* 
SUr'aRA. 22 48 1 



R. A of merid. 
Sun's R. A. 



App. time at ship 
App. time at Green. 

Long, in time 
.Without £q. 2d diff. 



31 
24 


33 32 
32 26 


7 
13 


.1 

29 


6 
49 



6 28 43 
6 28 54 



91^ W 45" W. 
97 13 30 W. 



Error 2 45 W. 

TABLE I. 

OOBRECTION FOR THE OBLIQUE 8EMI-DIAUETBR. 

For Argument A, 



• 


For A A 


—J 


d 


For h h- 


-J d 


For h h — s i 


i 




* 


° A 


o 


A 





A 


° A 


^ o 


X 


. : 

• 




89 924 


1 


176 


59 


71 


31 29 


29 94 61 


6 


• 




88 954 


2 146 


58 


72 


32 28 


28 95 62 


5 


J 


» 


87 972 


3 128 


67 


74 


33 26 


27 95 63 


5 


* 


■ 


86 984 


4 


116 


56 


75 


34 25 


26 95 54 


5 




» 


85 994 


5 106 


55 


76 


35 24 


25 96 65 


4 


f 


1 

t 




B 




54 


77 


36 23 


24 96 66 


4 


• 




84 2 


6 


98 


53 


78 


37 22 


23 96 67 


4 


1 




83 9 


7 


91 


52 


79 


38 21 


22 97 68 


3 






82 14 


8 


86 


51 


80 


39 20 


21 97 69 


3 


. 




81 19 


9 


81 


50 


81 


40 19 


20 97 70 


3 


1 




80 24 


10 


76 












; \ 




79 28 


11 


72 


49 


82 


41 18 


19 98 71 


2 


; 


• 
t 


78 32 


12 


68 


48 


83 


42 17 


18 98 72 


2 


_ 




77 35 


13 


65 


47 


83 


43 17 


17 98 73 


2 


. 


■ 


76 38 


14 


62 


46 


84 


44 16 


16 98 74 


2 


'•' ! 


t 


75 41 


16 


59 


46 


85 


45 15 


15 98 75 


2 


" . 




74 44 


16 


56 


44 


86 


46 14 


14 99 76 


I 1 


1 




73 47 


17 


53 


49 


86 


47 li 


13 99 77 




J 


1 


72 49 


18 


51 


42 


87 


48 13 


12 99 78 


■l 


If ! 


• 


71 51 


19 


49 


41 


88 


49 19 


11 90 79 




s 


70 53 


20 


47 


40 


88 


50 19 


10* 99 80 








69^55 


21 


45 


39 


89 


51 If 


9 99 81 


1 




r 


68 57 


22 


43 


38 


90 


52 10 


8 100 82 





t 


1 


67 59 


23 


41 


37 


90 


53 10 


7 100 83 







y 


m 61 


24 


39 


36 


91 


54 9 


6 100 84 





'A\i' \ 




65 63 


25 


37 


35 


91 


b5 9 


5 100 85 





n 


? .. . . 


64 64 


26 


36 


34 


92 


56 8 


4 100 46 





"i 


-•• . ••■" r--«»k.- ■-" 


63 m 


27 


34 


33 


92 


57 7 


3 100 87 





- 




62 67 


28 


33 


32 


93 


58 7 


2 100 88 









61 69 


29 


31 


31 


93 


59 7 


1 100 89 









60 70 


30 


30 


30 


94 


60 6 


100 90 








SPH£KKJii: -nrtGdSOMETRY. 



Ittt 



TABLE II. 

CORRXCTION FOB TBB OBLIQCB BBMI-DIAMBTBB. 

DIMINUTION OP THB SBMI-DIAHBTBB. ' ' 

Argument A {h)-JfA {k—t)-\-A{d). 





























' M 












AlTiiude. 


1 


furaofA 


5' 


-gr- 


7° 


8° 


1!! 


10° 


11" 


12° 


H- 


16- 


Ilf 


E 


w 


w 


0" 


25" 


19" 


l4" 


7r 


9" 


8" 


"a" 


IF 


4" 


3" 


"F 


2" 




1" 


20 


24 


18 


14 


11 


9 


7 


6 


5 


4 


3 


2 


2 




o 


40 


23 


17 


13 


10 


S 


7 


6 


5 


4 


3 


2 


2 







60 


31 


16 


12 


9 


8 


6 


6 


S 


3 


3 


2 


2, 







70 


20 


15 


12 


9 


8 


a 


5 


5 


3 


B 


9 


9l 







80 


19 


J4 


11 


8 


7 


6 


5 


4 


3 


2 


2 


2 


1 





90 


17 


13 


10 


8 


7 


B 




4 


3 


2 


2 


2 







100 


16 


12 


9 




6 


5 


4 


4 


3 


2 


2 


1 







no 


14 


10 


8 


6 


5 


4 


3 


3 


2 


2 


1 


1 







120 


11 


9 


7 


5 


4 


3 


2 


2 


2 


1 


1 


] 








130 


9 


7 


5 


4 


3 


3 


3 


2 




1 




l: 








la-i 


7 


6 


4 


3 


2 


i 


2 


1 


1 


1 


.1 


(1 








140 


6 


s 


4 


3 


2 


2 


1 


I 


1 


1 


I 











145 


5 


4 


3 


2 


2 


1 


1 


1 


1 

















loO 


3 


3 


2 


2 


1 


1 


1 


1 




















155 


3 


2 


2 


1 


1 


1 


1 























160 


1 


1 


1 



































170 






































o 





178 


1 


1 


1 








o 























fl 


180 


2 


1 


1 


1 


1 


1 


























182 


3 


2 


2 


1 


1 


1 


1 























184 


4 


3 


2 


2 




1 


1 




1 


1 














186 


5 


4 


3 


2 


2 


2 


1 


1 


1 






1 







188 


7 


6 


4 


3 


3 


S 


2 


1 


1 






1 






190 


9 


7 




4 


3 


3 


2 


2 


1 






1 







191 


10 


8 


6 


4 


4 


3 


3 


2 


2 






1 







192 


11 


y 


7 


5 


4 


4 


3 


3 


2 


2 




1 







193 


13 


s 


7 


5 


5 


4 


3 


3 


2 




2 


1 






194 


J4 


10 


8 


6 




4 


4 


3 


2 


2 


2 


2 







195 


15 


1] 


9 


a 


6 


5 


4 


4 


3 


2 


2 


2 








196 
197 


17 


la 


10 

11 


7 
8 


6 
7 


6 

e 


S 


4 


3 
3 


3 
3 


2 
2 


2 

2 


- 


z 


19 


14 


108 


21 


16 


J2 


9 


8 


7 


R 


5 


3 


3 


3 







— . 


199 


23 


17 


13 


10 


8 


8 


6 


5 


4 

















200 


25 


19 


14 


}}_ 


9 


— 


"i"" 


— 


— 


— 


— 


— 


- 


Alt 


[El 


^ 


^T 


3= 


9° 


10= 


u<n 


I4":16° 


18" 


20» 


30- 


45' 



['-• m i"' 



M-^jwroewreTsmuansf 



TABLE ni- 
EquATioNS OP Sbcond Difference for Three Hour 

anand OiflWaitt," 



l»1.l UUi 



^ 3, 
^a lis naft 



._U 



il 



P 




SPH£AI€AI^ TUGQNOMETRY. ]M 

la the practioe of lunars four pc^raons are freauently employed in 
making toe observations, the first to take the distance, the second 
to take the altitude of the sun or star^ the third to take the altitude 
of the moon, and the fourth to write down the observadons. One 
person^ however, may make the whole himself, according to the 
Allowing method, which was obligingly communicated by Uiat dis- 
tinguished practiod navigator Captain Basil Hall. Speaking of his 
own praGtiGe, he says, — " I always take all mv altitudes and dis- 
tances with the same instrunfent. First the altitudes of the sun, 
then tfiose of the moon, then several distances ; next the altitudes of 
the moon, then those of the sun, and interpolating by proportional 
kgaritbnis for the id t itud e a at the mean time of the distances.* At 
night I never take an altitude, unless it be about twilight, when it 
ca]3L be done with accuracy and ease." 

** The method which I use to connect lunars and chronometers is 
not v^enr general, but infinitely the best, and ought to be univesrsally 
adoptoil^ as it renders all allowance for the distance nin in the in- 
terval of Mtile or no consequence." 

" jiie use of lunars at seal conceive is, in a great degree, to 
pheck the chronometers : the method by lunars being infiaUible, 
though not very nice ; that by chronometers being fidlible, but as 
nice as possible. So tiiat a number of lunars are necessary to check 
II chronoineter, and the object is to bring the whole of. such lunars 
to bear rigorously on the chronometer without makinjg use of the 
logboard." 

^^ This will be best illustrated by an example. At noon, or any 
other hour during the day most convenient lor taking a lunar, I ob- 
serve a set, or ha& dozen sets of lunars with the sun, carefully not- 
ing what the chronometer shows, but without taking any account of 
the actual time. At any other hour when the sun is near the prime 
vertical, or most suitable for determining the time, I take altitudes 
expressly with this view, from which 1 discover the error of the 
same chronometer used for the lunars. Again, during the night I 
take lunar distances with the stars, on both sides of the moon if pos- 
sible, at the moments most favourable, but never mind the exact 
time, '.only carefully recording what the chronometer shows. Now 
by the sights for absolute time I ascertain what was the error of 
the chronometer on apparent time at that meridian, and this same 
error, 20orrected for rate during the interval, I apply to each of the 
differfnt times by the chronometer when the lunars were taken. By 
this means I get the apparent times due to the meridian, on which 
the absolute tune sights were taken, with as much accuriicy as i£ 
the wholes lunars and all, had been taken at that fixed meridian. 
The distances give the several times at Greenwich, and dius they all 
coDciir in settling the difference of time, between the first meridian 
and that diosen for taking the time, with a view of seeing what 
longitnde the dironometer gives. Hence, if there had been an un- 
men ^current of some miles an hour of which no account could 
possifely be taken, still the result would not be vitiated thereby, 
but ail the lunars would be found to contribute to the sayne end^ 
thus tasking* jbcoording to Dr Wollaston's simile, the moon serve the 
purp^ of a great Greenwich clock in the heavens. After having 

^ . - •'■!.- . : ■ • 

_2 . ' _--■■ ^ - - - - --- , , ^ _ _ 



I 



i . * This ii liflpf ir to the method giren in Nonets Navigation. 



F 



determuied tW tfik lortjfitude aAd eynta' of th« ohninom«tvrs^wfi% 
within a few dsyB-eail of the land, I run the remainder of the voyage, 
fa i great degree, by the chronometers alone." 

(Xtt'^i; ^\v"' ■ OnJndinglhelfOngUude. 
VIMlJQ .\-' II, Br CHROBOMETEllS. 

^'^^^'foregc^g method of findinE; the longitude by iunara is very 
T&hiHbie at sea, on account of the frequent opportunitiea which oe- 
ettr ifor observation. About the time of new moon^ and in unsteady 
weather, the necessary observatiohS for the practJco of this mHIMd 
cannot be dbtaitied, end the dead reckoning is not to be depended 
on for any length of time, therefore recourse must be had to other 
.methods. 

On account of the very high degree of perfection to which 
chronometers have been brought, the longitude determined by a 
mean of three or four of these delicate machines nterits- gr^atr aoBA> 
dence. If the rate of a chroDometer be determined on shsr^t.91 
rather perhaps on board in the situation it is intended to occupy 
4W*9g *^^ voyage, where the various causes which act upon it, am) 
fte likely to alter 'its rate, are in operation, it is likely this rate wiH 
temam pretty uniform for some time, and the amount of the gain or 
IftSj Ijieing allowed -for on the time indicated b^ it ut any.fuUu« 
period, the true time may be obtained at the mendian of tbo placv 
where its rate and original error was determined, with aa much accu- 
racy as if it had been adjusted to go accurately to meaa solar tiina 
on diat me^dian. Hence, it is obvious, that if the original error, 
^d^e gain or loss in 24 hours, called the daily rate, of a chrcHUk^ 
Betar>'be known, on any meridian, such for example as that at 
<|reenwich ; by making proper allowance for these, the mean time aX 
Greenwich may be readily known to such a degree of accuracy aa 
the going of the chronometer will warrant. • 

l. It !£ now only necessary to find the apparent time at ship^ by an 
stitvde of any celestial body properly situated, bj' some of the me- 
mbds'already given; to which the equation or time being taken 
tKica the Nautical Almanac and properly applied, the result will Ii* 
the mean time to be compared with that 3t the given meridian to 
now the longitude of tiie ship. 

^.Thetate of a chronometer is readily obtained, by observing dailv, 
^possible, the altitude of one or more ccyestial objects near tOe 
^nme vertical, from which the mean time may be accurately deteri 
mined, and, being compared with that shown by the chronometer, 
its' gain or loss in 24 hours, and also its error on the day of the last 
observation, called the original error, will become known." 

E±. 1.— Near Falmouth, in latitude 50° 8' 48" N., and longitude 
20" lO- W., at about Hf 47" 20*, the following altitudes of the sun's 
lower limb were taken, with an artificial horizon, in order to jiscer* 
tain the daily rate of a chronometer previously set to Greenwich 
time. The observations were made with s sextant of which the in- 
dex error was-l- 1' 30", the barometer 29.6 inches, and the thcrmomei 
ter 56" Fahrenheit. ■ ■* 



* ThcK woulil be more accuniely perftinned od thore by usniR an utiSdil bi 
widthe meihodofcquil sltliudct. la tbS* coaa vockci chwiivnicut ibaubltM:..^ 
pinfed, 10 bt conipued with IhoH on bOKd, whtdi nu|{hl to be bi nuiDCtOiU •■ fM- 



spHEBioiL nuararoMETRY. 



M6 



W10P35* 87»48'46" 
IS 45 88 4 30 
14 68 88 90 16 






^ 



8 18 3 



13 30 



ate. 1»* 8' 



Z. D. 70 67 log. «S.92160 
dier. 60> log. 0.90400 
bw. 29.6 log. 9.00417 
ther. 66» log. 9.80974 



Uma 19 12 40 38 4 80 

I. E. + 1 30 

9!38 6 



r 168".4 

s3'4S".4 
•un'* pacalln 8^'.l 



2.S198S 



m 



19 

Thw ilFtdiiMudi 
IioogftaaeiB tinw 

QnonWicli tioM 
BuDyntriBtioii 

VMj^ipvt to IV ^\r 
Bee. ftt DOOD, May Itt 

Svttlf ttdnced declination 
ObaervBd idt sun's /. /. 
SemidisnietcT 
. Refiwcnon 



3 

1» 47*90' 
+ 90 10 



19 7 80 

14 19 
16 8 49 

15 93 8 



D.L. 
P.L. 

P.L. 



0419081 
1M080 

1.09941 



34 46 40 
70 43 42 

106 29 22 
36 58 2 



True altitude 

Sun's trfte dec. 
t^tude 

Sffirence 
Zenith dist 

OI1V&. 
SfiffpipeQoe 

r*. 

^me from noon 



Afqpisrent time at Falm. 
Equation of time 

Mean tfaM Hi Falm. 
Thne b J ehronemetsr 



16023' 8" N. 
50 8 48 N. 



secant 
secant 



i9« y 0" 

+ 16 63.8 
— 9 48.4 
+ 8 .1 

19 16 18 

0.015860 
O.J98900 



half52<»44'41''8ine 9.900884 
half 17 59 1 sme 9.489699 



t- 



ChKOBOiiieCer for Falm. 



2P*36"93'.8 
2 

6 12 47 .6 
34 

18 47 12.4 
— 3 10.9 

18 44 1.6 

19 19 40.0 < 

28 444> tut 



19.690693 
9.7997B6 






sm 



y HTtllSTBODUfiTION. ! 



PTF 



i Again, on the lltli a£ Kitty, 1834, the altitude of die Sim's lower 
limb taken with tiie same instruments as before, the index error be- 
Ong constant, -was 18° 9' 50", when the chronometer showed 18" 
,67"° Sff. This gives the mean time at Falmouth 18^ 30°. 23'.5, and 
ttbe eFEoi of the chronometer for -die meridian of the place S?" 3Sf.B. 
■Whence, on May 1st, the error was . . 28° 4415 

' Utb . . . 27 32.5 

^The loss 



1 12 



1 ten days is 
e day it is 7.2 

ITence the daily rate is — 7.2 

■; It is to be observed, that the altitudes should be taken nearly at 
the same time of the day, otherwise an allowance must be made for 
the rate during the interval. 

1. On the 22d of May, 1824, in latitude 32° 36' N., and longitude by 
account 16" 40' W., the altitude of the sun's lower limb at sea was 
37° 24', when the chronometcB showed S" 12° 24'.5, the height of 
the eye being 20 feet ; required the longitude ? 
-*Inae »er. watcli 5" 12° 24'.5 Daily rate . ' 7.2 



^Original error 


-2844.6 LoninUlday. 


1 1.8 

eoJsTo 


I«99in 11 daysS 


4 43 40.0 
!'■ + 1 21 Or 


'Creenwich M. time 4 43 1 
•Alt sun'8 1. 1. 39" 26' dec. 
"Cor. table XIII. + 10 cor. for 5' 




Ttoe alt. 


39 36 cor dec. 


20 a»K. 


'True alt 


Z.D. 


09 32 








'Latitnde 
'Sum 


32 36 secant 
141 44 


0,074466 




70 62 ,^g^: ...... . 

^' '« ..-..i^.,,«-,l.d,non„ 

1^50"'.39' sine 
2 


9.615666 
9.715186 

i.:- jL_J 

19.333585 


1. 


9.66676S 


'App. time 
'Eq. of time 


3 41 IB 
— 3 40 


179 


'Mean T. at .hip 
M. T. at Green. 


3 37 38 

4 45 1 





Long, in time 1 7 23 = 16= 51' W. 

For. the usual computationa at «e« it is uniieceM^;^,tq; ff^ tbe 
calculations farther than the ueareit minute. . ~.i- '■ ..i.^i« .t......i)'.-,|.,> 



SPHEBMXili' TSIGOVOMETRY. '107 

vi'aL;OiQrtitlM»irllh ^ifksikh^, ^aM»faiham^\tm the meiUiMi of 
ftitfM ii iul^ Jwrchttmwirtttr^ ml* Hie' didly^ntfe 

MBe<:d^bCittiriAMir/ tlie *ob8ert«d rititadr of *tiie am'it 'lower Ifnb 
tMiii^)4S?i7^'ilOf'; ioid the height ofthe i^ 90 Ibet^ 'nH|virtd tbe 
*WPtede? . 

1 v.. t*» Jim.— 83» 25' E. 

,^S^.X}|i-the 16th August, 18S8, in latitude 39> 90" S., the meui of 
seirfral altitudes of Antares west of the meridian was 14^ Hy, the 
jiq^t of the eye being 12 feet, and the mean of the timea per 
qnlftdti IP 41" 38> p. M., which had been compared with mean tvne 
/^ the Cape of Gcx>d Hope -on tbe^ 22d of Jun^ and was foiuid tQ be 
)^ )Q^ 38* too 9low> ^d gaining .9^*61 9 day ; required the longitude 
of the ship? 

.i:'.;^.(i. SQUATION TO BlIUAL' AXiTtTUnna* *■ '* 

n In ordinary cases the error and rate of a chronometer niay be de- 
Itfmined by single altitudes ; but when ffreat accuracy is re<|aired 
equal altitudes are very superior, especieSly when a transit instru« 
ment cannot be obtained. On this account various taU^ h^y^^bfiin 
CWVputed to fadUtate this operation^. tbQUgh it is bdieved few of 
thieuL.afford great advantage m actual wactiCQ* To those who would 
Metier. juch a table, that of D. Joa^ S. C^rquerOjnrgiyqa.Ui»d(hc 
uirteenth volume of the Journal of Sdence^.is perhaps the most 
commodious and exact By this means, however,- tablea W4Mi)4. fae 
multiplied to any extent without giving much ad vantage, •on^jaC'^ 
count of the inconvenience of taking proportional parts;. ifVftd iS^Bi 
thia -consideration it is often better to give an easy practical rule, 
veqdirinff the use of the ordinary tables, where neither double ep« 
trieoi different signs, nor proportional parts are necessary. 

TThe equation of equal altitudes is a correction for the cb^ng^ pf 
4QcUlMti<Hi of the celestial body during the interval of ob^eryatiiQi, 
tQ,]>e a{i^lied to the middle time between the instants showjDki)iy«a 
chronometer, at which, cm a ffiven day> that body has equal alti- 
tudes ; to find the true time by the chronometer when the object 
was upon the meridian. 

Rule* 

^ r Xo the coeine of half the interval between the times of observa- 
tioOL addthe cotangent of the latitude, the sum, rejecting 10 in the 
Jildez#,«jll be the tangent oiarc^rst, the <Hfference between which, 
^pd;|j^:polar distance, will be arc tectmi. 

^]3^ow to the constant logarithm 5.364517^ add the cotangent of 
■half the elapsed time, the cosecant of arc &rst» the cosecant of the 
^fhr distance, the sine of arc second, the lo^rarithm of the elapsed 
tune in minutes, the logarithm of the daily variation of the declji;iat3on 
in seconds, the sum wiU be the logarithm iof the equation of equal alti- 
tudes in seconds of time, which, when apfdied to noon, is addifivj^A^ 
the polar distance is increasing, and subtractive if it is decreasing. 



dple, though perhaps in the detail Mmewhat kne timpio. ' '- - •''((' u- -^iv. 



If tb« equtf^oH' is Mpplied to taimnam, it ia addiftteitAcptHsr-^tf' 

tance is decreatitig, a.nd subtructive if the polar (lutance is incrvas- 

Et. ] .-^11 the 23d of March, 1809, at Pisa in latitude 43° 43' 11'^ : 
N. equal Bltitudes of the planet Venus were taken before andraftes 
transit, the elapsed time bietweeii wliich was 8'' 50°' ; required the 
equation of equal altitudes wlien het declination was 20° 42" 40''N., 
and her daily variation •f-2l>' 5" or + 1205" increa^ng, and ootue- 
quently the pcdar diglance deereasing? r . ^ _i 



Latitude 43" 43' cot. 0.019462 
IJ.^. J. ,.. ..4^25" tos. 9.605032 


C.L. 

cot. 

cosec. 


6B«45ir 

9.ee4«8 


XSr '«^r'ii^--"ur.: 9.624494 
Pol. diM. 69 17 ccecant 

-1', .-l ,■ V 

Atca 46 37 .™ 
El.p.«im» 8' 60- = 530- log., . 
Dai5:irM-. dec. 20- 5" =; 1 205": log. 


a4.]iiio 

O.02903(r 
9.860202 

IB 



Eq.B*All8 — IS'.gg ... . U13685 

Omibtractive, because the poUx distance is decreasing and is to be 
applied to HOOH. 

£x. 2 — On the afternoon of the 17tb of September, 1810, altitudes 
ofthe sun were observed ut Marseillea, in latitude 43" 17'5fffJN^ 
and equal altitudes were taken on the forenoon of the 18tfa, after an 
interval of 21" 50"', the sun's declination for the 17th at midmght 
being 2" 14' 23" N., and daily variation of declination — 23'.,|^ 
= ^^ 1394" ; required the equation of equal altitudes ? 

Am — Equation ofequal altitudes — 136".70. i jf 

Or aubtractive, for the polar distance is increasing, and is to be ap. 
pMed to midnight. 

Ect. 3.— At Florence, in latitude 43° 46' 40" N., on the 8tfe of 
April; lfl09, equal altitudes ofthe planet Mars were taken at an in* 
terval of 8" 20°' when his declination was 5° 9' 40" 8., decreasing at 
thb*M*<>f 6'38" daily; required the correction for tbe planet's ^ii- 
pc^iiff'pasflage? 

^ni.— Equation of equal altitudes— 5M96. ' ' '"■ 

Or subb-active, because the polar distance is decraasmg, and i»lo he 

■ trunoit - ^'5 



sp{>Iled to the superior transit. 



, Bif the San- — The sun is in general the most convenient 'objettiftfr 
determining the error of a chronometer >>y equal altitade^t «id ihte 
ferehoon and afternoon of the same civil day are often p»i«n»d. 
Jhougb the evening and succeeding morning may sometimea be nSo 
nloycd with advantage. 'I'T 

In the morning when the sun is more than two hotirs distant finifa 
die meridian, in mean latitudes, let a set of observations be taken 
^i^the corresponding tiiqes by & chmnotneter. In the (ifteMfliAi 

— ^ — ■ -_ : -L^e 



i.,^>^< 



**'?'^ fciii Si'iitieein the cii^paisilon, Ii ihttat ilu^iMancc of tht o^eaifaoi ttfi 



ifdMU 



SPHERICMEi ilKHMWlMETBY. Mi 

<Mtarrt4mltBtiBti whmftklB.m^tMlMi «t Ae jmna. aUiliiAW) WVitfl 
hw— afcf lama dofm- oprio«t» HA cy net qK w d ipy dtitade> :>> :r 

rfow half the sum of any two times, answering to the same iillL* 
tad*, irilllft^eaippraxiiittte time of mini. .F&4 tke-paeani cf fU 
tlie thueaafr.noan in this manner from each corresponding pair of 
observation'; to whidi the eouation of equal altitudes being appliedf 
the result will be the time or appar^it noon, or the instant that the 
Sim -a oentve ifl en the meridian oy the chronometer. The diSSurenoe 
between this and noon is the error of the dircsuwnntyy whi^fi wi}) 
hp^^l^,ag slow according as the time of noon thereby is greater or 
Ijf^likui twelve hours. i • '\'i 

Ex. i:— On the 29th of January^ 1826, in latitude 6?^ 9" N., the 
fiiUoTifin^ equal altitudes of the sun wt^e observed ; jrequired the etj 
ror of the chronometer ? 

* . 1 1 ■ 

Altitudes. Times A.M. Times P. M. 

.. ' a^- i' 2P 36- 8- • 2^ 66- ^a- '* 

:, ;5;fi 36 8 . 64-4a, i 

'* 8-K 37 9 ' 68 48^'? 

38 9 62 41 

39 10 51 4d .> 









•7*'. -■-■■!■ 



1 


35 


44 


SI 

a 


37 
53 


8^ 
41.6 


6 


16 


33ie 



16 «..,.p 

9, 68,^4if0b 
SI 37 8A 



Xft^pi^time 6 16 32 « Sum 24 80 .69.^.4 



t 9> '^ 



H. E. T. 2 38 16.4 Half 12 Ifi 25il 

San^s dddinatiDn at noon, on merid. Greenwich 17^ W \6t* 8.- 

Daily variation or decrease of polar distance — « 16 16 N. 

BStude . 67° 9' cot 9.810026 C. L. 6.364617 

^. fi, t. 2^ 38* cos. 9.887406 cot OX«389fli \ 

Arch 26 29 tan. 9.697431 coseci 0360796- 

Pol. dist 107 59 cosecant . aW176S- • 



ibc2. ' ' 81 30 sine 9.9M209 

El. time 4'* 16 .5 = 316".5 log. 2.600874 

Daily var. dec 16' 1 5'' = 975'' log. 2.989006 

mj^Jt^-"^:^. 1.305471 

flilf swn oi^jffipKoxilviate time of noon 12^ 16- 25'.2 

JEEfas^tim or ^ual altitudes , . ~ . ^^ 

Time of apparent noon by chronometer • 12 15 '5.0 

dOflUi^m^luaeJWith <^ntracy sign -^; 13 27-9 

ftm» of mean noon by chronometer^ . 12 1 37 -1 

Hence the chronometer was 15" 5' fast for apparient noon, and T" 

ST**! ^Mt for. mean time. ^. _ ., 

Ex. 2. — On the 24th of July, at Pendennis castle near Falmbiith, 

te latlttkle.60P 8' 48" N., Dr Tiarksi, with a sextant of tax inches 

i^^Mr'by Mr'QViAightoii^ and an artificiftL^heMOii, (v^ether^ltb a 

■. ''-V ~<t\ "i.v. .v 



IM T3 

dtooDonoeter by Morice, found the.douUe altdtude of the mm'S'Mi 

■^i. , ,. ^.^^. ,»,c^i -'"■™'°'13'A.M.,and4"25''5'.3P.Mlj 



per limb t 
required the t 
■ftme after no. 



ig" 47' 20', at 8 
ae of apparent n 



;i by tile chronometer ? 



I 



r approximate nc 
)T elapsed time 

Hslf elapsed time 

The declination of tbe s 
Dwly variation 12' 39" 
ititiide 60° 9' 



*1 S 



■ aras- 


13 i) 






'•24 54 


18. 3 


12 27 


9.15 


7 65 


5^,30 



3 S7'S6115 



I, at noon S4th, is 19° 58' nearly. 
, or increasing the polar distance. 



cot. 9.92J,503 
COB. 9.71 



5.364517 
9-770148 






&4;"^q:aib..f9'.844 
"to afiproxiniate nooii 



9.864716 
2.677607 
3.S80242 

0.993168 
IS? 27" S-.IS 
+ 9.844 



aiy'ya*:-fl6c l^ 39" = 759" log. 

log. 

Add tlie equation of eqii^ altituaes 

Apparent noon . . . 13 27 18.994 

£jr. 3.— On the 24th of July, 1822, at 3" o" 38- .7 P. M., and 25th 
July, at 9" 49" 59*.7 A. M. at the same place, tlie double altitude of 
the sun's upper limb was 93° 40' ; required the apparent time of 
midnight by the chronometer? 
Tune after noon, July 24th 



24 

^^H Sum 

^^^H Half sum, or approximate midnight 
^^^H ^apaed time 



3" 5™3ff.7 
21 49 59.7 



Half elapsed time . 9 22 10 ^ 

Declination at midnight 19° 52' N., daily variation 12' 39" S. 
_ Or increasing the polar distance, and the equation is thereforaii^»> 
tive for midnight. 



1 



H. »:t. c « asr» cot; 9«d74(i« iw. • 

Arcl. w 32 47 tan. 9.806809 oof^ • .M,,,^^ vOdSlMM 
PoLdisi 7P 8 cosecant ;i.. O.Q9i3iM3 




Arc3. 37 21 sine . . . ft789961 

felap.time 1^44" =1124- log. . a060766 

Diufyvana. dec. 12^39''= 769" log, •Mfx..=c. ".. WHOSMBi 



Mu«i. iiU.r-^M log. . .nn n^ c Mttm 

VVohi approximate midnight « 19^87*40**SP 

t|i0equation of equal altitudes » liirr.i-q^.SIBi^M 




Apparent midnight ' y ' '18 9^ 9(^ 

^ Proccfjedm^ in this'inianiito till a considerable number of ol^ 
tioAs flBPe made^ the error of <i dbocp^oiflibter max be determiilM. 
gteat aeeuracy. If this ctironometei^' is comparM with any ^ 
number of them, all their errors SBfl rates maj be fpund as luui, 
ddh^fyDr TiaAs. ' ' *^ '-' ^^ -^ , ^ 

■ - iTie same thing may be done by thestars, though rather leu dcih- 
veniently. v ,,; 

'-^ni^'fellowing method of comparikljg a ditonotneter ^^^0^9 
tiiiMB %jr Dr Tiarks, communicated by Captain .Sasil J^i^^ ^^ 
wiH' be ibund very useful. ' * 

^TTheilifference of a chronometer from the m^n ;|we ft,.*.l^lyl> 
ti^iig'luioiNrQ at three different instants, to find that .di|[^^Eenpe ^ 
anj^-rateniiiediate instant with a proper regard to the chang^'clf rite 
#hicli may have taken place betweoi tfie flrsi and secohoT ^nd b£ 
tween the secopd and third times. 
" liC^ the diflerence at the " *^ '' 

■:... •■^.>*!fsi;«-J.*4o ■■ :- '■' '■■■ ■''" 

So that b is the difference between the first and neeotfd ttatei til 
l^l^mmoipetery and c the difference between the seconded diiid 
ati^ ^ the same chronometer, the state of a chronometer, (namely^ 
A^/ difference from the mean time of a given place), at die moment 



1.rl 






_-■ «««--^-. .^*.vT^r-ar- r-i 



iKi^ilijpWriwilianiiibotii-iiM-jiottftiWr '^ ' '' 



• •- ^"» - « 



EXAKPLB. 



-t^^.^^^?*^?? 9^* chronometer frfpq)^ O^e BJ^ftP tjpif,!^.Sg^^ 
pliui^ utrkl^^own on the following days \. .,-. v ■■. \t :o iw;. 



Auirust 9'.5243 
I_81 -4J-46 



- it-i^j :||)THG1>UCT10N. 

Differences. Days. 

21.8903 Difference between 1st and 2d=21.8{KB 



4.6104 



1 and 3 ='. 



= 21.8903 
= 2ff.4007 




j'+i"= 48.2910 
It is now required to find the state of the chronometer for August 
■"""'■ '"" """7. Deducting August 9*.5&45 from 
iterval / = 7''-9394. 
(' =21.8903 



J7th, at 11" 7° 44- = 17''.4637. 
AuKUBtl7'-4637we have the ' 
■■ ■ ■■' =48.2910 



('+/" = 



i = 7.9394 log. 0,899788 (" = 20.4007 
t'+t'--^ =40.3516 log. 1.605861 i'xCMog. 



fxf or Aenoimai 



um. log, 2.505649 
itor, log. 2-761868 



K 



-L_>^(C+ ("—(}, log. ».74378i;orfaclorof 6. 
log. 0.899788 i" =26.4007 



7.9394, 
(' ='21.8903 



= 13 9509 log. 1.144602 i"— i'= 4.5104 




log. 2.044390 f (("— n denom. log. 2.0751 
log. 2.075831 



9.968559, or factor o 



August 9tb, 12^ 35™ chronometer slow SI" 57'.35 

Slst, 9 57 ... 64 10.33 

Sept. 4th, 22 12 . . . 54 39.16 

What is the difference, August 17th, U" 8". 
b = 132*.98 log. 2.123782 c = 28'.83 

factor b . log. 9.743781 factor c 



which ii negal 
Diff. 
132*.98 b 



:1 



log. 1. 459846 
log. 9.908559 



(./)6=+73'.72 log. 1.867563 
(/)c=— 26.83 



cm- := + 46.90 to be applied to t 

the time a. 

August 9tb, 12^ 35" chronometer alow for M. T. 

correction 

Chronometer Blow for mean lime 
On AttgoK I7tt, at 11" 7" 44'.* 



(/)c=— 26«2]og. I.ffi8«)ti 
error of the chronometer Mi 




spifERttiAti 'titm,mtK-n.Y. 



1U 






fi»^|QHDy(A 



UfdiMi Fnoii™ 



1'' ;o.'«41667 10' 
3 |0.0tia3"" " 

3 lo. 125000130 

4 0.16666740 
o a208a33fiO 

6 0.25OOOW i 

7 0. 291667 2 

8 0. 333333! 3 

9 0. 375000 4 

10 0.416667 5 

11 0.468333 6 

12 0.500000 7 

13 '0.541667 8 

14 0.583333 9 

15 0.6250001-^ 

16 0.666367 lo- 
ir 0. 70833320 

18 0.75000030 

19 0.791667^40 
30 0.83333350 
21 0.875000 1 
32 0.916667 2 
23 0.958333 3 
34 1.000060 4 



For 12'' 

double that 

for 24^ 




ErplanattM. , .^ . .\ 

■TWHcI. contuns the decimBl fraction of a day of 34^. It is use- 
t^tat Rndioy what 'part aC at dav any Dumber a£ hoots, ^nut^, j 
and seconds are, ana conReqnently may be conveniently employMl ■, 
in many calcalations where duly differences are necessarily involved,' 
■iiHh tti tbo daily* nU of a cIdck, tiie chaaee of whicN !> an« ^r«^ 
nomber of hours, &c may be thereby readily obtained. It is slai^vGry ,' > 
uMtiirfa tile preceding irn«Aoil of ooinpanng cfarano0iettts, «ndet)iaT j\ 
p^Hiofefes. ' 

^Ewjle II. serves the same purpose when an hour is taken for 
ui^ K^ is useful in several astronamiaLcipantHUcc' - "ii:tca'-:inii.'i 

Table III. is supplementary to the gdi*rit Hhbln; Vi Ma^^ttPi^ 

to nnniMrt dim- jntii degrmn if \^kh ):him ff or 90°. B u t as 6* answers 



il4 ■■"'■iMfHBiAbritiiV.'J^inc ^ 

by the proportional parts at the bottom to every single aetfonJ. 
Whence it la only necessuvy lo convert the decimal part of the Ihne 
into degrees by this table to tomplete the whole. 

III. BY OCCDLTATIONS AND ECLIPSKB. 

The luoon in her periodical revolution frequently passes between 
tlie earth and a fixed star, of whicli she iniercepts the spectator's 
,.view, thu3, producing what, is, called an oeciiUalion. 
.,,' Since the instant o.f dJaai^earBnce and reappearance of the star 
■>. can he ascertained v^ithout the use of any instrument liable to error, 
... 'the longitude may be tleternjiiied more accurately by aii observation of 
this jihentnnenoii, tJian by alunar distance. An observer poHsesse<I ol' 
an ordinary telesctqje, a chronometer, and an instrument to deter- 
mine its error and r«te,* can readily make the observations; and the 
.,,iiecessiiry ,i;alculfltiona are far from difficult. Severid riUea have 
,,. ,beeD proposed fur this purpose independent of the method of deter- 
..i~(nining tnB parallaxes by tfie nonagesimal, and compar^jtively much 
.,, more .simple. Of these, Dr Inman's of Portsmouth, wJiiuli we shall in 
...ithe mean time adopt with some alterations, appears to us tlie most 
-i. iC«i»eiueofc 

. I . , At the JHfitant of tlie disappearance or reappearauce of the star, 
I, the apparent right ascension and declination of the point of the 
,1 moon's limb in contact with the star is the same as the right ascen- 
.,i ,eion and declination of the sUu*, which can be obtained wllb great 
I, ..facility and accuracy from table;. The apparent right asceiuiun and 
.,^ A^qlination of this point being corrected for parallax, its true right 
„ lapcensioii and dechnatjou will be determined. Now since tbc 'iqi»- 
-..{tanue of this point from the moon's centre, which is equal to, her 
Il..9eaudianieter, and the declination of the centre for the estimated 
./.. tftn^ at Qr^enwich, may be found by the Nautical Almanac, tbe ^e 
right ascension of the moon's centre ia easily computed. ,Sho>Uld 
.fitthere be aft uncertainty in the estimated Greenwich lime amounting 
u)ito aboutone minute, tlie operation must be repeated, till the fsti- 
/ tqated and comjiuted Greenwich lime be very nearly the same. 

Ru/f. 
] By applying the estimated longitude in time to the observert ap- 
parent time, tile reduced Greenwich time to the nearest minute'will 
jr, be obtained. 

To this time take from the Nautical Almanac the sun's R. A:,' the 
moon's H. A. and their declinations corrected tor second differences, 
,_ together with the variation of declination for Itf, for the purpode i^ 
H repeating the operation when supposed necessary; and the mooD's 
, aemidiameter, and the horizontal parallax corrected for the sphere^ 
.|dfll Ugure of the earth. ■ ' 

j^' . Take also the moon's R. A. for 3" after the first estiraated'^time 
' 'corrected as formerly. , j..ji^ 

_ Find from the Nautical Almanac, or from otlier tables, the Wpa- 
rent R.A. and D. of the observed fixed star; and reduce the given 
latitude for tlic spheroidal figure of the earth. ' " 



• I<tb<t«bHrrMlMiawaDwdaiRM«. ari«ll(HrMniij]HMl>a)n*dn£« iht.«»t^ of Uir 

mmometur beiwren ihf diispprnmnA' nnJ impjifiitBrra t>f thv W Mid (he run ill* ilu 
fp. ns In luiian. 



SPHEBIGAJL XRI0OVOMET&Y. 115 

To thi^appa^pent time add the sun's R. A., and fiom the suip, in^ 
cxeasod if necessarv by 24^ subtcact thestar's R. A. ; the remu&dfr, 
iiPless than 12^, n^ill.be the hour an^le; if greater than 12^, its com- 
plement to 24^ will be the hour angle. 

Now write down the proportional logarithm of tlie reduced hori- 
zontal parallax under the numbers (1), (2), and (3). Under (1) and 
(2) put the secant of the reduced latitude ; under ^3) the cosecant 
of the same ; under (1) the cosecant of the hour angle {a), and take 
the sum of these. 

Below the sum of the three logarithms under (1) put the constant 
logariihm 1.17609^ and the costne of the star's decunation; at the 
same time under (2) put the cosecant, and under (3) the secant of 
the same; the sum ot these three logarithms under (1) will be the 
proportional logarithm of arc Jirst, or the parallax in R. A. in time, 
nearly ; one half of which (b) is to be subtracted from the hour an- 
gle (a), giving (fl-^6), the corrected hour angle. 

Unaer (2) put the secant of the hour-angle thus corrected. The 
sum of the logarithms under (3) will be the proporttonal logariihm 
of \hejtrsl part of the parallax in declination, ana that under (2) the 
second. The first part must be applied with such a sign as to dimi- 
nish the star's distance from the elevated pole : the second must be 
applied with the same sign as the first, if the hour-angle and polar 
mstanc^ are the one greater, and the other less than 90*^ or 6^ ; other- 
wise ^th a contrary sign. The result will the true declination of 
the observed point of the moon's limb. Take the difference between 
this true decimation of the observed point and the declination of the 
moon's centre, found from the Nautical Almanac> under which put 
the moon's horizontal semidiameter properly corrected; and ti^e 
the sum and difference. Add together the proportional logarithm 
of this sum and difference, and take half the sum^ to which add the 
costne of the mean of the two declinations just found, the sum will 
be the proportional logarithm of the moon's semidiaitaeter in R. A. 
nearly. 

Under (4^ put the constant logarithm 1.17609, the first sum Un- 
der (1), and tne cosine of the declination of the observed point, the 
sum wUl be the proportional logarithm of the exact parallax of R: A. 
in time. This being added to the star's R. A. when west of the meri- 
dian, but subtracted if east, will give the true R. A. of the point ob- 
served. To the true R. A. thus obtained add the moon's semidia- 
meter in R. A., or subtract it therefrom, according as the reappear- 
ance or disappearance of the star has been observed, and the result 
will be the true R. A. of the moon's centre deduced from observa- 
tion. 

If this differs considerably from theR. A. taken from the Nautical 
Almanac, alter the moon's declination by as many seconds as will 
make a corresponding variation in the first R. A. such as the Nauti- 
. cal Almanac would ffive for the same alteration in declination. Re- 
peat the operation till this is the case, and the last R. A. will be that 
required. 

tinder this put the moon's (1) R. A. taken from the Nautical Al- 
manac for the Greenwich time, and then the moon's R. A. tliree 
hours after, or the (2) R. A. Take the difference between the first 
and second^ and the difiierence between the second and third. Then 
from the proportional logarithm of the first difference subtract that 
' of Ae second^ t)ie remainder will be the proportional logarithm of a 



1%^ -nn-AtwanBomcaomAU't^ 

portion of time , which must be added to the Greenwich time when 
the first K. A.' is greater than the second ; otherwise aubtracted ; and 
the result wiE be the Greenwich apparent time. The difference be- 
tween this bimI the apparent time of the observer will be the longi- 
tude in time. 

fia. l._On the 3<l of March 1823, at Bahia, in latitude 12° 5?' 
17" S.J and longitude by estimation 38° SC W., the reappearance of 
Antares from the dark limb of the moon was observed at 15'' 3ff° 

; Required the true longitude ? 
Bahia, March 3d, 15" 30" C.S Moon's 1st R.A. 244° 27' 29".75 



I JLou. in time 



2 34 



2d R.A. 246 6 16 .82 

Dec 25 55 16 .6 S. 

var.forIO'+ 0.63S. 

hor. S.D. 14 50 

eq. par. 54 26 .5 

red. to 13° — .5 

red. par. 54 26 .& 

titude is-sri^'s. 

d. to 13° 8. — 5 
id.lat. 12 52.17 



fiw angle 1 51 36.88 

o^ As it is convenient that the work should foQow from beginning 
l9 ^n4 in regular order, that of the foregoing example has been 
fiansferred to the two following pages, and to avoid unnecessary 
waste of roan), the remainder of this has been filled with the follow- 
ing ejtample for exercise ; — 

£:&a^On the 26th of May, 1822, at San Bias in latitude 21 "32' 26" 
%., aadlongitude by estimation 105^" W., at 9" 22'° 41'3, A.T. the hn- 
■■iersioD.of«Leoni 9 was observed by Lieutenant H- Foster, then Mas- 
ter'B Mate of his Majesty's ship Conway ; what was the true longi- 
tude? 

,^„,._105°]a'27"W, 




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£4r. il--On Uig 9ptli of July JKSa at RillJf«i0irPiJnhtM&9>' 
W W 8., and Ma9i4^^ta^ 1^^^ ^* ^ Sbi^SSmkii 
X Sagittarii bch^di^tUe mi^'a dark ^h-^fr^m^hmfffAMiOi'^ 
9«.2; What was die true loi|#tiwik?; ' ^* * J "^ 

Rio Janeiro, July 2b, , & 49^ l^'a lib kjk. ^ fteSt' 

Lon.intime . S fi3 ^ ' 2d lUL ^74 Si 48 

— — - ^ var-inia* 4- 6.44 

Est Greenwich time 9 42 - dm. 8i5 80 21^ : S. 

TothUiime. ¥tr. itt-lC -Xi- , 0^6. 

Sun's RA. T 57"a« 7 >.; - Itor. W). 14' 6*' 

Stir's RA. 18 17^r^ , ;, tt|.par. 53 68 

Dec. 26° 30 31''.0 ' ^ to 23° — 2 

App. time ff*49» 9' .2 ? c 

Sun's ItA. 7 6723.7 ; *d. par. 6366 

latitude 22<'64'10'' S. 

Sum 14 46 32 .9 reduc. — 8 12 

Star's R.A. . . 

red.kt S^ 1|$ 6i S. 
Diff. 



14 46 
18 17 


32 J9 
7 3 


aO 39 
24 


26 .6 



Hour angle 3 30 34 .4 - ^^ 

It is hardly necessary to give the variation of the sun*8 R. A. and 
D. in 10*, as it is very smalH and as the true time must differ but a 
few seconds from the estimated, on repetition the longitude cannot 
vary much on this account ! 

Exs ^ — On the 3d of January, 1826^ at Port Bowen, 4& lil^tbde 
73<^ Id' 40"' N., and longitude by estimation 6^ 66" W^ the iiwner* 
sion of z G^eminorum or the 4th magnitude was observed at 6^ 14" 
23'.26 M. T., and the emersion at T H" 12'.17 M. T., by Lieutenant 
Henry Foster, R. N. ; what was the true longitude ? 

Ans. — By immersion the longitude is 6^ 5&^ 48*, and by emer-^ 
sion it is 6»» 65" 35* W. ' 3 

It was intended, if room would have permitted, to pidf the whole 
of the calculation on one paffe^ and^ tfiqei^'sot doneJbere^^iajareadi-i 
ly enough be so placed by me calculalpir* This little attention .ought 
not to be slightea, as a neat form, like a convenient formiHa, wilfbe 
found of some service in acoorate computations* 



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All eclipse of the jjuii depends upon the sanie cause as an occid^ 
tstion, his light being intercepted by the body of the moon passing 
between him and the spectator. The beginning and end of a. solar 
eclipse is easily observed by a telescope of' moderate power properly 
prepared, when the point of contact of the limba being nearly 
known, and the rule for computing the longitude is similar to that 
now given for an occultation. If the semidiameter of , the ni'o<ui 
passing through that poin^ of the sun and moon, apparently m, con- 
tact, be supposed to be produced to the centra of the spn, as seen 
from the observer, and conceiving this centre to be at tlie,di8^fti|Ce 
of the fixed stars, so as to have uu sensible parallax, then U )s mani- 
fest, that the rule for an occultation must apply by substituting ter 
the moon's Bemidiameter the sum of the sun ana moon's semidiam^- 
tergj. considering the sun to be at the same distance as ^he motni 
when seen from the earth's centre, — that is, subtracting the aug- 
mentatioti for the sun's semidiameter as if it were the qioon's htm 
it, as found in the Nautical Almanac In the supposition jfipt 
made, the sun's centre was supposed to liave no parallax : but, as it 
has a hwizontal parallax of about 8".7> in fiiiding the apparent plape 
we ceimot proceed exactly as for a fixed star. The sun's rig^f ^- 
cension and declination, as seen from the centre, must be ta^n fmn 
the Nautical Almanac, which, corrected for parallax, fi'il).,give't|ie 
apparent right ascension and declination, thus reducing the cAse of 
a, solar eclipse to a similarity with that of an occultation. The ap- 
parent right ascension and declination of the sun's centre must now 
be corrected, using the horizontal parallax of the moon in the com- 
putation. This would evidently give the same true place as if, talc- 
ing the right ascension and declination of the sun's centre from the 
Nautical Almanac, we considered these elements as apparent, and 
corrected them for parallax, instead of the moon's horizontal paral- 
lax employing the difference between the horizontal parallaxes of 
the sun and moon. 

WheHce the true right ascension of the point answering to the 
sun's centre is obtained, and consequently, as formerly, the true 
right ascension of the moon's centre, from which the Greenwich ap- 
parent time is determined. The apparent time of the observer is 
found by means of a ciironometer, whose error and rate have been 
determined by double altitudes if possible, if not, by altitudes both 
to the east and west of the meridian. 

Rulf. 

By applying the estimated longitude in time to the obser- 
ver's apparent time expressed astronomically, the Greenwich time 
will be obtained to the nearest minute. For this time take from 
the Nautical Almanac the sun's right ascension and declination, the 
sun's semidiameter thminished by the augmentation, the moon's 
right ascension and declination, aemidiameter and horizontal paral- 
lax corrected for the spheroidal figure of the earth, and diminished 
by the sun's horizontal parallax. Take also the moon's R. A. for 3 
hours after the first R. A,, or estimated Greenwich time. 

Find the hour angle, which, in the afternoon, is the observ er^a 
apparent time, and in the morning its complement to 24 houra^ ^^H 

Employing the moon's diminished horizontal paralUx, corrett^^| 



■un's right sscension and declJn&tion, as if for some point on the 
moon extended, proceeding' as (brmerly, only putting the sum of the 
sun's semi dill meter, diniliiUliMl by aucnientation and the moon's 
semidiaineter, instead of the maon's seinMiamet«r alone;'; If the re. 
suiting Greenwich time differ from 'the estimated, th«' nin's R.A. 
and declination munt be corrected for the difference, repeating the 
operation as often as necessary, till the GrMnwkh time by o 
tation and e ' 



Sir.— On the 7tli of September, 1880, at the Royal Nafil CoUeg«, 
Porismouth, in Intitude SO" 48* 3" N., and longitude by eatima^i 
i* W., the end of a solar eclipse wM obseiVed at ff" 1^ 66* ; reqniiu 
ed the true longitude > 

Ports. Sept. 7- 3* 13^ Moon's fl) B. A. IW'64'^'' ■ 

Lon. intfme 4 rar. in iff 4*. 4 '■■ 



^) R. A. I6fr 19 96 



jBst. O. T. ' 3 7 dec. 9 91 4 H. 

to tb» tiifie. ' var. in lO* 1 iA4 8l 

iBuns R. A. 11* 4- iS'.eO hor. 8. D. 14 49 .T ■■ 

■'Var. in JO* ' 0.026 equa. par. ftS 06 A . 

Dec. S'Sffay-N. red. tolat. _ -e .3 ■ 

"Var. in KV . ■ .16 ■ 

"'Sertudia. 15 54 . 8 red. p». 

"Hor. par. 8.6 tnn'a hor. par. 

'"Alt.'^ nearly 

.^'Apfa'. tinle3"l2"65' = H.A. dlflference 

''■ r' latitude 

"'' Reduction 

■ ' Red.Iat. 

. '■ Sun's 8. D. 

f ' ' Aug. to 30° 





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On the 7th of July^ 1823, at Dunglass House, the seat of Sir James 



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,. Ji^7tli,; .:ir*65~34' July 7th, Mean Time, 17" 56" 84' 1 

§ E#loii;iikT. ,^+ 9 30 equ. of time to Iff* 1" — 4 28.7^ 
" Bq. of T. at noon, — 4 >51 

• — 5*— apparent time at D. 

>w, A]^rox. G. T. - ]8t ^43 
\^Ot. iS^lDjiarly. 

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Var. in IC, 
Sun's dec. 
Var. in W, 
Sun's §. D. 

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8 .54 
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SPHERIOAL TKIOiWDliETRy. 19B 

IV. BY THE moon's TEANSIT. 

The method of finding the longitude bj the calmination of the 
moon aiid stars^ is now considered very convenient and accurate. 

Sipee the observations require a transit faiatmoient, and the dock 
used with it generally shows siderial time ; the difference of the 
timea it supposed to be siderial time. If it is not, it must be redueed 
to siderial time by Table XXXI. If the moon had no motion, the 
difference of times between her transit and that of a fixed star would 
be the same at both places. 

The difference of the differences arises firom^ and is eaual to the 
increase (I) of the moon's right ascension in time, in toe bitervd 
, between the passages over the meridian at eadi place. 

Hence, if the increase (N) of the moon's R. A. in one hoar of 
siderial time be known N : I : : P : X, the angle described by the 
western meridian in the interval of the passages of the moon. 

This is equal to the difference of longitude + 1* 

Hence, the difference of longitude is equal to X—- 1 = jj~^' ^7 

the Nautical Almanac the moon's right ascension is given at every 
noon and midnight ; whence its increase in an hour of siderial time 
may be found nearly in the middle of the interval including the ob- 
sertations. 

Assume the difference of longitude = L' as nearly as can t>e esti« 
mated, and compute the increase (£) of the moon's R. A. in the si- 
derial time Jy, tnen 

IT / 

As £ : I: : L': X s: -p-, (1) and the exact (Hfference of longitude 

^t * |y ■ I (2). But this exactness is only necessary when the places 

differ considerably in longitude. 

The moon's limb is observed by a transit instrument, and not the 
cantre, which makes some little difference when the difference of 
longitude is considerable. 

When great accuracy is required, it would then be necessary to 
make an idlowance for the moon's alteration of distance, that chan- 
ges her Mparent diameter, and also for change of declination, which 
changes her semidiameter in R. A.* 

Exj^-^vme 13th, 1791, the following observations of the passage 
at the moon and « Serpentis were made at the observatories of Green- 
wich and Dublin ; required their difference of Longitude ? 

At Greenwich, R. A. )'8 1st limb 16»» 5" S-SS at fi»» 36" App. T. 

R. A. • serpentis 15 33 34.70 

1st Difference 28 31.18 

At Dublin R. A. )'8 1st limb 15"* €r 12-.49 

• serpentis 15 33 36.91 

2d Difference- 27 24.42 



* For a mare comdktftseliitioD of thituMtliod, see ArBdpUef*! Artide in Um 
■t nuinber of the l>at!lfai Phflosophica) Jomnal, and Mr^aOy^s Meraob in flit 
Transactions of the Astrononiical Society. 



m TirrawmtoouarioN. \ud hI'' 



Difference of lat and 2d differences 



1 



i.44 = 16'3ft".8. 



W 



Aa the places do not differ much in longitude, it is unnecessary ta 
reduce apparent to mean time- 

This difference It)' 36" .6 is the increase of the moon's R. A., in 
the interval of its pasaages of the meridians of the observations of 
Greenwich and Dublin. 

By the Nautical Almanac, we find the following differencea of the 
right ascensions of the same limb of tlie moOn, and the star at about 
the same time. 



■ Jjm» 12, midnigiit 213° l.V 
',,,'13, noon 220 3B 

13, midnight 228 11 



J^if- 



H, 



7°! 
7 42 ( 



7 33\ 
if 



r 37-. 



U; midnight 243 43 ■ - ■ 7 4() 

If the places differ much in longitude, tlie motion in B, A, sbjoHJid 
ube calculated to seeonjs, though, m the present case as tlje aectfmi 
differences are sufiicieiitly uniform, the mean first difference con- 
taining the interval will be sufficiently accurate for the rate of in- 
crease in 12 hours at the middle time. 

Hence, by formula (1) 7° 37'.5:16' 



and when the difference of longitude is n 
1568'.42 + - 



i:;12'':x = 1568'.^, 
iderable^-H-.,-^^ = 



' lf> 
tween the 



\33 = 6" 16' 3o " W. 
be the increase oi' the n 



isits, then i + 



ion's E. A, during the interval be- 
— } r must be used when the dif- 



ference of longitude is considerable. 

It would extend this article toomuch to give Baily's or Brinkley's 
methods, which ai'e more accurate and complete, and can only Be 
fully treated in a work on astronomy- ' 

In the foregoing example the (lifterence of R. A. between- the 
moon and star was determined at both places by observatifln; Wt 
for ordinary purposes that at Greenwich may lie found by the HJi 
ticai Almanac. '*■ 

OF THK TRANSIT INSTRUMENT. 

A transit instrument is a telescope properly place<i in the meri- 
dian for the purpose of observing the times at which the ce^etial 
bodies pass this circle. If the clock or ctironometer by which the 
time is marked be adjusted to show uderial liine, tlien their ifight 
ascensions will be found. This is perhaps the best method of de- 
termining the rates of chronometers. 

The telescope is fitted to an asis, of which the ends tapered into 
uoints turn in notches, from their shape called Vs or Ys. This^sis 
, lUiQiuIehoUoWjiOpposite one ofthe eixle of which t« placed ai«pp 
for illuminating trie wire* in night observations. 



SPHEBICAiL TBIGOKOMETKV. ISl 

These vnren, genenliy fiire in number, aire placed in the telescope 
equidistant firom esdb cmer, and pefpendteskr to the h orizon, hav- 
ing also a horizontal wire biaecdng them, near or upon which the 
transits are observed. 

When properly adjusted^ the middle vertical wire coincides with 
the meHdian, and the instant that the centre of my heevcfil j body 
passes this wire, is called its transit. The other parallel wires are 
intended to correct or verify the observation by taking a mean be- 
tween the transits over the first and iasl, the second and focbtb, 
and comparing it with the third or meridian wire ; or, what is more 
correct, a mean of the whole called the reduction of the wires. 

There are five principal adjustments necessary in placing a tran- 
sit instrument, tmree relative to the telescope and two to the axis. 

1. The wires should be set perfectly vertical — ^This is verified by 
observing that any distant object cut oy a wir& does not change its 

Sosition relative to that wire, on moving the instrument up and 
own. If it does, the wires' must be all turned till the object is 
kept upon them, when moved through their whole extent, and the 
a4]usitment is then c(Hnplete. 

2. The telescope shomd have no parall^ix, — ^When any distant ob- 
ject is bisected oy the horizontal wire, if, on moving the eye up and 
dbWn aHttle, the object should appear to separate fVora the wire, 
€tie instrument is said to have a parallax. This must be corrected 
by pliEicing the object and eye glasses at sudh a distance from each 
/miir, that their foci may meet in the point of intersection of the 
wires. When the object-glass has been properly fixed by tlie in- 
Ittrdment-maker, the observer has only to adjust the eye-glass. 

3» The line of coUimation slumld be correct,* — ^Thia ia known by 
bisecting any object by the meridian wire, and if, on reversing the 
^us, the object still remains bisected as before, the line of collima- 
uoii is correct. If not, it must be adjusted by means of the small 
screws in the sides of the telescope. This is effected by easing the 
oae screw and tightening the other till the error appears one half 
diminished, when the axis is again reversed, and the operation is re- 
' pleated till the adjustment is properly effected. 

4. To level the axis, — This is performed by means of a screw 
placed under one of the Y^ or notches, which raises or depresses 
that end of the axis at pleasure, while the true horizontal position is 
ascertained by a spirit-level. 

5. To bring the telescope to the meridian, — This is accomplished by 
means of a horizontal, screw acting on one end of the axis, by which 
it is moved back ward or forward till its proper position is obtained. 

As tlie problem of bringing a transit instrument into the meridian 
is one of considerable difficulty, it is proposed to treat it at some 
length. 

I : ^... i . : . To take a Transit. 

'^^^'Xfiih the latitude of the place and the declination of the object 

^"cljiihibute its meridian altitude. 

'**.' Wheh it is known to approach the meridian, elevate the telescope 



The line of coUimatlon is an imaginary straight line supposed to join the centn; of 
ketioni df Ae «4)}ei^^Iltss» atfd the intersection of the meridian and horiMntal wire 



v-ttimetioni 
in the centre of the teleMopo»" . '-.■ 



E 



132 ■ INTRODDCTION. '" 

to the given ftltititde by the circle attached to the end of the axis. 

NoWj because the telescope inverts objects, the object will Appear to 

Come into the field of view from the west and move towu^ila the 

east. 

■' Mark the time of transit 

save the eye when the sun ii 



FRUM TBS eaii£NWlCU OBasBVATlON^. 




■ 1-- 






18t(L 


Vnra. 


Iteduc. 


4' 


Kov 


I 


II 


111. 


IV. 


V. 


3d 


r.4 

22.6 


2(p.9 
55.2 


2]*55°'38'.5 
29 27.5 


5-.6 
0.0 


15'.2 
32.6 


27.56 


• Aquarii. 


4th 




18.4 


21 56 37.2 


55.7 


14.1 


37.16 


B AquariL 




6th 


51-29-.4 
53 45.0 


51-48'.6 
63 4.3 


14" 52'" 7V6J52" 26'.7l52" 
14 54 23.4154 42.5^6 


46'.0 
1.6 




Sun'ilL. 
Sub's 2 L. 



I By taking the n 

'That of the 3d will be 

4th 
' 8th, botli limbB 

By the Nautical Almanac the si 
't'ight ascension that day was 



. The error of the clock on the 8th i 
Suppoae the observation had bee 
"'iluddle one only, then 

To 
' Add Bemidiameter, Table XV. 



ai" 56'" 38M0 
29 37.16 
14 6» 14£0 



i 

slow, o 




14 S4 , 4.W 
— 49^ 


made 


.ith 


lie wire, » ihi 

14' 62- T-e 
+ 1 7J( 










14 53 16.8 
14 53 IS J 



. Difference only , — 0,2 

, The error of the clock may readily be determined from the Btars, 
; if one of those whose true placeearegiven in the Nautical Almanac is 
.; Qbserved. Otherwise the corrections miut be api^H^d from aj^ro- 

Sriate tables- 
bserved transit on 3d . " " 21'' 55" 3ff,30 

- Aquarii R. A. by tables 21 5tf 24.35 



l,',vEcTin^>of clock by the star slo- 

«, loss in 4.71 siderial days 
,,,0r the daily loss is 

■('■' 



, on the 3d 
On the 8th 



— 46.05 



3.15 
0.(f7 



SPHEBXCAL TRIGONOMETRY. 1^3 

' " ¥6 l^mtf A TllAMSrr ntSTBUltSMT ihto thb msbidiam. 

■ !.i ■ .. ■, r. ■ ;v. 

t 

' To peffotfmdm problem, the time should be accurately determined 
by an altitude near the prime vertical^ or still better by equal altitudes 
as aheady explained. Bring the telescope to any celestial object when 
nearly passing the meridian^ and^ by turning the horizontal screw> 
make the middle wire bisect the object at the instant of its transit, 
then is the instrument in the meridian. 

Should the object be the sun^ as it cannot be accurately bisected, 
either limb must be observed when on the meridian by allowing for 
the time his semidiameter takes to pass the lneridtan« This isfband 
niost accurately in the Nautical Almanac, or, if it is not at hand, 
from Table XV. 

Tojtnd ike TSme that any Star takes to past from one tvire to ano- 
ther in a Transit Instrument, that of the Equinoctial being known. 

Rale. — To the cosine of the star's declination add the proportional 
logarithm of ^e time at the equinoctial, the sum is the proportional 
logarithm of the time by the given star. 

Ex.~On the 10th of April, 1826, by a transit telasoope which 
gave 26'.4 for the passajee of a star on the equinoctial ' from wire to 
wire ; what would be the time by Antares, having 2CP 7B."d^ituU 
tion? 

Declination 26^ 2^ cosine 9.95364 

Time Q5rA P.L. 2.62867 



Raducedtime 28.26 P.L. . 2.58221 

Or this would be more readily performed by considering the se* 
conds minutes, and converting the decimals into thirds to be esti- 
mated seconds, thmi the answer will come out in minutes and seconds 
to be estimated seconds and thirds. 

Declination 26° 2' cosine 9.05354 

Time 25'. 4, or 25°» 24' P. L. t).85044 



28.27>or28 16 P.L. a80398 

Hence the star's expected time of approach to the other wires be- 
comes known after its contact with the first is observed. 

One of the most convenient methods of fixing the transit telescope 
in the meridian in mean northern latitudes is by means of Polaris. 

It is required to set a transit instrument by Polaris, on the Ist of 
tt^rdij iw6, at Edinburgh, in latitude 55*" bf 21" N. Bv a reference 
^9 Qie Nautica!! Almabac its altitude at its superior transit wiU be 57^ 
M, and at its uiferibr 84** 21' ; and its R. A. is 0^ 68" 12'.20. It must 
^ tb^rfore pass the meridian about 2^ 8", and 14^ 8°" at die altitudes 
/mied above, which serve as a guide to advertise the observer to be 
jMrepared. 

JNow let the clock be regulated to siderial time, and when it 
Isbpws 0^ 68" 12'.2 make the middle wire bisect Polaris, then will 
_jthe instrument be in the meridian. If, however, the time first as- 
sumed was not known with sufficient accuracy^ the error of the 
\ dock can now be found very nearly by the transit of the sun or a 
star. By repeatedly observing Polaris, and correcting in this man- 
ner, the instrument will at last be truly in the meridian. This may 
be verified in several ways. One of the most general methods is by 
observing that the semirevolutions of circumpolar stars are equal, sup« 



w^ 



INTUODUCTlp^^. 



posing that the rate of the clock is uniform. Should the obnetver 
not choose to trust to that, he may select two eircumpolar stai-s 
whose right ascensions differ nearly 12^ as it requires in this case 
only • few minutes perfect regularity in the clock, Take the differ- 
ence between the transits of eircumpolar stars by the dock, wTiich 
are nearly in the same azimuth, tlie one above the other below the 
pole; repeat the operation 12 hours after successively, when the 
Btars have reversed their positions, and if there be a variation iu 
their differences, it shows a deviation in the instrument, which may 
be corrected by substituting half the difference for the error, ana 
repeating the trial by approximation till the adjustment is complete. 

If some of tliose stars whose apparent places are given in the 
Nautical Almanac be selected, the operation will be comparatively 
easy. These in pairs are ; I, « CassiopeiBE and ) Ursse Majoris ; 2, 
Polaris and ^ Ursee Majoris ; 3, Polaris or m Arietis and s Dracoms ; 
4, Capella and « Herculis ; 5, ff Tauri and fi Draconis ; 6, fi Aurigte 
and y Draconis ; 7, Pollux and y Aquilse. No doubt some of these 
can only be so observed in very high northern latitudes ; and, there- 
fore, recourse must be had in some instances to other tables, such 
as those of Dr Pearson," 

It sometimes happens that an observer has not a command of 
the whole meridian, especially if he has not an observatory properly 
adapted to the purpose, yet may find it necessary to take transits for 
the regiilMtions of docks or chronometers. In this case recourse 
must be had to the sun, and to pairs of high and low stars having 
nearly the same right ascension. Having, by the sun and a goop 
fratCh or chronometer, placed the instrument nearly in the meridian, 
observe the transits of two stars having nearly the same right aacen- 
sion, but differing at least 30" or 40" of declination. Now if the 
interval between their passing the meridian in siderial time be ex- 
actly equal to their difference of right ascension, the instrument is 

■ ^ni\y placed ; if not, it wants correction. 
If, when the latitude is N- and the stars S. of the zenith, the high- 
t star come first to the meridian and the interval between the tran- 
8 be too great, it deviates towards the west ; if too small, towards 

\ ibe east. 

But if the lowest star come first to the meridian, and the interval 

I between the transit be too great, it deviates towards the east; if too 

I Bmall, towards the west. In either case there is required a correc- 

I tion, which may be computed in the following manner: — 

1 Rule- — ^To the secant of the star's declination add the sine of the 

difference of the latitude and declination, if they are of the tame niam, 

J or the ane of their jutn, if they are ot different names; of the sum of 

1 3Phich find the natural numbers. To the logarithm of the sum of 

^ese add the urithtnetical complement of the logarithm of their iit- 

nirence, and the logarithm of the difference between the excess of 

|. |lie right ascension of one star above that of the other, and the pb- 

Ifirved interval of time between the transits, the sum will be the lo- 

F|prithm of an arc in time. 

EciyHalf thesumof theescess of the right ascension of the o 

■ -^^ — « the other «Hd the Ibregmng arc, *ill be the deviation aiJJ 



* Perimjii the catalogue in ihc Nauiical Almuisc might be exlcnilc. 
n marcjudiciuui. toT eiBmpIv, the jilaccB uf aome of the niiiallcr 
ght be properly ixchangrd tor ftlhcf circiimpolBi or high and low " " 




SPHERXCAL TBIGONOMETRY. 136 

ItfWMt star/ and half the. difference between these will be the devk* 
tibh at the hifFheBt. 

- The deviation in time at each star being now known, the instrih- 
ment may bie easily rectified by either, or t>oth of them cm the f<dU 
hxvHog night, or still more readily br a third star on th« same even- 
ijag ; or; if the telescope is sufficiently, powerful to diow stars in the 
day, all the corrections may be pmormed at any time in a few 
Saccessive hours. For the deviation of one star being known, that 
^ another may be computed by the following — 
■ Rule,'^To the logarithm of the given deviation add the cosine of 
the corresponding star's declination, the secant of the declination of 
l!he third star, the cosecant of the sum of the latitude and declination 
of the first star if they are of difftrtnt names,, or of their difference if 
tliey are of the saiAe name^ and the sine of the sum of the latitude 
and declination of the third star if they are oidijferent namesj or of 
their difference if they are of the same name; the sum of these will 
be the logarithm of the deviation in seconds of time at the third star* 
Ex.— On tiie 1 st of March, 1826, at the observatory of Edinburgh, 
in latitude 55"^ 57' 81'^ N., I observed the transiu of Capdla, and 
Rigel, on the same evening, about a quarter past 6, and found the 
interval between the two transits 2'.5 less than the diffSerence between 
their true apparent right ascensions^ as given in the Nautical Alma- 
fiac ; required the deviation of the instrument at either star, and 
also at a third, as Sirius ? 

Latitude SS^^S^N. 65'57'N. 

Dec. of Capella 45 48 N. sec. 0.156664 Rig. 8 25 S. sec. a004703 

Difference 10 9 sin. 9.246069 sum 64 22 sin. 9.955006 



1. Nat. number 0.2528 9.402733 ..^.959708 

2. Nat number 0.9114 

75 

Sum 1.1642 log. 0.065953 

Difference 0.0586 ar. co. 1.181478 

Diff. ofR.A.and\2.g , 0.397940 
Obs. interval J ® 

Arc in time 4 .42 log. 1.645346 

Sum 6 .92 half = 3*46 = the deviation at Rigel. 

Difierence 1 M half = .96 = the deviation at Capella. 

Now since the highest star comes first to the meridian, and the 
interval between tke transits is too short* the deviations are easterly. 

if the stars had been between the zenith and the north pole, the 
deviations would. have been westerly . 

Since it has be?n found necessary to fix the instrument mb soon as 
possible, we shall proceed to compute the deviation at the diird star, 
w)iich can be easily done, as we nave an hour and three quarters 
nearly to perform uie calculations and complete the arrangements; 
thus: 



Declination of Ripcl (rt) raff 3. fosine 
Latitude {/>) 65 57 N. 

DeciinfltloiiofSirius(c) IS 29 9. secant 

First Miffl, or, (n+6) — 64 23 posecani 

SecondMuu,w(A+c) =72 20 
Devlution at Bigel 3'.46 



3.7745 



log. 
log. 



pmsmfM 



0.0449) 
9.9792* 
0.5.390" 



a57«8M 1 

After having corrected the instrument by means of Sirius, I obr J 
served the transita of Castor and Procyon, and again those <f J 
Procyon a/id Pollux, and found the interval of time to agree (vitlij 
their difference in right ascension, from Tvhich I concluded, thi^l 
in the apace of about three houra I had placed my transit iattra>.a 
ment exactly in the meridian. 

As it is ratlier a difficult operation to fix a transit instrument » 
curately in the meridian, these operations should be repeated a G< 
aiderable number of times to insure the utmost possible accnracyal 
After the observations prove satisfactory, a meridian mark may (M j 
put up in a horizontal direction at a considerable distance, wiA 1 
which the central wire may be frequently esamined and rectified | 
previous to any very nice observation. This mark may 1 
various constructions, such as a copper-plate with a hole in, it, ■ 
as a small segment of light may be seen on each side of the v' 
tical middle wire, or a small notch in a building, or evE 
at some distance. A thin slip of brass or copper painted blac 
with white lines or divisions at every inch, and numbered throiu' 
out, will also be found very convenient, and by knowing its a 
tance the deviation upon it may be computed.* 

The transit instrument being now properly rectified, it will I 
found the m.ost accurate of all for determining the error and rate V 
a. clock or chronometer, by taking the transit of the sun 
daily, and marking the dilf«'ence regularly in a column prepared'B 
that purpose. If a star be observed, siderial time must be reducS 
to mean solar time by Table XXXI. when necessary. 

Ex. 1. — The observed times of the suns passing the mei 
the observatory were as followt ; — What was the original err 
last day of observation and the daily rate? 



m 



2 25 l(i.6 

3'0 25 5.4 

4 24 54.0 

5 24 42.0 

6 24 29.8 



Of 12™ 40'.7 
12 28.5 

12 le.o 

12 3.0 
11 



0^ 12- 46'.4 
12 48,0 
12 40.4 
12 51 .0 
12 52.5 



11 35.6|0 12 54.2 



+ 1-.6 

+ 1.4 

+ 10 

+ 1.5 

+ 1.7 



Wean daily rate is therefore 
■nd the original error at noo 



5j7,8 

+ 1.56 
a the 6th of March, 1826, 
0M2" 54*2 fast 



■Bai-dcviiUin 



sec all. X COS. die X d1»- dlfF. of limi: y IS, to ndiui I. 
by cuDtricling ibc diuoeur of die objnUKlua by some contHv* 
( lou pioHH, the meridian ra«k ni»y bo only ■ few feet duinnL — »« hii 



SPH£BICAL .TBMKNNOMETBY. 



117 



Hflooe its error, Buppomfg the rttetoNouun uQifionn, may, at any 
moderately distant future tmie^ be detcrmiBed. 

Ex. 2^--Qn the same eveniijffs tha star Kigd passed the meridian 
as follows :«-Required the daily rate and the origmal error on the 
sixth at the time of observation^ about 6 o'dock in the evening ? 



itei 


OlM.T1ai& 


jMirOucor 


nmoiiumua 


IMyBMa 


MBDu 


8un'>Ttuuit 


tart IVantlt. 


aMnUUme 


Bfarehl 


6^ 48- 66'.6 








3 


6 39 3.3 


3"54'.3 


8-56'.9 


+ 1.6 


3 


6 35 7.8 


3 54.5 


3 56.9 


+ 1.4 


4 


6 31 13.4 


3 64.4 


3 55.9 


+ l-S 


5 


6 37 19.3 


3 54.3 


3 66.9 


+ 1.7 


6 


6 33 35.0 


3 64.3 


3 56.9 


+ 1.7 



5\7A 



Rate bv the star 
By the snn^ 




SJ&14 



Mean rate by both 

Sun's R. A. at noon^ on the 6th, 
Prop, part of daily var., to 6^, 

Reduced R. A. 

Star's R. A. by Nautical Almanac 

Apparent time of transit 
Equation of time 

Mean time of transit of star 

lime of transit by chronometer on the 6th 

Error of chronometer, fast by star. 
Allowing for change of rate in 6^, by sun. 



+ 1^7 
+ 66.6 



93 7 
6 6 


16.9 
13 J 


6 68 
+ 11 


56.6 
36.6 


6 10 
6 33 


31 .S 
36.0 


13 
13 


6311 
54.6 


8.4 



Mean error at &" fast 
With a daily rate of 



12 54.2 
+ 1^7 



As opportunities may not occur daily for celestial observations, it 
is in that case necessary to compare a chronometer with a good clock, 
the rate of which can be depended on, and is occasionally ascertain- 
ed by the heavenly bodies. 

Ex. 3.— -Given the daily difference between a chronometer and a 
clock, the rate of the clock being occasionally determined by celes- 
tial observations ; to find the error and rate of the chronometer ? 



(s) 



INTRODUCTION. 




,«. 


Ma.n n™^ 


tctt- 


'ls:-S,';r|«BK«..| 


3 

4 
5 
6 


+ M • 
+ 8.9 
+ 9.4 
+ 9.8" 
+ 10.1 
+ 10.5 • 


+ 2'.5 




+ 1'.7 
+ 1.9 


+ 3.8 
+ 5.2 
+ 6.5 
+ 7-9 


+ 12.7 
+ 14.6 


+ 16.3 
+ 18.0 
+ 19.8 


+ 1.7 
+ 1.7 
+ 1.8 


+ 9.3 



6i+ 1 



4 



Mean daily rate +1-76 

And on the 6th at noon, the original error was fast 19'.8 
Hence the error of the chronometer may be found at any moder- 
ate distance of time, so far as its steady rate can be depended on. 

The clock was examined by celestial observation, only where the 
asterisks are placed, or on the 1st, 4th, and 6th, and these are suffi- 
cient to ascertain, with the requisite precision, the rate of the chrono- 
meter when the clock is good. It is in a somewhat similar manner 
that the prize thronometers are tried at Greenwich. 

Table of the variations of the sun's R. A. and dec. in 1" for every 
month in the year. 



January 
February 
(March 
April 
May 
June 
July 
August 
Sept- 
October 
Novem. 
Decembe; 



0".0029 
0.0027 
0.0025 
0.0026 



0.0028 
0.002G 
0.0025 
0.0026 



0".008 N. 
.0J4 N. 
.016 N. 
.014 N. 
.009 N. 
.000 
.006 S. 
.013 S. 
.016 8. 
.015 S. 
.010 S. 
.002 S. 



This table will be useful wlien the chanjfe of llie 
1). for a few seconds only is wanted. 



MM»SWMM¥>ti. IM 



»-•■ 



. ' . V. PART ni. . 

J ■ • 1 

' MENBUEATIOH^ SUBVBYmei &C 

1 Sbction I. • 

■ ■' ■ - * 

Metuuratum of Snrfaoes» 

Meniiiraticm is die application of Arithmetic to Geometry, by 
which the valves of geometrical mairnitudes are obtained in num- 
bers. 

In this case some determinate magnitude of the same kind with 
that to be measured is assumed, as unit^ and the number of times 
this unit is contained in the given magnitude is the measure of that 
magnitude. _] 

See Leslie's Geometry, Book V. Prop. XXV. 

1. To find the area of a paralUhgram, multiply the' length by the 
peniendicbiar breadth.' 

^ Triangle. — Multiply the base by the perpendicular sfttitnde ; 
half the product is the area. Or take half the prodiidt^'of tiie'two 
aides and tiie natural sine of the contained angles ' Of'trhen tiie 
three sides are given^ multiply half the sum of we tfai'ee' sides, and 
the differences between tiiat half sum and the tluree aides together, 
tiie square root of tiiis product will be the area. This may be per£onn- 
ed readily by logarithms. 

3. Tropezttmi.*— Multiply the base into half the* sum of the per- 
pendiculars. • 

4. Trapezoie?.— Multiply half the stmt of the patfaUel sides by the 
perpendicular distance oetween them. 

5. Irregular Polfvirofi.— Divide it into triangles^ find their areas^ 
the sum of these wifi be the area. 

6. Regular Pofygpn. — ^Multiply the square of tiie aide given into the 
proper multiplier for areas from the table, page 14di for that purpose, 
and the product will be the area. Or, divide the polygon into tri- 
angles ; find the area of one of them by some of the foregoing rules. 
Multiply tills by the number in the whole polygon, the product is 
the area. « . - > 

7- Circ^— /The diameter is to the circumference as 1 to 3.1415923S36^ 
or 1 to 3.141693 nearly. 

The circumference ifi to the diameter as 1 to 0.318309. 

The area is equivalent to the square of the diameter multiplied 
into 0.785398. 

The area is equivalent to half the diameter multiplied into half 
the circumference. 

. 8. Circular Arc. — ^The length of a circular arc is equivalent to the 
radius of the circle multiplied by 0.0174533 and by the number of 
d^rees in the arc 

Or^ from eight times the chord of half the arc subtract the chord 
of the whole arc, one third of tiiis remainder is the length of the 
arc nearly. 

9. Circular Sector, — The area is equivalent to the radius multipli- 
ed into half the length of the arc. 

10. Circular Segment, — Multiply the square of the radius by either 
half the difference of. the arc of the segment and its sine, or by half 
their sum, according as the segment is less or greater than a semi- 
circle, and the product will be the area. 



Ml INTBODUCTION. 

11. Parabola. — The area is equivalent to two-thirds of the pro- 
duct of its base and altitude. 

12. Ellipse. — The area is equivalent to the product of the trans- 
verse axis into the conjugate axis multiplied by 0.785398. Peri- 
phery. — Multiply the square root of half the 8uni of the squares of 
the two axes by 3.141593, the product will be the periphery nearly. 

Examples for Exercise. 

1. Required the area of a square of which the side is 5 feet 9 
inches ? ^n*.— 33.0625 feet. 

2. Required the area of a rectangle, if the length is 1375 links and 
the breadth 950? Jns.—13" ff W. 

3. Required the area of a rhombus, of which the length of the 
side is 12.24 feet and height 9.16 feet? 

A«t.— 112.1184 square feet. 

4. Required the area of a rhomboid, of which the length is 7 feet 
9 inches, and heights feet 6 inches? 

Ana.—2T 1'" 6^^. 

5. Required the area of a rhomboid, of which the adjacent sides 
axe S535 and 1040 links, and the contained angle 30° ? 

^n«.— 13" 0' 29". 

6. Required the area of a triangle, of which the base is 1225 links 
and altitude 850? 

7. Required the area of a triangle, of which two of the sides ne 
30 and 40 and the contained angle 28° 57' 18'' ? 

.4nj.— 290.47366. 

8- Required the area of a triangle, of which the three sides are 20, 
30, and 40 feet ? .4n*.— 290.4737 square feet. 

9. How many acres are there in a triangle, of which the three 
aides are 380, 420, and 765 yards ? 

Ans.—9"- 0' 3&\ 

10. A ladder, 50 feet long, being placed in a street, reached a 
window 28 feet from the ground on one side ; and, by turning it 
over, without removing the foot, it reached another window 36 feet 
high on the other side ; required the breadth of the street ? 

/ins.— 76,1233 feet. 

11. How many acres are there in the trapezium, of which the dia< 
gonal is 475 links, and the two perpendiculars falling upon it on op- 
posite sides, 225 and 360 links respectively. 

Ans.—13" 2' 25' . 

12. Required the area of a regular hexagon, one of whose equal 
sides is 14.6 feet and the perpendicular from the centre 12.64 feet. 

Aiis.~5o3.633 feet. 

13. If the diameter of a circle be 17. what is the circumference? 
.^n».— 53.4072. 

14. If the circumference of the earth be 34850 miles, what is the 
diameter ? Ans.—'J910. 

15. If the chord of an arc be 30, the height or versed sine 8, what 
is the length of the arc ? 

An3.—35f,. 

16. Required the length of an arc of 67° 17' 44''.8; the diameter 
of the circle being 35 feet? 

^fij.— 12.5, which is equal to the ra<Iiu«. 



1 



MtmfXCtAttOS. 



141 



17- Required the area of a circlej of which the diameter is 161 
feet? Jw.— fil.1798. 

18. Required the radius of a drde in yards^ of which the area is 
an acre? Ans, — 39^^*. 

19. The diameters of two circles are 16 and 10 ; what is the area 
of the ring formed between these two circles, the centre being com- 
mon to b^ ? Ans. — 122.5224. 

20. Required the area of the sector, whose height or raised sine is 
4 and the diameter of the circle 16 ? 

Ans 33.5103. 

21. Required the area of the segment of a circle, of which the 
chord is 16 and the diameter of the circle 16} ? 

J«*.— 70.7083. 

22. Let ABCD be a four-sided field, and from the side AB to 
the points C, D^ let fall the perpendiculars PC and QD. Now the 
measure of AP is 110 links, PC is 352 links ; AQ is 745 links, QD 
is 595 and AB is 1110 links ; required the area of the field ? 

Ans.^S*^ 3'- 35P-. 



TO FIND THE ABBAS OF CIBCULAB 8BGMBNT8. 

jRu/e.-— Divide the height of the segment by the diameter, and 
find the quotient in the column of heights in the following table : 
Take out the corresponding area in the next column on the right- 
hand; and multiply it by me square of the circle's diameter, for the 
area of the segment. 



TABLE OF THE ABBAS OF CIBCULAB SEGMENTS. 



Height 



.01 
.02 
.03 
.04 
.05 
.06 

.07 
.08 
.00 
.10 



Area of the 
Segment. 



.00133 
.00375 
.00687 
.01054 
.0146( 
.01024 

.02417 
.02944 
.03502 
.04088 



Height. 



Area of the 
Segment 



.11 .04701 



.12 
.13 
.14 
.15 
.16 

.17 
.18 
.19 
.20 



«««•*• ^8S^?1h-«»». 



.05339 
.06000 
.06683 
.07387 
.08111 
.08863 
.09613 
.10300 
.11182 



.21 
.22 
.23 

.25 
SIS 

.27 

.281 
.29 
.30 




Area of the 
Segment 



.31 

.32 

.33) 

.34 

.35 



.20738 

.21667 
.22603 

.23547 
.24408 



Height 



.37 .26418 

.38 .27386 

.30 

.40 .2033 



.41 
.42 
.43 
.44 
.45 



Area of the 
S^ment 



.36 ^25455 .46 .35274 



.47 
.48 
.40 



.30319 
.31304 
.32293 
.33284 
.34278 



.8627^ 
.37270 
.38270 



.50 .39270 



£x, 1. — Taking as an example the chwd 12, and the radius 10, 
or diameter 20. 

And haying found the perpendicular from the centre upon the 
chord = 8; Vxea 10 — 8 = 2. Hence, by the rule, = 2 -<- 20 = -1 
the tabular height. This being sought in the first column of the 
table, the ccnrresponding tabular area is found = 'O4088i Then 
-04048x20« = -04088 x 400 = 16.352, the area. 

The use of the following tables will be readiJy understood, from 
considering that the areas of similar figures are as the squares of 
their like dimensions, and their soliditibs as the cubes. 




1. Prism. (1.) Surface. Multiply the perimeter of ona end by 
the length OT height, the product ■bHU be the surface of the sides. 
To thisadd the areas of the two ends, and the sum will be tlie whole 
surfaced i 

(2.) Solidity or Capaeiii/. Multiply the area of the base by the 
height, the product will be the solid content. The same rules de- 
termine the larface and capacity of a cylinder. 

2. Pyramid or Cone. (1.) Surface. Multiply half the perimeUt 
of the base by the slant height. To this add the surface of the base, 

;tli« Hum is the whole surface. ■ ■ 

(2.) Capacily. Multiply the area of the base by one-third the 
fievpendiculac height. 

3. Frwslunt of a Pyramid. (1.) Multiply half the sum of the pe- 
rimeterg of the two ends by the slant heigi^ht. To this add the areas 
of the two ends, the sum will be the whole surface. 

(2.) Capacily. Add a diameter or side of the greater btftse to one 
of the less ; &om the square of the sum subuaot Uie product of these 
two sides ur diameter ; multiply the remainder by a third of the 
height, and this last product by the proper number for tl»e circle, 
■785398, or polygon, the last product will be the content. 

4. Sphere. (1.) Surface. Multiply the square at' the diameter 
by 3.l4l593, the product ia the surface. 

(2.) Capacily. Multiply the cube of the diameter by 0.5236, or 
the cube of the circumference by 0.0]6B87. 

5. Spheric Segmcttt. (1.) Surface. Multiply the circomftreiioc 
of the sphere by the height of the segment. . 

, (2.) Capacity, or c = 0.523(i k<' (3d— <SA), in which d is the 
diameter of the sphere and k the height ; tx c = U.5236 A' (3 H 4 
■/t"); in wbiifh r is the radius of the base of the segment and A ita 

,, 6. Paraboloid, or itolid formed by ithe rotation of a parabola 'about 



Capacily. Multiply the base by 
tlie coot*>nt. 
. 7- Sphc'viti, or aolid lopmnl by titt rev^uti 



haigbt, half iho pr«chH^ U 
ellipse about 



MENSUBATION. 



143 



Capacity. Multiply the >qutfe of the revolving axis by the fixed 
axis, and the product by 0.6236^ the result will be the content. 

8. EegtUar, or Platonic bodies^ as they are sometimes oalled, are 
contained under like» equal, and regular plane fi^pures, of which the 
solid angles are all equal. The names and descriptions of these bo* 
dies, together with their multipliers, the side of each bdng unity, 
are contained in the following tables :— > 

Suffaces and Solidities of Regular Bodies, the Side being Uniitf, or 1. 



No of 
Sides. 


Name. 


SuiSm. 


Sulidity. 


4 

6 

8 

12 

20 


Tetraedron 

Hexaedron 

Octaedron 

Dodecaedron 

Icosaedron 


1.7320508 
6.0000000 
3.4641016 
20.6457288 
8.6602540 


0.1178513 
1.0000000 
0.4714045 
7.6631189 
21816950 



Thediam.of««phere|T5MmBy »» »n-iTh*tMy te ciret^ 

beinglithe»l<ferf«»cri«>«l fi> «» scribed about the 
^ ' ^ere« ii tiquaie, fa 



Tetraedron 

Hexaedron 

Octaedron 

Dodecaedron 

Icosaedron 



That fa equal 
tu the sphere. 



0.816497 
577350 

0.707107 
0.525731 
0.356822 



2.44948 
1.00000 
1.22474 
a661o8 
0.44903 



1.64417 
0.88610 
1.03576 
0.62153 
0.40883 






Examples for Exercise. 

1. Required the solidity of a cube, of which the side is 5 feet 3 
inches ? Ans. 144y'^ feet. 

2. What is the solidity of a block of marble, of which the length 
is 10 feet, breadth 5{ feet, and depth 3^ feet ? Ans. 201^ feet. 

3. Required the solidity of a prism, of which the base is a hexa* 
gon, each of the equal sides being 1 foot 4 inches, and the length of 
the prism 15 feet? Ans. 69.282 feet. 

4* Required the convex surface of a cylinder, of which the cir- 
cumference is 8 feet 4 inches, and length 14 feet ? Ans. 116| feet 

5. What is the solidity of a cylinder, of which the length is 5 ft. 
and diameter of its base 2 feet ? Ans. 15.708 feet 

6. The diameter of the base of a right cone is 4^ feet, and the 
slant height 20 feet ; required the convex surface ? Ans. 141.372 
feet 

7* Required the convex surface of a frustum of a right cone, the 
drcumference of the greater end being 30 feet, that of the less 10 
feet, and the slant height 20 feet? Ans. 400 feet. 

8. What is the solidity of a triangular pyramid, of which the 
height is 30, and each side of its base 3 ? Ans. 38^97* 

£ What is the solidity of a cone, of which the circumference of 
the base is 40 feet, and its height 50 feet ? Ans. 2122 feet 

10. What is the solidity of the frustum of a cone, of which the 
diameter of the greater end is 5 feet, that of the less 3 feet, and the 
perpendicular height 9 feet ? Ans. 115.454 cubic feet. 

11. What is the solidity of a frustum of a square pyramid^ one 
side of the greater end being 18 inches, that of the less l^ inches, 
and the height 5 feet ? Ans. 16380 cubic inches. 



Mt 



INTRODUCTION. 



UUIXi 
:htl» 

1 

dace 



12. Required tho convex superficieB of a sphere, of which the dia- 
meter is 17 inches P Ans. 907'^ square inches. 

13. Required the solidity of the same ? Ans. 1 48868 cubic feet. 

14. Kequired the solidity of the earth, considering it as a perfect 
sphere, of which the diameter is 7910 miles P Ans. 259136798136 
cubic miles. 

I 15. What is the solidity of the segment of a sphere, of which tfap 
I diameter of the base is 20 feet, and its height 9 feet? Ana, 1796. 
cubic feet. 

Section III. 
Surt'eTfing. 
In land surveying, the iustruments commonly employed fot 
ordinary purposes are — ' 

1. Gunter's chain, and ten iron pins. 

2. Cross-staff', and signal staves. 
3- Field-book, or paper. 

p 4. Case of mathematical instruments. 

5. Plotting scales. 
I 6. ParalleT ruler, and beam compassc 

7- A small quadrant, if a theodolite is not at hand, to reduce 
the hypotenusal to their horizontal measure. 
It would exceed our present limits to describe all these, as 
as some others, which may however appear perhaps in a work 
posed with that view. 

All Example of Lading Off a Field- 
Having set up poles at A, E = 

,B, C, and D, so as with the 
different dotted lines to re- 
tduce the body of the field 
(to a quadrilateral form, and 
(drawn a sketch of it, into 
'which the measures when 
ftaken may be inserted ; be- 
jgin at any point A, measur. ^ 
ang the successive distances 
iA a, A c, &c., on the chain- 
jline A B, and the corre- 

[tiponding offsets ab, cd, &c., and marking them as in the figure 
js complete circuit A B C D A of the field and the diagonal A C are 
Iflieasared ; these afford data for planning it, and computing the 
iarea. For the various portions may be considered either as trapes- 
lOids or triangles, whose contents may be ascertained by the rules 

Siven for that purpose," The area computed in this manner will 
e 2.4295 acres, or 2 ac, I ro. 28.72 po., though it is better in gene- 
^ral to retain it in acres and decimals. It is necessary tu take an ac- 
t of the roads, dikes, ponds, &c., of which tlie contents must all 
jbe stated distinctly by themselves when a whole estate is surveyed. 
-In the cose of the Mle of crops, that in tillage only must be mea^ur^ 
Required the plan and area of the field, fimm the following field- 
book, in which the angles were measured, with the pocket4xu, 
sextant, an<l the distances with the chain, begbining the operations 
at the gate near the south-east corner f 

■rvcyan fint conmnicl in M«tmK plan, from whith, by tale and 
ia ubuined wlih mlScknt practilon : ind Ihii ii at lesM ■ gnoi mt- 
ihf rmtilt hy compulwinn. 




emnvtyiNo. 



145 



FieidJBooL 






Hedge. 



Deadrigffs* or 
Crosshali lands 
on the south or left 
hand 



Boundary. 



Hedge. 

Hardacres land 
on the north 
or left hand 



To Ist, or 





88 148 

01rt99*46'8Or'W. 

900 

940 

701 300 

34 400 

480 

510 

44 660 

736 
810 



Bemark. tbmcbdA* 
line bears netriy west 
along the north lide 
of Bitterick Sjrke. 



lOQ 




S 
5 
3 

10 




900 
400 
000 
800 
860 
866 



elSIWWW 


100 
200 
964 
350 
456 
544 
700 
756 



The diain-line bears 
nearly north. 



The chain-line bears 
nearly east. 



< -it 



©4tiil0I*>28'0^8. 
100 
200 
300 
400 
500 
600 

Area i= 6.14537 ac. 
or 6 ac Q ro, 23 po. 



The chain-line bears 
almost south along 
the road from Oreen- 
law to Eccles. The 
diagonal from Ist 
to 3d^ measuring 
1053 links, was also 
taken, that the area 
might by the three 
sides of tne triangles 
be a check upon that 
determined mmi us- 
ing tite angles. 



-If thinre ire dikes, ditches, or fencea of any kind, they must be mefr- 
anrad during the survey, and their amount stated. Also plantations, 
raads, conmions, lakes, ponds, &&, must he til tttrveyed and d a s se d 
separately from the arable land, for these we cannot here enter 
into detul. 



This place is mendoned in Sir Walter Seott*s Mi&stieisr of tbe Scottish Bocfa. 



.a^ INTHODUCTION. 

' '"' '■■'■' Levelling. "■ " - ' ' ■ -.- 

It ig^jftennecesaary to ascertain the difference of elevation oSoae 
point above another, for the purpose of conveying a stream of vater 
to drive machinery. This may be performed in several waySj but 
the readiest and niost acurate is by means of a spirit'level of the best 
(^instruction. It must be accompanied byapole, or rod divided into 
ffet> and at least hundredths of a foot. On this rod a sliding vane 
ifl fitted, capable of moving easily up and down, and having a dark 
other well-^fined mark upon it, by which the telee- 
levels the sight, may be directed. The slider 



strong 1; 
cope. 



it be moved 



upw 



r downwards on the rod, till the mark 



Coincide with the intersections of the cross hairs in the focus of the 
telescope. When this is accomplished, and the level being properly 
adjusted, the height in feet and hundredth parts is to be carefully 
read off and marked in a book for the purpose. Now, by means of 
a chain or measuring tape, let the pole-bearer place it at equal dis- 
tances, alternately on each side of the level, such as about one or two 
hundred yards, if convenient, if a level with a good telescope be 
used. If an ordinary level with a plain sight be used, the distance 
must be reduced to as many feet The heights taken with the 
telescope turned towards the place whence the observer set out, are 
c^led the back observations ; and those taken towards the place 
where he means to finish, are called fore observations, for the sake 
of distinction. Since the pole is always placed at equal distan- 
ces from the level, no allowance need be made for the curvature 
of the ear til.* 



EXAMPLE. 





iience the difference of level on a sloping height of 2100 linka of 
Gunter's surveying chain, or 2100x006 = 1380 feet, is 19 feet. 
When a spirit-level exactly adapted to this purpose, is not at hand, 
if there is a theodolite to be had, it will perform the operaxi^a, 
&toagh it is not quite so convenient. 

, ■ Tlie diffnttice of level is aboul B inches in a mile, which increMea a* the *quiu« <rf 
Atadfgtancc. The diffbreaw of te*d in /mj ntlovins for Bc&ictlaii, i* | af .lb* .ti|wiiv 
of the dutancc in English milt'E^. 



BULBS AND TORMVUE, 147 

In cftse of levelling for canals^ tJie process is not difTerent, only the 
canal is earned fm an ezactlevel, by judiciously choosing the situation 
winding round rising grounds^ conveying it across ravines' by aque- 
duct lAridges^ and allowing it to descend at particular points^ by means 
of locks, lloads ouffht to be carried along a level line as nearly as pos- 
sible, and only having gentle acclivities and declivities. T%is may 
be readily obtained by following routes somewhat circuitous in un- 
even parts of the country, taking the advantage of ravines, water 
courses, and the sides of lakes ; for a greater distance on a road neariy 
level, is productive of less expense of animal strength, than by pas- 
sing over considerable elevations. All very quick turns in the road, 
particularly when entering upon a bridge, ought to be avoided, as 
the danger from centrifugal force, whi(3i may be readily estSmated 
by the formula. Part III., Sec. IV., is considerable. The justice of 
these remarks may be readily appreciated by considering many parta 
in most of our public roads wnich have hitherto been constructed 
upon the very worst principles, having been entrusted to what are 
called practical men, who are frequently the mere slaves of custom. 

f 

Section IV. 

Rules and Formulas, 

When two angles of a plane triangle are known the third may 
be found, consequently, for general purposes, it is unnecessary to 
measure the third angle. But when great accuracy is required, or 
when the sides on the surface of the earth are large, they become 
spherical arcs, and then the third angle should always be measured 
as a check upon the results. In conducting geodetical operations, 
the triangle should be so chosen, if possible, as to produce the 
most accurate conclusions. To diminish the probability of error, 
the following rules should be observed : — 

I. When one side only of a triangle is to be determined, the mea- 
sured base should be nearly equal to the required side. 

II. When two sides of a triangle are to be determined, the triangle 
should, if possible, be equilateraL 

III. When the base cannot be equal to one or to both the required 
sides, it should be as long as possible, and the two angles at the 
base equals and not less than tyienty or thirty degrees.* 

IV. When the centre of the instrument cannot be placed in the 
vertical line occupied by the axis of a signal, the observed angles 
must be reduced to it by an appropriate formula. Let C be the cen- 
tre of the station, such as a tower, P the place of the centre of the 
instrument, by which the angle subtended by A B at 
P is to be measured. Let the angle A P B be ob- 
served, and the distance C P be measured, it is re- 
quired to find C, the measure of the angle A C B ? 
Suppose APB = P, BPC = ;), CP =^rf, A C = 
D and B C = D'. 

Since the exterior angle of the triangle A P I is 

e^^ual to the sum of the two interior and opposite C P 
ai^le8>- A IB =:. P+I A P, and of the triangle 6 I C, the exterior 
angle A I B = C+C B P. ]\Iaking these two values of A I B 
equals by. transposition, we have G— P = I A P^-C B P. But 

' * For a ^esaoDatsaldm of these piopertiei. Me voL lll^ of Uutton'H Ccnusse of 
Mathematics. 




IKTHODUCTIOM. 
~ tin trianglea CAP, C B F give sin. C A P Ar;BiiU'I A[P & 
Jl sin. A P c=^ ^--(^+P)- ., .-..CBP^JI «„. B P C = 

— fj— ^- And since the angles C A P, C B P, are, by hypodiCHS, 
always very small, their sines may be substituted for their arcs, 
y., ■■■ ' ■ which in seconds becranes 
Y ^, > ; or R" being the length of an arc in 
lius, or 206264".8, then C— P = R- J x 
The nee of this formula cannot be embar. 
Msing, provided the signs of ain. p, and sin. (P+p) be properly 



hence, C— P = 



rfsin_(P + p)_ 



Bin. 1" I 



■(P+rt_ 



seconds equal to the 
J sin.(P+ p) _.sin-pl 



attended to, as is illustrated by the t'ollowiog example: — Let the ob- 
•erved angle P be 43° 52" 49".44. p =: 264° 41' 24^ d = 10.706 feet, 
D = 57508 feet and D' = 66750 feet, required the reducUon? 

do (2.) 

I»g.R" . 5.314426 
log. d 1.032860 

+ 6.34728S —6.347885 

foe' a4'A^"}~« 8931 18 sine ;> 264' 41' 24" . — 9.9{H)]32 

ar. CO. log. D+5-340272ar. CO. log. D' +5.175549 

(1.) —30" 246—1.480075 (2.) + 3.3".187 + 1.530966 

(1.) —.30 .246 



P . . . . 43° 52' 49 .440 „„,, 

C .... 43 52 52,381 1' 

When signals are circular or polygonal towers, various methods 

nay be employed to find the true angle, from a due consideration of 

I the nature of Uie case, which, to any one possessing a knowle<ige of 

I Ae elements of geometry, will readily occur. 

V- The angles measured in an inclined plane, should be reduced 
to the horizontal plane. 

In this case the altitudes must be also observed, and then there is 
I fbrmed a spherical triangle, of which the three sides are given to 
tempute tile angle at the nenith, which may be performed by the 
I rules of spherical trigonometry. 

■ VI. A spherical triangle being proposed, of whidi the three sides 
ire very small compared with the radius of the sphere ; if from each 

I «f its angles, one-mird of the excess of the sum of it» three angles, 
I kbove two right angles be subtracted, the angles so diminished 
I may be taken for the angles of a rectilineal triangle, whose side* are 
f equal in length to those of the proposed triangle. 

" To find the spherical excess when the three sides are given in feet. 

■ 1. Rule.—To the constant logarithm J.349380, add ihe logarithn* 
I- of halfthe sum of the three sides, the logarithms of the three diffier. 
t «MeB between tbew sides and that half sum, half tbe sum of these 
V-M« logaifthms will be ^e logarithm of the apherical «xces« in ae* 
I conds. 



aULBS AND WOBMUUE. 14B 



3. To i3b» legmthm cf tlie ares of tha triangle UkaQ at a pbaia 
one in fysUpdd the constant ]^unthn(i4M746BO; .theav>n is ipelom 
garithm ofthe excess above Iw* in secondst 

3. If the base and perpendicular of a triangle be given. To tlia 
logari&m of die base m feet^addthe logarithm of the perpendicular^ 
am the constant logarithm 0^73660; the sum will be the logarithm 
of the spherical excess in seconds. 

The q[iherical excess amount3 to <me second for an area of 70 
English square miLes> whence^ if the area in square nules be known, 
4^ flph«rical excess may be readily obtained ny dividing it by 78. 

Vn. To reduce a base on an elevated level to that at the surfiiDe 
of the sea. 

If^t r represent the radius of the earth, correnKMiding to the base 
b at ihe level of the sea, and r+a the radius referred to the level of 
Ifbe measured base B; then it is envious that r-i^a : r ; : B : 

b = Bx-f-. Hence, B— 6 = B— B-4^= Bx-^- =5? By 

r+a r+a f+o '■' ^'• 

f — — — 3 + &c\ But the radius of the earth being very great in 

comparison of the difference of level a, we have the correction ) sufi* 

o 

ciently accurate^ by retaining the first term. Hence, ) =z B x - . 

Rule. — By logarithms. To the logarithm of the measured iNise in 
fisel:^ add the logarithm of its height above the sea, and the constant 
logarithm 2.680110 ; the sum will be the logarithm of a number of 
feet which^ taken from the measured base^ will be that at the level 
of &e sea required. 
' Vttl. To determine the horizontal refraction f^om observation; 
Rule. — ^From the measure of the intercepted terrestrial arc, sub« 
tract the sum of the two depressions at its extremities; half the re- 
mainder is the refraction. If by reason ofthe smallness of the coh- 
tained arc, one of the objects has an elevation instead of a depression, 
then the depression must be taken from the sum of the contained 
arc and elevation; half the remainder is the refraction. 

FOBMUIiA. 

R = e=|=:^=''-i^ (1.) 



If — d' becomes an elevation, it changes its sign, and becomes 

-fie, and in that case R = ^^^r — (ft.) 

• « . . . 

•The ex^cjb quantity qC terreatrial refraction is very variable* It la 
estimated by t)r Maskelyne at one-tenth of the intercepted aire, by 
Delamhre at one-deventh, by Gteneral Mudge at one-twdith, and by 
Legendre at one^^fourteenth at a mean state of the atmondiere* In 
peculiar circumstances it varies very considerably from this; as from 
one-8i:dii to one-eighteenth of the contained arc. 

IX. To find the angle made by a given line with the meridian. 

With a good instrument measure the greatest and least angular 
distance of the pole star from the vertical plane in which the given 
Uae is situated ; half the sum of these two measures will the angle 
gea[gbied« ■..;■■.'. . ^ it , . 

"This may also bfldone^ though less aaeu£ait0y> by eo«ipu$i|i|^,^ 
asimuth of th6 aini^;«r a' star, wlmi^iii l^e line> vom'Ml|)4Mtv^ 
taken for that purpose. ^^ •. 



■0 INTHODUCTION. 

X. In addition to what has already been said relRttre to fifrf. 
iiig the latitude of the place, we may add here, that the same 
thins may be very accurately obtained, by observing the greateat 
and least altitude or zenith distance of a circumpolar star, and cor- 
recting them for the effects of refraction ; half the aiim of the al- 
titudes, thus corrected, will be the latitude, or half the sum of the 
aenith distances will be the colatitude. 

XI. To determine the ratio of the earth's axes, and their actual 
magnitude from the measure of a degree of the meridian in two 
given distant latitndea, supposing the eartji a spheroid generated by 
flie rotation of an ellipse about its minor axis,* 

Let d and d' be the measure of two degrees, d being the least, or 
that nearest the equator, / and I' the latitudes of their middle painta, 
t the semitran averse axis of the meridian or radius of the equator, 
c the semiconjugate or semipolar axis, e the excess of the equatorial 
radius above the polar aemiaxie, and r° = 57°,2957795, the number 
of degrees in an arc are equal to the radius. 



Then, • 
And 4 



i.(/' + Oxsin.(i'-0 



in. (/'+OXsin.(/'-0 ■ 

' If — =f, ellipticjtyorcompresaion, i=— 

When ^ is nothing, or when one of the degre 
from&nnttla (!•) 



_ r- (f-,0 



■ • (1.) 

■ ■ (2.) 

B at the egdatbr 



Therefore, the excess of the degree in any latitude above this de- 
gree at the equator, when divided by the square of the sine of the 
latitude, should always give the same quotient ; or the excess of the 
degrees of the meridian above the degree at the equator, should be 
as the squares of the sines of the latitudes. 

-tita - ,k.„ j._, -If.;,. (,.+,,« 



Since e 



"3sin.{/'+0sin.(/'— /)' 
Bin. (i'+O .._... 

If d' and d are two contiguous degreesj 



, then d' — d = - 



0.017453, d'- 



1. (2/-t-l°)Bin. I", ar 
J _3 ex 0.017453 . 



J that f = 1+ 1°, then 
[le sine of one degree is 



(2'+l-) 



(6.)t 



The contiguous degrees therefore differ, by a quantity ptapcr- 
tional to the sine of twice the middle latitude. The difference ii a 
maximum when 2 /+ P = 90°, or when the middle latitude is 4S^ 

From five tlifferent measures combined so as to produce the most 

I accurate result, Mr PJayfair found 1 = 0.0032 = 513-5 nearly, and 

I the equation representing the degrees of the merSdian setting «tt> 

' ftx>m 45", will be ,:.,, :< 

D = 607S9.4J2— 290.576 cos. 2 /f .... (7.) 
I in fathoms, or, 

• PlajWr'a Oullinca of Nalural PhUosnphy, Vol. II. Art. Sfl. 

t Using lognrilhnw, iT— ri >c C. L. 1.0081715 + log. ain. (8 (+!•) Ii 
or*— J=C. I« 1.7Be68fB+lo([. sin. (J 1+1°) in fcft, where c= lllAS.arathi 
or 609Ai.B feet respectively, and d b GOiSO faihotna, or SSSI60 feel. 

t Jn toi«» D = 57011— !«.65 cos. 8 (. 




RULES ANB FCNUCULiE. 



Ifil 

(ft) 



• D >= 0».04^^.%S99 COS. 2 i 
in SngHsh mSes. 

Hence, e = 11158.8 = JL2.680 

; < = 3486858.8 = 3969.349 

^ c =s 3475700.0 = 3949.669 

The radius of curvature for the parallel 

fath* =: 3956.009 milesb The circumferenoe of the meridian ia 
therefore equal to the product of the mean degree at 45^ by 360 =s 
24855.84 miles; and tne circumference of the equator is 34896.16 
mHes7<nr about 40 miles more than tike preceding. 
A geographical mile ia therefore 1012.6 fathomsj or 6075.6 feet* 




the same point, then, « = -5 (J^ — d) 8€c. * L 



' • 



(10.) 
(11.) 



_^ (D-^rf) tan. « /. 

2Dco8.«/""'2lF^*^''^ ^^^'^ 
For ezerdse the following measures of degrees of latitude vte 
given. 



■7 or • ^ 



Observen. 


Ui, 


Degreeiln 
Ttif. 


Deductiopa. 


Bouguer 

Condamine 

Tiambton 

Lacaille 

Mason 

Boscovich 

.Delambre 

Mudge 

Swanberg 


0° 0' 0" 


12 5 ON. 
35 18 OS. 
39 12 ON. 
43 1 ON. 
46 12 ON. 
52 220N.« 
66 20 ON. 


£6753 
56749 
56761 
67037 
66888 
56979 
67021 
57069 
57I68 


Radius of the equator 
3271691 toisak 
Semipolar azjs. 
3260964 toises, 

q"J 6Y3074O toises. 

1 toise = 1.949037 metre. 

1 French foot = 144 lines. 

1 £nglishfoot-135.0731iQe8. 



Let these be solved by the foregoing theorems, and the various 
consequences drawn. 

- i&r- i^-«*Li the Philosophical Transactions for 1795, D the d^^ree 
perpendicular to the meridian, is given equal to 61182 English fa^ 
t}umi^;is6085Uandrc=:50^4'N. By formula. . (12.) 
Ml . .^.* ... I 



1 



'lEx: 2.— The length of a degree in latitude 52^ 2^ 20^ N. is 57074' 
MUK8> dutt: in •II'^.O' N. is 56755 toises ; required the elliptidty by 
>rmula (2.) ? 



.^•? i ■ ?y- ■<• 



>%• 



-•\ -11.. ■, . *.' t ••yi' 

■ . ; J ■ . • * 







SB INTRODUCTIOir. ' 

i* z= 52* 2' 20" const, log. . 
I =n 

l'+l=63 2 20 cosecant, 0.049969 

/' — I =41 22 cosecant, 0.182718 

d' = 57074 

d 1= 56755 nr. co. log 5.345996 

#— i 319 1(^. 2.503791 

« = 0.0032(ffl log 750fi^^H 

If L = the length of a degree of longitude, then ^^^ 

I.= ^Al+. ™-"'+|.'.in."). . . . (13.)' 

If the value of the degree ia wanted in toiaea, fathoms, or feel, the 
Second member of this equation muat be multiplied by the semitrans- 
verse axis in the same measure. 

Ex. 1. — Required the length of a degree of latitude at Edinburirh, 
in56'N.? 

By formula (7), B = 60759.472+290.676 x sin. 22° =^ 60759.427 
+ 290.576 X 0.374607 = 60868.3 fathoms. 

Ex, 2.— Required the length of a degree of longitude in latitude 
86™ N., the ellipticity being ^^ ? 

Byfonnula(13),L = ^5|-./l+^x0.68694};arL= 

6.009760 X 1.00229 X 20918750=204635 feet, or, taking in the secoBd 
term mentioned in the note, it is 204648 feet. These formulie are 
iseful for fixing the latitude and longitude of a particular point when 
referred to some object whose situation has been well (^termined, 
such as many places in Britain are by the trigonometrical survey 
In this case any amateur observer may verify the latitude and longi- 
tude of his observatory deduced from his own observations, by a 
comparison with some point well settled in that work, when pro- 
perly connected by trigonometrical operations. Even by taking a 
few angles with great care, the situation of a particular point may be 
well settled bv spherical trigonometry, as in the following esample 
communicated by Captain Hall. 

Ex. 2,— Given the latitude of the Staff on North Berwick Law. 
' ,W3'8"N., Iffligitude 2° 42' 11" W., and Uie latitude of the Isle of 
I gf May light 56° 11' 22" N., longitude 2" 32' 47" W. ; the angle at 
[ North Berwick Law, between the Isle of May and Dunglass Tower. 
J. fi7° 41' 1", that at DunglasE, between the Isle of May and North 
I Berwick Law, being 37° 20' 13" ; required the latitude and longi- 
tude of Dunglass Tower ? 

Ans. Lat. fiS" 56' 31"-7 N., Long. 2= 21' 42" W. 



B • If grot accurtcj i» not 
Wtity withtn the pirtnihciis- 




RITLK8 AND FOKMULi«. l^ 

If p be the length of a degree perpendicular to the raeridiaDy i the 

equatorial radius^ c the semipolar axis» t^-^zzid the difference of 

these^ r^ the length of an arc in aegrees equal to radius, or 57^2957705, 

1 > 1 1 . ^ ^ i + d Bin. • / , ^, , 

and i the latitude, then p = -5 nearly. . . (14) 

Ex. 6.— If t = 348B850 fathoms, d = 11160 fathoms, and / = 56^ 

. 3486850+ 11160 X -68694 anoni 4^ .u 
Aenp = ^y^gyg^g =60991 fathoms. 

If p be the measure of a degree of a great circle perpendicular to 
a meridian at a certain point, m that of the corresponding degree on 
the meridian itself, and o the length of a degree on an ol)1iquc arc, 
making an angle a with the meridian, then 
p m m 



P—ip—'^) sin. ^ a J _Pz:^8in ^ a ' ^^^^ 

P 
Ex. 6.— If ;} = 61182 fathoms, m r- 60850 fathomB, and a = %V 56' 

53'', therefore 
_ 60850 _ 60850 _ 60a50 _ awnr at 

"*"" i__^2^^098a38^^~^-^^^^^^""^'^^^ "" ' 
61182^ 

length of the oblique degree in fathoms. 

For an extension of this subject, see Mr Ivory on tlie properdes 

of a line of the shortest distance traced on the surface of the oblate 

spheroid, in the sixty-seventh volume of the Philosophical Magazine. 

It is rather too long and difficult to be inserted in tnis place. 

SECTION V. 

Rules and Forimihv, 

SPECIFIC GRAVITY. 

The difference between the absolute weight of a body, ami it:^ 
weight when entirely immersed in a fluid, is the same with the 
weight of a quantity of the fluid equal in bulk to the body. 

If W be the weight of a body in vacuo, (which is nearly the same 
as that in air,) and W its weight in water, then W — W is the weight 
of a quantity of water equal in bulk to the body ; and since the weight 
of any body divided by an equal bulk of water, measures the specific 

gravity, S, of the body, then S = ^yr^jnr, • . (1) 

The specific gravities of bodies are determined by the hycbostatic 
balance^ the hydrometer, &c. described in books on Natural Philo- 
sophy. 

To compute the specific gravity of air under given circumstances. 

It is shown in Playfair's Outlines, vol. I. § 333, that if the elasti- 
jcity or tension at the freezing p<»nt, be denoted by unity and x, 
any number of degrees above that point, then the elastic force/ at 

. that point, will be/= (1.375) "^^o^ Fahrenheit's scale, or 

log./:^ j^ X log. (1.375) = j^ X 0.138303 . (2) 

This also gives the bulk of gas in like circumstances. But the 
specific gravity is reciprocally as the bulk, therefore the reciprocal 
cf the bulk or the natural number answering to the arithmetical 
complement of the log.y^ will be die specific gravity of permaneiiV 




L' 



■^ INTUOUL'CrUlN. 

elastic (tUids. Thns let the bulk and specific gravity of i 

P. =: 1, then at C2° F. they will be 1.036, and 0.9652 respectively. 

Prom the experiments of GayLusgac, it maybe shownthat 0.4546 
will be the specific gravity of aqueous vapour, when compared witli 
atmospheric air, at 32" P. Now, when the temperature is given, 
the specific gravitj rf aqueous vapour is directly as its temperature, 
and the tension being given, the specific gravity is reciprocally ae 
its bulk, tjie specific gravity s of aqueous vaponr, (that of water be- 
ing 1), in saturated air at any temperature/, and elastic force /i (from 
Dalton's table) will be obtained from the following formula, the bt- 
rometer being at 30 inches. 

,- 04545 x^ x-?5?- - i5=^ (31 

If it be not saturated, and (' being the dew point 
10/ 448+f_ Wf / l~l' . 

~448+( 44a+i'~448+( V ^ U8+t'/ ^' 

The quantities in this expression are all known except/, whidi is 
to be taken from any good table, such as Dalton's or Ure's. See Table 
11., page 48. 

if, therefore, *' be the specific gravity of air fully saturated "with 
moisture, a the specific gravity of dry air obtained from formula (2), 
and s the specific gravity of aqueous vapour in saturated air, d^ved 
from formuIjE (3), then from the law of expansion discovert by 

Dalton and Gay Lussac, that o ^ -f»! p being the barometric 

preBsure,/the elastic force,Hnd v the volume, 

'■ = '+'-¥o , (°) 

If t' be the clew point, and s" be the specific gravity, according to 
the actual state of the atmosphere, 

^" = ("+'-«) ('+4-BT?) ■ . . m 

in which n and j are got from the following table, page 165, and/ from 
Dalton's. 

£x, — Required the specific gravity of air saturated with moisture, 
at 92' P. ? 

By formula (2), ~ x0.138303^^x0.138303=0.046!Ol,ar. Co. 
of which is 9.953899. To this the natural number is 0.89929=0. 

But by formula (3), * = ^^- ^0.02782, and ^ = 0.0450-.?, 
Now,*' = n+,_^ by formula (6) ; therefore, «" = 0.89929+ 

0.02782—0.04502=0.88209 the specific gravity of air saturated with 
moisture, at 92° F. If the air is not saturated. Suppose 87=" P. the 

dew point represented by /', then the factor I +j^^—, in formuU 
■ Uanicll and TredgoU, coniend dint thU fonnula (houlit baM-t/, Tlic didctCDce 



RULES AND FO&MUJL.C. 155 

(6i, becomes 1+;^:^=1 + jig. therefore, 0.88209 + ^-:^?- = 

0.88209+0.00617=:0.89aa6, the specific gravity of «ir in the gifen 
circumstances, that of dry air at 32° F. being uqity. 

It is shown in Plavfair's Outlines, vol. I., art. 256, that if the 
specific gravity of air be called m, that of water being 1 ; if W be 
the weight ot any body in air, and W its weight in water, then 
W+m (W — ^W') is its weight in vacuo very nearly. In a mean state 
of the atmosphere at 30 inches of the barometer and W F. ai ss 
0.00122 nearly, which may be reduced to any other temperature by 
the foregoing formula (4), and to any other pressure by multiplying 

El 

30 . 

If s be the specific gravity of a body ascertained by weighing it 
in air and water, and m the specific gravity of the air at the time 
when the experiment was made ; the correct specific gravity s\ or 
that which would have been found if the body nad been weighed in 
a vacuum instead of air, or 

^ = *+m(l— *). (7) 

Where the body is heavier than water, this correction is subtrac- 
ttve ; when lighter it is additive. 

Ex. — ^The weight of Captain Kater's experimental pendulum wan 
carefully determined in air, by Barton's balance from the Mint, and 
found to be 66904 grains. The trough, which had been previously 
placed under the pendulum, was then filled with distilled water, aiid 
the weight of the water displaced was 9066 grains. The small por- 
tion of iron wire which was immersed in tne water was rareiullv 
noted ; the weight of the wire by which the pendulum was suspend- 
ed was 56 grains, and the weight of water equal in bulk to thnt part 
of the wire which was immersed was 2..5 grains. The temperature 
of the water was 6Q° F., that of the atmosphere (i2' F., and the ba- 

rometer 29.9 inches. Now since * == -,, w being the weight 

in air, and m' that in water, then 

66R4fi 
» = g^^ = 7^7552 at 20.9 bar. and 62« F., and a' = 'J/^^oT^J f 

0.00120678 (1—737552) =7-36783 at 68*' Falirenheit. 

But the specific gravity of water ^-at 68" is -99936, thnt ni &2^ l>e 
ing 1 ; and, therefore 

] X y=: q;^^ X 736783 = 737254 at 62^' F. 

Biot's experiments give at 30 inches bar., and 60' F , the specific 

gravity of air 0.00122, or ^^, water being 1. 

MrS. Rice, horn Sir G. Shuckburgli's experiments^ deduces 

O.OOJ2085ji not differing much from Biot's, and generally supposed 

the more correct. According to Gay Lnssac. the expansions of fluids 

.375 2 
fr9m 32*' to 212^ F. is 0.375, whence j^ =^ for 1° F. 

Now suppose c = the first correction of the length of the pendu- 
lum, c' the second, / the measured length of the pendulum, j) the baro- 
metric pressure, the standard being 30 inches ; and > / the difference of 
ieniperature from the standard, then 




INTBODUcnON. 



If "=: the corrected length of the pendulum I, from ; 
Captain Kater's experiments atLonflon in air,tlien/'^i+— 
s being the specific gravity of the pendulum. 

Whence ■; = |S - 826, and J i = 69=.62 - 6i 



4B0 



Hence by fortnul* (10) i' == / + 39.13284 X g^g X ygygsJ* 

/+ 0.00633. 

It is now only necessary to correct for the height above t&e mb. 
which is 92.5 feet. 

The correction for this heiglit found by tlie formulaj which will 
presently be givenj is 0.00023. 

Hence V = 39.13284 + 0.00633+0.00023 = 39.13940. In thi= 
(See no allowance is made for the hygrometer. Now if the air were 
supposed half saturated with moisture, since Captain Kater dues nut 
j^ive the state of the hygrometer, and the mean between Biota and 
Rice's specific gravity of air taken, the true length would come out 
39.13938, whicli differs from Captain Kater's result by 0.00009 in 
excess. 

It is shown by writers on mechanics, that when the eemiarc de- 
scribed by a pendulum is 1°, the time lost by oscillating in a. circu- 

lai', instead of a cycloidal or infinitely amall arc, is rSVoJ '" ^^c'' se- 
cond, and that in different small arcs of the same circle, the time 
lost varies nearly aa the square of the arc ; hence if a pendulum 
makes V vibrations in 24'', when vibrating in very small circular 
arcs, of which the mean at the commencement and termination of 
each experiment is «£ degrees, it would, in the same time, make v+ 

hifinitely smaU vibraliom. Hence to correct the oscillations ol' 

A pendulum for die arcs of vibration, multiply the square of the 

tntan are when it makes 

Daily 86000 oeoillations l>y 1.637 1 

aeioo 1.639 

86200 1.641 

8G300 . 1.643 }- (A) 



1.645 
1.647 
1-649 J 



- (i.i— 86500 

,i;i 8C600 

Since the force of gravity varies directly as the length of the pen- 
dulum, or inversely as the squares of the number of vibrations, and 
the diminution of the force of gravity, arising from the buoyancy 

ol' the atmosphere, is - past; therefore if w be the number of vibra- 
tion in air, and V those in a vacuum, then 



nULKH AND FUIIMI'L.K. ]67 

V = t/+c, and hence c = ^r — nearly. 

Jm " 

In Captain Kater's experiments at Unstj the Hpecific gravity of the 
pendulum^ to that of air^ was as 7099 to 1, hence — :=-yrr— , and 

^. . V 86090.77 aivr i 
therefore ^= Tii^ =6.07 nearly. 

If n' be the number of oscillations performed in 24* by the expe- 
rimental pendulum^ n the true number^ e the expansion for a change 
of one degree Fahrenheit, t the standard temperature, and t' the ob- 
served, then 

In Captain Kater's pendulum ^=0.00001 of an inch nearly, whence 
n = n' + i »' X 0.00001 (<' — /). 

Hence if v = 86058.82, t' = 71 "0 and i = 62«, the number of vi- 
brations at the latter temperature are n = 86058.72+ i X BUa5a72 x 
aOOOOl X 9.6 = 86062.77. 

To reduce the length of the pendulum from any height to the 
levdi of the sea, the true length being denoted by /, the observed by 
r, the height above the sea by a, and the radius of the earth by r, 
then 

l^f+^JLl .... (12) 

Some allow one-third for the effect of the dense strata iminediate- 

4 a/' 
ly under the pendulum^ in which case / = Z' + -^ — (13) 

In a similar manner « ^ i/ + -= — (14) 

2 v' a 
At Unst -s- — = 0.06, therefore 

86090.77 + 6.07+0.06 = 86096.90 = the number of oscillations of 
thependuluxn in a mean solar day at the level of the sea in vacuo. 
, Thes^ formulae are sufficient for most purposes. Biot has, how- 
ever, demonstrated, that if c be the correction in seconds for the 
mean arc of vibration, n the number of oscillations, M the logarithmic 
modulus, a the arc of vibration at the commencement of the interval, 
and b that at the end, then 
^ _ n^ sin, ja+b) sin, {a — b) ,,. 

"""" 32Mlog.(i) ^^^^ 

These arcs being small, their lengths will not differ sensibly from 
their sines, whence if a and b are given in degrees, the lengths of 
these arcs will be 0.0174533 a and 0.0174533 b, and M = 2.302585, 
these values being substituted for a, b, and M, equation (15) will be- 

'^'^"^ mmio^^og^lg. by ""'^ "'^'^'^^ logarithm,, we 
finally, haye log. c = ^log. n' + log. {a + b) + log. {a — b)] — 
. JC. L.*53i83611 + log. (log. a — log. b). (16) 



.:!■ 



158 



INTHODL'CTIOX. 



To apply lliis to practict let u 
marked E, and we have o=P.31 and 6=1' 
a + 6 = 2.30 log. 

n" = 86056.47 log. 



Sura 

Constant ]ogaritlim 
Log. a = l''.2I . 0.082785 
6 = ] .09 . 0.037426 



Diff. 



0.045359 log. 2.6566a') 



™CB) 




(A — B)=log. c = 2*.165. 

Hence n = n' -j- c = 86056.47 + 2.165 := 86058.635. Capt*.^ 
Kater thinking %his an unnecessary reftnementiu practice, multiplier 
the square of the mean arc by 1.638 Table (A) ; thus 1.15 X 1-15 
Xl.ra8 = 2'.166 nearly the same as before; and, by select ing the 
proper numberj this is sufficiently correct for almost any purpose, 






iichn 



If the length of a pendulum oscillating seconds of mean time ui 
one place or point on the earth's surface be knoivn, its length at 
another place, where the same invariable pendulum makes a dJAer- 
ent number of vibriitions, may readily be found- For if / be the 
length at the first place, I' that at the serand, v the number of vibra- 
tions at the first place in 24 hours, and r' that at the second, then as 
is shown by writers on mechanics,* I: I' : :v^ :v'^ . (17) 

consequently if three of tliese be known the fourth may be foinid. 

As this is rather laborious, an approximate rule may be obtained 
sufficiently correct for most purposes where the difference of osdlla- 
tigns does not exceed 30 or 40, or in an arc of five or si> degrees. 
If AXi represent a small variation of the length of the pendulum, 

and A N that in the number of oscillations, then A L, — — :-^iv-' 

, andAN = ^^-t. .... (18) 

Let 3 L be the variation of L for one degree of Fahrenheit's ther- 

' mometer, and n the number of degrees of change of temperature, 

[ ttr this then A L = n J L X L, and A N = i N h S L (i9) 

^ .Since the variation of braas from expansion is nearly 0.00001 inch 

r 1- Fah. A N = 0.432 », and A L = -^^^^ (20) 



"100000 
ExAstPLj: I. 
,, Captain Kater found the experimental per 
:_ !.•:•. ..!„ ^.i' ■II' O" VT oofi^^ Ro :ii.>: 



n made at Loudon 

IIb latitude bV 31' 8' N. 86061.52 oscillatious at 6S° Fah. in a mefto 
solar day, while at Unst in latitude 60° 45' 28" N., it made 860tHl.eD 
oscillations in the same time ; require<l the length of the pendulum at 
Unst, that at London being 3!>.13929 inches P 



m 



RULES AND l«X)KMUI.yli:. J59 

Here 86096.90 -- 86061.52 = 36.38 = A N. Now A L = i^r4^ 

formula (18) = — g^^^ — = 0.03217, consequently 39.13929 

+ 0.03217 = 39.17146 inches, the length at Unst. 

Ex, 2.— Captain Hall found an experimental pendulum^ making 
86235.98 OBCiUations at London at 62° Fah., made 86101.34 osdlla- 
tions at Glalapagos at the temperature of 68°. Hence from the num- 
ber of oscillations at London (since 68° — 62°=6°,) we must subtract 
(formula 20) 0.432 X 6 = 2.59 oscillations from that at London, which 
becomes 86233.39. 

Now by formula (17), as the places are very distant, r^ :«'*:: /: 
t : : 39.13929 : 39.01951, the length of the pendulum at Galapagos. 

Of late the figure of the earth has b^en determined with great ac- 
curacy by means of the pendulum. It is demonstrated by the 
theory or gravitation, that the length of the pendulum is augmented 
from the eauator to the pole, proportionally to the square of the 
me of the latitade, in such a manner that if the length of the equa- 
torial pendulum be represented by z, and its absolute variation from 
the'eouBtor to the pole by ^, then /, its length in any other latitude, 
L will be represented by the following equation : — 

I = 2 + y sin.* L . (1) 

If we have two equations of this form, in which / and L are de- 
termined by observation, we can obtain the values of z and ^. 

/ = z + ^ sin.* L 
/'=:z+^8in.« L' 

hence j^ = sin. (L' + L)~sin. (TTHL) ' (2) 

Andz = / — y sin.* L • . (3) 

Consequently — represents the diminution of gravity from the pole 

to the equator. 

Now by the doctrine of central forces if ^ denote the centrifugal 
force ; at the circumference of a circle to diameter unity ; r the ra- 
dius of the given circle in which a body revolves ; t the time of re- 

4t * r 
volution^ and g the gravitating force, then/= — . But by the 

theory of the pendulum, if/ is its length, ^ == x-* / ; hence 

/=757=(Tjr/ • (4) 

The ratio of the centrifugal force to gravity may be expressed by 

Y^r~h} and the ellipticity of the meridian or flattening of thie earth 

is from theory equal to f * of the ratio of the centrifugal force to 
ffravity, diminished by the fraction obtained from dividing the dif- 
Serence of the lengths of the pendulum at the pole and equator by 
its length at the equator. Wherefore if i denote the ellipticity. 



* This fraction is obtained by approximation, and is not perfectly correct By tak- 
ing in the quantities of the ieeond order, the d^fnidty would Tsry about g^- ftom the 
first approximation. It is difficult to solve the equations involving these. ^^SL^Wn - 
ever, no error should be allowed, if possible, to affect t\ve ^t\«\ Te«.\3\\.^^ 'W^.^V-aX -^xv- 
avoidably belong to the observations. 



INTRODUCTION. 



, _ , ,, / 11 

By substituting the value of/ from equation (4) 



^ 



As / in these investigation a denotes the time which the earth takes 
to perform a rotation about its axis, or 86 164". 0908 ; i (' = 
1856062632, r, the radius of the equator, is 20918750 feet, /, the 
length of the equatorial pendulum bv numerous obserTBtions, ia 
, gj.^^^^ and.y = 0.?O712 inch. 



t 39.013 inchi 
Whence i — 0. 

By combining 
found I = 0.0033; 



(6) 
great number of tlic beat observations I have 

compute the length of the 



"300 



From these we may get a formula 
■ pendulum at any latitude. 
' -Commencing at the equator I =. 39.013+0.20712 sin." L . (A) 
'Settingoutfrom46", /=: 39.11656 — 0.10356 cos, ^ L . (B) 

Ex. — Required the length of the pendulum at Leith, in latitude 
65= 58* 39" N. ? 

nj.— 39.1555 inches. 
' - «nce g =■■ ■ / = 32.2 feet. 

Hence the length of the pendulum and force of gravity may be 
found at any latitude. 

But the force of gi-avity may be found more readily by a particu- 
lar formula for that purpose. 

Since g is equal to 32.172 feet, or 9.6058 metres at 45", then G at 
any other latitude will be 

G=ff (1—0.00268 cos.« L) . (?) 

Or = 32.172(1—0.00268 COS.* L) in feet. 
Let L be the length of the sesagesimal pendulum and / that of the 
French decimal- metrical pendulum, then 

1 = 52.74079/ ... (8) 

of Sir George Shuckburgh's scale, 

. or L = 52.740564/ (9) 

of Bird's Parliamentary Standard of 1758. 

~ et y be the velocity of sound at 30 inches of the English barome- 
._., 60' of Fahrenheit's thermometer and 14" of Mr Goldinghain's 
hygrometer which he used at Madras, also let * be the change of 
I ,|relocity for a variation of one inch of the English barometer, fi for 
I 'that of one degree of Fahrenheit's thermometer, y that for one de- 
\ a/rte of Mr G's hygrometer, « the velocity of the wind, iind * the 
I ^gle which the direction of the wind makes with that of the sound, 
I'iuid V the true velocity under given circumstances, then 
fy = v+^W~p) + fi (f-t) + y 0,' -/,) + . COS. ? (10) 

r "in which ;» = 30 inches, ( = 60° Fah. h = 14° hygrometer, and p', I' 
I ind A', the observed states of the barometer, tliermometerj and hy- 
I'grometer, respectively, 

I From an examination of Mr Goldinghara'a experiments at Madras, 
■ fl have found « = 18.8 feet, ^ = 1.14 feet, and y = 2.87 feet The 
^, values of* and f not being stated in any set of experiments which 
I have seen, have not been exactly verified. They must be known, 
bowe\er, at the time of computing the velocity as they undoubtedly 
it0ect it. Without these it becomes 
V=UOO+ J88(// — W) + l,\'l(,l'— 'JOPH^^XC*' — 14°J (IJJ 



KULES AND FOBMULiC:. 161 

Required the velocity at Port Bowen, the Bar. being at 30.398 in. 
Fahrenheit's Ther. — £i8°.5., the state of the hygrometer^ and veloci- 
ty and exact direction of the wind being unknown ? 

Ans, — ^995.19, differing about 19 feet from observation from want 
of the other parts of the data. 

Or, if V be the velocity, t the temperature^ f the elastic force of 
vapour by Dalton's table for the dew pointy obtained by Danieirs 
hygrometer, or otherwise by formula, page 53, p the barometric 
pressure, X the latitude of the place of observation, and « cos. p the 
same as before, 

V= {104.0885 + 0.10831 (< — 32°)} (l +_/_.J (10.2738 — 

0.01378 COS. 2 ;i) + « cos. <p, in English feet. . (12) 

Ex,— On the 19th of July, 1826, in mean latitude M^ N., longi- 
tude 3° lO' W., several experiments were tried on the velocity of 
sound, when the guns on Edinburgh Castle were fired in honour of 
his Majesty's coronation. They were made on the coast of Fife at 
the distance of 42546 feet, the barometer standing at 29.96 inches, 
th& thermometer at 72^9 the dew point by Danielfs hygrometer, or 
by a thermometer, having its bulb moistened with tissue paper, (piwe 
53) at 66°, the velocity of the wind by an anemometer was 15 mi&i 
per hour, or 22 feet in a second, making an angle of 60° with that 
of the sound ; required the true rate per second and the difference 
between theory and experiment, when the arithmetical mean of a 
number of experiments gives 37*448 seconds for the time elapsed be^ 
tween seeing the flash and hearing the report ?* 

V = {104.0885 + 4.3324] (l + -^^) (10.2738+ 0.1136) + 
22 X 0.5 = 108.4209 X 1.004 xia3874+ 11= • 1141.715 

42546 

Experiment gives = . , , 1136.189 

37.448 

Difference . . . . + 5.526 

or excess of the formula. 

In a river or open canal, let v be the velocity of the stream mea- 
sured by the inches it moves over in a second of time ; r a constant 
quantity, called the radius of the section, and obtained bv dividing 
me area of the transverse section of the stream expressea in square 
inches by the boundary or perimeter of that section, diminished by 
the superficial breadth of the stream expressed in linear indies. 
Also let A be the length of an open canal or of a close pipe ; ^ the 
difference of the level of its extremities, d the diameter in the case 
of a pipe, h the height of the water in the reservoir above the , 
upper orifice of the pipe, and h' the height above the lower orifice, 
at which the water stands in the cistern into which it is emptied. 

Now let — = t or the sine of inclination and = k. ■ 

The formula for the velocity of water in pipes, per second, will be 

V =5 J32806.6 dk + 0.023751 J* — ai54113 . (13) 



* If a seriM of expariments aze made by a gun at each end of the measured baae, the 
geometrical means of the times should be taketti See BuileHn de SfMmu for 1836. 

0^1 



ffte IMTROnUtTION. 



.^^^=0.0(m2=k, therefore 

t>=)32806.6d A +0.0207511*— 0.154113=46.9 inches the velocity 
per second. 

In rivers anil other canals, the formula is 
ii=5:{328G6.6ri+0.023751!i— 0.154113 . . . (14) 

These formula have been simplified, and are tolerably correct. 

Suppose V, d, 3, aiid A, are all expressed in feet, 

t=60J — h nearly the velocity in feet, per second. . (15) 
Let D be the discharge per minute in cubic feet, then 

/"\i ... 



"•C^^)* 



(■«) 

Ta find the fall in a river caused by obstruction, such as the piers 
of abridge, &c. 

Let V be the velocity of the stream in feet per second, b the whole 
breadth of the channel in feet, c the contracted breadth between the 
obstacles, and/ the fall, then 
- (/25 6\" , 1 t)« 1.42 6"— c« , , 

■^={(2n)-i}64=~M7^- ^"^"y"^"^'' - ^'^^ 

Let, as is nearly the case with the old London Bridge, 
i.=3i, 6=926, c=200, 

Hence/=i:^|^-X««=0.46xl03^=4.73 feet, or 4 ft. 8\ 

iocbeq \>y the formula, while that by experiment was 4 feet 9 inches. 



TO FIND THE TONNAGE O 



MNON METHOD. 



Rule. — If the vessel is a ship of war, let fall a perpendicular fnHn 
the fore-side of the stem, at the height of the hause holes ; but if a 
merchantman, the perpendicular ii to be let fnll from that part of 
the fore-side of the stem which ia at the Bame height above the Iteel, 
as the wing transom: also let fall another perpendicular from the 
back of the main post, at the height of the wing transom. Ffsd 
the distance between these two perpendiculars, from which subtract 
three-fifths of the extreme breadth ; and also, the product of the 
height of the wing transom above the upper edge of the keel, by 2^ 



inches, and the remainder is the length of the keel for tonnage, 
the logarithm of which, add the logarithm of the breadth, and tti 
the half-breadth, and the constant logarithm 8.02687 i* 



jecting 10 from the indc.v, will be the logarithm of the tonnage re- 
quired. 

Ex. — Let the length between the perpendicular at the fore-part of 
the stem, and the back of the post, belOOfeet: the extreme breadth 
27i feet, and the height of the wing transom 15 feet. Required the 
tonnage? — Aut. 321 tons. 

* The orithmeliul compkinciit of the luuuilhni oT 114, being the conimoD diriMt 
fur flnding (be tuiiMKe- Thf> mnbod U fu frran bciag coirecl. .See papcra on A'aua/ 
^rrl,il,i4,tri; pMMui by Mnipin »nd Creiia-, (1, B. MTiittBlifr, Jxindon. IBM, 



163 



TABLES OF SPECIFIC GRAVITY. 



SOLIDS. 



Platina . . 20722 

Gold, pure^ hammered 19362 

Guinea of George III. 17.629 

Tungsten . . 17.600 

Mercury, at 32^ Fahren. 13.598 

Lead . 11.352 

Palladium 11.300 

Rhodium 11.000 

Virgin Silver 10.744 

Shilling of George III. 10.534 

Bismum, molten 9.822 

Copper, wire-drawn 8.878 

Red Ccmper, molten 8.788 

Molybdena 8.611 

Arsenic . . 8.308 

Nickel, molten 8.279 

Uranium 8.109 
Steel from 7769 to 7 816 

Cobalt, molten 7812 

Bar Iron . . 7.788 

Pure Cornish Tin . 7291 

Ditto hardened 7299 

Cast Iron 7.207 

Zinc 6.862 

Antimony 6,712 

Tellurium 6.115 

Chromium 5.900 

Spar, heavy 4430 

Jargon of Ceylon . 4.416 

Oriental Rub^ 1 4.283 

Sapphire, Oriental 3.994 

Ditto Brazilian 3.131 

Oriental Topaz 4019 

Oriental Beryl 3.549 
Diamond . from 3501 to 3.531 

English Flint Glass 3.329 

Tourmalin 3.155 

Asbestus 2.996 



Marble, green Campanian 
, Parian 



-, Norwegian . 
-, green Egyptian 



Emerald 

Pearl 

Chalk, British 

Jasper 

Coral 

Rock Crystal . 

English Pebble 

Limpid Feldspar 

Glass, green 

, white 



-, bottle 



Porcelaine, China 

, Limoges 

Native Sulphur 

Ivory 

Alabaster, 

Alum 

Copal, opaque 

Sodium 

Oak, heart of 

Ice . 

Potassium 

Deech 

Ash . 

Apple-Tree 

Orange-Wood 

Pear-Tree 

Linden-Tree 

Cypress 

Cedar 

Fir . 

Poplar 

Cork 



Sulphuric Acid 
Nitrous Acid . 
Water from the Dead 
Nitric Acid 
Sea- Water 
Milk 

Distilled Water 
Wine €vf Bourdeaux 



LIQUIDS. 

1.841 
1.550 
Sea 1.240 
1.218 
1.026 
1.030 
1.000 
944 



Burgundy Wine 
Olive Oil . 
Muriatic Ether 
Oil of Turpentine 
Liquid Bitumen 
Alcohol, absolute 
Sulphuric Ether 
Air at the Earth's sur 



2.742 

2.837 
2.728 
2.668 
2.775 
2.752 
2.784 
2.710 
2.680 
2.653 
2.619 
2.564 
2.642 
2892 
2733 
2.385 
2.341 
2.033 

1.917 

1.874 

1.720 

M40 
973 
950 
930 
866 
852 
845 
793 
705 
661 
604 
598 
561 
550 
383 
240 J 



991 
915 
874 
870 
848 
792 
7I6 
about If 



1. Since a cubic foot of wAter, at the temperature of 40° Fahren- 
hdtj ireighs 1000 ounces avoirdupois, or 62i pounds, the niimbers 
in the preceding tables, omittifag the dechml points, exhibit ^r&t^ 



INTllODUCTION. 

nearly the reapective weights of a cubic fool of the several Bubitancet 
in avoirdupois ounces. 

2. If the weight of a body be known in avoirdupois ounces, its 
weight ill Troy ounces will be found in multiplying it into -911^. 
And, if the weight be given in Troy ounces, it will be found in 
avoirdupois by multiplying it into I'OS?!' 



Atmospheric air' 
^^apour of hydriotic ether 
il of turpentine 



Hydriotic acid-gas 
^tuo-silicic acid-gas . 
V^apour of sulph. of carbon 

sulphuric ether 

::hlorine . 
Fluo-boric gas . 
Capour of muriatic ether 
Sulphurous acid-gas . 
"yanogen , 

t'apour of absolute alcohol 
Nitrous oxide . 
bonic acid 



10000 
5-4749 
5 0130 
4-4430 
3.5735 
26447 
2-5860 
2-4700 
2-3709 
2-2190 
2-1920 
l-8(Hi4 
1-6133 
1-5204 
151!)6 



Muriatic acid-gas 
Sulphuretted hydrogen 
Oxygen -gas 
Nitrous -gas 
Olefiant-gas 
Azote, or nitrogen-gas 
Oxide of carbon 
Hydro- cyanic vapour 
Phosphu retted hydrogen 
Steam of water 
Aranioniacal -gas 
Carburetted hydrogen 
Arseniated hydrogen 
Hydrogen -gas . 



1-2474 
11912 
M03e 



0.9t 
t)-9l 
0-9476 
O-87O0 

or— 

0-5967 
0-5550 
0-5290 
0073S 



, hence Gas S. G. x 0'[)012I = S. G. 



: gravilif of Distilled WaUr at different tcmperalures, thai at 
62° being lakeii as unili/. 

'llJoi0273Fj 



1.00000 
100018 
100035 

1-00O50 



1-00064 
1-00076 

1 ■00087 
1-00095 



1-00107 36 

loom 38 

1-00113 40 I 



MISCELLANEOUS C' 



UTATIONB AND £XP£ItIUENTH. 



The pendulum vibrating seconds of mean solar time at London' 

a vacuum, and reduced to the level of the sea, is39'13g3 inches 

I Cequently the descent of a heavy body from rest in one second of 

I fame, in a vacuum, will be 193.145 inches. The logarithm 2-2858828. 

A platina metre at the temperature of 32°, supposed to be the ten 

millionth part of the quadrant of the meridian, 39 3708 inches. The 

J-atio to the imperial measure of three feet, as 109363 to 1, the loga- 

[ Tithm 00388717- 

The following stindards, accurately measured, give these resulta; 
Sen. Lambton's scale, used in the Trig. Surv. of India, 35-99934 inc? 
Sir G. Shuckburgh'a scale (which for all purposes \ 
may be considered aa identical with the impe- V35-99998 
rial standard) . . . . ) 

I iGen. Roy-B scale .... 3600088 

I (Royal Society's standard 36-(K)135 

\ Barasden's bar ... . 3600249 

1 Weight of a cubic inch of distilled water in a va- \ 

cuum at the temp. 62°, as opuMed to brass Mog, 2-40S6430< 
weights in a vacuum also, 253-723 grains ) 






BULES AND FOBMVIJE. 



196 



'"|log. l-78fi0887 
mair.tl 
t of the Vlog. S 



;. 2.4081857 



ConKquently a cubic foot 62*38^ pounds ■ 

dupois .... 
Wdght of B cubic inch of diitilled water in 

6^ of temperature with a mean height a 

barometer 252.456 grains 
Consefjuently a cubic foot 62'3862 pound* avoir. 1, , 

dupois - • i 
And an ounce of water 1-73298 cubic inches log. 0-2387924 
Cubic inches in the imperial gallon, 277-276 log. 2-4429] 24 
Diameter of thecylindercontainingagallonatonel, , n-onno 

inch high, 1878933 /'"K- ^ -^^^^ 

SPECIFIC GHAVITY OF DEY AND SATURATED AIB. 

That M 30 in. Bv., aitd 32* F^. being I. 



g!t 


■iSSS;:- 


bss.. 




SpKlBc Gnv. 
of D.J Air. 


SjHsUlcGnT. 

o/siUintAJi. 


3ii» 


1.00000 


0.99750 


Ir 


0.93996 


0.93164 


33 


0.99824 


0.99568 


68 


0.9.3829 


092968 


34 


0.99647 


0.99385 


69 


0.93664 


0.92772 


85 


0.99471 


099203 


70 


0.93499 


0.82676 


36 


0.89294 


0.99021 


71 


093333 


0.92380 


37 


0.99119 


098839 


72 


0.93168 


0.92184 


38 


0.98944 


0,98664 


73 


093004 


0.91988 


39 


0.98769 


0.98470 


74 


092839 


0.91702 


40 


0.98595 


0.98286 


75 


092675 


0.91596 


41 


0.98420 


0.98101 


76 


0.92511 


691400 


42 


0.98246 


0.97917 


77 


0.92347 


0.91203 


43 


0.98073 


0.97731 


78 


0.92184 


0.91006 


44 


0.97900 


0.97645 


79 


0.92021 


0.90811 


46 


0.97726 


0.97358 


80 


0.91859 


0.90609 


46 


0.97563 


a97172 


81 


0.91656 


0.90411 


47 


0.97381 


0.96986 


82 


0.91634 


0.90213 


48 


097209 


0.96798 


83 


0.91373 


0.90013 


49 


097038 


0.96610 


84 


0.9J211 


0.89814 


50 


0.96866 


0.96421 


85 


0.91060 


0.89615 


51 


0.96695 


0.96233 


86 


0.90889 


0.89416 


52 


0.96524 


0.96045 


m 


0.90728 


0.89216 


53 


0.96354 


0.95855 


88 


0.90667 


0.89014 


54 


0.96183 


0.96666 


89 


0.90408 


0.88813 


55 


0.96013 


0.95475 


90 


0.90248 


0.88611 


56 


096843 


0.96286 


91 


0.90089 


0.88410 


67 


095674 


0.96095 


92 


0.89929 


0.88208 


58 


0.95504 


694902 


93 


0.89770 


0.88006 


59 


0.96336 


0.94710 


94 


0.89612 


0.87803 


60 


0.95168 


0.94518 


95 


0.89453 


0.87602 


61 


0.94999 


094326 


96 


089296 


0.87401 


62 


0.94831 


094134 


97 


0.89137 


0.87199 


63 


0.94864 


0.93940 


98 


088979 


0.86995 


61 


094496 


0.93746 


99 


0.88821 


0.86790 


66 


0.94.329 


093552 


100 


0.88664 


0.86586 


66 


094162 


0.93338 


110 


0.87110 


0.84329 



On Hub aubject aec BM't Traili de Phgtigm, voL 1., i^ xi 



^^B-Me 


INTRODUCTION. ^ 


■ 


^^M EXPANSIONS Oy 


SOLIDS, AND LIQUIDS AT DIFFEUEH^^I 


^H 


TEMPERATUHES, prom 32" TO 212° Fah. 


1 


^H 


Glass tube, linear . 1.000822 


^^^H 


Plate gla 


a, . . 1.000878 






Deal. 


1.000808 




^^^H 


Platina, 


1.000911 


^^^1 




Cast iron 


1,001110 




^^^H 


Steel, 


1001213 


^^^1 


^^^H 


Iron, 


1.001249 


^^^1 


^^^H 


Gold, 


1.001458 


^^^1 


^^^H 


Copper, 


1.001 7^6 


^^^1 


^^B 


Brass, 


1.001873 


^^1 


^^^ 


Silver, 
Tin, 


1.002003 
1,002372 


■ 




Lead, 


1.002858 




^^^^P 


Zinc, 


1.002976 


^^H 


^^^H 


Mercury 


volume, . . 1.018100 


^^1 




Water, 


1.0446b0 




^^^^P 


Alcohol, 


1.1050OO 


^^H 


^^^B 


Fixed Oils, . 1.075000 


^^1 


^^V TABLE FOR COMPUTING THE FLKXIBU.ITY AND STUKNGTH 


^H 




OF TIMBEB.* 






1 


«^.of,h..L.d.f 


rv'. 


Vfllue 


VahieofE, 


V^....f V.,..„f 


V^utof 




Teak . . . 


745 


818 


9657802 


2462 ': 2488 


15550 


^^^M 


Poon . . . 


579 


596 


6759^00 


2221 


2266 


14707 






Eng. Oak . . 


969 


598 


3494730 


1181 


1205 


9836 




^M 


Do. Sp 
Canac6 


ec. 2. . 


934 
872 


435 
588 


5806200 
8595864 


1672 
1766 


1736 
1803 


10853 
11428 




in Oak 


^^^H 


Dantzic Oak . 


750 


724 


47S5750 


1457 


1477 


7386 






Adriatic Oak . 


993 


610 


3805700 


1583 


1409 


8808 




^^^H 


Ash ... . 


760 


395 


6580750 


2026 


2124 


17337 




^^^V 


Beech . . . 


696 


6J5 


5417266 


1556 


1586 


9912 




^^^ 


Elm ... . 


553 


509 


2799347 


1013 


1042 


5767 




' 


Pitch Pine , . 


tJ60 


688 


4900466 


1632 


1606 


10415 




!'"" 


Red Pine . . 


657 


605 


7359700 


1341 


1368 


10000 






New Eng. Fir 


553 


757 


5967400 


1102 


1116 


9947 




^^^g 


Riga Fir . . 


753 


588 


5314570 


1108 


1131 


10707 




^^^H 


Do. Spec. 2. . 


738 





3962800 


1051 


1081 








Mar Forest Fir 


696 


588 


2581400 


1144 


um 


9539 




^^^H 


Do. Spec. 3. . 


693 


403 


3478328 


1262 


310 


10691 




^^^H 


Larch . . . 


53] 


411 


2465433 


653 


890 








Do. Spec. 2. . 
Do. Spec. 3. . 


522 

556 


518 
518 


3591133 
4210830 


832 
1127 


850 
1149 






^^H 


765,5 




■ 


Do. Spec. 4. . 
Norway Spar . 


560 
577 


518 
648 


4210830 
5832000 


1149 


1172 


7352 
12180 




1474 


1492 


■ 


• I 


roui liiidow OD lilt Slren(,-ih of Timber. 






^^^^ 


^^^^^^k 




■ 



RULES AND FOSMULiE. 107 « 

• * ■ 

Solution of Practical Problems, from the preceding Data. 

Pbob. I. — Tojind the Strength of Direct Cohesion of a Piece of Tim- 
ber of any given Dimensions, 

Rule^ — ^Multiply the area of the transverse section, in inches, by 
the value of C, in the preceding table of data, and the product wiu 
be the strength required. 

^o/^. — If the specific gravity be not the same as the mean tabular 
specific gravity ; say, as the latter is to the former, so is the above 
product to the correct result. 

Ex. — ^What weight will it require to tear asunder a piece of teak 
3 inches square, the specific gravity being ^45 ? — Ans, J 99*95 lbs. 

F^OB, II. — To compute the Deflection of Beams Jixed at one End and 

loaded at the other with any given Weight, 

Rule 1. — ^Multiply the tabular value of E by the breadth and cube 
of the depth of the given beam, both in inches. 

2. — Multiply also the cube of the length in inches by the given 
weight, and that product again by 32. 

3. — Divide the latter product by the former, for the deflection 
sought. 

Ex, — ^An ash batten, 3 inches square, is fixed in a wall^ and pro- 
jects from it 4 feet. If a weight of 200 lbs. be hung on its extre- 
mity, how much will it be deflected ? — Ans, IJ inches. 

Note. — The same rule will apply> when the weight is distributed 
throughout the length, by multiplying the second product by 12 in- 
stead of 32. 

pROB. III. — To compute the Deflection of Beams, supported at each 
End, and loaded in the Middle with any given Weight. 

Rule 1. — Multiply the tabular value of £ by the breadth and cube 
of the depth, both in inches. 

2. — Multiply also the cube of the length, in inches, by the given 
weight in lbs. ; then divide thp latter product by the former for the 
deflection sought. 

Ex, — A square beam of English oak, whose side is 6 inches, is 
supported on two walls, 20 feet distant, and is to be loaded at its 
middle point with 1000 lbs., what will it be deflected ? — Ans. 1*8 inch. 

Note. — If the beam be fxed at each end, the deflection will, with 
equal weights, be two-thirds of that found by the above rule. 

Pbob. IV. — To compute the Defection of Beams supported at each 
End, and loaded uniformly throughout their Length with a given 
Weight. 

Rule. — Compute the deflection the same as in the last problem. 
Multiply that result by 5, and divide the product by 8, and the quo- 
tient will be the answer. 

Ex. — A uniform bar of Adriatic oak, 2 inches square, is rested 
upon two props, distant 24 feet, how much will it be deflected by 
its own weight, its specific gravity being 9^, or 60 lbs. to the cubic 
foot ? — Ans. 9^ inches. 



168 INTKODUCTION. 

To apply Uiis to practice let us asaume Kater'ti 2th ex])ennient' 
inarkeJ E, and we havE a=:l''.2! and 8=1 ".09, whence ' ' ' 



fl + 6 = 2.3U log. 
a — fi = 0.121og. 

n' = 86056.47 log. 



Constant logarithm 
Log. a = 1°2I . 0.0827a'". 
6 = ] .09 . 0.037426 



0.361728 \ 
1.079181 J(A>' 
4.0^785 I 



4.04O274 (B) 



Diff. 0.045359 log. 2.656663 

Sum(B) . 4M0274 

0-335420 , 

(A — B)=log. c = 2M65. 

Hence n = n' + c= 86056.47 + 2.165 = 86058.635. Captwo' 
Kater thinking this an unnecessary refinement in practice, niiiltipliQ||' 
the square of the mean arc by 1.638 Table (A) ; thus 1.15 X 1 J§; 
X 1 .638 = 2M66 nearly the same as before ; and, by selecting ^^' 
proper number, this ia sufficiently correct for almost any purpose, 
and much more simple. , 

If the length of a pendulum oscillating seconds of mean time ^, 
one place or point on the earth's surface be known, its length «t 
another place, where the same invariable pendulum makes a differ- 
ent number of vibrations, may readily be found. For if / be the 
length at the first place, (' that at the second, ti the number of vibra- 
tions at the first place in 24 hours, and d' that at the second, then U 
is shown by writers on mechanics,' I: I' : :v^ :v'^ . (17) 

consequently if three of these be known the fourth may be found, 

As this ia rather laborious, an approximate rule may be obta' 
sufficiently correct for most purposes where the difference of osi 
tions does nut exceed 30 or 40, or in an arc of five or six deg 
If dL represent a smaU variation of the length of the pendu 



and A N that in the number of c 
ANAL 



;illations, tlien A L, - 



and AN = 2^-. . (1^'^ 

Let 3 L be the variation of L for one degree of Fahrenheit's tbo^ 1 
mometer, and n the number of degrees of change of temperaturt^ I 
for this then A L = a * L x L, and A N = ^ N « 3 L '"" ' 

Since the variation of brass from expansion is nearly O.OOOQl ii 

li>r 1- Fah. A N = 0.432 n, and A L = j^j^gjj 

Example I. 
Captain Kater found the experimental pendulum made at 
In latitude 51' 31' &" N. 86061.52 oscillatJons at 62° Fah. io 
solar day, while at Unst in latitude 60" 45' 38" N., it made 
oscillations in tlie same time; required the length uf thependali 
Unat, that at London being 3U,ia929 inches f 



•n 11., In ll 



:i fermulmimd ca 



RULES AND l«X)KMUI.y1i:. J59 

Here 86096.90 — 86061.52 = 36.38 = A N, Now A L = M.^ 

formula (18) = ~ 430484 5"~ " 0.03217, consequently 39.13929 

+ 0.03217 = 39.17146 inches, the length at Unst. 

Ex. 2.— -daptain Hall found an experimental pendulum, making 
86235.98 OBCiUations at London at 62"^ Fah., made 86101.34 osdlla- 
tions at Galapagos at the temperature of 68°. Hence from the num- 
ber of oscillations at London (since 68° — 62**=:6°,) we must subtract 
(formula 20) 0.432 X 6 = 2.59 oscillations fk*om that at London, which 
becomes 86233.39. 

Now by formula (17), «8 the places are very distant, v^ : v*^ ::l: 
I':: 39.13929 : 3901951, the length of the pendulum at Galapagos. 

Of late the figure of the earth has b^n determined with great ac- 
curacy by means of the pendulum. It is demonstrated by the 
theory or gravitation, that the length of the pendulum is augmented 
from the eauator to the pole, proportionally to the square of the 
me of the latitude, in such a manner that if the length of the equa- 
torial pendulum be represented by x, and its absolute variation n'om 
the eouBtor to the pole by v, then /, its length in any other latitude, 
L will be represented by the following equation : — 

1= 2 + y sin." L . . (1) 

If we have two equations of this form, in which / and L are de- 
termined by observation, we can obtain the values of z and tf. 

/ =: z -|- y sin.* L 
/' = 8 + ^sin.« L' 

^^""^^= sin. (L-+L)~in.(/--L ) ' (2) 

Andz = / — y sin.* L . (3\ 

Consequently — represents the diminution of gravity from the pole 

to the equator. 

Now by the doctrine of central forces it\/ denote the centrifugal 
force j AT the circumference of a circle to diameter unity ; r the ra- 
dius of the given circle in which a body revolves ; t the time of re- 

4»" * r 
volution, and g the gravitating force, then^* = — . But by the 

theory of the pendulum, if / is its length, g^zir^ I ; hence 
4 f r 

/=7«T"(J)M • • (4) 

The ratio of the centrifugal force to gravity may be expressed by 

Y-~->, and the ellipticity of the meridian or flattening of the earth 

is from theory equal to f * of the ratio of the centrifugal force to 
ffravity, diminished by me fraction obtained from dividing the dif- 
Serence of the lengths of thependulum at the pole and equator by 
Hs length at the equator. Wherefore if i denote the ellipticity. 



* This fraction is obtained by approximation, and is not perfectly correct By tak- 
ing in the quantities of the seccHid order, the eil^[>ticity would Tsry about g^. from the 
first approximation. It is difficult to solve the equations involving ihefte. ^XSQ^Ym^-^ . 
ever, no error should be allowed, if possible, to affect t\ve ftT\«\ Twro\\.^^ \i\3X^\«X xa^- 
avoidably belojif^ to the observations. 



^^V IC6 


INTRODUCTION. "^^^^J 


■ 


^^^1 EXPANSIONS OF 


SOLIDS, AND LIQUIDS AT DIFKEBENt 




TEMPKRATURES, fkom 32- to 212" Fah. 






Glass tube, linear . 1.0*10822 




Plate gia 


9, i.mo87a 






Deal, 


1.000808 






Platina, 


1.000911 






Cast iron 


1.001 110 






Steel, 
Iron, 


1001213 
1.001249 


^_ 




Gold, 


1.001438 


^^^H 




Copper, 


1.001796 


^^^H 




Brass, 


1.001873 


^^1 




Tir*"' 


1.002009 
1.002372 


■ 




Lead, 


1.002858 






Zinc, 


1.00297B 


^^^H 




Wercury 


volume, 1.018100 


^^^H 




Water, 


1.0446f)0 






Alcohol, 


1.105000 


^^^1 




FisedOih, . . 1.075000 


^^1 


^^H TABLE FOR COMPUTING THK FLEXIBILITY AND STUENGTB 






OF TIMBER.* 








"""•^"Cr"""' 


gr. 


'j'ijr 


Value or E, 


1 


VJ„., 




Teak . . . 


746 


818 


9667802 


2462 


2488 


15550 




Poon . . . 


579 


696 


67592(10 


2221 


^66 


14787 






Eng. Oak . . 


969 


698 


3494730 


1181 


1205 


aim 






Do. Spec. 2. . 
Canadian Oak 


934 


435 


5806200 


1672 


1736 


10863 






872 


688 


8595864 


1766 


1803 


11428 






Dantzic Oak . 


756 


724 


4765760 


1457 


1477 


7386 






Adriatic Oak . 


993 


610 


3HB5700 


1683 


1409 


8808 






Ash ... . 


760 


395 


6680750 


2026 


2124 


17337 






Beech . . . 


696 


615 


6417266 


1556 


1586 


9912 






Elm ... . 


653 


609 


2799347 


1013 


1042 


5767 






Pitch Pine . . 


660 


688 


4900466 


1632 


1666 


10415 






Red Pine . . 


657 


605 


7359700 


1341 


1368 


10000 






New Eng. Fir 


553 


757 


5967400 


1102 


1116 


9947 






Riga Fir . . 


753 


688 


5314670 


1108 


1131 


10707 






Do. Spec. 2. . 


738 




3962800 


1051 


1081 








Mar Forest Fir 


696 


588 


2581400 


1144 


1168 


9539 






Do. Spec. 2. . 


693 


403 


3478328 


1262 


310 


10691 






Lardi . ■ - 


631 


411 


24654:13 


653 


890 








Do. Spec. 2. . 


522 


518 


3591133 


832 


850 








Do. Spec. 3. . 


556 
560 


518 
518 


4210830 
4210830 


1127 
1149 


1149 
1172 


7655 
7352 




Do.gpec. 4. . 




Norwav Spar . 


677 


648 


5832000 I 1474 


1492 


12180 






• i 


raw Rwlow on tlic Strength 


jfTimbe 




_ 


■ 



RULES AND FORUVUE. l(fj i 

Solution of Practical Problems, from the preceding Data, 

Pbob. I. — To find the Strength of Direct Cohesion of a Piece of Tim- 
ber of any given Dimensions, 

Rule^ — ^Multiply the area of the transverse section, in inches,, by 
the value of C, in the preceding table of data, and the product wiU 
be the strength required. 

}fote, — If the specific gravity be not the same as the mean tabular 
specific gravity ; say, as the latter is to the former, so is the above 
product to the correct result. 

Ex, — ^What weight will it require to tear asunder a piece of teak 
3 inches square, the specific gravity being 745 ? — Ans. J 99*95 lbs. 

P^OB. II. — To compute the Deflection of Beams fixed at one End and 

loaded at the other with any given Weight, 

Rule 1. — Multiply the tabular value of E by the breadth and cube 
of the depth of the given beam, both in inches. 

2. — Multiply also the cube of the length in inches by the giveii 
weight, and that product again by 32. 

3. — Divide the latter product by the former, for the deflection 
sought. 

Ex, — An ash batten, 3 inchies square, is fixed in a wall, and pro- 
jects from it 4 feet. If a weight of 200 lbs. be hung on its extre- 
mity, how much will it be deflected? — Ans, 1} inches. 

Note, — The same rule will apply, when the weight is distributed 
throughout the length, by multiplying the second product by 12 in- 
stead of 32. 

Pbob. III. — To compute the Deflection of Beams, supported at each 
End, and loaded in the Middle with any given n eight. 

Rule 1. — Multiply the tabular value of £ by the breadth and cube 
of the depth, both in inches. 

2. — Multiply also the cube of the length, in inches, by the given 
weight in lbs. ; then divide thp latter product by the former for the 
deflection sought. 

Ex. — A square beam of English oak, whose side is 6 inches, is 
supported on two walls, 20 feet distant, and is to be loaded at its 
middle point with 1000 lbs., what will it be deflected ? — Ans, 1*8 inch. 

Note, — If the beam be fixed at each end, the deflection will, with 
equal weights, be two-thirds of that found by the above rule. 

Pbob. IV. — To compute the Deflection of Beams supported at each 
End, and loaded uniformly throughout their Length with a given 
Weight. 

Rule, — Compute the deflection the same as in the last problem. 
Multiply that result by 5, and divide the product by 8, and the quo- 
tient will be the answer. 

Ex, — A uniform bar of Adriatic oak, 2 inches square, is rested 
upon two props, distant 24 feet, how much will it be deflected by 
its own weight, its specific gravity being 9^, or 60 lbs. to the cubic 
foot ? — Ans. 9i inches. 



^^B MS 


INTRODUCTION. ''^^^^| 


■ 


^^H EXPANSIONS OF 


SOLIDS, AND LIQUIDS AT DIFFEKENT 




TEMPERATUHES, prom 32=> to 212° Fah. 


■ 


^H 


Glass tube, linear . 1.000822 


^^^H 


Plate gk 


s, . . 1.000878 






Deal, 


1.000808 




^^^H 


Platina, 


i.oooyii 


^^^H 


^^^H 


Cast iron 


1.001110 


^^^H 


^H 


Steel, 
Iron, 


1001213 
1.001249 


m 


^^^^M 


Gold, 


1.001458 




^^^P 


Copper, 


1.001796 


^^H 


^^H 


Brass, 


1.001873 


■ 


^^H 


Tin, 


1^002372 


S 


^^^^M 


Lead. 


1.002858 




^^^^P 


Zinc, 


1.002970 


^^H 


^^^^K 


Slerciiry 


volume, . ],0]8i00 


^^H 




Water, 


1.04461)0 




^^^^1 


Alcohol, 


1J06000 


^^H 


^^^B 


Fixed Oils, . . 1.075000 


^^1 


^^V TABLE FOR COMPUTING THE FLEXIBILITY AND SI 


IlENGTH 


^H 




OF TIMBEB.' 






1 


N..,.^.j,.w„aof 


?.^v 


SS 


..1.,.,. 


\.IUQ0f|\.1u.Df 


.*.» 




Teak . . . 


745 


818 


9667802 


2462 


2488 


15550 


^^^H 


Poon . . . 


579 


696 


6759200 


2221 


2266 


14787 




^^^H 


Eng. Oak . . 


969 


698 


3494730 


1181 


1205 


9836 




^^^H 


Do. Spec. 2. . 


934 


435 


5806200 


1672 


1736 


10853 






Canadian Oak 


872 


688 


8695864 


1766 


1803 


11428 




^^^m 


Dantzic Oak . 


756 


724 


4785760 


1457 


1477 


7386 






Adriatic Oak . 


093 


BIO 


3885700 


1583 


1409 


8808 




^^^H 


Ash ... . 


760 


385 


6680750 


2026 


2124 


17337 






Beech . . . 


696 


015 


5417266 


1556 


1586 


9912 




^^^1 


Elm ... . 


553 


509 


2799347 


1013 


1042 


5767 




^^^1 


Pitch Pine. . 


660 


688 


4000466 


1632 


1066 


10415 




^^^H 


Red Pine . . 


657 


805 


7359700 


1341 


1368 


10000 




^^^H 


New Ens. Fir 


553 


757 


5967400 


1102 


1116 


9947 




^^^H 


HigaFir . . 


753 


688 


5314570 


1108 


1131 


10707 






Do. Spec. 2. . 
Mar Forest Fir 


738 




3962800 


1051 


1081 






^^^H 


696 


588 


2681400 


1144 


1168 


0539 




^^^H 


Do. Spec. 2. . 


693 


403 


M78328 


1262 


310 


10691 






Lardi . . . 


631 


411 


2405433 


653 


890 






^^^H 


Do. Spec. 2. . 


522 


518 


3591133 


832 


850 







^^^H 


Do. Spec. 3. . 


556 


518 


4210830 


1127 


1149 


765,5 




^^^H 


Do. Spec. 4. . 


560 


618 


4210830 


1149 


1172 


7352 




^^^1 


Norway Spar . 


577 


648 


5832000 j 1474 


1492 


12180 




^H 


• i 


roin Barlow oa llw SidMikUi of TLnibci. 






^B _^ 






^ 


^1 



RULES AND TORUVUE. 107 i 

Solution of Practical Problems, from the preceding Data. 

Pbob. I- — To find the Strength of Direct Cohesion of a Piece of Tim- 
ber of any given Dimensions, 

Rule^ — ^Multiply the area of the transverse section, in inches,, bv 
the value of C, in the preceding table of data, and the product will 
be the strength required. 

Note. — If the specific gravity be not the same as the mean ti^bular 
specific gravity ; say, as the latter is to the former, so is the above 
product to the correct result. 

Ex. — ^What weight will it require to tear asunder a piece of teak 
3 inches square, the specific gravity being 745 ? — Ans. J 39*95 lbs. 

F^OB. II. — To compute the Deflection qf Beams fixed at one End and 

loaded at the other with any given Weight. 

Rule 1. — Multiply the tabular value of E by the breadth and cube 
of the depth of the given beam, both in inches. 

2. — Multiply also the cube of the length in inches by the giveii 
weight, and that product again by 32. 

3. — Divide the latter product by the former, for the deflection 
sought. 

Ex. — ^An ash batten, 3 inches square, is fixed in a wall, and pro- 
jects from it 4 feet. If a weight of 200 lbs. be hung on its extre- 
mity, how much will it be deflected? — Ans. IJ inches. 

Note.-r-The same rule will apply, when the weight is distributed 
throughout the length, by multiplying the second product by 12 in- 
9tead of 32. 

Pbob. III. — To compute the Deflection qf Beams, supported at each 
End, and loaded in the middle with any given weight. 

Rule 1. — Multiply the tabular value of E by the breadth and cube 
of the depth, both in inches. 

2. — Multiply also the cube of the length, in inches, by the given 
weight in lbs. ; then divide thp latter product by the former for the 
deflection sought. 

Ex. — A square beam of English oak, whose side is 6 inches, is 
supported on two walls, 20 feet distant, and is to be loaded at its 
middle point with 1000 lbs., what will it be deflected ? — Ans. 1*8 inch. 

Note. — If the beam be fixed at each end, the deflection will, with 
equal weights, be two-thirds of that found by the above rule. 

Pbob. IV. — To compute the Deflection qf Beams supported at each 
End, and loaded uniformly throughout their Length mith a given 
Weight. 

Rule. — Compute the deflection the same as in the last problem. 
Multiply that result by 5, and divide the product by 8, and the quo- 
tiait will be the answer. 

Ex. — ^A uniform bar of Adriatic oak, 2 inches square, is rested 
upon two props, distant 24 feet, how much will it be deflected by 
its own weight, its specific gravity being 960, or 60 lbs. to the cubic 
foot ? — Ans. 9^ inches. 



INTKODUCTION. 





Krant. 


Glass tube, linear 


1.060822 


Plate glass. 


1.000878 


Deal, 


1.000808 


Platina, 


1.000911 


Cast iron. 


I.OOUIO 


Steel, 


1 001213 


Iron, 


1.001249 


Gold, 


1.001458 




1.001796 


Brasa, 


1.001873 


Tin, 


1.002372 


Lead, 


1.002858 


Zinc, . 


1.0029715 


Blerciiry, volume. 


1.018100 


Water, 


1.04461)0 


Alcohol, 


J. 106000 


Pised Oils, 


1.075000 



"■°'-5.--^ 


<'E. 


ufL-r 


,-..,. 


Value or VJu. of 


"r" 


Te«k . . . 


745 


818 


9057802 


2402 24S8 


15550 


Pooii . . . 


579 


600 


6759200 


2221 2266 


14787 


Eng. Oak . . 


969 


690 


3494730 


1181 1205 


1/836 


Do. Spec. 2. . 


934 


435 


5806200 


1672 1730 


10853 


Canadian Oak 


872 


688 


8595804 


1766 ' 1803 


11428 


Dantzic Oak . 


750 


724 


4705760 


1457 1 1477 


7386 


Adriatic Oak . 




010 


3H85J00 


1583 


1409 


8808 


Ash ... . 


760 


395 


6580750 


2026 


2124 


17337 


Beech . . . 


696 


0J5 


6417206 


1566 


1586 


9912 


Elm ... . 


553 


609 


2799347 


1013 


1042 


5767 


Pitch Pine. . 


060 


688 


4900460 


1032 


1666 


10415 


Red Pine . . 


057 


605 


7369700 


1341 


1308 


10000 


New Ene. Fir 


553 


757 


5967400 


1102 


1116 


9947 


liisaFir . . 


753 


688 


6314570 


1108 


1131 


10707 


Do. Spec. 2. . 
Mar Forest Fir 


738 




;1902800 


1051 


1081 




696 


688 


2581400 


1144 


llfB 


9539 


Do, Specs. . 


693 


403 


3478320 


1262 


310 


1U691 


Lardi ■ . . 


53] 


411 


2405433 


ff-,3 


890 




Do. iJpec. 2. . 


522 


618 


3591133 


832 


850 





Do. 6pec. 3. . 


556 


518 


42108:10 


1127 


1149 


7655 


Do. Spec. 4. . 


560 


518 


4210830 


1J40 


1172 


7352 


Norway Spar . 


577 


648 


6832000 


1474 


1492 


12180 



' I'laui Jlulon aa llic Eitrenglti uf Tiuibei. 



RULES AND TOUUVUE. 107 

Solution of Practical Probleins,Jrom the preceding Data. 

Pbob. I- — To find the Strength of Direct Cohesion of a Piece of Tim- 
ber of any given Dimensions, 

Rule^ — ^Multiply the area of the transverse section, in inches, bv 
the value of C, in the preceding table of data, and the product will 
be the strength required. 

Note. — If the specific gravity be not the same as the mean tabular 
specific gravity ; say, as the latter is to the former, so is the above 
product to the correct result. 

Ex. — What weight will it require to tear asunder a piece of teak 
3 inches square, the specific gravity being 745 ? — Ans. J 39*95 lbs. 

PffOB. II. — To compute the Deflection qf Beams fixed at one End and 

loaded at the other with any given Weight. 

Rule 1. — Multiply the tabular value of E by the breadth and cube 
of the depth of the given beam, both in inches. 

2. — Multiply also the cube of the length in inches by the given 
weight, and that product again by 32. 

3. — Divide the latter product by the former, for the deflection 
sought. 

Ex. — ^An ash batten, 3 inches square, is fixed in a wall, and pro- 
jects from it 4 feet. If a weight of 200 lbs. be hung on its extre- 
mity, how much will it be deflected? — Ans. IJ inches. 

Note. — The same rule will apply, when the weight is distributed 
throughout the length, by multiplying the second product by 12 in- 
9tead of 32. 

Prob. III. — To compute the Deflection qf Beams, supported at each 
End, and loaded in the middle with any given Weight. 

Rule 1. — Multiply the tabular value of E by the breadth and cube 
of the depth, both in inches. 

2. — Multiply also the cube of the length, in inches, by the given 
weight in lbs. ; then divide thp latter product by the former for the 
deflection sought. 

Ex. — A square beam of English oak, whose side is 6 inches, is 
supported on two walls, 20 feet distant, and is to be loaded at its 
middle point with 1000 lbs., what will it be deflected ? — Ans, 1*8 inch. 

Note. — If the beam be fixed at each end, the deflection will, with 
equal weights, be two-thirds of that found by the above rule. 

Pbob. IV. — To compute the Deflection qf Beams supported at each 
End, and loaded uniformly throughout their Length with a given 
Weight. 

Rule, — Compute the deflection the same as in the last problem. 
Multiply that result by 5, and divide the product by 8, and the quo- 
tkBDt will be the answer. 

Ex, — A uniform bar of Adriatic oak, 2 inches square, is rested 
upon two props, distant 24 feet, how much will it be deflected by 
its own weight, its specific gravity being 960, or 60 lbs. to the cubic 
foot ? — Ans, 9^ inches. 



^^^V- U6 ^^^^^1 


^^H EXPANSIONS OF SOLIDS, AND LIQUIUS AT DlFrEUgj^^l 




TEMPERATURES, pbom 32= to 212° Fah. 


H 




Glass tube, linear . 1.000822 


■ 




Plate glass, . 1.000878 






Deal, . . 1.000808 






Platina, . . l.(K)09]l 


^^^1 




Cast iron, . . I.OOIUO 


^^^H 




Steel, . . . 1001213 


^^^1 




Iron, . . 1.001249 






Gold, . . 1.001458 


^^^1 




Copper, . . 1.0017y6 


^^H 




Brass, . . 1.001873 


^^1 




Silver, . 1.002002 






Tin, . . 1.002372 






Lead, . . 1.002858 


^^^M 




ZiDc, . 1.00297fi 


^^^H 




Mercury, volume, . 1.0)8100 


^^^1 




Water, 1.0446b0 


^^^1 




Alcohol, l.JOoOOO 






Fixed Oils, . . 1.076000 


^^1 


^^^ TABLE FOR COMPUTING THE FLESIBIUTY AND STllENGTH 




OF TIMBER.' 






""'"-'" "^ 


^Z- 


Value 


v.... 


V^ueof V.,u..,f 


nr"' 




Teak . . . 


745 


818 


9657802 


2462 2488 


15550 




Poon . . . 


579 


596 


6759200 


2221 


2266 


14787 






Eng. Oak . . 


969 


59» 


3494730 


1181 


1205 


9836 






Do. Spec. 2. . 


934 


435 


5806200 


1072 


1736 


10853 






Canadian Oak 


872 


588 


8595864 


1766 


1803 


11428 






Dantzic Oak . 


756 


724 


4765750 


1457 


1477 


7386 






Adriatic Oak . 


993 


610 


3805700 


1583 


1409 


8808 






Ash ... . 


760 


395 


6580750 


2026 


2124 


17337 






Beech - . . 


696 


615 


5417266 


1556 


1586 


9912 






Elm ... . 


553 


509 


2795)347 


1013 


104-2 


5767 






Pitch Pine . . 


660 


588 


4900466 


1632 


1666 


10413 






Red Pine . . 


657 


605 


7359700 


1341 


1368 


10000 






New Enff. Fir 


553 


757 


5967400 


1103 


1116 


9947 






Riga Fir . . 


753 


588 


5314570 


1108 


1131 


10707 






Do. Spec. 2. . 
Mar Forest Fir 


738 




3962800 


10.51 


1081 








696 


588 


2581400 


1144 


1168 


9539 






Do. Spec. 3. . 


693 


403 


3478328 


1262 


310 


10691 






Lard) . . . 
Do. Spec. 2. . 


53] 
622 


411 
51B 


2465433 
3591133 


653 


890 
850 














Do. Spec. 3. . 


556 


518 


4210830 


1127 


1149 


7655 






Do. Spec. 4. . 


560 


518 


4210830 


1149 


1172 


7352 






Norway Spar . 


577 


648 


5832000 1474 


I4fl2 


12180 






• i'roni I!»rlow on ihg SweDBlh of Timhet. 


■ 



RULES AND TOUUVUE. 107 i 

Solution of Practical Problems, from the preceding Data. 

Pbob. I- — Tojlnd the Strength of Direct Cohesion of a Piece of 7tm- 

ber of any given Dimensions, 

Rule, — ^Multiply the area of the transverse section, in inches, by 
the value of C, in the preceding table of data, and the product will 
be the strength required. 

Note. — If the specific gravity be not the same as the mean tabular 
specific gravity ; say, as the latter is to the former, so is the above 
product to the correct result. 

Ex. — ^What weight will it require to tear asunder a piece of teak 
3 inches square, the specific gravity being 745 ? — Ans. J 39*95 lbs. 

V^OB. II. — To compute the Deflection of Beams Jixed at one End and 

loaded at the other with any given Weight. 

Rule 1. — Multiply the tabular value of E by the breadth and cube 
of the depth of the given beam, both in inches. 

2. — Multiply also the cube of the length in inches by the given 
weight, and that product again by 32. 

3. — ^Divide the latter product by the former, for the deflection 
sought. 

Ex. — ^An ash batten, 3 inches square, is fixed in a wall, and pro- 
jects from it 4 feet. If a weight of 200 lbs. be hung on its extre- 
mity, how much will it be deflected ? — Ans, IJ inches. 

Note» — The same rule will apply, when the weight is distributed 
throughout the length, by multiplying the second product by 12 in- 
9tead of 32. 

Prob. III. — To compute the Deflection of Beams, supported at each 
End, and loaded in the middle with any given Weight. 

Rule 1. — Multiply the tabular value of E by the breadth and cube 
of the depth, both in inches. 

2. — Multiply also the cube of the length, in inches, by the given 
weight in lbs. ; then divide thp latter product by the former for the 
deflection sought. 

Ex. — A square beam of English oak, whose side is 6 inches, is 
supported on two walls, 20 feet distant, and is to be loaded at its 
middle point with 1000 lbs., what will it be deflected ? — Ans. 1*8 inch. 

Note, — If the beam be flxed at each end, the deflection will, with 

equal weights, be two-thirds of that found by the above rule. 

» 

Pbob. IV. — To compute the Deflection of Beams supported at each 
End, and loaded uniformly throughout their Length with a given 
Weight. 

Rule. — Compute the deflection the same as in the last problem. 
Multiply that result by 5, and divide the product by 8, and the quo- 
tkBDt will be the answer. 

Ex. — A uniform bar of Adriatic oak, 2 inches square, is rested 
upon two props, distant 24 feet, how much will it be deflected by 
its own weight, its specific gravity being 960, or 60 lbs. to the cubic 
foot ? — Ans. 9^ inches. 



INTRODUCTION. 

Prob. Y.~To compute Ike ultimnle Deflection of Beams or HuAi. 
before their Rupture. 

Note. — Tlie beams are supposed to be supported at each end. 

Rule. — Multiply the tabular value of U, in the preceding table 
of data, by the depth of the beam in inches, and divide the square 
of the length, also in inches, by that product, for the ultimate de- 
flection sought. 

Ex. — A square inch rod of ash, 6 feet long, is broken by a weight 
applied to its centre: how much will it be deflected before it breaks? 
Ans. 13'1 inches. 

Pbob. VI. — Tojind the ultimate transverse Strength of any rectangx- 
' lar Beam of Timber, Jixed at one End and loaded at the other. 
Rule I. — Multiply the value of S, in the preceding table of data, 
by the breadth and square of the depth, both in inches, and divide 
that product by the length, also in inches, and the quotient will be 
the weight in lbs. This is approximative. 

Rule II. — ]. Take the ultimate deflection 8 times that of the hist 
I problem, and divide the deflection by the length, which will give 
'he sine of the angle ; whence, by a table find the secant. 

2. Multiply the secant by the breadth and square of the depth in 
inches, and the product again by the value of S' in the table of data. 

3. Divide this last product by the length in inches, and the quo- 
tient will be the answer in lbs. 

Ex. 1. — What weight will it require to break a piece of Mar forest 
fir, fixed by one end in a wall, and loaded at the other ; the breadth 
being 2 inches, depth 3 inches, and length 4 feet ? — Ans. 518 lbs. 

Pros. VII. — To compute the uUimale transverse Strength of any reet' 
angular Beam, when supporledat both Endx and loaded in the Centre. 
Rule I. — Multiply the tabular value of S by 4 times the breadth 
and square of the depth in inches, and divide that product by the 
length, also in inches, for the weight. 

Rule II. — 1. Compute the uliimate deflection by Prob. V. ; snuare 
that deflection, and divide it by the square of half the length ot the 
beam, and add the quotient to 1, for Uie square of the secant of de- 
flection ; which multiply by the length in ihches. 
2. Multiply the tabular value of S' by 4 times the breadth, and 
I the square of^the depth; and divide that product by the former an- 

Ex. — What weight will be necessary to break a piece of larch si- 
I milar to the third specimen, the length being 8 feet 4 inches, the 
I breadth 8 inches, and depth 10 inches ; being supported at each end, 
I —■'" 'ed in the middle ?—^nj. 36676 lbs. 



MATHEMATICAL TABLES. 



TABLE I. 

THK MILB» ANB PABTS OF A lilliB IM A OSSUK Ot UmOITODB 

AT XVXBT DBOBXB OF LATITUDE. 



D.L. 


MUea. 


D.L. 


Miles. 


D.L. 


Milei. 


D.L. 


MOes. 


D.L. 


MUm.|D.L. 


Mikt. 


1 

% 
S 

4 
5 


59.99 
59.96 
59.92 

59.85 
59.77 


16 
17 
18 
19 
20 


57.67 
57.38 
57.06 
56.73 
56.38 


31 
32 
33 
34 
35 


51.48 
50.88 
50.39 
49.74 
49.15 


46 

47 
48 
49 
50 


41.68 
4a92 
40.15 
39.36 
38.57 


61 
^% 
63 
64 
65 


29.09 
2&17 
27.24 
26.30 
25.36 


76 
77 

78 
79 
80 


14.52 
13.50 
12.47 
11.45 
10.42 


6 
7 
8 
9 
10 


59.67 
59.55 
59.42 
59.26 
59.08 


21 
22 

23 
24 
25 


56.01 
55.63 
55.23 

54.81 
54.38 


36 
37 
38 
39 
40 


48.54 
47.92 
47.28 
46.63 
45.96 


51 
52 
53 
54 

55 


37.76 
36.94 
36.11 
35.27 
34.41 


M 
67 
68 
69 
70 


24.40 
23.U 

22.48 
21.50 
20.52 


81 

82 
83 
84 
85 


9.39 
835 
7.31 
6.27 
5.23 


11 
12 
13 
14 
15 


58.89 
5aG8 
58.46 
58.22 
57.95 


26 
27 
28 
29 
30 


53.93 
53.46 
52.97 
52.47 
51.96 


41 
42 
43 
44 
45 


45.28 
44.59 
43.88 
43.16 
42.43 


56 
57 
58 
59 
60 


33.55 
32.68 
31.80 
30.90 

3aoo 


71 
72 
73 

74 
75 


19.53 
18.54 
17.54 
16.54 
15.53 


86 
87 
88 
89 
90 


4.19 
&14 
2.00 
1.05 
0.00 



TABLE II. 



LOGARITHMS OF NUMBERS. 



Ha 1 100 Log. 0.000000 ^2.oUo66 1 


No. 


Log. 


Na 


Log. 


No. 


Lofr 


Na 


Lofr 


No. 


hog. 


1 

9 

4 
5 

6 
7 
8 
9 
10 


0.000000 
0.301030 
0.477121 
0.602060 
0.698970 


21 
fi 
23 
24 
25 


1.322219 
1.342423 
1.361728 
1.380211 
1.397940 


41 
42 
43 
44 

45 


1.612784 
1.623249 
1.633468 
1.643453 
1.653213 


61 
62 
63 
64 
65 


1.785330 
1.792392 
1.799341 
1.806180 
1.812913 


81 

82 
83 
84 
85' 


1.908485 
1.913814 
1.919078 
1.924279 
1.929419 


0.778151 
0.845098 
a903090 
0.954243 
1.000000 


26 
27 
28 
29 
30 


1.414973 
1.431364 
1.447158 
1.462398 
1.477121 


46 
47 
48 
49 
50 


1.662758 
1.672098 
1.681241 
1.690196 
1.698970 


66 
67 
68 
69 
70 


1.819544 
1.826075 
1.832509 
1.838849 
1.845098 


86 
87 
88 
89 
90 


1.934498 
1.939519 
1.944483 
1.949390 
1.954243 


11 
12 
13 
14 
15 

16 
17 
18 
1ft 
20 


1041393 
1.079181 
1.113943 
1.146128 
1.176091 


31 
32 
33 
34 
35 


1.491362 
1.505150 
1.518514 
1.531479 
1.544068 


51 
52 
53 
54 

55 


1.707570 
1.716003 
1.724276 
1.732394 
1.740363 


71 
72 
73 
74 

75 


1.851258 
1.857332 
1.863323 
1.869232 
1.875061 


91 
92 
93 
94 
95 


1.959041 
1.963788 
L968483 
1.973128 
1.977724 


1.204120 
1.230449 
1.255273 
1.27^754 
1.301030 


36 
37 
38 
39 
40 


1.556303 
1.568202 
1.579784 
1.59106& 
1.602060 


56 
57 
58 
59 
60 


1.748188 
1.755875 
1.763428 
1.770852 
1.778151 


76 
77 
78 
7ft 
80 


1.880814 
1.886491 
1.892095 
1.897627 
1.903090 


96 
97 
98 
99 
100 


1.982271 
L986772 
1.991226 
1.995635 
2.000000 













■2 


A Table of Ixigaiithms of Numbers from I Vt 100,000. 




¥77 


N. 1 1 


L * 


3 1 4 


5 


6 


7 


9 1 9 |D 








(100 UU 8 


001301 


001734 


002 16U 


002598 


003089 


00346 i;CH)3a9 


^ 






41 


1 438J 4751 


5181 


5609 


6038 


6466 


6S94 


7321 


774fi 


817 


42! 






82 


8 8600 SoiS 


9451 


9876 


01 0300 


010724 


011147 


011370 


011993 


016114I481 






lt\ 


30I«83T'oI3859 


013680 


014100 


4521 


4940 


5360 


5T79 


6197 


eeiBi48 






165 


4 70331 745! 


7866 


6884 


8700 


9116 


9338 


9947 


02O36I 


080775 


41 






(OS 


5 081I89;0!1603 


082016 


022*38 


088841 


083252 


0836S4 


084075 


44B6 


4896 


41 






HI 


6 5306 


5715 


61851 6533 


6942 


7350 


7757 


8164 


8571 


8078 


VM 






888 


7 9384 


9789 


03O!95;03O600 


031004 


D3140S> 


031818 


032816 


038610 




«H 






330 


S 0334S4 


0338?6 


4887 




5089 


5430 


5830 


6230 


6629 


70W 


Ud 






3TI 


9 7t«6 


788, 


8883 


8680 


B0I7 


9414 


9811 


040807 


010602 


04O998|M 






110 041393 


0417B7 


0481n2 


048576 


018969 


043368 


U4375S 


044148 




0*49»2{3OT 




SB 


1 i383 


5714 


6i05 


C495 


6885 


7875 


7664 


8053 


8442 


683(1 


39e 






Ti 


8 9818 


9606 


9993 


050380 


0507G6 


051153 


051538 


05192* 


058309 


Q58694|S« 






113 


3 053078 


053*63 


053846 


1830 


4613 


4B96 


5378 


5760 


G142 


e5!4J«! 






150 


4 6905 


7886 


7666 




8426 


B805 


9165 


9563 


9942 




S7S 






IBS 


5 0eo6D3 


n6]075 


061452 


061829 


082806 


068582 


068958 


063333 


063709 


4083|SJ 






ite 


6 1458 


48^8 


5806 


5580 


5B53 


6326 


6699 


7071 


7443 


7815|3)a 






863 


7 8188 


8557 


S928 


9298 


9666 


070038 


070407 




071145 


071514 


S7<l 






301 


8 071888 


0788SO 


078617 


078985 


073332 


3718 


4085 


4*51 


4816 


£168 


36a 






^ 


9 5547 


5918 


6876 


6640 


7004 


7368 


7731 


609* 


6437 


B8I9 


S03 






180[>7B181 


079543 


079904 


080866 


Oa06«<i 


080987 


081347 


OBI 707 


0H81H.-J 


08242ei3WI 




35 


1 088785 


083144 


083503 


3961 


4819 






5291 


56*7 


6004 


357 






69 


2 0360 


6716 


7071 


7426 


7781 


8136 


6490 


6845 


9196 


S668 


Si 






104 


3 9905 


09O85B 


090611 


090963 


091315 


091667 


092018 


098370 


092781 


093071 


35i 






138 


4 093488 


3772 


4182 






5169 


5518 


5866 


6814 


6568 


349 






173 


5 6910 


783? 


7604 


7951 


8898 


8644 


6990 


B335 


9681 


00086 


348 






iOB 


6 I0U37I 


100715 


101059 


101403 


101747 


102091 


102434 


102777 


103119 


3462 


3tS 






84g 


7 3804 




4*87 


4828 


5I0B 


3510 


5B51 


6191 


6531 


6871 


Ml 






ST7 


8 7810 


7549 


7888 


8887 


8365 


8903 


9841 


9579 


0916 


10253 


iSt 






311 


9110590 


110926 


111263 


111599 


1U934 


112270 


nS605;118B40 


113275 


3609 


33i 








130113943 


114877 


I146I1 


iXiSS 


115878 


115611 


lt5943|llU27HjlI660a 


I694U 


133 






38 


I 7871 


7603 


7934 


8265 


B595 


8988 


9856 9566 9915 


180844 


330 








2 180574 


180903 


181231 


121560 


181888 


182816 


122544182871 1S319H 


3484 


S»« 






9S 


3 3858 


4178 


4504 


4830 


5156 


5181 


S80B 6131 


6456 


6781 


vti 






128 


4 7105 


7489 


77S3 


8076 




8782 


9045 9369 


9690 


30012 


38; 






160 


513033* 


130655 


130977 


I3IS98 


131610 


131939 


138860132580 


138900 


3819 


Wl 






IBS 


6 S539 


3858 


4177 


4496 


4814 


5133 


54.W 5769 


6096 


6403 


318 






!!5 


7 6721 


7037 


7354 


7671 


7987 


8303 


8618 893* 


9849 


9564 


tv 






25T 


8 9879 


140194 


110508 140888 


141 136 


1*1451 


141763148076 


H8SS9 


48701 


SM 






389 


9 143015 


3327 


3639 3951 


4263 


♦574 


4885 5196 


6507 


5818 


Sit 






140146188 


146438 


14U74B 147058 


14736! 




l*7yB5 14889* 


148W13 


48911 


V» 




30 


1 9219 


9527 


9B35 150148 


1504*9 


1 50751 


151063 151370 


151676 


519B« 


WJ 






60 


2 158888 


158591 


158900 3805 


3510 




4180 4424 


4728 


5031 


S05 






00 


3 5336 


S640 


5943 6846 


6549 




7154 7*3 J 


7759 


b06l 


SO! 






ISO 


4 8362 


8664 


8SG5 9266 


9567 


9868 


160I6B 160469 


1607 69 


61068 


01 






14S 


5 I613BB 


161067 


161967162866 


162564 


168863 


3161 3460 


3758 


4054 


893 






17S 


6 4353 


4650 


49*7 5841 


55*1 


3838 


G134 6430 


6786 


7088 


197 






»0B 


7 7317 


7613 


79081 8803 


6497 


8798 


9086 93B0 


9674 


9968 


m 






839 


8 170868 


170555 


170848171111171434 


171726 


178019172311 


178603 


78S94 


a. 






gS9 


9| 3186 


3478 


3769! W60l 4351 


4641 


4938! 588* 


5512 


4801 


I9t 






140176091 


176381 


176670 1769591177248 


1775S6 


II78J5n8ll3[17B40l| 


786e« 


M 




i% 


1 . 697J 


9864 


9S52i 9639ll80126 


IB0413 


180699 


lB09B6llBliI2 




W7 






Sfl 


3 1B1844 


188189 


1BS41 5, 182701 


2985 


3270 


3555 


3839] 4123 


Utll 


tm 






64 


3 4691 


4975 


5859 5548 


5823 


6108 


6391 


6614 6956 


7839 


(ft 






lie 


4 7521 


780? 


808 4 8366 


6647 


8988 


S809 


9*90 9771 


90051 


eel 






139 


5190338 


10O618 


190R98:igll7l 


191451 


191730 


192010 


1921:89 198567 


S«4b 


71 






187 


6 3185 


3403 


3681 3959 


4837 


4514 


47B2 




53J0 


4683 


71 






IU5 


1 5900 




6153 678B 




7261 


7556 


7H32 


6107 


8388 


78 






8*3 


B 8657 


893* 


B80e! 9481 




80002B 


800a03 


800577 


800830 


01184 


7i 






841 


9 801^97 


801670 


801943 208816808488 


87HI 


3033 


3305 


3477 


■itm 


71 




Fr 


N. 


1 


< 1 3 I * 


i 


6 


^j~ 


H 


» 


>; 













P.>-IK. 





> 


2 


3 


1 


5 [ G 


_7_ 


9 1 9 ft 






16( 


soiuu 


iota^i 


804663 


aowai 


-05201 


805475 8057Mi 


206016 


2U6296 2U655u:7I 




2fi 




6886 


7091f 


7365 


7634 


7904 


8t73| hill 




8979, 9217 a 




SI 


i 


93I5 


9783 


210051 


210310 


310596 


210853211121 


ill388 


8I165121I9IIM 




IS 


3, 


21! 188 


2U454 




299(1 


3858 


351B 3793 


4049 


4314 45TI M 




1(U 


4 


484* 


5109 






5902 


6166 6130 


6691 


695], 7811 M 




131 


S 


748* 


7747 


8011) 


8873 




879M; 9060 


9323 


9395' seMH 




Ul 




38010B 


220370 


^20631 


820992 


881153 


S814I 1881675 


281936 


888196 !ti454H 




183 


7 


8716 


8976 


383G 


3196 


3755 


4<I15: iiU 


4533 


4793, 5031 M 




309 


8 


S3U9 


S5G8 


5886 


6094 


6348 


6600. S'i5h 


7115 


7372^ 7G30U 




33S 


9 


7887 




8400 


8657 


9913 


91T0i 91EL. 


8698 


(l!i:i9 23019SI» 






mi 


gao+tB 




230960 


231215 


831470 


231721831979 838231 


232488 832742 M 




*5 




2896 


3850 




3757 


1011 


1261 1517 


1770 


5ii23| 6878 H 




JO 




s. 


6781 


6033 


6885 


6537 


6799 7041 


789J 


7514' 7;MH 




74 


3 


8297 


8549 


8599 


901B 


Si93. 9550 


9800 


210lU(.8403odH 




S9 


4 


St05i9 


840799 


811018 


241897 


841516 


2*1795 242014 


218893 


251 1; 279UH 




191 


fi 


3033 


3886 


3531 


3788 


4030 


4217, 1585 


1772 


5019. 52Mi4 




149 


B 


&515 


5759 


6006 


6858 


6199 


6745 G991 


7837 


74981 7:18 N 




1T4 


7 


7973 


8219 


B161 


8T09 


8934 


U198| 9*13 


969; 


9938 850 176 M 




108 


8 


250420 


850664 


850909 


251151 


2^1395 


851638 231891 


832125 


2523661 8S10M 




2!3 


9 


2953 


3096 




3580 


3M8B 


lOGll 1306 




4790i 5031 U 






180 


23^73 




835755 


855996 


85623T 


256477,856719 




257 198,837 IMS iJ 




S3 


1 


7679 


TSK 


8158 




9637 


8977 9116 


9355 


93941 9Miil3 




41 


i 


260071 


! 60310 


260518 


860797 


861025 


861263 861501 


861739 


8G1976.26M14JH 




10 


3 


2451 


2688 


2985 


3162 




3636 3973 


4109 


4346 






94 


4 


4818 


5054 


5290 


5525 




5996 6238 


6467 


C702 


eBJ7|tij 




111 


£ 


7172 




7641 


7875 


8110 


8341 9579 


8818 


9016 


98T9kil 




140 


6 


oa\5 


9746 


S990 


2I08I3 


870416 


270879 270912 


871 144 


271377 


271SOBJia 




164 


J 


871848 


272074 


872301. 


8639 


2770 


3001 3233 


3161 


3696 


_8H?M 




isr 


& 


4158 


4389 


4680 


4850 


5081 


5311 5512 


5T78 


6002 




311 


9 


64S2 


6698 


6921 


7151 


7380 


1609 7838 


S0(i7 


8890 






190 




278982 


879811 


279139 


879667 


279B95 280123 


890351 




307i|tti 
5338|SHi 
757««»J 




23 


1 


281033 


891261 


881488 


281715 


281948 


8821 G9 


2396 


8G8! 






44. 


8 


3301 


358; 


3753 


3979 


1205 


1131 


4656 


♦B98 


6107 




67 


3 


5557 


5798 


6O07 


6238 


6156 


6691 


6905 


7130 






H9 


4 


7808 


8086 


6219 


8173 


9696 




9113 


9366 




9912 »m| 




til 


i 


890035 


290857 


290180 


890708 


1^90985 


i9lllT 


291369 


B91591 


891813 


298031888 




JS3 


e 


2856 


2478 


2699 


8920 


3141 


3363 


3581 


3804 


4085 


1210 821 




lis 




44G6 


1691 




5187 


5317 


5567 


5787 


6007 


6826 


6146 880 




178 




6G65 


6381 


7104 


7323 


7512 


7761 


7979 


8199 


9416 


8633 218 




aoo 




8853 


9071 


9289 


9507 


9785 


9943 


300l6i;3OOJ79 


.100595 


300913l*ia 






iiJo 


301O3U 


301 84 J 


301464 


301681 


301899 


308111 


302331,308517 


302764 


302980 21 T 




ii 


1 


3196 


3412 


3629 


3844 




1275 


4491 4706 


4981 


5130 815 




4S 


2 


5351 


5566 


5781 


5996 


6211 


6186 


6639 6H54 


7068 


7292216 




fi3 


3 


7496 


7710 


7984 


9137 


8351 


9564 


97781 8991 


9804 


9117 1213 




84 


4 


9630 


9943 


310056 


31086s 


310181 


310693 


310906:311119 


311330 


311548 21? 




105 


S 


31175i 


311966 


2177 


2399 


2600 


8918 


3023 3831 


3115 


3636811 




137 


6 


3S6J 


4018 


1289 




4710 


4920 


5130. 5340 


5551 


5760210 




lis 


7 


5970 


6180 


8390 


6599 


6809 


7018 


78871 7436 


7646 


7951809 




IBS 




806:1 


8878 


8181 


869a 


9898 


9106 


9311 9522 


9730 


9938!2<B 




IBO 


9 


320146 


320354 


320568 


380769 


380977 


381191 


3213911321599 


321805 


323018,807 






210 


3«819 


388486 


38263^ 


322839 




323852 


323159 32^665 


383871 


384077 8Ue 




20 


1 


4888 


1488 


1694 


1999 


5105 


5310 


5516 5781 


5926 


6131 205 




40 


! 


0336 


6511 


6715 


6050 


7155 


7359 


7563 7767 


7972 


8176 804 




Bl 


3 


8380 


8583 


8787 


8991 


9191 


9398 


9601 9805 


130008 


330211 203 




fli 


4 


330414 


330617 


330819 


331082 


331223 


33142T 


331630 331832 


8U31 


2*36 20« 




101 


5 


2t38 


2640 


2848 


3011 


3816 


3447 


3619 3950 




4253 201 




Itl 




4454 


4655 


4856 


5057 


5857 


3458 


5G68 5839 


6059 


6860 201 




J41 


7 


6460 


6660 


6860 


7060 7260 


7159 






9257tOI] 




Ifii 


8 


8156 


8656 


8855 


9054 9853 


9151 


9630 9919 


340017 


34O246llB0 




16« 


B 


340414 


310612 


340811341039 341237 


311135 


341638 311930 


2029 


8825;i98 




rs 


N.| 


» 


2 1 3 1 * 


=5= 


6 1 7 


8 


-^1^ 





L 



A, A Table of Logirithms of Numbers from 1 to 100,000. 




P.P. 


N.| 


—J— 


8 


3 


4 1 5 


6 


7 


B 1 9 










3«483 


312680 


318817 


3430 14 


343818 


313109 


343806 


343808 


sussg'sum 






IS 


1 


4398 


438') 


4795 


4981 


3178 


5374 


6570 


5766 


5962 


615T 






3!J 


8 


03.53 


65*9 


6744 


S939 


7135 


7330 


7585 


7780 


7913 


BllO 






SB 




BIOS 


8300 


8G94 


8889 


9083 


9878 


947! 


9666 


9860 


350054 






T7 


4 


350348 


•550+48 


350636 


350889 


3S10S3 


361816 


351410 


351603 


351796 


1999 


19; 




96 


S 


8183 


8375 


8568 


!7B1 


8954 


3147 


3339 


3338 


372* 


3916 


193 




11B 


6 


4108 


4301 


44S3 


46H5 


4970 


5068 


5860 


5438 


3613 


5834 


191 




i:)5 


T 


e086 


6817 


6403 




6790 


6981 


7172 


7363 


7554 


7744 


191 




!.5* 


8 


7935 


8185 


831 B 




8696 




9076 


92GG 




0646 


190 




in 


9 

aao 


9835j3600!5 


360815 


3B040t 


360503360793 


360978 


361161 


36I330|3ei539 


las 




3«l7a8 361917 


3li2l05 


368891- 


368*88,368671 


3G8859 


363018 


363836 36:1484 


lig 




IB 


1 


3B1S 


3800 




4176 


4363 


4551 


4739 


4986 


6113 


5301 


isa 




37 




ilBa 


5675 


5868 




6836 


6483 


6610 


6196 


6983 


7169 


18J 






3 


7356 


75*! 


7789 


7B16 


9101 


9897 


9473 


9659 


884S 


9030 


1B6 




7i 


4 


9816 


9401 


9597 




9959 


370143 


370388 


370513 


370698 


3TOSS3 


iBi 




9S 




371 068 


3TI853 


371437 




371 806 


1991 


2175 


8360 


2544 


8788 


181 




110 




8913 


3096 


3890 




3647 


3931 




4198 


4388 


4665 


IB 




1?9 


7 


4748 


4938 


5115 




5491 


5661 


5816 


6089 


6812 


6331 


IB3 




1*7 


P 


6577 




6948 


7184 


7306 


T4BS 


7670 


7858 


803* 


8816 


IB 




im 


fl 


9398 


8580 


8761 


8943 


0184 


9306 


9481 


9668 


9849 


39003^18 






sto 


380ail 


380398 


380573 


380734 


3809.14 


391115 


381896 


38147b 


381656 


381 »37 


IT 






I 


8017 


8197 


!37T 


i;SJ7 


8737 




3097 


3877 


S456 


3636 


190 




35 


E 


3813 


3995 


4174 


4353 


4533 


471! 


4891 


5070 


6!49 


5488 


179 








5606 
7390 


5785 
7509 














7034 
8811 


7818 

BDBS 


I7it 
178 




71 


4 


7746 


798' 


810: 


887! 


8456 


863 1 








S16R 


9343 


9680 


9698 


9873 


390051 


390889 


390405 


390598 


390759|1J7 




lOG 




390933 


391118 


39I88H 


391 484 


3B1641 


1817 


1993 


2169 


2345 


2581 


176 




Ul 




8697 


8973 


3048 


3884 


3400 


3375 


3751 


3986 


4101 


4.77 


176 




148 


e 


4458 


4687 


4808 


4977 


5158 


.5380 


6301 


3676 


5830 


6015 


176 




IS9 


9 


6199 


6374 


0349 


6788 


6896 


7071 


7845 


ni9 


7592 


7766 


114 






850 




^9114 


3S88H7 






39B808 398981 


399151 


399328 


399501 


173 




IT 


1 


9674 


9947 


400080 


400198 


400365 


4005381400711 


4O0BM3 


401036 


401288 


17t 




34 


9 


401401 


101573 


1745 


1917 




!!6I 


8433 


860J 


8T77 


2949 


lit 






3 


3181 


3598 


3464 


3635 


3807 


3978 


4149 


43!0 


1498 


4663 


171 




68 




4834 


3O05 


61T6 


3346 


3517 


6688 


3839 


60!9 


6199 


6370 


171 




85 


B 


6540 


6710 


6881 


7051 


7881 


7391 


7361 


7731 


7901 


8070 


110 




lU! 


a 


8840 


94I( 


8579 


8749 


8918 


908: 


9857 


9426 


9i9J 


9764 


169 




119 


1 


9933 


110108 


410871 


410440 


H0609 




41094b 


(11114 


111293 


411451 


161 




136 




411680 


17HH 




8184 


8893 


8461 


8689 


8796 


29G4 


3131 


168 




lfi3_ 


9 


33ni 


3467 


363S 


3H03 


39T0 


4137 


4305 


4478 


4630 


4S0ff 


187 




860 


4149T:* 


415140 


415307 


413474 


4156*1 


413808,415974 


IIGILI 


41630b 


41<i4I* 


167 




16 


1 


6641 


6801 


6973 


7139 


7306 


7478 


7639 


7901 


7970 


8135 


W 




33 


8 


631H 


SiGT 


963; 




8964 


9189 


9893 


9460 


B6S5 


9791 


m 




49 


3 


P9;h 


180181 


480886 


180451 


480616 


480781 


48094.^ 


181110 


4818T5 


481439 


lU 








481601 




1933 


8097 


saei 


8486 


8590 


2734 


8918 


30fl8 


164 




Bi 




3846 


3410 


3S74 




3901 


4065 


4288 


4398 


4555 


4719 


164 




OH 


6 


4888 


3045 


5808 


3371 


5534 


5897 


5860 


6083 


6186 


6349 


I<b 




llfi 


1 


esH 


6674 


6831 


6999 


7161 


7384 


7486 




T911 


7973 


161 




131 


S 


8133 


6897 


8459 


8681 


8783 


8944 




9868 


0489 


0691 


in 




US 




97.58 


9914 


430075 


430836 


430398 


430559 


430780 


430881 


431018 


131803 


ISl 








4ai3B4 


431585 


43168?! 


431846 


438U07 


432167 


438328 


43848S 


438649 


438bOfl 


i«i 




16 


1 


JSGB 


3130 


3890 


3450 


3610 


3770 3930 


40mi 


4849 


4409 


160 






8 


4569 


4789 


4«I88 


5049 


5807 


5367 3586 


5695 


5944 


6004 


1S» 




*t 


3 


6163 


638! 


6481 




6799 


6957 7116 


7875 


7433 


7592 


161 






4 


T75I 


7909 


8067 


8886 


8391 


8518 9701 


8859 


9017 


9173 






79 




9333 


9491 


S64B 


9a0b 


9961 


4*0188 44087(1 


440437 


440591 


1107.it 


t5t 




flj 


6 


44D909 


441061; 


441184 


441381 


44]S»B 


1695 1852 


8009 


8l6(j 


2383 


137 




HI 


7 


8480 


8637 


8793 


8950 


3106 


3863' 3419 


3570 


3732 




157 




186 


8 


4045 


4!01 


4357 


4513 


4669 


4«85' 4981 


SI37 


5893 




IM 




148 


9 


£604, 


6760 


5915 


6071 


6286 


6398^ 6537 


6692 


_6848 


70(W 


tS5 




frKs. 





1 


~r~ 


"~5~ 


~T^|~5~| "6~ 


7 


-^ 


II 


'K 





A Table of Logarithms of NumberK from I lo 100,000. 5 




^ " " " " 


N. 





I 


2 


3 


4 I 5 1 6 1 7 1 H 1 9 |D. 






880 


1*7158 


147313 


M7+6e 


447683 


4(7778 4479^3.448088 418848 448397 448 558il5^ 




IS 




8T0fl 


SSGl 


9015 


9170 


9.38l! 9*78' 96331 9787 9941450095 


154 




30 


S 


4408*9 


450403 


450557 


450711 


150865 451018451178451386,451479 1633 


164 




46 


3 


1786 


1910 


8093 


8847 


8IOO; 8553 


8706 


8959 


3018 3165 


IX. 




ei 


4 


3318 


3471 




37 !7 




4835 


438T 


4540! 4698 


16S 




7S 


i 


4845 


4997 


5150 


5308 


5454' 56l>6 




5910 


6068; 6814 


I6( 




91 


6 


636fi 


6518 


6670 


6881 


6973' 7185 


7876 


7489 


75791 7731 


158 




106 


1 


788* 


8033 


8184 


8336 


84971 8638 


9799 


8940 


9091 9848 


151 




IM 


» 


939! 


9543 


9694 


984.< 


9995,460 146460896 


460447:460597.460748 


161 




1ST 


9 


430898 


461048 


4011B8 


481348 


461499 1649; 1799 


1948 


8098| 8848 


IfiC 






m 


*6g398 


468548 


468G97 


46884J 


468U97 


463146,463896 


463445 


163594 


4637*4 


150 




15 




3893| 


4048 


4191 


4340 


4490 


4639; 4788 


4936 


5085 


5834; 


149 




to 


t 


5383 


5538 


5SS0 


5H89 


6977 


6186 


6874 


6483 


6571 


6719 


119 




** 




6808 


7016 


7164 


7318 


7460 




7766 




8058 


8800 


148 




S9 




8347 


8*95 


8643 


9790 


8939 




9833 


9380 


9587 


9C75 


148 




73 


6 


98!8 


9969 


470116 


470863 


170410 


470557 


470704 


470851 




471145 


147 




88 


e 


imai 


4T1438 


1585 


1738 


1978 


8085 


8171 


8318 




8610 


141 




103 


7 


8756 


2903 


3049 


3195 


"3^41 


3487 


3633 


3779 




4071 


146 




lie 




4sia 


4368 


4508 


4653 




4944 


5090 


5835 


5391 


5586 


146 




13! 


9 


5671 


iSlG 


5968 


6107 


6858 


6397 


6548 


6687 




6976 


145 






300 


477181 


47788«; 1.77411 


477555 


477700 


477844 477989 


17S133 


478i!7W 


479488 


\U 




14 




8566. 


97H aS5J 


8991 


9143 


9887 




9575 


9719 


B863 


144 




i» 


! 


480007 


tB01flll|K0894 


1-80438 


180588 


1807SS 


180869 


481018 


131156 


481899 


144 




43 


3 


1443 


15S6| lias 


1878 


8010 


8159 


8308 


8445 


8588 


8731 


14S 




ST 


4 


*8T4 


301(1' 31.^9 


330S 


3445 


3587 


3730 


3878 


4015 




I4E 




71 


S 


43001 


4448 




4787 


4869 


5011 


6153 


5895 


5437 


6579 


J48 




66 


B 


.3731 


5863 


6005 


6147 


C8B9 






67H 


6854 


6997 


148 




99 


T 


71S8 


7880 


7481 


7563 


7704 


7845 7986 


8187 


8869 


8110 


141 




114 


8 


8Sil 


8698 


8833 


8974 


9114 


9855 


9396 




9677 


DSIB 


141 




lis 


6 


99SB 


490099 


400839 


490380 


490580 


490661 


490901 


490911 


191091 


191888 


I4€ 






310 


i9I38» 


491508 


491648 


41.'17:-;8 


491988 


49 806 8 


498801 


1U8,1H 


198481 


198681 


14(1 




14 




8760 




3040 


3179 


3319 


3458 


3587 




3876 


4015 


139 




ta 


a 


4155 


4894 


4433 


4578 


4711 


4950 


4999 


518b 






I3S 




41 


3 


5544 


5693 


5888 


5960 


6099 


6238 


6376 




6633 


6791 


139 




55 




6930 


7068 


7806 


T341 


7483 


7681 


77.59 


7897 


8035 


9173 


138 




6» 


5 


9311 


8448 


8581, 


8784 


8868 


8999 


9137 


92 7 i 


9418 


9550 


138 




83 


6 


9687 


9884 




500099 


300836 


300374 


'^005 1 1 


300618 


500783 


500988 


137 




P7 


1 


501059 


301196 


501313 


1470 


leoT 


1744 


1880 


aoij 


8154{ 


8891 


137 




110 


8 


g4»7 


8584 


!700l 


8637 


8973 


3109 


3848 


3388 


3618 


3635 


136 




1» 


S 


3791 


3987 


4063 


4199 


4335 


4471 


1607 


4743 

30609; 


4878 


5014 


136 






3!U 


5U5l 50 


505886 




505557 




505888 




506831 


506370 


136 




IS 




6505 


6640 


6776 


6911 




7181 


7316 


7451 


7586 


7781 


135 




II 


i 




7091 


8186 


9860 


8393 


8530 


8664 


6199 


8934 


9068 


135 




40 




9?03 


9337 


B4I1 


9606 


9740 


9874 


510009 


51011S 


."; 10877 


510411 


134 




53 


4 


510545 


510679 


510813 


510947 


511081 


511815 


1349 


149L' 


1616 


1750 


134 




86 




188; 


8017 


8151 


8884 


8*18 


8551 


8684 


8818 


8951 


3084 


133 




80 


e 


3818J 


3351 


3484 


3617 


3750 


3983 


4016 


4149 


4888 


4415 


133 




93 


7 


4548 


4B81 


4813, 


4946 


5079 


58U 


5344 


517U 


5609 


5741 


133 




IOC 


8 


5974 


6006 


6139 


6871 


6403 


6535 


666B 


6800 


6938 


7064 


138 




IW 


MO 


7196 
5)8.il4 


1388 


7460 


7598 


7784 


7855 


7997 


8119 


8851 


8398 


138 

131 




51B046 


516777 


519909 


519044^ 


519171 




519434, 


519561 


519697 




13 


1 


9S2B 


9959 


380090 


580881 


38035a 


580195 


580615 


520745 


580976 


581007 


131 




»6 


g 


581 13a 


581 8B9 


1400 


1530 


1661 


1798 


1988 


8053 


8183 


8314 


131 




39 


3 


iiMr 


8375 


8705 


893^ 


8966 


3096 


3886 


3336 


3486 


3616 


130 




53 


4 


S74b 


3876 


4006 


413f, 


4866 


4396 


4586 


4656 




4915 


ISO 




S5 


5 


6045 


5174 


5304 


5434, 


5563 


5693 


£888 


5961 




6810 


189 




78 


6 


6339 


6469 


6598 


6781 


6956 


6985 


7114 


784S 


7378 


7301 


18S 




91 


T 


7630 


7759 


7SB8 


8016 


8145 


8874 


8408 


8531 




8:8S 


189 




104 


8 


8917 


9045 


9174 


931)8 


9430 


9550 


9687 


9915 


994L< 


530078 


ise 




U7 


9 


530200 


S30388 


530456 


530594 


5307 1'i 


530940 


53091)8 


.531096 


531888 


1351 


189 




^ 


^7 


~0 


1 


3 


3 


4 


^^ 


~6~ 


~T~ 


d 


=5= 


dT 





























1 Tnble of Lognnlhms of Numliera from 1 to 100,000. 




B 






?rpT^ 





1 


8 


3 


1 


5 


6 


7 


8 


9 


D. 




"~ 


34i 


S3I4T9 


531607 


^173' 


^iir2 


531990 


^^fi! 


332845 


538378 


538501 


532687 


m 




13 




(754 


!8B2 


3009 


3136 


3261 


3391 


3518 


361 


3178 


3S9B 


117 




25 






4153 


1280 


4407 


1331 


4661 


4781 




5011 


S167 


187 




39 




5i94 


542 


5347 


5671 


SBOO 


5927 


6053 


6179 




6438 


m 




SO 




6558 


6685 




6937 


lOflS 


7189 


7313 


111 


7567 


7693 


18 




63 




T9I0 


7915 


8071 


8191 


8388 


8448 


8514 


8e!iB 


8825 


8951 


lit 








9076 


9!0! 


9327 


9152 


9578 


B70S 


9829 


9951 


510079 


510301 


125 




88 




3403S9 


540455 


510380 


310703 






31IO80 


511205 


1330 


1151 


m 




101 




1579 


1701 


1829 


1953 


8078 


82o: 


2321 


8168 


8576 


8701 


Its 




IIS 


335 


!8S5 


8950 


3071 


3199 


3323 


3447 


351 


3696 


3880 


3941 


181 




344068 


5*4192 


344316 


511110 


314561 




344812 


611936 


515060 


545183 


lit 




IS 




630* 


5431 


5555 


5678 


5802 


5926 


6019 


6172 


6896 


6419 


18 




at 




651^ 


6666 


6789 


6913 


io;)6 


7 IS! 


7882 


7105 




765: 


in 








7775 


7898 


17021 


8111 


8867 




8512 


8633 


8158 




1(3 




19 




9003 


91S6 


9219 


9371 


9491 


9616 


9739 


9881 


9984 


350 tO< 


II 




61 




550828 


550351 


SS0173 




3S0717 


330840 


330962 


351081 


351806 


1388 


It 




73 






157! 


1694 


1816 


1938 


8060 


818 


2303 


8185 


2S47 


II 




85 






2790 


sail 


3033 


31 55 


3216 


3398 


3619 


3640 


3768 


18 




98 








4188 


4247 


136K 


4189 


4610 


4731 


1852 


4973 


18 




110 




509' 


5«15 


5336 




5518 


6699 


3880 


5910 


8061 


ei8f 


1! 


































360 


356303 


SSB483 


556S11 


356661 






557026 




567867 


557387 


t8U 




18 




7S0T 


76J7 


1748 


7868 


7988 


8108 


8888 


8319 


8169 


8589 


110 




U 




8709 


SB2D 


8918 


B068 


D188 


9308 


9188 


951B 


9667 


9787 


IM 




36 




9907 


360026 


360118 


360865 


560385 


.160501 


360681 


560713 


580883 


560988 


tl9 




48 




561101 


1821 


1310 


1459 




1698 


IBll 


19S6 


8055 


8174 


119 




59 




ass 


2112 


2531 


2630 


8169 


8887 


3006 


3125 


3844 


3161 


II» 




71 




3481 


3600 


3118 


3837 


39Sj 


1011 


4198 


4311 


4489 


*S1S 


lift 








4866 


4781 


4903 


60^1 


5139 


5857 


6371 


5491 


5618 


5730 


18 








5848 


5966 


6081 


6202 


6320 


6131 




6873 


6791 




Ufd 




107 


m 


7086 


7144 


786! 


737H 


7191 


1614 


7732 


7819 


7967 


8081 


m 
111 




5G8ffl)2 




368136 


368351 


,'h;m67 I 




36890. 


669023 


569 HO 


369851 




IS 




9371 


S491 


9608 


9725 


9818 


9959 


670076 


370193 


5J03O9 


510426 


IIT 




£3 




570543 


370660 


510776 


570893 


571010 


511128 


1843 


1359 


1476 


1598111 




35 




1709 


18!3 


J 918 


8058 


2114 


8891 


2407 


2583 


8639 


8755'1I6 




46 




!873 


2988 


3101 


3!8( 


3336 


3458 


3568 


3681 


3800 


3915;iltf 




58 




4031 


4111 


4263 


1379 


4191 


4610 


4126 


1811 


4957 


a078:llfi 








5188 


5303 


5119 


5531 


5650 


5165 


6880 


5996 


8111 


6886115 




8! 




6311 


6157 


6578 


6687 


6802 


6917 


7032 


7111 


7288 


7317 llli 




93 




7498 


TS07 


7722 


7638 


7951 


8066 


8181 


8295 


8110 


B58SI1( 




10« 




8639 


8751 


8868 


8983 


9097 


U212 


9321 


9111 


3555 


9669 


'I 






380 


S7978* 




380012 


380186 


380811 


580335 


580169 


58058^ 


6BOii97 


580811 








390993 


581039 




1267 


1381 


1193 


1608 


1788 


1836 


1950111 




!3 








2891 


8101 


8518 


8631 


2715 


8858 


2978 


308511* 




3i 




3199 


3312 


3486 


3339 


3638 




3879 


399! 


1105 


48l8tl:i 




45 




4331 


4144 


4337 


4670 


47B3 


4896 


5009 


5128 


6836 


53481113 




56 




5161 


55J1 


5888 


5799 


S9I8 


6021 


6137 


6850 




6476 lis 




68 




65BT 


6700 


6818 


0983 


T037 


7119 


7862 


7374 




7699111 




IS 




7711 


7823 


7935 


8047 


8160 


8872 


8381 


91B6 




8^80111 




BO 




8832 


8944 


9036 


9167 


9819 


9391 


9503 


9615 


9!86 


gF>38III 




lOS 




9950 


390061 


390113 


590284 


590396 


390507 


390619 


590730 


590842 


590953 


HI 






J90 


591065 


591 176 


391887 


591399 




591681 




591813 




598061 




11 






8288 


8399 


8510 


8681 


8732 


8843 


8951 




»l7fiTll 




St 




3286 


3397 


3308 


3618 


3729 


3810 


3930 


4061 


4171 


<888,1T1 




33 




4393 


4.W3 


4614 




4834 


1915 


6055 


5165 




s3sa'm 




4* 




5496 


5006 




5887 


6931 


6047 


6157 


6261 


6371 


6487'! 11 




S5 




659 T 


6707 


6817 


6921 


7031 


7116 


7266 


7366 


7176 


7Sf.lt! I U 




flS 




7C9S 


1805 




8081 


8131 


8843 


835-1 


816! 


8578 


SCHllin 




77 




8191 


8900 




9119 


9288 


9337 


9 we 


9651" 


9685 


9771 ide 




88 




9883 


9S!I2 


600101 


600210 


600319 


600428 


800537 


eooHM 


600755 


600861 IW 




99 




600973 


601082 


1191 


1299 


1108 


1517 


1683 


1734 


1843 


1951 


1 




rrr: 


^ 


~~0 i ] 2 


3 4 1 A 


6 


T 


8 f." ! 





A Table of LogsrithmE ot Numb«ra Avm 1 To 100,000. 7 




P. P.| N. 





1 


S 


3 


4 


5 


6 


7 


9 1 « [D, 






401 


60}060 


80! 169 


60**77 


608386 


608491 


60*603 


60*711 


eoms 


80*9»860S03«ia 




11 




SI 44 


3*53 






3577 


3686 


379* 


390* 


4010 411810 




fl 


t 


itts 


4334 


441* 




4658 


4766 


4974 


4988 


5089 5197 10« 


3i 




S30S 


5*13 


55*1 




5736 


5844 


5951 


6059 


6166 6*74101 




43 




6381 


6489 


6596 


6701 




9919 




7133 


7*41 7318 107 




53 




1*55 




7669 




7984 


7991 


B098 


BSOi 


63)8 8119101 




64 




85*6 


H033 


8710 


8817 


9954 


9061 


9167 


9874 


9381 918810) 








95941 9701 


9808 


9911 


S1008I 


610189 


610*34 


610311 


610M7,610554 lOT 




86 




610860'6 10767 


6108 73 


610979 


1086 


119* 


1396 


1105 


1511 1617 lot 




9S 


9 


nSS! 1889 


1936 


801* 


8118 


8*51 


g360 


8166 


8578! 8678 


5^ 






410 


612784 


61*890 


61*99(1 


81310! 


613807 


613313 


613419 


613585 


013630613736 




10 


1 


384! 


391T 


4053 


4159 


1*64 


4370 


4175 


4581 


4686J 4798 106 




»l 


t 


48S7 


4003 


5108 


5*13 


5319 


5181 


5589 


5631 


6710 5815,105 




31 


3 


5950 


6055 


6160 


6865 


6370 


6176 


8581 


6686 


6790 6895105 




4g 




7000 


7105 


7*lffl 


731 i 


7180 


75*5 


76*9 


7734 


7839 7943105 




St 




8048 


8153 


8*5T 




8466 


8571 


8676 


8780 


8884 8989105 




63 




8093 


9198 


930! 


9100 


9511 


9615 


9719 


98*1 


9989 6*003*104 




T3 




680138 


6*0*40 


6*0344 




880558 


6!06S6 


680760 


680B61 


6*0969 107*1104 




B* 




1IT6 


1*60 


13S4 




1598 






1903 


8007 8110'l04 




94 


9 


_m4 


i318 


24*1 


85*5 


86*8 


!73! 


*835 


8939 


304*1 3148 


04 
103 








6i32iV 


8*3353 


6*3156 


6*3569 


6*3663 


H8376B 


6*3889 


683973 


684078:6*4179 




10 




4!8g 




W.8B 


4591 


4695 


4798 


IBOl 


5001 




5810 


1031 


JO 




5312 


5415 


5518 


5681 


SJ84 


58*7 


59*9 


6038 


6135 


6*38 


103 




31 




6340 




6546 


6618 


6751 


6853 


6956 


7058 


7161 


7263 


103 




41 




7366 


7468 


7571 


7673 


7T75 


797B 


7980 


8088 


9185 


8897 


10! 


ii 




8389 


8491 


8593 


8695 


8797 


8900 


9008 


BIOI 


9806 


9308 


10* 
109 


61 




9410 


951* 


9613 


9715 


8817 


9919 


630081 


630183 


6308*1 


630386 


71 




6304S8 


B30530 


S30631 


630733 


630B35 


830936 


1039 


1139 


1841 


131*1 1 OS 




6! 




1*44 


1545 


1647 


1748 


1B49 


1951 


805* 


*153 


**55 


8356 101 




9i 


9 


!45T 


i559 


8060 


8761 


896* 


!963 


3064 


3165 


3866 


3367 


101 

101 








G33+S8 


633569 


633670 


633771 


83387* 


633973 


631074 


631175 


834*76 


631370 




10 




4477 


4578 


4679 


4779 


4B80 


4981 


6081 


5188 


5883 


5383 101 




20 




5184 


5584 


56B5 




5886 


59B6 


6087 


6187 


6887 


6396100 




30 




6486 


6588 




6789 


6B89 


6989 




7189 


7890 


7390'lOO 




40 




7490 


7690 


7690 




7B90 




8090 


8190 






100 




M 




8489 


8589 


8689 


8789 




8988 


9089 


9 188 


9887 


938: 


100 




60 




S186 


9586 


9686 


9785 


99B5 


9981 


640084 


640183 


610*83 


61038* 


99 




70 




640481 


640581 


B40880 


610779 


640879 


610978 


1077 


1177 


1876 


1375 


99 




80 


B 


1474 


1573 


167* 


1771 


1871 


1970 


S069 


8168 


8867 


*36r 


99 




90 


S 


!465 


*563 


86GS 


8761 


8860 


8959 


3058 


3156 


3*55 


3354 


99 






440S434^3 


643551 


643650 


613749 


643847 


613918 


644044:644143 


614 81*, 6 44310 






10 




4439 


4537 


1636 


4734 


483* 


1931 


5089 


51*7 


5*86 


5381 


Si 




19 


t 


54gS 


55*1 


5S19 


5717 


5BI5 


5913 


6011 


6110 


6808 


6306 


98 




n 


3 


6401 


650* 


6600 


6698 


6796 


6894 


699* 


7089 


7IBJ 


7885 


9B 




39 


4 


7383 


7481 


7579 


7876 


7774 


787! 


7969 


8067 


8165 


8868 


98 




46 


5 


8360 


8458 


8555 


8653 


6750 


8848 


8945 


9043 


9110 


9*37 


97 




M 


6 


9335 


B43S 


9530 


9687 


9784 


99*1 


9919 


650016 


650113 


850*10 


9^ 




68 


7 


650308 


650105 


6505081650599 


650696 650793 


650890 


0997 


1094 


1181 


9T 




78 


B 


1878 


1375 


1418 1569 


18661 1768 






8053 


8150 


91 




87 


9 


!g46 


*343 


*410' *536 


*633; *730 


8B86 


8981 


3019 


3116 

6540BU 


~Wa 






450 


653813 


653309 


653405,65350* 


85359s;6S3695,653791 


0538S8 


65398* 




ft 


I 


41T7 


»S73 


,4369 


4165 


456* 


4658 4754 


4950 


4948 


5018 


96 




19 


a 


5138 


5*35 


5331 


54*7 


5583 


56191 5715 


5810 


6906 


6008 


96 




18 


3 




6194 


6*90 




6488 


6577 


6673 


6769 


6884 


6960 


96 




38 


4 




715* 


7847 




7438 


7534 


7689 


778S 


7880 


7916 


98 




41 


d 


8011 


8107 


880* 




8393 




8581 


S67H 


977* 


8870 


95 




»T 







9060 


9155 


9*50 


9346 


914] 


9536 9631 


97*6 


!1881 


95 




66 


7 


9916 


660011 


660106.660801 


660*96,66039! 


660496 660581 


660676 


esorn 






7S 


8 660885 


0960 


loss, 1150 


1845 133B 


1434 15*9 




1718 






85 


9 1813 


1907 


*008| 8096 


SI91 **86 


*390. *175 


8569 


*063 


SI 




Kp. 


mT-o- 


1 2 1 3 1 1 1 5 i 6 r 7 


s 9 1,.: 









, 


1 


1 




» 


A Table of LcB«rithmB of Numbers from 1 to 100,000. 


■ 


n 


e7K 


N.| 


1 1! 1 a 1 ♦ ( 5 1 ti 1 7 1 S 1 8 lO. 






i60,66%7m 


668H52 668»47 




663830663321 063419,663518,663607' 94 






B 




3701 


3795 


3999 


3983 


4078 


4178 


4866 




4454 


♦548 


H 






19 




4648 


4736 


4830 


4984 


5018 


5118 


6806 


6899 


5393 


5*97 








29 




5S8i 


5676 


5TS9 


5962 


6956 


6050 


6143 


6837 


6331 


e*84 


9* 






37 




flSlB 


66U 


6705 


6799 


6898 




7079 


7173 


7866 


7360 


»« 






16 




7453 


7546 


7640 


7733 


TB26 


7920 


8013 


8106 


S199 


8893 


98 






56 




B3B0 


6479 


8578 




8759 


8858 


6945 


9038 


9131 


988* 


93 






es 




9317 


9410 


9503 


9396 


9689 


9788 


,9876 


9967 




670153 


9S 






T* 




6J0846 


670339 


670*31 


670624 


670G17 


670710 


070808 


670896 




1080 


93 






B* 




1IT3 


1865 


1358 


1451 


1513 


1636 


1788 


1981 


1913 


9005 


93 








(178098 


678190 


678883 


078375 


678167 


678560 


678638 


672744 


678B36 


678989 


9? 




9 




3oei 


3113 


3805 


3897 


3390 


3488 


3574 


3666 


3738 


3850 


« 






19 




3948 


4031 


1126 


4818 


4310 


4108 


♦191 


4586 


♦677 


4769 


at 






ST 




4861 


4933 


5045 


5137 


5829 


6380 


5418 


3303 


5595 


669- 


98 






36 




ane 


5870 


6962 


6053 


6146 


6236 


6329 


6419 


6511 


6608 


91 










6694 


6785 


6976 


G969 




7151 


7818 


7333 


748* 


1516 


91 






5S 






7699 


7799 


7881 


7978 


8063 


8154 


8815 


8336 


64S7 


9 






6i 




6518 


H6U9 


8700 


8791 


8998 


8973 


9061 


9135 


9246 


933 


9 






73 




9489 


9319 


9610 


9700 


9791 


SBB8 




090063 


680154 


680815 


9 






98 




680336 


6B018G 


680511 


6B0607 


680699 


680789 


680879 


0970 


1060 


113 


91 






180 


681 t4l 


681338 


691428 


691513 


681603 


691693 


681781 


691874 


691964 


6««U63 


90 






9 




8145 


8835 


8386 


8416 


8606 


8596 


8696 


8777 


8867 


t957 


00 






IB 




3047 


3137 


3287 


3317 


3407 


3497 


3687 


3677 


3767 


3957 


90 






87 




3947 


40M7 


4187 


4817 


4307 


4390 


14B6 


4576 




♦756 


90 






36 




48*5 


4935 


5085 


5114 


5804 


6891 


5383 


6473 


5663 


6658 


90 






4i 
5* 




6742 

6636 


6B31 


6981 


6010 


6100 


6189 


6819 


6368 
7201 


6*68 


6617 


89 






















J MO 








63 




7588 


7618 


7707 


7796 


7966 


7975 


6061 


8153 


B848 


6331 


89 






78 




8480 


8509 


H599 




8776 


9865 


9963 


B042 


9131 


9880 


89 






61 




»3oe 


939H 


9166 


9573 


9664 


9753 


9841 


9930 


690019 


690101 


89 






*9C 


B90196 


SU0<85 


B903T3 


690408 


690650 


690639 




690HI( 


690905 


690993 


99 








1081 


1170 


1839 


1347 


1135 


1624 


1612 


1700 


1769 


IBT7 


BB 






16 




1965 


8053 


8142 


3830 


8319 


8*06 


8194 


8583 


8671 


»759 


BB 






>S 




!8tT 


8935 


3023 


3111 


3199 


3867 


3376 


3463 


3551 


3639 


SB 






35 




3787 


3915 


3903 


3991 


4078 


4106 


♦831 


4348 


4130 


♦517 


BB 






U 




460S 


4693 


4781 


4968 


4966 


5044 


6131 


6819 


S30J 


6391 


B8 






fi3 




£488 


6569 


5657 


5714 


3938 


6919 


6007 


6094 


0188 


6269 


B7 






68 




6336 


6144 


6331 


6618 


6706 


6793 


6B80 


6968 


7065 


7)48 


87 






70 




T8S9 


7317 


7404 


7491 


7678 


7005 


7752 


1839 


7986 


BOU 


87 






79 




6101 


9188 


8875 


9368 


9449 


B536 


8688 


8709 


9190 


eU83 


87 






ioi 


698970 




69M111 


699^31 


699317 


699101 


HU9I91 


699578 


69966* 


699151 


87 




9 




9838 


9984 


700011 


700098 


700194 


700871 


700358 


700111 


700531 


700617 


87 






IT 




T0OT04 


700790 






105O 


1136 


1882 


1309 


1393 


1498 


86 






80 




1568 


1654 


nil 


1887 


isrs 


1999 


8080 


817* 


8868 


gsu 


se 






34 




8131 


8517 


8603 


8669 


8775 


8B01 


29*7 


3033 


3119 


3803 


BS 






43 




3891 


3377 


3163 


3349 


3633 


3781 


3807 




3979 


♦066 


B6 






i! 




4151 


4836 


4388 


4408 


4494 


♦679 


4663 


4751 


4831 


♦988 


86 






60 




5009 


6094 


5179 


6265 


5350 


5130 


6383 


6007 


6693 


5T;9 


B6 






69 




5864 


6949 


6035 


6120 


6806 


6891 


6376 


6462 


6517 


6638 


W 






17 




6718 


6803 


6888 


8974 


7059 


1144 


7829 


1313 


14O0 


7*96 


85 






sli 


I07S70 


707655 


707140 


707886 


70791 1 


!0799b 


rOBOMl 


708166 


708861 


lUHSSd 


86 






8 




8481 


8506 


9691 


6676 


6761 


6840 


B931 


9015 


BlOO 


9196 


B5 






17 




9870 


9355 


9440 


9584 


9609 


9694 


SI 79 


9963 


991b 


710033 


B5 






tl 




710117 


710808 


710887 


710371 


710466 


7105*0 


710685 


710710 


710794 


0879 


«5 






3* 




0B63 


1048 


1132 


1217 


1301 


IS86 




1551 


1639 


118* 


84 






48 




IflOT 


1998 


1976 


8060 


2144 


8889 


8313 


8397 


1181 


1566 


M 






«) 




S6S0 


873* 


8818 


8908 


8996 


3010 


3154 


3838 


3323 


34U7 


«4 






fi9 




3491 


3575 


3669 


3748 


3826 


3910 


3994 






4846 


84 






87 




♦330 


4414 


4497 


4581 


♦605 


47*9 


183S 


4916 


6000 


iOBl 


M 






76 




6167 


5861 


6335 


54IB 


6508 


6586 


6669 


5763 


6836 


5980 


« 






i= 


























L. 


^p 


N.I 


1 


1 1 3 




6 


6 


^ 


^ 





I 


d 



A Ttble of Logsritfamt of Namben ttofo 1 to 100,000. 9 




P^N, 


1 I 


2 


3 


4 


5 


8 


' 


B 


9 


D. 






Bq 


T16003TltH>S1 


716170 


71685* 


716337 


16*81 


716S0* 


716589 


7 1867 1 


710754 


83 




6 




em 


G9!l 


700* 


7088 


7171 


786* 


7338 


7481 


7iO* 


7597 


93 




IT 




7S7I 


7754 


7837 


79!0 


S0O3 


8086 


9169 


9S63 


8336 


8*19 


63 




85 




SJOi 


8S8S 


9668 


8751 


BSSi 


8917 


9O00 


9083 


8166 


92*9 


93 




35 




S331 


9414 


9*97 


9590 


9663 


97*6 


9888 


9911 


9994 




83 




*l 




7101 59 


T!Ot*! 


7!0385 


720407 


780*90 


780573 


780655 


780739 


780S8I 


0903 


S3 




to 




0988 


106B 


1151 


1833 


1316 


1399 


1481 


1S6S 


16*6 


1788 


88 




M 




1811 


1893 


1975 


8058 


!l*0 


882! 


8305 


2387 


8*69 


855! 


8! 




m 




ie34 


8716 


8799 


8981 


8963 


30*5 


3187 


3809 


3891 


337* 


8t 




T« 




3t5e 


3538 


3680 


3708 


3781 


3866 


39*6 


4030 


*M2 


4191 


9! 

98 






rau 


Tiim 


784358 


784U0 


78M8! 


78*60* 


784685 


7/47G7 


7*48*9 


72*831 


785U13 




B 




408S 




5!58 


5340 


5*88 


6503 


6595 


5607 


5748 


6830 


88 




IB 




S81* 


6993 


6075 


6156 


6838 


6380 


6401 


6493 


666* 


66*0 


Bt 




J4 




6Tg7 


6809 


6990 


0978 


70,W 


713* 


7S16 


7897 


7379 


7*80 






as 




7S41 


7683 


7704 


7785 


7866 


7948 


9089 


8110 


8191 


8873 






40 




B3M 


8*35 


8516 




8679 


8759 


9941 


998! 


9O03 


908* 






4S 




9165 


9846 


9387 


9*09 


9489 




9651 


9732 


9913 


9893 






if 




BB7* 


730055 


730136 


730817 


730898 


730379 


730*39 


730510 


730681 


730702 






« 




73078! 


0863 


09*4 


1084 


1105 


iisr 


1866 


13*7 


1*89 


1509 






w 




1589 


1669 


1750 


1830 


1911 


1991 


8078 


815! 


8833 


S313 






uo 


TitW* 




738555 




73i715 




73*976 


738US6 


733037 


I33I17 


^ 




e 




3197 


3878 


33» 


3*38 


3618 


3599 




3769 


3839 


39191 80 




IS 




3999 






4840 


*390 


4400 


**80 




4640 


*780 


SO 




34 




4800 


4880 


4960 


50*0 


51 SO 


6800 


6819 


6359 


643B 


5619 


80 




3« 




5599 


5679 


5759 




5918 


5998 


6078 


6157 


6837 


6317 


80 




M 




6397 


6476 


6556 


663J 


6715 


6796 


6874 


6964 


7031 


7113 


80 




4S 




7193 


T!7! 


7358 


7*31 


7511 


7580 


7670 


77*8 


7889 


7908 


78 




e« 




79H7 


8067 


81*6 


6885 


8305 


938* 


9*63 


95*3 


868! 


9701 


79 




64 




8781 


8960 


8939 


9018 


90B7 


9177 


9856 


9335 


9*1* 


9*93 


79 




7« 




957! 


S65I 


9731 


9810 


9^m 


9968 


7*00*7 


710186 


740805 


74088* 


79 






iso 


T4«3tf3 


7*0**8 


7*U5il 


740600 








710915 


740U9* 


7*1073 




e 




11S8 


leso 


1309 


1398 


14*7 


1546 


168* 


1703 


1788 


1960 


79 




ifi 




1939 


8018 


8096 


8176 


!3S* 


!33! 


8411 


8*89 


8669 


8647 


79 




S3 




!785 


280* 


!988 


!B61 


3039 


3118 


3196 


3876 


3353 


3*31 


78 




SI 




3510 


3588 


3667 


37*5 


BBSS 


3908 


3980 


4068 


*13l 


*816 


78 




M 




4!Ba 


4371 


4**9 


*S!H 


4606 


408* 


4768 


4640 




*9B7 


79 




4.T 




5075 


51S1 


6831 


5309 


6387 


6*66 


65*3 


6621 




6777 


78 




M 




5955 


6933 


6011 




6167 


08*5 


6383 


6*0 


6*79 


6556 


78 




«3 




Gi;31 


8TI! 




6868 


69*5 


7083 


7101 


7179 


7856 


733* 


-19 




11 


Stti 


7*1! 


7199 


7567 


76*5 


7788 


790(1 


7978 


7955 


8033 


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78 
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7*8866 


7183*3 


7*8481 


749498 


749576 


749653:748731 


7*8800 


748985 




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90*0 


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9195 


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9350 


9*87 9504 


9588 


9669 


77 




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9991 


9968 


750045 


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7508O0;740!T7 


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750*31 


77 




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750508 


750596 


760G63 


750740 


0817 


0994 


0971 


1049 


1185 


180! 


77 




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1876 


1356 


1*33 


1510 


1667 


1664 


1741 


1818 


1995 


197! 


77 




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818S 


880! 


8879 


8366 


8*33 


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2596 


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87*0 


77 




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8970 


30*7 


31!:! 


3800 


3877 


3363 


3*30 


3500 


77 




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3583 


3660 


3736 


3813 


3889 


3966 


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4119 


4195 


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77 




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4348 


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450 


457S 


465* 


4730 


4807 


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4960 


5036 


76 




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5189 


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53* 


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5570 


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6718 






6940 


7016 


70981 71681 7!4*l 7380 


76 




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7700 


7775 


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8830 


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991* 


98'JO B366! Bin 9517 9598 


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97*3 


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140! 1477 


1658 18!7| 1708! 1778' 1863 


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5 


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763653 


763787 


76380 




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76410 




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4386 


4400 


4475 


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4998 


5072 


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582 


5896 


5370 


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5580 


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5668 


5743 


SB 18 


5998 


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601 


6115 


6190 


6864| 6338 


71 






90 




6413 


64B7 


6568 


6636 


6710 


678S 


6859 




7007 


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74 






87 




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7830 


7304 


7379 


7453 


7587 


760 


7675 


7749 


7883 


7* 






44 




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7978 


8046 


818( 




8868 


8342 


8416 


8190 


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9718 


87S6 


88o0 


8934 


900B 


9098 




9830 


9303 74 






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9451 


9585 


9599 


9673 


9746 


9980 


989* 


9968 


770048 74 






6T 




770115 


770189 


770863 


770336 


770410 


7704B4 


770557 


770631 


770705 


07T8 74 






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77lt*b 


17188(1 


771893 


771367 


7714*0 


7715t4Jl4 




7 




1587 


1661 


1734 


1808 


188 


1955 


2088 


2108 


8115 


8848 


73 






15 




838! 


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8468 


8548 


2615 


8688 


8768 


8835 


8908 


898 


73 






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3055 


3188 


3801 


3871 


3348 


348 


3494 


3567 


36411 


3713jra 






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37BB 


3860 


3933 


4006 


4079 


4158 


4225 


4898 


4371 


4444^73 






3S 




4517 


4590 


4663 




4809 


4882 


4955 


5088 


5100 


51731 IS 






41 




5846 


5319 


5398 




5588 


5610 


5683 


6756 


58S» 


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73 






51 




5974 


6047 


6180 


6193 


6865 


6338 


6111 


6183 


6556 


9689 


73 












677+ 


6846 


6919 


6998 




7137 


7809 


7888 


735* 


73 






66_ 




74l!7 


7499 


7578 


7644 


7717 


7799 


7862 


7934 


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9079 


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778884 




178368 




178513 




778B58 


77B730 


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8947 


9019 


9091 


9163 


9836 


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9380 


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9885 




780029 


780101 


780173 


780845 


78 






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J80317 


790389 


780461 


780533 


780605 


780677 


0749 


0881 


0893 


0965 


71 






89 




1037 


1109 


1181 


1853 


1384 


1396 


1468 


1510 


1618 


168* 


78 






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189S 


1971 


8048 


8111 


8186 


88S8 


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1401 


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43 




2473 




8616 


8688 


8759 


8831 


2908 


8974 


3046 


311) 


78 






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3189 




3338 


3403 


3475 


3511 


3618 


3689 


3761 


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71 






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3901 


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4046 


4118 


4199 


4861 


4338 


4403 


4475 


4546 


11 






6fi 




4617 


4689 


4760 


4831 


4908 


4971 


5045 


5116 


5187 


5851 


71 
71 






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J8i3au 


785401 


7[t5*78 


J85543 


7B5615 


785690 


785757 


7858(8 


7B5BH9 


755970 








6011 


6118 


6183 


G85t 


6385 


6396 


6167 


6539 


6609 


6880 


71 






14 




6751 


6888 


6893 


6964 


7035 


7106 


7177 


7849 


7319 


7390 


11 






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7460 


7531 


7608 




7744 


7815 


799S 


7956 


8087 


809H 


71 






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8169 


8839 


831U 


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8*51 


8522 


8593 


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35 




8875 


8946 


9016 


9087 


9157 


988H 


9899 


936! 




9510 








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9788 


97B8 


9863 


9933 


790004 


790074 


790114 


790815 


70 






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790356 


790486 


79019B 


790567 


790H37 




0778 


0948 


0918 


70 






57 




098n 


1059 


1189 


1199 


1869 


1340 


1410 


1480 


1551 


1680 


70 






64 




1691 


1761 


IB3I 


1901 


197i 


8011 


8111 


2181 


28S8 


8388 


70 






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79S3B)j 


798468 


798538 


798608 




798718 


798018 


79ias8 




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3831 


3301 


3371 


3441 


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3581 


3651 


3781 


70 






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3790 


3860 


3030 




4070 


4139 


4809 


1279 


4349 


4418 


70 






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4558 


1687 


4697 




483li 


4906 


4976 


5045 


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TO 






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5185 


5854 


S384 


S39S 




5532 


5608 


5678 


5741 


5811 


70 






35 




5880 


5949 


6019 


6088 




6287 


689) 


6366 


64.')6 


SA05 


68 






43 




6574 


6644 
7337 


?L>6 


6788 


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6021 
7614 


6990 


7060 
7758 


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7198 


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7406 






7890 


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8098 


9167 


8836 


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8513 


9588 


69 






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B7B9 


8859 


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9065 


9134 


B203 


9878 


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7995*7 


J9961D 


799685 


7U9751 


799883 


799892 


TWbl 




7 




1(00019 


800098 


B00167 


800836 


800305 


800378 


800148 


BOOSll 


800580. 


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0854 


0983 


0998 


1061 


1189 


1198 


1866 


1335 


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147! 


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1609 


1678 




1915 


1884 


1958 


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8*86 


8895 


8363 


8438 


8400 




8637 


8705 


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34 




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8818 


8910 


8979 


3047 


3116 


3194 


3858 


3381 


3389 


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41 




3457 


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3594 




3J30 


3799 


3867 


3935 


4O03 


4071 


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4i^ 




4139 


4808 


4876 


4344 


4418 


4480 


464B 


4616 


468S 


♦753 


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4889 


4951 


5085 


5093 


5161 


5889 


5*97 


536S 


M33 


W 






01 


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5501 


5S6B 


5637 




5773 


6841 


5908 


5976 


6014 


6118 


68 




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P. P 


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6 1 7 


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806723 


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6858 


6986 


6994: 7061 


7129. 7197 


7264 


7332 




7167 


68 




13 


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7535 


1603 


7670; 7739 


7806; 7873 


7941 


9008 


8076 


8143 


BB 




»0 


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8211 


8819 


B34S: 8414 


8181 


8549 


8816 


8084 


8751 


8818 


67 




87 




8886 


8953 


90«ll 9068 


9156 


9223 


9890 


93.^8 


9125 


9192 


67 




33 


a 


9560 


9827 


9691 9762 


B829 


9996 


9961 


H1003J 


81009ft 


910165 


6T 




iO 


B 


610^33 


810300 


9 1036T 810431 


810501 


810569 


910636 


0703 


077( 


0837 


B7 




4.1 


7 


0904 


097 


1039 


1106 


1173 


1810 




1374 




1509 


67 




fil 


e 


15T5 


1842 


1709 


1776 


1843 


1910 


1977 






2178 


67 




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9 


8345 


231* 


2379 


2445 


8518 


K79 


2646 


87i; 


8790 


8847 


67 

67 






li60 


BlgaI3lHI2BH0 


81304T 


813114 


B1JI81 


813217 


813311 


91338 


813448 


813511 




7 




3S8I 


3848 


3714 


3781 


38*8 


3914 


3981 


4018 


4114 


4181 


67 




13 


! 


4216 


4314 


4381 


4417 


4514 




4647 


1714 


4790 


4917 


OT 




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3 


4913 


4980 


5016 


5113 


5179 




5318 


5379 




5511 


66 




£6 


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iSTB 


5644 


5711 


5777 




5910 


5976 


6012 




6175 


fi« 




33 




6*41 


631)9 


6374 


6440 


6506 


6573 


6639 


6705 


6771 


6839 


66 




40 


6 


eaoi 


S970 


7036 


7108 


7169 


7835 


7301 


7367 


7433 


7199 


66 




46 


7 


756S 


7631 




7764 


7830 


7996 


7968 


9088 


8091 


8160 


66 




«3 


B 


8886 


8898 


83SH 


8484 


9490 




8682 


8689 


8754 


8820 


86 




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9 


8985 


8951 


9017 


9093 


9149 


9215 


9881 


9346 


9412 


9116 


66 






seo'Mnsu 


9IB610 


019676 


819711 


819807 


819973 


819939 


980004 


820070 


820136 


66 




6 


ilsgosoi 


B20867 


320333!b20399 


880164 


880530 


880595 


0661 


0727 


0792 


6B 




13 


i 


0958 


0984 


0999 


1055 


1120 


1186 


1251 


1317 


1392 


1419 


66 




19 


3 


1514 


1579 


1645 


1710 


1775 


1841 


1906 


1978 


2037 


8103 


66 




ae 


4 


ei68 


8233 


2299 


8364 


2430 


8495 


2560 


2626 


2691 


8756 


66 




3* 


5 


8888 


2897 


2952 


3019 


3083 


3148 


3813 


3279 


3344 


3*09 


66 




39 


e 


3*74 


3539 


3605 


3670 


3735 


3800 


3865 


3930 


399ff 


4061 


66 




15 




41 S6 


4191 


4856 


1381 


4386 


4451 


4516 


4591 


4646 


4711 


65 




sa 




4T76 


4841 


4906 


4971 


5036 


6101 


5166 


5831 




5361 


65 




jSfl 


9 


5«S 


5491 


5556 


5821 


5686 


5751 


5815 


5880 


5946 


6010 








670 


920075 


88G140 


926201 


8:6269 


986:t31-926399 


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65 




« 




6783 


6787 




6917 


698 1| T046 




7175 




7305 


65 




13 




7369 


7434 


7199 


7S63 


7628 7692 


7757 


7881 




7951 


66 








8015 


8090 


8141 


9209 


8273' 9338 


8102 


8467 


9531 


8595 


64 




36 






8184 


8789 


9853 


9918 8982 


9040 


9111' 9175 


9239 


64 




38 


S 


9304 


9369 


9138 


9197 


9561 9625 


9690 


975*^ 9818 


9882 


64 




39 


6 


9B17 


830011 


830075 


930 L39 


830204830869 


830338 


930396630460 


830525 


61 




is 


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930589 


0853 


0717 


0791 


0845 0909 


0973 


J037 1102 


1166 


61 




AI 


a 


1830 


1894 


1358 


1422 


1186 1550 


1614 


16781 1748 


1806 


61 




J*l 


g 


1810 


1934 


1998 


2062 


2126 2189 


2253 


2317 2381 


2415 


64 






BSO 




932573 


8J8H37 


93270U 


838764 9328*8 


838998 


838956833020 


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1 


3147 


3811 


3876 




3402 


3466 


3530 


3503 


36S7 


372 ll 61 




13 


K 


3184 


3849 


3912 


3975 


1039 


4103 


4166 


4230 


4294 


4357 


64 




IS 


3 


44*1 


4494 


1518 


1611 


4675 


4739 


1902 


4666 


4989 


4993 


64 




SS 


4 


5056 


5120 


6183 


5247 


5310 


5373 


5137 


5500 




5627 






SI 




S691 




5817 


6881 


5914 


6007 


6071 


6134 




6261 


6S 




38 


6 


63!4 


6397 


6151 


6514 


6577 


6641 


6704 


6767 




6994 






44 


7 




7080 


7093 


7146 


7210 


7273 


7336 


7399 




7525 


63 




SO 


S 


759M 


7658 


7715 


7778 


7841 


7901 


7967 


9030 


8093 


9156 


63 




ST 


9 


8819 


8288 


8315 


8108 


8171 


9534 


8597 


9660 


8723 


9786 


63 










83H9I2 




939038 


839101,9:19161 




939889 83 »35]! 


839415 




6 


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9478 






9667 


97291 9798 


9855 


9919 9981 


810013 


63 




11H 


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840106 


910169 


840232 


910894 


940357 840180 


810188 


840545 840S06 


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3 


0733 


0J96 


0859 


0981 


0984 


1046 


1109 


1172 


1231 


1297 


S3 




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1359 


142i! 


1185 


1547 


1610 


1672 


1735 


1797 


1860 


1982 


63 




31 


S 


1995 


2017 


2110 


2r72 


2835 


2897 


8360 


2422 




8547 


62 




S9 




2b09 


2672 


2731 


2796 


2859 


2921 


2983 


3046 


3109 


317U 


62 




4i 




3S33 


3295 


3357 


3420 


3192 


3641 


3606 


3669 


3731 


3193 


62 




■50 


S 


3tf53 


3918 


3990 


1012 




4166 


4889 


4291 


4353 


4415 


61 




'37 




4477 


4539 


4601 


1661 


4786 


4788 


4950 


1912 


4914 


5036 


62 




ftp; 


T 





1 


2 


3 


4 


. . 


7 


8 


9 


a' 



















'•■'•■ 


N. 


. 


1 


3 


4 1 5 


e , , 


8 


9 


a 






700 


84SU98 84518(5 


8458!! 


145884 




845470 


8 45 538 


8*560* 


Bi56M 


68 




6 


1 


a7ia 


6780 


594! 


6904 


5966 6088 


6090 


6161 


0213 


6876 


68 






18 


i 


8337 


6399 


S161 


6583 


6585 66*6 


6108 


6770 


6832 


6994 


68 






19 




6955 


7017 




7141 


7808 7864 


7386 


7388 


7*49 


7511 








*5 


4 


7573 


7631 


7(196 


7758 


7819 7991 


7943 


8004 


8066 


8189 


68 






31 




8189 


8»S1 


831 J 


9374 


8435 849T 


8559 


8680 


9692 


8743 


68 






37 




8905 


88flri 


8988 


8989 


9051 9118 


917* 


9835 


9897 


9359 


61 






43 


7 


9419 


9481 


951! 


9604 


9665 9786 


9768 


9919 


9911 


9978 


61 






SO 


8 8500331850095 


850156 


950817 


850879 850310 


950101 850168 850584 860595 


61 






S6 


t> 064S 
710 851853 


0707 


0769 


08!iO 


099 1| 0958 


1014 


I075i 1136. 1197 


11 






aims 


851381 


851448 


851503 851 564 


951686 851747 951809 


<l 




6 




1870, 1931 


199S 


205S 


811* 


8175 


8836 


8897 


83.>8; Ul» 


61 






\t 


t 


2480, 2541 


860? 




8784 


8795 


8946 


8907 


896a .1029 


SI 






16 


3 


3090 3150 


3?li 


387) 


3333 


3394 


3455 


3510 


3517 3637 


61 






ti 


4 


3B98; 3759 


3820 


3881 


3911 




4003 


4184 


1195 4845 


81 






30 


S 


43061 4367 


4488 


4489 


4549 


4610 


4670 


4731 


47981 4852 


6) 






37 


s 


4913 4974 


5034 


5095 


5156 


5816 


5877 


6337 


5398) »159 


61 






♦3 


T 


S519'. 5580 


5640 


5701 


5761 


5S88 


6898 


5943 


6003 6064 


61 






49 


8 Ginl fiias 


6? 45 


6306 


6366 


6487 


648T 


66*8 


6608 6668 


se 






S£ 


91 67!9. 6789 


6950 


6910 


6970 


7031 


7091 7158 


7!1« T81! 


60 








7i0Bi733g,851393 


857453 


957513 


857574857034 


857694 857755 957915 857975 


60 






6 




7935 7995 


8056 


8116 


8176 8836 


88971 8357J B4IT| B4i7 








n 


i 


8537 8597 


8657 


8719 


8778 8838 


8998: 8958 9019! 9078 








18 


3 


9138 9198 


9859 


9318 


9379 9439 


9499] 955S| 9619' 9679 


60 






!l 


4 


9739I 9799 
860338 81)0399 


9859 
860458 


9918 
860518 


9978 800038 
800578 0637 


860098 960158 8008 19 8608 IS 


% 






30 


5 


0697 




0817, 0817 




36 




0837; 099B 


1056 


1116 


1176 1836 


1896 


135A 


\415^ 1475 
8018 (078 


«t 






42 


5 


1534. 1594 


1654 


1714 

8310 


1773 1933 
8370 8*30 


1993 
8489 


1958 
8549 


to 










8851 


8609, 1608 w 






54 


9i 87*81 »7H7 


8947 


8906 


89G6 3085 


3085 


3144 


3804' 3863> 








T30,8e33a3.8e338! 


S63M8 


863601 


863561,863680 




963739 


863799 863958 


59 








1 


3917 


39T7 


4036 


4096 


4155 


4814 


4874 


4333 


♦398; ♦451 


St 






li 


i 


4511 


♦570 


4630 


4669 


♦749 


♦908 




4986 


4995' 5045 


ss 






IB 


3 


5104 


5163 


5888 


5888 


5341 


5400 


6159 


6519 


6579 6637 








*4 




5696 


57S5 


5814 


5874 


5933 


5998 


6051 


6110 


6169; 6888 


w 






*9 




6i87 


6340 


6405 


6465 


6584 


6583 


6648 


6701 


6760 6819 


iB 






33 




6876 


0937 


6996 


T055 


7114 


7173 


7838 


7291 


7360 7409 


59 






41 


T 


7467 


75«6 


7585 


7644 


7703 


7768 


7981 


7880 


7939 7998 


S« 






47 


8 


8056 


8115 


81 74 


9833 


8898 


8356 


8*09 


8468 


8527 8586 


5> 






i3 


9 


86441 


8701 


8768 


9981 


8879 


8939 


8997 


9056 


9114 SMS 


59 

5V 




740869*3! 


B69!90 


869349 


869408 


969460 869583 


869584 


969648 


969101 9697S0 






i'sToioi 


B87 


9935 


9994 


870053,87011 


870170 


970888 


670897 9703+5 


ts 










97046! 


S70S81 


870579 


0639 


(1696 


0755 


0813 


0878 0930 


58 








3 


0989 


1047 


1106 


1164 


UiJ 


188 


1339 


1398 


1456| 1515 


68 






!3 


4 


1573 


163 


1690 


1748 


1800 


1965 


1983 


1981 


80*01 8098 


68 






!9 


5 


eisto 


»glfi 


8873 


8331 


8389 


8448 


8506 


85641 26881 3681 
3146; 3804: 3868 


» 






3S 


6 


J739 


gJ97 


8955 


891S 


8978 


3O30 


3088 


M 






41 


7 


33*1 


33IH 


3437 


3495 


3553 


361 


3669 


3T27 


37951 38*4 


H 






46 




3908 


396U 


4018 


4076 


♦ 131 


4198 


4850 


♦309 


4366 4424 


tt 






Si 




448? 


4540 


4598 




4! 14 


477S 


4830 




4945I 5O03 


M 




8740BI 


BI511D 


BI5177 


B7583a 


875893|b7535 


97540! 




81568*816688 


39 




e 


1 


5G40 


5698 


5756 


5813 


5971 


5989 




6045 


610! 


6160 


68 






11 


i 


6il8 


6176 


6333 


639 


6449 


B50 


6564 


6688 


6680 


6737 


6S 






17 


3 


6795 


6SS3 


6910 


6969 


7086 


7093 


7141 


7199 


7856 


7314 


59 






t3 


4 


7371 


7t!S 


748T 


7544 


7608 


765S 


7717 


771* 


7938 


7888 


6» 






t» 


& 


T94T 


8004 


8068 


ens 


8177 


8834 


9898 


8349 


9401 


e4«* 


67 






34 


6 


85*1 


857 


8637 


8694: 8758 


8809 


8866 


8984 


8981 


9039 


57 








1 


9096 


9153 


911 


9869 0385 


9393 


9440 


9*91 


9555 


9618 


57 








8 


96fl9 




9784 


98tt! 9899' 9956 


880013 


980070 


880187 


990185 


47 






Jll 


e 


88084! 


980299. H80356,880413 89047 1 1880518 


0585 


064! 


0699 


0746 


l!1 




FFiTq— 3— 


1 { S 1 3 1 « { 3 1 6 




9 


E 



ATableofLogtriEhmaofNumbertftoral to 100,000. IS 




P.P. 


N.| 


1 1 8 1 3 1 4 


5 


6 1 7 1 8 1 B 1 D. 






7«)8tt(J81* 


'»B0ti'71 B80988«S0B85 8BI018 






17 




881099 981166 881813 881871 8913W 




6 


1 


138^ 


1448 


1199 


1556 


1613 


1670 


1787 


17841 184II 19B8 57 




11 


S 


1956 


2018 


8069 


8126 


3183 


8810 


8897 


8351 fill 8189 57 




17 


s 


tm 


2581 


8638 


8096 


2752 


8909 


8886 


8B83 8980 3037 51 




13 


4 


S09S 


3150 


3807 


3864 


3381 


3377 


3*3* 


3491 3518! 3606 67 




iS 


S 


3661 


S71B 


3775 


3838 


3888 


3915 


4008 


4059 *II5| 4178:57 




34 


e 


4ii9 


1285 


4318 


4399 


1155 


4518 


4569 


4023 4688 4739157 




40 


T 


479i 


4858 


4909 


4965 


5088 




6135 


6192, 6218| 6305' 47 




■K 


8 


i36l 


5*19 




5531 


S5S7 


5614 




5757 6813 6870 67 




£l_ 


B 


5BS8 


5983 


6039 6098 


61 S« 


6809 


6865 


638l| 637b| 6*3156 






770 


B86i91 


HH6517 886601,886660 


986716 


886773 886889 


886885 8869*8 8869981 56 




6 


1 


7054 


7111 7167 


7823 


7280 


7338 


7398 


7**9 


7305 


7561 56 




11 


1 


7617 


7674 )730 


7786 


T818 


7899 


7B65 


8011 


8067 


8m' 56 




17 




9179 


8236 8292 


9348 


eioi 


9160 


8616 


8373 


8689 


8686. 6« 




a 




8711 


8797 8853 


890B 


8965 


902 i 


90T7 


BIS* 


9190 


Si*s;s» 




w 




9308 


9358 9111 


9470 


9586 


9388 


B638 


9694 




98041 H 




34 




9802 


99I8I e97189003O,H90U8S 


890141 


890197 


900853 




890365 H 




SB 


7 


904*1 


890177 8905;i3 


05891 0645 


0700 


0766 


0811 


0968 


0984; 66 




IS 


e 


0980 


1035, 1091 


1I17| 1203 


1259 


131* 


1370 


1128 


148S « 




&0 


9 


1537 


1593^ 1619 


1705| 1760 


1816 


1878 


1988 


1983 


8030 


« 








9^095 


»92 ISO 892806 


8922621892817 


898^73 


8B8189 


992181 


892610,898595 




6 


1 




87071 2768 


881S 


8873 


8989 


8983 


SOlO'i 3096 


3151 56 




11 


! 


3201 


3868' 3318 


3373 


3129 


3481 


3540 


3593 


3661 


3706 66 




la 


3 


37S« 


3817 3813 


3988 




4039 


1091 


4160 


4205 


4861165 




II 


4 


431 B 


4371 4187 


4*88 


4538 


4593 


4648 


4701 


4759 


4814 55 




il 


i 


4H70 


4985 4980 


5036 


5091 


5146 


5201 


5857 


5318 


6367 45 




53 


a 


sua 


5478 5533 


5588 


5611 


569B 


6754 


6809 


5984 


6980 66 




36 


7 


^915 


60301 6095 


el+0 


6195 


6861 


6306 


6361 


H416 


6471 66 




44 


e 


6ste 


6591 6636 


6692 


67*7 


6808 


6857 


6918 


6867 


7088 66 




49 


_? 


7077 


Tl«! 7187 


T81! 


7I9J 


735! 


7407 


7468 


7517 


7578 54 






780 


HSIBSI Holes? S97m 


S97792 


8978*7 


BB7902 


»97957898012;898U(I7B99188 56 




d 


1 


8176 8S3i| ssae 


8311 


8396 


8*31 




9561 


8616 8670 66 




11 


t 


8725 BTBO] B83i 


8990 


89** 


9999 


0054 


9109 


9164 9818 66 




16 


3 


B«73 9388 9383 


B137 


9*92 


95*7 


9602 


9656 


9711 9766 63 




K 


4 


9S81 9875 99-.iO 


9B8S;B00n39'900UB4. 


900119 


900803 


900869900318 65 




il 


B 


9003fi7 00043S 900*76 


D00531 


03861 0640 


0695 


0749 


0804 


0869 6S 




33 




0»13 0968 1092 


1077 


1131 1186 


1810 


1895 


1349 


1404' 55 




3S 




ms 1513 ir,6i 


1628 


16761 1731 


I78S 


1940 


1994 


1918' 54 




44 


a 


2003 80*7 all* 


2166 




8389 


2384 


2138 


il98 54 




♦L 


9 


gS47 S601 SB55 


8710 


8764 2819 


2873 


8987 


8981 


3036! 64 






too 


903090)103114 9031 9U 


903853903307,003361 


903116 


B0347( 


80J521 


903618, 64 




i 


1 


3633 


3697 37*1 


3793 


3819 


3901 


3968 


4018 


4066 


4120 64 




11 


t 


4174 


4229 1883 


1337 


4391 




4*99 


4553 


4607 


466I( 64 




16 


3 


4716 


4770 482* 


4978 


4938 


4986 




6094 


5118 


5808 


64 




St 


4 


62S6 


5310 536* 


5118 


5*72 


6586 


5580 


5634 


6688 


5718 


54 




37 


a 


6798 


5830 5901 


5958 


6018 


6066 


611B 


6173 


6*87 


6281 






31 


6 


6335 


6389 6*13 


filBT 


6551 


660* 


6658 




676b 


6880 


64 




38 


J 


8871 


G927 6981 


7035 


7099 


71*3 


7196 


7250 


7301 


7338 


64 




43 


8 


7*11 


7185 7519 


7673 


7686 


7680 








7893 


54 




*6 


9 


7919 


8008 8056 


9110 


9163: 8217 


8870 


8321 


8378 


8131 


64 




81( 


9084BS 90MS3H 908592 


908t>*6 


908699 9097B 


908807 


908HB0I90891* 


909967 


5* 




5 




902 Ij 9071 9188 


918 


98351 9889 


9318 


9396 


9119 


9503 


64 




11 


i 


9S46| 9610 9663 


9716 


9770 9823 


9877 


9930 


9984 


910037 


63 




16 


3 


910I9IS1014491019T 


91026 


91030*910358 


91011 


610*84 


910318 


0571 


63 




21 


4 


0684 


0679 0731 


0784 


0838 0891 


0914 


0B98 




1101 


63 




£6 


S 


1158 


1211 1861 


131 


1371 1*84 


1477 


1530 


I5S4 


1637 


63 




SI 


6 


169C 


1713 1797 


185D 


1903 IB56 


8009 


20H3 


8116 


2169 


53 




sr 


7 


828 


2275 8328 


838 


8435 2*88 


(511 


8591 


8617 


8700 


63 




41 


a 


8753 


8B0G 2859 


2913 


8966 3019 


3078 


3125 


3178 


3231 


53 




48 


9 


S8B1 


3337 3390 


8113 


3196| 3519 


3608 


3666 


3709 


3761 


63 




^F 


N.1 1 1 1 8 


3 1 4 1 6 


a 


^f- 


8 


9 


dT 





14 A Table of Logarithma of Numbers iVom 1 to 100,000. 




f-i- 


Jli 


"„ 


_»_!_!_ 


3 


4 1 5 1 6 1 7 1 fi 1 9 |D. 






BiU 




9I^S5i,9T39i0 


913973 


914086,914079 914132 914184914237 914890 


S3 




s 




iSf! 


439 








4608] 466C 


4713 


4766 


4819 


53 




11 


; 


4a7S 


4985 


4977 


5030 


5083 


5136, 5189 


581 


5894 


5347 


53 




10 




54O0 


545; 


550, 


555H 


561 


5664 


6716 


S76S 


5882 


£875 


53 




!1 




5P!? 


S9HD 


6033 


609a 


6138 


ei9 


6213 


6296 


6349 


6401 


63 






5 




650 


6559 


6618 


6664 


671 


6770 


6822 


6 875 


6987 


53 




3S 




69S< 


7033 


70ri5 


7138 


7190 


7843 




7318 


7100 


7453 






37 




8030 


7SS8 
8083 


761 

6135 


7663 
8188 


7716 
8840 


7768 
88D3 


7820 
6345 


7873 
8397 


7985 
8450 


J978 
8502 


SI 
S8 




♦? 


S 




4M 


H 


S55. 


860 


8659 


B7i2 


8764 


8816 


8869 


898 


8973 


9086 


68 






BJoaTaoTw 


91BI3U 


SlBlSa 


9Tm5 


919887 


919340 


919398 


91944 


019196 


919549 


58 




5 


1 960 


9653 


S706 




9810 


9868 


9914 


9967 


980019 


920OTI 


58 




10 


a.BiOiaa 


9S0176 


9808*8 


980880 


980338 


980384 


980436 


980189 


0541 


0593 


S! 




16 


3 


0fi45 


0697 


0749 


080 


0853 


0906 


0958 


1010 


1062 


Itl4 


58 




gl 


4 


116' 


leis 


1870 


1388 


1374 


1426 


1478 


1530 


1588 


1634 


SI 




8S 


5 


I68R 


1738 


1790 


1848 


1891 


1946 


1998 


8050 


8102 


8154 


58 








iiO'. 


S35a 


8310 


8362 


8414 


246b 


8518 


8570 


3688 


8674 


58 




36 


7 


2785 


8T77 


8889 


8881 


2933 


8685 


3037 


3089 


3140 


3192 


58 




4! 




38« 


329ti 


33441 


3399 


3451 


3503 


3555 


3flOJ 


3658 


3710 


58 




47 


» 


3Jt!S 


3814 


3865 


3917 


3969 


4081 


+072 


418 


417(1 


4888 


S2 






934;)3 


984383 


9i*43t 


924486 




9 « 1589 




924693 


924744 






5 


1 


4796 


484a 


4899 


4951 


5003 


505. 


5106 


5157 


5!09 


5861 






10 


i 


531! 


5364 


5415 


5467 


5518 


SS70 


5681 


5673 


5725 


ST 76 


48 




IS 


3 


589H 




5931 




6034 


6085 


6137 


6188 


6240 


6891 


SI 




W 


4 


63 tS 


fi394 


6445 


6497 




6600 


6651 


6708 




0803 


51 




25 


Sl 685 


6908 


6950 


7011 


7068 




7165 




7268 


7319 


31 




31 


e! J370 


7Wi 


747;i 


758+ 


7570 






7730 


7781 


7832 


51 






7, 7BH;f 


7935 


7986 


8037 


8088 


8110 


8191 


8248 


8893 


8315 






41 


8| B30I* 


SU7 




8549 


8601 


8658 


8703 


8754 


8805 


8857 


51 




4fi 


9 «i«W 


8959 


9010 


9061 


91 [^ 


9163 


9815 


9866 


9317 


9368 


51 
5i 






8j0 9a!mM 


a*9l,70 9*95«l 


118957? 






989725 


9.?B776 


929887 




5 


I itD:«j 


9981.93003! 




930134 


930185 


930836 


930897,930338 


930389 


51 




10 


S 9304)0 


930491 


0348 


0592 


0643 


0694 


0745 


0796 


0817 


0898 


51 




IS 




0949 


1000 


1051 


110.' 


1153 


1201. 


1854 


1305 


1356 


1407 


Sl 




go 


4 


1458 


1509 


1560 


1610 


1661 


1718 


1763 


IBH 


1865 


1915 


SI 




u 


S 


I9liti 


aoi7 


8068 


2118 


2169 


8280 


3871 


8J28 


8378 


8483 


51 




31 


6 


8474 


8524 


8575 


8686 


8677 
3183 


8787 
3834 


8778 


8889 


8879 


8930 


51 




36 


7 




3031 


3088 


3133 








3137 


51 




41 


8 


3487 


3538 


3583 


3639 


3690 


3710 


3791 


3811 


3898 


3043 


Sl 




•6 


9 


3993 


4044 


4094 


4145 


4195 




4896 


4347 


4397 


4448 


51 






860 ssi*9a 


934549 


934599 


934650 


934700 


934751 


934801 


934858 934908 


934953 


W 






I 


50U3 


5054 




5154 


6805 


5255 


5306 




5106 


5437 


SO 




10 


a 


5507 




5608 




5709 


6759 






5910 


5B60 


so 




15 


3 


6011 


6061 






6812 


6262 






6413 


6463 


50 




» 


4 


6514 


6564 


6614 


6665 




6765 


6815 




6916 


6906 


50 




!5 


d 


7016 


7066 


7117 


7167 


7817 


7867 


7317 


7367 


7418 


7468 






30 


fl 


TSI8 


75G8 


7618 


7668 


7718 


7769 


7819 




7919 


7969 


M 




3i 


7 


8019 


8069 


8119 


8169 


8819 


6869 


8380 


8370 


8420 


6470 


50 




iO 


e 


8580 


8570 


8680 


8670 


8780 


8770 


8820 


8870 


8980 


8970 


SO 




45 


9 


9080 


9070 


9180 


9170 


9280 


9870 


9380 


9369 


9419 


9469 


50 




870 9:19619 


939569,939819 




939719 




939819 


939869 939918 


939908 




i 


19t4XllH 


94OO0B91O11H 


940168 


940818 


940867 


940317 


940367 940417 


940167 


SO 




10 


S 


0516 


0566 


0616 


0666 


0716 


0765 


0815 






0961 


SO 




15 


3 


1014 


1064 




1163 


1813 


1863 


1313 




1418 


1468 


so 




20 


4 


1511 

200S 


1561 
8058 


1611 


1660 
2157 


1710 
2807 


1760 

8256 


1809 
8306 


8353 


1909 
8405 


1958 
8455 


40 
SO 




ss 






30 


6 


(504 


8554 




2653 


8708 


2758 


8801 


2851 


2901 


8950 


SO 




35 


7 


3000 


3049 


30B9 


3149 


3198 


3847 


3897 




3396 


3445 


4> 




40 


8 


34S5 


3544 


3593 


3643 


3698 


3742 


3791 


3841 


3890 


3939 


4» 




4ft 


9 


3»«> 


4038 


408S 


4I3T 


4186 


4836 


4885 


4335 


4384 


4*33 


49 




ft: 


n: 


1 . 


8 


'~~a~ 


* 


5 


e 


"~7 i- 


« 1 


X 









p.p 


W 





1 


_?_ 


3 4 


i 


6 


7 


8 


» 


D. 






mi 


944463 


B44532 


)t4591 


941631 


9446B( 


944T29 


944779 


944888, 


944877 


944927 


^ 




6 


1 


49TS 


50!S 


^50T4 


5184 


6173 


5222 


5272 


5321 


S370 


5419 


49 




10 


> 


£489 


5519 


S56J 


5616 


5685 


5715 




5813 


5802 


5918 


49 




IS 


3 


5961 


6010 


6059 


6108 


61 57 


6207 


6856 


6305 


6351 


6403 


49 




JO 


4 


645« 


6501 


6551 


6600 


6619 


G6B6 


8747 


6798 


6945 


689* 


IS 




t* 




6913 


6992 




T090 




7189 


7239 


78B7 


7336 


7384 


46 




id 


6 


7434 


7483 


7532 




7630 


7679 


778S 


7777 


7986 


7876 


49 




34 




79!4 


7973 


B022 


8070 


8119 


6168 


8817 


e26fi 


8315 


6304 


49 




S9 


e 


8tI3 


St6! 


851! 


8560 


8609 


8657 


6706 


9755 




9953 


49 




t*_ 


9 

S90 


8808 


8951 


8999 


9049 


90B7 


9146 


9195 


9841 


B292 


934 1 


4S 
49 




949390 


949*39 


949488 


949536 


949585 


949634 


949683 


949T31 


949780 


949889 




B 




gSTS 


9926 


9975 


950024 


950073 


950121 


950171 


950819 


950267 


950318 


49 




10 


i 


950355 


95041 1 


950462 


0511 


0580 


0608 




0706 


0754 


0903 


49 




la 


3 


0851 


0900 


0949 


0997 




1095 


1143 


1198 


1840 


1899 


49 




W 


4 


1338 


1386 


1435 


1483 




1580 


1689 


1677 


1786 


1775 


49 




li 


5 


19S3 


1872 


1920 


1969 


2017 


8006 


8114 


2163 


8811 


S260 


48 




t9 


6 


»08 


SSS8 


2405 


8453 






2599 


2647 


8698 


8714 


48 




3* 


7 


mi 


«841 


2BB9 


2938 




3034 


3093 


3131 


3190 


3288 


48 




39 


8 


S«6 


33ZS 


3373 


3421 


3470 


3518 


3566 


3615 


3663 




48 




M 




905 


3T60 


asitB 


3856 


3905 




4001 


4019 


4098 


4146 




48 




954Si3 


954291 


9S4339 


954387 


954435 


954484 


954538 


9545B0 


93482S 


9546T7 


48 




S 


1 


47*5 


4773 


4821 


4869 


491 J 


4968 


5014 


5068] 51 Iff 


5159 


49 




10 


t 


5S0T 


5855 


5303 


5351 




5447 


6495 


55431 5598 


5640 


48 




14 




5688 


5736 


5784 


5832 


5880 


5928 


5976 


6084 


6078 


6i2o;4e 




19 


4 


G16S 


6216 


6865 


6313 


6361 


6409 


6457 




6553 


66i)ll 49 




!4 


5 


G649 


6697 


6745 


6793 


6940 


6888 


6936 


6984 


7038 


70S0; 49 




19 


8 


71 88 


7176 


7224 


7872 


7380 


7368 




7464 


7518 


7559 


49 




34 


T 


7607 




T703 


7751 


7799 


794T 


789* 


7942 


7B90 


803B 


49 




3B 




S08f 


8134 


8181 


8289 


8277 


B325 


9313 


8481 


8468 


B5I8 


48 




il 


9 

BlU 




8012 


9659 


8707 


8755 


8803 


9950 


689B 


894t 


8994 


49 




9M0lT 


959089 


959137 


959185 




959280 


959388 


9S9J75 


959423;95a47i 






s 




9518 


9566 


9614 


9661 




9T57 


9804 


9858 


9900 9917 


49 




9 


! 


9995 


960042 


960090 


960 I3H 


960185 


960233 


960281 


96O328i96O3;6;960t83 


49 




14 


3 


980471 


0518 


0566 


0613 


06B1 




07S8 


080*1 0851 




48 




19 


4 


0948 


0994 


1041 


1089 


1136 




1231 


1279 


1326 




47 




!3 




mi 


1*69 


1516 


1563 


1611 


1658 


1706 


1763 


1801 


1848 






SB 


6 


1895 


1943 


1B90 


2038 


2085 


2138 


2180 


2223 


827J 


8322 






33 




236B 


2417 


2464 


asn 


2559 


8606 


8653 


8701 




87951 47 




36 


8 


g843 


2890 


8937 


8985 


3032 


3079 


3186 


3174 


3881 


3S6B 47 




48 





3316 


3363 


3410 


3457 


3504 


355* 


S599 


3640 


3093 


3741 4T 




063TH8 


9(i3S35 


963888 


963929 


9(i397T 




964071 


964118 


964165 


964818 41 




5 




4280 


4307 


4354 


4401 


4448 


4195 


4548 


4690 


4637 


4694 47 




g 


i 


4731 


4778 




49T8 


4919 


4966 


5013 


5061 


5108 


6155 47 




14 


s 


520t 


5249 


5896 


5343 


S3B0 


5437 


5484 


5531 


5578 


5625 47 




19 


4 


5671 


5719 


5566 


5813 


5860 




5954 




6043 


6095 47 




IS 


S 


814* 


6189 


6836 


6283 


6389 


6376 


6423 


6470 




6564 47 




«a 


8 


6611 


6658 


fiTOS 


6758 


6799 


6845 


6992 


6939[ 6986 


7033 47 




33 


1 


T080 


7127 


7173 


7220 


7267 


7314 


7361 


7408, 7454 


7501 M 




36 


6 


7548 


7595 


764S 


7688 


7735 


7788 


7829 


787S T988 


7969 47 




4! 


9 




8062 


9109 


8156 


8803 




■8896 


83*3l 8390 


8436 47 




























B3t 


968183 


968530 


968576 


9BB683 


9'i8670 968116 


968763 


968810,968856 


968a03 47 




S 




8950 


8996 


9043 


9090 


91361 9193 


9829 


9876 


9323 


936» 47 




9 


i 


9416 


9463 


9409 


9556 


96081 9649 


9695 


9748 


9789 


9835 47 




14 


» 


98B2 


9S28 


997S 


970021 


970068,970114 


970161 


970207 


970864 


970300 *T 




18 


4 


9T0347 


970393 




0486 


0S33I 0579 




0678 


07 la 


0765 48 




S3 




031! 


0839 




0951 


0997 lOU 


1090 


1137 


1193 


1289 48 




!3 




1876 


1322 


1369 


1415 


146l| 1509 


1554 


1601 


1647 


1693 46 




32 




1740 


17B6 


1832 


1879 


1985' 1971 


20(8 


2064 


8110 


8157 46 




37 


e 


2203 


2249 


2295 


8342 


2388 2434 


8481 


8587 


8573 


2619 46 




41 


9 


2666; 2T12 


8T58 


2604 


8951 2997 


2943 


8989 


3035 


3062 4« 




Kf 


z: 


""o~|~i~ 


=rn 


^T^ 


* fi 


~8~ 


7 


~8~ 


9 ft 





























16 A Table of Logarilhma of Numbers from 1 to 100,000, 




ivp. 


N, 





1 


8 


3 


1 


5 i 6 1 7 1 8 1 « la 






So 


9J3I!B 


973J74 


973880 


973866 


973313 


973359 


y73403 973431 


973*97,973443 1*! 








1 


3i90 


3636 


36B2 


3788 


3774 


3B80 


3866 


3913 


3959 


40<K,46 






9 


i 


4051 


4097 


4143 


4189 


4235 


4881 


4381 


4374 


4480 


4466 4« 






14 


3 


4518 


4558 


4604 


4650 


4696 


4718 


4768 


4834 


4«80 


4986' 46 






19 


4 


4«7S 


501B 


S0G4 


5110 


5156 


5803 


524S 


(894 




5386146 






i3 


a 


543! 


5478 


5584 


5S70 


5618 


5668 


5T07 


5733 


S799 


5845] 46 






» 




5891 


6937 


5933 


B089 


6075 


6181 


6167 


6818 


6858 


G304146 






3* 


7 


6350 


6396 


6448 


6488 


6533 


6379 


0685 


6671 


6717 


6763 


46 






3T 

*1 


a 

9 


6808 


6854 
7318 


6900 
7358 


6946 


6998 
7419 


7037 
7495 


7083 


7189 

7566 


7173 

7638 


7280 
1678 


46 
46 
M 








97J7M 


a717<i9 


y?7B15 


97786i 


977906 


977958 


9779B9 


978043 


97B089 978135 




4 


1 


8181 


B886 


8878 


8317 


B3fl3 


8409 


8434 


8500 


6516 


8391 


46 






9 


» 


8637 


8683 


8788 


8774 


B819 


8865 


6911 


6936 


9002 




46 






13 




9093 


S13M 


9184 


9830 


9275 


9381 


9366 


9412 


9451 


9403 


16 






Ig 


4 


954S 


B501 


9639 


B6B5 


9730 


9776 


9B8I 


9867 


9918 


9958 


4« 






** 




980003 


990049 


98O094 


980140 


990183 


980831 


980876 


980388 


930367 


980418; 45 






87 




015S 


0503 


05*9 


0594 


0640 


0683 


0730 


0776 


0881 


0667 45 






3J 


7 


0912 


0937 


1003 


1018 


1093 


1139 


llBl 


1289 


1875 


1380 44 






Sfi 


a 


1366 


1411 




1501 


1547 


1398 


1637 


1683 




17731*4 






40 


9 


1BI9 


1S64 


1909 


1934 


2000 


2049 


2090 


8135 


8181 


3886 


^ 








988*71 


188316 


9B8368 


988107 


9B2458 


9B8497 


988513 


988488 


9B8633,9»8e78 




4 


1 


ms 


8769 


8814 


8839 


8901 


8949 


8991 


3010 


3083 


3130 45 






9 


i 


3175 


3880 


3865 


3310 


3356 


3401 


311S 


3491 


3536 


3581 1*5 






IS 


3 


3686 


3671 


3716 


3768 


3807 


3852 


3897 


3948 


3987 


4038|45 






18 


* 


4077 


4188 


4167 


4818 


4257 


4308 




4398 


4437 


4488 43 






2t 


5 


4587 
4977 


4372 
503S 


4<>li 


4668 
511! 


4707 
3157 


4738 
5808 


4797 
5817 


4818 
3892 


4687 
5337 


49381*5 
6388, 45 




a7 


a 




31 




5486 


5471 


5516 






5631 


3696 


5741 


3786 


M30) «3 






36 


8 




5980 


5965 


6010 


C056 


6100 


6141 


6169 


6834 


6879|*5 






40 


9 


6384 


6369 


6*13 


6458 


6303 


6518 


6593 


6637 


6688 


6787 


il 








9BU778 


986917 


B868B1 


986906 


9S(i9SI 


986996 




987UB3 


987130 9B7I75 




4 




7819 


7864 


7309 


7333 


739S 


7413 




753! 


7577 


7688 










2 


7666 


7711 


T7S6 


TflOO 


7845 


7B90 




7979 


8024 


8068 


U 






13 


3 


8113 


8157 


8808 


8847 


8891 






8185 


8170 


8314 


44 






IB 




S55B 


86Q4 


8648 


8693 


8737 


87B) 


8826 


B871 


8916 


6960 








a 


fl 


BOOJ 


9049 


9094 


9138 


9iB3 


9887 


9878 


B316 


036 1 


9405 


45 






ii 


6 


9450 


9494 


9539 


8583 


9688 


967S 


9717 


9761 


9806 


9850 


41 






31 




9895 


9939 


9983 


990088 


990078 9901 17 


990161 


990806 


99O8i0;99O894 


44 






36 


8 990339 


9903B3 


990188 


0478 


0516 0561 


0603 


0630 


0691 D738 


U 






40 


9 078^ 


0887 


0871 


0916 


0960| 1001 


I04B 


1093 


11371 1188 








^59lm 


991870 


991315 


991339 


99I4U3;991448 


991198 


991336 


991580 99 m5 


4* 






1 1SS9 


1713 


1758 


1BU3 


1846 


1890 


1935 


1979 


2083 2067 


It 








S 2111 


8156 


iSQI 


2841 


8288 


2333 


8377 


8181 


8465 S509 


44 






13 


3 S5it 


8598 


8648 




8730 


8774 


8819 


2863 


890T 29£1 


44 






18 


4 !99S 


3039 


3083 


3U7 


3178 


3816 


3860 


3304 


33*8 3398 


44 






Si 


5 3*36 


3180 


3584 


3568 


3013 


3651 


3701 


3715 


3789 3833 


44 






is 


3S7T 


3981 


3965 


4009 


4053 




4141 


4183 


4889 4873 


4* 








7 4317 


4361 


4405 


4419 


4493 


4337 


4581 


4685 


466Sl 4713 


t* 










4801 




4889 


4933 


4977 


3081 


5065 


510B: 41^ 


41 






40 


y 

990 


5196 


5810 


58B4 


5388 


5378 


5116 


5460 


5304 


5517| A39I 


4i 




996635 


995679 995783 


995767 


995811 


993854 


995B98 


995948 9959B«eS<iOS0 






4 


1 


6074 


6117 


6161 


6803 


6819 


6293 


6337 


6380 GU4^ Mm 


4* 








i 


6518 


6555 


6599 


6643 


6687 


6731 


6774 


6B18 6668 SSOS 


44 






13 


3 


6949 


6993 


7031 


7OB0 


1184 


7168 


1818 


7853 78S9I JS43 


44 






18 


4 


7386 


7430 


7474 


7517 


7561 


J605 


7646 


7698 


7T3tf 1779 


41 






n 


5 


7883 


7867 


7910 


7951 


7998 


8041 


8083 


8189 


61 7S 8816 


44 










8859 


8303 


6347 


8390 


8134 


8477 


8381 


8464 


8606 »ast 


4« 






SI 


7 


8695 


8739 


8788 




B869 


8913 


6956 


9000 


9043 9087 


44 






S5 


8 


913 


9174 


9818 


9861 


9305 


9348 


9398 


9435 


9479 9SK 


44 








9 


9M5 


9609 


9668 


9696 


9739 


9783 


9826 


9870 


&913 »M7 


t$ 




ftp 


NT 


6 


1 i 8 


3 


4 


3 


6 


7 


8 B ■ 


^ 


^^ 





K„. 


Poiim. 


2 48 45 


Pont*. 


^. 1 






Oi 


u 










St 


6 3T 30 










8 t6 IS 


»I 






N.b.B. 






11 15 
14 3 45 












16 58 30 


1 1 












19 41 15 








K.N.B. 


N.N.W. 




»30 
85 IB 45 


It 


S.S.E. 








18 T 30 












30 56 15 








N.E.b.N. 


N.W.b.N. 




33*5 
36 33 45 




S.B.11.S. 


S.W.I>.S. 






SO !2 30 














41 11 lA 








N.H. 


N.W. 




45 
47 49 45 


n 


S.G. 








50 37 30 






















N.B.b.B. 


K.W.UW. 




56 IS 
S9 S45 




S.E.b.£. 


S.W. b. W. 






61 5S 3C 














64 41 15 








B.N.B. 


W.N.W. 




6T SO 
70 18 45 


l\ 


E.S.B. 










73 T 30 












75 56 15 








B.b.N. 


W.b-N. 




TS 45 
81 33*5 


u 


B.b.3. 








8* «»sn 




















E«t. 


W«t. 


8 


90 




Banl. 


Weit. 



Points. 


Sine 


Cosine. 


Tangent. 


Counf-. 


Secant. 


c™«% 


Pointa. 





1 


0.00001)1) 
e.6!)0796 
a991303 
a.l665S0 


lU.OOOOOO 
9.999177 
9.997904 
9.995274 


0.000000 
8.691319 
8.993308 
0.171847 


11.308681 
n.006608 
I0.B28753 


10.000000 
10.000583 
10.003096 
1000*736 


Infinite. 
11.30930* 
11.008696 
10.833180 


8 






9.890836 
0.38557 1 
9.46ga31 
9.6S74R3 


9.991574 
9.9867afi 
9.980885 
9.973911 


9.89866? 
9,398785 
9.481939 
9.553647 


10.70I33B 
10.601815 
10.518061 
10.4tBS53 


10.(IOtl48H 
10.013314 
10019115 
10.086159 


10.700761 
10.61*189 
10.337176 
10.478513 


1 

6 
6 
6 




H 


9.58? B40 
9.630993 
9.67338T 
9.11 1050 


9.965615 
9.956163 
9.B4S430 
9.933350 


9.617324 
9.674889 
9.787957 
0.777700 


103B8776 
10.335171 
10.878043 
10.888300 


10.031385 
10.043837 
10.054370 
10.066650 


la«17160 
10.369008 
10.386613 
10.888950 


6 

H 




l\ 


9.744739 
9.7TS0?7 
9.808359 
9.S8708* 


9.9198*6 
9.9048S8 
9.888tS5 
9.869790 


9.834893 
9.870199 
9.9U173 
9.957895 


1 a 175107 

10.189801 
10.085837 
10.048705 


10.080151 
10.095178 
10.111815 
10.130810 


10.855861 
10.831973 
10.1916*1 
10.178916 


5 




* 


9.8*9485 


9.849483 
Sine. 


io-oooono 


1O.0OU000 


10.150515 


IO150SI5 






1 C-me. 


Coi^nR. 




CoseE. \ S«™>^. \ \ 










B ^' ■■ 

















18 Table V. Logarithmic Sines, Tangents, 




Hour, or 


[fcgrM. 






"■ ■■ 


J. 


Sice 


COKCV 


T«.8. 


D.9.T 


Co«ng. 


S6a»L 


d: 


Co«nc 1' 


5^ 




o"o 


( 


0.000000 




[1.0UOOUO 




InEmte. 


10.000000 




10.000000jM!« >i 




4 




S.463TS6 


13.536274 


B.463726 


101717 


13.536274 


oooooo 


00 




H 




9 




764756 


235241 


T64756 


293484 


235244 


oooooo 


00 




i! 






1! 




940947 


059153 


940847 


208231 


059153 


oooooo 






*fi 






16 




7-065786 


12.9342U 


7.065786 


161517 


12.931214 


oooooo 


00 




41 






to 




I6JB96 


S37304 


162696 


13I96S 


937304 


oooooo 


00 


000000 55 


40 






ti 




141811 


758123 


241878 


111577 


768122 


000001 


01 


9.999999 54 


3«l 




te 




3088M 


691116 


309825 


99653 


691175 


OUOOOl 




99999953 


S2 






St 




366816 


633184 


366817 




633183 


000001 




99999952 
999999]ai 


28 






36 




417969 


582038 


417971 


76263 


582030 


000001 


01 


11 






40 




463725 


536275 


463727 


6B99S 


636273 


000002 


01 


999999 


50 20 






4i 






494892 


505120 


62981 


494880 


000002 




999998 


49 16 






4B 




548906 


457094 


542909 


57934 


457091 


000003 


01 


999897 


ia iH 




6» 




577668 


422332 


517672 


53642 


42232B 


000003 


ot 


999997 


47 8 






56 




609853 


390147 


6098S1 


49939 


390113 


000004 






4e 4 




1 


i3,7.b8981U 


l2.3li01B4 


7.639820 


46715 


12.360180 


10.000004 




9.9»99»6 4A{9S 




4 




667845 


332150 


661049 


439B2 


332151 


000005 


01 


999995 44| K 






9 




694173 


305821 


694179 


41313 


305821 


000005 


01 


999995 43 


** 






li 




718997 


28IO0a 


719003 


39136 


280997 


000006 




999994 48 


46 






16 




74gl77 


2375!; 


742484 


37128 


257511 


000007 




999993] 4 


**\ 






m 




764154 


23524«l 


764161 


35136 


23523! 


000007 


01 


999993 


40 


U 






u 




785943 


! 14057 


785951 


33673 


21104B 


000008 


01 


99999S|39 


» 






*b 




806146 


193954 


800155 


32116 


1938*5 


ooooos 


01 


99999139 


Si 






3! 




825451 


17454! 


825460 


3080G 


174540 


OOOOIO 


01 




ft 






3fl 


>4 


8*3834 


15606l< 


843944 


29548 


166066 


000011 


02 


999989JS( 


t 






40 


15 


861662 


13833a 


861674 


28389 


138326 


000012 


02 




ti 






44 


ifl 


87B69S 


12130.5 


879708 


27318 


121292 


000012 


02 


S99988 




l« 






48 


*7 


895085 


104915 


893099 


26324 


101901 


000013 


02 


999987 




M 






6i 


!8 


910879 


089181 


910894 


25400 


089106 


000011 




899966 




e 






AS 
"S~0 


30 


926119 


0738B1 


926134 


24539 


OT3866 


000015 


08 


999985 


S 


1 

5tt I 




7.S40842 


12.05915! 


7.940958 




12.05914^ 


10.000017 


02 


9.999993 




4 


31 


955082 


04491* 


955100 


22981 


044900 


0O0U18 


02 


999QS8 




U 






8 


32 


969970 


03113(1 


969989 


22214 


031111 


000019 08 


999991 




{: 






II 


33 


Oet233 


017761 


982253 


21609 


017741 


000020 02 


999980 




tt 






IS 


34 


995 I9S 


004902 


995219 


20992 


001181 


000021 


02 


999979 




* 






CO 


3i 


&0O7787 


11.992213 


*. 007809 


20391 


11.992191 


000023 


08 


999977 




U 






24 


36 


020021 


979979 


020045 


19832 


919955 


000021 


02 


999916. 




» 






28 


3T 


031919 


968081 


031945 


19304 


968065 


000025 


02 


999975 




s 






St 


38 


043501 


95649 » 


043527 


18802 


956473 


000027 


02 


999973 98 


t» 






36 


39 


054781 


945219 


05*809 


1B326 


945191 


00002B 


02 


999918 


21 








40 


40 


065176 


934224 


065806 


11813 


934194 


000029 




999971 




ti 






44 


41 




923500 


076531 


17443 


923469 


000031 02 


99996S 




t( 






4« 


4( 


086965 


913035 


086997 


17033 


913003 


000032 02 


90996B 




1: 






£1 


43 


097183 


902917 


097217 


16640 


90278; 


000034 08 






i 






&S 


44 


107167 


892833 


107202 


16267 


992797 


0OOU36 03 


999964 










T-0 


43 8.116986 


11.883074 


1.116963 


15909 


11.883037 


ia000037 03 


9.999963 




5j— i 






4 


4« 


126471 


873520 


126510 


15567 


873490 


000039 03 


S99961 




« 






8 


47 


135810 


864190 


135851 


15240 


864149 


000041 03 


999959 




£ 






\l 


4fl 


144953 


855041 




11926 


866004 


000012 03 


999958 




« 






18 


49 


153907 


846093 


153952 


14624 


846048 


00001*03 


999956 




l< 






»0 




162681 


837313 


162727 


14334 


837273 


000046 03 


999954 




K 






2« 


fit 


I1I280 


828720 


171328 


14056 


829672 


000048 03 


99995S 




• 






n 


«» 


179713 


820287 


179763 


13789 


820237 


000050 03 


999950 




» 






3> 


63 


187995 


812015 


188036 


13530 


811964 


000052 03 


999918 




f 






36 


54 


196102 


8039 9H 


196156 


132B2 


803B44 


000064 03 


999916 




P 






40 


U 


204070 


795930 


201126 


13043 


795871 


000066 OS 






V 






4t 


56 


2U8B6 


798105 


211953 


12918 


788047 


000058 04 


999941 




l< 






48 


57 


219581 


780419 


219641 


12509 


780359 


000060 04 


9999*0 




It 






fit 


59 


227134 


772861 


227195 


12374 


772805 


000062 04 


999938 




f 






t6S9 


2345ST 


765443 


234621 


12166 


765379 


000064 04 


999936 




4 






* 060 


241855 


758145 


841921 


11966 


T580I9 


000086 04 


999934 





S« ( 


1 




Coung. 




T..8. 


COKC. 


9» 


Sine, i • 
cpta. 


»■ » 




8 


J 


"-/r/i^S! 


[• 1 15" 
2 30 


3560 1 !■ 1 15"" 


ri ', 


P.M. 

or' 


mi 1 a 1 *o jiwiiH) 


3 1 15 






y 


1 




i 




1 



and Secants. 



Table V. 



19 



;>=l= 



yHoor. 



or 



1 Degree. 



Sine. 




4 
8 
12 
16 
20 
H 
28 
32 
36 
40 
44 
48 
52 
66 



0*8.241855 



1 
2 

3 
4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 



5 



15 



416 
8 
12 



17 
18 
19 
20 



16 
20, 

24<,2l 
2822 
3223 
3624 
4025 
4426 



249033 
256094 
263042 
269881 
276614 
283243 
289773 
296207 
302546 
308794 
314954 
321027 
327016 
3329241 



4827 

52i28 
5629 



8.338753 
344504 
350181 
355783 
361315 
366777 
372171 
377499 
3812762 
387962 
393101 
398179 
403199 
408161 
413068 



Cosee. 



11.758145 8.241921 



750967 
743906 
736958 
730119 
723386 
716757 
710227 
703793 
697454 
691206 
685046 
678973 
672984 
667076 



Tang. 



D.S.T.| Cotong. 



249102 
256165 
263115 
269956 
276691 
283323 
289856 
296292 
302634 
308884 
315046 
321122 
327114 
383025 



11.661247 8.338856 



6 0.308.417919 



431 
832 
12*33 
16;34; 
20,35 
2436 
28*37 
32 38 
36'39 
4040 
44*41 
48*42 
5243 
5644 



422717 
427462 
432156 
436800 
441394 
445941 
450440 
454893 
459301 
463665 
467985 
472263 
47-6498 
488693 



655496 
649819 
644217 
638685 
633223 
627829 
622501 
617238 
612038 
606899 
601821 
596801 
591839 
586932 



344610 
350289 
355895 
361430 
366895 
372292 
377622 
382889 
388092 
393234 
398315 
403338 
408304 
413213 



11965 
11770 
11582 
11400 
11223 
11052 
i0885 
10724 
10568 
10416 
10268 
10124 
9984 
9849 
9716 



1 1.582081 
577283 
572538 
567844 
563200 
558606 
554059 
549560 
545107 
540699 
536335 
532015 
527737 
523502 
519307 



7 045-8.484848 



8 



4'46 
8*47 
1248 
16!49 
2050 
2451 
28.52 
82 53 
36 54 
4055 
4456 
48 57 
52:58 
56*59 
060 



m. s. 



488963 
493040 
497078 
501080 
505045 
5089741 
512867 
516726 
520551 
524343 
528102 
531828 
535523 
539186 
542819 



Coeine. 



9588 
9463 
9340 
9222 
9106 
8993 
8883 
8775 
8670 
8567 
8467 
8369 
8274 
8180 
8089 



Secant. 1 D. 



11.75H0T9 
750898 
743835 
736885 
730044 
723309 
716677 
710144 
703708 
697366 
691116 
684954 
678878 
672886 
666975 



^.418068 
422869 
427618 
482315 
486962 
441560 
446110 
450613 
455070 
459481 
463849 
468172 
472454 
476693 
480892 



11.515152 
511037 
506960 
502922 
498920 
494955 
491026 
487133 
483274 
479449 
475657 
471898 
468172 
464477 
460814 
457181 



Secant 



8000 
7912 
7826 
7748 
7660 
7580 
7502 
7425 
7349 
7276 
7203 
7132 
7063 
6995 
6928 



04 
04 
04 
04 
04 



10.000066 
000068 
000071 
000073 
000075 
000078! 04 
000080 04 
000t)H2'0l 
0000851 04 
000087"; 04 
000090 04 



Cosine. 



000093 
000095 
000098 
000101 



11.661144 
655390 
649711 
644105 
638570 
633105 
627708 
622378 
617111 
611908 
606766 
601685 
596662 
591696 
586787 



04 
04 
04 
05 



10.000103 
000106 
000109 
000112 
000115 
000118 
000121 
000124 
000127 
000130 
000133 
000136 
000139 
000142 
000146 



8.485050 
489170 
493250 
497293 
501298 
505267 
509200 
513098 
516961 
520790 
524586 
528349 
532080 
535779 
539U7 
543084 



Cotang. 



6862 
6798 
6735 
6673 
6612 
6552 
6493 
6435 
6379 
6323 
6268 
6215 
6162 
6110 
6059 
6008 



11.5819321 
577131 
572382 
567685 
563038 
556440 
553890 
549387 
544930 
540519 
536151 
531828 
527546 
523307 
519108 



05 
05 
05 
05 
05 
05 
05 
05 
05 
05 
05 
05 
05 
05 
05 



9.999934 6U 
99993^ 59 
999929 5S 
999927 57 
999925 56 
999922 55 
999920:54 
99991S 53: 
999915 52 
999913 51 
999910-50 
999907:49 
99990548 
99990-2'47 
999899146 



56 



9.999897145 
999894 41 
999891 43 
9998H8 42 
999885 (1 
999882 
999879 
999876 
999873 
999870 
999867 
999864 
999861 
999858 
99985 i 



10.000149 
000152 
000156 
000159 
000162 
000166 
000169 
000173 
000177 
000180 
000184 
000188 
000191 
000195 
000199 



11.514950 
510830 
506750 
502707 
498702 
494733 
490800 
486902 
483039 
479210 
475414 
471651 
467920 
464221 
460553 
456916 



06 
06 
06 
06 
06 
06 
06 
06 
06 
06 
06 
06 
06 
OQ 
06 



DO 



9.999851 
999848 
999844 
999841 
999838 
999834 
999831 
999827 
999823 
999820 
999816 
999812 
999809 
999805 
999801 



Tang. 



5 Hours, 



10.000203 
000207 
iD00210 
000214 
000218 
000222 
000226 



07 
07 
07 
07 
07 
07 
07 



0002311 07 



000235 
000239 
000243 
000247 
000252 
000256 
000260 
000265 



Cosec. 



or 



07 
07 
07 
07 
07 
07 
07 
07 



9.999797 
999793 
999790 
999786 
999762 
999778 
999774 
999769 
999765 
999761 
999757 
999753 
999748 
999744 
999740 
999735 



iO 
39 
38 
37 
36 
35 
34 
33 
32 
31 

30 
29 
28 
27 
26 
25 
24 
23 
22 
21 
20 
19 
18 
17 
16 

15 
14 
13 
12 
11 
10 
9 



Sine. 



54 



53 



8 
7 
6 
5 

4 
3 
2 
1 




i\ 
5(i 
5.' 
4^ 
41 
40 
3t 
Hi 
2> 
2( 
20 
1< 

-: 

6\ 

4^ 

41 

4(1 

3(' 

3- 

2^ 

24 

21. 

Ui 

U 

_4 

~~0 
56 
52 
4» 
44 
4(1 
36 
32 
2h 
24 
20 
16 
12 
8 

_4 

(i 
56 
52 
48 
44 
40 
36 
32 
2h 
24 
20 
16 
12 
8 
4 




111. 8. 



88 Degrees. 



P. P. to 



8 or 



// 




1200 
2400 
3600 



I 



1» 

2 

3 




1200 

24(yo 

3C00 
















18 Table V. Logarithmic Sinet, TangenU, 


Hour, or Degree 


o 




Sm^ 


Co.cc 


■i'-B- 


0.8. T 


':"""«■ 


Sfcub 


D^ 


Cari 




~ 


Sbooooo 


TnKi^liT 


3.000000 





InGn.Le. 


10.000000 


~~ 


laoo 




i 


: 


6.463726 


13.536274 


9.463726 


501717 


13,536274 


000000 


00 


00. 




8 


! 


764756 


235214 




293464 


235214 


000000 




00 




IK 


3 


H40S47 


059153 


910B47 


208231 


059153 


000000 


00 


00 






■ 


7.065786 


li.934211 


7.0657S6 


161517 


12.934214 


oooooo 


00 


00 




10 




162696 


837301 


162696 


131969 


83T304 


000000 


00 


00 




Si 




241877 


758123 


211878 


111577 


75B122 


OOOOOl 


01 


9.99 




S8 


7 


30B824 


691176 


30S825 


99653 


691175 


OOOOOl 


01 


99 




3S 


B 


SeOBlG 


633184 


S66S17 


85254 


633183 


OOOOOl 




S9 




36 


9 


417068 


682032 


417971 


76263 


6S2030 


OOOOOl 


01 


99 




40 


10 


463725 


536275 


463727 


6B988 


536273 


0000U2 




99 




44 


II 


5051 IB 


491882 


505120 


62981 


1948B0 


000002 




69 




4B 


12 


512906 


457094 


542909 


57934 


457091 


000003 


01 


S9 




5! 


13 


57766S 


122332 


577672 


53612 


422328 


000003 


01 


99 




56 
1 


14 

1^ 


609853 


390117 


609857 


19939 


390113 


OOOOOl 


01 


99 


7.6398 lU 


12.360181 


7.639820 




12-360180 


laoooooi 


w 


9.99 








667845 


332156 


667819 


138B2 


332151 


000005 


01 


99 


1 


g 


11 


094173 


30SS27 


694179 


4137.1 


305S2I 


000005 


01 


99 


I 


U 


18 


718997 


281003 


719003 


39136 


280997 


000006 




89 


I 


16 


19 


748477 


257523 


742484 


37128 


2575161 


000007 


01 


89 


I 


SO 


20 


764754 


23524& 


764761 


35136 


235239 


000007 


01 


99 


I 


24 


21 


785943 


21*057 


785951 


33673 


2140*9 


000008 


01 


89 


■ 


!B 


22 


B06146 


193851 


806155 


32176 


193915 


000009 




99 


I 


Si 


23 


825451 


174549 


825460 


30806 


17*510 


OOOOIO 


01 


08 


I 


36 


24 


643934 


156066 


843944 


29518 


156056 


OflOOIl 


02 


S9 


w 


40 


is 


861662 


138338 


861674 


28389 


139321 


000012 


OS 


99 




44 




878695 


12130. 


87S708 


27318 


121292 


000012 


02 


99 




4B 


27 


895085 


1049 15 


895099 


26321 


101901 


000013 


02 


90 




6t 




910879 


089121 


910891 


25400 


089106 


000011 


02 


9Bl 


■ 


't~d 


30 


9261 19 
7.940842 


073881 


926131 


24539 


07386(J 


000015 


02 


9« 


12.059158 


^9i0858 


23731 


12.059112 


10.000017 


W 


Tm. 


1 


i 


31 


955082 


01491b 


955100 


229B1 


014900 


000018 


02 


98 


■ 


s 


32 


968870 


031130 


96B88U 


22271, 


031111 




02 


99; 


I 


It 


33 


962233 


017767 


982253 


21609 


017747 


000020 


02 


98 


■ 


16 


34 


095198 


004B02 


995219 


209B2 


0047B1 


000021 


02 


99: 




!0 


SS 


8.007787 


11.992213 


8.007309 


20391 


11.992191 


000023 




98: 




24 


36 


020021 


979979 


020045 


19832 


97995i 


000021 


02 


9U: 




is 


3; 


031919 


968031 


0319*5 


19304 


968055 


000025 


02 


99: 




3S 




043501 


956199 


043527 


18902 


956473 


000027 


02 


991 




38 


3! 


054781 


91521S 


054809 


18326 


945191 


000029 


02 


891 




40 




065776 


934221 


065606 


17873 


934194 


000029 


02 


9ft 




44 


41 


07S5O0 


923500 


076531 


17443 


923169 


000031 


02 


89: 




46 


42 


086965 


913035 


086997 


17033 


913003 


000032 


02 


901 




5« 


43 


0971tt3 


908817 


097217 


10640 


902783 


000031 


02 


081 


1 


fi6 


44 


107167 


892B33 


107202 


1B267 


892797 


000036 


03 


901 


■s~o 


U 


8.1 16926 


11.883071 


1. 116963 


15909 


11.883037 


10.000037 


03 


9.991 


1 


4 


46 


126471 


873529 


126510 


1556T 


873490 


000039 


03 


081 


1 


fl 


47 


135810 


664190 


135651 


152*0 


861149 


000011 


03 


981 




11 


4B 


144933 


855047 


144996 


14926 


855004 


000042 


03 


99! 




10 


4S 


1.S3907 


846093 


153952 


14624 


816048 


OO0044 


03 


091 




to 


SO 


162681 


S37319 


16(727 


14334 


837273 


000046 


03 


991 




24 


51 


171280 


828720 


171328 


11056 


B28672 


000049 


03 


801 




te 


51 


179713 


880287 


179763 


13788 


820237 


000050 


03 


991 


L 


3t 


53 


187985 


812015 


1B8036 


13530 


611964 


000052 


03 


991 


■ 


ss 


54 


196102 


60389f 


196156 


13282 


603844 




03 


m 


■ 


40 


55 


204070 


795930 


201126 


13043 


79587* 


000056 


03 


991 


r 


44 


56 


211895 


788105 


211953 


12812 


5880*7 


000059 


0* 


»m 




48 


y. 


219581 


780419 


219G11 


12509 


780359 


000060 


0* 


99! 




6S 




227134 


772806 


227195 


12374 


772805 




04 


991 




M 


5t 


234557 


765443 


231621 


12166 


765379 


OOOOfi* 


04 


90) 


i_ 


241855 


758145 


211921 


11965 


75B079 


000066 


04 


881 


^H 


"^r 


Geiul. IColang. 




T,ng. 


c™=. 





_aa 


^^B 


^Houn, 





89 D 


Egnn, 




'I /5" 1 3560 


1' 


15" 


SS«I 1 1' 1 15" 


\—i 


^^^^H I SO 1 im 


* 


30 


\ IWO \ 4 \ ■iO 


ii 


^■^L ' ' 


'"WHO 


3 1 


V. 


\ lOG'iX \ -i \ V. 




^1 


■ 










1 



r' 










18 Table V. LogBrithmie Sines, Tangenw, | 


Hour, 01 


ODfBMfc ^ 


nTTJ 




Sloe. 


COSK. 


T.nti. 


D.S.T 


Coang. 


Secant. 


D. 


Corine. | ' 


SbJ'J 


"o"^ 


1 


0.000000 




0.000000 


__ 


.n6m«. 


lO.OOOOOO 


„ 


labooooo 


m15«-T) 




4 




6.463786 


13.336874 


S.463786 


501717 


1153687* 


000000 


00 


oooooo 


59 


M 




8 




764736 


835844 


764756 


893484 


835244 


oooooo 


00 


oooooo 


3S| td 




It 




940847 


OJ9153 


9*0847 


808831 


05915: 


000000 


00 


oooooo 


51 


M 




16 




7.06i786 


18.934814 


7.065786 


161517 


18.934814 


oooooo 




oooooo 


36 


U 




« 




168B96 


83730* 


168696 


[31969 


837304 


oooooo 


00 


oooooo 


551 td 




2* 




841877 


738183 


811878 


111577 


738182 


000001 


01 


9.999999 


5* 


» 




W 




308884 


69U7G 


308883 


99653 


B9U75 


OOOOOI 


01 


999999 


53 


SI 




32 




366816 


633184 


366817 


85851 


633181 


OOOOOl 


01 




38 


tl 




36 




417368 


388038 


417971 


76863 


588030 


000001 


01 


999999 


41 


ft 




40 




463785 


336873 


463787 


68988 


536873 


000002 


01 


999998 


SO 


tt 




14 




£03118 


49*888 


505180 


68981 


49188U 


000008 


01 




49) 15| 




48 




3*8906 


437094 


348909 


5793* 


457091 


000003 


01 


99999' 


4i II 




Bi 




377668 


488338 


377678 


336*8 


488388 


000003 


01 


999997 


47 t 




£6 
1 




009B53 


3901*7 


609857 


10939 


390113 


000004 


01 


999996 


41 ' 




7.0398 III 


18.380 IS* 


7.639880 


16715 


18.360181 


10.000004 


01 


9.999996 


U 


mTS 1 




4 


16 667845 


338136 


667849 


13888 


338151 


000005 


01 


999995 


44 


■9 1 




B 




GB4173 


305B87 


694179 


11373 


303881 


000003 


01 


S9993S 


.1 i\ 




IS 




718997 


881003 


719003 


39136 


880997 


000006 


01 


999991 




16 




748177 


8S7583 


748484 


37188 


857316 


000007 


01 


999993 


4 


M 1 




80 




764T54 


835846 


764761 


35136 


835839 


000007 


01 


990993 


401 M 1 




84 




T83943 


814057 


785951 


33673 


814019 


000008 


01 


999998 


se 


t 




!S 


Si 


806 US 


1B383* 


806133 


38176 


193845 


000009 


01 


99999 


38 




S8 


83 


885431 


17451H 


883460 


30800 


174540 


000010 


01 


099990 


37 


« 1 




se 


!4 


843S34 


156066 


843944 


895*8 


156056 


000011 


08 


999989 


36j t4 1 




40 


*5 


861668 


138338 


86167* 


88389 


13B386 


000018 


08 


999988 


U 


M 1 




44 


86 


878693 


18l30i 


878708 


87318 


181898 


000018 


08 


999988 


N H 1 




48 


8T 


6S308S 


1049 IS 


895090 


86381 


10*901 


000013 


08 


999987 


ul \i 1 




fit 


88 


910879 


089181 


910S91 


85400 


089106 


OOOOll 


08 


999986 


32 


* 1 




fiS 


t9 


986119 


073S8I 


986134 


84539 


073861 


000013 


08 


9999BS 


31 


A \ 


T^ 


3( 


7.940818 


18.059151 


7.940858 


83734 


18.0591*1 


10.000017 


08 


9.9999S3 


MjftTil 1 




4 


31 


S3S088 


011918 


933100 


88981 


0*4900 


000018 


08 


999991 


19 






8 


38 




031130 


968889 


88874 


031111 


000019 


08 


99g9»l 


tt 






It 


33 


088833 


017767 


988853 


81609 


0177*7 


000080 


02 


999980 


ST 






16 


S4 


993198 


00*808 


993819 


80988 


004781 


000081 




999979 


10 






to 




8,007787 


11.998813 


1.007809 


80391 


11.998191 


000023 


08 


9999T7 


13 






14 




080081 


979979 


080045 


19838 


979955 


000084 


08 


999978 


14 






S8 




031919 


968081 


031945 


19304 


968035 


000085 


08 


999973 


23 






3S 




043501 


956*99 


043387 


18808 


956473 


000087 




999973 


11 






36 


3S 


054781 


943819 


05480B 


18386 


945191 


000088 


08 


999978 


11 






40 


40 


064776 


93488* 


065806 


17873 


934194 


000089 


08 


999971 


H 






44 


41 


076300 


983SO0 


076531 


174*3 


98346S 


O0003I 


02 


999969 


IB 






4B 


48 


0H6S63 


913035 


08699T 


17033 


913003 


000038, 08 


999968 


u 






6t 


43 


097183 


B08S17 


097817 


15640 


908783 


000034' 08 


999966 


IT 






se 


U 


107167 


898833 


107808 


16867 




000036 03 


999964 


l« 






TH 


43 


8.116986 


11.883071 


1.11 6963 


15909 


11.883037 


la0O0O37 


M" 


9.999963 


Ti 


vr 




4 




186471 


87 338 S 


186510 


15367 


87349D 


000039 


03 


999961 


i« 






8 




133810 


864190 


135851 


15840 


861 lit 


000041 


03 


999939 


19 






It 




144953 


855047 


14499b 


14986 


833001 


000012 


03 


999958 


II 






16 




133907 


8*6093 


133958 


11684 


84604* 


000041 


03 


999936 


11 






to 




1626SB 


8373 IS 


168787 


14334 


637873 


000046 


03 


99995* 


10 






S4 


SI 


171880 


888780 


171388 


1*036 


888678 


000048 


03 


999952 


II 






ta 


51 


179713 


850887 


179763 


13TBB 


880837 


000050 


03 


999930 


8 






31 


as 


181983 


818013 


188036 


13530 


811964 


000058 


03 


999948 


T 






36 


u 


19610* 


B03898 


196156 


13888 


803844 


000034 


03 


999946 


e 






40 


M 


104070 


793930 


801186 


13043 


795874 


000056 


03 


999944 


c 






U 


u 


111893 


788105 


811953 


18S18 


788047 


000058 


04 


999941 


4 






M 


$T 


tlB381 


780419 


819641 


18309 


780359 


OOOOGO 


0* 


999910 


a 






U 


» 


117134 


778866 


887193 


18374 


778805 


000068 


04 


999938 


1 






M 


fiff 


13*357 


765443 


83*681 


*«79| 


000084 


01 


999936 








4 


to 


»1B53 


75B14S 


8*1981 




3 


000066 


01 


999934 


_.»« 1 


. 


r~ 


Cdne. 1 Senxt. | Cotu- 






Cowc. 


— 


Si„^ 


^lOl 




Sama, 


7 




OD 


^'"^-ylJ 


^L__ 


ii^i ism.. 


I'Ul 


^^^^i ** iggf^^^^^ 


4Jh 


^iil 



t^ 


















-^ 












— ; ohsif; sr — — ~sDi 


p- 






__t 




Sine. 


D. 


CoKc 


T««. 


D. 


Coung. 


S«int. 


D. 


Cotine. 




"■ *■ 









1 


S,94089t, 


8*C0 


11.059701 


8.941958 


8481 


11.058048 


10.001656 


l9 


9^983*4 


bS 


10 (J 






4 


1 


9*1738 


!394 


OSH262 


9*3*0* 


8*13 


056596 




19 


998333 


>8 


56 






fl 


i 


843! 1* 


8397 


056886 




2405 


055118 


001678 




998321 


iH 


58 






18 




94«)0G 


8379 


055394 


9*6895 


8397 


053705 


001689 




996311 


57 


48 






J6 


i 


91603* 


8371 


053966 


8*773* 


2390 


058866 


O0170O 


19 


998300 


56 


41 






«0 


5 


947*56 


8363 


0585*4 


9*9168 


8382 


050838 


001711 




99828(1 


55 


10 






«4 


e 


94S87* 


8355 


051 126 


950597 


837+ 


019*0; 


001723 




998877 


St 


3€ 






se 


1 


950887 


83*6 


0*9713 


952021 


2366 


0*7879 


00173* 




998861 


53 


31 






32 


8 


951 69f 


83*0 


0*830* 


9531*1 


8359 


0+6539 


001745 




99825£ 


58 


28 






36 


E 


953100 


8332 


01S900 


95*856 


2351 


0*5144 


001757 


19 


9962*3 










♦O 


10 


954*99 


S38S 


015601 


956267 


83*4 


0*3733 


O0I768 


19 


99883S 


50 


20 






44 


11 


95589* 


2317 


0*4106 


957674 


8337 


012386 


001780 




998280 


(9 


16 






48 


18 


957881 


8310 


0*8716 


959075 


2389 


010925 


001791 




996209 


Vi 


18 






AS 


ts 


958670 


8308 


041330 


960*73 


2322 


038527 


001803 


18 


998197 


17 








&6 


14 


B60O58 


8895 


0399*8 


961866 


2311 


038134 


00181* 




898196 


16 


4 






.w 


ii 


I.96U89 


8888 


11-038571 


i.96325S;2307 


11.0367*5 


10.001826 


T9" 


9.99817* 


U 


39-0 






4 


IG 


968901 


8880 


0371 9E 


961039 


2300 


035361 


001837 




99816S 


U 


St 






S 




96*170 


2873 


035830 


9S6019 


8293 


033981 






998151 


13 


52 






le 


18 


96S53* 


2266 


034466 


967394 


2886 


032606 


001861 


80 


998139 


18 


*e 






: le 


IS 


966893 


2259 


03310T 


968766 


8279 


031234 


001872 


20 


9961 8f 


*1 


44 






80 


80 


9688*9 


2858 


031751 


970133 


2271 


028967 


00188* 


80 


996116 










84 


11 


969600 


22*5 


030*00 


971196 


8265 


089501 


001896 


80 


998101 


J9 


3« 






!S 




970917 


8238 


029053 


872655 


8257 


0271*5 


001908 


20 


998098 


IB 


3i 






3S 




978889 


8831 


027711 


971809 


i851 


08579! 


001920 


80 


99809(1 


17 


88 






31 




973688 


8821 


0263T8 


975560 


284^ 


08**10 


001938 




98806E 


16 


81 








26 


97196? 


2817 


085038 


976906 


8237 


023094 


00191* 


20 


998056 


U 


80 








S6 


976 !93 


82Ift 


023707 


976818 


2830 


081752 


001956 


80 


9980*1 


U 






E 


48 


n 


977619 


8803 


022381 


979586 


8223 


080114 


001968 


20 


998032 


13 


18 




Si 


88 


978911 


2197 


081059 


930981 


2217 


019079 


001980 


20 


P96O20 


18 


e 




jfc 


56 




980849 


8190 


019J*1 


988251 


2210 


017749 


001992 


80 


998008 


11 


4 






«~« 


30 




2183 


11.018*87 


S.983577 


8201 


11.016*83 


10.00200* 


80 


y.99799( 


to 


38 






4 


SI 


9889B3 


8177 


01711T 


9B1899 


B197 


0151OI 


008018 


80 


99798* 


!8 


M 






8 


18 


984189 


81T0 


015911 


986817 


2191 


013783 


002088 


80 


88797! 


28 


5! 






u 


a 


986491 


2163 


014509 


987538 


2164 


012*68 


0080*1 


80 


997959 


27 


48 






la 


34 


9BG789 


8157 


013811 


988912 


il7S 


011156 


002053 


20 


887917 


26 


44 






80 


Si 


988083 


2150 


0119IT 


980118 


ei7i 


009851 


002065 




997ii3J 


25 


40 






14 


36 


9B937* 


2141 


010686 


881151 


2166 




008076 


21 


99792S 


84 


36 






86 


S7 


990660 


8136 


009340 


992750 


2156 


007850 


008090 


81 


8»791( 


23 


38 






38 


St 


981943 


8131 


008057 


98*015 


2158 


005955 


002103 


81 


9978S7 


28 


H 






36 


IS 


893888 


2185 


0O677B 


995337 


2146 


00*663 


008115 




997885 


81 


14 






40 


k 


99449T 


8II9 


005503 


996681 


mo 


003376 


002128 


81 


897871 


20 


n 






44 


41 


98S7S8 


2II2 


004232 


897908 


213* 


002098 


0081*0 


21 


997860 




16 






4S 


(! 


B97036 


8106 


00896* 


999186 


2187 


000818 


0021.53 


81 


997847 


18 


18 






58 


« 


999899 


2100 


001701 


9.000*65 


2181 


ia998535 


008165 


81 


997B8i 




8 






A6 


14 


999S60;S09* 


000*41 


001738 


8115 


998262 


008178 




997988 




4 






13-0 


Vi 


kOOOSieiSOttS 


10.999181 


).O03O07 


810M 


la886983 


10.008191 


IT 


9.997809 


n 


sTU 








4 


ie 


0080698088 




004872 


2103 


995728 


008203 


21 


997797 




M 








ft 


47 


003318 8076 




00553* 


8097 


994466 


002816 


81 1 99TT81 


1! 


62 








IS 


4t 


00*563*070 
00580518064 


995*37 


006798 


8091 


993208 




81 997771 




49 








16 


4S 


99*195 


0080*7 


8085 


991953 


008218 


21 1 99775t 


M 


41 








SO 


S( 


0070*1 ZD58 


998956 


009298 


2080 


890702 


008255 


81 


9977 IS 


10 












51 


0068788058 


991728 


0105*6 


207* 


889154 


002268 


21 


997732 












18 


58 


009510 80*6 


990*90 


011790 


8068 


988210 


002281 


21 


997719 




3i 








S8 


Si 


01 0737120*0 


98926S 


01 3031 12068 


986969 


00889* 


81 


997706 












as 


M 


011968,8034 


988038 


01126e|2O56 


995732 


602307 




997693 




81 








40 


U 


0131 88 8089 


986818 


015508 


8051 


98**98 


008320 


88 


99768(1 












44 


56 


01HOD80?3 


9B560( 


016738 


20*5 


98326b 


008333 


22 














4fl 


57 


015613,8017 


99*387 


017959 




8820*1 


0023*6 


82 


<I97651 












58 


56 


01688*8018 


983176 


019183 


8033 


980817 


008359 


81 


9976+1 












56 


59 


018031,8006 


981969 


020103 


2088 


979597 


002372 


21 


997688 












!4 


60 


019835 8000 


980765 


021620 


2023 


97838( 


002396 




99761* 




ma 




























"^ 


n 










Cosine. 1 1 


Scam. 


'C^^V. 




Tang. _ 




= Sm.. 






iHout^ or »4IJ^»«. i 




s 




P. P. to 


1- 


15" 


387 




15-' 


330 1 1> 14" 3 J 


t 






a 


30 


655 


a 


30 


661 i 30 ' ' 








ioc" 


3 


45 


S68 


3 


45 


998 ^ % <« 











r 




18 Tmle 




1 


OHour, o[ 


OD^BTM. ■ 1 


STT 


' 


SiDC 


COKC 


T„,^ 


D.S.T 


ro»ng. 


Secant. 


D. 


c™n=. |_l|m._fc 




"o 


0.000000 


Infinile. 


KOOOOOO 




InHnite. 


10.000000 




iaoooono60« < 




4 


1 


6.463T26 


13.536271 


9.463726 


501717 


13.536271 


000000 


00 


000000 59 t> 




6 


9 


764TJ6 


235211 


764756 


293164 


235214 


000000 


00 


000000 58 tl 




IS 


3 


9i084T 


059153 


940817 


20B23I 


059153 


000000 




000000 57 Ifi 




16 


« 


7.085786 


12.934211 


7.065796 


161517 


12.934214 


000000 


00 






H 




162696 


837301 


162696 


131969 


83730+ 


000000 


00 


000000 53 « 




1\ 


( 


241877 


T5B123 


2+1879 


111577 


738123 


OOOOOl 


01 


9.999999 51 » 




If, 


T 


308821 


691176 


308825 


9S653 


691175 


000001 


01 


999999 53 |I 




3! 


9 


366816 


633184 


366817 


65251 


633163 


OOOOOl 




999999 52 M 
999999 61 It 




36 


9 


4ITg68 


582032 


4I797I 




582030 


OOOOOl 


01 




40 


10 


463T25 


536275 


463727 


699BB 


536273 


000002 


01 


999998 50 n 




44 


11 


M5119 


491862 


505120 


62961 


49+88U 


000002 


01 


999998 48 It 




48 




542906 


157091 


542909 


5793+ 


457091 


000003 




999997 48 ll 




5i 


13 


5T7G69 


422332 


577672 


43612 


+22328 


000003 


01 


999997 M I 




56 


14 


609853 


390117 


609857 


+9939 


390113 


OOOOOl 


01 


999990 4e :< 


1 


15 7.63BB1U 


12.360181 


7.639820 


16715 


12.360180 


10.00000+ 


01 






4 




667845 


332156 


667819 


13892 


332151 


000003 




B9099J 44 t 




6 


IT 


694173 


3058J7 


691179 


11373 


305821 


000005 


01 


999995 43 1 




li 


IB 


718997 


2Bt003 


719003 


39136 


260997 


000006 




999994 42 II 




IG 


19 


742177 


a575i3 


71248+ 


37128 


257316 


000007 




99999341 i 




!0 


iO 


76475+ 


835216 


764761 


35136 


235239 


000007 


01 


999993 40 t 




g4 


ii 


7859W 


21+057 


78.^961 


33673 


21101S 


000006 


01 


99999838 % 




«B 


ti 


8061+6 


1B395+ 


806155 


32176 


193945 


000009 


01 


999991 38 « 




31 


83 


825+51 


17451! 


625160 


3080S 


17+510 


000010 


01 


999990 3T ft 




36 


«* 


643934 


I5606t< 


843944 


29518 


156056 


00001 1 




999989 36 1 




40 


25 


861662 


13833a 


861671 


28389 






02 


999988 35 V. 




44 


26 


eT3695 


12130.> 


878708 


27318 


12129! 


000012 


02 


999988 34 |I 




48 


2T 


695085 


101915 


893099 


2632+ 


10+901 


000013 


02 


999967 33 I: 




5t 


2B 


910379 


089121 


910BB1 


25100 


089106 


OOOOIl 


02 


999986 as 1 




£6 

I 


29 


926119 


073881 


92613+ 


24539 


073866 


000015 


02 


999985,31 < 


30 


7.91084? 


12.059158 


r.940958 




12.059112 


I0.0O00I7 


W 


9.999983;30.W~a 






31 


955082 


01+918 


955 100 


2298 1 


01+900 


000018 


02 


999962 


2S i 




8 


n 


968870 


031130 


968899 


22271 


031111 


000019 




999961 


26 I 




li 


i3 


992233 


017767 


982253 


21609 


017717 


000020 


02 


999990 


87 f 




16 


34 


995198 


001802 


995218 


20982 


004791 




02 


999979 


ifl 1 




to 


S5 


8.007787 


11.992213 


8.007909 


20391 


11.992191 




02 


9999T7 


2f i 

84 1 




24 


36 


020021 


979979 


0200+S 


19932 


979955 


0000211 02 


9l>9976 




28 


ST 


031919 


968081 


0319+5 


19304 


968035 


000025 02 


999975 


K % 




3! 


38 


043501 


956199 


013527 


18902 


956473 


000027 02 


999973 


28 1 




36 


39 


05+T81 


945219 


051609 


19326 


915191 


00002B; 02 


999972 


*> \ 




40 


40 


065776 


93122+ 


065806 


17873 


934194 


000029 


02 


99997 1 


to i 




44 




076500 


923500 


076531 


17413 


923+69 


OO0031 




999969 


19 1 




48 




086965 


913035 


086997 


17033 


913003 


000032 


02 


999968 


U 1 




6t 








0S7217 


16610 


902783 


000034 


02 


999966 


11 |l 




56 


M 


107 167 


692833 


107202 


16267 


892797 


000036 


03 


999964 


is| ,. 




3 


45 


8.II692t> 


11.883071 


1.116963 


15909 


11.883037 


la000037 


53 


9.999963 


14 5T < 




4 


46 


ie6471 


673529 


126510 


15567 


8T3490 


000039 


03 


999961 


14^ « 




8 


4T 


135810 


861190 


135851 


15210 


6611+9 


000041 


03 


999959 


13 * 
11 i 




1! 


48 


144953 




111996 


11926 


655004 


000012 


03 


999958 




16 


49 


153907 


846093 


153952 


14621 


6460+8 


00004+ 


03 


999956 


11 




in 


50 


162691 


837319 


162727 


11331 


637273 


0000+6 


03 


999954 


10 




14 


51 


171880 


928720 


171328 


11036 


828678 


000018 


03 


999952 


s 




18 


52 


179713 


820267 


179763 


13799 


820237 


000050 


03 


999950 


C 




St 


53, 


187985 


812015 


199036 


13530 


611961 


000032 


03 


999946 


J 




3« 


H 


196102 


803898 


196156 


13282 


603811 


000054 


03 


999946 


fl 




40 


S5 


204070 


795931 


201126 


130+3 


793871 


000056 


03 


999944 


1 




44 


56 


211895 


78810; 


211953 


12912 


788017 


000058 


04 


999941 


4 




48 


ST 


119581 


780119 


219641 


12509 


780359 


000060 


04 


999910 


J 




6t 


S8 


22713* 


772868 


227195 


12371 


772803 


OOOU62 


01 


999938 


1 I' 




H 


S9 


234557 


765*13 


234621 


12166 


T6537B 


000061 


04 


999936 


> ' 




4 o;6o 


24IB55 


T59t+5 


811921 


11965 


T58079 


OO00B6 


01 


999934 


OM \ 




*^=== 










= 


— - ^==ial J 




». 1.1 • I CorirB. 


Becuit. ICotang. 










Sine. 1 ' !» « 1 
cgres. 1 


fiH<r 


un. «( 


~89I] 

16" 


/ 


■.-.-,' J' M 


"1 3560 


1' 


Is'' 1 


lUlJ 


I '--'IL^ 


1 TIIO 


t 

3 


30 
45 






30 


1 ' 








WBr 






^H 



r 


^^ 


™ 


^^^ 


^™ 






^^^ 












ao Table V. 


Logarithmic Sines, Tangents, 




OHour, 




8IlcM». 




n. s. 




Sme. 


D. 


COBK. 


T-g. 


D. 


Coang. 


BtaukU 


D \ Cosine. 




ra, 1. 


"8~0 





).54SB19 


S004 


riTiSTlBl 


1.543084 


(iola 


IJ.456916 


10.000805 




9.999735 


60 


5n 




4 




5464S! 


S955 


453578 


516691 


5968 


45330a 


000869 


07 


999731 


SB 


» 




8 


K 


fi49995 


;S06 


450005 


550269 


5914 


449738 


000274 


07 


999720 


58 


K 




18 


3 


fiSSSSf 


58591 




553817 


5886 


416183 


000818 


08 


99978? 


51 


if 




16 


1 




5811 


448916 


557336 


58 IB 


448661 


000883 


08 


99971? 


£6 


4 




80 


5 




5765 


439460 


560929 


4773 


439172 


000287 




999713 


55 


to 




et 


6 


563999 


5710 


436001 


564?91 


5727 


435709 


000898 


09 


999708 


SI 


it 




SB 


7 


567431 


5074 


438503 


567787 




438873 


000296 


OS 


99970* 


sa 


^ I. 




3S 


S 


570836 


5630 


489164 


571137 


5639 


486863 


000301 


08 


999698 


M 


X 'i 




36 


9 


fiTliU 




4857S6 




5505 


485480 


ooosoa 


06 


999694 


51 


I' . 




40 






5541 


488434 


577977 


5558 


488183 


000311 


08 


099689 


50 


* 1 




14. 


11 


590898 


5508 


41 Blot 


581808 


5510 


418798 


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999685 


M 


h t 




4H 


le 


£84193 


5160 


415807 


581514 


5168 


415496 


000380 


08 


999680 


W 


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53 


13 


587469 


5419 


418531 


587795 


5187 


418805 


000385 


09 


9996tS 


11 


t 




S& 


14 


59072 i 


5379 


409879 


591051 


5397 


408949 


O0O331 


08 


999670 


14 


. 


g 




i.i9394S 


5339 


11.40605S 


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11.405717 


I0.U0U335 


08 


9:999664 


i 


sTl 




4 


16 


59715S 


530O 


40884f 


59749S 


5309 


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000340 


OS 


099660 


u 


5 




B 


n 


60033S 


5861 


39966( 


600677 


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ooasi5 




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396511 


603839 


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398101 


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999650 


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le 




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999645 


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It 




£0 


io 


609734 


5149 


39086fi 


61D091 


5158 


399906 


000360 


09 


999641 


Ki 


K 




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n 


618883 


5118 


387177 


613189 


5181 


386811 


000365 


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» 


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615B9I 


5076 


38410!! 


616862 


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383738 


1100371 






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61B93T 


S041 


38100^ 


619313 


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390687 


000376 


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a 


68196! 


5006 


ST8038 




5015 


377657 


000381 


09 


999619 


36 


11 




40 




684985 


1078 


375035 


686358 


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3T4648 


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09 


999614 


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ID 




44 


te 


887948 


1838 


378058 


688340 


4947 


371660 


000392 


09 




M 


11 




49 


ti 


830911 


4904 


36908S 


631308 


4913 


368898 


000397 






» 


li 




£2 


w 


833844 


IBTl 


368146 


631256 


4980 


365744 


000403 




999597 


U 


i 




£S 


!S 


8367J6 


4839 


363824 


637181 


1818 


362811 


000408 


U9 


999598 


11 




nro 


30 


).e39690 


4806 


11.360380 


i.640O93 


lBl6 


11.369907 


1O.0O04H 


09 


S.99958t 


10 


i5~i 




4 


SI 


64S563 


4775 


357437 


048988 


1784 


35701 S 


000419 


09 


999581 


n 


b 




e 


3S 


645489 


4743 


354575 


645953 


1753 


354147 


000425 


09 


9995 Tj 


« 


K 




1! 


ss 


848874 


4718 


351 78( 


649701 


1788 


351896 


000430 


09 


B995lfl 


87 


« 1 




le 


34 


851108 


1888 


S46S9F 


651537 


1691 


348463 


000436 


OS 


999564 


M 


1 




so 


SB 


653911 


4658 


346089 


654358 


4661 


31564S 




10 


999556 


85 


It 




S4 


36 


656708 


1688 


343898 


657119 


1631 


348851 


000417 


10 


9995A3 


H 


* 1 




28 


37 


659475 


4598 


340585 


859928 


1608 


340078 


000143 




999547 


83 


9. 1 




3S 


38 


6S8830 


1563 


337770 


668689 


15T3 


337311 


000159 




999541 


18 


i 




SB 


3a 


884968 


1535 


335035 


666133 


1544 


334567 


000465 


10 


09953fi 


11 


1' 




40 


40 


667699 


4506 


338311 


668160 


1516 


331840 


000471 




990fi8( 


90 


» 


■ 


44 


4t 


670393 


447S 


389607 


670S70 


UBH 


389130 






999584 


tft 


K 


4B 


»! 


8730B0 


4151 


3afl92( 


673563 


1461 


386437 


000188 


10 


899516 


IB 


11 




fi! 


18 


675751 


1184 


38184! 


676839 


1134 


383761 


000488 


10 


999511 


n 


t 




fie 


U 




1397 


38l69ji 


678900 


1417 


381101 


000494 




999506 




■ 




m 


45 


i.681043 


1370 


1I.3IS95J 


^691544 


1380 


11.318451 


10.000500 




9.999501: 




m 






46 


683665 


1344 


31633; 


684178 


1354 


315889 


000507 


10 


999193 




St 




e 


4t 


686872 


4318 


31378t 


686784 


4388 


313216 


000513 


10 


999187 


13 


II 




1* 


48 


688863 


1898 


311137 


699381 


1303 


310619 


00051! 




999181 




« 




16 


49 


69I43S 


1867 


30956S 


691 B63 


1877 


308037 


000585 




999175 


11 


44 




20 


30 


693999 


4848 


300008 


694589 


1852 


305171 


000531 




9B9169 


10 


40 




S4 


51 


606543 


1217 


303457 


697081 


1888 


308919 


000537 




999463 




SI 




19 


M 


699073 


1198 


300987 


69B617 


4203 


30038a 


000511 








» 




32 


53 


T01599 


1169 


893411 


702139 


4179 


2B7861 


000550 








e 




36 


54 




4111 


895910 


701646 


4155 


195344 


000557 


11 


999443 




84 




40 


55 


706577 


4121 


89348S 


707110 


1138 


292960 


000563 


11 


999437 


5 


W 




14 


5( 


709049 


1097 


890951 


709619 


1108 


890388 


000569 


11 


999431 


4 


|i 




48fiT 


711507 


1071 


8X8493 


718083 


1095 


£87917 


000576 


11 


999424 


3 


Jl 




Sg58 


713958 


4051 


286018 


714534 


1068 


895465 


00058! 


11 


999119 


1 


t 




Aebss 


716393 


4089 


283617 


716B78 


1040 


883028 


000589 


11 


99S411 I 


4 




u o,ao 


718800 


1006 


881800 


71939B 


1017 


860604 


000596 


11 


999404| 


W ■ 


/ 


m. ».|"' 


rcotine. 1 


Secant. ICotang. 




Tang. 1 Co«c. 




Sine. 1' 


sir~» 


6 Hran, or B 


~Dit 


irea. 




r-^v'"/ > 1 - 


781 
1418 


!■ 1 15" 

2 1 ao 


1 788 1 1- 1 15" 


I !■•- 


p. to 


L 


""' / 3 1 4. 


2163 


S 45 


\ »u-i \ ^ \ ^ \ ^ \ ■ 


or" 





and S6Cftiita> 



19 




4 

8 
12 
16 
20 
24, 
28 
32 
36 
40 
44 



k 



OHour, 



or 



Sine. 



48 
52 
56 



13 



( 



9 

10 
11 
12 
18 
U 





4 
8 
12 
16 
20 
24 
28 
32 
36 
40 
44 
48 
52 
56 



16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 



D. I Cotea. 



7188004006 
7812043984 
723595 3962 
725972 3941 
7283373919 
7306883898 
7330273877 
7353543857 
737667 3836 
739969 3816 
742259 3796 
744536 3776 
746802 3756 
7490553737 
751297 3717 



1518.753528 3698 
755747 3678 
757955 3661 
760151 3642 
762337 3624 
7615113606 
766675 3588 
768828 3570 
770970 3553 
773101 3535 
7752233518 
777333 3501 



14 O.30 8.785675 3434 



431 

832 
1233 
16 



34 
36 



20 

^* . 
2837 

3238 

3639 

40 40 

44^41 

4842 

5243 

56|44 



446 



8 



47 



12481 
16 



49 
50 
51 



7794343484 



781524 
783605 



11.28120C 3.719396 



278796 
276405 
274028 
271663 
269312 
266973 
264646 
262333 
260031 
257741 
255464 
253198 
250945 
248703 



3467 
3451 



787736 
789787 
791828 
793859 
795881 
797894 
799897 
801892 
803876 
805852 
807819 
809777 
611726 
813667 



3418 
3402 
3386 
3370 
3354 
3339 
3323 
3308 
3293 
3278 
3263 
3249 
3234 
3219 



15 045 8.815599 3205 



20 

^* - 

2852 

3258 
36 54 

40,55 
44'56 
48'57 

52 58 



16 



56 




59 
60 



m. 



817522 
819436 
821343 
823240 
825130 
827011 
828884 
830749 
832607 
834456 
886297 
838130 
839956 
841774 
843585 



244253 
242045 
239849 
237663 
235489 
233325 
231172 
229030 
226899 
224777 
222667 
220566 
218476 
216395 



Tang. I D, 



Tablk V, 
31>cgrccfc"^ 



721806 
724204 
726588 
728959 
731317 
733663 



4017 
3995 
3974 
3952 
3030 
3909 
3889 



735996,3868 
738317:3848 
7406263827 



742922 
745207 
747479 
749740 
751989 



11.246472 8.754227 



11.214325 8. 
212264 
210213 
208172 
206141 
204119 
20210G 
200103 
198108 
196124 
194148 
192181 
19022 
188274, 
186333 



Cosine. 



3191 
3177 
3163 
3149 
3135 
3122 
3108 
3095 
3082 
3069 
3056 
3043 
3030 
3017 
3005 



11.184401 8. 
182478 
180564 
178657 
176760 
174870 
172989 
171116 
169251 
167393 
165544 
163703 
161870 
160044 
158226 
156415 



Secant. 



756453 
758668 
760872 
763065 
765246 
767417 
769578 
771727 
773866 
775995 
778114 
780222 
782320 
784408 



3807 
3787 
3768 
3749 
3729 



.786486 
788554 
790613 
792662 
794701 
796731 
798752 
800763 
802765 
804758 
806742 
808717 
810683 
812641 
814589 



816529 
818461 
820384 
822298 
824205 
826103 
827992 
829874 
831748 



3710 
3692 
3673 
3655 
3636 
3618 
3600 
3583 
3565 
3548 
3531 
3514 
3497 
3480 
3464 



3447 
3432 
3415 
3399 
3383 
3368 
3352 
3337 
3322 
330? 
3292 
3278 
3262 
3248 
3233 



3219 
3205 
3191 
3177 
3163 
3150 
3136 
3123 
3110 



8336133096 



835471 
037321 
839163 
840998 
842825 
844644 



Cotang. 



3083 
3070 
3057 
3045 
3032 
3019 



Coung . I Scm nu j D. | 

1 



Codn& 



11.280604 
278194 
275796 
273412 
271041 
268683 
266337 
264004 
261683 
259374 
257078 
254793 
252521 
250260 
248011 



11.245773110 
2435^17 
241332 
23912b 
236935 
234754 
232583 
230422 
228273 
22613-i 
224005 
221886 
219778 
217680 
215592 



11 



.213514 10 

211446 

209387 

207338 

205299 

203269 

201248 

199237 

197235 

195242 

193258 

191283 

189317 

187359 

18541 1 



n 



.183471 
181539 
179616 
177702 
1 75795 
173897 
172008 
170126 
168252 
166387 
164529 
162679 
160837 
159002 
157175 
155356 



Tang. 



10.000596 
000602, 
000609, 
000616, 
000628; 
000629 
000636 
000643 
000650 
000657 
000664 
000671 
000678 
000685 
000692 



,000699 

000706 

000714 

000721 

000728 

0007351 

000743 

000750 

000758 

000765 

000773 

0007H0| 

0007Hh| 

000795 

000803 



.000811 
000819 
000826 
000834 
000842 
000850 
000858 
000866 
000874 
000882 
000890 
000898 
0<J0906 
000914 
000923 



10.000931 
000939 
000947 
000956 
000964 
000973 
000981 
000990 
000998 
001007 
001016 
001024 
001033 
001042 
001050 
001059 



Cosec. 



2 
2 
2 
2 
2 
2 
2 
2 
2^ 

2 
2 
2 
2 

2 
2 
2 
3 
3 
3 
3 
3 
3 
3 
3 

3 
3 
3 
3 
3 
3 
3 
3 
3 
3 
3 
3 
4 
4 



9.999404 60 
999398 59 
999391 
999384 57 
999378 56 
999371 
999364 54 
999:i57|53 
99935 
9993 
99933 
999329|'i9 



81 



D1. f 



48 U 
56 
6i 




999322 



999315 47 



999308 



9.999301 45 
999294 44 



999286 
999279 
999272 



43 
48 
41 



999265 40 



999257 



lU) 



999250 :i8 
999242 37 
999235 36 
999227 35 
999220 34 
999212 33 
999205!38 
999197 31 

97999189 
999181 



30 
29 

999174 88 



999166 
999158 
999150 
9991] 



42 24 
999134 23 
999126 22 



9991 
9991 i 



999102 19 
999094 18 
999086 17 



999077 



9.999069 
999061 



999053 13 
999044 18 
999036 11 



999027 
999019 
999010 
999002 
998993 
998984 
998976 
998967 
998958 
998950 
998941 



Sine. 



48 



46 



44 

40 
36 
32 
2H 
84 
20 
16 
12 



i7 ( 



87 
26 
25 



18 21 
10 20 



16 



15 
14 



16 



56 
52 
48 
44 

40 
36 
32 
2ti 
24 
20 
16 
12 
8 
_4 


56 
52 
48 
4i 
40 
36 
32 
28 
24 
20 
16 
12 
8 
4 



45 



10 
9 
8 
7 
6 
5 
4 
3 
2 
1 
044 






5 Hours, 



or 



P. P. to 



8 or 



// 



2 
3 



15" 

30 

45 



515 
1030 
1544 



1* 

8 

3 



15" 

SO 

45 \ 1551 



>17 I 1' I 1 
)34 I 2 \ 3 

551 \ ^ \ 



517 
1034 



86 D e grees. 

5"' 




56 
58 
48 
44 
40 
36 
38j 
28 

!* 
20 

16 

12 

8 

4 





01. 8. 




r- 












' 


aa TiflLfi V. 


Logarithmic Sines, Tanger 


ta. 


1 


1 


OHour, 


or 




4Degr™. ~I 


n. s. 




Sine. 


D. 


Cora. 


3i!8:_ 


ID. 


Coang. 


Seoinl. 


D.| C^in... 1 


' m. 1 


n 


1 


«.e43&H5 


3005 


11.156415 


j.84464^ 


m 


U. 155336 


idoOHiSB 


15 


9.99S94I 


^U 1 




4 




845387 


399* 


154613 


846455 


3007 




001068 




99tt93S 


9 5. 




S 




847183 


*980 


138817 




8995 


1S174( 


0OIO73 




sssan 


8 i; 




1! 


s 


848B71 


2967 


1S1029 


850057 


*9se 


149943 


001086 




998911 


7 41 




16 


■ 


BS0T.51 


8955 


149249 


831846 


8970 


14BI34 


0011)93 




99B905 


6 4) 








65S52S 


*94a 


1474T3 


853628 


*B58 


146372 


001104 




B0889« 


3 U 




2 


i 


■ 854891 


2031 


145709 


835403 


8946 


144597 


001113 




99S881 


1 31 




ZH 




8fi604B 


S919 


143931 


857171 


8935 


148889 


001122 


IS 


99S878 


3 31 




3? 


B 


BSTeOI 


8B08 


142189 


86B932 


8983 


141068 


0011311 15 


998869 


8 81 




36 


9 
10 


859540 
861893 


8896 
8884 


I40+S4 
138717 


860686 
86*433 


8911 

8900 


1393U 

137567 


001140 




998860 
99SS5I 


1 M 

•0 « 


001149 










86S()I4 


8873 


1369SS 


864173 


2888 


135887 


001 I 59 




998S41 


9 11 






18 


864738 


8861 


135868 


863906 


8877 


134094 


001 168 




99H832 


« 1: 




5! 


13 


866455 


2860 


133545 


887632 


88S( 


138368 


001177 




998S8S 


7 ( 




ae 


14. 


8S8165 


2839 


131835 


869351 




130649 


001187 




9998 IJ 


( 


IT 


Ifi 


t.8GB86S 


2888 


11.13013a 


!,87I064 


8843 


11.128936 


10.O0I196 




9.998aM 


1.5*3 1 






IB 


871565 


8817 


188435 


878770 


8838 


18783( 


001205 




999795 


A X 




8 


17 


873855 


8806 


186743 


874469 


gB81 


185331 


001813 




9997S. 


^ a 1 




]? 


16 


874338 


8795 


1251168 


876162 


8811 


1*3838 


0OI224 




998776 


* « 




16 


IB 


876615 


3784 


183385 


877849 


8800 


misi 


001834 




9987fl( 


I « 




so 


M 


879285 


8773 


1*1713 


8795*9 


k;s 


120471 


001243 




998757 


•0 * ! 




24 


81 


819949*763 


180051 


881802 


1 18798 


001253 




9997*7 


9 » 




SH 


as 


8el60T|«7S8 


118393 


882869 


9768 


117131 


001868 




99973* 


» » 




38 


33 


688a5H174! 


I I 6748 


■884530 


8JS8 


H5470 


001*;* 




9987*t 


7 K 




36 


g4 


884903*731 


115097 


886 183 


8747 


113813 


001*82 




998718 


« » 




M 


as 


8885438791 


113*39 


887833 


8737 


112167 


001898 




99B70B 


A «J 




4t 


95 




111886 


889476 


2787 


110384 


001301 




99669! 


4 ll 




48 


«7 


889H0l|8700 


110I9B 


891118 


9717 


108888 


001311 




S9S68! 


! 1 




St 


ts 


8914*18690 


10857S 


892748 


9707 


107838 


001381 




9986t( 


i 




6e 


ss 


893035,86S0 


10C9B5 


894366 


8697 


105834 


001331 




998fi*! 


1 _< 




30 


).894643;Se70 


11.103357 




8687 


11.104016 


10.001341 


17" 


9,99863! 


S4iii 






31 


890246*660' 


103TA4 


89759^ 


aS77 


10*404 


001331 




S98fliS 


9 1 






e 


38 


8978481*651 


102138 


899803 


8667 


1007BJ 


001361 




9!>8eS! 


! i 






1! 


33 


899i38|8B4I 


100368 


B00803| 


9658 


099197 


001371 




99861! 








16 


S» 


901017*631 


096983 


902398 


8648 


09760! 


001381 




99901! 








80 


S3 


90859686** 


097404 


903987 


8638 


09601; 


001391 






5 f<] 






84 


36 


904169 8618 


095831 


905570 


2689 


09443( 


001401 




99850! 


1 ' . 






W 


37 


905736*603 


09(864 


901147 


86*0 


099853 


001411 




99958! 


3 1 






32 


9» 


907*978893 


09270; 


908719 


2610 


091*81 


001422 




99SSII 


• 1 1 






SB 


38 


908853*584 


091117 


910*85 


2601 


089715 


00143* 




99856! 








4fl 


10 


9I0404B57fl 
9110498566 


089591 


siieifl 


8398 


088154 


001448 




996 5Sf 


It 






44 


41 


088031 


913401 


85S3 


086599 


00145* 




999£it 


i 






4^ 


43 


913488 8656 


086518 


914951 


8574 


08504S 


001463 




999531 


8 






AS 




9150888347 


0S497H 


916495 


2363 


08330: 


001473 




9985ST 


7 f 






50 


44 


916350 8538 


083430 


91803412336 


OBI96( 


001484 




998511 


e M 






15-0 


41 


i.9180738a?9 


11.081987 


1.9 19568:8341 


11.08043J 


10.001494 


S 


9.99850( 


54i' ( 






4 


M 


9195S1 8520 


08M09 


98109618338 


078904 


001305 




99H4SJ 


4^ i' 








17 


9*1 [038318 


078897 


9**61918530 


077381 


001515 




998485 


3 f. 








49 


989(!toa303 


077396 


9841361*581 


075864 


001386 




99S*7i 


8 « 








W 


984US!24B4 


075888 


9*36*9l8S12 


074351 


00133S 




999164 


^ 








SO 


9i5609l848S 


074391 


9*71 5618503 


078844 


001547 




998453 


a jti 










9g7100iS477 


072900 


P88638;8495 


07134! 


001553 




999*4* 


? p- 








fl! 


9(858718 469 


071418 


930165*486 


069843 


001569 




998431 


» 31 








£3 


D300fl8l*1«0 


069932 


931647 8478 


068353 


001579 




9984*1 


> 








d* 


9313441243* 


068136 


9331348470 


D6686( 


001 590 




998110 


a* 








53 


9330IJ{2443 


066995 


931616 8461 


063384 


001601 




9W839J 


s 








6fi 


B3448l|84S5 


065519 


936093 8433, 


063907 


001618 




S9»38e 








93fl94S;84*7 


064038 


1)37365,8443 


0fil43i 


001683 




998377 


>it 




fialsfi 


9873988418 


O6J60i 


9390382437 


06096i: 


001634 




9983«e 


B 




iffss 


938830,8411 


061130 


940194 8430 


039501 


0016*3 




998351 


4 




!o_o|bo 


04O896J*«I3 


059704 


94195**481 


038048 


001656 




998144 


Wi 


iT-rr 


Cwine, 1 1 Secant. 


Co„.> 


:r^-r 


Dwc. 


^^ 


Sine. 


^ "^ 




SHoun, 






85 


Depcei. 1 


i / 


:-:°/ i' / ir 


fiOl 


? \ '£ 


IS 


r^T 


^i i-\xA 


m. 


-- / 3 1 45 


120* 


3 \^ 


\ Mm 


\ » \ iS.\ _» \, 




m 






1 


^^1 



■ 








and SMtnU. Tuli V. 


93 






Houi. 


ot 


5D. 


«TC(& 




"■ * 




Siiw. 


D. 


Cawc 


T-w. 


D. 


Coung. 


S««l. 


D. 


C0dD<. 




O- K 




» < 


1 


I.9W1BG 


iio 


n7oa9w' 


8.94195! 


m¥ 


11.058048 


ia001656 




rggiiu 


ii 


io U 










041 T3B 


(39 


05986! 


943404 




056596 


001667 




999333 


» 


M 








■ 


9*3174 


iS» 


05688fl 


94485! 


!*oa 


0551*9 


001678 




999381 


W 


51 








3 


944608 


!37e 


05539< 


94619S 


839 


053703 


001699 


19 


9093 II 


ST 


48 








4 


94SOS4 


137 


0539Sfl 


947734 


8390 


051166 


001700 


19 


998300 


W 








3 


* 


94T456 


!363 


OSli*^ 


949168 


8381 


050838 


00171 


19 


99989!> 


>5 










6 


B48874 


a35i 


051 lie 


95059 


8371 


049403 


0017*3 




998177 


5* 


36 






gs 


T 


950187 


134S 


049713 


95808 


1366 


04797B 


001734 






W 


38 






Si 


e 


951 69G 


a34e 


048304 


95344 


8359 


0*6559 


001715 




9BB855 


» 


88 






36 


fl 


953100 


133 


046900 


954856 


1351 


0451*4 


00175 


19 


0M849 


il 


84 






M 


IG 


954499 


831 


04.5501 


956167 


8344 


043733 


OOlTfiB 




999831 


^ 








44 




953S94 


131 


04410G 


9676T4 


8337 


0*1316 


0017811 






U 


II 






i8 


li 


957194 


13 IC 


04!Tle 


9S90T6 


8310 


0*0915 


00179 


19 


998109 


M 


1! 




t 


«i 


IS 


9M6T0 


130 


041330 


960473 


831! 


039517 


0111803 


19 


998197 


17 


H 




1 


H 


14 


96005! 


119 


03994B 


961B66 


8314 


038131 


001814 


19 


998181 


16 


4 






(T-o 


IS 


1.981419 


IIBS 


11.038571 


1.06315S 


8307 


11.036744 


10.00'i8!6 


19 


9.998174 


15 


39 ( 


i 


4 


l( 


961801 


1180 


03T199 


064639 


8300 


035361 


001837 


19 


998161: 


14 


56 




1 


8 


IT 


9e4ITC 


1173 


035830 


966019 


1893 


033981 


001819 




9981 SI 


13 


51 




1 


1! 


It 




1166 


034466 


96T394 


8886 


031606 


001961 


10 


999139 


tl 


4« 






16 


IS 


B66S93 


8859 


033107 


969766 


1879 


03113* 


O0IM71 


80 


99918) 


il 


4* 






to 


io 


969149 


115! 


031751 


070133 


1171 


08986T 


00198* 


80 


998116 


to 


10 






u 


81 


969600 


8143 


030400 


971496 


1165 


08850* 


001896 


10 




19 


36 






ta 


a 


9T0947 


1138 


019053 


9T1B55 


1857 


0871*4 


001908 


80 


99809! 


19 


3* 








13 


9T!16^ 


113 


01771 


9T480B 


1851 


015791 


001980 




99808( 


17 


»fi 






38 


}« 


9T361S 


188 


08637! 


9J5560 


181* 


0144*0 


00193! 




QB806) 


ts 


84 






♦e 


15 


9T49fll 


111 


085038 


976906 


1837 


08309* 


001911 


10 






80 






4 


1< 


976193 


88] 


083707 


9T884B 


8830 


011751 


001956 


80 


Mfmou 


H 


16 






48 


IT 


8TT6I9 


iin 


01138 


0T95B6 


1113 


080414 


001988 


80 


998031 


13 


11 






ai 


» 


9Tfl94I 


119 


081059 


9809! 


811 T 


O190T9 


001980 


10 


smnii 


S8 


8 




( 


36 


19 


990159 


1190 


0197*1 


981851 


8110 


017749 


001998 


80 
80 


998008 
9,997991 


y 


* 






it 1 


is 


1.981573 


1193 


11.018487 


a.9B35TT 


il04 


11.016413 


laooiooi 


10 


38 






SI 


98gS83 


11 T7 


017117 


984809 


8197 


015I0I 


008016 


10 


9S7984 


99 


56 




n 


s 


SI 


984199 


11 TO 


0I5BI1 


986117 


8191 


0137H3 


001018 


80 


997978 


i8 


5! 






12 


S3 


985491 


8163 


014509 


98T538 


11B4 


011468 


008011 


10 


99795! 


17 


4f 






16 


M 


986789 


8157 


013111 


0B8841 


ilTB 


011158 


001053 


10 


997947 




44 






10 


a 


988093 


8150 


011917 


990149 


il71 


00995! 


001065 


11 


997MJ 




40 






n 


S6 


9S93T4 


1144 


010686 


991451 


1165 


0095*9 


008078 


11 


997981 


84 


3( 






« 


IT 


990660 


1138 


009340 




8158 


OOT150 


008090 


It 


997911 


83 


3! 






3* 


Id 


9B1943 


1131 


00805T 


991015 


8158 


005955 


008103 


11 


997897 


a 


88 






36 


is 


993111 


8115 


O067TR 


99533T 


1146 


001663 


001115 


11 


997685 


81 


84 






40 


40 


99+497 


1110 


005503 


996614 


11*0 


0033T6 


001 118 


81 


997878 


m 


ID 






44 


*1 


995768 


8118 


00483! 


997909 


1134 


00809! 


0081*0 


11 


997861 


19 


16 






49 


ti 


991036 


1106 


001964 


909188 


8187 


000811 


0011.43, 


« 


997947 


19 


I! 






58 


M 


999199 


8100 


001 TOl 


9.000465 


8181 


ia999535 


001165 


81 


997835 


IT 


8 






fifi 


** 


9995601094 


000441 


001738 


8115 


998168 


001178 


!1 


99781! 


16 


4 






S 


U 


9.000816 8088 


10.999184 


9.003007 


1109 


10.99fi99J 


ia00819l 


11 9.997809 


T5 


37 




i 


46 


0010691081 


997931 


00*87! 


8103 


995718 


008803 


81 


997797 


14 


56 




1 


fl 


IT 


00331 BJtOTI 


996688 


005531 


109T 


994466 


001816 


11 


99TT94 


13 


58 




1 


It 


18 


OOS805'l06( 


99543T 


006798 


1091 


993108 


008889 


11 


9J7771 


18 


4E 




B 


]6 


u 


994195 


008047 


10B5 


991953 




!) 


997758 


11 


44 




1 


ao 


so 


OOT04410S8 


991956 


00919B 


1080 






81 


99TT4J 





40 




B 


■4 


SI 


0091T9,1051 


99171! 


010546 


1074 


989454 


601868 




997T31 




36 




B 


IS 


SI 


0095102046 


990490 


011790 


1068 


989110 


001181 


81 


B9771! 


9 


31 






3g 


S3 


0IOT3T104O 


989163 


013031 1068 


986969 


00189* 


11 


997706 


7 


8E 






36 


S4 


0119B!,S034 


088038 


01 4168 1056 


985731 


608307 


88 


997693 


6 


81 






40 




0131 Bl 1089 


9B6819 


015501 


1051 


98*198 


eo!380 


1! 




5 


10 






44 


56 


ouioo'ioaa 


995600 


016738 


1045 




001333 


81 


997667 


* 


16 






4a 


57 


OI561310I7 


g8438T 


017959 


10*0 


99!0*l 


008346 


18 


997654 


3 


11 






Bi 


5B 


0168141011 


983176 


019183 


8033 


98091 T 


001359 


!8 


91)7811 


1 


8 






BB 


5S 


0190311006 


081969 


010403 


808S 


979597 


001371 


88 


997618 


1 


4 






!« 


60 


01913S'lO00 


9B0T65 


011680 


8013 


979380 


001396 


88 


997614 


j; 


96 


\, 




*"• 


~ 


CoBnt 1 1 


8«.o. 


Coisng. 


~ 


T.ng. 


"css^r 


Siat. 1 




mTT 






5HDun, 




84r,ii««. — 






P.r.. 


!■ 1 15" 
1 30 


317 

655 


8 


15" 
30 


330 1- j 15" 1 3 („ 

eoi \ * \ -w \ fe \ 

09t \ X \ \b \f ^ \ 


e.Aa 






.or 


S 45 


991 




45 


^ 












11 


■ 


t 
















^M 



r 










94 Table V. 


Logarithmic Sines, Tangents, \ 


Hour. or 


6 D 


«g"«. 1 


m. B. 


4L5i!!^ 


D. 


Couc 


Tars. 


^ 


_C0Ung^ 


SmanL 


D 


CoidDe. 




■n. ■ 


M~0 


( 1.01983. 


loo 


ia98076 


9.0316ao|a08' 


10.91839 


10.00838 


"88 


9.097614 


S 


m1 






1 08043 




97956 


038834 


801 




00839 


88 


997601 


S 


51 




S 


S Oil 83 


198! 


978368 


084014 


30! 


.97595 


00841 


tt 


997588 


» 


St 




13 


3 oaass 


[984 


977175 


oasgsi 


800 


97174 


00848 


St 


997574 


5 


48 






4 03101 


197H 


9J59B1 


086165 


200 


9T3545 


00343 


n 


997MI 


H 


44 






oasao 


1973 


9J1797 


087655 


199 


97831 


00345 


28 


997M7 


S 


40 




a* 


08638 


196 


973614 


088832 


199 


971148 


00846 


23 


991534 


S 


3li 




f8 


03756 


196! 


078433 


030046 


198 


969951 


00348 


83 


997580 


s. 


3t 






02B7U 


1957 


9T1256 


031837 


197 


968763 


O0249 


83 


991507 


s 


at 




36 


03991 


195 


970083 


033485 


197 


967575 


00850 


23 


997493 


s 


31 




*0 


03108 


1947 


96891 


033609 


196 


96639 


00258 


83 


901480 


« 


80 




4* 


03885 


191 


967743 


034T9I 


196 


965209 


002534 


23 


991466 


4 


le 




4S 


03348 


1936 


966579 


035969 


195 


96403 


00851 


83 


997458 


it 


li 




Si 


03458 


1930 


965418 


037114 


195 


968856 


00356 


23 


9974S9 


1 


e 






03571 


1985 


964359 


038316 


194 


961684 




83 


997*25 


U 


i 






























B'~o 


>.03fi99 


1980 


10.963104 


9.039485 


1943 


10.960515 


10.00858S 


83 


9.997411 


4. 


iSI 




4 


OSBOM 


1015 


961958 


010C5I 


1938 


959319 


■ 002603 


83 


997397 


44 


t 




9 


03919 


1910 


960B03 


011813 


1933 


95B187 


00261 


S3 


997383 


13 


S: 




la 


04034S 


1905 


9596S& 


042973 


1988 


957027 


00363 


83 


997369 


4! 


41 




le 


04UBi 


1699 


958515 


014130 


1983 


955810 


003615 


83 


997355 


4 


4 




80 2 


otasaa 


189a 


95737i 


045384 


1918 


951716 


O0865S 


83 


997341 


4U 


U 




24 8 


043T6! 


1B89 


956838 


016131 


1913 


953566 


008673 


84 


997387 


» 


3 




!sa 


01489^ 


1B84 


95510i 


017588 


190S 


952418 


002697 


81 


997313 


(8 


S 




38 a 


01608f 


1879 


B53974 


048737 


1903 


951873 


00810 


84 


99729! 


IT 


81 1 




36 3 


047151 


187S 


9i88« 


049869 


1898 


950131 


008715 




997283 


M 


«- 




4oa 


04837!) 


1870 


051781 


051008 


1893 


948998 


008739 


84 


997871 


is 


31 i 




41 a 


ai940C 


1866 


9S060{ 


053114 


186S 


947856 


O0871S 


81 


997857 


» 


|. ■ 




4Ba 


05051 S 


1B60 


949481 


0S3877 


1884 


946783 


008159 


21 


991841 


33 


1: 




5S2 


05163. 


1858 


94836i 


0S440? 


1879 


915593 


008772 


21 


99728i 


52 






56? 


058749 


1850 


947851 


055535 


1874 


944465 


003786 


ai 


9978 U 


SI 


• 


i6-53 


1.053863 


1815 


10.946111 


9.05B6S9 


1870 


10.943341 


10.002B01 


21 


S.99710« 


» 


14 a 




4 3 


051966 


1841 


945034 


0577BI 


1865 


948319 


003815 


84 


997185 


!9 


5 




BS 


05607 


1836 


943986 


05B9O0 


18B0 


941100 


002830 


21 


997 1 T( 


K 


S 




las 


057173 


1831 


942881 


060016 


I8SS 


939984 


008814 


84 


997151 








1G3 


058871 


188T 


911788 


061130 


1851 


93887( 


008859 


24 


997111 




4 




W3 


059367 


1883 


94063S 


068210 


846 


93776( 


008973 


84 


991 I SJ 




1 




a4S 


060460 


1S17 


S3954C 


063348 


84a 


936653 


003888 


24 


991118 




9 




88 3 


061&51 


ISIS 


938449 


064453 


837 


S35547 


008908 


84 


991099 




3 




3? 3 


osaess 


1808 


937361 


065556 


833 


934414 


0089 J 7 


85 


S91083 


r8 


2 
1 




36 3 


063784 


1804 


936871 


066655 


8BB 


933345 


008938 


!5 


99106( 21 




40 i' 


OBISOC 


1799 


935194 


067752 


824 


933348 


008947 


85 


99705! 20 


2 1 




444 


065885 


1794, 


931115 


068BIC 


819 


031134 


003961 


15 


997030 III 


I 




484 


OflfiSOa 


1790 


933036 


069938 


815 


93006! 


002976 


85 


99702< 19 


1 ' 




ia4, 


068036 


1786 


931964 


071027 


810 


98B973 


008991 


25 


9910ns IT 






5e4 


069107 


ITBl 


93089) 


073113 


806 


987881 


003006 


25 


996994 


6 




87 0*. 


S.O7017G 


1777 


10.939834 


0.073197 


l03 


10.98680; 


10.0030811 35 


9.996979, 


1W^ 




44- 


071843 


1 778 


93e75f 


071378 


797 


98518! 


003036: ?5 


99696^141 54 
995919^13 5 
99693«I2 4 
99691911 4 
996904*10 4 




S4 


018306 


1768 


987694 


075356 


793 


984644 


003051 


85 




184 


073366 


1763 


986334 


076132 


189 


933568 


003066 


25 




16 4 


074484 


1759 


985j7( 


0T7505 


181 


92849^ 


003081 


85 




80 5 


0754flO 


1755 


084SM 


078616 


780 


921184 


003008 


25 




84 S 


076533 


ITSO 


983487 


079644 


776 


980356 


003111 


85 






88 5 


07TB931716 


938111 


080710 


778 


919890 


003126 


25 


B9687J 8 3 




33 5. 


078631 


1748 


981369 


081773 


16T 


918237 


003148' 25 


996858 7 t 




35* 


079676 


1138 


980384 


088833 


76.=! 


911167 


003151' 85 


996819 6 S 




40 SJ 


080719 


1733 


9198S1 


083891 


759 


91610S 


003178 85 


99082^ & a 




44 5< 


0B17S9 


178B 


918341 


084917 


753 


915053 


003198 86 


99681»J i> U 




435 


088797 


1785 


911803 0860001 


751 


914000 


003Z03| 26 


S9679T S 11 




aas 


083838 


1781 


916168 OB 7050 




918950 


003818 86 


996I98J 2 ■ 




se& 


084664 


1717 


B15136 0880981743 


0119OE 


003834 36 


996166} 1 4 




S 06 


085894 


1713 


914106 089114 1738 


910856 


003ai9! 86 


99675l| 


038 


m. a. 


~Con>.«.l 


"^ 


Secant CotEnp. 


T«ng. 


Cosec 1 Sine. 1 


fn. 1 


SHoHis. 


or 


83 


Degrta. 




./ 


'^■,i''/ I / SO 


87T 1 1- 1 15" 

fi54 1 a 30 


I aSO 1 1' 1 1 
\ 66V I 8 3 


rt ? 1 


';L^^ 


/"" / a t 4.5 


931 1 3 1 45 


\ *AV \ 3 \ ^. \ -v^ I 


If " 








T^ 


^H 



and Secants* 



Table V. 



S5 



OHoar, 



or 



m. t. 



98 



m. 




4 

8 
12 
16 
20 
2i 
28 
32 
36 
40 
44 
48 
52 
56 



Sine. 



29 




4 

8 
12 
16 
20 
24 
28 
32 
36 
40 
44 
48 
52 
56 



019 

1 

2 

3 

4 

6 

6 

T 

8 

9 
10 
11 
12 
13 
14 



1.0658941713 
086922 1709 
0879471704 
0889701700 



15: 



7 Degrees 



30 
4 
8 
12 
16 
20 
24 
28 
32 
36 
40 
44 
48 
52 
56 



15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 



3019 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 



|31 045 

4{46 



8;47 
12148 
1649 
2050 
2451 
2852 
3253 
36i54 
4055 
4456 
48^57 
52:58 
56 59 
132 0:60 



089990 
091008 
092024 
093037 
094047 
095056 
096062 
097065 
098066 
099065 
100062 



9 



.101056 
102048 
103037 
104025 
105010 
105992 
106973 
107951 
108927 
109901 
110873 
111842 
112809 
113774 
114737 



1.115698 
116656 
117613 
118567 
119519 
120469 
121417 
122362 
123306 
124248 
125187 
126125 
127060 
127993 
128925 



1696 
1692 
1688 
1684 
1680 
1676 
1673 
1668 
1665 
1661 
1657 



1653 
1649 
1645 
1642 
1638 
1634 
1630 
1627 
1623 
1619 
1616 
1612 
1608 
1605 
1601 



COMC. 



ia914l06 
913078 
912053 
911030 
910010 
908992 
907976 
906963 
905953 
904944 
903938 
902935 
901934 
900935 
899938 



10.8989-M 
897952 
896963 
895975 
894990 
894008 
893027 
892049 
891073 
890099 
889127 
888158 
887191 
886226 
885263 



Tang. 



9.089144 1738 
090187 1735 
091228 1730 
092266 1727 
093302 1722 
091336|1719 
0953671715 



D. 



096395 
097422 



1711 
1707 



098446! 1703 



099468 
100487 
101504 
102519 
103532 



1699 
1695 
1691 
1687 
1684 



9.129854 
130781 
131706 
132630 



1597 
1594 
1590 
1587 
1583 
1580 
1576 
1573 
1569 
1566 
1562 
1559 
1556 
1552 
1549 

1545 
1542 
1539 
1535 



1335511532 
134470ll529 
135387.1525 
136303:1522 
137216!l519 
138128 1516 
139037jl512 
139944 1509 
1408501506 
1417541503 
1426551500 
1435551496 



10.884302 
883344 
882387 
881433 
880481 
879531 
878583 
877638 
876694 
875752 
874813 
873875 
872940 
87200 
8710751 



10.870146 
869219 
868294 
867370 
866449 
865530 
864613 
863697 
862784 
861872 
860963 
860056 
859150 
858246 
857345 
856445 



9.104542 
105550 
106556 
107559 
108560 
109559 
110556 
111551 
112543 
113533 
114521 
115507 
116491 
117472 
118452 



9.119429 
120404 
121377 
1 22348 
123317 
121284 
125249 
126211 
127172 
128130 
129087 
130041 
130994 
1.31944 
132rtJ)3 



1680 
1676 
1672 
1669 
1665 
1661 
1658 
1654 
1650 
1647 
1643 
1639 
1636 
1632 
1629 



Cocang. 



ia91085€| 
90981 
908772 
907734 
906698 
905664 
9046: 
903605 
902578 
901554 
900532 
899513 
898496 
897481 
896468 



10.895458 10. 
894450 
893444 
8924 Itl 
891440 
890441 
889444 
888449 
887457 
886467 
885479 
884493 
883509 
882528 
881'-' 



J48 



9.133839 
134784 
135726 
136667 
137605 
138542 
139476 
140409 



1625 
1622 
1618 
1615 
1611 
1608 
1601 
1601 
1597 
1594 
1591 
1587 
1584 
1581 
1577 



1574 
1571 
1567 
1564 
1561 
1558 
1555 
1551 



1413401548 



142269 
143196 
144121 
145044 



1545 
1542 
1539 
1535 



145966.1532 
146885|1529 
1478031526 



10.880571 
879596 
878623 
877652 
876683 
875716 
874751 
873789 
872828 
871870 
870913 
869959 
869006 
868056 
867107 



Secant 



ia003249 
003265 
003280 
003296 
003:U2 
003327 
003343 
003359 
003375 
003390 
003406 
003422 
003438 
003454 
003470 



26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
27 
27 
27 
27 



10 



.003486 
003502 
(K)3518 
003535 
003551 
(K)3567 
003583 
()03(>00 
003616 
0();5()32 
003649 
003665 
003()82 
003698 
003715 

^)0;i73'l 
003748 
003765 
003781 
003798 
003S15 
003832 
003849 
003866 
003H83 
003900 
003917 
003934 
003951 
003968 



10.866161 
865216 
864274 
863333 
862395 
861458 
860524 
859591 
858660 
857731 
856804 
855879 
854956 
854034 
853115 
852197 



Cosine. 



Secant. 



27 
27 
27 
27 
27 
27 
27 
27 
27 
27 
27 
27 
27 
28 
28 

28 
28 
28 
28 
28 
28 
28 
28 
28 
28 
29 
29 
29 
29 



29 
29 
29 
29 
29 
29 



10.003985 
004002 
004020 
001037 
004054 
004072 
0040891 29 
004106 29 
004124 29 
004141 29 
004159 
004177 
004194 
004212 
004229 
004247 



29 
29 
29 
29 
29 
29 



l_Coniie. I^kn- fc 
9T99675TioS~~0 



996735159 
9967 
996704157 
996688(56 
996673 55 
996657 54 
996641 53 
996625 52 
996610 51 
996594 50 
996578 49 
996562 48 
996546 47 
996530 46 



9.996514 45 
996498 44 
996482 43 
996465 42 
996449 41 
996433 40 
996117 39 
996400 38 
996384 37 
99(i36S 36 
996351 35 
99633.5 34 
996318 33 
996302 32 
996285 31 



9.996269 30 
996252 29 
996235 28 
996219 27 
996202 26 
996185 25 
996168 24 
996151 23 
996134 22 
996117 21 
996100 20 
996083 19 
996066 18 
9960^9 17 
996032 16 



9.996015 15 
995998 14 
995980 13 
995963 18 
995946 11 
995928 10 
995911 9 
995894 8 
995876 7 
995859 6 
995841 5 
995823 4 
995806 3 
995788 2 
995771 1 
995753 



29 



56 
58 
48 
44 
40 
36 
38 
88 
84 
80 
16 
18 



31 C 
56 
52 
48 
44 
40 
36 
32 
88 
24 
80 
16 
18 
8 



30 a 

56 
58 
48 
44 
40 
36 
38 
28 
84 
20 
16 
12 
8 




56 
58 
48 
44 
40 
36 
32 
28 
24 
80 
16 
12 
8 
4 




5 Hours, 



P. P. to 

■ or" 



1^ 


i5f' 


840 


1» 


2 


30 


479 


2 


3 


45 


719 


3 




1 — 




96 Tabls V. 






OHtnit, 


or 


SD 


jta. 1 


n. ■ 




Sine. 


n. 


( Co«c 1 Tang. 


■^ 


CDUng. 


Scant. 


H 


Conne. 


' n. t. 


^t 


~: 


SJMSS5 


iW 


10.856115;9.1478O3|l52 


10.85219 


10004247 


30 


9.9967 ik 


sow~o 




t 




144153 


1193 


85554 


149TI 


152 


8512B 


001265J 30 


99573. 


S9 56 




fi 




145349 




85465 


11963 


152C 


85036 


0012B3I 30 


99571 


S« 52 




IS 




116243 


1487 


85375 


1505441151 


84945 


00130 


30 


09569 


57 U 




18 




UT136;i481 


85286 


151 4541 SH 


84854 


001319 


30 


99568 


56 M 




20 




U80S6'I1B1 


85197 


] 52363' 151 


61763 


004336 


SO 


99566 


55 44 




24 




I4SQI5'147S 


85109 


153269il50e 


84673 


001351 


30 


99,564 


H 36 




SB 




149802 1475 


850198 


1.54171:1505 


84582 


004372 


30 


99662 


S3 32 




3t 




1S0686.1472 


849311 


1550771502 


81192 


004390 


30 


9956U 


S8 8B 




36 




]51.;69 1469 


84B43 


155978 1499 


84402 




30 


99659 


51 21 








1524511166 


847549 


150977 1196 


8IS123 


001187 


30 


99557. 


M 8C 




*4 




153330 1463 


846670 


15777.5 1193 


B4282 


004115 


30 


B9555 


19 It 




4H 




l5420Sil4G0 


845792 


158671 1190 


84132 


001163 


30 


99553 


16 It 




5S 




155093 145T 


844917 


159566 14«7 


81043 


001191 




99551 


IT 1 




66 




155057 1454 


841043 


1601,^71181 


93951 


004499 


31 


99550 


laj 1 


B~0 




). 156830:1 451 


10843170 


9.161347,148 


10.938653 


10.004519 


31 


9.99548 


15 27 1 




4 




1577001418 


842300 


le2236|l478 


837764 


001536 


31 


996464 


H « 




8 




158569 1415 


81I43I 


1631231475 


836977 


001554 


31 


99544t 


13 51 




1! 




15913.^ 


1U2 


94058S 


164009,1473 


635992 


001573 


31 


995*»7 


M 4 




16 




160301 


U39 


839G99 


164892 1470 


835108 


004591 


31 


99540B 


H 4 




20 80 


161161 


1436 


938936 


I6577l|ll67 


834226 


004610 


31 


995SM 


to « 




i* 


21 


162025 


1433 


837975 


166651 


1461 


833346 


004628 


31 


995ST! 


19 31 




iS 


22 


162885 


1430 


837115 


167532 


1461 


832469 


004647 


31 


995363 


Wi SI 




32 


23 


183743 


142T 


836257 


16B409 


1459 


83159 


004666 


31 


996331 


S71 t 




36 


24 


161600,1424 


835400 


169281 


1455 


830716 


004684 


31 


995316 


36 a 




40 


iS 


1651511422 


834546 


170157 


1453 


829813 


001703 


31 


99629T 


M 2 




44 


26 


1663071419 


933693 


1T1029 


1450 


82897 


001788 


31 


995*76 


1* 11 




4S 


27 


1671591116 


832841 


171899 


1447 


82810 


001710 


31 


995*60 


n 11 




5i 


2S 


I6BO08I113 


831992 


172767 


1444 


827233 


004759 


3* 


995241 


SI 




56 


29 


lReS56{l410 


831144 


173634 


1448 


826366 


00477B 


32 


9962!« 


SI i 


iTo 


3oaA69i6iHwf 


10.830298 


9.174499 


1439 


10.825501 


10.004797 


«" 


9.99610^ 


win 




4 


31 


17051711105 


829163 


175362 


1436 


esiesB 


004816 


32 


996184 


u c 




S 


32 


17I3B911402 


928611 


176224 


1433 


823776 


004836 


32 


995164 


S 5 




IS 


39 


I72230;i39B 


8277T( 


177084 


1431 


822916 


004854 


3! 


996144 


!7 « 




16 


34 


1730701396 


826930 


1T7B42 


1428 


822058 


004673 


38 


995l2t 


tS 4 




io 


35 


173908^1394 
I71T141391 


826092 


178799 


1125 


821201 




se 


99510S 


IS 4 




S4 


36 


8252S6 


170655 


1123 


82034.^ 


004911 


38 


99608S 


M 3 




ia 


37 


1755781388 


B24122 


180509 


1120 


819492 


004930 


38 


S960TO 


13 S 




32 


38 


1761111386 


823589 


191360 


1417 


918610 


004949 


32 


995051 


» t 




36 


3fi 


177242 IS83 


822758 


192211 


1415 


817799 


004968 


32 


995031 


n 8 




40 


40 


IT80J213B0 


821928 


183059 


1412 


816941 


O049M7 


32 


9BS01! 


W 81 




41 


41 


1T8900 137J 


821100 


1B390T 


1409 


816093 


005007 


32 


99499; 


19 le 




49 


4t 


179726:1371 


820274 


184752 


1107 


815248 


O0S02S 


32 


994971 


18 It 




Si 


4< 


1B055I1372 


8IS149 


185597 


1404 


614403 


005013 


32 


094955 


IT 1 




56 


44 


lB137l|l3G9 


8IB626 


186439 


1402 


613561 


005065 


38 


994936 


16 4 


a 


45 


9.182t9ti.l3a7 


10.817801 


9.197280 


1399 




10.005081 


Sj 


9.9949H 


rjii < 




4 


46 


183016 1364 


816984 


I9BI20 


1396 


811880 


005101 


S3 


994696 


14 « 






17 


lB3834ll361 


816166 


189958 


1393 


811042 


006183 


33 


994877 


iS s 




12 


48 


1846511359 


815319 


199791 


1391 


810206 


005143 


33 


994957 


12 4t| 




16 


19 


1854661356 


814531 


190629 


1389 


809371 


006168 


33 


994839 


11 41 




20 


SO 


1962801353 


813720 


l(>I462il396 


608538 




33 


991918 


« 




24 


SI 


197092 1351 


812908 


1922941384 


807706 


005208 


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806876 


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195606 1374 


801394 


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608570 


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197853 


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366 


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10.005380 


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804075 


2013+5 


1356 


798655 


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33 991591 




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803281 


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1354 


797841 


005110 


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797089 


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9.811815 


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81*989 


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79OU08 


815780 


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78770! 


81BU8 


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781858 


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818926 


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813818 


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786188 


819710 


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814579 


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785481 


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1301 


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888830 


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9.99*00! 


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774944 


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176394 


889773 


1875 


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831714 


1816 


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1852 


761880 


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838444 


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838878 


1850 


761128 


006428 


37 


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833178 


1818 




839622 


1848 


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840371 


18*6 


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1807 


765375 


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764S31 


8*1865 


184S 


758135 


006516 


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993181 


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1803 


763987 


848610 




757390 


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836795 


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763S03 


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8 


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1195 


761017 


843579 


1838 


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839670 


1193 


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1230 


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Swant 


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9,38747411035 


10.61 858 f 


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661587 


3880951033 


671903 


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680934 


388T15I038 


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660348 


389314; 1030 


ei066( 


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669430 


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998 


679570 


331187 1086 


668613 


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333191 


971 


676806 


3330331083 


666967 


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84 






to 


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3g3780 


976 


678880 




1081 


666331 


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334859 


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1017 


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1016 


663901 


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10.67330( 


9.336708 


1015 


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10.010003 




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678719 


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668689 


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67 1 556 


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661473 


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964 


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339133 


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660867 


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988 


6701Q1 


339739 


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660861 






969880 


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981 


669684 


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1007 


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46 


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330T53 


960 


66984T 


340948 


lOOB 


659038 


010196 


46 


980804 


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331389 


958 


6B8671 


341558 


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658448 


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989177 


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668097 


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651645 


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635448 


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10.66466; 


1.345733 


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10.854843 


10.010*16 




9.989381 


to 


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31 


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946 


664094 


346333 


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633641 


010417 


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653031 


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337043 


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3475*5 


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651859 


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940 


660693 


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650078 


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64948f 


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659561 


351106 


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648694 


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936 


659004 


351697 


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341558 


933 


658441 


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64771; 


010729 47 


999271 


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348119 


934 


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0107311 47 


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343839 
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6.56761 


351053 


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999186 


16 


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10.656aOJ 


t.354640 


977 


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9.989 I3T 


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616 


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010989,49 


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68 i 


6 412058 


787 


5879*8 


4870*1 


843 


67895B 


014989 


56 


995011 


t 


e 




56 i 


9 41858* 


796 


687471 


4875*7 


843 


67816: 


015032 


56 


994976 


1 




I 


!iO Ofi 


« 418996 


78S 


687004 


48H058 


B48 


5718*81 


016056 




984941 





c 


nT^i 


■ C„in^ 




SeauL 


Cnwnc. 




^f^SiT- 


CoKC. 




Siae. 




oi!"* 


filio 




1 ^^^ 


75 D« 


greet. 




r-"/ i- 1 


-ir 

30 


188 1- 1 15" 

fit a M 


1 1^ k 


P. 10 


LV-" / 3 / « 


369 , S I *6 


A^ 


\ » ^ « < «.i- 


Zju 


■i 




g^ 


■ 


■i 




■ 


ill 


^ 


■ 


^1 


1 


m 





and Secants. 


Tjiiii V. 


in 




1 Kmc. 


or 




_^^15D.«r«* 






n. M' 


1 Sine. 


D. 


CCM. 


T.nK. t D. 


Coun^ 




^ 


Codnc 






0^ 


'■4I«<I96 


785 


10.587001 


9.488058,848 


10.57 1949 iaoiio'56 




99si94i 


Kl 


10 




4 


1 


41346T 


784 


586533 


428557 841 


671443 


Oiiono 


57 


»fl4«l( 


N 


M 




8 


f 


4139^ 


783 


586062 


439062 840 


570938 


OlSESl 




9!)4H7( 


>^ 


SI 




1! 


a 


414408 


783 


585598 


429566' 839 


570 W4 


015158 




944>44! 


J7 


4« 




Ifi 


4 


41*e7B 


788 


585188 


430070 838 


569930 


015192 


57 


984808 


;fl 


44 




iO ^ 


41.431T 


781 


581653 


430573 838 


569487 


015886 


57 


•184-71 


w 






St ( 


4IiBIS 780 


fi8HB5 


431075 837 


588925 






9S474C 


54 


36 




i» 1 


416283'" 


683717 




568483 


015a!14 




984700 


M 


32 




St t 


41675! "8 


583249 


438079 8H5 


5679SI 


015328 




flH40Ji 


52 


28 




38 S 


417817 'T7 


588783 


432580 834 


567480 


015362 


57 


98463( 


51 


81 




40 in 


417684 '"6 


5H23U 


433080 833 


568920 


01439- 


57 


984603 


M 


JO 




4111 


418150 TIS 


581850 


433580 838 


5fl648C 


015431 


57 


984569 


IS 


16 




48 li 


418615! "i 


581385 


434080 832 




015465 




984535 


48l 


18 




St\3 


419079 


773 


580981 


434579 831 


565181 


015500 




884500 


171 


> 




5614 


419544 


773 


5B045( 


435078 830 


56198! 


015534 


57 


9844S6 


la 


4 




1 Old 


).480007 


778 


10,57999S 


9.435576 8*9 


lu.564424 








45 49 I 




4'16 


4804T0 


771 


57953C 


436073 828 


563927 


015003 






44 


JtO 




e,n 


480933 


770 


579067 


436570 888 


563430 


01,11137 


58 


»f<43S3 


43 


58 




12 It 


481395 


769 


57B605 


4.37067 827 


562933 


015672 


58 


984328 


48 


♦s 




1616 


481857 


768 


578143 


437563 886 


502437 










44 




K)!0 


428318 


767 


677682 


438059 825 


561941 


014741 


.■>8 


981259 


ui 


40 




!4 8l 


488778 


767 


67722S 


438554' 824 


561446 


01577H 


58 


9848(4 


19 


30 




8e«! 


483838 


766 


576762 


439048 883 


560958 


015811 


58 


984190 


w 


38 




as 83 


483697 


765 


57630- 


4311543, 883 


560457 


015845 


48 


984 U5 


IT 


18 




36 84 


484156 


764 


575844 


440030, 888 


659964 


015880 




9841(0 


W 


2 




iOtS 


484615 


763 


575385 


440529 821 


559471 


015915 




984084 


U 


20 




44 SI 


4250T3 


768 


574927 


441088' 820 


658978 


015950 




9840SI 


11 


10 




48 27 


485530 


761 


57447( 


441514 819 


558486 


01S983 


48 


981015 


J3 


18 




a!« 


485987 


760 


574013 


4*8006! 819 


557994 


on:oi9 


58 


98398 


12 






56 89 


486443 


760 


573557 


U2497; 818 


557503 


016054 


T58 


9M394f 
». 983911 


Jl 


58 'V 




i 031 


U2fi899 


75a 


10.573101 


I0.44898H 817 


10.557018 


10.O1U0M9 




431 


487354 


758 


572646 


443479 816 


556581 


01612, 


5-* 


Pfl3S75 


a 






8 38 


487809 


757 


572191 


443968, 816 


556038 


016160 


59 


9838 «1 


IS 






183; 


428263 




571737 


4444S8 815 


655542 


016195 






rt 






16 34 


428717 


755 


511283 


444947! 814 




01683( 






26 


44 




80 3i 


489170 


754 


57083(1 


445135 813 


554565 


016285 


59 


98373.5 


IS 


it 




8436 


429683 


753 


670377 


445923 818 


554077 


016300 


59 


D83700 


H 


Sb 




28 37 


430075 


752 


509985 


446411! 813 


553589 


016331 


59 


983061 


a 


32 




88 3M 


430527 


758 


569473 


446898 811 


653108 


010,171 


59 


983029 


li 


(8 




36 39 


430978 


751 


569082 


447384] 810 


5528 10 






98359 


i\ 






4040 


431489 


750 


668571 


447870 809 


558130 




59 


983558 


10 






44 41 


431879 


749 


568121 


448356' 809 


551644 


OI647J 


59 


983523 




18 




4B4t 


438389 


749 


567671 


448841! 808 


551159 


016513 


59 


983487 




12 




5843 


432778 


748 


567822 


449386 807 


550874 


016548 


59 


983458 








56 44 


433886 


747 


566774 


449810 806 


5501 9C 


016584 


59 


983416 




■i 




S 045 


I.433a75: 746 


10.566323 


9.450894 BOO 


10.44n0t 


10.0161. 1! 


59 


9.983381 








44« 


434128145 


S6S878 


450777 805 




016655 


59 


9B334S 




51 




841 


4345691 74* 


565431 


451260' 804 


548740 


010691 




B8;i3U' 








18 48 


435016 744 




451743 8(13 


548857 


016787 




B, •13 2 73 




*» 




ie4s 


4354<J8|743 


604438 


452284 802 


£17775 


010768 


60 


983238 




41 




»H 




564092 


458706' 808 


547294 


010798 


60 


983202 




40 




14 51 


436353' 741 


563647 


453187,801 


5+6813 


016834 


60 


983iaS 




8' 




test 


436798 740 


£63802 


453688 


800 


546332 


016870 


60 


98313' 




3£ 




38 53 


437842! 740 


568758 


454148 


7^9 


515853 


0169M 


60 


osrmi 




tt 




38 54 


437GS6 739 


562314 


454688 


799 


545372 


016912 


60 


9S305S 








40 55 


438 r 29 738 


561871 


455107 


798 


i44B93 


016978 


ao 


083022 




K 




44 5« 


438578 737 


561488 


455486 


797 


644414 


017014 


60 


93898r 




1< 




48 57 


439014 736 


560986 


456064 


796 


543936 


017050 


60 


B8895( 




11 




5858 


439456 736 


560544 


45654! 


7!»6 


513458 


017080 




982914 




e 




56 59 


439891 735 


560103 


457019 


795 


54;9ai 


017188 


60 


9H2a7B 




4 




4 060 


410338 734 


559662 


4i749(i 


794 


513504 


01 7158 


^' 


98284* 




)(! (1 




n. ». 'i Corinc 1 


""Stcani. 


tolling^ 




r.^er- 


(kiicfc 




Siie. 




^ 




4 Houn, 






DT 




74 




iii:t. 






"■-H ^' 1 jr 








Hi 


i' \ 


\ ^^ 




\ 


283 1 8 


30 


246 


, 2 \ -it. \ VI \ 


■^w'^V 


""'"I .1 / 4S 


311 ( 3 




3r,R 


\ ^ \ ^b \ %^ ^ 


^ A 



^' 




1 




40 T*bi.eV. 




ts. 






1 Hour. 




88 Dtgr™ 






D. 1 


J_^!!5:_ 


D. 


Co«c 


T«B. 


3l 


C<..«g. 


1 SeauU. 1 D. Ccwne. ' ' 


ni. >. 






e8~u 


0.9.iT.i57a 


ItT 


iaMH4l5 


Jt.6064lO 


606" 


10.393590 10.032B:i*| 85 9.967166 60 


3a~o 










fi73HHx 


.5*0 


486112 


606773 


606 


39388 


038885; 85 


967115 59 


56 










ST4S0r 






6071.'! 7 




392863 


032936 


85 


9670N4 58 


58 






li 




57*51 J 


519 


42S188 


60751)0' 605 


y92500 


032987 


85 


96701357 


48 






](> 




i74S8l 


518 


425176 


607863, 604 


39813 


ir.aoia 




966961 56 


4* 






80 




firai36 


519 


48W61 


608225, 604 


391775 


033090 


US 


06691 055 


40 






84 




575447 


518 


424i5: 


608588 


601 


391418 


033141 


85 


966859'S 


36 






sa 




SJSISH 


518 


484812 


608950 


603 


39iOS0 


033192 


83 


966908 53 
966750 58 


32 






32 


B 


5TG0S9 


sn 


483931 


609318 


603 


3906H8 


03384 


96 


89 






3S 


e 


576379 




423621 


60" 671 


603 


390326 


033895 


96 


966705 S 


n 








1(1 


a7fiMBS 




423311 


610036 


60! 


3899lil 


033347 


B6 


B06653 50 


80 






44 




47H999 


516 


483(X>1 


61039? 


608 


389603 


033398 


86 


96660249 


11 






43 




57730!! 


516 


488691 


610759 


602 


38984 


033450 


86 


966550 4a 


1! 






53 




577610 


515 


422388 


611120 


601 


388880 


033501 


86 


9664B9 47 


« 






SB 




577087 


SJ5 


48807^ 


611480 


601 


388520 


033553 


86 


966447 4l1 


_J 




ir~o 




B.57a83» 






}.U1184J 




3877Sy 


I U. 03360 S 


86 


9-960.19; 4. 


31 ( 






4 




578545 




421455 


618201 


600 


03365S 


86 


966344 m 6i 






s 




57B8S:i 


513 


421147 


018361 




3B743fl 


033708 


86 


966298 43 


a: 






IS 




57916? 


513 


420838 


612981 




387079 


033760 


86 


9fi684C 48 


t» 






16 




579470 


513 


13053 U 


613881 


599 


386719 


033Br8 


86 


966188 4 


u 






20 




579777 


512 


420283 


613611 


599 


386359 


O33804 


86 


966136 10 








91 


21 


580085 


518 


419915 


614000 


598 


386000 


033915 


87 


9660H5 19 


3( 








S8 


S803B; 


511 


419608 


611359 


598 


385641 


033967 


87 


966033 W 


31 






3g|!3 


580699 




419301 


614718 


598 


38588^ 


031019 


87 


9659S1 17 


81 








iSIOOJ 


511 


418H95 


615077 


59T 


38192: 


031071 


87 


965989 3e 


81 






■loaa 


5B13I3 


510 


41868( 


615435 


597 


384565 


034124 


87 


965876 M 


80 






41 


88 


58161^ 


510 


41838S 


61579:1 


597 


384207 


034170 


87 


965921 M 


16 






4H 


87 


S8I984 




41807<i 


616151 


596 


383849 


034228 


87 


965778 13 


11 








8S 


58828!) 


509 


417771 


616509 


596 


38349 


0318801 87 


964780 38 


( 






5( 


89 


588535 


509 




el6b67 


596 


383133 


_034333 87 


965668 


s 


4 
S0~6 




JO 


3( 


).5B8B40 


508 


10.417160 


9.617884 




10.388776 


10.U3'13B5, 87 


9.965615 






* 


31 


583145 


508 


41fiS5J 


617582 


595 


382418 


034437 97 




St 






B 


38 


5S3449 


507 


4163,51 


617939 


595 


382061 


034189 87 


965511^ 


5! 






U 


S3 


583754 


SOT 


416246 


618895 


591 


381705 


031512 87 


S6i4SS8T 


*t 






16 


3* 


58405)^ 


506 


415942 


618652 


594 


381348 


031594 87 


96540^16 


4t 






80 


35 


584361 


506 


415639 


6190U8 


591 


380992 


031647 88 




40 








3fl 


584665 


5(16 


415335 


619364 


593 


380636 


034699 se 


96530l2i 


36 






88 


37 


594968 


505 


415038 


619781 


593 


380279 


034752 88 


96524Mi3 


38 






32 


3t 


585378 


505 


414728 


62O0T6 


693 


379984 


034803 88 


965195 Jg 


88 






3G 


39 


585574 


5()4 


414186 


680132 


598 


379568 


034857 88 


TA 


tl 






40 


40 


685877 


504 


414123 


620787 


598 


3798 li 


034910 S8 


20 






41 


41 


586119 


503 


413821 


681142 


592 


378856 


034963 88 


965037 


19 


16 






48 


42 


586488 


503 


4135h 


621497 


591 


378503 


0330 16^88 


964981 


16 


IS 






5! 


43 


586T83 


503 


413217 


621858 


591 


379 1 4i 


035069 88 


964931 


17 


8 






£G 


44 


58T083 




412915 


688807 


590 


37779: 


035121 


s 


964879 


16 


* 




11 


*5 


».5B73a(i 


UJf 


10.41 Kb 14 


). 6 88561 


590 10.3774391 


10.0;J5171 


9.964886 




85-; 






4 


46 


587688 


501 


412318 


682915 


590 


377085 


035287 88 


964773 


14 


H 








*1 


597989 


501 


4120U 


623269 




376731 


035880 88 


964780 


13 


Si 






law 


588889 


401 




623683 


589 


376377 


035334' 89 


964H6t 


12 


48 






16'4S 


588590 


500 


♦ 1141(1 


623976 


589 


376021 


035387 89 


964613 


It 


41 






80.S0 


588H90 


500 


4I11K 


68*330 


588 


37S(i7( 


035440 89 


961560 


10 


VI 






sVsi 


581)190 


499 


4108 LO 


624683 


588 


375317 


035493, 89 


9643</7 





3> 






gS:5S 


58l)t8!l 


499 




685036 


588 


374964 


035546 89 


961*5* 


8 








38153 


589789 


499 


4ioaii 


625388 


587 


3740 Li 


035600 




961*00 


T 








3(>'&1 


590088 


198 


409918 


685711 


587 


37t2Sll 


03365:{ 




9643*7 












5B0387 


198 


409613 


626093 


587 


373907 


035706 


»9 


Q 64291 


5 


80 






*ifi6 


590fi8ti 


197 


409314 


686445 


586 


3735Si 


035760 


89 


96*2*0 


* 


lu 






48 


a; 


590984 


497 






586 


37380:: 


035813 


89 


86*187 


3 


12 






5! 


Si 


591 i8V 


+97 


408718 


62J149 


586 


378851 


035867 


89 


96*133 


8 


H 






ifi 


6!l 


591580 






627501 


585 


378499 


035U2:j 


89 


96*081 




4 






ig 


90 


591818 


+!)R 


408l«t 


687H5i 


BV.'i 


378 14H 


03597* 




96*026 





ga 






sr^-T 


TvHuiiT 


= 


SecimT 




















c^uii^- 


^TlSi^ 


Cosec." 




Sine. 








-I Hour.. 


or 


6 


1* 


no. 


• 




"•ri ;■ ! r 


70 
153 


1< 
8 


IS" 
30 


1 i?a 


8 







13 

2(J '^■ 


p. to 




1 41" ' ^ ' " 


no 


3 


*S 


868 


3 


45 


39 ■ 


or" 








■ 


^^^^^^^^^^^^^^^_ 




i 




■ 




^^^^^^^^^^m 









And SecsDls. 


TjiH V. 


41 




IHour, 


or 




!3Degr«. 






■■■ ■■ 




Sine. 1 D. 


C0«c ' Tuig. 0. 


Ccitmg. 


S«»ni. D. 


C«iT.^\- 


n. « 




ii~i 




il^SISTSW 




ioTiT^m 


10.03597-1 89 


9,96 HIJ8^ 


W ( 






1 


632176 495 


407824 628203 585 


371797 


036028 89 


963972 


i9 


S( 




8 


2 


S9*173 495 


407527 628554 585 


37141) 


OJ609l|89 


96391! 


SH 


6J 




12 


3 


592770 495 


407230 628905 581 


371095 


016135 90 


903865 


S7 


48 




IB 


i 


SD3067, 491 


106933 629S5J 584 


370746 


030189 90 


96381 1 


56 


♦! 




iii 




ii)t3453 494 


40663T, 639S06 ,583 


370391 


036849 90 


96375; 


S5 


40 




!4 


e 


593(159 493 


406341 629956 583 


37001* 


036296 90 


963701 


34 


3t 






1 


£93955 493 


106015 630306 5S3 


3691.04 


036350 90 




53 


Si 






a 


5B4S51 493 


405749: 630656 583 


389341 




963591 


38 


tt 




36 


e 


594517 492 


403433, 631005 592 


368995 


036438 90 


96351! 


31 


21 




4O10 


594842 49a 


405159 


631355 582 


368615 


036512 90 


963 ►88 


■'0 


21 




ill! 


595137 491 


4U1863 


631704 582 


368296 


036566 90 


963134, 


19 


ll 




*»,U 


595132 191 


404368 


632053 581 


387917 


0366x1' 90 


963379 


l» 


18 






595727 491 


404273 


632401 581 


307599 


036675 90 


963323 


17 


8 




se'u 


59602 1| 490 


403979 


632T50 581 


36725( 


036729 


90 


963271 




4 




i3~0ji 


b.596315 490 


10.4036859.633098 590 


1U.36690' 






15 








596609 489 


403391 


633117, 380 


366553 


0369371 90 


963163 


11 


M 




flll7 


596803 189 


403097 


633795 580 


366205 


0369931 91 


963108 


13 


62 




I«I8 


597196 489 


402804 


631143, 579 


365857 


036916' 91 


963054 


18 


49 




16IS 


597490 488 


402510 


631490 579 




O37U0liBl 


96299* 


1) 


44 




iO,»] 


5977831 *88 


402217 


634838 579 


365I6i! 


037 0J5' 91 


962915 




40 




9*;!! 


5D80J5 


48T 


401925 


635193 578 


384813 


037110 91 


962^9( 




36 




i»ii 


698368 


1S7 


401632 


635532, 578 


361469 


037161 91 


962836 




32 




32 «3 


598660 


4ST 


401340 


635879, 578 


364121 


037219 91 


962781 


IT 


19 




■iG\U 


598952 




40101^ 


636226 .^77 


363774 


037i!73 91 


96*727 


16 


21 




io'is 


5992 V4 


186 


40U75S 


636572 577 


363428 


037328 91 


962672 


iS 


SO 




4*S6 


599S.3G 


485 


100164 


636919, 577 


363081 


0373831 91 


962617 


31 


16 




48 27 


599827 


485 




837265 577 


362735 


037139 


91 


962362 


93 


1! 




5SiS 


600118 


485 




637611 376 


362389 


037492 




9625!)8 


M 


1 




ssw 


8O(H09 


481 


399591 


6379361 576 




037317 


91 


962433 


]1 


* 




u o.aos.eMioo 


484 


10.399300 


9.638302, 576 


10.361698 


10.O376Oi 


9,96231-8 


ii 


26- 




431 


600990 


484 


399010 


63S647 


675 


3613.i3 


037657 


92 


962313 




6( 




S3t 


6JI280 


483 


398720 


638992 


575 


361008 


037712' 92 


962288 


28 


62 




laaa 


6>I57( 


4S3 


3981.iG 


63933T 


575 


3S0663 


037767, 92 


962233 


87 
86 


49 




l<i34 


6 )I960 


492 


398 lU 




571 


360318 


037822' 92 


968179 








6 12150 


1!J2 


3fl785( 




574 


3599J.- 


037H7;; 93 


96212 




10 




!4SI 


a02+:i9 


482 


397581 




.574 


35962S 


037933 92 


962067 




38 




!B37 


602728 


481 


397*7! 


6107161 573 


359281 


037998 92 


962012 




32 




32 3! 


6U30I7 


481 


396US3 


6110(^0 513 


S.iSIH 


038043 02 


961957 




88 




36 39 


6033OJ 


481 


396 'i9i 


611101 573 


358596 


038099 9t 


961902 




n 




40 40 


603594 


480 


396406 


641747, 572 


358233 


0381SV 9? 


9818W 


21 


K 




41 4J 


6036BZ 


480 


396118 


642091 572 


357909 


03H209I 92 


96 [79 


19 


IS 




4M14! 

5t ia 


601170 


179 


395830 


612134 572 


357561 


038265 92 


961735 




12 




604457 


479 


39551! 


642777 572 


357223 


038320 92 


SB1680J1T 


8 




asu 


601715 


479 


395255 


643120 571 


356880 


039(76 


93 

93 


961624|lB 






iS o^a 


1.805032 


478 


laauiseH, 


9.B43463 571 


10 356537 


10.038131 


9.061569:15 


*5~i 




416 


60)319 


479 


394691 


613800 


571 


356191 


039487 93 


96151314 


56 




817 


605606 


478 


391394 


611148 


570 


3538Si 


0385121 93 


961458 13 


5! 




12 48 


605892 


177 


39410» 


644490 


570 


35551( 


038399' 93 


961102;i2 


48 




ll>49 


60'il79 


177 


39382 1 






355 16f 


0396541 93 


96131611 


41 




iOiO 


606165 


476 


393535 


615171 


509 


354921 


O3ri710 


93 


98129010 


40 




24 Al 


60(i75 


476 


39324S 


6155 16 


569 


354481 


039765 


93 


961835 


9 


36 




28 52 


607036 


476 


392964 


645957 


569 


3341 4J 


038921 


93 


961 1 79 


8 


32 




■ 32£ 


607:122 


475 


392678 


846199 


569 


353901 


038877 


93 


961123 


7 


8t 




sew 


607607 


175 


392:593 


61G5ie 


568 


353160 


0389.33193 


961067 




21 




405.5 


807892 


471 


392108 


618881 


6S8 


353US 


038989 93 


961011 




20 




44|o6 


608177 


ITl 


391823 


6172^2 


568 


35?77B 


039015 


03 






1( 




4B|i7 


608161 




391539 






352438 


039101 


93 






li 




£2S» 


60B74J 




391255 


647903 




352097 


039157 


94 






s 




S(i|s9 


6D9U29 


473 


390971 


648213 


5B7 


351757 


039*11 


91 






4 




)G OfiO 


609313 


173 


390687 


648593 


560 


351417 


039271, 


94 


B6073U 


2 


24 II 




srrn 


Cmiic. 




Secant. 


Cuung. 




■'■""B' 


V.o6ecr 




""siSTT 


aiTT 




iHourt, 


or 




60 De, 


recs. 




P. P. IQ 


8 
3 


15" 
30 
45 


73 
145 

(18 


2 

3 


15" 
30 
45 


«:>9 


!• 1 1.^" 


5i l- 


P. lo 


\ 



1— 


iSfl Table V. 


Logarithmic Sines, Tangents, 


( 










14 De 


^ 1 


V. B 




Sine. 


D. 


Cdmc. 


J^^ 


U- 


Comng. 


Secant. 


D. 


Corine. 


l^L^ i 


56""i 


"o 


9.3H3BT5 


844 


ia6163a5 


9.396771 




10.603889 


10.01309b 


52" 


9.9869m 


iO 


4 . 








394188 


943 


615816 


397309 


896 


608691 


013127 


53 


936873 


S» 


5( 




t 


8 


384687 


848 


615313 


397846 


995 


008154 


013159 


53 


996S41 




« 




IS 




39S19I 


841 


61480F 




B94 


601617 


013191 


63 


986909 


» 


4 




16 


4 


38S697 


S40 


61430; 


399919 


893 


6DI091 


013838 


53 


986TTE 


K 


* 




80 


S 


396801 


939 


613799 


399455 


898 


600545 


013251 


63 


9867*6 


iS 






a 


e 


396704 


H3S 


613896 


399990 




eoooio 


013286 


53 


980714 


4 


3 




sn 




38T807 


937 


618793 


400524, 


890 


599*76 


013317 


S3 


98669! 


3 






38 




387709 


836 


618891 


40105g 


889 


598948 


0133+9 


53 


996651 








36 


9 


SHeaio 


835 


611790 


401531 




5B940B 


013381 


63 


9866IS 


t 


■» 






10 


388711 


834 


611889 


408184 


987 


597876 


013413 


53 


986587 


BO 


t 




4-t 


n 


389111 


833 


610786 


408656 


886 


597344 


013445 


6S 


986565 


IE 


I 






18 


38971! 


838 


610889 


403187 


985 


596813 


013477 


53 


986583 


W 


1 




sa 


13 


390810 


931 


609790 


403718 




5B6882 


013509 


63 


986491 


K 


; 




56 


14 


390708 


8S0 


609898 


404849 


993 


595751 


0135*1 


53 


986459 


a 




iT 


la 


9.391206 




10.609794 


).404779 


998 


10.595821 


10.013573 


S3 


9.9B648T 


iS 








ifl 


391703 


887 


608897 


405308 


991 


594692 


013605 




986395 


u 






e 


17 


398199 


8se 


60780 1 


405S36 




594164 


013637 


64 


986363 


a 






18 


19 


398695 


835 


607305 


406364 


979 


593636 


0I366B 


5* 


9S6331 


u 






Itl 


19 


393191 


884 


606809 


406898 


978 


693108 


OtSJOl 




BS6K99 


»} 






go 


SO 


393685 


S83 


60R315 


407419 


977 


592581 


013734 


5* 


986866 


w 






24 


81 


39U7B 


8SS 


605B21 


407945 


876 


598055 


013166 


64 


9863S4 


» 


1 






88 


3946T3 




60.5387 


408.171 


815 


591539 


013799 


5* 


986808 


» 




38 


8S 


395166 


880 


60.1834 


408997 


ST4 


591003 


013931 


64 


986168 


« 


II 






84 


S95658 


B19 


604348 


409581 


874 


590479 


013863 


64 


980137 






40 


8^ 


396150 


818 


603S50 


410045 


9T3 


589956 


0I38S6 


54 


996104 






41 


M 


896641 


817 


603359 


+10569 


87! 


699431 


013928 


54 


99607a 




■ 41 




4H 


87 


397 138 


817 


608968 


411098 


871 


688908 


013961 


64 


986039 




■* 31 




S2 


se 


8ST881 


816 


60837B 


411615 


970 


689385 


013993 


64 


98600' 




' 11 




S6 


89 


39S111 


815 


60199G 


418137 


969 


58786: 


01*026 


54 

54 


S»S91. 




-^n 


sa 


30 


1.398600 


814 


10.60 I40C 


9.413658 


968 


10.587348 


ia01405tj 


B:99mi 






4 


SI 


399088 


813 


60091! 


413179 


B6T 


586881 


014091 


56 


985901 




I 




9 


3S 


399575 


818 




413699 


806 


586301 


014124 


55 


ilpiSlfa 


8tt 


i 




I! 


9J 


40006! 


811 


69993t 




965 


585781 


014157 


56 


9MJI-4-J 


i: 






J6 


34 


400549 


810 


699451 


414738 


86* 


595868 


014I8D 


55 


!lt-.;fiil 


ic 






80 


Si 


401035 


809 


59896; 


415857 


96* 


58414: 


014823 


55 


HftJ778 


ij 


* 




84 


36 


401580 


808 


599480 


415776 


963 


68488£ 


014255 


56 


9SJ7+^ 


81 


3 




88 


37 


408005 


807 


597995 


416893 


868 


583707 


01*888 




'Js.;7i3 


« 


3 




38 


3f 


408489 


80S 


59T511 


41fl8H 


961 


S8S19t 


014381 


55 


9flju;y 


ii 


1 




86 


39 


408978 


805 


597088 


41 7386 


860 


598674 


014351 


55 


9i*.^.(i l-ti 


n 


t 




40 


U 


403455 


904 


59654J1 


417H43 


859 


fi9816i 


014387 


55 


osjoia 


n> 


8 




U 


41 


403938 


803 


590068 


418358 


858 


681648 


014480 




^^.;5So 


19 


1 




48 


4i 


404480 


90S 


595590 


4188T3 


857 


681187 


014453 




ilhJ5i; 




I 




52 


4; 


404901 


901 


595099 


419387 


956 


580613 


014486 


55 


0^.55H 


17 






5B 


44 


4053B8 


8D0 


594619 


419901 


856 


580099 


014580 


65 




16 


— 


59 


45 


9.405M6S 


■799 


10.594138 








10.014553 




4 


4C 


406341 


798 


593659 


4809S7 


S54 


67907! 


014586 














47 


406830 


T97 


593180 


481440 


853 


578560 


01.1619 


56 


9858a 








18 


W 


407S99 


796 


598701 


481968 


858 


57804t 


014653 


66 


rasH 








16 


49 


407777 


795 


598883 


488463 


851 


677537 


014696 


56 


9e6Sl< 








80 


50 


408S54 


79* 


591746 


482974 


860 


577086 


U14780 


56 


B858« 


1( 






2+ 


51 


408731 


794 


591869 


483494 


84B 


576511 


011753 




98584J 


« 






88 


51 


409!O7 


793 


5S0J93 


483993 


849 


676007 


014797 


56 


98521] 


fl 






38 


S3 


4096S8 


798 


69031B 


484503 


8*9 


675497 


011880 


56 


99619(1 


1 






36 


54 


41015) 




589843 


48501 1 


847 


6749B9 


014954 


56 


Us5t*l 


■ 








Si 


410638 




5S936) 


4855IS 


846 


674481 


011887 




99511! 


1 






44. 


5S 


411106 




589894 


4S6087 


S45 


5739T3 


011921 


60 


9S.W71 


4 






48 


ST 


41157B 


798 


588481 


486534 


844 


573*66 


01*965 


56 


9B50W 


s 






S8 


58 


418051 


787 


5979iB 


487011 


943 


S7E9SB 


014989 


66 


995011 


I 






56 


5B 


418584 


786 


6S747( 


♦276*7 


843 


57245; 


015028 


56 


9B497f 


1 






iO_0 


60 


41!996 


T8S 


5870041 


488058 


fliS 


671948 


015056 




9849+4 





-ifl 


n. >. 


~ 


Codne. 




Seoinl. 


Cow- 




T»s. 


Cosec. 


^^ 


Sine. 


7 


sTa 


5 H0UC6. 










75 Dc 


g™._ Ml 




--«; i' [ r 


\S!" 




aso" 
«eo 


'■ 




,.«JI 




i- 




kJ^ 




^ 


aj 




■ 


I 






^1 










■ 




^ 


^M 





unJ Secanli. 


T*i.LE V. 


— 




ih™. 


*t 




tiDtpta. 






14 


C 


i.(iU94g 


451 


10.37*053 


rSesSTii 


"550 


Coang. 
10.331337 


SCCML 
10.013784 


D. 


Cosne. 








9:957876 


BO 


n 






J 


6!S!ig 


451 


373791 


669O03 


519 


330998 


018783 


m 


057317 


59 


68 






i 


GSS49CI 


451 


373510 


669333 


549 


^30668 


013843 


98 


957158 




S9 




IS 


3 


6SGT60 




373340 


B6966I 




33033B 


013901 


98 


957099 




41 




ib 


4 


SS1030 




373970 


669991 


548 


330009 


043961) 


99 


957O10 


66 






so 


i 


6ST300 


450 


373700 


6703?( 


548 


389680 


Oi;i019 


98 


9569MI 


55 


40 




ei 


6 


6ST5J0 


449 


373i:«I 


670649 


548 


339351 






956981 


54 


36 




ss 


7 


eSTSlr 


449 


373160 


670977 


548 


339033 


043138 


99 


956863 


i3 


38 




3! 


8 


OSS 109 


449 


371891 


67130B 


547 


338691 


043197 


99 


956803 


53 


88 




S6 


I 


GS33T8 


448 


3716*3 


671634 


517 


3383tifi 


043356 


99 


9567*1 


51 


84 




to 


I( 


e2S647 


44s 


3713.53 


671963 


617 


338037 


013316 


99 


956691 


50 


SO 




44 


u 


6SB91b 


44T 


3710M 


673391 


547 


33770S 


043375 


99 


956635 


49 


1( 




48 


1! 


esgiR5 


447 


370815 


673619 


546 


337381 


04343* 


99 


95666( 


48* 


V. 




Si 


1! 


flg94.13 


447 


370547 


67S917 


546 


33705: 


04319* 


99 


95650G 


11 






66 
U 


14 


6S97S1 
I.6S99HS 


446 
446 


370379 
10.370011 


673371 
1.673603 


546 

546 


386731 
10.3SH39t 


043553 
10.043613 


99 

99 


956147 


16 






9.956381 


45 


I9~i 




4 


16 


630JS7 


446 


369743 


873939 


545 


386071 


043673 


99 


956337 


14 








n 


630514 


446 


369476 


674357 




335743 


043733 




956861 


43 


61 




la 


18 


63079? 


445 


36U3W 


6745:^4 


515 


335416 


013798 


100 


956808 


It 


41 




IG 


IS 


631059 


445 


36S91I 


674910 


544 


S35090 


013853 


100 


9561 4H 


41 


4< 




30 




G313?6 


445 


368674 


6753'i7 


514 


33476; 


013911 


101 


9560SS 


40 


4< 




Si 


!1 


631i9;t 




368407 


675564 


544 


334436 


043971 


too 


95608! 


» 


34 




iS 


S3 


631859 




368141 


675890 


51.4 


3341 1(* 


044031 




98596! 


W 


31 




33 


S3 


e3SlS5 


444 


367873 


676317 


543 


333783 


011091 




955901 


17 


18 








63339? 


443 


367609 


676513 


6*3 


333457 


04*151 




955848 


M 


8' 








63SSa9 


443 


367343 


6768S9 




323131 


044311 


100 


955789 


15 


K 






SI 


632983 


443 


367077 


677194 


543 


338806 


044371 


100 


955739 


U 






4( 


?7 


6331S9 


143 


366911 


677530 


543 


383480 


044331 


lOU 


955669 


33 






&1 


S8 


633454 


4*3 


3G6546 


677846 


543 


3SS154 


044391 


100 


B5580S 


t! 






as 


S9 


633719 


44S 


366S81 


678171 


543 


381839 


041453 




955541 


11 






i2~0 


30 


D.G339H4 


441 


10.366016 


9.67819f 


543 


i 0.331.504 


I0.O445I8 


Too 


9,95648t 


10 


m 




4 


31 


634849 


441 


365)51 


67S831 


541 


331 MS 


044573 


101 


9554*8 


39 


61 






31 


£34314 


440 


365486 


679146 


5(1 


330854 


044633 




955368 


i8 






1! 


X 


634778 


440 


3653JS 


679471 


541 


380539 




101 


955307 


87 


4i 




16 


34 


635048 


440 


364958 


679795 


511 


Siosai 


044753 


101 


9558*7 


i6 


44 




SO 


31 


a3i3D6 


439 


3B4694 


680130 


540 


319880 


044814 




955186 




40 




ti 


3( 


SS&STOl 


439 


364430 


e804U 


640 


319556 


04*874 




955136 




3(1 




ia 


,31 


635834 


439 


364166 


680768 


540 


319331 


044935 




955065 




3! 




3S 


38 


63G09T 


438 


363M! 


681093 


540 


318901 


044905 




955005 




3t 




36 


39 


636360 


438 


3B364( 


681416 


539 


318584 






95*944 


ii 


31 




40 


W 


636633 


438 


363377 


681740 


539 


318860 


045117 




95*883 


80 








41 


63GfiS6 


437 


363 1 14 


683063 


539 


3179371 


045177 


101 


9548S3 


19 


16 




1( 


42 


637 US 


437 


S6!R5t 


683387 


539 


317611 


045338 


101 


95 47 6i 


18 


11 




AS 


43 


63T411 


437 


363589 


68g710 




3178m 


045399 


101 


95*701 




( 




5S 


44 


637673 
I.83T93S 


437 


363337 
10.36306^ 


683033, 
r.683356 


638 
638 


316967 
10.316644 


10045431 


101 
101 


964640 
9.954579 


16 
15 


m 




a 




< 


41 


6381 97 


436 


3618(R 


883679 




316381 


045.183 


108 


964518 


14 


51 






M 


638458 


436 


36154S 


884001 


537 


313999 


045543 


103 


954457 


13 






1! 


48 


638730 


435 


S6I390 


684334 


637 


31567t 




103 


954396 


13 


48 




16 


49 


63S9H1 


435 


361019 


684616 


537 


315351 




954335 




41 




SO 


£0 


639243 


435 


3607 5t 


684968 


537 


315035 


045 7 36' 103 


954374 


10 


V 




ii5l 


639503 


434 


361.I497 


635890 


636 


314710 


0*5797|108 


95131! 


s 


3i 




ts 


Si 


639764 


434 


360336 


685613 


636 


31438* 


045648103 


954153 


8 






3S 


SS 


640081 


434 


359971 


68593* 




314066 


045910 


108 


9540y( 




£8 




36 


M 


640384 


433 


359716 


686355 


536 


313745 


045971 


102 


95*039 


6 


31 




40 


u 


640544 


433 


3594se 


686577 


535 


313483 


016033 


103 


953966 




30 




44 


u 


640804 


433 


3591 BG 


686898 


535 


313103 


04609* 


108 


953906 


4 


16 




V 


fil 


641064 


433 


358936 


687319 


536 


313781 


046155 


103 


9538*5 




It 






Sf 


641334 


433 


358676 


687540 


536 


313461 


046317 


103 


953783 




e 




£1 


S9 


6*1583 


433 


358 U 7 


687861 


534 


31313! 


04637S 




953783 


1 


4 




14 




64184? 


431 


358158 

SCCML 


688183 
Cotang. 


534 


311818 


016310 
Coscc. 


103 


953661 





16 




Sine. 




sn 




4Hoins, 


— ■ -~Ti — 64 


Drg»«. 






]• 


IS" 


66 




—15^ 


81 


— 1. 1 15"! 15 p 


p. to 






2 


30 


138 


3 


30 


163 


S ) 30 I 30 l*^ 






lot" 


3 


45 


199 


3 


45 


tiv 


^ -i \ te \ ^^ \ 


^ 









ii T»HI.E V. 


Lc^arithmic Siues, Tangents, 




1 Hour, 


ot 




86Ucftret«. 




m. «. ■! 




^ 


C«et 


TanK. 


D. 


CnwDg. 


Seomt. ID. 






m. ( 




i*~0 


= 


^4iaM 


wT 


40.358151 


.688188 


IS 


10.311811 


0.016340,103 


9.95366( 


SO 


16^0 










64«101 


431 


357893 


688508 


531 


31149i 


01640lll03 


953599 


5B 


Bt 










6483110 


431 


357640 


688883 


531 


311177 


016+63103 


953537 


58 


5) 










648618 


430 


35738! 


689113 


533 


310857 


0+6585 103 


953473 


57 


4f 






id 




848877 


430 


3571 S3 


689+03 


533 


310537 


0165871103 


933113 


56 


41 






80 




643135 


430 




689783 


533 


310817 


046648103 


9S3358 


S5 


40 






2* 




643383 


430 


356607 


690103 


533 


309897 


046710103 


953891 


54 


Si 






iS 




643650 


489 


350351 


6911+83 


533 


309577 


046778 103 


953888 




31 






33 


8 


61-390B 


489 


356095 


0907+8 


538 


30985S 


046834 


03 


953166 


S* 


88 






36 


S 


644165 


489 


35583i 


691068 


538 


308938 


046896 


03 


953104 


51 


8 






40,10 


6M423 


488 


365577 


691381 


538 


308619 


040958 


03 


953048 


50 


81 








6146ti0 


488 


355380 


69170O 


53] 


3OS3O0 






958980 


19 


11 






48 1! 


644936 


48S 


3SS064 


698019 


531 


3079:41 


017088 


04 


958918 


48 








«13 


645 193 


487 


3S4807 


698338 


531 


30766' 


047115 


01 


958853 


H 








5614 


645450 


487 


354551 


098656 


531 


307344 


017807 


04 


968793 


45 


m 




tS Oli 


S.645706 


487 


10.354894 


9.698975 


IsT 


10.307023 




9.958731 




4;l( 


645962 


486 


35+03* 


693893 


530 


306707 


047331 


104 


958669 


U 


51 






8'l7 


646818 


486 


353788 


693618 


S30 


306381 


017391 


104 


938606 


13 


51 






lait 


646+74 


486 


3535a< 


693930 


530 


306071 


0+7+56 


10+ 


958514 


K 


VI 






!6ig 


64S789 


485 


353871 


6948+8 


530 


305758 


017519 


101 


958181 


+1 








so go 


646981 


485 


3S30IG 


691566 


589 


305434 






958419 


10 


ID 






nsi 


647840 


485 


358761 


694883 


589 


305117 


0476+1 


104 


958336 


3!^ 


3< 






s»ii 


64T494 


484 


358506 


695801 


589 


304799 


047706 


104 


958891 


ie 


3! 






3*g3 


647749 


484 


358851 


695318 


589 


304188 


0+7769 


104 


95823J 


37 


» 






3GJI 


648004 


484 


351996 


695836 


589 


301104 


0*7832 




9SS16B 


36 


81 






40 ai 


648*58 


484 


351748 


696153 


588 


303847 


017894 


105 


938106 


is 


til 






4t'?6 


6485 I 8 


483 


351+88 


696470 


588 


303330 


0479S7 


105 


95201! 


» 








4887 


648766 


483 


351834 


696787 


588 


303813 


O48O80 


103 


9519* 


J3 








5? 28 




483 


330980 


697103 


580 


302897 


048083 


105 


95IS17 


H 








5«;2g 

IB 030 


64987' 


488 


350781 


697+80 


587 


30858( 


048146 


105 


951854 


31 


1 




9.649581 


488 


10.350171 


9.697736 


587 


ia308864 


10.018809 


105 


9.951791 


3t 


m 




*;3i 


64978 


488 


330819 


69B053 


587 


301947 


0+8878 




95178t 


89 


Si 






938 


650U34 


+88 


3+996f 


698369 


587 


301631 




105 


951665 


i» 


51 






IS 33 


650887 


+81 


349713 


69B683 


58S 


301313 


01R39e 


105 


95160: 


87 


«t 






IS' 31 


65Ui39 




3+9461 


69900 


586 


300999 


048161 


103 


951539 


86 


44l 






ao'ss 


650798 


481 


319808 


699316 




300681 


048381 


103 


951 Mr 




40 






S436 


651044 


480 


348956 


699638 


586 


300368 


018588 


103 


95141? 




3(1 






«B37 


651897 


480 


348703 


699947 


586 


300053 


018651 


106 


951349 




38 






38 3B 


651549 


480 


348+51 


700863 


535 


i9B737 


016711 


106 


951886 




89 






36 39 


65J8mi 


419 


348800 


700578 


585 


89918^ 


0+6778 




951888 




81 






40 40 


658058 


419 


317948 


700893 


585 


899107 


018S41 


108 


951159 




lu 






4441 


658304 


419 


317696 


701 808 


584 


89879$ 


oisgoi 


106 


95I09< 




16 






4S4g 


658555 


418 




701583 


584 


898+77 


018908 


10 


951038 




If 






£843 


65880b 




347194 


70183 


584 


898163 


049032 


10 


9.50968 




8 






sn'*) 


6S:K>5 


418 


31694^ 


7081S 


584 


89T848 


019095 


10 


05090J 




4 




17 43 




^418 


I0.31669S' 


9.70816 


584 


10.8975:14 


10.0(9159 


10 


9.950811 




m 




*4(i 


65355P 


417 


346448 


7087 Wl 


583 


897880 


019288 


10 


93077B 




58 






B:47 


6538(W 


417 


346191 


70309 


583 


89690 


019886 


10 


95071+ 




51 






U,4« 


65*05 


417 


345941 


703+0 


583 


8965U 


019350 


10 


950650 




48 






16|4fl 


654303 


416 


345691 


70378 


583 


89687 


019114 


10 


950386 




41 






80 au 


654S58 


416 


3454+8 


70403 


588 


895904 


019478 


10 


950588 




to 






!4S 


6S4tKrt: 


416 


345198 


70+351 


588 


895G5( 


0495 1« 




950158 




36 






88 5 


65505 


416 


344918 


70166 


588 


89533 


019606 


10 


93039 




3] 






38 53 


6A530 




34169; 


70+97 


588 


895083 


019670 


10 


950330 




a» 






36i 


65555 






70589 


6i8 


8917 If 


0+9734 


10 


93086G 




i 






405 




415 


344195 


1056U 


581 


89+39 


01979S 


10 


950808 




K 






41 M 


65005 


414 


343916 




581 


89108 


019S68 


10 


930138 




It 






48'5 


65fi30 


414 


343698 


70688 


58 


89377 


019981 


!0 






1 






Hiri 


65655 


414 


313119 


70SS+ 


58 


89345 


019990 


10 


930010 










5UiS 


65679 


413 


31380 


T0685 


581 


893146 


051*5i) 


10 


919943 










la OGG 


65704 


413 


318953 


70716 




898834 


05011!' 




949881 


< 


l( 1 
































b, *.'-• 






~si^;nr" 


CW.I^ 


1~T-mkT~ 






~^^ir 








4 1 ..r^ 






6 


-D 








83 

187 


8 


"15" 
30 


158 


5' ' 


5" 


31 


^ 


P.(0 


ij 


■L'"' 


3 


45 


saa 


I » " 


*T 




'^- 


1 


m 




K 






I 


■ 


I 



I 



and SeeeDta. 



Table V. 



46 



1 Hour, 



or 



«7 



48 




4 

8 
12 
16 
20 
21 
28 
32 
36 
40 
44 
48 
52 
56 



49 




4 
8 
12 
16 
20 
24 
28 



50 



Sine. 



9.657047 

1 657295 
9 657542 

3 657790 

4 658037 

5 658284 

6 658531 

7 658778 

8 659025 

9 659271 

10 659517 

1 1 659763 

12 660009 

13 660255 

14 660501 



15.9 

16 

17 

18 

19 

20 

21 

221 



3212:) 
3624 
4025 



44 

48 
52 
56 




4 
8 
12 
16 
20 
24 
28 
32 
36 
40 
41 
48 
52 
56 



51 



26 
27 
28 
29 



660746 
660991 
661236 
661481 
661726 
661970 
662214 
662459 
662703 
66294C 
663190 
663433 
663677 
663920 
664163 



D. I Cowe. I Tmog. 



30 9 

31 

32 

33 

34 

85 

36 

37 

38 

39 

40 

41 

42 

43 

44 





4 

8 

12 

16 

20 



45 9. 

46 

47 

48 

49 

50 



24151 
28 



.664406 
664648 
664891 
665133 
665375 
665617 
665859 
666100 
666342 
666583 
666824 
667065 
667305 
667546 
667786 



32 
36 
40 
44 

48 

56591 



52 
53 
54 

55 
56 
57 

58 



668027 
668267 
668506 
668746 
668986 
669225 
669464 
669703 
669942 
670181 
670419 
670658 
670896 
671134 
671372 



413 
413 
412 
412 
412 
412 
411 
411 
411 
410 
410 
410 
409 
409 
409 

409 
408 
408 
408 
407 
407 
407 
407 
406 
406 
406 
405 
405 
405 
405 



404 
404 
404 
403 
403 
403 
402 
402 
402 
402 
401 
401 
401 
401 
400 



52 0.60 671609 396 



Cosine. 



400 
400 
399 
399 
399 
399 
398 
398 
398 
397 
397 
397 
397 
396 
396 






10.34295.' 
34270i 
342458 
342210 
341963 
341716 
34146S 
341228 
340975 
340729 
S4048S 
340237 
839991 
339745 
339499 



10.339254 
339009 
338764 
338519 
338274 
S38O30 
337786 
337541 
337297 
337054 
336810 
336567 
336323 
836080 
335837 



70716ti 
707478 
707790 
708102 
708414 
708726 
709037 
709349 
709660 
709971 
710282 
710593 
710904 
711215 
711525 



10.335594 9. 
335352 
335109 
334867 
334625 
33438S 
334141 
333900 
333658 
333417 
333176 
332935 
332695 
332454 
332214 



.711836 
712146 
712456 
712766 
713076 
713386 
713696 
714005 
714314 
714624 
714933 
715242 
715.551 
715860 
716168 



716477 
716785 
717093 
717401 
717709 
718017 
718325 
718633 
718940 
719248 
719555 
719862 
720169 
720476 
720783 



10.33197! 9. 
331733 
331494 
331254 
331014 
330775 
330536 
330299 
330058 
329819 
329581 
329.342 
329104 
328866 
328628 
328391 



Secant. 



721089 
721396 
721702 
722009 
722315 
728621 
722927 
723232 
723538 
723844 
724149 
724454 
724759 
725065 
725369 



D. I 



520 
520 
520 
520 
519 
519 
519 
519 
519 
518 
518 
518 
518 
518 
517 



CotMg. I 



Secant. 



10.292834I1(X050119 



292522 
292210 
291898 
291586 
291274 
290963 
290651 
290340 
290029 
289718 
289407 
289096 
288785 
288475 



517 
517 
517 
516 
516 
516 
516 
516 
515 
515 
515 
515 
514 
514 
514 



514 
514 
513 
513 
513 
513 
513 
512 
512 
512 
512 
512 
511 
511 
511 



10.288164 10. 
287854 
287544 
2872'U 
286924 
286614 
286304 
285995 
285686 
285376 
285067 
284758 
284449 
284140 
283832 



725674 508 



511 
511 

510 
510 
510 
510 
510 
509 
509 
509 
509 
509 
508 
508 
508 



10.283523 
283215 
282907 
282599 
282291 
281983 
281675 
281367 
281060 
280752 
280445 
280138 
279831 
279524 
279217 



050184 107 



050248 
050312 
050377 
050U2 
050506 
050571 
050636 
050700 



050765 108 



0508:iO 



050895 108 



050960 
051025 



.051090 
05115.5 
051220 
051285 
051350 
051416 
051481 
051546 
051()12 
051677 
051743 
051808 
051874 
051940 
052005 



D. 



107 



107 
108 
108 
108 
108 
108 
108 
108 



108 



108 
108 



108 
108 
109 
109 
109 
109 
109 
109 
109 
109 
109 
109 
109 
109 
110 



Coeine I'm. •. 



m 

6 5 



9.94988116012 
949816 S9 
949759 58 
949688 57 
949623 56 
949558 55 
949494 54 
949429 53 
949364 52 
949300 51 
949235 SO 
949170 49 
949105 48 
949040 47 
948975 46 



10.278911 



ia052071 
052137 
052203 
052269 
052335 
052400 
052467 
052533 
052599 
052665 
052731 
052797 
052864 
052930 
052996 



ia053063 



878604 
878298 
877991 
277685 
277379 
877073 
276768 
276462 
276156 
275851 
27.5546 
275241 
274935 
274631 
274326 



053129111 
053196111 
053262111 
053329 111 
053396 111 
053462111 
053529111 
053596111 
053663 111 
053730118 
053797112 
053864112 



053931 
053998 
054065 



110 
110 
110 
110 
110 
110 
110 
110 
110 
110 
110 
110 
HI 
111 
HI 



111 



.9489 lU 45 
948845 44 
94878U 43 
948715 42 
948650 41 
948584 40 
948519 S19 
948454 M 
948388 37 
948323 36 
948257 35 
948192 34 



U 
56 
52 
48 
44 
40 
36 
32 
28 
24 
80 
16 
12 
8 
4 



948126 33 
948060 32 
947995 31 



9.9479^9 30 
947863 29 



947797 
947731 
94766^ 



947600 25 
947533 24 
947467 23 
947401 28 
947335 21 
947269 20 
19 



947803 
947136 
947070 
947004 



947136 18 

947070 17 

14 



118 
112 
112 



9.946937 15 
946871 14 
946804 13 
946738 12 
946671 H 
946604 10 
946538 9 
946471 8 
946404 7 
946337 6 
946270 5 
94620S 4 
946136 S 
946069 8 
946002 1 
945935 



11 U 
56 
52 
48 
44 
40 
3(i 
32 
2i* 
24 
20 
16 
12 



88 
27 

26 



10 
56 
52 
48 
44 
40 
36 
32 
28 
84 
80 
16 
12 



9 
56 
5S 



4( 
36 
32 
28 
24 
20 
16 
12 



8 ( 



Cotang. I I Tang. | Coeec. i I Sine. 



4 Hours, 



= 



m. Si 



or 



62 Degrees. 



P. P. to 
• or" 



1» 

8 
3 



15" 

30 

45 



61 
181 
188 



1« 

2 

3 



15" 

30 

45 



77 
15i 



u 



15^' I 16 

. 30 I 33 



P. P. to 




^ 










^^^ 


' 


39 Table V. 


Logarithmic Sines, Tangents, 




"~r 


Hour, 


or 


14 De 


<^ 1 


m. E. 




Sine. 


D. 


Cdkc 


Ll-g^ 


D, 


Ctang. 


Seoot. 


D. 


Corine. 


iH^lJ 


sr"o 


( 


9.383675 


844" 


10.ei632S 


D.396771 


89ir 


10.603889 


10.013096 


58 


a.986904 


io 


* < 




* 




384192 


843 


615618 


397309 


696 


608691 


013187 


53 


986873 


S9 


6< 




8 


s 


38468? 


8*8 


615313 


397846 


895 


608154 


013159 


53 


986841 


S8 


Si 




n 


2 


385198 


84! 


614808 


398383 


994 


601617 


013191 


53 


986809 


S7 


48 




16 


4 


385097 


840 


614303 


398919 


893 


60IO91 


013228 


53 


986778 


M 


44 




80 




386801 


839 


B13799 


399155 


698 


600616 


0138S1 


53 


9867*6 


55 


40 




ai 




386704 




613896 


399990 


991 


600016 


013886 


53 


996714 


H 


31 




2H 




38 7807 




618793 


400581 




fig947( 


013317 


53 


996685 


53 


31 




aa 


S 


38TT09 


836 


618891 


401058 


669 


69991i 


013319 


53 


986651 


M 


8t ' 






g 


388810 


8W 


611790 


401691 


888 


598109 


013381 


53 


B86619 


il 


81 1 






10 


38ST11 


834 


611889 


408124 


B8T 


697876 


013113 


53 


966687 


w 


» 1 






11 


380811 




610789 


402666 


886 


597311 


013*45 


53 


986655 


M 


le i 




49 


18 


389711 


838 


610889 


403167 


865 


696913 


013177 


53 


086583 


t» 


I ^ 




sa 


i; 


390810 


831 


609790 


403718 


884 


6962B! 


Q13509 




B86191 


M 


1 1 




56 


14 


3B0T0B 


830 


60929! 


404819 


883 


695761 


013511 


53 

63 


086159 


to 

15 


TH 1 


ST 


15 


ilislioe 




10.608791 




"982" 


10.595221 


la013673 


9.986427 




4 


16 


3B1T03 


98T 


606297 


405308 




694832 


013605 


53 


986395 


U 


5t 




S 


17 


39819D 


886 


607801 


405836 




59*164 


01363T 


54 


986363 


43 


S. 1 




1* 


18 


308693 


385 


60730i 


406361 


979 


593636 


01366B 


54 


086331 


48 


« 




16 


19 


333191 


884 


606609 


406892 


878 


593108 


013701 


54 


966899 


41 


** 




80 


20 


393685 


B83 


606315 




877 


692581 


01373* 


54 


986261 


u 


V ' 




ai 


8! 


394179 




605881 


4079*5 




S980S6 


013766 


54 


986834 


JB 


31 




!8 


88 


394873 


881 


605327 


40B471 


873 


591529 


013798 


54 


986808 


)» 


3 




38 


83 


395166 


880 


6018:14 


408997 


874 


591003 


013831 


64 


9B61GS 


H 


a 




36 


84 


3S365B 


819 


604342 


409581 


674 


59047 S 


013863 


54 


986137 


36 


8 




40 


85 


396150 


818 


603850 


410015 


973 


68995; 


013996 


54 


986104 


35 


« 




41 


8( 


396641 


BIT 


003359 


410569 


878 


669431 


013988 


54 


9Bfi07S 


H 


1 




48 


87 


397132 


BIT 


608868 


411092 


871 


58890! 


013961 


54 


96603f 


38 


I 




53 


W 


397681 


816 


602379 


411615 


870 


fi88385 


013993 


54 


986007 


a 


; 1 




56 


89 


398111 


816 


601889 


418137 


869 


587BG3 


014086 


54 


985974 


jii 


'a 1 


S8~ii 


3t 


».398e00| 




10.601400 


1.418668 


868 


10.587348 


10.011058 


9.9B594S 




4 


SI 


399088 


BI3 


60091S 


413179 


867 


fiB6821 


011091 


66 


9B6flOi 


!9 


5 




P 


38 


399575 


818 


600*85 


413699 


86S 


586301 


01*124 


55 


»B587t 


» 


s 




18 




400068 


811 


599938 


411819 


866 


586781 


011167 




9fl584S 


!7 


4 




16 


34 


400549 


810 


599451 


414736 


861 


585868 


014189 


55 


985811 


U 


4 




80 


35 


401035 




598965 


415257 


664 


59*743 


014228 


66 


985T7f 


85 


4 




84 


36 


40158<> 


808 


S98480 


41S775 


863 


584825 


011855 


66 


9Si1*f 


a. 


3 




88 


37 


408005 


80J 


597995 


416293 


868 


58370T 


0112S8 


65 


985713 


S3 


3 




38 




408189 




597511 


416810 


861 


583I9( 


014381 




985679 


82 


f: 




S6 


39 


4O80T3 


605 


597081 


41738r 


860 


6886T4 


011351 




985846 


!1 


t 




40 


4( 


403455 


804 


596545 


4i7aia 


959 


588168 


014397 


66 


9856 IS 


80 


8 




41 


11 


403938 


603 


596068 


418366 


B6S 


581648 


014180 


65 


985680 


Ift 


]i 




4B 


48 


404480 


808 


595580 


418873 


957 


581 181 


014453 


65 


9855*1 


LA 


1! 




fig 


H 


404901 


601 


595099 


419387 




590613 


011496 




965SH 




1 




fie 


tt 


405388 


BOO 


5916 IB 


419901 


866 


580099 


014520 


£i 


9H548G 
9.9BJi47 
9aJiU 


It 


1 ( 

.If 


i9~~0 






798 


10.594138 
593059 






10.579595 
579073 


10.014553 
011586 




'. 


u 


40634J 


48098; 


854 


66 




it 


*7 


M(i880 
407899 


797 


B93180 


481110 
481968 


863 

868 


576660 
£79048 


011619 
01*653 


56 

56 


995.S47 




51 


796 


S98701 




u 


49 


40777J 


T95 


593883 


422163 


851 


57753T 


011696 


66 


985314 




M 




so 


SO 


408854 


794 

79* 


591749 
591869 


488974 
48348* 


860 
819 


577086 
576516 


014780 
014763 


56 
66 


985884 
985844 




41 . 


SI 


51 


408731 




86 


51 


40980T 




590793 


483993 


648 


576007 


D11IB7 


66 


986811 




9 




38 


S3 
54 


409689 


798 


690318 
66981; 


484603 
485011 


846 
847 


5754J7 
574989 


011820 
014851 


56 


UBfilBfl 
98514c 




a 


IIOIST 


T9I 




40 


5! 


410eS8 


T90 


589368 


486619 


816 


674481 


011887 


56 


D86nS 




« 




44 


51 


411106 


789 


6888941 


4860 ST 


945 


5739T3 


014981 


56 


9S5UTI 




- If ■ 




48 


ST 


41 1579 


789 


589481 


486634 


844 


ST3466 


014956 


60 






u 




fi8 


58 


418058 


787 


6879*8 


487011 


813 


678969 


011969 




985011 




1 




56 


5P 


418581 


786 


687176 


487647 


643 


57845; 


015028 




98*97t 


1 


4 


I 


iO 


60 


418996 


783 


587001 


48B058 


848 


57194^ 


015066 


66 


984944 




C 


m. L 


T 


Codnt. 




Secant." 


CMMg. 




Tuig. 


C.otcc 


'~~ 


~&ii^ 


=r 


sr~» 


SHour*. 






graa. 




1 I' 1 IS'" 


188 1- 1 16" 
814 8 50 


1 130 1» 15" 

860 8 30 


—8-1 
16 


" 


"Z 1 


fc /i„;-7 ' 1 » 


p 


P.I* 


- 


nr" 1 




m » I « 


\ «»V , * -* 


85 1 " 


J ■ 





[~ 






^^^" 






^ 


« Table V. 




w. 






a Hours. 






30 De, 














m. M' 


Sine. 1 D. 


CoKt 1 T«4.. 1 D. 


ColHDg. 


s™nl. D. 1 C™oe. I/ 


s 






















■(To 


( 


1.698U70 394. 


iaSOIoaO 9.761439, 48S 


10.239561 


iaOM!4e9 121 


9.9375;(1^|60 






4 




699189364 


30OHII 


761731 496 


8598S9 


062542,18! 


93745D^9 


5tl 








! 


699401 364 


300593 


T6!0!3 486 


237977 


06!et5,l!! 


D373B5 JH 


5( 






I! 


3 


899628, 36* 


300314 


762314' 486 


237686 


062689 1!2 


937318 57 


4^ 






16 


4 


fl99BU 363 


300156 


7 62606 1 495 


237394 


062762 1!8 


93123a se 


4" 






80 


£ 


TOOOtiS 363 


2999:(B 


768997, 495 


237103 


062815 122 


937 I6J U 


40 






!* 


6 


T00i80 363 


299720 


763189 485 


236912 


068908122 


93T>l9i M 


3C 






28 




700498 363 


899502 


763179, 495 


836521 


062981 188 


9370 IS S3 


3* 






38 


S 


700T16 363 


899284 


763-r70 496 


836830 


063051183 


936948 S! 


tl 






3S 


! 


700933 362 


899067 


764061 4S5 


235939 


0631*8122 


93687! i 








4010 


701151 86! 


898949 


764352 491 


23564e 


063801 18* 


936799 V 










71)1368 362 


29963! 


764643, 484 


235357 


0632751*2 


936785 49 








481! 


701585 36! 


!98415 


76*933 484 


2350ri7 


063348123 


036658 te 








BS13 


T0180! 361 


!98I9S 


76588*' 484 


834776 


063122 1*3 


936579 W 








iBl4 


70?01B 361 


297991 


7655141 481 


231486 


063495,123 


936505 16 






I Uli 


3.702!a6 361 


IO.!97?6t 9.765905, 4S+ 


10.834195 


10.063569,1*3 


9 936431 45 


sn 






4' III 


10^45! 361 


897518 


766095 48* 


833905 


(163643183 


B3635I H 








817 


70!rt69, 360 


897331 


766395 483 


233616 


063716 123 


936294 13 








lixf. 


702885! 360 


297115 


766673 483 


233325 


0S3790l*3 


938210 it 








IflilS 


7031011360 


896899 


7609H5 483 


833035 


063861123 


936136 11 








90 i( 


7053 IT 360 


896683 


767235, 483 


23*745 


063939 123 


936061 40 


* 






!t:8i 


7035:13 359 


896467 


767545 483 


832455 


064012 12:4 


9359a9 IB 


» 






89,« 


703T49, 359 


896851 


767834 483 


832166 


061086123 


035914 t4 








3*.!3 


703964 359 


896036 


768121 492 


831876 


064160,123 


935BKJ il 








36,i1 


7041 791359 


895881 


769414 498 


831591 


061834 124 


93576H » 


*4 








704355 359 


895605 


769703 4n2 


231897 


06i;i09184 


935691 U 








t4.'8h 




S95390 


768992 49! 


83100* 


0S1388 181 


B3Snl) II 








48 iT 


704885' 3Sfi 


895175 


769281 492 


230719 


064457 184 


B355U S 








£3» 


705010 359 


294960 


769570 492 


!3043( 


064S31I84 


935468 « 








56'S9 


7052o4| 358 


99WiB 


769860! 491 


2301 4lJ 


064605 184 


SSS39S U 






X^BO 


» 705*69 




10.891531 


9.770148,481 


10.289951 


10.064690; 124 


9.935J» n 


M~5 




431 




3S7 


891317 


770437 491 


229563 


064751184 


»:1584<J 19 


M 






8 38 






291108 




289214 


061829 124 


93517129 


5! 






IS'33 


706118 






771015' 491 




064903184 


935097 ei 


la 






ig;s4 






893671 


771303! 491 


228697 


061979184 


935082 M 


** 






so'si 




356 


293461 


771592! 481 


289408 


0630581 121 


934946 M 


40 






!4;36 


70«753 


356 


29:1247 


771B80' 480 


229180 


065127 


1*4 


934873 M 


36 






S8ST 


706967 


356 


293033 


778169:480 


28793* 


06580? 


125 


B3479t 23 


3* 






3«:3* 


707180 


355 


892880 


772157 480 


227543 


065877 


125 


93*783 K 


» 






SG'SS 


T0T393 


355 


898607 


? 78745 480 


287255 


065351 


185 


93464S n 


«t 






4040 


70TfiOS 


355 


292394 


773033. 480 


286967 


065**6 


125 


93457* to 


sti 






41^41 


707BI9 


355 


298191 


7733211 480 


886679 


065501 


1*5 


934498,19 
9344**111! 


I« 






484! 


70803! 




291968 


773fiU8l 479 


226392 


085576 


1*5 


II 






SiM 


70S!i5 


354 




773896 479 


226104 


065651 


125 


9343411 17 








^S 


709458 


354 


291542 


7741«4l479 


285816 


065726 


185 


93*274 16 


' 




). 70*670 






9.7744711479 


10.886529 


10.063901 


125 


9.93*196 15 








V44 


70888* 


353 


891119 


774759; 479 


885841 


065977 


185 


93*123 14 


56 






84T 


709094 


353 


890906 


7750161 479 


284954 


065952 


185 


9340 W 13 


W 






1!48 


70930B 


353 


SB0694 


775S33 479 


281667 


0660*7 


185 


93397: IB 


40 






I«4B 


709518 


353 


890188 


775621 478 


281379 


066102 


186 


933998 11 


4« 


1 




!0'5< 


70B730 


353 


290270 


7T59U81 478 


28409! 


066178 


120 


93382! 10 


40 








709941 


Si! 
SS8 


890058 
289847 


77SI95 
776182 


478 


883805 


066*53 
066329 


126 

186 


B33717 9 
933671 H 


3« 




478 


S!3518 


ll 




3!|S3 


TII136' 


35! 


899636 


776769:478 


!8323l 


066401 


126 


933596 T 


tt> 


1 




36ls4 


T10575 


352 




777055; 478 


28<945 


086480 


m 


93358II « 


9* 






40|5.4 


7 1 0786 


351 


88921* 


777342 


478 


882656 


066.555:1 86 


9334U « 


. 2U 






44l.iH 


710997 


351 


889003 


777689 


477 




066631 


186 


933361 4 


in 






48 


hi 


711208 


351 


888792 


777915 


477 


S820Bi 


066707 


186 


933*9: 3 


11 


1 




Sf 


58 


7ll41fl 


351 


B885B1 


778201 


477 


281 79S 


06b79J 


186 


983817 2 


t 






£<I 


S9 


7116^9 


350 


28H371 




477 


281512 


066959 


126 


9331 11 1 






L 


4 


(JU 


711838 


;*50 


288161 


778771 


ill 


g2122( 


0S6n3* 


186 


9331166 


511 t 




inTT 


' 1 C«ine. 




"SriomtT 


Coung. 


— 


■I'-ng. 


COMC- 




Sine. ' 


m. i. 


i 


3 Hours, 


or 




5 


^ 


lecL 


1 


,^^"; ' 1 » 


107 
1 I61_ 


!■ 1 15" 
8 30 


1 72 
141 


;■ 1 '£ 


" h. " 


r;" 


'i 




-i 


3 1 


4S 

1 


Ui 


i6 
■ 


lii- 


■ 


-ill: 


i 


i 



E 




and Secants. 


Table V. 


«t 


- 


Z att™«. 






33IXsn». 








SintL 


"dT 


c™. 


jr^ng^ 


^ 


Crtuig. 


SeeuL 


D. 


Caiat. 


' u. % 






1.73610 


381 


10.26389 


9.8(8517 




iaie748 


ia0764OI 


13" 


0.9l3e» 


^iTi 




♦ 


7SC30 


324 


80369 


818791- 


161 


1B780 


07649 


IS 


983Wt 


« St 




9 


TSGig 


384 


26350 


813070 


401 


18093(1 


07S57 


!37| 98348 


X 58 




38 


T36BB 


383 


80330 


913347 


460 


1B065S 


07665 


137| 92334 


W 48 




10 


736B8 


383 


26311 


813623 


160 


18637 


07673 


13 


98386 


56 44 




30 


T3Tog 


383 


86892 


8la89!> 


460 


18B10 


07681 


13 


98318 


35 40 




a* 


737!T 


383 


86878 


8I4I7S 


460 


18588 


07690 


13 


983oe 


U 36 






737*6 


383 


8B2S3, 


914458 


460 


18534SI 


07698 


13 


98301 


>8 32 






T3766 


388 


86833 


814788 


400 


18587 


07706 


13 


982«3. 


« SB 






J37HiS 3!* 


86211 


Biaooi 


460 


18499 


0T711. 


13 


92895 


>1 t* 






73B0t9l 3B2 


26195 


815879 160 


18472 


07723 


IS8( 98»r6f 


V H 






TSBSi 


388 


86175 


815555 


459 


18M4i 


07731 


ISfi 


SiSWf 


n 16 






738434 


382 


80156 


815831 


459 


184169 


07739 


i38{ ataeoi 


H 18 




k "|l 


7396! 


381 


261373 


816107 


459 


183993 


077 1B( 


1381 S28681 


IT H 






m^ 


7896*0 


381 


S6IIHC 


816383 


459 


1H3H1B 


07756 


13B 


92243^ 


w! 4 




S^T 


RssoU 


Sir 


10.26098 


9.8(6058 


"459" 


10.193318 




l3B 


9.wa83« 


M4T 




■lit *I 


739206 


381 


860T91 


8 16933 




193067 


OT773t 


138 


92287 


H 50 




7S939B 


381 


26060 


817809 


459 


198791 


0778 1 


138 


98219 


a fit 




T39SB0 


SBt 


200410 


817481 


459 


182316 


077B9 


13B 


9»210 


18 49 




7397R3 


380 


860817 


81776!) 


459 


IB821 


07797 


138 


02202 


11 44 




739BT. 


380 


200085 


81B035 


458 


181963 




ISB 


98J91C 


10 40 




740167 


380 


259833 


819310 




1S1G90 


07811 


13B 


98195 


W 36 




7403.5H 


380 


25961 


818585 


458 


181415 


07882 


139 


98177 


« 38 




*-a 38 8 


7*0550 


319 


259450 


818860 


458 


181110 


0783(1 


139 


92168 


*T 29 






740742 


319 


!5985f< 


8I9I35 


458 


1B0SB5 


07839 


139 


98160 


le 24 




■»«l 40 S 


740934 


319 


259066 


8(9410 


458 


I80S90 


07817 


139 


92158 


u 36 




»<l «!( 


74118,5 


3i9 


8588 7o 


81S681 


458 


180316 


07855 


139 


92141 


M 16 




K^! 4SS 


741316 


319 


85S681 


819959 


458 


180041 


07801 


139 


98135 


J3| 18 




B* 1 Bi,if 


741 iOS 


318 


259498 


H80231 


458 


179706 


07978 


139 


981271 


H 8 




■■* 


' fi6;gt 


741699 


318 


258301 


B80508 


457 


17940? 


07881 


139 


92119{ 


It 4 




»«L4 0,3( 


».74laS9 


31H 


10.25811 


9.880783 


457 


10.179217 


!a07989 


139 


9.921107 


ioiTl 




t J 43 


7480H0 


5!S 


857020 


821057 


457 


178949 


07897 


139 


921083 


id M 




I f 83 


T4mi 


SIB 


857789 


981338 


457 


178068 


O790G 


iio 


980939 


!B 52 




t x lis. 


748468 


317 


857538 


981606 


457 


17B391 


079(4 


(40 


9808.56 


K 48 




V M 1«!34 


74S6S! 


317 


8S731S 


981860 




179180 


079Sif 




920771 


16 44 




. ■ 


!03 


748848 


317 


857 15t 


888151 




177816 


079318 


140 


'98n«8ti 


15 to 




' ■ 


!tM 


T43033 


317 


856967 


882489 


457 


177571 


079396 


140 


980B0 


|4v 36 




« 


!H3 


743883 


317 


856777 


888703 


457 


177897 


079180 


lit 


B805t( 


ta 32 




d 


3«3lj 


743413 


310 


256587 


828977 


*50 


177083 


07950 


140 


910431 


a s) 




■ 


:k;39 


743608 


31G 


256398 


983850 


456 


17fl75C 


079648 


140 


92035! 


H 2 




■ 


4O;40 


713793 


316 


83680H 


883384 


456 


176476 


079738 


140 


B8086B 


0^ 8( 




■ 


4H 


743982 


316 


25G01H 


B23798 


450 


176202 


079816 


140 


88018, 


19 1( 




■ 


4H4I 


74H71 


316 


855B8B 


881072 


*66 


175989 


079901 


140 


92009 


IB 18 




■ 


52,43 


744361 


315 


855039 


824345 


456 


175655 


079985 


140 


98001.) 


IT 8 




■ 


£S'44 


744350 


315 


2551St 


884019 


456 


173391 


080069 


ill 


91BB3I 


IS 4 




ui 


15 (1« 


a.7 14739 


315 


10.855261 


9.984993 




10.175101 


10.0HOI54 


141 


9.919841 


13 45 1 




If 


4 46 


74498S 


315 


25507S 


885166 


*56 


171834 


080838 


HI 


91B762 


'4| 56 






K4J 


7451 17 


315 


254BB! 


825139 


«5 


17450[ 


080,383 


141 


919677 


iS 58 




■ 


1S4& 


74S306 


314 


854691 


884713 


455 


1742K7 


0B0IO7 


141 


919393 


li 48 


f 


i«4S 


7.4i*94 


314 


854501 


885986 


455 


174014 


080192 


141 


9IB50( 


U 44 




soao 


745683 


314 


254317 


980859 


453 


173741 




141 


91918* 


to 40 




8VJ1 


745871 


3U 


8511 2S 


82653? 


455 


173468 


080661 




B1933H 


» 33 




8SS! 


746060 


311 


853940 


886905 


455 


173195 


080740 


141 


91925) 


8 3i 




3*53 


7t6J46 


313 


8S375S 


887078 


455 


178928 


080831 


Ul 


919169 


7 8( 




56 Si 


746436 


313 


253561 


887351 


455 


172619 


090015 


141 


919095 


6 21 




to Si 


746684 


313 


253376 


987684 


455 


178376 


OS 1000 


141 


BI9O00 


5 80 




44 M 


746818 


313 


253IB8 


887997 


M4 


178103 


081085 


I4i 


91891, 


4 1( 




4Ba7 


746999 


313 


853001 


888170 


454 


171830 


081170 


48 


BiSfOn 


3 12 


• 


jsae 


747187 


318 


258813 


88811? 


154 


171558 


08 1:55 


48 


918745 


8 B 


t 


£6 $9 


747374 


318 


£58686 


8287(5 451 




081311 




BiaC514 







16 oeo 


747568 


318 


258138 


9X9987 434 


I7I0I; 


081481 


41 


usa= 


JtH ' 




sm^ 


Coiinc. 


-n S^cnu 1 


Cotung. 


~T^^ 


a^J 




Wfc * 




li Moun, 














P. P. to 


1' 
t 


15" 

30 


95 


1' 
t 


IS- 
30 




















Bor 


3 


45 


143 


3 


^&i 










_ 


i 



^ 






38 Table V- Logarithmic Sines, Tangents, | 


OHour. or 14 De 


grees. | 


rn. I 




Sine. 


D. 


Cossc. 


TanK. 


D. 


Cotnng. 


SeauiL 


D. 


C«hne. 


l45l_H 


W~i 


( 


WBMTfi 


ill 


ladiesa; 


a 336771 




10.003889 


10,013090 


58 


B.S869m 


H) 


4 1 






1 


3S41S8 


813 


616611 


397309 


890 


608691 


013187 


53 


08687! 


» 


a 






! 


384687 


648 


61531) 


387840 


895 


608154 


013159 




986841 


S8 


6! 




18 




SSfilSS 


811 


61480) 




694 


001617 


013191 


63 


980909 


S7 


« 




16 


4 


385C97 


810 


61430; 


398919 




601081 


013282 


53 


986T7) 


•S 


* 1 




20 


S 


386801 


839 


613793 


399*35 


898 


600515 


013851 


53 


986746 


iS 


« 




n 


6 


386704 


838 


613896 


399990 


691 


60001 


013286 


S3 


986714 


>4 


36 








387807 


837 


61879! 


400584 


890 


599176 


013317 


63 


986683 


S3 


31 




3a 


8 


387709 


836 


618891 


401058 


889 


698948 


013319 


63 


986661 


it 


88 




36 


9 


388810 


835 


611791 


401691 




598409 


013381 


S3 


986619 


11 


8 




iU 


10 


398T1I 


831 


61ia8( 


408124 


887 


60T8T6 


013113 


S3 


986S87 


» 


» 




44. 




389^11 




610789 


408656 


886 


537314 


013415 


63 


986665 


10 


» 




49 


Ig 


389711 


838 


610886 


403187 


885 


596813 


0131TT 


S3 


986S83 


16 


11 




62 


13 


390810 


831 


609791] 


403718 


881 


596888 


013509 


63 


986191 


M 


i 




se 


14 


3H070S 


830 


60989i 


401819 


883 


595751 


013541 


53 


986159 


(a 


■ 


iT 


II 


9.391106 


888 


10.608794 


9.101778 


888 


10.595821 


10.013573 


53 


3.986*87 


Is 






4 


16 


391703 


887 


608897 


406308 


881 


691692 


013605 


63 


986395 


u 


s 




8 


17 


398199 


896 


607801 


405836 


880 


591164 


013637 


64 


986363 


w 


■ 6 




IS 


IE 


Si)869j 


885 




40G364 


879 


593B36 


013669 


64 


980331 


u 


4 




Ifi 


19 


3931B1 


881 


606809 


406898 


878 


69310) 


013701 


64 


986899 


41 


41 




20 


!0 


393685 


883 


sosaii 


407419 


677 


598581 


013731 




980861 


m 


4C 




S4 


21 


39H79 


888 


605831 


40T915 


876 


59805£ 


013760 


61 


986834 


M 


3 




86 


eg 


394673 


8Z1 




408171 


875 


69152B 


013796 


64 




H 


3 




3S 


S3 


395166 


B!(» 


eoiarsi 


10B997 


8T4 


691003 


013931 


54 


98616! 




81 




36 


84 


S9S6S6 


819 


BftBlS 


4095S1 


871 


590178 


013863 


64 


980137 




8 






ifi 


396150 


818 


603860 


410043 


873 


589955 


013836 


64 


98S10! 




8 




41 


aa 


396641 


81T 


603351 


4105GS 


87! 


589131 


013986 


64 


986078 








48 


87 


397138 


817 


60886) 


411098 


871 


69B90B 


013961 


54 


966038 








62 


88 


397681 


816 


608378 


411616 


870 


668385 


013393 


64 


986007 


« 






fie 


89 


398111 


815 


601889 


418137 


869 


aB7affi 


014086 


54 


98597' 


11 


■ 1 




30 


1.398600 


"bIi 


10-601401 






10.687318 


10.014058 


51 


9.9859U 


M 


3 




4 


31 


399088 


813 


600918 


113179 


667 


686821 


014091 




98590' 


iO 


M 




S 


38 


399S75 






113099 


666 


686301 


014184 




985871 


ta 






18 


Ba 


400068 




59993) 


411819 


866 


686781 


014157 


53 


9B59*: 


i1 


il 




16 


31 


400319 


810 


699451 


411738 


661 


585868 


011189 


55 


1185811 


i6 


4 




go 


3fi 


401035 


809 


59896; 


415257 


664 


68471^ 


014828 


55 


98S71t 


15 


« 




u 


36 


101380 


808 




116776 


863 


58428i 


014855 




98a74j 


i4 


s 




if 


ST 


40800S 


807 


59799^ 


416893 


868 


683707 


011288 


35 


98571! 


S 


s 






SI 


403489 


806 


697511 


110810 


801 


683190 


01138 1 


55 


98567! 


IS 


a 




36 


30 


408978 


805 


597088 


417386 


800 


688674 


011351 


55 


B856K 


11 


t 




40 


Ml 


U131fifi 




596515 


117812 


859 


688158 


0L138T 


55 


985013 


M 


» 1 




44 


41 


403938 


803 


596068 


118358 


868 


681648 


011190 




963380 


19 






48 


4S 


4044Sff 


sot 


595580 


118873 


867 


581187 


011453 




995517 








fi! 


4f 


401901 


601 


595099 


1193S7 


856 


580013 


011186 


55 


995511 








ae 


44 


40S388 


80O 


594618 


119901 


836 


580099 


011590 


55 


995190 


10 
15 


i I 
1 1 


59 


iS 


[>.40580i! 


■799" 


10.594138 








10.014553 




i 


*S 


10'>34l 


T98 


693659 


180987 






011586 


56 


995111 


14. 








« 


4069?O 


TOT 


59316(1 


481140 


853 


578560 


011619 


36 


995361 


IS 


et 




n 


li 


407899 


790 


699701 


421968 


858 


678048 


011053 


56 


995317 


18 


K 




16 


19 


lOlTTI 


795 


598883 


422163 


851 


577537 


014686 


56 


985314 


11 


U ' 






50 


408851 


791 


691716 


489974 


860 


5TT08G 


014780 


50 


985«8e 


la 


* 




at 


SI 


10BT3I 


794 


59186! 


483184 




570516 


011753 


66 


395817 


s 


9 i 




38 


Si 


409807 


793 


690793 


483993 


848 


570007 


014787 


56 


985818 


6 


3 




32 


5J 


409088 


798 


590318 


484503 


848 


575497 


014820 


56 


9e5t8e 


1 


* 




36 


Hi 


410137 


791 


689813 


48S01I 


817 


5749SB 


014831 


66 


985141 


a 


' M 




40 


si 


410638 


730 


58936) 


485519 


816 


671481 


014887 


50 


9951 U 


a 


« 




44 


Sfl 


11 1106 


Tes 


588891 


486027 


815 


573373 


011981 


66 


K850II 


4 


U H 




4S 


37 


lusie 


788 


589481 


426634 


8U 


673«( 


014963 




99504J 


a 


u ■ 




fit 


SB 


41»fit 


787 


667948 


487011 


813 


678969 


014983 


56 


985011 


8 


9 




56 


it) 


418584 


788 


5874T6 


487547 


813 


678153 


015028 


56 


984971 


1 






fO 


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118996 


T85 


687004 


488058 


818 


fi7191)i 


015056 


56 


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C«in. 




Seoul U 


cSTogT 




Tang. 


Ccaec. 




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a Houn, 


75 Degree. | | 


„ „..l 1' 1 IS" 


182 
814 


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1 130 1 1- 1 15" 1 8 
860 1 2 30 Hi 




^J"" J S 1 *S 




366 


3 45 


\ m \ a I 45 t SSI •" III 







and SecflTtti. 


TiSL* V. 


Si 




IHmir, 


or 




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ii-tr 


Sine. 


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Cotrt. 


T«,g. \ D. 


Cowng. 


P«int. 


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10-597004 


S.1W058| 81! 


10.571919 


I0.015()"S6 


57- 


B 981944 




UTo 




4 




413461 


794 


596533 


4!9557 941 


57I44.'^ 


01 SOW 


57 


991910 


M 






e 




4I39W 


793 


5«G068 


439063 8«> 


670938 


0151S1 




98487f 




a 




18 




4I440S 


783 


585592 


489S6fi 8119 


571) Ctl 


015159 






S7 


iS 




16 




414876 


798 


5S51si 


43(K)7n 93H 




DI5I9V 




981«0fi 


i« 


44 




80 




4I53» 


791 


584653 


+3057:1 93H 


569187 


0I58!6 


57 


981771 


H 


40 




84 




4I58I5 


790 


BHH85 


43! 0-5 837 


5«S!)8.1 


OI5!80 




nstiio 


i* 


16 




«8 




4I6!!)3 


779 


SK3-17 


431J77 836 


58H1K 


015894 


57 


98170« 


^^ 


38 




31 




41B7SI 


778 


583149 


4381)7!) 8:15 


5679!1 


0153!( 


57 


9X467* 


5> 






Sfl 




417gl7 


T77 


58878! 


43S5S0 834 


567180 


015368 


57 


9846W 


51 


81 




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417684 


776 


5H83lf 


4330SO 833 


5G6!)!r 


01.^:19' 




904603 




«0 




i41t 


4181 £0 


775 


5H1R50 


433580 838 


.W6480 


fli5i:ii 


57 


9815(:B 


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Hi 




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413615 


774 


591 38S 


434(>t<0 H:<8 


5S,',9!0 


015465 


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984535 


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1! 




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773 


580981 


434 J79 931 


565181 


01S50( 


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98450(1 


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MM 


419544 


773 

778 


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43,50-M 930 


561 9J8 


015534 

Tu.()i5,iii[. 


57 

M 


981466 

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T^jfi 


I.4!0(K1T 


10.579993 


1.435576 ma 


10.561*)! 1 




4l( 


480410 


771 


5T9530 


436073 888 


563987 


015603 


58 


981397 


14 


56 




8V 


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770 


579067 


436570' 888 


5SS43C 


0I.M:t7 




981:163 


W 


5! 




1 8 If 


4S139.5 


769 


579605 


437067 887 




015678 




9943!R 


I! 


4h 




16 li 


481 B57 


768 


57814; 


437563 886 


56!41- 


015701 


58 


n-i4';94 


11 


44 




20 8( 


48831 R 


767 


577681 


438059 985 


561941 


01. '.711 


M 


98U59 


40 


4(1 




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4S817( 


767 


577888 


438,554 981 


561446 


015771 




984!!4 


19 


36 




WH 


4I3S3( 


766 


576768 


43!)01S 823 


66095! 


Ol.iBIO 


59 


!)Hll»n 




38 




38 C 


M3U7 


lej 


57630: 


439513 833 


560457 








itI 


88 




36l< 


4U1II 


T64 


5TS844 


410036 88! 


559961 


015880 




S81I8D 


J8 






40U 


484615 


763 


575385 


440589 881 


559171 


015915 




984095 


15 


80 




4486 


485073 


76! 


57t9i7 


441088: 8S0 


558978 


015950 


58 


B8405' 


U 


16 




49 8r 




T6I 


57447( 


441514 919 


558496 


015985 


09 


98I0I5 


13 






sstt 


485981 


760 


57101: 


44800S 819 


557994 


016019 




98398) 


8 




sets 


486443 


760 


57^557 


44!497 HIS 


S57S0r 


016054 


,-.s 


9H3StF 


11 


4 




T-oa 


>.4t6899 


769 


lu.S7;nui 


[l.41!98M, 817 


10.557018 






!), 983911 




58 




4'SI 


481354 


758 


578646 


443479, 816 


556581 


Olfil8.i 




9H:IS75 




56 




ea 


4RB0a 


rsT 


578I9I 


443969 816 




OKilKO 


59 


9838 Kl 


» 


a 




1831 


4tfl86S 


756 


571737 


44U5B 815 




016195 


59 


993S0J 


i7 


48 




IB 34 


488T1T 


755 


57189; 


4419171 911 








983771 


16 






30 3S 


4EB170 


754 


570930 


44,Sl:t5 913 


551565 






983735 


15 


40 




8431 


489683 


753 


570377 


415883 81! 


551077 




59 


■ltl37O0 


H 


3! 




89 31 


4300T5 


758 


5G9985 


416111 819 




016331 




093681 


S 


3! 




38 3S 


430287 


75? 


569473 


446803 811 


55:noi 


016,i;i 


59 


U8S68t 


Ig 


!B 




36 38 


430978 


751 


56S08S 


417384' 810 


53^616 


oieiof 


j9 


983591 


!1 


81 




4«40 


4314(9 


750 


568571 


447970' 809 


S58130 


01611! 


5n 


98355f 


?0 


10 




444i 


431879 


749 


568181 


449356 809 




016177 


59 


691583 


LB 


I' 




484! 


4383i9 


749 


567671 


418811' 808 


55115E 


016513 


59 


983197 


18 


I! 




58 43 


438778 


7*8 


567882 


44i)386 807 


550671 


016548 


59 


9834.5SI 


17 






BS44 


433g«8 T47 


566774 


419BI0 806 


550190 


016581 


59 


9834If 


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4 




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K.43367 51 746 


10.566385 




!0.5+»701> 


ia016.il& 


59 


9.MB3381 








446 


434188 745 


565878 


450777 905 


519883 


01665,5 


5U 


BB334S 




5'i 




847 


434560 


744 


565431 


451 !60 901 


518740 


016691 


59 


9H;f30< 




6-' 




1848 


435018 


744 


561984 


451743 803 


548857 




60 


9*073 




*» 




10« 




743 




45!8!5 80! 


547775 


01670! 




983!3t 








W« 


435908 


748 


564098 


45!70S'90! 


517291 


016798 




B8S80I 




4( 




14 U 


4363531 741 


563647 


453187.801 


51GB13 


016834 


60 


983166 




8( 




ssft 


436798 740 


56380! 


4S3668I 800 


54B332 


016870 




98313(1 




Si 




SSK 


437848 J to 


568758 


451148; 7»9 


51JHi! 


016906 


ao 


B8:i091 




at 




S6M 


437686, 739 


568314 


451688 709 


54.«78 


01694! 


60 


98305B 




!i 




40W 


438189, 738 


561871 


455107] 798 


51t893 


016978 


60 


993081 


I 


811 




44H 


439478 737 


56I4!t 


455586 


797 


511114 


OI70I4 




983981 




U 




48 57 


439014 738 


560986 


456064 


796 


543936 


017050 


80 


B8305C 




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58 58 


439456 736 


560544 


45654! 


796 


513459 


017086 


60 


98S911 




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46 59 


43S897 735 


560103 


457019 




512981 


0I7i!8 


60 


988a7S 


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4 080 


440138 731 


559668 


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791 


51Y501 


017158 


60 


98281* 




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8.;c»d(. 


CuUng. 


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Tsng. 


i:u..c 




Sme. 




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74 


Dfgte... 






114 

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311 


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3r.R 


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Sin 


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or 


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p. 


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SHint. 


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734 


iai59e68 


1,457496 


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10548504 


ia0i7159 


60 


9.98884 


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441818 


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559888 
559788 


457973 
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017195 
017231 


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61 


98880 
98876 




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51 






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54155 


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441658 


731 


5583*8 


458985 


798 


541075 


017867 


81 


98873^ 


57 


48 






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448096 


731 


557904 


459400 


791 


54O600 


01730* 


61 


98869 


56 


41 






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448533 


730 


557466 


459975 


790 


540135 


017340 


61 


9986m- 


53 


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448973 


789 


657087 


460349 


790 


539651 


017376 


61 


98862 


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788 


556590 


460883 




539177 


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443847 


787 


55615! 


461897 


T88 


638703 


017449 




9a855 


58 


86 






36 


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444884 


787 


6557K 


461770 


788 


638830 


017196 


61 


9S8S1 


5 


81 






40 


10 


4.44780 


786 


555880 


4688*8 


787 


537759 


017583 


61 


99817 


SO 


8tl 






44 




445155 


785 


55*845 


46871* 


786 


537896 


017559 


61 


998*4 


19 


16 






48 


12 


44S5B0 


78* 


654411 


*63196 


785 


53691* 


017596 


61 


988404 


18 


It 








13 


44C035 


783 


6S3975 


463658 


785 


536348 


017633 


61 


98836 


17 


( 






ifi 


14 


446459 


783 


553541 


4641 ?8 


784 


63587! 


017669 


61 


98833 


46 


1 




5 


15 


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iM 


10.553 lOT 


9.464599 


783 


10.635401 


10.017706 


61 


0.98889 


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56 U 






4 


16 


4*7386 




538674 


465Q69 


783 


534931 


017743 


61 


988857 


U 


51 






S 


n 


447759 


780 


558i4I 


465539 


782 


534461 


017780 


68 


9S2i2C 


13 


5 






19 


18 


448191 


780 


551809 


468008 


781 


533998 


017817 


68 


98818: 


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19 


448683 


719 


551377 


466*76 


780 


53352* 


01785* 


68 


99814 


41 


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to 


80 


449034 


718 


550946 


4669*5 


780 


5S3055 


017891 


6! 


9881 (M 


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4494S5 


717 


650515 


467413 


779 


538587 


017988 


69 


988078 


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3 








w 


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449915 


716 


550085 


487880 


778 


638180 


017965 


68 


988035 


W 


31 








3! 


83 


450345 


716 


649655 


468347 


778 


631653 


018008 


68 


98 1998 


17 


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36 


84 


450775 


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6*9885 


468814 


777 


S3U96 


018039 


68 


98196 


» 


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40 


M 


451804 


714 


5*8796 


469880 


776 


530780 


018076 68 


981934 


t5 


81 








44 


86 


451638 


713 


6*836! 


469746 


775 


530854 


018114 68 


aS188lS 


M 


IE 








U 


87 


458080 


713 


6*7940 


470811 


775 


689789 


0181511 68 


9918*9 


ra 


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ft! 


8S 


458488 


718 


5*7518 


470676 


774 


589381 


018199,68 


9K1818 


« 


1 








S6 


89 


458915 




547095 


471141 


773 


589959 


018286 


63 


981174 
9.981737 


!l 

10 


1 
64 


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30 


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ia5*e65& 


9.t71S05 




10.588395 


10.018263 










31 


4S3763 


710 


5*683! 


4;80B9 


778 


587938 


0I83OO 63 


981700 


19 


31 










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454194 


709 


5*5801 


47253? 


771 


687468 


018338; 63 


991601 


W 


S 








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33 


454619 


708 


5*5381 


47899.) 


771 


587005 


018375:63 


981685 


n 


« 








16 


34 


455044 




6*4951 


473457 


770 


586513 


018*13' 63 


981597 


K 


i 








to 


ia 


455469 


707 


5**531 


473919 


769 


586081 


01 8131 163 


991549 


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24 


36 


455893 


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6**107 


474391 


769 


525619 


018*98 63 


9915U 


84 


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37 


45«316 


705 


5*3694 


47*948 


769 


585159 


018586 63 


9914T4 


13 


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3i 


36 


456739 


704 


643861 


475303 


767 


524697 


019564 63 


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8! 


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86 


31 


457168 


70* 


548839 


475763 


707 


581837 


018501 


63 


991399 


81 


t 


1 






40 


40 


457534 


T03 


642*16 


476883 


766 


583777 


018639 


63 


08136 


80 


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41 


458006 


708 


611994 


476683 


765 


523S17 


018677 


63 


991383 


19 


16 








46 


48 


458487 


701 


541573 


477148 


765 


528858 


0187 [5 


63 


991895 


18 


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43 


458848 




64115! 


477601 


764 


588399 


018753 


63 


991 84T 


17 


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44 


459268 




64073! 




763 


5810*1 


01879! 


63 


981800 


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9.450688 


■fiSF 


10.5*0313 


9.479517 


763 


10.521 4gf 


10.018829 


63- 


9.98 UTl 


13 


53^ 








4 


46 


4601 OS 


699 


539898 


478975 


768 


681083 


019867 




991133 


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460587 


698 


539473 


479438 


761 


5S056f 


018905 64 


OS 1095 


IS 


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18 


41 


460946 


69T 


539051 


4798B9 


761 


580111 


018043 64 


991057 


18 


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16 49 


461364 




538631 


4B0345 


760 


519655 


018981 64 


981019 




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461788 


695 


538818 


4SD80I 


759 


519199 


019019 G4 


99098 


10 


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468199 


695 


537801 


481857 


759 


5197*3 


019058 64 


980941 


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16 








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468816 


604 


537394 


181718 759 


518888 


01 90981 6* 


98O0O4 


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38 63 


4C303S 


893 


536988 


488167' 757 


51783; 


019134 6* 


980869 


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36 S4 


4634*8 


893 


536558 


488681 


757 


517379 


019173^ 6* 

010811 61 


990887 


6 


t 








40 5i 


463864 


698 


536131 


483075 




516985 


08079( 


5 


K 








h',S6 


46*879 


691 


535781 


493589 


735 


516*71 




99075(1 


* 


U 








48 £T 


464«94 


680 


53530^ 


493988 


755 


516019 


019888 64 


990711 


a 


II 








52'M 


465108 


090 


53469! 


484*35 


754 


515565 


019387 64 


S8067! 


t 


1 








sesB 


46538! 


689 


63447F 


4948S7 


753 


61511; 


OI936S64 


98063J 


1 






1 




e o|eo 


4659S5 


889 


634065 


485339 


758 


514661 


010104] 64 


9S05»« 





St t, 




=^i=? 


^sri 


'*~ 


Secant. 


Couuig. 




■•""B- 


C««. 1 


Sine. 


" 


i3 




iHoutt. 




73 D., 


!«». 1 




L / 


^„^"/ '■ 1 - 


106 
813 


!■ 1 15" 
i SO 


1 118 1 1- 1 15" 
1 831 1 8 30 


rr 


p. 


i- 




mJ- 


'"Is / .5 


319 


3 ,\ W 


\ ail \ ^ \ W I 19 


* 


sL 




mr 


■ 


k 




^ 


■ 







md Secants. Tjiili V. 


151 




)Uoun. 


Or 


36 D. 


p-. 




D. t. 




Sioc. 


D. 


COKC 


T.,«. 


_5^ 




Sout. 


dT 


Coont 


-'-■ -i 




fO 





9.158591 


MOT 


10.841401 


9.845827 


«9 


10.1S47f3 


10.096635 


W 


f^mi 


JO 


'io~~i 




* 




758778 


300 


841288 


845196 


448 


154504 


096781 


147 


913276 








s 




758952 


300 


1410t» 




449 


154836 


086813 




913197 


t8 


Si 




18 




759132 


300 


840868 


8460.33 


448 


153967 


OB6901 


14B 




57 






16 




759312 


300 


240688 


946308 


448 


15369B 


086990 


149 


913010 


S6 






to 




759498 


300 


240508 


846570 


447 


153430 


097078 


149 


912922 


55 






Si 




769678 


899 


84U38B 


946839 


447 


153161 


097167 


148 


918833 


51 


3. 




IS 




759859 


899 


240 lU 


847107 


447 


152893 


087856 


148 


918741 


53 


31 




3! 




760031 




83996D 


847376 


447 


158684 


087315 






58 


8a 




3(i 




760811 


299 


839789 


947644 


447 


158356 


087434 


148 


918566 


51 


84 








760390 


299 




847913 


447 


152097 


087 5i3 


148 


912471 


M 


20 




U 




T60569 


299 


839131 


848181 


447 


161819 


087618 


149 


91239t 


19 


16 




48 




7G0748 


899 


239252 


849449 


447 


151551 


087701 


149 


91889S 


18 


11 




S3 




760987 


899 




8-48717 


U7 


151893 


087790 


149 


918810 


17 


a 




56 




761106 


2 98 


838891 


849986 


447 


161014 


097879 


149 


912121 


16 


4 




!1 




9.761885 


ioa" 


lU.23B7ii 


f.849i51 


447 


10.150746 


10.087969 


149 


9.918031 


IS 






i 




7SU64 


899 


838536 


B49528 


447 


150478 


099058 


149 


911942 


14 


56 




a 




761642 


897 


238358 


849790 


446 


150210 


08H1 47 


149 


911853 


13 






u 




761821 




838179 


950058 


416 


149948 


088831 




911763 


18 






16 




761999 


897 


838001 


B50325 


446 


149675 


089326 


149 


911674 


11 


44 




?0 


SO 


768177 


897 


837S23 


8.50593 


446 


149407 


088116 


149 


911681 




40 




H 


21 


76235b 


297 


837644 


85U86I 


446 


149139 


088505 


149 


91149; 




36 




aa 


82 


762534 


296 


837466 


851129 


446 


148871 


088595 


149 


911405 




32 




3« 


2'. 


762718 


296 


237288 


851396 


446 


148604 




160 


9I131S 


M 


29 




3G 




762889 


896 


837111 


851664 


446 


148336 


09977-1 


150 


911886 


16 


84 




40'.«i 


763067 


296 


23693; 


851931 




148069 


08B86* 


150 


9U136 


(5 


80 




u\ie 


T63245 


896 


236755 


858199 


446 


147801 


098954 


150 


91l04t 


}4 


16 




4S 


S7 


763428 




236578 


852466 


446 


147534 


099044 


150 


910951 




11 






2^ 


763600 


895 


236400 


853733 


445 


147867 


089134, 


160 


910961 








S6 


89 


763777 


295 


83638: 


853001 


445 


146999 


089884 


150 


910776 


« 


' 




ji 


30 


9.763954 


295 


10.83604( 


9.8538eti 


445 


10.146738 


ia089314 


150 


9.910696 


iO 


39 




4 


31 


7G4131 


295 


835839 


853535 


445 


146465 


099404 


UO 


9105!)( 


19 


5S 




8 


3i 


764308 


895 


235E9! 


853802 


445 


146198 


099494 


150 


910508 


28 


68 




1? 


33 


764485 


894 


835515 


854069 


445 


145931 


099585 


150 


91041J 


81 


48 




lb 


Jl 


764668 


894 


235338 


854336 


445 


145664 


089675 


151 


91032^ 


26 


44 






35 


764839 


294 


235162 


954603 


445 


145397 


089765 


151 


910835 


86 


40 




gl 


3G 


763015 


294 


834985 


854870 


445 


145130 


089856 


151 


910141 


i4 


36 




29 


17 


765191 


894 


234809 


855137 


445 


144863 


089946 


151 


910054 


i3, 


32 




33 


3( 


765367 


894 


834631 


855404 


445 


144596 


090037 


151 


909963 


i2 






30 


39 


7655 U 


893 


834456 


BSS671 




14432S 


090187 


161 


909873 


il 


84 




40 




765780 


293 


23428( 


BS5938 




144068 


090818 




90978i 


20 


80 




11 


41 


76589b 


893 


23410* 


856804 


444 


143791 


090309 


151 


909691 


19 


16 




*8 


is 


766078 


293 


833928 


856471 


444 


14352! 


090399 


151 


909601, 


18 


1! 




£8 




766847 


893 


233753 


856737 


444 


14386E 


090490 


15] 


9095 [( 


17 


S 




JW 


u 


766423 


293 


833577 


857004 


444 


1429S6 


090681 


151 


909419 


16 
15 


4 
37 




B~0 


u 


1.768568 


898 


ia833*01 


1.857870 


444 


1ft 148731 


10.080672 


168 


9.90938t 




4 


u 


78S7J4 


898 


233211 


857537 


444 


148463 


090T63 


152 


909831 


14 


£S 




S 


47 


788949 


298 


233051 


857803 


444 


142191 


090954 


[58 


909146 




58 




1! 


4« 


767124 


898 


832876 


858069 


444 


141931 


090945 


158 


90905i 


18 


48 




IB 


49 


767300 


292 


232700 


958336 


444 


J41664 


09103b 


158 


909964 


tl 






to 


SO 


767475 


891 


g3858i 


858608 


443 


141398 


091187 




908973 


10 


40 




24 


51 


767649 


291 


232351 


868868 


443 


141132 


091219 


158 


903791 


9 


36 




is 


S2 


767884 


891 


232176 


659134 


443 


140966 


091310 


152 


908690 


9 


32 




3! 


53 


767999 


291 


83800! 


85S400 


443 


140600 


091401 


152 




7 


29 




se 


54 


76S173 


291 


231827 


859666 


443 


140334 


091493 


158 


908507 


6 


84 




40 




768348 


890 


831658 


859932 


443 


140068 


091684 


163 




5 


80 




44 


56 


768523 


290 


231418 


860199 


443 


139803 


091676 


153 


908321 


4 


16 




48 


5T 


768S97 




231303 


860464 


443 


13S53Q 


091767 


153 


90823; 


3| 


u 




S3 


58 


768871 


290 


831189 


860730 


443 


139810 


091859 


1S3 


908141 


8 


a 




50 


59 


769045 


890 


830955 


860995 


443 


13900; 


091951 


153 


90804S 


1 


4 




14 


!2 


769819 


290 


230791 


861261 


443 


138739 


092042 


153 


90795f 





36 




nTTl" 


Cosine. 




SecBdL 


L'qung. 




Tm- 


Co«e. 




Sine. 


~ 


m. ■ 




3Ht>un. 


or 




i 


gro* 


' 




P. P. to 


1' 

i 


IS" 
30 


44 

m 


1- 
8 


16" 
30 


1^; I i' I 




.^.-B 


\ 


lot 


3 


45 


133 


3 


46 


2M \ ^ \ «. 


*"■ \ 



I-" 










^^ 


ae 


T.Btf V. 








IHour. 




19D(SiM«. 






iTT 


' 


Sine. 


D. 


COKt 


T..,. 


D, 


Catang. 


8Msnt. 


°L 


Count. 




^r~i 


rri 


"o 


^48998* 


648 


lasioui) 


9.511776 


"TTe" 


ioTiSSaii 


iih081T94 


"68 


9^79856 


id 


l»~0 






1 


490371 


648 


S0962S 


518806 


716 




021835 




979165 


&9 


.M 




6 


t 


4907 59 


647 


509841 


5i«635 


715 


48T365 


081976 


68 


978184 


58 


58 




11 


3 


191147 


646 


50935: 


513064 


714 


486936 


08I9E7 


69 


978083 




W 




IG 


* 


491535 


646 


508165 


513493 


714 


486S07 


081958 




9T804! 


56 








6 


491982 


645 


508018 


513981 


713 


486079 


081999 


69 


979001 


55 


*0 






6 


49SS08 


644 


507692 


514349 


713 


485651 


022041 


69 


97 7959 


51 


s« 






7 


491695 


644 


S0T3O3 


fil4777 


712 


485223 


022082 


69 


977918 


S3 


St 




3! 


t 


493081 


643 


508919 


51 5204 


713 


484796 


028183 


69 


977877 




88 




36 


9 


4g:f466 


64S 


506534 


515631 


711 


484.369 


022165 


69 


977835 




U 




40 


10 


493951 


642 


506149 


516057 


710 


483943 


088806 


69 


977791 




2C 






11 


494Wfi 


641 


505761 


516484 


710 


483516 


022848 


69 


9777.52 


49 


K 






1« 


494631 


641 


505379 


516910 


709 


483090 


022289 




97771J 


48 


It 




5S 


13 


495005 


640 


504995 


517335 


709 


488665 


028331 


69 


97 7669 


47 






56 


14 


495388 


639 


50+612 


511761 


108 


488239 


088372 


69 


B77688 




4 




13 


U 


'.495779 




IO.50422I 


9.518185 


709 


10.461815 


la022414 




9.y77S»ti 


*5 


47 






IS 


496154 


638 


503846 


518610 


707 


48139(1 


088456 


70 


9175*4 


U 


U 








496537 


637 


50346: 


519034 


106 




02S497 




97I.«0:t 


*3 


5i 




12 


IS 


496919 


637 


503081 


519458 


106 


480548 


088539 


70 


911461 


*8 








19 


497301 


636 


502699 


519882 


705 


480118 


082581 


70 


917419 


41 


41 






!0 


49T68! 


636 


508318 


520305 


705 


4T969i 


022623 


70 


917377 


40 






ti, 


!1 


49X064 


635 


501936 


520788 




479878 


088665 


70 


977333 


39 


S« 






H 




634 


501556 


521151 


704 


♦79849 


0887 U 7 


70 


977893 


M 


3i 






83 


49Se!5 


634 


501 17i 


521573 


703 


♦78481 


028749 


70 


977251 


*r 


2) 






(4 


4S9S04 


633 


500791 


521995 


703 


479005 


088791 


70 


977809 


ie 


2t 




U 


a 


i99584 


632 


500416 


528*17 


702 


477583 


028833 


70 


977167 


IS 


M 






ie 


499963 


632 


500031 


522838 


708 


47716! 


088815 




97TI85 


M 


If 






a? 


50034! 


631 


499658 


523259 


701 


476741 


082917 


JO 


977(183 


» 


11 






iS 


500721 


631 


499219 


523690 


701 


476320 


082959 


70 


977041 


12 


8 




Si 


«s 


S0I099 


630 


49890! 


52*100 


700 


475900 


083001 


TO 


9T6999 


11 




m 


3fl 


►.MUTB 


629" 


10.49^52' 


9.584520 




10.47549( 


10.0230*3 




9.Dlt>9&7 


10 


Ml * 






31 


501854 


629 


498141 


534939 


699 


475061 


083096 


70 


976914 


19 


51 1 






3» 


jOtSSl 


628 


497766 




699 


♦14641 


023128 


71 


976878 


N 


51 






X 


502B07 


628 


497393 


525778 


699 


♦74228 


023170 


71 


97Gg3( 


17 


« 






34 


S0W84 


687 


497016 


526197 


697 


♦73803 


083813 


71 


97H7B1 


86 






*0 


3J 


50S360 


626 


496640 


586615 


697 


473385 


0?I38S5 


71 


9767 *i 


15 


*C 




14 


36 


fi0373i 


626 


♦96265 


527033 


696 


♦7896r 


023299 


71 


976708 


84 


3( 




ts 


37 


504110 


625 


495890 


527451 


696 


472519 


0233*0 


71 


97666( 


!3 


31 




s« 


31 


504485 


625 


495515 


587868 


695 


♦78138 


08S383 


71 


916617 


12 


W 




3( 


39 


5048SO 


6S4 


495140 


588885 


695 


♦71715 


023*26 


71 


91G574 


U 


t. 








S05S34 


623 


494766 


528702 


694 


♦71898 


083469 


71 


976532 


to 


n 




*l 


♦1 


505608 




494392 


589119 


693 


♦70981 


023511 


71 


976*89 


19 


ii ' 




18 


4i 


505981 


622 


494019 


529535 


693 


470465 


083554 


71 


976i4( 


li 


i: ' 




S2 




506351 


622 


493646 


529950 


692 


♦700.70 


083596 


71 


976401 


17 






SB 


44 


506727 


621 


493873 


Sfi03'iS 


692 


469634 


083639 


71 


916361 


|6 




iii) 


4J 


l».5l)709fl 


B20 


la492yoi 


I.530I81 


697 


la*698l9 


[U. 08368 8 




9.9I«3tB 


15 


in 






4fi 


507471 


620 


492529 


531196 


691 


♦68804 


083725 


71 


976»7i 


14 


J 








507843 


619 


498157 


S3I611 


690 


♦68369 


083168 


78 


976831 


13 


1 




\i'A6 


5092U 


619 


491796 


638025 


690 


♦6791S 


083811 


78 


976189 


12 


« 




1619 


508585 


618 


491415 


538419 


699 


♦67561 


023854 


78 


916146 


11 


4 




snsa 


508956 


618 


491044 


632853 


689 


467147 


083897 




97610; 


10 


t( 




24 il 


5093*6 




490674 


433266 


688 


♦66734 


0839 10 


781 976080 





SI 




sslit 


509696 


616 


490304 


533619 


688 


466321 


023983 


78 1 976017 


* 


3J 




3! 


S3 


510065 


616 


48993^ 


5340921 68T 


465901 


024086 


78 


975974 


7 


21 




36 


54 


510434 


616 


4?»566 


534504' 687 


46S4»e 


084070 


72 


975B30 


6 


li 




40 


53 


510903 


615 


499191 


534916 


686 


♦650S4 


0141 13 


78 


975887 


5 


8f 






5S 


511172 


614 


488888 


5353^8 


686 


♦64672 


014156 


72 


91584* 


4 


H 




48 


57 


511540 


613 


488460 


535739 


685 


♦648«t 


084200 


78 


976800 


3 


I 




ii 


58 


511B07 


613 


486093 


536150 


685 


4838M 


084243 


72 


975757 


S 


1 




SB 


59 


512875 


61! 


487725 


536561 


684 


♦63*39 


084296 


78 


975714 




( 


u 


in 


8(y 


512642 


612 


481358 


536972 


684 


46308S 


021330 


72 


975670 


(1 


44 < 


iiT^ 


=T 


"ciiiMr 


l~s*™~ 


"UittS^ 




I'^B- 


=C^= 


~ 






iZ^ 




4 Houn, 




7 


V) 


reti. 




;-."/ 


f 30 


94 
199 


1- 1 15' 


\ i°5 [ i- [ 


^n 


P 




M 


3 / 4.5 


8-13 


a _» 


\ SIS \ » \ >!. \ »l 




L 






] 







md Secanu. 


TiluV. 


-^ 






LHouT. 


Dt 




19 De 


gre-. 




^1^ 


Sine. 1 D. 


CD«t 


J^ 


B. 


Coruig. 


Stauii. 


A 




1^^ 




i5~i~o 


9rsl8IJ48l6U 


10.48735( 






10403086 


10.08*330 




997567* 


SC 


4* ( 






1 


S13009|eJl 


486991 


537388 


683 


468018 


081373 


73 


975687 




6t 






8 


si3^75 an 


486685 


6377D4 


683 


468808 


081*17 


73 


B75S8; 




51 




1! 


3 


513741 6!0 


486859 


538808 


688 


*«179f 


084161 


73 


975539 


57 


46 




16 


4 


SUI07 609 
£14478 609 


4BS893 


438611 


698 


461381] 


084504 


73 


976*g<S 




41 




80 




485 68f 


439080 


691 


4609BO 


08*5*8 


73 




55 


40 




H 


6 


JI4837| floe 


485161 


43B489 


681 


460571 


08*598 


73 


976408 


$4 


30 




88 


T 


115808 608 


48479f 


439837 


690 


460163 


08*635 


73 


975305 
975381 


^ 


31 




3? 


8 


ilSSfie! 607 


484431 


440845 


660 


459755 


084679 


73 


S? 


88 




36 


9 


SliQSO 607 


484070 


540653 


679 


459317 


OZ4T83 


73 


B758T7 


SI 


84 




\0 


10 


iI6BBt 608 


483706 


541061 


679 


458939 


0847S7 


73 


97583; 


» 


SO 






11 


S16857 eOS 


4S331f 


641468 


678 


463538 


084811 


73 


97S18S 


19 


u 






IB 


517080 60S 


48S98t 


54IH76 


078 


458185 


084865 


73 


9751*^ 


18 


18 








S173B8| 604 


188616 


6*8881 


677 


45771 B 


08*899 


73 


97510 


17 


9 




SG 


14 


51774s' 604 


488855 


4*8688 


877 


457318 


08*943| 73 


975057 


16 






iT~ii 


IS 


)filB107|603 


10.481 89;i 


^.543094 


67U 


ia45690U 


ia081997 


73 


9.975013 


a 


43 




^ 


16 


S1S4B6 603 


481538 


443498 


670 


456601 


085031 




B7496l» 


u 


56 






17 


S18889 608 


481171 


5*3B05 


076 


450095 


085075 




97198J 


13 


58 




11 


18 


519190 601 


480810 


544310 


075 


455690 


085180 




97488(1 


18 


4)j 






IS 


519S5I 


601 


488445 


5M716 


674 


*i6885 


085164 




97483t 


11 


44 




SO 


80 


519911 


600 


48008E 


645119 


674 


464981 


085808 




97479S 


10 


4(1 




81 


81 


5B0871 


600 




545584 


073 








B7474i 


19 


30 




SH'El 


580631 


689 


47936E 


4*4988 


073 


451078 






9I4703 


» 


38 




38;S3 


520990 






546331 


678 


453669 


085341 1 74 


9716 S 9 


)7 


BN 




aun 


581349 


598 


478651 


6*673.^ 


078 




08538b 


74 


97*011 


iO 


SI 




40«i 


5!n07 


599 


478803 


547138 


671 


458868 


085130 


74 


974570 


iS 


20 




4*26 


588066 


597 


477934 


a*764U 


671 


458460 


085*75 


71 


H74685 


3* 


le 




i^!*' 


588484 


596 


477578 


4*7943 


670 


458057 


085519 


74 


97148 


33 


11 






588781 


596 


477819 


548346 


670 


451655 


08566* 


74 


974436 


a 


d 




iti\ii 


583138 


695 


47686! 


5487*7 


069 


451853 


085609 


74 
75 


974391 
9.974317 


n 

id 


4 
18 ( 




lB^i3( 


B583495 


596 


10.476405 


9.549149 


loB 


10.450851 


10.085653 




4J31 


58384! 


591 


476US 


54B560 


698 


460150 


Oi5698 


75 


574308 


%» 


ie 




8 




484808 


694 


47579 S 


549951 


668 


450346 


085743; 76 


971857 


!8 


48 




IB 




584564 


593 


475436 


3S0368 


667 




086788; 76 


974818 


87 


48 




lb 




584980 


693 


475090 


550758 


667 


4*984^ 


085833 


75 


B74167 


16 


44 










SS8 


471785 


561158 


666 


*488*8 


085878 


75 


97*128 


15 


4( 




8i 




585630 


591 


47437fl 


551658 


666 


4481*( 


025983 


75 


97*077 


a 


36 




8S 




535981 


591 


474016 


661958 


666 


448048 


025968 


75 


974038 


a 








39 


S863S9 


590 


473661 


658361 


666 


447649' 


680013 


75 


9739S7' 


(8 






30 




58(i693 




473307 


558750 


664 


447850 


086058 


76 


973948 


M 


U 




40 


40 


587046 


589 


478954 


653149 


064 


4*0851 


086103 


76 


BT3897 


80 


id 






U 


587100 


589 


♦7860B 


65354B 


66* 


4*6458 


086149 


75 


973968 


19 


u 






42 


587753 


SSS 


478847 


553916 


663 


446054 


086193 


75 


B73807 


IS 


li 




i8;t3 


589105 


588 


471896 


66*344 


663 


4*5666 


030839 


75 


973701 


17 


R 




S6|i4 


588458 


687 


471548 


551741 


608 


445859 


086884 


76 


973710 






19 


ts 


I.S8H810 


587 


10.M119( 


9.555139 


668 


10.4448S1 


IO.08H389 


"76- 


9.973671 




4 




589161 


586 


470836 


S55536 


661 




086975 


76 


973685 


1* 


Sfi 




8 


47 


589513 


686 


470487 


655933 


661 


4*4067 






973580 


13 


58 




18 




52^884 


5flS 


470131 


556389 


600 


443671 


086465 




973535 


18 


« 




18 




530815 


585 


469783 


556786 


660 


443876 


086511 




B73489 


11 






snlMi 


530S65 


584 


46B-135 


557181 


069 


448879 


086556, 76 


973444 


10 


40 




8t'5l 


530915 


584 


469085 




S59 


4l!t8i 


086608, 76 


973398 


9 


30 




88*8 


531865 


583 


46873.' 


557913 


659 


418087 


080618 78 


B73358 


8 


38 




3853 


531614 


588 




iaSSes; 658 


4*1098 


O80O93; 76 


973307 


7 


88 




36 Bt 


S319S3 


588 


468037 


553708, 053 


441898 


02673B 76 


973861 


8 


2* 




405S 


638318 




467088 


559097 657 


44aB')3 


08678£ 70 
026831 76 


973816 


5 


SO 




44 £6 


5386S1 


581 


467339 


559491 667 


44A509 


973169 


1 


16 




4857 


433009 


580 


466991 


6598BS, 66S 


*40115 


088876 76 


B73I84 


3 


18 




i8i( 


S33357 


580 


46S643 


5S0879I 656 


439781 


0869881 76 


973078 


8 


8 




iBSff 


533704 


S79 


46689G 


560673 665 


43938T 


080968 77 


973032 




* 




80 0\6I 


634048 


678 


46S91« 


561066! 056 


43893* 


027014 77 


978986 





40 




ai. i-l ' 


CcKlne. 




SenuL 1 Covaig. 1 


Tang. 


CgB«. 1 1 Sine. 




"^^~^ 






4 Hauf. 












P. P. lo 


!■ IS-- 
1 30 


1J8 


3 


30 


1 100 
800 


IVIS'V^.\; 





\ 


■ or 


3 44 


g68 


3 


46 


1 aoi 


\ i \ \5 \ -M. N 


,nlA 









a Table V. 


Logarithmic Siues, Tangen 


B, 


-J 




1 Hour, 


or 




26 Dec 


r^ 




n. t. ' 


Sine. 


dT 


Co.«. 


Ttag- 


a. 


Coi-dg. 


Seoint, [ 0. 


CMine. 




n. t 




'*~0~Q 


sma 


M 


K).3SH15H 


.688182 


m 


10.311811 


0.016340,103 


.95366( 


60 


= 






4 1 


642101 


431 


357B99 


688502 


534 


31149) 


016401103 


953599 


5S 


« 






S ! 


6 423d 


431 


357640 


688823 


534 


311177 


016463,103 


953537 


59 


51 






la 3 


648618 


430 


35738! 


689143 


533 


310857 


016525103 


9531T5 


47 


4t 






IS 1 


64*877 


430 


357123 


689463 




310537 


046587,103 


953113 


56 


41 






io s 


fi43l32 


430 


356865 


689783 


533 


310217 


046S48103 


953352 


55 


« 






S* 6 


643393 


430 


356607 


690103 


533 


309893 


O46710IO3 


g532B<i 


^* 


Sfl 






Sfi 7 


643610 


429 


3S6351 


690423 


533 


309577 


016772,103 


953228 


S3 


3! 






Si e 


613908 


4!9 


356092 


690742 


532 


309258 


046834103 


953166 


52 


tt 






36 9 


64411J5 




355835 


691062 


532 


308938 


046896 


03 


B53101 


51 


2 






40 10 


644423 


428 


355577 


691381 


532 


308619 


046958 


03 


953012 


W 


t( 






44. U 


«i4flW0 


428 


35532U 


691700 


531 






04 


9529811 


19 


1' 






481! 


64493G 


428 


35S061 


692019 


531 


307981 


047082 


01 


652918 


18 


1: 






52 13 


64S1B3 


427 


354807 


692338 


531 


30766! 


IHTI45 


04 


952965 


17 


t 






M14 
4S~0T1 


6*i450 


427 


354S5I 


692656 


531 


307341 


047207 


04 
IW 


952793 




' 




0.645706 


427" 


10.354294 


B.6B29T5 




10307025 




.952731 


45 






41G 


6459Gi 


426 


354038 


693293 


530 


306707 


047331 


lOl 


952669 


U 


Si 






SI7 


646218 


426 


35378! 


693612 


530 


S0638B 


047394 


104 


952606 


13 


61 






1816 


646474 


426 


353526 


693930 


530 


306070 


0474S6 


104 


9525H 


W 








16 18 


646T29 


4!5 


3S3871 


69424H 


530 


305752 


017519 


104 


952481 


(1 








so' go 


646B81 


425 


353016 


694566 


529 


305434 


017581 


104 


9524191 


to 








ii.n 


647240 


425 


36276( 


694883 


529 


805117 


(H7644 


104 


952356 


19 


3C 






ti.ii 


647494 


424 


352306 


695201 


529 


3047 9B 


01T70S 


104 


952294 


18 


31 






3ig; 


647749 


424 


352251 


695518 


529 


3044BJ 


017769 


104 


952331 


37 


» 






38 24 


648004 


4!i 


35J99ti 


695836 


529 


304164 


017832 




952168 


16 


U 






40 li 


649*58 


424 


35174! 


696153 


528 


303847 


047891 


105 


952106 


M 


» 






4V2e 


648512 


423 


351488 


696470 


529 


303531 


047957 


105 


9520(3 


(1 








48 87 


648766 


423 


351234 


696787 


528 


303213 


018020 


105 


951981 


U 








fiJ.SH 


6490*0 


423 


350980 


697103 


528 


302897 


018083 


105 


951017 


w 








as;*!. 


649874 


4!! 


350726 


69T420 


527 


302580 


(118146 


105 


951854 


S) 




1 


16 30 


1.649527 


4!2 


10.35017; 


9.69773(1 


527 


10.302264 


IO.0482O9 


105 


B.S51791 


UJ 


14 I 




Vsi 


649781 


422 


350218 


698053 


52T 


301947 


01827! 


105 


951798 


29 


56 






e'38 


650U34 


422 


34»96( 


698369 


527 


301631 




105 


951865 


38 








12 33 


6502S7 


421 


349713 


698685 


52b 


301315 


048398 


105 


95160 


BJ 


4« 






l(i;34 


650539 


421 


349461 


699001 


526 


300999 


048161 




951539 


!6 


44 






!0 3fi 


650192 


421 


349308 


699316 


526 


300884 


018521 


105 


95lt7f 


35 


18 






£43(1 


651041 


4!0 


34895G 


699632 


526 


300368 


018588 


105 


95141! 


21 


3 








6iU!t? 


420 


348703 


699947 


526 


300053 


018651 


106 


951349 


M 


3 






3238 


651519 


1!0 


348451 


T00263 




i9fl73J 


048714 


108 


9SUBS 


1! 


M 






se'as 


6ilB00 


419 


348200 


700578 


525 


29942* 


018778 


ion 


B5122 


21 


« 






40,40 


6i»052 


419 


347919 


700893 


525 


299107 


018841 


106 


95115 


to 


21 






444 
48*2 


6St3U4 
652555 


419 


347696 


701 208 
701523 


524 

524 


29878i 
298477 


048901 
018968 


106 
106 


95109 
95103 


19 


1 
1 
















5«41 


652M>6 


418 


347 194 


701837 


524 


2a8163 


019032 


106 


95096S 












65:iU5I 


418 




70215! 


534 


29T848 


0(9095 


106 


95090 


lb 






« Oli 


^.65J308 




10.34669* 


B.7U24«6 


524 


10,297534 


10019159 


106 


B.950B1 


15 


l3- 




4'4G 


653558 


417 


346442 


10278U 


523 


297220 


019222 


106 


95077 


14 








847 


653808 


417 


346194 


703095 


523 


29690 


049286 


108 


95071 


13 








12 4B 


651059 


417 


345941 


703409 


523 


29659 


049351] 


106 


95065 




U 






16|4a 


654309 


*16 


34569 


703723 


623 


29627 


049114 


108 


95058 


li 


*t 






20 «1 




416 


3454*2 


70403h 


522 


29596 


019178 


lOT 


95052 


10 


m 






24 5 


854808 


416 


345192 


704350 


52! 


29565 


04051? 


lOT 


95015 


6 


at 






ess 


65505b 




3U942 


T04663 


522 


29533 


01960(3 


107 


95039 


8 


SI 






32 S 




415 


344693 


7049T7 


522 


29502 


049670 


107 


P5033C 


1 


n 






3Si 


655 55 h 


415 


344444 


705290 


522 


29171 


01973 


107 


95026 




» 






40,6 


055805 


415 


344195 


T05603 


52 


29439 


049798 


107 


95020 


J 


St 






44'i 


656054 


414 


343916 


705916 


52 


29408 


049^62 


107 


95013 


1 








4H'j 


65630! 


414 


343698 


706228 


52 


29377 


049921 


■07 


95007 


3 


1 






Sis 


65655J 


414 


343449 


70054 


52 


29315 


019991 


107 


95001 


2 








5e,i 


656769 


413 


34320 


70885 


52 


29314 


osmsa 


107 


91994. 


1 








W OGC 


65704? 


413 


342953 


T07l(i6 


520 


292831 


050119 


107 


91988 





It 




rr--^ 


■c^SiiT- 




^i^Hit- 


COIWR. 


i~T'^ir~ 


Coirt. 




~sTn»r" 


~ 


ST^ 


4 Hour,, 


or 




6 


D^ 


grces. 




«^"/ ^' / L' 


J «i 
I2T 


! 30 


\z 


I * 


5" 
30 


^^ I" 


p. 10 


[ / "" 1 a 1 45 


i ,90 


3 i^ 


\^ 


^^^^^^*^^ 


Sh 


J 




K 


^ 




■ 


1 



■ndSeeiDta. 



Tablb V. 



IHon, 



SI 



^ne. 



D. i CdMC 



.5543S9 

554658 

5543187 

555315 

555643 

555971 

556299 

556626 



548 

548 
547 
547 
546 
546 
545 
545 



40 



556953, 544 
557280 544 
557606 543 
557932; 543 
558258 543 



558583 
558909 



542 
542 



L559234 
559558 
559883 
660207 
560531 
560855 
561178 
561501 
661824 
562146 
562468 
562790 
563112 
563433 
563755 



1233 



t6 03019.564075 
564396 
564716 
565036 
565356 
565676 
665995 
566314 
566632 
566951 
567269 
567587 
567904 
568222 
568539 



4040 
4441 
4842 
5243 
5644 



541 
541 
540 
540 
539 
539 
538 
538 
637 
637 
636 
636 
636 
635 
635 



ia445671 
445342 

445013 
444685 
444357 
444029 
443701 
443374 
443047 
442720 
442394 
442068 
441742 
441417 
441091 



Tng. 



9.684177 
584566 
584932 
685309 
585686 
586062 



D. 



629 
629 
628 
628 
627 
627 



Cotug. I Secut D. Coanc I ' Inii 4. 



586439, 627 
686815, 626 
587190 626 
687566 625 



687941 
688316 
588691 
589066 
589440 



626 
625 
624 
624 
623 



ia4l582a 
415445 

416068 
414691 
414314 
413938 
413561 
413185 
412810 
412434 
412059 
411684 
411309 
410934 
410560 



ia0t9848 81 



10.440766 9.6898141 623 ia4l0186 10.030580 82 



117 0.451^.568856 
669172 

669488 
569804 
570120 
670435 
670751 
671066 
571380 
671695 
672009 
672323 
572636 
572950 
573263 



3253 
3664 
4055 
44{56 
4867 



62 

56 





68 
59 
60 



634 
634 
533 
533 
632 
632 
631 
631 
631 
530 
530 
529 
529 
628 
528 



440442 
440117 
439793 
439469 
439145 
438822 
438409 
438176 
437854 
437532 



436567 
4362451 



573575 521 



528 
627 
627 
526 
626 
525 
525 
524 
524 
523 
523 
523 
522 
522 
521 



10.435925 
435604 
43528 



590188 
690562 
690935 
591308 
691681 
692054 
692426 
592798 
693171 
693542 



437210 693914 
436888 694285 



694656 
595027 



A 



434964 

434644 

434324 

434005 

433686 

433368 

433049 

432731 

432413 

432096 

43177 

431461 



^.595398 
595768 
596138 
596508 
596878 



623 
623 

en 

622 
622 
621 
621 
620 
620 
619 
619 
618 
618 
618 
617 



597247 615 



617 
617 
616 
616 
616 



029897 
029945 
029994 
030043 81 
030091 81 
030140 81 
030189 81 
030238 81 
030286 81 
030335 81 
030384 82 
030433 82 
030482 88 
030531' 82 




409812 
409438 
409065 
408692 
408319 
407946 
407574 
407202 
406829 
406458 
406086 
405715 
405344 
404973 



030630 82 
030679.' 82 
030728 82 
030777 82 
030827 82 
030876 82 
030925 82 
030975 82 
031024 82 



031074 
031123 
031173 
031223 
031272 



83 
83 
83 
83 
83 



10.404602 10.031322, 83 



969957 
969909 55 
969860 54 
969811 53 
969762 52 
969714 51 
969665 50 
969616 49 
969567 48 
969516 47 
969469 46 



9.96942( 4535 

969370 44 

969321 

969272 

969223 41 

969173 40 

969124 39 

969075 !I8 

969025 37 

968976 36 

968926 35 

968877 34 

968827 33 

968777 32 

968728 31 



• 



697616 
597985 
698354 
598722 
699091 
699459 
599827 
600194 
600562 



616 
615 
614 
614 
613 
613 
613 
612 
612 



ia431144 
430828 
430512 
430196 
429880 
429565 
429249 
428934 
428620 
428305 
427991 
427677 
427364 
427050 
426737 
426425 



9.600929,611 



601296! 611 
601662; 611 
602029; 610 



I 



40423 

40386 

403492 

403122 

402753 

402384 

402015 

401646 

401278 

400909 

400641 

400172 

399806 

S9943{ 



602395 
602761 
603127 
603493 
603868 
604223 
604588 
604953 
605317 
605682 
606046 
606410 



610 
610 
609 
609 
609 
608 
608 
607 
607 
607 
606 
606 



031372 

031422 

031472 

031521 

031571 

031621 

031671 

031722 

031772 

031822 

031872 

031922 

031973, 84 

032023' 84 



83 
83 
83 
83 
83 
83 
83 
83 
84 
84 
84 
84 



10.399071 
398704 
398338 
397971 
397605 
397239 
396873 
396507 
396142 
395777 
395412 
395047 
394683 
894S18 
393964 
393590 



ia032073,'84 
032124 84 
032174*84 
032226' 84 
032275 84 
032326 84 



032376 
032427 
032478 



032629 85 



032579 
032630 
032681 
032732 
032783 
032834 



84 
84 
85 



9.96867 
96862 
96857 
968528127 
968479226 
96842985 
968379 24 
968329 23 
968278 22 
968228 21 
968178 20 
968128 19 
968078 18 
968027 17 
967977 16 



9.967927 



15 



967876 14 56 
967826 13 
967775 12 
967725 



85 
85 
85 
85 
86 
86 



XI 

967674 10 
967624 
967573 
967622 7 
967471 
967421 
967370 
967319 
967268 
967217 
967166 



33 



6 
6 
4 

3 
2 
1 



52 
48 
44 
40 
36 
32 
88 
24 
20 
16 
12 
8 
4 



I 



32 



^^ 






38 Table V. 


Logwithmic Sines, Tangents, | 


IHout, 


or 


80 D< 


■greefc | 


D. t 




Sine. 


D. 


COKC 


T«.K. 


D. 


_Colang^ 


Secanl. 


D. 


C«iae. 




a. t 




iO 


( 


9.534058 


m 


10.465945 


».56106e 


655 


10.438931 


10.087014 


71 


9.978996 


ii 


w=. 




4 


1 

2 


534399 
53*745 


577 


465601 


5614S9 
50 195 


654 
664 


43861 
43814H 


027060 
087106 


77 
77 


972SiL 

972B94 


59 
i8 


5! 


577 


465256 




12 


3 


5350D2 


577 


464908 


562244 


663 


437756 


087152 


77 


97284t 


57 


4t 




16 


4, 


fi3543fi 


576 


46*568 


562636 


flSS 


437364 


087198 




97880S 


56 


41 




ao 


a 


635783 


576 


46*217 


663028 


653 


436978 


08781-5 




978765 




*C 




84 


1 


53GI29 


575 


463S7L 


56341 U 


658 


436391 


087291 




972709 


H 


3« 




88 


7 


53R474 


574 


463526 


50381 


652 


436189 


087337 




97g66S 


53 


31 




38 


8 


536818 


574 


463182 


564202 


651 


435798 


087393 


T7 


07Sen 


58 


tl 




36 


S 


537163 


573 


468837 


56*592 


661 


435408 


027430 


77 


972570 


SI 


» 




40 


10 


537507 


573 


468493 


564983 


650 


135017 


087476 




97838* 


W 


H 




4* 


11 


537S5I 


578 


462149 


665373 


650 


434627 


087588 




978*7t 


U 


II 




4a 


18 


538194 


678 


481806 


665763 


649 


43*237 


087569 


79 


97243 


IB 


1 




58 


1! 


538538 


671 


461462 


666163 


649 


433947 


081615 


78 


97 8395 


17 


_ 




56 


14 


53a8B0 


671 


46I12fl 


666542 


64B 


433459 


087668 


T8 


BT83SH 

9,97 5!29 


16 
15 




15 


U39883 


570 


10.460777 


1.666932 


648 


10.433068 


10.027709 






16 


639566 


570 


460435 


567320 


649 


432G80 


027755 


IS 


97224J 


V* 






8 


17 


63!) 907 


569 


46D0S3 


567709 


647 


43^891 


087808 


79 


978I9R 


13 






12 


IB 


610249 


569 


459751 


569098 


617 


431908 


027819 


78 


97815 


18 


H 






IS 


510590 


568 


469410 


569496 


646 


431514 


027895 


78 


972105 


11 


1 




80 


20 


540931 


568 


459069 


568873 


S46 


431187 


087912 


78 


97806S 


U> 


tl 




94 


81 


541272 


567 


45878S 


569261 


645 


430739 


027999 


IB 


B72UU 


10 


s 




2H 


82 


541G1S 


567 


459387 


560649 


645 


430358 


02903S1 79 


97196 


M 


3 




33 


83 


641953 


566 


458047 


670035 




489966 


028093; 78 


97191 


17 


81 




36 


84 


643283 


666 


457707 


570488 


61* 


489578 


088130 78 


971870 


16 


8 




10 


86 


548638 


565 


467368 


570909 


644 


429191 


02H177 79 


97182; 


U 


2f 




4+ 


26 


6129TI 


565 


457029 


571195 


613 


4!B806 


029-J81, 79 


97177(( 


U 


1 






87 


643310 


564 


466690 


571691 


643 


189419 


089271 79 


971729 


la 






52 


88 


613649 


561 


466351 


571967 


648 


48S033 


088313 79 


971688 


19 






5S 


2S 


643987 


563 


466013 


578352 


618 


487618 


088365 


79 


S716;15 


11 




ir~o 


30 


1.611325 


563 


la465S75 


3378738 


648 


lft487868 


10.028*12 




9.971581 


10 


38- 




*l 


31 


64M63 


568 


455337 


573183 


641 


486877 


088*60 


79 


97I5K 


in 






8 


32 


6*51X10 


562 


455000 


573507 


641 


486493 


088507 


79 


97149: 


IS 






12 




616338 


561 


464661 


573892 


640 


48610i 


028551 


79 


971441 


87 






16 


34 


616674 


561 


454386 


574876 


610 


485784 


089M08 


79 


971391 


•« 


* 




80 


35 


S46011 


560 


453988 


574660 


639 


485340 


0886*9 


79 


971;i5I 








84 


36 


64S347 


660 


453653 


575044 


639 


484956 


088697 


79 


n713n;l 


it 






8B 


17 


546683 


658 


*53:!n 


676427 


639 


484573 


0897*1 


79 


97 12.56 


ii 


3 1 




32 


3t 


647019 


559 


158991 


575310 




484190 


028792 


79 


971809 


n 








39 


5*7354 


558 


45864( 


576193 


638 


483607 


U29839 


79 


971 161 


>i 


1 




40 


4U 


517689 


658 


152311 


5J6576 


637 


423484 


029987 


79 


971U:i 


to 


2 




44 


41 


548081 




461976 


576959 


637 


423011 


08893* 80 


971061 


19 


1 




4B 


42 


64S369 


537 


451641 


57J341 


636 


4B865fl 


089998 80 


97101' 


IE 


1 




58 


13 


518693 


Siti 


451307 


577783 


836 


428277 


089030' 80 


97u9;t 


17 






56 


44 


S19087 


556 


450973 


579101 




421896 


029078^ 80 


aioDii 


18 






83 


« 


a549360 


555 


10,150U4C 


1.578486 


635 


10.12151* 


10.029 126; 80 


9.970874 


u 


sT! 






46 


649$g3 


555 


450307 


578967 


635 


48113: 


0891731 80 


97088? 


II 


5 






47 


560086 




449974 


679248 


63* 


48075: 


089821 80 


970779 


13 






18 


18 


550369 


551 


449611 


6T9629 


631 


420371 


089269' BO 


970731 


i8 


4 




16 




56US98 


563 


44930E 


680009! 634 


4199(1 


029317 90 


97068 


11 


4 




SO 




551084 


563 


448976 


580389 633 


419611 


0893^-5 90 


9706 ^5 


10 


41 




84 




651356 


552 


4*8644 


680769, 633 


419831 


089* [* 80 


9705W 


9 


» 




88 


5* 


551687 


552 


448313 


691149; 638 


119961 


029*62 


80 


S7033P 


8 


3 




32 


s; 


658019 


558 


447938 


581588 


63t 


419478 


089510 


80 


970491 




8 




36 


64 


652349 


651 


447631 


591907 


638 


41809: 


089658 


80 


970 Ui 


6 


8 




*0 


55 


jsseso 


551 


447381 


588886 


631 


417714 


02960ri 


80 


970391 


5 


1 




44 


46 


663010 


550 


446990 


59J665 


631 


417335 


08965; 


81 


9T03U 


* 


. 1 




4S 


i7 


463341 


560 


446659 


583013 


630 


416957 


08970;l 


81 


970297 


S 






S8 


58 


M3870 


6*9 


44633(1 


583488 


630 


4I667I 


029751 


81 


97081t 


i 








59 


564000 




♦*600( 


683800 


689 


416800 


029900 


81 


9708W 


1 






14 


M 


554329 


518 


445671 


68*177 


649 


41598; 


0898*9 


81 


«015i 




K ; 


1 


fc~i 




"c^iT 


~~~' 


SeCSQt. 1 CDUiOg. 1 


T««. 


Cotec. 


— ' 


anc 


T 


sn 




m 




U.g™. 1 


P. P. lo 






J. I 15" 


81 1 1' 1 15" 


i^^^U'un '.i\r.^A 


l.^-l ' 30 


169 1 a \ -M 


L '■ 


3 1 i.. 


853 1 3 \ *5 


\ 'm\ ^\ ^.^ 


vjfcjj^rr. I 1 


i 


. 










lir^ 


I 


^1 



and SeeuU. Tiblb V. 


:» 




1 Hma. a !l D<«n» 




H. 1 ' 


Sine. D. 


Cote. 


T«ig. 


dT 


CoUDf. 


SeciDl. 


D. 


Codnt 


i 


m. ■ 




r^i 


).Sj43«9' S18 


10.4*5671 


J.594177 


6M 


10.* 1582! 


ia0t98*8 


■JT 


9.97015i 


io 


in 




4 1 


ssiem 548 


445342 


584555 


629 


*lfi445 


089697 


91 


97010! 


S9 


A( 




8 ! 


iSMST 5*T 


445013 


594932 


628 


415068 


0299*5 


61 


970O5i 


S8 


51 




It fl 


S.5i3l5 547 


444695 


585309 


629 


414691 


08999* 


61 


970006 


17 


«e 




le i 


55S643 548 


444357 


585696 


627 


4143141 


030043191 


969957 




44 




to s 


SiS97l 546 


444029 


596062 


627 


413938 


030091 ' 81 


960909 




40 




i* 6 


35I)K>9 54S 


443701 


588439! 627 


*13561 


030140; 81 
03019d 81 


969860 




ae 




ia I 


sseete 545 


443374 


596815 626 


413185 


S6991I 




3) 




31 9 


596953, 544 


443047 


687190' 626 


412810 


030238 81 


96976! 


Sf 


29 




36 g 


557280 544 


442720 


5975661 625 


412434 


030886 81 


96B714 


51 


21 




40 I( 


557606 


543 


442394 


587941 625 


412059 


030335, 81 


969665 


•^ 


80 




4tlt 


557932 


543 


442069 


5B93I6 


625 


411884 


03038* 82 


969616 


t» 


16 




4811 


55Bi5S 


543 


4417*2 


£88691 


62* 


411309 


030*33 88 


969567' 


U 


18 




BIV 


556583 


542 


44141T 


589066 


624 


410934 


030492 62 


969518 


ir 


8 




A61J 


5ffi909 


542 


4410D1 


589440 


623 


410566 


030531J 88 


96946B 


IS 


4 




IS Oil 


D.5S9234 


541 


10.440781 


9.58981* 


623 


10.41018^ 


10.030480; 82 


B.U69420 


Ea 


sj-o 




41< 


559558 


541 


440442 


590198 


623 


409818 


030630 


82 


969370 


u 


M 




eiT 


559893 


£40 


440117 


59056! 


6ig 


409439 


030679 


82 


969321 


IS 


52 




U18 


560207 


6*0 


439793 


590935 


628 


409065 


030788 


88 


969878 


42 


48 




16 IS 


SG063I 


539 


439469 


591309 


622 


40669! 


030777 


8! 


96022; 


41 


44 




SO go 


fiS0S55 


539 


430145 


591681 


621 


40931!) 


030927 


98 


969173 


10 


40 




mi 


561178 


538 


4388(2 


592054 




407946 


030976 


92 


969124 


J9 


36 




if «! 


561501 


53B 


438*99 


592426 


620 


407574 


030925 
030975 


82 


969075 


W 


3S 




3iS 


5616!* 


537 


439176 


59279H 


620 


407202 


82 


969025 


17 


28 




3e!4 


5621*6 


637 


437954 


593171 


619 




031021 




96H9T6 


16 


24 




40 W 


562466 


536 


437532 


593542 


619 


40615B 


031074 


93 


96B986 


U 


SO 




4-t.J6 


5B2790 


536 


437210 


59391* 


618 


406096 


031183 


83 


968877 


M 


16 






563112 


536 


4368B9 


59*295 


618 


405715 


031173 


93 


B68887 


13 


18 




5g|29 


563433 




436567 


594656 


618 


405314 


031283 


83 


968777 


it 


( 




56 29 


563755 


535 


436245 


595027 


617 


404975 


031278 


83 

83 


966788 


)1 


i 




te 03( 


9,564075 


534 


10.435925 


S.595398 




IO.4046O1 


10.031322 


9,968678 


10 


3* 






58*396 


534 


435601 


595769 


617 


401838 


031372 


83 


S69689 


29 


56 






564716 


533 


435284 


596138 


616 


403862 


031122 


83 


968579 




5! 






565036 


f33 


431964 




616 


403492 


031472 


83 


968588 


i7 








565356 


532 


434614 






403122 


031521 


83 


968479 28 








565676 


532 


43*324 


597247 615 


402753 


031571 


83 


96848925 








565995 


531 


43*00i 


597616,615 


408394 


031621 


83 


96835 9 


i* 


3( 






566314 


531 


433686 


597985 615 


402015 


031671 


93 


969389 


t3 








566632 


531 


433368 


699351 


614 


4016*6 


03178! 


83 


96821 S 








36J38 


566951 


530 


43301B 


598722 


61* 


*OI278 


031772 




969288 




2* 




40 40 


567269 


530 


432731 


599091 


613 


*O09O9 


031822 


84 


968178 


!0 


20 




4441 


567597 


529 


43211: 


599459 


613 


*0OS41 


031872' g4 


966129 


19 


16 




ieUi 


567904 


529 


432091 


599927 


613 


40017; 


031922; 84 


968078 


19 


12 






56822S 


528 


431 77t 


600194] 612 


399806 


031973 8i 


968027 


17 


« 




56\u 


569539 
li.56i«B56 


528 
528 


431461 
10.4311*4 


600562' 612 
9.600929 61 1 


399439 


032023 84 
10.032073 84 


967977 
9.967987 


16 
15 


* 

33 ( 




10.399071 




*\ie 


56917? 


527 


430828 


601296 611 


S9B704 


032124 84 


""^If 


1* 


56 




84T 


559489 


52T 


4305li 


601662 611 


398338 


03217*84 


967826 


13 


52 




12 46 


569304 


526 


430l9f 


602029 610 


397971 


032825 84 


067776 


12 


4t 




16 4g 


570 ISO 


526 


12998( 


602395 610 


397605 


032875 8* 


B67725 


II 


44 




tO.M 

S'dii 


570 (35 


525 


429565 


602761; 610 


397239 


038386. 8* 


96767* 


10 


40 




S70751 


MS 


42S249 


603127:609 


396873 


032376 84, 


967624 


9 


36 




571066 


524 


429934 


603493 


609 


396507 


038427 


84 


967573 


8 


38 




3tS3 


571390 


524 


429620 


603958 


609 


3961*5 


038479 


65 


967528 


7 


29 




36 54 


571695 


523 


429305 


601223 


608 


395777' 


032529 


85 


967*71 


9 


24 




40'5J 


572009 


523 


42799] 


604588 


609 


395*18 


038579 


85 


967481 


S 


20 




44'i( 


578323 


523 


421677 


601953 


607 


395017 


032631 


85 


967370 


* 


It 




4B'£7 


572636 


522 


427364 


605317 




39*683 


038681 


85 


967319 


3 


1! 




stlss 


572951 


522 


427050 


605682 




39*31& 


032732 


65 


96786B 


8 


8 




asm 


573263 




426737 


60601.6 


606 


393954 


032783 


85 


967817 




4 




K 06(1 


573575 


521 


426425 


60S410 


606 


393S9ol 


032634 


85 


967166 


^ 


38 






























srrn 


Coainc. 




9««ni. 






T«W. 


Cosec. 












4 Houn. or 69 D( 


^"- 




P. P. to 


1. 15" 


80 


li 15" 


93 1 1- 1 15" 


[ il [^ 


P.„ 




2 30 


160 


2 30 


185 1 2 I 30 




I or" 


3 45 


240 


3 45 


aift \ a \ -^.s \ ?■■> \ 


'""Ji 



^^ 






^^^™ 






48 Table V. 


Logarilhraic Sines, Tangenls, j 


2Hau«. 


or 




30 De 






5!l_^U 


Sine. 1 D. 


CosHi 1 Tsng. 1 D. 


Clang. 


Jieonl- 


D. 

W. 


9.9375lliSUIiO ( 
93715a>M 5« 





(1 


1.69BU70, 381 


10.301030,9.781439,486 


10.238561 


10.062169 




4 




699189 3S4 


300811 


761 731 496 


2S9269 


068518122 




9 


! 


69S40TI 364 


300593 


762023 496 


2379J7 


068615,188 


93739^ S9 51 




1! 


3 


69H6g6, 364 


300374 


762314' 4S6 


237686 


062688,122 


93731a S7 W 




IB 




699844 3G3 


300156 


768606) 485 


237394 


062762122 


93723a »« 4" 




SO 




700068' 363 


899938 


768897 485 


237103 


062815 128 


937165 Si H 




8* 


e 


7(M)iB0 363 


299780 


763188; 485 


836812 


068903 182 


93709* il 38 




i8 


7 


700498 363 


299502 


763*79, 495 


836521 


062981 




937019 S3 3! 




3? 


(■ 


TO0716 363 


299284 


763770; 483 


236230 


063051 


122 


936916 57 II 




3a 


g 


7009:53 36? 


299067 


764061 485 


235939 


063128 




936978 S 


1' 




401(1 


701151 362 


298840 


76*358 494 


235648 


063201 




936799 --0 


« 




Ult 


7111368 368 




16*643 484 


235357 


063875 




936785 19 


1< 




481* 


70IWS 368 


898il5 


76*833, 484 
76522* 484 

765514' 484 


g350ri7 


063349; 123 


936658 16 


i; 




M)4 


701802 
70S019 


361 


298 1 98 


834776 
231486 


063122123 
063495183 


g:t6579 11 
036605 16 


1 




297981 






10.29776*9.765805, 484 


10.33+ 195| 


10.063569,123 


9 936i:il IS 


mH 




4'1( 


'702*48 381 


897548 


766095; 484 


233905 


063643123 


936357 11 


■* 




sn 


7a2lje9 360 




7663S5 483 


233615 


063716 183 


9362HI IS 


51 




islis 




897115 


766673 483 


233325 


063790 123 


936210 1! 


« 




i6;i£ 


703101! 3fiO 




766963 483 


833035 


063861123 


9361.16 l( 


41 




ao?o 


705317,360 




767255, 483 


832745 


06393H18: 


9S606i IC 


« 




!*S1 


103i33 3S9 


896467 


7675*5 493 


8324S5 


064012,12:1 


9359BH 18 


3i 




8Pg! 


703718 359 


296251 


767834 493 


232166 


064096 183 


935911 W 


3: 




3? S3 


J0;i96t 359 




768181 482 


831876 


061160 183 


935B10 17 


M 




36i4 


704179,359 


295821 


769114 492 


231596 


06123l'l21 


9357(il U 


81 




40 S5 


704395 359 




T6B703 4M2 


831297 


0613UB 181 


935692 13 


e 






70461o| 358 


S9539( 


768992. 488 


231008 


081392 181 


935018 11 


\% 




48 g) 


704)J25 35A 


295171; 


769281 482 




081157 121 


935513 <& 


, 11 




H,*f 


705010 


35S 




769570 488 


230130 


061531181 


935469 M 


1 




ie'ss 


7038.54 


358 


29l'4« 


769860; 491 


8301111 


061605 184 


935.395 11 


« 


T-oso 


1 705469 


357 


10.29i.i31 


9.770148,481 


I0.229»fi( 


10.1«*690|I81 


9.9;l5:f20^ 


« u 




4 31 


T0S683 


357 


294317 


770137 481 


229563 


06175*124 


935216 i9 


M 




832 


705898 


357 


294102 


770726; 481 


229274 


061829 184 


035171 B8 


51 




1^33 


706112 


357 


893888 


771015,481 


228985 


061903 184 


935097 %1 


U 1 




lG'34 


706386 


356 


293674 


771303 481 


829697 


06*978124 


935082 i( 


44 




80 33 


700539 


356 


S93461 


7715921 181 


229108 


065052 124 


931918 tj 


4C 




*43e 


700753 


356 


893247 


771880 480 


288120 


065187 


124 


931973 Ik 


31 




S8'37 




356 


293033 


772169 480 


227832 


065202 


125 


934798 M 


31 




3g'3H 


707180 


355 


292820 


772157 480 


887543 


06527 T 


125 


934723 18 


» 




Se'SH 


707393 


355 


298607 


772745 480 


287255 


0C53S1 


125 


93464B 21 


81 




40U( 


707601 


355 


89239* 


773033 480 


226967 


065426 


185 


934574 iO 


80 




4V41 


707819 


355 


298161 


773321 480 


826679 


065501 


125 


93449t>>lS 


I« 




48'lt 


708038 


354 


291968 


773608 479 


886392 


065576 


125 


93*424 18 


II 




S2'43 


708245 


354 


291755 


773896 479 


226101 


065651 


125 


G3431B 17 


% 




_S6l« 


708458 


354 


8915*^ 


774181147.0 


885816 


065786 


185 


93*mjl( 


4 


3 a\i 


9.70S670 


35* 


1U.89133U 


9.7744711479 


la2855Jff 


1 a 06591)1 




9.B34198 15^7 U 




446 


J0B88B 


353 


891 UH 


7747591 479 


885241 


066877 


185 


B3412; 14 M 




84T 


709034 


353 


290906 


7750161 479 


821951 


065952 


125 


934019 13 « ' 




lg4H 


709306 


353 


8S0B91 


775333 479 


824667 


066027 


185 


83307: 12 4e 




IKiS 


709518 


353 




7756211478 


821379 


066102 


186 


933898 LI U 1 




!0'iO 


709730 


353 


2902TU 


77590H1 478 


824092 


066179 


126 


933828 10 ID 




El!il 


709941 


358 


£900^9 


776195 479 


823805 


066253 


126 


033747 S 31 




3«'S3 


710151 

710364 


352 
352 


8898*7 


7764M8 
776769 


479 


823518 


0663?9 
066401 


126 
126 


933671 9 M , 
9335011 7 » 1 


479 


S2:H31 




36M 


710675 


352 


289*25 


TT7055; 478 


S22943 


066180 


186 


8:ws« 6 W 1 




405* 


7107B6 


351 


2892M 


777312 


178 


882858 


066555 


126 


933415 4 , Ml 1 




44i,M 


710997 


351 


280003 




♦77 


822371 


066631 




B3.13IIS 4 ' l« 




48:ST 


71I80B 
711419 


351 


28HTP2 


777915 
77B201 


17T 

177 


882085 
881709 


066707 


126 
12(1 


B.i:i29: 3 It 
983;; 17 % e 


351 


288581 




SfiSS 


7nei& 




88^371 


778188 


477 


281512 




126 


933141 I • 


1 


i o|bi 


711839 


3.W 


8BISI6I 


778774 




a2l22< 


066131 


186 


933066 OSa 11 


SrT|"^("ciiiii^ 





"aJ^niT" 


Coung. 


= 


^■i5g.~ 


i:,Bsc. 




"Sioi^ ^mTT 


SHmn, 


nr 




59 D.tf««. 1 


"•'•; i' ) ^," 


107 


1- 1 15" 
S 30 


1 il! 


\y£\ s; -.:-.• " 


1 ,/-- 1 s i u 


161 


3 45 


\ »n 


1 a \ IS I s« •■' . . 














■ 




■ 


^H 




B 






.2 


^H 





«nd SccanU. 


■rA.i.v. ' 


TT 




!Hmm. 


*f 




43I3« 


PM. 






' 


SL,. ID. 


C^ 1 Tang. 1 D. 


a«j^ 


Seoul. {D. 


Comf. 


' 


o- * 




I=B 


"( 


S-S337B3) 886 


10.166817 9.9696.^61488 


10.030344 


10.135B73|196 


9.66418T 


50 


6 ii 






1 


833919 885 


J 66081 


969909 488 


030091 


135990196 


604010 


59 


s« 




S 


S 


834054] 885 


165946 


970168 488 


089 BS» 


136106|l9T 




58 


58 




1! 


; 


634189; 8^5 


165311 


970*16: 488 


089594 


1368861197 


663T71 




48 




16 


i 


834385, 885 


165675 


9706691*88 


089331 


136344! 197 


863656 


i6 


44 




io 


s 


634460, S8.S 


165540 


970988 488 


089078 


136168197 


863538 


U 


40 




SI 


6 


834595! !8S 


165405 


971175 *88 


089985 


136561,197 


86*119 


54 


36 








834730, 88S 


165370 


971*89 488 


086571 


136699197 


863301 


53 


31 








834865 885 


165135 


971688 488 


088318 


136817 197 


863193 


58 


Si 




3(1 


S 


B34999 884 


165001 


971935 488 


088065 


136936 197 


96306* 


31 


8* 




40 


10 


635134 884 


164866 


978196; 488 


087818 


137054 I9S 


6689*6 




80 




44 




835869 884 


16*731 


9781*1 488 


08J5SS 


137 173! 198 


868887 


*B 


16 




4e 


li 


835403 884 


164597 


978694 488 


081306 


137891,196 


668709 


18 


11 






13 


835538 884 


164468 


978946 488 


087058 


137*10] 196 


B68590 


17 


a 




H 


14 


H35678] 8?4 


16*388 


973801! 488 


0B679B 


13T589199 


868+71 


t6 


4 




S3~0 


li 


I.8358UTI 884 


10.16419t 


9.8734541 488 


10.0864*6 


10.137647 


199 


9.86835i 




"n 




4- 


16 


635941 884 


164059 


973707 488 


086893 


137766 


198 


868834 


Ui 


56 




B 




936075, 883 


163985 


973960 488 


086040 


137695 


198 


868115 


43 


51 




1? 


18 


836809 883 


163791 


974813 488 


085787 


138004 


198 


661996 


48 


48 




16 


19 


836343 


883 


163657 


974466' 488 


085534 


136183 


198 


881677 


tl 


41 






S6 


8364T7 


883 


163583 


974719,488 


085891 


139818 


199 


861759 


to 


« 






21 


6366U 


883 


163389 


974973 488 


085087 


136368 


199 


861636 


» 


3f 






ii 


B36745 


883 


163855 


975886 488 


08*771 


136181 


199 


86151£ 


48 


3S 




38 




836878 


883 


163188 


975479 488 


084581 


138600 


199 


861401 


17 


26 




36 




837 01 8 


888 


168988 


975738 488 




138780 


199 


861 SBC 


38 


8* 








837146 


888 


168954 


975985 488 


084015 


138839 


199 


861161 


JS 


81 






86 


83T8T9 


888 


188781 


976839 488 


0837fli 


138959 


19$ 


861011 


M 


1< 




48 


87 


837418 


888 


168588 


976491 488 




139079 




86098! 


)S 


li 




fi« 


es 


B3754G 


888 


168*51 


97674 U 488 


083856 


1391.1)8 


199 


66080S 


it 


a 




S6 


!9 


83J579 


888 


168381 


9769971 488 




139319 


800 


860664 


Jl 


4 






30 


9.837818 


888 


la 168 168 


9.977850, 488 


10.088750 


10.139138 


8O0 


9.6605i;i 


iO 


~6~1 




4 


31 


837945 


888 


168055 


977503 488 


088497 


139559 




86011^ 


89 


66 






6380T9 


881 


161988 


977T56 488 


088844 


139679 


800 


86038! 


88 


6S 






838811 


if.\ 


161789 


976009 488 


081991 


139798 


800 


960805 


87 


4E 






838341 


881 


161656 


9T886S 428 


081738 


139916 


800 


86O065 


86 


41 




g?31 


838*77 


881 


161583 


976515 488 


081+65 


110036 


800 


859968 


85 


4( 




838610:881 


161390 


979769! 488 


081838 


1401591800 


959618 


il 


St 




838748,881 


161856 


9790811 488 


080979 


140a79|801 


859781 




38 




32,3. 


83aa75j 881 


161185 


979871! 488 




I10399;801 


859601 




28 




36 3i 


B3900T 


881 


160993 


9795871 488 


080473 


140580:801 


85948(3 




2* 




40 4( 


839140 


880 


160650 


979760 488 


080880 


tl064o!801 


659360 




8( 




4441 


83B878 


880 


160788 


980033, 488 


019967 


140761 


801 


859839 




16 




4S4S 


83910* 


880 


160596 


980866, 488 


019714 


110691 


801 


859119 




1! 




da 


839536 


880 


160464 


980539 488 


019168 


1*1008 


801 


958999 




8 




5644 




880 


160338 


980791! 481 


019809 


141183 


801 


859877 




4 




\S~0^ 


1.639800 880 


la 160801 


9.9610*4 481 


10 016956 


10.141844 


808 


9.858756 








446 


639938 


880 


160O6S 


961897 481 


O187o:l 


141365 


80S 


868635 




56 




847 


8*0064 


819 


1S993< 


961S50 481 


016450 


111486 


808 


858514 




58 




1848 


8*0196 


819 


139604 


961803:481 


018191 


141607 


808 


858393 




48 




164! 


840386 


819 


159678 


968056 


481 


017911 


141189 


808 


H56878 




1* 




!051] 


640159' 819 


1595*1 


988309 


481 


01T691 




808 


956151 




40 




8451 


640591! 819 


15a*0D 


988568 


*SI 


017438 


141971 


808 


656089 




36 




SS5! 


840788 819 


159878 


988914 


481 


017186 


1480981808 


857908 




38 




3a'53 


64085 V 819 


159146 


983067 


481 


016933 


148814 808 


857786 




28 




36|S4 


8409B5 819 


159015 


983380 


481 


016680 


118335,803 


857665 




24 




40,S5 


8*1116,818 


158664 


963573 


*81 


OI618T 


148457 


803 






8( 




4456 


B11847 


818 


158753 


983886 


481 


01617* 


118579 


803 


8574 8f 




16 




4S|37 


8*1378 


818 


158681 


964079 


481 


015981 


148700 


803 


837300 




18 








811509 


819 


156491 


984331 


481 


0156B9 


148888 


803 


857176 




( 








841640 


819 


158360 


981584 


*81 


015416 


1489*1 


803 


857056 




4 




M 


60 


841771 


819 


158888 


98*937 


481 


015163 


143056 


803 


B36934 




4 ( 




STi; 




cori.. r 


Sewnu 


■cHSTnir 




T««. 


Cosh:. 




~Sb^ 


~ 


sn 




" — 3 Ho"iSir 






K 


Di 


tree,. 




p. p. JO 


1- 

8 


30 


33 
67 


8 


15" 
30 


63 

1S7 


1' 1 15" 
8 \ 3ft 




V^ 


\ 


Bor 


3_ 


45 


100 


3 


45 


1»0 


* \ ^S \ W> \ 


*"■' ^ 







V. 


















&0 Table 


LogarKnmic oines, langencs, | 


S Hnut), 


or 








11. s, 




Sine. 


_U^ 


Cuuc 


S-^ 


D. 


Coluig. 


a«imt ID. 


C^^ 


' m. ( 


"9~ 




J.7?**I0 




10.875790 


9. 7957 89 


Te's' 


10 20481 


10.071380,132 


}.i)mK 


»«-( 








184418 


337 


875588 


796070 


468 


S0393C 


071638,138 


98834S 


t9, S 








men 


336 


875386 


79635 


468 


80361 


07173 


132 


B88265 


B S. 




1 




liiSlh 


336 


8751 H4 


796638 


468 


8033SB 


07181 


132 


98816! 


17 * 




1 




785017 


336 


874983 


796913 




80308 


071896 


138 


S2B104 


>6 4 




! 




785819 


336 


874781 


T97194 


46S 


80280* 


07197 


132; 9880J 


(5 40 




( 




7854S0 335 


874580 


797475 


468 


808685 


07805 


138 927S4* 


».] » 




g 




725688 335 


S7437B 


797755 


468 


S0884J 


07813 


1321 B*786 


a! 3 




3 




78SB83' 335 


874177 


798036 


467 


£01964 


078813 


138 


98778 


i8, tf 




3 




786084.; 33S 


273976 


798310 


467 


801684 


07889 


138 


98770 


>I 11 




40 




786885 335 


87377S 


79B596 


467 


80140 


07837 


132 


92762 


50 80 




* 




7864SB; 33+ 


87337' 


798817 


407 


801183 


07845 


138 


93754 


43 U 




*S 




786686 






799157 


467 


800843 


078S3(; 


133 


987*7 


48 1! 








52 




786887 


334 


873173 


799437 


467 


800563 


O78610 


133 


92739 


« ( 








T87087 


334 


8789T3 


79971T 


467 


800883 


072690 


133 


92T3I 


10! 




B (1 




>. 7 87888 


334 


10.878778.9.799997 


466" 


10.8(10003 


10.072769 


133 9.92783 


iiin 




4 




T874SB 


333 


878578 


800877 


466 


199783 


072B4S 


1331 B871S 


w; ii 




B 






333 


878378 


600557 


466 


199443 


078989 


133 


93T07 


43 ii 




IS 




T!7888 




878178 


800836 


466 


199164 


073008 


133 


9S699 


48 48 




16 




789087 




87197: 


801116 


466 


198884 


07308 


133 


98601 


« 41 




20 




78B887 


333 


871773 


801390 


466 


198604 


07316S 


133 


98683 


w! « 




ii 




788487 


338 


8715T3 


801675 


466 


198325 


073*4 


133 


9i675 


19, 36 




SH 




786686 


338 


871374 


eoi9.s^ 


466 


I0BO45 


073389 


133 


92667 


18, Si 




3! 




78B885 


338 


S71175 


80883+ 


465 


197766 


073409 


133 


92659 


17, 8r 








789081 


338 


870976 


808513 


465 


1SJ4S7 


073439 


134 92651 


36 81 




40 




789883 


331 


870777 


S0e79'' 


465 


19780B 


073569 


134 88643 


15 V 




M 


8B 


789488 


331 


870i78 


B03<t7a 


465 


196988 


073649 


134 98635 


14 16 




4S 


n 


789681 


331 


870379 


80335 


463 


196649 


073730 


134' 926870 


ik i; 1 




£i 


SB 


789880 


331 


8701S( 


803630 


465 


190370 


073810 


134' 986190 


jJ . 




£6 


S9 


730018 


330 


869938 


803908 


465 


196098 


073890 


134' 986110 


n' . 1 


10 




9.730817 


330 


10.869783 


9.804187 


465 


10.195813 


10.07397 


134 9.9260i« 


JOM 1 




4 


31 


730415 


330 


869585 


804406 


464 


195534 


07403 


134 985919 


l» it 1 




8 


38 


730613 


330 


S693H7 


80474. 


464 


195855 


074138 


131' 92586= 


es 5! 




1! 


33 


730B1I 


330 


869 1S9 


805033 


404 


191977 


074212 


134 9857Ht4 


91 4S 




lb 


31 


731009 


389 


868991 


805302 


464 


194698 


071893 


134| 925707 


is 41 

83 40 

84 sa 






sa 


731800 


389 


g6e79J 


8055S0 




191480 


074374 


134 985:i8u 




ii 


36 


T3I401 


389 




805859 


464 


19414 


074453 


135 925345 




g« 


37 


73160! 


389 




806137 


464 


193863 


074535 


135 985463 




3S 


SB 


731799 


329 


863801 


806415 


463 


193585 


074016 


135 925384 




30 


Af 


731D96 


388 


868004 


806693 


463 


193307 


07469J 


13S; 985303 


III u 




40 


IC 


738193 


388 


867807 


806971 


463 


193029 


07477B 


135, 983888 


io te 




49 


41 


733390 


388 


861610 


807849 


463 


192751 


074859 


135 98514 


t9 h 






li 


738587 


383 


867413 




403 


198473 


074940 


135 9250«0 


1^ 1) , 






43 


738784 


388 


SB78I6 


807805 


463 


192195 


075021 


133 S24979 


17 1 




mi 


44 

45 


738980 


387 


867080 


B080M3 


463 


19191^ 


073103 


135 984897 


le! 


9.733171 


387 


ia2668^,a.B0836l 


463 


10.191639 


10.075181 


135 9.984nlfi 


nM~o 




4 




7S.-WT3 


387 


866B87 




468 


191368 


075863 


136 9.4735 


li- ss 




8 


*7 


733««9 


387 


866431 


308916 


463 


191084 


075346 


136' 984641 


13 ii 




1! 


4M 


733785 


387 




809193 


462 


I908O7 


076488 


136 984572 


12 U 




16 


4fl 


733961 


386 


86603' 


809471 


468 


190589 


075509 


I3fil 98«91 


IL 4t 




zo;ai 


734137 


38« 


S6SS43 


B09748 


m 


190252 


075591 


136 924409 


lU 4U 




S451 


734353 


386 


865647 


810025 


468 


189973 


075678 


136 98438)^ 


9 3« 






Si 


734449 


386 


865451 


810302 


468 


18969} 


076751 


136 984246 


e 31 




3! 


13 


734744 


385 


885836 


610330 


462 


189480 


075836 


136 934161 


7 »l 




36 


S4 


734939 


385 


865061 


810857 


■lea 


189143 


075917 


130' 9840S.' 


6 11 






55 


735135 


385 


864865 


811134 


461 


1 88861 


075999 


130 984001 


5 80 




4» 


5(j 


T3«3S0 


385 


864670 


811410 


161 


188590 


076081 


136; 98391<l 


4 If 




48 




735485 


385 


864475 


811637 


401 


188313 


076163 


136] 983837 


3 11 






58 


735719 


384 


864881 


8i 1964 


461 


188036 


076841 


137 S837W 


2 1 






S!l 


735914 


384 


86408( 


HI 8841 


461 


18775S 


076387 


137 983673 


1 * 




11 ojfiu 


736109 


384 


863891 


818517 


461 


18748: 


07640'. 


1371 983591 


048 t 




STTI" 


C«me. 


_ 


Scciini. 1 Cotmg. 


1 Taog. 


Cosec. 1 


1 bme. 


~.mr~i 


3 HoutB, 






57 


5" 


,«. 


7 


f -w i' ( ^r 


"• 1 " \ 


l5" 
30 


\Z\Ul 


isn 


'M 


/■ ■ 13 1 U 


118 1 3 \ *S 


\ li«B \ ^ \ «. \ «1 1 


■« J 




"^^l 





miStmntM. 


T4I.L8 V. 


s* 


IHcor. 


at 




11 DegnH. 




-Sne. Id. 


Co^ 


T«ng. 


dT 


cc^e^ 


Stoat. 


D. 


Come 


j_ 


!"•"* 


I.Ut3f9 Uw 


iatws7i 


(.584177 


6(9 


10.41581; 


10.019848 


w 


S.97015J 


JO 


H ( 


SJMW 54S 


44534! 


584555 


6(9 


41 544 J 


019897 


8[ 


97010; 


sg 


M 


J jlUS?! 5*7 


445013 


5849S( 


62a 


415068 


0(9915 


81 


970055 




51 


555315; 5i7 




59i309f 6ifl 


414691 


0(9994 


81 






48 


aSiflWI 548 


4i+S5T 


5958861 6(7 


414314 


030043; 81 


969957 


56 


44 


555BT1 546 


44tO!9 


58e06(' 6(7 


413938 


03OO9i;ei 


969909 


i5 


40 


55a«!>0l54S 


443701 


586439 6*7 




030140,81 


96986C 


54 


3* 


i566tS. 543 


443374 


586815 6(6 


413185 


030180-81 


969811 


W 


3S 


556953, £44 


443047 


5H719U 6(6 


41*810 


030(38; 81 


969781 


W 


18 


5ST?90, 544 


♦4STM 


5B7SfiG| 6(5 


411431 


030(86 Bl 


969714 


SI 


(1 


5JT60« 


543 


44g394| 


587941 1 fi(5 


411059 


030335 81 


969665 




(( 


697933 


5i3 


44(061 


588316 


613 


411684 


03038^81 


969816 


49 




UBtM 


543 


441741 




6(4 


411309 


030433 8( 


969J87 


48 


11 


UMBS 


54( 


441417 


589066 


6(4 


410934 


030481 8( 


969S18 


IT 


8 


e»ms 


548 


441001 


689440 


6(3 


4I056( 


030531 


8( 

w 


969469 


15 


4 
35 


ISSSiSi 


•sir 


10.440766 


1.689944 


6(3 


10.410186 


lu,030590 


9.9691(( 


Biassn 


541 


440442 


590188 


6(3 




03063O 8( 


96937f 


11 




£39883 


640 


4401 IT 


6905fi( 


613 


40943* 


030679 8( 


969311 


4- 




M0t07 


540 


439793 


590935 


6(1 


409065 


0307(81 91 


969(7 ( 




48 


Mt»3 


539 


439469 


591308 


611 


408S»i 


030TT7 


8( 


g69((; 




41 


S6DS35 


539 


43ai4j 


591081 


611 


40831t< 


030817 


8( 


9691 7S 


40 


40 


961 ira 


538 


438BH 


50(054 


611 


407940 


030876 


8( 


9691(1 


39 


3( 


^150 


63S 


43S4l)g 


5n4(6 


610 


407574 


0309(5 


8( 


9S907J 


W 


3* 




637 


43817(i 


5S!798 


0(0 


40710a 


030975 


81 


9690*5 


ST 


19 


■£*' 


fi3T 


437854 


593171 




406S1( 


0310*1 


8* 


968976 


16 






sss 


437531 


59354( 


619 


406458 


031071 


83 


9689(6 


S5 


10 




SS9 


4S7110 


593914 


618 


4060Be 


031113; 83 


968877 


34 


16 




SSS 


430688 


594(85 


618 


4057 U 


031173' 83 


968817 


« 


11 




KM 


436581 


594650 


618 


405341 


0J1((3;83 


B68777 


» 


8 




SSS 


4Sn4J 


5950(7 


61T 


401973 


031(71 


83 
83 


968118 


)1 


4 




aSS 


La43S»^ 


».5953W 


'elf 


10.401601 


10.0313(1 


g.96867« 


k 


34 




SS4 


43560' 


595708 


617 


40413J 


031371 


83 


969619 


(9 






SSS 


4S5X94 


596138 


616 


40398* 


0311** 


83 


968579 


(8 


51 




fSS 


434984 


596508 


616 


40349* 


031171 


83 


9685(8 


17 


48 




sst 


434641 


5988TB 


616 


40311* 


0315S1 


B3 


968179^ 


44 




at 


4343(4 


59Ii4T 615 


40(753 


031 STl 


83 


9684(9 


^5 






sal 


43400J 


597816' 615 


40(384 


0316*1 


83 


96B379 


11 


36 




SSI 


tsaan 


5979B5 615 


40(015 


031671 


83 


96831!) 


23 


31 




SSI 

590 


4306t 


5»SS5( 
fiHTSl 


614 
B14 


401646 
401178 


0317(1 
03177* 


83 
84 


968(78 
908119 


it 


(8 
(4 




433IHI 




S30 


4StT3! 


5M091 


613 


4OO90S 


0318(1 


81 


8681 TS 


to 


10 




sw 


43(41! 




613 


400511 


031871 


94 


968118 


19 


16 




00 


43(0ft( 


5998(7 


013 


400172 


031911 


B4 


968079 


18 


1! 




5t» 


431 7T( 


600194 61( 


399BW 


031973 81 


968017 


IT 


8 




5(8 


431461 
ia43ll4l 


60056(1 611 
9.6009(»;611 


3»943f 


03(0(3 


84 

84 


96T977 
9.9679(7 


16 

11 


4 
33 




10.399071 


10.03*073 




WHO 


601(96 611 


398704 


03*1*4 84 


967876 




56 




4USU 


60I6G( 611 


39833* 


03(171 gt 


967816 


1; 


51 




ttaiw 


e0I0!9, 610 


397971 


03*(*5 S4 


9677TS 


I( 


46 






601396.610 


397605 


03(375 g4 


06771^ 


1 


41 






60176 1| 610 


397*39 


0313*<) 81 


967674 


u 


K 


wS 


429(41 


6031(7 609 


396873 


03*376' Si 


967611 


E 


» 


Moi 


41893-11 


603193' 609 


396J07 


03*1*7; Bl 


967513 


S 


S 


■IjH 


amto 


6039531 609 


39611* 


031178' 85 


967S1S 


7 


« 


Mm 


4S8305 


604*(:j 608 


395777 


0315*9 85 


96741 


6 


3i 


Has 


4(7991 


604583 608 


395411 


03157y! 85 


9674* 


' 




nm 


4(7677 


604953 


SOT 


395017 


03*630, 85 






4(7364 


605317 


607 


394883 


031«HI 95 


9«73iS^5 


8 

4 
« 




427050 


60568( 


607 


391S1B 


03173*^ 85 






4(6737 


6060 tb 




393951 


siigSLStS^ 




•ns4sJ 


6O011O 


60H_ 


333590 




Sl~ 


'Cm^ 




Tang. 


cw^^^^JLL 






rT~" ■ 








— ^!Zi^ 


a= 









TTTTT 

J 3 I *s la 












^^^™ 


^ 




^ 


^^^ 


■ 












38 TlBLE V. 


Logarithmic Sinea, Tangents, | 


IHour, 






20 De 


gnei. 1 


m. ■. 


' 


Sine. 


D. 


COHW. 


Tang. 


D. 


Coung. 


Secant. 


D. 


Ccine. 


' 


D. * 




iO 


"o 


1.534032 


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10.46594S 


9.661066 


665" 


10.438934, 


10.027014 


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9.97 399t 


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534399 


577 


465601 


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654 


438641 


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677 


465255 


561851 


664 


438149 


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535092 


677 


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5622U 


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562636 


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576 


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563028 


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575 


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363419 


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571 


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652 


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636818 


574 


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77 


97361 


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21 




36 


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537163 


573 


462837 


56*392 


661 




087430 




97857 


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10 


537.507 


573 


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435017 


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637351 


572 


462U9 


565373 


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434627 


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77 


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48 


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538194 


578 


461806 


665763 


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431237 


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78 


97843 


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52 


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6385SS 


571 


461462 


666153 


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433847 




79 


97238 


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56 


14 


6388B0 


571 


46II26 


566512 


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087662 


78 


978336 


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9.639223 


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10.460777 


1.666932 


648 


10.433068 


10.027709 


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9.97289 


15 


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648 


432680 


087735 


78 


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567709 


647 


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569 


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568098 


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540590 


568 


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568*88 


646 


431314 


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540931 


508 


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541272 


567 


458728 


569261 


645 


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97201 


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541613 


567 


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569648 


645 


430362 


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4580*7 


570036 


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429963 


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97IDlt 


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542293 


566 


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429573 


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971870 


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563 


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570B09 


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429191 


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512971 


565 


457029 


571195 


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42BS0i 


02B224 78 


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27 


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456690 


571681 


643 


428419 


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97172! 


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643649 


564 




571967 


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42803! 


088313 7» 


971682! 


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572332 


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427648 


028365; 79 


971634 


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10.455675 


9.572738 


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10.427261 


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9.971581 


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31 


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562 


435337 


573123 


641 


426877 


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562 


455000 


673507 


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561 


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573892 


640 


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454326 


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640 


426721 


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971391 


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360 


433989 


574660 


639 


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560 


453633 


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97130- 


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37 


546683 


559 


453317 


5754S7 


639 


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97185< 


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559 


452981 


576910 


639 


424191 


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97I80t 


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358 


452646 


576193 


638 


423807 


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971161 


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411 


647689 


558 


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576576 


637 


423421 


088887 79 


97111: 


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557 


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576959 


637 


4230*1 


02B9:U 80 


971061 


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548359 


357 


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377311 


636 


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97101! 


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548693 


556 


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577723 


630 


122277 


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378101 


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10.029126180 


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6807691 633 


419231 


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581149 632 


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582286 631 


417714 


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583043 630 


416957 


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445671 


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415823 


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586439' 681 


413561 


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596915,686 


413185 


969911 


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556953, 544 


44304? 


587190' 626 


418810 


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96976! 


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857280 544 


442720 


561566 685 


412*34 


030286 91 


969114 


11 


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5576061 513 


442394 


58194ll 625 


412059 


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969665 


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557938 543 


4t!066 


599316 625 


411694 


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441142 


588691 


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969567 


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559583 


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441411 


589066 


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410934 


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569440 


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590188 


623 


409612 


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969310 


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617 


559SS3 


540 


440117 


590568 


6i2 


409436 


030619 62 


969321 


13 


58 




U 18 


BB0aO7 


540 


439193 


590935 


68! 


409066 


030728J 88 


96981! 


12 


49 




16 IS 


560531 


539 


439469 


591308 


622 


40869i 


03011 J 


68 


969283 


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5H08SS 


539 


439U5 


591691 


681 


40931S 


030887 


62 


969113 


to 


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561178 


538 


4396!! 


592054 


621 


407946 


030876 


88 


9691 !4 


w 


36 




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561501 


536 


438409 


598426 




407571 


030985 


68 


969075 


SB 


38 




3«K 


S6iea4 


S3T 


436116 


592199 


620 


407202 


030975 


82 


969083 


S7 


28 




36 !< 


56*146 


537 


437854 


593111 


619 


406829 


031024 


88 


968916 


K 


84 




40!. 


562469 


536 


437532 


59354! 


619 


406456 


031014 


93 


968986 


IJ 


20 




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562790 


S3S 


437210 


593914 


619 


406086 


031123 


83 


968877 




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48 21 


56311! 


636 


43l588t 


S94!83 


618 


405TI5 


031173 


93 


968687 




11 




i!» 


563433 


535 


436567 


594656 


619 


405341 


031283 


83 


968177 




f 




sa!9 


5fl375i| 


535 


436245 


595087 


617 


404913 


031818 


83 


9661 ttJ 


11 


4 




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I.5B40JA 


T3-4 


10.4359!S 


9.595399 


617 


10.404608 


10.031388 


9.966618 




34 ( 




431 


5643961 


534 


435601 


595769 


617 


404232 


031378 


83 


968689 




66 




632 


564716 


533 


435284 


596139 


616 


403862 


03142* 


93 


966578 




58 




1?^ 


5H3036 


.'33 


434964 


596508 


616 


40349! 


03147! 


93 


068528 


11 


48 




1G31 


J65356 


S3! 


434644 


396818 


616 


4031 !2 


031521 


93 


968479 26 


44 






565676 


53! 


434324 


59184T 


615 


402753 


031571 


63 


40 




24 31 


565995 


531 


434005 


591616 


615 


402364 


0316!! 


83 


968371 


H 


35 




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566314 


531 


433fi9( 


591985 


615 


402016 


031611 


83 




23 


38 




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531 


43336S 


599354 


614 


4016K 


031728 


63 


968279 


22 


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566951 


530 


433049 


598782 


614 


401878 




84 


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81 


81 




4O40 


567269 


530 


438731 


699091 


613 


400909 


031822 


84 


968118 


80 


80 




4^141 


567587 


529 


43241: 


599459 


613 


400541 


031978 


64 


968129 




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567904 


529 


438096 


699827 


613 


40011; 


031982 




96807f 


19 


12 




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563222 


528 


431778 


600194 


612 


39980* 


031973 


64 


966087 


17 


8 






569539 


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431461 


600568 


612 


39943( 


032023 


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64 


967977 


16 


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10.431144 


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10.399071 


10.032013 


9.967927 


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569112 


521 


430828 


6U1896 611 


398704 


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967876 




56 




569488 


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430512 


60166!| 611 


399338 


032174 


64 


961826 




68 




1246 


569304 


5!fi 


430196 


608029; 610 


397911 


032285' ei 


967173 


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570180 


526 


4!99B0 


602395 610 


397605 


038273 B4 


967723 


11 


44 




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510135 


525 


429566 


60216!; 610 


391239 


038326 Si 


961674 


10 






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570751 


525 


489249 


603127' 609 


396873 


032376' 64 


961621 




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8S52 


511066 


524 


486934 


603493 




396507 


032427 B4 


967573 




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428680 


603869 


609 


39614! 


032478 95 


967588 




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511695 


523 


488305 


604223 


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395777 


038529' 85 


961471 




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572009 


5!3 


481991 


604588 


608 


395412 


0325791 85 


961481 




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512323 


5!3 


487677 


604953 


607 


395017 


032630 


95 


967370 




16 




4e'57 


572636 




487364 


605317 


607 


394683 


038681 


65 


967319 




12 




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572951 


58! 


427050 


605682 


607 


394318 


038732 


85 


967268 




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S73!fi3 


521 


426731 


606046 


606 


393951 


038783 


85 


967217 








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581 


486425 


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606 


393S90 


032834 


85 


967166 





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3.15B36 
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t.esiii 

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i.i5939 

8.31387 

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i. 12938 

2.11697 
g.079lS 

1.9Bgi7 
l.954Si 



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1.3590! 
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1.31545 
1.33883 
l.33ggp 
1.32SS5 
I.3IS5I 



I.07W0 999*9 77455 

i.m 

1.06491 
1.06145 

1.05709 
.05456 

1.05115 
1,04777 



4771* 
47688 
4753? 
47448 
5309U 4735! 



6183S 
67608 
67549 
67406 

67864 S94eSJ5iBe5|47173 
59370 58794 470B3 



38081 
3794H 

3T8T7 
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S7733|33(ii(J 



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31661 
37589 

48036 37JISSmi 
41958 37 44<i3Jlii 
41979 37375! 33i 91 



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88913 1 
1.86330 
1.87755 
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1.86687 



1.04448 
1.04109 
1.03779 
I.034S1 
1.03186 
1.08803 
1.08498 
1.02164 



S6390 74643 65 130 



418O0 37;i03|33iB 



37832 331111 
37161 

3709033(13.! 



46464 414S5 371)8038910 



1.796S4 
1.77615 
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1.86587 
1.84988 
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.01884 

.00914 
1.00608 



1.13989 1.003O3 



36597 MSSJ 
36597 3«5« 
3G4&7 314J' 
36388 38 91 
363Ih'3I33I 
36«48'38!«i 
36179 38!M 
36110 Stl4l 



45850 40408 36040 ': 

45ir----- ■ 



1.80418 
1.19938 
1.19458 
1.IB968 
L185S4 
1. 18064 
1.17809 
1.17169 
1.10711 
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0.98887 S3614 



40S4G 3fil>MSIMl 
44991 40173 358333ISSJ 



4OO07 351«s3]8ti 
3.S6K 317(3 
35«S73niH 
3^A593163« 

39194 35491 3U15 
SMS) 31513 



39719 



S584S 49750 44403 9»«44 S53U3tl51 
*9MJ **3M S9M8 MWaaiaSi 



1.5A630 

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.58490 
1.5149 
1.50515 
1.49561 
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1.15836 
1.15404 
1.14976 
1.14551 

14133 
1.13717 

13306 



70966 

7081 1 

70658 

70505 

70358 

70100 S1989 S49B4 48998 

700«fl BISOS 54877 



4483« 394S4 3$8I8,aiS81 
44153 SHIS 9515191 lES 
39344 350B3bllU3 
3937CSM153II 
43903 39195 UBISSIDIS 



43655 38913 34T1. 
43573 SS89B 34671 
4S491 3BB85 34Slt|30I!! 



.45939 
•45079 
.44836 
1.43109 



.41011 
40949 
1.39494 
1.3B751 



1.11304 
1.109J5 
1.105i9 
1.10146 

1.09767 

1.09" 

1.09018 
.08618 

I.08S88 



0.08791 
98537 
98J8- 
9803! 
91781 
91533 
91865 



SS150 



681454 
B8707 
66561 
S8415 

J 7 99 7] 88869 



78368 



61168 54347 13448 

" 18350 

49859 
46167 

48076 
47985 



13387 38678 3UT8 











Degrte, at Hour. 






g 


Om 


I" 


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3n. 


V" 


5^ 


6« 


Tii 


gi pi 










2.2552 


1.9512 


t.77915i.e5321{l.55S3r 


1.4771 


rrior 


1.35818 l^la 






1 


4.0334 


24S0 




77375 85l4l| 5548 


4759 


40914J 35I29i 3002; 






S 


3.7S!3! 


2110. 


9170 


77335 649B 


5531 


4747 


1081 


3503* 8994 
34048 8986 






3 


5S63{ 


2340 


9135 


7700 


61788 55198 


4135 


40708 






4 


4313 


2872 


94001 


7086 


6160a 55055 


4123 


40600^ 31858 2918 






5 


3341 


2205 


9365 


766S 


01426^ 5191 S 


17113 


405031 341691 8B10- 






6 


KS5Z 


2138« 


9330 


1639 


H4849 54770 




4040 


34679 8968 








1883 


!073i 


9296 


76156 


6KJ13^ 51029 


46876 


403001 34589] 8944J 








1303 


2000 


9262 


7592 


0399 


51481 


46758 


10I9BI SlSOOl 2916 






9 


0T9I 


19457 


9229 


75696 


6312 


54341 


4664fl 




3441 1| 893H 






10 


j.0331 


2,18933 


I.919M 


1.75167 


1.0351 


1.5180b 


1.46522 


1.39996 


1.313831 1.293W 






11 


?.99«03 


18217 


9161 


15239 


6331 


51066 


16405 


39895 


3183« 8982 






I! 


9ii8 


17609 


9128 


75012 


6320 


53987 


16889 


39791 


31146 89I4f 






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BiBie 


IIOIO 


9095 


74181 


6303C 


53799 


16171 


39691 


3405M 8907 






u 


89730 


16419 


9063 


74562 


6295 


53619 


40055 


39593 


33970 M99 






IS 


B5T33 


15936 


9030 


7133B 


6268 


53511 


45939 


39403 


33998 2891 






16 


BiflSO 


15161 


8998 


71117 


6251 


53371 


45924 


39391 


33794 8883. 






n 


80?!)} 


14693 


8967 


73996 


6231 


53236 


45708 


39894 


33707 2875 






19 


7781. 


11133 


8935) 


73670 


6218(J 


53100 


15S93 


39195 


336 IS «86- 






19 


75467 


13580 


9904 


73157 


S801 


58963 


45419 


39090 


33532 !B50 






t73?3» 


2.13033 


1.8873 


1.73239 


1.6181J 


1.52881 


1.15364 


1.3S997 


1.S341S 1.28521 






81 


711 !0 


12194 


8S12C 


73023 


61678 


fi!692 


45850 


38899 


S335« (8146 






iS 


60100 


11961 


88111 


72807 


6151! 


52557 


45136 


38800 


33872 S8S69 






13 


67170 


11435 


8780^ 


78593 


61317 


52428 


45028 


38702 


33186^ n;9! 






il 


65321 


10911 


87506 


78379 


61188 


58888 


44909 


39604 


330991 88215 






15 


63518 


10100 


87208 


72167 


61019 


52151 


44796 


38506 


33013 88138 






!6 


61845 


09B93 


86907 


71956 


60851 


52081 


44681 


3S409 


32987' 8806 






87 


60200 


09390 


8661 


71745 


60691 


61888 


41571 


3931! 


3284»: 87994 






(8 


59621 


08894 


86316 


11536 


60589 


51755 


41159 


38215 


38756 


87909 






89 

30 


57103 


08103 


96021 


11388 


60381 


51623 


44347 


38118 


3267 1 


81831 






2.55630 


v. 07918 


1.85733 


1.71120 


1.60200 


1.5U9I 


1.14236 


1.39021 


1.38585 


I.8I75i 






31 


54*06 


07138 


85115 


70914 


600+5 


51360 


14125 


37925 


S8500 


I7E79 






3« 


5288? 


06961 


85159 


10709 


59895 


5122D 


4.1014 


37889 


38415 


876113 






33 


51191 


06191 


84813 


70501 




51098 


43903 




S2331 


81527 






3* 


50194 


06030 


81590 


10301 


59561 


S0968 


43793 


37631 


S881fi 


87451 






Si 


48936 


05570 


81309 


70O99 


59409 


50838 


43693 


37541 


32168 


87376 






30 


47718 


051 15 


81031 


69897 


59851 


50709 


43513 


37416 


S8077 


S730» 






37 


1652! 


01665 


83752 


69696 


59091 


50579 


43163 


31S5I 


SI 993 


87825 






36 


15361 


042S0 


83177 


69491 


58938 


50451 


43351 


37246 


31900 


tJISO 






39 


11236 


037 79 


83203 


69299 


58182 


50322 


43845 


31161 


31920 


, 87075 






iU 


.431 3M 


2.03312 


.82U30 


1.69100 


1.58687 


1.50194 


1.43136 


.37067 


.31T4S 


I.KCOO 






41 


42084 


oaani 


82660 


68903 


59112 


50007 


4308^ 


3607! 


S16S9 


S692S 




4! 


41017 


02192 


82391 


68707 


58317 


49940 


42020 


36678 


31575 


86830 




(3 


3B0S6 


0S06O 


82121 


08512 


58104 


19913 


18918 


36791 


31191 


86176 






41 


38991 


01639 


81959 


68318 


58011 




42701 


36601 


31409 


H701 






45 


38021 


01213 


81591 


68184 


67859 


49560 


42597 


36591 


31326 


86(27 






46 


37067 


00812 


81332 


67938 


57706 


49135 


12490 


36504 


31811 


86553 






47 


36133 


00101 


81071 


677*0 


57554 


49309 


18383 


36411 


SI16I 


M479 






46 


35218 


00000 


80811 


61519 


57403 


49184 


48876 


36318 


31019 


H105 






40 


31323 
.33445 


L99600 


80551 


67359 


51253 


49060 


42170 


36285 


3fl»Bf 


C633I 






^9203 


.80297 


1.61170 


.57103 


.49930 


.42004 


.30133 


:3MT5 


rg»i 






51 


32585 


98810 


80013 


66981 


66953 


48912 


4I95S 


SOOMI 










S! 


31742 


98121 


79790 


66791 


56804 


48698 


41833 


35918 


30751 


MIlOj 






i3 


30915 


98035 


79538 




56656 


19565 


41117 


35856 


30670 








54 


30103 


97632 


79297 


66181 


56509 


49412 


41642 


35765 










55 


29306 


91213 


79030 


66236 


50360 


49320 


41538 


35613 


30507 


tM91 








29524 


96897 


78791 


60051 


56813 


18107 41131 


3558! 


SOllfl 


SWIS 






S7 


27755 


96524 


78515 


65868 


56087 


49076 11329 3549l| 


30315 








58 aiowl 


9BI51 


78300 


05085 


55921 


47931 41885 SS400 


sossS 








M Z6tST\ 


95189 


78057 


65503 55775| 4T8331 4ll2l| 3530D| SOISSJ 


tswaj 







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ODegr«,<«OHoar. 




J 


10^ 


11" 


.^ 


13^ 


ni 


I5i 


16" 17^ 


18^ 


19^ 






l.SSfi!7 


1.1I3B8 


1.17609 


1.14133 


1.10911 


1.07918 


1.051151.08488 


1.00000 


0.97658 




1 


S515S 


81388 


175491 


11077 


10883 




05070 02140 


a99960 


97614 




« 


853B3 


81857 


17489 


11088 


10811 


078*8 


05085 08397 




97576 




3 


(S3 1 1 


81191 


17429 


13966 


10764) 


07771 


01990 08355 


99880 


97538 




i 


!Sg39 


81186 


17369 


139(1 


1070H 


07786 


04935 02312 


99939 


9750C 






86167 


81060 


17309 


13855 


10657 




04990 02870 


99799 


97462 






85095 


80905 


17849 


13900 


10605 




0184S 0888B 


D9759 


97484 






85081 


80930 


17189 


13715 


1055+ 


07588 


04S0n 08185 


99719 


97396 




e 


84958 


80865 


17189 


13690 


10503 


07531 


04755 02143 


93679 


97348 




9 


84Sgl 


8OS00 


17070 


13635 


10*58 


07186 


04710 O8101 


99610 


97310 




10 


1.8*1*09 


1.80735 


1.17010 


1.135B0 


1.10100 


1.07138 


1.01665 1 .080^9 


0.99600 


0.978J3 






84738 


80570 


16S51 


13585 


10319 


07391 


04680 08017 


99560 


97235 




1! 


84667 


80605 


16B9I 


13170 


10898 


07343 


04576 01974 


99580 


97197 




13 


84596 


80S1I 


16838 


13415 


10247 


07895 


04531 01938 


99480 


97159 




U 


84586 


80176 


167T3 




10197 


07218 


01486 01 890 


99441 


97188 




IS 


8U55 


80118 


16714 




10146 


07800 


01448 01918 


99101 


97084 




IS 


tl384 


80318 


16655 


13851 


10095 


07153 


01397 01806 


99361 


97017 




17 


843U 


80281 


16596 


13197 


10014 


07105 


01353 01764 


99388 


97009 




18 


81844 


80819 


16537 


13118 


09S94 


07058 


01308 01783 


99282 


96978 




IS 


84ns 


80155 


16178 


13088 


09943 


07011 


04864 01681 


99243 


96934 




SO 


1.841U3 




1.16419 




1.09893 


1.06964 


1.04820,1.01639 


0,99803 


0.96997 




!1 


81033 


8008B 


16361 


12979 


09842 


06916 


04175 


01597 


99164 


96859 




i! 


83963 


19961 


16308 


18925 


097M 


06869 


01131 


01556 


99124 


96888 




83 


83B91 


19900 


16843 


12671 


09711 


06922 


04087 


01514 


99095 


96784 




U 


83884 


19937 


16185 


18817 


0969 


06775 


01013 


01178 


99015 


B6747 






23761 


1S773 


16187 


18763 


09G41 


06728 


03999 


01431 


99006 


96710 






83(iB5 


19710 


16068 


18709 


09591 


06681 


03955 


01389 


98967 


96673 




27 


83616 


19647 


16010 


18655 


09S10 


06634 


03911 


01318 


99928 


90635 




!8 


83548 


19584 


15958 


18601 


09490 


(16588 


03867 


0130b 




9S599 




!8 


S34T7 


19580 


1SB9+ 


18549 


0914« 


06511 


03823 


01265 


98819 


96561 




30 






1.15836 


1.18194 


1.0939O 


1.06191 


1.03779 


1.01283 


aS981l 


0.96,384 




31 






1S778 


18410 


09311 


06447 


03735 


01182 


98771 


9848T 




82 


83871 




15781 


18387 


09291 


06401 


03691 


01141 


99732 


96450 




S3 


83808 




1S663 


12333 


00841 


06351 


03617 


01100 


99693 


S6413 




34 


83133 




lS60o 


12880 


09191 


06308 


03604 


01059 


99654 


96376 




S5 


83065 


19144 


1554^ 


12827 


09118 


06861 


03560 


01017 


98615 


96339 




3fl 


889B7 


19031 


15490 


18173 


09092 


06815 


03516 


00976 


98S76 


96.308 






88988 


19019 


15433 


18180 


09048 


06168 


03473 


00935 


98537 


96265 




SB 


88860 


19957 


15375 


18067 


08993 


06182 


03189 


00894 


98498 


96888 




39 


8a?9a 


18895 


15318 


12014 


08943 


06076 


03386 


00853 


98459 


96191 






1.88784, 


ns833 


1.15261 


Liissi 


1.08894 


1.06031) 


T:03318 


1.00818 


0.98181 


0,96154 




41 


18657 


18771 


15801 


11909 


08815 


05983 


03899 


00771 


98382 


961 IT 




48 


88589 


18709 


15147 


11855 


09796 


05p37 


03856 


00730 


98313 


96081 




43 


88581 


18647 


15090 


11808 


08746 


05891 


03818 


00689 


98301 


96044 




44 


88154 


18583 


15033 


11750 


09697 


05845 


03169 


00648 


98866 


96007 




45 


8838G 


18S23 


11976 


11697 




05799 


03186 


O0EO7 


98887 


95971 




46 


88319 


isies 


14919 






05753 


030S3 


00567 


98189 


05934 




47 


88158 


18400 




11592 


08550 


05707 


03039 


00526 


98150 


95897 




46 


821B5 


18339 


14aOG 


11539 


08501 


056G2 


08996 


00485 


98111 


95961 




49 


88118 


18878 


14750 


11487 


09158 


05616 


08953 


00445 


98073 


95981 




S5" 


1.88051 


M88i7 


1.14693 


1.11135 


1.09103 


1.05370 


1.03910 


l.O0'KI4 


U.9803S 


0.95788 




Al 


?1981 


1S155 


14637 


113S2 


08355 


05581 


08867 


00363 


07B96 


95751 




« 


81918 


18091 


11581 


11330 


08306 


05479 


02921 


00323 


97938 


95715 




as 


SIS5I 


18033 


145^4 


11278 


08857 


05433 


02781 


00888 


97919 


95678 




» 


81TH5 


17913 


11168 


1182(1 


08809 


05399 


0?739 


00842 


97881 


95648 




AA 


8 me 


17918 


, 11118 


11171 


oaieo 


05312 


02696 


Op808 


97843 


95606 




AS 


sissa 


17851 


1*356 


11128 


08112 


05897 


0SS53 


00161 


97S05 


95569 




6T 


21586 


177S0 


14300 


11070 


08063 


05251 




00181 


97760 


95533 




SB 


81580 


17730 


14811 


iioia 


09015 


05106 


085(T9 


oooso 


97788 


95197 




fift 


81151 




14189 


10966 


07966 


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s i-i 1^ 21. a& lu ax 


A ^ 











Degue. or Hobt. 


a 


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aim 


2am 


23^ 


21-n 


25i^ 


28" 


27" 


88^ 


89^ 


^ 


BMH 


93305 


91285 


99354 


8T506 


85733 


84030 


98391 


80811 


79887 






OSSBH 


fl3S7l 


918S« 


893*3 


97*76 


8570* 


8*002 


88364 


60766 


798e3 








S53SZ 


93^31! 


ei!i9 


89898 


97446 


85675 


83974 


82337 


80760 


79838 






3 


95318 


9380* 


91166 


egseo 


87416 


858*6 


83946 


68311 


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95!60 


83188 


91154 


89H9 


87386 


85618 


83919 


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& 


95344 


93133 


81181 


89197 


87356 


65589 


63891 


82257 


80662 


79163 








95108 


93099 


9108B 


69166 


87386 


65580 


83863 


82230 


80857 


79138 






1 


95178 


93065 


91055 


89135 


87296 


85531 


83835 


88804 


80631 


79113 






8 


95136 


930,30 


01023 


89103 


87888 


85502 




68177 


60605 


79088 






9 


05 100 


93996 


90990 


89078 


87236 


85473 


83780 


82150 


60579 


79063 






95064 


02968 


9096T 


89011 


67206 


85*45 


83752 


8818* 


6055* 


79039 






D50S8 


9S98H 


90925 


69010 


67176 


85*16 


63785 


82097 


60528 


7901* 






li 


9499! 


9S894 


90892 


88979 


87146 


85387 


83687 


82070 


80508 


78989 






i3 




98860 




8994T 


87116 


95359 


83670 


8804* 


90477 


78S6* 






14 


94981 


9?8!5 


90887 


8g916 


870S6 


85330 


83842 


88017 


60151 


79939 






IS 


9488S 


BS791 


9079* 


88885 


67056 


85301 


93814 


81991 


80425 


79915 






IS 


94949 


9ST5T 


90768 




B7026 


85878 


83567 


81964 


80400 


T8890 






IT 


94313 


8?783 


90789 


8B883 


96996 


6524* 


83559 


81 939 


6037* 


79865 






IS 


9*778 


9?e8S 


80697 


88798 


96967 


85215 


63538 


91911 


80349 


78940 






IE) 


94743 


92665 


9086* 


88761 


8693J 


85187 


8350* 


6188* 


80383 


78916 






94708 


926S1 


00638 


9B730 


8UB07 


65158 


Tsm 


61859 


80297 


79791 




«1 


94671 


9S597 


90599 


88699 


86877 


85129 


93**9 


61838 


80872 


78166 






sa 


9463a 


92S5* 


90567 




868*6 


85101 


83*88 


81805 


80848 


787*8 






S3 


94600 


98520 


90.535 


88637 


88818 


65078 


8339* 


91779 


80221 


J9717 






!4 


94564 


93486 


90508 


88806 


86769 


85044 


63387 


9175! 


80195 


78693 






S5 


94589 


92452 




88575 


66759 


85015 




81786 


80170 


78668 








94493 


93418 


90438 


885*4 


86729 


84997 


83312 


91699 


90144 


796*3 








94458 


98385 


9040S 


89513 


86699 


94956 


63285 


81673 


80119 


78619 






S9 


94423 


98351 


00373 


86482 


86670 


84930 


93SST 


81647 


8009* 


7959* 






89 


9438T 


98317 


9034] 


8B451 


88640 


84902 


93830 


81680 


80068 


79570 




30 




08883 


903U9 


88*80 


8881 1 


8*873 


83803 


81594 


80043 


79515 




31 




928SO 


90S7T 


88390 


86591 


84845 


93175 


815G8 


80017 


78581 






Si 


91 291 


92216 


903*5 


88359 


8655! 


61916 


83148 


81511 


790S8 


78*96 








94446 


98193 


90813 


89389 


86588 


84788 


93181 


81515 


79967 


78178 






31 


94iU 


93149 


90181 


89897 


86493 


84760 


83094 


81489 




79**7 






3S 


941T6 


98115 


90148 


B9287 


66463 


84738 


830G6 


81463 


79018 


78423 






36 


04U1 


92088 


90116 


B8236 


8643* 


84703 


93039 


81*36 


79891 


78398 








94105 


92048 


900B* 


88805 


86401 


84675 


830U 


81410 


79865 


7937* 






sa 


94070 


98015 


90058 


89175 


88375 


646*7 


68985 


81384 


79810 


763*9 






^ 


04O35 


919S1 


Booao 


881*4 


863*6 


64619 


88959 


81358 


78815 


78385 






94000 


91948 


"89989 


88114 


86316 


84590 




61332 


797SO 


7830U 






83965 


91915 


899ST 


88083 


96887 


84568 


88003 


91305 


7976* 


78878 






♦8 


93930 


91881 


89985 


88058 


96859 


84531 


88876 


81879 


79739 


I885t 






43 


93895 


01848 


89893 


89088 


66288 


84506 


829*9 


91253 


79714 


78887 






44 


938G0 


91815 


89881 


87991 


86199 


64478 


82828 


81887 


79689 


78203 






45 


S3B2S 


91791 


80989 


87961 


86170 


84450 


B8795 


9 1801 


79663 


78 170 






46 


93791 


SI748 


89797 


87930 


86UI 


84481 


63768 


81175 


79638 


78 IS* 






47 


93T56 


81715 


B9766 


S79O0 


96111 


S1393 


827*1 


81149 


70613 


76180 






48 


93TS1 


91882 


89734 


87970 


86088 


94385 


62714 


81183 


795B8 


T8I06 






19 


936S0 


91848 


80708 


87839 


86053 


84.337 


82687 


810BT 


79563 


7fiOBl 




SO 


93H51 


916"l5 


89870 


87809 


88084 


84309 


88660 


81071 


7953b 


7805T 




SI 


B38I7 


91588 


89839 


B777H 


85995 


84891 


88633 


91045 


79513 


780S3 






53 


935S2 


91549 


8960T 


87748 


85965 


84253 


82606 


B10I9 


79488 


7B0OB 






53 


9354T 


91S16 


89575 


97718 


65936 


84225 


82579 


60993 


70463 


77984 






5* 


93513 


01463 


8954* 


878B7 


85907 


81197 


82558 


80967 


79437 


77960 






55 


93478 


91450 




8T657 


65678 


H4169 


82585 


80941 


79*12 


7793< 






5G 


934t3 


91417 


89*81 


87627 


95849 


84141 


82498 


80915 


70387 


7791t 






57 


93409 


913S4 


80*49 


87597 


SSB80 


84114 


98*71 


80889 


7036! 


7786S 






58 


93374 91351 


89417 




85791 


84086 


82445 


80963 


79337 


77963 








93340 91318 


89386 




85762 


9*059 


92419 


80937 


79318 


77-*39 






tSopuriionB! Put to 


.1 .2 .3 .1 ,5 .6 .7 .8 .9 


1 i 




3 6 y 18 15 18 !1 3* 8T 




L 




H 





I^oportional Logorith 


im. TabliX. 


tl 




ODi;pet.OT 


OH(HV 








3oi 


3l»> 


38^ 
"750"i8 


33" 


2V> 


SSn 


ssi 


3T» 


^ 


3S- 




TTBIS 


76391 


73676 


78379 


71180 


69897 


-6T75? 


67549 


66481 




TITfll 


7«36S 


74990 


73654 


7835iS 


71100 


69877 


68698 


67530 


8640S 




7TI67 


76344 




73638 


7(337 


71079 


69857 


68668 


67511 


863Mi 




77743 


76381 


74944 


73S10 


78316 


71058 


69837 


68619 


67492 


66364 




TTTIB 


76808 


74988 


73588 


78891 




69817 


68689 


67*73 


66S17' 




77695 


76875 


74899 


73566 


72873 


7IOI7 


69797 


68609 


67454 


6a3M> 




77671 


76851 


74877 


73544 


78858 


70997 


69777 


68590 


67435 


68314 




77647 


7G8SS 


74854 


73583 


78831 


70976 


69756 


68570 


67416 


68891 




77683 


76805 


74838 


73501 


78809 


70955 


69736 


6B5SI 


67397 


68173 




71i»9 


76ISI 


74809 


73179 


72IB8 


70935 


69716 


68531 


67378 


68»54 




7757S 


76 158 


74787 


73457 


78167 


70914 


69696 


68518 


67359 


Sim 




77551 


76135 


74764 


7343.5 


78146 


70894 


69676 


68498 


67340 


61817 




77587 


76118 


74748 


73413 


78185 


70873 


69656 


68473 


67381 


66199 




775ai 


76089 


74719 


73398 


78103 


70858 


69836 


68451 


67308 


6S1S0 




77479 


76065 


74B97 


73370 


78088 


70838 


69616 


68431 


67883 


6(168 




774M 


76048 


74674 


73318 


78061 


70811 




68115 


67864 


68143 




77431 


76019 


7465* 


73326 


78940 


70791 




68395 


67M5 


611 85 




77407 


75996 


74689 


73305 


78019 


70770 


69557 


68376 


67886 


68106 




77385 


75973 


74607 


738B3 


71998 




69537 


683*6 


67*07 


68088 




77358 


75950 


74585 


73861 


71977 


70789 


69517 


68337 


67189 


61070 


SU 


77335 


75987 


74568 


73839 


71956 


70709 


69497 j 68318 


67170 


eeoM 


ai 


77311 


TS903 


74540 


73819 


71935 


7U688 


69477 1 68898 


67151 


66033 


sa 


77888 


75880 


74517 


73196 


71914 


70668 


69157 , 68279 


67132 


68011 


£3 


77864 


75857 




73174 


71898 


70B47 


6B437 


68259 


67113 


65996 


Si 


77840 


75834 


74473 


73153 


71871 


70687 


69117 


68810 


67094 


64878 


2S 


77816 


75811 


74460 


73131 


71850 


70606 


69397 


68881 


670T5 


85959 


!6 


77198 


75788 


74488 


73109 


71889 


70586 


69377 


68^1 


e7oa9 


68911 


87 


77169 


75765 


74406 


73088 


71808 


705Q6 


69358 


68188 


67038 


64983 


tfl 


77145 


75748 


74383 


T3U66 


71787 


70415 


69338 


66163 


670I» 


64901 


89 


77181 


75719 


74361 


73044 


71766 


70585 


69319 


68143 


67000 


64986 


30 


7!01t7 


75696 


74339 


73083 




70504 


69898 


68184 


6G9S1 




31 


77074 


75fl73 


74317 


73001 


71784 


70484 


69878 


68105 


66963 


65S49 


sa 


77050 


75650 


74894 


72980 


71703 




69858 


69086 


66041 


64S3I 


33 


77086 


75687 


74878 


73958 


7i6ea 


70443 


69839 


68066 


66885 


65813 


Si 


77008 


75604 


74850 


78936 


71663 


70483 


6Uai9 


68047 


66906 


65781 


Si 


7(i970 


7558 1 


74«B 


78915 


71611 


70 403 


69199 


68089 


66887 


65776 


30 


7(1955 


75559 


74205 


78893 


7 1610 


70388 


69179 


680O9 


66869 


64758 


37 


76931 


75536 


74183 


72878 


71599 


70368 


69159 


67999 


66850 


64739 


38 


76008 


75513 


74161 


73850 


71578 


7034t 


69110 


67970 


66831 


64781 




7GSe4 


75490 


74139 


73889 


71557 


70381 


69180 


67951 


6681* 


65703 


Ji) 


7H8BI 


75467 


74117 


78807 




703OI 


69100 


67938 


66794 


64694 


41 


7Ge37 


75444 


J 4095 


78786 


71515 


70881 


69080 


67918 


66775 


65666 


fj 


76813 


75481 


71078 


78761 


71494 


70860 


69061 


67993 


66756 


65648 


4.:) 


76790 


7S39S 


74050 


78743 


7147S 


70840 


69041 


67974 


66737 


05630 




76766 




74088 


78781 


71153 


70880 


69081 


67955 


66719 


05618 


JS 


7UT43 


75353 


71O06 


78700 


71439 


70800 


69008 


67936 


66700 


65594 


i6 


76719 


75330 


739S4 


78678 


714n 


70179 


68988 


67916 


66681 


65474 


*T 


76696 


75307 


73968 


78857 




70159 


68968 


67797 


66663 


64S5T 


te 


76678 


75895 


73910 


78636 


71369 


70139 


68918 


67778 




64539 


4S 


7664B 


75868 


7391B 


78611 


71349 


70119 


68923 


677 5S 


666U 


644II 




76«!5 


75S39 


738a« 


78593 


71386 




69903 


67740 


66807 


64503 




7GG08 


75^16 


73871 


78571 


71307 




69884 


67781 


66589 


64191 


ia 


7657B 




73858 


78550 


71886 


70059 


68861 


67702 


66570 


64166 


as 


76355 




73830 


78589 


71865 


70038 




67688 


66551 


64149 


fit 


76531 


75148 


73808 


7850J 


7184^ 


70O18 


68885 


67663 




65430 


S5 


76508 


75185 


73786 


78186 


71884 


69999 


68805 


67614 


66511 


65118 


5B 


76485 


75103 


73764 


78165 


71803 


69977 


6B785 


67685 


66495 


65394 


S7 


16461 


75080 


73748 


78113 


71183 


69957 


eB786 


67606 


66477 


64376 


ss 


76138 


75058 


7:1780 


78188 


71168 


69937 


68716 


67587 


66158 


65347 


-^9 


76414 


75035 


73698 


72401 


71141 


6991 T 


69787 


67569 


66139 


64339 


PfapOEIiooBl l-nri to 


.1 


.S 


3 .* 


.5 .6 .1 


Ol 


unths of " nr t. 


8 


4 


i f 


10 13 15 1 





^ 




^^ 










Degree, ot Hour. 




■; 


40" 


4I™ 


4gi 


l^i 


Iti 


45" 


IG" 


47" 


48" 


49" 







fissai 




63802 


68180 


61188 


60806 


59851 


58317 










1 


653m 


64S31 


63185 


68164 


61166 


60 190 


59830 


59308 


57383 


56493 








B5»B5 


61814 


631 68 


68117 


61119 


60171 




58887 


57373 


56478 






i 


6586T 


61196 


63151 


68130 


01133 


60158 


59804 




67359 


58463 








Ci!49 


61178 


63133 


68113 


01116 


60148 


59189 


58856 


57343 


58419 








65*31 


61161 


63116 


68096 


6II0O 


B0186 


59173 


5884 


67388 


58431 






6 


65813 


61113 


63099 


68090 


61033 


flOllO 


50157 


58885 


57313 


56419 








65195 


64185 


63088 


68063 


61067 


60094 


59141 


58810 


57898 


66401 






a 


65177 


61108 


63065 


68046 


61051 


6O078 


59186 


58191 


5T8B3 


56380 






9 


65159 


6109U 


63018 


68089 


61031 


6U06I 


59110 


58179 


57868 


56375 




IC 


65141 


61073 


63030 


68018 


61018 


60045 


59(194 




57853 


56360 






65183 


61055 


63013 


61998 


61001 


601)89 


59079 


68 1« 


57838 


66315 






IS 


65105 


6iai8 


6S996 


61979 


60985 


CO013 


59063 


58133 


67883 


56331 






13 


65087 


61080 


6S9J9 


61968 


60969 


59997 


59017 


58118 


67208 


5631 6 






It 




61O08 


68968 


61915 


60958 


59981 


59038 


58108 


57193 


56301 






15 


65051 


639BS 


68945 


61989 


60936 


59965 


59016 


58087 


57178 


56887 






10 


650;» 


63967 




61918 


60980 


59919 


59O0O 


58078 


57103 


66878 






17 


R5015 


63950 


68910 


61895 


60903 


59933 


58985 


6805G 


67148 


66857 






18 


6499T 


6393S 


68893 


61878 


60887 


59917 


58969 


68041 


67133 


58843 






19 


64979 


6391 5 


68876 


61868 


608TI 


59901 


58954 


58080 


57J18 


68888 






64961 


63S97 


68859 


61815 


60851 


59885 


58938 


5H0U 


57103 


6i«8I3 




81 


64913 




68818 


61888 


60838 


59870 


58988 


67995 


57083 


56199 






88 


64985 


6386! 


68885 


61818 


60888 


59851 


S8907 


67980 


57073 


5H1B4 






83 


64907 


63845 




61795 




59838 


5889 1 


57905 


67058 


5B169 






S4 


61889 


63887 


08791 






59888 


58875 


57949 


57013 


66155 






85 


61871 


63810 




61768 


60773 






57934 


57088 


56140 






£6 


61853 


6^798 




6I74i 






58811 


57919 


67013 


fisisa 






87 


61835 


63775 


68739 


61788 




59771 


58889 


57901 


509BS 


58111 






SB 


61818 


63757 


68788 


61718 


00724 


597SB 


59813 




509S3 


5(KlftS 






89 


61800 


637411 




61U1I5 


60708 


59718 


5B79B 


57873 


59968 


680141 






61788 




68688 








58782 


57H58 


ibOaii 


6(illU7 




31 
3« 


61761 
61716 


63688 


68671 

68654 


61668 
61615 


60675 
601359 




58766 
58751 


57813 
57887 


56938 
56983 


seosi 

6601T 




59691 




33 


64788 


636TO 


68631 


61688 


60618 


5967H 


58735 


57818 


50 908 


50083 






31 


61710 


63653 


68680 


61618 


60686 


59663 


58780 




66B93 


5G008 






35 


64B98 


63635 


68603 


61595 




59617 


68 704 


57788 


56879 


55994 






36 


64B75 


63618 


68586 


61579 


60594 


59631 


580S9 




56804 


55HT9 








616,57 


63601 


68569 


61568 


60578 


59615 


58673 


57751 


511849 


55965 






38 


61639 


63583 


68558 


615(5 


60561 


59599 


58658 


57736 


66834 


65B60 






39 


61681 


63586 


68535 


61589 


60515 


59583 


58018 


67781 


56819 


S6B35 




40 


61^03 


03518 


68518 


01518 


60589 


^59567 




57706 




55981 




41 


6158(1 


63531 


085(11 


01196 


60513 


59551 




57691 


567t'9 


55906 






48 


61568 


63511 


68181 


oii;9 


60496 






57675 


50771 


55898 






43 


61550 


63(96 


68168 


611S3 




595^0 




67060 


56759 


6S87J 






44 


61533 


63479 


68151 


61446 






58565 


57645 


5674.-. 


55868 








61514 


B3i6« 


68434 




601(8 




58519 


57630 


56730 


6584S 








61197 


63144 




01113 


60138 


69478 


58531 


57615 


66715 


65B3S 






4T 


61179 


63+97 


68100 


61396 




59157 


58518 


67600 


56700 


6S81B 






48 


61(61 


63110 


68383 


61380 


60399 


59111 


58503 


57581. 


56685 


35804 






49 


61113 


63398 


68366 
68349 


61363 


60383 
603S7 


59185 
59109 


58178 


57569 
57551 


64670 
56656 


55790 
55T75 






61186 






51 


6140B 




68338 


61330 


00351 


59393 


68156 


57539 


56041 


557UI 






SS 


64390 


6^340 


68316 


61311 




59378 


58111 


67581 


511686 


55746 






at 


64373 


0:1383 


688H8 


61897 


60319 


59308 


58485 


67509 


66611 


6573* 






e* 


61355 


G3306 


88888 


01881 


00303 


59346 


5841(1 


67194 


56596 


55717 








61337 


63889 


68865 


61861 


60886 


59330 


58395 


57479 


6S588 


54T«3 






iK 


6i^?0 


63871 


68848 


61818 


60870 59311 


68379 




66567 


55088 








63851 


68831 


61831 


60854 S989S 


58364 


57418 


56558 


55S74 




AS 




6:i83T 


68814 


61815 


60838 59883 


58318 


57(3:1 


56637 


55«5A 






642S7 




68197 


61198 


60888 59867 58333 


57418 


5658» 


UGtt 




leotbs of "or 


t [a 


.1 .8 . 


i 1. ^ ^ 1 — :»~m 


^H 




6 8 10 11 IS a^^ 


IP 


^^^^^^^^^^^^^^^^^^^^H 





Proportional Lngarith 


ni, 


TisLi X. 


7S 






OB<«rce,or 


OHmu. 














SOW 
5.^630 


il" 


&t!^ 


S3" 


54^ 


S5« 


56" 


,^ 


aSM 


59^ 




547 70 


53927 


53100 


52888 


"61491 


40709 


49940 


49184 


48412 






5561B 


54756 


53913 


53086 


5227* 


51478 


50696 


49987 


49172 


49430 






£5601 


54742 


53899 


63072 


58261 


51465 


50683 


49914 




48418 






55387 


54728 


53883 


530J9 


58818 


51458 


50670 


49902 


491*7 


4»405 






5^57! 


51714 


63871 


53015 


52834 


51438 


50657 


49889 


49135 


48393 






55516 


54699 


538ST 


53031 


52881 


51426 




49876 


49182 


48381 






55343 


54685 


53843 


53018 


58208 


51118 


50631 


49864 


49110 


48369 






5J5!9 


54G71 


53830 


53001 


52194 


61399 


50618 


49951 


490U7 


48356 






55515 


54657 


53816 


52991 


58181 




50S05 


49839 


49Ut<J 


483*4 






5SJ0O 


54643 


M802 


52971 


52I6T 


51373 


60592 


49826 


49078 


18338 






55186 


51629 


53788 


62963 


68154 


51360 




49813 


4S060 


48320 






55 Wl 


S46I4 


5377* 


52950 


52141 


51346 


50566 


49800 


49017 


48307 






55457 


54600 


53T60 


52936 


5*187 


513S3 


50554 


49788 


49035 


48895 






S544Z 


54586 


53746 


58928 


52114 


51320 


50541 


49775 


411023 


49883 






sswe 


54572 


53732 


52909 


52101 


J13CT 


60528 


4B762 


49010 


48871 






5.5414 


54553 


53719 


528B5 


62087 


S18B1 


60515 


49750 


48998 


49858 






55399 


51544 


53705 


52992 


68074 


51291 


60508 


49737 


48995 


48246 






55.1B5 


545S0 


53691 


3286B 


52061 


51269 


60189 


49721 


48973 


48234 






55310 




538J7 


5285S 


52047 


5125.1 


60176 


49712 


48960 


48828 






55356 


54501 


53G63 


58811 


52034 


518*2 


60461 


49699 


489*8 


48210 






35;ua 


54497 




58827 


52021 


51829 


"50451 


49687 


*C936 


49197 






55387 


54473 


53636 


58814 


52007 


61815 


60438 


4967* 


48923 


4UI85 






55313 


54459 


53S22 




51994 


51808 




19661 


48911 


48173 






a5!99 


51445 


53608 


58787 




51189 


50418 


496*9 


489B8 


48161 






55234 


51431 


53594 


52773 




S1I76 


50399 


49636 




481 4B 




!5 






53580 


58T60 


51954 


51163 


50387 


19623 


4887* 


48136 




!G 


65?SS 




53567 


52740 


519*1 




50374 


19611 


49861 


4918* 




87 


55241 


54389 


53553 


52738 


51927 


51137 


50361 


49598 


481^49 


48112 






5s-ig; 


54375 


53539 


52719 


51 9U 


51124 


50318 


49586 


18938 


48100 




S9 




54361 


53525 


52705 


51901 


51111 


60335 


495 7 3 


4898* 


48089 




30 


55198 


54347 




52692 


6188H 


51098 


60328 


49560 


48912 


49076 




31 


5-,I8t 


54332 


5:)499 


52678 


51874 


51085 


50310 


49548 


48799 


48063 




sa 


55169 


54318 


534«4 


52605 


51861 


51078 


50897 


49535 


48787 






33 


55155 


54304 


53470 


58651 


518*8 


51059 


50294 


49583 


48775 


48039 




34 


55141 


51290 


53456 


68638 


5 1 835 


51046 


50271 


49510 


48768 


*8027 




35 


55187 


54276 


63142 


52684 


51821 


5:o;t3 


50858 


49408 


48750 


48015 




36 


55 IIS 


54262 


53429 


52611 


51808 


51020 


50246 


49185 


48737 


49003 






55098 


54248 


53115 


52597 


51795 




60233 


49*78 


48785 


479B0 




38 


.^5084 


54731 


saioi 


58584 


51791 




60220 


49*60 


48713 


17978 




39 


55069 


51220 


53397 


58570 


51769 


~S096^ 


50807 


49147 


48700 


47966 






550S5 


54206 


63374 


58557 




601 Ml 


19435 


49688 






ii 


55041 


54102 


53360 


58513 




50955 


50182 


49422 


49676 


47S42 








541 78 


53Wfl 


52530 


S172S 


50942 


50169 


49*10 


48663 


47930 




*3 


550 li 


51161 


53332 


52516 


61715 


60929 


60156 


49397 


48651 


47918 




44 


54<>nH 


51150 


53319 


52503 


61708 


60916 


50113 


49385 


48639 


479U6 




*5 


54994 


54136 


5330J 


62199 


SI 689 


509O3 


50131 


4B378 


48686 


47893 




M 


54S69 


54122 


S32BI 


52476 


51676 


50890 


50118 


49360 


48614 


47881 




«I 


54955 


51109 


53279 


52468 


51662 


50877 


50105 


493*7 


48602 


47869 




46 


54<I41 


54094 


53264 


58149 


51619 


50864 


SOD92 


4B334 


48690 


47857 




46 


5(pa7 


54080 


53850 


58136 


51636 


50861 


5O08O 


49382 


48577 


47845 






5491 S 




53^36 


52422 


51623 


60938 




49309 


48S6S 


4783:( 




SI 


51899 


51052 


53223 


52409 


61610 


50885 




49297 


49553 


47881 




St 


548BI 


5103B 


53809 


52395 


61596 


50812 


500*1 


49894 


48510 






S3 


54870 


54081 


53195 


58382 


51.W3 


50799 


60029 


49278 


48528 


47797 




54 


54955 


51011 


53182 


6S3B8 


51570 


60786 


50016 


49869 


48516 


47786 




BS 


54K41 


5399T 


53168 


58355 


61557 


60773 


5U003 


49847 


48S03 


47778 




56 




5'J993 


53151 


583*2 


51541 




49991 


49231 


48191 


47760 




5T 


54913 


53969 


53111 


5*329 


51530 


60747 


49978 


49822 


48479 


47719 




5tl 


a4799 


53955 


53127 


58315 


51517 


50734 


49966 


*9809 


49107 


47736 




A9 


54784 


S3B11 


53113 


52301 


61501 


50721 


49958 


49197 


18454 


47784 




Proporlionnl Pan to- 




.8 


3 .4 




.6 




8 S 




umh. c,f "or B. 


1 


3 


4 5 


« 


B 


9 


12, 









^ 




74 Table X. Proportional LogBiitl 










1 Degree, oi 1 Hour. 


'^ 


0- 


]i 


8^ 


3" 


4« 


5" 


ei 


■jm 


ei 


gi 


loi 


11" 




47m 


469!) 4 


46888 


♦5M3 


44909 


Tme 


«573 


48980 


48176 


1184! 


41017 


40101 




1 


47700 


46988 


46876 


♦5598 


44998 


44885 


43568 


48909 


48866 


11638 


11007 


♦0391 






> 


47688 


46971 


*6i6S 


45570 


44987 


♦4811 


43551 


48898 


18865 


41681 


10997 


♦OSfll 






3 


47678 


46959 


46853 


4B659 


41975 


41803 


♦3540 


48887 


18844 


41611 


♦0986 


40371 






4 


4Tfi64 


46947 


46811 


♦5547 


♦♦864 


44191 


43680 


18877 


♦8834 


11600 


40976 


♦0361 






6 


47G5S 


46935 


46830 


45S36 


44953 


44180 


43518 


♦8986 


♦8883 


♦1590 


10966 


40350 






e 


47640 


46983 


46819 


45584 


44941 


44169 


43507 


♦8855 


♦8813 


41679 


♦0956 


4O340 






7 


4TG?B 


46911 


46806 


45513 


41830 


♦♦156 


43496 


48941 


♦8808 


41569 


♦0945 


4OS30 






B 


47616 


46899 


4619J 


45501 


41819 


44147 


43495 


♦8833 


48191 


41559 


♦0935 


403M 






fl 


47604 


46988 


4«193 


45490 


4180^ 


♦4136 


43471 


♦8883 


48181 


♦1548 


40914 


40310 






10 


47598 


46976 


46171 


45478 


4+796 


44185 


43163 


48818 




11538 


40B14 


♦O3O0 




11 


4T5B0 


46B84 


46160 


45167 


44795 


41114 


4S458 


48801 


♦8159 


♦1587 


40904 


40869 






IZ 


4T568 


46953 


46149 


45456 


44774 


44108 


434+] 


48790 


♦1149 


♦1517 


40894 


40879 






13 


475^6 


46940 


46137 




44768 


44091 


43131 


18780 


♦8138 


41506 


♦0SB3 


40169 






It 


47544 


46988 


46185 




♦4751 


44080 


43480 




48118 


41196 


40673 


40159 






li 


4753! 


46S17 






44740 


44069 


43109 


18759 


48117 


41485 


40963 


WHS 






16 


47580 


46805 


46108 


45410 


44789 


44058 


48398 


48747 


48106 


41476 


4095! 


40«39 






iT 


4750a 


46793 


46090 


45398 


♦4717 


44047 


43397 


48737 


48096 


41464 


409+1 


40986 






IS 


47498 


46791 


46078 


45397 


44706 


44036 


4M76 


48781 


48083 


41454 


40BS8 


40C18 






19 


47*94 


46789 


46067 


♦5375 


41695 


44085 


♦3365 


49715 


♦8075 


41443 


10981 


40808 








"46759 




45364, 




410I4 


43351 


48701 


48064 


♦1433 




40199 






47460 


46746 


46044 


♦5353 


44fi78 




43343 


48693 


43053 


414i3 


4O801 


40IBS 






Si 


47449 


4673* 


46038 


45341 


♦♦661 


4399! 


43338 


18683 


18043 


41418 


40791 


40178 






S3 


47436 


46788 


46080 


45330 


♦♦650 


43981 


13381 


48678 


♦8038 


41408 


♦0780 


40168 






u 


47434 


46710 


46009 


45319 


♦4639 


43969 


♦3310 


48661 


4808! 


41391 


♦0770 


40157 








41413 


46699 


45997 


45307 


44687 


13959 


43300 


48651 


♦8011 


41391 


40780 








te 




46GB7 


45B86 


4589S 




43947 


43889 


48S40 


♦8000 


41370 


407+9 


40137 






!T 






45974 


15894 


♦4«05 


43938 


43878 


♦8689 


♦1990 


41360 


40739 


40187 






!H 


4737( 


46663 


45968 


45873 


44594 


♦3W5 


43867 


♦8618 


41979 


41350 


40789 


40117 






£9 


473G4 


46658 


45931 


4S861 


44.583 


13914 


43856 


♦8609 


41969 


♦1339 


407I9 


401 07 




TO 


4735* 


iesu 


45939 


45850 


4*571 




43345 


TW57 


41958 


♦1389 


407 00 






31 


473W 


466g8 


459*8 


45838 


44560 


13898 


43834 


18596 


41918 


♦1319 


40699 


4008 7 








473S8 




45916 


45887 


44519 


13891 


43883 


18575 


41937 


♦ 1308 


10688 


40076 








47316 


46604 


45905 


4SS16 


44538 


♦3970 


43818 


♦8565 


41987 


11198 


40679 


40066 






3t 


47304 


46593 


45893 


45804 


4M86 


♦3869 


43808 


♦8551 


41916 


41887 


10667 


♦O056 






35 


47»9i 


465B1 


♦5891 


♦5193 


♦♦515 


43918 


♦3191 


48S43 


41905 


♦1877 


♦0657 








S6 


47890 


4656S 


45870 


45188 


♦♦504 


43837 


43190 


48533 


41895 


♦1866 


10617 


4003( 






37 


47S6B 


46557 


♦58 SB 


45170 


44493 


43886 


43169 


48588 


11884 


♦1856 


10637 


WOW 






38 


47g56 


46546 


45947 


4.5159 


♦♦498 


13815 


43158 


48511 


♦1871 


♦1846 


40686 


tans 






H9 


47S14 


46534 


4!83A 


45117 


44470 


43801 


43147 


48500 


41863 


41835 


40616 


4U0M 






4m§ 




4588' 


45136 


41459 


13793 


43136 


48490 


Tissa 


41185 


♦0606 


Sm 






4;S!0 


46510 




45185 


♦1448 


13788 


♦3186 


48479 


♦194! 


41114 


40596 


SD9R5 








47808 


46499 


45801 


15113 


44437 


43771 


♦3115 


48109 


4183! 


41804 


♦0565 


3»75 






43 


47196 


46487 




♦5108 


44486 


43760 


♦3104 




♦1811 


41194 


♦0575 


3S9G5 








47185 






45091 


44411 


13749 


43093 


4811: 


41811 


41183 


40565 


30955 






45 


47173 




4576S 


45079 


44403 


13739 


43088 


48136 


41900 


♦1173 


4055S 


30945 






46 


47 J 61 


4645! 


♦5754 


45069 


44398 


13787 


43071 


18486 


41789 


41168 


40S++ 


3S935 






47 


47149 


46440 


45743 


45057 


4439 


♦3716 


43O60 


48415 


41779 


♦1168 


40534 


39915 






4S 


47137 


46489 


♦573 


450*5 


14370 


43705 


43050 


48101 


41768 


♦lilt 


40684 


3Mtfi 






49 


471*5 


46417 


45780 


45034 


44359 


♦3694 


43039 


41394 




♦IISI 


40514 


3990« 






SO 


47113 


464415 


4S70H 


♦Si«8 


44347 


♦3683 


43088 


18383 


♦1747 


♦1181 


10503 


aSBM 






SI 


4710 


46393 


45697 


45011 


44336 


♦3678 


13017 


1837! 


♦1737 


41111 


10493 


3MB5 






M 


4708S 


463B! 


45695 


♦5OO0 


♦4385 


♦3661 


43006 


18368 


41786 


♦1100 


10493 


3S974 




■ 


53 


47077 


46370 


45674 


♦4988 


44314 


♦3650 


48995 


18351 


41716 


♦!O90 


40473 


3VM4 






54 


470G6 


46358 


1566! 


♦4977 


♦4303 


43639 


48985 


48310 


41705 


41O80 


10463 


3B9M 






55 


47054 


4634b 


♦565 


♦4966 


44891 


♦3688 


48974 


48330 


41695 


41069 


40458 


39S44 






50 


470ti! 


46335 


45639 


♦4955 


448M 


43617 


48963 


48319 


41681 


41059 


♦044! 


3M34 






57 


47O30 


46383 


45688 


♦4943 


448SS 


43606 


11958 


48308 


4167 


41018 


4013! 


39884 




1 




47019 


4631 


45616 


44938 


44858 


43595 


18941 


48198 


41683 


♦1039 


4011! 


3P9I4 






59 


47006 


46300 


15605 


44981 


4484 


43584 


41931 


41187 


41663 


+10B8 


40418 


SBMH 






.1 .1 .3 ♦ .5 .6 .1 






nfcrs. 


1 8 3. 4 5 T 9 9 It 








d 



Proportional Logarithms. Table X. 75 


1 Degree, or 1 Hour. 


It 

8 



1 

2 
3 

4 
5 
6 
7 
8 
9 


18m 

39794 
39784 
39774 
39764 
39754 
39744 
39734 
39724 
39714 
39704 


13^ 


14^ 


15^ 


Iff^ 


17°» 


18i 


19"^ 


20^ 


2lm 


22a 


2S» 


39195 
39185 
39175 
39165 
39155 
39145 
39136 
39126 
39116 
39106 


38604 
38594 
38585 
38575 
38565 
38555 
38545 
38536 
38526 
38516 


38021 
38011 
38002 
37992 
37983 
37973 
37963 
37954 
37944 
37934 


37446 
37436 
37427 
37417 
37408 
37398 
37389 
37379 
37370 
37360 


36878 
36869 
36859 
36850 
36841 
36831 
36822 
36812 
36803 
86794 


36318 
36309 
36299 
36290 
36281 
36271 
36262 
36253 
36244 
86234 


35765 
35755 
35746 
35737 
35728 
35719 
35710 
35700 
35691 
35682 


35218 
35209 
35200 
35191 
35182 
35173 
35164 
35155 
35146 
35137 


34679 
34670 
34661 
34652 
34643 
34634 
34625 
34616 
34607 
34598 


34146 
34137 
34128 
34119 
34111 
34102 
34093 
34084 
34075 
34066 


33619 
33611 
83602 
33593 
33585 
33576 
33567 
33558 
33550 
33541 


10 
11 
12 
13 
14 
15 
16 
17 
18 
19 

20 
21 
22 
23 
24 
25 
26 
27 
28 
29 

30 
31 
32 
33 
34 
35 
3() 
37 
38 
39 

40 
«l 
42 
43 
44 
45 
46 
4? 
48 
49 

50 
51 
52 
53 
54 
55 
66 
57 
5H 
59 


39i>94 
39684 
39674 
39664 
39653 
39643 
39633 
39623 
39613 
39603 


39096 
39086 
39076 
39066 
39056 
39046 
39037 
39027 
39017 
39007 


38506 
38497 
38487 
38477 
38467 
38458 
38448 
38438 
38428 
38419 


37925 
37915 
37905 
37896 
37886 
37877 
37867 
37857 
37848 
37838 


37351 
37341 
37332 
37322 
37313 
37303 
37294 
37284 
37275 
37265 


36784 
36775 
36766 
36756 
36747 
36737 
36728 
36719 
36709 
36700 


36225 
36216 
36207 
36197 
36188 
36179 
36170 
36160 
36151 
36142 


35673 
35664 
35655 
35646 
35636 
35627 
35618 
35609 
35600 
35591 


35128 
35119 
35110 
35101 
35092 
35083 
35074 
35065 
35056 
35047 

35038 
35029 
35020 
35011 
35002 
34993 
3^4984 
34975 
34966 
34957 


34589 
34581 
34572 
34563 
34554 
34545 
34536 
34527 
34518 
34509 


34058 
34049 
34040 
34031 
34022 
34014 
34005 
33996 
33987 
33978 


33532 
33524 
33515 
33506 
33498 
33489 
33480 
33471 
33463 
33454 


39o93 
3958? 
39573 
39563 
39553 
39543 
3953:^ 
3952J'. 
39513 
39503 


38997 
38987 
38977 
38968 
38958 
38948 
38938 
38928 
38918 
38908 


38409 
38399 
38389 
38380 
38370 
38360 
38351 
38341 
38331 
38321 


37829 
37819 
37809 
37800 
37790 
37791 
37771 
37761 
37752 
37742 


37256 
37246 
37237 
37227 
37218 
37208 
37199 
37189 
37180 
37171 


36691 
36681 
36672 
36663 
36653 
36644 
36634 
36625 
36616 


36133 
36123 
36114 
36105 
36096 
36086 
36077 
36068 
36059 
36050 


35582 
35573 
35563 
3555i 
35545 
35536 
35527 
35518 
35509 
35500 


34500 
34491 
34483 
34174 
34465 
34456 
34447 
34438 

34420 


33970 
33961 
33952 
33943 
33935 
33926 
33917 
33908 
33899 
33891 


334 1>5 
33437 
3:U28 
33419 
33411 
33402 
33393 
33385 
33376 
33367 


3949.S 
3948^^ 
3947:' 
39*64 
39454 
39444 
39434 
39424 
39414 
39404 


38899 
38889 
38879 
38869 
38859 
38849 
38839 
38830 
38820 
38810 


38312 

38302 
38292 
38282 
38273 
38263 
38253 
38244 
38234 
38224 


37733 
37723 
37713 
37704 
37694 
37685 
37675 
37665 
37656 
37646 


37161 
37152 
37142 
37133 
37123 
37114 
37104 
37095 
37085 
37076 


36597 
36588 
36578 
36569 
36560 
36550 
36541 
36532 
36522 
36513 


36040 
36031 
36022 
36013 
36003 
35994 
35985 
35976 
35967 
35957 


35491 
35481 
35472 
35i63 
35454 
35U5 
35436 
85427 
85418 
35409 


31948 
34939 
34930 
34921 
34912 
34903 
34894 
34885 
34876 
34867 


34411 
34403 
34394 
34385 
34376 
34367 
34358 
34349 
34340 
34332 


33882 
33873 
33864 
33856 
33847 
33838 
33829 
&S820 
33812 
33803 


33359 
33350 
33^11 
33333 
33324 
33315 
33307 
33298 
33289 
33281 


39394 
39384 
39374 
39364 
39354 
39344 
39334 
393H 
39314 
39301 


38^00 
38790 
38781 
38771 
38761 
38751 
38741 
38731 
38722 
38712 


38215 
38205 
38195 
38186 
38176 
38166 
38156 
38147 
38137 
38127 


37637 
37627 
37618 
37608 
37599 
37589 
37579 
37570 
37560 
37551 


37067 
87057 
37048 
37038 
87029 
37019 
37010 
37001 
36991 
36982 


36504 
36494 
36485 
36476 
36467 
36457 
36448 
36439 
36429 
36420 


35948 
35939 
35930 
35921 
35911 
35902 
35893 
35884 
3.5875 
35865 


35400 
35391 
35381 
35372 
35363 
35354 
35345 
35336 
35327 
35318 


34858 
34849 
34840 
34631 
Si822 
34813 
34804 
34795 
34786 
34777 


34323 
34314 
34305 
34296 
34287 
34278 
34270 
34261 
34252 
34243 


33794 
33785 
33777 
33768 
33759 
33750 
33742 
33733 
33724 
33715 


33272 
33263 
33255 
33246 
33237 
33229 
33220 
38211 
33203 
33194 


39291 
39284 
39274 
39264 
39254 
39245 
3923.') 
39225 
39215 
39205 


38702 
38692 
38682 
38673 
38663 
38653 
38643 
38633 
38624 
38614 


38118 
38108 
38098 
38089 
38079 
38069 
38060 
38050 
38040 
38031 


37541 
37532 
37522 
37513 
37503 
37494 
37484 
37474 
37465 
37455 


36972 
36963 
36953 
36944 
36935 
36925 
36916 
36906 
36897 
36888 


36411 
36401 
36392 
36383 
36374 
36364 
36355 
36346 
36336 
36327 


35856 
35847 
35838 
35829 
35820 
35810 
35801 
35792 
35783 
35774 


35309 
35300 
35291 
35282 
35273 
35264 
35254 
35245 
35236 
35227 


34768 
34759 
34750 
34741 
34732 
34723 
34715 
34706 
34697 
34688 


34234 

34225 
34217 
34208 
34199 
34190 
34181 
34172 
3416* 
34155 


33707 
33698 
33689 
33681 
33672 
33663 
33654 
33646 
33637 
33628 


33186 
33177 
33168 
33160 
33151 
33142 
33134 
33125 
33117 
33108 


i'roportional I'art to tenths 
of " or a. 


.1 .2 .3 .4 .5 .6 .7 .8 .91 
*1« 3 44/567 81 



r 






^™ 










70 Table X. Proportional LogorithniB. 




1 Dtgm, ot I Hont. 


■^ 


.4^ 


jji 


26" 


27° 


28- 


29^1 


SO- 


31^ 


3*" 


^ 


34^ 


ssi 


~0 


^3559 


32585 


32077 


31575 


31079 


30588 


SO 103 


296*3 


29148 


28679 


28214 


81 155 






33091 


3a,'.77 


32(169 


31587 


31071 


30580 


30095 


*9615 


89141 


28671 


29*07 


87741 








33082 


32.568 


320B1 


31559 


31063 


30572 


300S7 


*9607 


S9I33 


28663 


28199 


27710 






3 




32560 


32059 


31550 


31054 


30564 


3O079 


29599 


*9I25 


28656 


88191 


81732 






4 




3*551 


32014 


31542 


31046 


30556 


3O071 


29591 


29117 


88648 


28184 


87724 






5 




32543 


32035 


31534 


3103H 


30518 


30063 


89583 


29109 


88640 


28176 


21717 






6 


330 IB 




320*7 


31526 


31030 


30539 


30055 


*9575 


29101 




28168 


87709 






1 




325*6 


32019 


31517 


31021 


30531 


30047 


29567 


29093 


28685 


26161 


27708 






8 




3*517 


32010 


31S09 


31013 


305*3 


30039 




29086 


28317 


2S153 


81894 






9 


330*8 


3*509 


32002 


31501 


31005 


30515 


30031 


29552 


29078 


28609 


89145 


87686 




10 


335T3 


32S00 


31993 


3U92 


30997 


30507 


30023 




29070 


88601 


28138 


87619 






33005 


32492 


31985 


31484 


30989 


30499 


3O015 




29068 


*8593 


28130 


87611 






IS 


32996 


32483 


31977 


SI 476 


30980 


30491 


30007 




29054 


28586 


28132 


87664 






13 


32987 


32475 


31968 


31467 


30972 


30183 


29999 




29046 


88578 


88114 


21658 






14 
Ifi 


32979 
32970 


32466 
3*458 


31B60 

31951 


3145! 


30964 


30475 
30466 


29991 
29983 


2951* 

89504 


29038 
29031 


88570 
8856* 


28107 
29099 


87648 
87641 














16 


3296? 


32MB 


31943 


31442 


30918 


30458 


29975 


294B6 


29023 


*8555, 


88091 


27633 






IJ 


32953 


32441 


31935 


31434 


30939 


30450 


29967 


89488 


29015 


88S4T 


28084 


27626 






IB 


32944 


3*432 


31926 


31426 


30931 


30442 


29958 


89480 


29007 


*853B 


*907fl 


87G1B 






19 


32936 


32424 


31918 


31418 


30923 


30434 


29950 


29472 


28999 


88531 


*80fl9 


21610 




20 


3892T 


32415 


31909 


31409 


30915 


30426 


2994? 


89164 


^8991 


28524 


88061 


87603 




n 


32919 


32407 


31901 


31401 


30907 


30418 


28934 


29456 


28984 


8851b 


te053 


ST596 






it 


32910 


32398 


31893 


31393 


30898 


30410 


29926 


89448 


28976 


88508 


88045 


27589 






23 


32902 


32390 


31884 


31384 


30B90 


30102 


29918 


89441 


28968 


28500 


28038 


87580 






H 


38893 


32381 


31876 


31376 


30882 


30393 


39910 


89433 


28960 


88493 


88030 


27S78 






25 


32884 


32373 


31667 


31368 


30874 




2990* 


89425 


28952 


28485 


28028 


17565 






26 


32876 


3*365 


31859 


31360 


30866 


30377 


29894 


20417 


28944 


28477 


88015 


27567 






!7 


31867 


32350 


31851 


31351 


30857 


3036B 




8940S 


28937 


18469 


88007 


87850 






SH 


32859 


32348 


31842 


31343 


30849 


30361 


29871 


S9401 


28929 


28462 


27999 


81518 






ta 


32850 


3233S 


31834 


31335 


30841 


30353 


29870 


29383 


28921 


8U454 


87892 


87534 




30 


32842 


32331 


"ITele 


3l32b 


30833 


30345 


*986* 


^9385 


28913 


28446 


27984 


815*7 




31 


32833 


32322 


31817 


3IS1S 


308*5 


30337 


S9854 


29317 


28905 


3843B 


87976 


8751 9 






31 


32824 


32314 


31809 


31310 


30817 


30329 


29846 


89369 


28897 


88431 


87869 


21513 






33 


32816 


32305 


31801 


31302 


30808 


30321 


S9838 


29361 


*8890 


88483 


21961 


87504 






34 


32807 


32*97 


31792 


31293 


30800 


30313 


29B30 


2S354 


28882 


88415 


21953 


87497 






3i 


32T99 


32288 


S1TB4 


31285 


30782 


30305 


298*2 


89346 


88874 


28 W7 


27946 


27489 






36 


32190 


32*80 


31776 


31277 


30TB4 


30296 


29814 


89338 


28866 


28400 


27938 


87481 






37 


32T82 


3*271 


31767 


31269 


30776 


30288 


29806 


29330 


28858 


88392 




87414 






38 


32TT3 


32*63 


31759 


31260 


30768 


30280 


2S79B 


80382 


28851 


28384 


319*3 


27466 






39 


32765 


3*255 


31750 


31*52 


30759 


30272 


29790 


89314 


28843 


28376 


81915 


8H58 




JO 


32756 


32*46 


31742 


31*44 


3076T 


30264 


"*97B* 


29306 


28835 


28369 


27908 
27900 


87451 




41 


32747 


32*38 


31T34. 


31*36 


30743 


3025G 




29298 


298*7 


28361 


21444 






iS 


32739 


32*2B 


31725 


31227 


30735 


30248 


29767 


89290 


28819 


88353 


87882 


81436 






43 


3*730 


3*221 


31717 


31*19 


30727 


30240 


29759 


89282 




88346 


27895 27429 






44 


327« 


3221* 


31700 


31*11 


30719 


30232 


29151 


28275 




88338 


27977 2T481 






45 


3*713 


32204 


31700 


31203 


30710 


30224 


89743 


29*67 


28796 


28330 


87869 


87418 






46 


32705 


32195 


3109* 


31194 


3070* 


30216 


29735 


29259 


88788 


8338* 


27962 


3T40e 








3*696 


32187 


31684 


31186 


30694 


30*08 


29727 


29*51 


28780 


28315 


27954 








48 


3S68fl 


32178 


31675 


31178 


30686 


3O200 


89719 


89243 


2977* 


28307 


87846 


11 






49 


32679 


32170 


31667 


31170 


30678 


30192 


89711 


89235 


28765 


29299 


27830 




60 


32671 


3210* 


31659 


31161 


30670 


:J0184 


*9703| 29227 


28757 


28*9* 


87931 87376 




5t 


32662 


32153 


31650 


31153 


30662 


30175 


29695! 29*19 


28749 


28284 


81884 17368 






68 


3*6»4 


32145 


31642 


31145 


30653 


30167 


29687 


29*11 


28741 


!8!76 


81816 21360 






53 


3*645 


3*136 


31634 


31137 


30645 


30159 


29679 


28204 


28733 


28268 


27809 87353 






£4 


3(636 


32128 


316*5 


311*8 


30637 


30151 


29671 


89196 


28726 


88861 


2790I 


27:1*5 






£5 


S262S 


321*0 


31617 


31120 


306*9 


30143 


89663 


29188 


28118 


88)53 


21793 


21338 






M 


32619 


S2I11 


3I6U9 


31118 


3062 1 


30135 


29G55 


29180 


28110 


28245 


*7795 


1T33D 
873S 








3«Sll 


32103 


31600 


31104 


30013 


30127 


29647 


29172 


28101; 


28*38 


27718 






SS 


3260* 


3*094 


31592 


31095 


30605 


30119 


89639 


891S4 




28230 


2T170 


87316 






as 


a*M4 


3*086 


31584 


31087 


ao.'^as 


30111 


29631 


89156 


!8687 


38282 


27763 


3T3C(1 




Fr«portion.i Part lo i*oihs 


.1 2 .3 .4 .5 .8 .7 . 


.9 






1 2 * -3 4 5 6 fl 7 




L. 




^ 




■ 



Proportional Logarithms. Tabie X. 77 


1 Degree, or 1 Hour. 


// 

8 


36^ 


37*i 


38« 


39« 


40« 


4li 


42^ 


43^ 


44m 


45« 


46« 


47« 




1 

2 

3 

4 
5 
6 
7 
8 
9 


27306 
27293 
27285 
27278 
27270 
27262 
27255 
27217 
27240 
27232 


26850 
26843 
26H35 
26828 
26820 
26813 
26805 
26798 
26790 
26783 


26405 
26397 
26390 
26382 
26375 
26368 
26360 
26353 
26346 
26338 


25964 
25956 
25949 
25942 
25934 
25927 
25920 
25913 
25905 
25898 


25527 
25520 
25513 
25506 
25498 
25491 
25484 
25477 
25460 
25462 


25095 
25088 
25081 
25074 
25066 
25059 
25052 
25045 
25038 
2.5031 


24667 
24660 
24653 
24646 
24639 
24632 
24625 
24618 
24610 
24603 


24244 

24237 
24229 
24222 
24215 
2420ft 
24201 
24194 
24187 
24180 


2.3824 
23817 
23810 
23803 
23796 
23789 
23782 
23775 
23768 
23761 


23408 
23401 
23395 
23388 
23381 
23374 
23.367 
23360 
23353 
23.346 


22997 
22990 
22983 
22976 
22969 
22963 
22956 
22949 
22912 
22935 


22589 
22582 
22575 
22569 
22562 
22555 
22548 
22542 
22535 
22528 


10 
11 
12 
13 
14 
15 
16 
17 
18 
19 


27225 
27217 
27210 
27202 
27195 
27187 
27180 
27172 
27165 
27157 


26776 
2fi768 
26761 
26753 
26746 
26738 
26731 
26723 
26716 
26709 


26331 
26323 
26316 
26309 
26301 
26294 
26287 
26279 
26271 
26265 


25891 
25883 
25876 
25869 
25861 
25854 
25847 
25840 
25832 
25825 


25455 
25 U8 
25440 
25433 
25426 
25419 
25412 
25404 
25397 
25390 


25024 
25016 
25009 
25002 
24995 
24988 
24981 
24973 
24966 
24959 


21596 
24589 
24582 
24575 
24568 
24561 
24554 
24547 
24540 
24533 


24173 
24166 
24159 
24152 
24145 
24138 
24131 
24124 
24117 
24110 


23754 
23747 
23740 
23734 
23727 
23720 
23713 
23706 
23699 
23692 


23339 
23333 
23.326 
23319 
23312 
23305 
23298 
23291 
23284 
23278 


22928 
22922 
22915 
22908 
22901 
22894 
22888 
22881 
22874 
22867 


22521 

22515 
22508 
22501 
22494 
22488 
22481 
22474 
22467 
22461 


20 
21 
22 
23 
24 
25 
26 
27 
28 
29 


27150 
27142 
27135 
27127 
27120 
27112 
27105 
27097 
27090 
27082 


26701 
26694 
26686 
26679 
26671 
26664 
26656 
26649 
26642 
26634 


26257 
26250 
26242 
26235 
26228 
26220 
26213 
26206 
26198 
26191 


25818 
25810 
25803 
25796 
25789 
25781 
25774 
25767 
25759 
25752 


25383 
25376 
25368 
25361 
25354 
25347 
25339 
25332 
25325 
25318 


24952 
24945 
24938 
24931 
24923 
24916 
24909 
24902 
24895 
24888 


24526 
24518 
24511 
24.504 
24497 
24490 
24483 
2U76 
24469 
24462 


24103 
24096 
24089 
24082 
24075 
24068 
24061 
24054 
24047 
24040 


2.S685 
23678 
23671 
23664 
23657 
23650 
23643 
23636 
23629 
23623 


23271 
23264 
23257 
23250 
23243 
23236 
23229 
23223 
23216 
23209 


22860 
22854 
22847 
22840 
22833 
22826 
22819 
22813 
22806 
22799 


22454 
22U7 
22440 
22434 
22427 
22420 
22413 
22407 
22400 
22393 


SO 
31 
32 
33 
34 
35 
36 
37 
38 
39 


27075 
2706? 
27060 
27052 
27045 
27037 
27030 
27022 
27015 
27007 


26627 
26619 
26612 
26605 
26597 
26590 
26582 
26575 
26567 
26560 


26184 
26176 
26169 
26162 
26154 
26147 
26140 
26132 
26125 
26118 


25745 
25738 
26730 
25723 
25716 
25709 
25701 
25694 
25687 
25680 


25311 
25303 
25296 
25289 
25282 
25275 
25267 
25260 
25253 
25246 


24881 
24874 
24866 
24859 
24852 
24845 
24838 
24831 
24824 
24817 


24455 
»4448 
24441 
24434 
24427 
24420 
24413 
24405 
24398 
24391 


24033 
24026 
24019 
24012 
24005 
23998 
23991 
23984 
23977 
23970 


23616 
23609 
23602 
23595 
23588 
23581 
23574 
23567 
23560 
23553 


23202 
23195 
23188 
23181 
23175 
23168 
23161 
23154 
23147 
23140 


22792 
22785 
22779 
22772 
22765 
22758 
22752 
22745 
22738 
22731 


22386 
22380 
22373 
22366 
22359 
22353 
22346 
22339 
22333 
22326 


40 
41 
42 
43 
44 
45 
46 
47 
46 
49 


27000 
26992 
26985 
26977 
26970 
26962 
26955 
26947 
26940 
26932 


26553 
26545 
26538 
26530 
26523 
26516 
26508 
26501 
26493 
26486 


26110 
26103 
26096 
26088 
26081 
26074 
26066 
26059 
26052 
260U 


25672 
2.5665 
25658 
25650 
25643 
25636 
25629 
25621 
25614 
25607 


25239 
25231 
25224 
25217 
25210 
25203 
25196 
25188 
25181 
25174 


24809 
24802 
24795 
24788 
24781 
24774 
24767 
24760 
24752 
24745 


24384 
24377 
24370 
24363 
24356 
24349 
24342 
24335 
24328 
24321 


23963 
23956 
23949 
23942 
23935 
23928 
23921 
23914 
23908 
23901 


23546 
23539 
23533 
23526 
23519 
23512 
23505 
23498 
23491 
23484 


23133 
23127 
23120 
23113 
23106 
23099 
23092 
23086 
23079 
23072 


22724 
22718 
22711 
22704 
22697 
22690 
22684 
22677 
22670 
22663 


22319 
22312 
22306 
22299 
2229? 
222S6 
22279 
22272 
22265 
22259 


50 
51 
52 
53 
54 
55 
56 
57 
58 
59 


26925 
26917 
26910 
26902 
26895 
26887 
26880 
26872 
26865 
26858 


26479 
26471 
26464 
26456 
26449 
26442 
26434 
26427 
26419 
26412 


26037 
26030 
26022 
26015 
26008 
26000 
25993 
25986 
25978 
25971 


25600 
25592 
25585 
25578 
25571 
25563 
25556 
25549 
25542 
25534 


25167 
25160 
25152 
25145 
25138 
25131 
25124 
25117 
25109 
25102 


24738 
24731 
24724 
24717 
24710 
24703 
24696 
24689 
24681 
24674 


24314 
24307 
24300 
24293 
24286 
24279 
24272 
24265 
24258 
24251 


23894 
23887 
23880 
23873 
23866 
23859 
23852 
23845 
23838 
23831 


23477 
23470 
23464 
234.57 
23450 
23443 
23436 
23429 
23422 
23415 


23065 
23058 
23051 
23044 
23038 
23031 
23024 
23017 
23010 
23004 


22657 
22650 
22643 
22636 
22629 
2262.3 
22616 
22609 
22602 
22596 


22245 
22239 
22232 
22225 
22218 
22212 
22205 
22198 
22192 


Proportional Part to tenths 
of "org. 


.1 .2 .3 .4 .5 .6 .7 .8 .91 
1 1 2 3 3 4 5 6 6| 



^■^ 




■ 






I 




1 Dtgne, or 1 Hour. ^ 








; 


4-m 


49m SO"" 


Sim 


580 


SS'" 


61^ 


BS" 


56^ 


57-^ 


59i 


59i 




"o 


831B5 


81785 21388 






M8i9 


19837 


19457 


19081 


19709 


T9339 


17973 




1 


a«l7B 


21778 21381 


80988 


80599 


20213 


19830 


19141 




19708 


1933S 


IT9G6 






t 


88171 


21771 81375 


80988 


20593 


8O807 


19H24 


19U5 


19069 


19696 


18387 


17960 






3 


mas 


81765 SlSfiS 


80976 


80596 


80800 


IBB IB 


19439 


190S3 




1S391 


17954 






i 


S315S 


8I7S8 8l3fi8 


20969 


80580 


80194 


19811 


19138 


19056 


1868* 


18315 


17949 








18151 


81751 81355 


£0968 


80573 


20187 


19805 


19186 


19050 


19679 


1930S 


179*8 










21745 2134!) 
81738 21348 


80956 
80949 


20567 
20580 


20181 
80175 


19799 
19198 


19120 
10413 


19044 
19038 


18678 
18665 


1B302 

18896 


179S6 
17030 






^ 


881St« 






8 


88131 


81732 81335 


80943 


80554 


80168 


19786 


19107 


19038 


IB659 


18980 


179*4 









52 185 


21785 81329 


80B36 


20S47 


20162 


19780 


19401 


19085 


18653 


19SB* 


17919 




JO 


88118 


8171B 813*8 


"20S30 


80541 


2UI55 


19773 


19305 


19010 


186*7 


19*7B 


17»!2 




11 


iilU 


21718 21316 


20983 


80534 


20149 


19767 


19398 


19013 


18641 


19878 


17906 






I! 


88105 


81705 21309 


80917 


20588 


20143 


19761 


10392 


19007 


19634 


18266 


I790n 






13 


88098 


81699 21303 


80910 


80588 


20136 


19754 


10376 


19000 


18688 


18859 


1T894 






It 


22091 


21698 2l89r 


80904 


80515 


80130 


19749 


19369 


18994 


18682 


18851 


1T987 






15 


88084 


21685 81889 


80H97 


8050B 


80183 


107*8 


19363 


1998B 


1B616 


19*47 


17991 






IG 


S8I)78 


21678 81283 


80SBI 


20508 


20117 


19735 


10357 


18982 


18810 


182*1 


179TJI 






n 


8SU71 


81678 2I27B 


80881 


20 196 


80111 


19729 


19351 


18970 


1B604 


19835 


17969 






IB 


8806* 


81665 81270 


80879 


80189 


80101 


19783 


19344 


18969 


18507 


19889 


17863 






19 


92058 


81fi59 81263 


20871 


80493 


80098 


1971b 


19339 


18963 


18591 


1B8S3; 17857 




80 


88051 


81658 21857 


80865 


80476 




19710 


19338 


18957 


1958-.| 18217, 17Hil 




81 


SS044 


21645 


81850 




20470 


80085 


19704 


19386 


19951 


19570 18810 1JB4J5 






it 


28038 


81639 


81843 


20858 


20464 


80079 


19697 


19319 


18944 


18573 18801 


17939 






S3 


88031 


81638 


81837 


80845 


80457 


20072 


19691 


19313 


IB93B 


18567 18198 


I7BW 






H 


88024 


81626 


81230 


80839 


20451 


80066 


19686 


19307 


1B938 


1B56D 18198 


17927 






S5 




81619 


21884 


20838 




80060 


19679 


19300 


18986 


18554 19196 


17881 








88011 


81618 


21217 


2082b 


80438 




19678 


19894 


18920 


185*9 19190 


17915 






87 


88004 


81606 


81811 


20S19 


80131 


80047 


19666 


19288 


18913 


185421 19174 


1T80S 






m 


8108B 


81599 


81204 


80S13 


8048a 


80040 


19659 


19882 


19007 


18536; 19168117903 






89 


81991 


81598 


21198 


20806 


80118 


80034 


19653 


10873 


1 8901 


19530 19168 


177M 




30 


81984 


81596 


81191 


20900 


80418 


80O2B 


19847 


19860 


lHa95 


19583' 1IS15S, 11790 




31 


SI979 


81579 


8118* 


80793 


80*06 


80081 


19640 


19803 


18889 


1851? 18I4»| 17784 






31 


81971 


81573 


21178 


20787 


80399 


8O015 


19634 


19257 


18992 


19511] 181^ ITT7S 






33 


81964 


815G6 


81171 


80780 


80393 


80009 


19629 


19850 


18876 


18,505, 18137 17778 






31 


81958 


81559 


2116. 


20774 


20396 


80008 


19621 


198** 


18e7( 


1949S, 18131 17766 






35 


8195 


21553 


81158 


80767 


80380 


199B6 


196 IS 


19239 


18864 


1B193 18185 177M 






36 


21944 


21616 


21152 


80761 


80373 


19989 


19609 


10831 


18857 


18487 18119 17754 






3J 


81938 


21510 


8II4j 


20754 


80367 


19983 


19608 


19225 


18851 


IBIBO 18113 


177W 








21931 


815^3 


21139 


80748 


8036 


19977 


19596 


19819 


18945 


lBl74i 18107 


177*2 






39 


21084 


21526 


81132 


20741 


20351 


19970 


19590 


19813 


18939 


191681 18100 


17736 




♦0 


81918 


81580 


8II86 


80735 


20348 


19964 


T9584 


19206 


"18833 


1B162| 18094 


1773H 




41 


8191 


21513 


8III8 


80788 


80341 


19959 


19577 


10800 


1B986 


18156 18089 


17781 










81S07 


81112 


80782 


80335 


1995 


19571 


10194 


19980 


16150| 18082 


117\)> 






♦3 


81891 


21500 


81106 


20715 


20328 


19945 


19565 


19198 


19914 


181-13 


18076 


17718 






4t 


8 ISO 


81493 


21099 


2O709 


20322 




19559 


1919 


18908 


18137 


18071 


1770* 






*5 


2198 


81497 


81093 


20702 


80316 


19932 


19558 


19175 


19802 


18131 


18084 


1771)0 






ie 


81878 


21480 


2108r 


8O09r 


80309 


19986 


10548 


19169 


lb795 


18425 


18059 


176S4 






47 


8197 


21474 


2108(1 


20690 


80303 


19919 


10539 


19163 


18789 


JB*19 


18058 


1768M 






48 


81804 


81467 


21073 


80683 


80296 


19913 


19533 


19156 


1B783 


18113 


I804r> 


IT688 






49 


81858 


81460 


2106 


20677 


80890 


1990 


19587 


19150 


18777 


18107 


18010 


I767B 




50 


818S 


81454 


8106(1 


20670 


80894 


1990(1 


T9580 


101*4 


1BJ7 


18400 




nnaa 




51 


81844 


814*7 


8105 


80664 


80277 


19894 


19511 


19139 


1876* 


1839* 


18081 


17663 






5! 


81638 


81441 


2104 


80S57 


8087 


19888 


19508 


1913 


18759 


18389 


18021 


17657 






53 


2183 


81434 


8104 


8065 


80864 


1988 


19502 


19123 


19752 


18388 


1801 fi 


17651 






5* 


8188 


81487 


8103 


806*4 


20858 


19875 


19495 


19119 


19746 


18376 


18009 


17 Sit 






5i 


2l81t^ 


81481 


2102 


80639 


8086 




19489 


19113 


18740 


19370 


18003 


I7C8S 






5fi 


8181 


21414 


8102 


8063 


808«a 


198681 19483 


19106 


18733 


1836* 


17997 


1J6S3 






57 


8I80£ 


21408 


8101 


80685 


80839 


IB856 19476 


19100 


18727 


18357 


17991 


17l!«7 






58 


81798 


81401 


8100 


20618 




IB849 1P470 


19091 


1878 


18:151 


17985 


176S1 






59 


21191181395 


8100 


20618 


80886 


199*31 i9M4 


19098 


18715 


18345 


17B79 


176(5 




ProponianBl I'art tu Ignths 


.1 .2 .3 .1 .5 .6 .i .8 .9 




of " or it 


1 1 2 3 3 4 5 e < 




L^ 


■ ^^^^^" 


B 











ftomnionsl Locflrithms. 


Taili 


X. 


« 












iDegna^oitHoim. 




s 


0" 


1" 


gi 


gii 


4i 


J» 


6" 


7^ 


M,™ 


lOil" 


11" 






11809 


17**9 


16891 


I6S37 


16185 


15936 


15490 


15117 


T4So6;T*?59 


Tirs! 


13800 




1 


11603 


17**3 


16885 


1G53I 


16179 


15830 


15484 


15141 


14801 14403 


111*7 


1379S 




! 


1T597 


17837 


16879 


I658S 


10173 


15985 


15479 


1513G 


11795' 11*57 


14188 


13789 




3 


17591 


17831 


16873 


165 19 


16168 


15819 


15473 


15130 


14789: 11151 


1*116 


13784 




4 


17585 


17885 


1686B 


1 65 13 


16168 


15813 




1518* 


1*791 14446 


141 1 1 


13779 




S 


17579 


17819 


1696! 


16i07 


10156 


15807 


15*61 


15118 


14778 14*40 


14105 


I37T3 






1TS73 


17*13 


16856 


16501 


16150 


16808 


15*50 


15113 


117781 14435 


11100 


13767 






17567 


17807 


I6S50 


16496 


161*4 


15796 


15*50 


15107 


11767, 11**9 


11091 


13761 




6 


175«l 


17801 


16B44 


16190 


1613B 


15790 


15*44 


15101 


147611 1**83 


HOBS 


13756 




S 


17555 


17i9i 


16838 


1618* 


16133 


1579* 


15439 


15096 


14755 


14118 

liiie 


140B3 


13750 






175*9 


171B9 


16838 


1647B 


16187 


15778 


15*33 


T5O90 


14750 


14077 


13745 




11 


l7S4:i 


IT 183 


16986 


16*7* 


101*1 


1S773 


15487 


1508* 


1171*; 1*407 


14078 


13739 




11 


17537 


17177 


16880 


16166 


10115 


15767 


15181 


15079 


14739 


14401 




13731 




13 


17531 


I717I 


lesu 


16460 


loioa 


israi 


15416 


15073 


14T33 


14395 


1100 1 


I378B 




li 


iTsas 


17165 


16808 


16454 


16103 


15755 




15067 


11787 


14390 


14055 


13783 




IS 


17519 


17159 


16808 


16449 


16098 


15749 


1540* 


15061 


1*788 


14394 


11019 


13717 






17513 


17153 


16796 


16443 


16098 


15741 


15399 


15056 


14716 


14379 




13712 




IT 


17507 


17147 


16791 


16437 


10086 


15738! 15393 


15050 


1*710 14373 




13706 




IB 


17501 


17141 


16 785 


16*31 


16080 


157381 15387 


ISOlt 


11705 Il:t07 


11033 






19 


17495 


17135 


16779 


10485 


16074 


15786 
15781 


15391 


1S039 


1*099 11308 


11087 


13095 




«u 


17+BU 


17189 


16773 


16119 


1606B 


15375 


150:i3 


116931 1*350 


110** 


13690 




SI 


1T4B3 


17183 


16767 


16413 


16063 


157131 15370 


15087 


116881 1*351 


14018 


13684 




a 


17477 


1711J 


18761 


16407 


10057 


15709115364 


15082 


11098 


1434S 


14011 


13879 




83 


17*71 




16755 


16108 


16051 


IS703 


15358 


15016 


14076 


1*339 


I40U5 


136 T3 






i7tes 


1710S 


167*9 


16396 


16015 


15697 


153S3 


15010 


14071 


14331 


IIOW 


13G68 






17*59 


17099 


167*3 


16390 




15698 


15347 


15005 


14005 


14389 


13994 


13088 




aa 


17453 


17093 


16737 


10394 


16034 


15686 


15311 


1*999 


146S9 


14383 


13988 


13857 




31 


17447 


17097 


16731 


1837B 


16088 


15690 


1S335 


14993 


14654 


11317 


13983 


13651 






17141 


I70BS 




16378 


16088 


tS6Tl 


IS33U 


11989 


14618 


14311 


13977 


136*6 




SB 


17435 


17076 


16781 


16366 


16016 


I56S9 


15381 


1498* 


14613 


14306 


13978 


13640 




30 




17070 


TTel* 


10361 


16010 


15663 


15318 


T4976 




T^OU 


13906 


13835 




31 


I74S3 


17064 


16708 


163SS 


16005 


150S7 


1S318 


14971 


11031 


14895 


13961 


13689 




3« 


17417 


17058 


16708 


1G3*» 


15999 


1S651 






14680 


14889 


13955 


13084 




33 


1741 1 


17058 


1G696 


16343 


15993 


15046 


15301 




146*0 




13950 


13819 




3i 


17*05 


170*6 


16690 


16337 


15987 


ISOiO 


15895 


1*951 


1*614 


11878 


13944 


13613 




35 


1T39B 


17040 


16094 


16331 


150B1 


15634 


15890 


149*8 


14609 


14872 


13939 


1360T 






17393 


1703* 


16678 


16385 


15975 


15688 


15891 


14948 


14603 


11867 


13933 


13602 




37 


17387 


17088 


16678 


16310 


1S970 


15083 


15878 


14937 


1*599 


14261 


13927 


13596 




38 


17381 


17088 


166G6 


1631* 


1S96* 


15617 


1S878 


11931 


14598 


14856 


13982 


13591 




39 


17375 


17016 


16660 


16308 


15958 


15011 


15867 


14985 


14586 


14850 


13916 


13585 




lU 


171169 


"nolo 


16655 


16308 






Tim 


14919 


1*591 


14844 


13M11 


13580 




tl 


17363 


17004 


16649 


16896 


15916 


15591 


1585S 


14914 


I457S 


14839 


13905 


13574 






17357 


1699B 


16043 


10890 


15941 


1553* 


15850 


14908 


U569 


14833 


13901 


13S69 




43 


173S1 


16998 


18637 


10884 


15935 


15598 


15*44 


1*908 


14561 


11888 


13994 


13563 




44 


173*5 


1698G 


16631 


16879 


15989 


15598 


15835 


148B7 


1*558 


14888 


13889 


13558 




45 


I733S 


I6fl9n 


10085 




15983 


I55T0 


15S3I 


14891 


11553 


14817 


13893 


13558 




46 


17333 


16974 


10019 


10867 


15917 




15887 


14985 


14517 


11811 


13978 


13547 






173i7 


16D6B 


10613 


10861 


1S918 


15555 


15*21 


14860 


14541 


11805 


13878 


13S41 




4B 


173gl 


13963 


16607 


16855 


15906 


155SS 


15315 


11971 


1*536 




13806 


13536 




49 


173 15 


16957 


16602 


10819 


15900 


I55S3 


15810 


14809 


1*530 


1*191 


13861 


13530 




ao 


17309 


"16951 


16590 


10843 


l58St 




laSi* 


"14863 


1*584 


14199 


13655 


135*5 




51 


17303 


16H4S 


16590 


16838 


I58S8 


15548 


15198 


14857 


14519 


14183 


13B50 


1351 9 




5! 


lUOl 


16939 


16584 


10838 


15883 


15530 


15192 


14958 


14S13 


14177 


13344 


ISS14 




53 


17891 




16578 


10886 


158T7 


15530 


15187 




1*509 


14178 


13839 


13509 






17285 


1698; 


16578 


16880 


15871 


15585 


15181 


I*84( 


14508 


14166 


13833 


13503 




aa 


17819 


169^1 


16566 


16814 


1S965 


15519 


15175 




14496 


14101 


1388B 


1S49T 




so 


17*73 


16915 


16560 


10808 


15959 


15513 


15170 


11889 


14491 


141S5 


13888 


13498 




57 


17!67 


1690B 


165S4 




15854 


1S507 


15164 


14883 


11185 


14150 


13B17 


13488 




59 


17^61 


16903 


16549 




15848 


IS508 


15158 


14818 


14480 


14144 


13811 


13481 




sg 


17*55 


l(iB97 


16^*3 


16191 


15948 


1.^90 


15158 


14812 


14*74 


I4I38 


1380G 


13175 




'' 


tii|>uiiiunul Pitt to tiiiitbs 


.1 -a .3 .4 .5 .0" .7 .8 .9 
1 18 2 3 4 4 5 5 





80 


T*Bf 


eX. 


Promr 


onal LiMjarithmB. 


















1 Degree., or S Hours. 




~0 


Ijii 


IS^^ 


14"! 


15" 


IS" 


17m 


19^ 


igi 


20^ 


81" 


2gi 


gai- 




1347 


ni4 




T849" 


1817. 


1185 


7153" 


1188 


lOBl 


10605 


1089 


09994 




1 


134S 


1313 


1291 


1248 


18168 


118S0 


1153 


1188 


1090 


10600] 1089 


09989 




8 


1345 


1313 




12483 


18163 


1 1845 


1152 


11215 


1090 


10595^ 1088 


09981 




3 


13*5. 


1318 


1880 


12478 


1215 


tl83S 


1158 




1089 


105BO| 1088 


09978 




4 


1344 


13180 


1879 


1847 


12158 


11834 


I131S 


11805 


1089 


10585 


1027 


09973 






1344 


I3I1J 


18790 


1246 


18147 


11829 


11513 


11800 


10889 


10580 


1027 


09988 






1343 


131 US 


1278 


IS46 


1814 


11981 


11508 


11195 


10883 


10575 


1086 


09963 






1343 


13104 


1277 


12456 


12136 


11818 


11503 


11189 


10878 


10569 


1086 


09958 




8 


1348 


13099 


18774 


1845 


1213 


11813 


1149 


1I1B4 


10873 


10564 


1025 


09953 




B 


1343 


13093 


1876e 


18440 


18185 


IISOB 


11498 


11178 


1086S 


I05S9 


1025. 


09948 




10 


134li 


130SB 


12763 


liiM 




71808 


11487 


TTm 


7086a 


10554 


1024 


09943 






1341C 


130H8 


1875- 


1243^ 


12115 


U797 


11488 


U16B 


10858 


10549 


1024 


09938 




U 


13*0 


13077 


18752 


18430 


18110 


11798 


11476 


11163 


10852 


10544 


1083 


09933 




13 


I33DS 


1307 


12747 


18484 


12104 


11797 


1147 


11158 


10847 


10539 


1083 


09928 




U 




13066 


1274 


18419 


12099 


1178 


11166 


11153 


10848 


10S34 


1022 


09B83 




15 


I3S8S 


13061 


12736 


18414 


18094 


11776 


1146 


11148 


10837 


10588 


1082 


09918 




IS 


1338 


13055 


18730 


12408 


18088 


11771 


11456 


11113 


10838 


10523 


1081 


0S9IS 




17 


13377 


13050 


12725 


18403 


12083 


11 765 


11450 


11137 


10987 


10518 


I02I8 


09908 




18 


1337 


13044 


12780 


18397 


18078 


11760 


11445 


11132 


10821 


10513 


IU20 


OB903 




19 




13039 


18714 


12392 


18078 


11755 


11440 


11187 


10S16 


10.i08 


1080S 


09898 




K 


T^60 


13033 


12709 


18387 


18067 


11750 


7il35 


11128 


1081 


10503 


10197 


09893 




«1 


13355 


13088 


18703 


18381 


18068 


11744 


11189 


11117 


10806 


101981 10192 


09887 




2S 


13349 


13083 


18698 


12S7fi 


12056 


1 1739 


11484 


11111 


108O1 


10493 10186 


09898 




«3 


13344 


13017 


12693 


18371 


12051 


11734 


11119 


11106 


10796 


10487 1018 


09877 




Si 


13338 


13012 


18687 


18365 


18046 


11729 


11414 lllOi 


10791 


10492; 10176 


OW78 




35 


13333 


13006 


186B2 


12360 


12041 


11783 


11408 11096 


10785 


10477 




09967 




i6 


I33JB 


13001 


12677 


1 2355 


12035 


11718 


11403 11091 


10780 


10178 


10166 


0936) 




87 


13M8 


18995 


12671 


1834S 


12030 


11713 


11398' 11085 


10775 


10467 


10161 


09857 




SU 


13317 


18990 


12B66 


18344 


18085 


11709 


113931 1 1080 


10770 


10162 


I015S 


09852 




iS 


13311 


18985 


18660 


18339 


18019 


11708 


113871 11075 


10765 


10157 


10151 


09847 




30 


"isaoT 


18979 




12333 


18014 




II3M8|11U71 


10760 


10152 




09»48 




31 


13300 


18974, 


18650 


18388 


18009 


11698 




10754 


10446 


10111 


09837 




3* 


I3!95 


18968 


18644 


12383 


18003 


11686 


11378! 1105! 


10749 


10441 


10136 09838 




33 


13289 


18963 


18639 


18317 


11999 


11681 


11367,11054 




10436 


101311091*87 




3* 


13iH4 


I8BS7 


12634 


12312 


11993 


11676 


11361 11049 


10739 


10431 


10185 09882 




35 


13878 


18958 


18628 


12307 


11987 


11671 


11356 IIOU 


1073* 


10126 


10120 09817 




36 


13JT3 
1386J 


18947 


12623 


18301 
12296 


11988 


11665 


11351 11039 


10789 
10784 


10421 


101 IS 09813 












11660 


11316 11034 




10110 OiiaiT 




38 


13S6S 


12936 


12818 


18891 


11978 


11655 


11340 11089 


I07I8 


1011 1 


10105 09801 




3» 


13847 


18930 


18607 


18885 


11966 


11650 


11335] 11083 


10713 


10406 


10100'OS797 




40 


I385t 


129*5 


1201)1 


12880 


11961 


"ITS44 


11330 llOlb 


10708 


10100 


10095, 0!fT92 




41 


13840 


12920 


18590 


12275 


11956 


11639 


11385 


11013 


10703 


10395 


10090 09787 




43 


13840 


12914 


12390 


12269 


IIBSO 


11634 


11380 


11009 


10698 


1O390 


10085 09T88 




43 


l3J3i 


18909 


18585 


18864 


11915 


11629 


11314 


I10O2 


10693 


10385 


IOO90 


09777 




44 


I38B9 


12903 


issan 


18859 


11940 


11683 


11309 


10997 


10688 


10380 


10075 


09778 




45 


13884 


1 8898 


12574 


18853 


11935 


U6I8 


11301 


10998 


10688 


10375 


10070 


09709 




46 


13818 


12892 


18569 


12248 


■ '■929 


11613 


11899 


10937 


10677 


10370 


10065 


09701 




4T 


13813 


12887 




:8243 


11984 


11609 


11894 


10988 


10678 


10365 


1005B 


09756 




4S 


13807 


128S2 


18558 


18837 


I19I9 


11608 


11289 


10977 


10667 


10360 


1005 1 


09T5I 




49 


13808 


18876 


if 553 


12838 


11913 


11597 


1 1883 


10971 


10668 


10355 


looig 


0974s 






"131 U7 


"1*877 


12548 


18227 


U908 


71532 




10966 




10;»49 


10044 


09741 




51 


131B1 


18865 


18542 


12881 


11903 


11587 


11873 


10961 


10658 


10341 


10039 


OBTSe 




Si 


13186 


I88C0 


18537 


I821G 


11897 


11581 


11267 


1095S 


10646 




10031 


09731 




J3 


I318D 


18855 


18531 


12211 


11898 


11576 


11862 


10951 


10641 


lO'iSi 


10089 


097 tS 




54 


13175 


18849 


18526 


18805 


11887 


IISTI 


11257 


10945 


10636 


10329 


10084 


09781 




55 


13I6B 


12814 


12521 


18200 


1 1888 


11.566 


11258 


10940 


10631 


10324 


10019 


Ol*7]« 




5rt 


13164 


18838 


U5I5 


IS IBS 


11976 


11560 


11847 


10935 


10686 


10319 


10011 


09711 




57 


13158 


18333 


18510 


18189 




1I55S 


11841 


10930 


10621 


1031* 


10009 


O9700 




58 


13153 


18888 


12305 


18184 




11550 


11236 


10925 


10616 


10309 


10004 


09701 




59 


13148 


18882 


18499 


18179 


11860 


11545 


I123I 


10920 


lOulO 


10301 


09999 


09696 






.1 .2 .3 .4 .5 .6 .7 . 


-fl 











Piojiortiauiil I^aritlimi. 



I 00380 0908i 087B6 



I 09370 
6 OR-ies 

I 0036 
6 09356 
1 0n35l 



1 oauii 0' 

! 091136 U 

r OB631 O 

t ()96ifi 

t 096il D 

i OSlJlB 

1 09611 0! 



I- 09S71 

) 09566 

i 09561 

i OBiiO 



I was 

t 095*0 
i 09SI. 



I 0MT5 
I 094;u 
i 0SU5 



own 

0B07» 
09067 
090G! 
09057 
09053 

T 0B0*« 
6 09037 
I 09033 



1 (IB809 
6 OSiOl «: 
1 ndl99 



9 0M738 
3 08727 
B OBTi! 



09281 
09216 II 
09*71 U 
09SK6 O 
09261 
09*56 
09i5l I 

09i*l 
09*36 
09*31 
09*iS 
09**1 
09* 1G 



08i77 Of 
0Bi7* Ot 
Oal67 0) 
OBM* Ot 
09457 Ot 

"osisgrn 

OBMT Of 

0844* Ot 
0843S Ot 
08433 Ot 
081*fl Ot 
094*3 Ot 
08418 Ot 
08413 01 
09408 01 



4 0810! 
9 0809T 
B 08384 0909* 



6 0B94rt 
OS943JOS648 
089391 08G43 
08934^ 08639 
089*9 08633 
099*« 096*8 
Ot)919 096*4 
08914^0861 
U8909 0861 
08904 OB609 
088991 08604 

I Ott«94i OB5M9 



09166 
09161 
09156 



1 U9U5 

i 094*0 

; 09415 

i 09410 



091*7 

091** 
09tl7|u 
0911* 
O9107 
0910* 
090971 



08053 
08019 
08044 



6 0t<UI5 
I 080 LO 
6 08005 
1 06000 



9 OSS75 
S 08570 
0B565 
B 08560 
08555 



5 08550 
08515 
5 08510 



08*57 
08^5* 
09*48 
09*43 
08*38 
08233 
082*8 
09*23 



8 07630 

3 076*5 

8 076*0 

4 07615 

9 07610 
4 07601) 
9 OTSOI 
4 07 596 
n 07591 



5 07567 
1 0756* 

6 07559 
1 07553 
6 07548 
1 07543 



I* 0751. 
8 07511 
3 07505 

8 07500 

3 07496 

9 07191 

4 07186 
9 07481 
4 07476 
9 0747* 
4 07467 
07462 
6 0715? 



6 07419 

« 07414 

7 0741O 
* 07405 

7 07400 
a 07393 

8 0739 J 
3 07386 
8 07381 
■3 07376 
« 07371 
i4 07367 

.4 07357 
19 07352 
■* 07318 



OTOU 

07039 

07031 

07030 

070*5 

070*0 

0701' 

0701 

07006 

07001 

I16997 C 

0699* 

06987 



07101 
07096 
07091 
O7087 
07082 
07077 
0707* 



06485 



06475 
064T1 
06466 
06461 
0645T 
067331 0645* 



06677 
0667* 
06667 



06916 
0S91* 
06007 



6 063iS 

* 063** 

7 06317 

* 0631* 



06579 

5 06574 

06569 

6 06564 

1 06SSU 

6 06555 
! 06550 

7 06545 



3 06531 

8 065*7 
3 065** 

9 06517 



06*Si 
1)6180 
05*75 
06S71 

06*66 
06*61 
06*57 
06*5* 
06*47 



L 























8S TiiBis X. rroportnuMi i^oBanttiniB. 




2 Degree), or i Houra. 




J 


m' 


37'^ 


3«^ 


3F)i^ 


40"' 


4!'" 


48" 


43,'n 


4,4m 


isi- 


46" 


*li 




"0 




05937 


056B8 


"5388 


05 US 






01308 


01013 


03779 


03516 


03856 






08! 10 


05933 


05657 


05383 


05111 




01571 


O1304 


01038 


03774 


03S1! 


03851 








05938 


05658 


05378 


05106 


04836 


0456T 




04031 




03506 


038H 




3 


()8!UI 


05RS3 


05648 


05374 


05108 


04831 


04568 


04895 


04030 


03766 


03503 


03343 






0619H 


Oi919 


05643 


05369 


05097 


04887 


04S,M 


04!91 


04085 


03761 


0349S 


03838 




i 


0819? 


0S9U 


05639 


05365 


05093 


04B33 


04553 


0-1886 


01031 


03757 


03495 


03831 




6 


OS 187 


059 10 


05634 


05380 


OSOBB 


0*818 




0438! 


01016 


03753 


03490 


0300 




J 


06I8« 


0,5905 


05630 


(«3S8 




01813 




04371 


01018 


03118 


034S6 


OSJM 




s 


OKI 78 


05900 


(IS6!,i 


UJ3SI 


05079 


048O9 




01373 


04O(lS 


03111 


034R8 


03111 




9 




05898 


05620 


05347 


05075 


04804 


04531 


01369 


04OO3 


03739 


03477 


0881J 




10 




05891 


05616 


05343 


O5O70 


04800 


04531 


04361 


"^999 


0373i 


03473 


DStll 






Ofil64 


05887 


05811 


0533T 


05066 


04795 


01537 


04880 


0:1991 


03731 


03469 


03iOfl 




IS 


06159 


05888 


0580T 


05333 


06061 


0479 1 


04533 


0435S 


03990 


03786 


03461 


03801 




13 


06155 


05877 


05803 


053!8 


05056 


04786 


04S1B 


043SI 


03986 


03788 


OSI61 


03199 




U 


06150 


05873 


05S9T 


05384 


05068 


04788 


04513 


O1810 


(I39BI 


03717 


03(55 


03IS5 




15 


0614,5 


05968 


0.W93 


0531 9 


OSOW 


01777 


1509 


01848 




03713 


03451 


03»1 




18 


06 [41 


OSB64 


055fi8 


05315 


OS043 


04773 




04837 


03978 


03T09 


03 («7 


03IB6 




17 


06136 


05B59 


05594 


OS3I0 


OS038 








03968 


03704 


0344* 


03188 




IB 


06131 


05854 


05579 


05306 


05034 


04764 


0«9S 


01839 


03963 


03700 


03438 


03118 




IS 


06187 


05850 


05575 


05301 


05089 


04759 


04191 


04881 


03959 


03696 


03434 


03113 






oeigs 


05845 


(15570 


05897 


05085 


04755 


01186 


04330 


03955 


03691 


0348S 


OTtn 






06in 


058H 


05565 


05893 


O50S0 


04750 


041S3 


04315 


03950 


03687 


03483 


03185 




ii 


06113 


05836 




058BS 


05016 


04746 


0*478 


01811 


039+6 


03083 


03411 


oaifio 




83 


06103 


05831 


05556 


05383 




04741 


04473 


04806 


03941 


03078 


03416 


03i«a 




it, 


06L04 


05BKT 


0555! 


OSSTB 


0500T 


04737 




01308 


03937 


0307* 


03418 


03131 




!5 


06099 


05338 


0S54T 


05!T* 


05003 


0473! 


04464 


01198 


039K 


03669 


OMOB 


03147 




aa 


06094 


038-IB 


05543 


05869 


0409B 


04739 


0416( 


UII93 


03988 


03665 


03103 


03m 




!T 


06090 


0S9I3 


05538 


05865 


04993 


0*7M 


04455 


04189 


03981 


03661 


03399 


03ISB 




38 


06085 


05808 


05533 


05860 


04989 


04719 


04451 


04184 


imiif 


036Si) 


U3!ia5 


(UlSl 




89 


06080 


05801 


05539 


05856 


04984 


04714 


04 H6 


01180 03HIS 


0365* 


033M 


WW 




30 


oeoTS 


05799 


05534 


05851 


04980 


047 10 


0444a 


Oin5lo:l9ll 


osm 


033M 


03m 




31 


060T1 


057S5 


05530 


05847 


04975 


04706 


04437 


04171 


039OS 


03043 


03391 


03N1 




3i 


06067 


05T90 


05515 


05348 


04971 


0470 


01433 


01167 


03908 


03639 


0S37T 


0311) 




33 


0606 i 


05785 


05511 


05!3S 


04966 


04697 


OltS!- 


04168 


03897 


03034 


03373 


03 U3 






OeOS7 


05T8I 


05506 


05333 


04963 


04693 


0443 




03893 


03630 


01369 


OJIM 




3.? 


06053 


0ST76 


05501 


05388 






04430 




03HB9 


05686 


033S1I 


03 104 






06048 


057J3 


05497 


05!!4 


04953 


046B3 


04415 


04149 


03884 


03f 1 Oil [ 111 HI 




37 


06043 


05767 


0549! 


05819 


04948 


046T9 


04411 


04144 


03880 


01 1 




06039 


05768 


05488 


0.5315 


04944 


04674 


04406 


04140 


03 75 


01 


39 


OfiOH 


05758 


05483 


05310 


04939 


04670 


01403 


04136 


03871 







«l 


01)030 


05153 


0547B 


058011 


04935 


01665 


01397 


04131 


03^ 




41 


0BO85 


05J49 


054T4 


05801 


0+930 


04661 


04393 


04137 


03868 




43 


OfiOJO 


0ST4+ 


05170 


05197 


0498b 


04B56 


01388 


OH! 


03858 




43 


06016 


05T39 


05466 


0519! 


04921 


04658 


01384 


01118 


038S3 






OCOll 


0ST3J 


05460 


05188 


04917 


04641 


01380 


01114 


03849 






06008 


OS 730 


05456 


05IB3 


0491! 


01643 


04375 


01109 


03815 






06Wi 


OST36 


05451 


05179 


04908 


04038 


01371 


Olio 


OJSK 






0599T 


OSTBl 


05447 


0517 ( 


04903 


04634 


04386 


04100 


03B3r 






0.5993 


05717 


05448 


05170 


04899 


01689 


0136! 


01096 


03833 




49 


(ISflWt 


0571! 


05438 


05165 


ai>i9i 


046S5 


04351 


04091 


01H37I 


ao 


DflSBS 


osjoT 


05«3 


05161 


OWUO 


04631 


0435 


^wT 




di 


03979 


05703 


05489 


05156 


04BB5 


04616 


04348 


OlOHi 1 


53 


0SD74 


OSfiDB 


05434 


05151 


04S9I 


04818 


01341 


Okj 1 


iS 


(159^0 


05694 


05419 


05147 


01876 


04S0T 


04310 




5« 


0.5965 


05889 


05415 


0514? 


04878 


048U3 


0(335 


0(or9 


is 


059 '^O 


0S6B4 


05410 


0S13B 


04887 


04598 


04331 


01065 f ja 1 


51 


05956 


0588O 


0540B 


0513: 


01863 


04594 


01336 


0)081 796 1 


37 


0.595 


056T5 


05401 




0t858 


01589 


0(338 


01056' 03J98 ■ 1 


iO 


0S94T 


05671 


05897 


05 IS! 


0*85 1 


04585 


0*317 


Ok) 8 037H8O! 1 


89 


0594* 


05666 


0539! 


05180 


1)4849 


04580 


0<313 


OWIT 037rt3lt3 ll 






.1 .8 ^ .4 5 6 7 « J. 


^ ofors. 


1,18 8 113* 









■* 


" 


MiH 


*" 






^ 




^ 


^ 








^^ 




^^ 


^ 












Hi TABLE XI. 


TABLE XIII.— Correciion to be added to the Obserred 








Allilude of the Sun's Lower Limb, wlien taken by • 






the Horizon. 


Fore Obaeryalion, to find the True Alutude. 




H.« 


ntp^ 


» 


Dipof 


Alt 


Hdgbt of tbe Kye above the Sea ia KeeC 




10 1 1« 


14 1 16 { 18 








Wf 


se 


39 




Ftn. 

1 


1 


Fkc 
l(i 
164 


3 aa 

4 * 
4 S 
4 9 

4 n 


.1 


.^.H 


3., 


3. 


8.H 


S.5 


' 


■ 


1.8 


1.6 


1.4 


1.? 


1.0 0.8 












1 i^ 




5.? 


4.! 


4.1 


4.' 


4.0; 3.: 


3.5 


3.; 


3.1 


V> 


1-6 


8.4 8.i 








m 




ff.< 


6.( 


.1,' 


.^. 


5.1 4.t 


4.r 


4.4' 4. 


3.!- 


r<i 








'^i 




H 


7.i 


6.1 


6.5 


6,8 


5.9 5.' 




5.3' s.o: \.i 


4,1 


♦.) 4.1 










To 


t:5 


~ 


j.a 


6.9 




"si 




ij 


5.1 4.9 

5.61 5.4 




11 


1 :«!> 

1 34 
I 39 
1 43 

1 47 


\'} 








6..* 






7.5 










fir 




ISi 


4 83 

4 !6 


II 


H.! 


M.I 


M.1 


7.i 


7.h 


7.' 


7.8 


«,! 


tf.'. 




6.5 


6. 


51 








».^ 


H.I 


M.: 


H.: 


H.r 


7.1 


T.I 


7.^ 


7., 


6.h 


fi.1 


G.5! 6: 








14 


H.^ 


H.( 


M.1 


H.! 


8.1 




H.8 


7.! 


V 


7.5 


7.^ 












Iff 


KM 


iir 


».: 


<t,( 


S.I 


H.I 


H.1 


8.4 


8.8 
8,6 


a.(i 


T.H 


JL5Jd 










IH 


1I),M 


10.4 


in.i 


B.H 


B.S 


fl.3 


B.I 


M,H 






i4 


4 £g 


WJ 


11.1 


10,1 


in,< 


10.1 


9.8 U.) 


fl^ 


1 1 


B,< 














n 


11.1 


ll.l 


10.1 


ro< 


to.i S.t 




■H 


Oi 












til! 5 




S 4 


iR 


11.1 


11.1 


1 1.0' 10. 5 


10.5 to.iiio.c 




9.J 


1,^ 


9.1 


S.B| 8.1 






*i 






30 
3S" 


l«.0 


nr 


ll.3|ll,0 
11.6,11.3 


10.810.5110,3 
11.0|10.7jl0.6 




9.1 


9.6 


9.4 


B.al 9.0 










103,10.1 


9.9 










4<l 


\t.!> 


t..t 


11,8111.5 


ll.3llI.O,lD.6 


IO.510.; 


10 1 










54|2!0 


30 




45 


IK.5 


tA 


\t.a\\i.\ 


ll.,' 


II.8II.1: 


IO.7I0.5 














s(i:i*.H 




liisjisic 


ll.l 




10.910.1 
















55 13.0 


i?.fl 


11.7 


U.5!ll.i 


II.0I10.I 


























611,13.1 


,a.i 


18.4:18.1 


\\.> 


11.6,11.S 


11.110.! 












34 


S 47 
3 53 


65 13.! 


8.. 


I8,.' 


18S 


M.! 


ll.l 


11.4 


11,811.1 


10.7 10.5 10.3 1U.I 






i* 




70,133 


Kt 




18.5 


\1.l 


ll.» 


M.5 


11.3 1 1.( 


10.8 10.6 10.4 lat 






2 i» 


36 


80 
DO 










18.1 
18.3 


11.9 
18.0 






11.0 10.9 ia6ia4 
11.1 lag io.nai 




136 


3.8 


18.Q'li.6 


n:8ii,6lii.3 




«* n« 


ji 




MonLh. 


Jar.. Keb. 










33 


S I! 


CoirecHon, 


+ 0'.3 +0'.8 


^-O^.l O-.O |_0'.l|_0'.» 










Monlh. 


.luly, Aug. 


Sepl. Ocu 1 Nov. 1 Dec 














CorreciioQ, 




— CI tCl 1 +0'.!| +0'.i 






3 17 


41 


fi Si 








1 86 
3 3! 


43 


6 52 

7 1 


served Altitude of a Fixed Star, to find the True Altiliidc. 




Ob» 
AlL 


HeiKlu of die Kye aboe tbe Se. in Vta. 




H 


10 


18 


14 


16 1 18 1 m 22 


84 1 86 88 


30 




Hi 


rs9 

3 iS 


*60- 


t 41 

B IB 


■'> 


l?,-! 


!.7 


,« 


13.3 


13.6 


118 I4.1I14.3 14.6 


14.8 15.0 15.8 


MA 








6 


10.! 


I.! 




11.1 


\t.t 


\S.i 


I8.7,18.S11S 


13.4 13.6 I3.e 


13.(1 












7 


U.t 


,0.i 


10,; 


KM 


ll.l 


11.; 


11.6 11.8 18.1 


18.318.518.1 


ir.fi 












S 56 


H 


H.1 


'1.' 




R.1 


lo.y 


\M 


10.7 10.9 U.8 




ii.f 








S 


«.a 


H.6 


8.« 


9.8 


9.5 


B.7 


io.oio.a!io.5 


ia7IO.»lI.l 
10.8 10.4 lOfl 
B.7| 9.9 10.1 


11,1 

IttT 
10.1 




TABLE X 
DIpatdiSer. Dis 




















fin«" 


11 


7.* 


T.6 




B.8 


8.5 


M 


9.0 8.8 9.5 












7.i 










B,6 B.8 9.1 


9.3 9.5 8.7 


g.a 












5.7 


6.1 


6 4 


6.7 


70 


7.8 


7.5 7.J 8-0 


8.8 8.4 a.6 


8.7 




,, Bd,k,»ffl,. 


Ey. 










































16 

li 


5,0 
4.3 
4.4 


5.4 

5.8 


5.7 
5.5 

5.1 


6.0 
5.B 
5.4 


6.3 
6.1 


6.5 






8.0 
7.8 
7.4 








1 


J^' 


iS 


H4 




e.8| 7.1 

6.4 6.7 




;-5 


fl.fl 


6.8 


6.9; 7.I| 




1 ( 




t 




U 


M 


30 


4.1 


4.5 


4.8 


5.1 






5.9 6.1 6.4 
5.6 5.U; 6.1 


6.6 6.S 7 
6.3 6.5 6.7 
6.1! 6.3 6.5 
5.9 6.li 0.3 


7.1 

64 
6.S 









40 


3.S 
3.6 

3.4 


4.0 


4.3 

4,1 


4.8 
4.6 
4.4 


5.1 
4.9 

4.7 










n 


14 
li 




ii^ 




B 


4.n 


5.8 5.4! £.7 










S 






n 




■A-t 


•■itt 


3.! 


4.) 


4.J 




5.0 5.8 5.5 


6.7 5.9, 6.1 


ft3 






n i 




4 




4 


H 


bit 


3.1 


S.5 


3.( 




4.4 


4,6 


4.9 


5.1I 5.4 
5.0, 5.3 


5.6 s.a' 60 


6.1 
6.1 




» 


4 






fill 




.tl 


3.1 


4.( 


■H 




4,8 




JH ■ 


H 


4 








65 


V.II 


*;■ 


3.t 


N.t 




4.4 


4.7 4.91 5.? 


5.4 5.6 5.) 


n,i) 






♦ ■ 


» 


4 






fi 


70 


i!.H 


3.i 


3..-, 


%i 


4.1 


4,f 




53 5.S 5.1 


51 






J ,1/3/* 






K 


K'J 


K-H 


S.1 


3.^ 


x: 


a! 


4,1 


4.4; 4.6 +.£ 


5.1: 5 3 5-5 


5.7 




. I 


Hil3liU\ 


i 


5 


90 


1!.4 


*.»! 3.\\ a.\\ a.i\ ^sV \.i 4,4.t 4.7 


4.9 5.1' 5.3I 


5.5 








d 








■ 


■ 


I 


■ 


■ 


^1 


^ 


■ 


■ 


^ 


^ 


^M 










^1 




^1 


I 



Itilli ''" 



.8! 3!.! 

.7 8 3!.! 

A i 3g.l 

.9 t 3!.( 

,3! 3?.' 



9.9»30iS 
f>.fl!1338 9 
r? B.9931T9 
9.994.831 
9.991788 



.1 8 89. 
.J 8 89.4 

.9« 8S.9 
.8 8 88. 



.996383 " 
9.997016 
9.99};i9 

.99H431 

cuSiuoiis 

.3 0.000886 

.HO0O15.U 

0.0O8851 



.4 8 8i.4 
,1 8 8S.0ai 
.8 8 84.GU. 



.0 a001854 
.G 0.001817 
.«O.O0A383 
.6 8 83.9 D.0OSTM : 



.fi8 83.G 

.!) 8 23.4 

.3 8 83.8 

,9 8 83. 



o.ooejHi 

0.006016 

0.006864 ' 

0.007043 ' 

0.007173 J 

38 83.0.0.007837 



.5 8 83.0 

.8 8 83.1 

.1 8 83.8 

.7 8 B3.3 

15 8 8iU 

.;(8 83.S 

.3 8 2i.l 



007818 ■ 

aoo709a ' 

.006910 ' 

.006675 ■ 



.7 8 86.7 
.4 8 87.8 
:0 8 87.7 



0005463 
O.OOlfilS 

OOt-tOi - 
0.003705 

.003010 
0.002333 
0.00161? 
0.000898 

olSSniig 

.8 9.999398 
.7 9.999633 
9.99TH98 , 






.3 8 38.4 
9! 38. 
.4 8 38.8 

.7 g aa.9 



B.993697 
9.993388 
9 993081 



TABLE XVI. 
■ Parallax id Altitude, &c. 



(kOO 
a 0.15 
0.30 
6 0.U 



=.88 8.WI B.75l«.ei 4 



1 — 


"" 


~ 












™ 














«G- 




Ta 










Fihrenhe 


■« Tl 




-TdT 


W 


W-!' 


Diff 


z. a 


ii 


L^S^ 


Diff 


Z. D. 


It 


Log.!/ 


Di£ 





0.00 


aoooo 




10 


10.30 


1.0189 


72 


80 


21.26 


1.3877 


38 




in 


an 


S.*304 


3011 


10 


10.47 


1.0?0l 


72 


10 


21.45 


1.3313 


39 






80 






1761 


80 


10.65 


t.0273 


71 


81 


81.65 


1.3354 


39 






» 


0.S 


0.7070 


1219 


30 


iaB2 


1,0344 




30 


21.84 


1.3393 


38 






40 


aes 


9 838.i 


969 


40 


11.00 


1.0414 


69 


40 


88.03 


1.3431 


38 






m 


o.ej 


9.9394 


79 




11.17 


1.0483 


69 


50 


82.23 


1.34fifl 


SS 






1.0 




"e?o 


TI~0 


11.35 


1,0558 


6S 


21 


88.48 


1.3507 


37 




n 


MS 






10 


11.53 




66 


10 


S2.SS 


1-3544 


3S 






i(j 


1.36 


0.133a 


518 


SO 


11.71 


1.0684 


66 


80 


28.81 


1.35S8 


37 






30 


1.83 


a 1817 


437 


30 


11.89 


1.07 50 


63 


30 


83.01 


1.3619 


3T 






4(1 


1.70 


08304 


414 


40 


12.06 


1.0815 


64 


40 


83.81 


1.3656 


ST 






io 


\.% 


0.8718 


379 


50 


18.84 


1.0879 


68 


51 


83.40 


1.3093 


36 












12 


12.48 






22 1 


23.60 


1.3729 


3T 




10 




0.3444 


38! 


10 


18.80 




61 


10 


8380 


1.3766 


36 






K 


8.3H 


0.376(i 


.■iO 


80 


I a. 78 


1.1064 


60 


20 


84.00 


1.3808 


39 






30 


2.SS 


a4087 


380 


a 


12.96 


1.1184 


60 


30 


24.80 


1.3838 


36 






40 


S.7! 


0.4347 


863 


40 


13.13 


1.1184 


58 


41 


84.40 


1.387* 


3S 






fly 


%m 


0.4610 


250 


50 


13,31 


1.1248 


58 




84.60 


1.3909 


3d 




3 


9 a.OB 


l).48«0 


835 


13 


la49 


1.1300 


57 


23 


84.80 


1.394s 






10 


3.83 


0.5095 


82' 


10 


13.67 


1.1357 


57 


10 


85.00 


t.30Sl 


34 






tu 


3.40 


0.S3I9 




80 


13.65 


1.1414 


55 


2C 


85.80 


I.401S 


S4 






so 


aar 


0.3530 
0,S783 


3o: 

193 


30 


14.02 
14.80 


1.1584 


55 




15,41 
85.61 


l-404» 
1.4084 


35 
31 


1 




3.T4 




« 


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O.S988 


180 


51 


14.38 


1.1578 


56 




85-81 


1.4118 


33 




4 U 


4.0b 


0.fllla 




14 


14.5b 


1.1034 


TT 


81 


86.01 


I.4I51 


34 




10 


4,(6 


0.0890 


171 


10 


14.74 


1.1686 




10 


88.81 


1.4185 


34 






!0 


4.43 


0.e»6J 


165 


80 


14.93 


1.1740 


53 




2S.42 


1.4219 


34 






3U 


4.un 


aG6S6 


158 


30 


15,11 






30 


86.68 


1.4853 


33 






to 


4.77 


0.6784 


153 




15.29 


1.1845 


58 


40 


86.8.S 


1-4286 


33 






iu 


4.U4 


aG937 


US 


50 


15.48 


1.1897 


50 


50 


27.03 


1.4319 


33 




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fi.ll 


0.70ti(l 


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15 


15,66 


1.1947 


51 


85 


27.84 


1.4358 


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0.7ttB 


139 


10 


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1.1998 




10 


87.45 


1^3S5 


31 






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S.45 


0,7367 


135 


80 


16.03 


1,8048 


60 


80 


87.66 


1.4418 


33 






31 






131 


30 


16.81 


1.20BB 


49 


30 


87.86 


1-4451 


31 






40 


S.BO 


0.7633 


187 


40 


16.39 


I.8I47 


48 


40 


88.07 


1.4483 


31 






fiO 


S.9T 


O.770O 


188 


50 


16.58 


I.2I95 


46 


50 


83.88 


1.4.S15 


38 




ti.l4 


0.7882 


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16 


16.75 


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26 


28.49 


1.4541 


31 




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0.8002 


116 




16.9.1 


l.aiB7 


47 


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28.70 


1.4379 


3! 






so 


S.4H 


0.8118 


114 


20 


17-18 


1.2334 


40 


80 


88.P( 


1.4011 


32 






31 


6.6G 


0.8838 




30 


17.30 


1.8380 


46 


30 


29.13 


1.4643 


31 








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40 


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1.8486 


46 




89.34 


1.46TI- 


38 






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101 


50 


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1.8478 


47 


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7.34 


0.8557 
0.6659 


102 
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1.8519 
1.2564 


45 
45 


27 


89.76 


1.4736 
1.4768 


38 
31 


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0.8760 


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1.8609 


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30.19 


1.4799 


30 






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97 


30 


18.42 


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44 


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1.4889 


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40 


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95 


40 


18.61 


1.8697 


43 


40 


30.68 


1.4860 


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fitf 


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O.SOil 


03 


50 


18.79 


1.274(1 


44 




30.83 


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31 






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m 


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31.05 


1.4921 


31 




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0.9834 


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1.8826 


48 


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31.27 


l.495> 


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20 




0.9323 


87 


80 


19.36 


1.8868 


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31.49 


14988 


31 






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0.9410 


85 


30 


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1.2910 


42 


30 


31.72 


1.5013 


30 






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0.9495 


84 


40 


19.73 


1.895* 


42 




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1.5043 


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84 


50 


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39 


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0.9743 


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80 


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1 51 9i 


29 








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40 


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33.2J 


l.«821 


89 




/ 




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75 


50 


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1.3837 


40 


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33.50 


1.5830 


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857 


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1.94808 


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33.85 


1.5308 


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49.88 


1,69867 


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10 


9.91 


1.94464 


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1.5337 


ie 


80 


49.58 


1.69583 


857 


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30 


34.40 


1.5366 


89 




49.87 


1.69780 


857 


30 


10.77 


1.84917 


857 




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34.63 


1.5395 


88 


40 


50.16 


1.70037 


856 


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11.19 


1.85834 


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34.86 


1.5ia3 


89 


50 


50.46 


1.7089! 


847 


50 


11.6( 


1.8549( 


857 




31 


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1.5t5S 


89 


41 


50.75 


1.70550 


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1 18.08 


1.85747 


85(1 






3i.3g 


1.5481 


89 


10 


61.06 


1.70901 


851 


10 


18.46 


1.8600.i 


859 




SO 


35.56 


1.5510 


89 


80 


51.36 


1.71058 


853 


8ft 


18.99 


1,86861 


8^ 




30 


3i79 


1.5539 


89 


30 


S1.66 


1.71311 


853 


3ft 


13.33 


1.86488 


859 




40 


36.0* 


1.5566 


89 


40 


51.91. 


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40 


13.77 


1.86781 


8S8 




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36,26 


1.5594 


88 


50 


58.87 


1.71818 


858 


50 


14.80 


l,87U3Sl 


849 




as 


J 36.49 


l.66i? 




48 


U 58.57 




858 


58 a 


1 11.64 


1.87898 


HO 




10 


36.73 


1.5650 


89 


10 


58,88 


1.78388 


858 


10 


15.10 


1.87558 


161 




go 


36.97 


1.5679 


89 


80 


6ai9 


1,78574 


858 


80 


15,53 


1.87919 


ISl 






ST.ai 


1.5T07 






53,50 


1.78986 


858 


30 


16.01 


1.88090 


IGL 






37.44 


J. 5735 


87 


40 


53.81 


1.73078 


851 


40 




1,SS341 


tao 




M 


37.68 


1,5768 


88 


50 


54.18 


1.73389 


851 


50 


16.88 


1.89001 


■61 




33 


37.93 


1.5790 


88 


43 


54.4:1 


1.73590 


853 


53 


1 17.38 


1.99863 


iut 




10 


38.17 


1.5819 


87 


10 


54.75 


1,73933 


854 


10 


17.86 


1.89185 


868 






38.48 


1.5H4S 


88 


80 


65.07 


1.74087 


853 


80 


19.33 


1.89387 


!<3 




3i 


38.66 


1,5973 


87 


30 


55,40 


1.74340 


853 




18.8 


1.89650 


tea 




40 


saeo 


1,5900 


87 




55.78 


1-74583 


851 


40 


19.89 


1.89913 


86S 




SO 


39.15 


1.5937 


37 


60 


56.04 


1.74847 


853 


50 


19.76 


1.90176 


864 




31 


39.39 




~w~ 


44 


56.35 


1.75100 


858 


54 


1 8081 


1.90410 


M« 




10 


39.64 


1.5881 


88 


10 


56.6B 


1.7535? 


858 


10 


80.71 


1.90705 


165 




»0 


39.89 


1-6009 


87 


80 


57.08 


1-75604 


858 


80 


81.84 


1.90970 


16(1 




so 


40-14 


1.6036 


8T 


30 


57.35 


1.75S56 


858 


30 


81.75 


1.91836 


866 




to 


40.39 


1.6063 


87 




57,6! 


1.76109 


858 


40 


88.85 


1-91508 


867 




BO 


♦0.61 


1-6090 




fiO 




1,76360 


851 


50 


88.74 


1.91768 


867 




35 


40-89 


1.6116 




45 


58,36 


1.76611 


848 


55 




1.98031. 


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10 


41.14 


1.6143 


81 


10 


58.70 


1.76863 


858 


10 


83.78 


1.88301 


169 




10 


41.40 


1-6170 




80 


58-05 


I.J7115 


858 


to 


84.30 


1.98573 


869 




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41.65 




86 


SO 


59.38' 


1.77367 


858 


30 


84.93 


1.98841 


811 




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41.91 


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87 


40 


49,74 


1.77619 


858 


40 


85,36 


1.93118 


870 




AO 


4t.l6 


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86 


fiO 


1 0,09 


1.77871 


858 


50 


85,88 


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870 




36 


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56 


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1.6330 


86 


80 


1,15 


1.78688 


868 


80 


87,58 


1.94196 


873 




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1.6356 


86 


30 




1.78880 


858 


30 


88-07 


1.84169 


873 




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43.41 


1.639t 


86 


40 


1.9h 


1.79138 


853 




89.68 


1.91718 


874 






43.74 


1.6408 


87 




8,81 


1.79395 


858 


60 


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1.85016 


875 




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1,6435 


86 


47 


1 8.57 


1.796.37 


153 


57 


1 89.73 


1.95891 


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1.6461 


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1.79880 


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877 




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44,54 


I.64B7 


86 


80 


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1.6513 


86 


30 


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1.90396 


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30 


31,48 


1.96180 


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40 


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1.6539 


86 




4.00 


1,80648 


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40 


38,06 


1-06397 


tin 




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45.34 


1.6565 


86 


5[ 


4.43 


1,80908 


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38.65 


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10 


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45.BS 


1.6591 

1.6917 


86 
86 


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10 


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1.6643 


86 


80 


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31.46 


1.97516 


891 




80 


46.44 


1-6669 


86 




5,85 


1.81916 


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35.08 


1.97797 


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40 


46.7.? 


1.6695 


85 




634 


1.98170 


844 


40 


35.70 


1.98080 


888 




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46.89 


1.67tO 


86 


50 


6,78 


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40 


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1.6746 




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1.98678 


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59 


I 36.93 


1,98616 


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41.56 


1.6778 


86 


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1.88933 


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37.59 


1,98931 


885 




to 


4T.84 


1.6796 


86 


80 




1.83188 


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38.84 


1.99816 


887 




30 


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86 


30 


8.38 


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38.89 


1.99503 


887 




40 


49,4! 


1.8950 




40 


8.78 


1.83699 


855 




39.54 


l.9fl7BO 


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5U 


4«.70 


1.6876 


85 


60 


9.18 


1.83953 


855 




4080 


8,00078 


889 






48,98 


1.6901 


ts 


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60 


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396 




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10 


49.65 


8.22956 


407 








0.01 


8.55635 


772 


0.048|o.O 








4S46 


?.0»718 


899 


80 


51.85 


2.83363 


410 




80 




6.50 


8.56107 




aoii 


n.m 






30 


4T.18 


8.03016 


300 


30 


68.B7 


8.83773 


113 




30 




13.19 


8.67192 


797 


0.046 


w 






40 


47.S3 


8.03316 


301 


40 


5*.i3 


8.84186 


417 








80.09 


8.S7989 


811 


0.049 


[).U 






iO 


4a68 


8.03617 


■Ml 


50 


56.81 


8 84603 


119 








87.26 


2.59BO0 


684 


0.05 






STU 




8.039 If 




fan 






183 




i8"o 






8.59824 




(i.os: 




10 


M.81 


8.048!) 


301 


10 


nd.GR 




185 




10 




48.37 


8.604«2 


851 


0.057 


IHM 






8(] 


A0.99 


8.04585 


30.5 


80 


•t 1.43 


8.85970 


139 




20 




50.33 


8.61313 


861 


i}.oet 


U09 






30 


Jil.77 


8.04M30 


307 


30 


3.23 


8.86293 


433 




SO 




58.5B 


8.68179 


993 


0.063 


n.iu 






40 
flO 


fi*.57 
53.36 


8.05137 

s.osi4a 


308 


40 


5.06 
6.93 


8.88738 
8.87168 


436 

no 




40 


7 


7.19 
16.13 


8.63068 
2.63961 


999 
914 


067 
0.07 


O.IU 

oT 


1 


i09 


60 




I 54.17 




sTo 




3 8.83 


8.8760B 






J3"0 


T 


25.40 


8.64tl75 


931 






10 


fi4.9B 


8.06064 


Ma 


10 


10. TT 


8.29051 


M7 


oooa 


10 




35.05 


2.658U6 


949 


0.0T9|ai8 






SO 


,M.ai 


8.06376 


31! 


80 


18.74 


8.88498 


450 


0.003 


80 




45.10 


8.66755 


967 


0.084 




1 




30 


fifl.60 




315 


30 


14.75 


2.89949 


454 


0.00) 


30 




55.58 


8.67782 


986 


0.099 




i 






57.S0 


8.07O03 


315 


40 


16.80 


8.89409 


459 


9 004 




9 


6.50 


2.68709 


I0O6 


095 








50 


flB.3fi 


i.0731M 


317 




19.S8 


8.89860 


468 




5( 




17-90 


8.69714 


1086 


0.101 


01 


1 
i 


lii~0 


I a!*.*» 


8.07635 


31H 


tTo 


3 81.01 


8.30388 


167 


l).(H)5 


i*T) 




89.90 






107 






10 
20 


i ao9 

0.S9 


8.07953 
8.08873 


380 
381 


10 
80 


8ai8 

85.39 


8.30799 
8.31859 


470 
475 


0.00( 

aoo6 


10 
80 




48.84 

65.25 


8.78856 


1069 

1098 


O.l[4'0.l 


1 


o,i88|ais 




30 


i.eH 


8.09591 


383 


SO 




8.31734 


479 


0.007 


30 




8.B9 


8.739*8 


1115 


0.130 awl 






40 


2.B0 


8.08917 






39.95 


838813 


483 


0.O0B 


40 




83.16 


8.76063 


I13(^ 


0.139 


0-81 






30 


S.7I 


8.09841 


S86 


50 


32.30 


8.32696 


488 
493 




50 




38.18 


2.76808 


1165 


0.149 




1 




g 4.e5 


8.095H7 


387 




3 34.70 


2.33184 


UOUS 


iS I 


T 


53.91 


8.77367 


ii9i 


0-159 


IWJ 




10 


5.iU 


8.09894 


330 


10 


37.16 


8.33677 


497 


0.009 


10 


10 


10.35 


8.78558 


1219 


0.171 


0.26 






20 


6.54 


8.10884 


330 


80 


39.65 


8.34174 


508 


010 


80 




87.73 


2.79777 


184« 




a.>« 






30 


7.51 


8.10554 


338 


30 


48.81 


8.34678 


507 


0.010 






46.03 


8.81025 


1877 










40 


8.49 


8.ID9g6 


334 


40 


41.92 


8.35193 


51! 


D.011 




u 


5.30 


8.88308 


1309 


0.(13 


053 






50 


9.48 


8.11880 


335 


60 


47.48 


8.35695 


517 


11.011 






85.66 


2.83611 


13*0 




03li 
035 




1,0 


8 10.4B 


».U535 




T6 U 


3 5(1.81 


8.36818 


&i. 


;i,018 


itTo 


rr 


47.15 


8.84951 


137* 


oJgw 




10 


11.50 


!.llb98 


33S 






8.36735 


589 


0.018 


10 


18 


9 98 


8.86325 


1410 


a869 


0.43 








1253 


8.18231 


3*0 


80 


65.95 


2.37863 




aoi3 






33.97 


8.67735 




0.898 


41 






3( 


13.57 


?.iaa7l 


348 


30 


58.76 


2.3T796 


538 


n.013 






59.51 


8.89188 


1484 


0.317 


1.51 








14.6S 


8.18913 


34.5 


40 


4 1.74 


8.39331 




D.0I4 


V) 


13 


80.61 


2.90666 


1523 


a31i0.5i 






S( 


I5.7W 


i. 13858 


345 


fiOl 


4.79 


8-38B79 


551 


9,014 






55.40 


8.981B9 


15(15 


o.s7m.ei 






i \6.n 


8.13603 


34> 


J7 




8.394.30 


557 


11.015 




14 




8.93151 


KIOJ- 


S:iIo|ft« 




H 


n.BB 


8.13931 


319 


10 




2 3998T 


563 


0.015 






59.71 


8.95.}6g 


1654 




0.1i 






KO 


19.00 


8.14300 


35S 


80 


14 39 


8 40550 


569 


0.016 


80 


15 33 60 


8.97016 


1701 


0.49O 


0.83 






30 


80.13 


8.14658 


SjI 


30 


17 74 


241119 


576 




30 


16 


10.89 


8.98717 


1749 




0.91 






40 


2I.«S 


J. 15006 


35; 


40 


8119 


8 41695 


593 


9.017 


40 




50.8 


3.0O466 


1901 




1.01 






50 


88.43 


i.l5361 


i5i 


Sf^ 


8178 


2 48278 


689 


9,017 


50 


17 


33.6 


3.08867 


1955 


654 


l.Ii 






8 83.61 


Uius 


T 


18 


1 89 33 


2 48987 


696 


9.018 


mH 






3.0418* 


rsju 






K 


8191 


8.1607S 


3j8 


10 


38 04 


8 43*63 


603 


0.OU 


10 






tOfiOSl 


19H7 


798 


1.41 






go 


(6.08 


8.16440 


31.4 


80 


35 84 


2 440S6 


611 


0.019 


80 


80 




i.0199B 


2086 


a8fl7 


L.SS 






30 


87.85 


8.16801 


386 


30 


39 76 


8 4*677 


618 


0,080 


30 




59,6 


110024 


8089 


987 


1.71 








88,50 


8.17171 


3fl( 


40 


43 76 


8 45295 


626 


0.081 


40 


22 




3.18113 


8155 


LlOl 


2.01 






SO 


89.7B 


8.17539 


371 


50 


47 8H 


2 *5981 


63.^ 


0.02S 


50 


83 


S.9 


114268 


8881 


1.231 


129 




(i« (1 


i 31.04 


8.17910 


37i 


To 


4 52 18 


8 46556 


618 


0.08: 


iTo 






1.16499 


2895 


Taw 




10 


38.34 


8.19883 


375 


10 


56 47 


8 47198 


650 


0.081 


10 


85 


40.9 


3.18779 


83St 


1.551 (.91 






80 


33.67 


J.1965S 


i78 


8 


5 OSl 


9 47848 


S59 


025 


80 


27 


T.I 


3.21140 


(434 


1.719141 






3< 


3S.0I 


2.1903* 


i8l 


■10 


5 54 


8 4850T 


SG9 


0.086 


30 


28 40.8 


18.3574 


E50D 


1.977 SB; 








36.37 


2.19417 


ISJ 


41 


10 28 


» 49176 


677 


0.087 




TO 


83.8 


3.86093 


(561 


tlllUt 




/ -Ji 


37.76 


f.tOSUi) 


J85 


50 


15.16 


8 49863 








33 


ISO 


3 88G67 


8(67 






Jroof 


3At6^.!018S 


assjao 


20.19 


a.S054V\69(i\0-traC 


feoi*3V\l.&VX3l334 




tgWI-ll 








i 




■ 














■ 






■ 


^M 


^^ 


■ 


^M 


■ 


1 







TABLE XVnt 






H! 




TAB, XX.— ThmnoiortCT. 










PR.., W. 


Th. W_J 




Therniometer. 




-ro"^7i:oi7n3 


SO" 


C.OOOOO 








TABLE XIX. 


a 
1* 

13 


0.00164 

oooieo 


91 
5* 
S3 


9.99996 
9.9W9I 
9.MW7 




P^|Th.| i...,. 


ET 


"Tin 


Lo«. 






10- 0.l):iJ79 




30" 


0.000(K) 




10 




0.036BO 


9 


1 


9.99BI0 


Barometer. 


14 


0.00156 


34 


g.999B3 




to 




0,0SSB8 


IS 


t 


9.998*0 




IS 


0.001 SI 


55 


9.9997S 




t9 
39 




0.03ia* 
03386 


57 
36 


3 
4 


9.99130 
9.99640 




IS 

17 


aooi47 

0.00143 


S6 
57 


9.9flST4 
9-99970 




P. P.| Bar. 1 LOK. 




*9 




aositis 


i& 


i 


9-9935U 




i7.S 


9.96**1 


la 


0.0()I3» 


S8 


9.99965 




68 




affiOBi 


Si 


7 


9.99UJ0 
9.99371 






9.IJ637H 
9.9S5ae 


19 


000134 


59 


9.99961 




OtH)l:lO 




9.99991 




7S 




0.US9BJ 


n 


8 


B.99!9! 






9,9669* 


*1 


0.001*8 




P.999S3 




88 




0.0?900 


81 


9 


91)9193 






9 B6fl4a 


88 

n 


ooon7 


6* 


9.99948 
9.99944 






IT 


(i-osBiia 




60 


9,99104 


+ 


Wo 






10 




0.0«70fl 


g 




S.99016 


1.S 




9.97158 


n 


0.00113 




9.999W 




19 






w 




P.9S9J7 


30 






15 


O.0OIO8 




9.999Si 




ti 




aotsu 


88 




e.9SB39 


46 




9.97466 


H 


0.00104 


66 


9.99931 




38 




aOHIH 


35 




9. OH 7.^1 


61 




9.976*0 


n 


000100 


07 


9.999*7 




te 




ao»3*3 


41 




9.98663 


78 




9.9777* 


18 


0.O0O9S 


flS 


».9Ba*C 








O.MS*7 
«.0813* 


62 




9.9B575 
9.98489 


fll 

m 




9.979*1 
9.98076 


*9 


aooo9i 


69 


S.g9»l8 




-■w 


0.00087 




9.9OT1S 




77 




0.02037 


TO 




9.98401 


m 




9.98**7 


31 


0.00083 


71 


9.M9M 




J8_ 




0.01913 


_T9_ 




9.9!(314 


m 




9.98378 


3i 


0.00078 


7* 


9 99904 






10" 






10" 


MHtil 




WAI 


9.1185*8 


33 




73 


9.99900 




A 






9 




9.98140 


15 






34 






9.99896 




19 






n 




9.98054 


29 






35 




75 


e.9!iegi 




S8 




auisoQ 


S6 






44 






3<1 


u.oooai 


76 


9.99S8T 




3S 






u 












3T 


0.00057 


77 


9.99883 




47 




0.01379 


43 




9.97795 


73 




9.99170 


38 


0.0005* 


78 


9.99878 




56 




O.OI?fl.i 

aoii9a 


^1 

AO 




9.97709 
9.97a23 


103 


1 


n.99417 
9.99503 


39 


0.OW4B 


79 


9.f)9S74 
9 9987t> 






0.00043 






la 




0.01039 


69 




9.97337 


liM 


ti 


9.99700 


41 


0-00039 


81 


9.99tl60 




*L 




aoioofi 


77 




9 9715* 


lai 




fl-39B55 


4* 
43 


0.00034 
0.00030 


S3 


9.99B6I 
9.99857 






ir 


0.1XIHI4 




80" 


K-mSrij 






0.000<»0 




A 




OOtWBB 


* 




9 97!9g 


It 




O.U0IW 


44 


0.000*6 


8V 


9998.13 




IB 




0.00730 


17 




B.971BT 


!» 




11.00*99 


♦5 


O.000it 


85 


9.99848 




is 




o.O(ib;(H 


iS 






43 




o.ooia* 


4fi 






999844 




3T 




OOOoW 


31 




9!97027 


57 






47 


U.O00I3 








4S 




aoo*55 


t! 




9.969 Vt 


71 




007 IS 


48 






9.9D835 




Si 




o.oo:i«3 


■W 




9 9im29 


Wi 




0.1)081)0 


49 






9.99831 




«4 

7t 




0.00278 


i9 
li7 




9.96775 
9.9ti69l 


100 
114 






50 




9.998*7 




P. P. io leiih. oi s Oegrte. 




83 




0,00090 


78 




9 96607 


m 




a01*84 


.1 .* .3 .4 .5 .6 .7 .« .9 






M 


0.00000 




*0 


9965?* 




31.0 


n.014*4 


— 011**3334 






EXPLANATION. 






nmputcd bj the roHawiog fonnula, viz. 


1 V P 




-i+fii.-su,"*. 








ihc Hp.n^ionorigiTcn 






height of the Kngliih tmomaa, ■^ ttw MiDpcnlnrc in the openair 
eter. il ths me» nfnai<m for 30 india mJ SO- j «,d ^ wid 




pmure napeclinJjr. 




ermining the eff«U of ehuige. in lh« ttmpenlure mil barometric 








Tmble XVII. eoDtsin 






^rithtn.cf_^X.43l. 




|-i Md Table XX. >he 























" 


" 














Auginentition of llic Moon's Semi- 
iliwuela in Aiiitudeand Zenitli Dist. 


Ktduction of the Moon's ParalljiK iii U» 

Spheroid- 






^ 


3 


14 30 


IS 


1S30 


IB 


16 30 


17 


LM. 


54' 


ay 


56 


57- 


S8' 


sy 


60- 


61' 






1 
! 

; 

t 

1 
e 

9 

11 
la 

13 
14^ 
IS 

it 

!3 
t\ 

Si 
i% 

n 

?B 

iO 

M 
Jl 


90 

B9 
BS 
87 
86 
S5 
tt* 
83 
82 
SI 


o.st 

0.*H 

0.71 
0.9S 
I.IU 

l.li 

1.6i 
1.8» 
2.11 


B 

0.7S 
1.00 

1,85 

i!75 
S.OO 
S.8S 


aoo 
0.8; 

O.B0 
1,07 
1.34 
1.60 
1.37 
8.14 
8.40 


aoo 

0.89 
0.5B 
0.86 
1.15 
1.43 
1.71 
8,00 
8,88 
8,5) 


0.00 
0,31 
0.68 
0.98 
1.83 
1.53 
1.83 

1A3 

8.73 


65 
0.97 
1.30 
1.68 
1.94 
8.8ti 
8.5S 
2.90 




3 

4 
5 

6 

9 


0.0 


0-0 



0. 
0. 
ft 
0. 
0. 
0. 
0. 


0.0 

o!o 

0.0 

ao 

0.1 

0^8 
0.3 


0,0 

ao 

ao 
ai 
ai 
ai 
a8 
a8 

0,3 


aa 

0.(» 
0.0 
00 

ai 

0.1 

ai 

a» 
ae 
as 


ao 

0,0 

ao 
ai 
ai 

01 

as 
as 
aa 


0.0 
0.0 

ao 

00 

ai 

0.1 

ai 
as 
a8 
as 






0.0 
0.0 
0,1 
0.1 
0.8 
0.2 


0.0 
0,0 
0.1 
0.1 

08 














79 

7B 
71 
TS 

?■! 
I3| 
78 
71 
70 
69 
6-6 

fi( 
6S 

I 

58 
SB 


«.5S 
S.81 
3.0* 
3.S7 

3.5f 
3.T3 

3.fli 
*.17 
4.40 


8.75 
3.00 

3.74 
3.S8 
4,88 
4.-»> 
4,70 


8.67 
8.94 

.3.47 

a73 

ago 

4.85 

4.51 
4.78 
5.08 


8,B5 
3.13 
3.41 

3.69 
3,97 
1.8S 
4.53 
4.80 
5.07 
5.35 


3.0.1 
3.33 
3 63 

3,9i 

4.S8 

4,81 
5.10 
5.39 

5.68 


a54 

3.8S 
4.1B 
4.1.9 
4.80 
5.11 
5.*8 
5.73 


10 

u 

18 

13 
14 
15 
16 
17 
IS 


0,S 

o,r 

0,7 
0.B 

1.0 


0.3 
0.1, 

0.6 
0,7 
0,8 
0,9 
1.0 


0. 
0. 

a 

(i! 

0. 

1. 

? 

1. 
1. 

!.*■ 

1. 
a. 

3. 
8.4 

8.5 


0.5 


0.3 

a4 
as 


a4 
as 

0.7 

as 

0.9 
I.O 

I.I 

1,2 


0.3 
0,4 

0.5 

a6 

0.7 
0.9 
0.9 
10 
M 
1.8 


0.4 
0,4 

ac 
ae 

0.7 

as 
as 

1.0 

1.1 

IS 






0.6 

a7 

0.8 
0.9 

1.5 

2.0 
8,1 
3,3 
8.4 

8.6 


a6 
a7 
as 

1.0 

1.1 
1-8 








1.1 






4.6* 
i.S* 
a-Oh 
i*h 

A.4» 
5.71 
5.9! 
6.1 » 

e.3i 

6.55 


+.94 
5 IB 
5,48 

5 89 
6.11 
6,34 
6.56 
6.79 
7,01 


5.87 
5-58 

6.87 
6.58 

e.7(, 

B 


5.89 
S,16 
6,48 
6,6H 
6.94 
7.80 
7.46 
7.78 
7.S7 


5.97 
6,86 
6.55 
6.B3 
7.11 
7.39 

7.93 
8.80 
8.47 


6,35 
6,65 
6.9S 
7,8.^ 

7,8* 
8,13 
a48 
B.71 
9.00 


80 
81 
88 
23 
84 
85 
86 
87 
88 
89 


1.8 
1.3 

l.t 

1.9 

8.0 
8.1 
8.3 
8.* 


1.8 
1.4 
1.5 
1.6 

8^0 
8.2 
8.3 
8.5 


1.3 
1,4 

1,6 
1.7 
1.8 
80 

e.i 

8.3 
8,5 
86 


1.3 
1.5 
1.6 
1.7 
19 
8.0 
8.8 
8,3 
8.5 
8.7 


1.4 
1,S 

1.6 
1.8 
l.» 
8.1 
S,3 
i.* 
«.i 


1.4 
1,6 
1.7 
1,8 
1.9 
8.1 
1.3 
1.4 
8.6 
8.S 






6.7J 
7.15 
7.55 

7.93 


7,ej 

BOB 
8,50 
8,00 
9.30 
9,68 
10-05 
10.41 
10.76 


B.63 
9,07 


8.88 
8.78 
9,80 
9,67 


8.74 
9.86 
9.7B 
1088 
10. 7H 
11.86 
11.78 
18.17 
18,61 
13-03 


9,88 
9.84 
10.39 
10.98 
11.48 
11.98 
18.44 
18.98 
13.38 
13.83 


30 
S8 
34 
36 

40 
48 

44 
46 
48 


2,6 
8.9 
33 
S.6 


8.6 
3.0 
13 

3.7 


2. 
3X 
3. 
3. 

4, 

4,5 
4.1 

6,( 


8.7 
3,1 
3.i 
3,8 
4,8 
4.6 
4.9 
5.S 
5.7 
6.1 


ai 

3.5 
3.9 
4.2 

4.6 
5.0 
5.4 
5,8 
6,8 


3.8 
3.6 
39 


8.9 
3.8 

3.6 
4.0 


3.(1 
3.3 
17 
4.1 
4.« 
4.9 
6.3 
6.7 
«.t 
6.5 






to 

»S 
U 

t« 

Jfi 

so 
Si 
s* 
s<> 

S8 


JO 


8,31 

a.67 

S.03 
D39 
9.7» 
10.05 


4.3 
4.7 
5.1 
5 5 
5.9 
6.3 


4.4 

4.8 

6.6 
6.0 
6.4 






9-93 10,SB 

10.34 11.08 
10.7411.4* 
I1.18I1.B5 
11.491 18.85 




4,4 






5.0 

£.4 
5.8 


4,8 

5,5 

5.9 






1H 
34 

J6 
St 


10.37 
10.67 
10.B5 
li.S! 
lt.4H 
ll.7« 
11.95 
1?.17 


11,48 
11. J2 
18.01 
18.8» 
18,56 
18.79 
13.08 


18,19 18.99 
18,5813.3* 

18.83' 13.67 
13.1813.99 

13.4011.89 
13.<I6 14.57 
13.9114.83 
14.14 15.08 
14.36jl5,30 


13.88 
14.19 

15.80 
15.50 
15.7B 
16.04 

16.88 


14.66 
15,06 
15.44 
15.79 
16.13 
10.45 
16,75 
17 03 
17,88 


58 

54 
46 
58 
60 
68 
64 
66 
68 


7.8 
7.5 
7.9 
8.8 

8.9 
90 


6^6 
7.0 
7.3 
7.7 
8.0 


6, 
7, 
7.5 
7.8 
8.8 


6.9 
7.8 
7.6 
7.9 
8.3 
8.6 
8,9 


7.0 
7,4 

81 

8,t 
8.8 
9.1 


7 1 

7*9 
8.8 
8,6 
8.9 
9.3 
9.6 
9.9 


1.8 
76 
9-0 
6.4 

8.7 
9.1 

D,i 
9.7 

la-i 


6.9 
7.3 
T.T 
fl.I 
S.5 
8.9 
9.8 
B.6 
99 

ia« 






9.2 


8,8 

9. 

9,4 
~9.fl 

9.8 
10, 
10.3 

la* 

10.6 
10.7 

10.H 

lae 

10,9 
10^ 






ti.iS13.it 


9,5 


97 






II 

14 

IB 
IB 

« 
ti 

86 


18 
16 

\i 

10 
8 
6 
4 


18,78 
U.S8 

13,0! 
13,14 
13.84 

13.33 
13.10 
13,46 
13 50 
1353 


!3.6i( 
13.79 
13^94 
14-07 
14.IB 
14.88 
14-36 
14.48 


14.55 15,51 

U,73|l,5.70 
14,89 1 5 B7 
15 03 16.08 
15.1516.15 
1S,85 16,8B 
15.3* 16,3.* 
15.41 16.48 
15.45 16,47 


16'70 
16.88 
17.04 
17-18 
17.S0 
17.39 
17.47 
17.58 
IJSS 
1T.S7 


17 51 
17.73 
17,98 
IB,09 
18,84 
18,36 
18,47 
18-55 
18.60 
18.63 

i8.ea 


70 
78 

78 
80 
88 
64 
B6 
88 
90 






9.8 
10,0 
10.8 
10.* 
10.6 
10,7 
10.9 
II.O 




10-4 

ia6 
laB 

ll-O 

IKS 

11,4 

11,4 

11.5 


lat 

IO.t< 
11,0 
11.8 
11.3 
11.4 
1.5 
11.6 
11-7 
11.7 


lai 

ll,A 

ii.i 

11.4 
11.5 
11.6 
II.7 
ll.B 
11.9 
11.9 






9.5 
9.7 

io!o 
10.8 
las 

10.4 


97 

9.9 
10.1 
10.8 
10.4 
10,5 
10.6 

lafi 

in.7 


10 
10 

10 
10 

II 
II 


8 

* 

9 

8 
8 
3 
3 




£ 




±S0. 


5.4S 


I6.S1 


lO.S,!*'' 


lU 


11 





TABLE XXIII. 

Loi^rithms of the Earth's Radii, in each 
Parallet of Latitude; the Equatorial Ra- 
dius being Uoitf, and Compression g js- 



LaL Log. R Lat. Log. R Lat. Log. R 



TABLE XXIV. 1 

Angles of the Verticil with the B . 
ilius ; 01 Reiluctioii of tbe Latiinde, 
in mch Parallel, the CompmsioB 
being ,ij. 



0" a.ooooooo ; 

1 19.99999!) a : 
998* : 



y.9999477 t 
93TS i 
9273 4 



1° 9.!)99fil0i ISO" i 



01 



9S9ST ( 

BS7tS 6 

BSK6 6 

95961 S 

95(1 ?3 66 
91 IS J 



87T3S 
S7S5i 
8737 S 



9.99937y4 1 
9354.;) ; 
93£9t 'i 



10 7.2 



10 



n 



10 SS.3 
10 3T.S 
10 4.fl.'V 

10 54.3 

11 l.« 
II 7.7 
II 13.S 



9 5T.4 

9 45. 1 

9 31.0 

9 18.3 

9 S.P 

8 tB.T 

8 3«.9 

e 16.6 



8711 4 

86!7 4 

847fi 4 

8318 _5 

9.9998153 5 

7993 & 

TiOS a 



79 
80 

■9.99'iI377 "i 
9103C 82 
9078S 83 
90Sii Bl 
9030i 8 
SOUSi 8 
87 



II i7.3 

II iS,2 

11 ».3 

1] 1S.G 



S.G I 

10 fii.T ! 

10 47.9 : 

10 39.4. 1 

10 30.0 I 

10 19-9 I 

10 9.0 I 

9 57.1 ■ 



4 41.0 
4 18.8 
3 5«.3 



TABLE XXV. 
For determiaing the Laiitade, at ani/ time, bj the Pole Star. 




^ = Z + f/txs.l — M eotui. Z + N : where i^ is •- the Latitude ; Z = tbe ZeniA 
Distance t p= l°40',aT 100'; I =^ the Horary Angle; '= tin the tint Quadrant ; = '~ 

in llie lecotid ; =1 — 12" in the thiid ; and = 24^ — (in (he fourth ; M and N hi „ 
Ae Tabular Quuidlto. The quiniily M is !=- J ^' ain.* I, and h atwaTS poaitive ; but tbe 

titf ti ^ ip' sin. If cdi. I. becomes Degatiie in die second and third Quadrants of 1. 

■n p (ihe Pillar Dislance) Huyments or diminishes l', tht Tabular Quantity must alaa be 
augmented or dimiiuilied by 0,02M ; and for any ddier quantity of vstiatioo in theti 
pruponioii. 



! TABLE XXVI. 

a flnd the Augm en lotion of ihe Moon's Semidiameter by the Altitude of the 
Nonagesimal, and the Apparent Distance of the Moon therefrom. 

PA RT I. 
nana. + app- du. Modti Tt. n 
. noha. — app. dia. MnOD ft- 1> 



i 40 (. 
5 »0 ( 
S I 



1(1047,0-610.8*1.01 
>lO-ttaMO.T4U!l 



HD.1! 



ieo.isu, 

XiO.IUO.liO. 

)*(i.0K0.ia 



■ 0.010.03 3. 

+ I 
13 0.01 + 0. 



SetridiamMer of the 



mD.OGOOta 
.Tr).18[).O8 0. 

i).lsa.[io. 

RJD-ifi 0.17 0. 

I! 0.31 o.«i a 

irJU.37 0.*3 0. 

« o.44[atg 0. 

13 O.«7 0.SO 0.330. 



li i.ot 
13 !s.l6 
1* ;a.3' 



l.g3 | l.l'J-t [O.B3|a.fii'o.48 0.al 

.60137,1 
1.7S 1.50,1.*.=. I 



1. 89 1. fid 1.32 
(.04'I.T51.4« 



in 0.750,50 0- 
IB0aiO.5tO. 
l7n,HT0.i8O. 



itO-ft 

ruD-ftj 



^aeoij-Boi. 

i3 0-7H 1-02 1. 





TABLE XXVn.— Equations .,f Secund Difieraicra for Twelve Houra. S3 






"^^m^ 


Statad Ditfbnnce. 












8- 


3' 


V 


^■ 


6' 


7' 


8' 


V 






1 f 


if. ( 


D 0.00 0.0 


11 lir 


) o.r 


0.0 


0.0 


ao 




0,1 


0.0 


0.0 








1 II 


11 JM 




1 13.; 


1 0.' 


J 0.f 


) I.t 






) 8.5 


) 8.S 


) 3.3 


> 8.7 






1 V( 




» 8.10 IfiS 


) !i.r 






1 2.1 




) 4.1 


1 4,i 


5 5.7 










1 M 


II H( 


I l?.00 SU 


1 3.5.! 


1 1.1 


) 8.1 


) 3.1 


) 4t 


) 6.1 


1 7.1* 


i 8.4 




1 10- 






1 u 


II yi 


) IS.7,0 ai I 


) 47.) 


1 ih 


> 3.1 


) 4.5 


1 h.; 


> T.l 


1 HI 












) »] 


11 K 


l!>-t|0 3fl.« 


5rt! 


) l.H 


3.!» 


) 5.M 


78 


CI 9.7 


11 


I 




15! 


17.' 










I noo *.5.t 


t HI 


1 t.\ 


J 4.li 


1 (J.) 


) 9.8 


) ll.f 




) 18.11 


) IB.S 


1 80.1 






11 


<) * 


) SB3 sa.i 


1 ig( 


1 8.1. 


J 5? 


1 7.1 


t 10.5 


) 13.1 


) I.W 


1 18.4 


) 81.1 


) 83.' 








41 


1 S9.6 S9.^ 


1 !8i 


1 3.1 


] 5.! 




) 1l.( 


) 14.^ 


) n.t 


) 80.7 


) 13.1 


) Ifi.' 






SI 






[ S.f 


1 38.i 


1 3.: 


1 ai 




) 13.1 


) ie.4 




1 83.C 


) 86.; 


) 89.5 






41 




1 3.1.! 


1 ii-t^ 




1 3.1 


1 7.8 J 10.t 


1 141 




1 81.5 


1 85,1 


) 88.1 








1 Sf 


II) It 


.18.8 


1 17.7 


1 .5H.! 


) 3.H 


7Bn ll.elO 155 


19.4 


1 8:1.3 


87.8 


31.1 


) 34.! 






f. I 


11) 1 


1 41.; 


1 rx: 


« 51 


) 4.>i 


) 8-3,0 I^.JlO ll,.i 


) io.f 


1 f.-kl 


) 89.8 










9. 1( 


» SI 


) 4I.1 


1 1?H.» 


a u.i 


1 4. 


) 6.90 ia30 I7.t 


) 88.8 


1 SH.h 


} 31.1 




I 39.! 






! «! 


» 41 


) 47.1 


1 ■AU 




J 4.: 


■> 9 40 I4.l[0 lai: 


1 83.5 


) 88.? 


) 38.9 


) 37.6 


) 48.; 






• -M 


n SI 


) 4e.£ 








1 9.B0 14.80 l!l.f 


( 84.7 


) m' 


I 31-6 


) 39,( ) 44.^ 






t V 


n 91 


) il.i 




i 35.; 


1 h.t 


J 10.40 15.60 80 1 


) 851 


1 31.1 


) 36.3 


) 41,/ 


)4fi.7 






t S( 


« 11 




1 *H.i 


S 48.; 


) 5.4 


IOH,0 Hi.a|0 81.ti 


) 87-1 


32.5 


37.9 


1 43.3 ] 4R.7 








H 1 


1 MS 


1 .'iil.,1 


a 48.; 


1 5.1 


1 11.3,0 1U.W0 8S.5 


) 88.1 


) 33.t 






1 50.f 






i 11 


H fli 


) 38.5 


1 M\.f. 


i stt 


1 .-i.! 


) 11.70 17.50 83.; 


) 89.1 


1 35.1 


) 40.S 


) 46.e 








i f< 


H 41 


; a« 


f 0.. 


3 0.1 


1 lil 


1 18.0,0 18.10 84.1 


) 30.1 


) 311. 


] 48.1 


) 4ai 








1 M 


H 31 


1 a,( 


• 4-1 


3 5.! 


) r,.: 


) >«.40 13.tiO i4.( 


) 3l.f 


) .37.8 


) 43.1 


) 4B.II 


1 55.( 






1 4( 


H fM 


1 3.5 


a 7.r 


3 IH 




1 18 7 19.10 2SS 


1 3l.t 


) 38.? 




) 50.1 


) 51.; 






1 » 


S l( 


1 5.2 


? 10.4 


3 15.R 




13.00 19.6,0 81i.l 


1 38.6 


39.1 


45.7 




) 58.7 






t Oj B ( 


1 d.i 


i {%-. 


3 ao.i 


1 «.■ 


) 13.3 80.110 8li.l 


1 3,*!.; 


1 41M 


) 16.1 


1 53,2 


; o( 






I so' T 1( 


1 9.8 


t IH.' 


3 87.1 


) «.! 


) 13.8 80,8 87.5 


) 34.5 


1 41.5 


) 4B.4 


1 551 


: !.: 






V 40| 7 «( 


1 11.; 


It n.i 


3 33.! 


1 7. 


) 14 3 81.4 88.S 


1 3.5.t Q 48.1 


) 49.S 


) 57.C 


1 4.! 






i 7 I 


1 12,1 


t ?.i.i 


3 3H.t 


1 7f 


1 U.BO 81.90 89.8 


1 36.5 43.1 


) Sl.C 




1 fi.l 








L 14.1 


' ^H? 


3 48. i 


1 7' 


) 1480 8880 89.6 


) 37.00 4+.^ 


) Sl.C 


) 59.E 








; iu' r. !o 


1 14.8 


i «e.s 


S 44,; 




a 15.00 88.40 8D.9 


) 37.40 44.9J0 S8.3 


.59.8 


[ J.: 






1 01 6 


1 i6.uig m.o 


3 45 ( 


1 J.5;o 15.0 88.5 30,0 


) 37.50 4.-..OiO 58.5 


1 0.0 


1 7.5 








ISeund DiRerenee. 


S>.-=f,S 






"M'nii'Bh"'" 


10" 


!0"|30" 


40" 


50" 


1" 


8" 


3" 


4" 


ft" 


6" 


7" 


8" 


9" 












" i ■' 
























h. m 








t 1 


lit 1 





0.0 


<).0 


0(1 


0,11 


0.0 





0,0 


0.0 


0,0 


11.0 




OO 





1 


84 






1 1' 


11 Al 


(M 


III 


11. IJ 


(fl 


0.3 


III) 

















0.0 


n.i 


0,1 


8( 


83 40 










1)1 


II, 't 


11.4. 


(if, 


0,1 








00 




01 


1 


0,1 


1 


0.1 












II HI 


Il.» 


0.-1 


0.1! 


0.H 


1.0 










(M 


<U 




o.s 


0.8 




83 








11 VI 


<I,H 




ll.H 


l.n 


13 


0.0 


0.1 


0.1 


0.1 


<l.l 






0.8 


0,* 


1 8( 


a 40 






1 51 


II 11 


VA 


0.H 




1.3 


l.« 


0.0 


0.1 


0.1 


0.1 


0.8 


0.8 


0.8 


0.S 


0.3 


1 V 


88 20 






. 1 


IJ 1 


114 


0.H 




1.5 


l.H 


0,0 


0.1 


0.1 


0.1 


0,1* 


Of 


0.3 


0.3 


11.3 


t 1 


^8 U 






; II 




«>1 


rtl 


1i 


l.H 


!>? 







1 


Of 


Oil 


01 


01 


0.4 


4)4 


8 81 


81 40 






VI 






1 II 




»ll 




OO 




01 




0* 


O-l 




4 


04 


8 41 










111 H( 


0.1 






?.« 


8.7 


0.0 




0.8 


n.t 


0.M 


a.1 


a4 


0.4 


0,5 


3 1 


81 






. * 






I.K 




8.4 


3.0 


0.1 


0.1 


0.8 


0.¥ 


(1.S 


11.4 


0.4 




0-5 


3 21 


80 40 






I BO 


111 11 


ill) 


i-:( 


1.U 




3.8 


0.1 


0.1 


0.5( 


0.3 


0.3 


0.4 


0.4 


0.5 


0,6 


3 40 


80 20 






•( ( 


ID 1 


1).'; 






a.M 


:4.5 


1 


lU 


OV 


l\'\ 


«.H 


0.4 


0.5 






4 1 


80 






f II 


H .11 


1J.7 


i-i 


v.v 


.S.I1 


3.7 


lU 


01 


0.1* 


OH 


<l,4 


<I4 


0.5 




0,1 


4 21 


19 40 






i* fl 




<)H 


I.H 


y.n 


;i.i 


.1« 


0,1 


0* 


l).f 


0'^ 


0.4 


0.5 


5 


ll,li 


(1.7 


4 4< 


19 80 






■ !i( 


1 HI 


rt-* 


in 


1'..1 


S-^ 


4..I 


1 


<\t 


ni" 


ot 


0.4 


0.5 


0,fi 


0.7 


7 


5 1 


19 










rti 


1 7 


?,« 


•i.s 


4,3 








03 


0.4 


0.5 


on 


0,1 


Ott 




18 40 






)• JO 


a 11 


0.9 


1 8 


2.1 


3.*! 


4.5 


0.1 


0.8 


a3 


0.3 


0.4 


0.5 


0.8 


0.7 


0.8 


5 40 


18 20 






* 1 


!» 1 


0.0 




88 


3.M 




0.1 


0.8 


0.3 


0.4 


0.5 


0.fi 


07 


0,7 


0.rt 










t II 


H .lil 


i.i> 


I.H 


J-H 


S.H 


4.11 


0.1 


o,a 


0.3 


0.4 


0.5 


0.(i 


41.7 


0.8 






17 40 






1 w 


H 44 


10 


)!.0 


H,ll 


4.11 


.1.0 




o.t 


0.H 


0,4 


0,5 


416 


0,7 


0.8 






17 80 






1 :<i 


H ■^ 


1.0 


ii.1 


HI 


4.1 


!i.f 


1J.I 


{\.i 


0.3 


04 


0.5 


Oil 


n.T 


0.8 




■7 ( 


17 1} 






1 41 


K »1 


I.I 


?.l 


H? 


4? 


5 4 


0,1 


M. 


0.3 


0.4 


n.s 


Oil 


or 


0.H 




1 21 


16 40 






t SO 


8 10 


1.1 


s.i 


3.a 


4.3 


5.4 


0.1 


0.S 


0.3 


0.4 


0.5 


0.4J 


0.8 


0.!! 


1-0 


7 40 


IB 80 






















0,* 












09 












I it 


7 41 


l.X 




;4..i 


4fl 


.1.8 


lUl 




0.3 


0.5 


0.rt 


0.-T 


08 


o.« 


1.0 


8 41 


15 80 






\: 41 


1 Kl 




¥,4 


H.H 


4.8 


5.H 


0.1 


11.8 


0.4 


0-1 


o.n 




08 


1.0 




9 81 


14 40 






S ( 


7 1 


1.* 


f.4 


H.7 


4.H 


li.l 


IM 


0.8 


4 


0.5 


0.<i 


0.1 


0.9 


1.0 


1. 1 


W "^V. Q 






S jfl 


(1 41 


l.V 


y.A 


8.7 


4.0 


fi.y 


0.1 


0.8 


114 


ti.5\ll.6\0.l\(S.9\V'a\\\\V<i <tf*\-J.>R.1 




W( 


6 1, 


1.2 


?.i 


ai' 


&0 


fi.S 


0.1 


0.«lo■'^'\oA\^>.a\^i.^^«.*\^a\>■•^V\ 




I 


« 01 e oij.il is 1 3.11 6.0 


e.s 


ai 


o.a\^^.*\o-5\n.«\^^,^\^^.'A^.ftViA^ 



^ 



r 


^ 




^ 


9( 




TABLE XXVm. 
















(1 


On> 


im 


8" 


3" 


ic 


S" 


fim 


-T- 


8>» 1 B- 


lO"- 


JV-^IJ^ 







OOO 


1.96 


7.8S 


„;, 


31.41 


49.09 


7o"6a 


fl6.80 


185.65 159.08 


196.38 


837.64 898.69 






1 


0.00 


8.03 


7.99 


17.87 


31.68 


49.41 


71.0' 


96 66 


186.1 7159.61 


196-97 


8-18.96 MS 4« 






8 


0.00 


*.10 


8.18 


19.09 


31.94 


49.74] 


71.47 


B7.18 


186-70160.80 


197.63 


!3a.9BSe4.SS 






3 


0.00 


8.16 


8,85 


18.87 


38 81 




71.86 


97.59 


187.83160.791 


199.89 


839.70 KtS.04 










8.83 


8.39 


18.47 


38 47 


50,40 


78.86 


98.01 


1 87.75 161.3» 


188-94 


840.48,!BiM 






S 




8.30 


8.58 


18,67 


38.74 


50.74 


78.66 


B8.5I 


188.89161-991 


199.60 


841.1 6;SM.6! 








o.ot 


2.3H 


8.66 


19.87 


i3.0I 


51.07 


73.06 


98.97 


188.81168.58 


800.86 


841-87,887.41 






7 


a03 


8.45 


B.S(1 


19.07 


J3.87 


5J.40 


73.46 


99.44 


18S.34 163.17 


800-93 


84(.60Sa&tO 






8 


0.03 


8.S« 


8.94 


19.88 


S3.54 


5'1.74 


73.86 


99.SO 


189.97 163.77 


801.59 


843.33*88^ 
!i4.06'8BS.7« 






9 


o.ot 


8.60 


B.09 


I9.4H 


3a88 


58.07 


14,80 


100.37 


130.41 164.37 


808.85 




[0 




e.6T 


9.88 




34.09 


mTT 






130.94 164.97 


80898 


844.79.aw.se 




11 


OOT 


8.T5 


9.36 


19. BO 


34,36 


58.75 


75.07 


101.3] 


131.48 165.57 


803.68 


8*6.68 89L38 






IS 


0.O8 


8.83 


9,50 


80.11 


34.6* 


J3,0B 


75.47 


101.78 


138.01 166.17 


804.85 


846.86891.18 






13 


0.09 


8,91 


S.fiS 


;0 38 


31.BI 


53.43 


75 88 


108.85 


138.55 166.77 


804.98 


846-Sig,S9t.Ba 






14 


0.11 


899 


9,78 


80.53 


3S.1B 


53.77 


7e.!fl 


108.78 


133.09 187.3? 


805.59 


847 T8 (93.79 






15 


0.18 


3.0T 


9.94 


80.74 


35.46 


54.18 


76.70 


103-80 


133.63167.98 


806.86 


!4B.4« 894.58 






IS 




3.1S 


loos 


80 95 


35.74 


54.46 


77.10 


103-67 


134^17 168.59 


806.93 


849.19 !B&3» 






17 


0.16 


3.83 


10.84 


ai.I7 


36.08 


54.81 


17.51 


104.15 


134.71 169.19 


807.60 


819.93,896.18 






18 




3.3! 




81.38 


36.30 


55,15 


77.98 


104-6^ 


136.85169.80 


809 87 


850.67 


!!)6.99 






19 


O.J0 


3 40 


10.,5l 


81.60 


36.59 


5S.,W 


78.31 


105,10 


135.79170.41 


809.95 


85141 


897.79 




!U 


o.*g 


3.4» 


1089 


81.88 


36.87 




78.75 


105.59 


136.34 TtIToS 


809.68 


858.16 


89e.M 




gl 


0.84 


3.58 


10.H4 


88.03 


37.15 


56.80 


79.17 


106.06 


136.89171.63 


810.30 


858.89 


!M.40 






gj 


0.86 






88.85 


3T.44 


56.55 


79.58 




137.43 178,84 


810.98 


853.63 


aoo.!i 






!3 


o.« 


3.76 


1115 


88.48 


3778 


56 90 


BO 00 


107.03 


137.99 178.86 


811.66 


864.38 


301M 






(4 


0.31 


3.es 


U-31 


iJ.70 


38.01 


57.85 


80.48 


107.51 


138.53 173.47 


818.34 


855.1! 


30I.S3 






S5 


0.34 


3.S1 


1147 


88.98 


38.30 


57.61 


90.84 


108.00 


139.08 174.09 


813.0! 


855.87 


301.65 






!6 


0.S7 


4 03 


11.63 


8;tl4 


38 59 


57.96 


S1.86 


108.48 


139.63174.70 


81170 


856.fi! 


30X48 






ST 


0.-U) 


4.13 


11,79 


83.37 


3M.88 


58.38 


91.69 


109.97 


140.18175,38 


814,38 


857.37 


304.87 






2S 


0.43 


i.ii 


L1.05 


f3.60 


39.17 




88.10 


109.46 


140.74176.94 


815.07 


868.lt 


305.01 






«9 


o.4e 


4.38 


18.11 


83,88|3fl.4? 


59. OJ 


88.53 


109.95 


141.89 176,56 


815.75 


8S8.BT 


30594 






0.*B 




18:87 


!!4.05;39.78 




98.95 


110.44 


141.B5i77.lB 


816.44 


859.6i 


300.78 




31 




4.SS 


18.41 


81.88 






83J9 


IID93 


148.40 177.80 


817.18 


860.37 


SOT.M 






3! 


0.68 


4fi! 


18.60 


84.51 


40.35 


60.15 


8i.Bl 


m.48 


148.96178.48 


817.81 


861.1! 


308.SS 






33 


0.59 


4,78 


18.75 


81.74 


40,65 


60.48 


84.83 


111.91 


143.68,179 05 


818.50 


861^9 


sosLie 






34 


0.63 


4.8! 


18.94 


84-98 


40.95 


60.84 


84.66 


U841 


144.08179.68 


819 19 


!e8.64 


310^ 






35 


0.67 


4.»8 


13.10 


85.81 


41.85 


61.81 


85.09 


118.90 


144.64190.30 


819.89 


863.39 


3iaw 






36 


0.T1 


5.03 


13.87 


85-45 


41,55 


61.57 


85.58 


113.40 


145.8018093 


880.58 


864.16 


3ll.6t 






37 


0.J5 


5.13 


ia*i 


85.68; 41.85 


61.94 


85.Bb 


113.BO:i4S.77|181.56 


881.17 


8S4^Si 


11S.W 








0.79 


5.84 


13.88 


85.98 


48.16 


68.31 


96.39 


114.40148.33188.18 


881.97 


IS&«7 


itxao 






39 


0S3 


5.35 


13.79 


86.16 


48 45 


68.08 


96.88 


I14,90[ 146.90 198.88 


888.66 


E66j4: 


SI4.II 










Tu 


13.96 


86.40 


iHi 


63:05 


97.8U 


116.40147.46,183.45 


813.36 


367:!C 


314.95 






41 


0.9« 


5 56 


I4sl4 


86.64 


43.06 


63.48 


91.70 


116.90149.03184.09 


884.06 


atnso 


31B.7ft 


1 




4E 


a9G 




14,31 


86,88 


43.37 


6179 


99.13 


116.10148.60184.78 


884.76 


!88.7! 


il«.8l 






»3 


1.01 


5.79 


14.49 


87.18 


43.68 


64.16 


99.5? 


116.81149.17 185.35 


886.46 


4S8.M 


318,44 






*♦ 


1.06 


5.90 


14.67 


87.37 


43.99 




99.01 


117.4l!l48.74 185.99 


8!6.1fl 


(TD.C6 


llfttT 






45 


1.10 


6.01 


14.85 


87.61 






89.46 


117.98, 1. 50.31 1186.63 


886.96 


871.03 


919.lt 






40 




6.13 


tfi.03 


87.86 


U.61 


65.!9 


99.flO 


119,43.150.88187.87 


8S7.S7 


871.79 


sie.»4 






47 




6.84 


tfi.81 


88.10 


44.98 


65.67 


90J4 


119.94,151.46 187.91 


889.87 


878.67 


380.16 






4S 




6.36 


16.39 


88.35 


45.84 




90.79 


119.45 168 03188.55 


888.88 


87334 


}tt.«8 






49 


1.31 


8.W 


1S.5(« 


89.60;4S.55 


66.43 


91.83 


Il9.1l6158,6l'l68.l9, 


888.69 


874.11 


J8845 




m 


T3ir 


e.6o 


ls.76 


88.95:45.87 


66.9! 


91:68 


18047153.19 199.83 


830.40 


874.88 


SS&M 




51 


1.4! 


fi.T8 


15.95 


89.10 


46.18 


67.19 


98.13 


180,98153.77 190.47 


831.11 


na.6t 


381.18 






5! 


1.48 


6.B4 


16.11 


89.36 


16.50 


67.58 


98.57 


lai..50 154,35191.1! 


831.91 


876.43 


dit-n 






53 


1.53 


6.B6 


16.38 


89.61 


16.88 


67.96 


93.08 


188.01 '154.93' 191.76 


8;}8.S3 


877.81 


S8WtS 






A4 


1.59 


J.09 


16.51 


89,86 


17.14 


68.35 


fl3.47 


188.53 155.61 198.41 


833.84 


87T.SB 


38e.M 






55 


1.05 


7.81 


16.70 


10.18 


47.46 




B3.93 


1!3.06:156.0B: 193.06 


833.9S 


879.77 


3tT.50 






56 


1.71 


7.34 


16.89 


30.38 


47.79 


69.1! 


94.38 




156.68183.71 


834.67 


(T9.A6 


SfS.S6 






57 


1.77 


7.4T 


J 7.08 


30.64 


48.11 


S9.51 


94.83 




157-86 194.36 


835.38 




SCftIO 






58 


t.83 


7.59 


1T-8B 


30.89 


49.43 


69.90 


9.S.89 


184.61 


157.95185-08 


836.10 


891.11 


S3ao> 






59 


I.BO 


7.78 


17.46 


SMS 


49.76 


70.S9 


95.75 


183.13 


lSrt.43|l 85,87 


836.88 


ms3 


!80J.» 
"a 16' 




/ ^ / 


o.oi am 


0.03 


QM 


006 


TTot" 


6.09 0.16 


o.i'i"To.i'8 al4 


roiT 




/ ■« / 


aoi OM 


0.06 


ao9 


ai8 


0.14 


o.n \ 0.W 


0.88 0.85 a8« 


o.ao 


0.3S 




. 


/ .« / 


rog 0.06 


0.10 


0.U 


ai8 


0.81 \ 0.46 \ 0.311 \ 0.^ \ 1.41 \ «.\\ 


0.4S OM 




1 ( 


r .» /o 


oelo.m 


CIS 


0.ltt 


0.24 


0.e9\o.3V\ 0.w\o.«.\0.«i \»,Vo\(l*« 


XS& 


J 


t 






s 


■ 



T. XXVIII 


TABLE XXIX.— Reduction to Mthn- Solstice- 9S 




P»BT 11. 


Obi 


quity of the Edipdc 83" 87 


40-- 








Arg. Reduc. 


DUE ^i^„, 


Arg. 


Reduc. 


Diff 


s 


A-K. 


Rriac. 


Diff 








+ 


"o o'o o"oo 


0.03 0.0000 


10 


1 11.71 


2.41 


[).0GS1 


80 


4 46.93 


1.80 


0.8808 








lolo aos 


0.0(1 0.0000 


10 




S.4S 


[1.0673 


10 


1 51.6:t 


18 


0.2650 







0.00 


300 o.oe 


0.100.0001 


80 


1 16.57 


S-19 


0695 




4 56 4- 


1.88 


0.8694 




30 


0.00 


300 O-IS 


0.14 0.OOO8 


30 


1 19.06 


2..W 


[).0717 




5 1.35 


1.92 


0.8738 




1 


aoo 


400 0.3? 


0.18 0.0003 




1 S1.59!8..57 


n.O710 


40 


5 6.8J 


4.96 


0.2783 




1 SO 


ftOO 


50 50 


0.8S 
0.26 


0.0005 


50 


1 81.16 8.61 


[).n763 


St 




SW 


n.8BS8 




t 




1 00 0.7* 


0,0007 


U 




0.O78J 


il ( 


5 16.!:? 




0Wt73 




t 30 


aoo 


10 o.ss 


D.89 0.0009 


10 


1 t9!lS2'6g 


0.0811 


1( 


S 81.87 


5.0J 


0.8919 




3 


0.00 


SO.O 1.S7 


0.34 0-0011 


80 


1 3S.1 1 8.7g 


D.Dfl36 




5 86.34 


5 12 


D.S965 




8 30 


0.00 


30 l.Bl 


0,38 0.0014 


30 


I 34.8:18.77 


0.0960 




5 31.46 


S.16 


OSOlt 




* 


0,00 


400 1.99 


0.4S 0018 




1 37.60 2.8 


[).0BS6 


40 


5 36.6? 


,i.80 


0.3059 




* 30 


Ofll 


son 8.41 


0.46 0.00S8 


SO 


1 40 41 2.85 


0911 


50 


5 41-82 


,^.2 


3106 




S 




S 0,0 S.MT 


0.50,0. oose 


IS 


1 13.26 2.89 


0.0937 


12 


5 *7.0h 


S.8P 


31S1 




10 




100 137 




10 


1 46.15 S.93 


0.0963 


10 


S 58.31 


S38 


0.3208 




so 




SOO 3.91 


0.57 0036 


SO 


1 49.08S.96 


0.0990 


SO 


5 57.66 


S.3, 


0-3S50 




30 


0.01 


300 4.48 


D 68 0.0041 


30 


1 58.04 3.01 


0.1017 


30 


6 3.01 


S.40 


0.38l>9 




40 


0,01 


400 5.10 


0-66 0046 




1 55 05 3.05 


0.1014 


40 


e B.41 


S.41 


0.3349 




*0 


OOJ 


500 S.76 


0,69|0,00ai 


SO 


1 58 10,3.09 


0.1072 


50 


6 13.85 


5.18 


0.'W99 




6 


TuJT 


3 0,0 fl,4S 


0-74 0.0059 


13 


3 1.19.ai8 


0.1100 


13 


6 19.3.1 


i.a 






10 


0.01 


100 7.19 


0. 7 b' 0,0066 


10 


8 431 


3.17 




10 


6 8181 


5.5ft 


3198 




SO 


0.01 


SOO 7.97 


0.91 lo 0073 


SO 




3.81 




80 


6 30. H 


S61 


0.3549 




30 




300 8.7S 


D.8fip.O0SO 


30 


2 10.69 


3.85 




SO 


6 36.00 


5.6 


0.3><DO 




*0 


O.OS 




0.89O.0088 


41 


2 13.94 


3.88 


n.I81B 


10 


6 11.'.4 


S.67 


n.sesi 




so 


0.0% 


50(0 la53 


n.94|0.0096 


50 


S 17.B8 


3.33 


0-1846 


SO 


6 47 31 


5.7* 


0.3-O3 






o.oa 


* 0,0 11.47 


970.0104 


14 


2 20.55 


3.36 


0.1876 


21 






asisi 




10 


oos 


100 1S.44 


1.08 0.0113 


10 


8 S39I 


3.41 


0.1307 


10 


6 6(*.78 


S.B( 


0.3807 




80 


0.03 


SD'o I3.4B 


1,06 0.0188 


SO 


8 87-38 


3.45 


0.13:18 


80 


7 4.5c 


5.B4 


a-3H«> 




SO 


0.03 


30'0 11^5S 


1.09,0,013? 


30 


2 30.77 


X18 


0.1369 


30 


7 10.12 


5.87 


[1.391.1 




40 


0.03 


40 1S.61 


1.14!o.01« 




8 34-85 


3.53 


0.14O1 


40 


7 16.89 


5.BS 


0.3967 




SO 




SO'O 16.75 


1.Ib|0.0I58 




S 37-78 


366 


a 113:1 


50 


7 22.21 


S9S 


4011 






■qo? 


5 00 17.93 


LSI 


0.0!(i3 


15 


2 41.31 


X§I 


1.1 46,5 


85 


7 Sa,16 


-iOll 


0.11175 




10 


0.0 1 


100 19,1* 


1.86 


0.01 7 1 


10 


a 4195 


3.64 


0.1198 


10 


7 34.16 


6.03 


0.113O 




ao 


O.05 


SOO S0.40 


1.89 


u.oie5 


20 


2 48-59 


J.69 


0.1531 


SO 


7 10.19 


BOB 


D1I85 




so 


0.05 


300 81.69 


1.34 


aoiflT 


30 


8 58.28 


3.78 




30 


7 46.27 


i.11 


0,4i4(t 




40 


0-05 




1.37 


0809 


40 


2 56.00 




0.1598 




7 58.38 


6.1.i 


0.4890 




so 


0.06 


50|0 S4.40 


1.4S 0.0821 


5( 


8 59.77 


^80 


0.1638 


50 


7 59.S3 


6.20 


0-4358 




9 D 


ooT 


6 00 2S.H8 


T:5s!o.0834 


ir-0 


3 3.57 


3.85 


1U67 


i6 




oTj 


0.4408 




10 


0.07 


100 27.S7 


1.49 


0.O817 


1( 


3 7 42 


3.88 


0.1 70S 


10 


8 10.96 


6.88 


aU6S 




to 


0.07 


SOO SB. 76 


1.51 


0.0861 


SO 


) 11.30 


3.98 


0,1737 


SO 


B 17.81 


6.31 


0.4SU 




30 


0.08 


30 30,30 




0.0874 


30 


3 15.83 


1.97 


1773 


30 


6 83 SB 


6.35 


0.157 » 




40 


O.OS 


400 31.37 


1-68 


0SH9 


40 


3 19-19 


1.00 






S 89,90 


6.40 


0.4637 




50 


0.00 


5I)|0 33.49 


1.65 


0.0303 


51 


3 23.19 


+.05 


0-1815 


SO 


8 36.30 


8.43 


0.4695 




10 




7 


35.14 




0.0818 




3 87.84 


1.08 


0.1 Baa 


27 




5:47 


I147S1 




10 




10 


36.83 


1-73 


0.0333 


10 


3 31.32 


118 


0.1919 


10 


B 49!20 


6.52 


0.4913 




30 


0.1 1 


SO 


38.SS 


1-78 


0-0319 


20 


3 35.44 


4.16 


0.1957 


80 






04873 




30 


Oil 


3D 


40.34, 


1-81 0.0366 


30 


3 39.60 


1.81 


0.1991 


30 




6.59 


1.193! 






ais 


40 


4S.I5 


l.SS 0.0391 


40 


3 43.81 


4.84 


0.8033 






8.63 


0.199:1 




50 


a 13 


50 


4400 


1.B9 0.0399 


50 


J 4B05 


4.88 


0-8071 


50 


9 15.411 


G.S8 


0.5053 




n 


"ou 




1) 45.89 


1.94 0.0111) 




3 52.33 


4.33 


0-8110 


88 


9 S8.17 








10 




a 


D 47.83 


1 97 0.0134 


10 


3 56.66 


4.36 


0.8119 


10 


88.88 




(5176 




so 


O.IS 


80 


1) 49.80 


8-01 D-015S 


8(1 


4 1.02 


4.40 


0.21 89 


80 


9 35,03 










oie 


30 


[) S1.81 


S.05 0.O170 


30 


1 5.12 


114 


0.2S89 


31) 


9 42,12 


6.83 


0.5899 




40 


17 


40 


53.81 


2.09 D-OIBS 


40 


4 9.B6 


1.48 


0.8869 


40 


9 49.25 




0.5361 




SO 


0-18 


50 


55,95 


a, 13 0.0508 


50 


1 14-31 


4.53 


as3io 


-J2 


9 66.18 


R.Ol 


0.5181 




U 


0.79 






8. 1« 0.0527 






4 5<i 


0.8351 




3.0.i 




0.5187 




10 


0.S1 


U 


I 0.86 


S.81 0.0547 


JO 


4 83.43 


4.60 


0.8393 




9.98 


6,99 


O.SSS0 




so 


0.8! 




I 8.47 


S.85 [1.0567 


80 


4 88.03 


4.64 


0.8135 




1697 


7.01 


0.561 1 




30 


0,S3 


30 


1 4.78 


S-89 0.0587 


30 


1 32.67 


4-68 






8101 


7.OJ 






40 


0.S4 


40 


1 7.01 


433 0.0009 




1 3T.35 








31.08 




0.5713 




50 


0.S5 


50 


1 934 


8.S7 Q.06i9 






1.76 


0-2563 




38.19 


7-15 


0.5808 




13 


n,!7 


10.0 


1 11.71 


0.0651 


80_0 


4 4fi.93 


"■*^- 


pB 0\\o V&3A N:iJ««i\ 









TABLE XXXr. 




To chsnge mean Solur into Sidereal Time. 


To change Sidereal into mean Solar Time. 






Add 




^iM. 






D.J.. 
1 


3 55.908 


SIT 


sss. 


s^' 


".^ 








, 




1 




0.164 


, 


OOOl 




( 


1 T fi3.1ll 


8 


0.389 


» 


0.006 




T 51.8U 




0.338 




0.00J 




3 


1 11 *9.eiii 


S 


49^ 


3 


0.0OS 




11 47.78 i 




0.191 




0,OW 




4 


1 1.1 4fi.88! 


4 




4 


0.011 


4 


15 43.631 


4 


0.655 


4 


OOll 




5 








5 


0.014 


S 


19 39-S4<! 


5 


0S19 


3 


0.011 




e 


) i3 39.331 




0.996 








83 35.41! 








aoi6 






1 8J 33.H9! 


7 


1.150 




0.019 




87 31.35t 


7 




7 


0.019 




s 


1 31 38 4k 


K 


1.315 




0.088 


8 


31 87.86) 




1.311 


H 


0.088 




9 


J 35 SB.O0 






9 


0.085 


9 


35 83 17i 


9 


1.474 


9 


0JI8i 




10 


1 39 *S Sfiil 


10 


1.B43 


10 


0.087 


10 


3S 19.080 


10 


1.636 


10 


ofin 














0.030 












0030 












18 






. 17 10.891 






18 


0.031 




13 


) 51 15.88* 


13 


8.136 


13 


0.036 


13 


1 51 e.Ho; 


13 


8-130 


IH 


uoSi 




U 


1 5S l|.7m 


14 


8.300 


14 


038 


14 


1) 55 8.711 


14 


8-894 




OffH 




Ij 


) 59 8.?I4« 


15 


8.464 




011 








8-681 








16 








ill 








Ih 




0.0 M 






I 7 l.tS! 






17 










8.785 




ftlHS 




18 


1 10 5HO0I 






1« 






10 46.31^ 


IH 


8 949 


18 


0.0t» 




19 


1 14 .545fi- 


19 


3.881 


19 


0.053 




14 48.851 


19 


3 113 


IB 






80 


1 18 5I.I8( 


80 


3 886 


80 


055 


80 


18 38.161 


80 


3.877 


80 


O.Oil 




n 


1 a 47.6T( 


^1 


3.150 


81 


0.058 


SI 


n 3i.0H( 




3.110 


81 


I'Mi! 




fi 


1 86 44 83i 


99 




8^ 








it 


3.h04 


88 


OOfiO 












81 






30 85.88- 


Kt 


3 768 


?;t 


11.063 




84. 


1 34 37 341 


84 


3.943 


81 


O.O'iB 


84 


31 81. 7M 


S4 


3 938 


84 






SS 


1 38 33.90f 


85 


4 108 


85 


0.0R9 


85 


38 17.70( 


85 


4096 


36 


aoM 




i« 


1 48 3U 45' 


n 




86 




86 


48 13.IJ0I 




1(59 




OOTl 




8T 


1 48 *T.Otl 






87 


0.076 






87 


4.183 


tl 






8S 


1 50 83.5'>) 






88 








in 


4.5S7 


8S 


0.076 




!9 


1 54 811.1?; 


89 


4.7(1* 


?9 


080 




54 1.33i 


89 


4.751 


SB 






an 


1 58 }S.iiSC 


30 


4.9i« 


30 


0,082 


30 


57 57 84 


30 


41)15 


.30 


HO'S 






i 8 IM 831 


31 






0(W5 




! I 53 14b 


31 


5 079 


31 






38 


8 S 9.7BJ 


38 


6 857 


38 


0.099 




• S 4U.05I 


38 


5.8^8 


38 


0.047 




33 


8 10 6.a« 


33 




311 


091 


33 


! 9 41 96 


3:1 


6.40S 


H4 






3i 


8 i4 8.fll)-( 






It 


O091 






34 


5.670 








as 


? 11 5«.4& 




5.7.50 




097 








6731 


3S 


OOUH 




Sol Hr> 


m L 


36 


5.9U 


3b 


lOO 


iU. Hr-, 


IB •■ 


36 


5.898 


3« 


0098 






9.H565 


37 




37 


0.103 


1 


9.889 


37 


6.0S8 


37 


0.101 




i 


19.713 




6.818 


38 




8 


19.659 




6.885 


38 


aio4 




3 


8U.5S9 




G.407 




0.108 


3 


89.4b>8 


39 




39 


n.io« 




4 


3!).4ae 


10 


fi.,«l 


4J) 




4 


39.318 


111 


6 .i5:i 


40 


aios 




& 


4U 2B8 


41 


6.735 


41 


0.111 




4a. 147 




6.717 


41 


am 




6 


59,139 


48 


6.90t 


4i- 


0-1 16 




58.977 


18 


6.881 


48 


0.115 




7 


1 S.995 


43 


7-004 


43 


0.119 




1 8,806 


i:i 


7.041 


43 






8 


1 ia858 


44 


7.88.- 


44 




B 


1 ltl.636 




7.81)8 


44 


aito 




9 


1 88 7()M 


45 


7.39t 


45 


0.185 


9 


1 8H.465 


45 


7.378 


45 


0.183 






1 3>t.5e5 


4fi 


7.55T 




188 








7.538 


46 


0.186 






1 48.481 


47 


7.788 


47 


0.131 


11 


1 48.184 




7,699 


47 


ait8 




IS 


1 58.878 


48 


T.83« 


4rt 


0-133 


18 


I 57.951 


48 


7 864 


4H 


0.131 




13 


t 8.134 


49 


8.05C 


49 


0.136 


13 


8 7.7H3 


49 


8.1)87 


49 


0-131 




1* 


8 17.091 


50 


fl.814 


50 


138 


14 


i 17.613 


M 


8.191 


50 


ai3i 




15 


8 87.M47 


51 


9,37H 


51 


a 141 


15 


8 87.448 




8 355 


51 


U13« 




16 


a 37.704 


58 


fl.S42 


.W 


0.141 


16 


8 37.878 


M 


8,519 


,W 


ai4i 




IT 


8 47.5fiO 


53 


e.705 


51 


0.147 


17 


8 47.101 


61 






V.115 




18 


8 67.417 








ai6o 


18 


I 56.931 


51 






147 




IB 


3 7.873 


55 


9.03'i 


55 




IB 


3 6.760 


."li 


HOIO 


6S 


0.150 




io 


3 17.130 


51 i 


9.81H 


66 


0.155 


80 


3 16.5B0 




9.174 


56 


ai.u 




81 


3 8S.9e7 


5; 


9.36< 


57 


0.157 


81 


3 86.119 


51 


9 338 


57 


(I.I5* 




M 


3 36.B44 


M 


9.58* 


58 


0.159 


88 


3 36.819 


58 


9.508 




158 




« 


3 4fi.7(IO 




9.69S 


59 


0.168 


83 


3 46 078 












84 


3 ili.55« 


HO 


■t 8.jfi 


60 


ai64 


84 


3 65908 


<iO 


9.889 


GO 


0.161 




/ \''^' 




■ RLAi 









EXPLANATION OF THE TABLES. 



TABLE I. — The Miles and Parts of a Mile in a Degree of Longi^ 
iude at every Degree of Latitude, supposing the Earth to he a Sphere. 

The first column of this table contains degrees of latitude, the se- 
cond the miles and hundredth parts of a mile in a corresponding de- 
gree of longitude, — of these the remaining columns are a contmua- 
tion. If the given latitude consists of degrees and minutes, a pro- 
portional part of the difference between two contiguous degrees, the 
one greater and the other less than the given latitude must be appli- 
ed to the miles, &c. corresponding to either of the adjacent degrees, 
by addition or subtraction, according as it is greater or less than 
the given latitude. 

Example 1. — ^Required the number of miles in a degree of longitude 
at the Isle of May, in latitude 56° 11' 22'' N. 

Milcfs in a degree of longitude in latitude 56^=33.55 
in latitude 57 =32.68 



Difference .... .87 

Then W : 11' 22" : : "87 : 165, which, subtracted from 3a55, 

fives 33.385 ; the measure of a degree of longitude in latitude 56^ 
1'22". 

Ex. 2. — Suppose the error of a chronometer to be half a mi- 
nute, after a voyage from Leith to the West-Indies and back, how 
many geographical miles would that amount to at the mouth of the 
frith of Forth, near the Isle of May ? 

Since 1® of longitude is lequal to four minutes of time, then half 
a minute will be the eighth part of a degree, and } of 33.385=4.178, 
or about 4^ miles. 

Ex. 3. — What is the distance in geographical or nautical miles 
between Stockholm in longitude about 18^ £., and Peters- 
burdi in longitude 30° £., the common latitude being 60° N. 
neaSy? 

30°— 18°=12°, and 12 x 30=360 miles nearly, since at 60 one 
degree is 30 miles. 

TABLE II. — Logarithms of Numbers. — Part I. contains the loga- 
rithms of all numbers from 1 to 100, inclusive, with their proper 
indices prefixed. Part II. contains the decimal part of the loga- 
nthms of all numbers from 100 to 10,000, without their indices. 
The indices are easily supplied by the computist, being always one 
unit less than the number of integers in the ffiven natural num- 
ber. The index of the logarithm of a number in which there 
are any integers is always positive ; but, if the number be properly a 
fraction, the index is negative, usually marked by the sign — either 

a 




2 EXPLANATION OP TirE TABLES. 

before, or more generally above the index. If the firat effectire 
figure of the decimal fraction be adjacent to the decimal point, the 
index is 1 ; if there be one cipher between them, the index is 2; if 
two ciphers, the index is 3 ; and, in general, the number denoting 
the place of the first signilicant figure from the decimal point will 
be the negative index. Instead of negative indices, their arithmeti- 
cal complementa are frequently used, especially by those unacquaint- 
ed with the first principles of Algebra. 

The decimal parts at the logarithms of numbers consistijig of the 
same figures are the same whether the number be integral, frac- 
tional, or mixed, which may be illustrated as follows : — 
Numbers 546800 Logarithms 5-737829 

54680 . . 4.737829 

5460 . . 3.737829 

546.8 - 2.737B29 

54.08 . 1.737829 

5.468 . 0-737829 

0.5468 1.737829. o 

0.05468 . 2.737829,0 

0.OO5468 . . 3.737829, o 

0.0006468 . 4.737829,0 , 

Problem I, — Tojind the Lognrllkm of any given Number 
RuIjE. — If the given number be under 100, its logarithm is found 
in the first page of the table immediately opposite to it. 

If the number consist of three fiffures, find it in the first column 
of the following or second part of the table, oppo^te to which, and 
under or above 0, is its logarithm. 

If tile 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, 
or above it at the bottom, will be found the logarithjn required. 
To this prefix the proper index, and the whole is completed. 

If the given number exceeds four figures, find the difference be- 
tween the logarithm answering to the first four figures of the given 
number, ana the next immediately following; multiply this differ- 
ence by the remaining figures in the given number, point off as 
many figures to the right hand as there are in the multiplier, and 
the remainder added to the logarithm, answering to the first four 
figures, will be the logarithm required nearly. The logarithm of a 
vulgar fraction is found by subtracting the logarithm of tlie deno- 
minator from that of the nnmerator ; and that of a mixed quantity 
is found by reducing it to an improper fraction, and proceeding b« 
before; or the vulgar fractions may be reduced to decimals, and 
the logarithms found as usual. 

£x. 1.— What is the logarithm of 56? 

In the first part of the table, opposite to 56, and under N. 
is 1.748188. 

£a:.— What Is the logarithm of 366? 

In the second part of the table, opposite to 366, and under 0, is 
2.663481, supplymg the index. The first two figures are understood 
to be supplied in the blank space, till the change takes place at 57 ; 
and this must be attended to throughout the whole of this table, >s 
vt'ell as several othcCs that follow. 
Ex. 3. — ReqoJred the logarithm of 7854 ? 
Opposite to 785, and under 4 is 3.a%091 



J 



EXPLANATION OF THE TABLES. 

Ex. 4. Required the logarithm of 100176 ? 
The log. of 1001 is 000434 
1002 is 000868 



« 



The difference is 434 

Then 434 x 76 is 32984. From this cut off two figures, because 
the difference has been multiplied by two figures, 76> and it becomes 
329.84. If the figure next the decimal point is less than 5, the 
whole may be rejected ; but, if greater, increase the figure before 
the point by unity, and consequently, in the present case, 329.84 
would become 330. Whence to 000434 

Add 330, and supply the index 330 

And the log. of 100176 will be 5.000764 

In general the difference may be taken from the right-hand co- 
lumn, under D, unless the logarithms vary very rapidly, which hap- 
pens only near the commencement of the table, as in the preceding 
example, where the difference under D is 432, the mean difference 
of the whole line, instead of 434 by actual subtraction. This would 
cause a difference of two units, in the last decimal place, less than 
that found above, or the logarithm would turn out to be 5.000762, 
instead of 5.000764. 

To facilitate the method of obtaining proportional parts, there has 
been added to these tables an additional column on the left-hand side 
of the page, under P. P. In the column under N, the two first 
figures are omitted, and the third alone retained, by which means a 
regular series of the arithmetical digits, beginning with 1 and ending 
with 9, are obtained between each bar, or line across the page. 
Hence the proportional parts corresponding to the mean difference 
within the space marked out by each pair of cross bars, answering to 
any of the nine digits^ can be placed opposite to each, which, in these 
tables, has been accordingly done. By this means the logarithm 
corresponding to any number extended to five or six places of figures, 
may be very readily obtained with sufficient accuracy, excepting, 
perhaps, when it fails in the second and third pages, where the dif- 
ferences vary rapidly. 

Ex, 5. — Required the logarithm of 546876. 

Log. of 546800 is 5.737829 , or 5.737829 
Prop, part for 70 56 , or 56 

for 6 48, or 5 

.^ or 

Log. of 546876 is 5.7378898, or 6.737890 
If the number consists of one figure more than four, or five figures 
altogether, the proportional part may be added at sight. 
Ex. 6. — Required the log. of ^^ ? 
Log. of 15 is . . 1.176091 

17 is . , 1.230449 



Log. of If is therefore . . 1.945642 or 9.945642 

Required the log. of 7f^ or V> or 7.625-^ 

Log. of 7.625 is . 0.88224 

Required the logarithms of 24, 56, 102, 546, 7854, 78653, 
54.4768, 97685.46, 0.001546, 0.176804, 0.00043689, 3f, f ff, 7684, 
4857if, 39766^11, 8546iV? 



4 EKPLaNaTION Ob' THE TABLES. 

pROBLKu lI.~-To ^tid Ike Number answering to any given LogO' 

Find the logarithm next leas than that given in the colunin mark- 
ed at the top, and continue the sight along that horizontal line till 
a logarithm the same as that given, or as near as possible, be found; 
then the three first figures of the corresponding natural number ipill 
be found opposite to it in the side-column, and the fourth immedi- 
ately above at the top or below at the bottom of the page. If the 
I index of the given logarithm be 3, the Jbiir figures thus found are 
integers; if the index be 2, the three first figures are integers and 
the fourth is a decimal, and so on, as may be easily understood by 
consulting Problem I. If the given logarithm cannot be exactly 
found in the table, and if more than four figures be wanted in the 
corresponding natural number, then find the difference between the 
given and the next less logarithm. To this annex on the right-hand 
as many ciphers as there are figures required above four in the na- 
tural number. Divide the whole by the difference between the 
next less and next greater logarithm, and the quotient annexed to 
the four figures formerly found will be the natural number required. 
The same thing may be done by the table of P. P. by subtracting a 
part corresponding to each unit from the difference between the 
given logarithm and the next le»s, and annexing these units succes- 
aively in order to the number previously found. 

Ex. 1. — Required the natural number corresponding to the loga- 
rithm 2.495544 > 

This logarithm is found opposite to 313 and under 0, and, as the 
index is 2, then 313 is the number required. 

Ex.2. — What is the number answering to the logarithm 3.828338? 

The logarithm is found 673, and under 5, therefore, since the in- 
dex ia three, the natural number is 6735. If the index had been 2, 
then it would have been 673.5, or the natural number must always 
consist of one integer (if there are integers) more than the index 
expresses. 

Ex. 3. — Required the natural number answering to the logarithm 
2.627980? 

The natural number corresponding to this is 4246 ; but the index 
being 2, one cipher must be prefixed, from what has been said in 
Prob. I., and it becomes 0.04246. 

Ei. 4. — What is the number answering to the logarithm 5.687956? 

The nearest less logarithm than this is 687886, corresponding to 
which will be found the number 4874. The difference between 
687956 and 687886 is 70, to this annex two ciphers, and it becomes 
7000, which being divided by 89, the difference of the columns 
found under D gives 79- This being subjoined to 4874 gives 
487479, the number required. Or the same may be performed thus ; — 
Original log. 5.687956 

487400 corresponds to 5.687886 



70 



Diff. in P. P. . 70 



remainder as dift. 7 

for , . .73 



or in all &Ji76, differing only one unit in the last place from J 
former number. 



EXPLANATION OF THE TABLES. 5 

IjOOABITHMIC ARITHMETIC. 

Pboblex m^— -To perform MuUipUcatum by Logarithms. 

RaLB. — ^Add the logarithms of thcf factors^ and the sum is the lo- 
garithm of the product. 

If there are both negative and affirmative indices^ their sum is 
taken according to the rules of algebra ; or the arithmetical comple- 
ments ci the negative indices may be used^ rejecting the tens in their 
sum. 

The arithmetical complement of the logarithm of any number is 
found by subtracting the given logarithm from 10^ or by subtract- 
ing each of its figures beginningat the left-hand firom 9, and the 
last effective figure from 10. When the arithmetical complement 
of the index alone is wanted, it is found by subtracting it froln 10. 

Ex. l.-^Mult^ly e664 by 836. 

T?-«f^,p /o5^ logarithm S.817169 

factors I ggg logarithm 2.922206 

sum 6.730375 

5487000 corresponds to 6.739335 

diff. in P. P. 40 

giveft 500 for . . 40 

I' ) ' ■ 
or in all 5487500^ which agrees as nearly with the real product 
5487504i as tables extending to six places of decimals will give. 

Ex. 2«^Multiply the numbers 4a68, 0.534, and 0.007685 together 
logarithmically. 

i 43.68 log. 1.640283, or 1.040283 

Factars-| 0.534 log. 1.727541 — 9.727541 

( 0.007685 log. 3.885644 — 7885644 

Product 0.179254 1.253468 9.253468 
Problem IV.— -To perform Division by Logarithms. 
Rule.—- From the logarithm of the dividend subtract the logar- 
ithm of the divisor, the remainder is the logarithm of the quotient. 
Ex. L—Divide 5486 by 96. 

Dividend 5486 log. a739256 
Divisor 96 log. 1.982271 

« 

Quotient 57.146 1.756985 

40 

Ex. 2.— Divide 0.07856 by O.008482. 

Dividend 0.07856 . . log. 2^5201 
Divisor 0.003482 . log. a541829 



i^M 



Quotient 22.5617 1.353379 

39 



33 
)9 



14 



6 EXPLANATION OF THE TABLKS. 

Problem V. — To perform Proportion bif Logarithms. 
lluLB. — From the 3um of the logarithms of the second and tliird 
terms, subtract the logarithm of the first term ; the remainder wOl be 
the logaritlim of the answer. Or, instead of subtracting the loga- 
rithm of the first term, its arithmetical complement may be added 
to the other two, which, in many cases, is more convenient- 

Ex. — A merchantman distant twenty miles, going at the rate of 5 
knots or miles an hour, is pursued by aprivateer, sailing at the rate of 
I 7 miles ; after three hours chase the breeze freshened, the merchant- 
man's rate wa^ increased to 6 knots, and the privateer's to 10. In 
-what time will the privateer come up with the merchantman ? 

As the privateer gained 3 miles an liour on the merchantman, at 
the end of the first A hours, the distimce between them is obviously 
14 miles. During the remainder of the chase the hourly gain of the 
privateer was 4 knots. Hence, 

As the hourly gain 4"' ar. to. log. 9.397940 ^H 

Is to the disUnce 14'" log. 1.146128 ^H 

So is 1' log. 0.000000 -^H 



To the time required 3^5 or 3" 30™ 0.544068 
Consequently, from the time the breeze freshened, the privateer 
would come up with the merchantman in three hours and a halt^ or 
in six hours and a half from the commencement of the chase. 

Problem VI. — To perform Involution bg Logarithms. 

RcLB. — Multiply the "logarithm of the given number by the index 
of the power, and the product will be the logarithm of the power 
required. 

Ex. 1.— What is the square of 64? 

Given number 64 log. 1.806180 

Index ol'the power 2 



Square 4096. . 3.612; 

Ex. 3.— What is the third power oi'24 ? 

Given number 24 log. 1.38021 1 
Index of the given power 3 



Third power 13824 



4.140633 



125 

Problem VII. — To perform Evolution bi/ Logarithms. 

RuLB. — Divide the logarithm of the given number by the 
the root, supposed to be expressed by an integer, as, for example, 
square root by 2, the cube root by 3, and the quotient will be the lo- 
garithm of the root. 

If the given number be a decimal, and the arithmetical comple- 
ment of the negative index be useil, then prefix 1 to that index for the 
square root, 2 for the cube root, 3 for the fourth root, &c. 

If the index of the root be expressed by a fraction of which the 
numerator is not unity, then multiply the logarithm of the gi yea 
number by the numerator, and divide it by the denominator cu j 
index. 

£*. 1.— What is the square root of J296 ? 

Given number 1296 log. 3.112605 
Square root 36 1.556302 



I 



EXPLANATION OF THE TABLES. ^ 

Ex. d.— Required the cube root of 0.0009261 ? 

Given number 0.009261 log. 3.966658, or 7.966658 

Cube root 0.21 1.322219, or 9.322219 

What is the fourth root of 0.00007634 } 

Given number 0.00007634 log. 5.882752 
Given index . . \ 

Log. of the root 0.0934734 2.970688 
In this example, because the index of the root 4 is not contained 
in the negative index 5 a certain number of times exactly, the loga- 
rithm 5.882752 is resolved into its equivalent 8+3.882752, and the 
product of this by \ is 2.970688 the logarithm of the root required. 

TABLE III. — The Angles which every Point and Quarter Point of 
tJie Compass makes with the Meridian, 

This table is useful for reducing the points of the mariner's com- 
pass to degrees, and conversely. It is divided into seven columns ; 
in the two first and two last columns are contained the names of the 
several points ; the third and fifth contain the corresponding points 
and quarter points reckoned from the meridian ; and the fourth the 
degrees, minutes, and seconds, answering to them. Its use is ob- 
vious. 

TABLE IV. — Logarithmic Sines, Tangents, and Secants, to every 
Point and Quarter Point of the Compass. 

In performing calculations relative to navigation, it will be found 
convenient to taJse the logarithmic sines, tangents, and secants, from 
this table, thereby saving the trouble of reducing them to degrees, 
&c., by the preceding table. The manner of using it is easy, and 
will be readily understood from the explanation of the table which 
immediately follows. 

TABLE V .^-^Losarithmic Sines, Tangents, and Secants. 

This table contains uie logarithms of the natural sines, tangents^ 
and secants, to each degree and minute of the quadrant in the usual 
manner. To facilitate Calculations in which time is involved, the 
degrees and minutes have been converted into time at the rate of 15*» 
to an hour, and annexed at the top and bottom of the page and in two 
additional side-columns.* These, together with proportional parts 
to each second of time, or to every fifteen seconds of a degree, at the 
bottom of each page, will, it is hoped, render this table still more 
eai^ and general in its use than those of a similar kind usually given. 

The degrees are numbered at the top of the table, in a direct 
order, from 0" to 45*>, and, at the bottom of the table, in a retro- 
grade order, from 45^ to OO*'. The minutes are contained in two of 
the marginal columns. The minutes in the left-hand column be- 
long to me degree at the top of the page, and those in the right- 
hand column belong to the degree at the bottom. In like manner, 
the minutes and seconds of time in the first left-hand column belong 



This table will therefore convert degrees into time, and conversely. 



s 



EXPLANATION OF THE TABLES. 



to tlie hour at the top, and those in tlie right-hand column belong to 
the hour at the bottom. Tu promote perspicuity, it ia recommendetl 
to mark minuter and seconds of the circle always by accents, ami 
those of time by m and s, as is done in the tables. 

Froblsm I. — Tojind the Sine, Cosine, SfC. answering to any given 
Degree or Minute. 

Rui^E- — Find the given degrees at the top of the page if less than 
45**, and the minutes in the left-hand column ; opposite to which, 
and under the word sine, cosine. Sec. is the number required. But 
if the given degrees be greater than 45° and less thkn 90°, find 
them at the bottom, and the required sine, cosine, && will be 
found above the word sine, cosine, &c. opposite to the given number 
of minutes in the right-hand column. If the given arc esceed 90°, 
find the sine, cosine, &c. of its supplement, or, which comes to the 
same thing, and will be more easy in practice, to find the sine of an 
arc above 90°, reject 90°, and take the cosine of the remainder. To 
find the cosine of an arc above 90° reject 90°, and take the sine of the 
remainder. The same method may be pursued for the tangents and 
secants botli for arcs and lime, recollecting that 90° corresponds to 
^. 

Ex. 1.— Required the log. sine of 23" 28'? 

Under the word sine in the page marked 23° on the top, and op- 
posite to 28' in the left-hand column, is 9.600118, the sine retiuired. 

Ex. 2. — What is the cotangent of 55° 57'? 

In the page marked 55°, at the bottom and opposite 67' in the 
right-hand side-column, is 9.829805, the cotangent of 55° 67'. 

Ex. 3.— Required the secant of 125= 40' ? 

The supplement of 126° 4^ is 64° 20', the secant of which is 
10.234280, or, which comes to the same thing, the cosecant of 35° 40^ 
the escess of 125° 40' above 90° is 10.234280, the secant required. 
Hitherto the given arc has been supposed not to exceed 180°; but, 
in several astronomical calculations, it frequently happens that arcs 
through the whole circle arc employed ; consequently, if the arc 
lie between 180° and 270°, diminish it by 180°; if ^between 270° 
and 360°, take its explement to 360°, and take the logarithmic sines, 
&c. as before. Otherwise, for the log. fine, &c. of an arc bet:ween 
iJ70° and 360°, take the log. cosine, &c, of its excess above 270°, and 
fbr the log. come, &c. of an arc between 270° and 360°, let the tine, 
etc. of its excess above 270° be taken. Anil for the log. sine, &c. of 
an arc between 180° and 270° let the log. sine of its excess above 
180° be Uken. Thus the log. suie of 300° 28' is the log. sine. See. of 
30° 28', the excess above 270° ; and the log. sine of ^0° 1 8' is the 
ume as that of 40° 18', and so on. The same may be dune when 
time is employed, recollecting that 6^ corresponds to 90°, 12° to 180°, 
18'' to 270^, and 24° to 360". 

Problbm W.—Tofuidike Sine, TangeiU, ^c. of an Arc ctpressedm 
Degrees, Minutes, and Secmds. 

1iuj.s. — Find the sine, tangent, &c. corresponding to the given 
degree and minute, and also that answering to the next greater mi- 
nute, multiply the difference between tJiem by the given number oi' 
seconds, and divide the product by 60; then the quotient added to 
the sine, tangent, &c. of the given degree and minute, or subtracted 
from the cosine, cotangent, &c. will give the quantity required near- 
ly. To facilitate tiiis process the difference, to 100", has b een 
given in the column marked 0. Multiply this difference hy ^^M 



EXPLANATION OF TH£ TABLES. 9 

number of seconds^ cut off two figures from the ri^t^ and add the 
remainder to the sine^ tangent^ &c. of the given degree and minute, 
or subtract it from the cosine^ &Cv and the quantity required will be 
obtained nearly. 

Ej:. 1— Required the \og. sine of 23° 27' 40" ? 

Log. sine of 23^" 27' is 9.599827 
23 28 is 9.600118 



Difference 291 

Seconds 40 



60111640 



194 
Log. sine of 23° 27' 9.599827 

^Proportional part for 40" 194 



Log. sine of 23° 27' 40" is 9.600021 
Or difference under D.^ and opposite 27'^ is 485 
Multiplying by 40', and . .40 

Striking off two figures on the right gives 194,00 
The same as before. 
If no very great precision is required, then the proportional part 
for the nearest fifteen seconds may be taken from the small table at 
the bottom of the page. 

Ex. 2.— Required the logarithm tangent of 2** 24" 46* ? 
Log. tangent of 2^ 24" 44' is 9.864180 
Proportional part for 2* is 132 

Log. tangent of 2^ 24" 46- is 9.864312 
Ex. 3.— .Required the secant o£&' 46" 36'? 
The cosecant of its excess above 6^ or 3** 45" 36", gives 10.079396. 
Required the sine of 20** 44*" 56' ? 

The cosine of 2 44 56 is 9.876236 being the sine of 

20^ 44" 56-. 

Problem III. — To find the Sine or Tangent of a small Arc, less 
than three Degrees. 

1. To find me siixe. 

To the logarithm of the arc reduced to seconds, with the decimal 
annexed, add the constant quantity 4.685575, and from the sine sub^ 
tract the third of the arithmetical complement of the log. . cosine, or, 
which comes to the same thing, one third of the secant ; the remain- 
der will be the logarithmic sine of the given arc. 

2. To find the tangent. 

To the logarithm of the arc in seconds and constant quantity 
4.685575, add two-thirds of the secant, the sum is the log. tangent 
of the given arc. 

Ex. 1. — ^What is the log. sine of the sun's mean horizontal paral- 
lax, supposed to be 8".68? 

Logarithm of 8".68 is 0.938520 
Constant . 4.685575 

One-third of sec. 8".68 is 0«000000 



Log. sin of 8",68is 5.624095 



)0 



EXPLANATION OF THE TABLES. 



Or, since in very small arcs the sine and tangent are each verr 
nearly equal to the length of the arc, when it does not exceed KK. 
and the length of an arc of one second is 0.0000048481368 ; multi- 
ply the length of one second by the number of seconds and parts 
of a second making the index positive by the former rules, and the 
sine or tangent, will be obtained, thus, — 

0.0000048481368 x8".68=0.0000420818274; the log. of this i« 

5.624004, the log. sine or tangent required. -^h 

Ex. 2.— Reouired the tangent of 1° 24' 36".46.> dH 

To the constant logarithm 4.685575 ^^H 

Add log. of 1" 24' 36".4C=5076.46 3.705561 ^H 

And 1x0.000132= 88 ^^ 



„ »ng. ' 

Problem IV. — To find ike Degrees, Minnies, and Seconds amwer- 
ingto ant/ given log. Sine or Tangent. 

TliiLB. — In its respective column find the nearest sine, tangent, &c. 
to that given ; and take the degrees from the top or bottom of the 
page, according as the quantity is found in a column, with the pro- 
per title at the top or bottom ; and the minute is found in the same 
horizontal line, in the left or right hand marginal columns, accord- 
ing as the quantity is found in a column titled at the top or at the 
bottom of the page. 

Ex. 1, — Required the arc, or degrees and minutes corresponding 
to the log. sine 9.5846p5 ? 

This is found in a column marked sine at the top under 22 de- 
grees, and opposite 36 minutes, or 1 hour, 30 minutes, and 24 se- 
conds of time. 

Ex. 2. — What is the arc in degrees or t 
tangent 10-358430, making use of the tabli 
the bottom of the page. 

Given log. tangent 
ee' 20' d" corresponds to 



le answering to the tog. 
of proportional parts at 



And 30 



Difference 



Hence 66 20 30 is the arc required. 
Or, 4" 25" 20" answer to 



And 



2 to 173, or nearly 



177 



i 



Hence 4 25 22 is the time nearly. 
Or to 177 add two ciphers, and divide by 572, the number under 
D. and opposite to 10.358253, or rather by 573, the number above it, 
as the form in which the tables are printed requires, and we have 
66° 20' 31" very nearly ; and this method must be followed in all 
similar cases. 

Pboblem V. — TofiiidOie Degrees, Minutes, and Seconds answering 
to the Logarithmic Sine or Tangent of a very small Arc. 

Rule. — To the given log. sine ad'd the constant 5.314425 and one- 
third of the corresponding secant, the sum, rejecting 10 in the index, 
■ will be the logarithm of the number of seconds in lie required arc 

To the given log. tangent add the constant 6.314425, and fnntj ' 
sum subtract two-thirds of the corresponding secant, rejecting ' 



hIM 



£XPIiANATION OF THE TABLES. 11 

the index^ the result will be the logarithm of the seconds of the re- 
quired arc: 

Ejb, l—- •Enquired the arc whose log. sine is 6.497655 ? 
Constant . . 5.314425 

Given log. sine 6.497655 

^ of 0.000000 is 0.000000 

Log. arc 64".8756 1.812060. 

Or r4".8756 

Ex, 2.— What is the arc whose log. tangent is 7*164440 ? 
Constant . 5^14425 

Given log. tangent 7*164440 

I of 0.000000 is 0.000000 

Log. arc 301^207 2.478865 

Or 5' 1".207 

TABLE VI.— jVa^vro/ Sines, Tangents, Secants, and versed Sines 
to every Degree of the Quadrant. 

The method of taking out the numbers required from this table 
will be readily comprehended from what has already been said rela- 
tive to the precedh^. When minutes or seconds occur^ proportional 
parts must be taken by means of the differences found by actual sub- 
traction. 

£a:.— -What is the natural sme of 5« 48' 56" ? 

Natural sine of 5° is . . 087156 

Prop, part of diffl 17372 for 48' 56" is 14168 

Natural sine for 5<' 48' 56" 101324 

TABLE VII.— itfrndtowaZ Parts to every Decree of the Quadrant. 
The degrees are found under the letter D^ and the meridional parts 
imder M. P.^ and when minutes and seconds occur^ proportional parts 
of Ihe difference must be taken in the manner shewn above. 
jELv.— 'Bequired the meridional parts answering to 45° 36' ? 
Meridian parts to 45° 3929.9 

Prop, part of diff. 85.7 to 36' is 50.6 

Meridian parts to 45° 36' is . 3080.5 

TABLE Vni.'^Traverse Table, or difference of Latitude and 
Departure. 

This table contains the measures of the sides and angles of right- 
angled plane trian^es^ the distance being represented by the hypo- 
tenuse^ and the difference of latitude and departure by the legs or 
«ides about the right angle, and the course and its complement by 
the acute angles. Hence, if any two of these be known, except the 
two acute angles, the rest are found by inspection. The course is 
given in degrees or points in. the two exterior marginal columns, the 
distance is found at the top or bottom of the page, according as the 
course is less or greater than four points or 45? ; and the difference 
of latitude and departure is found in colunms under or above these 
words respiectively. 

If there are minutes in the course, proportional parts may be taken 
where great accuracy is required, odierwise they may be omitted if 



12 EXPLANATION OF THE TABLES. 

less than W , but, if more than SC, the de^ees in the course most 
be increased by J". The distances 1, 2, 3, 4, &c. at the top and the 
bottom may be accounted JO, 20, 30, &c., or 100, 200, 300, &c if 
the difference of latitude and departure be increased in the same pro- 
portion by removing the decimal point a corresponding number of 
places to the right. If the distance consist of several effective figures, 
the difference of latitude and departure must be found for each 
figure separately, and the sum of the resuitii taken. 

Problem I. — The Course and Distance being given, tojind the Dif- 
ference of Latitude and Departure. 

Find the course in rieht or left hand column, and in a line with 
it, under or above the given distance, the difference of latitude and 
departure will be obtained. 

Ex. 1.— A ship sails N. N. E. 60 miles^ what difference of latitude 
and departure has she made? 

Coutae, Dist. DifF. Lai. DepBiture. 

2 points . 60 . 55.433 . 22.961 
Ex. 2.— A ship sails S. E. b. S. ^ S., or S. S. £. ^ E. 244 miles, re- 
quired her difference of latitude and departure ? 

Couiw. Dist. DiE I.au DepartuFe. ^^m 

2i points . 200 . 176.38 94.28 ^M 

40 . 35.277 ■ 18.856 ^H 

4 3.5277 1.8856 ^H 



t 

L 



244 215.1847 ■ 115.0216 

Ex. a— A ship sails 300 miles S., 54° 30' W., what ia her differ- 
ence of latitude and departure f 

54° . 300 . 176.34 . 242-71 ^M 

55 . 300 . 172.07 . 245.75 ^H 

Mean 54* - 300 . 174.20 244.33 "^ 

When several courses and distances are fpven, the results must he 
placed in a table, the sum of the several northings and southings, 
eastings and westings taken, and placing the less sums under the 
greater, the differences will shew how much the ship has, upon the 
whole, changed her situation, and in what direction she has moved. 

TABLE IX.— Diurnal Logarilhms. 
This table, to which I have ventured to give the title of Diurnal 
Iiogarithms, is useful for making computations in which time is con- 
cerned, particularly for reducing the right ascension and declinadan, 
&c. of the sun or moon to any intermediate time between those times 
the Nautical Almanac, where the proportional parts to daily 
squired. It has two eetsofarguments, the one answer- 
moon's place is given in the Nautical Almanac for 
. midnight ; the other corresponding to 24'' for the 



differences 

to 12^ since thi 

every noon am 

Rule. — To the logarithm from this table corresponding to the 
Greenwich apparent time add the proportional logarithm (Table X.) 
of the variation on the given day for 24'' or I2^ as the case may be, 
the sum will be the proportion^ logarithm of the part of it for the 
given time, which, added to or subtracted from the number corre- 
sponding to the preceding noon or midnight, according as it is in- 
creasing or decreasing, will give its value at the instant i • ■ -^ 




EXPLANATION OF THE TABLES. i3 

Ex. L— Required the sun's right ascension March 20th^ 1896, at ^ 
20^ 4ff" 40* apparent Greenwich time. 

Greenwich time . 20^4^40- D. L. 0.06262 

Change of R. A. in 24* 3 38!2 P.L. L69457 

Prop, part for 20^46" 40^ 3 9.0 L75719 

R. A. at preceding noon 23 67 42.0 

R. A. at 20^ 46~ 40* 51.0 

Ex. 2. — Required the moon's declination September the 15th^ 
1826^ at 7^ 4ff" 30» P. M. apparent time on the meridian of Green- 
wich ? 

Moon s declination at noon . 2** 7' 8"S. 

at midnight 9 19 N. 



Sum = diff. in 12 hours 2 16 27 

App. tune 7»^ 48" 30» diurnal log. . 18662 

Change of dec in W, 2" 16' 27" prop. log. 12030 



Change in 7* 48» 30«+l«28 47 prop. log. 30692 
Dec. at noon . — 2 7 8 



Dec. at y* 48" 30' — 38 21 S. 

When the differences are v^y irregular, a correction on that ac- 
count becomes necessary. This will be exemplified in the explana- 
tion of Table XXVII. 

TABLE X. — Proportional Logarithms. 
This table is chiefly useful for facilitating the method of finding 
the apparent time at Greenwich, answering to a given central^dis- 
tance between the moon and the sun, a fixed star or a planet, by the 
assistance of the Nautical Almanac. It is extended to three hours 
on account of the distances being given in various ephemerides to 
every three hours of time. As degrees and hours are similarly di- 
vided, it answers equally well for either, and is marked accordingly. 
To this table proportional parts have been added at the bottom of 
each page to every tenth of a second, which may be useful where 
great accuracy is required. The table is veryuseful in calculations 
where sexagesimal divisions are employed. The method of taking 
out the log. of any quantity will be readily understood from what 
has already been said. 

TABLE 'Kl.'^Depression or Dip of the Horizon. 
The dip of the horizon is an angle contained between a horizontal 
line passing through the eye of the observer, and a line from his eye 
to the visible horizon, when these lines are in the same vertical plane. 
This table contains the dip answering to a free unobstructed horizon, 
and the numbers corresponding to the height of the eye are to be 
subtracted from the observed altitude when taken by the fore obser- 
vation, but added to it in the back observation. 

TABLE XII.— Z%e Dip at different Distances from the Observer. 

If the land is not sufficiently distant to afford a free horizon, it 
may be sometimes necesary to obtain an altitude referred to the sur- 
face of the sea at some known or estimated distance. Under such 
circumstances the dip may be taken from this table. 



EXPLANATION OF THE TABLES. 



^ 



TABLE XIII.— Correction io be added lo the observed Alliludt of 
the Sun'a lower Limb when taken by afore Observation tajind the lr«t 
Altitude. 

This table was computed by the author a good many years ago 
for the purpose of combining the usual corrections, namely, dip, re- 
fraction, parallax, and semidiameter. The variation of the sun's 
8emidiament«r from 16'. is given at the bottom of the table, which, 
unless considerable accuracy be required, may be neglected. The 
arithmetical complement of the numbers from this table to 32', will 
be the correction to be subtracled when the upper limb is observed. 

TABLE XIV.— Correction to be aubtracted/roj« the observed Al- 
titude {fujixed Star lo ^find the true. 

This table is similar to the last, and contains the sum of the two 
corrections, dip and refraction, to be subtracted when the fore ob- 
servation is employed. 

TABLE XV.— This table, taken from the Nautical Almanac for 
, will answer for most purposes for a considerable number of 
years to come. It contains the time of the sun's semidiameter passing 
the meridian, the sun's semidiameter, hourly motion in longitude, 
and the log. of the sun's distance from the eaith, for every sixth day 
in the year. 

The time of the sun's passing the meridian is useful for reducing an 
observation of a passage of the preceding or subsequent limb over 
the meridian taken with a transit instrument, to that of the centre. 
The semidiameter of the sun is necessary to reduce an observation 
of the limb to that of the centre, whether in altitudes or angular dis- 
tances. It is also useful for determining the index error of a sex- 
tant, or the exactness of the scale of micrometers. 

The hourly motion is useful for computing eclipses. The log-, of 
the sun's distance is requisite in the calculation of the places of the 
planets and comets, and for some other purposes. 

TABLE XVI.— rAe Sun's ParaUax in Aliilude and Zenith Dit- 

The author computed this table from a mean of the determinations 
of Delanibre from the observations of the transit of Venus over the 
sun's disk in June 1769. He found the mean hotiEontal parallax 
to be 8".68. It is hoped it will prove useful where great accuracy 
is required. 

TABLE XVII.— Mean Refractions. 
For the elements of this table the author is indebted to the liber- 
ality of Mr Ivory, the most distinguished mathematician in the Bri- 
_ tish islands. On comparing it with that given in the Trans- 
actions of the Royal Society of London, it will be seen that it has 
been expanded considerably, so as to render its application more easy by 
giving the mean refraction, and its logarithm for every IIK from the 
aenith to the horizon, subjoining the differences of the logarithms for 
the purpose of computing proportional parts more readily. 

TABLES XVni, XIX, and XX,— These Ubles are employed 
to correct the precethng according to the state of the barometer and 
thermometer, as shown in the explanation at the bottom of page 89 



EXPLANATION OF THE TABLES. 



15 



of the tables. In the seventh line firom the bottom of that page^ 

after thermometer^ there should have been added^ '' or 0.002083 for 
one decree of Fahrenheit/' that used in the construction of the table. 
Ex. J . — Required the mean refraction for 21° 40' of zenith distance 
or 68^ 20" of altitude ? 

S)posite to 21° 40' in table XVIL, and under h, will be found 
.21^ the refraction required when the barometer stands at 30 
inches^ and the thermometer at 50°^ and this is sufficient for most 
purposes when great accuracy is not required. 

Ex. 2. — Required the true refraction when the zenith distance is 
70° 41'.7, the barometer 30.045^ and thermometer 34° ? 

Zenith distance 70° 40^ log. ii Table XVIL 2.21752 

1.7 . , . 68 

Thermometer 34° Table XVIII. 0.01472 

Barometer 30.0 Table XIX. 0.00000 

.045 ... 6 

Thermometer 34 Table XX. . 70 



Log. r • . 2' 61".27 = 171"27 

Observed refraction 2 51 ,50 



2.23368 



Error of the table — .23 
Ex. 3.— Let ^ = 87° 42^ 10", thermometer 35®, and barometer 29.5 
inches, what is the true refraction ? 



^ = 87° 40^ 0" 
2 10 
Ther. 35° 
Bar. 29.5 
Ther. 35 

Loff. r' 
^x(35°-^°) 
— 0.606 X-.15 
^X (29.5—30,0) 
X 1.04 X— 0.5 



log, >^ 



17' 16".81 = 1036".81 



\ 



= + 9.09 



3.00466 

390 

0.01379 

9.99270 

65 

3.01570 



= —0.52 



r = * . . 17 25 .38 

Observed refraction 17 26 .50 



Error of the table — 1 .12 



Examples for Exercise, 





Z. 


D 


1 
• 


Bar. 


Therm. 


Ob*. Ref. 










In. 


In. Out 






1. 


70° 


4Sf 


30" .0 


29.686 


46^ 44.17 


2' 


44".83 


a. 


76 


55 


81:2 


20.686 


40 37.10 


4 


8.98 


a 


81 


27 


18.6 


29.924 


61 58.19 


6 


1 .90 


4. 


83 


58 


6.7 


29.810 


36 29.95 


8 


48 .52 


6. 


86 


14 


^.0 


29.174 


47.75 


12 


4 .20 


6. 


87 


23 


44.0 


30.000 


60 56.08 


15 


32 .80 


7. 


88 


39 


32.0 


29.800 


38 34.40 


23 


7-94 


8. 


80 


26 


51 .4 


29907 


39 3a46 


30 


16 .60 



Jbrror. 

+ 1".51 
+ 1 .86 
+ 1 .55 
+ .53 
+ .28 
— 1 .15 
—15 .70 
—39 .70 



16 EXPI^NATION OF THE TABLES. 

Hence at moderate zenith, distances the error of the table ia ■mall, 
sometimes + itnd at other times — . Prom 70" to about 85°, the error 
is generally +, but from 85° to 90° it becomes — , and is consider- 
able near the horiaon We may tberefore infer that the horizontal 
refraction, 34' 17" -5. given by the table in a mean state is, in general, 
too small, though, from the uncertainty and irregularity to which it 
is subject, it is very difficult to estimate accurately its true quantity. 
Perhaps from the irregularity of temperature in various parts of a 
line near the surface of the earth through which the ray of light 
must pass to reach the eye of the observer, it will be impossible ever 
to assign the true quantity of the horizontal refraction under given 
circumstanceH. In fact, no instrument, as yet, has been employed 
to ascertain the etfects of aqueous vapour floating in the atmosphere, 
on the exact quantity of the horizontal refraction; and we anspect 
that the barometer and thermometer alone ore inadequate to that pur- 



TABLE 'S-'^l.— Augmentation of the Moon's Semidiameter in Ah 
titude and Zenith Distance. 

The apparent magnitude of any object being in the inverse ratio 
of its distance, and as the moon is nearer the observer in the zenith 
than in the horizon, by the earth's radius her apparent semidiameter 
must be greater in the former situation than in the latter. This 
table contains that increase corresponding to six different values uf 
the semidiameter, at different degrees of altitude. If the quantity 
is not found to the accuracy required by inspection, it may be deter- 
mined by proportional parts in the usual manner. 

TABLE yiXn— Reduction of the Moon's Parallux in the Sphc- 

As the earth differs somewhat considerably from a sphere, the 
eccentricity being about jln, it follows that Uie equatorial parallix 
must be greater than that at the various intermediate latitudes from 
the equator to the pole. This table contains the quantity to be But>- 
tracted from the equatorial parallax given in the Nautical Almanac 
to reduce it to what it ought to be at any other latitude. 

TABLE XXIII.— Logon(Aw« of the Earth's Radii i« eacA Pa- 
rallel of Latitude; the Egaatorial Radius being Unit, and Covi- 
presHon jj^g- 

This table will be found useful in some nice observations in astro- 
nomy, where the spheroidal figure of the earth must be taken into 
account. 

Example. To Greenwich in latitude 51° 28' 38" the radius is 
9.9991 lil. 

TABLE '^iXlV .—Angles tiihich, the vertical to an^ point of the 
Earth's surface, makes rvilh the Radius drarenfroia that mint la the 
centre, or, as it is usually called, the Reduction of the Latitude to 
,h of compresiAon. 

This table is useful in several astronomical observations, such a» 
the computation of eclipses, occultations, &c. 

EiflMpfe.— The apparent latitude of Greenwich is 51" 28' 38"A 
■ ■^- ' *1 tot- 




that reduced to the centre .' 



J 



EXPLANATION OF THE TABLES. 17 

Latitude 51'' 28' d8''.4 

Reduction — 11 10 .8 



Reduced latitude 51 1? 27 6 
From this table the reduction of the altitude may hit obtained by the 
following rule : 

To the secant <^ the azimuth reckoned from the meridian of an 
opposite name from l^e latitude^ add the proportional logarithm of 
the reduction of latitude^ the sum will be the reduction of the alti- 
tude^ to be reckoned positive when the azimuth is less than 90°^ and 
negative when greater. 

Example, — Required the reduction of altitude corresponding to an 
azimuth of 36'* ^' in the latitude Greenwich 51" 28' 38" N. 
Latitude 51" 28' 38" Secant 0.20563 

Reduction of alt. 11 10.8 Prop. log. 1.20683 

Reduction of lat 6 57.8 Prc^. log. 1.41246 

In computing time^ &c.^ if the reduced latitude be used^ the re- 
duced altitude must be employed also ; but^ in general^ unless abso- 
lutely necessary in such computations as that of timCj it is easier not 
to employ either of these reductions. 

TABLE XXV. — For determining the Latitude at any time hy the 
Pole Star. 

This table was computed by Mr Littrow of Vienna, and will be 
found veiy useful for determining the latitude of a place by the pole 
star. A uiU explanation is given at the bottom of the page immedi- 
ately under the table. 

Ex 1. — In latitude 5&* N. nearly, the zenith distance (Z) of 
the pole-star, by an astronomical circle, was found to be 35**. 20' 50", 
when its apparent polar distance {p) was 1° 36'.7, and the star just 
14** 26" 56' from the time of upper culmination ; required from these 
data the exact colatitude of the place of observation. 
Now 14»»26'°56' gives M r= 3".23, and N = — 0^ O' 0".48 
And 31".23 x — 3'.3 x 0.«h=:— 2".06=.— 3.3 x .02 M 
Then 31".23 — 2".06=29'M7=M', log. 1 .4649 
Cot.Z36«»21' 0.1491 



1.6140=— 41 .12 



Cos. t 14* 26" 56'=9.9039 
p W/J log. 1.9854 



—1.8893 = — 77'.5= —1 17 30.00 



*■■'*. 



—1 18 11 ,60 
Z 35 20 50 .00 



Colatitude 34 2 38 .40 

Latitude 55 57 21 ^60 

Edinburgh, \(Hh January, 1896. 

Op the Caltonhill, near the Observatory, with one of Trough ton's 

reflecting circles on a stand, and an artificial horizon, the autnor, at 

about ten o'clock, p. m. observed the following double altitudes of 

jthe poUr star, when the sympiesometer stood at 29.86 inches, and 



'<» EXPLANATION OF THE TABLES. 

thermometer at 42* Fahrenheit ; required the latitude of the place of 
obHerration. 

Siderial Time. Double AUiludes 

After Transil. Wiih Aru Horizon. 

#" 22°" a* U3= IC 50" 

4 23 35 113 10 55 

4 24 30 113 10 55 

4 25 40 113 10 50 



App. alt. or halt" 



App. zenith diat. or comp. 33 24 33 

Now by tables 17> 18, 13, and 20, compute the re&actioii. 
Zenith diBt.=33'' 24' 33" log. 3s (17) 1-5860 

Thermometer 42° Fah. (18) 0.0073 

Barometer 29.86 inches (19) 9.9980 

Thermometer 42° Fah. (20) . 0.0003 



113 


U 





113 


10 


54 


56 
90 


36 



27 




Log. r=39".05 

31. zenith distance 
raction 



1.5916 
33° 24' 33" 
+ 39.05 



True Kenith distance . 33 25 12.05 

Now 4" 24" 3ff gives M=72".973, and N=+0° 0' 0" 

Then 72"-973 x — 3'.3 x 0.02=— 4.81 6 =3.3 x iXi M. 



Natural number 
Cos. (=4" 24" 36' 
p 96'.7 log. . 

Natural number 



.29=2.014045=— 
9.6Wi751 
1.985426 



D5 J 

I 



)'.19=1.592177 = + 39 11.40 



1^ or colatitude 34 2 40.73 

Latitude . 55 57 19.27 N. 

From a trigonometrical measurement he also founil the latitude 55"* 
57' 20".? N., supposing with Captain Kater the latitude of the flag- 
staff in Leith fort to be 55° 58' 39" N. 

TABLE XXVl.—DelambTeJtr»t calculated this TahU for Jndii 
the augmentation of the semidiameter of the Moon in solar kcU} 
oeeultaliont, without computing the altiiude. It is used asfoUons : 

To the altitude of the nonagesimal in signs, add the distance of the 
moon from it, and from that altitude .;u&(racf the moon's distatice &oni 
it ; then take the equations from this table, Part I. answering to the 
sum and difference, and take the »\an of these, regard being had to the 
si^g. To this add the equations corresponding from Part II. If 
the observation be that of an occultation. the equation answering ' 



ndine 
s ai3 



I. If J 



EXPLANATION OF THE TABLES. 19 

the true latitude and parallax in latitude of the moon is to be taken 
from Part III. In a solar eclipse this part vanishes. Then enter 
Part IV. with the sum of the former equations in the first vertical 
column, and the horizontal semidiameter at the top ; and take out 
the corresponding number, which being applied to the former aggre- 
gate, according to its sign will give the augmentation of the moon's 
semidiameter. 

Ex, — Let the altitude of the nonagesimal be 55° 18', the apparent 
distance of the moon from it 14° 42', the moon's true latitude 24' 2" 
S., the parallax in latitude 35' 40", and the horizontal semidiameter 
15' 30" ; what is the augmented semidiameter ? 

Altitude of nonagesimal 1* 25° 18' 

App. dist. of moon from it 14 42 

Sum 2 10 PartL + 7".70 

Remainder . 1 10 36 I. +5.33 



+iao3 

PartIL+ 0.17 
Moon's true lat. 24' 2" S., and par. in lat. 35' 40" Part III.— 0.12 



Sum ..... +13.08 
To moon's semidiameter 15' 30", and Sum 13".08 Part IV 0.82 



Augmentation ..... 12.26 

Semidiameter .... 15' 30.00 



Augmented semidiameter .15 42.26 

TABLE XXVII.-^Equations of Second Differences for twelve 
Hours, 

In computing the moon's place from the nautical almanac for any 
given time by proportion, a correction resulting from the moon's un- 
equal motion must be applied to the proportional part of the moon's 
motion in longitude or latitude, answering to the given tinn^ after 
noon or midnight. This correction is contained in the table, the 
arguments, of which are the mean of the two second differences a£ the 
moon's motion at the top, and the apparent time after noon or mid- 
night in the respective side column. This equation must be added 
to, or SUBTRACTED from, the proportional part of the first difference 
of the moon's motion in twelve hours, according as that difference is 
decreasing or incbeasing. 

Hence the correct change, corresponding to the given interval, will 
be obtained. 

If the given second difference is not found in the table exactly, the 
sum of the equations answering to the several terms, which make up 
the second difference collectivdy, is to be taken. 

This table may be applied in the computation of the place of a 
planet. And as the sun's declination varies somewhat irregularly 
about the solstices, a column has been added to the lower half of the 
table on the right side for differences in twenty-four hours, to deter- 
mine the exact declination for any given time where great accuracy 
is required. 

Ex, 1 .—Required the moon's declination on the 1 5th of September, 



i 



20 EXPLANATION OF THK TABLES. 

1826, at 7'' 48°' 'Sff p. m. apparent time on the meridian of Greefl- 

In the explanation of Table IX. tills is found to be 0° 38' 21" S. 
by proportion ; it is only now required to find the correction depend- 
ing on second differences. For uiis purpose two declinations must 
be taken out preceding the given time, and two after it, from which 
the mean second difference must be found. 

The Moon's declination, 
1826, First Dif. Sec. Diff. Mean. 

Sept. 14th at midnight is 4° 23' 24" S. „„ ,-;, ,(.„ 

Ifithatnoon 2 7 ^ ^4 \% ^ O* 1" -U- 

15th at midnight 9 19 N.q |? V 19 ■** 

Ifithatnoon 2 24 2? N.'^ '^ ** 

If the first differences first increase and then decrease, or vice versa, 

half the difference of the two second differences is the mean, instead 

of half the sura, as would have been the case had the differences 



b regularly increased or decreased. 

1^ fn t.Tlift rnsp thp pniiAHnn must Yh 



In this case the equation must be added or subtracted, according as 
the^rj( first difference is greater or less than the Ihird first differ- 
Now to 30" and 7'' 48i'" the equation is 3" .4 
to 4 . . .4 

The whole equation is . 3 .8 

Which, according to the rule above, must be added to thepropor- 
tional part formerly found under the explanation of Table IX. ; thai 
is, to 1° 48' 4?" we must add 4", and the true proportional part be- 
comes . . . . -I- 1" 48' 51" N. 
And declination at noon being . — 2 7 3 



The true declination is . . . — 18 1? S. 

Unless the declinations are all north or all south, it is almost unneces- 
sary to use the equation of second differences. 

Ex. 2. — Required the moon's right ascension on the 20th Novem- 
ber, 1826, at 9" 36"- 30" p. m. ? 

The Moon's right 
1826 First Dif. Sec. Diff. Mi 

Nov. 19th at midnight is 116" 20' 7" «o iw An,, 
20th at noon 122 31 4? ^^^ 

20th at midnight 128 41 36 ^ 
ajstatnoon 134 .'lO 7 

App. time 9^ 36'" 30* Diurnal log. 

Change of dec. 6° 9' 49" Prop. log. T 
Or -h by 60-6' 9".82 Prop. log. / 

Prop, part 4' 56'.12 Prop, log, 1.56196 

Or . 4°56'7"-2 

In this example we have considered the degrees qiinutes, the 
nutes seconds, and the seconds have been converted into a decimal 
by dividing by 6, since the change of declination exceeds the limits 
of the table. This comes to the same thing as dividing by 60; 
but any other aliquot part might have been taken, — siich as a balf, a 



I 31 



L4654;i 



1 



EXPLANATION OF THE TABLES. 21 

third, Stc. provided the proportional part be doubled^ trebled^ &c. as 

derived from this table. 

Now to 9^ 3^ 30° and 1' the equation is (K 4".6 

and 30" 2 .3 

and 4^ . .4 



Amount of the whole equation is T ,2 

Which must be added to 4° 5& 7"'2, because the first differences are 
decreasing, consequently the corrected proportional part is 4° 56' 
14.''4. 

Therefore, if to the right ascension at noon on the 20th, that is 

to 122° 31' 47" 

There be added . 4 65 14 .4 



The true right ascension required is 127 28 1 .4 

Ex, 3. — Required the sun's declination at noon, on the 20th of 

June, 1826, at Otaheite, in longitude 9»* 58°» W. ? 

Sun's declination at noon 23P 27' 11" N. 

Time 9^ 58" diurnal log. 0.38166 

Var. 0' 25" prop. log. 2.63548 

P.P. 10".4 . • 3.01714 -h 10.4 

First I>iff' Second DifF. Mean, 
Diff. for 19th 51 ^ 

20th 25 Si 25 + 3 .0 

21st 1 ^ — : 

True declination 23 27 24.4 

In this example the argument in time is found in the right-hand 
column in the lower half of the table. In lunar distances the ap- 
proximate time found by proportion after the hour given in the 
nautical almanac must be quadrupled, which, being used as an argu- 
ment, will give to the mean second difference Uie true equation, 
amounting^ in some cases> to about 6" in distance^ or 3' of longitude. 

TABLE XXVlll.— Reduction to the Meridian, Parts I. and II. 

In the course of the great trigonometrical survey lately performed 
in France, the repeating circle was much used in the determination 
of latitudes and other operations. Latitudes were determined by ob- 
serving repeatedly, near noon, the altitudes or zenith distances of 
a celestial object, reducing those taken off the meridian by appro- 

Sriate formulse or tables to what they would have been on the meri- 
ian. This method may be successfully practised by smaller instru- 
ments,'— such as Troughton's reflecting circle, or even a good sextant ; 
and Dr Brinkley, with his large eight-feet circle in the observatory at 
Dublin, takes three or four observations each day as near noon as 
possible, which are afterwards reduced to noon. 

To facilitate these operations, this table has been computed. Part 
I. by Delambre, and Part II. by Schumacher. 

Ex, 1. — ^Application of the preceding table to observations o£ the 
star Arcturus at the observatory of Dublin, on May 12th 1820, made 
with the eight-feet circle, having three microscopes, one on the right 
side of the instrument, one at the bottom, and one on the left. 

The latitude of the observatory from numerous observations of Dr 
Brinkley, corrected by his own very accurate table of refractions, 



2S 



EXPLANATION OF THE TABLES. 



which arepeculiarly adapted to hit observatory, is 53° 23' 13^.46 
Mean N. V. D. of Arcturui for 1820 09 52 31 

Mean right aacenaion 211 51 51 .6 

Place of moon's node 11" 29 26 



TioM by Clock. 

h* ID* %» 

13 56 28 

14 28 
14 9 51 
14 14 52 



Micro*. 



Z. D. Bottom 

MiCTOMOptti 



RfgHt 
Micros. 



oftheUiree 



I 



ReAiction. 



49.7 
31.7 
50.6 
38.0 



33 19 50.5 E. 
33 17 32.6 £ 
33 14 54.5 W. 
33 16 41.0 W. 



43 

471 
45.0 

31.7 



33 



19 54.83 
17 37.13 
14 50.03 



37^ 

3777 
3774 



Inter, 
BaronMC«r 19.67 Ext. 

Tune of OfaMr- 



Ther. 6l5 



16 36.90 I 37.77 



Me«i. 33 17 14.72 37.775 



Time tiT Ktar«t 
TnuMit by Clock, 
h. ro. •. 



14 
14 
14 
14 



7 
7 
7 
7 



33 
3.3 
33 
33 



Tfttlon. 

h. ID. I. 

13 56 28 

14 28 
14 9 51 
14 14 52 



DIffneoce. 
h. m. I. 

10 353 
6 353 

2 47.7 
7 48.7 



Rcductioo. 



Parti. 

220"'.] 
85 .22 
15 32 

119 30 



Part II. 

0".12 
.02 
.00 
M 



Sum'i 



440 .44 I .18 



110 .11 I .045 



Now, if the tabular quantity in Part 1. be called m, and that in 
Part 11. be called n, the latitude x, the declination i, the approxi- 
mate zenith distance z, the declination and zenith distance bemg + 
if north, and — if south, and the true zenith distance Z ; 

, „ cos. A COS. J . /COS. A cos. J\f ^ _ 

then Z = 2 ; — = — . f«+ ( ; — ^ — ) cot. Z*n 



or Z = 



sin. Z 
cos. A cos. ), 



sin. Z 



sin. Z 

cos. A COS. )\ 



(COS. A COS. e\ 
tn rin'"Z — ^°^ ** nearly. 



In the formula it is supposed that the latitude of the place and de- 
clination of the star, and consequently its zenith distance, are previ- 
ously known ; but in all cases where the latitude alone^ or the de* 
dination alone, is known, z must be substituted for Z in the formula, 
and then the resulting reduction, which will not differ materially from 
the truth, when appBed to z will give Z and a very nearly correct; 
after which, the operation pointed out by the formula, must be re^ 
peated with Z and A as if they had been previously known. This re- 
petition which, as appears by the following example, is easily per- 
formed, will give the reduction correct enough for all observationg 
made near the meridian; but, if the horary distance be.n^eat, a se- 
cond repetition may be necessary, though scarcely when the obser- 
vations are kept within the extent of our table, and, unless from 
necessity, they should not be taken more distant, as in that case, 
a small error in the time will produce a considerable error in the 
senith distance. On this account observations very distant from 
the meridian are not to be recommended, as they may tend to vitiate 
those made near it. 



EXPLANATION OF THE TABLES. 33 

A 63^ 23' 13^' COS. 9.776644 

> 20 7 28 COS. 9.972641 

z 33 . 17 16 cosec. . 0.260664 (a) cot 0.182732 

0.008739 X 2 = 0.017478 

m 110.11 log 2.041787 n 0.046 log. 8.663213 

38 2d, cor. + .0713 8.863413 



1**, Cor. — 112".35 (e) 2.060664 (c) 

or — 1' 62 .35 380 

2d, Cor. + .071 



134 



—1 52 .279 
z 33^ 17 14 .720 

2' 33 15 22 .441 
Ref. + 37 .776 

2" 33 16 .216 (/) cosec. a260794 (J>) 

240 (ft—a) 



— 112 .41 {d) 2.050804 {c+{b—a)] 



— .06 {d—e) 766 



«"' 33 16 .166 J/— (d-^) } 38 
This result scarcely differs from Dr Brinkley's, which is 33^ 15' 0".17, 
to which the aggregate of precession, aberration, and nutation, 
amounting to — 13".53, being applied, gives 33° 16' 46".64 for the 
mean zenith distance on January 1, 1820. 

Ex. 2. — ^At Maranham, August 28, 1822, Captain Sabine took the 
following observations of the star « Lyrse with a repeating circle of 
fix inches in diameter, the barometer being 29'" .96, the wermome- 
ter 80" Fahrenheit, the chronometer. No 423, fast 2*" 65"" 69*; the 
star, whose right ascension was 18^ 30" 67^.4, was on the meridian, 
at 8** 4°» 36» mean time, and at IP I'" 34* by the chronometer.* 



* This example is extracted ftom Captain Sabine*s work on the determination of the 
length of the seconds pendulum at various points of the earth's surfiitce, lately publish, 
ed at the expense of the Board of Longitude. It is a work highly to be reconimended, 
for perusal, to those likely to be employed in such experiments in future, as it contains 
valuable examples of all the requisite operations likely to occur in such researches. 



2tA0, 



10 49 40 8 54 



JJiVl^NATION OF U'HE TABLKS- 



305.090.23 
155.51 0.06 
64..^i'0.01 
28.85 0.00 
1.100.«0 
a940.00 
55.150,01 
1 16.91 0.03 



+s|+ i 
— 2LI 4 

=f " 

-8— 
+4 + 
-8~.. 
+ 7+ 6 



^ r First Vemit 

J Second 
t 1 Third 
£ I^Pourtli 

Mean 



r 167" 11' 50" 

11 30 

12 10 
11 40 

167 11 i?:* 



{First Vernier 136" 35*^1}"' 
Second 34 30 

Third 35 30 

Fourth 35 



Mean 
Index 
Level 



> 41° Iff 22" log.3«1.70813 
Ther. 80 F. . 9.9736? 

Bar. 29.95 9.99926 

Ther. . 9.99870 

r 4r5* ■/-■'" _* I.fi7976 



',31: 45': COS. 

37.38, (COS. 
10 22 cosec. 



m 92" JM leg. 

l*i owv^l69".08 
or — r 49 .08 

2d cor. + .07 



9.999578 
9.892776 
0,181555 («) cot 



Obs. Z D 

Cor. 

True Z D 
Star's dec. 

Latitude 



136 35 
192 48 12 5 
16 J 

8)329 22 56 

41 10 22 
+ 47-84 

— 1 49 .0] 



0.073909 X 2 = 
1.063835 n, 0.0425 log. 



0.147818 
8.628389 



— J ' 



2.037744 0.068 



318 



is unnecessary to repeat the operation in this case, as the difference 
the result would only be 0".04, making the latitude 2° 3J' 43.' 



TABL-E Xyil\.— Reduction lo either Solstice, the ObHquUif of 



k 

^^^^Htdtan altitudes, or zenith distances near either solstice. If the 
^^^^B longitude were three 

■1 



nee I 

en- I 



J signs exactly at noon, the operation 



<to reduce the actual obaervations to which they would have 
under these circumstances. To accomplish this object, this 
has been constructed. In the table the obliquity is supposed 
•33° 27' 40", and the reduction is the difference between this quant 
and the sun's declination at the seversl points of the eclipti 




EXPLANATION OF THE TABIiKS. 36 

responding to the ohaerved right ascensions. With the diferenoes 
ana variation f<n* IW change of obliquity the table may be adapted 
to any time within the limits of the table's variation of obliquity. 
Both quantities will thus be additive till the year 1835. The table 
is esrtended to 30°^, and consequently observations may be reduc^ 
by it for about seven days before and as many after the solstice. 

Ex, 1. On the 15th of June^ 1826, the sun's declination was ob- 
served to be SB"" 18' 51'^7» when the. right ascension was 6^ 35°^ 
51'.4^ and the obliquity 23'' 27' 39"', what was the reducticm to the 
solstice? 

Tabular obliquity 83<> 27' 40^' 
6b O"" 0* Estimated obliquity 23 27 30 

5 32 51.4 : 

Excess 1 

27 8.6 = distance from the solstice^ 
27 0.0 gives .8' 42".73 

8.6 gives 5 .564 

V'.0 var. obi. gives .005 

Reducticm 8-48^209 

Sun's declination 23^28 51 .7 



True obliquity . 23 27 39 .999 
By operating in this way for several days near either solstice^ the 
true obliquity may be obtoined from a mean of a number of observa- 
lions, and d(tti8equ6nfiy IScely very near the truth. It may be ob- 
served, however, that the sun's latitude from Delambre's tables, taken 
with a omtrary sign^ should be applied to the obliquity determined 
in this manner. 

JSflP. 2.^ — ^I had commenced to determine the obliqui^ of the eclip- 
tic from the totality of the Greenwich observations by Dr Maskelyne, 
and had proceeded so far when I was anticipifted by Dr Brinkley. 
I used the Frendi table of refractions, Delambre*s table of reducfimi 
depending on the sun's longitude instead of the R. A.^ which, being 
rather more convenient in practice, is made the argument here. Tlie 
longitude and latitude of tne sun were cottipateq from ]>elambre's 
Tables, and, as the methods are analogous, any one who can compute 
by the longitude can readily also use the right asoension, and the 
fQllowing example is given as an illustration of either. 



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TABLB XXX.~To change Mean Solar mie SUkreai Time. 

At a clork Mgulated by sidereAl time if indupensable in every ob- 
■enratory^ it it neceMary to ccnivert solar into adereal time in order 
to know bj the dock when any phenomena, such aa edipses, occhI- 
Utioni, &c^y calenlatecl in mean solar time ahoald take place. Hiis 
table is eaifloyed finr that purpose, as will appear by tne following 
example. • 

An in un er si mi of # Aqnarii by the moon took place on January 5, 
1824, at fl^ 40*60^, apparent solar time by the maridKan of Green- 
wich; what will be tne time by a sidereal clock which ahoit 
H)" 0* when the point Aries is on the meridian, and her error dut 
day 9» .54 fiut ? 

In this case the clock would be a right-ascension clock ; sad ? 
she went true would show the right ascension of the celeatialbpditi a 
they passed the meridian when observed by a transit instrumesi 
Now on the 6th of January, }S24, the sun's right aaoenaidn at dood 
is 19** I'^ST'.O, the sane as would be shown by a clock truly r^ 
lated. < ; 

Bat aa the dock waa 80* JMk ftut <m that day this omakf be addal 
to give the time shewn bv the dock, that is, she shows J19^ 2^19M 
at noon. As the immersion happened at 3^ 4^ 60" P, If • wA^iw^t 
be converted into sidereal time, and added to the preceding tihf^ 
th^ time shown by the clodk> so that an astronomer may beprefiaicd 
to observe It 

This operation may be accomplished by the table. 

Time, AcceUraiion, 

fi^ CF» tf gives 0"29^JM» 

46 TJ5ff] 

50 0.138 



3 46 60 37.264 

Hence to the time show by the clock 19h 2» 13^i54 
There must be added 37:^ 

And . 3 46 50 .00 



Whence the time shown will be 23 49 40 .JBO 

TABLE XXXI.— 7o change Sidereal into Mean Sbiar. Time. 

Thu table may be useful for finding the rate of a clock or chrono- 
meter. As the transit of a fixed star advances 9^ 66*.90B-da]ly on 
mean solar tiifie« if the ppssaae of a star be observed with a transit 
instrument each day for several successive davs, or the disnipearsDce 
of a star during several suoceasive niffhts benuida finsd^^Sj^t^sadi 
as the vane of a fltoeple or the body cithe steeple itself^ neariy inthe 
meridian^ the position of the eye of the obs^er bdn^ also fixed, 
the rate of the clock beeomes known on sidereal, and cbnse^^uently, 
by this table, on mean solar time. 

Required the retardation on lO* 5^48™ .56* of ddereal^me? • 






EXKLAVAXION OP TSB *T14BIJaii SV 

0*5^ 0-0" 49.147 

48 . 7864 

66. . . 0.163 

" , 1 6 4866 --0 40 .16J44 

10<>6 48 66.000 

Me«B 8ol«r time » 10 6 8 9^5% 



T A!BLE XXXi;.— 7(^ convert Mean Time, into ParU^ if 'the Equqtm-. 
'This fkble Haay f>e useful for converting intb'de^ees, &q; ^e 
hojursy minutes^ &c. shown by a clock or chronometer re^rulAted ac- 
cording to meiui time; and tne method' of us^* it will be reaclilj 
'vinderstobd from the examples to the t^o preceding tables^ and that 
oir Iplaptaih Rater's in the appendix. 

TABLE XXKIIL— Lengths ^ circular Arcs. 
The method of using this can be no difficulty to those acquaintBd 
iwitli the preceding tables, as they are employed.in a similarmaBOiier. 

\ TABLES iXXXiy., XXXV., XXXVi:, XXXVII., XXXVIII^ 
^t^XIX., XL., XLIi, and XLII. are Abridged from a series of tablet, 
lg^!ltor Fi^ows^ astronomer at the Cape of 6opd Hope, and were 
traiikaitted to the Adihiralty, along with an approximate catalogue 
of stars which he had formed there, and are very convenient for 
finding at once the aM\>unt of the correctlohs for precession, aberra^ 
tion, and nutation for any given observation, both in right ascension 
and declination. In addition to these, howev^, another table must 
be computed annually. Since the tables are only given to every ten 
minutes of right ascension, proportional parts are added for every 
single minute as far as 6 indicated by the figure in the place of tens 
in the side column. If the odd minutes exceed 6, the proportional 
part mu^t be taken at twice, or the complementary proportional 
part to the next minute of even tens, must be applied wiw a contrary 
sign when necessary. 

To understand the method of applying these tables is premised 
the following 

Synopsis : — 

Coofttnti. 

Table XXXIV. ::=— r.3962 sin. R. A. tan. dec.+S'.OeTB =a 

XXXV. =1— 1.36008ih. R.A.=j[),andj9X8ec. dec.2=:6 

XXXVI. = — 1 .2300COS. H.A. = 9,and gxsec. dec.=r c 
XXXVIL =+ 0.6430coB.R.A.=r#,and«xtan.dec. = cr 

• XXXVIII. £7— >- 90 .0436 cob. R.A. = annual precession = t! 

XXXIX. t= — 20 .2660 cos. R. A. = js", and j/ x sin. dec. = V 

* XL. = + 18 .6800 sin. R. A. ±i ^V tod ^xsin. d.+K = t^ 

XLI. = + 8 .0659 cos. dec. =: . • =: r 

XLIL =— 9 .6480 sin. R. A. = ^^ =1^' 

1 »^ . 1 1 , sin. 3 sin. 2 
Annual Table, part Ist = <: 5 -^ — . . s= A 

part 2d = 0.93046 (cos. « — cos. 2 Q + 

2 
•^ cos. 2 0) . . = D 

where t is the time elapsed since the commencement of the year 
when the sun's mean R. A. is supposed to be Iff* 40™. 



30 



EXPLANATION OF THE TABLES. 



Table of sloes of bud's longitude at the time of culmination t= B 
Tablepf ctj^nesofthe game . . , =C 

Then the vrhole correction in R. A. = Ao +B b +Cc+J}d (1) 
indec. =A«'+B 6'+C c'+Drf' (2) 
Ex. — Required the corrections of Foraalhaut in right ascenaon 
and decUnation fiw July 20th, 1824, at the time of his passing the 
meridian of Greenwich, the R. A. of the star being 22" 48"', and de- 
clination 30" 33' South. 

The sun's longitude for this time is 118" 19', of which the natoral 
sine ia .881 = B, and the cosine is .473 = E. A and D must be 
taken from an annual table, or computed from the formuls given 
above for that puTpose. 

' Then from table XXXIV., &c. take the proper numbers for the 
R. A. of the star, and complete the multiplications iiidicated by for- 
mula (1) the sum of the results will be the total correction in R.A., 
and those by formula (2) will be that in declination. ' <, 



Thus Table XXXIV. .414 
^ 590 

37 

I44+ 
Constant . 3.068 


B 


C 


A 


dT 


.881 
-418 

.352 
. 9 
. 6 

-1$ 


.473- 
1.178— 


.901 + 
3.312 + 


■11S+ 

.61 + 
.06 


471 

3 


2.981 
3 


06 


2.^84+ 
ran. .dec 


... 558+ 

. .1 ,, 


■068+ 
■S?0 


Nat. .ec. dec. 


1.161 




, 


034 


923 
02 
55 

1 


040+ 






i 


1.071 + 
2.984+ 
0.04«+ 

4.095 + 


= .v 


. .'i' 


1 rections in right ascension. 


=-+