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iSEARCH UMUHES
MATHEMATICAL
AND ^ ,
ASTRONOMICAL TABLES,
FOB THB USB OP
8TUDBNTS OF MATHEMATICS^
PRACTICAL ASTRONOMERS, SURVEYORS, ENGINEERS,
AND NAVIGATORS ;
WITH
AN INTRODUCTION,
COKTAlVIirO
ION
THE EXPLANATION AND USE OF THE TABLES,
ILLUSTKATIO BY
NUMEROUS PROBLEMS AMD EXAMPLES.
BY WILLIAM 6ALBRAITH, M,A^
TKACHBB OF MATBBMATICS IN ■DINBUBOH.
EDINBURGH :
PUBLURBD BY
OLIVER & BOYD, TWEEDDALE-COURT ;
GEO. B. WHITTAKER, AND J. W. NORIE & CO.,
LONDON.
1887.
oeiL
f
I
CNTKBBD IN STATIONERS* HALU
» « . • •
• • » .
OI.IVXE h BOTD, PAIHTEHS.
TO
Sib GEORGE CLERK, of Pennycuick,
BART., M.P., F.R.S.,
ONE OF THE LORDS COMMISSIONERS OF THE ADMIRALTY,
Sis,
The following Work, which you have allowed me
the honour of inscrilMiig 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 been 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,
Sib,
With the utmost respect,
' Your most obedient servant,
WILLIAM GALBRAITH.
Edimbxjbgh, Nw.^ 1826.
TO
Sir GEORGE CLERK, of Pennycuick,
BART., M.P., F.RS.,
ONE OF THE LORDS COMMISSIONERS OF THE ADMIRALTY,
&c, &c., &c.
Sis,
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 been 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,
Sib,
With the utmost respect,
' Your most obedient servant,
WILLIAM GALBRAITH.
Edinburgh, Nov., 1826.
PREFACE.
The application of the mathematical sciences to practical pur-
poses has of late made great advances in accuracy and precision.
The perfection also which astronomical and geodietical operations
have reached, and the extreme delicacy of construction to which
instruments have been carried, require correspondent improve-
ments in the methods of computation and reduction ; and, there-
fore, convenient tables of moderate expense must be of great
value to those engaged either in the details of practice, or the
business of instruction.
There are two classes of tables chiefly in use; one either
large and expensive, or attached to expensive works, and which
therefore can with difficulty be procured by the generality of
purchasers ; the other so limited and defective as to be totally
unfit for constant reference. It has been my study to hold a
middle course between these two extremes. By making such
additions to the usual tabliss as to render their application more
easy, without greatly increasing their bulk ; by selecting the most
useful from larger collections ; by supplying some new tables, and
amplifying the practical rules, several very laborious processes
have been rendered more simple and precise, while the requisite
accuracy for the nicest purposes has been strictly preserved.
In most of our initiatory works for popular instruction, the
processes and examples are unfortunately conducted in such a
manner as to be comparatively of little advantage in actual prac-
tice, and, consequently, what has been learned in youth, must,
in a great degree, be forgotten in manhood, while new methods
are then to be acquired.
vi PREFACE.
To remedy this inconvenience, I have selected some of the
most approved modes of treating the problems frequently re-
quired by Astronomers, Navigators, and Engineers, from the
works of persons 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 Eater,
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
amplifying some operations and rendering others more accurate.
The explanations will, it is hoped, be found full and explidt,
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
minutiae a second time. Still, however, there may be some parts
which require to be expanded, in order to be more readily un-
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.
In the first I have shortly described the nature, and investi-
gated the more simple series for the computation of Logarithms.
I have generally, however, only given the more important rules
jn words at length, without investigation, so as to be readily com-
PR£FAC£. vn
preheiided by persons who have acquired a knowledge of the
elementary principles of mathematics. In fact, the demonstra-
tions can only be understood by those who have obtained a tdier-
able knowledge of the elements of geometry and algebra, and,
^ce 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-
gendre^s Elements of Greometry in order to study that science, hf
will find it to contain also very el^ant investigations of almost
all the useful properties in Plane and Spherical Trigonometry.
On this account, I have only ^ven 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 the 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 either 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,— 4in 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 Eater 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
formulae for that purpo^e^ in order to initiate^ as far a;& ^go^^<&^
viii PBEFACE.
our Cadets and Midshipmen in these interesting researches ; as
such higher objects of pursuit, not only invigorate their faculdes,
but inspire them with enthusiasm for the attainment of profes-
sional renown.
The fourth part contains the necessary Explanation of the
Tables.
I have thus endeavoured to collect, into as small a space as
possible, the greatest quantity of useful matter naturally con-
nected with the subjects treated in the work ; but with what
success I must allow the. public to determine.
WILLIAM GAXBRAITH.
Edinburgh, November^ 1826.
.1 if :■
' • .1 •. ■> 11 .1
• M \
V.
■ CQNTJEOTS
INTRODUCTION.
Pagft
PAHT I. PROPERTiBS of LOGARITHMS «i.. 1
ConftmctkMi: of Logsritfams.. ;....«..<.... 4.^*,i 4
'Trig oa o m ettical Lines, called Sines^ Ac. «*»... .^ B
'Multiples and Powers iff Arcs... 12
PLANE TRIGONOMETRY .«.., ,, 15
lu Applicarian to Sailuiga ia. Natvigalion.,,.*, f.o*» 24
Its Application to the Mensuration of Height and Pittaqoes....... 25
Iti ApfficatioA to tlM Detecmination of t)ie Lines and A«gl9> of Re.
{^ttbr Fortresies^ and a TaUe of their Measpns ww^ 43, 44
Meaaurenfint of Altitudes.... •^••^•, 44
II. SPHERICAL TRIGONOMETRY, Ac.
Definitions, Principles, and General Properties....... 56
Solution of Spherical Triangles, with their Stereographic Projection 63
Napier's Rule of the Cireular Parts 64
Maskdyne^s Rules for determining the Latitude and Longitude,
ftom the Right Ascension, Declination, and the Obliquity of
the£cliptic,&c.. 68
Solution of Oblique-Angled Spherical Triangles . 73
' On finding the Latitude by Observation. J... ^ 81
On Finding the Longitude by Observation.
1. By Lunars 89
2. By C hf oB dmotc rs. .•««..^<..», 104
Equation to Equal Altitudes 107
3. By Occultations 114
4. By the Moon?s Transit. ,129
Of the Transit Instrument *139
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 ClocA: or Chronometer by the
Transit Instrument 136
m. MENSURATION, SURVEYING, AND FORMULA, dec.
Mensuration of Surfaces r 139
Mensuration of Solids 142
CONTENTS.
Paet IIL Land Snnreying 144
Lerelliiig 146
RULES and FORMULA 147
The best Form of Triangkt 147
To reduce Angles to the Centre of the Station 147
To compute the Spherical Excess 148
To reduce a Measured Base at any Height to the Lttel of the Sea. 149
To determine the Horizontal Refraction by Rule or FormnlA 149
To find the Angle made by a given Line with the Meridian 149
To determine the Elliptidty of the Earth by the Measuzement of
Arcs 160
To determine a Degree of Latitude 152
To determine a Degree of Longitude • 162
To determine an Oblique Degree. • 163
Sperific Gtarity ; 163
To detennine the Specific Oravity of Air, Dry, saturated with
Moisture, and according t9 the actual State of the Atmosphere... 164
To determine the Specific Orayities in vacuo 166
To detecmiiM the J^ibcts oif the Buoyaiicy of the Atmosphere on
the Pendulum • 166
Cofieetioa of Pendulums vibrating in Circular Arcs 166
^(T^ption of Vibration iot Buoyancy • »••, 166
,.,,., .for Expansion f • 167
£brHeiflihtab9vetbeS«i^...,.,... 167
Determination of the Length of the Pendulum at difbrqnt Points
on the £arth*s Surface ;.. 168
Determination ef the Figuioof the Earth by the PendnlniD 169
Compansonof the English and IVfndi Pend«luiB8..v»«>— 160
Velocity of Sound 160, 161
Vclodtyof the Disdiatgeof Water-pipes^ RiTsrs, anj€a—lB..161, 162
FallinaRtvevcausedby ObstMstioaintheStiaam^ 162
Tonnago of Ships.- ^ ....^ «..4. 162
Strength of Timber • .^., 166, 167, 168
CONTENTS
or
EXPLANATION OF THE TABLES, Ac.
Pkge. Page,
Exp. Tab.
Table I. Miles of Longitude at any Latitude.. 1 1
II. Logarithms of Numbers 1 1
Logarithmic Arithmetic 6
III. Angles whidi every Pohit and Qnaiter Point of the
Compass makes with the Meridian 7 17
IV. Logarithmic Sines, Ac to every Point and Quarter
Point of the CompssBS 7 17
V. Logarithmic Sines, Tangents, &c to Dqprees 7 18
VI. Natural Sines, Tangents, Secants^ sad Veislnes, to
every Dcsree of the Quadrant 11 63
VII. Meridional Paris to every Degree of the Quadrant.... 11 64
VIIL Traverse Table 11 64
covnumk
.%
,«fr)T4^LE IX. Diurnal Logarithms 12 66
X. Proportional Logaritfams • 13 68
•* XI. Dip of the Horizon 13 84
\^ XII. Dip at different Dutancei 13 84
^^< XIII. CorreetionofiheSun*s AldtttdeatSea 14 84
XIV. CerijBctionofa8tHr'a AltHode 14 84
XV* Svii^B Sonidiameter, fte 14 86
XVI. Son's Parallax in Altitude 14 86
'^ XVII. Mean Refractions by Mr iTOiy 14 86
XVIII. )
XIX. > Subsidiary to XVII 14 89
XX.J
''' XXI. Augmentation of the Moon*a Semidiameter in Alti-
tude, andZ.D. 16 90
-^ XXII. Reduction of the Moon's Parallax on the Spheroid... 16 90
XXIII..Logaridimsofthe£arth*s Radii on the Sphermd 16 91
XXI V. Reduction of the Latitude. : 16 91
XXV. For determming the Latitude by the Pole Star 17 91
XXVL Augmentation of the Moon*8 Semidiameter by the
Nonagesimal 18 92
XXVII. Equation of Second Diffoenoes for 12 and 24 hours... 19 93
XXVIII. Reduction to the Meridian 21 94
XXIX. Reduction to either Solstice 24 96
XXX. To duuige Mean Solar into Sidereal Time 28 96
XXXI. To change Sidereal into Mean Stdar Time 28 96
' ( XXXli. To convert Mean Time into Parts of the Equatcr 29 97
XXXIII. Lengths of Circular Arcs 29 97
'*' XXXIV. ToXLVIII. For computing the Corrections of the
Fixed Stars 29 98
XLIXi Mean Obliquity of the Ediptic 32 103
L. And LI. Corrections of the Obliquity 32 103
LII. And LIII. Solar and Lunar Nutations of the Equi-
noxesinTime 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 33 106
LVII. Sun*s Declination for 1828 33 106
LVIIL Equation of Time for 1828 33 107
LIX. Coirection of Longitude by Chronometers 34 107
LX. Latitudes and Longitudes of Places.... 36 108
LXI. To convert Space into Time 36 109
LXII. To convert Timeinto Space 36 109
LXIII. Useful Numbers in Calculation 36 110
LXIV. And LXV. To find the Time and Height of High
Water 36 111
LXVI. And LXVII. Tables of Equation of Third and
Fourth Differences 36 112
LXVIII. Table to find the Latitude by the Pole Star 87 112
MISCELLANEOUS TABLES IN THE INTRODUCTION.
Tabls I. Signs of Trigonometrical Lines. 12
II. Multiples and Powers of Arcs 12, 13, 14, 16
III. Measures of Forts 44
IV. Depression of Mercury in Glass Tubes 48
V. Elastic Force of Aqueous Vapour (Dalton) 48
VI. Logarithms of the Bulk of Gas at different Tempera-
tures... , 49
7
X CONTENTS.
Base.
Paet IIL Land Snnreying 144
Lerclling 146
RULES and FORMULA 147
The best Fonnof Triangkt 147
To reduce Angles to the Centre of the Station 147
To compute the Spherical Excess 148
To reduce a Measured Base at any Height to the Lerel of the Sea. 149
To determine the Horizontal Refinction by Rule or Farmnl» 149
To find the Angle made by a given Line with the Meridian 149
To determine the Elliptidty of the Earth by the Measuzement of
Arcs 160
To determine a Degree of Latitude 152
To determine a Degree of Longitude......... 152
To determine an Oblique Dc^e...... 153
Specific Gravity ; 153
To detennine the Specific Oravity of Air, Dry, saturated with
Moisture, and according t9 the actual State of the Atmosphere... 154
To deteimine the Specific Gravities in vacuo,,,, 155
To detecmiiM the J^ec|s of the Buoyancy of the Atmosphere on
the Pendulum 156
Correction of Pendulums vibrating in Circular Arcs 156
(lorreption of Vibration for Buoyancy ^•••.. ,,», 156
.,.,,MfM for Expansion •••y. ..,.•• 157
«....£brHeiflihtab9vetbeS«io.Mv-M 157
Detei^nination of the Length of the Pendulum at diJFer^t Points
on the Earth's Surface 158
Determination ef the Figure of the Earth by the PeadnlniD 159
Comparisonof tbeEnglMiandlVfndiPendvlunu....,.^... 160
Vdocity of Sound 160, 161
Vcaodty of the Disdiaige of Water>pipes, Rivers, and Ca—ls.. 161, 162
Fall in a Rivenr caused by Obstnietioa in the Stream^ 162
Tonnago of Ships.- , ^..t.,, «... 162
Strength of Timber ^., 166, 167, 168
CONTENTS
or
EXPLANATION OF THE TABLES, &c.
... Page. Page
Exp. Tab.
Table L If ties of Longitude at any Latitude ,.... 1 1
II. Logarithms of Numbers 1 1
Logarithmic Arithmetic 5
III. Atigles whidi every Point and Qnaiter Point of the
Compass makes with the Meridian 7 17
ly. Logarithmic Sines, Ac* to every Point and Quarter
Point of the CompssBS ; 7 17
V. Logarithmic Sines, Tangents, &c to Degrees 7 18
VI. Natural Sines, Tangents, Sccaatt, sad Veisines, to
every Dcsree of the Quadrant 11 63
VII. MeridionafParts to every Degree of the Quadrant.... 11 64
VIIL Traverse Table 11 64
CX)irTSliTSb id
.^]T49Li: IX. Diurnal Logarithms 12 66
X. Proportional Logaiithms 13 68
XI. Dip of the Horizon 13 84
XII. Dip at differont Diatancet 13 84
XIII. Correetion of the 8un*8 Altitude at Sea 14 84
XIV. Cwnection of a 8tar'i Altitude 14 84
XVi 8«ii'*8 Sonidiameter, dte 14 86
XVI. Smi't Parallaz in Altitude , 14 86
XVII. Mean Refractions by Mr iTOiy 14 86
XVIII. )
XIX. > Subsidiary to XVII 14 89
XX.J
"^ XXI. Augmentation of the Moon*t Semidiameter in Alti-
tude,andZ.D * 16 90
^ XXII. Reduction of the Moon's Parallax on the Spheroid ... 1 6 90
XXIIL. Logarithms of the £arth*s Radii on the Spheroid 16 91
XXI V. Reduftion of the Latitude. .1 16 91
' • XXy. For determining the Latitude by the Pole Star 17 91
XXVL Augmentation of the Moon*« Semidiameter by the
Nonagesimal 18 92
XXVII. Equation of Second Differences for 12 and 24 hours... 19 93
iXXVIII. Reduction to the Meridian 21 94
XXIX. Reduction to either Solstice 24 96
XXX. To duuige Mean Solar into Sidereal Time 28 96
XXJtI. To change Sidereal into Mean Solar Time 28 96
' XXXli. To oonvert Mean Time into Parts of the Equator 29 97
XXXIII. Lengths of Circular Arcs 29 97
■ ' XXXIV. To XLVIII. For computing the Corrections of the
' ' Fixed Stars 29 98
X'LIX. Mean Obliquity of the Ecliptic 32 103
L. And LI. Corrections of the Obliquity 32 103
LII. And LIII. Solar and Lunar Nutations of the Equi-
noxes in Time 32 103
LIV. Right Ascensions and Declinations of Stars for 182&.. 32 104
LV. Decimal Numbers for each Day in the Year 32 104
LVL Sun's R.A. for 1828 33 106
LVIL Sun*s Declination 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 36 108
LXI. To convert Space into Time 36 109
LXII. To convert Timeinto Space 36 109
LXIII. Useful Numbers in Calculation 36 110
LXIV. And LXV. To find the Time and Height of High
Water 36 111
LXVI. And LXVII. Tables of Equation of Third and
Fourth Differences 36 112
LXVIII. Table to find the Latitude by the Pole Star 87 112
MISCELLANEOUS TABLES IN THE INTRODUCTION.
Table I. Signs of Trigonometrical Lines. 12
II. Multiples and Powers of Arcs 12,13,14,16
III. Measures of Forts 44
IV. Depression of Mercury in Glass Tubes 48
V. Elastic Force of Aqueous Vapour (Dalton) 48
VI. Logarithms of the Bulk of Gas at different Tempera-
tures • 49
7
»•
an OONTKNT&
Fage.
Table VII. Logarithms of the Effect of Latitude on Barometric
Ahitndes 4i, 60
VIII. CorTCetion of the Oblique Semidlsmetef in Lnnan by
Dr Young 100, 101
IX. Equation of Second Difference for Three Htmn or for
OK and 160" 102
X* Correction of Apparent Tine dependingnpon the Equa-
tion of Second Difference and -the vfaiation of the
Distance between the Moon and die Sun, or a fixed
Star, in Three Hours. 102
XI. Of the Decimal Fractions of a Day 113
XII. Decimal Parts of an Hour 113
XIII. To convert Decimals of Time into Degrees at the rate
of fifteen Degrees to an Hour 113
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 141
XVf Polygons 142
XVIL Reguhir Bodies 143
XVIII. Table A. for Correcting the Number of Oscflktions
for the Arc of Vibration 156
XrX. Tables of Specific Gravity 1«3, 164, 166
XX. Expansions of Solids and Liquids ^..-.. 166
XXI. Table for computing the Strength of Timber ....^ 166
XXII. Table for Correcting Lunars for Spheroidal Figute of
the Earth. Explanation of Tables 43
XXIIL Table for finding the Latitude by the Pole Star.. 43
. I
t I -
INTRODUCTION.
PART I.
OF LOGARITHMIC AND TBIOONOMBTBICAL TABLES.
Section I.
I Of the Properties of Logariikms,
l.-XoGABiTHMs are a series of numbers^ originally invented by Baron
Napier^ for -tbe^ purpose of facilitating arithmetical calculations.
Tins end is attaineid by their enabling us to perform the operations
of '.mukfplication by addition^ of division by subtraction, of involu-
tion* W multiplication^ and of the extraction of roots by division.*
9. It is evident that any two series of numbers^ the one being in
ar^thmeticid arid the other in geometrical progression^ possess these
prdperties^ thns^ for example^ let the
Ar. series be D 1 2 3 4 5\«
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 1000, 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, the second, or square root of 10,000 the
number under 4, is 100, the number under 2, and so on.
Napier called the first series the logarUhms of the corresponding
numbers in the second.
3. Since the two series may be assumed at pleasure^ we may have
as many different 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 particular cases, their peculiar
advantages.
On considering these series, it appears that the logarithm of 1 is
* The identity of this process with that perfonned upon the exponents of quantities
in the corresponding operations of algebra, will be obvious to those who have acquired
the rudiments of that branch of mathematics.
2 INTRODUCTION.
0^ and that of 10 is 1^ and hence the logarithms of all numbers be-
tween 1 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 less than 2, that is^ they are 1 with some fraction an-
nexed^ and so 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 mtegral 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 us assume the two following series :
r*, r^, r«", r^', &c. . . (1)
^ , y , . y, y". &c. . (2)
in which r is some given number greater or less than unity^ and x,
X*, x^^y x'" y &c. any variable quantities chosen in such a manner that
r*==y, r*':=zy', r^'-=!U"y r^'-zzy'"^ &c., then the several exponents^
Xi X*, x"y X'", &c. of the series (1) are called the logarithms of the
corresponding terms in the series (2).
Thus if ^, y, y, y"\ &c. be a series of numbers such that r*=:y,
r"=y, r''=y', /^"irry^ &c., then ar=log. y, ar'=log. y', ar"=log. y,
»"'=log.y", &c.
6. For the purpose of adapting the series (1) to the series of na«
tural numbers 1^ % 3, &c. the given number r must be greater than
unity^ the first index x must be equal to 0^ and the several indices
or', V', x'", &c. must continually increase. For, since by the prin-
ciples of a^ebra, x°:=lf whatever r may be, -this series will increase
from 1 to infinity ; and by properly adjusting the values of x', x",
«"', &c. it is evident that the several quantities r*', r*", r"", &c.
may be made to coincide with the numbers 2, 3, 4, &c. For ex-
ample, let rsrlO; then, girtee 10°=1, find 10^=10, the indices of 10,
which HFOuld give 10*', 10*'^ 10*'", &c. equal to the numbers 2, 3,
4, &c., must be fractions between and 1. If we take the number 3
we haye 10^=3.16 nearly> from which we infer that a fraction (ar'J
somewhat less than ^ or 0^5, being made the index of (r) 10, woula
give lO^'ssS. This ftturt;ion is found by calculation to be '47712 ;
hence 10*'^''^*' =3; therefore, when r=10, the logarithm of 3 is
.47712.
. In like manner, if we assume the numbefr 5, whose logarithm is to
be found in place of that of 3, we have 10^=4.64 whence a fraction,
«(»)' somewhat greater than f, or .606 being made the index or
exponent of 10, would give 10*^ =5. This fraction more accu-
rately computed is found to be .69897^ that is, when r=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°=sl, the logarithm of 1 must
be 0. Tms constant quantity r from the powers of which the na-
tural numbers are formed, is called the radix or base of the system
to which it belongs.
8. In the general equation r'zzy, (art. 5.), let us make x vary
and observe the correspondent variations of ^.
LOGARITHMIC TABLES.
If r is greater than 1^ on making ^^O^ we have ^:=] ; when xszl
then y=:r or the logaridim of the base 18=1 ; in proportion as s in-
creases from to infinity^ y will increase from 1 towards r, and after-
wards to infinity^ so that if we suppose x to pass through all the in-
termediate values^ in following the law of continuity, y will increase
alao in the same manner, though much mcure rapidiy.
If we put for x^ negative values, we shall nave yzsar", or
y'=^' 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-
tively y 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
Briggs on their celebrated meeting at Edinburgh, when conversing
on the propriety of changing the logarithmic scale.
If r is less than 1 we shall make r=i-^y h being greater than 1 and
we have S^^jz or ^=:6* , according as x is positive or negative. We
fall here upon the same case, with this difference, that x is positive
wheny is less than 1, and negative when y is greater than 1. This
nro]>osal Briggs made to Napier, but immediatly abandoned it on
Namier suggesting that mentioned above, which was finally adopted.
If r=:l, we have y=:l whatever x may be.
We may then say generally, that provided r is not unity, there
can always be found a value for x, which renders r* equal to any
given number y. The constant use that is made of the properties of
ue equation y:=r* requires the denominations of its parts to be fix-
ed in order to avoid circumlocution. Hence as before remarked, x
18 called the logarithm of the number v, the invariable number r is
called the base and, finally, the logarithm of a number, the power to
which the base must be raised in order to produce that number.
With regard to the base r it is arbitrary, and when we write
x=log. y to show that x is the logarithm of the number y or that
y=zr', the base r is alway understood, because when once chosen it
IS supposed to remain fixed. If it should be changed the new base
ought to be indicated.
9. From these principles are derived several properties.
1^. In every system of logarithms, the logarithm of 1 is and that
of the base r is 1.
2^. If the base r is greater than 1, the logarithms of numbers
greater than 1 are positive, the others are negative. The contrary
takes pj^ji^ce if r is less than 1.
3°. The composition of a table of logarithms consists in determin-
ing all the values of x when y is made successively equal to 1, 2, 3,
&C. in the equation y=r*'
If we suppose r?z=^ on making
x=,0, (, 2#, 3(, &c. . . . n^
We find y=ly ft, ft\ f»>\ &c. . ^»
The logarithms therefore increase in progression by differences,
while the numbers increase in progression by the product or quo*
tient, according as /m is an integer or a fraction.
The ratios are the arbitrary numbers ^ and /». We may, therefore,
regard the systems of values of x and y which satirfy uie ec\yv.'a>L\citv.
4 INTRODUCTION.
y^sf', as dasfled in these two progressioiiB^ which coincides with
what has been already said in art (2.)
10. We shall now demonstrate algebraically the various propertiea
of logarithms.
Let N and n be any two numbers belonging to the series (1) ; and
for example^ let Nsst* and «=sr^> then N n-izr* x f^rsr*^', but, by
art 6, the logarithm of r^ is or+x'rslog. r'+log. f^=log. N+log.
In like manner, if n, n', n" be any set of numbers in the series (1)
it might be shown that the logarithm of nx«'X«", &c.=log.
n+log. »'+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 — =~ir * ^^^ *^® logarithm of r»"^=:a?— a:'; therefore,
N
the logarithm of — r=a?— a:'=.log. r» — ^log. r*'=log. N — ^log. n ; hence
it appears, that tl)e logarithm of the quotient of any two numbers is
equal to the difference of their logarithms ; and that the logarithm
of a fraction ( — j is equal to the logarithm of its numerator minus,
the logarithm of its denominator.
If N be less than n, then log. N — ^log. n is negative ; therefore,
the logarithms of all proper fractions are negative.
12. Let N=r' be raised to the wi^ power, then N*»=:f*"; but the
logarithm of r*~ is=9iix, hence the logarithm of N**=:»m;=:w log. r*
Sim log. N ; for the same reason, since ^ N=NOT=:rOT, the logarithm
of /N=— =— ^^ — : from which we infer, that the logarithm of the
HI** power of any number is found by multiplying its logarithm by .
m, and that of me mf^ root of any number, by dividing its logarithm
by HI.
Section II.
Of the Construction cf Tables of Logarithms,
13. Let r* express generally any term of the series, (1), and let N
be the corresponding number, then r*=N. Hence to find the loga-
rithm of N is merely to solve the equation r'zzN where x is the un-
kno¥m quantity. In order to accomplish this purpose let r=l + h
and N=l + «, then extract the y* root of each side of this equation,
and we obtain ^1 + 6 V=^l + nV, which by expansion gives
•+>+,- (^■) (!>;- (,--■) (H Q+--
' +!«+} (i-o (iVi 0-0 (h) (S)+^
Now suppose y to be indefinitely great with respect to x and 1,
X 1
then will- and - vanish in reference to — 1, — 2, &c., so that
-—1 and 1 will each become equal to —1 : —2, ^2, each
y y y y
LOGA&ITHMIC TABLES. 5
equal to -—2^ Sec, Sec*, hence rejecting 1 from each side of the equa-
tion we have
- (ft— i b^ + i 6'— i 6* + &c.)=- (n-4 n^ + i «'— i n*+&c.)
1. , /I . N w— An« + J »'— i»*+&c.
hence ^, the log. (1 + ^) =^ ^.^ j ^,11 ^4^^^ .
but fi=N — 1 and 6=r — 1, therefore, by substitution, the above ex-
pression becomes
(N— 1) — i (N-l)« + j (N-iy— i (N-iy+&c.
(r— 1)— K.r— l)« + i('— l)'-i('— l)*+&c.
^^ ^(r-l)-i (r-^ (r-l)« + ^ (»— l)'-i (r_l)«+&«-=
This quantity M, which evidently depends upon the base r, is
called the modmus of the particular system of logarithms to which it
belongs. As it is obvious the series n — Jw*+J n' — } »*+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
paorticular 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
1 ,
(r-1) — i ir-lY + ], (r-l)'-i {r-iy+Scc; ^''
Log. (l + ,i) =M (n— i n« + ^ n«— i n*+ j— «*— &c.) . . (3)»
In the above for n put — n, and then
Log. (1— n) =M (—n—i r«— J «»— i n*— ^ n»— &c.) . . (4)
Subtract (4) from (3), then log. (1 + n) — log. (1 — ii)=±1og.
J±|=2M(n+Jn' + i«* + ^«' + &c.) (5)
Let Nm^i-' — , then n=z ^. . , hence
1 — n N + «
Log. N=2 m{ (§=-}) +J (g=-J)' + J (1^})' + ^"} • («>
Again let »==-Tr= — r, then ^ =?rrf — r, hence by substitution in
^ 2N — 1 1 — n 2N — 1 "^
formula (5)
w- nzt=2 ** (2N-4 +3 (2 N-1) + rm-iy "•" *'*'•) *^'
Log. N-log. (N-1) =2 M (23^-^ + 3(.23j-,y, +6(2tr)*
+ &c); andlog.N=2M(jji3+3-^J-3^,+g-^^-+&c.^
+log. (N— 1) (7)
^^y' ^ rS=-^' *^«° "=211+1 *"'* ^•'s- (^+^) =
^*^(2iri:i+3(2irM)'+6(N+i)»+«^*=-)+l°g-N- • (8)
• By means of this formula the logarithm of a quantity exceedmc vmVU "V^n «^ NCt-^
snian fraction may be readily found.
6 INTBOBUCnON.
the log. of IsaO^ this last series which ccmTerges verv
ra^idly^ will give the logarithms of all the natural numb«r8> with
facility in succession. To these theorms nught 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 arbitrary^ let
it be so assumed that , =-r — ^-^ -,^ ^ . . ^.-= — r— - or M ehall
(r— 1)— i (n_l)« + ^ (r— l)»-,-&c.
be equal to 1, that adopted by Napier. Taking series (8) we have
since «
Log. 1 = (art. 6.)
2 = 2 ^i+l4+X + &c. to 8 terms) . =0.6931472
3 = 2(i+gL+^^4.&c.)+log.2 . . =1.0986123
4 =2 log. 2 (art. 12) =1.3862944
6 = 2(i+g^+A_+&c.)+log4 =1.6094379
6 = log. 2+l*og.3(art 10) . =1.7917595
7=2(r3+3W+^m7+^^'+^^^-^ • =i-^»ioi
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
&c.
In this manner the Napierean logarithms of all the natural num*
hers may be found. As their accuracy, however^ depends . upon
thoie immediately preceding, being derived successively from each
other, it would be necessary to check the computations in the actual
construction of a table of logarithms by some independent fcnmula,
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
=/and M=l, we have /=«— ^n+Jn^ — i»*+> &c. ; reverting this
series, and l + », or N=1+/+^Z* 4-^-5-/3+5-5-3- 1^, &c. Now let
/=:1, then the number whose logarithm is 1, that is, the base
r=l + l+i+2^+2^^+,&c. =2.7182818. To prevent confu-
sion, however, we shall always designate the base or radix of this
system by R^ retaining r for that of the common logarithms. Hence
R=2.718,281,82846.
Hiese are also called hyperbolic logarithms from their application
to the quadrature of the hyperbola ; but this designation is improper,
as any system may be similarly employed.
17. When we have the logarithm of a number N for any particular
value of r, the base, we can readily obtain the logarithm 01 the same
number in every other system. Since, art. (5), when the base is r
we have r*=N, we shall likewise have R^zrN when the base is R,
in which a? is different from X, therefore, R^= r».
LOGARITHMIC TABLES. 7
Now taking the logarithms relatiTely to the syttem whose base is n
then
but Lrm^zx by hypothesis^ and LR^ =X LR, art (12), whence X LRszx,
or X=— s- But if R is the base^ X will be the logarithm of N in the
system having that base^ and designating this by L.N to distinguish
it from the other, we shall have Jj.N=zj^ , . (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
KNx/.R=/,N . . . (13)
Hence in every system the logarithm of any number is the product
of its Napierean logarithm by the logarithm (n R^ called the modidus.
/. N
Also since =^r==/. R, there exists between /. N and L.N a constant
L.N
ratio represented by Z.R
/.N
Since we have by formula (12) L.N=^, as N=10, then art (15)
2.3025851 = i, or M = oonio^ = 0.4342944819, and 2M =
M^ 2.3025851 '
0.8685889638 .... (14.)
18. It is now easy to construct a table of common logarithms
whose base r=10, for by formula (13) we have /.N=^R xL.N, but
tR=:M = 0.4342944849; consequently /. N=: 0.4342974819 xL.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. (N+l)=0.86858896(^l-j4 3^2J^3+5P^^ &c)
+ log. N, and making N successively 1, 2, 3, &c.
Log.l=: ... 0.0000000
2=-86858896(i+^+^+,&c.) . =0.3010000
3 =-86858896(i+ ^+ ^+, &c.) + log. 2 =0.4771213
4 = 2 log. 2. =0.6020600
5=-86858896(^+gl3 +^^+> &c.) + log. 4 =0.6989700
6=log. 2+log.3 . . . =0.7781613
7 = S6858896(^ +g^3 + _J_+,&c.)+log. 6. =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=2.7182818 to r=10, at the 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 tVve \(Vfc«k. c?L
8 INTRODUCTION.
changing tke radix, and had computed- logarijtlims on .a plan some-
what less commodious^ by making the logarithms of integers nega?*,
tive^ and those of fractions positive^ whilch^ upon a personal com-
municatibn with Lord Napier^ he rejected^ and finally adopted his
lordship's views. He soon afterwards published the first thousand
logarithms of this kind under the title of Logarithmorum CltUias
Prima.
Section III.
Of the Trigonometrical Lines, called Sines, Tangents, Sfc.
20. The Egyptians and Chaldeans began to study astronomy at a
very early period. As the determination of the relations and distances
of 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«
nitudes^ and the distances of various points in space. About six
hundred years before the Christian era, Thales measured the heights
of the pyramids in Egypt by means of their shadows ; a metnod
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, wmle 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 qf 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 sides, 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 determine, 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, horizontal, or oblique.
Several series of triangles of the kind now mentioned have been
TRIGONOMETRICAL TABLES. 9
actually computed and arranged in tables under the designation of
trigonometrical tables.
These were not accomplished at once> but were the improvements
of successive ages. Hipparchus^ about 150 years before the Christian
era, supposed similar triangles to be inscribed in circles^ and employ-
ed in his computation the chords subtending the arcs measuring
them in sexagesimal parts of the radius. Nearly 300 years after-
wards, Ptolomy, in his MtyuXn 2i;»T«|if, recomputed the chords^
but in his Analemma employs the half' 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 tenth 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-
gress has been made. At that period Miiller invented the tangents,
and shortly after Maurolycus produced his table of secants. These
were all in natural numbers to a given radius now generally taken
at unity^ and, therefore, their application was in many 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.
Uipparchus, who has been followed by most of the moderns, em-
ployed the circle to measure angles. He supposed the whole circum-
ference to be divided into 360 equal parts each 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<:ircle, a minute with one accent,
a second witJi two accents, &c. Thus 57° 17' 44".806, denotes 57
d^ees, 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 5^°.295^^95, 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 one-'
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 latdy adopted the centesimal division, which, in many cases, is
preferable to the sexag^imal. The whole circle is divided into 400 degrees, each de-
gree into 100 minutes, and the centesimal diviuon is continued. Hence the semicirclo
contains 200 decrees, the quadrant 100, and the ratio of the centesimal to the sexage-
simal is as 9 to 10.
To caavaci sezagesfanal dc^grecs into centesimal add | of the axe to itse\£.
The converse is effected bj BubtnctiDg j^g of tlie arc from itself.
10 XNTBODUCTION.
21. If two straight lines intersect one another in the ocntre of a cir-
cle, thfe BTC of the circmnference intercepted between them is called
the meiMUife of the contained angle, whatever be
thi^ rudius of the circle, since the arcs are pro-
portional to their l-adii. Thus, the arc AB or A'B',
IS the measure of the angle ACB, and is expressed
in degrees, &c.
92. The complement of an arc is its difference from
a quadrant, its supplement, its difference from a
semicircle, and its explement, its defect from the "^
whole circumference. Thus if AB be any arc, then BD is the com-
pleikient, BE the supplement, and BDEFA the explement.
The same thing holds with regard to the angles of whidb the arcs
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 BGA, measured by the arc BDEFA, the
^mlement or difference fVom four right angles.
23. The sine 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 upon a ra-
dius or diameter passing through the other.
24. The versed sine or versine of an arc is that part of the diameter
intercepted between its sine and the circumference.
25. 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 drawii
from the centre passing through the other.
26. The secant of an arc is die straight line drawn from the cen-
tre to the extremity of the tangent.
27* It is usual to express the sine, 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 Ihe radius OB, and
draw BG, AK perpendicular to AO or AD, and
HB, CI perpendicular to CE; then BG is the Hj
^ine, BH or GO the cosine, AG the versine, CH the
coversine, DG the suversine, and HE the sucoveV"
sine of the arc AB. Also of that arc AK is the tan*
gent, CI the cotangent, OK the secant, and OI the cosecant,
29. Since the diameter which bisects an arc, also bisects the chord
of that arc at right angles, therefore, the sine d£ an arc is equal to half
the diord of twice the arc. Thus BG=i BF=half the chord of the
iu*c BAF, the double of the arc AB.
SO. In the right-angled triangle 0GB, BG< + OG«=OB«, that is,
the squares of the sine and cosine are together equal to the square
of the radius.
31. The trianffle OGB being similar to OAK; OG : GB :: OA : AK,
or the cosine ofan arc is to the sine as radius is to the tangent.
32. Also the triangles OGB, OAK being similar, as before,
OG : 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 a mean
proportional between the versine and suversine.
34. Again, AD : AB : : AB : AG, or the chord of an arc is a mean
proportional between the diameter and versine.
TRIGOHOHETUCAI' TABLES. 11
Cor. — Since AB'=AD . AG, then, because AD is constant, AB"
variei U AO, or (i^ AB}taAG, that ii, tha b^Uus of the «« Tvie*
dirM^yas the Ternne, or invendy as the coune, of twice the arc.
35. 'nietrisnjK^fl04KuidICOar4^iailar,tberefaTeAS:9AO::
QC ; CI ; consequently the radras ig a m6att proportidnal ttttwMn
the tanxent and cotangent of an arc.
dS. Hx the application of algebra to
o geometry, where the trigone^
memcai iinea are ampioyea, it is necessair to trace their changes in
the several quadrants 6f the circle, since it is obvious that the sanoe
lines treatao of above, may be applied to each. ^
In the first quadrant AC, if the sine BQ and cosine
QO be supposed potiihe, then the sine B'Q^ on the
•une side of the diaineter A A', and in the same di-
rection, still remains positive ; but the cosine OC
having changed its position with respect to the -^ ~^
ecntre O, or diameter CC, becomes ttegatwt.
In tlio third qoadrairt, the cosine AC and sins
QfW, having bodi (dtanged their posidona, are
both ntgiOive. ta the foiuth q^uatbant, the cosine V^
havimri'Muawd its arifpnalmnium, OG is now pOiUiee, whila the
mt 6B'", remaining as in the thurd quadrant, is negative. The taa-
ftnta and secants depending upon the sines and cosines have Aur
MMt dctennined accordingly.
^Thnm article 30; to 36 and inclusive, R being radios, kc. we idttain
}■ dn. =: (R* — cos.*)* 7- tan. = — ■ '■
a ^ _RxcoB.
JL COS. = (R«--(in.«)* 8- '
8. tan. =i(see.f— R«)t S.aec si-
ft*
4. cot. srfcosec."— R")* 10. cosec = ^
'■ sin
5. sec = (R«+tan.«)i 11. versine =d^—
«. COMC, K CE"-t:«oL")* 13. coyarfc =|^^
If radius be supposed unity, then
sin.
l-sin. =(l-«*«)i 7. tan. =: —
COS.
Icoa. sr (1— wn.")* 8. cot. = ^
3. tan. = (sec«— 1^ 9. See. ^ Si
4 cot. =£(coBec<.^I)' to. eoie«. cs J_
6. sec. =(l+tan.«)*- 11. »ersbw=^.™:!_
' 1+cos.
«. coaec. = (1 + cot*)* 12. covers.= ,^^
l+sm.
■ Id te sboTC wwd-oni, B^ hw besn antitwd ntu 1% whidi Di>T eMtti ^ sccgiilM
12 INTROMJCTION.
sin
37* Noiw^ since (7) tan. =---; then it follows from the principles
.of algebra^ that when the signs of the sine and cosine are like, the
'sign of the tangent h positive, aiid when unlike, the sign of the tan-
gent is negative' In like manner^ the signs of the cotangent^ secant^
^ and cosecant may be determined from formulas (8)^ (9)^ and 10).
Table of the Signs of Tngnometrical Lines.
Quadrants. ^ Sine. Cosine. Tangent. Cotangent. Secant. ' Cosecant. ^
1591+ + + + + +1
26 10J+ - _ - _+l
37 11)-- + + - -I
4 8 12, &c. I — + — — + —J
Of the Multiples and Powers of Arcs.
38. Inmost treatises on gisometry, nuch 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
two or three different works. We shall only give a few of the re-
' suits most generally useful, referring to those works on' geometry
and trigonometry where the requisite information may be obtained.*
If a and b are two given arcs of a circle of which the radius is
unity, then
sin. (a+6)=8in. a cos. 6 + sin. b cos. a . (1)
COS. (a+6)=cos. a cos. 6— sin. a sin. b . (2)
sin. (fl — 6)=sin. a cos. b — sin. b cos. a . (3)
cos. {a — i)=:cos. a cos. a-j-sin. b sin. a . (4)
If we divide these equations, the one by the other in succession, that
is, (1) by (2), and (3) by (4), then
, _ _^ sin. a COS. 6+ sin. b cos. a
tan. (a+6)=: 7 ; ; — r • • (5)
' cos. a COS. 6 — sm. a sm. b '
J, f^ iL\ sin. a cos. b — sin. b sin. a ,^v
tan. (a— 6)=^ -_— , — _. — . . (6)
COS. a cos. 6 + sm. b sm. a
I>ividihg the two terms of the second numbers by cos. a cos. b, and
substituting tan. a and tan. b for their values in terms of the sine
and cosine
tan. a+tan. b
^^•(«+^)-l_ten.fltan.6 ' ' • (7)
, ,. tan. a — tan. b
tan. (a — b)-=:.—— 7 r • • • (8)
^ ' 1+tan. fltan. 6 ^ ^
expressions which give the tangent of the sum and of the difference
of two arcs in terms of the tangents of these arcs.
If we make a=zb in the preceding formulae, they give
sin. 2 az=2 sin. a cos. a, . . . . (9)
cos. 2 a=:cos.®fl — sin.* a . . (10)
^ 2 tan. a
tan. 2 a=^_^^,^ .... (11)
• Those we would more particularly recommend are the treatises of Gregory,
Wobdhouse, Lardner, and Cagtioli. ih Kelly *8 Spherics is a' very good treatise for
teaching the practice of the stereographic projection of spherical triangles.
TRIOONOM£TaiCAL TABLES. 13
expressions which give the sine^ cosine^ and tangent of twice the
arc in terms of the sine, cosine, and tangent of the simple arc.
39. Returning to equations (1), (2), &c. we have by addition and
subtraction
sin. (a+b) + sm, (a — b) = 2 sin. a cos. h . (12)
COS. (a+b+coB. (a — b) = 2 cos. a cos. b , (13)
an. {j.
sin. {a+by—4axL (a — b) =: 2 sin. b cos. a . (14)
COS. {a — i)=cos. (fl+6) =: 2 sin. a sin. b . (15)
Let(a + ^)=«^ a^^d (a — b) =t?, then by addition and subtraction
fl=^ («+t^)^ ft=i (« — ^)^ consequently the preceding formulas
become
sin. w+sin. t? := 2 sin. i («+v) cos. i (« — «) . (16)
sin. tt — sin. t? = 2 cos. i (w — v) cos. X (w+v) • (17)
COS. w + cos. t; = 2 cos. i (w+v) cos. \ \u — v) . (18)
COS. V — COS. w =: 2 sin. | (w+w) sin. ^ (« — 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 u + sin. p ___ tan. % {u+v )
sin. u — sin. v tan. ^ {u — v) ' * ^ /
If we multiply these equations member by member, observing to
substitute sin. 2a=:2 sin. a cos. a, formula (9), then
sin.^w — sin.*t? = sin. (u+v) cos. {u^v) . (21)
COS.* V — cos.^w = sin. («+ v) cos. \u-\-v) . (21)
Since sin. 2 a=i2 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 a=:] — 2 sin.* a, and cos. 2 a=2 cos.* a — 1
. ^ 1 — COS. 2a 1 + cos. 2a
whence sm. * a =: s > ^^^ cos. * a = o • (22)
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=«, then a=^M formula (22), these formulae become
1 COS. u 1 + COS. u
sin.*4M= — "2 , cos.*^w= g— . (23)
and dividing each corresponding number successively, they give
1 COS. u
tan.* it.=i^-^^^ . . • (24)
1— ten.*^M
^^ ^^«- « =l + tan.*it. • • • . (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 . ^26)
COS. 3 a= — 3 cos. a sin. aH- cos.^ a . (27 j
These may be put under the form
sin. 3 a=cos.^ a(3 tan. a — ^tan.^ a)
COS. 3 a=:cos.^ a(l — 3 tan.* a)
1.2.3.4.6
tan.'^ a,.,&c
14 INTEODUCTION.
In general n being my integer,
sin nasscos.** a s « tan. a-*- ' i 6 a ' t an.^ ^+
•} (29)
( n(n-l) «(>i-^l)(ii-2){«-S> ^ )^,
. iwi=cos.no^ 1 — Y-a-Han.«fl+ -^ — ^ 2.3 4 — ^^'^»^<^' f^)
The coefficients of the different terms are those of the n^ power pf
the binomial, whence these series may be collected under the follow-
ing form :
sin. n ^=^^ "I COS. a+ ^-^sin. fl f -g .nj \ cos.<^- V-i «i»- « f (30)
cos. na=ii{cos. a+y^ sin. a }**-!- ^ {cos. a — »J—1 sin. it }"(3l)
These formulae, by development, will give the two foregoing series,
and are thus easily verified.
41. It may be snown* that if x represent any arc •
"^•* = *-"L2:g- + 12:3:23 -^1233337+'*'^ • (32)
cos.ar=l_j^ + 1.2.3.4 "~ 1.2.3.4.6.6 +> ^^•
In these expressions the arc x is supposed to be divided by the
radius, which is here taken for the unit of length, and consequently
jf . sin. X
if we wish to restore it we must write — in place of x and
r * r
instead of si^ x in the two members of these equations.
These formulae might be carried much farther than can be intro-
duce d into this place. Most of them may be seen by consuliing the
books already refehred to, but above all the antihfsis irifiniiorum of
Euler.
Tables ofMuUiples and Powers qf Arcs.
1. • 2.
f being thel
sin. a = *, * -J sine of the >- cos. a = (1—^*)
( arc a. I
sin.2a:7 2«(l— sin.^)^ cos. 2a=i:l— 2«»
sin. 3 a = 3j— 4*^ cos. 3 a = (1— 4*«) (l-^«)i
an. 4 a = (4*— 8*5) (1— *«)i cos. 4 a = 1— 8*« +&*
sm. 5 a =16*5.^20*5 + 5*, &c. cos. b az=: (1..12*«+16*4) (l-i «i) &c.
3. 4.
tan. o = <,^|I^^^°«,^^n cot a = cot.
' I tangents. J
o 2/ ^ o cot.«— 1
tan. 2 a =i:- cot. 2 azz —, ^
1— <« 2 cot
..^ Q ^ St—t^ ^. Q cot*--3 cot
tan. o a =: cot. o a =^ ■ ■ - ■ , ■
1—3 <« 3cot. «— 1
tan.4a = /t^^^ cot 4 « =.^:^=« ^^
1— 6<« + <* 4cot5— 4cot
fo« K ^_5<— 10<5 + /5 -^^ . . ^_cot5— 10cot.« + 6cot f.^
tan. a =- — __ L— -, &c. cot. 6 a = — . — -— — ' . - , &c.
1— 10<«+5<4 5cot.*— 10cot« + l
* Woodhouse^s Trigonometry, third edition, page 245u^Oregory, page 42 and 50.
TRIGONOMETRICAL TABLES. U
5. 6.
sin. 4 =810. a cos* a:=cos. a
2 8in.'a=:l — cos. 2a 2 cos. ^ a =1-1- cos. 3 a
4 sin.' a = 3 sin. a — sin. 3 a 4 cos.^a = 3 cos. a + cos. 3 a
8 sin.^ a=3— -4 cos. 2a+ cos.4a 8 cos.^ a =3+ 4 co6.2a -(- co)s.4a>&c.
42. Having given a short abstract of the more useful formulae re-
lative to multiples and powers of arcs^ we shall now proceed to shew
the method of constructing the tables of sines^ tangents^ &c.
When the radius of a circle is unity, the semi circumference is
a]416026536 nearly. Now there are 180° or 10800' in a semicircle,
ccmsequently, if the former be divided by the latter, the result will
be 0.0002908882, the measure of an arc of one minute, which, as
die 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 formulae, (32), and
(33), art. 41 the sines and cosines may be obtained tlurough the
whole quadrant.
Thus let the arc fl=l', and, therefore, sin. a:=0.0002908882. Let
fl=5^ then 5 X 3.14I5926536_q ^gyc^g^^ ^^^ ^^^^j^ ^^ ^ ^^ ^^ ^^^
a:=+ 0.08726646
g>5
— -1.2:3 = ■" ^-^^iiwe
+ 0^4.5-= +0.00000004
x^
therefore, x — r23+ 12 3 45* &c. =0.0871 5574= the natural sine
of 5**, the logarithm of which is 8.740206, the log. sine the same arc.
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 smes are computed, the cosines of the same arcs toay be
found £rom art 41, formula (33), or from art. 35, formula (2), the tan<>*
gents and cotangents, from formula (7) and (8), and the secants and
cosecants from (9) and 10).
Section IV.
Of the application of Tables of Sines, Tangents, Secants, S^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 ;
As the side opposite to the first.
Is to the side opposite to the second.
These terms may be taken alternately, inversely, &c.
44. When one of the angles is a right angle, then the pre-
ceding rule may either be applied, or a modification of it derived
hom the properties which are peculiar to right-angled triangles.
16
INTRODUCTION.
In right-angled triangles^ it is usual to call that side subtending
the right angle the hypotenuse, and the other sides which contain the
right angle the legs, or the one the base and the other tibe perpendi'
cular.
Then if one of the sides of any triangle ABC, be assumed equal to
the radius, the names of the other sides must be determined by art.
28^ as follows : —
Radius
Tangent C
The names of the sides being thus known when three of the parts
of a triangle including a side are given, the rest may be found by
the following rules : —
I. — To find a side.
As the name of the given side.
Is to the name of the required side ;
So is the given side.
To the required side.
II. — To find an angle.
As the side made radius.
Is to the other given side,
So is radius.
To the name of this side.
Any side may be made radius to find a side, but one of the given
sides must be made radius to find an angle.
In the solution of plane triangles, it must be recollected 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 may 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 sum of the other two is also 90°, and
the one is the complement of the other.
CASE II.
45. In a plane triangle when the two sides and contained angle
are given.
I. As the sum of the given sides.
Is to their difference ;
So is the tangent of half the sum of the opposite angles.
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 I.
Or, after having found half the sum and half the difference of the
angles, the remaining side may be found without determining the
actual angles, as proposed by Thacker in 1743, and recommended by
Professor Wallace, in the Edinburgh Philosophical Transactions, in
the following manner :
PLANE TBIOONOMJETRY. 17
II. As the sine of half the difference of the opposite angles^
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 ^ach 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 opera-
tion, then Gauss' table for finding the logarithm of the sum and dif-
ference of numbers from their logarithms, without first determining
the natural numbers themselves, would be some advantage, though
it was not thought sufficient to warrant an insertion of it among £e
tables.
The following method of resolving this problem is convenient, 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 les^ side, the
remainder will be the logarithm tangent oi 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 opposite angles, the sum, rejecting 10 in the index, will be the
log. tangent of half their difference, from which the angles themselves
may be found.
CA8B III.
46. 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 opposite anffle.
It may perhaps be convenient to call the longest side the 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 ue 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 Greometry. 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 sides,
Aen half the sum of these four logarithms will be the log. sine of
half the required angle.
ni. To the arithmetical complements of the sides containing the
required angle, add the logarithm of half the sum of t\ve t\vtee %\^e««
18 INTRODUCTION.
and the loffarithm of the difference between this half sum and the
side opposite the required angle ; half the sum of these four loga-
rithms will be the log. cosine of half the required ancle.
rV. To the arithmetical complement of the logariumi of half the
sum of die three sides^ add the arithmetical complement of the dif-
ference between half the sum of the three sides and the side oppo-
site the required angle^ and the logarithms of the differences between
that half sum and the sides containing the required angle ; half the
sum of those four logarithms will be the log. tangent of half the re-
quired angle.
It may be remarked that these three last rules will^ in general, be
the most commodious in practice, though, in particular cases, eadi
mayhave its peculiar advantage when great accuracy is required.
when the required angle does not exceed 90% Rule II. may be
used, when it does. Rule III. may be employed ; and in either case
Rule IV. will give correct solutions. These observations depend
upon the variation of the trigonometrical lines in certain parts m the
curde, as, for example, near 90°, the sines vary very slowly, so that
the true value of an arc cannot be obtained by our ordinary tables,
while the tangents always vary by such perceptible quantities as to
leave no doubt of the real value of the required arc. These remarks
may be easily verified by examining any of our tables extended to
six or seven places of decimals.
Of the Construction of Triangles.
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 fircm a
scale of e^ual parts, and the angles are laid down by a scale of chords,
or more Conveniently by a protractor.
EXAMPLBS.
CASE I.
48. 1. Given the angles and hypotenuse of a right-angled triangle,
to find the base and perpendicular.
Let the hvpotenuse AC of the right-angled triangle ABC be 288y
and the angle A 39^ 22^ ; it is required to find the
sides AB and BC.
Construction. — In the indefinite straight line AB J^^
take any point A, and by a protractor or scale of
chords, make the angle A equal to SO'' 22f ; from
any convenient scale of equal parts take AC equal ^
to 288, and from C draw CB, perpendicular to AB ;
then ABC will be the triangle required. In order to simplify and
preserve uniformity, the an^es may, in general, be denoted by the
capital letters A, B, C, and the opposite sides by the small letten
II, h, c. The sides a and c being measured by the same scale from
which b was taken, will be found to be 182.7 and 222.?.
PLANE TUGONOMSTEY. 19
Cala^aium
1. By natiunl mimbers, $ (48).
To find a.
As im. B : am. Ai: b: a, or a= — -, — »-
sin. B
1 : (HSMaSl 2 : 288 : 2:®^il^=182.e73=
To find c.
And aia, B : ain. C, or cos. A : : 6 : c
1 : 0.773103 :: 288 : 9lIP^^^^^
zzc
1
2. By logarithms.
To find a.
As sin. B^ <nr radius 10.000000
Is to ain. A 38'^ 22^ 9.802282
So is & 288 2.459302
To a 182.673 2.261674
To find c.
As radius 10.000000
Is to COS. A 30^ 22' 9.888237
So is 6288 2.469392
Toe 222.663 2.347629
nie solutions may be varied by assuming any of the sides for ra^
dina, according to art (44), and verified by Gunter's scales.
2. Given the angles and one side, to find the hypotenuse and the
cAer side.
Let die side AB be 758, and the angle C 39° 26"; to find the angle
A, ttid the sides BC and AC.
Ams^BC is 921.7, and AG 1193.36, and the angle A 50" 34'.
Cansiructum, — ^From a scale of equal parts make AB equal to 7^^
the angle A 50° 34', the complement of C, and draw BC at right
anglea to AB ; produce AC and BC till they meet in C; then ABC is
the triangle reauired, and a and b measured on the same scale from
which c was ta&en will be found to be about 922 and 1193 respec-
tively.
3. Given the hypotenuse and one side, to find the angles and other
Let die hypotenuse AC be 544, and die base 464; to find the angles
A, a and c, and the side BC.
Jiu^^Tbe angle A is 8P 28', diough C is 68« 32^ and BC 284.
C0fw<rtic^tofi.M— Make AB eaual to ^ frcmi a scale of equal parta,
and filom B draw BC perpendicular to AB, then from die centre A
at Cha distance AC equal to 544 describe an arc intersecting BC in
C« join AC, and die triangle is constructed. The angle A being
meaanred 1^ a pfotxactor or scale of chords, will be found to be 31''
Sff, consequently C is 58^32', and the side BC 284 from the same
scale by wnich die other sides were laid down.
4. Given the bpse and perpendicular, to find the angVeft axidYi^^^^
tcnuae*
20 INTRODUCTION.
Let the base AB be 558, and the. perpendicular BC 456; required
the angles A and C and the hypotenuse AC.
An8.—A 39'' 15' 21", and b(f 44' W, and AC 720.692.
Construction. — Make AB equal to 558, and draw BC perpendicular
to AB and equal to 456, join AC, and the triangle is constructed.
The angle A will measure 39j^°, and the hypotenuse will be about
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. Giyen the 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 the
sides AB and BC.
^iw.— AB 708.76, BC 443.34.
Construction, — ^Draw the indefinite AB, at A
make the angle BAC equal to 38° 40', and from a
scale of equal parts make AC 532, at C draw CB
making the anffle ACB equal to 92° 46', it will cut
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.
Ans.— The angle C is 61° 14', the angle A 48° 33', and the side
BC 203.22.
Construction,— ''bleik.e 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«
tions required, unless from some known circumstances one of them
must be adopted in preference to the other.
Thus in the oblique-angled triangle ABC there are given AB 318,
BC 195, and the angle A 32° 40'.
Ans.— The angle B is 61° 50' or 118" 20', the angle C is 85» 40^
or 29°, and the side AB is 360.246 or 175.15.
Construction, — ^Make AB equal to 318 from any
convenient scale of equal parts, the angle A equal
to 32^ 40', and with the centre B and distance equal
to BC 195 describe an arc cutting AC in C or C ;
ABC or ABC will be the triangle required.
CASE II.
49. Given two sides and the contained angle, to find the other
angles and the third side.
la the triangle ABC let the side AB be 920 and AC 500, and die
contained angle A 39* 52' ; required the angles B and C, and the
third side BC.
Ans.^B is 20° 58' 50", C 113^ 0' 10", and BC is 60051.
PLANE 'nUGOMOMfTRY.
31
Construction. — ^Make AB equal to 990, at the
point A make the angle BAG equal 90^ 52^, and
AC equal to 500;. join BC; ABC is the triangle
required..
By Calculation, art. 45, 1.
As AB+BC 1420
IstoAB— BC 420
So is tan.i (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"
Istosin.A36 52
So is AB 500
To BC 600.31
Or by art. 45, II. and III.
As sm. A (B— C) 41^^ 35' 10"
Is to sin. 4 (B+C) 71 34
So is AB— BC 420
ToBC 600.31
As COS. 4 (B— C) 41° 35' 10"
Is to COS. I (B+C) 71 34
So is AB+BC 1420
B
ai52288
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. Oiven the three sides of a triangle, to find the aneles.
In the triangle ABC^ there are given AB 800, AC S^, and BC
B02 ; to find the angles. '
ConHruction. — ^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 'by^^'^-^a
distance 320 describe an arc, in like manner, .
with the centre B and distance BC 162^ intersect, j^
the fcnrmer arc ia C ; ABC is the triangle required.
In the solution of this question, if me angles A or B are first to be
detomined, then rules II. or IV. § 46, will 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 give the angle with all the requisite
accuracy in nice operations.
vnwmiisfiojst.
AB
AC
BC
By Calculation.
Bi^hMU, ■
800
330 ar. co.
563 art co.
Sum
1683
Half 841
let diff. 521 log.
Sd diff. 379 log.
•
•
Sum
• •
• •
Half I
W V 64".4 1
3
sin.
C 128 3 48 .8
AB
AC
BC
RULB III.
800
330 ar. CO. ,
663 ar. co.
•
Sum ]
1683
m
Half
Diff.
841 log.
41 log.
• •
• •
Sum
•
• •
Half I
B40 1/ 64"^
9
ooa.
C 128 3 48 .8
AB
AC
BC
800
380
663
RciB IV.
Sum
1882
Half
Istdiff
Sddiff.
3d diff
841 ar. co.
41 ar. CO.
621 log.
279 log.
• •
•
•
•
Sum
Half 64" 1' 64''.7 tan.
3
C 138 3 tf .4
7.494860
7360364
3.716838
9.446604
19.907566
9.963778
74M850
7350964
2.924797
1.612784
19.282694
9641347
7075904
a387316
3.716883
2.446604
* >
20.624862
10.312431
From these solutions it appears that the first and second differ about
1'' from each other^ while the second and last only differ 0'^4
PLANC «miaONOM£TBY. >B
Had the anffle C been nearer 180"^ the first and second tolntiona
might perhapsbave differed more contiderablyj while the teoondand
third wonld^have agreed more nearly. Hence it is clear that the
poper mles^ when great nicety is required^ must be diosen accord-
mg to the nature of the angle.
EXAMFLSS FOB EXBaCISB.
51. 1. What angle will one foot subtend at the distance of Mtw
miles? Ans.^'.1S.
2. The hypotenuse of a right-angled triangle being 6473 feet, and
the acute angle adjacent to we base, 29^ 5(y 5&', what are the base
and perpendicular ?
^fB^._The base 4746.064, and the perpendicular, 272a53a
3. If the base of a plane triangle be 3o4, and the other two sides
2B8 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 ?
^fw.— Perp. 139.4274, segments 230.4 and 153.6.
4. Therearetlu-ee towns. A, B, C, so situated that the bearing of B
and C from A forms an angle double that of A and G from B, and
that of A and B from C double that of A and G from B, or
the angle opposite b is double of that opposite o, and the circuit
round all the three is just one hundred miles ; what are their rela-
lative distances from each otlier in succession ?
Ans.—19Jm^, 35.6861, and 44.5066 miles.
5. In the right-angled triangle right-angled at B, ^ven the base
AB 70, and the sum of the h3rpotenu8e and perpendicular AC and
BC 200, to find the hypotenuse and perpendicular, and the remain-
ing angles?
^M— The angle ACB is 37*^ 16', AAC 5V 24', and AC 112^2,
and BC fl7.6ft
6. In an oblique-angled triangle ABC let the side BC be 532^ the
ai^le BAG 110<* SO', and the sum of the sides AB^ AC 637 ; requir*
ed the ai^^es 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 834.7.
7. In the oblique-angled triangle ABC, let the side BC be 250,
the angle BAG 96° 50^, ^so the mfference between the sides AB and
AC 106; required the angles ACB and ABC, together with the
sides AB and AG' ?
Ans.— ACB is 57^ 55', ABC 25*» 15', and AB 213.4, and AC 107.4.
& Given the base 214, the vertical angle 49° 16^, and the sum of
the other two sides 459 ; to find the sides and remaining angles ?
Ans^— The acute an^le is 33° 44' 48", the obtuse angle is 9P
50' 19^\ the side opposite the acute angle is 176*754 snd the side op-
posite die 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
ai^es and sides.
^nx.^— The vertical angle is 94° 22' 28", the remaining angle is
55"* 51' 82", the base is 50708, and the other side 421.08.
10. Given the base 1514, the vertical an^le 75"" 24' 50", and the
perpendicular 973.41 ; required the remaining sides and angles.
iw^— The sides are 1208 and 1172, and the ang^ea are W 4! U"
and 48" 31' 6*' vupeetirdy.
M INTBODUCTION.
fiSL The variout MiUngs in nayigaidoii are only jtbe apptieatiam of
trigonometry in particular circumstances.
The course is the. angle formed between the meridian and the
point on which the ship sails^ the distance is the hypotenuse^ and tiie
difference 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 P B C
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 4
the departure. A^in, if the ship sail south easter- ^
ly, AF is the distance, AE the different latitude, £F the de-
parture, and FAE the course. If, however, AE' be the meridian
difference of latitude, E'F' is the difference of longitude, E'AF'
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
Mercator's sailing.
Parallel, middle latitude, and oblique sailings, may readily be ex«
Slained on similar principles, though these can only be completely
iscussed in regular treatises on navigation.
See Mackay's, Norie's, Riddle's, Inman's, or Robertson's Na-
vigation.
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 ?
Ans. — Difference of latitude 81.48, departure 5445 miles, and the
latitude come to 46o 9' N.
2. A ship from latitude 48° 32' N. sails between north and west
till her departure is 54 miles, and then finds herself in latitude 49^
54' N. ; wnat course did she steer, and what distance did she run ? j
i^itf.— Course 32^ 22' N. W.^ and distance 9ai8 miles.
3. Coasting along shore I saw a cape bearing N. E. by N. After
standing N. W. 20 miles the same cape bore E. N. E. Required the
distance of the ship at each station.
Ans, — From the first station 33.26, and from the second 35.31
miles.
4. Required the course and distance from Caithness point in Scot-
land, in latitude 58° 46' N. longitude 3° 17' W., to New York in
North America, m latitude 41° 5' N. and longitude 74° 15' W.
Am Course 68° 32' or W. S. W. nearly, and distance 2899.2
miles.
5. A ship from latitude 60° 24' N. and longitude 43° W. saila
between South and West till she is in latitude 56° 30' N., and has
made 226 miles of departure ; required her course, distance, and
longitude ?
Jn«.— Course S. E. nearly, distance 325.4 miles, and the longitude
of the ship 35° 47 W.
6. Required the course and distance between the Isle of May, in lati-
PLANK TUOONOMRTRY. 9S
tade 68° W N. longitade 9» 33^ W., and Heligoland in latitude fi^
ir N. longitude 7** 59^ £ ^
Am-^Coone & 71**^' E- and Diat 377 milm.
7- A Aipftom the Isle of May sailed on die fbllowing true coiunes ;
required ner situation?
COUMS.
Duu
Diff.
l^t.
Sqputui*.
8.£.
40
N. S.
E. W.
1
283
28.3
s» s* hi.
50
46.2
19.1
|N.£.
20
14.1
141
S. £. bm S-
60
1 49.9
333
K« o* £!•
200
76.5
1843
W» o« o»
15
2.9
14.7
N.N.W.
20
18.5
7-7
iN.E.&.N.
76
63J2
42.2
iE.S.£.|E.
60
95.8
14.6
5&2
S. 71^ B.
378
218.4
380.0
22.4
Diff. of Lat.
•
95.8
22.4
C
123.6
357.6
2° 3'S.
j
jLatleft
1
•
•
56 12 N,
1
'Lat. in
54
t 9N.
Hence the ship is about 3 miles south of Heligoland light
Section V.
AppUcaium i^ Plane Trigonometry to the MensuraHon of Heights and
Distances,
SSL One of the most important applications of plane trigonometry
is the mensuration of heigots and distances. The data are some of
the sides and angles of a triangle. The sides are measured by rods,
linea^ tapes> or chains^ constructed according to the degree of accu-
racy required ; and the angles are measured by some angular instru-
ment Boch as the quadrant^ sextant^ reflecting circle^ repeating circle^
or theodolite. The repeating theodolite is perhaps^ in general^ the
moat conTenient of all for taking the necessary angles^ and the chain^
properly constructed^ the best £r 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
higUy useful^ when accompanied by the box-measuring tape. One
orScnmalcalder's surveying compasses will also be found very commo-
dious in military and nautical surveying. A complete description* of
these inatrumenta would fiur exceed our limits, and their use is best
* Those 1^ wish for written descriptions may consult Jones' edition of Adam's
Gsomdricd and Giaphical Essays, already mentioned, loot's Tiahtf d*Aalionomie
Fhrriqae, Ddambn'a Astron o mic, Base du Systeme Metrique, Woodhoaie*B^ Vinee'a^
and Peanon*B TnatiBea of Attrommiy.
96 INTBODUenON.
leamt under the miperintendence of a master. In gaiitnil, k maj
be reiiiarked^ that an allowance must be made for the height of tiie
eye above the horizontal plane ; and when the bate above-mentioned
is inclined to the horison^ it mnst be reduced to it according to the
given inclination^ though in nice operations the baie is selected so as
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 exact horizon-
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 precautions as 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 iii the Trigonometrical Sur-
vey of the ^tish islands under Uie 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 Systeme de
Metrique Decimal,. and in Piussant's Geodesie.
Example I.
To determine the distance of a tower, inaccessible by reason of an
intervening river, I measured, on a horizontal plane, the base AB,
500 yards, and at each end took the angle included between the
other end and the tower, which were 60° 60' C
and 76*^ lO' respectively : What is the distance
of the tower from each end of the base ?
In the annexed figure,
AB = 500
CAB = 50° 56'
CBA = 76° 10', and consequently _
Angle C = 180^_(A+ B)=53° 54' G^
Hence, sin. C 53° 54' 9.907406 9.907406
Is to AB 500 2.698970 2.698970
So is sin. A 50° 56^ 9.890093
To BC 480.46 2.681657
So is sm. B 75^0" 9.985280
To AC 690.2 2.776844
The perpendicular or nearest distance C d may, if required, be easily
founathus:
As radius 10.000000
Is to AC 698 2 2.776844
So is sin. A 60° 66' 9.890093 .
To Cd 464.45 2.666937
Remarks,^-^These distances might have been determined without
an instrument to measure the angles. Thus, suppose that, in the
* Thcfe aie leveral methods of approximating to the heights of objects by means
of mimm, shadows, stafik, {geometrical squares, and Gunter's quadrants ; but as they
are seldom used where much accuracy is required, they are omitted here.
PLANE TBIOONOMETRY. 97
contiiiiution cyf the base AB^ and the lines CA> CB^ the four distances,
AD, AE, BF, BO, were taken all equal to 100 feet, and D£ measur-
ed 86, and FO 1^ feet, the respective chords, to a radius of 100
&et» of the exterior angles DAE, fBG, which are equal to their Ver-
tical interior angles GAB, CBA. Now, since half the chord is the
sine of half the angle, we have <f\f{j=*43=8in. ^ A=d25^ 28', and
A=50^ 5&. In like manner, sin. iB— 61=37** 35', and B=76*' lO',
which results agree with the former.
Noie 1. — ^The number 100 was chosen for the sake of simplicity ; but
any other convenient number may be adopted, taking care to divide
half the measure of the chord by it.
Noie S. — ^The same thing may be accomplished when the sides of
the triangles bear any proportion to each other, by finding firom
them the angles DAE, FBG. Also the supplements EAB, /UBO' of
the ori^pinal angles may be found in the same manner, or otherwise
by joining AO and BE.
Example IL
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 by the other end, and a tree close by the oppo*
site side, to be 53*^ and 79^ l^ ; what is its perpendicular breadtn ?
^iw.— 105.89.
Example III.
In order to find the distance between two trees A and B, which
could not be directly measured on account of a pool of water which
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 the angle ACB between the two trees 55° w.
Required their distance.
180° (y
Angle C 55 40
A+B 124 20
HA+B 62 10
As BC+AC 1260 3.100371
Is to BC— AC 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°2r53'' 9.971203
Is to BC 672 2.827369
So is sin. C 55 40 9.916859
To AB 592.96 2.773025 ♦
Example IV.
In the trigonometrical survey of Britain^ Colonel Mudge found,
from computations depending on former operations, that the loga-
ridun of tiie number expressing the distance between Cheviot and
Cross Fell in feet was 5.4654017, and between Cheviot and Wisp
Hfll .6.2072278, and the angle contained by these, corrected for
• In some of the examples the computations hi proportion are perfdnntd ^"5 cobk^hl-
ing the sineM of the an^^ with the siaes, a method sometimes mote ca«^ U> \s«^Tuenu
9B IMTBOJDUGTION.
wgbnkA mbotmi, was 53^80" IB'* Reqnifed the other angles^ and
At dtiataica between Wis]^ Hill and GroM Fell, without firit findioff
Ai Talne of the given mdes in natural numbers.
iliM^The angle at ^isp HiU is 87'' 14' ^'.
Cross Fell 30 15 46
The distance of Wisp Hill from Cross Fell 2350ia6 feet.
Example V.
In order to determine the height of a tower, I mea-
sured In ft direct line AB 366 feet on a horizontal
(Aane. I then took the angle Cab 37° 3(K, the height
Aa of my instrument being 5 feet. Required BC the
lidght of the tower.
4nM.bC =280.84.
Add Aa 5.00. ^
Height BC =285.84.
Example VI.
Walking alon^ 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 Jp
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.
5o&«/i<wi.— Because the angle CBD=CAB + ACB,
CBD— CAB=ACB=50«>39'— 33°30'=17°9',hence ^ ^
sin. C : AB : : sin. A : BC ; and in the right-angled triangle DBC are
now given BC and the angle CBD^ to find DC and tS), 521 and
427*2 feet respectively.
Example VII.
A solution of this problem^ more easy and commodious in practice^
may be obtained thus :-—
LiOt CD represent any object whose height is to be determined ;
at the points A and B observe the angles of elevation^ and measure
the distance AB^ the points A>B,C^ and D being in the same plane^
See preceding figure.
For in the triangles ABC, CBD^
sin. ACB : AB: : sm. A: BC,
and R : BC : : sin. CBD : CD, from which we have sin. ACB : AB
XBC: : sin. Axsin. CBD : BCxCD or sin. ACBxBCxCD=
sin. A X sin. CBD x AB x BC ; radius being unity.
„ ^^ sin. A X sm. CBD x AB , . ,
Hence CD=. . . ^^ ; or, makmg the terms homo-
geneous, and substituting cosec for -;-, ^
R» X CD = sm. A X sin. CBD x cosec. ACB x AB.
That is, to the sines of the observed angles of elevaticm^ add the
cosecant of the difference of these angles, and the logarithm of die
measured distance; the sam, rejecting SM) from tiie inaex» will be the
height of tlie object
Let the angles of elevation be 55® 54/, and 33^ 20^ re^pectiveW,
and the distance between the stations 100 feet Required the
height of the object
FI^ANE TBIGON()M£T&Y 9Q
Angles of elevation ^ 33 20 sine 0.739975
IMfference
Distance
22 34cosec
100 feet
118.5
5.5
124.0 feet
10.415942
2.000000
Height
Height of the eye
Height of object.
2.073979
ExAMFIiE Vni.
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 top
(^ which I took the angles of depression of the two objects; that oiP
the most remote being 24° 48', and that of the nearest 58° SO'. Re-
quired their distance from the tower, and firom each other.
ilit^.— 139.842 feet.
Example IX.
Wanting to know the distance between two boats lying at anchor
in a strai^t line from a light-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 theur distance ?
Jn^.— 129.5286 feet.
Example X.
From the top of a hill I observed two mile-stones on a horizontal
road, which ran straight from its bottom, and took their respecdye
angles of depression below the horizontal plane passing throiufh the
place of my eye ; that of the nearer mile-stone was 36° ly , aiul
that of the more distant 15° 26". Required the height of the hill.
ilM.~780.I7 yards.
Example XI.
In order to find the height of an obelisk standing on the top of a
regularly sloping hill, I measured from its bottom a distance of 40
£eet, and then found the angle formed by the inclined plane, and a
line from the top of the obelisk 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 ^
ifMw^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 in-
strument to the top of the obelisk, and found it 41 ; but after mea-
sorinff downward m the same sloping direction 54 feet farther, I
found the angle formed in like manner to be 23*' 45'. What was die
height of the obelisk itself, and that of its top above the last place
of observation, supposing the angle formed by die inclined plane and
the horizon to be 21'' 54'^ ?
ifii^.— 51. 86 feet the height of the obelisk, and 83.51 above the last
place of observation.
30 INXBODUCTION.
EXAMPLB XIII.
Being on a horizontal plane^ and wanting to know the height o£ a
tower on the top of an inaccessible hill, I took the angle of eTeyati<Hi
of the top of the hill 40^, and of the top of the tower 51'' ; then mea«
soring 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 heiffht of the tower ?
ilitf.-— 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 58^, and of the top of the hill 25° ; but could not, as in last
example, measure a sufficient distance directly from the castle. I
tibererare measured in an oblique direction 52 ■ yards, making with
the casde an angle of 72° lO', at the farther end of which the angle,
in the same manner, was 64° SO'. What was' the height of the caa^
tie?
iln^.^-^.464 feet.
Example XV.
Wanting to ascertain the height of a tower standing upon a hill, the
height of the hill, and the horizontal distance from uie nearest place
of observation, on account of the nature of the ground I proceeded
as follows :— *
At A I took the angle
GCK3°38', andGCE2°6';
then havinff set up a staff AC
equal in height to the centre
of the theodolite, I measured
1810 feet up the sloping
ground AB in a direct line wiu
the tower, keeping the points
K, E, C, B, in the same ver-
tical plane. At B I took the
anffleFDC=BAI=l? 54', I A
and EDF=1° 32'. Required the height of the tower, the height of
the hill, and the horizontal distance from the first place of observa-
tion.
1. In the triangle DCE, are given the side DC=1810 feet, the
angle ECD 175° 2^, EDO 3» 26', and DEC 1° 12^; to find CE=5176.89
feet*
2. In the triangle CKE, the angle K=86° 22', G£K=02° 44',
KCE=:0° 54' and G£=5175.89; hence EK=81.463 feet
d. In the triangle CGE, the angle GGE=2° 44', and CE=:51 75.88;
hence CO=AH=5170 feet ; and GE=24&826.
4 In the triangle ABI, AB=:1810, the angle BAI=1° 54'; hence
AI=:1809 feet, and BI=60.011 feet
If EK, the height of the tower, were only wanted, it may be found
thus:
* In calculations where the same number is used which has been found £rom previ-
ous computBtioD, its log. should be reserved from the first to be used in the next, &c.
PLANE TBIGONOMETBY.
31
Sin. DEC : DC : : sin. CD£ : C£=DC sin. CDE. cosec. DEC,
sin. K : CE (=:DC. sin. CDE. cosec. DEC) : : sin. KCE : KE, and
R^KE=DC sin. CDE. sin. KCE^ sec GCK. cosec. DEC.
By logarithms.
CDE 3^ 26'
sin.
sin. KCE 0^ 54'
sec. GCK 3° 38'
cosec. DEC 1° 12'
log. DC 1810
EK 81.463
8.777333
8.196102
10.000674
14.678023
3.Ste7679
1.910061
Example XVI.
At the top of a castle which stood a hill near the sea^shore^ the angle
of depression of a ship's hull at anchor was 4° 52'; at the bottom of
the castle the angle of depression was 4° 2'. Required the horizontal
distance of the Ycsssel^ and the height of the hill on which the castle
stands above the level of the sea^ the castle itself being 64 feet high.
Ans. — 4373.75> and 308.4 feet respectively.
Example XVII.
From a window in the lower part of a house^ nearly on 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 directlv above
tibe former, the same angle of elevation was 37° 30'. Required the
height and distance of the steeple.
iii«.— 210.44, 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 {iLme be 6° 36', and that of CO 5° 22'; what are the horizontal
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 (X>. Again, in the triangle AQO right-angled at G,
are given the angle OAG and the side AO ; to find AG=660.3 and
OG=76*4. Lastly, in the triangle COB, right-angled at B, are known
CO and the angle OCB ; to find CB 600.7> and OB 56.4, and OG—
OH=76.4— 66.4=20 yards nearly =HG=CP, the difference of the
hdffhts of the stations, supposing AP to be horizontal. Now in the
right-angled triangle AFC are given AC a nd CP, to find AP=s
{(AC+ CP) (AC-^P) }* = V430rxa»0 = ^167700 = 409.6 yards.
Hence the sides of the horizontal triangle APG are ^ven, to find the
angles, which maybe determined by Case iii. Plane Trigonometry, to
be AGP=:37** 3r 29", GAP=63° 19' 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 most common cases.
Example XIX.
The height of the mountain called the Peak of Teneriffe ^^^^^.^ ^Qxmdi)
7
33
INTBODUCXION.
barometrioallT» by the methods described in Gregory's Mechanics^
Vol. I. book 5, to be 12^356 ffeet, or 2^ English nules, and the ragle
of depression of the horison, finm the mean of a great number of
observations^ 1° 58' 12'^ ; it is required to determine the diameter of
the earth, supposing it to be a perfect sphere.
Am.— 7913.6 miles.
Let G be the ^centre of the earth, the circle BTG
a vertical section passing through the centre, AB
die hei|i^ht of the Peak, AT the tangential line
drawn from its top to the visible horizon, and AD
a line perpendicular to a plumb-line hanging free-
ly : also, let BE, a tangent to the earth's surface at
B, meet the other tangent AT in E. Then, in the
triangle ABE, right-angled at B, there are given
BAE the ocxnplement of DAT, the angle of depres-
non=88^ 1' 4Sf', and AB=2.34, hence R : AB : : tan.
A : BE : : sec. A : AE. But since the triangles
CBE, CTE, are ri^ht-angled at B and T^ have the side CB=:CT,
and CE common, they are (Leslie's G^om. I, 22, or Hutton's Geom.
theo. 34, cor. 2) equal, and therefore BE=ET ; hence, AE+BE
=:AE + ET=AT. In the triangle ATC, right-angled at T, we have
R : AT : : tan. A : TC, the radius of the earai. '[uie operaticm thus
performed occupies but small compass^ which may auU be fartlifSC
■iwrtened. For since tan. A+ sec. A=tan. (A-f-J comp. A) we sba^^
fay incoi^rating the proportions from which AE, BE, aiid CT ii€
deduced, have .^i:'. ^'}
R« CT=AB tan.(A-hicomp. A) tan. A ; ,
or, log. GT=log. AB+log. tan. (A-f-^comp. A) -flog. tan. A — 20, in
the index.
fhe logarithmic computation is as follows-:-
Depression 1° 58' 12"
Half r 69 6
Comp. depress. \ 88 148 tan.
Sum 89 54 tan.
Height of Peak 2.34 miles, log.
Earth's semid. 3956.8
2
■ * .1
.■: ■»■
11.4834iEi53
11.7646436
0.369215lf
3.5973447
tf
Diameter 7913.6
Distance 136.1 . 2.1338595
If AT were required, we have only to take radius (10) from the sum
of the two last lines, and the remainder, 2.1338595, is the log. of
136.1, the distance sought.*
Note 1. — ^This method of determining the earth's radius, though
elegant in theory, is useless in practice, at least where any thing
more than an approximation is wanted, by the great irregularity of
the horizontal refractions.
Note 2. — ^When the diameter of the earth is known, and height of
the object given, the distance of the visible horizon may be easily
found; for, Euc. IIL 36. AB.AG— AT«.
• See I>r O. Gregory's Trigonometry.
PLANS 'FSIOOirOlMnSTRY. »
8y logarhhinB*
AB SM Ug. . 0.388816
BG T^iaa
i«db
AB+B6sA6 7916.94 log. ^808603
4267719
As before 136.1 miles, log. 2.133869*
Note 3. — ^The depression of the horizon, or the dip, as it is called
St sea, is the angle DAT contained between the trne aiid. visible
horison. For if an observer, whose eye is situated at A on the deck
of a renaei, takes the altitude of a celestial object with Hadley's
quadrant or sextant, by bringing that object to the surface of me
water at % instead of the true horizon AD, the altitude is evi-
dently tc^ great by the angle DAT=TCA. This may be csScuiaM
far the usual formulae of trigonometry for that purpose ; but as It
wSl, at any probable altitude, be a small quantity, those which give
Ae cosine or secant of its value are not sufficiently correct; Ibr
irldch reason we shall give the following method :— -
(BQ+AB)xABi-AT«, (Euc III. 36.), hence BGxAB+AB«-AT*,
or SBCxAB4-AB«=AT^ and AT« beinff, at any probable ^eT».
tkm, but a small quantity in comparison ofAC, it may be safbly rriH
l^ected ; therefore V(^^^ X AB)=AT. But CT(=-BC) : R : : AT
[^(fflCx AB)] : t^ C=.un. DAT ^ B V(gC.AB) ^^^^^
Now Aace -rg^ is a constant quantity, and BC being taken in gant^
lal at 3066 miles =20687688 feet, hence the log. of -g^ is 12^11^
fend tan. DAT=4(12.98114+log. AB). Since, in the present case,
tte arc jomy be substituted for its tangent^ the radius, therefore, be*
csmea 57* 17' 44''.8=206264''.8; and we have log. DAT in seconda
=xK3.e6099+log. AB in feet).
Thm dh) is affected by terrestrial refraction, which is verjr varidble,
iai Wy.-^ffwent authors it is estimated at different quantities. Dr
MaAehrne estimated it at one-tenth of the whole ; M. Delambre, one-
deveutOy and Col. Mudge, one-twelfth. See Dr Hutton's Coarse,
voL m. page 13a
£r.F— Required the dip, the height of the eye being 40 feet, and
esthnating ue terrestrial refraction at -f^.
Constant log. 3.60999
Height of eye 40 feet 1.60206
6.21205
403^.6 log. 2.60602
Refiraq. sub. ^ 33 .6
Dipt 370=6' lO'^
* See alao the method bj Leslie in his G eometr y.
ir Ths dip in n^tttes is equal to the square root of the bsl|(nt\ik {Mi iM^
84 INTBODnCTION.
Nole 4— Since AB x BG+ AB«=AT«, therefore
AB(AG+AB)=AT«, andAB = g^ijg. (1.)
Now^ if AB is the unknown quantity^ and being small in comparison
AT*
of BG, it may be found approximately by making, first, AB's: -^^
nearly, substituting this value of AB' for AB in formula (1.), and
"^rmj^' • • • (^)
which will be sufficiently correct for most purposes. If not, the
operation may be repeated till it is so.
This is useful in determining the height of an object considerably
dutant.
Now, the mean diameter of the earth is about 7912 miles, or
1775360 feet =GB, of which the logarithm is 7.620920, and it9
arithmetical complement is 2.379080; therefore to twice the log:
of AT, in feet add the constant log. 2.379080, the sum, rejecting
tens in the index, will give AB', wmch will be sufficiently correct if
AT does not exceed 1000 feet. If more distant, the operation mufft
be repeated. This correction must always be added to heights de-
termined geometrically as the usual instruments give their eleva^
tim only above the tangent AT.
Example XX.
Given the angles of elevation of anv distant object, taken at three
places in a horizontal straight line, wnich does not pass through the
P€4nt directly below the object ; and the respective distances between
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
the object F above that plane; A, B, G, the three places of observa-
tion ; FAE, FBB, FCE, the respective
angles of elevation, and AB, BC, the
given distances. Then, since the tri-
ani^ AEF, BEF, CEF, are all right- »r
aDffled at F, the distances AE, BE, GE^
YTiU manifestly be as the cotangents of the
angles of elevation at A, B, and C ; and
we must determine the point E, so that
these lines may have that ratio.
Construction,
To effect this geometrically, we must take BM, or AC produced,
equal to BC, BN equal to AB ; and make
MG : BM (=BC) : : cot A : cot B, and
BN (=4B) : NG : : cot B : cot C.
With the lines MN, MG, NG, construct the triangle MNG ; and
join BG. Draw AE so, that the angle EAB may be equal to MGB ;
this line will meet BG produced in E, the point in the horizontal
plane falling perpendicularly under F.
*?>8** Demonstration.
"By the similar triangles AEB, GMB, we have
AE : BE : : MG : MB : : cot. A : cot. B, and
BE : BA (=:BN) : : BM : BG.
Therefore the triangles BEC, BGN are similar; consequently
7
PLANE TRIGONOMETRY. 35
BE : £C :: BN : NG :: cot B : cot. C. Whence it is obvious that
AE, fi£, C£, are respectivelf as cot A^ cot B^ cot C.
Calculation.
In the triangle MGN are given all the sides, 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
angle EAB. Hence, in the triangle AEB are known the side AB,
and all the angles; to find AE and BE. And then EP=AE .tan.
A=BE . tan. B.
Analytically,
Let AB=r, BC=j; also let the cotangents of the angles FAE,
FBE, FCE, be denoted by the letters a, b, c, respectively.
Then, putting EF=j?, we have, to radius 1, I : a:: x: ax=AE,
l:b:i X : 6«=BE, 1 : c : : a? : car=CE ; and on AC from E, letting
fidl the perpendicular ED, we have (Euc. II. 12) a« ar«=6* x^ +
r«+2r.BD; hence BD=.^l^!r:|!.?!=!:L. In like manner CD =
^ = BD— BC=BD-^: whence BD=5-^^=^^^-±l.-
Therefore ^ ■ — = . Hence «« =
rs^ + rs^ ^ I rl{T+s^
Otherwise thus ;
If AB and CB be conceived to be bisected in M' and N', and ED la
perpendicular upon AC, which are however omitted to avoid com-
plexity in the figure; then, (Leslie's Geometry, II, 21.) AE«— BE*
=AB X 2M'D, and CE«— BE«=BC x 2N'D ; therefore, AE» X BC
-BE«xBC=ABxBC+2M'D, and CE« x AB— BE« x AB=AB
X BC X 2Na>. Adding equals to equals, and AE» X BC+ CE» X AB
-^C X BE«=AB X BC X AC; consequently AE* X BC+CE» X AB
=AC xBE« X AC X AB X BC.
If ABrdBC, then AE* + CE''=:2AB*+2BES the line EB being
drawn from the vertex E of the triangle ACE, to any point B in the
bMe. Pat AB=D, BC=£/, EF=:«, and then expressing algebraically
the foregouig theorem.
■ The equation thence resulting is,
dx^ cot «A+Dar« cot «C=(D+rf)a?« cot. «B+(D+£0 ^^^
Hence, transposing all the unknown terms to one side of the equa-
tion, dividing by the sum of the coefiicients, and extracting the
«lu-reroot, weBhaUhavex^ y^^^ «A+ W »C^D+ri)cot.«B -
Thus £F becoming known, the distances AE, BE, CE, are found
by multiplying the cotangents of A, B, and C, respectively, by EF.
Cor.— When D=£i, or D-|-^=2D=2d^, the expression becomes
x=<l-s-V(i cot.*A+^cot«C— cot*B), which is pretty well suit-
ed to logarithmic computation. The rule may, in that case, be thus
expressed. — ^Double tne 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 <tf elevation : then half the log. of this remainder subtracted
38
INISOPUCTION.
fioom. the Uag. of ib» meaaofad distanee between the firit and eeeond^
or the aecond and third, sftatmif will be the Icig. of die height of the
oinect.
X I^t AB=60 feet, BC 72 feet; angle FGEszSO^ 23', the
angle FBE=:40^ 33', and the angle FA£=:90» 48'; required the
dii^cea AE, BE, G£, and EF, the height of the object
Jns^AE^lSdm feet, B£=:l 10.84 feet, €£=78.51 feet, and
BF=;9484 feet
2, Let the three angles of elevation be 36^ 50^, 2P 24", 'and 14%
and the two equal measured distances 84 feet ; required the height
of the object Ans.r^£^964: feet
SXAMPLB XXI.
Given the angles of elevation at which an object b aeen horn thffe
given points in a horizontal plane ; to find its position and attitude.
Let A, B, and C be the three
points of observation, and D
the bottom of the perpendicu-
lar from the given object to
the hori;^ontal plane. It is evi-
dent that the horizontal di^
tances AD, BD, and CD are
proportional to the cotangents
of uie vertical angles at the
stations A, B, C, ; let these co-
tangents be respectively de-.
Qottd by. the L, M, and N.
Divide AB internally and ^
ei(ternally at the points £ and
F in the ratio of L to M; and
the lines DE and DF joining
in th^ vertex D must bisect
hiternally and externally the
angle, whence EDF is a right
angle, and contained in a se-
micircle; wherefore on EF de*
scribe a semicircle. In the same manner, divide CB internally umA
externally at G and H* in the ratio of M to N, and on 6H deaerihe
a semicircle. The point D common to both semicircles muat oecur
in their intersection.
From this construction the trigoncmietrical calculation is v^adOy
deduced. For L+M : M : : AB : BE and L— M : M: : AB : BF;
whence D£ =
BE+BF EF
2
= -o~' <^ radius KE is found. In like
manner N+M : M : : CB : BG, and N— M:M: : CB: BH, conae.
BG4-BM
qnently DI = X . In tl e triangle IBK, the sides BI and
BK, with dieiY included angle, = ABC, are given ; and, tfaerefbre.
* See Leslie's Geometry, fourth edition, page 27d. To avoid extending the figure too
moch, the point H, which should be in continuation of BI, in the same way as BF is in
continuation of BK, as well as the lines joining D£ and DF, is omitted.
PLANE TBIOONOMETKY. Sfj
tibe ai^le BKI and the baae IK are found. A^n^ all the aides of
die triangle IDK being giTen^ the anole IKD is fi>and. Hence^ in
the triangle BDK the whole an^e BKD and its containing sides are
given ; and, therefore, the base BD, or the horisontal distance firom
die station B, imd consequently its altitude, is determined.
It is obvious, that the opposite semicircles will likewise, hv their
iatersection, give, on the other side, a second position IK for that
point. In practice, however, this ambiguity could be easily remor-
ed It may be remarked too, that the point D may fall either with-
in or without the triangle.
If- the object be seen at the same elevation from all the three
pointB, the arcs of the circles will evidently become tiungents, which
Disect at right angles the sides of the triangle ABC. T^e projection
D of the ot^ect on the horizontal plane, will then be the centre of
the^rcle eircumscribing that triangle ; and, therefore, the radius or
distance AD may be found by prop. 18, book VI. Leslie's Geometry^
as shown in the notes, page 347*
If the three points of observation should lie in the same straight
line, the centres of the determining circles will occur in that line or
its extension ; and hence the process of calculation will be greatly
abridged, and will coincide with the foregoing proposition.
Exanmle. — ^Let the angle of elevation of the object at A, be B€P 45%
that at B 58^ 15', and that at C 46<'45' ; also the side AB 24 yards,
AC 38, and BC 50. Required its height ?
Hence L = cot 50^ 45', M = cot ES^ 16', andN = cot 46^ 46'.
From the given sides the angle ACB = 27" 36' lO"', ABC = 47'' 9"
22", and BAG lOd^ 15' 28". Also L = 0.8170343, M = 0.6188188,
N=04M07061 ; therefore, BE = 10.343, and BF = 74928, whence
KE = ^6355, and BK = 32.2925. In Uke mamier, BG = 19.846,
QH = 96.123, hence DI= 57.9845, and IB == 38.1385. Frbmtiiese
the ai^le 1KB = IT 11' 24'', and KIB 55" 39' 14^ ; and the side IK
= 23.D77- J^ow firom the three sides ID, IK, and KD, the angle
IKD =^ lOj** 10' 26". To this, by applying the angle LKB by ad^
ditioti and subtraction we obtain the angle BKD' = 184" 21' 60",
^ind BKD =.AKD = 29" 59' 2".
From the sidei^ BK and KD, and the contained angle BEID, are
ftrand the Iknrie KBD = 102" 16' 39", and KDB =: 47" W IV's
ifroni whibh BD =. 21.8065, and the height of the objed: 36it4yard&
Should the point D' be the foot of the perpen^cular, the auj^
KBD' = 2" ^and KD'B = P 52' 50", and BD' = 74^6; whence
the heiffht above D' will be 121 yards.
EXAMPLB XXIL
Otherwise thus :
GKven the aiu^les of elevation of the object from three points in
thj9 9aiiie plane forming a triangle, of which the sides are known, to
find, die position of the object referred perpendicularly to that pl^
and^its altitude above it
.. .pcMi^rMc<jo»y— The perpendicular from the object to the pfame
38 INTBODUCnOBT.
may fall either within or withoat the tri-
angler .^botti ca9^^ let A, B, and C be the
pomtft of obd^rvation^ and tt, /ij and y the
angles of elevation at these points respective-
ly. Join A^ B, and C, and on AB produced,
if necessaij^ make AE equal to AC^ and AD
to A^ jom ED, and upon it construct the
triangle EDF so that cotangent /S : contan-
gent yS : : AE : £F, and cotangent /8 : co-
tangent y : : AD : DP. Join AP, and from ^ ^' C
B curaw BG, making the angle ABG equal to the angle ATE, and
join CO. The point O in which the straight lines BG and AF iii«
tersect each other will be the point at which a perpendicular let
fall from the object would meet the plane, thus ascertaininjp the'
position of the olnect, from which, and the given angles, its alti-
tude may be found.
Demonstration, — It is obvious that the straight lines drawn ftcfos
each of the points of observation to the point at which a perpendi-
cular let fall from the obiect meets the plane, ought to be in propor-i '
don to the cotangents of the angles of elevation at these pmnts re-
spectively. The proposition therefore resolves itself into this. To-
find a pomt 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 the construction just given.
SoMion, — In the triangles ABG, AFE, the angles at B and F are
equal 'by oonstruction, and the angles BAG is common to both/
tneee two triangles are therefore similar. And AG : BG : : AE : EF'
il.QOit. m :'Cot fi. Hence EP= r Again AG : AE : a'
cot « ^ .' '.•?
AB : AP or AG : AC : : AD : AF ; and as the angle at A is com*'
inon to the two triangles AGC, and ADF ; these triangles sin^lar,
consequently AG : CG : : AD : PD : : cot. • : cat y, whence Fl>==:
ABxeot^y-^
—
cot. m
The triangles ADE, ABC having the sides AD, AE of the one
equal to the sides JiBy AC of the ower, and the angle at A, common
to both; are eiqiial, and the side ED is equal to the side BC There-
fore in the triangle ADE, the three sides are given, and those of the ^
triangle PDE are already found; whence the angles AED and
FED, and eonsequently the angle AEF may be obtained; and from
the angle AEF, with the sides AE snd EF, the angle APE 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 knoMm.
£jrai}tp&.— Let the side AB be 80 feet, BC=119, and AC=]4D,
also die angle at A or «=50^, that at B or /8=60°, at C or y=56° ; '
required the height of the o^ ect.
From these EP=96.329, l5P=66.758 ; the angle AED=34*» 48*,
EDA=87' 6' 23", EAD=87**6' 23^ EAD=58° 5' 37", GEP=34°
6' 67", APE, or ABG=70« 37' 8", PAE or AGB=40° 28' 16", BG
=55.673 ; and the height 96.392 feet.
PIJkNE TBIGONOMETRY. 30
BZAXPLB XXIII.
¥rom a conyenient station P^ there could be seen three objects A'^
B, and G, whose distance from each other were Afi=8 nEiiles^ AC±=^
milesj BG:=4 miles; I took the horizontal angles AFC=::33^ 45^
BPCs=22° dC. It is hence required to determine the respective
distances of my station from each object. Here it will be necessary^
ag iUnstrative and preparatory to the computation^ to describe the
manner of
Construction,
I>raw the given triangle ABC from anv convenient scale. From
the point A mraw a line AD to miJce with AB
an angle equal to 22° 3(K, and from B a line BD
to make an angle BDA equal to 33° 45\ 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
the cirde again in P^ which is the point re-
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 =BAD. So 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 side AB j t^
find cme of ue other sides AD. Take BAD from BAG, 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, BCPrzBCA— ACD; and
PBC=ABC+PBA=:ABC+ sup. ADC. Hence, the three required
distances are found by these proportions.
As sin. APC : AG : : PAC : PC, and : : sin. PGA : PA ; and,
lastly, as sin. fiPG : BG : : sin. BCP : BP. The operation at length
is as under :
By Rule IL^ Case iii., we have
SuL i BAG = jy^ = VA=i=-26=sin. 14*» S» 39", and
BAG = aff* 57' 18".
Sill, i ABC = yj^ =i J10rr-3962847t=8in. 23» 17' 1'% and
ABC sr 4e» 34' 3".
Sin. i ACB = J^^=Ji=yiO--790m4^an. 62» 14f 19"*, and
ACB = 104" 28' 30".
DAB= 22° 3^ CAB=28<' 57' 18" 180° 0' 0"
DBA 33 4£ DAB=22 30 . J>AC= 6 27 18
Sum fi6 16 DAC= 6 27 18 ADC+ACD=173 32 42
180 KAI>C+ACO)= 86 46 21
ADB 123 45
INTKOmJCTlON.
As sill. ADB 123P 46' ar. co.
btoAB . Smiles
80 bsiiLABD 33^45'
To
ADIog. . •
AC 6 miles, log. 4- 10
Arc . 4»>18' 7" tan.
Subtract 45
Remainder 3 18 7 tan.
^ADC+ACD)=:86 46 21 tan.
KADB— ACD) 45 39 17 tan.
• ACJD 5= 41 7 4
ACD 41 • T 4" sin. 9.819678
APC 33 45 ar. cosin. a.265a610
Sum 74 59 4
180
PAC 105 7 5(i sin.
AC 6 miles log.
PA 7.10199 mfles
0.7781613
0.8613801
PC 10.«»25 miles
ACB=104<' 2ff 39^'
ACD= 41 7 4 BCP+BPC=
BCP= 63 21 35 PBC:
As sin. BPC 22^ 30' 0" ar. co.
Is to BC 4 miles
Soissin. BCP 63" 21' 35"
0.0801536
OJM0O9OO
*aM«i
io.osoied7
a7611283
11.3467967
10.0099260
0.36«3ei&
9.9840740
0.7731513
IJildOBOi
lao^or W
85 iSl 35
0.4171003
O.tK»060O
9.9612594
94 825
To PB 934385 miles 0.6704797
The computation of problems of this kind, however, may be a little
shortened by means of die following
General InvesHgaihn.*
Put AC=fl, BC=6, APC=P, BPC=P', ACDsC, and let there be
taken for unknown quantities PAC=:x^ PBC==y. The triangles PAC
and PBC give
Sin. APC : sin. CAP : : AC : CP, and
Sin. BPC : sm. CBP : ^ BC ; CP; that is^
Sin* P : sin. x :: a: • . "V :=CP, and
sm. P
Sin. P : sin. y : : 6 : ^.SHJfzsCP.
•^ sm. P'
Hence. —, — ^ = -: — ^ : which may be reduced to a mxL P sin.
sin. P sin. P' '
sin. P sin. y=0.
* See LacToix Trigonometiie, and Qregorj^u Trigonometry.
PJJlN£ TBIGONOMETRY. 41
JfLiitm quadrUateral ACBP, we have CBP=:S60^— APC—BPC
^<3»w*^C AP, or v=3eO^P--P'— 0--ar. ,
.rj^i^ato^— P—P'— jC==R, then we shall have y=IU^« ; and cwi-
^gquentljUA sin. P' sin. x — b sin. P (sin. R cos^ j;— cos. R. sin. a;)=:0.
i?:«SL^u;j. by sin. x, there results, a sin. F-^ sin. P (*n. R ?®' *
,1.1..
sin. X
R}=0.
■^ifiA-:.- ' • , COS. X ^ a sin. P+6 sin. P cos. R
Whence we have -; = cot x = ^ . — . — = .
sm. X sm. P sin. R
This expression separated into two parts, we have
' •'- '■ .' a sin. P' . COS. R
cot. a? = -7 — ; ^—> TO H — ; n5 > OTi '
h 8in. P sin. R sin. R
COS. R / a sin. P , . \
cot X = -; — = It—-, — g = + A J ; or, ' . .\t
sm. Ryo sin. P cos. R / ' ' ' ■
_, / a sin. P' . , \ , ,
cot. X = cot R ( T— ; — T> o + 1 1 ; ^T^> lastly,
Vo sin. P cos. R / ^ ^: f .
cok.'*"=sV8in. P' cosec. P cosec. R cot. R-fcot R. » * " '
o . .
Hence, x being thus determined, we get^ from the equatipn ^==:
Rr— ix ; and CP from either of the expressions given above.
We shall now apply the foregoing formula to the solution of the
quiBtilii last proposed. < ; )/
'Ivr i ExAXPLB XXIV.
Here o = 6 P= 33^46' 0" PAC=j:
6 = 4 F= 22 30 PBCs=j^ -
■ ,.ACB = 104 28 39 found by computation
r 160 43 39
360
<\ J.
R=199 16 21
cot jr =~ sin. !P' cosec. P, cosec. R cot. R-f-cot. R : or, ..
cot ^ = cot.R ( ^';^'pe!l.R +l) and using logarithms '
we have a' = 3 log. 0.4771912 '
6' = 2 ar. CO. 9-66W700
F = 22° 30^ ff' sin. 9.5828397 ■
P = 33° 45' 0'' ar. co. S. 0.2562610 "
R whose cps. is neg. 199 16 21 ar. co. C. 0.0250452
— 1.09458 log. 0.0302371
+ 1.00000
--. 0.09458 log. a9767993
cotR + 199° 16^21'' 10.4563594
cot^ — 105 8 10 9.4321587
As sin. 33° 45' 0" ar. co. 0.2552610
Is to ain. « 105 8,10 9.9846660
So is 6 0.7781513
To PC 10.4251 1.0180783
Whence the rest may be found.
^^^
48 INTKOJDUCXION.
in using these ftHramlae great attention must be paid to the signs
of the quantities.
Example X&V,
Suppose the objects A^B^C^ are seen from D^ and have their
distances AB 7} miles, BC 12 miles, and AC 8 miles, ibm angle
BDA 25°, and CDA 19° ; it is required to determine the distances
DA, DB, DC.
^it^.~DA 10.0286, DC 16.7867> DB 149005 nules.
Example XXVI.
Suppose the objects A, B, C, are seen from D, and have their
distances AB 8 miles, BC 13, and AC 7i; the angle BDC bemg
17'' 47' 19''. Required the distances DA, DC, and DB.
Ans.—DB 12, DC 22.85, and DA 20 miles.
Example XXVII.
If, AB be 8, AC 7*2> a»d BC 12 miles, and the angle ADB 107""
50^ 13". Required the distances DA, DC, and DB.
-4iw.— DB 5, DA 4.892, and DC 7 miles.
Example XXVIII.
Let the objects A, B, C, be in a straight line; and their distances
AC a626, AB 12, and BC 8.374, the angle ADC being 10^, and
BDC 25°. Required the distances DA, DC, and DB.
Ans.—DA 9.471 1> DC 10.861, and DB 16.8485.
Example XXIX.
Let the objects A, B, C, as seen from D, be within the tria^^ ;
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.
Am.— DA 1.372, DB 5.523, DC 8.0ia
Example XXX.
A ship from Bombay in latitude 18° 57' N, sailed S. W. by S.
224 miles. Required the latitude come to, and the departure.
Ans. — ^The dinerence of latitude is 186.2, and the departure 124.4
Latitude of Bombay 18° 57' N.
Diff. of lat. 186 miles z^ 3 6 S.
Latitude come to . 15 51 N.
Example XXXI.
Having occaiddtk to travel through i^ counties of Kent and Sur-
rey, I perceived the fort built by Lady James, on Shooter's hill,
which bore from me N. N. E. ; and after going 20 miles in a
W. N. W. direction, I perceived the fort again, which now bore
N. E. by Ek Required my distance from it at each station.
Ans. — ^29.93 miles, and SO miles.
Example XXXIL
From a sbi^ At sea, I observed a point of land to bear £. bv
S., and after sJsdOAff 12 miles N. E., it bore 8. E. by £. Reauued
the distance of the last place of observation from the point ot land.
Am.'^^Q miles.
PLANE TRIGONOMETRY. 43
Example XXXIU.
Sailing N. N. W. at Ae rate of 6 knots an hour^ at 8h. jf, u. I dis-
covered two light-houseSj the northemmogt of which bore N. N. E.
and the other E. by N.^ and at lOh. 30m, the nojrthemmost light
bore E. N. E.^ and the other E. S. E. The bearing and distance of
the lights from each other are required.
Cofevfa/tan.— In the triangle ACD are given the side AC equal to
15 miles^ the angle ADC 3 points^ the interval between £ oy N.
and E. S. E. and the angle CAD 4 points^ the distance between
a S. E. the opposite point to N. N. W., and £. S. E. ; to find CD =
19.09. Again, in the triangle ABC are given AC as before equal to
15 miles^ the ai^le ABC equal to 4 points^ the interval between
N.N.E. and E.J^TE. 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 ffiven the sides CB, CD, equal to
21.21 and 19.09 respectively, and uie included angle BCD 5 points,
the interval between N. N. E. and E. by N. ; to find the angles CDB
= ey aC, CBD = 56° 15' = 5 pomts, CBE = BCN = 2 points, and
the distance BD = 19.09.
Example XXXIV.
The side AB of a pentagon being 180 toises, the face of the bas-
tion AC 50, the normal or perpendicular KL 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.
SohUian^Uere ^ = ^ =90=AK. "^ ^-^ ®
Hence, in the right-axigled triangle
AKL, AK (90) : R : : Kli (30): tan.
LAK=18»26'. Because AB is the side iN^
ofaregular pentagon, wehave-^ =
72»:=AOB,and ^ = 3ff> = A0K,
whence 90^— S6«=54f = EAK, and 54<>— 1»> 26'=:35^ 34" = EAC,
which heaag doubled is Tl"" B", the salient angle PAC or DBR. Join
BC> lii0ift will ABC be a triangle in which are given AB, AG, and
their contained angleBAC ; to find ABC=6'' 48'. Now sin. ABC (9*
40r) : AC(50) : : sin. BAC (Iff* 26^) : BC = 13^.52, equal to the line
of defence AH or B&. In the triangle BCO, ABG^ABC=18<'
aO'^a* 48" = 110 3gf ^ CBQ, Because BC = BG, we have
Again, because AB and EF are parallel, and AH, B6 equal ; we
have the angles BAH, ABO, AHE, andBGF all equal, that is, each
equal to 18° 26'.
In the triangle CGH, we have the angle CGB+BGH=84'' 11'
+ 18° 2Cr = 1(»' 37' = CGH ; 180°— (CGH+ CHG) = 180°— (102°
37'+lff'29^ = 5»>57' = theangleHCG; and the sideCHrzAH
-AC = 133.62-60=: 83.52 =CH. Then sin. CGH (102° 37'):
CH (83.62) : : sin, CHG (18° 26') : the flank CG or DH = 27062 : :
an. HCG (58* 570 : the curtain GH =73.323.
44
INTEODUCTION.
TABLE OF THE MEASURES OF THE PKINCIPAL LINES AND
ANGLES IN REGULAR FORTRESSES, FROM FOUR TO
TWELVE SIDES INCLUSIVE.
•
Names of Sides and Angles.
Names of Polygons. 1
Square Pentag
Hexag
Hepta.
Octag.
Nonag. Deo^.
Undeo. Dodecj
Exterior side, in toises
180.
180.
180.
180.
180.
180. 180.
180. 18QL 1
Radius of exterior side
127.3
153.1
180.0
207.4
235.2
263.1
291.2
319.4
347.7
Inleri(Hr side
115.5
123.9
130.6
136.2
140.0
142.9
144.3
146.3
148.1
Radius of interior side
81.7
105.4
130.6
157.0
183.0
208<9
233.4
259.7
286.1
Capital
45.6
47.7
49.3
50.5
52.2
54.2
57.8
59.7
61.7
Normal
22.5
27.0
30.0
32.0
34.0
36.0
39.0
.41.0
43.0
Cartin
78.0
77.1
76.4
75.9
75.3
74.7
73.7
71.4
69.3
Flank
20.3
24.5
27.3
29.2
31,1
33.0
35.8
37.0
88.1
Pace ....
50.0
50.0
30.0
50.0
50.0
50.0
50.0
51.0
52.0
133.0
134.2
135.1
135.8
136.4
137.2
138.2
138.2
188.2
Dem]goi]g8 ...
18.7
23.4
27.1
30.2
32.4
34.1
35.3
37.4
39.4
Angle of Oie Cmtre
90«»0'
7200'
60»0'
51«26'
45O0'
40»0'
36»0'
32«44'
soo a
Angle of the Polygon
Angle of the Curtm
90
97 1
108
98 21
120
99 15
128 34
99 47
135
100 21
140
100 54
144
10143
147 16
102 15
150
102 46
Angle of the Shoulder
111 3
115 3
117 39
119 21
121 3
122 42
125 9
126 45
128 18
Angte of Bast, or Flanlc. Angle
6156
74 36
83 8
89 26
93 36
96 24
97 8
98 16
98 56
Diminished Ande
Exterior Flanking An^^e
14 2
16 42
18 26
19 34
20 42
2148
23 26
24 30
25 32
15156146 36|143 8
140 52
138 36
136 24
133 8
131
128 56
Breadth of Foes, in Toises
15 1
16 1
17 1
18 1 19 1
20
21
22
23
APPENDIX.
BABOMBTRIC MEASUBBMENT OF ALTITUDES.
Having given a pretty full view of the method of measuring the
heights of objects geometrically, we shall here subjoin that of deter-
mining them by the barometer, thermometer, and nygrometer.
That the observations may be carefully and properly made, the
persons who undertake them should be provided with two portable
barometers of the best construction, filled with mercury of the same
specific gravity, on which, by means of a vernier properly adapted to
the scale, the height of the mercurial columns may be read on to the
500th part of an inch; each barometer being fitted up with an at-
tached thermometer, set in the wooden frame in the same manner as
the barometer tube is. The ball of each thermometer would be best
if nearly of the same diameter as the barometer tube. Besides
these, they must also be provided with two other thermometers cle-
tached from the barometers. Of these barometers, one, with its at-
tached and detached thermometers, is to be placed in the shade at
the top of the eminence, while the other remains below. Let them
continue in their places at least a sufficient time for the detached
thermometer to acquire the temperature of the air, that is to say, till
the contained fluid is stationary. Then the observer on the emi-
nence must note down the height of the mercurial column in the ba-
rometer, as well as the temperatures exhibited by the attached and
detached thermometers ; and, at the same time, the other obsen'er
must make like observations upon the instruments below. It\ in
PLANE TRIGONOMETRY. 46
this manner^ three or four sets of observations be taken^ at each sta-
tion^ after short intervals of time^ and the mean of the results fur-
nished by these sets respectively be taken^ the probability of error in
the ^true altitude deduced by the following rules wiU be much
diminished. When our third method of computation is adopted^ two
of Daniell's hygrometers 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 heiffhts are too smaU, when the observations have been made
near sunrise or sunset^ or when the wind blows fresh from the
south ; and that^ on the contrary^ the computed results are too great,
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 dieduced
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 different temperature^ this 87 feet must be
increased or diminished by about 0.21 of a foot for every degree of
difference of the temperature from 32^.
4. JStverj 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 of 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 f<s^ 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 A.
Thus is the approximated elevation very nearly.
m
IV. Multiply the difference } of thp mercurial temperatures by
2.833 feet^ and add this product to the approximated elevation if tlie
upper barometer has been the warmest ; othenvise subtract it ; then
wm the resulting sum or difference be the corrected elevation.
Qr^ this rule may be expressed by the following formula^ where d
is the difference between S&° and the mean temperature of the air^ D
is the difference of barometric heights in tenths of an inch, m is the
* One perwm may perform the whole operation with one set of instrumentfl. by mak-
ing the observBtionB two or three times alternately at the top and bottom, and taking a
mean of the ranilts at eadi station.
46 INTBONJCnON.
mean barometric height^ i the diflfereace between the mercurial tem-
perature^ and S b tte correct elevatioii.
m
For aix exampk^ suppose that the mercury in the baromet^ at the
lower statioii was 29.4 inches^ its temperature 50^ of FahrenheitTs
thermomet^^ att(| ^e t^siperature of the air 45^; the height of the
mercury at the upper station 26.19 hiches^ Ha temperature 40> and
the temperature <» the air 89^
Here D = 294 — 251.9 = 42.1
k := 87+(19x-21)= 89.1
fi9 = K^.4+ 25.19)= 27.295
^ — r= approximal:? elevation = 4I23>24
tn
Correction for temp. mere. 4 X 2.833 = 11.33
Correct elevation in feet 4111.91
Do in fathoms ..... 685.32
II. Dr HtUUm's Method.
1. Observe the height of the barometer at the bottom of any height
or depth intended to be measured^ with the temperature of the quick-
silver by means of a thermometer attached to the barometer^ and also
the 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 ok
depths 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 same tempera-
ture^ if it be thou^rht necessary^ by correcting either the one or other ;
that is, augment we height of the mercury in the colder temperature,
or diminish that in the warmer, by its ^^^^^th part for every degree
of cKfference of the two. The altitudes so corrected being denoted
by M and m.
3. Take the difference ofthe common logarithms of the two heighta
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 fkthoms in whole numbers.
4. Correct the number last found for the difference of temperature
of the air as fbllows : 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 ^}7th part of the fathoms above
found, and add them if the mean temperature be above 31% but sub-
tract diem if the mean temperature be below 31^; and the sum or
differaice will be the true attitude in fathoms ; or, being multiplied
by % it will be the altitude in feet
Same example.
Tkemumeters.
Detached
45
39
Mean 42
Attached
50
46
Diff.
BoroMeicfS'
29.4 lower
25.19 upper
PLANE TRIGONOMETRY. 47
As 9600: 4 :: 29.4 : .0123
Mean 42 corr. .0123
Stand 31 M= 29.3877 log. 4681656
Diff. 11 m=: 25.19 log. 4012282
435 : 11 : : 669.374 : 16.924
Corr. 16.924
'The altitude sought 686.298 fathoms.
Let the state of the barometers and thermometers be as follows
to find the altitude.
Thermometers.
Detached.
67
42
Barometers,
Lower 29.68
Upper 25.28
Altitude 719.897 fathoms.
AtUiched.
57
43
Method III.
The foregoing methods have been found from experience to give
results tolerably correct in ordittarv circumstances, though they de-
viate considerably from the trutn in peculiar cases. To obviate
this^ as far as possible, we have given another method, which, it is
hcmed, will prove very accurate.
in tiiis case let B be the height of the English barometer at the
lower station, b that at the upper, t, the temperature by Fahrenheit
at the lower, and f that at the upper, L the latitude of the place of
observation, y* the elastic force of vapour at the lower, and J'' that at
the upper, and H the height of the oneplace above the other in feet, then
|(i.:
2 ''
H=:flD000 1(1.375) 180 (1 + 0.00268 COS. 2 L) x
{^'^ B+6fl+6-0001(<^0i)^^* ^b{l + 0.000l(u^)]—if')}'^^^
The fiictors (1.375) ^^0 ^nd 1+0.00268 cos. 2L, may be re-
dooed into tables; and^ if given in logarithms, they wiU be very
lemSSly applied. If tiie centigrade thermometer be usedj then
H =s 06345.6 \ (1.^75)^ (1 + 0.00268 cos. QL)^
\ B+6{(+awoi8(i-^)} ^ Wi+ox)ooi8(t-^)}~tf y I "^ ^
In which case also B, 6,^ ahdf^ may be given in reference to the
Vrench standard metre.*
l%te log. of the constant 60000 feet may be employed with ad-
vanti^^ being 4.778151.
tf fiaplace's constant 18393 metres, or 60345.6 feet, be taked, the
oonstftOt logarithm would be 4.780646, and the factor 1 + 0.00268
Oos. 2 2i must be nsed.t
* Hee iBiot'i TnaU de Physiqae, Tome I. p. SSI.
t At l#ft|»lAoi's ccfeiftalit J0 paiu^s ih^ mote accurate, it inay \>e u««& \iv V>\\\ c»m»»
48
INTKODUCTION.
BAROMETRIC TABLES.
TABLE I.
TABLE OF THE DEPRESSION OF IfEBCUBY IN GLASS TUBES.
Deprankms by
Dbm.
Ivory.
Laplace.
Young.
In.
In.
In.
In.
0.06
0.29494
0.2964
0.10
0.14028
0.13040
0.1424
0.15
0.08628
0.08538
00880
0.20
0.05811
0.05798
0.0589
0.25
0.04075
0.04117
0.0404
0.30
0.02916
0.02965
0.0280
0.35
0.02110
0.02165
0.0196
0.40
0.01534
0.01591
0.0139
0.45
0.01117
0.01174
0.0100
0.50
0.00835
0.00868
0.0074
0.60
0.00443
0.00462
0.0045
0.70
0.00228
0.00244
0.80
0.00119
0.00128
This table is to be used only when two barometers, differing con-
siderably in their internal diameters^ are employed.
The expansion of the volume of mercury for 1° Fahr. = 0.000086^
more correctly than 0.0001^ though the difference in the nicest baro-
metric observations is almost insensible.
TABLE II.
MB DALTON's table OF THE ELASTIC FOBCE OF AQUEOUS VAPOUB.
Barometer 30 Inche*.
-
Teaaap.
Force.
Fone.
Temp.
Fatcft
Tfemp.
F01C& iTemp.
tOKM.
Inches of
Fahr
Inches of
Fahr
Inches of
Fahr
Inches of
Fahr
iDcbe* of
faur.
Mercury.
0.064
A mmmMm ■
Mercury.
A <M1*«
Mercury,
f mi&*
Mercury.
A CH1»«
Mercury.
0°
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
4&
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
1.210
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
1.280
9
0.087
29
0.180
49
0.363
60
0.698
89
1.320
10
0.090
30
0.186
50
0.375
70
0.721
90
1.360
11
0.093
31
0.193
51
0.388
71
0.745
91
1.400
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
a796
93
1.480
14
0.104
34
0.214
54
0.429
74
0.823
94
1.690
15
0.108
35
0.221
55
0.443
75
0.851
95
1.580
16
0.112
36
0.229
56
0.458
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.'ffiO
78
O940
98
1.740
LR.
0.124
39
0.254
59
0.507
79 1 0.971
99
1.800
PLANE TRIGONOMETRY.
49
TABLE III.
LOGARITHMS OF THE BULK OF GAS^
Prom the formula j^ x log. 0.1383027, in which x is the number
of degrees above 32° Fahrenheit.
hreoip.
log. Bolk.
Temp.
Log. B.
Temp.
Log. B.
Temp.
Log. B.
CP
r.975413
25"
1.994622
50^
0.013830
75°
0.033039
1
.976181
26
.995390
51
0.014599
76
0.033807
2
.976950
27
.996158
52
0.015367
77
0.034567
3
.977718
28
.996927
53
0.016135
78
0.035344
4
.978486
20
.997695
54
0.016904
79
0.036112
5
.979255
30
.998463
^
0.017672
80
0.036881
6
.980023
31
.999232
56
0.018440
81
0037649
7
.980791
32
0.000000
57 0.019209
82
0.038418
8
.981560
33
0.000768
58 0.019977
83
0.039186
9
.982328
34
0.001537
59
0.020745
84
0.039954
10
35
0.002305
60
0.021514
85
0.040723
11
.983865
36
0.003073
61
0.022282
86
0.041491
12
.984633
37
0.003842
62
0.023050
87
0.042259
13
.935401
38
0.004610
63 0.023819
88
0.043028
14
.986170
39
0.005378
64 > 0.024587
89
0.043796
15
.986938
40
0.006147
65
0.025356
90
0.044564
16
.987706
41
0.006915
66
0.026124
91
0.045333
17
.988475
42
0.007683
67
0.026892
92
0.046101
18
.989243
43
0.008452
68
0.027661
93
0.046869
19
.989911
44
0.009220
69
0.028429
94
0.047638
20
.990780 45
0.009989
70
0.029197
95
0.048406
21
.991548 46
0.010757
71
0.029966
96
0.049174
22
.992317 47
0.011525
72
0.030734
97
0.049943
23
.993085 |48
0.012294
73
0.031502
98
0.050711
24
.993853 1 49 1 0.013062 1 74
0.032271 1 99 1 0.051489
P.P. 1 -2 -3 -4 -5 -6 -7 -8 -9
to tenths 77 153 238 307 384 461 538 615 691
TABLE IV.
L06ABITHMIC VALUES OF 1 +0.00268 COS. 2L.
I4t.
Log.
Lat 1 Log. 1
Lat. Log .
Lat.
Log.
0»
0.001162
13»
0.001045
2e» 0.000716
39P
aooo24@
1
0.001162
14
0.001027
27 0.000684
4a
0.000202
2
0.001160
15
0.001007
28 0.000651
41
0.000162
3
0.001166
16
0.000986
29 O.OOO6I7
43
0.000122
4
0.001151
17
0.000964
30 I 0.000582
43
0.000081
B
0.001145
18
0.000941_
31 0.000546
44
0.000041
6
0.001138
19
0.000916
32
0.000510
45
0.000000
7
0.001129
20
0.000891
33
0.000473
46
V»\fo^JvuV
8
0.001118
21
0.000864
34
0.000434
47
9.999919
9
0.001106
22
0.000836
35
0.000398
48
9.999878
10 0.001093
23
0.000808
36
0.000360
49
9.999838
11 0.001078
24
25
0.000778
37
0.000321
50 9.999798 1
12 1 6.001062
0.000747
38
1 0.000281
v^^
50
INTRODUCTION.
TABLE IV.— Continued.
Lat.
52"
Log.
Lat.
Log.
Lat
Log.
Lat
Log.
9.999719
62^
9.999349
,72"
9.999059
82°
9.998882
63
9.999679
63
9.999316
73
9.999036
83
9.998871
54
9.999640
64
9.999284
74
9.999014
84
9.998862
55
9999602
65
9.999253
75
9.998993
85
9.998855
56
9999566
66
9.999222
76
9.998973
86
9.998849
67
9.999527
67
9.999192
77
9.998955
87
9.998844
58
9.999490
68
9.999164
78
9.998938
88
9.998840
59
9.999454
69
9.999136
79
9.998922
89
9.998838
60
9.999418
70
9.999109
80
9.998907
90
9.998838
161
9.999383
71 i
9.999084
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
feet.
Bar. Att. ther. Det. ther. Dew point.
Leith Pier 29.567 55.25 54°.0 50°.0 / =0.375
Arthur's Seat 28.704 51.75 50.5 48.5 /'=0.357
Fah. ther. 54°.0
50.5
3.5 * /+/'=0.732
28.704 X 0.0001 X 3.5=0.010 nearly, and
6=28.704+0.010 =28.714
Sum 104 .5 Constant log. of 60000 feet
4.778151
Half 52.25 log. B . . . 0.015367
B=29.567, B— j/=29.567-0.062=29.505 log. 1.469895 153
b =28.714, 6— 1/''=28.714-0.059=28.655 log. 1.457201 38
Difference
i+'g6=^+rar=i«i2^«»"g-
.0126941og.2.103462
138
0.006181
206
25
H =799.32 feet
H' =802.66
2.902721
10
Defect =3.34 feet II
The operation^ when Laplace's constant is used, would be as fol-
lows :
* The / and ^ in the denominators of the fractions in the fonnula should have been
r and r^, the temperatures of the attached, to distinguish them from those of die de-
tached thermometers.
PLANE TRIGONOMETRY
51
liaplace'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.
H =803.12
H'=802.66
Excess = 0.46 foot^ or 5^ inches
Example II.
Required the height of the Peak of Snowden above
quay nrom the following set of observations ?
Bar. Att. Ther. Det. Ther.
Caernarvon Quay 29.984 56.5
Snowden Peak 26.271 42.75
Ck>nstaht logarithm
Correction for latitude 53° 4'
^^-^t^^9\m log. B.
B—I/ =29.920 log. 1.475962
6'— i/'=26.262 1.419328
Difference^ 0.056634 log.
55.25
43.00
H = 3561.2 feet
H'= 3555.4
4780646
9.999566
0.015367
153
38
2.103462
138
0.005181
206
25
2.904782
70
12
Caernarvon
Dew Point.
. 50^25
41.00
4.780646 ^
9.999679
0.013062
77
15
4
31
2.753047
0.004751
247
33
3.551592
Excess 5.8
Example III.
Captain Sabine found the height of a hill at Spitzbergen^ deter-
mined geometrically^ to be 1644 feet ; required its height barometxi-.
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
Reduction to 32° F. . —0.0200
Capacity — 0.0561
Capilla^ action (Young) +0.0280
Index . -. +0.1960
Attached ther. 39°.75
DeUched 34 .90
Dew point 34.00
By Danieirs hygro-
meter.
True height
+ 0.1479
29.8214
62
INTBODUCTION.
Observed height of the barometer^ &c. at the top.
In.
Barometer, (diam. of tube 0.15) 28.0075 Attached ther. 36».4
Reduction to 32° F. --0.0105 Detached . 35 .4
Capacity . —0.0445 Dew point DL 35 .6
Capillary action (Young) + 0.0880
+0.0330
28.0405
True height
C(Histant logarithm ....
Correction for latitude about 80^
B -.J/=29.8214r-0.0357=29.7857 log. 1.474008
h — J/'=28.0405-^.0375=28.0030 log. 1.447204
Difference 0.026806 log.
Mean temperature J — - = 35.2 log. B
1 4- /+/ - 0.2144-0. 225_ 0.439 ™.q ,
^+ 5+6-— 5p6— -^+57^=^*^^^ ^^«-
4.780646
9.998907
2:428135
0.002305
153
0.003029
247
3.213488
Bair. H = 1635
Geo. H = 1644
Difference *— 9 feet
By another set of obiservations.
In. In.
Barometer^ at bottom . 29.8304, at top . 28.0624 corrected.
Attached thermometer 39°.4 . . 35°.2
Detached 35.4 . . 34.2
Dew point 35.4 . .34.2
Constant logarithm
Correction for latitude 80°
B—l/ =29.8304— 0.0374=29.7930 log. 1.474115
h — |/'=28.0624r-0.0360=28.0264 log. 1.447569
4.780646
9.998707
Difference 0.026550 log.
35*4+34.2
Mean temperature of the air ^ — = 34.8 log. B
Bar. H = 1618 3
Geo. H = 1644.0
Biff. — 25.7
By the first set oi'experiments H
By the second . .
Difference
2.424065
0.001537
615
0.003099
268
3.209067
979
1635 feet
1618
PLANE TEIGONOMETRY. . 53
Captam Sabine thinks there is some error in the second set of ex-
periments^ arising £rom the circumstance^ that Mr Foster^ his assistant^
was obliged to hold the instruments to prevent their agitation by the
wind.
It is proper to remark^ that Captain Sabine finds 1644.58 for the
first and 1630.66 for the second set of observations, as stated in the
Philosophical Transactions of the Royal Society of London^ but the
particular formula he used is not mentioned. The usual for-
mulae given by Roy^ Shuckburgh^ and Laplace may five the
height more near the geometrical method in certain cases, such as in a
mean state of the atmosphere, than that which we have given^
though there is no doubt but that the circumstances which have in-
duced US to give a new method^ involving considerations not usual-
ly attended to in such measurements, are 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 by Daniell's hygrometer.
If that instrument is not at hand, the dew pomt may be found by two
good thermometers, one of which has its ball covered with moistened
tissue-paper, as proposed by Mr Anderson^ Rector of the Academy of
Perth^ who also gives a formula for the barometric measurement of
altitudes, in which in some of the corrections I have been antici-
pated.
Let F, the elastic force of vapour by Dalton's table be thus re-
duced toy* according to the difference between the naked and covered
thermometers, then/=F— ^'^^^P =:F— 0.00092^^ xp, in which
} / is the difference between the temperatures of the thermometers, and
p the barometric pressure.
Now let ^ be the elastic force at the dew point, then
' _ f F- ^-0.00092p?< ,
Here t', the temperature of the dew-point is unknown, l)ut may be
determined, first approximately from the numerator of the formula,
and then substituted in the denominator, and a second approximation
obtained, which will generally be sufficiently correct.
To exemplify this, let the thermometer witn the dry ball show 60" F,
and that covered with moistened tissue paper .51^
T— # cft ot, 8 J
Now if liie barometer be at 30.4 inches we have from the numera-
ting of formula(l)/=0.624— 0.00092x8^x30.4=0.524— 0.238=
bSSBS. Thisy corresponds, by the table of Dalton to 42? nearly,
wUch being substituted for t' in the denominator of the formula
0.286 0.286 ^^_^ i.- u 4i u •
^'^^ *= 1+0.0021(60-^ = 1:6378 = ^'^^^^ "^^""^ ^"^^^ ^^^"
f=z41^.3f the dew point. This is perhaps one of the best methods
of determining the point of deposition, as the instruments are not,
like the hygrometers of Deluc and Saussure, liable to be deteriorated
by time, and besides, may still answer other purposes which none of
the usual hygrometers can.
Cor.-*From the same principles, may be derived a formula to
determine the weight of moisture in 100 cubic inches oi ^yc ot
54 INTRODUCTION.
W = 1 4,0 0021 r<^—32^ ^* *^^ freezing point. When ^=:i-2756 and
^'=41.3 we get from the expression W=0.1837 grains when the iair
is completely saturated with humidity. But when the temperatures
are 60« and 4P the W= i ^o.002H^41) =^'^767 grains in 100
cubic inches. Perhaps this method may be conveniently compared
with Mr DanielVs, 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 gives a depression of 1°
temperature Fah. and an elevation of 390feet gives a depression of 1°F.
of the dew point. Hence, if t be the temperature and V the dew point
^^ AH . _ AH
Method IV.
For ordinary heights, such as those usually met with in Britain,
the following method, requiring no tables, wmch is somewhat simpler
and more easily recoUected than Dr Robison's, is subjoined.
Let B be the barometric altitude at the lower situation, and b 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=.13100^5±^|^?="^ in feet.
Bar. in. Att. Ther. Det. Ther.
JBo:.— Leith Pier 29.567 55°^ 54°
Arthur's Seat 28.704 51 | 50 ^
28.704 X 0.0001 X 3.5=0.010, and 28.704+0.010=28.714=6
1 + / — 't^^«.32o\ X 0.00246 = 1+ 20^ X 0.00245=1 .04961, hence
«=13100x^^xl.04961 = mteeL
Height by levelling . 803
Difference .... 2 feet.
Examples for Exercise.
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 ?
J«j.— 14.52369.
2. K the base of a plane triangle be 40, and the other two sides 20
and 30, what are the segments of the base made by a line bisecting
the vertical angle? Arts, — 24 and 16.
3. The hypotenuse of a right-angled triangle is 19630040, and one
of the legs 19630000 ; required the two acute angles ?
^n*.— 6' 56^4, and 89° 53' 3".6.
4. If the sides of a plane triangle be in proportion to each other as
the niunbers \, J, and ^ ; what are the angles ?
iiwj.— 117° 16' 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 TRIGONOMETKY.
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, Edinburgh, to be 60 feet, and that the figure placed upon
the 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 under 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 Point, and the Lizard respectively from the
following data :
i Plymouth to Lizard 60 \
The distances from -c Lizard to Start Point 70 > miles.
. ( Start Point to Plymouth 20 )
Plymouth 1 i North
Lizard > bears from Edystone -l W. S. W.
Start Point j (E. byN.
( Lizard 53.04 )
Ans,-^¥xom Edystone to < Plymouth i^ ^^ v
(^ Start
Thermometers,
Attach. I Detach.
38 31
41 35
8. Barometers,
Lower 29.45
Upper 26.82
14.33 > miles.
1736 1
Required the Altitude,
Ans, — 409.61 fathoms,
or 2458 feet, by Hut-
ton's method.
EXAMPLES BT THE FRENCH MEASURES.
Dew
Point.
OtMHervcr
Humtx>ldt.
Quindiu,
Pac. Oc.
Chimb.
Pac. Oc.
ISamuieter.
Attached
'Ilienuuiuetcr.
J.
0^.50981 8 20°.0 cent.
.762944 25 .3
O".37727610°.0cent.
.762000125 .3
Detached
Thci uiumcier.
18°.75 cent.
25.30
' — P.Ocent
-1-25.3
16°.0 cent.
20.0
0°.0 cent.
20.0
Latitude.
5° ON.
H=3543"«
H=
45' N.
=5925"
Calculation of the last Examphhy Method III.
Constant 18393 metres log.
(1375) 100 = ii5^ X 0.138303 =
^ ^ 100
Latitade 1^ 45' log. _ .
B— J/=:0.759114 log. jl.880307
6— J/' =0.377471 log 1.576884
4 264653
0.016389
0.001161
Difference
0.303423 log.
^ + 5+*-^ + 1.136585 - ^-"^^^^
1.482048
0.008467
H=5925 4 metres . . . 3.772718
Or 19441 English feet, the height of Chimborazo above the \eve\ o^
the Paci6c Ocean.
56
PART II.
SPHERICAL TRIGONOMETRY.
Section I.
Definitions, Principles, and General Properties.
1. Spherical Trigonometric is that branch of mathematics by which we
are enabled^ in all cases, where three of the six parts of a triangle
formed by arcs of great circles in the surface of a sphere are given,
to compute or determine the other three.
2. In plane trigonometry the knowledge of the three angles is not
sufficient for ascertaining the sides ; for in that case the relations
only of the three sides can be obtained, and not their value ; where-
as, in spherical trigonometry, when the sides are circular arcs, whose
value aepends on tiieir proportion to the whole circle, that is, on the
number of degrees they contain, the sides may always be determined
when the three angles are known. Among other remarkable dif-
ferences between plane and spherical triangles are,
(1.) That in the former, two known angles always determine the
third ; while in the latter they never do.
(2.) The surface of a plane triangle cannot be determined from a
knowledge of the angles alone ; -vmile that of a spherical triangle al-
ways can.
3. A sphere or globe is a round body formed by the revolution
of a semicircle about its diameter, which remains fixed.
4. The centre 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
semicircle revolves.
Proposition I.
6. If a sphere be cut by a plane, the section will be a circle.
Let the sphere AEBF be cut by the plane ADB ; then will the
section ADB be a circle. Draw the chord, or
diameter of the section AB, perpendicular to
the .section ADB, and through the centre C
draw the axis of the sphere ECGF, which will
(Euc. III. 3.) bisect tne chord AB in the point
G. 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 the plane
ADB, it must be perpendicular both to GA and
GD. Hence CGA and CGD are two right-angled triangles, having
SPHERICAL TRIGONOMETRY. 57
the perpendicular CO common, and the hypotenuse CA equal to
the hypotenuse CD, being both radii of the same sphere ; there-
fore their third sides GA, GD, are also equal. In like manner, it
may be shown, that any other line drawn from G to the circumfer*
ence o£ the section AD6, is equal to GA, or GB ; and consequently
that section is a circle.
CoT'^ — ^If a sphere be cut by a plane through the centre, the section
is a circle, haying the same centre with the sphere, and equal to the
circle by the reyolution of the half of which tne 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, haying all its
points equally distant irom the centre of the sphere, and is conse-
quently tne curcumference of a circle, haying 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.
Car. — All great circles of the sphere are equal ; and any two of
them bidect each other.
They are all equal, because they haye all the same radii, as has
just be<Ni shewn, and any two of them bisect one another ; for, as
they haye the same centre, their common section is a diameter of
both, and therefore bisects both.*
8. The pole of a ^reat circle of the sphere is a point in the surface
of the sphere eqmdistant from eyery 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 incSnation of the planes of, or tangents at the
point of intersection to, these ffreat 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
ksBS than a semicircle.
11. A right'Oneled spherical triangle has one right-angle; the
sides about the nght-angle are called legs, and that opposite the
rifflitpangle is called the hypotenuse.
i2» A quadrantal spherical triangle has one side equal to a qua-
clrant, or 90°.
13. An obUque^BJi^ed. spherical triangle has none of its angles
right.
14. Spherical triangles are also called equilateral, isosceles, or sea*
lene, according as they haye three sides equal, two sides equals or all
the three sides unequal.
15. Two arcs, or angles, when compared together, are said to be
alike, or of the same affection, when boUi are less, or both are greater
than 90^. But when one is less, and the other greater than 90°, they
are said to be unlike, or of different affections or characters.
16. Eyery spherical triangle has three sides and three angles;
* Hence the Inten^ctioiiB of the dzounfereoces of two great cu^qLm tie Viio'^cfc!&\&
Ametiically oppothe to each other.
66 INTRODUCTION.
and if any three of theae aix parts be given, the otlier tbT«e may be
found. ■
17.Abuu Uapartofthesnrikce of 4 sphere contuned by Vte
■emicircuniferenGei of two great circlei.
la A imaU cinle of the Bphere Ja that who« plane doe* not pua
dirough the centre of the spnere.
19. The small circles of the sphere do not fall under the coasMtor-
ation of spherical trigonometry, but such only as hare the tome
centre with the sphere itself. And hence it is that sphwieal tngo-
nometry is of so much use in practical a*trononv> the apparral
heavens assuming the shape of a concave spl>«e whoee centre » toe
■ame aithe centre of the earth.
20. The tidet of a spherical triangle are all arcs of gnwt cu«lea>
which, by thrir intersection on the auriace of a sphere, ooiutitnte
that triangle.
31. If ABDO be a greet circle of the
sphere whose centre is C and PCP* a die- '"
meter of the sphere perpendicular to its
I^aue, the points P, P'^ are the polefl of that
circle. And if the small circle abed be
perpendicular to PP*, we call P,?* the
poles of that sm&ll circle also.
23. Hie great circles PAP', P6P', pass-
ing through the poles P, P' of the great
circle ABDG, are called secondaries to that
circle.
Proposition U.
53. If two arcs of circles meet eafh other they make two anglM
^hich are together equal to two right-angles.
Let the arc AB meet the arc CD in the point
B ; then will the two angles ABC, ABO be equal
to two right-angles. For, suppose the arc BE
to be peipendiciilar to CD, dien the angles
EBC, EBD are right-angles. ^
And since the angle EBD is equal to the angles ^^
BBA, 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, are also equal to two rigtit-
angles.
Paopdbition III.
54. If two arcs of a circle intersect each other, the vcftjcall, or
apposite angles, will be equal.
Let the two arcs, AB, CD, intersect each other in
E, then will the angle AEC be equal to DEB, and
AED to CEB.
For since the arc AE meets the arc CD, the
aneles ABC, AED are together equal to two
rignt-angles, (Prop. 11.) C^
And because the arc DE meets the arc AB, the
an^s DEB, DEA ore also equal to two right-angles.
Taking away fnnn each the conunon angle AED, and the re>
spheeical thigokometry. 59
mainiAg img)e^ AEC will be equal to DEB. In the same manner
tt ma;^ beproved that the angle AED is equal to CEB.
Cww— Hence if any number of arcs of circles intersect each odker,
ttll the angles formed about the point of intersection are together
^ual to four right-wangles.
P&OPOSITION IV.
3& The arc of a great circle^ between thie pole and the circum-
fbrenoe of another great circle^ is a quadrant.
Let ABC be a great circle^ and P its pole ; if PC, an arc of a
great drde^ pass ibrough P and meet ABC in C^ the arc PC is ^
quadrant.
Let the cirde, of which PC is an af c, meet ABC again in A^ and
let AC be the common section of the planes
of these great circles^ which will pass through
B> the centre of the sphere : Join PA^ PC.
Because AP=PC> (def.)^ and equal straight
lines in the same drde, cut off equal arcs^
the arc AP = the arc PC ; but APC is a
semidrde^ therefbre the arcs AP^ PC, are
each (^tiiem quadrants.
Cor. 1. if P£ be drawn^ the anffle AEP is a right-angle; and
PE, being at rigliti^uigles to every Ime it meets with in the plane of
the drde ABC, is at right-angles to that plane. Thererore the
straight line drawn from the pole of any great circle to the centre
i>f the sphere is at right-angles to the plane of that drcle ; and^ con^
Tersely, a straight line drawn from the centre of the sphere perpen-
dicular to the plane of any ffreat circle, meets the surface of the
sphere in the pole of that circle.
Cor. 2. Tlie 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 pmnts but these can be poles of the circle APC.
Proposition V.
26. If the pole of a great drcle be the same with the intersect <m
of other two drdes, the arc of the first drcle intercepted betweeh
the €»ther two, is the measure of the spherical angle wnich the same
two drdes make with one another.
Let the great cirdes AP, BP, on the surface o€ the sphere of
^hich the centre is O, intersect each other in
P, and let AB be an arc of another great
GJrcle of the pole as P, AB is the measure of
lihe spherical angle APB.
J<»n PO, AG, BO ; since P is the pole of
AB, PA, PB are quadrants, and the angles
POA, POB are right; therefore the angle
AOBis the inclination of the planes of the cirdes
PA, PB, and is equal to the spherical angle
APBi but the arc AB measures the angle
AOB> therefcve it also measures the spherical angle APB.
Car. 1£ two axes of great cirdes, FA, PC, which intersect each
<Aher ki P, be each of them quadrants, P will be t\ie i^o\e oi xVv«
60 INTRODUCTION.
great circle which passes through A and B, the extremities of these
arcs. For since the arcs PA and PB are quadrants, the angles
POA, POB are right-angles^ and PO is therefore perpendicular to
tlie jj^Ane AOB^ ttuit is, to the plane of the great circle which passes
through A and B. The point A, therefore, is the pole of the great
circle which passes through A and B.
Pbopositiom VI.
27. An angle made by any two great circles of the sphere is
equal to the angle of inclmation of the planes of these circles.
Let BAE be a spherical angle made by two great circles CBA*
CEA ; then will this angle be equal to the angle
of inclination of the planes of those circles. For,
take the arcs AB, AE, each equal to 90S or a 'f
quadrant, and through the points B, E draw
the arc of the great circle Bb, 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, and the
lines DB, DE are each perpendicular to the^ common section AG ;
consequently BDE is tne angle of inclination of the planes CBA^
OEA. But since DB, DE are equal, being radii of the same sphere,
the angle BDE, 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
CEA, each perpendicular to the common section AC, the angle
HFG, which is equal to the angle BDE, will also be equal to 3ie
angle BAE.
Cor, The angle BAE made by two great circles of the ^here
BA, EA, is equal to the angle n A m, formed by two tangents mrawn
froin the angular point A, one in each plane, these tangents being
each perpendicular to the diameter AC.
Proposition 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 those
circles.
. Let AEB, CED be two great circles, and P, P' their poles ; theti
will the arc PP' be equal to the angle of their p
inclination AOC or BOD.
For, since P is the pole of the circle AEB, C^
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, the
remaining arc, PP', which is the distance of
two poles, is equal to CA, the measure of
the angle of inclination aOC.
Pboposztion VIII.
29. The circumference of a secondary is at right angles to the cir-
cumference of its ereat circle at the point of intersection.
The direction <tt the circumference of a great circle at any point
SPHERICAL TRIGONOMETRY. 01
being the same as the diameter of its tangent at that pointy the angle
OBT, (figure prop, v.), is a right^angle^ BT beinff a tangent to fiP
at the point B. JPOB is also a right-an^le, and the arc JrB is in the
plane rOB, therefore the direction of uie 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. '
Car, 1. — ^If a. great circle^ PBP'^ be perpendicular to ABC, and
BP^ BP' be taken each equal to a qu^ant^ or 90% P^ P' are the
poles of the circle ABC.
Car. 2. — ^If any two great circles^ PAP', PBP', be perpendicular to
the circle ABC, they meet at the poles P, P' of that circle.
Proposition IX.
90. In an isosceles spherical triangle the angles at the base are equal.
Let ABE (fig^ure pr^. VI.) be a spherical triangle, having the side
AB equal to the side AE, the spherical angles ABE, ABE are equal.*
Cor. 1. — Hence, if two of the angles of a triangle be equal, the
sides opposite to ^em are likewise equal.
Csr. 2. — ^A perpendicular drawn from the vertex of an isosceles
spherical triangle to the base, bisects both the base and the verticid
angle, except when the two sides are quadrants ; in which case there
are an indefinite number of perpendiculars.
Pboposition X.
31. If the three sides of one spherical triangle be equal to the
three sides of another, each to each, the angles which are opposite
the equal sides are equal.
Proposition XL
.32. If two sides and the included angle of one spherical triangle
be equal to two sides and the included angle in another, these two
tr ingles are equaL
Proposition 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 circles,
-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
duee angles c^anodier, each to each, the sides which are opposite
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 demomtntions, whidi may be seen in PIayfiur*i or Lraendre^i Geometrr^ are
emitted, as thcj woa|d swell this work too much, but may pechav* «P9«» Vn. Skictfm
complete tzeatise on trigonometry that has been long meditated.
« IKTRODUCTIOS.
Proposition XV.
36. The sum of any two sides of a spherical triangle is gi'eatei'
than the third side ; and the difierence or any two sides is less than
the third side.
Cor, — The shortest distance between any two points on the sur-
face of a sphere is the arc which passes through these points.
Proposition XVI.
37* The greater side of any spherical triangle ii^ opposite to the
greater angle^ and the less side to the less angle.
And^ in a similar manner^ it may be shown that the less side is op-
posite to the less angle^ and the less angle to the less side. «.
Proposition XVII.
38. The sum of the three sides of any spherical triangle is less
than the circumference of a circle, or 360^ ; and the difference of
any two sides is less than 18(r.
Proposition XVIII.
39. The sum of the three angles of every spherical triangle is
greater than two right-angles, or 160°, and less than six, or M^P.
Cor. — ^The sum of any two angles of a spherical triangle is great-
er than the supplement of the thurd angle.
For the angles A+B+C, being greater than two right-angles, olf
than ACB-J-ACG, if ACB or C be taken away, the sum of the t*e-
maining aaglet A+Bi 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 their opposite
Aneles 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 be equal to, greater, or less than 180^, the sum
of the opposite sides AB and AC, will also be equal to, greater^ or
less than l?XP.
Cor. 1. — ^If each side of a spherical triangle be equal to, greater^
or less than 180^, each of the angles will, accordingly, be right,
obtuse, or acute ; and conversely.
Cor, 2. — Half the sum of any two sides of a spherical tdangle is
of the same kind as half the sum of their opposite angles.
Proposition XX.
41. In any riffht-angled or quadrantal spherical triangle, the 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
triangk, which in this case is right-angled, the similarity wiu be
the same as before.
Proposition XXI.
43. In any right-angled spherical triangle the hypotenuse is less
or greater than 90^, according as the two Tegs, or tbe two angles, at
a )eg and its adjacent angle, are alike or unlike.
SFHEBICAL TKIOONOMETRY. 68
Section II.
Solution of Spherical Triangles.
Having given a view of the general principles and properties of
spherical triangles, the solution of the various problems in spherical
trigonometrv 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 stereographic, in which all
the circles of the sphere are represented by straight lines or circles.
Of the Stereographic Projection qfthe Sphere.
Dbfinitions.
I. To jfrafect an object^ as it is commonly called, is to represent
every point of that object upon the same plane, as it appears to die
eye m a certain position.
II. That plane upon which the object is projected is called the
plane qfprqfection, and the point where the eye is situated, the pro^
jecting pMrit,
IIL The stereographic 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.
V. By the semitangent of any arc is meant the tangent of half
Aat arc.
VI. The line of measures of any circle of the sphere is that diame-
ter of the primitive, produced indefinitely, which is perpendicular to
the line or common section of the circle and the primitive.
VII. The prof ection, or representation of any point in th6 sphere,
is die polnft m which the straight line drawn nom it to the project-
iDg pomt intersects the plane of projection.
Theorem I.
Every great circle of the sphere, which passes through the pro-
jecting point, is projected in a straight line, passing trough the
centre m the primitive; and every arc of it, reckoned from the other
pole c£ the primitive, is j^rojected into its semitangent.*
CoTm 1. — ^Every small cirue, which passes through the projecting
point, is projected into that straight line which is its common secticMi
with the primitive.
Cor. 2. — ^Every straight line in the plane of the primitive, sand
produced indefinitely, is the projection of some circle on the sphere
passing through the projecting point.
Cor. 3w-— The stereographic projection of any point on the surface
^^>i^i*i*i****fc*—»»-**M*^M^^^^^^W I » ■ i*^^i**»«^-^t^—^..*«« «■ fi^*—^— ——*■»— ^ijL
* For the investigatioii of the properties of this method of piajficUon, «e& Ox^gvni''^
or Keith's l^cettlBes of Tr\g€0(mietry, tafdWeii^ ]Aathemanc8.
64 INTRODUCTION.
of the sphere, is distant from the centre of the primitive by the 8e«
mitangent of the distance of that point from the pole opposite the
projecting point.
Theobem II.
IRverj 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.
Gn", 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 die
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.
Theorem IV.
The centre of projection of a small circle, perpendicular to ihe
primitive^ is distant from the centre of the primitive by the secant oP
the distance of the circle from its nearest p<Me, and the radius of pro**'
jection is the tangent of the same.
Theorem V.
The projections of the poles of any cirde 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 two
great circles, intercept equal arcs upon them.
Theorem VII.
If, from either pole of a projected ^eat circle, two straight Mnes
be drawn to meet the primitive and we projection/ ihey will.l^tar**'
cept corresponding arcs of these circles. ■''-
Solution of Righi'AHgled Spherical Triangles.
The floludon of right*angled spherical triangles may be accdih-'
plished by formulse investigated expressly for that purpose. We kr^
mdebted to Napier, however, for a comprehensive rule of great iUl-
vantage to the memory, by reducing all the theorems employed in the
aoluticm of right-angled triangles to two. This is called tne ruk of t^
circular parts, and is perhaps one of the happiest examples ofdrfi-
ficial memory that is known.
Definitions.
I. If in a right-angled spherical triangle the right-uigle be set
aside* and the five remaining parts of the triangle akoe be Gonsader-
ed, oonsifitang of the three sides, and the two oblique ancles, Umq^
the two sides containing the right^mgle, and tibie oompleita'eMi of
SPHF.RICAL TRIOONOICETBY. M
tha otiber tfarMf maaakj, of the two tng\t», and of the l>7PotcaiiM^
II. When, of the we circular parta, any one ii taken for tW
middle paii, then, of the remaining four, the two which are iinine-
diatdy ac^cent to it on the ri^ht and left are called adjacent parU i
and t£e other two, each of which is aeparated from dw middleyait
1^ an adjacent part, are called ofipoMte «ri#. '...::(
IUb arrangement bang Blade, the Muution is obtained by the M-
lowii^ ■ ■ ^
Tbbobbm. '-
In ai^ iwht-angled •pherical triaogle, the rectangle under the'n^
dini, and the sine of the middle part, is equal to the rectuigle un^
the tangenli of the adjacenl parts ; or to the rectangle under the co<
BiNxa of the ofpositk parts.
Iliia theorem, or rule, may be easily remembered, by rcmarknig,
that the flrrt vowel* in tine, tangent, otuine, are respectively the aanie
•a the flrat in middle, a^actnt, oppotUe,
or, Rxam. mtdi^rect. tan adj. = rect. cos. op."
It ia usual to convert the equation under consideration into an
'logy having the unknown <juantitf for the last term, thtm|^j to
m acqiiainted with ^{cbra, it would be more convenient to make
Imetbe first term ofan equation, and the remaining terms, <xim-
bfaicd properly according to me rules of dgebra, the losL
Oiveo three of the nx j^arts, as, for example, the h vpotenuae ^i>d
one of the angles of a nght^ngled spherical triangle, to find' the
ndes ■"'! the remaioiiig angle-
On the first of May I^ the sun's loiwitude was 1' 10° 32* 12",
and ihe obliquity of the ediptic 33° 3?' W' ; required the right as-
cendon aoddedUDationft
Ani.—Vi.. A. 2* 32- 2?.3 ; dec. 14= W 4?" N.
CoMlriK^tott.— With the chord of 60° describe
the primitive circle EPQI" on the plane of the
solstitial Golure, and draw the diameters £Q
and PP' at right-angles to one another, then
wiO EQ represent ue equator, and PP' the b|
polar axis. Lay off from Sesame line of chords
E e=23° 27' 40", the obliquity of the ecliptic,
and draw tiie diameter el representing the
ecliptic, at right-angles to which draw pp',
mBap,ff are uie poles of the ecliptic. Prom
iftff are uie poles of the ecliptic. Prom the line of ssmi-
ta, (Theorem I.), lay off the sun'a longitude 1* 10* SSf -\^,
..^- 931. i on the ecliptic, from A to C, then C will be the place
(tf ittie aun, and » c m a parallel of declination. Through the pointa
FCP'.draw a circle of right ascension, cutting the equator EQ
tuwenta, i
« of the jMiii, (hen,
iplcnunt, thatit, for
unoomj SUV !■* mi-
K w UKgHDO, Ennn ideu ir^~~'~~ ' '
lisamsbtiw^lr obtained Itn
luilf orthe ccUptie, n* ecmpuiad A
Miwuii' i^ined. itm^bea23cd,d»tdic(u^laec)tnAa,ni^4t«
u^
aO INTBOMJCriON.
at right-angles in B, then will AB be the right McennoQ, BC. the
declination, and BCA the remaining angle or an^e of poiitloii, as
it ia fometiaies called, which, in astronomy, is seldom of much uiie.
. CUcMJaitbiiw'— In the triangle ABC there are given ACss4IOP
SSf la^', and the angle BACszTA^ 27' 40", to find EG, the distance
of the sun from the equator EQ, or the declination, as it is usually
called. Now, since in spherical trigonometry the sinei of the mdet
are proportional to the «tiira of their opposUe anglu^
Therefore,
As sine ABC or radius 10.000000
Is to sine BAG 23» 27' 40'' . 9.600021
So is sine AC 40 32 12 9.812870
To sine BC 14 69 47 . 9.412891
To find AB we may employ the method of the circular parts.
In <he triangle ABG are given AG and the angle BAG, to find AB
the right ascension. Now, since the side GA, the angle GAB, fmd
the side AB are all connected, that which stands in we middle or
the angle A is called the middle part, and the sides AG and AB ad-
jac^it to it on each side are called the adjacent parts.*
Consequently Rxcos. A = cot AGx tan. AB; and resolving
Xjoi^ into an analogy, as is frequently done in this country, we have.
As cot. AC 40^ 32' 12" . . 10.067939
Is to radius ... 10.000000
So is cos. A 23 27 40 . 9.962526
To tan. AB. ^ 32- 2r.3 9.894587
or, since cot. : R : : R : tan., or tan.= — -^ to radius unity (§ 35> page !!•)
As radius 10.000000
Is to tan. AC 40° 32' 12" . 9.932661
So is cos. A 23 27 40 9.962526
Totan.AB 2^32»27'.3 9894587
the same as before.
To those acquainted with algebra, it is better, after the manner of
foreign mathematicians, still to retain the form of an equation thua^
tan. AB = — , . ^ - = cos. A X tan. AC, the radius beiiiir re-
cot. AC • ' .
S resented by unity ; in which case ten must be rejected in the mr
ex.
. To log. COS. A 23^ 27' 40" 9.9^536
Add log. tan. AG 40 32 12 . 9.932061
Sumtan. AB 2^3a»27.'3 9.894687
To find the anffle ACB, since the parts under consideration are
itill all connected, AC standing in tne middle is assumed as the
middle part, and tiie angles A and C are the adjacent parts, whence
< * It may be remarked, that if the parts are aU eennected, that wfiich stands In the
middle is called the middle part, and the other two arc called the adiiacent parts. If
•two oidjr are connected, and one stands bj itself, then this is called the middle part, and
the other two are called the opposite parts.
SPHERICAL TRIGONOMETRY. 67
RxcaB.JiC == cot Axcot C, and cot. C = ' ^ sscoa.AC XtML
cot. A
To Ictt. COB. AC 40° 33" 12"' .... 9.880006
Add tog. tan. A 23 27 40 . 0.037406
*»m.
Sam ^ CDt. C 71 44 42 .2 . 9.518304
Or the comp. 18 15 17 •8, is called properly the anffle of pbti-
tion, sometiines useful in computing the parallaxes in Bolar edipeet
and eecultataoni of the fixed stars and planets by the moon.
BjF assuming different parts of the triangle ABC for the middle
pari; 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* lO*' 0' 45", the obliquity of the
ecliptic 23° 27' 36'' ; required the right ascension and declination ?
AH9.—B.. A. 4»» 34" 7'.6; Dec. 21° 59' 34" N.
2. August 12th, 1827, the obliquity of the ecliptic being 23° 27'
36", the sun's right ascension will be 9^ 25*° 29'.3 ; required his
longitude and declination }
1«.— Jjongitude 4* 18° 56' 28", Dec 15° 9' 32'; S.
3. Oitthe lOth November, 1828, on the meridian of Greenwich,
the eunTs right ascension will be 15^ 2°" 32'.7> and declination 17^ 14"
l2" S.; required the sun's longitude and the obliquity of the
ecliptic?
Ans. Longitude 7* 18^ 6' 7", and obliquity of the ecliptic «3«
27' 34".
4. On the 2d of March, 1828, when the sun's declination was
T fi' 18" 9^, and obliquity of the ecliptic 23° 27' 35" ; required his
longitude and right ascension ?
Jiwr-Lpngitude 11' 11*" 56' 34" ; R. A. 22'' 53"" 24'.
Pboblem II.
Wbcn the celestial object is not upon the ecliptic, as the moon, or
tiie planets, and some of the fixed stars, the right ascension and de^
dinatiionr are found by the solution of two right-«ngled triangles.
1. On the 17th of January, 1826, at noon, on the meridian of
GreeBiHch,-the moon's longitude was 1' 11° 5' 14", and her latitude
2^ iV d^ N. ; required her right ascension and declination, the
ofaiiqaity of the ecliptic being 23° 27' 40" ? To resolve this eziunple
it iv necessary to employ two right-angled spherical triangles.
Del.iIm} foregoing figure, the longituoe of the moon or any' star S,
is AD, liie latitude DS, the obliquity of the ecliptic BAG, uie right
aoeension AB and declination BS. Now, supposing a line drawn
fxQKaAto 6, there would be formed the right-angled sphoical trian-
gie ADS, right-antfled at D, of which AD and DS are given to find
thfi angle DAS and the side AS. If the position B of the star is
mkh&ui llie ecliptic, then to the obliquity a£ the ecliptic B AC, add
the apgle DAS, the sum will be the angle BAS ; but if S is witklH
the ecliptic, that is between it and the equator, subtract the angle
DAS -m>m the obliquity BAC, and the remainder will be the angle
BAS. Since the side AS, and the angle BAS, are now known, AB
niieiruAt wkemunom, and BS the declinatioa, may be found.
Cahnlatiim. — By the rule of the cirqular pafts». fiiTsX AD accv^\^'9k
68 INraODUCTIOW.
are given to find AH, and since the last is separated fitna the two
first by the oblique angles, it will be the midme part, and'Af) -fSnA
DS are the opposite parts ; therefore, R x cos. AS = cos. DS x|cos*
Aiy,*6r 66s". AS = cos. DS x cos. AD to radios anitj. *
Tolog. cos. DS 2^34' 3'' . 9.900664
Add fog. COS. AD 41 5 14 9.877^^
Log. cos. AS 41 9 11 . 9.87a7fiB
Again^ to find DAS^ since the right angle does not separate the
parts, DA standing in die middle is called ttie middle part, and the
aide DS and the angle DAS are the adjacent parts, hence B^ain.
DA = tan. DS X cot. DAS, and, therefore, cot. DAS =' — ■■■ ^ =r
tan« Do
sin. DA X cot DS, consequently
To log. cot. DS 2^U' Sf' . . ■ . 11.348322
Add log. sine DA 41 6 14 . . . 9.817634
Sum=log. cot. DAS 3 54 14 11.165956
TothiaaddOb.;Bc.23 2? 40
Sum = angle BAS 27 21 64
Hence AS and BAS are now known, to find AB and BS.
. First to find AB.'. In this ease the parts are connected $ therefere
BAS is the middle part, and AB ana AS are the adjacent parts,
whence
H X COS. BAS = tan. AB X cot. AS, or tan. AB = * * rsr-, and
cot AS
tan. AB =: cos.- BAS x tan. AS, hence
To W. cos. BAS 27° 2r 54" . 9.948460
Add tog. tan. AS 41 9 11 , 9.941505
Sum = log. tan. AB 37^ 49' 5" 9.889965
Or in time R. A. 2*^ 31" 16*.3
' To find BS, the angle BAS and side AS are connected, and BS is
difjoined, whence R x sin. BS = sin. AS X sin. BAS, <Hr since the
sines of the ddes are proportional to die sines of their (^pposite an-
gles;
As sine ABS or radius ..... 19J00O00O
Isti^sineiAS 4P 9* 1" . . • 9.818874
Stf lysine BAS 27 21 54 . , , a6^M34
To sine Dec. BS 17 36 26 N. . . 9.480706
* The foregoing method is general and applicable to any part of the
ecliptic^ provided proper attention be paid to the ntuation of the
'cJMestial object wfch respect to the ecliptic and equatdr.- As'thiapro-
blem and its converse is of frequent occurrence m practical astsono-
'my, rules and formulae, and even tables, have been' formed for the
purpose of facilitating the computations. The follo^ngrtilesr given
'By flie late Dr Maskelyne, wwl be found very eonveniie^t for this
puifpose.
Froblsm II.
■ ■ ■ ■ ' ^ ■ I ■
Given the right asi^ension, the declfnatianf and the obliquitv of the
ecliptic, to find the longitude* aild. latitude. .
SPHERICAL TBIOONOM£T£Y. 60
Let RA denote the right aacemion, O the obliquity of the ecliptic,
aad D.the dedinatiaiu
TtDii iX—«n. RA =tan. A, North or South as the declination iv.
CSall O in the first six signs of RA South or S. and in die last ax.
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 = tan. 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°.
Tafou A+cosec. A-|-cos. B+tan. RA = tan. Ion. of the same kind
as BA, unless B be more than 90^, when the quantity found of the
same kind as RA must be taken from twelve signs.
Ifon. being nearer III. and IX. signs than O and VI. signs.
. Sin. lon.+ tan. B = tan. lat. of the same name as B.
Xiini.' nearer O and VI. signs, than III. and IX. signs.
Tan. Lon.+cos. Ion. + tan. B =tan. lat. of the same name as B.
EZAMFLB.*
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° 61' 68" N. tan. 8.699633
RAslO^ 90" 3P sine 9.636560 tan. 9.563908
A 8° le' 50^' N. tan. 9.162973 sec. 0.004551
O 23 27 40 S.
B 15 10 50 S. cos. 9.984575 tan. 9.433407
Lon. 160 20 17 tan. . 9.553034 sine 9.526946
Lat. 6 12 59 S tan. a960443
Pbobleh III.
Given the Idnffitude and latitude of a celestial object, and the ob-
liqmty of the ediptic ; to find the right ascension and declination.
Tan. Lat.^^^ne Lon.=tan. A, North or South as the latitude is.
Call O North in the six first signs, and Soath in the six last signs.
A + O = B, as before.
A being less than 45% sec. A+cos. B +tan. lon. =Tan» RAof the
same kind as the longitude, unless B be more than 90^, when the
cnumtity found of the same kind as the longitude must be subtracted
mm twelve signs.
Abeinff more than 45% tan. A + cosecant A-f cos. B+tan. Ion.
= tan. RA of the same kind as the longitude, unless B be more than
■90^j when the quantity found of the same kind as the longitude must
be subtracted nom 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 VL signs, than IIL and IX. signs,
ian. RA+cos. RA+tan. B = tan. Dec. of the same name as B.*
• These rules may, in general, be depended upon, except in peculiar circumstances,
wfaii;^ acmldenuion of the ^gnxewiU enable the computer to correct, as when the lon-
gitude, or RA, ftn upon PP% or pp', &c.
See Ut Abiam Robert8on*s niper k the Phil. Tnois. fbt 1S\A, Y«iefc\S&^i[j^Vdtil«K
want of xoan} caniiDf Iw^ven Aem.
INTHODBCTION,
Od tlie lit of JaniWT, ISSOt the mean longitude ot iIk Star Po-
nial^af tTM 11" 1° 19* 34", the mean latitade 21" 8*45" S. j rei^iiired
the rleht ascennon and declination, the obliquity of the ediptjc b^.
Lit. 21" ff 45" S. tan. 9.586721
hoa.aSl 19 34 Bine 9.6B1082 tan. 9-737901
A= 38 49 26 S. tan. 9.1
0=23 27 46 S.
, B=: 62 17 12 S. cosine . 9.6674S8 tan. 10379685
11A==341: 65 14 tangent . . 9.6138X9 sine 9.^1831'
Dec. 30 34 21 S. . tan. 9.771416
Examplet for Exercite.
1. The mean longitade of « Arietis, on the 1st January, 1820, was
1* S° 8' 48"j Mid mean latitude 9' 67' 34" N. when the obliquity of
ibe ediptic was 23" 27' 46" ; what was the right ascension and decli-
nation?
Ant.—n. A. 1" Sr 3-; Dec 22" 36' 24" N.
2. Required ths right ascension and decliniition of Pollux, when
the longitude was S* 20° 43' 58", the latitude 6" 40- 17" N. the ob-
liquity of the ecliptic being 23° 27' 46''?
Ans.~-R. A. T 34" 17.5' ; declmaHon 28° 27' 8- N.
a The mean longitude of Spica Virginia is 6" 21" IS" 50", latitiide
2f3'24"S. and the obliquity of the ecliptic 23'''27'46"; required
the right ascension and declination ?
- ^M.— R. A. IS- 16- 43.5'; decimation 10° 13' 4" S.
4 The mean right ascension of « Aquilse is ]&■ 4Sr, and declina-
tWn 8° 24' 4" N. the obliquity of the ecliptic being 23" 27' 46' ; re-
quired the longitude and latitude ?
. ^nr.— Longitude 9" 29' 14' 14", Latitude 29" 18' 36' N.
' 6. Required the longitude and latitude of « Pegasi, of which the
right ascension is 22^ SS™ 4^, declination 14" 14' 21", the obliquity
of the ecliptic being 23» 27' 46" ?
^n*.— Longitude II' 20° 58' 47", Latitude 19" 24' 36" N.
PaoBLEu IV.
, Givra the latitude of thejilace, and the sun's declination, to -find
liiB altitude and azimuth at 6 o'clock.
1. At Edinburgh, in latitude 55" 57 20*' N. on the 21et of Jun^,
UQ6, the aun'B declination was 23° 27' 36" N. ; required his altitude
^d azimuth at 6 o'clock in the morning or evening, hia declination
being supposed to remain the same. ' \
Con»/rurtton.— Describe the primitive HPON on the plane of Uie
meridian. Let HO represent the horiioRj ZN
the prime vertical at right angles to the former,
Make OP, from a scale of chords equal to thela-
titudeofthe place, North in therirtsentinstance; .
draw PP', the six o'clock hour circle in this case, tlC
and at right angles to it draw the equator EQ ; ^
describe the small circle »nt at the distance of
23° 27' 36" from the equator, representing the
/xanllel of declination, and it will cut the six
u'clock liour circle PP' in P, the sun's place
SPH£B{CAL TBIOONOMETRY. IX
Through Z^ F, and 1^ de«eribe the a^ifnuth circle ZFN cutting the
horixon in I), then .FD is fhe aUittde^ FZ,the seiiith distance, and
the angle FZP, or .its measure, the arc DO, is the azimuth ; conse*
quently, the things given and required fSdl in either of ti^e triangles
FZP, or FDA, whi(£ are supplemental to each other. For, since OP
is the latitude, PZ is die colatitude, AF is the declination ; coiiibe*
quently, FP is the polar distance, DF being the altitude, FZ must be
toe senith distance.
Cahulaium* — ^In the right-angled spherical triangle FP2^ ngfat^
angled at P, FP and PZ are given, to find the angle FZP and FZ; or
in the, triangle ADF» riffht-angled at D, there are given the ai^e
FAD, equal to the latitude of me place, and AF, the sun's dedina*
iion^.to nnd DF, Uie altitude, and the side AD the asimuth*
By the rule of the circular parts FP, PZ, and PZF, are all con*
necUd* therefore PZ is the middle part, and PZF and PF are the
a^jacentt parts, vhere
R X sine ZP = taa. PF x cos. PZF, or
H X COS. lat. = cos dec. x cos. azimuth, therefore
cos. azimuth = -^ — = cos. lat x tan. dec.
cos* dec*
To log. cos. lat 66° 6? 20" 9748061
Add log. tan. dec. 33 27 36 .^ 9.637472
Sum = log. cos. az. 76 20 88 9.386633
Again, Xxy find FZ the coaltitude, the same things being given,
II X COS. FZ = COS. ZP X cos. FP, or sine alt = sine lat X sine dec
To W. sine lat ^"^ 67' 20" . 94)18347
Add k>g. sine dec. 23 27 36 . . 9.600002
Sumslog.8]nealtl9 15 40 9.518340
Problem V.
«
Given tibe latitude of the place, and the sun's declination, to find
the altitude and hour when tne sun is due East or West
. .. EXAMPLB.
At Edinburgh^ on the 21st June, 1826, what was the sun's alti-
tude whi hour when due East or Wes^ the declination being WSf
■27'38"N. . '
In the last figove, let ZAN mieet the jMursllel n m in K, and sup«
pose a circle to oe drawn through the points PKP; forming the tri-
angle ZKP^ right^mgled at Z, then ZK is the coaltitude, and ZPK
the hour from noon ; hence
B X COS. PK = COS. ZP X cos. ZKj or
cos. ZK = ■ ' „^ = COS. PK X sec. PO, or
COS. ZP
J
, sine alt =^ sine dec. X sec. lat
Dec. 23° 27' 36" sine 9.600002
Lat 66 67 20 sec 0.081663.
./
r<: : Alt 88 43 66 sine jB.681656
R X COS. SSPK s tan. ZP X eosi PK, or
, . ■ . COS. T=rco8. lat X tan. dec •. .
72 INTEODlTCTION.
Ltt B6^ &T 20^ COS. 9.899714
Dec. S3 37 96 tan. 9.037472
Time 4''61"4«' cos. 9.467186
Fiom noon, that is, at 7" 8" 12* a. m., and 4^ 51"' 48" F. v.
This problem is of considerable utility to ihe navigator and prac-
tical astronomer, for the purpose of determining time accumtdiy
when an altitude instrument is used. As the change of altitude, oa
whidi the accuracy of the determination of the time depends, is
quickest when the object is on the prime vertical, the most proper
tune for observing an altitude for tnat purpose is, therefore, when
the ol^ect is due East or West, as any small error in the observatioh
has then the least possible effect on the time. Other errors are also
in tlus'case in a great degree avoided, or at least considerably lessened,
particularly that arising from any small error in the estimated latitude
at the time of observation. To facilitate its application, tables, opr«
responding to the latitude and declination (which must be of the same
name with the latitude), have been given in books on Nautical Astro-
nomy, such as those of Mendoza Rios, Mackay, and Lax. When the
latitude and declination are of different names, the altitude must be
as near ^e horizon as is consistent with accuracy, so far as depends
upon the uncertainty of the horizontal refraction. Altitudes under
5^ should not be used when great accuracy is required.
Problem VI.
€Kven the latitude of the place and the sun's declination, required
his amplitude and ascensional difference.*
At iSdinburgh, on the 21st of June, 1826, from the data given, on
what point, and at what time, did the sun rise and set ?
In the triangle ABC, in the last figure, there are ffiven the angle
BAG, equal to the colatitade, and BC the sun's declination ; to find
AC and AB.
R X sine BC = sine AC X sine B AC, or
. _ sine BC . -,^ » a ^
sme AC = -; 5-^7^ = sme BC x cosec. BAC.
sme BAC
BC, or dec. 23° 27' 36" N. sine 9.600002
Latitude, 55 57 20 sec. 10.251939
AC, 45 19 33 sine 9.851941
CO, 44 40 27> in which case AC is the ampli-
tude reckoned from the East or West, to the Nortli and South, ac-
cording to the name of the decliniation, and CO is that reckoned from
the meridian, or from the North or South, according to the name of
the declination.
Again, in the same triangle AB is the ascensional difference, and
R X sine AB = cot. BAC x tan. BC, or sine AB = tan. lat X tan. dec.
Lat. 55° 57' 20" tengent, . 10.170286
I)ec 23 27 36 tangent, 9.637472
A. D. 2^ 39- 52" sine . . 9.807758
6
8 39 52 = time of setting.
3 90 8 = time of rising, the latitude and de-
* By the ascensional difiexeoce is meant the time before or after 6 o'clock the tan
rites or sets. By this problem, therefore^ the leng^^ of the day and night are deter-
asdaedy and the rariation of the mariner's compass.
SPHERICAL TRIGONOMETRY. 73
dinatioii being of ^e same name> or if instead of sine we read cosine^
then we would get the time of rising }f the latitude and declination
are of the same name^ and the time of setting if of different names.
This^ however, is only the'approximate time, as no allowance is made
fbr the effects of a change of decliiEiationj the . horizontal refraction
and parall^ in thie case of the sun .and planets. For these see
tts^kayon the longitude, or they may be found by the following
kile. / J^st, let the approximate time pe found. To this time let
Hike cteciination of the object be ' reduced. With it find the as-
centtoni^ difference 4^ formerly. Now, find the sum' and dif-
fjn^noe pfj'the natural cosine of tiie reduced declination and natii-
nfl'fune of^the latitude, which may be carried to fom* places of figut^
mily, i&^ being qufficientiy accurate for thip purppse, and take half
Cbe';siim of the logarithms of tiiese quantities, to which add the con-
i^ia^KlbJD^rit^ 7*1761, and the proponional logarithm of the diffe)"-
enixi bet;w^n tiie horizontal parallax and the sum of the horizontal
j^iftaCTpl^ and dip of the horizon, the sum, rejecting J.0 in th^ index^
vnSji be the ^proportional logarithm of the correction w^icb is to be
nibiracUd from the time of rising, or added to the time of setting. If
the horizontal . parallax is less than the sum of horizontal refraction
1^4 dip, otherwise the correction must be added in the first case^ and
fuiira^edi^ the second.
Example. -^
Required the time of rising and setting of the sun on the 1st of
Aj^ril, 1826, in latitude 33" 42' N., and longitude 16° 20' W. the
height of the eye, above the sea, being 28 feet.
Dec. 4° 28' N. cos. 990^
Lat. 83 42 N. sine 5548 '' -
Sum 15517 log. 4.1908
Diff. 4421 log. a6455
Dip to 28 feet — 5' 16" 78363
Hor..reiTac. — 34 17 — —
Parallax + 9 3.9181
■ const, log. 7-1761
— 39 24 P.L. a6o98
— 3" 10- P.L. . . 1.7540
X^ <MoftectxAi to be subtracted from the time of rising, or added
tdtiie'titail^ of setting. As the moon's horizontal parallax is ingena-
faH'gr^ocef thati the effects of dip and refraction, the correction thus
M>timi^ would haVe been appbed with a contrary sign. This me«
thod of determining time may sometimes be of use tvben a better
eMttnot'be obtained, 'and -in the case of the sun or moon, a mean of
twtitili^'i^ appearance of the upper and lower limb may be tokeou^
SdiiUfm qf Obliaue''Angled Spheriifal Triangles.
The different cases of oblique-angled spherical triangles may be
solved by the following theorems :— <
* To find the lisiiu^ and setting of a star or planet, the transit over the meridian must
be first computed as follows : — From R. A. of tine star subtract that of the sun for noon,
the remaindler is the approximate ffane of transit.' Reduce the R. A. of both to this time
>^ ^<l¥^BIlJm^tua«,^d subtract as .bci$^ and the itmaiader will be the true Ivkv^
of ttannV WQEn, pnvemr apS^ ^ the swaiqmid arc, wm glyc^ w\mii coRec\e\ ^«it
dip, &C., «hfi'tnie time off inu" itf settio/^.
74 INTRODUCTION.
Theorsm I.
In every spherical triangle
the sines of tne sides are pro-
portional to the sines of the
angles opposite to them^*
Or^ sin. AB : sin. AC : : sin.
C : sin. B.
Theo&em II.
In oblique-angled spherical triangles a perpendicular arc being
drawn from any of the angles upon me opposite side^ the cosines of
the angles at the base are proportional to uie sines ci the segments
of the vertical angle^ or cos. B : cos. C : : sin. BAD : sin, CAD.
Theorem III.
The same things remaining, the cosines of the sides are propor-
tional to the cosines of the segments of the base, or cos. AB : cos.
AC : : COS. BD : cos. CD.
Theorem IV.
The same construction remaining, the sines of the segments of tbe
base are reciprocally proportional to the tangents of the angles at
the base, or sin. BD : sin. CD : : tan. C : tan. B.
Theorem V.
The same construction remaining, the cosines of the segments of
the vertical angles are reciprocally proportional to the tangents of
the sides, or cos. BAD : cos. CAD : : tan. AC : tan. AB.
Theorem 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 of half the sum of the two
sides of the triangle, as the tangent of half the difference of those
sides to the tangent of half the difference of the segments of the base,
or ten. A (BD+CD):tan. J (AB + BC) :: ten.^ (ABcdAC) : ten. J
(BD <r CD).
When the three sides or the three angles are not the mven parte
of the triangle, to have sufficient data for the solution of l£e problem,
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 VU^
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 tengent of half the base
is to the tangent of half the difference of ite segmente : And the sine
of the sum of the two sides is to the sine of their difference, as the
cotengent of half the angle contained by the sides is to the tangent
* See Playfair's Geometer, artide SiAeiiGal TrigoBometzy, Prop- XXIV., or Legen-
dreHi Oeometrj, Brdde LAX VI., ana the following in order.
SPHERICAL TRIGONOMETRY. 75
of half the difierence of the angles which the same sides make with
the perpendicular^^or sin. (B+C) : sin. (B c/) C) : : tan. ^ BC: tan. '
(BD <D CD). And sin. ( AB + AC) : sin. ( AB ^ AC) : : cot. 4 A : tan.
(BAD dD CAD).
Thbobem VIII.
The sine of half the sum of any two angles of a spherical triangle^
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 aides opposite to them. And the cosine of half the sum of the
same angles^ is to the cosine of half their difference^ as the tangent of
half the side adjacent to them^ is to the tangent of half the sum of
the sides opposite^ or sin. A^A+B) : sin. i (A (/) B) : : tan. \ AB : tan.
4(BCar) AC). And cos. \ QA+B) : cos. ^ (A j)B) : : tan. ^ AB : tan. i
(BC tf) AC).
^ Coro22ary.— 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 the cosine of half their difference,
as the cotangent of half tSe angle contained between them, is to the
tangent of half the sum of the angles opposite to them,t or sin.
(AB+AC) : sin. \ (ABc/>BC) : : cot | A : tan. i (B odC) cos.
( AB X AC) : cos. i (ABudBC) : cot i A:tan. ^ (B+C).
Theorem IX.
It will be sometimes more easy in practice to compute an angle
from the three given sides by the following formulae and rules, than
by anjr of those already given : thus, suppose A, B, C, are the angles
as ben>re, and a, 6, c, the sides opposite ; then
Sin. i A = / Mn> I i (a+6 + c)— c| . sin. \ \ (a+6+c)--6 } ,j.
^ sin. h sin. c
Co«.iA= / ""' i (a+^+g) sin. \ \ (fl + 6 + c)—a \ .^.
* V sin. h sin. c
Tan A A — / «"' \ k (a+^ + c)— ^ \ - sm- \ k (fl+64-c)— g \ /Qx
lan. iA-^ sin. \ i (a + 6+c)-« } . sin. H (a + ^+O ] ^^
Rules 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 cosecants 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 ^e required angle.
II. Find the difference between half the sum of the three sides,
and the side opposite the required angle. Then to the cosecants of
the two oontainrng sides add the sines of the half sum and difference ;
half the sum of these four logarithms will be the cosine of half the*
reauired ansle.
III. To the cosecant of half the sum of the three sides add the
* Thit theorem fonnB Pnmoeition XXX. in Playfaii's Spherical Trigonometry,
where it is partly erroneous, it is also given in Mr J. VTallace^s edition of Brown*s
l^^^thfwfc Tahks. Enoneous rules and impossible triangles should always^ if vo8&v.
ble, be avoideil*-8ce the Fie&di Edition of CagnoU's TrigQiumistcy , %\Qi^^ V\^ «(A
t hegendn, § LXXXIIL
76 INTBODUCTION.
coBjecapt of half that sum diminhhed by the side oppofte the zeovAr^
ed angkj and the sinea of die same half sum dimim«hed bjr oBm ^
the aidea containing the required angle ; half the awn of these' fimr
logarithms will be the tangent of half the required angle; 8ae re-
marks annexed to Case m.^ Plane Trigonometry.
Theorem X.
^ Qiven two sides and the contained angle> to find the side oppoaite
that angle.
To twice the sine of half the contained angle^ add the ainea of the
two containing sides^ ^d from half the sum of these three k^^-
ithms subtract the sine of half the difference of the aides i the re-
mainder will be the tangent of an arc, the sine of which beimr aub*
tracted from the half sum of the three logarithnia already founds
leaves the sine of half the required side.
Thkossm XL
1/ Hie two sides and contained angle being given^ the third side may
be found in the following manner.
To twice the sine of half the contained angle add the sines of th^
two containing sides ; half the sum of these t&ee logarithms^ after xer.
jecting 20 in the index^ will be the cosine of an arc. Also fi^d half
the difference of the two containing sides.
To the sine of the sum of these two last arcs add the sine of thei<
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 80^^ thaorem. Xl«
may be employed when great accuracy is required.
Thbosbk XII.
The three angles of a spherical triangle being given^ to find the
sides.
From half the sum of the three angles subtract each of the an|dea
next the required side^ then to the cosecants of the adjacent angles
add the cosines of the two remainders ; half the sum of these ^>ur
logarithms will be the cosine of half the required side.
Thsobem XIII.
The same things being 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 logaritihnia
will be the cosine of half the required side.
Either o( these theorems may be employed^ which wiD give the
more accurate result.
Having stated the theorems on which the solutions in oblique-aii-
gled spherical triangles depend^ it is necessary to illustrate them by
exam^es which will chiefly consist of those applicable to the usual
cases 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 the Calton-hill^ in latitude
55''57'21" N., on the third of June, 1JB26, the following observa-
SFHEftlGiL T]U0QNOM£TRY.
77
tioiu or Mie so^ a lower lunb were taken in the manmiff
the time and aximuth, the barotnetcr. bebg at 20.56 in
thenBMmeler at M"" F. ?
by Watch. AUiiudes.
7*1~20' . 26° 51' 20"
2 18 . 26 59 30
3 25 ... 27 7 15
4 30 . 27 15 40
6 27 . 27 23 45
■■■■i
& 17
W««^
MeMM. 7 3 24
Olr obaerrted Z.D.
Z. D. m^ 52^.5 log. » tf
Thermometer 64° F. log.
Barometer 29.56
Thermometer 04.0 F.
35 37 30
27 7 30 Lower limb.
62 52 30
2.03692
9.96751
9.99358
9.99940
r = 106".5
Z. dirt.
Refraction
1' 46".5 log.
:62*' 62^30"
+ 1 46 .5
2.01741
True Z. D. 62 54 16 .5 of the lower limb.
Semidiameter -— 15 47 '&
IVue Z. D. 62 38 29 of the centre.
Approximate time^ June 2d^ 19^ 4"
Longitude in time add
Estimated Greenwich time
Dail jT vaziatiosi of dec.
Pw^ Dart to 17* 18"
DiBC'Jane 2d^
+ 12 West
19 16 D. L. a09503
T 4Si'' p. L. 1.36878
+ 6 11 P. L. 1.46381
22» 9 38N.
Reduced declination 22 15 49 N.
Polar distance 67 44 11
]. Now in the figure^ (p^^ 76)j there are given OP the latitude^
and eanaequently Z P die colatitude^ PK the polar distance^ and
ZK the lenith distance, the place of the sun being K near l^e
prina vertical^ as beinff most advantageous to determine the time
wiA aoenracy^ or the three sides of the triangle KPL ; to find the
angle ZPKp the time^ and the angle PZK me azimuth from the
soolheni meridian PEP. Thia^ Werefbre^ is solved by means of
theorem IX.
78
nmtoDucnoK.
Now the latitade being S6<> &T 21'% the ooUtitade isW^W
Z. D. 164 38 29
Colatitude 34 2 30 cosec. 0.261942
Polar dist. 67 44 11 cosec. 04)33647
Sum
164 36 19
sine
sine
•
•
« .sine
for apparent
■ «
Half .
First rem.
Second rem.
82 12 39
48 10
14 28 28
2^27-2'..
2
9.872208
9.397850
19.666647
9.777B84
Time firom noon 3d
4 54
12
42
Ato. time, a. h.
Time, by watdi
Watch slow
7 5
7 3
1
18
24
li
time
Again app. time
Equation of time
7 6
— 2
18
23
Meantime
Time by watch
7 2
7 3
55
24
Watch fast 29 for mean time.
2. To find the azimuth or the angle KZP, the point K being diat
in which the circles n m and ZlNcut each other^ there are given the
three sides of the triangle KPZ.
KP, or polar dist 67** 44' 11" :
VZ, or colatitude 34 2 39 cosec. 0.26]9£l
ZK, or Z. dist 62 38 29 cosec 0.051515
Sum
164 25 19
Half
IHfoence
82 12 39
14 28 28
45 7 41
2
sine
sine
COS.
B.
sin. or . .
9.995974
9.39785a
19.697281
. 9.84B640
N- 90 15 22
44 52 19
2
8. 89 44 38 E. or redeemed fitMn die
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, vhidi is the difierence betweeA'tli^
true and observed amplitude or azimuth.
SPHEBIGAL TBI60N0METBY. 79
To determine this, let the observer be supposed to look directly
from the centre of the card towards the point representing the true
azimuth ; then if the observed aximuth is to the l^ of the true asi-
mnth, the variation is auterly, but if to the right it is westerly to the
amount of the difference between them.
Thus let the true azimuth be S. W 44' 38"' E.
Observed . . . 65 24 38
Vaxiation St4 20 West
Or alxnit 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 proportional parts annexed to them are
used, the following method may be advantageously employed
for determining the time.
Rule. — ^When the latitude of the place and the declination are of
the same namey let their difference, but, if of contrary names, let their
sum, be taken. Under tfas 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, from which the apparent and mean
time may be obtained as formerly.
Latitude 55« 57' 21'' N. secant 0.251877
Declination 22 15 49 N. . secant 0.033605
42
Difi'erence 33 41 32
Zenith dist. 02 38 29
Sum 96 20 1 half 48" lO' Ol{* sine 9872206
Difference 28 56 57 half 14 28 28| sine 9.397821
233
19.555652
2»» 27*20' sine 9.777826
1 .05 P. P. 681
3
37
3]
.06
3
4
64
43
.10
34
June 2d, 19 6 17 ^ P. M.,
In the above computation the several proportional parts are set
down and summed aU together, which renders the operation some-
what more easy when our tables are emplojred. ^
Several vari«ti<ma may be made on the six things here proposed,
that mvj iflnre iw a' naefdl ekercise, which, by a referenoe to tK«
theoveBM and futeaateeady, given, will be easily perforoMd.
80
It-
' «.
•PlMNBlUUrlL
j.'l *• 1 • • ., I.f 'tl
■l ■'■ •■'.It't I."
Qi^^^ lafitiidje of ilj^ jp\ao^.'wd^^^^^ di^Uniit^fD j V ^
' t&'e £ime when tWilig^'l>^n& f^ - v' » * ». rv.
^'t what time will twilight 'bejpn md end. at Ifondw, ^l 'Jiatjj|(^
51
^j:*:
15° 14' N. ? ,, , . ,.^
In figure^ (page 70), suppose a parallel « w to the equator EQ to
be drawn ti.we distance ^15° 14^ above it, while another pmallel
to the horizon HO is drawn at the distance of 18° below it^ Uiese
two would cut one anothf^r somewhere between c and .m ip- ^,£prm-
ing the tria&ffle ZPS« in which ZP, PS, and ZS, arajp^yei) ^t^, find
the angle ZPS^ the angle between the meridian PEP and another
meridian mssing through the sun at the time he is 18° degrees below
the horijson^ Bis situation whe];i twilight begins and ends.
Z s 9J; zenltji distance 108° O'
P V or jpblar distance 7^ 46 ' cosecant 0.015534
PZ or colatitude 38 28 cosecant 0.206168
Sum ' .
221 14
sine
sine
cosine
the evepuig .. .
the nunmiiijt.
till.
Half ' .
Diffidence
• ■ ■
110 37
2 37
>
4»»58- &
2
9.971256
3.659475
ia852433
9.426216
443
Time from noon
Or at
9 56 12 in
2 4 48 in
pROBIiEAl
227
• ■ . ■
Given the right ascensions and dedinatioiis^ pt th^.longitu^Vand
ladtudiee of two ^lestial objects ; to find their angular distance. ^
In this probl^n there are given two sides and Vie contained axigle
to find its opposite side. The contained angle is the differ^ce be-
tween their right ascensions or longitudes^ and the containing sides
are the complements of the declinations or latit\idea. If the sun be
on^ of the objects^ as his latitude, is very small^ he may be supposed
to be -always in the ecliptic; then the triangle so formed wvl be
right angled if the longitudes and latitudes are used^ and the com-
putation becomes more simple. By nleans of this problem the lunar
distances in the nautical almanac are computed.
On the Ist of June^ 1828^ required the distance between thjejnoon
and « Pegasi^ at noon^ on the meridian 6f Greenwich^ the moon's right
ascension being 295° 23' 46", and declination 16° 11' 45'' S., the
star's right ascension being 22*» 5&^ 13'-85, or 344* 3' 28", and north
polar distance 75° 43' 2", or declination 14° 16^ 58" N.
344° 3' 28"— 295° 23' 46" = 48° 39' 42" the angle at the pole.
Instead, however, of following tiie operation derived from the spheri-
cal triangle, a more simple proctioal rule may be derived from' it ac-
cmsdin^ to theorem IX.
iTo Iwioe the sine of half the oontaineJ angle add the c e s i iietf of
the moon' and star's declinations^ 4aid tdke hidf the sum o^AhmB
SPHEBICAL TRIGONOMETRY.
81
tbree lofiiitfams. Fnnn this half sum subtract the sme of half the
sum of ttie declinations if they are of contrary names, or that of half
their difference if of the same name, the remainder will be the tan-
gent of an urc, the sine of which bdng subtracted from half the
flom of tlie tihree logarithms already found will give the sine of hidf
the required distance.
Biff, of R. A. 48° 30' 42"
Half
24 19 51
«ne X S
S.cos.
N. COS.
sine
tan.
sine
sine
b=19.a29804
Moon's declination
Btar's declination
16 11 45
14 16 58
0.982413
9.086364
30.198581
ftiim
30 28 43
K7I11J1 • •
19.599291 <a)
9.419717
Half
15 14 21^
56 31 18
28 27 29
2
Arc
Same arc
Half distance
10.179674
9.931316 (b)
True distance
56 54 58
Examples for Exercise.
1. Required the distance between the moon and sun on /uly 2d,
1828, at noon on the meridian of Greenwich, the longitude of the
sun being 3^ 10° 28' 44', the longitude of the moon II* IT 59^ 39",
and latitude 2° 51' 40"' N. ?
ilii#.^112° 27' 19" east of her.
2. Required the distance between the moon and sun on the 20th
January, 1828, at noon, the sun's longitude being 0" 20° 20^ 30",
that of the moon IP 17° 54' 42", and latitude 3° 24' 28" ?
SL Required the distance between the moon and « Aquilse, at
noon on the 10th of May, 1828, the right ascension of the moon be-
ing 0^ SB' 49", the declination 4° 44' W' N., the right ascension of
« Aqnilce in time, bdng 10^ 42" 25*.62, and north polar distance 81°
34'll''?
Ans,^:fJOP 54' 61'' west of her.
4. Required the distance between the moon and Aldebaran, at
nddnight on the 16th of December, the moon's R. A., being 32* 31'
30^, the decUnaticm 11° 18' 11" N., the R. A. of Aldebaran being 4|.
28- ».67, and N. P. D. 73° 50' 37".4 ?
iliM,--33° 21' 10".
Pbqblkm IV.
On finding the latitude by observation.
Tte moit simple practical method of finding the latitude, is from
the meridian altitude of a celestial body whose declination is known.
Bhoold the olnect be the sun, moon, or some of the p\«nfi\A, \\v&
aMtBfk or wmam distance d the lower or upper Wmbt^T \xaVi> ax^
82 . . INTSOBUCTIQN^. . . .
obttrfied^> and.by theapfitieation of tsevcral correetionA Afit of ithe
centre is Ubtainra. . • * ., .:• . i :i • ii ...... ■ .^w .1. ... <
iWhiir reflecting instrumentSj such as the sextant, repeatiiig cmde^
&c.tMth an artificial hatizan, are employed, the arp iseadc^ mnst^
fnliii'the principles of optics^ be halved brfore the oithcr .carrections
are applied.*
A meridian altitude of the sun, moon, or a planet taken> a!t land,
must be corrected for refraction, parallax, and semidiameter, and at
sea for ihe dSp of the horizon.t i «
Having found the true altitude, take its complement to 90",
which ffms the zenith distance, denominated north or souths accord-
ing as-we^observer is nortii or south of the object
r^owj 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.^ the same name with the greater.
j^x. L«-r^BdiQburgh Observatory, March 28th, 1825, with an arti-
ficial horiaon and one of Troughton's best sextants, the vendetr o£
which showed W, Captain Pringle Stokes, R. N. found tiie meridian
altitude^tiie sun's lower limb to be 73'' 32' 15'', the index errcnr being
+2' 26^^, the barometer standing at 29.66 inches, and Fahrenheit's
thermooaeter 56^ ; what was the latitude, employing tiie refractions
in the table in tiie nautical almanac ?
Obwjnred altitude T?"* 32^ ip"
Index eriror + 2 26
Sum . 73 34 41
Half . . . 36 47 20
Refraction to 29.66 and 56* P. — 1 15
Fwallax - . . +8
Semidiameter -j- 16 3
True altitude 37 2 16
Zenith distance 52 57 46 N.
Decimation 2 59 43 N.
liatitiide 55 57 98'
' E»^fL-^To determine from the observations of Captain BMrit-Hall.
ILN., taken June 4th and 6tii, 1822, the latitude of Siiti 'Bldil, thkt
bt estimation being about 21'' 32^' N^ and kn^itode ia&<* 16^}^ ^
7* 1" in time. • t V. .
To compute the sun's declination^ June 4th, 1833^ >-'^
Longitude in time T l"" D.L. 0.684ft8
Daily variation 6" 56"' P. L. 1^41433
Prop, part to T* 1" 2' 1".6 P. L. ■ 1.94841
Eq. to sec. diff.— -23" and 7* + 2 .4 "^^ !
Correct prop, part * 2 4 .0
Declination at noon G. 22° 24 41 .0
Sun's time dec. 22 26 45 .ON.
* 8— eatptogatkm of Table XXV.
f TsUet Jnll. and XIV. have been computed, esprawly fiov^tldf pupose Mk sea»
eombimDg the whole Jn oite^
SPHERICAL TRIGONOM£TAY. 83
To cfMnpufee the refraction^ the barometer beinf^ at 29.?^ indies,
and the thermodleter 86"" Fahrenheit, to merid. alt. I L WP.W^ jot
Z.D. PKKlog. I 0.0756
Thcr. 86*Pah. ... 0.9686
Bar. 29.75 . 9.9963
Ther.86^ 9.9984
r I'M .... a0388
Parallax 0''.2 (table 16)
- Face of the circle west.
p«.^i««o J ^«t Vernier 88^60' 0"
Obs. mend, idt san's LL 88 SO &
Son*B semidiameter -f 15 47^2
Befraction -*> I»l .
Parallwt . +• . 0.2
.'True alt. sun's centre . 89 6 61^3
90
Zenith dist 64 ta? S.
Declination 22 26 65.0 N.
>4i
Latitade with face west 21 32 36.3 N.
To compute the sun's declination, June 6th, 1822.
Longitude in tune ?** 1*" D. L. 0.53406
6 9"P.t. 1.46640
Prop, part to 7* 1", 1' 48" P. L 2.00048*
Bq. to sec. diff.— 24" and 7* + 2.5
» ' '
Correct prop, part + 1 50.5
Dee. -at noon G. 22 38 10.0
p^ii^;dee. at S. B. 22 40 0.5 N.
^ (MQmte the refraction, the barometer being 29.8 inch(M> and
the.tliei»ometer 85'' Fah., the meridian Z. D. bemg 1° 23'.5 nearly.
*/ft '» X> 23'.5 log. J926 Pardlar 0".M
Ther. 85^ log. 9.9694
Bar. 29.8 9.9971
Tliec..85° . . 0.90a5
i".44 0.1576
Face of the circle east.
T° \2d vernier 25
Obs. zenith dist. suns's /. /. 1 23 27:5 '
San's- semidiameter — 15 '47:6
Reaction — 14
Bvanax +0:5?
rrr
true tner. Z. D. . IT ^."^ ^.
9t
INXBO]>UCXION.
True mer. Z.D.
O^BcBnation
IfffiHiie, face east
face west
17 aoi8i &
23 40 0£ N.
21 33 213 N.
21 32 3&3
Mean latitude by sum 21 32 28.75
When the latitude is determined by an astronomical circle^ an ob*
servation is not supposed to be complete^ till the observer has re-
versed the circle^ by this- means Gomnining two sets of observ)itions«
with the face or graduated limb of the instruaaent ateemat^^ ai» in
this example^ towards the east and west.
San Bias, 30th May, 1822, the barometer being at 99.7ft inches,
Fahrefiheit'fr thennometer SS*, the chronometer, too fiut' for .mean
time 4*" 4*° 45*, Polaris on the meridian below the pole by dbroaome^
ter a4: 1^ S"" 41' and its true apparent N. P. D. l"" 38^ 28''«46. -. ^
taaaU
TmiefraDi
ItiivMerid.
ReSucSSTtorTscTrTCasr
CHroDOiiieteT.
East-
West
h ra 8
1 fl 6
1 7 61
1 .8 41
1 14 3
1 16 U
1 18 35
m. 8
2 36
50
5 22
7 30
6 54
6
ISfiVid.
13".27
1 .36
0.00
56.55
110 .44
192 .41
374.03
62.34
Alts.
70 3 345
3 SiJO
3350
19 56 19.0
56 lao
55 20.5
AWtaSn
:■ .\
}
19 56 Q^j^.
66:86^1
56 25.0
56 19.0.
56 lao
19 56 22.33
•Tf i
To compute the correction of altitude on account of the djuimnce
of the star ^om the meridian.
A 21° 33' 30" cosine
^ 28 21 30 cosine
Alt 19 56 22 secant
«. eStf'M log. .
9.06865ir
8.457118
0.096814
1.7««7«>
Cor. — J .77 log.^
The correction for pi|i^ IJL is in this caae insensible.
Td dempute the refraction.
ii fii I
OiMTSKft
.■ . . i
Z.D. 70^3'.6
Ther. 83 P.
Bar. 99 .78
Ther. 89
]ag.$
log.
log.
log.
a.2032fi
8.97115L
9.9944fi
r 147''.02
Or 2' 37".0»
Observed altitude
Refraction
Cdrrectioii
log.
• . 1
• •
2.18730
19° 66' 22"^ia
— 3 27.W
- 1.77
True altitude *
• •
19 53 53 .64
SPHEBICAL TRIGONOMETRY. 8Br
True akHiide below the pole 19 5d'53^.54N.
Polar difltande .... 1 38 28 .46 N.
Latitude from Polaris 21 32 22 .00 N.
from Sun . ' 21 32 28 .75
Mean 21 32 25 .37
Captain Hall makes it 21 32 29 .87
Differenoe — 1 .70
Whidi app e aiB to be occasioned by neglecting the application of
the equation of second difference in reducing the sun's oecllnation to
the place of observation.
it aeeflM unnecessary to extend our remarks farther with regard
to these observations/ more especially if the examples in the explaiuu
taanmi At table XXVIII. be consulted. If the observations are
taken at tea with a reflecting mstrument^ on the principles of Had-
ley's quadrant^ a correction must be made for the dip in addition to
these already given. . This may be taken from table XI. ; or the
t roe altitude may be still more readily found from table XIII.' or
XlV. sufficientiy correct for all the usual purposes at sea.
. J&rw 1. Mky fst^ 1825^ in longitude 64'' 25' W., the observed me-
ridiin'^titude of the sun's /. T was 48° 34' 30"^ the lenith being
nortbof the sun'^ and the height of the eye 14 feet ; what was the la-
tkude?
M»f 1st at ship, time ff" 0" Dec. Ist 15** 4' IV N.
Len^. intime 4 18 P.P. + 3 14
Or. time. May 1st 4 18 R. D. 15 7 33 N.
Observed Altitude 48° 34'.5
' Cor. to 48^^ 14 feet, and May +11.5
Tmeait. 48 46.0
g.D. . 41 14.0 N.
Dedintion . . 15;,7.6N.
'»
LatiM0 56 21 .6 N.
It ia unnecessary to push the calculations nearer than tenths of a
WUM9J M «ny observation taken at sea is, from the indistinctness of
die horiaon and the uncertainty of the horizontal refraction, unless
a dip section be used, liable to an error of at least one minute.
Examples far Exercise,
]; Oil the 1st of September, 1824, in longitude 54° W., the meri*
dii^ ahfitade of the sun's lower limb was 79^ 44' 15'' 8., the height
of t&e eye being 24 &et ; what was the latitude ?
JhliJliV 3(f.9 N.
the 1st of January, 1826, the meridian altitude of the star
Was 60* 41' S., the height of the eye being 24 feet ; what
was the latitude?
3.; Oil the 14th September, 1827, in longitude 103"* 18' £., let the
meridi^ altitude of the moon's lower limb be 51° 4' N., and the
heitfhrof the eye 20 feet ; required the latitude ?
^ Ans^lO* 48^.4 8.
80 INTBODUCTIOM, ;
4, Ottdie 39tfa SepUmber, 1827, in longituda 20° ^O* W., if the
obHTved moridian altitude of the moon'a upper limb b* 83° & N.,
and tba-heigbt-of the eye 16 feet ; required toe latitude i
^n*.— 21" 25'.7 S.
Ai the meridian altitude may, by the inter- .,
pontion <tf doodg, or other causes, be lost at
eea when a knowledge of the latitude is neces-
aar* &r the B>ftty of the ship, recourse mnit
be had. to other methods, particularly to that of Jil-
donble altitudes, and the time between them, " ""
08 being the mcst practicable.* Tliin me-;
thod reqiiirea solutions in three spherical triaji-
fles. In the triangle ZPS there are given:
'S the auD^B polar diatsnce at the time of the Acst observation.
PSf that at ue second, and the. angle S'FS roeaBured. by tli^
^ elapsed time ■ to'lwid the si^.S'8 andtbe/anfrle PS'S.t Ag^n
in the triangle ZS'S Utra'e are ^ven the zenith ^atanae^ 2^ «t 4be
time of iheltrBt observation, ZS' that at the second, and tbBrflWfe.S^S
already found to determine the angle .ZS'S.' But FS%. Mtg al-
ready ctnhputad, ZS*? mav be obtuned. Whmce there are; in the
triangleZS'P, the sides ZS', and PS.', and tho contained angle Zf*?;
to 4nd the aide ZP the oolatitude. Tha !s the T^i^lar method ^by
^hESBcal trigouometvy ; but if the polar diatuicc; PS be aiipposed.th-
remain the same, that at the midtUe time, b^^esn the obaerratioii^-
or, as ProfesBor Lax seems to think preferable, the same as at tha
tnne.af the greater altitude, and, by combining ihe eoilatiitilB of the
sever^ b^anffles in one, the ^eration becomes roore simple. Jin
order to render this method sipll more easy to practical seamen,
Seuirea proposed an approximate method by introduxnng' the ' Lati-
tude- by. accoimt, which, when properly restricted according to the
mlep of [Maakelyne or the tables of Jm, will generdly give the <ie-
sired result sufficiently correct for nautical purposes, and tiie com-
putatiops qwy ^ vc^ readily performed by the tables of Lynn. '
Whenthecommon tables are used, Mr Ivory's solution is the beet,
perticwlArly in the tbrro that Mr Riddle has given it, wliich wQ shall
adopt here.
land the spn's declination for the time of the greater altitude^ and
the true altitudes, reducing the lesa if necessary for the ahip'a run
to what it would have be^ had H been taken at the lame plttce with
the greater. This is accompUabed by observing tiie sun's bfwing
by compass, at the time of taking the leas altitude, and, finding the
angle contained between that ana the ship's course by composs, -eor-
rected for leeway if ^e makes any, in the interval between the ob-
servations. With this angle aa a coarse enter a traverse tahle^ and
the difference of latitude, answering to the distance run during jbe
elapsed time, will be tiie reduction of altitude.
If the less altitude be observed in the forenoon, the reduction o'f
altitude must be added to it, if the angle between the ship's cpnjrsc
and the sun's bearing be less than eisht points ; but if that angle' be
S eater tiian eight points, the reduction is to be subtracted from the.
)s altitude. If the less altitude be observed in the afternoon, the
■ On the salhority df m T«n diiiingBlshtd pntMical iHwiytoT,.lMai»tterl>«J|i.d
(louble sliitndea ace not of nten impoiUnce aa to gcnanllj suppoied.
f A eirde is supposed to pus through VIH P' rimlUr lo PFtP.
SPHEBtCAL TRIGONOMETRY. 87
reduction is to be 'subtracted from it^ if the angle between the
ship's course and the sun's bearing is less than eight points ; but if
greater, the reduction is to be added to the less altituoe. With the
corrected altitudes, the elapsed time, and the declination, the la-
titude at the time of the observation of the greatest altitude will be
founds which may be reduced to noon by means of the dead
reckoning.
1. Take half the interval between the observaticms, and call it
the half elapsed time.
3. To tihe sine of the half elapsed time add the sitie of the sun's
polar distance, the sum, rejecting always ten in the index^ will be
arcjlrsi^
3. To the secant of arc first add the cosine of the polar distance,
the ram will be the cosine of arc second, which will be of the same
affection or duonacter as the polar distance.
^.To the cosecant of arc nrst, add the cosine of half the sum of
die true altitudes, and the sine o£ half their diifference; the sum will
be the sine of arc third.
& Add together the secant of arc first, the sine of half the sum of
the true altitudes, the cosine of half their difference, and the secant
of are third, the sum will be the cosine of arc fourth,
61 HhB-difference of arc second and arc fourth is arcjiftk, when the
lenidi and the elevated pole are on the same side of the great^drde,
pasi iiig t itfoogh the places of the sun at the times of observation,
otherwise then: sum is arcji^h.
7. To the cosine of arc third add the cosine of arc fifkhy and the
sum will be the sine of the latitude.
Ex. l.--On the 6th of June, 1828^ in latitude 58'' N., and longitude
48^ W., by account, at 10^ 53"* 20* A. M. per watch, the altitudeof the
son's lower lunb was 52'' 20', and at P 17°" 8", the altitude of the
same limb was 52° 54', and the bearing per compass S. W. by W.
The ship's course during the elapsed time was S., the wind £.».£.,
and hourly rate of sailing 8 knots, and the ship making 1^ pts of
leo-way. Required the true latitude at the time of observaoon of
Ae greatest altitude, the height of the eye being 16 feet ?
Ship's apparent course S. or 0^
Xiee-way 1^
Ship's true course 8. by W. i W. = 1^ pts S. W.
Smrs bearing at 2d obs. S. W. by W. = 5 pts S. W.
' Confadbed angle ^
Interval between the observations = 2^ 23^ 4S^ = 2P*.4
. Distnioe run =: =:2'.4 X 8 = 19i} miles.
Nirir to course 3^ points and distance 19^.2, the cdfference of lati-
tude is 14^.84/. apd since the least altitude was observed in the
it^ara/boii^ and the.iingle between the ship's course and sun's bear-
ing is less than .ei|^t points, this reduction is subtrictive.
i*A<
* Should there be ukj doubt whether the zenith and elevated pole are on the same
- ilfc sfliw snikfrdide,pw8uur through the places of the sun, the latitude maj be com.
. patBd on bm iapporitfaw, which, being eompared with that by account, the true lati-
tads wiBL^tijLWiji|p1, ba readily diacoverad with Uttk additional tnmhU^ isn *\X wk «iA^
arc loora ana its eonne that wm require alteration.
S8
INTBODUCTION.
First observed alt.
Cor. taUe XIII.
63° 20' Second observed alt 51« 64r
+ 11 .2 +11:2
1. True alt.
53 31.2 Redaction , .
2 True ak.
-. 1441
52 60.4
1. True alt
3.
63» 31'.2
62 60.4
Sum
Difference
«. J 10^ 63" 20^
106 21 .6 ha\£B3P lO'.S = 5»> lO' 4ff'
40.8 half 20.4= 90 24
Time
Long. W.
Elapaedt. 2 23 48
H.B.T. I 11 64
App. time
Dauy variation
Prop, part
Dec. at noon or i6th
Reduced dec.
Polar dist.
on 6th at
2* 5" 20*
6' 66"
D.L.
P.L.
31
220 41' 17" N.
22 41 48 N.
67 18 12
W 53- 20- A. M.
3 12
14 6 20 A.M.
2 6 20 P. M.
1.06030
1.48320
2.64350
9.4»MMiiii. li>ll»64^ H.E.T.
9.9660W tin. 67 12 12 pol. dist. cot. 9.586422
9.454469 sin. 16 S2 S7 aic Ist sec. 9.018302 cosec.
66 15 52 arc 2d 008. 9.604784
33 22 8 arc 4. cos.
32 52 44 arc 5. cos.
Latitude 57 5 51 N. arc 6. tine
0.018382
9.903874
9.999993
0.000034
9.921763
sec arc 1
rin. 58<» 10" 48^ COS.
COS. 20 24 sin.
secSdO 42 58 sin.
3d cos*
0.545824
9.777846
7.771187
8.096457
A QQOOtUt
9.^MUM
9.994070
In this example the computation is carried to seconds^ but sudi a
degree of accuracy is imnecessary at sea.
2. On the 6th of March^ 1827^ in latitude 00^ N. by account, and
longitude 105^ E., the altitude of the sun's lower limb was obienred
to be lO^* 42" at 40^ 4'" 20^ in the forenoon, his centre bearii^ S. S. £.
by compass, and at P 32» 36" afternoon it was 21<> 8". l%e ship's
course during the elapsed time was N. W. by N., sailing at the nQe
of 9 knots pear hour, and the height of the eye 16 feet Required
the ship's latitude at the time of taking the greater altitude ?
Ans.—W 37' N.
a August 31, 1827^ in latitude 12^ 40^ S. by account and longi.
tude lOS** £. at ll"" 13~ 30* A. M., the altitude of the son's lower
limb was 66^ 9" 80"', and at 1*" 15" 12* P. M. it was es^" 0" 15^, bear-
SPHERICAL TEIGONOMETRY. 89
ins at Ae^mJtam time N. W. 4 W. Ihmnp the elapted' tk^e l^e ship
w«9aaiUng 8.W. by W. at the rate cvf 4 knots per famiH aiftd the
heijirht of the observer's eye was 98 feet. Required the latitude at
4h# tiuie of taking the first altitude ?
*'• ProbjubmVI.
Onjinding ibe Longitude.
I. BY LUNARB.
SUbe -Ae rotation of the earth about its axis is performed in a day,
the sun appears to pass over 36(r in 24 hours^ and^ consequ^iefiitly,
^'^ei*t5'''mone hour; therefore^ it is obvious^ thalt the ^^fference of
time between any two places will give the differenjce of longitude be-
tween those places*
A variety of methods have been proposed Tar, dfteiyttjnjgtig the
i^l^tiide of a place^ but almost all of them depend upon one gene-
til prMcMe^ me comparison of the relative times unoer tpo.Sfer.-
ent meridians ; so that> if the time on two different mciridfanf be
kno^n^ \ht difference of thesis times turned into degrec^^ at tn.e. rate
of IS^ "to an hour^ will give the difference of longitude between itKeso
meridians. .,
An the sun apparently moves from the east towards the West^
it is evident^ that all places lying to the eastward oj^ any meridian
will hare noon^ or any other nour^ sooner^ or if westward^ 1a^'» b}'
the precise time the suit takes to pass from the meridian of the one
place to that of the other, 'tlence^ if the time on the meri^an of
Greenwich^ the place from which our longitude is reckoned^ and
that of any other place at the same instant oe kiK^wo^ tl^e longifcfi^c
of the latter place from Greenwich is also known, by turning die
d^jflTei^ience of time into d^rees, at the rate of 15"^ to an 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, planetiB, and fixed stars near the ecliptic, as
jthfi^^fff the moon>; the (uumal motion of that object being at a mean
MCe' Aout 19^ If. ' Hence, her distance from these bodies is con-
tii^lfldly chaqging in proportion to the time, and an error of 2'' in
tliie distance between the moon and any of these bodies will produce
■ I 'tattjor of about 1' only of longitude. Of all the various modes,
iMjK/'which have been proposed to determine the longitude at sea,
ittj^fKrobable the method by lunar observations will continue to be
the most practicable. It appears also from the numerous observa-
t><9!9llJ4|f9^ made by teveral of our moat distitiguiahed navigators,
that a aenes of lunars taken at land with good InatrHmernts, wt)1,
,w]tieii,{[mt nicety in- the remiisite obaervationa knd calculations is
'^ tjp^'giva the langitoae with singular accuranr.
^jtflTfuiiente generally emploved are a good curonometer for
"ig otaierrtuana taken at dmerent limes with one another,
I qimjcanta for obtaining the altitudesy and a sextant or ve-
.fjrde. fop taking tibe distance. Thcee inatrumenta ere all
4^ our uauid treatises on navigation and nautical astrd-
iiQiny*
kXCi4Afll«M!ltlV-^ter .be at eauffitient< ^distance tern thrmerfdisn at
'tij^MV tlie disiUnM, ^^ tni^ altftad^ of eidler itf ifteae
^_ liMH&'liO. eam|iutt die apparent time at^ the tfeo^v «cvdL^3toMk
c^pared with the Greenwich time, derived from tY\e\uwaT ^\?\».wcft.
90
i2JTRODUCTION.
will give the longitude. The same thing may be obtained from the
moon's altitude^ but less readily, as her right ascension 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.*
Ex. 1.— September 24, 1827, in latitude 48° 50' south, and longi-
tude by account 120° west, at 8** 18" 30* A. M., the following obser-
vations were made to obtain the true longitude ; the height of the
eyes of the observers being 30 feet above the surface of tne sea, the
angular instruments 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", the altitude of the sun's lower limb 22° 4' 15",
and the altitude of the moon's upper limb 6° 6' 0".
Time at ship 23^ 20^ 18"^ 30* To this time by estimation.
the sun's semidiameter is 15' 59"
the moon's 16 2
augmentation " 2
hor. parallax 58 49
reduction to lat. 49° S. — 7
Longitude m time 8
Ext Green, time 24 4 18 30
Obs. dist. n. /. 44° 33' 45"
Sun's semidia. + 15 59
Moon's semidia. + 16 2
Augmentation + 2
App. cent dist. 45 5 48
Alt sun's /./. 22° 4' 15"
reduced parallax
58 42
alt. moon's u. L6° 6'
Z. D. 67 55 45 log. $ 2.15567 83 54 1. ^2.70124
Thermometer 66°.0 F. 9.99104
Barometer 29.4 E. 9.99123
Thermometer 60 .0 F 9.99957
r = 137".25
Or 2' 17''.25
For tile sun
9.98184
or =
— 0.1 04 X (60— 50) = 104 X 10
+ 0.15 X (30— 29.4) = 15 X -6
2.13751 r' = 482".3
8'2"3
— 1 .04
+ 0.09
9.98184
2.68306
True refraction for the moon
= 8 1 .35
Alt sun's I /. 22° 4' 15"
Dip to 30 feet— 5 27
21 58 48
Semidiameter + 15 59
Alt. moon's u. I.
Semidiameter augm.
App. alt. 22 14 47 App. altitude
6^ 6' 0"
— 5 27
6 33
— 16 4
5 44 29
* The neceswiry computations are leadily and very accurately performed, according
to the mlet ctf spneriotl trigonometry from the tables contained in diia work. There
are Mvetal collections of tables, such as those of Mendosa Rfos, Laa^ LynD^ apd
Thomson, which, for general practice at sea, by abating something of rigorous accun-
ey, vender the calculations more shnple. Some of them, however, are rather 'bulky
and expensive.
SPHEBICAL TRIGONOMETRY.
91
App. alt. 2Sr» 14^ 47
Refraction — 2 17
App. altitude
Renraction
Parallax
+
Sun's T. alt. 22 12 33
Alt. moon's tt. /. &* 6' 0"
Red. par. 58 42
Parallax in alt.
Moon's true alt.
Par. in alt.
58 22
5« 44' 29
— 8 1
5 36 28
+ 58 22
6 34 50
Secant 0.6024?
P. L. 0.48663
P.L. 0.48010
The reduction of the apparent to the true distance is effected bv
the solution of two sphenod triangle. First the angle at the zenith
is found from the triangle formed by the apparent zenith distances
and apparent distance. Next the true distance is computed from the
angle at the zenith and the true zenith distances^ and these two may
be combined in the following manner.
App. dist
App. alt.
App. alt. )
46° 5' 48"
22 14 47
5 44 29
secant
secant
cosine
cosine
cosine
cosine
• •
cosine
sine
sine
sine
0^42' 9"
1 37 15
P.L.
P.L.
PL.
0.033593
0.002188
Sum
73 5 4
Eklf
Diffecence
36 32 32
8 33 16
9.904949
9.995141
True alt ©
True alt )
22 12 33
6 34 60
9.966522
9.997129
Sum
Half
Arc
28 47 23
14 23 42
27 1 66
19.899515
9.949757
Sum
Difference
41 25 37
12 38 13
22 21 54
2
9.820638
9.339993
19.160631
IDist
1
9.580316
True jdist.
Di8t.«ta^
6
44 43 30
44 1 21
45 38 36
P 18" 1'
3
0.63048
026738
Time past 3"
Preced. time
0.36310
Appros. time 4 18 1
n
INTRODUCTION.
Dist. at noon 42° 24' 13''
3^44 1 21
6 45 38 86
9 47 15 58
1*< Dif.
P37' «'
1 37 16
1 37 22
M
Meatt.
•- ■ *.
r
7
+7
7/
r:
■•..^
(V 1»» 1') X 4 = 6*» 12" 4', to which and second difference 7" we
get (from table XXVII.) 1" of motion^ that at a mean rate gfVcAli
Mtonds of time. ^
This^ from the en>lanation of the table^ because the first differ-
enoes are all increasing, must be subtracted from the approximate
distance, and consequently added to the approximate time.
To the approximate thne 4*" IS"" 1*
Add cor. from sec. diff. -f* ^ *
True T. at Greenwich 2* 4 ]8 3
Ootmutadon of the tim^ derived from th^ figure in page ^0, *
Theorem IX., page 75 alter the examples in page 78*
Sun's T. alt.
Son's pol. dist.
iMUatttde
Sum
22" }2f 33"
89 40 33 cosecant
48 50 secant
160 43 6
0.000007
ai8l60B
Half
iMir.
80 21 33 cosine
58 9 sine
I Time from noon V 50" 4ff^ sine
2
Time from noon
3 41 37 .2
24
9.22394}
9.92912^
19.334686
9.667343
Apj>.time 23^20 18 23i>
App. T. Green. 24 . 4 18 3 .0
Lon. in time
Lynn gives
Difference
7 59 40.0=119^65' 0"We8t.
= 120 4 46"W. i^in his cc^ious
■ \ nautical tables.
9 45
Ex. 2.— On September 12th, 1823, in latitude 26^ 30' Ijf., laQ|n.
tilde by account 24'' 30' W. at 5^ 34"" P. M. by wat<ih, the akitiKle
* The eauation of second difference happens to be small in this example. It may
amount to o seconds of distance, 12 seconds of time, or 5' of longitude in some cases.
The correction of seomd difference is taken from the usual tables and its effecta esti-
mated according to the moon's mean motion. It is perfomMBd more cooectly, however,
by means of Tables 3d and 4th immediately following this article, whidi have been com-
puted by the author expressly for this purpose.
SPH£BICAL TBIOONOM£TKY.
of the sun^s lower limb was 7^ 37^ that of the moon's lower limb
WM 95^ 35", the distance of their nearest Ihnbs dS*" 19' 58^ the ban>.
meter being 30.28 inches, and the thermometer ^2^.4 Fdirenheit, the
height of the eye being 25 feet ; what was the longitude ?
Time per watcn 5^ 34"
Longitude 24'' W. in tune + i 26
Estimated Greenwich time
Ifooa^s semidiameter at noon
GoRectkm for Ghreenwich time
Augmentation
True semidiameter
Alt of sun's /. L r 37'
7 10 P. M.
1 4' 62" parallax 54' 31 "
— 2 correc.for7^10"— 5
14 50
+ 8
equatorial par. 54 26
reduc. for lat. — 2
14 58 red. hor. par. 54 24
Moon's 35'' 35'
Z. D^
82
23 log. I 2.61313 Z. D. 54 25 log. I 1 90970
PJP.3' 260 P.P. 6' 133
4 log. 9.98020 9.98020
28 log. 0.00289 0.00289
114 114
Thermomaer 72 4 log. 9.99902 9.99902
Thermometer 73
Barometer 30
r
Or
^Mf»X +22.4:=:
+-09X+-28 =
398'M 2.59898 r 78''.3 log. 1.89428
6' 38'^1 or 1' 18".3
— 1 .3 alt moon's /. /. 35'' 35' secant 0.06977
+ .0 red. hor. par. 54' 24'' P. L. 0.51967
Alt sun's L L
lip to 25 ft
6 36 .8 parallax in alt. 44 14 P. L. 0.60944
37' 8" alt of moon's /. /.
4 68 dip. to 26 feet
7 32 2 semicfiameter
Semidiameter + 15 56 app. altitude
refraction
parallax
moon's true alt.
App. alt
Reiractl(
». alt
[cm
Parallax
7 47 58
».
6 37
+
9
1.
0.60944
36"
35'
0"
—
4
58
38
30
2;
+
14 68
36
45
mm^
1
18
+
44 14
36 27 56
Son's true alt 7-41 SO
Observed distance
Bodi's Semidiameter
' Moon's semidiameter
App. central dist.
95<' 19' 58'^
+ 14 58
-f 15 56
95 50 52
* I »
M
. INTAOUUCTION.
App. ditt
Sun's app; alt.
Mooq's app. alt.
95°5a'52"
7 47 58 .
35 45 1
secant
.secant
Sum
139 23 50
Half
Difference
Sun's true alt.
Moon's true alt.
69 41 55
26 8 57
7 41 301
36 27 56/
cosine
cosine
cosine
cosine
oiNMoaa
0.090672
Sum
Half
Arc
Sum
Differepce
44 9 26
22 4 43
56 14 50
78 19 33
34 10 7
cosme
sine
sine
Ha]f (list.
True dist.
True distance
Dist. at 6^
9
Time past 6^
Time of first distance
47 52 13
2
95 44 26
95° 44' 26"
95^ 8 17
96 30 13
sine
9.540277
9.953107
9.996075^
9.905378
19.489539
9.744770
9.990922
9.749450
19.740372
9.870186
0°36' 9" P.L.
1 21 56 P.L.
V 19- 25' P.L.
6
0.69716
0.34181
0.35535
Approximate app. time at Green. 7 19 25
To find the corection for second difference.
1*/ Diff.
P22' 3"
1 21 56
1 21 49
Mean,
-r
— 7
r
MDiff.
Dist. at 9^ 93° 46' 14'
6 95 8 17
9 96 30 13
12 97 52 2
To the approximate time (P 20") x 4^ or 5*" 20"> and the mean
second difference — 7'^ the equation from Table XXVII. is 0^.9 or
about l'',' which^ since the second difference is negative^ oueht to be
added to the proportional part of the distance computed by even
proportion for the approximate time^ and consequently it muat be'
subtracted from the approximate time, or in general this correction
for the time must be applied with a contrary sign to that which is
employed when correcting an arc^ or with the same sign as that of
the second difference.
Now 1° 21' 56" : 1" : : 3** : 2^ of time nearly.
Or this operation may be performed by proportional logarithms^
thus.
Equation of sec. diffl 1" P.L. 3.03342
Variation in 3^ P 21' 56" P.L. 0.04181
Equation of 2d diff. 2" P.L.
2.69161
SPHERICM. TBIOOKOMETRY.
»6
From approximate apparent time
Subtnct equation now found
T Id" 86'
— 2
True apparent time at Greenwich 7 19 23
To find the apparent time at the place of observation.
The reduced declination is found as in the explanation of Table
IX. and XXVII., then
I^tude 26» 30" (y N. secant 0.048200
Declination 4 17 7 N. secant 0.001214
Difference 22 12 53
Zenithdi8t82 18 30
Sum 104 31 23 half 52 15 42
Difference 60 5 37 half 30 2 48
sine
sine
App. time
Greenwich time
2»»47" 4"
2
5 34 8
7 19 23
sine
9.808075
9.699582
19.647080
9.823540
Longitude in time ] 45 15 W. = 26<' 18' 45'' West
Or about two miles less than Mr £. Riddle makes it in his trea-.
tise on navigation, a very useful work, combining theory with prac-
tice, a method too much neglected in the present plan of nautical
instruction.
Ex. 3.— On the 14th of June, 1827, in latitude 28° 31' 10" N.,
and longitude 144° W. by account, at about 20^ 32", the distance
between the sun and moon was observed to be 97° 22^ 40" ; when
the altitude of the sun's lower limb was 44° 36^ 40", the altitude of
the moon's upper limb was 35° 38' 20", the height of the eye being
20 feet ; required the longitude, the barometer being at 29. w inches,
and Fahrenheit's thermometer at 68°.
Estimated time, June 14th, 20^ 32*"
Longitude 144° W. in time + 9 36
Approxiinate Greenwich time, June 15th, 5 8.
Tothia time moon's semidiameter is 15' 37'' hor. par. 57' VS!'
Augmentation to 36° alt . + 9 red. to lat. 284"" . — 2
Correct 'semidiameter
Alt.of8un'8/./.44°36'.7
15 46 cor. hor. par.
moon'8«./.35°^.3
57 16
Zenith dist. 45 23.31o.^l.77198Z.D.
Thermometer 68 9.98401
BihiAieCer 29 .7 9.99563
ThetnkMEneter 68 9.99922
54 21 .7 lo. ^ 1.90987
9.98401
9.9956B
9.60932
r56".3
1.75084, rrir'.4
\««K1^
INTMODUeriON.
Sun's iemidiameter 15' 46''
Parallax in alt.
6
moon's alt. SS"" 3» sec. a06M4
hor. par. 57' 16" P X. 0.40737
Alt. sun^ /. /.
Dip. to 20 feet
Semidiameter
App.aJt,
Refraction
Parallax
44° 36' 40"
— 4 26
+ 16 46
44
48
-«.
56
+
6
par. in alt. 46 32
moon's alt. u. L
• • •
semidiameter
app. alt.
refraction
parallax
P.L. Q.58741
35"^ 38' 20"
— 4 26
_ 15 40
35 18 9^
- 1 ml
+ 46 «'
36 a 22
97" 22' 40"
+ 15 46
+ 15 46 .
Appar^Uti central distance 97 54 12
Now to compute the correction of the oblique semidiameters^ by
l)i: Y^Vig's method, there are given.
d = 97'' which by table I. gives A =
« = 45
#' = 36
True alt. 44 47 10 true alt.
Observed distance of nearest limbs
Sun's semidiameter
Moon's semidiameter
178
s 44
k—A'=i 53
= 924
= 84
8
i - <k
90
14
Aft these give in Table II. 1'' for the sun and l'^' for the moon/tl^
2^ in all^ it is necessary to subtract them from the apparent or nvtr
tme distance when they are so small.
97° 54' 12 ''
44 48 secant
35 18 8 secant
Apparent distance
Sun's app. altitude
Moon's app. altitude
Sum
•
•
•
•
;ude
•
178 20
Half
Slfference
Sun's true alt.
Moon's true altil
89 10
8 54 2
44 47 10
36 3 22
cosine
conne
cosine
cosine
Sum
80 50 32
Half
Arc
40 25 16
82 30 Q
cosine
XHfcr^nce
122 55 16
42 4 44
sine
sine
0.088248
8iS406i7
9.907648
■■■«?
9.n0Bte'
^i.Jli' I-
19.750153
SPBERICAL T&IGONOMETRY.
Halfdist.
48° 35' aSi" sine
2
19.750158
9.876076
97 11 7
Cor* for oblique semidia. — 2
Truedist
I>i8t.at
Bitt.at
97 11 6
ff» 97 18 52
9^95 47
0^ T 47 ' PX,. 1«36411
1 31 52 PX. 0.29211
0«»15-I6' P.L. 1.07200
6
Dirt, at S** 98** 51' 7"
6 97 18 52
9 95 47
12 94 15 31
6 15 15
l** 32' 15"
1 31 52
1 31 29
23" Mean.
23 —23"
To 15" and 23" the equation of second difference is 1'^ whichj
for a variation of P 32' nearly, gives 2* of time to be subtracted^
whence the true time is 6^ 15'' 13" of the 15th of June> or 30^ 15°* 19
after the noon of the 14th.
True altitude
Pdar distance
Latitude
To compute the time.
44" 47' 10"
66 41 10
28 31 90
cosecant
secant
cosine
sine
sine
0.086993
0<M;6193
omn
139 59 40
Wf . . .
Sifirence
69 69 60
25 12 40
P40"35'.3
2
9.534111
9.629364
19.256600
9.628330
100
Time from noon
3 21 10.6
24
821
199
; • • I
Jipiktime J4th
i^fpi fiaw^at Greenwidi
20 38 49.4
30 15 13 X)
oe.
•*■
9 36 23.6 =144 6^ W.
IjfBgHude
4 On the 29th of March, 1826, in latitude b6^ 12' S., and longi.
taBTby account 97'' W. at about 7* 20" P. M., the observed distance,
bt^^fHwn. die moon's nearest limb and the star Fomalhaut, was, from
ameah of ^ve sets of observations, 61^ 56' 30'-; the observed altitude
^C^iOMKm's lower limb was 32° 4'; the observed altitude of the
a 'If l& ; the barometer being S9.3 inches, the thermometer 42^ F.,
I'tiie height of the eye 20 feet : what was tiie true longitude ?
98 INTBODUCTION.
MA^tiHae. T ^ moon's equatorial hor. par. 68" 14'' sun's 9"'
LoM. In time 6 38 reduction for lat. 56^ 8. — 8
' t • \ *■■ ■ t
Est O. time 13 80 reduced hor. pur. 58 6
moon's semidiameter 15 52
augmentation to 32^ + 9
augm. semidiameter 16 1 - '
Now to correct the obliaue semidiameter by Dr Young's meohxl
lirom Tables I. and II. we nave
i ^ ee^ gives A= 5 in Table I.
y = 32
^.
100
A r= 50
= 81
A-^= 44
^ 84
8um
Observed distance
Itoan -8 aug. semidiameter
•
170 give Cor. -^ in table II.
+ 16 1
App. central distance
Alt. of star 6^10'
alt.
of
as IS 31
moon's /. /. 3S^ 4'
Z. D. 83 44 log. I aeonO, Z. D. 57 56 log. 4 1.96844
Thermometer 42^"^ 0.00730
Bttometer 29.2 in 9.98826
Thermometer 43^ 0.06034
9:99590 9.99590 9J90CHW
r' r= 48S'^4 2.08700 r=:r32''log, UWH
Or =» 6.4 : . V
--O^M X —8= + .8 Moon's alt. 32<> 4' secant 0.07)90
0^.14 X -8 = + .1 Hor. par. 58' 6" PX. O.^llO
.lif =87 .a Par. in alt. 49^ 14'' PJL 0JSM»
Alt. of star 6« 16' 0" alt of moon's II. 32* 4' 0«
XBp:to20feet — 4 26 dip. to 20 feet — 4 2K
Apjp. alt. 6 11 34 31 59 94
Imaction .— 8 7 semidiameter 4- 16 .1
•■ • 1 » * <■*.* I
True alt. of Star 6 3 27 app. alt centre 3ff IS^'Sft
retraction .— J[ 3)
par. in alt. + 49't4
•MMH
true alt centre . 33 ^J
• . I •. . . J .
•-'i
* t.
SPHEEICAL TRIGONOMETRY.
App. dist 6»' IS" 31''
Starts app. alt 6 11 34
Moon's app. alt. 32 15 35
Sum 100 39 40
Half 50 19 50
Die 11 52 41
Star's true alt. 6 3 27
Moon's t. alt. 83 3 17
secant
secant
cosme
cosine
cosine
cosine
Sum
Half
Are
Sum
Diff.
39 6 44
19 33 22
38 5 49
57 39 11
18 32 27
cosine
sine
sine
31 13 ^ sine
2
0.t)03B4S
0.072816
9.805067
9.990600
9M7MB
9.923322
r
19.791915
9.895957
9.920766
9.502400
19.429166
9.714583
True dist.
Dist at
62 96 la
12^63 10 41
15 61 41 45
0* 44^28"
1 28 56
P.L.
P.L.
P30"0^
12
P.L.
0.60724
0.30621
0.30103
13 30 at Greenwich.
— 42"
— 45
~43".6 or — 44" nearly
Precedii^ hour
Amroximate app. time
64® 4w 19 .lo oMi/ QfH/
61 41 45 J ~ u
60 13 86 * ^ ^*
Now to approximate time 1** 30*^ and second difference <«— 44"^^ the
equation of second diVerence is 5".5> to which and variation 1"* 29'
nearly inJ3 hours, the final eouation in lime is about 11* to be sub-
tracted.^ ' Whence from 19" 30" this equatiea of II* beinj^ mbtracted^
t!he true tppaient time is 13^ 29" 49* at Ckeenwich.
To compiute the apparent time at ship.
Star's true alt 6^ 3' 26"
Pdardiit. 59 27 40 cosecant 0.064853
Udtadp; 56 12 aecant 0.254694
P^l
Ipqr*': . 60 51 33 cosine 9.687492
iM - . 54 46 7 sine 9.912309
19.919348
9.959674
54
lai 43 6
60 61 88
54 48 7
cosine
sine
4" 22- 46i*
a
sine
iist. £. 8 45 31
100
INTBODUCTION.
Star's merid. distance E. ff" 45*° 31'
Star's a A. 22 48 1
R. A of merid.
Sun's R. A.
Ai^ time at ship
App* time at Green.
Long, in time
Wimout Eq. 2d diffl
31
24
33 32
3Q 26
7
13
1 6
29 48
6 28 43
6 28 54
97^ IC 45' W.
97 13 30 W.
EiTor . 3 46 W.
TABLE I.
OORREOTION FOB THE OBLIQUB SEMI-DIAMBTBR.
For Argument A»
For h h — * d For A k — * d
« A
89 924
88 954
87 972
86 984
85 994
84 2
83 9
82 14
81 19
80 24
79 28
78 32
77 36
76 38
76 41
74 44
73 47
72 49
71 51
70 53
11 72
12 68
13 65
14
15 59
16 56
17 58
18 51
19 49
20 47
69 55
68 57
67 59
66 61
65 63
64 64
63 66
62 67
61 69
60 70
21 45
22 43
23 41
24 39
25 37
26 36
27 34
28 33
29 31
30 30
« A
31 29
32 28
33 26
34 25
35 24
36 23
37 22
38 21
39 20
40 19
For A A — s d
49
48
47
46
45
44
43
42
41
40
82
83
83
84
85
86
86
87
88
88
41 18
42 17
43 17
44 16
45 15
46 14
47 14
48 13
49 12
50 12
39 89 51 11
38 90 52 10
37 90 53 10
36 91 54 9
35 91 55 9
34 92 56 8
33 92 57 7
32 93 58 7
31 93 59 7
30 94 60 6
28 95
27 95
26 95
25 96
24 96
23 96
22 97
21 97
20 97
62
63
54
65
66
67
68
69
70
5
6
6
4
4
4
3
3
3
19 98
18 98
17 98
16 98
15 98
14 99
13 99
12 99
11 99
10 99
71
72
73
74
2
a
2
2
75 2
76
77
78
79
80
1
1
1
1
1
9 99
8 100
7 100
6 100
5 100
100
100
100
100
4
3
2
1
81 1
82
83
84
85
46
87
88
89
100 90
2.
SPHERICAL TRIOONOMETllY.
TABLE II.
CORRECTION FOB THE OBLIQUE SEHI-DIAHETKR.
DIMINUTION OP THB BBMI-DIAHBTBB.
Argument A (h) + A (A— *)+A((0.
Altilude. j
■umofA
5'
6-
7-
8»
9-
10-
n.
!Z
14'
16^
W
20°
30°
45^
0"
23"
19"
TF
11"
~r
i"
6"
5"
T
3"
3"
2"
~V'
I"
20
24
18
14
11
9
7
6
5
4
3
2
2
<t
40
23
17
13
10
8
7
6
5
4
3
2
2
60
21
16
12
9
8
6
5
5
3
3
2
2
70
20
15
12
9
8
6
5
5
3
3
2
2
80
19
14
11
8
7
6
5
4
3
2
2
2
90
17
13
10
8
7
6
5
4
3
2
2
2
100
16
12
9
7
6
6
4
4
3
2
2
1
no
14
10
8
6
5
4
3
3
2
2
1
120
11
9
7
5
4
3
2
2
2
1
1
130
9
7
5
4
3
3
2
2
I
1
]
135
7
6
4
3
2
2
2
I
1
1
140
6
6
4
3
2
2
1
1
1
1
141
6
4
3
2
2
1
1
1
1
150
3
3
2
2
1
1
1
1
155
3
2
2
1
1
i
1
160
1
1
1
fl
n
170
178
1
1
1
fl
180
2
1
1
1
1
1
183
3
2
2
1
1
1
1
184
4
3
2
2
I
1
1
1
1
186
5
4
3
2
2
2
1
1
1
188
7
6
4
3
a
2
2
1
1
__
190
9
7
5
4
3
3
2
2
1
191
10
8
6
4
4
3
3
2
2
102
11
9
7
5
4
4
3
3
2
2
193
12
9
7
5
5
4
3
3
2
2
2
194
14
10
8
6
5
4
4
3
2
2
2
2
195
15
11
9
6
6
5
4
4
3
2
2
2
196
17
13
10
7
6
(1
5
4
3
3
S
3
197
19
14
11
8
7
6
5
5
3
3
2
2
198
21
16
12
9
8
7
6
5
3
3
3
199
23
17
13
10
8
8
6
5
4
200
25
19
14
11
9
-
-
—
-
—
—
^
—
—
Alt.^..
5°
"?~
?•
3«
9=
10
11=, 12
14=' 16" 18°
20'
30''
45<
INTRODUCnOK.
TABLE III.
Equations ok Second Diffbbbncb por Three Hours.
°- yuvuvuvuui.» :i:i;ui;ui:i;il
SPHEBICAL TRIGONOMETRY. 103
In the practice of lunars four persons are frequently employed in
making the observations, the first to take the distance^ the second
to take the altitude of the siin or star^ the third to take the altitude
of the moon^ and the fourth to write down the observations. One
person^ however^ may make the whole himself^ according to the
following method^ which was obligingly communicated by that dis-
tinguished practical navigator Captain Basil Hall. Speaking of his
own practice^ he says^ — '^ I always take all my altitudes and dis-
tances with the same instrument. First the altitudes of the sun^
then those of the moon^ then several distances ; next the altitudes of
die moon, then those of the sun, and interpolating by proportional
logarithms for the altitudes at the mean time of the distances.* At
night I never take an altitude^ unless it be about twilight^ when it
can be done with accuracy and ease."
" The method which I use to connect lunars and chron(»neters is
not very general, but infinitely the best, and ought to be universally
adopted, as it renders all allowance for the distance run in the in-
terval of little or no consequence."
'^ The use of lunars at sea I conceive is, in a great degree, to
check the chronometers : the method by lunars being infallible,
though not very nice ; that by chronometers being fallible, but as
nice as possible. So tJiat a number of lunars are necessary to check
a chronometer, and the object is to bring the whole of such lunars
to bear rigorously on the chronometer without making use of the
logboard."
" This will be best illustrated by an example. At noon, or any
other hour during the day most convenient for taking a lunar, I ob-
serve a set, or half 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, frora which I 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-*
aible, 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
toe chronometer on apparent time at that meridian, and this same
error, corrected for rate during the interval, I apply to each of the
different times by the chronometer when the lunars were taken. By
this means I get the apparent times due to the meridian, on whicn
the a)>«olute time sights were taken, with as much accuracy as if
the whole, lunars and all, had been taken at that fixed meridian.
The distances give the several times at Greenwich, and thus they all
concur in settling the difierence of time, between the first meridian
and that chosen for taking the time, with a view of seeing what
loDgitade the chronometer gives. Hence, if there had been an un-
seen eiurent of some miles an hour of which no account could
pomriMy be taken, still the result would not be vitiated thereby,
bat all the lunars would be found to contribute to the same end,
thus making, according to Dr WoUaston's simile, the moon serve the
purpose of a great Greenwich clock in the heavens. After having
This is si. *■' r to the method given in Noric's NaviRalion.
104 INTRODUCTION.
determined the true longitude and error of the chronometers when
within a few days sail of the land^ I run the remainder of the voyage>
in a great degree, by the chronometers alone."
On finding the Longitude,
II. BY CHRONOMBTBR8.
The foregoing method of finding the longitude by lunars is very
valuable at sea, on account of the frequent opportunities which oc-
cur for observation. About the time of new moon, and in unsteady
weather, the necessary observations for the practice of this method
cannot be obtained, and 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 merits great confi-
dence. If the rate of a chronometer be determined on shore, or
rather perhaps on board in the situation it is intended to occupy
during the voyage, where the various causes which act upon it, and
are likely to alter its rate, are in operation, it is likely this rate will
remain pretty uniform for some time, and the amount of the gain or
loss, bemg allowed for on the time indicated by it at any future
period, the true time may be obtained at the meridian of the place
where its rate and original error was determined, with as much accu-
racy as if it had been adjusted to go accurately to mean solar time
on that meridian. Hence, it is obvious, that if the original errbr^
and the gain or loss in 24 hours, called the daily rate, of a chrono-
meter, be known, on any meridian, such for example as that of
Greenwich ; by making proper allowance for these, the mean time at
Greenwich may be readily known to such a degree of accuracy as
the going of the chronometer will warrant.
It is now only necessary to find the apparent time at ship, by an
altitude of any celestial body properly situated, by some of die me-
diods already given ; to which the equation of time being tidcen
from the Nautical Almanac and properly applied, the result will be
the mean time to be compared with that at the given meridian to
show the longitude of the ship.
The rate of a chronometer is readily obtained, by observing dailyv
if possible, the altitude of one or more celestial objects near toe
prmie vertical, from which the mean time may be accurately deter-
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 die last
observation, called the original error, will become known.*
Ex, 1.— Near Falmouth, in latitude 50* 8' 48" N., and longitude
20" icy W., at about Iff* 47" 20^, the following altitudes of the sun's
lower limb were taken, widi an artificial horizon, in order to ascer-
tain the daily rate of a chronometer previously set to Ghreenwich
time. The observations were made with a sextant of which the in«
dex error was+ 1^ ^', the barometer 29.6 inches, and the diermome-
ter ^eP Fahrenheit.
* These would be more accurately performed on shore by using an artiiclal horiBMi
and the method of equal altitudes. In this case a pocket chronometa should b« am*
ployed, to be compared with those on boaxd, which ought to be as numerous M pos-
sible.
SPHEBICAL TEIOONOHETRY.
7MMf if CArMNMMter. DmMe AU. alt. l^"" 3'
106
W Iff- 36-
12 45
14 58
87^ 48' 46"
88 4 80
38 20 15
8 18 3
13 30
Z. D. 70 571<«. 12^150
tlier. Se^ log. 9.9940D
bar. 29.6 log. 9.9941,7
ther. 66'' log. 9.99974
Mteana 19 12 43 38 4 30
I.E.+ 1 30
r 163".4
= 2' 43".4
son's parallax 8'M
2iaa8&
2.38 6
19
Time at Faknouth
iKntgiMle Ui time
Qfi^enwich time
Ilkuly variation
Pm^ part to 19*. 7i"
Det- ft noon; Hay Ist
- ' ■
Salf i il^uccd dediQation
OtMshrved alt. sun's /. /.
3
Iff* 47" 20*
+ 20 10
19 7 30
17' 68"
14 19
15 8 49
15 23 8
D.L.
P.L.
P.L.
o.oaesi
1.00080
Poallak
Txioe fOtftude
Siobmb tme dec.-
l4Btitude
cHnaTCBce
Zenith dist.
Sum.
15*> 23' 8" N,
60 8 48 N.
secant
secant
19« 3' 0"
+ 15 53^
— 2 4B.4
+ 9tl
19 16 18 '
0.015860
0.1989A)
34 45 40
70 43 42
105 29 22
35 68 2
half 53° 44^ 41" sine 9.900684
halfl7 S9 1 skie 9.48Mii9
1*ne
»'36-33'.8
2
5 12 47-6
24
19.599593
sine 9.799796
Apparent time at Falm. 18 47 12.4
Equation of time •— 3 10 .9
IfcBK time at Falm.
Thne br dnronometer
Cbroiunietef for Falm.
18 44 1 .5
19 \i 46 .0
38 44.6 fast
^o^
106
INTEODUCTIOM.
^ Again, on the 11th of May^ 1834^ the altitude of the aim'a knrer
limb taken with the same instruments as befcnre^ the index enor be*
5mjf constant^ was 19'* 9' 60", when the chronometer showed Iff*
67^ 59. This gives the mean time at Fahnouth 19* 20^-23^.5, and
iihe error of the chronometer for the meridian of the place 27" 32'.5.
Whence, on May Ist, the error was 28" 44^^
11th . . 27 32^
Hie loss in ten days is . * . 1 12
Or in one day it is 7^
Hence 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 S2^ 36' N., and longitude by
account 16° 40' W., the altitude of the sun's lower limb at sea was
37^ 24', when the chronometer showed 6** 12" 24'.6, the height of
the eye being 20 feet ; required the longitude ?
Time per. watch 5** 12" 24'.6 Daily rate •
Original error — 28 44.5
Loss in 11 days 5'
4 43 40.0
+ 1 21
Loss in 11^ days
Or
Greenwich M. time 4 45 1
Alt. sun's /. /. 39« 26^
Cor. table XIIL + 10
dec.
cor. for 5*»
True alt.
IVneaU.
PoLdist
Latitude
39 36
39** 36'
69 32
32 36
cor. dec.
cosecant
secant
Sum
141 44
'
Half
Bifil
9
70 {>2
31 16
cosine
sine
— • ■
1»» 50" 39*
2
sine
App. time
£q«oftime
3 41 18
— 3 40
\ 1 . ,
+ 8.
Mean T. at ship 3 37 38
M. T. atOreen. 4 46 1
ao- 9ft 17.
e» as >
a038318
0074466
... o
9^16666
9.716186
19.338686
9.668708
683
. • ■ j« .* A
Long, in time 1 7 23 ;=; Iff' 51' W.
For the usual computations at sea it is unnecessary to push the
calcnlations farther than the iMkrevt minute. '- '
SPHERICAL TRIGONOMETRY. 107
• SL Oftthe. llth iif Odober^ 18SI4^ at noon, on the meridian of
Qnttmich, 'm dhMnometer wai 11"* 19*.4 fast, and the daily nfle
was + 4M. On Ac 2l8t of October, at ff» 42°> lO* A. M. by tfaii
Inae- ^iibnometer, liie observed altitude of the son's lower limb
wal(42° 17' 9(K', and the height of the eye 20 feet; required the
km^tude?
Jiw.— 33° 26' E.
a On the 16th August, 1828, in latitude 38^" 2(y S., the mean of
several altitudes of Antares west of the meridian was 14^ 29^, the
i^e^ht of the eye being 12 feet, and the mean of the times per
wa t ch IP 41"" 38* P« M., which had been compared with mean time
at tbe Cape of Good Hope on the 22d of June, and was foimd to be
1^ 1^ 3ft too slow, and gaining 3^.54 a day ; required the longitude
of the ship?
Ans^n^' d& E.
BQUATION TO EQUAL ALTITUDES.
In ordinary cases the error and rate of a chronometer may be de-
termined by single altitudes ; but when great accuracy is required
eqaal altitudes are very superior, especialLy when a transit instn^-^
n&nt ctonot be obtained. On this account various tables have beeQ
cooapnled to facilitate this operation, though it is believed few of
tihiBm afford great advantage m actual practice. To those who would
prelbi nich a table, that of D. Josef S. Cer^uero, given in the
thirteenth volume of the Journal of Science, is perhaps the moat
commodious and exact By this means, however^ tables would be
lOidtiplied to any extent without giving much advantage, on ac?
count of the inconvenience of taking proportional parts ; and froiai
this consideration it is often better to give an easy practical ruUyi
reqidfinff the use of the ordinary tables, where neither double eh«
trie%.4inBrent signs> nor proportional parts are necessary.
The equation of equal altitudes is a correction for the change of
dipdHrtation of the celestial body during the interval of observationl
to iM -applied to the middle time between the instants shown by a
chronometer, at which, on a given day> that body has equal altir
tudes ; to find the true time oy the chronometer when the obj<ict
-WMM upon the meridian. .
Rule.*
Ta the cosine of half the interval between the times of observa-
tiapc tiiditke cotangent of the latitude, the sum, rejecting 10 in the
indaZy.irill be the tangent of arcjlrst, the difference between which,
mtptt- ^i^polar distance^ will be arc second,
T.JNowto the constant logarithm 5.36451 7^ add the cotangent of
||jil£tlie elapsed time, the cosecant of arc first, the cosecant o£ the
polar distance, the sine of arc second, the logarithm of the elapsqd
tixna in minates> the lomrithm of the daily variation of the dedinatieiJ
in seoonds, die sum wiU be the logarithm of the equation of equal alti«
todea in seaaadsjof time, which^ when appHed to moon, is addUife H
the polar distance is increasing, and subtractive if it is decreasing.
, ♦ * I Hr^ «-.*_«._. .1-. •■■■V^
f*a.ov Mr Ridd]e*g Navigatioa fbr a ai]nyw.TulB, •ulQgjBraAVii'^vsftM
ciple, tbovi^ ptthapt in tbe detMSl aomewhat less simple.
1«6 INTSOUUCTIDM..
If Ui/e e^uatioii is «4pplied to vimi^ht^ it li AAttiPf If thtfMlir db-
tmceiM deontwng, and stbMctiViB if die polar diatanee ia inoraMh--
. JIip. l-^-^On the 23d of March, 1809, at Pisa in ladtudt 49° 49' 11''
N-e^oal altitudei of the planet Venus were taken befhre.and after
transit, the elapsed time between which was 8*" 50°^ ; required die
equation of equal altitudes when her declination was MP iOf 4ff* N.,
and her daily variation +20' &' at +1205^' inereasiiig, and ocnne-
quently. the. polar distance £fe(?reaMii^ ?
Latitude 43^" 43' cot 0.019462 C. L. 5.364617
H. E. T. 4*' 26" COS. 9.605032 cot. 9.649468
Are 1.^ 22*" 50' tan. 9.624404 cosec. 0.411110
Pol. dist 69 17 cosecant 0.029690
Arc 2. 46 27 sine 9.860202
£ln>. thne » BOr ^ 530" lo«, . 2.724276
Daily yar. dec. 20" 5" = 1206'' log. aO80987
Eq.S. Alts — 12'.99 . l.l]958ff
Or subtractive, because the polar distance is decreasing and is to be
applied to noon.
Ex. 2— On the afternoon of die ]7di of September, 1810, aldtudeii
of ^e sun were observed at MarseQles, in udtude 43*^ 17' 50" N.«
and equal aldtudes were taken on die forenoon of the 18th, after an
Interval of 21^50^, the sun's declination for die 17di at midnight
being 2* 14' 23" N., and daily variation of declination r- 23' 14"
= — 1394'' ; required the equation of equal altitudes ?
Ans, — ^Eauation of equal altitudes — ld6'.70.
Or subtracuve, for the polar distance is increasing^ and is to bfi ap-
pli^ to midnight.
., Ex. a^At Florence, in latitude 43" 46' 40" N., on die BOi of
Apr^, 1809, equal altitudes of die planet Mars were taken at an iiv-
terval of 0* 20^ when his declination was d"" 9' 40" S., decreasing at
die rate of 6' 38^' daily ; required the correction for die planet's su-
periior passage ?
Ans. — Equation of equal altitudes — 5M96.
Or subtractive, because the polar distance is decreasing, and is to be
applied to the superior transit.
TO FUn> THE BBBOB OF A CHBONOHBTBB BY EQUAL ALTITI7DS8.
By the 6'«n.«— The sun is in general the most convenient object iot
4etmnining die error of a chronometer by equal altitudes, and tiie
linrenocm and afternoon of the same dvu day are often p r e figffed,
diough die evening and succeeding morning may sometimes be em*
ploved with advanti^
^1 the morning when the sun is more dian two hours distant from
die meridian, in mean latitudes, let a set of observations be taken
die eorrespcmding times by a chronometer. In die aft er nDon
* Bypolar distanoe in the computation, is meant ^ distance of the object from the
§ k ott i § d Mfe, whidk may be citfaor idMna to the nsidi or Math pote, Memisa to the
aune of the Ikdtude,
SFHERXOJlL tbuqovqhetby. uki
n fc wj'w Ji ilimMrti utoi the wa comet to tke Mme idthmh^ mtk^
inm-mmk jtaai^ down mpedle its 4wi i ta|wwi ding iitetirfte> '
Now half the «iim mBinj two timM, mswmng to tho MttO «ltlu
tpdoy wiU bf the approximate time of xukod. FmA the mew of aU'
the times of.noon in this manner from^ each corresponding pair of
<A>serva1ions ; to which the equation of equal altitudes being applied,
jtlie result will be the time of apparent noon, or the instant tnat t)ie
son's centre is on the meridian by the chronometer. The difference'
between this and noon is the error of the chronometer, which witt
be,&0t or slow according as the time of noon thereby is greater or
Ion than twelve hours.
JKrll.— On the 29th of January, 1826, in latitude 5T ^ N., the
ftUowing equal altitudes of the sun were observed ; required the ev«
for of the chronometer ?
AUUndes. Times 4^ M. Time* P.IC
. S** 5' . 2P35" ff ., 2^55"*48;
8 10 36 8 . 54 ^
• - *1S 37 9 63 42
. . » 20 38 9 , 52 41 ...
8 ^ 39 10 . ; 51 40 ■
Timet 4- M.
21" 36"
' »
36
8
37
9
38
9
39
10
35
44
21 37
8JI
2 53
41.6
5 16
32.8
15 8 r
Henu. 21 37 8JI 2 53 41^
31 37 8.8
Blipeedtimtt 5 16 32.8 Sum 34 30 60 .^
H.E.T. 2 38 16.4 Half 12 16 255
Sam's dedlnation at noon, on merid. Oreenwioh l?"" 59^ W &.
Dajly yariation or decrease of polar distance *— 16 15 N.
XiMtitude 67'' 9" cot 9.810025 CL. 5.364517
H. B.T. ^3ar COS. 9.887406 cot 0.083899
Arc I. 26 29 tan. 9.697431 cosec. 0.350726
Pol. dist 107 59 cosecant O.0Q1763
-"ta
Arc 2;— 81 30 sine 9.995203
Bl. tune 4»»16 .5 = 316».5 log. . 2.500374
ihrily Titf . dec. 16' 1 5" = 975" log. 2.989005
>«•
Sq. Teqn^ alts — 20^.2 1,305471
Balf sdm or approximate time of noon 12^16* 26'.2
B^wSoP of equal altitudes > . — 20 J^
Timeof apparent noon by chronontet^ 12 15 6.0
MiipMim ^time with oontrary sign -i- 18 97 -9
w^amtm
Time of mean noon by chronometer . 12 1 37*1
Hence the chronometer was 15" 6' fast for apparent noon, and 1^
37^-1 ^Mt for mean time.
JSia, 2. — On the 24th of July, at Pend^nnis castle near Falmoutl^
in latitude 50'' 8' 48"' N., Dr Tiarks, with a sextant of ten inches
radius by Mr Troughton, and an artificial horizon^ toge^et V\\)^ ^
• ( 1. ' A I.I
UA uxBaMrenoK.
vri^fdiifed the timje of apparent noon. by die. cfaronometer ?*
Xime aflttr noon 33d per dmmomeCer ' 20^29*13
34 . 4 26 '6.
Sam . 24 54 IQ. &
Half snm^ or approximate noon . 12 37 9*1$
Btffier^ce^ or elapsed time 7 55 62...S5
Half clapfied time • . 3 57 K:w
The dedina^on of the sun^ at noon 24th» is 19° 58^ nearly. . .
Ihaij Tariation 13' 39^' S.^ or increasing the polar distance.
Latitade BO^V cot 9.921503 C.L. &J3mi7
H.B.T: ' ^58" cos. 9.7(X>4e9 cot 9.770148
Aret '— 9Sf>BT tan. 9.020973 cosec. .a40eOlff
PoLdist 70 2 cosecant 0.036982
Arc 2. 47 5 sine . . 9.864716
£Um.timiB " 7^ 56- = 47f log. . 2.677007
BaflyTar.dec. 17 39"' = 759'Mog. 3.880243
Eq.eq.alts.+9".844 log. "' . 0.9f^lfi^
To apprditintefe hood 1?'37- 9'.15
AdUtEe equation of eqtudaltitaaes . + ^.^M(i
- » » .... >- ■ .
Apparent noon 13,37 18^9^
Ex. a-On the 34th of July* 1833^ at 3^ 5- 38>.7 P. M., and 35A
July, at 9^ 49" 59^.7 A. M. at die same place, the double altitude of
the sun's tipper limb was 93*^ 40^ ; required the apparent tiihe of
midnight Ixy the chronometer? , r «
Time afiber. noon, July 34th ^« 3^ B^BB^.f
24 iSl 49 59.7
Sum • •
Half 8ttm» or approximate midnight
£liq>sed time . ...
Half elapsed time . 9 23 10 «6,
beciUnation«t midnight 19^ 53^ N., daily varialio^
Or increaainff the polar distanocj^ apd V^a a^iii^yon i^ t^iffryfwf >Hf gai. ' >
live for midni^t.
• See a Report oa ChiMMiiietrical 0bMrvmtiQiit to vxmM the lcko'gi(£d^ <^ m^
24
55
3^4
w ay
49S
18
44
a^ijo
SPHEBICAL TBIOWOBCETBY. Ill
LMitad*. 50° 9* «ot 9MIB(» C. L. 5.364617
H^&X 9^ as* COS. 9.887406 cot. . Ag|388|^
ArcL 32 47 tan. 9.808909 cosec. a9664ii
PoLdist 70 8 cosecant .^ . 0.036649
Area. S7 21 sine 9.782961
Elap-time Iff* 44" =1124" log. . . 3.050766
DailTvam.d6cl2'39''= 759^' log. •: 3.880848
JE9. eq. alt*. ^S»J4 log. , •. 1.4554U
Fram approximate xoidnight • • 19^ 37" 49*4)0
Sioibtnct die equation of equal altitudes — - .' 9$Jl4
Apparent midnigiit /JL2 37 3066
^ Proceeding in this manner till a considerable number of obtam*
tions aire madcy the error of a chipiometcr maj^ be determinefl.^di
great accuracy. If this chrononieter is compared with any ipveii
number of them^ all their errors aond rates may be fimnd as has been
dime by Dr Tiarks.
Tlie same thing may be done by the stars, thought rather less ooni>
^Iw idlowing method of qomparing a chronometer iirith; mean
tnaie by JDr Tiarks, communicated by Captain Basil Hall> Jt N.,
iriO be found very useful. . .
TftOB difference of a chronometer from the mean time at ft pkUHp
b(9|M;1iuow9 at three different instants, to find that difierence for
-eirt^ M fe ruled iatfe instant with a proper regard to the chunfe ofiMlt
^rnlc^ may have taken place between the first and second^ and be^
tween die second and third times.
'£eC die difference at the
Liilwf'o =a
So'that h is the diflerence between the first Hiixd second tti^tes of
tfae joipv|Doaieterv and c the difference between the second and tlilrd
0tefCi^af die seme chronometer, the state of a chronometer, (namely,
ita jdifiBrence from the mean time of a given place), at die monittit
* \p(i^'^ + r (r-^0 J ^■*" f'xr^ff*
Iff &' to diln «', j^^ i» poriiw apd ^^^^^^negaiiuf,^^
if ^ i g iie ii r then l^ bodi are paiitive.
or
EXAMPLB.
^1^10 diffarence of a chiODometer Scm t]ie mugk tinM «f mvattio
place WM known on the following day* : •..-/, - 1*
M t ftf W Ol lWA' IO Kk
I ah . t
.'•^^ IDiffenmdet. Bkft:
^!!^v^'m& S1-8P08 iMAnence between lat and ad=21J90d
i*r"M.MS 4-SlO* . 1 «nd» =26.4007
««ie»Jo =00' " •" .
^ = 21.8903
:<"= 26.4007 ■'■■' '•'■'
j:<"=4a2»10- ■ - .
It ir now required to find the state of the chronometer foil^Aiunist
tyOt, at ll" 7" 44' = ]r.4S37. Oednctins AagUrt £^.Bm^imm
' AilMi8bi7Vl6S7 we have the interval t ±i 7*.raM. t'
^+(''ar4&2910- I': s^31.89iD0 kv. iSMBfii
r ar '7-0304 log. 0.899788 /"= 26.4007 lqg.t^l616
/+<''-< =140^8616 iog. I.fi06j961 I* x ^, log . , S.261868
«x(t'+i'W>»«rnuialo(^;a*(S6646 , \ ', ?!
4^X<^ocde9an)&uit«rk I09, 2.^61868 , ! , . ;.
<tt 7.83M, Kf. 0800788 <r'' :± 26»4007 ..log. j^i&1616
^=.i».8gQ3 (" 0^21.8069 . , r'i
f.m4 K 18«MB legl l.M460» <".*«'« it^lOiK^ log- ^^64316
flfuneniMnr kg. MHMSOat^ H^'-^O dmuw. log*
^tendiiiHiator ki* &075891
tit-T)
»■• 1 r,^
J/tTvx ^- 9.068569, or factor tif '«' ^tiWSttt Is AM«Sye,
because f U less than tf. . - v . -t . ^
dist, 57 ': - M 10 .8& . . la^M ^
Sept, 4thy22 12 64 39.1tf • • «-88 <?
What is the difference, August 17th, 11** 8".
6=: 132*^ log. 2.123782 cs28'.83 log. 1.469845
factor b log. 9.748781 £ictor c log. 9.968659
' ' u.
(/)6=+7».7a>g, 1.8876«af (/)tor-a»» kg.' 148810*
(/)C=— 26.82 '^ -^ ^Onr - ^'M
cor. sS -4- 4SJBI^ Sfi he ^appli^ (6 ttic^ error of tli« diitaalMlarl in
the tune a. ^ '•'■ v.-'lu^.c^i
August 9th, 12** 36" chronomeWt slow fifr M; T; •'^^' ' » •§» ifl>Wfi*
correction +o- 4ihMf
r mean time . . 62ii44i2S^;
©Npwoija^dgw
On Aujgust 17tn^ «
. .1.
'I'M T?:i 'Jt a-
* For RoimI's method of cowacting ilie enoc in rate of a chip|ioaMF, ^, JV^ •
./Ijiiraaoiiiiey v<d, JII., or Myer*8 tnmalatioD of this, page 95. ' ' f
8PHEBIOAL TRIOOMCaiETRY.
lU
r Hi
DkIbhI nictloiM of • Dnj of 94^.
T.DtdmaL
T.OecliiML
mdell.
DMimal Fteto
an Hour.
TaOmtlh
conyertL
TUMiiito]
1 19^ to an Hour.
1* k'«4iee7 io-N.oo6044fio-i
2 lO. 0633^0 .01388^'
3 0. laSOOOSO 1 .020833
4 0.180067140
5 la
I
7 !0. 2910671 3
8 10.333333 3
250000
!0. 291067
a333333
0.375000
0.416007
1 0.
12 0.500000
13
14
15
10
17
18
32
23
M
rll I I
4
5
6
7
8
9
0.541667
0.583333
0.025000
0160000710'
0.706333^
^ 0.75000030
19 0.79180740
90 0.83333350
0.875000 1
a 916667 2
0..958333I 3
1.
.001
.002778
.003472
.004167
.004861
.00555^
.00625
!40
1
2
3
4
5
6
7
8
9
For 12*
I doable that
I for 84^.
4
5
6
7
8
9
.0001] 6110^
.00023
.000348230
.000464^40
.000580|50
.000012 1
.000023 2
.000035 3
.000046 4
.000058 5
.000069 6
.000081 7
.000092 8
.000104 9
edmaL I
1666675
.333333 ft J2
.50000020.3
.666667.0 .4
.833333X).5
.016667X>.6
.033333|0.7
.050000V).8
.066667|U .9
.0333330 .01
.100000^ .02
.116667» .03
.133333J) .04
.150000X) .05
.».06
Arc
— — — jV .lAI
.00277«9.97
.005556V) .08
.008333X).09
.0111 Ilk) .001
.013889(0 .002
.000278^ .0031
.00055^1
iooiiii
.0013891
.0011
.0019^
.0041
.005
.006
.007
.008
.009
Explanation.
TflUal. contains the decimal fraction of a day of 24**. It is nMm
Atfbr findiiig what part of a day any number of hours, minutes,
and aeconda are, and consequently may be convenientlv employed
tn nuray calculadons where daily differences are necessarily involved,
soeh m die .daily rate of a dock, the change of which, in any given
mmiber ef hours, &c may be thereby readity obtained. It is also very
i ncft d-ia die preceding method of comparing chronometers, and other
wtim II. aerves the same purpose when an hour is taken for
unit, and is useful in several astronomical <^>erations.
Table HI. is supplementary to the general Table V. which serves
to convert time into degrees u less than 0" or 90°. Butas 6^ answers
ta:9lt» 12^ to 180^ and Iff" to 270^, this toble wiU easily be applied
to M^ or Mf^f die whole drde to every four eeconda e€ \n«L«» «Dt^
i*Wiiigii08''it90 only liseefisary to eonvert the deeiia«Jl pairiof |^;jtiy|pe
in<x)'degjffe€sib;^tmflitabl0to coBoplete t^ '^-t .n .!t ^.^ii'u
^' - III. BT OCOUI^TATIONB AMD BCIilPSBB. - :><'\
The ihoon in her periodical revodutioti ^equently psM^s- Jbftn^n
, the eardi 4nd a fixed star^ of which she interotpte the>{8pecUtknr'8
yiew^ thus^ producing what is called an occultatum4 -i ■■* ^.('
Sinci^ the instant of disappearance and reappe»«ieeiof <tiid star
tan be ascertained; without the use of any in8trumttd^liablflf.tOieEv6r^
thelongitu^e may bedetermined more accurately by finob8eKratMKu>f
this pheinocrienon^ than by a lunar distance. An obeeryer pofMBSBftof
an ordinary telescope^ a chronometer^ and an instrumeMt <oo<delH(r-
mine itv etror iihd rate,^ can readily make the obaevvationsjv«iiiid*ldie
.pecessary calculatioBS-are hr from difficult. Several ;j-|]de^ llBve
' been plbposed for this purpoee independent of the ma&pd^ deter*
inining tbe paraUaxes- by the nonaffesimal^ and^ <U»aparatit^ mnch
more simple. * Of these^ Dr Inman^ of Portsmouth^ which w^shi&in
the mean time adopt with some alterations, appears to us the nost
convenient. .': -.
At the TBstant of the disappearance or reappeax^nce of the ftxur«
the apparent 'right ascension and dedination of the point- tof 'the
niooh's limb in contact with the star is the same as the right -aaeen-
' sioh and dieclination of the star, which can be obt^hed with gt^at
facility' ated accuracy from tables. The apparent right asceniien land
declination of this pmnt being corrected for paraQaX) its true light
'jfdscensioii and declination will be determined. Now since ^ the^ldis-
t^ce oif^this point from the moon's centre^ which is equals to her
s^idiaknieter^ and the declination of the centre for the eatlniated
ti^e at dreenwich^ may be found by the Nautical Almanac, Aetrue
right Ascension of the moon's centre is easily compute^. IShdbld
there be an uncertainty in the estimated Greenwich time anumnting
to about one minute, the operation must be repeated, till the esti-
mated bnd computed Greenwich time be very nearly the sahie.
i. ^«f&-
By appl3dng the estimated longitude in time to the observer's Ap-
parent time, the reduced Greenwich time to the nearest minute will
be obtained.
To this time take from the Nautical Almanac the sun's B. A^.the
moon's R. A. and their declinations corrected for second diffev^Dces,
together with the variation of declination for 10% for the purpq^ of
repeating the operation when supposed necessary ; an4 Dunoon's
semidiameter, and the horizontal parallax corrected for th^ ^b^oi-
dal fiffure of the earth. ^^,^
Ta£e also the moon's R. A. for 3*" after the first jwtiyr^f j-H jt'"^
c o rrec te d as formerly* . i : . . v^-
Find from the Nautical Almanac, or from other tables,. ^be^.mpap-
retit R. A. and D. of the observed fixed star; and reduce ^thp[ given
latitude for the spheroidal figure of the earth. j . iV "
■ •■" ■ ■■■■.■ ■■. '.■/■
n: — t ' HM i - .. ■ , , ■ y y,u i.^,4 •
, * If the observations are made at sea, an aHowaace must be tbiftii fbr & mffm the
cbrori^nieter beiweeta th^ disapi^hasnce find reappearance of the starm* thsdu^of tbe
shipf B8 in lunan.
SPHEMCUUL TBIOeVOMETBY. US
'^^^^9tf^iblmtff as m ti ilme !»i*^ie mii> 1L A-v oid'fmii' the nun, /fiti-
if 1ms thm 19"^ wmbethehoar mffle; if greater than iS^/iteiOMb-
plement to 24^ will be the hour angle.
Now write down the proportional logarithm of the reduced hori-
Bontal parallax under the numbers (1), (2), and (3). Under (1) and
:^%ipaat'th0>aaxiJitf of the reduced ladtude; under (3) the cosecant
^dieirfaiBf.| mttder (1) the cosecant of die hour angle {fl), and take
the sum of these.
i:Bebw the eun ^the three Icyarithms under (1) put the comiiimt
ifl^^nfAm 1.19609, and the cosine of the star's declination i at.t^e
«Hie tinae under (2) put the cosecant, and under. (3) the secant of
■'4l»smme; the sum of these three logarithms under (I) will be.tfte
pnlpoeHomd logarithm of arc Jirst, or the parallax in ]^ A. il^tjme,
rMHrljTcr one lulf of which (6) is to be subtracted from the hqux aa-
' '1^ (^ giving (a — 6), the corrected hour an^.
': Uniet (X) put the secant of the hour-angle thus correpted. ,/M^e
-> mofL dCthtttogarithms under (3) will be the proportional logarithm
oilbe^rst part of the parallax in declination, apa that under. (SVthe
\mmmdi Toe first part must be applied with such a sign as to 4)iiau-
niah the star's distance from the elevated pole : the second must be
anjpiiad witb the same sign as the first, if the houTrangle. and polar
^.'mtiuam afethe one greater, and the other less than 90^ (^€1'*; oji;^-
• iriae with a contrary sign. The result will the true declination of
. . the observed point of the moon's limb. Take the j^fierence betw^n
' tiua true decimation of the observed point and th^ declination or the
fnoaa'scenitre, found from the Nautical Almanac, . under which put
- the moon's horizontal semidiameter properly : corrected*, ^f^^^^ke
^ 'itbn sum and difference- Add togetner the proportional l^f^hm
i. ^«f ^tfaift sittu and difference, and take half the sum, io whi^cli fHf^ the
MHHoifmeof the mean ef the two declinations just fou^d, |t^ .siyoi'will
' vkfgffbe proportional logarithm of the moon's semidiaweter in jBl A.
;.;: ■ ' flwsny*
.\. Under. (4) put the constant logarithm 1.17609, tjbe $rst ^upi/un-
der (1), add nie cosine of the declination of the observed pomh the
sum wHl be the proportional logarithm of the exact parallax of R. A.
m time. This being added to the star's R. A. when west of the meri-
'*ltiitii, H/at subtracted if east, will give the true R. A. of the pcSnt ob-
aetffed; To the true R. A. thus obtained add the moon's semidia-
mf^ter in R. A., or subtract it therefrom^ according as the reappear-
'-'''anfce tir disappearance of the star has been observed, and the result
■-^J »#il! be the true R. A. of the moon's centre deduced from observa-
^' 'iidtt.- ■■: ■
%]n^;:(If A^ ^ynSers considerably from the R. A. taken from the Nautical
^^^^"^TAttBOtAc, 'alter the moon's declination by as many aeconda no will
make a corresponding variation in the first R. A. such as. the Nnuti-
'^"((iai'-AltBat»c would give for the same alteration in deidinadoil. Re-
peat the operation till this is the case, and the last R. A. wiid be that
-iiq^fecf^fed.
'r ' '.'^' ^Mer tills put the moon's (1) R. A. taken ftOxA die Nautical Al-
manac for the Greenwich time, and then the moon's Rb A.:tthree
hours after, or the (2) R. A. Take the difference between the first
and second, and the difference between the second and third. Then
••<• toari4iMltdMond> Ae remamoer.wjill be the pivportumot^XogQi^^ ^^ ^
m
A TAu\9ifsn(itiP(iMmAm
pntimk 9i ibm which muafc be oclfWto the Oreewruii tfade^ji^eii
a^jbit R. At A ^tcE than the Mcoiid; «tbenrii^ii&<raetel^ind
Ili^5«nilt wa| f^ie Greenwich apmreat tune, fl^e difierendt^Si-
twSn this acud Ithi apparent time of the observer will be the kngt-
. tii4B in time.
j|r. l.--On the 3d of March 1823, at Bahia, in latitude 12^ ffj'
1T^&., and longitude by estimation 38^ 3(K W., the reaiipeara^ee •f
Antares from the dark umb of the moon was obaervea at Ifi^. 90^
0^.3 ; Require the true longitude ?
fiahia, Marc&'Sd, Iff^dfTOr.S Moon's 1st R.A. S44<' 27' SO^'J^
2 34
IkifL in time
i«
O.Jest&Ene-: 18 4
9dHhkiMke.
tkm'iKA. 22** 60" 58*.64
AntatesRAt 16 18 85.8
p^l :^. ^ 26 1 50.1
Anp^time "^ Iff'SO^O.S
BB^iraa. ^ 22 sa 58.64
Sum . 88 26 58 .84
AntareslLA. 16 18 '85;6i2
Diff. 2t 8-^.12
2d RA,. 246 6 U( .82
Dee. 25 «6 ^^ S.
▼«r.fcrla»+ > 9-SlS^.
hor.«.D. 14 mxr
54 26 75
— J5
eq. par.
red. to 19*
red.^ar. 64
latitude
red. to 13^ 8.
red. lat.
12« 57' 17" 8.
— 60
W 68 17,
24
JiiL^
rVe<^ rai 36.88
jCrp- ' ■* - "T »
^1N;^.cdhvemeiilt that the work should fcSlow firoia lieginnUig
iq fsnA in'riegular order, that of the foisegoiQg example' Jia^ been
irahslerrHL to the two following pages, and to avoid unxiecessary
waste of rhom^ the remainder of this has been filled witli the follow-
ing example for exercise :-— ■ ^.. "*
EkSi^-^On the 26th of May^ 1822, at SanBlas in latitude 2VSSf^'
N., ^dlongitude by estimation lOSJ? W., at 9** 22" 41'.3, A.Ti Uie xm-
tnen&ij^pf « Leonis was observed by laeutenaiit H. Foster, th«Q Maa-
l^r's ^iffe of his Majesty's ship Conway; what W4S the true lon^
tadef " '
^n*.-.106° 18' 27'* W.
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Ex. ft^-On^ SKMi of Jtdy 189SL at Rio^Jj
X Sagittarii UU^«^
mo Janeiro, Jul^^ #^ i^ "^ 'm^aafs 111
Lop. in time c> ro * S^ 0; : ^
Bat Oreenwidlf%ni€> 9 42
To this time. S "
Sun's RJV. 7^67*fi».7
Star's RA.
I>ee.
App. time
Son's ILA*
Sum
Star's ILA.
IHff.
lflU7 7 .3
26<»30 31".0
7 57 23 .7
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18 17
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|iid.par. 53 58
liMtude 22<>54a0^ S.
fiMuc. — 8 19
f.; • t- '^ ! 55 *^ O
11 '-^^ <^ I
*-. 1 ''■•..
Hour angle S 30 34 .4 U {U .
It is hardly necessary to give the variation of the sun*s R. AT and
D. in 10^, as it is very smalH and as the true time must differ but a
few seconds from Ae estimated^ on repetition the longitude cannot
viary much on this aocoimt -^ a
Ex. 4.---On the 3d of January, 1826^ at Port Boiven^^ UtfS^de
73«> IS' W N., and Imwitude by estimation 5^ 56" W, th i^oiier-
sion o£ X OeminonEm of the 4th magtdtude waa ohaexvef^iLtii 1^
23'.26 M. T./and ifid emersion at T H"" 12'.17 M. T., by lieutenant
Henry Foster, R. N. ; what was the true longitude ?
iliM^x— By immersion the longitude is 5^55''48^ and byemar-
aionitis5*^55-35«W. 8
It was intended, if room would ha^ pennitted, to pi^the whole
of the calculation on one paffe, and^ HSi^glGaot daneherq^BiaTMdi*
l]f enoB§^ be so nlaced by the calculafttnr. ThiftHtd&tteB hotigm At
not t6l>e dightea, as a neat form, likira coiivetfentwniAa, ffilfbe
found of some service in acoorate conqputationa.
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(•An ecHpsd of the son depends upcni: the same 'CftfueiriM'Bik^oebak'
tseion^ hifi light being intercepted hy the body of ^^w m0aii:piaini9
between hrm and ^e spectator. The beginning ahd^eiHLrbf n rtiuhr
eclipse is easily observed by a telescope of moderate power 'prd p^ a i y
prepared^ when the point of contact of the limbs ' being nen^
known, and the rule for computing the longitnobtis siniilarr1;o<tlut
now given for an occultation. If ^e semidiamef^r' bfi>tlle Hi4»on
passing through that point of the sun and moon, apparehtly'a» omMf
tacti be supposed io be produced to 1^ centre of toe sun; ^a^ -^ibei
from the observer, and conceiving this centre to be at the-fdiithiDiBe
of the fixed stars, ^ as to have no sensible parallax, thenri^^h aoiaiifl'
fest,Hhat the rule for an occultation must apply by substittitingi£air
the moon*^ semidietmeter the sum of the sun Mid moon's semidiame-
t^s, cbiisidering the sun to be at the ssttne distance as the knoA
when seen from the earth's centre, — ^that is, subtracting '^e aug-
mentati^ fet the sun's semidiamecer asif itiwere the mfootfs-fipoHl
it, ad^fontidin the Nautical Almanac. In' the supposMon jsA
made, th^ sun's centre was supposed to have no pariailax ; but, as ft
has a iiorizontal parallax of about 8'^7> in finding the apparent plaefe
we cann^ proceed exactly as for a fixed star. The sun's riglitrtail
censibn -and declination, as seen from the centre^ must be tafceifi frtkk
the Nautical Almanac, which, corrected for parallax,' w<ili:g{4'e mSth
appaifent Tight ascension and declination, thus rediificin^ tbe'caim^
arsolar eclipse to a similarity with that of an occultation. The ap-
Earent right ascension and declination of the sun's centre must now
e corrected, using the horizontal parallax of the moon in the com-
putation;^ This would evidently give the same true place as if, tak-
ing the Hgbt ascension and deciinadon 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 eninploying the difference betweisn the horizontal parallaxes of
the sun and moon.
Whence 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 chronometer, 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.
Rule.
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 diminished by the augmentation, the moon's
right ascension and declination, semidiameter and horizontal paral-
lax corrected for the spheroidal figure of the earth, and diminished
by the sun's horizontal parallax. Tak^ 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 observer's
apparent time, and in the morning its complement to 24 hours.
j&2Sp]oying the moon's diminished horizontal parallax, correct the
SPHERICAL TRI60K0METEY.
1S8
sun's right ascension, aad decUnatian^ as if for some point on the
moon extended^ proceeding as formerly, only putting the sum of the
sta^ii^semidiamirter, dimniished by augmentation and the moon's
Sprnkliifniekei) ktotcadi^the moon's semidiameter alone. If the re^
lahing Qrdtirwich time differ from the estimated, the sun's B. A.
8iid<i^ei4ination must be corrected for the difference^ repeating the
9tnr«tioii «s often as necessary^ till the Grreenwich time by compu-
lation «nd estimation agree.
■.'•£m,-^*4)n the 7th of September, 1820, at the Royal Naval College,
Pit fji oath, in latitude 50^ 48' 3'' N., and longitude by estimation
1^ Wv the «nd of a solar eclipse was observed at 3*" 12°" 55* ; requir-
ed ithe tnie iMigitude ?
FMb. Bepu 7. 3^ 13^ Moon's (1) R. A. 166« 64' 47"
]yiiil«'iiiitniie' 4 var. in 10*
(2) R. A.
dec
var. in lO*
hor. S. D.
bbO.T. 3 7
VIM. To this time.
fioBTs^ILA. 1P4»13-.(K)
Viar. in Mf 0.025 * equa. par.
Dm. >. . 5''Sa'22'' N. red. tolat.
Vah^in la .16
SkaamHim 15 54 . 8 red. par.
fioti JMT. 8.6
MLmtnuaAv
ikf^ikm9^l9P' 56' =: H. A. differoice
!p mI I latitude
'■ <i( I.- (ill • > Reduction
sun's hor. par.
At.*, 'tf -
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r-
Red. lat.
Sun's 8. D.
Aug. to 30°
Red. S. D.
Moon's S. D.
Sum
4". 4
168 13 26
6 21 4 N*
1 .84 8.
14 42 .7
53 66 .0
— 6 .3
*
53 40 .7
8 .6
50
53 41 .1
48 a
11 15
50
36 48
15 54.8
— 7 S
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14. 42 .7
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17^51
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5.4
On the 7th of July> 1823, at Dunglass House, the seat of Sir James
HtilL Bfj^tf, in lati^de 55*^ 56' 32^' N., and longitude by e8ti^lation >o
9" W W.; CaptflS[i Basil fi^U'^Sl. N., observed the end of^ solar ^
; ed^se at 17^55'^34M nie£n;ti^; required the true lon^^de of ':2^
Duaglaas? - t, • * o —
Julj 7tl|| i ,17^ 55- 34' July 7th, Mean Time, 17** 55^ 34' 1 .
Es|£lon^T. r+ 9 30 equ. of time to 18M'' — 4 28.7 ^
Eq.VT. atnoon, ~ 4 21
"■■ ■ apparent time at D.
Ai^ox. G. T. 18 43
Or is tlneerly.
• ^^ hour angle,
Totfaistitfie^
Sun's R. A.. 7^^2!.65
Var.inlO-, - Oj93
Sun's dec. 22<' K' 40^^^
Var.inlO-, — " (^.05
Sun's S. D. 15' ^".5
\ Jkt^.ito^'' ill. -^ 6 .5
;<J3ofcp.i>. Z
Hor. pur.
Latitude;
!
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15 40 .0
8 .54
55'' 56* 32"
Reduction, - — 10 40
Rei. law
4S 45r 5^
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Moon*i
9(1)R.A.
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108 J6 |6 .* ^^
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23 48 1S5 .:^
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SPHERICAL TRIGONOMETRY. 129
^ IV. BY THE moon's TBANSfiT.
9^} T^ikitjitiod of finding the longitude by tbe cnlmiqatioii of the
j^'^taocnf it^cTttarSj is how considered very convenient and accurate.
-'z Sivsdthe observations require a transit instrument^ and tbe dpck
*^ «s^ whh it generally shows siderial time ; the difference of the
iintfKhjqpposed to be siderial time. If it is not> it must be lieduced
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 wonli
be4he same at both places,
f fhe difference of the differences arises from> and is equal to the
I . intftiwisc (I) of the moon's right ascension in time, in the interval
[ r befeipwu me passages over Uie meridian at each pkoe.
! r Henccj if the increase (N) of the moon's R A. in one hour of
' ' nderitl time be known N : I : : l*" : X^ the angle described by th^
■ wflitsrn meridian in the interval of the passages of the moon.
TtA» is equal to Ae difference of longitude + 1. '-
Hfliioe» the difference of longitude is equal to X — I = i^— -I* By
die^irtical Almanac the moon*s right ascension is given at everjr
BOtB ai^ midnight; whence its increase in an hour of siderial tim^
HM^ befoiind nearly in the middle of the interval including the ob-
.Amine the difference of longitude = L' as nearly as can be esti-
mflletfj ind compute the increase (E) of the moon's R. A. in the si*
deiM tim# L', then
IL'
jta^.: I;: : !>: X r= -=-, (1) and the exact difference of Ifpgitude
H/
^rf^^— 1(3). But this exactness is only necessary when the plao^
Cfier. considerably in longitude.
I iim moon's limb is observed by a transit instrument^ and not the
0BBir»9 which makes some little difference when the difference of
ionntude is considerable.
When great accuracy is required^ it would then be necessary to
spudMLan dlowance for the moon's alteration of distance* that chan-
fei.her Mpparent diameter, and also for change of declination^ which
dumges her semidiameter in R. A.*
■ JBdV/— Jane 13th, 1791, the following observations of the passage
of tht moon and « Serpentis were made at the observatories otOreen-
vich and Dublin ; required their difference of Longitude ?
At Chreenwich, R. A. )*8 1st limb 15^ 6" S* 52 at 9h 36" App. T.
R. A. « serpentis 15 33 34.70
1 St Difference 28 31.18
u R. A. )'s 1st limb 15^ 6™ 12'.49
Kserpentis 15 33 36.91
9d Difference 27 24.42
« $ ^'f iBW| eoBi|dete solatioD of thu nulhod, jce J^ Brisklej^f Article in the
hd wnmi oC the JMUin Philotophieia J4iunial, and 1lf»- BidlyH McxfloVx Vxi <cft
TEundioDB of the AetroBonikal'Sodee^.
130 INTliODUCTION.
Differenqe 2T 24/42
Daily rate ofclock—16*.88 prop, part +052
2dcor.diff. 27 24.74
m!1
difference of Ist and 2d differences 1 6 .44 = 16" '^Kh,
A» the places do not differ much in longitude^ it is uttnecessMiylkl
i*educe apparent to mean time. : ■ . ; k .
' '- This diflTerence 16' dO'^O is the increase of the mokm's R. A:; hi
the interval of its passages of the meridians of the ohsermtionB-of
Greenwich and Dublin. .^ .i
" By the Nautical Almanac^ we find the following differences 'oFthe
right ascensions of the same limb of the moon^ and the star at^aAKliit
the' same :time. -• k .*.
Diff\ ■ ■•• ' •■» *""»
June IS, midnight 213*^ 16' *to oo^ ■•«
13, noon 220^ 38 ' ' ' i oil
13, midnight 228 11 • • I %\ .o o»/f^
14, noon 235 53 ' * ' i ^ ' ' ' ' ^^^
14i midnight 243 43 • • 7 ^ .
If the places differ much in longitude^ the motion in It A^ should
he calculated to seconds, though^ m the present case as -the secoad
differences are sufficiently uniform, the mean first diff^l«noe coti-
taining the interval will be sufficiently accurate for Xht nte of in-
j^reaiBe in 12 hours at the middle time. ::-..- ^il.K..
Hence, by formula (1) T 37'.5 : 16' 36".6 iil^xx^ \b«»M»
4nd \f hen the dijQ^ence of longitude is not considerable x -fr - ^\ A. =
156».42^ + ^ . ^ = 20" 12'.77» consequently 2»" 13^.77—1'" ^44
= 25" 6'.33 = 6« 16' 35' W. , . . V ;. .
If ) r be the increase of the moon's R. A. during the ii](t?rxai,be-»
tween the transits, then x + ;r^ — > r must be used whetl tft^'dtf-
365 .^ . .,,. .
ference of longitude is considerable.
It would extend this article too much to give Baily*s or Brinkley'B
methods, which are more accurate and complete^ and caa only lie
fully treated in a work on astronomy.
> In the foregoing example the tufference of R. A. between the
-moon and star was determined at both places by obsesyationi • hut
for ordiniary purposes that at Greenwich may be found by the Nau-
tical Almanac. . ' .
OF THE TRANSIT INSTRUMENT.
A transit instrument is a telescope properly placed in the meri-
dian for the purpose of observing the times at which the celestial
bodies pass tnis circle. If the clock or chronometer by which the
time is marked be adjusted to show siderial time, then their right
• ascensions will be found. . This is perhaps the best method 6! de-
termining the rates of chronometers.
The telescope is fitted to an axis, of which the ends tapered into
points turn in notches, from their shape called Vs or Ys. This axis
^ IS made hollow^ opposite one of the ends of which is j^aced a lamp
for illuminating the wires iii night observations. . . ."'. " ."i '
SPHERICAL TBIG0N01^JKTllV^ 131
Thede wires, generally five in number^ are placed in the telescope '
equidistant from each other^ and perpendicular to the horixon, hav-^
ing also a boriaontal wire bisecting them^ near or upon which the.
transits are observed.
Whe^ properly adjusted, the middle vertical wire coincides with
the menoian, and the instant that the centre of any heavenly body
paasfsa this wire, is called its transit. The other parallel wires are
mtended to correct or verify the observation by taking a mean be«
tween the transits over the Jirst and last, the sbcomd and youRTH,
and oomparing it with the third or meridian wire ; or, what is more
c(NTect, a mean of the whole called the reduction of the wires.
lliere are five principal adjustments necessary in placing a tran-
dt imstrument, 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 by a wire does not change its
portion relative to that wire, on moving the instrument up and
down. If it does, the wires must be all turned till the object is
kept upon them, when moved through their whole extent, and the
adju!<^ent is then complete,
2. The telescope should have no parallax. — ^When any distant ob-
J0GtiB bisected by the horizontal wire, if, on moving the eye up and
down a little, the object should appear to separate from the wire,
the instrument is said to have a parallax. This must be corrected
by placing the object and eye glasses at such a distance from each
outer, that their foci may meet in the point of intersection of the
IfiBCB. When the object-glass has been properlv fixed by the in-
strument^maker, the observer has only to adjust the eye-class.
.^ Ti^e Une of coUimation should be cotrect,* — ^This is Known by
bisecting any object by the meridian wire, and if, on reversing the
axis, die object sdll remains bisected as before, the line of comma-
tion 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
one screw and tightening the other till the error appears one half
dkninished, when the axis is again reversed, and the operation is re-
peated till liie adjustment is properly effected.
4. To level the axis. — This is performed by means of a screw
placed under one of the Ys or notches, which raises or depresses
^bM end of the axis at pleasure, while the true horizontal positioiris
ascertained by a spirit-level.
- '6. To bring the telescope to the meridian. — This is accomplished bv
nfiieanB of shoHsontal screw acting on one end of the axis, by which
if is moved baickward or forward till its proper position is obtained.
As the problem of bringring a transit instrument into the meridia/n
is one of considerable difficulty, it is proposed to treat it at some
lengths
— 2b lake a IVansit,
. With the latitude of the place and the declination of the object
tsgmpute its meridian altitude.
: When it Is known to approach the meridian, elevate the telescope
■■■ t
. * Thf line of collimation is an imagiiutry Btraigbt line 8up]>0Kcd to join tiiei centre of
refnctkmB of the object glass, and the intersection of the meridian axiCi VioTvioivVaSL Vvk
in the centre of the uAcscope.
132
INXaODUCTION,
to the gcwon latitude by the ckcLe .attached to ibfi eniioC t^Me axis.
Now^ becaase the telescope inverts objects^ the object will appear to
oMtte'iiitb the 'field of view from the west and mofe i«wajn|%.the
east* 'f .■ ;».\ /ii*
'Mark die time of transit over each wire, using a darHj^a^to
saVe the eye when the sun is observed* • ■,■1 :,•»»
FROM THS OABKNWICH OBSERVATIONS.
1SI&
Noir.
Wiret.
I.
3d
II.
III.
1'.4
22.6
20'.9i2P55«38'.5
0.4
55.2
29 27.6
IV.
18.421 55 37.2
5*.5
0.0
55.7
V.
Redlfc,
Unrest
]5*.238*^
32 .5 27 SO
.f
,» -
!lf.:lM ^•
■r.,- «
tfsAl^uarii.
14.137.1$*Aqijvi?ii.
Cassiop.
n"
rff^^i 61"2e?.46l'P48%5
^r |fia 45jQ|53 4>3
14 54 23.^
652
im
454
2©'.762" 4e'.Q
42ii^ 1^
X
SuntalL.
By taking the means as directed.
H I,,,. oy laiung me meaus «s uirecvcu. ' '
^itliat of the 3d will be 2P55"^aS!!io
4th .... 29 27.I6
8th, both limbs 14 53 iSgfO
^ By the Niiutical Almanac the sun's \ 14 54 4 70
}
-•I i '^'l; '
ri^ht ascension that day was
The error of the clock on the 8th is slow^ or .*—;•. 4AJIO
''^'Soppose the observation had been made with one wiN^.aa ihe
fi^die one only, then • <. 1.
To .... 14** 53" 7*-6
Add semidiameter. Table XV. . +17^
Transit of centre
.^SiWd of the. whole
14 53 15^
14 53 16.5
.DifTeiirmce only — OiJ
, V • J^ld^en^ of^the clock may readily be determined friim the stjurs^
i)f <me: p£ tl^o^ whose true places are given in the JH^autical A)manac is
j^lifienf^^^S: Othen^ise the corrections must, be applied ftofU apipro-
pciate.tablesi.,
Observed, transit on 3d
m Aquarii R. A. by tables
2P6ff^3ff^
21 56 SCIJ35
* ■ ■ -
J{SrKW.fff dock by the star slowj on the 3d
On the 8th
%084 iu 4.71 sideriial days
Ot^ th^ daily loss is
— ft. 05
- , mvo
■ ' "'3.15
0.67
I ■ '
■ in - il «
I.' i". "I/Ofl vir-.
■;.-,., ..^ ^
' .«.- •fll'^M ■ rf
; ■■ • rr' t
■4- : r
( ;m»" 1 J- I f
. • i«»i^
i« '
SFUKHJUJAL XUIGONOMETRV. 1£I3
"'''' ^jHflM n VhAMAPrnfBTRimEKT into TBB MBRIDIAHr '
:;; 7r»-)ar|ii ;.f. « r .,1 . . . j*
*<t*flr JM^ftirlkitMt'pMsleiii^ the time should be accuraAely debnrnsn^d
by an altitude near the prime vertical, or still better by equal altitudes^
as already 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,
.jBljtJIfir Hmb must be observed when on the meridian by allowing for
ihe time Yjob semidiameter takeis to pass the meridian. This lis fblffid
most accurately in the Nautical Almanac, or, if it is not at hand^
from Table XV .
Ln.t iyfl^jSnd tie Time thai any Star takes to pate from one wire to ano^
qtkerina Tr^eit Inttrument, that of the Equinoctial being inofvn.
^ " ISwK?.— To.the cosine of the star's declination add the proportioBal
^itajifillithtn of the time at the equinoctial, the sum is the proportionai
.liSgiiitiibm of the time by the given star. - ..
'! * JBr.^-^n the 10th of April, 1826, by a transH telescope whiofe
dLgeakSStAJSx the passage df a star on die equinoctial from wire to^
wire; what would be ue time by Antares, having 26° 2^ S. declimu
tum?
I>ecl2iiation 260 2" cosine 9.96354
Tkhe . 25'.4 P.L. . > 2.62897
BoAiced tibke 28.26 P.L. . 268221
- Pr t^ would be more readily performed by oonsidflring t)|9 se«
ooiida minutes, and ccmverting the decimals into' thirda to. ibe ifjivti-
mated, seooada, then the answer will come out in minutes and seconds
ia'Ue estimated seccmds and thirds.
]>edinatikm 26<'2' cosine . 9,86364
Time 26\ 4, or 26» 24* P. L. . OSBOU
T • 28.27. or 28 16 P.L. . : 0.80398
Hcpce the star's expected time of approach to the other wires be-
oopies [known after its contact with the first is observed. ' ' ''^
' ^ One of the most convenient methods of fixing the trahUttelt^itc^ie
lid the meridian in mean northern latitudes is by means of Polaris.
It. is reouired to set a transit instrument by Polaris,' on tb6' M 'of
Ee pass the meridian about 2** 8°*, and 14*" 8°" at' the altfttiiAes
ibove> which serve as a guide to advettise the ob^fV^ tti^be
, r^^ow let the clock be regulated to siderial timCj, and when it
iisljgws 0^ S^ 12'.2 make the middle wire bisect Polilris, then' -v^
J^k instrument be in the meridian. If, however, the time first as-^
Bi^ed was not known with sufficient accuracy^ the error of the
'plock can now be found very nearly by the transit of die sun^ 'o^f a
star. By repeatedly observing Polaris, and correcting in this nian-
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 circumpolax stars ace ec!^«\, «>x(*
J 34 INTliOWJCTION.
posinff ^t^tii^ r^tQ of the clock is imifonn. Should the kAntetrtT
not choose to trust to that^ he may select two circun^pelAr'^tiMv
wjifi^. right MQepfliosiA differ yearly 12^, sm it requires io this'kaAse
aii\y 41 few nuQUtes perfect regularity in the clock. Take ibefdiffi^
enQe: between, the transiu of circumpolar stars hj the cAook^^Mik^
j^te nearly in the same azimuth^ the one above tne Qther b^erw th^
pd^ ; repeat the operation 12 hours after successively^ when, thb
stars have reversed their positions, and if there be a variation in.
their differences, it shows a deviation in the instrument, wMch'lllsJr
be corrected by substituting ha]f the difference for die error, ^d
repeating the trial by approximation till the adjustment id oomplel^
. 4f some of those stars whose apparent places are given 3B tte
Nautical Almanac be selected, the operation will be cmmMatiVciv'.
easy. These m pairs, are ; 1^ « Cassiopeiae and ) Ursse Mi^ons ; 2^
Polaris and {^ Urss Majoria ; 3, Polaris or « Arietii and 'd I/rftijonftt^;
4, Capella and m, Herculis ; 5, <3 Tauri and /6 Draoonis ; 6, i8 Auti^
and y Dracpnia ; ?» Pollux and y Aquilse. No doubt somd'<if tbese
can only be to observed in very high northern latitudes ; and, there-
fore, recourse must be had in some instances to other tabled- endi-
a« those of Dr Pearson.*
Itsometimet happens that an observer has not aioimtmand^of
t)ie whole meridian, especially if he has not an observsfany p'^P^'^Jf
adapted to the purpose, yet may find it necessary to take transits wr
die regulations of clocks or chronometers. In this case' reeonrea
must be had to the sun, and to pairs of high and low stars hi ~ '
nearly the same right ascension. Having, by the sun and a
watch or chronometer, placed the instrument nearly in the merit
dNerve the transits of two stars having nearly the same righf^siittb-
sionu but differing at least 30^ or 40° of declination. Now if the
interval between their passing the meridian in sideriid'tiiQe berexv
actly equal to their difference of right ascension, the instarttmeik ia
truly placed ; if not, it wants correction.
■ Ii, when the latitude is N. and the stars S. of the zenith, the higft^
est star come first to the meridian and the interval between the tiiahlw
sits be too great, it deviates towards the west ; if too small, towards
the east.
But if the lowest star come first to the meridian, and the interval
between the transit be too great, it deviates towards the east; if tifio
small, towards the west. La either case there is required a correc-
tion^ which fnay be computed in the following manner : —
Rule. — ^To the secant of the star's declination add the sine a£ the
different of the latitude and declination, if they are of the same ndM,
or ^e sin(s of thdr Auim, if they are ol different names s of the sum of
w&ich find die ^ natural numb^. To the logarithm of the sum of
i^ese add tbe arithmetical complement of the lo^parithm oi their dif-
iE^rence, and'lllie logarithm of the difference between the excess ^of
the right ascension of one star above that of the other, and the-^b'^
served interyiii of time between the transits, the sum will be thole^
gantiun of^jx arc in time. /.
'jtlall^the.^um of the excess of the right ascension of the one star
above the other and the foregoing arc, will be the deviation at the
* Periiaps the catalocue in the Nautical Almauac mi^^ht be extended and the selec-
tion more judicious, hot example, the places of some of the smaller stars in Orion
might be properly exdiangcd for either circumpolar or high and low stars.
SPHERICAL. TRIOONOM£TRY.
135
]fiim^MBMi,MnA half the difibrence betweenthese will be the-deviftt
tiqi|.at,4»^hi|Sfae8t. - '
r^.ji^^^fii^ovk in time at each star beinjf now known, the imthi-
meat .nuMF be easily rectified by either, or both of them on the fol-
}lff^iisig itight, or still more readily by a third star on'the saine evm^
^^;^;er>]f the telescope is sufficiently powerful to show stari in th^
d^fp'Mtthe corrections may be performed at any time in a fM
faooaasive hours. For the deviation of one star being known, that
i^faus/Qmt may be computed by the following —
tj,.i2«iff.«— To the logarithm of the given deviation add the cosine of
t^i^xyesponding star's declination, the secant of the declination of
ijie wrd star, the cosecant of the sum of the latitude and decUnaticm
^f ,the firat star if they are of different names, or of their dSStirence if
^^ev.ape of the same name, and the sine of the sixm of the latituQ^
ana dedikiation of the thiid star if they are of different names, or of
lj)ieif. difference if they are of the same name; the sum of tliese wm
bie llie logarithm of the deviation in seconds of time at the Aird star.
. JE<r»-^On die 1st of March, 1826, at the observatory of EdifabUrgb^
^ latitude S5P 67' SI" N., I observed the transits of CaneHa and
Jligel, on the same evening, about a quarter past 6, and fotfnid the
^terval between the two transits 2*.5 less than the difference b^ween
Ijh^, true apparent right ascensions, as given in the Nautical Alma-
nac ^ nequired the deviation of the instrument at either star, aitd
41a^atA third, as Sirius? '
i;Mitade 65«6rN. 55*67'N.
5^ q£ Capella 45 48 N. sec. 0.156664 Rig. 8 25 S. sec: 0.094703
10 9 sin. 9.246069 sum 64 « *slA.'9.95S0fe
.: . -■ .7 .. «
iS^feisepoe
Ji^ M^ i^iumber a2628
j). Nat;.,piimber 0.9114
9.402783
- • V
76
.9.989706
i • ■>
1.1642 log. a065953
0.6586 ar. co. 1.181478
Jt^liTfrenQe
- «■• I
Arc in time
Sum
PiArtaoe
4.42 log. 1.645346
6.92 half s 3*46=: the deviation at Itigel.,
1.92 half==0.96£=thedeviatiohAfCii|k^
Now aince the highest star comes first to the mericBah,' am the
intarrtl iMtween the transits is too short, the de^tions are easterly,
•iiOftib^' stars hiMi been between the zenith and iihe north polel' the
AflfftediMis would have! been weHerly.
< ic8ittide' it has bieen found necessary to fix the instruttient as soon as
piaiflllld^ wis shall proceed to compute the deviation at the thirdjstar,
which can be easily done, as we nave an hour and three quarters
iirtarlyto' perform die calculations and complete the arrang^j^ts;
- jdix ^ji!J i)i»«« t/j>. I •■
iftii' f ■■• 'Til'', Ti;ji..i-
■I., '
136
INTRODUCTION.
Declination of Rigel (a) 8' 25' S.
Latitude (b) 55 57 N.
Declihaticm of Siriiis (r) 16 99 S.
First 8um^ or^ (<'+^)
Second sum^ or (6-fc)
Deviation at Rigel
64 22
72 26
3-46
cosine
secant
cosecant
sine
log.
Deviation at Sirius 3 .77^ ^og.
. 9.9MM7
Oj01«B6
0.044006
9.979200
0.530076
0.676854
After having corrected the instrument by means of Sirius^ I ob-
served the transits of Castor and Procyon^ and again those of
Procyon and Pollux^ and found the interval of time to agree with
their difference in right ascension^ from which I concluded^ that
in the iqpace of about three hours I had placed my transit instru-
ment exactly in the meridian.
As it is rather a difficult operation to fix a transit instrument ac-
curately in the meridian^ these operations should be repeated a con-
siderable number of times to insure the utmost possible accuracy.
After the observations prove satisfactory^ a meridian mark may be
put up in a horizontal direction at a considerable distance^ with
which the central wire may be frequently examined and rectified
previous to any very nice observation. This mark ma^ be of
various constructions, such as a copper-plate with a hole m it^ so
as a small segment o£ light may be seen on each side of the veiv
deal middle wire, or a small notch in a building, or even a post
at some distance. A thin slip of brass or copper painted blacky
with white lines or divisions at every inch, and numbered throiudh-
out, will also be found very convenient, and by knowing its ms-
tance the deviation upon it may be computed.*
The transit instrument being now properly rectified^ it will be
found the most accurate of all for determining the error and rate of
a clock or chronometer, by taking the transit of the sun or stara
daily, and marking the difference regularly in a column prepared for
that purpose. If a star be observed, siderial time must be reduced
to mean solar time by Table XXXI. when necessary.
£x, 1. — The observed times of the sun's passing the meridian of
the observatory were as follows : — ^What was the onginal error on the
last day of observation and the daily rate ?
March
1888.
OIm. Time.
Sun's Transit
1
2
3
4
5
6
0^25
25
25
24
24
24
27M
16.6
5.4
54.0
42.0
29.8
Mean Time.
Ai^ Noon.
Qh i2« 40'.7
12 28.6
12
12
11
11
16.0
3.0
49.5
35.6
Chronometer too
0^ 12" 46*.4
12 48.0
12
12
12
12
49 A
51.0
52.5
54.2
Dul7B«te.
+ 1'.6
+ 1^4
+ ].«
+ 1.5
+ 1.7
Nf ean daily rate is therefore
And the original error at noon, on
is
5|7.8
+ 1.56
the 6th of March, 1826,
0^ 12" 54-2 fast.
• Hot. deviaticm ss sec. alt. x cos. dec. x obs. difF. of time x 15, to radias 1. On
Captain Kater's plan, by contiacting the diameter of the object^lass by some ccmtriv-
anee for that purpose, the meridian mark may be only a few feet distant.— See his pa-
per on the Floating Collimator.
SPHEBIGAL TRIGONOMETRY. 1^
^BmtmiiU error, suppoaiiig the rate to remain umfgniv A^X*^f(f, WV.
moderately distant futore time^ be determined. ^. J^'^^ ^
fdBl0i dht^On the same evenings the star Eigel passed the ifienfm^,
as follows :— Required the daily rate and the original error on t!fie
sfai|h4rt.ihe time of observation^ about 6 o'clock in the evening ?
Obt. Time.
8on*s Transit
6»» 42- 56'.6
6 30 2.3
6 36 78
6 31 13.4
6 27 19.2
6 28 26.0
DtdlyDUE of
Star's Transit
DUE of Mean and
Siderial Time.
3™ 54\3
3 54.5
3 54.4
3 54.2
3 54.2
3" 55'.9
3 55.9
3 55.9
3 55.9
3 56.9
mdljIUta,
I
+ 1.6
+ 1.4
+ 1.5
+ H
51 7-ft"- :.-•>*«
!»-.
IBM br the
' %^ the sun
,-* ■
+ 1^
2i3.14
ifeail rate by both
S^y R. A. a^ noon^ on the 6thj
'^^ ~. part of daily var., to ff".
T :.:•■.
+ 1^7
23^ e»21-.4
-h 55.5
B^dace4tlvA.
S^tr^is' ft. A. t>y Nkiitidll Almanac
^parent time of transit
^nation oTt^e .
Sfiean time of transit of star
Ylun^ of transit by chronometer on the 6th
Error of chronometer^ fast by star^
Allowing for change of rate in 6^^ by sun>
93 7
5 6
10.9
12.6
5 58
+ 11
06.6
36.6
6 10
6 38
31.9
25.0
12
12
53 .B
54.6
8.4
12 64.2
+ 1^7
Mean error at ff" fast
WidrailaHy rate of
As opportunities may not occur daily for celestial observationis^ it
is in that case necessary to compare a cnronometer with a good clock,
the rate of which can be depended on, and is occasionally ascertiun-
ed by the heavenly bodies.
£x. 8.— ^iven the daily difference between a chronometer and a
clcf^, the rate of the clock being occasionally determined by celes-
tial observations ; to find the error and rate of the chronometer ?
fi
(B)
I-:i
mJ .• ■%-»■■
-■J ?.- ■ ..
M. •
138
INTRODUCTION.
Oock before
Chr<XLdilfen
Chron. bef<»e
1826.
Mean Time.
from Ckxdu
Mean Time.
DaUrlUte.
May 1
+ 9.5 •
+ 2'.5
+ ir.o
2
+ 8.9
+ 3.8
+ 12.7
+ 1'.7
3
+ 9.4
+ 5.2
+ 14.6
+ 1.9
4
+ 9.8 *
+ 6.5
+ 16.3
+ 1-7
5
-1- 10.1
+ 7.9
+ 18.0
+ 1.7
6
+ 10.5 •
+ 9.3
+ 19.8
+ 1.8
5i+ 8.8
Mean daily rate ^ ] .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 titne^ so far as its steady rate can be depended on.
The dock was examined by celestial observation^ only where the
asterisks' are placed^ or on the Ist^ 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 chronometers are tried at Greenwich.
Table of the variations of the sun's R. A. and dec. in 1* for every
month in the year.
Month.
Var. in R. A. for
1 Second.
Var. in Dec for
1 Second.
January
February
March
April
May
June
July
August
Sept.
October
Novem.
December
0'.0029
0.0027
0.0025
0.0026
0.0028
0.0029
0.0028
0.0026
0.0025
0.0026
0.0028
0.0031
0".008 N.
.014 N.
.016 N.
.014 N.
.009 N.
.000
.006 S.
.013 S.
.016 S.
.015 S.
.010 S.
.002 S.
This table will be useful when the change of the sun's R. A or
D. for a few seconds only is wanted.
MENSURATION. 139
PART III.
MEKflURATION^ SURVEYING, &C.
Section I.
Mensuration of Surfaces,
Mensuration is the application of Arithmetic to Geometry^ by
which the values of geometrical magnitudes 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 g^iven magnitude is the measure of that
magnitude.
See Leslie's Geometry, Book V. Prop. XXV.
1. To find the area of a parallelogram, multiply the length by the
perpendicular breadth.
2. Triangle. — Multiply the base by the perpendicular altitude ;
half the product is the area. Or take half the product of the two
sides and the natural sine of the contained angle. Or when the
three sides are g^ven, multiply half the sum of the three sides, and
the differences between that half sum and the three sides together,
the square root of this product will be the area. This may be perfcurm-
ed readily by logarithms.
3. Trapezium, — ^Multiply the base into half the sum of the per-
pendiciilars.
4. Trapezoid. — ^Multiply half the sum of the parallel sides by the
perpendicular distance oetween them.
5. Irregular Polygon, — ^Divide it into triangles, find their areas,
the sum of these wifi be the area.
6. Regular Polygon, — ^Multiply the square of the side given into the
proper multiplier for areas from the table, page 142, 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 this by the number in the whole polygon, the product is
the area.
7< Circle. — ^The diameter is to the circumference as I to 3.1415926536,
or 1 to 3.141593 nearly.
The circumference is to the diameter as 1 to 0.318309.
The area is equivalent to the square of the diameter multiplied
into 0.7853^.
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
degrees in the arc.
Or, from eight times the chord of half the arc subtract the chord
of the whole arc, one third of this 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.
140 INTftOBUCTION.
11, ParaboitU'^The area is equivalent to two-thirds of the pro-
duct of its base and altitude.
12. EUipsc^-^The area is equivalent to the product of the trans-
verse axis into the conjugate axis multiplied by 0.785398. Peri'
j»Aer^.— 'Multiply the square root of half vie sum of die squares of
the two axes by 3.141693^ 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 ? Ans. — 33.0625 feet
2. Bequired the area of a rectangle^ if the length is 1376 links and
the breadth 950 ? Jim.— 13*' 0" lO".
3. Required the area of a rhombus^ of which the length of the
side is 12.24 feet and height 9.16^ feet?
Jn^.— rll2.1184 square feet.
' 4. Required the area of a rhomboid^ of which the length is 7 ^^^
9 inches^ and height 3 feet 6 inches ?
Jm.— 2r* 1*" &*•
5. Required the area of a rhomboid, of which the adjacent sides
are 2535 and 1040 links, and the contained angle 30^ ?
Ans.—I2r (T 29^
6. Required the area of a triangle, of which the base is 1225 links
and altitude 850 ?
Jns 5« O' 33P .
7* Required the area of a triangle, of which two of the sides are
30 and 40 and the contained angle 28° 67' 18'' ?
J«*.— 290.47366.
8. Required the area of a triangle, of which the three sides are 20,
30, and 40 feet ? Jnj.— 290.4737 square feet
9. How many acres are there in a triangle, of which the three
sides are 380, ^, and 765 yards ?
Ans.—9^ ff 30^.
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 d6%et
, hjgluQn the other side ; required the breadth of the street?
Jjtfi— 76.1233 feet
11. How maxrjT acres are there in the trapezium, of which the dia-
gxmal is 476 linKs, and the two perpendiculars fidUng iipoB it on op-
posite sidoa^ 5225 and 360 links respectively.
Ans.—ldT' 2' 25^.
12. Required the area of a regular hexagon, one of whdee equal
sides is 14.6 feet and the perpendicular from the centre 12J04 feet.
ilM.--653.632 feet
13. If the diameter of a circle be 17 > what is the circumference?
Ju*.— 53.4072.
14. If the circumference of the earth be 24850 miles, what is the
diameter ? Ans,— 791 0.
15. If the chord of an arc be 30, the height or versed sine 8, what
is the length of the arc ?
Ans. — 35 J.
16. Required the length of an arc of 57° 17' 44".8; the diameter
of the circle being 25 feet ?
Ans. — 12.5, which is equal to the radius.
MJENSUAATION.
141
17- Required the area of a circle^ of which the diameter is 15i
feet? Jw.— 81.1798.
18. Required the radius of a circle in yards^ of which the area is
an acre ? Jn*.— ^^.
Id. The diameters of two circles are 16 and 10 ; what is the area
of the ring formed between these two circles, the centre being ccnn**
mon to both ? Ans 122.5224. •
20. Required the area of the sector^ whose height or raised sine is
4 and the diameter of the circle 16 ?
^ii*.^83.5103.
21. Required the area of the segment of a circle, of which the
chord is 16 and the diameter of the circle 16f ?
Jnj._7O.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 362 links ; AQ is 745 links, QD
is 595 and AB is 1110 links ; required the area of the field ?
yiiw.— 3^ 3'- 35^
TO FIND THE AREAS OF CIRCULAR SEGMENTS.
jRtf/^.— Divide the height of the segment by the diameter, and
find the quotient in the column of heights in Uie following table :
Take out the corresponding area in the next column on the right-
hand ; and multiply it by we square of the circle's diameter, for the
area ci the segment.
TABLE OF THE AREAS OF CIRCULAR SEOHENTS.
Height.
Area of the
Segment.
.01
.02
.03
.04
J05
.06
.07
.06
.09
M
.00133
.00375
.00687
.01064
.01468
.01
.0241
Height.
Ares of th(
SigmeDt.
Hoioh* lArea of the
"®*8**^ Segment ^
.047011
.05:
.11
.12
.131
.14
.15 .07387
.16 .08111
.17 .08853
.18 .09613
.19 .10390
.20 .1118S
.21
.22
.23
.24
.25
.26
.27
.281
.29
.119901
.12811
.13646
.14494
.15354
.16226
.17109
.18002
.18905
Height
.30| .198171
.31
.32
.33
.34
.35
.3^
.37
.38
.39
.40
Area of the
Segment
i20738
.21667
.22603
.23547
.24498
.25455
.26418
.27386
.2835
.2933
Hel^t
6
3
i
.41
.42
.43
.44
.45
.46
.47
.48
.49
.50
Area of the
S^imuit.
.30319
.31304
.32293
.33284
.34278
.35274
.36272
.3727(
.3827(
.3927(
£x, 1. — Taking as an example the chord 12, and the radius 10,
or diameter 20.
And havinff found the perpendicular from the centre upon the
chard = 8; Uxen 10 — 8 = 2. Hence, by the rule, = 2-1- 20 = -1
the tabular height. This being sought in the first column of the
table, the corresponding tabular area is found = '04088. Then
•04048 X 20« = -04088 x 400 = 16.352, the area.
The use of the following tables will be readily understood, fVom
considering that the areas of similar figures are as the squares of
their like dimensions, and their solidities as the cubes.
142
INTRODUCTION.
TABLK OF POLTGOK
IS.
NACf
Sido.
3
4
5
6
7
8
9
10
11
12
Names.
Multipliers for
areas.
Radius of circum.
circle.
Factors for
sides.
Trigon
Tetragon, or Square
Pentagon
Hexagon
Heptagon
Octagon
Nonagon
Decagon
Undecagon
Dodecagon
0.4^30127
1.0000000
1.7204774
2.5980762
3.6339124
4.8284271
6.1818242
7.6942088
9.3656399
11.1961524
OJ>773503
O.707IO68
0.8506508
1.0^00000
1.1523824
1.3065628
1.4619022
1.6180340
1.7747324
1.9318517
1.732051
1.414214
1.175570
1.000000
0.867767
0.765367
0.684040
0.618034
0.563465
0.517638
Section II.
Mensuration of Solids.
1. Prism, (1.) Surface, Multiply the perimeter of one end by
the length or height^ tne product will be the surface of the sides.
To this add the areas of the two ends, and the sum will be the whole
surface.
(2.) Solidity or Capacity, Multiply the area of the base by the
height^ the product will be the solid content. The same rules de-
termine the surface and capacity of a cylinder.
2. Pyramid or Cone, (1.) Surface, Multiply half the perimeter
of the base by the slant height. To this add the surface of the base^
the sum is the whole surface.
(2.) Capacity, Multiply the area of the base by one-third the
perpendicular height.
3. Frustum of a Pyramid, (1.) Multiply half the sum of the pe-
rimeters of the two ends by the slant height. To this add the areas
pf the two ends, the sum wiU be the whole surface.
(2.) Capacity, Add a diameter or side of the greater base to one
of the less ; fi*om the square of the sum subtract the product of these
two sides or diameter ; multiply the remainder by a third of the
height^ and this last product by the proper number for the circle,
.785366, or polygon, the last product will be the content.
4< Sphere* (1.) Surface, Multiply the square of the diameter
by 3.141593, the product is the surrace.
(2.) Capacity, Multiply the cube of the diameter by 0.5236, or
the cube of the circumference by 0.016887.
5. Spheric Segment, (1.) Surface, Multiply the circumference
of the sphere by the height of the s^ment.
(2.) Capacity, or c = 0.5236 h^ i3d^2 A), in which d is the
diameter of the sphere and h the height ; or c = 0.5236 k^ (3 r^ +
h^y; in wjhich r is the radius of the base of the segment and h its
height.
6. Paraboloid, or solid formed by the rotation of a parabola about
its axis.
Capacity, Multiply the base by its height, half the product is
the content.
7. Spheroid, or solid formed by the revolution of an ellipse about
one of Its axes.
MENSURATION.
143
Capacity. Multiply the square of the revolving axis by the fixed
ixis^ and the product by 0.52B6, the result will be the cootent.
8. Megulnr, or Platonic bodies^ as they are sometimes called^ are
x>ntained under like, equal, and regular plane fibres, of which the
solid angles are all equal. The names and descriptions of these'bo«
dies, together with their multipliers, the side of each being unity,
are contained in the following tables :•—
Surfaces and SoUdUitt of Regular Bodies, the Side being Unity y or 1.
No of
Kdes.
Name.
Suxftce.
Solidity.
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
2 1816950
Thediam. of a sphere
being 1 ; the side of a
That may be in-]That may be circum-
scribed ill the scribed about the
sphere, b {iqoare, is
Tetraedron
Hexaedron
Octaedron
Dodecaedron
Icosaedron
That is equal
to the sphere,
is
0.816497
577350
0.707107
0.525731
0.356822
2.44948
1.00000
1.22474
0.66158
0.44903
1.64417
0.88610
1.03576
0.62153
0.40683
Examples for Exercise,
1. Required the solidity of a cube, of which the side is 5 feet 3
inches ? Ans. 1441^^ feet
2. What is the solidity of a block of marble, of which the length
is 10 feet, breadth 5f feet, and depth 3^ feet ? Ans. 201^ feet.
3. Required the solidity of a prism, of which the base is a heiiii
gon, each of the equal sides being 1 foot 4 inches, and the length of
me prism 15 feet? Ans. 69.282 feet.
4. Required the convex sur£Eu:e of a cylinder, of which the cir-
cumference is 8 feet 4 inches, and length 14 feet ? Ans. 116} feet.
. 6. WfaAt is the solidity of a cylinder, of which the length iiSi 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, <nd the
slant height 20 feet ; required the convex surface ? Ans. 141.372
feet
7* Required the convex surfiu^ of a frustum of a right cone, the
circumference of the greater end being 30 feet, that of the less XO
feet, and the slant height 20 feet? Ans. 400 feet.
8. What is the soHdity of a triangular pjrramid; of which '^e
heiffht is 30, and each side of its base 3 ? Ans. 38.97>
£ What is the sdlidity of a cone, of which the cireamferenpe 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 whicli Vhe
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 pjrramid, one
side of the greater end being 18 inches, that of the less 15 inches,
and the height 5 feet ? Ans. 16380 cubic inches.
144 INTBODUCTION.
13. Required the oonyex miperfideB of a sphere^ of which the dia^
meter i$ 17 indies ? Ant. 907*W square inches.
la Sequiz^ the solidity of the same ? Ans. 148868 cubic feet.
14h Required the solidity of the earthy considering it as a perfect
sphere, of which the diameter is 7910 miles ? Ans. 259130798130
cubic miles.
15. What is the solidity of the segment of a sphere, of whidi the
diameter of the base is 20 feet, and its height 9 feet? Ans. 1795.4244
cubic feet.
Section III.
Surveifing,
In landsurveying, the instruments commonly employed for the
ordinary purposes are —
1. Gunter's chain, and ten iron pins.
2. Cross-staff, and signal staves.
3. Field-book, or paper.
4. Case of mathematical instruments.
5. Plotting scales.
6. Parallel ruler, and beam compasses.
7* A small quadrant, if a theodmite is not at hand, to reduce
the hypotenusal to their horizontal measure.
It would exceed our present limits to describe all these, as well
as some others, which may however appear perhaps in a work pro-
posed with that view.
An Example of Laying Off a Field,
Havinff set up poles at A, ' ^ ^
B, C, and D, so as with the J?-T'^^»'^i*^
different dotted lines to re- Jl/^ ' ^'^'''"^^ ? , „^
duce the body of the field /^-<. %^^ ^V
to a quadrilateral form, and / ® 2 ***•••... * I* I
drawn a sketch of it, into Ul "^^"•••.. '•«^??
which the measures when j^ ^ o ©'''■^^-g.^ ^^\
taken may be inserted ; be- / qlT 5 S *!....^.S!.Vj!r;.;^^^
gin at any point A, measur- b .-s;---^-: l^^^"^
fng the successive distances ^ss.^ ^j ^ ^^
A a, A c» &c., on the chains ^ g S
line A B, and the corre- '^ S
sponding offsets ab, cd, &c., and marking them as in the figure till
a complete circuit A B C D A of the field and the diagonal A C are
jtneasured; these afford data for planning it, and computing the
krea. For ihe various portions may be considered either as trapea«
Olds or triangles, whose contents may be ascertained by the rules
ffiven for that purpose.* The area computed in this manner will
Be 2.4295 acres, or 2 ac. 1 ro. 28.72 po,, though it is better in gene-
ral to retain it in acres and decimals. It is necessary to take an ac-
icount of the roads, dikes, ponds, &c., of which the contents must all
be stated distinctly by themselves when a whole estate is surveyed.
In the case of the sale of crops, that in tillage only must be measiiredi
Required the plan and area of the field, from the following field-
book, in which the angles were measured, with the pocket-ibcnL,
sextant, and the distances with the chain, begrinning the operations
at the gate near the south-east comer ?
* Many Undsurveyors first construct an accurate plan, from which, by scale and
compass, the area is obtained with sufficient precision ; ana this is at least a good me-
thod of checking the result by computation.
SUHVEYINO.
145
tield^Bodk.
MaMtaMlik
MM
T.
fle^ige.
Dcadrigtfs* or
Crosshail lands
on tlie south or left
hand
88 143
0l8t99^46'S(y'W.
73 200
83 240
70 300
34 400
480
510
44 050
42| 736
810
'^•^
*ii
Remark. The
line bears nearly WjMi
along the north $ide
of Ktterick Sy&e.
i^tai^>-^MW
2
5
3
10
©2d 66° 43' 30" N.
200
400
600
800
860
866
Boundary.
Hedge.
Hardacres land
on the north
or left hand
w^swww
50 100
66 200
83 264
30 360
66 456
5 544
ISOi 700
The chainF-line bears
nearly north.
12
S8
To O 1st, or
75^
The chain-line bears
nearly east.
©4thlOP23'0"S.
lOO
200
30O
400
lOef 500
143 600
Area s= 6.14597 ac.
or ac ro. 23 po.
The chain-line bears
almost south along
the road from Green-
law to Eccles. The
diagonal from Ist
to Sd, measuring
1053 links, was also
taken, that the area
might by the three
sides of tne trianffles
be a check upon mat
determined ntim us-
ng thie angles.
dSc0iy ditches, or fences of any kind, they must be
^dofi^g the Surrey, and their amount stated. Also planMaoni,
roads^ oommons, lakes, ponds, &c, must be all surveyed and dftned
separately from the arable lamL For these we cannot faexe enter
intfr dated.
■M.
* This place It mentioDed in Sir Walter Scott'i Minstielty of tbe Soottiab Betdct.
146
INTBODUCTION.
Levelling.
It Ib often necessarv to ascertain the difference of elevation of one
point above another^ tor the purpose of conveving a stream of water
to drive machinery. This may be performed in several waysj but
the readiest and most acurate is by means of a spirit-level of thf best
construction.' It must be accompanied by apole^ or rod divided bata
feet> and at least hundredths or a foot. On this rod a sliding vane
is fitted/ capable of moving easily up and down^ and having a dark
strong line or other well-(&fined mark upon it^ by which the teles-
cope^ or in common levels the sights may be directed. The slider
must be moved upwards or 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 hundiredth 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 tne levels 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
called the back observations; and those taken towards the place
where he means to finish^ are called fore observations^ for the sake
ot distinction. Since the pole is always placed at equal distan*
ces from the levels no allowance need be made for the curvature
of the earth.*
EXAMPLE.
Back.
Dist.
Links.
100
100
200
200
150
150
100
50
Height
on!
uebt Heigi
I\>lei onFo
Fore.
Height
1050
2
2100
Feet.
2.92
1.56
0.48
1.35
1.27
1.34
2.36
3.28
Feet.
4.68
3.79
5.63
4.86
3.74
2.56
3.94
4.36
14.56 33.56
14.56
19.00
Dirt.
Links.
100
100
200
200
150
150
100
50
Hence the difference of level on a sloping height of 2100 links of
Gunter's surveying chain^ or 2100 x 0*66 = 1386 feet, is 19 ftet
When a spirit-lev^ exactly adapted to this purpose, is not at hand,
if there is a theodolite to be had, it will perform the 'operati<Mi>
though it is not quite so convenient.
* The difference of level is about 8 inches in a mile, which increases as the'sqnare of
the distance. The dificrence of level in feet allowing for refhiction, is { of the square
of the distance in English milc^.
RULES AND FORMULiE. 147
In cftse of leyelliiig far canals^ the process is not different^ only the
enuil is cfldrried cm an exact level, by judiciously choosing the situation
winding rofand rising grounds, conveying it across ravines by aque-
duct bndMs, and allowing it to descend at particular points, by means
of tbicilu. Koads ought to be carried along a level line as neariy as pos-
8ifaile» atld only havinff gentle acclivities and declivities, liiis may
be readily obtained oy following routes somewhat circuitous in un-
et^ parts of the country, taking the advantage of ravines, water
odunes, and the sides of lakes ; for a greater distance on a road nearly
lerdly if productive of less expense of animal strength, than by paa-
Bxng over considerable elevations. All very quick turns in the road,
pHxtic|darly when entering upon a bridge, ought to be avoided, as
the danger from centrifugal force, whioi may be readily estimated
bj the fmnula. Part III., Sec. IV., is considerable. The justice of
these Telharks may be readily appreciated by considering many parts
in moat of our public roads wnich have hitherto been constructed
iip6n the very worst principles, having been entrusted to what are
csalled practieid men, who are frequently the mere slaves of custom.
m
Section IV.
Rules and Formuke.
When two angles of a plane triangle are known the third may
be found, consequently, for general purposes, it is unnecessary to
measure tibe 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 6e 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, ifpossible, 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 equal, and not less than twenty 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 ^rmula. 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- B,
served, and the distance C P be measured, it is re-
quired to find C, the measure of the angle A C B?
Suppoie AP.B = P, BPC=p, CP=i/, AC =
I>mdBG?=iy.
Siace die exterior angle of the triangle A P I is
equal to the sum of the two interior and opposite C P
angles, AIB = P+IAP, and of the triangle BIG, the exterior
angle A. I. B = C+C B P. Making these two values of A I B
equal, by transposition, we have C— P = I A P — C B P. But.
— ' ■■ ^ ■1 ■ ■■ ■ ■ I ■■11 11 III . ■
* For a dflmoDBtratioa of these properties, see voL 111. of HutUm's Ckrnne of
Mathemadet. ' - -
148 INTBODUOTIQN.
Um Kianglcf C A P, C B P give dn.CAPsaiikIAF =
^ on. A P C=?L'*5lg±fk . dn.CBP=5| rfn. B P C ?=
!L«gvE. And .ince the «4fle. C A P. C B P. «re, by lvpo««A.
alw«y« very small^ their sines majr be substituted for Aeir ttei^
liepoe, C— P = y^ ^^ -^ — g;-^ wUch in sacoodt becomes
gi_ 1 "°' (P+f) ~ "g;^ }.; or R^^ being tfaelepgA of m wlii
seconds equal to the radius, or 206364^8, then CU-P :^'Bfdx
f suL(P+ £j — ^^1. The use of this formula cmnot be ahbar.
assing, provided the signs of sin. p, and sin. (P+p) be pcoperly
atleudea to, as is illustrated by the following example >^Let4dl)e ol>-
served angle P be 43<' 59f' 4^".i/i. p = 264'' 41' 34^ d ^ 10.706 feet,
D =; 676& feet and D' =: 66750 feet, required the reduction^
(1.) (2.)
Log. R'' 5.314425
log. d 1.032860
+ 6.347285 *^6347985
308 aS'"i|'/ }-"9-883118 sine p 264° 41' 24" — 0.096132
ar. eo. log. D+ 5-240272 ar. QoAog. D' . . + 6.1755A
m^tm
(1.) .-30" 24a-l«480675 (2.) + a3M87 + 1.520066
(1.) — 30 .246
fttm
C— P = + 2 A41
P . 43*' 52' 40 .440
■*•••■
C 43 52 52381
When signals are circular or polygonal towers, various methods
may be employed to find the true angle, from a due consideration of
the nature of ue case, which, to any one possessing a knowledge of
the 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
formed a spherical triangle, of which the three sides are given to
compute the anffle at the zenith, which may be performs by the
rules of spheric£u[ trigonometry.
VI. A spherical triangle being proposed, of which the three sSdes
are very small comparea with the radius of the sphere; if from eadi
of its angles, one-third of the excess of the sum of its three angles,
above two right angles be subtracted, the angles so diminished
may be taken for die angles of a rectilineal trian^e, whose sides are
equal in length to those of the proposed triangle.
To find the spherical excess when the three sides are given in feet.
1. Rule.^^liO die constant logarithm 1.34^^, add the logarithm
of half the sum of the three sides, the logarithms of the three differ-
ences between these sides and that half sum, half the sum of these
five logarithms will be the logarithm of the spherical excess in se-
conds.
BULKS AND FCWMUUE. U0
& Tp tte kig«ri|Jim of tbe am of the triaogle tafcM ^^jiliine
fine in IbfC «dd tke constoit Warithm a074a8D; the Attn is tihe W^;
garithm of die excei0 abote 18ir in secondB.
3* If the bate and perpendicular of a triangle be^given. To the
bgvithv of the base m xeet^addthe logarithm of the perpendicular,
mi the constant logarithm 0^73660; the sum will be the logarithm
of the spherical excess in seconds.
He i|iherical excess anuHints to one second for an ai:ea of 76
Bnglish sc[uare miles> whence^ if the area in square miles be known,
the /nriwsxical excess may be readily obtained by dividing it by 76.
VII. To reduce a base on an elevated level to that at the sur£u»
gf tike sea.
hfit r represent the radius of the earth, corremonding to the bafe
h At the level of the sea, and T'\-a the radius reined to the level of
ti|0 measttred base B; then it is obvious that r^a \ r i iT^ i
6 = Bx-4--- Hence, B— 6 = B — B-4^=:Bx^— = Bx
r+tt r'\'a r+a
"- '——5 + &c Y But the radius of the earth being very great in
comparison of the difference of level a, we have the correction d suffi-
ciently accurate, by retaining the first term. Hence, } = B x -•
T
ICtf/Sp.— By logarithms. To the logarithm of the measured base in
fteb |bdd the logarithm of its height above the sea, and the constant
loffarithm 2.680110 ; the sum will be the logarithm of a number of
feet whidi, taken from the measured base, will be that at the level
of flie sea required.
Vni. To determine the horizontal refraction from observation.
12tf/e.-»-From the measure of the intercepted terrestrial arc, sub-
tract the siun of the two depressions at its extremities; half the re-
mainder is the refraction. If by reason of the smallness of the con-
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.
FOBMUUE.
^ — — 2 — 2 ^ ^^
If !-i^^ becomes an elevation, it changes its sign, and becomes
-fi eyiuid In that case R= -^^^^ — (2.)
TIbo iSMCt quantity of terrestrial refraction is very variable. It is
estJBUited by Dr Maskelyne at one-tenth of the intercepted arc. by
Selunbre at one*eleventn, by General Mudge at one-twelfth, and by
Legendre at one4burteenth at a mean state of the atmosphere. In
p^smiliar circumstances it varies very considerably from this, as from
CEDe-aixth 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
line is situated ; half the sum of these two measures will the angle
iMwived.-
This -nay ako be done^ though less accurately, by computing the
aumuth of tibe sun, or a star, when on the line, from an altitude
taken for that purpose.
160 INT&ODUCriON.
X. In additicm to what has already been raid relative to fiikL-
ing the latitude of the place^ we may add here^ that the ■ same
thinff may be very accurately obtained^ by observing the greatest
and least altitude or zenith mstance of a circumpolar star^ and cor-
recting them for the effiscts of refraction ; half the sum of the al-
titudes^ thus corrected^ will be the latitude^ or half the sum of the
zenith distances will be the colatitude.
XI. To determine the ratio of the earth's azes^ and their actual
magnitude from the measure of a degree of the meridian in two
^▼en distant latitudes^ supposing the ^urth a spheroid generated by
the rotation of an ellipse aoout its minor axis.*
Let d and df be the measure of two degrees^ d being the least, or
that nearest the equator, / and I' the latitudes of their middle points^
t the semitransverse axis of the meridian or radius of the equator^
c die semioox^ugate or semipolar axis, e the excess of the equatorial
radius above the polar semiaxis, and r° = 67°-29577d6, the. number
of degrees in an arc are equal to the radius.
Then, <? - 3 ^.^ ^^,_^^^ ^ ^.^ ^^,_^^ (1.)
^^I'^Sd sin. (Z'+O X sin. (/'—/) • • • - (2.)
e r^ d
If — =: I, ellipticity or compression, tz=i (3,)
* 1 — i — §icos.2/
When / is nothing, or when one of the decrees is at the eqoator
from formula (].) e zzl—^—, — ^j . , (4.)
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, sbomd 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.
sin. (f+0 (fi.)
If <{' and d are two contiguous degrees, so that t = /+1°, Uien
3 e
d^—d ^ —^ sin. (2 2+1°) sin. 1", and since the sine of one degree is
0.017463, «<'-d=?^^^^?^^^ sin. (2 /+P) . . (6.)t
The contiguous degrees therefore' differ, by a quantity propor^
tional to the sine of twice the middle latitude. The difference is a
maximum when 2 /+ 1^ = 90^ or when the middle latitude is 46^
From five different measures combined so as to produce ihe most
accurate result, Mr Playfair found f = 0J00QI2 =: oTk g nearly, and
the equation representing the degrees of the meridian setting out
from 46^ will be
D = 60759.472—290.576 cos. 2 /J .... (7.)
in fathoms, or.
• Playfair^s Outlines of Natural Pfailosophj, Vol. II. Art. 69.
t Using logarithms, eF^-d « C. L. 1.0084715 + log. sin. (2 l-^-l^) in fathoms,
or <i'— <I»C. L. 1.78662Sa4'log. sin. (2 2+1 <>) in feet, where e «. 11158.8 fathoms,
or 66958.8 feet respectively, and d zs 60460 fathoms, or S62760 feet.
X In toises D a 57011—272.65 cos. 2 L
RULES AND FORMUIJE.
151
P = 0e.(Mi-O.3299 COS. a /
in English miles.
Fithomi. Iflles.
Hence, ezs 11158.8= 12.680
i =r'348e858.8 = 3962.349
c = 3475700.0 = 3949.669
(&)
e
The radios of curvature for the parallel of 45'' = e-^=: 2481 279.4
fath. =: 3956.009 miles. The circumference of the meridian is
thftre ftfr e eoual to the product of the mean degree at 45^ hy 300 =
248S5^ nmes; and tne circumference of the equator is 24896.16
miles^ or about 40 miles more than the preceding.
A geographical mile is therefore 1012.6 fathoms^ or 6075.6 feet.
The seraidiameter or distance from the centre to the surface^ at any
latitude /, or r = ^ (1 — 1 sin. « /+ 1 1« sin. « / cos. « /.) . (9.)
If <f be a degree of the meridian at any point of which the lati-
tude is (y and D a degree of the curve perpendicular to the meridian at
the same pointy then^ ^ ^^~q (^"^) ^c* ^ ^* • • (10.)
i = r^ D— ^ (I>— ^) tan. « /.
t ^'•-2Dco8. «/
2D
(11.)
X sec. « /. (12.)
For exercise the foUowing measures of degrees of latitude are
given.
Bouffuer
Concuanine
Z<8UDton
Haaon
Boseovicfa
Ddambre
Mtidge
Swanberg
Lat.
DMreeain
Tbifles.
0^
12
35
39
43
46
52
66
5 ON.
18 OS.
12 ON.
1 ON.
12 ON.
220N.
20 ON.
56753
56749
56761
57087
56888
56979
57021
57069
57I68
Deductiona.
1
Radius of the equator
3271691 toises.
Semipolar axis.
3260964 toises^
Q = 5^30740 toises.
1 toise = 1.949037 metre.
1 French foot = 144 lines.
1 Englishfoot=135.0731mes.
Let these be solved by the foregoing theorems^ and the various
odkiaequences drawn.
i&r. >1. — ^In the Philoso]3hical Transactions for 1795^ D the degree
perpendicular to the meridian^ is given equal to 61182 EngHsh ft*
thodui; d = 60851, and / = 50^ 4' N. By formula. (12.)
331 1
* ^ g x " gil82 ' ^ '^' ^ ^"^ ^' " rSii '^^■^^y V «»d «n«ch too gretft
jJBa. aL^The length ^ a degree in latitude 52^" 2^ 20^^ N. is 57Q74
toises, that in 1 1"" O' N. is 56755 toises ; required the elliptidty by
(Viraiula(2.)?
1 ■ .
162 INtRODUCTION.
/' = S2° 2" 20"' const log &.fiaifi79
/ =11
r+l z=:63 2 20 cosecant, 0.040069
I'— I = 41 22 coMcant, 0.1827T8
d = 56755 ar. co. log 6.346996
d"-^ 319 log 3406091
f == 0.003202 log. 75Q63fiS
33 1
*" = 10066= ms "^^y-
If L =£: the length of a degree of longitude, thtn
L = ^1^1 + . sin. «/+!•« sin. * 0- • • . (l3-)»
If the value of the degree is wanted in toises, fathoms, or feet, the
ficflsond member of this equation must, be multiplied by the semitrans-
verse axis in the same measure.
Ea. 1. — Required the length of a degree of latitude at Edinburffh,
in5^*N.?
Byfbnniila (7), D =: 60769.472+290.576 x sin. 23" a 6O760>I37
+ 290.576 X 0.374607 = 60868.3 fathoms.
Ejf, 3.— 'Required^the length of a degree of longitude iu latitu^
56'* N., the ellipticity being ^g^ ?
By formula (13), L = ^^^^ I +^ xb.68694 }; orL^
0.009760 X 1.00329 x 20918760=204636 feet, or, taking in the aeoQOd
term mentioned ifi the note, it is 204648 f^et. These fomiidn are
itseful for fixing the latitude and longitude of a pafticukr point wlMSli
referred to some obje^ whose situation has been well ae^SarvusMi,
such as many places in Bi4tain are by the trigonometrieal mxrrej
iki ibh case any amateur observer may Verify the latitude and loi)gih
tude of his observatory deduced horn his own obsetvalaonay by a
comparison with some point well settled in that work, when pif^
perly connected by trigonometrical operations. Even by takhsg a
few angles with great care, the situation, of a particular pcnnt may bt
well settled bv spherical trigonometry, as in the foHowing ^Uttple
cokiununicat^a by Captain Hall.
Ex. 2.— Giv^n the latitude of the Staff on North Berwick Lmt,
500 3^ g// ^^^ longitttde 2^ 42" II'' W., and the latitude of the lale ot
of Mny. light 56» W 22'' N., longitude 3" 32' 47'' W.> tfasaule at
HcitA Berwi A Law> between the Isle of Mary aad^Ditnglssf TWwer,
87^ 41' 1", that at Dunglass, between the Isle of May and North
Berwick Law, being 37'' 20' 13" ; required the latitude and loDgt-
tude of Dunglass Tower ?
Ans. Lat. 55° 56' 31"7 N., Long. 3* 3l' 42" W.
* If great accuracy is not required, } 1 * sin. * /. may be omitted in ths quan-
tity within the parenthesis.
RULKS AND FORMULiB. 163
Ifp be the length of a degree perpendicular to the meridian^ / the
equatorial radiiu> e the semipoiar axis, i— c=i2 the difference of
theae^ r° the length of an arc in degrees equal to radius^ or 5T*>2d57J95,
«d / the latitude^ then p = —- — -^ nearly. . (14)
Ex. 5 If i =r 3486850 fathoms, di=z llieO fathoms^ and 1 = 60^,
ien p = «^«^,+ "y««^ = 60991 fathom.
'^ 57.29573
If jp 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, ana o the length of a degree on an oblique arc,
making an angle a with the meridian, then
p m m
Q ^^ * — . - ■
p— (p— «i) sin. « a J ^EnHLsin « a " ' ^^^^
P
Ex. 6.— If JP := 61182 fathoms, m r- 60650 fathoms, and a = Vi^'W
53^, therefore
6085 _ 60850 60860 «„^c^e 1.
'- 1-^x0.98038 -l^^^™^ -6^5^ = 61176.46. the
61182
length of the oblique decree in fathoms.
For aa extension of this subjeet, see Mr Ivory on the properties
of a line of the shortest distance traced on the surface of uie oblate
spheroid, in the sixty-seventh volume of the Philosophical Magasine.
It is rather too long and difficult to be inserted in this place.
SECTION V.
Ruhi and Formula!,
SPECIFIC GRAVITY.
The. daffierence between the absolute weight of a body, and its
wagbt when entirely immersed in a fluid, is the same with the
weight of a quantity of the fluid equal in bulk to the body.
uW be the weight of a body in vacvo, (which is nearly the same
aa thatinair^) and W its weight in water, then W — W isthe wdght
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 = ^^^^7 . (1)
Hie specific gravities of bodies are determined by the hydrostatic
balance^ the hydrometer, &c. described in books on Natural Philo-
To compute the specific gravity of air under given circumstances.
It la. shown in Playfair's Outlines, vol. I. § 333, that if the elasti-
eitjpriir tamion at the freesing point, be denoted hr unity and x,
any number of.d^rees above that point, then the elastic force jf at
that point, will be/= (1.375) jStTof Fahrenheit's scale, or
log./- jgg^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 Inilk' or the natural number answering to the arithmetical
complement of the log./ will be the specific gravity of permanently
1^ INVftODUerJQN.
elilBitic ftijlcls^ > Thu&let the bulk and Boeciflc gravity of «ir ^:^2^
1^. s£ 1; tfien at 52<' F. they will W:i,0a6, and a9S52're6peCtS?eljF^
. -Vr^roinillhiB expevimentt m Gay: Littaaac, it may be •howB.lliat 04(46
>^ill be the specific ffraviiy of 'aqueous vapour^ .when qompa^ e4 ^i^tli
atmospheric ailr^ at 32^ F. "Sa^f when the temperature is given^
the ^ftcific g^vity of aqueous vapour is direftljr as its tfrnpevftare,
and toe tension being, given^ tb0. specifiq gravity is i^procaUyas
its bulk^ the spedfic gravity s of aqueous vapour, (that of water li*-
ing l)j in saturated air at any temperature t^ and elastic forced (^om
ibalton's table) will be obtained from the following formula S|^^.'.}ia*
roaoieCe^ being at 80 inches.
, = 0.4546 x^X553:p=45g^ ... (3)
. . If it be not saturated^ and i' being the dew point
, ■7^+<^448+^""448+/^V ^448+<V . ' .^7
' The quJEUitities in this expression are all known except J> i^hidLis
to be taken from any good table^ such as Dalton's or Uire's^ See'T^Te
IL,liMre48.
If^ merefore^ y be the specific gravity of air fully saturated with
moisture^ a the specific gravity of dry air obtained from Hohnula (2)«
and 9 the specific gravity of aqueous vapour in saturated aifc-^ ddnvcd
£hM»;formulffi (3)^ thto from the law of expannba'discoveied by
Dalton and Oay Lussac, that v = — fW^ P ^% ^^ baropetzic
pressure^ f the elastic force^and v the volume,
*' = «+ ^0.4545 X 44QT7 — «) X^ ®' ^y amplication,
*' = «+*— ^ (6)
If I' be the dew point, and ^' be the specific gravity, according to
the Acfbual state of the atmosphere,
f"r(?+'-^)0+MT?) • • • ^(*^
in Which^i and s are got from the following table, page lfl5> andjf from
Dalton's.
JK^fp.-— Required the specific gravity of air saturated with mdisture,
at Sfy JF. ?
By formula (2), — x 0.138303 =.^ X 0.138308=0.046101, ar. co.
of whieh is 9.958899. To this the natural number is'0.99929=ssai
But by fqrmul<i (3), s = ~^^ =.0.02782,.an4 ^ = a04^.
Nmf,s^ =r a^s-^-^ by formula (5) p therefore, ^ =3 089999+
'0.03782—0.04502=0.88209 the specific (pravitv of ah* saturated widi
moisture, at 92° F. If the air is not saturated. Suppose 8T P- ^e
dew point represented by f, then the factor 1 + jjp — 7 in formula
' r '' ■' " ■ * ■' -" * ■■ ' ■ •! i . \ u - . _ . ■ 1 ■ . ■ ■• ■ '■ ., 1 , , ' , '
* DanieU and TzedgoU, contaid that.tkis fioicmiila abould.bc^^^r- The dlf^^ce
in a nurierste range, howewr, is not great. The elasticity in tbi example, wasnot
taken from Dialton. It is difRcnlt to obtain correct formuhe for tJhese researches.
BULES AN O VanHVhJR' 1 $6
(ej^^becomei. 1+^^=1 + -J-, therefore, ^.^09^^^
0JB86W9+ 0.00817=0.89026, the specific ^vit^ of air in the^riYen
cCn^mstances, that of diy air at Sz"" F. beiiiff lu^ty;-^
It is shown in Playfair's Outlines, vol. I., iui;. 256, that if the
t^e-gniyitr of air*oe called m, that of Water being 1 ; if W be
jp. weigtit of any bodj in air, and W its weight in water, then
-P'iNt'fW-'^W^) IS its weight in vacuo very nearly; In a mean state
of tike atmosphere at 30 inches of the barometer and 60^ F.msz
0.00192 iiearly, w;bich may be reduced to any other temperature by
t&e ftnr^ding formula (4), and to any other pressure by ma1tipl3ring
30 : " .
If « be the specific gravity of a body ascertained by weighing it
in air aoid water^ ^.afid m the specific gravity of the air at Uie utie
when the experiment was made ; the correct specific gravity ^^ or
that wh)ch wQuld have been found if the body nad been wei^jnl in
i vicnnin instead of air, or
Whete the body is heavier than water^ this correction is siibtrac-
tive ; when lighter it is additive.
£ri^«-Th9 weight of Captain K^ter's experimental pendukini was
careliilly detennmed ii^ air,. by Barton's balance firom the Mint, and
found to he 00904 grains. The trough, which had been previously
pU^ediinAdr-tfie' pendulum, was then filled With dtstill^ifat^/siffd
the Wd^t G^ the water displaced wn^ 9066 grainy. Tile Sttiall pc^^^
tion of If bn wire which was inurierstf'd in the water w*8 carwelttp
noted ; the weight of the wire by which the pendulum was fuspens^
ed was &6 grains, and the weight of water equal in bulk tothatjpart
of the wire which was immersed was 2.5 grains. Hie temperfektiil«
f^tfaie water was 68° F., that of the atmosphere 62' F., and the ba-r
ft/
romrteC 29.9 mches. Now since s sz — ^-r^> tf being: the weight
in-^^^knijv\thAt\nvrater,then
SL 7.87552 at 29^ bar. and eS^f F^, «iid s' s 7-37553 +
O.o6i30S7§ (1—7.37552) = 7-36783 at 68*^ Fahrenheit.
Bat dill specific gravity of water (rat eS"" is -99936, that at 62t.bQ-
ing 1 : and^ therefore
Y>r if ,i',z.,j^, i.. ., . ; . ^^^ . . ,
- X '^ = 7fo5oq« X 736783 = 7.37254 at «8* F.
^iot'qi^p^riments give at 30 inches bar., aiid 60^ F-, the spedfic
gravity. qf.a|r 0.00122, or ^^, water being 1.
Mil fti^Bice, from Sir G. Shuckburgh's expenments, deduces
O.OO]l206I& liot differing much from Biot's, find gqiiirflly supposed
the W^&hect. According to Gay Lussac, the exnmatons or fluids
fjcom^> ffi^ F. » .0.376, whence ^ =^ far !• F. .
r sup^se c ±= the first correction of Uie ■ length. ii9P; the pendu-
Ittfljly cK the. seQond, I the measured length of the pendulum, d the baro-
metric pressure, the standard being 30 inches ; and 1 1 the difference of
tempi^ratiire from the standilrd, t£si» ' ' '; \- -.^
' \
156 ' nrrsoDUcTfON.' ' >
p p
(«>
~ jj»~ — wr w
If If 1=2 the corrected length of the pendulum I, from ii mean of
Captain Kater's experiments at London in air^then/^^/-) — t—tttt (10), "
^ being the specific gravity of the pendulum.
Whence c = ^^ = 826, and > < = 69^62 — 62'' = 7^62, hence
^' = ~^l^ = 13. therefore c+c' = 839.
. 480
Hence by formula (10) /' = Z + 39.13284 x gL X yg^gy =
/+0.00633.
It is now only necessary to correct for the height above the ^ea^
wtii<4i is 92.5 feet.
Tjbie.>oorreptioQ for this height found .by the formula; which will
presently be given, is 0.00023.
Hence /" = 39.13284+0.00633+0.00023 = 39.13940. In thk
case no allowance is made for the hygrometer. Now if tlie air were
supiijpsed half saturated with moisture, since Captain Kater ^oes not
^ve the state of the hygrometer, and the mean between Biot'« and
Rice's specific gravity en air taken, the true length would come out
39.13938, which differs from Captain Kater's result by 0.00009 in
esKsess.
It is shown by writers on meohanics, that when the semiarc ' de-
scribed by a pendulum is P, the time lost by oscillating in a circu-
kr, inBtead of a cydoidal or infinitely smaU arc, » ^ m each^
cdfxd, and that in different .small arcs of the same circle, the tiase
lost varies nearly as the square of the arc ; hence if .a jpendiilum
makes v vibrations in 24^ when vibrating in very smaU ^circular
atcs, of which the mean at the C€rmmencement ana.termijQiitipa^of
each experiment is d degrees, it .would, in the same time^ inake,v+
injinitely small vibratioHs, Hence to correct the oacillaitioitf of
a pendulum for the arcs of vibration, multiply the square of the
mean arc when it makes > ...
paily 86000 oscillations by . 1.637
86100 . 1.630
86200 1.641
86300 1.643 \ (A)
86400 L645 Mr
86600 1.647
86600 L649 ^
Since the force of gravity varies directly as the length of tlie pen-
dulum, or inversely as the squares of the number of vibratidns, and
the diminution of the force of gravity^ arising from the buoyancy
of the atmosphere, is — past ; therefore if v be the number of vibra-
tion in air, and V those in a vacuum, then
RULES AND FOHMULiE. I57
V
V =5 h+c, and heuce c = = — nearly.
aw "^
In CSaptain Hater's experiments at Unst^ the specific gravity of the
pendulum^ to that of air, was as 7099 to 1, hence — ==ic=^ and
tn 7v99^
,,_. V 8609077 -^ ,
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, I the standard temperature, and t' the ob-
served, then
tt = n' + in*e(f — t) (11)
In Captain Kater's pendulum e=0.00001 of an inch nearly, whence
« = ft' + A Ji' X 0.00001 (<' — 0-
Hence if t; = 86058.82, <' = 71".6 and / = 62°, the number of vi-
brations at the latter temperature are 71 = 86058.72+ ^ X 86058.72 x
000001x9.6=86082.77.
To reduce the length of the pendulum from any height to the
leyel of the sea^ the true length being denoted by /, the observed by
1^3 the height above the sea by a, and the radius of the earth' b)r r,'
dben
Some allow one-third for the effect of the dense strata immediate-
ly under the pendulum^ in which case / = ^ + -^ — . (18)
In a similar manner v = t/ + -5 — (M)
At Unst -^'-^ = 0.06, therefore .
3 r
80090.77 + 6.07+0.06 = 86096.90 = the number of oscillations of
dii^jpietidulum in a mean solar day at the level' of the sea in. vacuo.
' ' Tlii^ formulfle are sufficient for most purposes. Biot has, how-
ever, demonstrated, that if c be the correction in secondB for the
nie4n arc of vibration, n the number of oscillations, M the logarithnuc
iniodalus, a the arc of vibration at the conunencement of the interval,
and b that at the end, then
n' sin." (fl+6) sin, (a — b) ' j^^.
3aMiog..(f) • • . ,: y*^
Theae arcs being small, their lengths will not differ .sensibly ftj^
their sizies, whence if a and b are given in degreef, the lengths iOf
these atCB will be 0.0174533 a and 0.0174533 6, and M = 2300865,
these values being substituted for a, b, and M, equation (15) will be-
~°* '^ = 2418^' lo^ll^^X ZiTb)' '^^ "^"P*^ logarithms, we
finally have log. c = Jlog. n' + log. (a+5) + log. . (a — b)\ —
{C. U £».d83611 + log. (log. a — log. i). . (16)
158 IN'JCSOUUCTION.
To apply this to practice let us assume Kater's 5th experiment .
markedE^ and we have a=:l°.21 And i=i**.09, whehcd" "
a + 6 = 2.30 log. . 0.961728^
a — ft =0.12 log. 1.079181 > (A)
n' = 86056.47 log. 4.da4785)
■T»"
Sum .... 437Q6M
Constant logarithm . 5.383611
Log.a = P.S!l . 0,082785
6 = 1.09 . 0.037426 ' •
Diff. 0.045359 log. 2.656663
Sum(B) 4.040274 . 4040874. (B)
0335^
(A — B) = log. c = 2*.165.
^e^c« »=:»' + c = 86056.47 ^i- 2 J65. = 86068.6B6: Captiun
S4t^ thinkinff th|s an unnecessary refinement in pjnctioe^ multnlie^
t^ BOiifire. pf we mean arc by 1.638 Table (A) ; thus l,X6x iJa
X l.'Co8 = 2*.166 nearly the same as before ; and^ by selectiiig .ite
proper number^ this is sufficiently correct for almost any purpose^'
and much more simple. •,-''-'•
If the length of a pendulum oscillating seconds of mescn time, at
one place or point on the eaith's surface be known^ its length at
another place^ where f he same invariably pendulum makes a difier-;
ent number of vibrations^ may readily be found. For if / be the
l^ctt^h at the first place, I' that at^ the second^ v the number of vibra-
tions at the first place in 24 hours^ and v' that at the second, then as
is shown by writers on mechanics,* /:/'::«*:»'* . .: (If)
consequently if three of these be known the fourth may be fbnnd;
As this is rather laborious^ an approximate rule niay be o^tfoOfH}
sufficiently correct forimost purposed where the difiPereiice qjT^oMilUf
tionadoefl not. exceed 30 or 40^ orm ftn e^ of^fiye oxi si:^ de^jrees.
If AL-r^eaent a: small variation of the I^d^^ of.tbepeBduluQi^
ind'A N'tiskt in th^.'iiumbier o{ oscillations^ then A L/-^ ■ i sir V
•'■■-■''■ tr" ■"■■'-'
and AN = 2-——. . . . (IB)
Let } L be the variation of L for one degree of FaUrenheit's tEer-
mom0t^> and n the nuniber of degrees of change of temperatiire^
fbr this then A L = ii ) L x L^ and A N = ^ N n d h , - . (19)
Sint^ the variation of btass from expahneaiis^BeariyOLOOOOLiacb
forloF^.A.N = 0.432«,andAL:;;;j^^: ,;. . (&)
ExAirpLE I: . . ..
Captain 'Katef found the experimental p^dolum xs^Axf at Lbn^O^
in latitude 51° 31' 8'' N. 86061.52 bsdUattbtis at 62^ Fak. m s^meui
solar day^ while at Unst in latitude 60° 45' 28^' N., it made 86006.90
oscillations in the same time ; required the length of the pendulum at
Unst, that at London being 39.13929 inches ?
* See Gregory** Mechanics, vol. I*, flectian IL, for thii and other formuUe and correc-
tiont more simple than those given here.
RUI4E8 AND ¥X)ltMUi^^:. J59
Here 86096.90—86061^2 = 35.38 = A N. Now A L = -,^^
formula (18) = ^^^^^— = 0.03217, consequently 39.13029
+ 0.03217 = 39.17146 inches, the length at Unst
JBdP. 9b-^^-^Cdptldn Hall found an experimental pendulum, making
-86385M oiKmU at London at 62*" Fah., macfe 86101,34. oadlliu
ti«n« at Cklapagoft 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 fWmi that at London, which
becomes 86233.39.
Now by formula (17), as the places are very distant, v* :©'*:: /:
t : : 30.13029 : 30.01951, the length of the pendulum at Galapagos.
Of late the figure of the earth has been determined with great ac-
curacy by means of the pendulum. It is demonstrated by the
jileaiy of gMvitation, thiit the length of the pendulum is augm^ted
from the equator to the pole, proportionally to the square' of the
$ifke of Ijie latitude^ in such a manner that if the length of the equa-
tlBviial pendulutn be represented by 2, and its absolute variation from
die.Mlial;or-to the pole by v, then /, its length in any other latitude,
L wiU be represented by the following equation : —
/= z + y sin.* L . . , (1)
If we have two equations of this form, in which / and L are de-
tenniaed by observation, we can obtain Uie values of z and y.
I z=iz + y sin.* L
^ = 2+ysin.«L'
^""^3f= sin. (L--hL)~sin.(^--L) ' (2)
And 2 = / — ^ 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
finrce ; «* the circumference of a circle to diameter unity ; r the ra-
diios of the given circle in which a body revolves ; t the time of re-
4t^ r
volutiop, and g the gravitating force, then/ = — -. But by the
theory of the pendulum, if / is its length, g=:iF^ I; hence
f-TTJ-JTyn ... (4J
TTie mio'^the centriftigal force to gravity may be expressed by
Y^-f, and the ellipticity of the meridian or flattening of the ebrth
IS nrom theory equal to f * of the ratio of the centrifugal force to
Myigr* diminished by the fraction obtained from divimng the dif-
fefcaai& of th« lengths of Uiependulum at the pole and equator by
Its ieagth at the equator* Wherefore if i denote the ellipticity^
*"nik'ftac3oB-is obtained bj appioximatioh, and is not perfect! j correct. Bj tal-
'ii%?te.te«|M»t|t|qs.of As tfobod order, tbecUiptieHy would vnj afaoat ,|. from the
firct appiozimatioD. It is difficult to boIyo tho.eqnatioiia ifivolving tlMe. Still, how-
cvfL pchflirar ij^^jba allowed? if poMible, to afiect the final results, bu.t what un-
«VMdtt»^lia6ii|pr ta the obwrtatlons.
160 INTBODUCl'ION.
By substituting the value oifftaOL equation (4)
• = ^Xiqr^-f • • (6)
Aa t in these investigations denotes the time which the earth takes
to perform a rotation about its axis, or 86164'.0006; \ t^ -sz
1856062632, r, the radius of the equator, is 20918750 feet, /, the
length of the equatorial pendulum by numerous observations, is
39^13 inches, or 3.25106 feet, and ^ = 0.20712 inch.
Whence 1 = 0.008638—^ .... (6)
By combining a great ' number of the best observationa I have
found I = 0.003333 = sgg nearly.
From these we may get a formula to compute the length of the
pendulum at any latitude.
Oommendng at the equator / = 30.013+0.20712 sin.< L . (A)
Setting out from 46'», I = 39.11656 — 0.10366 cos.« L . (B)
£«. — Required the length of the pendulum at Leith, in latitude
56<> SS' 39" N. ?
Ans, — 39.1555 inches.
Sinceg=ir« / = 32.2feet
Hence the length of the pendulum and force of gp-avity may be
found at any latitude.
But the K>rce of gravity may be found more readily by a particu-
lar formula for that purpose.
Since g is equal to 32.172 feet, or 9.8058 metres at 45", then O at
any other latitude will be
G = g (1 — 0.00268 co8.« L) . . (7)
Or G = 32.172 (1 —0.00268 cos.« L) in feet.
Let L be the length of the sexagesimal pendulum and / that of the
French decimal-metrical pendulum, then
L = 52.74079/ ... (8)
of Sir George Shuckburgh's scale,
or L = 52.740564 / . . . (9)
of Bird's Parliamentary Standard of 1758.
Let V be the velocity of sound at 30 inches of the English blutime-
ter, 60** of Fahrenheit's thermometer and 14" of Mr uoldingham'a
hygrometer which he used at Madras, also let'« be the change of
velocity for a variation of one inch of the EngliiA barometer, /S for
that of one degree of Fahrenheit's thermometer^ y diat for one de-
gree of Mr G's hygrometer, m the velocity of the wind, and p the
angle which the direction of the wind makes with that of the soand,
and V the true velocity und^r given cii^umstanees, then
V = u + * (p'— i^) + iS (<' — + y (A' — A) +«»oos. p (10)
in which |i = 30 inches, / = 60° Fah. A = 14" hygrometer, and »', /'
and A', the observed states of the barometer, thermometer, and hy-
grometer, respectively.
From an examination of Mr Goldingham's experiments at Madras,
1 have found « = 18.8 feet, ^ = 1.14 feet, and y =2.87 feet The
values of m and ^ not being stated in any set of experiments which
I have Seen, have not been exactly verified. They must be known,
however, at the time of computing the velocity as they undoubtedly
aflbct it. Without these it becomes
V = 1100+ 188.(p' — 30)+ 1.14(/'— 60") + 2.87(A'~14")(ll)
RULES AND VOBMUUE. IfU
#
Required the velocity at Port Bowen^ the Bar. being at 30.398 in.
Fahrenheit's Ther. — 38°.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, J' the elastic force of
vapour by Dalton's table for the dew point, obtayied by Daniell's
hygrometer, or otherwise by formula, page 53, p the Daroinetric
pressure, A the latitude of the place of observation, and *i cos. ^ the
same as before,
V= {104.0885+0.10831 (/-32°)J (l + ^ ) (10.2738 -
0.01378 COS. 2 A^ 4- «f cos. (p, in English feet. . (12)
Ex.— On the l9th of July, 1826, in mean latitude 56^ N., longi-
tmde 3° 10' 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,
tbe thermometer at 72% the dew point by Daniell's hygrometer, or
by a thermometer, having its bulb moistened with tissue paper, (page
53) at 66°, the velocity of the wind by an anemometer was 15 miles
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
niunber 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 x 10.3874+ 11 = 1 141.715
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 by dividing
oie area of the transverse section of the stream expressed in square
indies by the boundary or perimeter of that section, diminished by
the superficial breadth of the stream expressed in linear inches.
Also let A be the length of an open canal or of a close pipe ; i 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 A' the heiffht above the lower orifice,
at which the water stands in the cistern mto which it is emptied.
i ' h + i h'
Now let — = t or the sine of inclination and = it.
A A
The fonnula for the velocity of water in pipes, per second, will be
9 z= {32806.6 dk + 0.023751 ] * — 0.1541 13 (13)
* If BseriMof experhnentsuemadebyagiuateachendof themeaiuredbasej^^
/feooietrioBl means oi the times should be talran. See Bulletin de Sciences for 1R26.
(X)
1*1 iNtRdmJcl'iaN.
Sjc. Let ) =r 65 feet, d = 19 inches, a = 18300 feet, - =
65
[:2=0.00352=iS;, therefore
18300
i;={32806.6 d k +a020751}i~04 541 13=46.9 inches the velocity
per second.
In rivers and other canals, the formula is
t;={32806.6ri+0.023751Ji--0.154113 . . . (14)
These formulae have been simplified, and are tolerably correct.
Suppose V, dy }, axid A, .are all expressed in feet,
t;=504 — >-* nearly the velocity in feet, per second. (15)
Let D be the discharge per minute in cubic feet, then
D=:2356 d«(^)* (16)
To find the fkll in a river caused by obstruction, such as the piers
of A bridge, &c.
Let V be the velocity of the stream in feet per second, h the irhole
breadth of the channel in feet, c the contracted breadth between the
obstacles, and/ the fall, then
^ f/25 6\« , ) f;« 1.42 6«^tf« . , ,,-^
>={te)"M64=— 847^-^'^^'"^^^^^ • ^^7>
Let, as is nearly the case with the old London Bridge,
t)=:3j, &=926, c=200,
Hence/=^!::— ^-Xv«=a46xl0j^5=4.73 feet, or 4 ft. 8|
inches by the formula, while that by experiment was 4 feet 9 inches.
TO FIND THE TONNAGE OF A SHIP BT LOGARITHMS, ACCOBBING TO THE
COMMON METHOD.
Rule. — If the vessel is a ship of war, let fall a pelrpendicular from
the fbre-side of the stem, at the height of tlie hause holes ; but if a
merchantman, the perpendicular is to be let fall from that pafrt of
the fore-side of the stem which is at the same height above tbe keel^i
as the wing transom : also let fall another perpendicular from the
b^ick of the main post, at the height a( the wing traniapm. Find
the distance between these two perpendiculars, from ithicb subtract
three-fifths of the extreme breadth ; and also, the product of th|s
height of the wing transom above the upper edge oTthe keel, by 2^
inches, and the remainder is the length of the keel fo^ tannage. To
the lojgarithm of which, add the logarithm of the breadtfau itna that of
the h^f-breadth, and the constant logarithm 8.02687;* the sum, re-
jecting 10 from t&e index, wiU be the logarithm of the toii^nage re-
quired.
Est, — ^Let the len^h between the perpendicular ^t the fore-part of
the stem, and the back pf fh^ post, be )00feet ; the extreme breadth
274 feet, and the height of the wing transom 15 feet. Required the
tonnage ? — Ans, 321 tons.
* The arithmetical com]»lement of the Ipgarithm of 94, being the common divisor
for finding the tonnage. Tnis method is fkr from beinff correct. See papers on Naval
Arehiteetnre, published by Morgan and Creoze. 6. B. Whittaker, Linidon. 1826.
163
TABLES OF SPECIFIC GRAVITY.
SOLI08.
Platina . . . 20.722
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
Vir«n Silver 10.744
Shilling of George III. 10.534
Bismuth, molten . 9.822
Copper, wire-drawn 8.878
Red Copper, molten 8.788
Molybd!ena . . 8.611
Arsenic . . . a308
Nickel, molten . 8279
tJranium 8.109
Steel from 7769 to 7 816
Cobalt, molten 7.812
Bar Iron . 7.788
Pure Cornish Tin . 7.291
Ditto hardened 7299
Cast Iron . . 7.207
Zinc •. . . 6.862
Antimony 6.712
Tellurium . 6.115
Chromium 5.900
Spar, heavy 4.430
Jargon of Ceylcm • 4.416
Oriental Rub^ 4.283
Sapphire, Oriental 3.994
Ditto Brazilian 3.131
Oriental Topaz . 4019
Oriental Beryl 3.549
Piamond . from 3.501 to 3.531
BpgUah FHnt Glass 3.399
Tomttdin 3.155
Asbealiis 2.996
Marble, green Campanian
, Parian
-, Norwegian .
-, green Egjrptian
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
Beech
Ash .
Apple-Tree
Orange-Wood
Pear-Tree
Linden- Tree
Cypress
Cedar
Fir .
Poplar
Cork
Sulphuric Acid
Nitrous Add .
Wate^ fixmi the Dead
Nitric Acid
Sea-Wiatei-
Milk
Distilled Wifter
Wine of fiburdeaux
LIQUIDS
1.841
1.550
S^ 1.240
1.218
1.026
1.030
1.000
944
i*«toMrfbiifa
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
2710
2.680
2.653
2.619
2.564
2.642
2.892
2733
2.385
2.341
2.033
1.917
1.874
1.720
1.140
973
950
930
866
852
845
793
705
661
604
598
561
550
383
240
901
915
874
870
848
792
716
about 1
1. Since a cubic foot of water, at the teinperature of 40^ Fahren-
heft, wei|^8 IflOO oontees avoiidiipois, or 62| pounds, the numbers
in the preceding tables, omitting the decimal points, exhibit very
164
INTEODUCTION.
nearly the respective weights of a cubic foot of the several substance*
in avoirdupois ounces.
2. If the weight of a body be known in avoirdupois ounces, its
weight in Troy ounces will be found in multiplying it into -91145.
And, if the weight be given in Troy ounces, it will be found in
avoirdupois by multiplying it into l'()971*
Atmospheric air*
Vapour of hydriotic ether
oil of turpentine
GASES.
Hydriotic acid-^as
Fluo-silicic acid-gas .
Vapour of sulph. of carbon
sulphuric ether
Chlorine
Fluo-boric gas .
Vapour of muriatic ether
Sulphurous acid-gas .
Cyanogen .
Vapour of absolute alcohol
Nitrous oxide
Carbonic acid
• Air . . 0-00122 water
Water = 1.
10000
5-4749
50130
4-4430
3.5735
26447
2-5860
2-4700
2-3709
2-2190
21920
1-8064
1-6133
1-5204
1-5196
Muriatic acid-gas
Sulphuretted hydrogen
Oxygen-gas
Nitrous-gas
Olefiant-gas
Azote, or nitrogen-gas
Oxide of carbon
Hydro-cyanic vapour
Phosphuretted hydrogen
Steam of water
Ammoniacal-gas
Carburetted hydrogen
Arseniated hydrogen
Hydrogen-gas .
1-2474
M912
1-1036
1-0288
0-9784
0.9691
0-9569
0:9476
0-8700
06235
0-5967
0-5550
0-5290
00732
being ss 1, hence Gas S. G. x 00012t » & G.
Specific gravity of Distilled Water at different temperatures, thai at
62° being taken as unity.
70^
0-9.Q913
62°
1.00000
54°
1-00064
26°
"46°
1-00102
34°
68
0-99936
60
100018
52
1-00076
28
44
1-00107
36
m
0-99958
58
1-00035
50
100087
30
42
100111
38
64
0-999B0
56
100050
48
1-00095
32
40
1-00113
40
MISCELLANEOUS COMPUTATIONS AND EXPERIMENTS.
The pendulum vibrating seconds of mean solar time at London in
a vacuum^ and reduced to the level of the sea^ is 39*1393 inches ; con-
sequently the descent of a heavy body from rest in one second of
time^ 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
ratio to the imperial measure of three feet, as 1*093^ to 1^ the loga-
rithm 00388717.
The following standards^ accurately measured, give these results: —
Gen. Lambton's scale, used in the Trig. Surv. of India, 35*99934 inches.
Sir G. Shuckburgh's scale (which for all purposes \
may be considered as identical with the impe- >35-99998
rial standard) . . . |
Gen. Roy's scale .... 3600088
Royal Society's standard . . . 3600135
Ramsden'sbar . . . . 3600249
Weight of a cubic 4nch of distilled water in a va- \
cuum at the temp. 62°, as opposed to brass Mog. 2*4026430
weights in a vacuum also, 252*722 grains |
BVLE8 AND FOBHULiG. iOfi
Consequently a cubic foot 63-38^ pounda avoir- > , , _„-,«««
dupSii . . . . /■•"«• 1-7850887
W^ght of a cubic inch of distilled water in air at )
6^ of temperature with a mean height of the J-log. 2.4021867
boronieter 252.456 grains . I
Consequently a cubic foot 62-3862 pounds avoir- \ . . .TojnqiA
log. 02387924
.... log. 2-4429124
Diameter of thecylinder containingagallonatonel , , jv™m,«
inch high, 18-78933 . . |log. 1-2739112
And an ounce of water 1-73298 cubic inches
Cubic inches in the imperial gall<
SPECIFIC GHATITY OF DRY AND 8ATUBATED AIB.
That ■( 30 In. Bw., ud SS- F>hr. bdng 1.
a;
lfD.TAta.
SSSi
I>nip.
w;™:-
S^lnnLAlr.
32=
1.00000
0.99750
"er
0.93.<)96
0.93J64
33
0.99824
0-99568
68
0.93829
0.92968
34
0.99647
0.99386
69
0.93664
0.92772
.15
0.99471
099203
70
0.93499
0.82676
33
0.99294
0.99021
71
0.93333
0.92380
37
0.99119
98839
72
0.93168
0.92184
38
0.98944
098664
73
093004
0.91988
39
0.98769
0.98470
74
092839
0.91792
40
0.98O95
098286
75
0.92676
0.91596
41
0.98120
0.98101
76
0.92611
0.91400
42
0.98246
097917
77
0.92347
0.91203
43
0.98073
0.97731
78
092184
0.91005
44
0.97900
0.97546
79
0.92021
0.90811
45
97726
0.97358
no
0.91859
0.90609
46
0.97553
097172
81
0.91656
0.90111
47
0.97381
0.96986
82
0.91634
090213
48
0.9J209
0.96798
83
0.91373
090013
49
097038
0.96610
84
0.91211
0.89814
SO
096886
096421
85
0.91050
0.89615
61
0.96695
0.96233
86
0.9O889
0.89415
S9
0.96524
0.96046
87
0.90728
0.89216
53
0.96354
0.95866
88
0.90567
0.89014
54
096183
0.95666
89
0.90408
0.88813
55
0.96013
0.95476
90
0.90248
0.88611
56
0.96843
095285
91
0.90089
0.88410
57
0.95674
0.95096
92
0.89929
0.88208
68
0.95604
94902
93
0-89770
0.88006
69
0.95336
O94710
94
0.89612
0.87803
60
0.95168
094518
96
0.89463
0.87602
61
0.94999
094326
96
0.89296
0.87401
62
094831
094134
97
089137
0.87190
63
0.94664
0.93940
98
0.88979
0.86995
64
094496
0,93746
99
0.88821
0.86790
65
0.94329
093652
100
0.88664
0.86685
66
094162
0.93338
110
0.87110
0.84329
On lUi nibjcct s« Biat'i TraUi dt Phytiqitt, rol. I., di. >
INfBODDCTION.
Glass tub«, linear
lJXm21
Plate glass,
1.00087a
Deal,
Loooeoe
Pladna,
1.000911
Cast iron.
1.001110
Steel,
1 001913
Iron
1.001249
Gold,
I.0014fi8
Copper,
1.001796
Brass,
1.001873
9lW,
1.00S003
Tin, .
1.00SB73
Le«i,
1.002858
Zinc,
1.0IK976
Uensurv, volume.
1.018100
Alcohol,
i.ionm
Filed Oils,
1.07H)00
"•-•""
5S.
ssr
Value of E
VaJu..f
V.,.,.,
™V"'
Teak . . .
745
818
9657802
2462
2488
16650
Poon . . .
670
696
6759200
2221
2266
14787
Eng. Oak . .
969
598
3494730
1181
1205
9836
Do. Spec. 2. .
CanatEan Oak
934
435
6806200
1672
1736
10S63
872
688
8596864
1766
1803
11428
Dantzic Oak .
756
724
476S750
1457
1477
7386
AdritaticOak .
993
610
3885700
1683
1409
8808
Ash . . . .
760
395
6380760
2026
2124
17337
Beedi . . .
696
615
5417266
1656
1586
9912
Elm . . . .
653
609
2799347
1013
104-2
6767
Pitch Pine . .
680
688
4900466
1632
1666
10415
Red Pine . .
667
605
7359700
1341
1368
10900
New Eng. Fir
553
767
6967400
1102
1116
9947
Rig. Fir . .
7.53
588
5314570
1108
1131
10707
Do. Bpec. 2. .
738
3962800
1051
1081
Mar Forest Fir
696
.588
2681400
1144
1168
9539
Do. Bpec. 2. .
603
403
3478328
1262
310
10691
Larch . . .
Do. Spec. 2. .
Do. Spec. 3. .
531
622
656
411
518
518
2405433
359U33
4210830
653
832
1127
890
850
1148
765.5
Do. Bpec. 1 .
660
518
4210830
1149
1172
7352
Norway Spar .
577
648
6832000
1474
1492
12180
' From Btrlow on the Strecgtii of Timber.
BULES AND FORMULA. 107
Solution of Practical ProblemSy from the preceding Data.
Pbob. I. — To find the Strength of Direct Cohesion of a Piece of 7tm-
ber of any given Dimensions.
Rule. — ^Multiply the area of the transverse secticm^ in inches^ by
the value of C, in the preceding table of data^ and the product wiU
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.
Prob. II. — To compute the Deflection of Beams fixed at one End and
loaded at the other with any given Weight.
Rule \. — 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
weighty 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.
Pbob. III. — To compute the Defection 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 tlie depth, both in inches.
2.^^Multipily also the cube of the length, in inches, by the given
weight in lbs. ; then divide the latter product by the former for the
deflection sought.
ExJ- — A square beam of English oak, whose side is 6 inches, is
SHmMyHjed on two walls, 20 feet distant, and is to be loaded at its
raiadle point with 1000 lbs., what will it be deflected ? — Ans. 1 *8 inch.
Nde. — ^If the beam he fixed at each end, the deflection will, with
equal weig&te, 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 960, or 60 lbs. to the cubic
foot }'-^Ans. 9^ inches.
leS INTBODUGTION.
Pbob. v.— To compute the ultimate Deflection of Beams or Rods,
before their Rupture.
Note* — The 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. — Toflnd the ultimate transverse Strength of any rectangu^
lar Beam of Timber, flxed 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. — 1. Take the ultimate deflection 8 times that of the last
problem^ and divide the deflection by the length, which wiU give
the sine of the angle ; whence, by a table And 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 or 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 ^t the other ; the breadth
being 2 inches, depth 3 inches, and length 4 feet ? — Ans. 518 lbs.
Pbob. VII. — To compute the ultimate transverse Strength of any red-'
angular Beam, when supported at both Ends and loaded in the Centre,
« ■
Rule L — 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 11. — 1 . Compute the ultimate deflection by Prob. V. ; square
that deflection, and divide it by the square of half the length of the
beam, and add the quotient to 1, for the square of the secant of de-
flection ; which multiply by the length in ihches.
% Multiply the tabular value of S' by 4 times the ^readth, and
the square of the depth ; and divide that product by the former an-
swer in lbs. ^
Ex. — ^What weight will be necessary to break a piece of larch si-
milar to the third specimen, the length being 8 feet 4 inches, the
breadth 8 inches, and depth 10 inches ; being supported at each end^
and loaded in the middle }—Ans. 36676 lbs.
EXPLANATION OF THE TABLES.
TABLE I. — The Miles and Parts of a Mile in a Degree of Longi'
tude at every Degree of Latitude, supposing the Earth to be 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 continual
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 ac^acent 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^ IV 22!' N.
Miles 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 fhim 33.65,
gives 33u385; the measure of a degree of longitude in latitude 56^
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
°^^™7 geographicia miles would that amount to at the mouth of the
firith c^Torth, near the Isle of May ?
Since 1^ of longitude is equal to four minutes of time, then half
a minute will be me eighth part of a degree, and \ of 33.385:1=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-
burgh in longitude 30° £., the common latitude being 60° N.
nearly?
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-
rithms cxf 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 nimiber of integers in the given 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 OF THE TABLES.
before^ or more generally above the index. If the first effective
figure of the decimal fraction be adjacent to the decimid point, the
index is I ; 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 significant figure from the decimal point wi&
be toe negative index. Instead of negative indices, their arithmeti-
cal complements are frequently used, especially by those unacquaint-
ed with the first principles of Algebra.
The decimal parts of the logarithms of numbers consisting 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 5737829
54680 4.737829
5468 8.737829
546.8 2.737829
54.68 1737829
5.468 0737829
0^5468 1.737829, or 9.737829
0.06468 2.737829, or a737829
a005468 ' . 3.737^9, or 7 737829
0XM)05468 4.737829, or a737829
Pboblbh I.— -To^ndf the Logarithm of am given Number.
RujLB.— If the given number be under 100, its logarithm is found
in the first pa^e of the table immediatdy opposite to it.
If the number consist of three figures, find it in the first colvmn
of the following or second part of uie table, opposite to which, and
under or above 0, is its logarithm.
If the ^ven number contains four figures, the three first are to be
found, asbefore, in the side-column ; and under the fourth at the top,
or above it at the bottom, will be found the logarithm raquirra*
To this prefix the proper index, and the whole is completed.
If the given number exceeds four figures, find the difierence be-
tween the logarithm answering to the first four figures of the given
number, and the next immemately following ; multiply this wffear*
ence bv the remaining fibres in the given number, point off* aa
many figures to the right hand as there are in the multi|dier, md
the remainder added to the logarithm, answeri^ to the first four
figures, will be the logarithm required nearlv. ^e logarithm of a
vulgar fraction is found by subtracting the'l<^;arithm of the deno-
minator from that of the numerator ; and that of a mixed quantity
is found by reducing it to an improper fraction, and proceeding as
before; or the vulgar fractions may be reduced to decimals, and
the logarithms found as usual.
Ex. l.-«What is the logarithm of 56?
In the first part of Uie table, opposite to 56, and under N.
is 1.74818a
Ex.'^Wh&t is the logarithm of 366 ?
In the second part of the table, opposite to 366, and under 0, is
2.5ffi481, supplying 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 tiiis table, as
well as several others that follow.
Ex. a— Required the logarithm of 7854. >
Opposite to 785, and under 4 is 3.895091
EXPLANATION OF THE TABLES.
Ex. 4 Required the logftrithm of 100176 ?
The log. of 1001 18 000434
lOOSis
III;;!:
The ctifferenoe is 434
Then 434 x 76 » 38984. From this cut off two figares, because
the differsnce hat been multiplied by two figures, 76, and it becomes
339.M. If the figure next the decimal point is less than 5, the
whc^ may be rejected ; but, if greater, increase the figure before
the point by unity, and consequently, in the present case, 329M
would become 33a 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
caoac a dilTerence of two units, in the last decimal place, less than
that fbond above, or the logarithm would turn out to be 5.000769,
instcMK of 5*000764.
To facilitate ^e 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
figuns are omitted, and the third alone retained, by which means a
rwular series of the arithmetical digits, banning 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 spaee marked out by each pair of crossbars, answering to
any of the nine digits, can be placed opnosite to each, which, in Uiese
tables, has been accordingly done. 6y this means the logarithm
correiponding to any number extended to five or six places of figures,
be very readi^ obtained with sufficient accuracy, excepting,
ips, when it falls in the second and third pages, where the dif-
_. icef vary rapidly.
Ejc, 5.—- Kequired the logarithm of 546876.
Log. of 54I^M) is 5.737829 , or 5.737829
Prop, part for 70 56 , or 66
for 6 48, or 5
or
Log. of 546876 is 5.7378890, or 5.737890
If the number consists of one figure more than four, or five figures
altMOther, the proportional part may be added at sight.
Ex. 6L— Required the log. of \i ?
Log. of 15 is 1.176091
17 is 1.230449
Log. of U is therefore 1.945642 or 9.945642
Required the log. of 7i, or V* or 7.625?
Log.of7.&5is 0.88224
Requir^ the logarithms of 24, 56, 102, 546, 7854, 78653,
544768, 97685.46, 0.001546, 0.176804, 0.00043689, 3\, f f$, 768i,
485711,39766? 1 1, 8546iV?
4 EXPLANATION OF THE TABL1&
Problem II.— > To JlnA the Number anewering to anjf gken Loga^
rithm.
F^nd the logarithm next less than that given in iSbe column mark-
ed at the top^ and continue the sight along that horizontal line till
a logarithm the same as that given^ or as xiear as possible^ be found;
then the three first figures of the corresponding natural number will
A
be found opposite to it in the side-column, and the fourth
atdy above at the top or below at the bottom of the page. If the
index of the given logarithm be 3, the four figures thus found are
int^[ers ; if t£e index be 2, the three drst figm-es are int^^ers and
the fourdi 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 betwe^i 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 Warithm, 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 less, and annexing these units succes-
sively 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 imder 0, and, aa 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, therdTOTe, since the in-
dex is 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 (me integer (if there are integers) more than the index
expresses.
Ex, 3.*— Required the natural number answering to the logarithm
2.627980?
The^tural 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.
Ex. 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 :-->
Origmallog. 5.687956
487400 corresponds to 5.687886
DiffinP. P. ... 70
gives . 70 for . . 63
remainder as diff. 7
gives . 8 for -72
or in all 487478, differing only one unit in the last place from the
former number.
£XBLAMAXION OF THE TABLJKS. 5
LOOAUTHMIC ARITHMETIC
Pbobi^bm III.— To perform MuUiplication htf LogarUhnu.
RuitB.—- Add the logarithms of the factors^ and the sum is the lo-
garithm of the product.
If there are Doth negative and affirmative indices, their sum is
taken according to the rules of algebra ; or the arithmetical comple-
moita of the negative indices may be used> rejecting the tens in tneir
sum.
The arithmetical complement of the logarithm of any number is
found by subtracting the ^ven logarithm from 10^ or by subtract-
ing each of its figures begmning at the left-hand from 6, and the
last effective fiigure firom 10. When the arithmetical complement
of the index alcme is wanted, it is found by subtracting it from 10.
f jg;*. l_Multiply 6564 by 836.
T7.^««. /^^ logarithm 8.817169
jractors ^ ggg logarithm . 2.922206
sum . 6.739375
5487000 corresponds to 6.739335
diff. m P. P. 40
gives 500 for . . • . 40
or in all 5487500, which agrees as nearlv with the real product
5487504, as tables extending to six places of decimals will give.
Es. SU-Multiply the numbers 43.68, 0.534, and 0.007685 together
logarithmically.
i 43.68 log. 1.640283, <vr 1.640283
Factors^ 0.534 log. L727541 — 9.727541
I 0.007685 log. 3.885644 — 7-885644
Product 0.179254 1.253468 9.253468.
Pbobl9M 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. 1.— Divide 6486 by 96.
Dividend 5486 log. a739256
Divisor 96 log. 1.982271
Quotient 57.146 1.756985
40
45
Ex. 2— Divide 0.07856 by 0.003482.
Dividend 0.07856 . log. 2.895201
Divisor 0.003482 . log. a541829
Quotient 22.5617 1.353372
39
33
19
14
6 EXPLANATION OF TBLE tABLfiS.
•
Problem V.— 7b perfbrm Proportion 6y LogarUkms.
RuLE.^-^roin the sum of the logarithms of the fleoond and third
terms, subtract the logarithm of the first term ; the remainder will be
.the logarithm 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 a privateer, sailing at the rate of
7 miles ; after three hours chase the breeze freshened, we merchant-
man's rate was increased to 6 knots, and the privateer's to 10. In
what time will the privateer come up with the merchantman f
As the privateer gained 2 miles an hour on the merchantman, at
the end of the first 3 hours, the distance 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. co. log. 0.S97940
Is to the distonce I4r log. 1.146128
So is l"* log. 0.000000
MM
To the time required 3^5 or 3^ 30™ 0.544068
Consequently, from the time the breeae freshened, the privateer
would come up with the merchantman in three hours and a haHf, or
in six hours and a half from the commencement of t^e chase.
Problkm Yh^^To perform Involuium by Logarithvts.
RoLB^-^-Multiply the logarithm of the given numbeir by die bidex
of the power, and the product will be the logarithm of tfate power
required.
Ex. 1.— What is the Mfuare of 64.^
Given number 64 . .log. 1.806180
Index of the power 2
Square 4096. ^612360
Ex. 2.— What is the third power of 24 ?
Given number 24 log. 1.380211
Index of the given power 3
Third power 13824 4.140633
508
"l25
Pbobleu VII.— To perform Evolution by LogarUkms,
Rule.— Divide the logarithm of the given number by the index of
the root, supposed to be expressed by an integer, as, for example, the
square root by 2, the cube root by 3, and the quotient will be the lo-
garithm of the root.
If the ffiven number be a decimal^ and the arithmetical comple-
ment of the n^ative index be used, 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 ffiven
number by the numerator, and divide it by the denominator of that
index.
Ex. 1.— -What is the square root of 1296 ?
Given number 1296 log. 3.112605
Square root 36 1.556302
EXPLANATION OF THE TABLES. 7
Ex. 2.— RequiT^d the cube root of 00000361 ?
Oiren number 0.009961 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
Oiven index ^
Log. of the root 0X^934734 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 5.8827^ is resolved into its equivalent 8+3.882752, and the
produet of this by \ is 2.970688 the logarithm of the root required.
TABLE IIL — The Angles which every Point and Quarter Point of
the Compass makes tvith 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 fourtib the
degrees, minutes^ and teconds, answering to them. Its use is ob-
vious.
TABIiE TV.^'^Laearithmic 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 take the logarithmic sines, tangents, and secants^ from
this tabl9, thereby saving the trouble of reducing them to degrees^
&C., by the prec«»ing table. The manner of using it is easy, and
will be rea£ly understood firom the explanation of the table which^
immediately foUews.
TABLE V.^ — Logarithmc 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 fkcilitate calculations in which time is involved, the
d^rrees and minutes have been converted into time at the rate of 15^
to an hour, and ann^ed at the top and bottom of the page and in two
additional side-columns.* These, together with proportimial 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
eaay and general in its use than those of a similar kind usually given.
The d^prees are numbered at the top of the table, in a direct
order^ from 0<* to 46<*, and, at the bottom of the table« in a retro-
grade order, from 45^ to 9CP. The minutes are contained in two c€
me marffinal columns. The minutes in the left-hand column be-
long to ttie deffree at the top of the page, and those in the right,
hand oolamn belong to the degree at the bottom. In like manner,
die minutes and seconds of time in the first left-hand column belong
* This Ubk will thoDefbfe eomrert degree* into time, and oonyenely.
8 EXPLANATION OF THE TABLES.
to the hour at the top, and those in the right-hand a^umn belong to
the hour at the bottom. To promote perapicuity, it is recommended
to mark minutes and seconds of the circle always by accents^ and
those of time by m and s, as is done in the tables.
Problem I. — Tojlnd the Sine, Cosine, S^c, answering to any given
Degree or Minute*
KuLE. — 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, &c. is the number required. But
if the given degrees be greater than 45^ and less than 90^, find
them at the bottom, and the required sine, cosine, &c. 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 exceed 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 both for arcs and time, 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 required.
Ex. 2.— What is the cotangent of 55° 57' ?
In the page marked 55°, at the bottom and opposite 57' in the
right-hand side-column, is 9.829805, the cotangent of 55° 57^
Ex. 3. — Required the secant of 125° 40' ?
The supplement of 125° 40' is 54° 20', the secant of which is
10.234280, or, which comes to the same thing, the cosecant of 35° 40'
the excess of 125° 40' above 90° is 10.234^0, 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 are 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.
Sec. as before. Otherwise, for the log. sine, &c. of an arc between
270° and 360°, take the log. cosine, &c, of its excess above 270°, and
for the log. cosine, &c. of an arc between 270° and 360°, let the sine,
&c. of its excess above 270° be taken. And for the log. sine. Sic o£
an arc between 180° and 270° let the log. sine of its excess above
180° be taken. Thus the log. sine of 300° 28' is the log. sine. Sec of
30° 28', the excess above 270°; and the log. sine of 220° 18' is the
same as that of 40° 18', and so on. The same may be done when
time is employed, recollecting that 6^ corresponds to 90°^ 1^ to 180°^
Iff* to 270°, and 24^ to 360^.
Problem II. — Tojindthe Sine, Tangent, ^t?. of an Arc expressed in
Degrees, Minutes, and Seconds.
KaitE. — Find the sine, tangent. Sic. corresponding to the given
degree and minute, and also that answering to the next greater mi-
nute, multiply the difference between them by the given number of
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 this process the difference, to 100", has been
given in the column marked D. Multiply this difference by the
EXPI4AKATION OF TH£ TABLES. 9
number of second^ cut cS two figures from the right, and add the
remainder to the sine^ tangent^ &c. of the given degree and minute,
or subtract it fr<nn the cosme^ &c, and the quantity required will be
obtained nearly.
£r. 1.— Required the log. sine of 23"" 2T 40'' ?
Log. sine of 23° 27' is 9.5d0827
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 o£ 23° 27' 40" is 9.600021
Or difference under D., and opposite 27', is 485
Multiplying by ^IK, 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 tne small table at
the bottom of the page.
Ea:. 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. a— Required the secant of ff* 46" Sfl- ?
The cosecant of iU 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 bemg the sine of
20*' 44" 56-.
Problem III. — Tojind the Sine or Tangent of a small Arc, less
than three Degrees.
1. To find we sii^e.
To the logarithm of the arc reduced to seconds, with the decimal
annexed, add the constant quantity 4.685575, and from the sine suh"
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 very-
nearly equal to the length of the arc^ when it cbes not exceed 10%
and the length of an arc of one second is 0.0000048481368 ; mnlti-r
ply the length of one second by the number of seconds and part»
of a second making the index positive by the former rules, and the
sine or tangent, will be obtained, thus, —
0.00000&48ia68 X 8''.08=0.0000420818274 ; the log. of this is
5.624094, the loff. sine or tangent required.
Ex. 2.— Reoim-ed the tangent of 1° 24' 36".46 ?
To tne constant logarithm . 4.685575
Add log. of 1« 24' 36".46:F507a46 3.705561
And $x 0.000132= 88
Log. tang, of 1« 24' 36".46 . 8.391224
Problem IV. — To find the Degrees, Minutes, and Seconds answer"
ing to any given log* Sine or Tangent.
Rule. — In its respective colunm 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 colunm, widi 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 tided at the top or at the
bottom of the page.
Ex. 1. — ^Required the sac, or degrees and minutes corresponding
to the log. sine 9.584665 ?
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 time answering to the log«
tangent 10.358430, making use of the tables of proportional parts at
the bottom of the page.
Given Tog. tangent 10.358430
66^20' 0" corresponds to 10.358253
Difierence 177
And 30 to 173
Hence 66 20 30 is the arc required.
Or, 4*^ 25"* 20* answer to 10.358253
And 2 to 173, or nearly 177
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 fcorm in which the tables are printed requires, and we have
6GP 20' 31" very nearly; and this method must be followed in all
tiuiil^r cases.
Problem V. — To find the Degrees, Minutes, and Seconds answering
to the Logarithmic Sine or Tangent of a very small Arc.
Rule. — To the given log. sine sM, 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 ue required arc.
To the given log. tangent add the constant 5.314425, and from the
sum subtract two-thirds of the corresponding secant, rejecting 10 in
JBXH^ANATION OF THE TABLE8. 11
the iadoi, the result will be the logarithm of the seconds of the re-
quired, arc
Ex, li— Reqpiired the arc whose log. sine is 6.497655 ?
Constant . . 5.314425
Given log. sine 6.497655
i of 0.000000 is 0.000000
Log. arc 64''.8756 1.812080
Or r4".8756
Ejc. 2.-*What is the arc whose log. tangent is 7*164440 ?
Constant . 5.314425
<Hyen log. tangent 7*164440
I of 0.000000 is 0.000000
Log. arc 301".207 2.478865
Or 5' 1".207
TABLE Yl.'^Natural Sines, Tangents, Secants, and versed Sines
io every Degree of ike Quadrant
The method ca taking out the nmnbers required from this table
will be readily comprehended from what has already been said rela-
tive to the preceding. When minutes or seconds occur^ proportional
parts must be taken by means of the differences foundby actual sub-
traction.
J^.— What is the natural sine of 5"" 48' 56^' ?
Natural sme of 5"" is 087156
Prop, part of diff. 17372 for 48' 56" is 14168
Natural sine for 5° 48' 56" . 101324
TABLE VlL-^Meridional Parts to every Degree of the Quadrant.
The degrees are found under the letter u, and the meridional parts
undar M. P.^ and when minutes and seconds occur^ proportional parts
of the difference must be taken in the manner shewn above.
£a;.— Hequired 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 VIII.— -Traverse Table, or difference of Latitude and
Departure.
This taUe contains the measures of the sides and angles of right-
angled plane triam^s^ the distance being represented by the hypo-
tenuse^ and the difference of latitude and departure by the l^s or
aides 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
«iveii in degrees or points in the two exterior marginal columns, the
distance is found at the top or bottom pf the page, according as the
course is less or greater tnan four points or 45? ; and the difference
of latitude and departure is found in columns under or above these
words respectively.
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 tihan W , bnt> Sf more than W, the degrees in the coune must
be increased by I"". The distances 1^ 2, 3, % &c. at the topand the
bottom may be accounted 10, 20, 80, &c., or 100, 900, 900, &c. if
the difference of latitude and departure be increased in die same pro-
portion by removing tibe decimiu 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 results taken.
Problem I. — The Course and Distance being given, tojfind the Dif-
ference of Laiiiude and Departure.
Find the course in right 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 ?
Course. Dist. DifF. Lat. Departure.
2 points . 60 55.433 22.961
Ex. 2^A ship sails S. £.b.S. i S., or S. S.£. ^E. 244 miles, re-
quired her difference of latitude and departure ?
Course. Dist. Diff. Lat. Departure.
24 points . 200 . 176.38 9428
40 . 36.277 . 18^6
4 . 3.5277 1^8856
244 215.1847 . 115*0216
Ex. 3.— A i^p sails 300 miles S., 54"" 30' W., what is her differ-
ence of latitude and d^arture ?
Course. Inst. DifF. Lat. Departure.
54« . 300 . 176.34 . 242.71
55 . 300 . 172.07 . 246.75
Mean 64| . 300 . 174.20 24423
When sever^ courses and distances are given, the results must be
placed in a table, the sum of the several northings and southhigs,
eastings and westings taken, and placing die less sums under uie
greater, the differences will shew how much the ship has, upon the
whole, changed her situation, and in what direction Ae has moved.
TABLE IX. — Diurnal Logarithms.
This table, to which I have ventured to give the title of Diurnal
Logarithms, is useful for making computations in which time is con-
cerned, particularly for reducing the right ascension and declination,
&c. of the sun or moon to any intermediate time between those times
ffiven in die Nautical Almanac, where the proportional parts to daily
differences are required. It has two sets of arguments, the cme answer-
to 19P^, since the moon's place is given in the Nautical Almanac for
cfvery noon and midnight ; the other corresponding to 2P for die
sun.
Rule. — To the logarithm from this table corresponding to the
Greenwich apparent time add die nrcmordonal logarithm (Table 2L)
of die variation on the given day i^r z4^ or 12**, as the case may be,
Ae sum will be the proportional logaridim of die 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, wiU give its value at the instant required.
EZPLAHATION OF THE TABLES. 13
Ex. l-wBequirad the tun't ri^t awenaion March 90th, 1836, at
aO^ 40" 40^ apparent Greenwich time.
Greenwich time 20^ 40" 40' D. L. 0.06262
m fl
Change of R. A. in 24>' 3 38.2 P. L. 1.0M67
Prop, part for 20^ 40- 40- 3 9.0 1.75719
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 ISth^
1826, at 7^ 49° 30* P. M. apparent time on the meridian of Green-
wich ?
Moon's declination at noon 2? T 8"S.
at midnight 9 19 N.
Sum = diff. in 12 hours 2 16 27
App. time 71^ 48" 30> diurnal log. . 18662
dbange of dec in 12*', 2" W 2f' 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* —0 38 21 S.
When the differences are v^ irregular, a correction on that ac-
count becomes necessary. This will be exemplified in the explana-
tion of Table XX VII. ^
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
asaiatance of the Nautical Almanac. It is extended to three hours
aa account of the distances bein^ given in various ephemerides to
every three hours of time. As degrees and hours are similarly di-
videdj 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 or a second^ which may be useful where
great accuracy is reqiured. The table is veryuseful in calculations
where sexagesimal divisions are employed. The method of taking
out the log. of an^ quantity will be readily understood from what
has already been said.
TABLE 'XL'^Depreuion or Dip of the Horizon.
The dip of the horizon is an angle contained between a horixontal
line paaaiiig through the eye of the observer, and a line from his eye
to the visime horizon, whoi these lines are in the same vertical plane.
Thia table contains the dip answering to a free unobstructed horizon,
and the numbers correspcmding to the height of the eye are to be
sabtnMSted from the observed altitude when taken by the fore obser-
vatioBEiy but added to it in the back observation.
TABLE XII. — The Dip at different Distances from the Observer.
If the land is not suffilaently distant to afibra a free horizon, it
may be sometimes necesary to obtain an altitude referred to die sur-
face of the sea at some known or estimated distance. Under ituch
circumstances the dip may be taken from this table.
14 EXPLANATION OF THE TABLES.
TABLE XIII.-— Correciton to be added to the observed Altiiude of
the Sun's lower Limb when taken by afore Observation to find the true
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
semidiamenter 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 subtracted when the upper limb is observed.
TABLE XIV. — Correction to be subtractedyrom the observed AL
titude qf&Jixed Star to 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
1826, 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 earth, 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.— TAe Sun's Parallax in Altitude and Zenith Dw-
iance.
The author computed this table from a mean of the determinations
of Delambre from the observations of the transit of Venus over tiie
sun's disk in June 1769. He found the mean horizontal 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 autiior is indebted to tiie liber-
ality of Mr Ivory, the most distinguished mathematician in the Bri-
tish islands. On comparing it with that given in tiie Trans-
actions of tiie Royal Society of London, it wiU be seen tiiat it has
been expanded considerably, so as to render its application more easy by
giving the mean refraction, and its logarithm for every KK from tiie
zenith to the horizon, subjoining the differences of the logarithms for
the purpose of computing proportional parts more readily.
TABLES XVIII, XIX, and XX.— These tables are employed
to correct the preceding according to the state of the barometer and
thermometer, as shown in the explanation at the bottom of page 89
ESFLAKATION OF THE TABL£g.
15
of die tablet. In the aevetith line from the bottom of that pege^
after thennometer^^there should have been added, ** or 0.002063 for
one decree of Fahrenheit/' that used in the construction of the table.
Ex, 1 . — Required the mean refraction for 21^ 4Xy of ^enith distance
or 68° 20" of altitude?
Opposite to 21'' 40^ in table XVII., and under )l, will be found
V 23^.21, the refraction required when the barometer stands at 30
inches, and the thermometer at 50°, and this is sufficient for most
pnrnoses when £^eat accuracy is not required.
Ex. 2. — Required the true refraction when the zenith distance i»
70° 41'. T, the barometer 30.045, and thermometer 34° }
Zenith distance 70° 40" log. iB Table XVII. 2.21762
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' 51".27 = 171"27
Observed refraction 2 61 .50
2.23368
Error of the table — 0.23
Ex. 3.— Let ^ = 87'' 42^ lO'', thermometer 35o, and barometer 29.5
inches^ what is the true refraction }
1 = 87° 40" 0"' log, i$ a00466
2 10 . .300
Ther. 35" . . 0.01379
Bar. 29i; . . . 9.99270
Ther. 35 . . . 65
Log. r'
^X (85°-60°)
— 0006 X— 15
^'x (29^-30,0)
xlMx— 0.5
17' 16".81 = 1036".81
= + 9.09
=z —0.52
3.01570
r =
17 25.38
Obtenred refraction
17 26.50
Error of the table
— 1
.12
Examples for Exercise.
Z. D.
Bar.
Therm.
Obs. Ref. Error.
In.
In. Out
I. ^0' m 30" .0
29.686
46' 44.17
2*
44".83 + 1".61
S. 70 55 81^
29.686
40 37.10
4
8 .98 + 1 .86
3. 81 27 18 .6
29.924
61 58.19
6
1 .90 + 1 .65
4 8B 68 6 .7
29.810
36 29.95
8
48 .52 + .63
5. 86 14 42 .0
29.174
47-76
12
4 .20 + .28
6. 87 23 44 .0
30.000
60 56.08
15
32 .80 — 1 .16
7. 88 30 32 .0
29.800
38 34.40
23
7 .94 —16 .70
a 89 26 51 .4
29907
39 3a46
30
16 .60 —39 .70
16 EXPLANATION OF THE TABLES.
Hence at moderate zenith distances the afror of the table is small,
sometimes + and at other times — . From ^(f to about 85°, the error
is generally +, but from 85° to 90° it becomes — , and is consider-
able near the horizon. We may therefore infer that the horizcHital
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 eve of the observer, it will be impossible ever
to assign the true quantity of the horizontal refraction under given
circumstances. In fact, no instrument, as yet, has been employed
to ascertain the effects of aqueous vapour floating in the atmosphere,
on the exact quantity of the horizontal refraction ; and we suspect
that the barometer and thermometer alone are inadequate to that pur-
pose.
TABLE XXI. — Augmentation of the Moon's Semidiameter in AU
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 of
the semidiameter, at different degrees of altitude. If the quantity
is not found to the accuracy required by inspection, it mayHbe deter«
mined by proportional parts in the usual manner.
TABLE XXII. — Reduction of the Moon's Parallax in the Sphe-
roid,
As the earth differs somewhat considerably from a sphere, the
eccentricity being about 7^^, it follows that me equatorial parallax.
must be greater than that at the various intermediate latitudes from
the equator to the pole. This table contains the quantity to be sub-
tracted from the equatorial parallax given in the Pfautical Almanac
to reduce it to what it ought to be at any other latitude.
TABLE XXIII. — Logarithms of the Earth's Radii in each Pa'
rallel of Latitude; the Equatorial Radius being Unit, and Conu
pression ^Jq.
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
9.9991121.
TABLE XXIV. — Angles which, the vertical to any point of the
Earth's surface, makes with the Radius drawn Jrom that point to the
centre, or, as it is usually called, the Reduction of the Latitude to
y^Ts (^compression.
This table is useful in several astronomical observations, such as
the computation of eclipses, occultations, &c.
Example.— The apnarent latitude of Greenwidi is 51° 28^ 38".4,
required that reduced to the centre ?
it
EXTLANATION OF THE TABLES. 17
Latitude SI"" 2& 9&\4
Reduction — 11 10 .8
Reduced latitude 51 1? 37 6
From this table the reduction of the altitude may be obtained by the
following rule :
To the secant of the azimuth reckoned from the meridian of an
opposite name from the latitude, add the proportional logarithm of
the reduction of latitude^ the sum will be the reduction of the alti-
tade, to be reckoned positive when the azimuth is less than 90^^ and
nemdve when greater.
Example. — Required the reduction of altitude corresponding to an
azimuth of 3(r* ^ in the latitude Greenwich 5P 28' 38" If.
Latitude SI** 28' 38" Secant 0.20563
Reduction of alt. 11 10.8 Prop. log. 1.20683
Reduction of lat 6 57.8 Prop. log. 1.41246
In computing time^ &c.y 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 time, it is easier not
to employ either of these reductions.
TABL£ XXV. — Far determining the Latitude at any time hy the
Pole Star.
This table was computed by Mr Littrow of Vienna, and will be
found very useful for determining the latitude of a place by the pole
star. A roll explanation is given at the bottom of the page immedi-
ately under the table.
Ex 1. — ^In latitude bG^ N. nearly, the zenith distance (Z) of
the pole-star, by an astronomical circle, was found to be 35** 2N> 50",
whcm its wparent polar distance (p) was 1^ 36'.7^ &nd the star just
14^ 20°* 56r from the time of upper culmination ; required from these
data the exact colatitude of the place of observation.
Now 14!" 2^ 56- gives M = ^'.23, and N = — 0^ 0* 0".48
And 3r'.23x— ^3x0.02=— 2".06=— 3.3 X. 02 M
Then 31".23 — 2".06=29".17=M', log. 1 .4649
CotZ3ff»21' 0.1^1
Cos. t. 1^ 20- 56'=:9.9039
p 96^7 U^. 1.9854
1.6140=— 41 .12
—1.8893 = — 77'.5= —1 17 30.00
—1 18 11 .60
Z 35 20 50 .00
Colatitude 34 2 38 .40
Latitude 55 5? 21 .60
Edinburgh, \(Hh January, 1826.
On the Caltonhill, near the Observatory, with one of Troughton's
reflecting circles on a stand, and an artificial horizon, the author, at
about Um o'dock, p. k. observed the following double altitudes of
the polar star, when the s3rmpiesometer stood at 29.86 inches^ and
18
EXPLANATION OF THE TABLES.
thermometer at 42^ Fahrenheit ; required the latitude of the place of
obflervation.
Stderial Time*
After Tianiit
4ft 22°^ 30*
4 23 35
4 24 30
4 25 40
4 26 45
Means 4 24 36
App. alt. or half
Bauble Altitudes
With Art. Horizon.
113° 10' 50"
113 10 55
113 10 55
113 10 50
113 11
113 10 54
56 35 27
00
App. zenith dist. or comp. 33 24 33
Now by tables 17> 18^ 19^ and 20^ compute the refraction.
Zenith di8t=33° 24' 33" log. h (17) 1.5860
Thermometer 42° Fah. (18) 0.0073
Barometer 29.86 inches (19) 9.9980
Thermometer 42° Fah. (20) 0.0003
Log.r=S»".05
App. zenith distance
Renraction
True zenith distance 33 25 12.05
Now 4*» 24" 36- gives M=72".973, and N= + 0° 0' 0".57
Then J2".973 x — 3'.3 x 0.02=— 4.816=3'3 x -02 M.
And 72".973~4".816=68".157 log. 1.833510
Cot. Z=33° 25' 12" . 0.180535
1.5916
33° 24' 33"
+ 39.0&
Natural number
Cos. <=4'» 24" 36"
p 96'.7 log.
103".29=2.014045=
9.606751
1.985426
^01 43.29
Natural number .
Sum
Z
39'.19=1.592177 =
• • .
• * .
+ 30 11.40
+ 37 2a68
33 26 12.06
^ or colatitude 34 2 40.73
Latitude . 55 57 19.27 N.
From a trigonometrical measurement he also found the latitude 55°
57' 20".7 N.^ supposing with Captain Kater the latitude of the flag-
staff in Leith fort to be 55° 58' 39" N.
TABLE XXVL—DelambreJirst cakulaied this Table far Jinding
the augmentation of the semidiameter of the Moon in solar Eclipses am
occidtatians, without computing the altitude. It is used asjbllows :
To the altitude of the nonagesimal in signs^ a(/c2 the distance of the
moon from it^ and from that altitude ^6^nzc^ the moon's distance from
it ; then take the equations from this table, Part I. answering to the
sum and difference^ and take the siun of these^ regard being had to the
signs. To this add the equations corresponding from. Part II. If
the observation be that of an occultation, the equation answering to
EXPLANATION OF TH£ TABLES. 10
the true latitude and parallax in latitude of the moon is to be taicen
from Part III. In a solar eclipse this part vanishes. Then enter
Part IV. with the sum of the lormer equations in the first vertical
column^ and the horiaontal 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' W, 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 PartI. + 7".70
Remainder 1 10 36 I.+5^
+ 13.03
Partn.+ 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 XXVIL— -Egtiaftoftf 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 time after
noon or midnight. This correction is contained in the table, the
argumentfl^of which are the mean of the two second differences of 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 coUectiv^y, 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. li^— Required the moon's declination on the 15th of September,
20 EXPI^AKAXION OF THE TABLU.
1896» at 7'' 48^ 30* p* x. apparent time on the meridian of Oreen-
wich?
In the explanation of Table IX. thi» is found to be 0° 38' SI'' 8.
by proportion ; it is only now required to find the correction depend-
ing on second differences. For tiiis 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 Diff. Sec. Diff, Mean.
Sept 14th at midnight is 4<» dd' 34" S. go ^a^ la,,
15th at noon 2 7 8 S. 5 i« i?^ O' 1" q^,,
15th at midnight 9 19 N.^ , ? ^2 ^ ^
16th at noon 2 24 27 N.^ ^^ ^
If the first differences first increase and then decrease, or vice versa,
half the, ^fference of the two second differences is the mean^ instead
of half the sum^ as would have been the case had the differences
regcdarly increased or decreased.
In this case the equation must be added or subtracted, according as
the^r^f first difference is greater or less than the third first differ-
ence.
Now to 30" and T 48^" the equation is 3".4
to 4 .4
The whole equation is . d .8
Which, according to the rule above, must be added to theproporwi
donal part formerly found under the explanation of Table I A. ; that
is, to P 48' 47" we must add 4", and the true proportional part be-
comes . . + 1° 48' 51" N.
And declination at noon being •« 2 7 8
The true declination is — • 18 17 S.
Unless the declinations are all north or all soutii, it is almost imneces-
saiy to use the equation of second differences.
Ex, 2.— -Required the moon's right ascension on the 20th Noyem-
ber, 1826, at 9^ 36"' SO* p. m. ?
The Moon's right ascension,
1826 First Diff. Sec. Diff. Mean
Nov. 19th at midnight is 116® 20' 7" ^o 1 1 / Aivf
20th at noon 122 31 47 a q^ ^' ^^" i/qai//
20th at midnight 128 41 36 X ^ ^ 1 18 ^ ^***
2l8t at noon 134 50 7
App. time ff* 36°» 30- Diurnal log. .09653
^p. time IT iJO" dU" uiumai log. .infooa
Change of dec. 6*» 9' 49" Prop. log. f | ^^^^q
Or -^ by 60=6' 9".82 Prop. log. / ^'^^^
Prop, part 4' 56'. 12 Prop. log. 156196
Or 4^ 66' 7".2
In this example we have considered the degrees minutes, the mi-
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 odier aliquot part might have been taken,— Hsuch as a half, a
EXPLANATION OF THE TABLES. 31
third> At. pvovided theptoportional put be doabled, 4r^bkd, Ac. m
derived from this table.
Now to 9" 30" dCr* and 1' the equation is 0" 4"J^
and 30"' 3 .3
and 44 .4
Amount of the whole equation is J ,2
Which must be added to i"" 56' 7^\2, because the first differences are
decreasing, consequently the corrected proportional part is 4° 66'
Therefore, if to the right ascension at noon on the 20th, that is
to 12Sr3r47"
There be added 4 65 14 .4
The true i^igl^t 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 0*" 58" W. ?
Sun's declination at noon 23"* 27' 11" N.
Time tf" 59* diurnal log. 0.38166
Var. 0' 25" prop. log. 2.63548
P.P.O 10".4 3.01714 4- 10.4
First Dif. Seamd Diff. Mean.
Diff. for 19th 51 o^
20th 25 ^ 25 + 3 .0
2l8t 1 ^
True declination . . 23 2? 24.4
In this example the argument in time is found in the right-hand
odnmn in the lower half of the table. In lunar distances the ap-
prozinuite time found by proportion alter the hour mvesk i& me
nautical almanac must be quadrupled, which, being used as an argu-
matl^ will give to the mean second difference the true equation,
amomiting, in some cases, to about 6" in distance, or df of longitude.
TABLE XXVUI.— A!difc^ to the Meridian, Parts I. emd 11.
In ^ke course of the great trigonometrical survey lately performed
in Fnoice, the repeating circle was much used in the determination
of latitudes and other operations. Latitudes were determined by ob-
ssmng repeatedly, near nocm, the altitudes or zenith distances of
a celestial object, reducing those taken off the meridian by appro-
priate fbnnnlae or tables to what they would have been on the meri-
dian. This method may be successfully practised by smaller instru-
mcnta,i— such as Troughton's reflecting cime, or even a good sextant ;
and Dr Brinkley, with his large eight-feet circle in the observatcnpy at
Dublin, takes three or four observations each day as near ne<m as
poaaible, 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.*— AppUcation of the preceding table to observations of 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 mstrument, 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,
EXPLANATION OF THE TABLES.
which are peculiarly adapted to his obaeryatory^ it 58° 98^ 19^46
Mean N. R D. of Arcturus for 1820 69 53 31 ^
Mean right ascension 211 51 51 ^
Place of moon's node ' IP 29 26
Time by Clock.
Left
Micros.
Z. D. Bottom
Microscope.
Right
Micros.
Mem of the three
Microgcopgg.
Refinction.
13 56 28
14 28
14 9 51
14 14 52
//
//
49.7
31.7
50.6
3ao
33
33
33
33
Barometer 29.67
Inter.
Ext
19 50.5 E.
17 32.6 E
14 54.5 W.
16 41.0 W.
Then 52.5
48.0
Time of OlMer-
vation.
h. m. 8.
13 56 28
14 28
14 9 51
14 14 52
4.3
47.1
45.0
31.7
33 19 54.83
17 37.13
14 50.03
16 36.90
37.^
3777
3774
3777
Mean. 33 17 14.72 37-775
Time tit Staf*
Tranait by Clock,
h. m. a.
14 7 3.3
14 7 3.3
14 7 3.3
14 7 3.3
Difference,
h. m. s.
10 35.3
6 35.3
2 477
7 48.7
Reduction.
Parti.
220".]
85 .22
15 .32
119 .80
Part II.
0".12
.02
.00
.04
Sum's
440 .44
.18
110 .11
.045
Now^ if the tahular quantity in Part I. he called m, and that in
Part II. he called n, the latitude x, the declination i, the approxi-
mate zenith distance z, the declination and zenith distance bemg +
if norths and — if souths and the true zenith distance Z ;
cos. A COS. ^ . /COS.. A COS. ^Xg ^ „
^+ ( — I^-y — / ^*- ^'^
then Z = 2 —
or Z = z
sin. Z
COS. A COS. ^
sin. Z
(«i —
sm.
COS. A COS.
sin. Z
1)
cot. Z^n 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 ; hut in all cases where the latitude alone^ or the de-
clination 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 applied to ^ will give Z and A very neariy 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 observations
made near t)ie meridian ; but^ if the horary distance be great> 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
zenith 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.
SXHVikNATION OF THE TAfiL£8. 23
A 6S* 2^ la^' C08. 9.775544
> 90 7 28 COS. 9.972541
z 83 17 15 cosec. . 0.200554 (a) cot ai89739
0.008739x2= a017478
m 110.11 log 2.041787 n a046 log. ae53213
38 2c{, cor. +.0713 8.853413
1*^, Cor. — 112".35 (e) 2.050564 (c)
or — 1' 52 .35 380
2rf, Cor.+ 0.071
—1 52 .279
z 33» 17 14 .720
s/ 33 15 22 .441
Ref- + 37 .775
134
'r'* -
33 16 .216 (/) cosec. 0.260794(6)
240 (6— fl)
— 112 .41 (rf) 2.050804 {c+ib—a)]
_ .06 (d—e) 766
z"' 33 16 .156 f/_(d— e)} 38
This result scarcely differs from Dr Brinkley's, which is 33^ 15' 0'M7,
to which the aggregate of precession, aberration, and nutation,
amountini^ to •— 13".53, being applied, gives 33'' 15' 46^64 fat the
mean semth distance on January 1, 18^.
Ex. 2. — ^At Maranham, August 28, 1822, Captain Sabine took the
following observations of the star » Lyras with a repeating circle of
tax inches in diameter, the barometer being 29*" .95, the wermome-
ter 80^ Fahrenheit, the chronometer. No 423, fast 2^ 55"" 59*; the
star, whose right ascension was 18^ 30^57^.4, was on the meridian,
at 8" 4^ 35* mean time, and at 11^ 1" 34' by the chronometer.*
* Thlf example is extracted from Captain Sabine^s work on the determination of tbe
kngth at the seeonds pendulum at various points of the earth's surfiMe, lately puUish-
«d At the expense of the Board of Longitude. It is a work highly to be reco m mended,
tfK perusal, to those likely to be employed in such experiments in future, as it contains
nmMt examples of all tne requisite operations likdy to occur in such researches.
24
ESUnLANATION OF THE TABLES.
Chronometer.
h. m. g.
10 49
1O4B40
10 62 50
10 55 44
11 49
11 3 42
11 6 52
11 9 17
Honry
Aiiglcf
fg ^305.090.23
8 54
5 44
2 8
5 18
7 43
Means.
Redttcttoii.
P. L p. II.
//
n
0.06
155.51
64.54)0.01
3 501 28.85b.00
45] 1.10|0.00
8.940.00
55.15
116.91
8736.070.34
0.01
0.03
92.01 0.0425
+2+ 1
—4
— 1
—8
+4
k-8
+ 7
" I I I "
*"«-•;?• !Mt»,
il'jf
g fPirst Vernier 167^' I1^«A"'':
— 4
— 6
— 2
— 10
+ 2
— 10
+ 5
o J Second
g] Third
S LFourth
— 16.5
* 4P10'22"log.>n.70813
9.97367
9.99926
9.99870
Ther. 80 P.
Bar. 29.95
Ther.
r 4r'.84
1.67976
A 2^ 31' 46" COS.
} 38 37 38 COS.
9.999578
9.892776
2 41 10 22 cosec. 0.181656 (a) cot.
Mean
\m\\^i%
{First Vernier 136° 35'' "6^'"
Second . . 34 30
Third . . 35 30
Fourth 35
Mean
Inde^ -f
Level — p-
136 35
192 48 12 .5
16 .5
8)329 22 56
Obs. Z D
r
Cor.
True
Star
41 10 22
+ 47 .84
'8 dec... ,58.^7 37:35
Latitude
It.";
m Oa".01 log.
\H cor^l0e".08
or— 1' 49 4»
2dcor. + .07
0.073909 X 2 =
1.063835 n, 0.0425 log.
2087744 0.068
Ifrt:::;.;;,;:
^..
318
--. 1 49 .01
It is unnecessary to repeat the operation in this case^ as the difference
in the result would only be ^'SA, making the latitude 2^ 31' 43.''] 9
TABLE XXIX.— iZec^tfc/ton to either Solstice, the Obliquity of the
EcUptic being ^"^ 27' 40".
The obliquity of l^e ecliptic is determined by a number of meri-
dian altitudes^ or zenith distances near either solstice. If the sun's
longitude were three or nine signs exactly at noon^ the operation
would be very simple ; but as that seldom happens^ it is necessary
to reduce the actual observations to which they would have been
under these circumstances. To accomplish this object^ this table
has been constructed. In the table the obliquity is supposed to be
230 27' 40"^ and the reduction is the difference between this quantity
and the sun's declination at the several points of the ecliptic cor-
EXTUtNATlON OF TH£ TABLES. 25
reflpondinff to the observed right asceiiBionB. With the ditTerences
ana variation for lOO'' change of obliquity the t4ble may bejidapted
to any tune within the liimts of the table's variation ojf obliquity.
Both quantities will thus be additive till the y^ar 1835., Tlie table
is extended to SO^, and consequently observations may^ be reduced
by it for about seven days before and m many after the splstice.
Ejc, 1. On the 15th of June^ 1826^ the sun's declination i^as ob-
served to be 23° 18' 51'^?^ when the right ascension was 15*^ 25°^
51".^ and the obliquity 23° 27' 39^. what was the reduetioii to the
adstioe?
Tabular obliquity 23° 27' 40"
Qh Qm Q. Estimated obliquity ^ 97 39
5 32 51.4 h-
Excess 1
27 8.6 = distance from the solstice^
27 0.0 gives 8' 42".73
8.6 gives 5 .564
1".0 var. obi. gives .006
' Reduction . 8 48 .299
Sun's declination 23° 98 51 .?
True obliquity . 93 97 39 .999
By operating -in this way for several days near either solstice^ the
true obliquitji^ maj he obtained from a mean of a number of observa-
tionsy and oolueqaen^ likefy very near the truth. It may be ob-
MTved, however^ that the sun's latitude from Delambre's tables^ talsen
with a contrary sign^ should be applied to the obliquity determined
in this manner.
Ex, 2. — ^I had commenced to determine the obliquity of the ecUp-
tic from the totality of the Greenwich observatinns by Dr Maskelylie^
and had proceeded so far when I was anticipated by Dr Brinkley.
I used the French table of refractions^ Delambre*s table of reduction
depending on the sun's longitude instead of the R. A.^ which^ being
rather more convenient in practice, is made the argument here, llie
longitude and latitude of tne sun were computed from Delambre's
Tables, and, as the methods are analogous, any one who can compute
by the longitude can readily also use the right ascension, ana the
fdUowing example is given as an illastratian of either.
96
EXPLANATION OF THE TABLEft.
1
' ■* . ■*
•.
s
i
s
>i^ •
s
I
1
H
;H
^ .
s?S
;; ^; * j
'M \j M
09
to
8
3
I:
• •
P
' .' •'' :c
I
. i
w^
U3 b«
^1
&0)
s
CO i
i'' C-
'S ^ ^>o C9 Oft ^ eo
M 6i « « CO CO CO CO
CO
s
•^
«• ^ •• * _,
• 1
■is "SJ *c j£ as? « ^
*^
— •• -i - -^i *«^ •^ O?
.w — ^ a" C- i-? *Ci P .; t
>•.* .^•h«
&•■ ^a
2* ^ &
• • •
t '■•■• ^ 'J
1- :;: '.5 ^^ b? c^ -=-
^ " ? •
.IS
^«
id
Eh
©
T.
r ■»
EXPLANATION OF THE TABLES.
37
1}
a
^-_^ -»» <«
^* • • » ■ • •
i
u
I Jliiiiiii
J L
I
f -v
•
•
0^ cq
-i«i
»
to©
<6 ■
i mHfA
P
1^
•8
I
•^is.
•5
0)
«0
■•a ~ <3 J* "^
>^ I"! g
- s ^ -^ I «
9'SJg
IS1-8
.1 .-;-»■■.-
•* «t t * «^"*. "»1 •! "U e» •?
I s s § Si s Si ^
28 KXFU^ATION OF XHlg XA)I^|L
. TABJ^E XXX.-^7*o change Mean Solar into Sidereal Time.
Am m cbd regulated by sidereal time is iiidispensable in every ob-
servstdry; If i« neceasarr to convert sellur fn^ fttfefeal tfme di order
to know by iht dock when any .phenomeintv Mtfii as eclipsei, oceiil-
tationa^ ^^ ealctdaled in mean solar time ^ould take place. Tliis
table is edaj^oyed for that purpose^ as will ippear by the following
e^npnple.'
An immerddn df 4 Aquarii by the moon took place on January 5,
18B4, at 8^ "W" 60^, apparent solar time by the meridian of Green-
wifdbkj what wSl be tne time by a sidereal clock which shOwf
W)^ ^ when the point Aries is on the meridian^ and hev error that
dayW^-Mfast?
In Idua etm the clock would be a right-aacemon cilod ; an^' if
sh^ #ent tme woold show the right ascension of the celestial bbdies^as
tU^yposised die meridian when observed by a transit instrument.
]^w <m &e 5th of Jannary^ 1824^ the sun's right ascension at noon
is ltf» 1"3^.0, the same as would be shown by a clock truly regu-
lateA
But' M the dock was 30".54 fast on that day this nyiist. be added
tf^glre the tiive shewn by the dock^ that is^ sne shows 10^ 9^ 19.54
at noofn. As the immennpn happened at 3^ 40^ M" P. M. tUs lyust
be.cimverted into sidereal tjme^ and added to the preceding to ^ve
the tin^B shi^wii hj.tiie cloc)c> so that an astronomer may be prepared
toebMrveit., , ..,.-.-.
.lllis openlion myy be.fuabomplished by the tablel \ \ ^ ^
^Tnne, ' " Acceleration,
S'* V*. gp'' gives 0*" 29* .569
4fl *. 7.567
j6Q 0.138 "
.*■: I'i
.5«-"'-
»►■# . «
"l ;'- 3 4f^M; 37.264
. Um» to the iime'i|ho« by the clock IDk ^T* 18^^"
Tbeie must be added Sf-S»:
Whence the time shown will be 22 49 40 .j90
' TABLE XXXl^^To change Sidereal intaMeati'SdAt'Time;^
This table may be useful for finding the rate of a clock or chrono-
meter. As the transit of a fixed star advances 3"* 56*«908 daily on
mean solar time^ if the passage of a star be obiervied.'wi&' a transit
instrument eodi day f6r severid duqdefsive daysj or the disappearance
of a star during aeveeal'Siiooessive nights beninda feed el^ect, such
as the vane of a steeple or the b(idyOT the steeple Itself^ nearly in the
meridian^ the position of die eye of the observer being 'qIso fixed^
the rate of the dock bec(»nes known on sidereal^ and eenaequently,
by this table^ on mean solar time.
Required the retardation on 10^ 5^48"" 5& of ndereal time?
JP«c 10 <^4 we bwre (V 99" 19^.099 , .
OiSfi 0*0" ,,. . #j«r
■I I
48^0 7.8M
J$6 .. 0.163
■'.^
- fc
1 5 48 66 . — ». 4ft .l&aii
lOiQ 48 56.Q0O
.1.
<^m
Mean wdar time IQ 5 8 39.760
•
TABLE XXXn.—Tb convert Mean Time into Parttif the Eq^mr*
Tfaif table may be naefnl for converthig into degreen^ '&b. '^e
hours^ minutes^ &c. shown by a clock or chronometer r^ulatgd a'6-
cording to mean time; and the mediod of nfSng it wiR oe tefldily
nsdellstood ikon the examples to the two pireeiemng tablbb^ arid Akt
dTC^ptidn Hater's in the appendix.
TABLE XXXlll.— Lengths of circular Arcs.
The method of using this can be no difficulty to those acquainted
with the preceding tables, as they are employe^ in ft vinviWl^'iJBW'fi'*
TABLES XXXIV., XXXV., XXXVl.y XXXVII., XXXVIII;,
XZZIX., XL^ XLIt «nd XLH. are abridged from a series of tabled,
by Mr Failowf , aMionoiner at the Gape ' fti Good Hope, and weile
titonnitted to the Admiralty, along with -an approximate catalogue
of stars which he had formed there, and are very convenient ftw
im d^ng at once the amount of the corrections for precessicm, aberrii-
don, and nutaticm for any given observation, both in right ascensioti
and declination. In ad^tion to these, however, another table must
be computed annually. ■ Since the tables are only given to every ten
mitiutes 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 aide column. If the odd minutes exceed 6, the proportion^
park, muat be taken at twice, or the complementary proportional
part to the next minute of even tena, must be applied wim a contrary
sign whfn necessary.
To uniderstand the method tff applying these tables is premised
the following
Synopsis :—
Conitanti.
Table X&XIV. =— l'.33Ga 1^9. |t. A. tan. dec. +9.0678 =a
XXXV. =— l,a$0OaiB.ItA.=:;),and|»xsec. dec.= A
XXXVL =— l.2300c»R.A. = 9,and^Xsec. dec.= (j
XXXVIL =+ 0.6430co8.B.A. = «, and«Xtan.dec. = d
i . :}(XX VIII. = r- 20 .0436 COS. R. A. = annual precession = a'
tSXnL :=— 20 .2550 cos. RJV.=;:ji^ and jp'x sin. decs i^
..^UL ■ • sB-f> 18.5800flin.R.A. =:9',and 9'x8in.d.+r'= c'
XLI. =+ 8X)650cos. dec.=: = r
XLIL =— 9.6480sin. R.A. ==*' =d'
; ^ , , _ , sin. Q sin. 2
Annual Table, part 1st = i « ^^r — . . ^ A
part 2d = 0.93046 (cos. Q — cos. 2 Q +
2
.^r-cos. 2 0) . . . . =D
OL
where t is the time elapsed since the commencement of the year
when the sun's mean R. A. is supposed to be IB*" 40^.
30
JESLnUa^ATfON OF THS TAttUB.
TMmat dvttM of mu>'8> longitude at 1]lfetitDe4f-«idniiiAtioii = B
Table oiP'coi^es of the same := C
Then die whole correction in R. A. ;=: Aii +B b +C c+Dd (1)
in dec =Afl'+B6'+C c'+Dd' (2)
£r.— Required the corrections of Fomalhaut. in right ascenrfon
and decSnation for July 9Mi, 1834, at the time of his passing the
meridian of Greenwich^ the B. A. of the star being 28^ 4JBr, and de.
cUnation 90^ 33' SoudL
.The iun'a longitude itbr this time is 118° IS', of which the natiiral
sine is ^1 s=,£b <^d the cosine ia .473= S; A and D mnstjbe
taken from an annual taU^ or oooputed finom the formulae giten
above for that purpose. v
TUei^ from table XX^XV.^ &c take the prog^ numbers for JBie
K A.' of th^ star^ and cdmp&te the multipbttitioiia indicated by ftr-
miila (1) the sum of the results will be the total correctioii imft. A.>
and those by formula (2) will be that in declination. ,
1
lliua Table XXXIV. .414
690
*207
37
, Constant
244+
a068
Nat. sec dec
.1
»;
/ .■ ■» . ■ »
rections
in right
B
.881
.418
.352
. 9
. 6
-.987+
.566+
.dS3+
1.161
55
1
C.
IJTM
471
3
1.07*+ ^ *
9.964+ ±±;y
0.040+ =: ^
4.096+ =
ascension.
t
.h
M1+
3J12+
i
2.9B1
3
li
061
d
1L984+1 1<
» . ■ I MM' •'t»' ■•— *^-.. »
' SSff^f Tail, cfet
T. -.■
• ■ r
D:
liS
fflflH-
.066
t
-068+
mo
>,.
"0110+
^' i- - ' ">
.« -..J. 1^1
±Ae suM'tf ccAr-
'. ' '»\.i J ' i;
.- -r — -^
■ w'.l '. ■
XXHiANATION OF TSE TABLES. 31
Annual prceeMioii fbr Stf' 48", table XXXVIII. b r: 19^.088.
TV ' • ■
. ■ '■
T-.-» ■...;..■
■ 1,- 1 •■..'•■■* '
^rin. dMi
<■
.. 1 ■
! ■
= tn«
: ■. \.
' ■
I
1
fi
C
C
A.
D
.881+
19J64—
.473—
6.946+
.47s —
6.945+
4W1+
19.03^
.113+
3.981+
15.411
IJ41
.019
3.778
.486
Ml
3.778
.486
.031
I7.I66
019
0398
80
6
lft971—
3J85—
•
8J85—
17.176—
■ - . .. w V
14966—
.608+
7.137
.114
7.941—
&986—
. .17.176—
0.384+
S7".866—
^ sum of
cdrrectiin
u in dfjcUnatipn. 1
" ^kA in' this manner the total corrections for any number of stars
maj be readily computed.
' CABLES XLIIL, XLIV., XLV^XLVI., XLVII., and XLVIU.
negffofnnd fm the same purpose ai those above. By the former are
b^inmted more readily the correetions of a number of stars near one
a^^er tbui by tfie present, though they are convenient and very
accurate fbr computing the corrections for any single star.
; £«.— Required the true apparent ritfht ascension and declination
^« Aqufflse on the 1st of Jaaqtvy, 1628; the mean R. A. beinflr
10k 4ar 2ar.6 and decimation ff" S5^ 16" N. ?
1st, To find the Nutation in R. A. and Declination.
Lon. moon*! node 7 1 54
.^«i- "' -*■
BenMindtr
Sum
8 23 42 tab. XLIII — (X'.97, 5» tS" 42 UK XLIIL+8«.72
4 27 90 tab. XLIV. + 1 ^» 7 27 30 Ub. XLIV.-H) .69
DeclinatkNi 8' 25^ tangent
Prodoet, or part first
Long, moon*! nodcy part Moond
Nuution in R. A.
•f 0.12
.15
+ .018
+ 9.14
+ 9.158
nut. in dec-f 9 .41
0'.61
*'ff.u ti '4gtf flie AbcitaHion in R. A. and declHiatimi;^' 'y^{^yM{
Sdb'slon. 9 10 8 _ •nnn
\ •
Dddination of star 8'' 25' secant « .... • \M
■■■■■'■■ ■ \ I
RtAikbkier 4 » s » 15P 9S' Tsb.. 30,1^ •fF4ft>.t< j.liu
Si«n < + 3 i = 10 6 44 IVa>. X£t]|Lj -^iMn^ji,. ith
SUr?» dedination .8^26'. Sinr . ^ ..^ . < i.,^ fliU
Si^'s kmgltttd^ 9* 10 8 Part Ist . \rV;(^'^.> r.aVaA^|^
Sum 9 18 d9v Tahld S^I^yiJI^^;^ ) .28
RAnainder 9 1 43 Table XLyilt.' —0 .12 \
Alieriitidil lb «ecl£nation . . •Uii^pi'S:^*; jii(j Jo^i-j.it
M&4 IL A. - W^ 42- 2#.60 Declinatibti " '^ ; '^ #' 25<»MWK.
Niltatiioit3iiItA.+ . , , • HM Not Jn dec.- • ; - •4^ A.V.v ©f.^ii: j
Tfiw R. A.'^] 1^ 48 ^2i91 Tbfiie dee. .- '-• i*M« A >*
TAJBXiS XLIX. contains the mean obliquity qf iKli'^66li^ji»(yhe
aoounne obMorvaliong I <JmM ^tain; f^jgoiset'itfy^ldii^
]«W)tUf .dixi^wtni^ for .the purpoBe of conipuliiiff k ^^^^i^bknr
ti«iA»' .^" . ' ■ ■': / ■ * '■■■ ■' •" ^'^' '' ^'"
.r. ,r
' ^_ , • ■ . ^ ■ . ^ .*
"•= .^li 1 ^l(Jli;t^al^
TABIJS$. L<md LI. gtv? l(^ neauinrf oamaAiMa t6t
and the mode of applicati& iii ib^oiJii. ' '■ ' *'.'W*'^T'
* ■
TABLES LH. and LIII. contain Me Lunar and Sdar'^E^fAmu
of the Equinoxes in time, which are sometimes more convenient thaB
in space. -?.* \ r
TABBE'UV. contains the "Man Hgkt dscenwMs and d^MHtms
qfiifefd bf the principal Jifei Hars for lIKffi, tbgdUM^widi? ^lih-
aniiual variations for reducing them to any o^r tline re^itiwd; ><i
TABLE LV. — Decimal Numbers for each Day in the year. It is
useful wherever the fraction of the year is wanted/ a9 in reducing
the places of stars^ &c. to anv ^iven day in the year. This is ac-
complished by multiply ing^'dnte annual variation by the nunlber of
ULPJLANATION Oi*^ THE TABLES. 33
years and deciinal for the given day. The result applied with its
proper sign will give its mean place after the given time to which
the correctiolis for precession^ nutation^ and aberration^ being also
applied with their proper signs^ ¥rill give the apparent place at that
Ume.
TABLE LVI.— TAe Right Ascension of the Sun,
This table is adapted to leap year, particularly the year 1828,
and is only intended to answer the purposes of instruction when no
gMKi degree of accuracy is required^ and the Nautical Almanac not
aclnild*
In order to adapt it to common years^ one-fourth of the difference
beMFeen the given and preceding days is to be subtracted fiiom the
right asccAision in the table for the first after leap vear^ one-^half for
the second after leap year« and Ihreefourths for the third; and in
the months of January and February^ the right ascension is to be
taken f<»r the day following that given.
This table may be employed in finding the apparent time by the
ahitode of a 8tar> for finding the time of a star's transit when tnat is
re4afitied> fcNr^obtaining the latitude by a meridian altitude^ &3C.
L Tojind the Time €f Transit.
Rule.'^Frota the R. A. of the star^ increased by 24^ if necessary^
subtract that of the aun ; the remainder will be the approximate^ time
of transit. To this time apply the longitude of the given place in
time bj addition or subtraction^ according as it is rvest or east ; the r^
suU may be called the reduced time. To this reduced time coni-
Sute the right ascension of the sun^ which will be the sun's true
L A. at the time of transit. Now from the stax*^ right ascension
f<nr die given time subtract the sun's true H. A. ; the 'remainder
will be me apparent time of transit. ' j -■ ■ ■
. It. To find the ajpparent time of rising and setting or a Icnown
star^ the latitude and longitude of the place^ and the year and day
of ibe months being known.
Rule. — Find the apparent time of the transit of the star by the
preceding rule ; then find half the time of the continuance of the
star above the horizon^ by the method shown in Problem .VI. of
8|il|^pcal Trigonometry in the Introduction^ page 72 and 73^ whlch^
being applied to the time of transit by subtraction and addition^
wiU give the apparent times of the rising and setting of the star re-
qpectivdy.*
• ■ ■ • ■. > ,
TABLE LVIL-^Declination of the Sun.
This table contains the sun's declination for the noon of each day^
on the meridian of Greenwich for the year 1828, or leap year. By
thi» table the declination, sufficiently correct for many purposes, may
be fiovnd for other years. For the first year after lewp year, take
one-fourth of t^e difference between the declinations ^r the given
* Mx Thon^ Lvmi hag given, in his extensive collection of Nautical Tables, the
times of transits of oO princip^ stars for every day in the year, which, in many calcu-
latioBS, are very useful.
34 EXPLANATION OF THE TABLES.
and preceding days^ which is to be added to the dedinatioii ^'the
given da^^ if at that time the declination is decreasing , but snbtmcled
if increasing. In the second after leap year take the half^ in the thild
take three-fourths of the difference^ and apply this correction' in the
same manner as before; the, result will be the declination required;
And in the months of January and February the declination is to be
taken for the day following that given.
\ ■
TABLE LVlIl^Tke Etfuatum of Time.
This table contains the equation of time for 1828, or leap yext ;
and is to be found for any other year in the same manner as the de-
clination above explained.
Time, deduced from observations of the sun, is called apparel
time, to which the equation of time, being applied according to its
title in the table, gives mean time. Since a clock or chronometer \^
constructed upon the supposition of a uniform motion, this table
will be useful for ascertaining the rate and error on mean solar time.
Also, if a clock be regulated to mean solar time, the instant when
the Sim's meridian altitude ought to be observed to find the latitude;
is known by applying the equation of time to 12^, with a tMttwtj
sign to that in tne table. These applications will be more readily
understood by consulting the article aa finding the longitude by
chronometers in the introduction.
4
TABLE JJi^.'-^Correction of the Longitfide by Chnmomeieri.
This table is on the same principles as uiat given by Rossel in the
third volume of Biof s Astronomic Physique, only substituting for
the natural numbers their logarithms, as being more convenient in
practice.
Ex^At Tongatabou, on the 6th April, 1793, at 19* 53" 31'.44,
the daily rate of a chronometer was + 5*.24, with an original error
of + 1" 20^.93. The ship sailed from Tongatabou, and arrived at
Ballade harbour, on the 22d of April, when, by observation, the
daily rate was + 8'.56, and the error 1* 24°* 23'.71 fast for mean
time.
Daily rate at Tongatabou ... J^ 5^24
at Ballade -I- 8'.56
■' I
Sum .... 13.80
Half, or mean daily rate . . 6 .9
Difference of longitude between Tongatabou and Ballade by the
first daily rate of 5'.24 . . . . 20^ 24' 34''
Difference of longitude by the mean rate of 6'.9 20 17 55
Difference easterly . 6 39 S.
because the difference of longitude ought to be diminished.
From these data, what is the correction of the observed ■ longi*
tude, on the 17th of April, at 7** 34" ?
Correction of the longitude of Ballade for 16 days
6' 39"=399" log 3.600&7
Log. for 16 days. Table LIX., ar. co. . 7.86646
From 6th April to the 17th, or II days, log. Table LIX. 1.81954
Correction 3' 14" = 194" log. 2.28697
KKFLANATION OF THE TABLED 35
. ' -The OM fUB Cti on of the longitude of the 17th gives the place of ob-
aerwtion more easterly^ because Ballade ou^ht to be to the east of
the jpodtion calculated by the daily rate determined at Tongatabou.
■ Bmm the fint two logarithms are constant^ the correction of the
Imgitode for odier days in the same run^ is easily obtained by sub*-
stitudng for the last logarithm that from Table LIX. for the given
number of days elapsed from the time at which the rate was origi-
nally determined, and in this manner ought all longitudes to be cor-
rected in a long run, where the rate of me chronometer has expe-
rienced considerable alterations.
The same thing may be done without the table^ as in the follow-
ing example taken from Captain Hall's observations on the coast of
South America :—
<' San Blaa, West Coast of Mexico."
. " Conrections to be applied to chronometrical measurements of
the longitude of places between Acapulco and San Bias."
.'f The rate of the chronometer, by which the differences of longi-
tude was obtained, of places between Acapulco and San Bias, was
thfit dfitennined at Acapulco, or ± O'.O per diem,"
'f On arriving at San Bias, however, after an interval of 18 days
from Acapulco, the rate was found to be + ^'^ P^ ^7* It became
necesaary, therefore, to make a proportional allowance at interme-
diate places for the increase of rate, which increase may be ti^en as
uniform during the interval. This is effected by computing the
whole difference of longitude by the mean of the two rates j^(f.O
and 2'.6, namely 1'.3, and taking the difference between this deter-
mination and that by the first rate, whence are obtained 351" for the
accumulated error in longitude in 18 days' interval."
'^ Now the sum of a series of 18 terms in Arithmetical progres-
sion, having 1'^ for the first term, and 1'' for the common difference, is
351"
171, consequently -y^^ = 2".053 nearly for the daily iiMnrease.in
the error of longitude, and this multiplied by the sum of the terms
in the series beK)re designed, according to tne number of the days
elapsed since the rate was first determined, will give the respective
corrections in longitude, to be applied to those deduced by chrono-
meter, with the Acapulco rate. Whence we get 2^ IS'', for an inter-
val of 11 da3r8, to be deducted from the longitude of Colima, west
of Acapulco ; and the correction for an interval of 15^ days is 4' 21^',
to be taken from the longitude of Cape Corrientes, west of Acapulca
. TABLE LX. — Latitudes and Longitudes of Places. .
Tfaia table contains the latitudes and longitudes of a few of the
principal places in the world, given with all the accuracy in my
power. It also contains the time of high water at the times of new
and full moon, and the depth of the water at spring and neap tides,
which are necemwry to find the time of high water at any particu-
lar place OR a given day, as well as the depth of the water of
any tide, and at any hour of the tide, which may sometimes be
necessary. The height of the neap tides is seldom given in tide
taUea, though for these purposes the one should be given as well
as tke other.
Indeed, it were to be wished that officers of the Royal Navy, as
weQ at others, should carefully mark all these circumstances; so
36 EUUJ^ATION OF THE TABUSa
that a connplat^ . tida tabla> embraoing . aU tha OBoaNiaryriffata,
miglitatlaft b« formad. ,,>.-, /r^r\, fyst*
TABLES IftXI. land LXII. serye to convert apace int^.timfllflaid
fioavaiaiely, and their use is so ea^ to those aoquainted ^nthonatyf- of
the foregoiDg tables that any fartber explanation is umoeesaai^fH.
TABLE LXIII. contains a selection of useful numbers fi^^utotly
wanted in calculation/ which have their Ic^parithms and arithttllelJliSal
complements subjoined. - ^t^ ■'
TABLES LXIV. and LXV. are giVen for the purpose itsf commut-
ing the time and height of high water^ as well as its height 'af%iy
particular time of the tide^ at such places where iXie h^j^lAts at
spring and neap tides are known^ It is to be hoped that oui^ tiiry
officers will be enjoined to give^ not only the time and height lit' ti'ew
and full moon/ but also at the quarters^ to Ornish data for* Aese
tables. • ' ■
Ex. — Required the time and depth of high water at Leithj, on the
12th of December^ 1826 ; and also the depth 2*" before or 'After high
water, or about 4** from die nearest low water ? i r. .rn-
■ As the time of high water would be that on the followingl^inim-
ing, half the sum of the transits on the preceding and given days
must be taken, thus :— ''*• *-^ '
Moon's transit on the 11th . -KJ**"^'
12 10 60
Mean
Equation to 3^ west longitude
Reduced transit
Time of high water at new and full moon
Equation, Table LXIV.
10
27
+
1
10 28
2
20
+.
10
;\
True time . . 12 68
Or 68™, part noon of the 12th.
To transit 10^ 27" and parallax 54' (Table LXV.) in .which • is
the height of the spring tide, and b that of the neap, tiie multipliers
respectively are 0.676 a =0.676x16=10.816 %fc *i jv
and 0.176 b = 0.17ft)< 8= 1.408 £^ /^ v
Total height in feet =12.224 .^ ,
Now, with the time 2*» after the nearest high water, the multiplier
in the right-hand part of the table is 0.779. This multiplied by
12.224 gives 9.6 feet at that time o£ the tide.
TABLES LXVI. and LXVlI. contain the equation of third and
fourth differences, which must be applied in order to obtain the
moon's apparent place with great accuracy, especially in occultadons,
in determining the longitude by the moon's transit over the meri-
dian, &c. They are used in the following manner : — Take out of the
Nautical Almanac the three right ascensions, &c. preceding, and the
three following the given time, and deduce their first, second, third,
and fourth differences, also the mean of the two second differences
standing on each side of the given time, and the mean of the two
SXPI.4NATXON OF THE TABLK8. 37
'<Mirtli'!|iHfei«ticefij. Tlieii to the proportionid psrt ^the middle
first difference^ corrected by the equation of mMA secotid dIflRstoice,
by Table XXVII. apply the correction of the third difference from
i'3JablB-LXVI. answering to the middle third difference; dhff the cor*-
f rr^tlkm in Table LXVIL answering to die mean fonrth difierenoe^
and/die^veaiih will be the correct moon's place. These conrections
must be made according to the following rules.
> i rclfi ithe third difference be poskive and die time ftaAk ndatk or mid-
:rJHght-leM than six hours> the correction is positive; but if greater
than six hours^ the correction is negative. If the third difference be
nesrative the rule must be reversed.
le equation of fourth difference has the same sign as the fourth
rence itself.
[phese tables and rules were given by Mr Henderson in die 38th
Jito, of die London Journal of Science ; but we have not -room to ex-
,einpliij them here, though to those well acquainted with the appli-
cution of the equation of second differences there will be litde diffi«
Gulty.
.t*ABLE LXVIII. was drawn up by Captain Kater^ an4» being
easy in its application^ it will be found very conveoient at saa, for
wluc)i jit is duefly intended. t, . ; - ,
,^, JSi^On the 23d of June, 1826, in longitude 30^ W^ ^foUow*
uig aldtiides of the pole star were taken, the height of tfae^eye be-
ing^SOfeet; required the ladtude ?
Mean Times, Observed Altitudes.
8»*34»24* . 50° 38' 20''
39 50 40 20
40 44 50 22 10
Meavs . 8 38 3 50 33 3?
Lcmg* 2 Dipto20ft.~ 4 26
M. t. G. 10 38 3 50 29 11
Eq. T. — 1 37 Refraction— 48
- iiUl. .1
8
2
38
3
10
38
1
3
37
10
6
36
6
26
6
16
42
58
32
44
App.T. 10 36 26 T. alt. 50 28 23 tang. 0.0635
SiSffiK.A. 6 6 6 1st cor. + 54 23 log. A" 1.7584
R.A.ihbf'. 16 42 92 3d cor. + 1 10 log. 1.8419
R A.*if •
•'aJIf.D. 16 43 48
Latitude 51 23 56 N.
i: ;■: ■■• .1 t
98 JOMLklKATiaS OF THE TABUS.
.. • til 1 i-t • •
; ■ t ' 1! .
i ■ ■ » ■ .' • ■ >
I- ■ : I
APPENDIX.
On the Minute Corrections of Lunar Distances.
luf lunar observatians the corrections for the spheroidal dgimof,
the eaxth have been applied according to the method of Professor
Lax of Cambridge^ Dr Inman of Portsmouth, Mr Riddle of Green-
widi^ &c. by diminishing the equatoreal horizontal parallaK by the-
reduction for the latitude ; but unless the latitude and altitudes are
in like manner reduced^ which leads to a complex calcul/^tioiK th^
reHMtftare still inexact. The method here proposed is 'siniiu^'b^
that of Metidoza Rios^ requiring only a small table to facil?bft6'^^^i^
application. The table has been computed by my id^enious mehS,
Mr Thomas Henderson^ for an ellipticity of ^^ which se^ms to
agree well with the latest measures^ and to the mean horiaontal pa-
rallax 57^ which is sufficiently accurate for practical puvposes^ aa
the greatest error can hardly exceed V, and at a mean not above
half that qugnti^. This is within the limits of uncertainty alrking
from an error in the ellipticity^ which seems to vary between ^» and
Atp^^cmfirom the best measures^ the mean between vrYdch, ~^
baa been here adopted. No doubt such refinements are unnecessary
in the usual sea practice ; but as the lunar method^ which is still ca^
able of improvement^ can be practised with great success at lana/
it was lihought necessary to correct an erroneous rule^ which I be- '
lieve has been generally acted upon. For illustratiob we shall gfy^ '
Examnle 4th, page 97 of the introduction, corrected in this iaktinet^
as explained by Mr Henderson in the 40th No. of the London Jour-
nal of Science.
Rule, — When computing the parallax in altitude ; to the logarithm
of the earth's radius (Table XXlII.) add the secant of the moop'a -
apparent altitude, and the proportional logarithm of the ixiOQf^9i>
equatoreal horizontal parallax, the sum of these will be the propor-
tional logarithm of the moon's parallax in altitude to beemployea^nt;
computing the true distance. Now from half the sum of the moon's
polar distance, the sun's or star's polar distances, and the distance of
the moon from the sun or star, subtract the moon's polar distance,
and the distance from the sun or star respectively. Then to the con-
SXPI«ANATI0N OF THE TABLES.
»
stant logarithm 0.90103^ add the cosecant of the moon's distance
from the sun or star^ the sines of the two remainders^ and the log«
arithm of the number from the table (L) here given ; the sum of
these is the Warithm of the number of seconds to be always suh-
tra^ited from the computed distance^ while the number from the
table itself is always to be added to it to give the true distance on
the h3rpothesis of the earth being a spheroid of ^^^^ of ellipticity.
Ex. 1.— Latitude
Moon's alt.
Hor. par.
Par. in alt.
56° 12' 0" S. log. radius
32 4 secant
58 14 P. log.
49 28 P.L.
App. alt. of moon's centre
J^enraction
Parallax in altitude
I
GotrepCed altitude
Qbfliearv^' distance
tf CKm'a iHig. semidiameter
Ion for oblique semidiameter
9.99900
0.07190
0.49010
0.56100
32« 16' a5"
~ 1 32
+ 49 28
J ■. •
33
3 81
61
+
(16 30
16 1
A{iparacit central distance
61 12 31
Now with the apparent distance and alt}tudes> the star's thx^ alti«
tude aad tiie moon's corrected altitude^ compute the true distance as
ucmal^ which^ in this example, will be found to be 68* SG*' ]7|^'/ to
which the corrections for the spheroidal figure of the earthy obtained
S' the foregoing rule, must be applied,
oon's polar dist ' 69* 59^
Star's polar distance 59 28
Apparent distance 62 IS
tin;* u
Sqql..
cosecant
190 40 const log.
0.05320
oaoMs
jremainder
95 20
26 21
33 7
Numf^firaQi^TableI.+ 18".9
— 10 .4
sine
sine
log.
log.
9.64724
9.W47
lJnF646
'iii ' i I II I
1.01540
Smti 4- 8 .5 or about 8^'^ to be added.
Hence to 62" 26^ VJl" add 8^'', and the true distance
is ' 92^ 26^ 26^'
D.at ]»> 63 10 41 0« 44' 15" P. L.
'16 61 41 46 1 28 56 PL..
< . f •• ; . 1 1
1*1
1" 29» 34- P. L.
O.Q0036
030621
03031{»
46^ £!KFLANATION OF THE TABLES.
Pag-
la
'34'
13 a»
34
11
13 29
7 1
23
6
SquAtion of sec. diff.
App. Greenwich time
App. ship time
Long, in time 6 28 17 = 97^ 4' 16" W.
The earth being a sphere, it 18 =97 12 90
According to Lax's method = 97 13 30
So that the error on the spherical hypothesis, without allowing
for the equation of second difference in three hours, is +8^ 16'^
By Lax's method it is + 9 15
Ex, 2.— On the 28th of August, 1823, on the east coast of Green*
land, in latitude W 32^ 19'' N., longitude W 40^ W., Ciqitain Ba.
bine observed the distance of the sun and moon's limbs to be 100^
39' 4", the apparent altitude of the moon's centre 29'' 54' 48", the
sun's 19'' 52^ 34" ; the barometer 30.03 ; the thermometer 39^.4 ; and
the apparent time at the place of observation 20^ 44" 35f • Requir-
ed the true longitude ?
Calculating on the foregoing principles, Mr Henderson hMs^and
the apparent central distance to be 101'' 15' 5"> and jthe tme^iftance
came out to be 100^47'.;^'
Captain Sabine makes it 100 42 33
The apparent time at Gh'eenwich, corrected for th^ iqifjatiQSi of
second Afference to the true distance 100° 47' 25" is 21* Sfi^ 4S«
Time at the place of obserFation, . 20 44 35
Longitude in time, 1 15 10 W.
in degrees, ... 18° 47' 30"
If the true distance be calculated by diminiahiag the jequatore^l
horizontal parallax only, as directed by some authors, ue true diih
tance becomes 100° 47' 29", but allowmg it to remain iinpcrrected
for the latitude, the distaiice is 100° 47' 24". In geneniliihi^. cpi^
rection of lunar distances for the earth's ellipticity, is small, selobm
amounting to 10" of distance or 5' of longitude, in any. case that;,cai)
occur in practice ; and in any place within the tropics, the results on
the spherical hypothesis, may be considered almost perfectly correct.
On this subject Mr Henderson has remarked to mif, tpat " the
method prescribed by most authors, of allowing for the ef-
fects of the earth's spheroidal figure upon the luna^ mstan^esj. by di-
minishing the equatoreal parallax, is not altogether exact, but leaves
an error uncorrected, which, at its maximum under any particular la-
titude, is nearly one-sixtieth of the reduction of latitude, or ang\e of
the vertical with the radius. The greatest error therefore which 6m
possibly happen in any part of the globe, is under the parallel of
45^, where it may amount to 12". Under the equator ana poles thd
error is nothing.
*' If the equatoreal parallax be employed in the computation of
the true distance, the result is liable to a greater error. The maximum
error under any particular latitude, may be expressed by the hypo-
thenuse of a right-angled plane triangle, in which one sine is
EXPLANATION OF THE TABLES. 41
equal to the sustkih part of the reduction of latitude^ and the other
to the correction of the equatorial parallax. Under the parallel of
Lcmdon^ the wiaximum error is 14-. ' • .
When this work was nearly -leadT lor publication^ the author
learned that Captain Kater^ whose skill and experienoe as an able
panctical man command the utmost respeot, was in the habit of usiniy^
the direct method of obtaining the latitude by the pole star, aamucb
shorter and simpler than by vxe use of tables, and upon application
being made by a friend, who has interested- himself in the success of
this work. Captain Katar was so-obliging as to forward to the author,
the followinir example computed by the tables in this volume, which
had been n^mitteJhb Uslnspecdon in their propeu tlmmsh tlie
ptBisi nCWptain Kater transmitted, at the same tune, a amall table
coniahiing the tmgentaand secants to every W of the polar distance
of IpbUdtis^ which will answer for some years to come, andwill be found
toaive ibe oomputer some trouble.
SHicaolaliMi depends upon the following formulae :•»•
'^Mu u s£ tarn p X COS. t . (A.)
igU -J, ' V cos.«xcoe.s ^^.
QS^i!rr^^) = = c^' ^ X cos. z x sec. p. . . (p.)
"'Jhr^ork Oat6, R^^ent's Park, London, on the 33d of PMnnaijfj
1898, it 7" 42" 49", mean time, the altitude of ih«'pole'Mai'>w*S''l)&--
seHM^W'tli^tahi KMer to be 6P 58" 11^ i^ required the lllliC«d«?-
'IWi^'fliia tte Hitei solar time when ttte ist&tr wa« tipM '4«i ttie.
•sAto[>. K."A. (r 68"15.'2 App. m. BY* ^9 ir^
0*8 H. . A; iBl ttOon> . 22 21 18. 3 * iMfifac. — ■ 46.4
< ^ i **
2 36 66.9 True alt SI 67 95-ff
Diff^reiv»&omTableXXXI. ~ 26.7 z =38 2 34.4
2 36 31.2
Eqotttbn of time for noon, + 18 60.7
* P^ the Meridian, 2 50 21.9
T&i^ of 0)>servation, . 1 412 49.0
•lit
Mkffl^-R^*'*" **T *^' V 4 68 27.1 |=n?IJ|l?n"^
Mendifln m mean time, J ' ) Table XXXiI.
*'^'A^ r 36' 48^^ tan. . a4497ie cos. co. an . 0.000172
•;'- ^=;i78 18 47. cos. . 9.468097
■■ Vr-tJ 27 48 .2ttan. ! 7907813 cosine . 9.999986
' ;' '^ 2= 8* 2' 34.4'' cosine . 9.896278
(4^*^=38 58 .2 cosine 9.896436
I-.-
if',/' ipiSQ 28 46 .4
a" ;=b:Sl 31 13.6
■t. 9ki nM i t
V'l^ouiid by precept, page 10 of Explanation of the Tables.**
woiddbeto .
die sum of these logs, would be tne lo^.
finding the value of the log. tangent ot small arcs.
4a
JKXPLANATION OF THE TABLES.
To those not very familiar with such Computations ii may be usefid to
show the Manner of Calculathn,
As a rule in words at length might be Mnriceable in the eolotibn of fthbt Qkrob-
lein» to thow who are little eonveraant with faramlflBf it has been added.
' To the constant log. 5.314485, add the log. tangent of the star's polar distanee
ft and the log. cosine of the meridian distance <« in degreesy the sum of theast will
be the log. of the arc u in seconds. Now, to the log* secant p add the log. coeiiie «»
and the cosine of the senith distance s, the sum will be the cosine of(>^£«)y an
are whicbt being increased or diminished by the arc «, will be the colatitudt. <^»
'irhmce the latitude X is readily obtained.
• Constant logarithm, • • 5.3144S5
p=^ 1« 36^ iS'' tangent, - S,4AOl\7 iSCMit* aopOlTS
<s73 18 47 cosine, 9.i58097
K« 87 4&2.«1668^.8
(>^— «)b38 . a 58.2
^ :ss38 28 46.4
k ::s51 31 1&6
k38« r S4.''4
obsliie, 9.90S986
cosine, I^.8062t6
cosine, ^MOidS
In the ai^Ucation of m, attention must be paid to the i^n of ^. ibt <^ iScc^-
ing to its sitmOloD in the ciiode which the stkrdcscribei round- the poW in Ib^Snr-
nal revoJatioii. If < is iq the Orst 6r fburth quadrant, it iii 'addiflM s tmt 'if IiMIm
sejcovid or third, it is subtractive. .;..:.''
" ■}.' '■ -
■i
•• . I •
TABLE I.
1
OD Kooaat of the Spheroidal Figure
rf the Earth, iu Elliptidly being ,ij.
Br Mb HEirsEBaoN. 1
Ut.
Ul
5
10
U
JO
4S
30
&
10
15
80
85
.30
0=
a
an
a
ftO
0.
a4
no.
46"
16.
i6.a
Ifi.
l.tB
l.-i.
14)1
i
ai
il.
0.1
II.
o.<
II.-
48
16.
16.!
16.
IH.;
15 4
i
1.1
1.
I.I
I.
1.:
£0
17
17
16 •
e
t.
?-•
9.
»»■
a.
a.i
9.]
5S
IM.
17.1
■7.
17.!
Ifi.
l(i.'
15 1
B
I
3.1
:t
;<.
:ti
*J
?.;
54
IH.
IK.'^
H.
17.1
Ifil
10
4.
».a
a.
3.H
3.7
3.6
A*
56
18.
ia.>
18.
18.;i
17.'
17.1
16.4
4.
*.<
4.
4-'
4.1
58
H.
IH.;
9.
IH.;
IH.
17 5
It
6.
ii
i.
.I.;
.%.■/
6.1
4.t
60
9.
rw.;
9.
19.1
18.
ir.!
17.;
IS
IL
6.i
(i.
fi.
A.;
.'.A
62
80.
i-O.!
.9.
IS.5
IB.
ih:
17.5
7.
7-f
h.
Ii.t
H..
fi.i
6.1
6*
m.
NJ.^
80.
19.1
19.
IKh
«0
7.
T.B
7.
7.
7.M
7.1
6.;
66
to.
jias
8a
80.1
19.
18.!
18.:
88
n.a
M.)
T.l
7.<
6b
«l.
81.1
SO.
80.5
I».
19,8
184
S4
9.
H.;
!i
H.S
H.'
H.^
K.f
70
•>]
I'l.'i
yi.
W:
?0,
36
|IM
H,
fl.i
»..
9.1
H.;
78
it.
21.1
81.
!!.(
80.
19.1
18.8
28
tn.i
III.
Kj.;
KM
ft!
«.;
74
N.f
'1.8
Wl
19'
30
il.
lU
11.
11.U
ia7
10.S
9.^
76
88.1
81.4
to:.
80.1
19.8
iif.(
1.
M.l
II.'
11.1
io.;i
78
8?.:
1!8.;
88.'
81.6
Sl.i
?n.i
iB.a
34
■v..
W.'i
11-..
ti*.;
!)>.<
1 ]..'>
lU
80
K*-..
«>!..;
VV,
V.l.',
'1
f<H
19,'i
36
IS.
IS.?
l?!
m
??.i
"?.S
ft-'
?l,t
96
40
14.7
IM
14.<
118
14.4
13.1
14.S
lis
i3.a
I2.T
13.3
I2.t
12.7
84
86
i8.7
M.7
i8.6
28.;
81.!
81.3
21.4
80.5
80.(
19.6
19.7
88.4
82.(
4S
fi-l
■ A.!
41
14.1
14.;
Lit
S.S
88
fi.-
M.(
81..
80.1
19 7
14
15.B
I5.H
litf
IB.3I
14.9
143
90
88.8
88.7
88.5
81.5
8O.7I19.9 1
TAB
.E II.
t
or Finding the Luitiide bj the
PdeSti
ir.— By CATTiiK Katek.
PoLu
Disunce.
rangem.
p.p.
+
CiMiae
Ca.Ar.oi
Secani.
FiAti
Distimcc-
rwigent.
P.P.
+
CcHine
Go. Ar. or
Secant.
W
M^V
^OOOISS
" ay
L000166
.43309;
"- 71
1.000159
.41838i
"-= 7^
11
.4i3387(
^\S'
L000160
.44308!
1.0001 6T
:i
.43464J
-g3i
loooiei
f)
.44381
= 8g<
1.000168
)
.43£41l
-301
tOOOlSI
It
.144fi9i
= 301
1OOOI68
lO
43619:
= 385
-46!
■-00016!
.0
1.445355
= 377
==153
J,000Ifi9
y,
«wnf<i
■.00016!
0"
.416 IH
1
4377S1
-S3f
■000163
.44686'
-58(
1000170
1
,438aX
.000163
(1
.447611
-60^
.OiWlO
1
-693
.000184
,lie36»
-678
.000171
>
1
.000165
.449111
.000178
440797)
,1
).00016S
>0
t.444»e6t
i00017»
ERRATA AND ADDITIONS.
IKTBODVCTION.
Page
00 For Etm 799.32 ISeet, read H sb a01.16 feet Defect, fir 3M feet, read
' 1.50 feet.
63 SecxMid line from the top« fir his anistant^ read an officer of the Griper who
aniited him in making the observationfl.
84 For mXQg read sun, in the fifth line ttom the top. For ^-3' 27'^.02y read
— 2< 27'^02, in the ihizd line ftom bottom.
85 For Dip section, read dip sector.
88 For 67'* 12' 12", r«w2 67* 18' 12^, Une 20.
102 Table IV. to var. 1"* 3Sf and 2'^ of second difierence,/»f 2'.7, read S".?.
116 For a Leonis, read A, Leonis.
119 For X, from Parry's last Journal, I suspect n Geminomm must he read.
126 For sun's dec. 22*'35'40'^ read 22<* 35^ 45'^, on account of haying apjplied the
equation of second difference with a wrong sign. Long. 9^ 55" W.
128 For 18* 1' 0^.0, read IS"* 1- O-.O.
159 Ex. 2. Captain Hall reduced both his experiments to 68^ F.^ and therefore
my correction is erroneous ; it may^ however, serve as an example of the .num-
ner in which such a correction should be made.
160 For cos ' L.^ read cos 2 L.
EXPLANATION.
17 Example for reduction of altitude is wrcmg, but may be easily coneeted by
the rule.
21 Line 11,>- 4» 65^ 14M, read 4« 66' 14^4.
23 For 380 16' 0".17, read SS^ lO' 0".17, in 11th line ftom the bottom.
25 For 230 28' 5l".7, read 23<> 18' 61".7.
In the tables not stereotjrped there are two or three errors.
101 Table XLIV.^j^ R. A. Star^Lon. Moon's Node, read +
103 Table L,, for Moon's true Long, read Sun's,
104 Decimation Fomalhaut,^ 30«» 34' 24"^ read W> 31' 54^
Appendix to Explanation of the Tables, page 39, line 3d Arom bottom, for
92° 26' 26", read 62* 26' 26".
MATHEMATICAL TABLES.
TABLE I.
THS MILB8 AND PABT8 OF A MILK IN A DSOBKS OF LONOITUDB
AT SVEBT DBGBSE OF LATITUDE.
D.L.
Milfli.
aL.
MilM.
D.L.
MUei.
D.L.
Blflet.
D.L.
Blile9.|D.L.
Bfiki.
I
59.99
16
57.67
31
51.43
46
41.68
61
99.09
76
1459
t
59.96
17
57.38
39
50.88
47
4a99
69
2ai7
77
t&50
3
59.92
18
57.06
33
5a39
48
40.15
63
27.94
78
I2.4T
«
59.85
19
56.73
34
49.74
49
39.36
64
96.30
79
11.45
5
59.77
90
56.38
35
49.15
50
38JS7
65
95.36
80
10.49
6
59.67
91
56.01
36
48.54
51
37.76
66
94.40
81
a3»
7
59.55
99
55.63
37
47.99
59
36.94
67
9a44
89
a35
»
59.49
98
55.93
SR
47.98
53
36.11
68
99.48
83
7.31
9
59.96
94
54.81
39
46.63
54
35.97
69
91.50
84
6.97
10
59.08
95
54418
40
45.96
55
84.41
79
9a59
85
5.93
11
58.89
96
53.93
41
45.98
56
S3.55
71
19.53
S6
4.19
It
5&68
9T
5SL46
49
4459
57
39.68
79
18.54
87
&14
13
58.46
98
59.97
43
4aS8
58
31.80
73
17.54
88
9.09
14
58.99
99
59.47
44
43.16
59
8a90
74
16.54
89
1.05
IS
57.95
30
51.96
45
49.43
60
saoo
75
15.53
90
aoo
TABLE II.
LOGARITHMS OF NUMBERS.
— Hsm
■■■nM
ToR
Log.
'^A.iAAAA i
xrrm
No.
1
Log.
No.
Log.
No.
Log.
No.
hOf,
Na
^
0.000000
21
1.399219
41
1.619784
61
1.785330
81
1«909485
9
0.301030
92
1.349423
42
1.623249
62
1.799399
89
1.913814
S
0.477191
23
1.361728
43
1.633468
63
1.799341
83
1.919078
4
0.609060
24
1.380211
44
1.643453
64
1.806180
84
1.994279
5
6
0.698970
25
1.397940
45
1.653213
65
1.819913
85
1.999419
0.778151
26
1.414973
46
1.662758
66
1.819544
86
1.934498
T
0.845098
27
1.431364
47
1.672098
67
1.826075
87
1.939519
8
0.903090
28
1.447158
48
1.68124;
68
1.832509
88
1.94U83
9
0.954943
29
1.462398
49
1.690196
69
1.838849
89
1.949390
10
1.000000
30
1.477121
50
1.698970
70
1.845098
90
1.954943
11
1041393
31
1.491362
51
1.707570
71
1.851958
91
L959041
19
1.079181
39
1.505150
52
L716003
72
1.857339
99
1.963788
13
1.113943
33
1.518514
53
1.724276
73
1.863323
93
1.968483
14
1.146128
34
1.531479
54
1.732394
74
1.869232
94
1.973198
15
16
1.176091
35
1.544068
55
1.740363
75
1.875061
95
1.977794
1.904120
36
1.556303
56
1.748188
76
1.880814
96
1.9812271
17
1.930449
37
1.568202
57
1.755875
77
1.886491
97
L986779
18
1.955278
38
1.579784
58
1.763428
78
1.892095
98
1.991226
19
1.978754
39
1.591065
59
1.770852
79
L897697
99
1.995635
90
1.30103P
40
1.602060
60
1.778151
80
1.903090
100
[t.(K»Qfi^\
S A Tsble4ifLogaritbniiof>IumliBnfi(iinl to 100,000.
P^N.
1 1
8
3 4 1 5
6
7
8 i 9
100
000000 0004i*
U0UU68
001301
001734)0021 6«
008598
0030211
00346 i;U03M)
il
4331 4751
5181
5609
6038
p466
6694
7881
77*8
817
8!
i
8600 9026
010300
010784
011147
011570
011B93
01841.
134
3
0li837'0]3g59
OUIOO
4581
4940
5360
6779
6197
«li
165
4
7033] 7451
081 18B!o! 1603
7868
8884
8700
9116
9338
9947
020361
08077.
808
022016
082428
088841
083258
083664
024075
4486
4891
S47
6
5306
5714
6185
6S33
6948
7350
7747
8l64
8571
8B7(
!8a
7
B381
97fi9
030195
030600
031004
031 40t.
031812
038816:038619
03308
330
8
033424
033826
5089
5430
5B30
6230 0629
lot,
371
9
7486
7825
8223
8620
soil
9414
9flU
040207]010GOS
Q4099'
041 7B7
042188
048476
048969
04^368
U43755
U4414B
04493:
38
5383
5714
6105
6494
6885
7875
7664
8442
8931
75
8
9818
96U6
050380
050766
051153
061538
03869.
113
s
053078
053463
05384(
4830
4613
4996
5378
5760
6142
652'
IfiO
4
6905
7886
7666
8186
9185
9563
9942
060381
IBS
060698
0S107S
061452
061889
062806
062582
068948
063333
063709
408:
iS6
6
4458
4B32
5206
5580
6386
7071
7443
781,
gfi3
8186
8537
8928
9898
9668
070407
070776
071145
07151'
301
8
071888
078850
072G17
078985
073348
4083
4816
618;
338
9
A54-7
5912
6276
6640
7004
7368
7731
8094
681!
ilo
079181
08U866
UB1847
obIto?
088067
nsSii
SS
Q82785
083144
083503
3861
4576
4934
5291
5647
600'
S
6360
6716
7071
7488
7781
8136
8490
8845
9198
964:
104.
090858
090611
090963
091315
098018
092370
098781
093O7:
138
4
093428
3778
4122
4471
4820
5169
4518
6215
6561
1T3
S
6910
7257
7604
7951
8898
8B44
8990
9335
9681
100021
808
e
100371
100715
101059
101403
101747
108091
108434
102777
103119
346:
Si!
7
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34
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10 A Table of Logsrithras of Number* from 1 to 100,000. j
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6
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5021
5065
5108
5158
U
40
m
5196
58*0
5284
5389
5372
6416
6460
6504
5547
4591
*4
9t>5635
995679,995783
9959U
995854
B95B98
996948 S95986
996030
14
4
6074
6117
6i61
6805
6249
6893
6337
648*
6468
14
651 £
6555
6599
664S
6697
6818
6868
6906
*4
13
6949
6993
7037
7080
7184
7169
7818
7855
7299
1313
**
19
7386
7430
7474
7517
7561
7605
7646
7662
7736
7779
44
S8
7883
7867
7954
7998
8041
8085
8129
8178
8216
44
96
8859
8303
9347
8390
8434
6477
8681
8564
8662
44
31
8695
8739
978E
8886
8969
8913
9966
9000
9048
9067
44
35
B
9131
9174
9£18
9861
B305
6348
9398
9436
9479
9522
44
40
9565
9609
9658
9696
0739
978;
B8:ie
9870
6913
6957
43
'
m
X
I 1 »
-3-
4
5
6
~^~^
r=9-
n
TABLE III.
THB AVOLE8 WHICH EVERY POINT AND QUARTER POINT OF THE COMPASS
MAKES WITH THE MERIDIAN.
North 1
Points.
Points
South. 1
,
/ //
2 48 45
oi
oi
u
5 37 30
04
8 26 15
0}
N.I1.E.
N.kW.
1
1 J
1 i
1 i
11 15
14 3 45
16 52 30
19 41 15
1
i i
1 4
1 i
S. b. ISm
S.b.W.
M«N«B«
NscitWa
2
8i
2 i
22 30
25 18 45
28 7 30
30 56 15
2
24
2 i
S«O.Ei.
S.S.W.
N.B.I1.N.
N» W» b. N«
3
IS
33 45
36 33 45
3
34
S.E. b. S.
S. W. b. S.
39 22 30
34
3 i
42 11 15
3 i
V.U.
N.W.
4
n
45
47 48 45
4
S.Em
S.W.
50 37 30
4 i
53 26 15
4 i
N.E.b.B. ;
ri.W. bw W»
5
66 15
5
S.E. b* E.
8.W. b. W.
&i
59 3 45
&4
&i
61 52 30
54
^ i
64 41 15
^ i
B.N.B.
W.N.W.
6
67 SO
70 18 45
6
6 4
B( S« K.
W.S.W
73 7 30
6 4
6 i
75 56 15
6 i
B.bwN.
W.b.N.
7
7 i
78 45
81 33 45
7
E.b.8.
W.b.S.
7 i
84 22 30
7 S
87 11 15
7 i
East.
West.
8 |90
8
East.
West.
TABLE IV.
LOGARITHMIC SINES^ TANGENTS^ AND SECANTS^ TO EVERY POINT AND
QUARTER POINT OF THE COMPASS.
Piywilfc
n
1
u
1 1
2
2
8
2
4
i
I
s
Si
s f
aoooooo
8.690796
8.991302
9.166520
9.290236
9^5571
9.462824
9.527488
9.582840
9.630992
9.673387
9.711050
CoBuie.
9.744789
9.775027
9.802359
9.827084
9.849485
/
10.000000
9.999477
9.997904
9.995274
9.991574
9.986786
9.980885
9.973841
9.965615
9.956163
9.945430
9.933350
9.919846
9.904828
9.888185
9.869790
9.849485
Tangent
0.000000
6.691319
6.993398
9.171247
9.298662
9.398785
9.481939
9.553647
9.617224
9.674829
9.727957
9,777700
9.824893
9.870199
9.914173
9.957295
10.000000
OogiBA I Sine. / Cotang .
Cotang.
In6nite.
11.308681
11.006602
10.828753
10.701338
10.601215
10.518061
10.446353
10.382776
10.325171
10.272043
10.222300
10.175107
10.129901
10.085827
10.042705
10.000000
B
ange
Secant.
10.000000
10.000523
ia002096
10.004726
COMC
Infinite.
11.309204
11.008698
10.833480
10.008426
10.013214
10.019115
10.026159
ia034385
10.043837
10.054570
10.066650
10.080154
ia095172
10.111815
10.130210
10.709764
10.614429
10.537176
10.472512
10.417160
10.369008
10.326613
10.288950
10.255261
10.224973
iai97641
10.172916
mt \
10.160615 \ \0.\5»i5\b
CoBec
^ecaiTvV.
«ft Table V.
LogwithmicSinea, Tangenti,
^"™
' OH<n«.
lODigreu.
a. B-l '
Sme.
D.
Co«c.
T«ng. 1 D.
Colug.
Secant.
D. 1 Corine. [^
ST^
W
(
l).g3B870
1193
10.760330
9.846319:1830
la 753681
10.0066*9
37 ,».99335l|flO
80~i
*
1
240386
1191
759614
847057 1888
758943
006871
37
993389 »9
«
8
?
gillO!
11B9
75B899
84779*! 1886
75880G
006693
37
993307 S9
51
la
a
841814
11B7
758186
8465301824
75147D
006715
37
993885 57
4f
16
4
84SS26
1185
75747*
849264 1988
750736
006738
37
993268 S6
4*
2t3g37|ilB3
756763
849998 1880
750002
006760
37
9932*0 55
*i
St
2*3847,1181
7Jfl053
8S0730I918
749270
006783
39
993817 54
31
88
7
g*4fiS6lll79
755344
2SUBI 1S17
718539
006805
38
993195 S3
SI
3*
t
8433631 11 77
754837
852191 1S15
7*7809
006888
38
993178 S2
88
S
846069:1175
75393 1
8589201813
747080
006851
39
993149 51
M
4o;io
!4677Slll73
753285
853648 1211
746358
006873
39
993127 50
8(
«IlI
! 4747 8
758521
854374 1S09
745686
006896
38
993104 Is
11
48|1«
848181
116B
75191'
255100 1807
744900
006919
993081 It
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Si\l3
848883
1167
751117
855821 1S05
7141 J6
0069*1
39
993059 17
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849583
1165
750417
856647;i803
74345f
006964
993036 l(
t
1.850882
U«3
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10.748731
10.006997
38"
9.99301; 15
19 (
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850980
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7490811
857fl9o'lSOC
748010
(WOIO
998990 14
sin
8516T7
86871(i;n98
741890
007033
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992987 13
51
leliH
858373
115B
747627
2594291 1 196
740571
007056
39
9989*4 12
48
IBilS
853067
1156
74633:
860146:1194
739851
007079
38
998981 11
4-
8o;s(
853761
1151
746839
860863
1198
739137
007102
S8
99899B U]
K
S4ai
254453
1158
74554T
738488
007185
38
992875 W
se
S»!S
855144
1150
268898
1189
007148
998858 19
31
sa'sa
855834
744166
863005
U87
736995
998829 17
88
3g'!1
856583
1146
743177
863717
1185
736883
007194
39
992806 i6
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♦o;w
857211
1144
74878P
864129
HB3
735578
OOJ817
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80
4*i«6
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11*8
74810?
265138
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734868
007841
998759 J4
U
4B|ai
858583
1141
741417
865817
1179
734153
O0T8G4
39
992736 iS
11
SS88
859868
1139
7W732
86655S
1178
733145
007897
39
998713 «
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56,29
859951
1137
740049
867861
U7{i
738733
007310
39
992690 31
4
K
(.860633
1135
10.739367
9.867967
1174
10.73203;
10.007334
39
9.998666 30
18 (
i
31
861314
1133
738686
868671
1172
731329
007367
39
998645 89
5<
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38
861 9S4
1131
73B00a
869375
1170
730885
007381
39
998619 tS
Bt
la
33
868673
1130
737387
870077
1169
78998;
007404
39
998596 i7
41
16
34
863351
1188
736 64y
870773
789881
0O748B
39
998578 86
41
20
3S
864087
1186
735973
871*79
1165
72S58I
007451
39
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84
36
864TD3
1184
87817S
1164
787888
39
9U85852*
31
ae
37
805377
1188
734623
878876
llfli
727184
007439
39
998501 83
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1120
873573
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786*87
007588
40
99847888
88
36
39
88GT83
1119
733877
874269
U58
785731
007546
40
99815121
81
40
W
867395
1117
274964
1157
725031
007570
99813080
80
44
11
868065:1115
731935
875658
1155
7813*2
007594
40
99840619
4B
i!
8687341113
731861
876351
1153
783649
007618
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S983881B
IS
fiS
43
863408 1111
277043
1151
728957
007641
40
992369.17
998335)16
B
46
M
!70069|lllO
78993;
877734
1150
788866
007665
40
4
,
IS
IS
M70735;110B
10.123865
[1.27848*
10.781676
10.007689
40
3.998311 15
IT <
>
46
8T1400!]]0G
788600
873113
1147
720887
00 J 713
40
998867 1*
56
47
818064
1105
787936
879801
1145
78019B
007137
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9988S3 13
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18
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878786
1103
727871
280488
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719518
007761
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881174;iI41
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881858.1 14a
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007810
992190 10
40
U
874708
1098
725892
88254.2,1138
717458
007831
40
992 16G 9
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Si
875367
1096
72463;
883885 U36
716775
007958
40
998148 8
3i
32
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876081
1091
78.1976
88390JI136
7160S3
0078B2
41
998118 7
a«
36
SI
876681
1098
783319
BB158B 1133
71541S
007907
41
998093 6
Si
ss
877337
1091
78866;
88586S113I
71*732
007931
41
992069 5
80
44
S6
277991
1089
788009
8859471130
71405!
007956
9980*1 1
IE
48
876645
1087
781355
8866241128
T1337t
007980
41
B98020 3
18
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879297
1086
78070;
887301
1136
718699
009004
41
991996 8
8
56 59
279048
1081
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887977
1185
718023
008089
11
991971 1
4
14 O.fiO
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1088
713401
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1123
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TABLt T.
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CdIec
Ta^
ri.s.T.
11865
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Sccknt. ,11.
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STT.
8.21I»55
u.rsBiia
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l^75M0i9
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7 8388 E
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3150*6
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04«6
0533
0579
0686
0678
0719
0705
46
83
0818
0904
U961
1044
1090
1137
1183
1829
46
XB
1IT6
1388
1369
1415
1461
1508
1647
1693
46
S>
1740
1186
1838
1979
1985
1971
8016
8110
8157
46
;»7
41
8803
8849
8895
834S
8388
8434
8481
8587
8673
8619
46
8666
8718
8T5B
880*
8851
8897
S943
8989
3035
30S2
46
LE.
nT
1 1
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^3'1~4~i~r"
6 r 1 \ * \ * \-^
M T^um V.
JjOgsrithmie Sinn, Tangents,
I T«^-
IUU3
3 146(43
UTise
1480i61Wl{
1468021175
1 JOBBG 14Ti|
15156914691
Oiq 1J«4S1146S
1*3330 14C3|
1«H)BI4«0
155083 I45T
155957, 1 4.U!
on ».l5fl8SUl4:
1517001448
15«SG9l445
15943.''
180301
1S]16kl436
ISSOM
ia!es5
163743; 14!T
164600
165454l4i2
1663071419
167159 1416
16Br'08I413
16Se5S|l4IO
).169rO!|1407
170547|140S
17138911408
1Tgg30'139B
1730701390
173 908 1391
1747441391
175478,1388
176411 ir ■
177842, l:
1780781380
1789001377
1797S6'!374
1803511378
lB1374l3ii9
10.9S6t4«
855547
854651
853757
BSeB64
851974
851 03«
850198
B49314
848431
847546
84«6T0
84579!
844911
_8440B
10-843170
B4t3O0
84I43I
8405GG
83Dli99
838S36
837975
837115
836857
835400
834546
833693
838841
831 99!
1514
153^69, 1508
154174 1505
155077,1508
155978 1499
15B877 U9S
157775 1493
158671.1480
159565 1487
1604.^7 1484
10.830SO»tl9.
889453
888611
887770
886930
82609«
885256
B!44?t
8(3599
8(8758
881 988
9.18819B13H711
183016 1364
1838341361
S51 1359
185466 1356
186(801353
187092 13511
187903 1348'
lHH7iai340l
189510 1343'
190K5I341'
ISUS0133B'
19193.) 1336
I9!73l 1333;
193.i;{ll330
IIH33(iy88l
8153M
81453*
8137(0
818908
81809T
811(88
610181
809675
808870
flOi3067
807866
80(1 160
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.161347 1481
1688361478
1631(3 I
164008! 1473
164898' 1470
165774)467
166654 146
107538 146
1684091 1 45t
109(84 1455
170157
1710(9
1718!
174499
175368
1T6284
1T7018
1 78799
1 7965 5
180508
181360
18(811
I 83059
183907
I 84758
1B55D7
.187(801399
188180,1396
1889581393
18979 H39I
1906891389
11)1468 1386
19((94 1384
1931(41381
1939531379
1947801376
195601
19643011371
1972.53
198074:1366
I98S94
10971311361
851(88
850368
840456
84854C
847637
846731
845886
84498S
8440(8
843123
8488^5
841 32S
10.004(47
004(65
004283
(H)4TOI
8368 77
835998
835108
834(86
828101
8(7(32
8(6366
10.885501
884038
823776
8((9I6
8(20.it
8(1201
8(0345
819498
818640
S177t
816R4J
004573
004591
004610
004688
004647
004666
004684
004703
00478!
00*759
0.004797
004811
004835
004854
00*873
004898
001911
004968
004987
005007
005026
814403 005045
ia8 187(0 I0.0U5O81
005104
0051(3
005143
811880
6I104(
810(06
B093II
807706
806876
e06O47J 005(41
005(61
OOS300
0053(0
005310
■99675!
f)9S735k9|
995717 SHJ
9S5699 i1
995681 56 1
B95664 S5|
995646 54
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99.5610
995591 5
B95573 Sd
995555 *9
995537 i9]
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9.9»51ti8 45187
995464 44
9S5446 t3
9954(7 48
995409 41
to
9S5353 W it
995334 ITI
99531 1 }6
995297 iS\
905(7E 34
995165 (8
995146
9951(7 iQ
99510): !5
99503S !4
995070 i3
995051
995038 (1
995013 80
994993 19
994974 18
994955 17
994935
.g9491( I.
994896 14
994877
994857
994838
994818 10
99479)1
994779 8
994759 7
9U473B
994719
991TO0 4
994«S0 3
99466( 8
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3 \ 45 \ 6W \ ^ \ ^^ \ ^'- \** -
1-
and Secanis.
TAmi V.
~
Y Hour,
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to. ,»•!'
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Sin<. 1 D.
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Count
l?!lJ
9.1Bt33gl3SB
S^IBOJl:!
10-800887
iaoo5:;8o
33
9.99*«2(
so
81 1
i 1
I9£li9I3J6
I0SgS5ll3S3
804871
800581!
1359
799*71
005100
33
994000
S9
a
8 %
801075
201343
1338
798656
005*80
994580
59
K
n 3
196719 1381
803^31
8081 3S
1334
7978*1
005 4 40
34
994580
57
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16 .(
I9Tfill 1318
B08l8fi
808971
79708B
34
994540
50
44
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issaoeisis
eoibsi
803782
1349
796818
005181
S4
891519
55
4(
H 1
IBH»1 1«]3
B00901
«0tS9i(
1347
795*08
003501
34
99*499
51
36
18 1
lBflBI9131I
600181
SOiMOO
I3M
7946 0(
005521
3*
fi9U7B
53
31
sa 1
eo066fil30e
799334
806207
1348
T93793
005541
31
994469
S8
88
38 1
8014311306
7985111
807013
1340
798987
005568
9S44Sb
84
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808834 130*
7977Hfi
i07MI7
1338
792IB3
005582
34
994<'lt
80
44!U
803017 1301
7969S3
808B1D
I33i
791381
0O5fi08
34
994S98
19
18
4«ili
803797 1899
796203
809420
1333
790580
005083
34
»94377
18
11
fi*'l!
80tiTTI896
79518:
810820
1331
789780
005643
3*
89435J
17
e
i^u
80S3S4|l89il
79tG16
811018
1328
788988
OOiG'il
34
994336
*6
15
4
83
ii w.;i$.9.a»iMi|im
lU.7!l38tiS
9.211813
I32D
10.788185
iaou3;;s4
^ 9:99"4arfc
4.W
i069»K 1889
793091
818611
1321
7B73tifl
005705
34
99489i
1*
36
80T6T9,iae7
798,121
213105
1381
786595
005726
35
99427*
*3
58
leh*
806448.1 88*
7gl51{
814198
1319
78380)
005746
35
SS4854
it
48
fi
S09tS81H8
79077!
814989
I31T
785011
35
99483^
11
44
tI»S98i8M
TS0008
SI 4780
1315
781220
005789
36
994818
4t
810160 I8T8
T8984(
81S568
1318
783438
005809
3.5
991191
IS
3b
illiiH'llTS
788174
81T356
1310
78264*
005889
35
99*171
IS
38
S18S9l!l873
tB770f
818148
1309
781838
005850
35
89*150
17
88
813(J5S'18T1
T8694J
S1S986
1306
781074
005871
35
991189
16
84
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tisoiiilitae
780 IM
819710
1303
78089C
005898
35
99*108
15
V.
4«'M
8l42T918fl6l
785181
•90*92
1301
779508
005913
35
994087
14
48 «r
81^338' 1864
g|(iOBTl!B[
7846G8
281278
1299
778788
005934
35
99*066
IS
52'a
783903
888052
1897
777918
005935
35
99*013
18
JI6,Ei
216834,1259
783141
1894
777I7C
005976
35
994084
11
ta o85,B.«i76oa,ias7
10.788391
9.883607
UB2
10.776393
10.0«59"97
35
9.991003
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818363 18S3
781637
894388
1890
773618
0060 l»
35
993988
19
H
ea
3I911618S3
78088*
8t5136
1B8B
7749*4
0060*0
35
99^960
tn
tA
isn
8196681210
780131
886929
1886
77*071
35
993939
•7
48
10 «
880618 18(8
779382
886700
1884
77S30C
006088
993918
!6
44
son
881367 1!(6
77963}
827471
1881
778589
008103
36
993897
ti
«.
Sin
888115 1844
77788,1
828239
1279
771761
0OBI85
36
993873
■4
SS
!BM
8888811848
77718!
itmn
1877
77099;
0OS146
36
99383.1
a
St
3KM
8*3606 1839
776394
829773
1875
770887
006168
38
993838
»
«
36 31
88*349 1837
775(i51
B3053B
1873
769*61
006189
36
84
40'4I
885098 183.S
774908
831302
1831
768898
00621 1
30
993789
H)
4441
88583:)|l83a
77*167
8380(13
I86S
767935
0068381 36
993J68
16
4848
88657:) 1831
773127
938826
1867
767174
00623*1 36
993741
18
69U
88731111888
778(i89
833586
1865
768114
006275
36
993735
8
se*<
888018|1!86
771958
8343*5
1862
763653
U06897
30
36
99370;
*
t9 ois
lh88STU^m8J4.
10.77 ISIS-B-SSalOS
1260
10.764897
10.006319
9.993681
81
44&
8895(8 1288
770182
83S839
185B
764141
0063 10
36
993660
M
SM
830258
18!0
76 97 It
836614
1836
763336
36
993038
d^
18 46
830994.
1S18
769011
8373BS
1254
78S63S
00638*
36
993616
4t
16'4g
831714
1810
7G»g8a
8381 SOI
1268
761880
006 W>(.
37
993394
44
io«
8384*4
12U
7ri755*
838878.1850
76112(
006428
37
993578
40
8451
833178
76B88S
asfiesSiisiB
76037 f
006450
37
atf
29W
833899
1809
766101
840371 1846
75H62S
006172
37
993388
^
3(ja
834685
1207
765375
8111181844
7588Bi
006491
37
se
36 S4
>3aS49
1805
764631
8118U51218
73813S
006516
37
99318)
40M
836073
1803
763957
8436101810
757390
006538
37
993162
44 M
836795
1201
7'6350,!
8433511 12SH
OOOJhO
37
9S314<i
isjn
837313
1199
768183
2410H7;I886
75390:
006588
37
893118
18
5g,S8
83BSS5
1197
761763
814S3!>ll8a4
755101
ooouot
37
993396
(
5659
838953
1195
761 on
2433791232
751181
O00li?fi
37
993374
4
W Ofi(
839570
1193
78033(
84631!!] 1830
753881
000049
eBaa&\^ q
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^
iiTT:"^
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io4a
44 41
48 4i
Table V.
t*9StG\}
i43S37 i 1
943947 1 1
SMOfiS 1 1
84677511
S»T47B11
ilHiei 11
.850883 1 1(
(5167711,
858373 II,
£5300711^
853761 11.
854i.Ullj
85514411,
855834 11'
85658311-
85781111'
85789S11'
86ief
862673
863351
86*087
861103
86,5377
866051
S667S3
86738^
Lt^BTithmic Sines, TangenU,
10.760330
75S614
7JBH»9
757474
736763
75G053
7553+4
754637
753S3I
T53885
75858!
751 B
Z69402111I
87OQ0P |l 1
.!70735;TTi
e71«Mlll
8TS064,1105
878786:1103
K733BS "
874049,1099
874708,1099
875367 1 109 6
876081 1094
876681
877337
877S91
878G15
879897
879948
}.li91U 9.
749080
T4B383
747687
746933
746839
745547
744856
744166
743477
74878!
74810!
74I'H7
74073?
740049
737387
736649
735973
735897
731683
733949
733277
73860i
731935
731866
730598
789931
78860C
787S36
787871
786618
785951
785898
784633
783976
783319
788663
788009
781355
780703
780(153
713101
T.i>g. I I
.846319,1830
84705711
8477941886
846530 I 884
849864 1888
849998 1880
85073012:
851461 1317
858191 181
258930 1213
853648 ISi:
a.M3741S09
3S5I0O1207
255B341805
856547:1303
358710 119
859439 119
860146 1194
3G0BG3
361678 1 190
863398 1 189
1187
la753fl81
768943
758806
751 47C
760736
74887t
74«63S
747808
747080
746358
745626
744900
7441 7(
743453
).748731
748010
741890
JJ17 ID
864488 1 183
86513811'
865847 1 1
866655 1 1
2678611176
.73936719.867967 1174^
869375
870077
270779 1167
271479 1165
273178
373876 1 168
873573 111
874369111
8749641167
875658
876351 1153
8770431161
27773
.87H484I148
8791131147
279801
1143
3811741141
281858 1
888548 1 138
283285 1 1:
883907 1 135
384588 1 133
985268; 1131
885947,1
386624;! 188
38730111186
287977|ll2S
888658:11: "
Ci'lany. I
736883
735578
734868
73*153
733445
73873S
10.738033 ioTC
731339
730635
789923
789831
738531
7 87 828
787134
786437
785731
786036
784343
733649
788957
72286(
10.781676
730887
780199
19618
lSa2(
18148
:7458
00673W a
006760 3
006783 3
006805 3
10.0069B7 3
007010 3
007033 3
993178 53
993149 SI
993187
993104 49
9.993013 45
998990 ii
998967 43
998944 43
992981
998898 40
998875 *9
998853 38
9938BS 17
998806 46
998783 35
998759
993736
908713
008004
008039
008063
lO^l
9986191
99859687
998549J35
998635|24
99860lfe
993*788>
992164JSI
992430 2(
998406 li
992093 6
998069 5
998044 4
992080 3
991996
991971
991947
for" I ^ 30 3i0 8 30 \ 351 \ 4 \ TO \ \1 y-'''
.n.1 B»..M
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B.Ce0599
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ia7I910I
19.188651
15:7 iisi;
10.008053
il
97991943
IK
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4
1
881840
1081
71875!
889316
1188
008078
41
99192!
6
a
88189J
1079
718103
(89999
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7IOfl01
008in:i
41
991997
32
IS
88854*
1077
7171,58
890671
1119
7093gB
008127
991873
57
48
883190
1076
716810
a9l3l8;lllJ
708638
008151
41
99191*
>S
*4
80
s
883836
1074
T16161
8980131115
707987
Oft8l77
41
991823
53
40
24
6
881180
1078
71fi5!0
1986811114
70T31B
008201
11
991799
36
xe
7
885184
1071
714976
8933S0'U11
70fl65t
008116
41
991774
Si
3!
38
8
885766
1069
7 11131
894017 III!
70598:
008151
41
991719
28
S6
9
386406
71359!
291684 1109
703316
008276
41
991714
51
11
40
10
897018
1066
711S5!
89.53191107
704651
008301
41
991699
iff
2(1
4A
88168T
1064
718313
S960131106
703987
008328
42
991671
19
18
♦8
IS
B88S86
1063
711671
196677,1104
703323
008351
41
991649
18
SS
888961
1061
7II038
8973391103
708661
008376
41
991681
&
A6
14
889600
1059
710100
!9BOI>I'll01
701999
008401
18
991599
16
4
»-o
13
9.I90I3G
ID,S8
10.709764
9.«9HB6!|110O
10.70133f
ia0O8486
41
9.991574
s
13 U
4
16
890870
1056
709130
19113111098
700678
008431
48
991319
14
56
8
17
891501
1051
709198
199980:1096
7000!0
008476
48
991381
13
18
le
8B8137
loss
707863
300639 109.5
099361
008501
48
991*99
18
48
16
in
898768
1051
707133
301195
699703
008317
41
99147:^
41
44
»0
ao
»9339fl
1050
706601
3019,51
1092
698019
OOB332
U
991448
to
40
84
81
t910ie
lOlfl
705971
30ifi07
1090
008578
48
99141!
19
30
88
88
891658
1 016
70534!
303161
1089
696739
008603
48
991397
SB
31
38
13
J95J88
1045
701711
30391*
1087
69608*
009688
43
991378
J7
18
36
84
S9J91S
1013
704087
30*567
1030
695433
009634
43
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00581
147680 801
85837
a
3i
a
a4fi«S8
213
15331!
994441
481
005559
147753803] 8588*
!
36
39
846eiG
813
153181
994694
481
005306
147878,808 85818
81
(
846914
213
163051
994947
481
005053
1*8003 808
85199
W (fl
813
15S98B
B95199
481
001801
14B188'808
95 J 87
19 l4
«
847199
313
15*S01
99.'il5?' 481
001519
149853,809
B5IT*
18 I A
58
43
8413JJ
813
158673
995705J 481
001895
149378 809
85168
17 *
S8
44
847154
818
I58S1C
9959571 481
001013
148503
809
809
85119
W 4
hTo
S.847588
ITT
10.158418
9.996810. *8I
la 003791
9.951378
15 1 1
4
46
847709
818
158891
996463 481
003537
149754 809] 851846
t* ^
fl
47
847936
818
158161
996715 481
q0388J5
148879809 851181
1^ it
18
48
847954
818
158036
996969' 481
00303!
149U04 809 85099(11
li i»
16
411
848091
818
151 909
997881 *81
O0877SI
I49130'809' 850870
II M
80
M
846818
8L8
151788
997*731 481
008587
149855'80d: 85074J
Id 4
81
il
848346
818
997786 481
008874
149391809' 650GIS
88
S8
848478
151588
99T979 481
008081
1*9507!8I0' 850*93
s m
38
53
94H59S
811
151101
999831! 481
001769
1*9638 810; 95036h
' 5
36
54
848786
811
151874
998484 481
001511
U975S
810 8508+>
i Si
40
M
848858
811
151148
998737; 481
001863
149994
810 8501k
44
S«
848979
811
151081
998989' 481
OolOll
150010
210
849990
48
57
8M100
811
J 50994
999848; 481
oooja*
150136
810
84986<
i it
58
849238
811
150769
999495 481
000505
150868
810
849738
5S
849359
811
150641
9997*7 481
000853
150389
810
S496I1
90
60,
849485
811
1S05I510.00I100'481
00000(1
150515
e4948f
Mh
/
^-/7^.r
1 Secant, i CaUng. \
T^nfr
Ctwec.
^^
~si^
'If-
m
3 H™™.
^V«««.
r^^^l ' \ »
38 \ 1> \ \E.
■ \ ^ \ V \
va' \ -sSTTTviV;^
6t 1 3 \ ^
i\\w\l\-J»\wt\ ^v\\
z °' ; 3 1 «
98 \ 3 \ 4
. \ 1^ \ ^ \ ^ \ *
-!
L_i
TABLE VI. 6S
NATVXAL 5INM, TAITOKNTS, SECANTS, >ND VERBINE5, TO ITKHY
DEOREEOrXHe QUADRANT.
Arclflixa.
C™n6
TMKml. Coun. | SecMit. | Co«c |Vmine
Cormine
Arc.
OUOOOO Inflnile. ll.OOOOOO InHn.K. .OOOOOC
1.000000
W0°
017*SS,ST.8B996!l.OOOl5i 57.29869 000154
B9919I
031981 88.63685 1.0OO609 88.65371 0006(19
069987 I*.3O06T|i.O0*1«;11.33559O0?*.11
0S7156
996195
0B7189 I I.13005I 1.0038*0 1 1.173J I 003805
I.0075I08 80S50(
X)7454
878131
1.00H888 7.l8S89i
15838*6.313758
1.018165 a3984a!
nsaa
98*808
176387 4.671888
I,015487J5.75B77I
191380 5.1**554
[.OI871i
1.088311
830868 1.331 17t
[.086301
8*93*8 4.010781
r.0306H
»SBI9
96S9»t.
!679*9ja 738051
1.035*71
3 963703
1.0*088!
1887955
J39731
1.015698
3480304
04369;
707688
[.051 46i
3*438fl8.90iill
[,057681
-|r
34!()?0
93969;
363970 '8.7*7477
1.061178
8.983^04
1.071145
8.79Ul?t
987181
40*026
8,475081
l,0tn535
l.09636(
2.55930J
1.09*63(
8.15859;
ta
«!6I9
S0630t
*6630t
8.14*507
1.103378
48773S
8.050301
1.968611
I.l!838t
I.SS0787
I.13a57(
8.13O0W
1.1*3354
(.06866;
so
500000
866085
577350
I.7380S1
1.6648H(;
1.154701
1.911604
148833
484968
59
1.600335
1.17917S
1.8870Bt
1.539865
1.19836:
I,83607S
1.80681 (
1.788898
35
573516
819158
700808
1.48BI48
1.880775
''■'
1.8T63Si
1.83606(
1.387015
1.858136
I.6el6«
l.87994i
1.86901t
1.68*869
1.8B676(
40
648788
76601'
839 10(
1. 191751
1.305107
1.58185:
845891
343941
1.II0613
I.07i369
LMses;
1.494*71
■56855
48
l.3673«
1.466!7(
1.390164
1.43955;
45
707107
1.000000
1.000000
I.4I4814
Ate.
CoRoe.
Sine.
-oiSST
T..„..
COKC
Soant.
Covcn-
Veraoe,
Are.
HERimoNAL P
TABLE VIX.
■9 To EVERY DKOBEB o
1825.1,
M .( IM 603.1^0
I 60.0 11 6641 81
'U8O.0 18 785.38!
>|180.l 13] 786.8 83
^»40.!1 --.
V3oaiI5| 910.585
8|36a7 le.' 97t.TgS
3471.5 !i0|45g7.1 70 5965.Bb(
1958.0141 8701.^1 J569.8 Sl;4619.a ii:ei45.7ni 1
1353.7 38 WW.«48 8781.7ta 3665.8 68,4775.0 72 6331.eg! i
■ SOBfl-fflSMeailM 3763.8 1I3;490*.B 73 6534.*" '
glTl.fi'44 8915.8te4 3864.6 S4 5039.1 r46745.7fc
8844.3145 3089.9165 i96ft.a|p5;6\l%.W(l a^^^ll.S;
23IR0163115.6lf' "
1418633
1484.134
55a035
IdlB.
E VIII. Difference of Latitude snd Departur
n.f Lbi.
Dgp.
ia9994003U I.
i 0.99TS 0.0699 I.
50.99fiJ0.0RT! 1.
Di«. 8.
Pep.
.□ITSll.SB97a03i9lI.
.99B8 0.069« I.
1.9976 0.()9S1 e.
:. 99730. 1047 ?.
1.9951 '0.I39J
99!10.IT43!.
.99040.1
).10«j I.
I!19 I.
so.g90sai3SS i.
.9995 D.
.99e2 a.
,9964 9.14
.9959 0.
.B9?T 0.
.9etf6 a.
.9SS6 0.
1047 1.9976 at39( LS970 3.
suit S,970e
10 0.345;
il 0.8617 i7
rBa3438 t6
100.4358 )£
!1|4.9759 O,4»0I
II I.9t260.sn8 i4
0.6093 i3
U.69JS ^!
,989*0.1467
.9677 0.1564
100.98480.1736
ll,0.9SlS0.190e
0.9806 0.1951
li,0,97Bl O.Hn9 1.
13,0.97440 «t50 1.
140,9703 0.!419
p.97000.i430
I.9TB4
1.9T»'0.31t9
1.96960.3473 E.
.96330.3316 t.
0.440t
S.9SB7
!8,aS901 t.
15^ 0.9859 0.!58S|l.
16.0.96130?
10.9569 aSSOSll.
7H>.RJ63ae9M I.
:K0.»5iia3oaa l.
,0945503156 1.
D.94150.336B 1.
',aS9T,0.34ial,
0.9336 0.3584 1.
.9S7I0.3T46 1,
B.9g390.39«7
1.9618,
.9563a4l5^.
.9487 a4499|g.
1.94060.
1.94010.4860b.
a517C t.
3'^ D.»i05O.3907 I.
0.91350.4067 1.
ti l}.»063!D.4ZiH 1.
[1.90400.4176 1.
!« D.B988 0.4384 1.
il 0.89100.4540 1.
Mi"
{10.4693 S. 0509
.95440.5S09 3.939!
.94490.5784 1.9!6S
.94240.5953 J.Bt31
.93U0.6g37 S.9U6
,9i31 0.6T49 3.89T5
.01090. 7!5S i.S81!
.9101 0.7289 LSBOl
,89780.7765
5.8637
[P.5Sli t.
0.5806
0.5947
.8135
D.B45S
0.B551
1.7143
33|0.83B7JO.S44( 1.6773
0.6018 1.
afisso 1,
S4 ).8g9O0.S59g
35 18198
3fi :).S090lo.5S78
).eo3glo.
37[0 .i98e j
.788010.6157
.77710,6893
.TT300.
7660 0.S188
4l[0.T547i
42 I.I43la669l
},74I0 6716 I.
{43 3.73140.6S80 1.
|44[0.71»30.
.70710.7071
1.0898|8.S73t 1.5483
.5716
.5441
.5160
.4803
.4819
.4687
1.4387
4148
>cp. I I..I
DiiU 1.
.8838:0.8869 S.8450
e.870B 0.870S S.8g7B
e.SG89'a8T71 }.8858
U.98T1 1.8048
0.9767 17881
.0107 5.7668
1.0861 ].7fies
1.0751 ).7343
.8538
1-8366
e.8I46
88191
g.7816
8.T716
2.76 IS
e.7406
i.7189
i.7180
,6964
3.5640
.6730 1.3680
.6488
.6458
.6839 1.4644|3.4985
.5000 3.4641
34309
1.3431
,364<
1.3B93
.4148
!.I941
B.181:
Dep.
0.99«; 498400.
0.763^90810.95IO|79
a975a
1.0396 78
I.I84H IT
1.8096 74
1. 2149
1.8941 r.
I.37BS 14
1.4514
1.4G1!I 73
1.5451 r:
6 1.6878
il 1.6985 1.
IS 4.6679 1.
4.6359 1.
1.5307 46194
,3141 15958
1.6339 1.3547
e.4944 1.6667 3.3859
1.677f 8.3168
1.780J - -
4.6085
*.'4677
1.5315
t.5199
1.4S40
L4550
■S U147
16 t.4096
12 4.3731
10 1.3301
1.8886
1.8858
1.8408
1.1934
!3 1.1573
8,8366 L1458
,8054 1.1945
1.847C
S.1580
1.6860 3.1096
3.0920
.9038
,3640
,3314
.3190
1.2981
8.8641
,8894 8.007^^.9786
8.8889 8.0147 8.9638
8.046(r8.S854
8.0840l8.S774
118.8864
.913)
.9337 r,
8.0337 36
8.1131 35
a.l37t
1919 14
8.a70f 33
2.3474 38
2.3570
2.484t 31
2.50OC «0
8.570i
r.576! i9
8.649( 18
2.783! 17
8.7779
8.7960 K
8.667! 15
8.93RS 11
8.978i
3.00B1 a
3.0783 S8
i7 3.146f tl
10 3.1780
)2 3.8139 iO
[5 3.3803 19
17 3.3457 18 .
*3.35T( -
:M 3.4100 17
<. I Uep
Disu 3. I Dial, i.
J
for Degreea and Quorter-Points.
.3140 S.
0A19S i.
0.5899 ;.
.5S81 S.
[).6«IS|6.
0.731;
0.T31T
0.8S31
D.9T4S T.
.04040.
.«ggl I.
0.1S7II
asui]
0.U1I
9.975B 0.
9.961 SO.
9,951 8ja
15 0. ITU S9
19 0.3490 BS
(0 0.4907
iS D.fiS34 il
.istek.
1.1449
L170S S,
1,!41S i.94J0
S,79S1
S.7Stlg
S.7615
:.3a£e 7.
I.4JS4 7.
1.5T47 7.
1.693J
1.7009
I.HllT
.9134].
.90151.1
.8TB5 l.31t9g|B.
.Hi30 1.5863 J.
.B463l.5eOI S.
.8253 l.es33
.7950t.T99U B.7G93
1G!4| 1.9354 9.
.760g| 1.9*38 9.
.7274.a.0706
I9.891U 1.467^
.87691.51
1.5G«^
.nTS9.8ies[l.9D81T!
..755«9.807B 1.9509
I.B71!l9. 78151
?.IT73|b.T030;
g.lSti6
£.3994
!.07S1 ri
.TB76
S.74I'
i.737S
i.70fi3
).6731
>.S493
I.63S!
S.6015
.liS38
.7417
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1.8541
.9534
9.0S13 S.5908
5.7BHe
[..6911
S.46T«
.9!9a
3,0320
^.046e
3.1631
3.279U
i.358!
a.3941
i.fi086
i.G78E
7.6085 £.4721
5fl4S!.fi045
7.53242.G951
jiTSea i.
7.46S6 8,8669 i.
7.41758.9969
7.3910|3,0f
8.4UD7».
E-Giiqt
.5630
3.781 3I9-5106
3.930l[9.4S53
.D3201il.4154
.07b3{9.396e
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3.90gS
8.9837
3.0903 ):
3,8557
3.3683
iS 3.5837 BO
11
30 11968 3,0000 10638
3.7351
8.847! 7.
3.958f 7.
3.9989 J.
,1916 3.0686 T.
.lT7Sp.
1806
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S.1883 3.S9S7
8.7839 !.8370 3,
36319 9.0191
39453k
4.0)3 59K
4.8436|e
.0 3.907S 57
15 4. 06 74 SG
11 4.836S 05
19 4.37 5C
4.3837 S4
4.5399 B3
5 4.6947 ii
13 4,714(1
!3 4.848 1
)3 5.000( SO
S.U64 3.0846 S.0041
S.lt30 3.0003 i.0(H>8
3,1795 S.9363S.7091 S.
.0380 3.1679 5.8707
3.81EdS. 7(194 4.3571
4.9889 3.3334 S.8303|3.B8go|6.S518
1.8 3.3553
^B 3.4415
11 3,5367
13 3,5743
18 3.0109
4.8137 S.3891
4.7033 7.
4.7656 r.
4.8145 J.
18.5773
8.5717
B.4B05
b.38G7
6,000l|8,3147
5.033718.3904
5.168818.1915.
5.S90ll8.0908
4.6381
4.5963
4.5883
48 4.4589
4.4457
43 4.3881
■14.3l(
,5 4,3436
3.6940
3.7769
3,8064
38567
3.9363
4.0148
4.08941
4.0930;
4.1680
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4.4059
4.440}^
4.4995
4,5934
16839
S.3U4I
S.817B
.184]
4.9353 7.'
5,0346 S.9I
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17 4.700S 6.9176
15 4.7740*
it 4.B636 5,7547
17 4 9*97
,0931 &5401
.9943 5.661
.957l|A.T0gj
S, 1334JS. 1483le.894«& 78511
IS4|5,9045
.688316.038! 7.
.6666 6,0440 V
.5883 6.1380
.4741 16.851 9
.36406.3640
.8485'6.
.3530:6.
5.3785'6.
5.15
5.39
5.4464157
5,5557
5,5919 56
5.735BJ;
.8779 51
14 6.0168 51 _
II B.i56t a
6.8933 SI
116.3439
.6604 6.487S 50
,5471 6,
,4314 6.
,4095 6.
.5606 19
6.6913 18
6.7 15C
,3135{6.83(;C 47
9166 46
.0711
7.071
4.S I
I Rep, I
Diurad Logaritfanis.
1301 0SS03 03
l.380«l
I K I M3fi 1.37303
t t.aS133 I.3fl.»T
1 t.69lii \.3S90i
.07918
1.07SS8
1.0T800
i i.iS939
' (.3I3S7
1.064!>t 8»S£5|T709T
.5 1.06115
1 1.05799
1.0S4ie
1.33gI9
1.3eS8S I
1.31 951 1.0iT77 BB19a 76>lfl
1.15636
t 1.11697
1 I.07S1S
i 1.044*2
i l-Ollti
t 1.98117
! I.951«4
I.9ST91
l.9030d
1. 87 961
1.30103 1.03779 B7506 75S&G
.!9f04 1.
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1.18330 1.
I.S7Tfi5
1.87187
l.«66!T
l.gSOTl
.031i6
.01803
■0148!
.oust 96390 74843
SenO 74074 8MD4 58067
T7H15 6t
77635 61
77455 61
77876 61
68114
67980
67836
67691
67549
76910 67406
66841 59134 5159 1 tl
59016 51
58899 S!
5ST8l5i
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66005 58433 5X98T V
1 47711
•e 1761!
\6 4753«
114744!
16 53O90 47351
41356 31
41176 31
41197131
411 17 31
5g370{51794|47083 41038 31
.T4180C 37303 :UJ»
17 46464 41485 37010
MlOl 4
46113 4
460S6 4
18 41329 36878 m4J
> 1.85733
. 1.83614
1.81594
1.79664
■ 1.77B15
> [.76041
1-74339
1.71700
I 1.71111
I 1.69
85733 74339 tSStl
74171
74006
73841
ST744 61393 45853 lOHO 365173Ji!t
W 51»4 451B6 loses 3U57SI4S^
16 511H M«80 tOT86 3638831 '»
57403 SlOeS 45593 40T09
09401 B445C 73348 S45t5 57176
S70S5 S0BO6 45336 H
S631HSi.l31
36148 3«i61
3617931M
3ellU'31ftI
I 1.6:
114 1.
I 1.639851.
> 1.61688
) 1.61430
1.60H)G
' 1.59016
I.5T85B
I.S6T30
3 50109 45150 UMOl 360*a'3tIIM
^150611 45165 40315 3J
.18514
.18064
.17609
.17159
.16714
.16273
1.12404
1 1.14976
1.5!4S0 1.I4A54
\ 1.4956 1
' 1.48617
i 1.47711
I 1.4
1.14133
1.13717
.13306
.12898
.11494
I. lie
i l.45»39
I 1.45079
i 1.44136
I 1.43^100
\ I.4159T
> 1.41 Boo
i 1.410181.09391
' 1.40149 1.09018
I 1.39494 1.08648
' )l.3a75l|l.08181
1.11697
1. 11304
10915
10519
.10146
71061
T1903
71T45
715B8
71431
1176
mil
70966
7081
I0S58
70505
70351
70100
70048
69B9T
69T47
69597
56177 50035 44656
II 44740 39945 359t73nn:
39870 35559,31S9)
I0U571 39794 35491 Slill
U 44487 39719 35411 Slill
WT50 44403 39644 3«354'3IUI
i5 14310 39569 35186 '3i:^M
ji 44S36 39491 S5118,3iair
It 49U6 UtA3 3Mlg S51S1 SliC^
13 391B5 3494e3l01<
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t7 HB13309H|
7 48S0^436<5l3S97a 3474C 30
'0 4S81! 43513 38B99 M
91284 79138 S9208
91031
91781
91533
91185
91039
9U551
78915 69001
7873^68054
785456870T
7836168561
78179188415
T799TL68i
36678 34
3H604S4
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430Sg 38458 34tT»3Dlt]
|e0387 53615 4TB0
30 9 9 30
) MlOSIIiliBTt^
i i9743i6iBl 13090 901
r iDSBS idiSS !30iB !0<:
B iSStSi6\Siii991iiK
I i9M4 86188 889*6 199
r iR50i 86074 2ZH91 191
9 828*319!
i* 88T98 I9S
19 887*1 191
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82857
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6
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93099
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98351
90373
88498
96670
84930
83237
91047
90094
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92317
90341
88431
96640
84908
93830
81620
90069
79570
94352
92883
90309
86611
B3203
80043
79316
31
94317
98850
90277
86581
84845
83175
81568
8O017
78681
3i
94!Si
98816
90845
88339
81918
93149
91541
78196
33
94246
9219S
908 13
88389
86538
81799
93181
81515
79967
T8478
34
942 II
92149
90IS1
88897
86493
847S0
83094
9H89
79941
78447
35
94176
92116
90148
98867
96463
84T32
83066
91463
79916
78483
36
94141
92098
90116
99236
86434
84703
83039
61436
79991
78398
94105
98049
90084
88803
86101
816T5
93018
81410
79965
78374
94070
98016
9O058
88175
84647
88985
81384
79349
39
94035
91981
BO080
98144
9J619
88959
81358
79916
79325
40
94000
98114
86316
81590
tijmo
81332
79790
79300
41
9S965
9I9I6
98093
86297
94568
88903
81305
79761
79276
4«
03930
B1991
89985
98058
86258
94334
82976
81879
79739
78252
43
9JB95
91849
89993
99088
86989
94506
82949
81853
79714
79227
44
93860
91816
89961
87991
96199
841,78
82988
81887
79203
45
93923
91181
87961
86170
94450
98795
81801
79663
79179
46
93791
99797
87930
96141
84421
92768
8I1T5
79638
78154
47
93756
91715
89766
87900
96111
84393
98741
61149
79613
79130
49
937SI
91682
89734
97870
96088
94365
98714
91183
79588
73106
49
93696
91648
89708
87839
96053
"9^09
98687
91097
79563
7W061
W
93651
91615
89670
8780B
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92660
7953B
51
93617
91582
8777S
85995
84881
92633
91045
79513
79033
53
93588
91549
99607
87748
86985
84253
92606
91019
794B8
79009
S3
93547
91616
99575
87719
85936
94223
98579
90993
79483
77994
54
93513
91483
99544
87687
85™7
841BI
88552
80967
79437
77H60
93478
91450
99518
8T657
88525
80941
79418
56
93443
914IT
89481
87687
83849
8H4I
82198
80916
7938T
77912
57
93409
91384
89449
97597
85980
9H14
92471
80889
79368
77988
SB
93374
91351
89417
9T566
85791
84086
88445
90863
79337
77963
59
B3340
91318
89386
87536
85762
84058
82419
80937
79318
77S39
■1
.8
3 .4
.5
.6
.7 .B .9
tenth!
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8
9 \
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11 84 91
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75018
73676
783TB
TU80
69897
68707
67549
66421
1
T779I
76368
T4990
73054
78358
■7J100
69877
67530
66408
«
77T6T
16344
74967
73638
71337
71079
69957
68668
67511
66364
3
T7743
T638I
74944
73610
72316
7I05B
69937
68648
67498
66365
777 19
T68S8
74988
73588
78894
71038
GB81T
68689
67413
66347
a
7T69S
J 6875
74899
73566
72273
71017
69797
69609
67454
6632b
6
TTB71
76851
74877
73544
T8258
70997
69777
68590
67435
66310
7
77647
76888
73583
72831
70976
6975<i
69570
61416
66291
N
77683
76805
71838
73501
72809
70955
69730
68551
67397
68813
9
7]i99
76181
74809
73479
78188
70935
69716
68531
61319
66854
10
7T57S
76158
74787
7345T
78167
70914
69696
68518
61359
66836
77551
76135
7*764
73*35
78146
7089*
69676
68498
67310
66817
18
I768T
Tent
7474!
73*13
78186
70873
69656
6S4T3
6T381
66199
13
77503
76089
74719
JS398
78103
70858
69636
68451
67308
66180
U
7T479
7606S
74697
73370
72098
70838
69616
68434
67283
66162
15
7T4S5
76048
74674
73348
78061
70911
69596
68415
67264
661*3
7T431
76019
74658
73386
78040
70T91
69576
68395
672*5
66185
17
7T407
T5996
74689
73305
72019
70770
69557
68376
67826
6610S
IS
773SS
TS9T3
74607
73283
71998
70750
6953T
68356
67807
66088
!iJ
77359
75950
74585
73861
7197T
707 29
695 IT
69337
67188
660TU
7T3a5
75987
74568
■73r39-
71956
70709
69497 1 68318
61170
66051
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7T3II
75903
74540
73818
7193S
70689
694T7 1 69299
6T151
68033
n
77888
T5880
74517
73196
71911
70668
69457
68879
67138
66011
S3
77864
75857
74495
73174
71898
70617
69437
68859
67113
65996
84
77840
75834
74473
73153
71871
70687
69417
68240
67094
659T8
85
77816
758 11
74450
73131
71850
70606
69397
69881
67015
65S5S
86
77198
T5T88
74488
73109
71889
70586
69377
67056
65841
87
ni89
75765
T4406
73099
T1808
70566
69358
61038
65983
8fl
77145
75748
74383
73066
71787
70545
69339
68163
67019
6590*
89
77181
75719
74361
73041
71766
70585
69318
68143
67000
658B6
30
7709T
75696
"74339
7a083
T174S
70504
69899
68181
66981
65868
31
77074
75673
74311
73001
T1784
70*84
698T9
68105
66968
658*9
33
770J0
75S50
74194
78980
71703
70164
69858
68086
66944
65831
33
77086
75687
74878
72958
71692
70443
69839
68068
66985
85813
3(
77008
T5604
74850
78936
71662
70483
&I819
68017
66906
65794
35
76979
75581
74888
78915
11 611
70403
69199
68029
66997
65776
36
T6955
75559
74F05
72893
71680
70388
69179
68008
66869
65758
37
76931
75536
74183
78878
71599
70362
69159
67B89
66850
65739
S8
76908
75513
74181
78850
71578
70318
69140
67970
68831
65781
39
76884
TSt90
74 139
78889
71567
70311
69180
67951
66818
65703
40
76S61
75467
74117
78807
71536
70301
69100
87932
85685
41
76B37
75444
74095
78786
71515
70881
69080
67918
66TT5
65666
48
76813
75481
7HJT8
78764
71194
70860
69061
67893
66756
65648
43
7B790
75399
74050
78743
71473
70840
69041
678T4
66731
65630
44
76766
7537S
T408B
72781
71453
70280
69021
67955
66719
65618
45
76743
753S3
71006
78700
71432
70800
69002
67836
66100
65594
4«
76719
75330
73984
78679
7I1I1
70179
69982
67816
66681
65575
47
TSS96
75307
73968
78657
71S90
70159
6S968
6TJ97
66663
65557
48
76678
75885
739*0
78636
71369
70139
68948
67778
86644
65539
49
76649
75868
73919
72614
71319
70119
69983
67759
66685
65681
SO
76685
75839
72593
71328
70099
"68903'
677*0
66607
65503
51
76608
75816
73874
78571
71307
70078
68894
61781
66588
85484
58
765T9
75194
73H58
78550
71886
70058
69864
67708
66510
65*66
S3
76555
741 71
73830
T1265
70038
68944
67682
654*8
S4
76 131
75148
73809
78507
71845
70018
68885
67663
86532
65430
iS
76508
75185
73T6S
78486
71884
69998
68905
67644
66514
65418
S6
T649S
75103
73701
78*65
T1203
69977
69785
67685
66495
6539*
fiT
7«4«1
75080
7*443
7U83
68766
676(t6
68117
65376
S8
76438
75058
73 780
78*88
71163
69937
66*59
65351
«9
76414
75035
73698
78401
71141
69917
68727
66439
65339
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ional f
J of ■■ o
"'"
.1-
8
.8
4
S .4
9 e
.4 .6
10 W
.7
\1 \
^
■72 Table X.
ma.
—\
Degree, or Hour. |
a
40"
4lm
43"
43^
44^
45™
ie^
47"
48"
49=
6s;isi
e4!t»
63302
68180
6118!
60306
69851
59317
1
65T)3
6«31
63185
63164
61 166
60190
69336
5930!
6T388
66493
8
63!fl5
64! 1 4
63168
6! 1*7
61119
00174
59880
58887
57373
66478
3
652Gr
64196
63151
63130
01133
60158
69804
58371
67358
66163
6.5a+9
641T8
63133
63113
61116
60148
69IB9
58856
67343
66419
S
6.5831
641 61
63116
62096
61100
60126
59173
58841
673!8
66434
65813
64143
68080
61083
60110
59157
68386
67313
66119
7
6SIB5
641?J
6308S
63063
61067
60094
69141
69310
67298
£6401
S
65177
61108
63065
63046
61051
60OT8
58194
67383
56390
9
65159
64090
6304ft
63039
61034
00061
69110
58179
57368
66375
To
65 Ul
64073
6:tOJ0
6301!
61018
60045
69094
57263
66360
11
651 S3
64055
63013
61996
61001
60039
69079
58148
67338
66315
u
65105
6403K
6!flB6
61979
60985
60013
69063
58133
67!83
66331
13
66087
6WS0
6!919
61963
609G9
59997
69017
58118
57209
56316
U
65069
64*103
639B3
61945
00952
59981
58103
57193
IS
65051
63985
63945
61939
60936
59965
69016
58097
5T1TB
66!97
16
65033
03967
639!?
61913
60930
699.19
69O00
68012
67163
66873
IT
6J0I5
639.W
68910
618B5
C0903
69933
58985
6805 G
67148
66!67
IS
649B7
63893
61878
6088T
69917
68969
58011
67133
66343
19
64979
63915
63876
61863
60871
59901
5B954
58086
57118
6n8!8
SO
64961
63WH7
63859
61815
6U854
58938
68011
57103
61.813
!1
03880
G384!
618!8
60838
69870
58938
57995
57088
66199
n
649S5
flSMfig
63833
61813
60823
58907
57980
57073
6G194
!3
64907
01815
63808
61795
60805
59838
58891
57965
6705M
56169
!1
64889
6388T
68791
61778
60789
59833
SH8T5
57949
67013
66165
i5
648T1
63810
6176!
60 J 73
59806
68860
57934
57033
66140
64853
6S792
61745
6075G
69790
58844
57919
67013
66186
21
64835
63775
63739
61738
60740
59774
68839
57901
66998
66111
tti
64«1S
63757
6*733
6171!
69758
68813
57S88
56993
t9
64BO0
63740
63705
61695
60708
69742
58798
57873
57853
66968
66081
m
64TBi
-m-n
68S8B
61678
60691
697 2(,
59788
56953
66067
31
64764
63705
68671
0166!
60675
69710
68766
57843
56938
66053
3*
64746
63668
6!654
01616
69691
59751
57837
66983
5603 T
33
617S8
63670
6363!
61638
60642
59678
58736
5781!
6G908
66083
31
61710
63053
6!630
6161!
60636
69663
58780
57797
66993
66008
35
6469!
63635
63603
61595
60610
69647
58704
57783
66979
65994
36
64675
63618
63586
61579
60594
59631
586S9
57767
66864
56979
3T
64657
63601
63569
61568
60ST8
69615
58673
57761
66919
65905
38
64639
63583
635.5!
61515
60561
59699
58658
57736
56834
5.5950
39
64631
63561!
6!535
61539
60345
59683
58643
577!1
56819
55935
B3il8
63518
6151!
60589
59567
59687
66804
41
6458S
63531
63501
61496
60513
69551
58611
57691
66789
56906
*?
64568
63514
63484
6147B
G0496
69536
58598
67676
66774
65391
43
64550
63196
6!468
61463
60480
59590
58.580
67660
66TS9
65877
**
6453!
63479
6!451
61446
60461
69504
58565
57646
6674.5
6586!
45
61514
8346!
63434
6 1439
60418
69488
58549
57630
56730
56848
4(>
64197
63444
6S41?
61413
60132
59173
58534
57615
6671.5
56833
64479
63t!7
63400
61396
60416
59157
58518
67600
66700
55S19
64461
63410
63383
61380
60399
69441
58503
67591
66685
66804
49
64443
63392
63366
61363
60393
59185
58497
57569
5(1670
56790
64486
"6337S
63349
61347
60367
68478
67551
56658
56776
51
6«08
63358
6333!
61330
60351
59393
68456
57539
6S611
56761
64390
63340
68315
61314
60335
59378
68441
67584
66626
55746
53
64373
633!3
63398
61897
60319
59363
68435
56611
66733
M
64355
63306
68383
61381
60303
59346
5B410
57491
66596
55711
65
64337
63389
62365
61364
60886
59330
58395
67479
56SB8
55703
5S
643S0
63S71
63348
61248
60370
59314
59379
574U3
58567
65689
o7
64S03
63854
63331
61331
60854
59299
58364
57418
585,53
66674
59
64B84
63S37
62! 11
61815
60!38
59883
58348
67433
5S537
55659
69
64S0I
63!30
63197
61198
60233
59367
58333
57418
5662!
56bH5
Propor
iunal I'in to
.8 .3 .4
.6
8 .9
renfb
9 of "or a.
2
3 5 6
8
10
11 1
3 )4
Proporlionul Logaiithwi.
Table X.
■73
D«gi«. or
OHdut.
;'
W-
51»
68"
53i
jji
S5«
66^
57'!'
68"
6B"
~o
54770
539S7
53100
62299
61491
49940
49184
494(2
65616
54756
53913
53086
58874
51179
60698
49927
49172
19130
i
S5601
5474!
53899
53078
58261
51466
S0683
49914
49159
48418
3
S5SS7
54788
53895
63059
52248
51468
60670
49908
49117
4
S.557!
snu
53971
63045
52231
51439
S065T
49889
49135
48393
B
S55fl8
54699
53957
53031
32881
51185
50644
49876
49128
48391
6
55.543
51-685
53943
63018
58209
51118
60631
49864
49110
49369
7
S5SS9
SM71
53930
63001
52194
51399
60619
49951
49097
4835(>
B
5551*
S4857
53916
58991
58191
51396
60605
4983S
19095
49341
9
55500
54643
53908
6S977
52167
51373
50598
49926
49078
48338
10
554S6
53799
68963
58164
51360
49913
49060
49320
11
55471
51614
53774
63960
58141
5131S
50366
49900
49017
49307
IS
S5457
51600
53760
52936
58187
51333
60551
49788
49035
4889S
13
fiSt4a
51586
53716
58988
saiii
51380
60541
19775
49083
48293
14
6SHB
54578
53738
68900
52101
5130T
50329
49762
49010
49871
15
S5414
51558
53719
52895
52037
51291
50316
49750
18998
48858
16
55399
51514
53705
62888
58071
51881
6050*
49737
48996
48848
17
SSSB5
51330
53691
52868
620S1
51268
50189
49784
49973
48834
18
55370
51516
63677
5i8S5
62047
51255
60476
49718
48960
48882
19
iS35B
54S01
S3fi63
58841
58031
51248
60464
49699
48918
48210
aJ
5534a
54487
5:t619
68887
5a02I
51229
soisT
4b936
49197
55387
514T3
53GS6
62914
52007
51816
60438
49674
48983
49195
sa
55313
51459
53(i88
52800
51991
31202
50485
49661
1H911
48173
B3
55899
51115
53609
62787
519BI
51189
50412
49649
48909
48161
u
SSiSi
51131
53iSl
6f773
51967
51176
60399
49636
48086
49149
S5
55iW
54117
53S90
68760
51954
51163
50387
49623
48874
48136
16
55S55
51403
53567
62746
51911
51160
50371
49611
48961
49124
!7
55*41
54389
58738
il987
5 1137
50361
19598
48919
48118
88
5saa7
51375
53539
68719
51911
51121
60318
49586
48936
49100
S9
5Siii
51361
53525
62705
51901
51111
60335
49573
49984
49089
30
55J9!t
51317
53511
68tiga
51989
51099
50388
19660
48912
19076
Si
S.MHl
51338
53199
58679
51971
51085
50310
49548
49799
48063
39
54318
53194
52666
519G1
51078
50897
49335
48787
49051
33
55155
54301
53170
58651
51818
30284
49323
48776
49039
34
55141
51890
53156
58639
51935
51016
50871
49510
19768
48087
3S
55187
51876
53448
58681
51821
51033
50858
49198
18750
18015
3S
55112
SiiGi
53189
sac 11
51908
51030
60246
49135
49737
19003
37
55098
51818
53115
52597
61795
51007
30833
49178
49725
47990
38
i509t
54831
53(01
58581
51 791
50880
49160
48713
17978
SB
55069
51880
53387
58570
61768
50991
50207
49117
49700
17966
17951
40
55055
51806
53374
68557
51765
50968
50194
49135
4S6H9
41
55011
54198
53360
68513
51712
S09SS
50198
49422
48676
17918
55088
51178
53316
52530
51789
50912
50169
49410
48663
47930
43
55018
54161
63338
52516
51715
30989
30I5S
49397
48661
44
S4908
SlliO
53319
58503
51708
50916
50113
49395
48639
♦7906
45
549S4
54136
6330S
58189
51689
30903
S0I31
4937a
48686
17993
46
54969
54188
53991
52170
51676
50890
SOUS
49360
486 14
47891
47
54955
64108
53878
58462
51B6a
50877
50105
49347
48608
47989
46
54941
54091
53861
52149
31619
60964
50098
49334
48390
47957
49
54887
64080
6383(1
58436
51636
50851
50090
49388
48577
47915
£0
54918
61066
63236
62482
61683
50839
500BJ
49309
48565
4793:(
SI
54898
51058
53aa3
52409
51610
50H85
50051
49297
48543
47981
B8
54884
51036
53209
52395
51696
50813
50011
49894
48510
479IJ9
S3
34870
54081
53196
58382
51593
50799
50029
49272
48589
47797
S4
S4855
51011
53183
52368
51570
50786
60016
49859
48316
47785
S5
54841
63997
63168
68355
51557
50773
50003
49847
18503
47778
S6
51887
63983
53151
52348
51541
50760
49991
19234
49191
47760
S7
54813
539 G9
53U1
S8388
51530
50717
19979
49882
19179
17719
SB
54799
63955
53127
58315
51517
50734
49965
49209
4«Hi7
47736
S9
54784
53911
53113
52301
51504
60781
49958
19197
49434
47721
PMpot
lon»l l-.rc 1»
.1
.8
3 .4
.5
.6
.7
ttnlh
B Of " or J.
I
3
4 5
6
«
SI
'^ \1\
Tt Table X. Proportional Logarithiai.
1 Degw, or I Hour.
n
u
li
si
3"
4i
ii
6^
Jm
8=
Sm
10^
11»
41711
46994
46881
44B09
44836
43573
42920
42876
41642
41017
40401
47700
46988
46276
4558!
44698
44825
43568
48909
42866
41632
41007
40391
4;eee
46971
46865
45570
44887
44814
48551
4!89S
48855
41681
40997
40381
47676
4695!)
46253
45559
44875
44803
43640
42987
48844
41611
40986
40371
47664
46947
46241
45547
44864
44191
43589
4(877
4i!34
41600
40976
40361
4765!
46935
46830
45536
44853
441B0
43518
4!866
4!883
41590
40966
40350
47640
469?3
46818
45524
44841
44169
43607
48855
4!813
41579
40B55
40340
470*8
46911
46806
45513
44830
44158
43496
48844
48802
41569
4U945
40330
47616
46899
46195
45501
44819
44147
43486
4!8S3
48191
41559
40935
40380
47604
46888
46183
4.S490
44808
44136
43474
4!683
48181
41548
40984
40310
4759!
46876
46171
45478
44185
43463
48918
48170
41538
40914
10300
47680
46864
46160
45467
44785
44114
4S458
48801
48169
41327
4090*
40889
47*68
46SJ8
46148
45456
44774
44108
43441
48T90
481*9
41517
40894
40279
47556
46S40
46137
45444
44768
44091
43431
48780
48138
41506
40983
40269
4T544
4688S
46125
45433
44751
44080
43480
48789
48188
41*96
40873
40859
4753!
46BIT
46113
45481
44069
43409
48758
48117
41485
40863
40819
46805
46102
45410
44789
44058
43398
48747
48106
41475
10858
10839
4J508
46793
46090
45398
44717
44047
43387
42737
48096
41*6*
40818
10828
47496
46781
46078
45387
44706
4*136
43376
42786
41454
4083*
4081S
47484
46769
46067
45315
44695
41085
43366
48715
48075
41443
40881
40808
46758
45304
44684
4401?
4335*
48704
48064
41433
4>J8U
46746
46044
45363
44678
44003
43S43
48693
48053
41423
40801
40I8B
46734
46038
46341
44G6I
43998
43332
48683
48043
41*18
40791
40178
47430
46788
46020
45330
43981
43381
4«678
48038
4140!
10780
40168
47*«*
16710
46009
45318
44639
43969
43310
48661
48088
40770
40157
4741!
46699
45997
45307
44687
4395B
43300
48651
42011
41381
10760
4014?
♦7400
46687
45986
45895
44616
43947
43889
48640
48000
41370
10749
40137
4738H
46675
45974
45884
43936
43878
48629
41990
41360
40739
40187
in
47376
46663
45968
45873
44594
43925
48618
41979
41350
40729
4011T
E9
47384
46658
45951
45861
44583
43914
43856
48608
41969
41339
10719
4010T
30
473St
46643
urn
45850
44571
43903
43845
48597
41968
41389
40708
4O09J
31
47340
46689
45088
45238
44560
4S892
43831
48586
41948
41318
40698
40081
3*
47318
46616
45916
45827
44549
43B81
48675
41937
41308
40688
4007S
33
47316
46604
45905
45816
44638
43870
4381i
48565
41B87
41298
40678
4O066
34
47301
46593
45893
45804
44586
43859
41916
41287
40667
40056
35
47298
46581
45881
45183
44515
43848
43191
4854!
41905
41277
10651
36
472SO
46569
45870
45188
44504
43S37
43180
48533
41895
41866
10647
40036
47?fiB
46557
45858
45170
4449^
43886
43169
4!588
41884
41256
40637
1O026
38
47856
4G546
4584T
45169
44488
43815
43158
48511
41874
41246
40018
39
472*4
46534
45935
45147
44470
43804
43147
42500
41863
41835
40616
4<K)0«
40
46588
45824
45136
4445S
43793
43136
48490
41B53
41885
"40600
39996
tl
47280
45812
45125
43782
43186
48479
41848
41!14
10596
3B9rtS
H
47208
46499
45600
45113
4443;
43771
43115
42468
41204
40585
39974
♦3
47196
464B7
45789
45102
43760
43104
48*58
41881
41194
10575
39965
47195
46475
45091
43749
43093
48447
41811
41183
10565
39955
47173
46464
45079
44403
4373S
43088
48436
41800
41173
405.45
3t»*S
47161
46452
45754
45068
44398
43727
43071
48486
41789
41168
10544
39935
47
47149
46440
45743
45057
44381
437X6
43060
48415
41779
41158
40534
39985
48
47137
46428
45731
45045
443T0
43705
43050
42404
41 708
41148
40524
39915
49
4TISS
46417
45780
4S034
44359
43694
43039
42394
41758
41131
40514
39905
■17113
4640S
45708
4508!
44347
43683
43088
42383
41747
10503
39895
47101
46393
456B7
4S01I
44336
43878
4S01T
48378
41737
41111
40493
39885
47089
4638!
45685
45000
443U
4S66I
43006
42368
41726
41100
10493
39974
53
47077
46370
45674
44988
44314
43650
48095
48351
41716
41090
40473,
3)864
54
47066
46358
4566e
44977
44303
4S639
uea5
48340
41705
41080
40463
398.M.
55
47054
4634b
45651
44966
44898
43688
48S74
48330
41695
41069
40458
39844
SK
47048
46335
45639
44955
448B0
43617
48963
48319
41684
41059
40448
39834
47030
46383
44943
44269
4360G
48952
48308
41674
410*8
40*32
39884
47018
46311
45616
44938
44258
43595
48941
41663
41038
40422
59
47006
46300
45605
44921
44!47
43584
48931
48887
41653
*1088
4041!
39804
.1 .8 .3 4 .5 .8 .7 .8 9
' of "or a.
laS45789lrt
iDtgrtcWl Hour.
a
Iji
,3^
ni
isi
ie«
Iji
19"
19"
5oi
81"
aai
83^
~0
39I9<
39193
38804
380*1
37446
36878
36318
3S765
35*18
34879
"3fi48
33619
39JHJ
39 IBS
38594
38011
3743(
36869
36309
3S76S
34670
34137
33611
8
397 H
391 75
38585
38001
37487
36859
36*98
35746
35*00
34661
34118
93608
3
39T6J
391 SS
38575
37991
37417
30850
36890
3S73T
35191
34C58
34119
33593
397 H
3915S
3856J
37983
37408
36841
36*81
35788
36181
3464!
34111
33565
39T44
39145
38555
3T9T3
36831
36871
95719
35173
34634
34108
33576
fi
39731
39)38
3SS45
37963
37389
36821
35710
35164
3*093
33567
39TU
39llfl
38536
3T954
373T9
36818
36*53
3STO0
35155
31616
3*09*
33558
39714
39116
msis
37941
37370
36803
35691
351*6
3*607
31075
33550
S
39T04
391D6
38516
37 934
37360
36794
36*34
3568*
35137
3*599
34066
33541
10
S9M9*
39096
MS06
379*5
37351
36784
36**6
3567:
351*9
34689
31058
33538
11
a»oa*
39086
38497
3T9IS
373*1
36775
36116
35664
35119
34681
31019
33684
18
3MT4
S90T6
38487
37905
3733?
367G6
36*07
35855
35110
9467*
34040
33516
13
3Me4
39066
38477
37836
373*!
36197
35646
35101
34563
34031
33506
H
9MS3
39056
38467
37886
37313
36747
36199
35636
35091
34654
3401*
33498
IS
9M«!
39046
38458
37877
37303
36737
361T9
356*7
35093
34545
34014
33489
IS
3903T
38448
37867
37194
367S9
36170
35318
35074
3*536
31005
33480
IT
3B0S1
38438
37857
37*94
86719
36160
35609
35065
3*5*7
339911
33471
18
SMI!
39017
384J8
37848
378T5
36709
36151
35600
35056
34518
33987
33483
19
SWO!
39007
38419
37838
3786S
38700
3614*
35591
35047
34609
33978
33454
W
39593
3C99T
38409
378*9
37*56
36691
36139
35H;t
35038
»SO0
3397(
33445
tl
3958?
38987
38399
37819
37*46
36681
3618!
35578
35689
SU91
33961
33437
IS
39ST3
38977
38389
37809
3TE3T
3667*
361 1«
35563
35010
31483
33951
334*8
n
39563
3S9ee
38380
37800
3T887
36663
36105
35551
35011
34474
33943
33419
*♦
3flM3
3S956
383T0
37190
37*18
36653
36096
35545
35O08
34465
33935
33411
IS
9SH3
38948
38380
37791
37*08
36644
3608<
35536
34993
34454
339*6
33408
IB
39533
38939
38351
37771
37199
36634
3607T
35587
34984
33917
33393
M
■I95S-!
399«
38341
37761
37189
366*5
96068
35518
349T5
33908
33385
18
99513
38916
38SS1
3775*
37180
36616
36059
35509
34966
33899
33376
»
SW03
38908
38381
3774*
37171
36606
8605(
35600
34957
344*(
33891
33367
30
WMK
xm
3B31I
37733
37161
36597
36040
35491
34948
34411
3398*
33359
31
3948:-
39889
38391
37783
3715!
36589
36031
35481
34939
34403
33873
33350
St
3MT9
38879
3TTI3
37148
36579
S6088
35478
34930
34394
33864
33341
33
39461
38660
38*8*
37704
37133
36569
36013
35*63
349*1
31386
3SB56
34
3945)
388S9
38173
37694
37183
36560
36U03
35*54
34918
34376
33847
33384
35
39441
38849
38*63
3T68S
37114
359 91
35145
34903
34367
33838
33315
36
39434
38839
38*53
376T6
37101
36541
36985
3613(
31991
31358
33819
33307
ST
39414
38830
38*44
37668
37095
35976
31885
34319
338*0
33898
38
39414
SBStO
38*34
37658
37085
35967
35418
31676
34340
3381*
33889
39
39404
SB8I0
38*14
37646
3707B
36613
35957
34867
3433S
33803
33881
4U
993M
^00
38*15
3T637
37067
35948
35400
34868
343*3
3379*
33871
tl
393131
38790
38*05
376*7
8705T
35939
35391
34649
34314
33785
33*63
K
39371
38781
38195
3T61B
370*9
36485
35930
35391
34840
34305
33777
33155
*3
39364
3877 1
38186
37608
37039
36176
359*1
353T*
34831
34896
33768
33146
H
39354
387S1
38176
3759S
370*9
36467
35911
36363
3481*
34W7
33759
i5
59341
387SI
38166
37589
37019
36151
35908
35354
34813
94178
33760
4e
38741
38156
37579
37010
35893
35345
34004
34170
9974*
»
39^4
38731
38147
37570
37001
3643!
35884
36336
34795
34861
33799
33811
46
S9S14
3813T
375S0
36991
364*9
35975
353*7
34786
S4*5t
39714
33803
4»
39304
38711
38117
37551
3698*
364*0
35865
35318
947T7
34843
33715
33194
SO
«9X«
mm
38118
37541
369TI
36411
35958
353TO
34788
34*34
WW
33186
51
39SM
38891
88108
37»f
36943
36401
35947
35300
34759
3i»5l
33698
33177
SB
HMU
37581
36953
3639*
35838
35*91
34760
34817
33689
33168
S3
139864
S86T3
S80B0
3TS13
36H4
36383
359*9
36881
34741
34*08
33681
33160
M
S9«S1
38«8S
88079
87403
3693S
S63T4
568*0
35*73
34731
34I99I
936T*
33151
U
S9M5
3B«33
38on
37494
36985
36364
35810
358S4
3478S
34190
33663
3914*
M
39C35
38S13
SBOSO
37484
36916
96355
35801
35854
34T15
341B1
33654
33134
»7
39815
38633
38O90
37474
36906
96346
35791
35145
94706
34171
33646
331*6
W
39X15
38CM
36040
97465
3SS97
36336
35783
35136
34697
34164
33637
33117
S9
39W5
38814
38031
3745«
36898
363*7
35774
35887
34638
34155
^sa'W.V^vw.
V
«*
flfor* f 1 g 3 t 4 !=. 6 T ■*
Proportional X.oguriiliins.
lIlcgiee,or 1 Hour.
3I»93
329gT
31909
mis
31!39S
3e3S0
32361
3!BB4|
3IBTG
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87801: 87315
87793; 8733B
8T795' 87330
87778 87383
87770 87315
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Proportional Logiirilhm
1 Degm, m 1 Hour.
I 87300 t
87S93 !
' a7!e.i 2
! S7S7B i
■ 8T*io a
i 87!fi2 a
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86649
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86590
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86567
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g 85513
8 85506
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26140
86138
86185
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3 86110
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8 26096
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86059
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83305 8
8389B 8
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83871
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83850
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83889
83823
83816
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g 22028
3 88932
6 88915
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24363 2
84350 8
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8*335 83914 83*9B
8 23*9t
1 23*8*
8*31* 8
8*307 8
8*300 8
8*893 8
84886 8
2*279 2
3*878 8
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'i 83177
i7 23470
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82488
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t 82*61
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22380
88373
88366
22339
88353
883(6
88339
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'22iU9
82318
88308
88299
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822ri6
888 7B
82272
82265
88259
82852
88845
82839
22838
88885
88218
28218
82205
28 1 OB
22192
Ftu[iorliDiial LogarithiiiH.
I 22131
insi
81738
BITS!
iins
SI118
1 8171!
S 8169fl
1 81^98
i 8808* 81(5S;
! 88078 8
r 88071 8
t 8806i 8
t 8805B 8
8l3Sa 81
8 1381 21
813T5 81
81368 81
813G3 81
8l35j 81
81349 81
813U 81
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81389 81
81388 81
8l31fl 81
81309 8(
81303 li
8IS9G 8(
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1 880ti
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I 88031
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i 8199B
I 81991
1 8I9S4
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i 81951
r 81944
' 81936
I 11931
I !198i
1 8139] S
i 81981 8
i 81878 8
31S8
S 81883 8
8 21876 8
S 81870 8
81863 8
8I8S7
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81837
81830
81881
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a 80S93
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i 80573
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8 805*7
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80800 II
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80181 1!
20168
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801*9 1
80143 1
8013G 1
8OL30 1
80183 1
80117 :
80111
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80098 1
80091 T
80085 1
80*57 i
8015! 2
804** 8
20438 2
80131 2
80483 2
80*18 2
804181
20400 2
80399 8
80393 8
i 81BI8
r 8tsn
: 81B05
81798
I 81791
1 8147*
4 814fi7
8 81460
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4 214*7
8 81**1
1 2143*
81487
81481
8(414
81408
81101
21395
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3 81099
7 81093
81086
21080
81073
81067
6 80J3S
9 80788
2 80788
80715
80709
80701
206 9G
20690
80348 I
80311 I
80335 1
80328 1
80328 1
80316 1
80309 I
81015 80685
81008 8061
20618
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80877 1
80271 I
8086* 1
20^51 I
80815 I
20839 1
19*4.6
19439
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19186
19*80
19413
19*07
19697
19691
19665
1967B
19989 1
199B3 1
19977 1
19919 ['
19913 I:
19907 I
17973
17966
17960
1795*
179*8
179*8
17936
19376 1
19369 1
19363 1
19357 1
19351 1
9 18647 1
! 186*1 I
I IB63* 1
1 18688 1
I 1B688 1
i IBG16 1
t 18610 1
S 1B60* 1
9 1B591 1
17906
1790(1
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17SB7
17881
17S7A
1786S
17Bfi3
17857
IB9S7 1
18951 1
189** 1
18938 1
IS938
18926
18980
185*8 1
185*8 1
18536 1
18530, 1
19869
19863
10257
19850
19814
1923B
19831
19225
19219
19813
19565 I'
19558 1
19558 1
195*6 1'
18505 1
18499 I
1BB64! 18493 1
7 18*87 1
I 18180 1
i 1B17* I
} 1B463 1
i 18408 I
> 1B456 1
} 1S450 I
I 18143 I
i 18137 1
1B131 1
i 18425 I
5 18817| 17SJl
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3 18804 1
17813
17887
17821
17815
17809
17803
8168 17797
8155. 17790
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81 43 1777ft
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8131 17766
8185 17760
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I 19183 I
19176 1
191T0 1
1 1 9464 I
8758 1'
; 18376
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8733 18364
7 18357
I 18:151
5 183*5
1769*
176B8
17688
17651
17615
17639
17633
17687
17681
I761B .
PropoTtiond Logarithmi. Tabu X. 73
gDqiHi,orSHim«.
^
o4
li
si
si
■4^
si
««■
7.^
,i|.i
loi
lli
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17149
16891
16537
16185
1583S
"15*90
15147
11806
1*4681
14133
13800
1
17603
17843
16885
16531
16179
15830
15(84
15141
iieoi
14483
14187
13795
2
1759T
17*37
16879
16585
16173
15885
16479
15IS5
11795
11467
11188
13789
3
17591
17831
16973
16519
16168
15819
15*73
15130
14799
I41S1
14116
13781
4
17585
178*5
16868
16513
16168
15813 15467
1518*
11781
11446
14111
13779
fi
17579
17813
I686g
1650T
16156
15807 15161
11778
14440
11105
13773
175 73
17813
16856
16501
16150
15808' 15456
15113
11778
14*35
14100
13767
17557
17807
16950
16496
161**
15796! 1S450
15107
14767
14489
1409*
13761
t
1758 1
17801
16844
18190
16139
15790
15144
15101
11761
14183
11088
13756
B
I75oi
171BS
1B93B
16484
16133
15794
15439
1S0B6
11755
11118
1*083
1*077
13730
IC
17549
16838
1647B
16187
15779
16133
15090
14750
11118
137*5
17543
17IB3
16986
16*78
16181
15773
15187
1508*
147WI 14107
14078
13739
Ij
17537
17177
16980
16*66
16115
15767
15*81
15079
14738 14101
11066
13731
13
17531
17171
ISHU
16460
16109
15761
14116
15073
11733 11395
1*061
13728
U
17585
17165
16808
1615*
16103
I57SS
15110
15067
I1787J 11390
14055
13783
lA
17519
17I5B
16808
16449
16098
15749
15404
15061
14788 14381
1*049
13717
16
17513
17153
16796
161*3
16098
16398
15056
11716 11379
11044
13718
IJ
17507
17147
16791
16437
16086
15739
15393
15050
14710| 1437;
13706
IB
17501
17141
16785
16431
160B0
16738
15387
1504*
14033
13701
19
174BJ
17135
16773
16485
16074
15786
15381
15039
14699 1436!
14087
13695
KU
n4»9
171*9
16773
16119
16068
15781
15033
11346
14088
13690
ii
17463
17183
I676T
16*13
16063
15715
15370
15087
11351
14016
13681
»
17477
17117
16761
16407
16057
15709
16361
15088
14345
140U
13679
KJ
17471
17111
16755
16408
16051
I5T03
15358
1501ff
14676
14339
14005
13S73
!i
I74e5
17105
16749
16396
16045
15697
15353
14010
14671
14331
14000
13868
!S
17469
17099
16743
16390
160,19
15698
14665
1438B
13994
1366!
^6
17*53
17093
16737
16384
16034
15686
15341
14999
14659
14383
13998
13857
ft
174*7
1TU87
16731
16378
16088
15680
15335
14993
11651
11317
13983
13651
ts
17441
17088
16378
16088
15674
15330
l*a6H
U649
14311
13977
1384S
S9
17*35
17075
167*C
16366
16010
15669
15381
11989
11613
14306
13978
13610
So
17429
17070
71614
16361
16010
15663
15318
TT637
14301
"13966
13635
31
IT4i3
17064
16708
16355
16005
15657
15318
1*631
14895
13961
13689
3(
17417
170S8
16708
16349
15999
15651
15307
14965
11686
14889
13955
13684
S3
17411
1705i
1669S
16343
15993
156*6
15301
11953
11680
11884
13950
13818
34
17405
1704G
16690
16337
15987
14895
14961
11611
1487B
13fll4
13813
iS
17399
17040
16684
16331
15981
15634
15800
11609
11878
13939
13807
36
17393
1T034
16678
16385
15975
15688
1588*
14918
1*603
14867
13933
13601
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I70*B
16678
16310
15970
15683
15878
11937
11599
14861
13987
I35BG
38
IT391
1708!
16666
16314
15617
15878
1*931
11598
14856
13988
13591
3»
1737 J
17016
16660
16308
15958
I56I1
15867
11985
11586
14850
13916
13585
io
1736S
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16655
IC30i
151158
15605
15i61
1*811
139 II
13580
41
1T363
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16619
16896
1591B
1SS99
15855
1191*
1*575
11839
13905
13574
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17357
16998
16643
16890
15941
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15850
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11569
11833
13900
13569
43
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16998
16637
1688*
15935
15583
16844
14908
11561
U88B
13891
13563
44
17344
16g8G
16631
16839
15989
15588
15838
1*897
11558
11888
13899
13558
45
17339
169BO
16685
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15983
15576
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11553
11817
13893
13668
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1T333
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15887
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14517
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13878
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17327
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16613
16861
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11880
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16963
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15906
15559
15815
1*874
14536
14800
13866
13536
40
17315
10957
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14869
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15877
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15187
14846
14509
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16578
168S0
15971
15585
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14166
139331 13503
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15175
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M
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16915
16560
16808
15859
15513
15170
14889
11491
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17867
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15507
15161
14983
11185
14150
13817 13486
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16903
165*9
16197
15508
15158
14818
11144
13811 13481
49 n2ii
16897
165*3
16191
15848
15*96
15158
11818
14*74
11138
13806 13475
1 !•■ uJJtWIluntil Pan lo tenlhi
.1 .8 .3 .4 .5 .6 .7 .* S
1 1 8 « a * ^ E.
k
Pio|M»rtioDaI LogarithmK
3 Degrea, or t tioUM.
13464 I
: t34J9 1
i I34S3 1
l3tiB I
. I3U! I
I I343T 1
13431 1
. t34}6 1
I ^3481 I
r 1341£
13410
' 13404
: 13390
. I33D3
. I33HS
. 1338!
1 337 7
. 13371
< 13:101
13300
13895
. i3ssg
. 13«M4
13278
13«73
13867
. 13i6%
< ]3'J57
1 13.'36
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I 13?U8
i 13197
13191
: 13186
I 13180
a 12105
18790
» 187^4
IS779
mii
IJTfiS
137U3
18757
1875!
li7*T
18741
18T36
l!730
18785
18780
18714
187l>a
18703
1869B
18(i93
i 13012 18<i81
S 13006 18682
i 13001 18671
! 18991
7 18990
1 1898;
12979
18914
18968
12963
18957
12958
12947
2911
8936
1 2930
2985
18BS0
8914
1S909
lt903
lS89a 12574
12494 1
12489 1
12183 1
18178 I
18118 I
18167 1
18462 1
18456 1
12451 1
18146 1
1841UT
18435
18130
18424
18419
Ii4l4
12408
12403
1 11889
1 11884
1181B
1I8I3
1539 11886
1534
5 11589 11815
11584
11518
11513 11800
18110 1
18104 1
12099 1
12094 1
I808B 1
18083 1
18U1S I
li-Olg 1
18067 1
12068 1
18056 1
12051 1
18046 1
I20U 1
12035 I
18030 1
11461 I
11456 1
11460 1
11442 1
18fi8S 1
186:j3 I
iifiu 1
12003 1
7 11998 1
11093 1
18991
12855
12840
128W
12838
12833
3 12829
i 12822
18569
18564
8558
8553
r 12548 1
i 12518 1
18537 1
12531
12526
12521
18515
18510
1105Q 1
11915 1
11910 1
1193S I
11089 1
2211 I
12205 I
12J00 1
12195 I
12180 1
113811 1101^
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113711 11065
6 11378,11059
11367 11054
11361 11019
11356 110U
11351 11039
] 113161 11031
' 11340,11088
11335 1 110 83
1 1330. 1 10~U
11385 11013
1 1 i«n 1 1 nnu
i 10951
I 10945
8 10940
I 10935
10605 1
10600 1
10595 1
10590 1
1U595 I
10580 1
10575 1
10569 1
10561 1
10559 1
10534 II
10528 li
10523 H
10518 1<
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10508 H
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10417 li
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10468 1 1
10436 1
10131 1
10486 1
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8 09973
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7 0LI89.1
8 09881
6 09882
lOOOO
10085
10090
10015
1 0010
10065
10059
09756
09751
00146
09741
09736
09731
09721!
09121
0IJ716
8 Degma. or 8 Hours.
;
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16»
sr«
88^
S8m 1 3oi
31"
SJm
33^
34=>
3»i
OOT91
09098
08796
08501
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07918
07630
07343
0T058
06775
0619*
096Sff
09S9S
09087
09701
09496
08801
07913
07685
07338
07053
06770
00499
2
096R1
093SD
ososa
087S6
08491
08199
07908
07680
07333
07049
06766
06495
3
09B7(I
09375
(19077
08791
0B486
09194
07D04
07615
07388
07044
06761
06480
4
09671
0937{
090 7 S
08776
0948?
08199
07899
07610
0738*
07039
06756
j
096fiff
09:i65
09067
08771
08477
08184
07891
07606
07318
07034
06758
06171
6
096el
OBSfil
09068
08766
0847Z
09179
07888
07601
0731*
07030
OS747
06466
7
09656
09356
09057
08761
08467
08175
0788*
07596
07310
07085
OS748
06461
R
09651
093-51
09058
08468
08170
0798O
07591
07305
O7O80
06738
06+57
»
oanw
oe.<{lfi
09O47
08751
08165
07875
07586
OT300
07016
06733
06458
10
wen
(JU3tl
0!I04!
08746
08158
08160
07970
07 sag
07885
06788
O0447
09038
09336
09037
08741
08447
08155
07865
07577
07891
07006
06784
064+3
[S
09631
(193a 1
09033
OB736
08*48
OBI 501
07860
07578
07886
07001
06718
06438
13
09fiSB
09386
09088
08738
08*38
08146
07855
075B7
07891
069D7
00711
06133
U
09681
09381
0TO83
08787
08*33
081*1
07851
07568
07876
06988
06709
06489
15
09616
09316
09018
08788
08*W
09136
078ir
07878
06987
06481
16
09611
0B3II
09013
08717
09*83
08131
07841
07553
07867
06983
0671)0
06419
IT
09606
093l)r
OilOOe
08718
08418
08186
07836
07.M8
07862
06978
00695
06+15
IS
09601
09301
09003
08707
09*13
09181
0J831
07513
07857
06973
06691
06+10
11
09JDe
0BS96
0899S
08708
08*08
08116
07887
07539
07853
06969
06686
06405
d)
09391
09*91
08993
08697
09103
08118
07988
0753+
01818
0696+
06681
06401
<i
09dS6
09886
OH98R
08698
09398
O9107
07817
07589
07843
06958
06677
06386
I!
09581
09881
089S3
08687
08391
08108
07818
07581
07838
06954
06678
06391
IS
095T8
08876
08688
09389
09097
07807
07519
07834
069*9
06667
06387
ti
09571
OB871
08973
08678
0938*
08098
07808
07515
0666:
06388
ts
09SM
09*fi6
08968
OB673
08379
08087
07788
07510
07881
069+0
06658
06 37 7
>6
09561
OflMl
08963
08666
0837*
08083
07793
07 505
07818
06935
00053
06373
»T
imsss
09856'
08958
0S663
08368
08078
07788
07500
07815
00931
06618
06368
IS
msM
09851
08853
08658
08364
08073
07793
07496
07810
06986
066*4
06364
»9
09545
09816
083+8
08913
08653
083581
08068
07778
07*91
07805
06881
06639
0635S
30
095*0
098* f
D8618
09355
08063
07774
07*86
078011
069 10
06631
oo:t6i
31
09535
09836
08939
086*3
08350
08058
07769
07*81
07196
06818
0663U
06350
it
09530
09831
0893*
086'i8
08345
09053
07764
07*76
07191
069D7
06685
063+5
33
09585
09886
089891 08633J 08340
08049
07759
07*78
07186
06808
06680
St
09580
09881
0898* 08689 08335
O9014
07751
07*67
07181
06336
35
09515
098 IS
0B9[9'0B621 O8330
09039
07750
07*68
07177
06893
06611
06331
36
09S10
09811
0891* 08619 08385
08034
077+5
07*5?
07178
0G898
066U6
06336
87
09305
09806
08909 08614 08380
O9089
07740
07153
07167
06883
06608
06388
33
095O0
O980I
08901 08609 08316
07735
07U8
07168
39
0»t9A
09196
08899! 0^604
08311
09080
07730] 07113
071. W
0887+
06588
06318
Ul
0949U
09191
0.-*H94| 08598
08306
08015
07786
07138
07153
06BU9
065nH
II630B
41
09185
09186
08889! 08594
08301
08010
077 Bl
07*33
07148
06965
06.^63
0630J
it
09480
091tll
08834
08589
08896
09005
07716
07*89
07143
06960
06578
06898
t3
09475
0917».
08879
08584
08891
08000
07711
07484
0;i3!l
06B55
06571
4i
094711
09171
08874
09578
08886
07995
07706
07119
07131
06950
06509
06889
4j
09465
09166
08869
08575
08888
07991
07703
0741*
07189
068+6
06561
06884
ie
O9460
09L61
08865
09570
08877
07687
07410
07184
068*1
06560
06880
tT
0P455
09156
08860! 08565
08878
07991
07688
07405
07180
06836
06875
•8
08450
09151
08S55 08560
09867
07976
07697
07400
07115
06838
06550
06871
a
aaus
09147
08850| 08555
088.68
07971
07688J 07395
07110
06887
06545
06866
w
OTiM
09T48
08H45
08550
08857
■STgSS
0J678
07105
06888
00861
11
09*35
09137
08840
09545
08868
07968
07673
07396
07101
06817
06536
06857
a
OS 430
09138
08835
08510
03249
07857
0766B
07381
07096
06813
06531
06858
a
091.85
09127
(;8fi30
08535
08213
0795?
07663
07376
O7091
0680M
08587
06817
H
0B*«0
09IS?: OB885
08530
08S38
07947
07658
07087
06803
06588
06843
U
09415
09117 08880
08586
08833
0794?
07367
07088
06799
06517
06838
JG
09410
09Ili 08915
08531
08888
07937
07649
07368
07077
0679+
06513
06833
06405
09107 08810
09516
08883
07933
0764+
07357
07078
06789
06509
00888
W
0B400
09108 08805
08511
09818
07S88
07639
07358
07069
06185
06503
06884
»
0W95
09097| OBBOO
08506
08813
07983
076341 07348
07063
06780
06499
06819
.1 .8 .3 .4 .5 .a -T A
\
of "or..
1 I 2 a 11 ^ ^ -A
8a Table X. Proportional I^uithmi.
2DBgi(«>,or«Houn.
;
38^
ST"
ssi
39°
40^
41"
48"
^
..i
46^
46"
47«
"o
MJ15
0583T
05389
05116
048*5
04576
04309
0*0*3
03179
M5T6
03866
I
MJIO
05933
05851
05383
05111
0*9*0
0*67t
04304
0*039
0S774
0S618
03851
8
O6J06
059if
05658
05379
05106
0*636
0466T
04300
0403*
03TTO
03608
03W
3
l)6!l)l
03933
05648
0537*
05102
04831
0*668
0*896
04030
03T6S
OSAOS
03148
4
06196
0S919
05643
05369
05007
04827
0*659
04291
040S6
0ST61
03WB
03838
0619?
0S9U
05639
05366
05093
0*882
0465S
048881
04081
0ST6T
03496
03C34
0618T
05910
0583*
05360
05099
0*818
045*9
01888'
04016
03T6»
0S490
03890
0618S
O.iSOS
05630
05356
O5084
04813
0*5*4.
04877
0*012
03T4B
03480
03885
e
061TB
05900
05685
05351
06079
0*909
043*0
04873
04O09
03744
03488
03881
9
OB 173
05896
05680
053*7
05076
04804
01536
04269
04003
03139
03477
03tlT
10
OUI68
OSt^
05616
053*8
05070
04531
04264
03999
03135
03473
03811
00164
05887
05811
05337
O6066
01387
0*860
03994
03731
0S4«S
03808
1!
06159
0588?
05607
05333
05061
04791
04.588
04866
03990
03116
08464
03104
13
05ST7
05608
05328
05066
01786
04518
04861
0S9B«
OSTt)
0S4«D
03199
I*
08150
05873
05597
05324
05052
04792
01513
0414S
vam
03T1T
03IU
15
oei4S
05868
O5503
05319
03047
04177
0*509
04848
OMTT
03TI3
03451
0S191
16
00 Ul
05864
05588
05315
050*3
0*773
0*601
0483T
039T2
D3T09
OSUT
081S6
17
06136
05859
0558*
05310
03038
0*769
01600
04833
03968
03T0*
0344*
D318>
IB
0S131
0585*
05579
053O6
0*764
01195
04289
03963
03700
03438
D31T8
19
06127
05850
05575
05301
05089
0*769
s
04221
03969
03698
03434
031 T3
W
(15510
05897
oTols
04756
04880
03936
03691
03429
"osIm
ii
05565
05898
05020
04760
04215
03960
03697
03425
03166
n
06113
05836
05561
05288
0SOI6
04716
04479
04211
03948
03692
03481
03160
aa
O6108
05831
05556
05883
05011
01741
0**73
04206
03941
03679
03*16
0316S
n
06 104
058 i 7
05SS8
0*737
04208
03937
03674
03418
03152
iS
06089
058S8
05274
0*738
01*64
04198
0393!^
03669
03408
03147
t6
06094
05819
05643
06269
0*998
04*60
01193
0392^
03665
03*03
031*3
!T
06090
05813
OSfiSB
05865
04993
01723
0*455
0*189
03924
03681
03399
03139
t»
06085
05803
0S533
05200
04989
01719
04*51
04184
03919
03856
0313*
39
OfiOSO
058O4
05589
05266
0*984
04714
0**46
01180
03916
03652
03390
03130
06070
OS6»4
05251
0*9B5
04110
04442
04175
wm
0364T
03386
O3120
31
06071
05795
05580
06S4T
049T5
04T06
04437
04171
03906
03643
03381
03111
3*
06067
05790
05615
05248
049T1
04701
04433
04167
03908
03639
OSSTT
03U7
33
06068
05785
06611
05238
04BB6
04697
0**29
01168
03997
03634
03373
03II9
34
06057
05781
05506
05833
04968
04698
04*81
0*169
0S993
OSS30
03109
35
06053
05176
05S0I
05228
04957
04688
04120
04153
03989
03888
OSS«
0310*
36
0577!
05497
052*4
0*953
04683
01415
041*9
03884
03621
03360
03100
05498
05819
0*948
04679
04411
0*144
03990
03617
03366
03096
oeos!
05768
0548 B
0*944
0467*
04406
04140
03975
03618
03341
03091
«
060(1
0(iO3U
0S758
05753
05483
05479
062 10
05806
04939
0*935
''^"'°
01102
01391
04136
0*131
03971
03887
03608
03604
03341
03348
0308T
03083
4]
06OS5
057*9
05474
05801
0*930
04661
04393
04187
03988
03699
4!
OGOgO
05744
0S470
05197
0*926
0*656
04388
04188
03959
03695
03334
OSOTl
43
06016
05739
06465
05192
0*921
04662
01384
04118
03591
03329
03070
44
06011
0S735
05460
05189
0*917
0*6*7
0*380
0411*
03849
03696
03385
03066
45
06006
05730
05183
0*912
0*643
01315
04109
03946
03592
03321
03081
OGOO?
05726
05451
01908
04639
01371
04105
03840
03579
03316
03057
OSilDT
05781
05447
ovm
04634
0*366
04100
03836
03573
03318
03068
48
05993
05717
05448
06170
04899
04689
0*368
01096
03838
03569
03308
03049
49
059SB
05718
0543B
06166
0489*
01686
04367
01091
03887
03564
03303
03044
50
05983
057OJ
05433
05161
01890
048(0
04353
01087
03823
03560
0J89«
03039
51
05910
05703
05189
05156
04895
04616
0434S
01083
03818
03356
03895
03036
Bi
05698
05484
05161
04981
04618
04344
OtOJ8
03814
03>51
0J890
03031
53
05694
05141
01876
04607
01071
03810
03547
0i8H6
03026
54
05415
04878
04069
03313
0.J882
03088
55
0S9B0
05410
05138
04867
045S8
01331
01065
O3B0I
OJ8J7
03018
56
05956
05680
05Mh
06133
04883
0159*
013i6
0W61
03796
03534
0J873
03014
ST
05951
05675
05 Wl
05129
04B68
0*589
04382
04O5S
03798
wi&sa
03209
03009
58
05947
05671
05397
05124
01951
045S5
04317
04052
03188
03526
03264
03UOS
59
0S942
05666
05392
06180
04849
0*580
01313
01017
03183
03581
03860 030Ut|
i
Propottioual Fan lo i.iahi
.1 .2 ,3 .4 .5 .8 ,7 .9 .«
/ or-.,..
01122333*1
Propottiaual I^nrithms. TaEI.e X. fi3
SD^«i,orSHoura.
"o"
48"
49"
Soi
Sli
fi8i.
53"
54"
55"
HP'
57"
58"
UOlBi
69™
08739
08188
08888
01971
01783
■oIiTa
01883
00976
00848
I
0899?
1)8731
0817H
08883
01970
01718
01468
01819
O0978
00721
00181
00839
S
0?S89
08730
08471
08819
01966
01711
01161
01815
00968
00171
00834
3
0!!)83
08170
08215
01968
017IO
01160
01811
00961
ooi7:i
oniSO
4
0S9T9
08781
08165
08811
01958
01456
009601 00714
0016!
00286
A
08975
08717
0846 1
08806
01953
01708
01803
00955 O07O9
00165
00888
6
(18713
08157
08808
01913
01698
01417
01199
00951 00705
00461
008(6
7
0!96C
08709
08453
08198
01915
01693
01413
01195
00917 00701
0015;
00814
8
0!9fi8
08704
08418
08191
01941
01689
01139
01190
00913 00697
00153
00810
9
OKBSS
08700
08U4
081 BO
01937
01685
01135
01186
0O939| 0069:^
00*49
00806
10
089S3
08896
08440
08185
01938
Oll3l
01188
00935
O0SH9
00445
01.808
11
0!94g
0869!
08136
08181
01988
01487
01178
00931
00685
00411
0J19T
IS
08945
08687
08131
081T7
01981
01678
01188
01174
00987
0OG8I
00436
00193
13
08940
08683
08187
08173
01980
01668
01418
Oil 70
00983
00677
00*38
00189
14
08936
08183
08168
01916
01664
01411
01166
00918
0067a
00)8^
00 185
15
08838
086 Tl
02119
OlBll
01660
01410
01161
00911
001169
00184
00161
(18987
0867O
081 J 1
08160
01907
01656
01406
01157
00910
00665
00180
00177
17
08983
08666
08110
08156
01903
01658
01408
01153
00906
0066(
OUllS
00113
IB
08919
08668
O8106
02158
01999
Oi647
01398
01149
00908
0065'
00118
001(9
19
08S15
08910
08657
08053
O8108
08397
08117
01895
oJbSo
01643
01639
01393
01389
01115
00^98
00^
O065i
00618
OO108
00 If 5
1.0401
O0161
El
0S90fl
08649
08393
08139
0188S
01635
0I3S5
01137
00390
006H
0(400
00157
it
0890*
08644
08389
08135
0188!
01631
01391
01133
00886
00640
00396
0OIS3
13
08897
08640
08385
08130
01878
01687
01377
01188
O0881
00S3I
00392
00149
84
08893
08636
08380
08186
01871
oieaa
01373
01184
00877
00H3!
00388
00145
25
08889
08638
08376
08188
01B69
016IB
0I3S8
0118(
00873
00381
0O141
g6
0S88*
08378
08118
01B65
01614
01364
01116
00869
00380
00137
87
08880 08683
08368
01861
01610
01360
01118
00865
00680
U03T6
00133
88
0S8J6 08619
08363
08109
01 606
01356
Olios
00861
00616
00372
00189
89
08878| 08615
083J9
08105
0185:
01 601
01358
01101
00857
00611
00367
00135
30
08867 08611
08355:08101
oTsTh
01597
01348
01100
O095;l
00607
00363
00181
31
0?B63 08608
08351 08097
OlSU
01593
01344
0109.1
0(J919
0O6O3
00359
00117
S8
08859 08608
08316 08098
OlBlrt
01589
01339
01091
00845
00355
00113
33
08S51 08597
0831S| 08088
01836
01585
013H5
01087
008(0
00595
00351
ooice
34
08850 OSS93
083381 08081
01838
01581
01331
01083
00831
00591
00347
00105
35
08eis| 08589
08334; 08(180
01887
01576
01387
0I07U
00838
00587
00313
00101
3fi
08811 08535
08389 08O76
01883
01578
01383
01075
00888
00583
0033)
S7
08837 0i5B0
08385' 08071
01819
01568
01319
0101!
00881
00579
00335
OU0B3
38
088a3 08576
023811 08067
01815
01564
01315
01067
00980
00575
00331
00089
nSB89 0857?
08317,08063
OlBll
01560
01310
01068
00816
00571
00387
OOOBS
(188841 0856H
083"i*]"08059
01806
0155(1
0130()
01058
"oiwTg
00567
01383 0'! 80
08880 08563
08308 08054
01808
01551
01308
OI05*
00809
00563
00319 00076
48
08559
08304 08050
OI79B
01517
01898
O105O
00801
00559
00315 00078
43
08811
08555
08300 0801(1
01791
01513
01891
00799
0U551
00311 00069
41
08807
08551
08895] 08048
U1790
01539
01890
01048
00795
00307 0006*
45
08803
08516
08891
08038
01785
0L535
01886
0103B
00791
00546
003O3 00060
46
03799
08518
02887
O8033
01791
0I53I
01881
01034
011787
00518
00899
0005S
47
08791
08538
08883
08029
01777
01686
01877
01089
00763
00.538
00895
00058
4B
08790
08533
08878
0808,5
01773
01588
01873
01085
00779
00531
00890
49
(18786
03589
08874
08081
01769
0I5I8
01869
0108
00775
00530
00886
00041
50
OaTHl
08585
08870
01764
01865
01017
00888
00O1O
51
08777
08581
08866
08018
01760
01510
01861
01013
U0767
00(78
00036
.58
08773
08516
08868
0800tf
01756
01506
01257
01009
00763
00SI8
0087^
00032
53
08769
08518
08257
08001
01758
01. W I
01858
010il5
00759
00514
00870
00086
51
08761
08508
08853
08000
01 748
01197
01818
01001
0075+
00510
00866
O0084
087fiO
08.501
08819
01995
01711
01493
01811
0099 J
00868
OOOSO
0875(1
08245
01991
01739
01489
01810
00992
O085W
00016
08810
01735
01836
0099h
00718
00 .'51.
00018
08191
08836
01983
01731
01838
00981
0O73h
ooi^ja
0025'J
OOU08
59
08713
08187
08838
019J9
01787
01470
01888
00980
O0T34
00816
00004
froponional fare to tenths
.1 .8 .3 .* .ti .6 .-V .%
i
of "or,.
11?*%^^
1*4 TABLE XI.
Depresiion or Dip oi'
the Horizan.
o'/ml
TABLE XII.
p at differ. Distances
roin ihe Observer.
TABLE XIII.— Cotreciion lo be added lo the Ohierved
Altitude of the Sun'i Lower Limb, when taken by *
Fore Obgervatioiii lo find the True Altitude.
Height of t b t Kje above ihe ijcn in Feec
i 2,5 2.3 i.l
} 3.T 3.e
r 5.4^ A.|l 4.S 4.0
tl_\2* '
1 :
; 9.* Si
IToTii fl.8
.7]0.4!l0.1
,oia7lo.*lc
1 11.010.7 H
1.3|ll.0I(
1 1.1
12.0(11.7 I
S1S.!|I1.9I
2.3 13.0 1
s as
I 4.2
I 4-9
i i.6| i.4
I G.I S.S
r 6.S 6.3
V 7.1 e.9
t is ! 7.4
• 8.0 T.B
i 8.Z| 8.1
' B.S 8.5
. 8.e 8.T
1 S 2| 6.0
311.0ia8
5U.i|lI.O
6U.3'lI.
i-a!i].g
70 13-3 IS
80 13.4 13.1:
.fi!l3.2 I:
10.710.5
10.9: l(
■0|l0.7
ll.l;l().B
11.211.0
I.3ill.(
1.411.9
11.6[ll.3
i 9.T
[fti; 10.3 10.1 i
).ejl0.4 10-2 10.1
lai 10.5 10.3 10.1
IO.Bia610.ilO.t
iaBiaoio.4
11.1 las 10.7,10.5
. 1 I + O'.i
Htislil of Jle Eye above the i
It,l{l4.3 I
13.7,1 g.91
6 11.81
10. 7' 10.9 1
10.0 10.2 1
0,5 U.tI
S.0 9i
8.6 e,8
i 7,7
n Vett.
24 1 2K <
"' I ' ~
1.815.019.2 \i
213.41.^01^
1 IJ.3 12.5 12.7 1:
5 10.710.911
010.2 10.410.6
9.Slai 1
9.3' 9.S 9.7 !
T.l, T^
e.a,\ 1.0. 7.3
6.G, 6.8 7.1
6,1, 6.4
5.9 6.1 e.4
i 5.6
7.3 7.S. T.7 :
6.9 7.l!
6.6' 6.H ia\ 'i
I* B i * fi i 8
I 5.6 S-B (
I, 5.5 5.7 i
. 5 3 5.5 t
Is
! 3S.4!).0!)33!9
! a*.? 9.<J 937 TB
9. 991831
.9917?!
9.99.i86T
9.99SBT9
TABLE XV!.
Sun's Paralliu in Altitude, Sie,
9.997016
,997719
9.9DH431
9991 TO
I 5.4 J 3 54.9
i i6.
.OULIOGS
.0008 !G
O.O0I5.54.
ooiiit
I 6.8 15 50.8
0.OU6S64
0.0OTO43
0.0OT173
a 23,00.007337
t ?:u)0ooi2ie
» S3.1 0.00709.i
! SASO.O0
8 83.
8 83.D
8 8t-.
8 84.5
8 8+.
.ohe ir.rlt ■lialo.ostTli
i.l«ft.6<i\%.;
so
t
Table XVII. Mean Refractions.
Fahrenheit
*8 Thennometer SOP. English Barometer 30 Inches.
Z.D.
1^
Log.J^
Diff.,
Z.D.
la
Log.^^
DiSt Z. O. 1
Ifi
Log.J^!
Dis:|
tf /
f //
u /
1 //
o /
/ //
1
0.00
0.0000
10
10.30
1.0129
72
20
21.26
1.8277 38 1
\(\
0.17
9.2304
3011
10
10.47
1.0201
72
10
21.45
1.3315
39
20
0.34
9.5315
1761
20
10.65
1.0273
71
20
21.65
1.3354
39
SO
0.51
9.7076
1249
30
10.82
1.0344
70
30
21.84
1.3393
38
40
0.68
9.8325
969
40
11.00
1.0414
69
40
22.03
1.3431
38
50
0.65
9.9294
791
670
50
11.17
1.0483
69
50
22.23
1.3469
38
1
1.02
0.0085
11 00 11.35|
1.0552
66
21
22.42
1.3507
37
10
1.19
0.0755
580
10
11.53
1.0618
%^
10
22.62
1.3544
38
20
1.86
0.1335
512
20
11.71
1.0684
m
20
22.81
1.3.582
37
30
1.53
0.1847
457
30
11.89
1.0750
65
30
23.01
1.3619
37
40
1.70
2304
414
40
12.06
1.0815
64
40
23.21
1.3656
37
50
1.87
0.2718
379
50
12.24
1.0879
62
50
23.40
1.3693
36
37
2
2.04
0.3097
347
12
12.42
1.0941
62
22
23.60
1.3729
10
2.21
0.3444
322
10
12.60
1.1003
61
10
23.80
1.3766
36
20
2.38
0.3766
.301
20
12.78
1.1064
60
20
24.00
1.3802
36
30
2.55
0.4067
280
30
12.95
1.1124
60
30
24.20
1.3838
36
40
2.72
0.4347
26^
40
13.13
1.1184
58
40
24.40
1.3874
35
50
2.89
0.4610
2.50
50
13.31
1.1242
58
50
24.60
1.3909
1.394.5
36
36
3
3.06
0.4860
235
13
13.49
1.1300
57
23
24.80
10
3.23
0.5095
224
10
13.67
1.1357
57
10
25.00
1.3981
34
20
3.40
0.5319
211
20
ia85
1.1414
55
20
25.20
1.4015
34
30
3.57
0.5530
203
30
14.02
1.1469
55
30
25.41
1.4049
35
40
3.74
0.5733
193
40
14.20
1.1524
54
40
25.61
1.4084
34
50
3.91
0.5926
186
50
14.38
1.1578
56
50
25.81
1.4118
33
4
4.08
0.6112
178
14
14.56
1.1634
52
2^4
26.01
1.4151
34
10
. 4.26
0.6290
171
10
14.74
1.1686
54
10
26.21
1.4185
34
20
4.43
0.6461
165
20
14.93
1.1740
53
20
26.42
1.4219
34
30
4.60
0.6626
158
30
15.11
1.1793
52
30
26.62
1.4253
33
40
4.77
0.6784
153
40
15.29
1.1845
52
40
26.83
1.4286
33
50
4.94
0.6937
149
142
50
15.48
1.1897
50
50
27.03
1.4319
33
5
5.11
0.7086
15
15.66
1.1947
51
25
27.24
1.4352
33
10
5.28
0.7228
139
10
15.84
1.1998
50
10
27.45
1.4385
33
20
5.45
0.7367
135
20
16.03
1.2048
50
20
27.66
1.4418
33
30
5.63
0.7502
131
30
16.21
1.2098
49
30
27.86
1.4451
32
40
5.80
0.7633
127
40
16.39
1.2147
48
40
28.07
1.4483
32
50
5.97
0.7760
122
120
50
16.58
1.2195^
46
50
28.28
1.4515
32
6
6.14
0.7882
16
16.75
1.2241
46
26
28.49
1.4547
32
10
6.31
0.8002
116
10
16.93
1.2287
47
10
28.70
1.4579
32
20
6.48
0.8118
. 114
20
17.12
1.2334
46
20
28.91
1.4611
32
30
6.66
0.8232
111
30
17.30
1.2380
46
30
29.13
1.4643
31
40
6.83
0.8343
108
40
17.48
1.2426
46
40
29.34
1.4674
32
50
700
0.8451
106
50
17.67
1.2472
47
50
29.55
1.4706
30
7
7.17
0.8557
102
17
17.86
1.2519
45
27
29.76
1.4736
32
10
7.34
0.8659
101
10
18.05
1.2564
45
10
29.97
1.4768
31
20
7.52
0.8760
99
20
18.23
1.2609
44
20
30.19
1.4799
30
30
7.69
0.8859
97
30
18.42
1.2653
44
30
30.40
1.4829
31
40
7.86
0.8956
95
40
18.61
1.2697
43
40
30.62
1.4860
30
50
&04
09051
93
50
18.79
1.2740
44
50
30.83
1.4890
31
8
8.21
0.9144
90
18
18.98
1.2784
42
28 C
31.05, 1.4921
31
10
8.38
0.9234
89
10
19.17
1.2826
42
10
31.27
1.495^
30
20
8.56
0.9323
87
20
19.36
1.2868
42
20
31.49
1.4982
31
30
8.73
0.9410
85
30
19.55
1.2910
42
30
31.72
1..5013
30
40
a90
0.9495
84
40
19.73
1.2952
42
40
3194
1.5043
30
50
9.08
9579
84
50
19.92
1.2994
42
50
32.16
1.507.^
29
9
9.25
0.9663
80
19
20.11
1.3036
39
29
32.38
1.5102
31
10
9.42
a9743
80
10
20.30
1.3075
39
10
32.60
15133
29
20
9.60
0.9823
78
20
20.49
1.3116
41
20
32.83
1.5162
SO
/ ^
P.77
0.9901
77
30
20.69
1.3157
41
30
33.05
1 51921
1.522r
29
/ *7
9,9S\ 0.991%
76
40
20.88
1.3197
40
40
33.27
29
/ 60^
10121 LOOSl
75
5C
20 (
i ^ 9S.$0
1.5250
29
u
O 01
W.S0ll.0l29
72
A^ ^\ ^aA'i\v.^'i^%V«|
T.fl
t XVII. B7
Pahnnbeit
Z. D.
"
L^»/
am
Z. 1).
n
Log>'
Diff.
Z. D.
»/
Log.it
Dlff.
30
3S.1!
I.5t79
89
40
-. — ir-
a 4&90
1.69010
857
60
I 6.58
1.9*808
IS8
83.BS
IJ308
89
10
49.88
1.69867
856
10
e.9i
1.8446*
867
34.19
1.S337
89
80
46.58
1.6958;
857
80
ia36
1.84781
869
34.40
1.6366
89
30
49.87
1.G6IS0
857
30
iaT7
1.84977
857
40
34.G3
1.A39S
88
40
50.16
1.70037
856
*0
11.19
1.85834
856
60
31.S6
I.fi4g3
89
50
50.46
1.70893
857
50
11,60
1,85*90
85T
31 n
[) 35.0»
1.5458
89
41
5a7S
1.70550
fir's
1 18.08
1.B5747
"86B
10
33.38
1.5*81
89
10
5106
1.70804
851
K
18.46
1.86006
859
eo
SiSfl
1.5510
8B
80
51.36
1.71058
853
go
18,86
1.86864
8SS
30
35.79
1.5538
30
51.66
1.71311
853
3(
13.33
1,8668!
859
40
36.03
l.SS6fi
89
40
51.96
I.T1664
864
40
13.77
1.86781
858
BO
sa-so
1.5594
89
50
58.87
I.718IB
858
54^
14.80
1.B703S
866
St
) 36.49
i.aiiif
Is
48 a
U iZ.S7
1.78070
858
68
nrei
1.B729B
860
10
36.73
1.5650
88
10
58.88
1.78388
85?
10
15.10
1.8756B
861
so
36.97
1.5678
89
80
53.19
1.7857*
868
20
15.55
1.97919
8G1
30
37.81
1.6707
30
53.50
1.78886
868
30
16.01
l.BBOBO
861
40
SJ.45
1.5735
87
40
5aei
I.7S078
851
*0
16.47
1.98341
860
6n
37.GS
1.5768
88
50
54.18
1.73389
861
50
16.98
1.S8G0I
868
33
i 37.93
1.5790
88
43 O
54.43
1.73iBtf
853
63
I 17.38
1.8B9b3
862
10
38.17
I.S8IB
87
10
54.75
1.T3835
854
10
17.96
1.89185
863
80
38.«
1.6845
88
80
55.07
1.74087
853
80
18.33
1.89397
863
30
3e.6Q
1.5873
27
30
55.40
I.T4340
853
30
18.81
1.S66SU
863
40
38.90
1,5000
87
40
55.78
I.T4S93
854
*0
19.89
1.89913
8G3
SO
39.13
1.59S7
87
60
56.04
1.7*847
853
50
19.76
1.90176
1.90440
864
866
34 D
38.39
1.5954
87
44
56.35
1.75100
54
1 80.84
10
39.64
I.5<I81
88
10
56.69
1.75358
858
10
80,74
1.90705
865
80
39.89
1.6009
87
80
57.08
1.75604
858
81.84
1.80910
868
30
40.14
1.6036
87
30
57.35
1.75866
868
30
81.75
1.91836
866
40
40.3S
1-6063
27
40
57.69
1.76109
858
40
88.85
1.91S0S
867
00
4ae4
1.6090
86
SO
58.08
1.76360
851
50
88.7S
1.91769
867
35
45
6836
UTeSn
868
1 83.85
1.98036
8W
10
41.11-
1.6143
87
10
1.76863
868
10
83,78
1.9830*
869
90
41.40
1.6170
87
80
69.05
1.77116
858
80
84.30
1.98STS
868
SO
4i.es
1.6197
86
30
68.39
1.77367
868
30
84.83
1.989*1
871
40
41.91
l.G!83
69.74
L7T6I9
858
85.36
1.93118
870
iO
48.16
1.6850
86
60
1 0.09
1.77871
858
60
85,88
1.93388
870
36
9 ItM
1.6876
87
46
I 0.43
1.7'8183
858
56
t 86.41
1.63653
871
10
48.66
1.6303
87
10
0.79
1.78375
853
10
86,96
1.0398*
878
90
48.9i
1.6330
86
80
1.16
1.78689
858
80
87,58
1.9*196
873
30
43.81
1.6356
86
30
1.78890
858
30
89.07
1.944G9
873
40
43.47
1.6388
86
40
1.86
1.79138
853
40
8&68
1.9*748
874
43.74
1.R40H
87
50
8.81
1.79386
858
60
89,18
1.95016
875
37
1.6435
86
1 8.57
1.79637
853"
57
1 89.73
1.95891
10
4fc87
1.6461
86
10
8.94
1,79890
863
10
30.31
1.95566
877
80
44.54
1.6487
86
80
3.31
1.80U3
853
80
30.90
1.958*3
877
SO
44.80
1.6513
86
30
3 69
1.80396
853
30
31.49
1.96180
878
40
45.07
1.6539
86
40
4.06
1.B0649
853
40
38.06
1.9G397
879
«0
4a.34
1.6565
86
4.43
1.80908
853
60
38.65
1.666IG
879
38
) 4i.ei
1.6591
86
48
1 4.80
1.B1I56
58
1 33.83
1.96956
280
10
4S.S9
1.6617
86
10
5.19
1,81409
854
10
3185
1,97835
881
80
46.16
1.6643
86
80
6.57
I.B1663
253
80
3*.*6
1.97516
891
SO
46.44
1.6669
86
SO
5.96
I.H1916
85*
30
35.0B
1,97767
883
40
46.7-8
1.6695
85
40
6.34
1.B8170
854
40
35.70
1.98080
888
fiO
46.99
1,67!0
86
60
6.78
1.8248*
864
SO
36.31
1.98368
89*
'SB O
47.87
1,67*6
86
49
1 7.11
1.B8678
866
69
1 36.93
1,98646
886"
+7.S6
1.6778
86
10
7.51
l.8893,<t
855
10
3J.6B
1.99631
886
80
47.8*
1.6708
86
80
7.91
1.83188
853
20
38-84
1.99816
887
30
4813
1.6S84
86
SO
8.31
1.934*3
855
30
38-89
1.96503
897
40
46.48
1.6850
86
40
a78
1.83698
855
40
39-64
1.99790
889
60
48.70
1.6BT6
86
50
9.18
1.83953
855
5(
40.7L<l'a.W»Ti\W,'»
\
40 q
48.99
1.6901
86
50
9,58
1.64809 ?5fi
\ W.%h\t,(»'ift*'SMJ\
fir
TAiLtXVH.
Mean Itefl^cliont.
Z.D.
%» -LogM
O.
Z.D
if
Lng, it
D.
dil
z.o
\l
u,.»
Diff
IL
517
0.01
m
1 40 85 8.00368
39n
JO
3 .■!9.ir
S.SOISS
389
m
5 30 19
8,60541
696
0.03D
♦ I.SS
8.ijoai8
f9l
10
40.S9
3 2057;
390
S5.36
3.51337
707
031
O.IM
!0
4S.SI
i nomfl
393
30
43.04
i 30993
393
3(
ao.7o
8.51944
716
0.033
»(
4S.90
8.01J41
39:
30
43.53
3.SI35B
396;
30
30.30
3.58660
787
0034
O.Ul
40
4:!.S9
!.0I5:)5
394
40
45 03
S.3175)'
39a
40
4i.se
S-53387
738
0.036
0.05
&)
_i*^
«.0)9^9
395
50
46,53
S.33I50
403
3.61135
749
0.038
OOi
^"i;
145 01
?.Oil^l
296
71
a 4li.08
2 3355S
404
tTo
5 53.79
3 S4N74
759
0.0 10
1(1
45.73
S.OSl?0
3I(B
10
49.65
•33956
407
10
6 0.04
3.S5635
773
0.013
so
4S.46
S.0a71M
399
SO
51.35
S.33363
410
so
fl.SO
3.56407
785
O.014|o,06
30
4T.I8
3.03016
300
30
53.87
3.33773
413
3t
13.18
3.57193
797
0.046
0.1)7
47.63
a.0331(i
301
40
54.53
35*186
417
40
30.09
3.67989
811
0.O4S
0.07
50
48.B8
S.03Bi7
:i01
5ri
56.81
3 S4a03
119
51
87.S6
3.5880O
884
1051
0.0«
63 II
I 49.44.
Z0391b
301
!3
S 67.B8
i.asos?
U3
tS 1
6 31.titl
8.596*4
aSH
0.05;i
10
50.81
3.01SS1
301
10
50.66
2.S5US
4S5
48.37
8.60463
851
0.057
0,09
*l)
50.99
S.015S5
305
ao
3 1.43
S.3587D
4(9
30
50.33
;.613[3
0.060
09
30
51.77
a.0443ft
307
sa
3.33
S.S6S99
43i
SO
53.59
8.6S179
833
1.1 1)
40
53.57
S.05I37
SOH
V
5.06
tSfi733
436
40
7 7.19
899
06-7
O.HI
SO
53.3(i
S.05I4,i
309
5tf
0.93
3.S7I68
50
16.13
3.63961
914
0.071
1 54.17
3,0J351
J3
3 8.83
3.37608
4«
r3"o
7 S5.40
8.B4B75
■53I
0.074
10
34.89
S.OGOtj'l
:II3
10
10.77
3.f80il
H7
003
ID
35.05
8.63H06
949
ao79
0.13
SO
.S5.81
3.06:J7(.
313
30
13.74
3.SS496
150
30
45.10
3.6fl7f5
967
l).(«4
D.13
30
5H.66
3.066H8
315
30
14.75
3.3891W
454
0.001
SO
55-58
986
0.089
0.13
40
67.50
S.0700H
313
40
16-80
3.S9103
458
004
40
a 6.,40
8.0870B
1006
095
0.11
SO
6H.S6
S.07aiH
317
51
18.88
3.S9960
15?
O.OOi
11.90
t.69711
10S6
a 101
0.15
IJl U
1 S».SJ
i 076Ji
I*
3 31.01
3.30333
467
O.OOS
iiT
8 39.00
8.70740
1047
0107
a.iu
10
i 0.09
2.079.^:f
3S(
10
S3. 18
3.30789
47(
O.00li
K
43.84
3.71797
1069
0.114 0,17
*0
0.9B
a.08S73
331
30
S5.39
S,31B5B
0.006
30
55.25
8,73856
1098
0.188 0.1H
30
1.8H
i.08594
3S3
30
S7.66
3SI734
479
0.007
30
8.73948
HIS
a ISO
0.S0
40
. s.eo
3.08917
331
40
39.95
338313
183
OOOt
!3:i6
3.76063
1139
ai3s
0.81
50
3.71
3.09311
3S6
SO
33.30
3.3S696
488
0.0«
5(
38.13
3.T6V03
Ilea
0.149
0.8£
% 4.65
3.095SJ
337
15
3.331«4
493
009
IS
9 53.81
8.77367
1191
0-159
0.3S
10
6.S[
3.098)14
330
10
37.16
3.33677
O-009
10
10 10.35
8.71^558
1319
0.171
036
SO
3.I0S31
330
stf
39,65
3.3 1174
SOS
OOIO
SO
37.73
S. 79777
1349
0184
038
30
7!si
3.10554
33!
so
3.34676
507
0.010
30
46.03
8.81085
1877
0.198
0,31
40
8.49
3.1088b
334
40
3.35183
513 0.011
40
11 6.30
3.83302
1309
0,813
0.33
50
9.48
3.118S0
33!
50
47.48
3.366B5
317|(i.0ll
50
35.66
8.83611
1340
0.3S9 3S
s laiB
ijTsIs
3rt
10
3 SiP.31
3.3UJ1S
Sti'j.Oli
(To
3.81951
l374
).34H0 3!i
10
11.50
!. 11893
33S
10
fi3JXl
3,36735
saelo.on
10
IS 988
2.86385
0.369 0.4:1
SO
ISSS
3.13331
34«
so
3.37t63
533a013
so
33.97
&87735
1447
0.893 0. 1.7
a
13.57
3.13371
343
30
S,37796
S38 0-O13
30
59.51
3.89183
I4S4
0.3170.51
14.63
S.li913
34-!
40
4 1.74
8,3833!
545 0.0H
40
13 36.61
3.90066
1583
0J4i0.5u
50
15.70
i. 13358
345
50
_JjT9
3 38879
65 i
0.014
50
55.40
3.93189
16M
a376a6«
03W3
34t
!7 (
S 39430
557
O.OU
(7
14 Str.04
3.93154
U.*100..^
17.88
3.13DJ1
349
10
u'.u
839987
563
0.015
10
.-.8.71
3.95 J6g
1654
0.44H
U.75
so
19.00
3.14300
3S3
80
14,39
8,40550
569
0.016
SO
15 33.60
8.97016
1701
atffo
[1.83
30
2ai:i
2.14653
314
30
17.74
341119
576
0.016
30
16 10,89
8.98717
1749
assB
aai
40
S1.S8
3.150OG
35.1
40
31.19
8-41695
583
0-017
40
60.8
S.0046B
WOl
0.593
l.Ol
50
gS.43
3.15361
S5(
50
S4.78
3.43378
589
0.017
50
17 33.6
3.03867
1855
,0.654
LIS
t Z3.61
i.I571!)
Wj
78
l 28.33
3.43867
596
Wo
18 19.6
3 04133
0.738,1.^6
S4.81
8.16078
363
10
32.01
3.1346-3
603
0.0 IB
10
19 9.0
(.06031
1967
0,789
1.41
SO
!e.02
3.16410
364
30
35.94
3.44060
611
0.O1D
so
30 2.3
(.07998
3036
0.8B7
1.59
30
g7.ga
3.16801
360
30
3.44677
618
0.03{
30
69-G
1,10021
3089
0-987
1.79
40
saso
3.17171
36(
4.3.7f
3.45395
636
40
32 1.7
3.IS113
3155
I.1U1
S.DS
50
W.76
3.17539
371
SO
3,4.^931
S35
U.023
50
33 8.9
1.11368
3331
LS31
inn
Ii9
rsToi
3.17910
79 U
4 52.13
3.46558
643
0033
<9
31 2I.«
lIH4t(9
S390
lU
38.34
2.183B-1
375
10
5li47
3-47 IS8
050
0.034
10
35 40.9
S.1B779
3361
l.S51;3.9a
33.67
i.l865ti
37!*
31
5 a94
3..47e4B
859
ooes
30
37 7.1
131140
3431
1.749,3.41
31
35.01
S.1903(>
381
30
454
3.13507
S69
D,03b
30
3H 40.9
S.S3S74
3509
1.97; 3.93
Ui
3S.37j2.l9m
38i
40
laea
3.49176
0.0s?
40
30 Si2
iSOOH:)
3694
S.8Hk,',1,
37.76 AiBsaokas
SO
13.16
i.isisa e«b\u.O'i«
mAI' \5.0
B8WiG7
e607
3.449,5.26
_o!
39.I6Ji.20lSSJ38ii'Sn
so. 19
2.50541 69G\o-Oidb0 ll\^i M .5 \?.-iV2a!>\ \t.*WiV>-*
k
TABLE XVIH
8.
•m., Log,
Tb. Ue_J
Therraoaieter.
lo-
U.0O173
^
CUXKXW
TABLE XIX.
ll
18
13
aooi69
0.00164
0,00160
58
53
9.gS99S
9.99991
g.999BT
^P.|Tb.| l..«.
[\P.|Th.
W-
10"
0.o:)J78
30"
aooooo
10
0.036HO
9
1
9.99910
Barometer.
14
0.00156
54
B.«B»e3
«»
0.03589
IB
8
9.99880
15
0.00151
55
29
39
i9
0.03*84
03386
87
36
9.99J30
9.99H40
999550
16
17
18
aooi47
0.00143
0.00 138
56
59
9.SB9T4
S.999T0
9.99965
P.P
_5'A
Lo..
87.5
9-96881
69
o.o;ii9i
0.030SI
fi3
6
9. 99160
9.99371
9.H6379
9.96536
19
0.00134
9.99961
20
aooi3o
To
9.9B95J
79
008997
7?
9
9.9S888
9.96698
81
aooise
61
9.99953
m
0.OSBO0
HI
9
9 99193
9.9681B
88
83
0.00181
aooii7
9.99948
9.99944
lo"
0.0iB03
~eo
9.99104
"+"
aslo
9.970U4
10
o.a£70fi
9
1
9.99016
15
9.97 15B
81
fl.00113
9.99910
19
O08li(l9
18
i
P.939!7
30
9.97313
85
O.0O1O9
65
9.99935
89
0-«?511
8li
3
9.M8839
9.97466
86
0.00104
9.99931
38
0.02tlB
35
4
9. fl87.il
61
9.97680
87
OOOIOO
48
007383
41
5
9.9^663
76
9.97778
88
0.00095
9.9998!
AS
67
0.08887
U.08I32
5A
9.99575
9.98488
91
106
9.97981
9,99076
89
0.00091
69
9.99918
-35-
0.00097
70
9.99913
77
0.08037
8
9.98401
188
9.99887
31
0.00083
71
St
0.01918
79
9
9.993U
137
9.9837B
38
33
0.0007B
0.00074
73
g!99904
9.99900
■30"
0.01B18
70
fl.9H88I
9.99588
9
0.01751
1
9.98110
9.09677
34
0.00070
74
9.99996
19
01660
n
9.93054
89
9.98886
35
0.00065
75
9.9B991
(8
0.0t5S«
86
3
9 97967
41
9.99975
36
0.00061
76
9,99897
38
anH7S
31
4
9 97BHI
59
9.99183
37
0.00O57
77
9.9988S
47
0.01379
43
S
9.97795
73
9-99870
38
0.00058
78
9.99978
SB
aol8B3
001198
5*
60
6
7
9.97709
9.97683
HO
9.99117
9.99563
39
0.00018
79
9.9997*
9.99910
40
0.00013
75
0.01099
69
9.975.17
9.99709
41
000039
9.99860
85
aoioofi
77
9 974.^8
138
9.99855
aooooo
48
43
aO0D34
83
9.99861
9.99937
40
oixTm
"wT
U^iiTSiTf
9
O00H88
9
1
9.B7298
U
11.00145
41
0.00086
9-99853
IB
O00T30
n
8
9.9T197
B9
0.00899
45
aooosi
85
9.99948
9S
aooflss
ts
3
9.97118
43
0.00 m
46
aooot7
8<J
9.99914
ST
aoasM
34
4
9.97087
57
47
O.00013
9.99940
46
aoo+55
48
71
o'o0718
49
9.99835
5A
00363
9-91)859
96
aooBiio
49
0.00001
89
9.99831
84
7*
0.00878
0.O0IBI
59
e
9.96775
9.06B91
100
114
0.0IU08
0.01 143
50
0.OUOO0
90
9.09887
-pT
P.lot.r..h,or-.n«t* 1
83
0.O0090
T6
9
9 96B07
189
0.01384
.1 .8 .3 .4 .5 .6 .7 .8 .91
io
aooooo
90
31.0
0.01484
_0 1 1 a 8 3 3 3 4I
EXPLANATION.
)(3/ + ^(,_50=)_^'(30 — f.); inwhicli r denota the mie Ttfricuon, p . 0.0()3T5
>bE Bip-n»ioiiof»giv.nv<JomeofairBtlh»mrfaieorthe earth tot nne.legreeof iho Mud-
p4->«r..p.cuvd,. ^^^ ^^^
Twhle XVIIl. «mlii[na the log»ri[hnn
«b\e XIX. Iht logaoihiDi of 1
^ l + ^(.-50-l''^
|j ; aad T.Ue XX. the logiirithm. of- ^^=j|^ x .431. {
3|.<
01S!tOl6 oiesoiT
aoo
ail u
iS ] g.S<)
•i li.S5
15 18.79
.713.0!
17 13.S4
iS 13.41
18.78 U
1S.SS1;
13.14 H
1^
I*.g8' 15.85
14-36|lS.34
I3.46JI4-4S 15.41
I3 50I4.46|1S.45
7*«f//«.*S;j5.48,
TABLE XXn.
Reduction of the Moon'i PoraUaK ill the
Spher "
Lm. 54' 55' i6'
90 Uo.5>o.i\\o-a\v\-\lvt-
TABLE XXIII.
^JfSLE XXIV. ii
LoKarithma of the Earth's Radii, in mch
Angles of the Verti(»l wilh the 11^
ilius; OT Reilucdnnof the Latitude,
Parallel of Latitude; the Equatorial Ba>
in each Parallel, the CotnpresBion
being ,i,.
(lius being Unity, and Corapreasioa ,S,.
.
Lat
Reduc
LiT
Redue.
^
Reduc
Lat.
Log-H
.at.
Log. B
Lnt. Log, It
0.0
30
~„
6^
9 57.1
lo5
9.9996*0*
60"
J.998915I
1
9.99999SJ
31
96181
88938
24,0
31
10 7.8
61
9 151
S
99B!
38
68
88780
8
4T.9
32
10 18.1
62
9 32.0
3
9960
9S78S
63
B95I2
3
1 lt.8
33
10 88-3
63
9 18.3
S930
31
95196
61
88308
1 35.5
31
10 37.8
6*
9 3.P
S
9890
35
95261
65
88111
5
I 692
33
10 46.4
65
8 49.7
98*3
30
93083
66
87918
6
a 88.7
36
10 54.3
66
8 32.9
1
9786
37
67
97738
7
2 16.1
37
11 1.4
67
9 16.6
S
9781
39
9*537
68
87552
9
3 9.2
n 7.7
68
7 59.6
9
SliiS
39
94291
69
87378
3 32,1
39
11 13.8
69
7 18.0
10
40
94044
70
87210
10
3 5*8
40
11 17.9
70
7 83.8
11
9.9999177
41
9:9993794
9.9987030
* 17.2
"iT
11 81.7
li
93TS
*8
935t:l
78
8(5896
12
4 39.3
18
11 81.7
e 46.9
13
9873
43
93891
73
B6T50
13
5 1.0
13
1! 86.9
6 26.8
916^
**
9303h
74
86611
11
5 22.1
11 88.8
6 6.0
15
9037
98786
75
86479
15
5 43.1
45
11 89.7
6 15.4
16
89(1>
*6
92533
8B3S6
16
6 3.9
16
11 28.1
5 24.S
17
B771
47
S888(
77
96840
17
G 81.1
47
11 27.3
5 8.9
IS
8687
*a
9808(
79
96131
18
6 13.7
49
11 83.8
4 41.0
19
8*T6
49
91776
79
86031
7 £.9
ID
11 22.3
1 18.8
*0
831(
50
91383
80
95940
80
7 81.6
60
11 1B.6
80
3 56.3
!1
9.9998153
51
"bT
9.9985857
21
7 39.7
51
11 14,1
3 33.5
!a
7983
fi8
9103(1
82
95782
88
7 57.3
58
11 9.8
82
3 'lO.*
«3
7803
S3
90783
93
85716
83
8 14.8
53
11 2.6
83
8 17.8
24
7621
54
903*8
81
95639
21
8 30.7
10 55.7
81
8 83.7
85
7*31
S5
90302
85
85610
8S
8 46.4
55
10 47.9
8 0.0
26
7*36
56
90U63
86
83370
!6
9 1.6
56
10 39.1
86
I 36.8
8T
7033
57
89831
87
B5539
87
16.1
57
10 30.0
67
1 18J
S9
6889
5S
89601
88
85317
28
9 29.9
10 ia.9
88
laa
!9
6618
59
89374
89
93304
29
9 43.0
59
ID 9.0
89
81.1
30
60
S915I
90
95199
30
9 55 1
60
9 57.4
90
0.0
TABLE XXV.
For deteimiiiiiig the Latitude, at any lime, by the Pole Star.
'
"INm MJN| / 1 M |N| * fM |N'|
yi
Tf
/ 1 M
N
1.DI.
" 1 " kk
^ "
h.m.\ " 1 " h.m. " 1 "
Tf-
h.m. "
) 00,000.0(«l (
3.85'ail
I 021.820.37 3 043.630.60
95.45
0.63
5 081.48
04
lOO-lTO.Oi 10
7.89 0. [4
1085.190.41 1047.4*0.68
S8.66
10 93.18
0.3
80 0.66 01 21
lo.ao'o.i'
80,8a7i;0.*( 805i.tlO.61
80
71.69
0.69
80 9k64
0.2S
301.1900:i iO
18.780.^3
3032.340.50 3054.930.63
30
71.49
0,55
5.78
0.8
40a.63;o.Oi 40
15.59 027
40 36.06 0.53 40 58.56 0.6i
1'7.06
40t
6.60
0,1
fio*.o9,aoB flo
IB a 10.3*
SO'39.83057 5068.07 0.64
51
79.3a
o!l«
608
7.10
0.0
1 os.e5;o.ii i
81.98 0.37
3 0,43.63;0 6( 1 o|65*5 0.63 5
81.42
041
6 0|9
7.86
0.0
^ = Z + pcOB.t — -M cotin. Z + N ; Bhere <; b = the Lntitudc; Z = the Zenith
DiiUinH;p=l''W,DrlOO'; ( = Iht Horary Angle! *=»( in the ftnl Quadrant; =18-.
— (intheEKond; = ( _ 18" in tiie third; and = 8i" _( in the fourth; M and N bdng
the Tabular (Joamiliea. The qoaniity M U=ip'idn.'i, and ia always posiiivc ; but the
iliianfety N = i f rin.=( on. (. becomea negatiie in ihe aecond and third Quadranu of (.
Wlien p (the Pular Ditlance) augmenti or diminlilua 1', (he Tabular Quantity tnuit also be
SS Tabi.1 X. Propoitional Logarithm*.
8 0egi<»,«tHoun.
;
ra
ST"
SB"
39"
4oi
41"
«i
43"
4ii.
45^
46"
47a
'S
08*15
0S93T
0566!
05388
05116
01845
04676
04308
01013
ram
rasie
03846
1
OS«IO
0SS35
05657
0SS93
051 II
04810
04571
04304
0*038
03774
03518
03851
t
oe«06
05928
0S6S8
0S37B
05106
0*567
04300
0403*
03508
03147
3
OGSUI
05923
05648
05374
0S1O8
0*831
0*568
04195
04030
03768
OS50S
03148
*
061 9B
05919
056*3
06369
05097
048SJ
04558
04891
04015
03781
0S499
03138
S
06198
05914
05639
053G5
05093
04822
0*553
04886
04081
03T5T
0S49S
03834
6
06187
059 10
05634
05360
05088
0(818
0*5*9
0488}
04018
0ST5S
03490
03830
7
OBI 82
05905
05630
06368
05084
04813
0154*,
04177
04018
03T48
03486
03215
e
06178
O5900
05685
05351
0M79
04809
o*su
04173
0400B
03T44
03481
03181
9
06173
05896
056K
0S34T
06075
0M04
04530
04IS&
0400!
03739
03477
03817
10
05616
05348
06070
04800
04531
04184
03999
03735
03473
oAsti
11
OB 1 61
05867
05611
05337
05066
0*795
01527
04160
03894
03T31
03489
03108
061S9
05998
05607
05061
01791
01522
04155
03990
037M
03484
03804
13
061S5
05877
0560?
05388
05056
01786
04518
04851
03986
03Tt8
03460
03199
U
oaiso
0S873
05,597
05384
05052
0*788
01513
04846
03981
03T1T
DS455
03195
IS
061 45
05869
05593
05319
05047
0*777
01509
04141
03977
03713
03451
03191
16
06U1
05864
05588
0S3I5
04773
04504
04I3T
03ST1
03T09
0344T
03186
17
06136
05859
05584
05310
04T6B
04600
04933
03968
0370*
0S44t
031S2
IB
06131
05854
05570
0S30S
04784
M495
04889
03963
03700
«I438
03178
IS
OBI 87
05850
05S7S
05301
04759
04491
04114
03959
DS89A
03434
031TS
to
061 a*
U5M*5
0529T
05085
01755
04186
04880
03955
^691
63489
03160
21
06I1T
05841
05298
05080
04750
04481
04816
OSS«
03415
03165
*8
06113
05836
05561
052BB
05016
04T46
0447B
04811
03946
OSMI
03160
i3
0610<j
05831
04556
05283
05011
04106
03941
0341S
0315S
n
oeioi
058ST
05558
068T8
05007
01737
0*169
04808
03037
03674
03418
03I5S
i6
06099
0£S(!
0U4T
05874,
05008
04738
01*64
04198
(»4ae
03147
16
011094
05818
05543
05869
01B98
04728
01160
04193
03403
03143
IT
06090
05813
00538
06866
01993
01723
0*155
04189
03984
03661
03399
03139
iB
06095
05808
05533
06160
0*999
0UI9
01151
04184
03919
03656
03396
03134
!9
060SO
05801
05589
05256
04984
01714
01146
04180
03915
03651
03390
03130
30
06076
05799
05681
05851
014*8
04176
03911
03647
03386
031*6
31
06071
05795
05520
05247
04975
0*706
04*37
04171
03906
03643
03381
03181
3>
O0O6J
05790
05515
05212
04971
0*701
04*33
04167
0390*
0363ff
03377
03117
0G06i
05785
05611
06238
04966
04697
0U89
04161
03897
036341
03373
03113
34
06057
05781
05506
06833
049S8
01698
0**8*
0*168
0389!
oaesi
03368
03108
3£
06053
05776
05501
06888
04957
04688
04480
04163
03889
0368*
03384
03104
36
060*8
0STT8
0549T
05184
04953
04683
04415
04149
03884,
03681
033«(
03100
37
06043
05767
05491
06819
04948
04679
044U
04144
03880
0381 T
03355
03096
38
06039
05TS>
05188
05815
019*4
04674
044OS
04140
03875
03611
ossct
03091
39
Dfl0:i4
DST5B
05483
05810
01HS9
01670
04*08
04136
03871
03608
03311
03087
W
06080
05TM
05179
05200
01935
01665
04397
04131
03867
03io*
0»U
03083
41
osots
OST«g
05471
05801
01930
01661
O4303
D411T
03861
0359S
03S38|
03078
48
oeoto
05T44
05170
05197
0192b
04656
04388
04)11
03856
03595,
0S3M
030Ti
4S
06018
0JT38
05465
01921
04652
04384
04118
0385!
03691
oasiB
030T0
44
08011
OSTSi
06160
01647
04380
04114
03849
03586
0331^
03065
4d
oeoos
06T30
054S6
01643
04375
0U09
03845
035B1
03311
03061
4a
ooooa
05716
06461
06179
01908
04638
04371
04106
03840
036TB
03316
03057
4T
05997
05TS1
06U1
06171
01903
01H31
04366
04100
03836
03573
03311
03058
48
05993
057 IT
06448
05170
04899
01689
01362
0109S
03838
03569
03308,
03048
49
Oe9B8
OSTIi
06438
05165
04891
01625
04357
04091
03817
03564
03303
M
05983
OiTOT
05433
06161
Ot890
0U97
mm
MtW
03039
ai
05979
0ST03
05«B
06166
04BB5
04616
OU^
01083
0381B
03556
03195
0303J
<i
05914
05698
03«*
05151
04881
04618
04341
01078
03814
OSSAI
OUW
03031
u
049T0
O5094
05119
05147
04876
0460T
043*0
0W71
03811
0J547
0iB8(
03026
»4
0596i
056B9
05116
051*8
04871
04603
04335
04069
03513
03018
6S
059E0
05684
05410
051 38
04B6T
04598
04331
01065
03801
03.J38
0J877
03018
sa
05956
05680
05106
05133
01863
01591
01320
D1061
033 J4
0J1T3
0301*
57
05951
05675
05101
05189
01858
01589
0*328
01066
03798
O.'U.'JO
032G9
03009
0594?
05671
05397
05184
04851
0*585
01317
01052
03788
03525
03264
03005
AS
OSSii
0566 G
05392
06120 01819
01580.01313
0*017
03783,
03521
032601 03001
of'or!'. |0ll!8S331
Proportional' Logarithms,
i Oegieu, H 2 Houn.
t 08979
i 0*975
i 02970
' 0!9a0
I 0!9bS
> OgRiS
onas
QiUB
700 omi
S OiSSH (I
H 08883
t 08819
U 08215
S 08811
1 02806
7 0880Z
OUlfl
G 01714
8 OI710
i 08987
r 08983
( 08919
} oaflis
r 08910 ■
I 0390G
J 08908
J 08897
I. 08893
S 08889
; 08884
1 08S63
! 08859
) 0885i
I 08850
• 08S46
1 08B*I
> 08837
08431
08481
02483
08419
08414
08H0
08406
08408
08397
08393
083S9
083B&
08380
08370
7 08378
3 08368
S 08363
083.'i9
OaSiS
08351
08340
0193 7
0193
3 01478
0144]
0146 IJD
||4G0
01(56
01458
01447 U
01443
5 01435
B OO7S0
8 0078 <
8 00788
1 009641 007II
0071'
5 00709
1 00705
7 0070!
3 00697
OtSi
00848
0093B
00834
00830
00886
00882
01988
01984
01980
01916
0816410191
0148T (
01488 <
01418 C
O1661J0U14
178 00931
.174 O0827
,17(1 00983
01907
08143
08139
08135
08130
08186
08188
01656
01658
0)647
5 01643 01393
8 01687
4 U1688
5 01618
B 01614 01364
00685
006ttl
00677
00673
OOGCB
O06S5
6 OIIG60
8 0U65'i
B 00658
0U4BI
0047'
00173
0046!
001G5
00461
oois;
004S^ 0081
:! ooug
00445
OOUl
OD436 9
00438 01
00) S,-' I
0012)
) 08799
I 0879*
I 08790
) 08786
9 0251
4 08568
02563
08559
08555
02551
08516
08548
085SB
03533
08589
08585
08581 C
08516 C
9 08518
1 01808
OI7.9t
01794
2 01790
8 01785
3 01781
4 01758
01748
6 01744
1 01 739
7 01 735
5 01731
9 01787
01319
01315
01310 ,0
013U6'o
01 308
01898
01894
01890
0I8S6
01881
01877
01873
3 0<ie49 0'
1 00843 0'
7 O084O Oi
3 00836 Oi
9 00838 0'
5 00888
1 00884 Oi
7 00363
3 00359
9 00335
5 0035!
1 00347
7 00343
3 00339
9 OO.'WS
3 00331
1 00387
01501 U
01497
01+93
0I4B9
5 01836
1 01838
6 01888
8 0O5(i7
8 00563
4 00559
9 01)554
I 00546
7 00542
3 0053B
9 00534
j 00530
T 0O5i6
7 0058S
3 00518
R 00514
4 00510
00506
6 00508
8 004U!
w 00 vn;
4 00489
O0II3
0bIL9
00105
OOlOl
00097
00093
00089
00085
OOSB, " 80
00319 00076
00315 00072
00311 00068
00307 00064
00303 O
00874
00870
00866
0O8S8
00858
00251
IMiM
00846
0005a
00058
O0O48
OOP 44
000 40
00036
O0O38
0008 B
00084
00080
9*
TABLE XXVriT.
Reduction to the Meridian. P* ht I.
a
Oa.
1»
go.
3".
^m
fi"
B"
T"
gm 1 90
10"
11- 1 18m
000
1.96
7.85
17.67
31.41
49.09
70.68
96.20
I8S.6S1S9.02
196.3!
237.64 882.68
0.00
S.03
7.99
17,87
SI.6B
49.41
71.07
186.17:159.61
196.97
838.86 883 46
0.00
!-I0
8.1!
18.09
31.94
49.74
71.47
97.12
126.7016080
197.63
838.98 881.85
0.00
2.IS
8.!5
ie.!7
3!2i
50.07
71.86
97.S8
187.83160.78
198.89
!39,70 285. 01
0.0i
!.!3
8.39
18.47
3! 47
5O40
7 2.26
98.01
187.75 161.39
198.94
?40.42 886.83
!.30
18.67
32.74
50.74
7!.66
98.51
128.28 181.98
199.60
241.15886.62
O.OS
8n6
18.H7
S3.0I
73.06
99.97
128.81 168.38
200.86
211.87,887.41
0.03
!.45
8.80
19.07
13.27
51.40
73.46
99.41
189.34163.17
800.93
218.60 888.80
0.03
!.5a
8.94
I9,8M
W.54
51. 74
73.86
99.90
129.87 163.77
801.50
243.33,886.99
0.01
!.80
9.0S
19.48
)3.82
52.07
74!h
I00..17
130.41 164-37
202-25
244.06,889.79
05
9.!!
19.69
ii09
liTi
74.66
100.84
130.91 164.97
S0292
844s 79,890.68
0,OJ
!.75
9.36
19.90
31.36
75.07
101.31
131.48 165.57
203.38
245.3! 891.38
0.08
9.50
!n.ii
34,64
53,09
101.78
132.01 166.17
204.26
246.«6 198.18
0.09
!.9I
9.63
^0 38
34.91
53,43
75 88
102,85
132.55166.77
804.fl2
846.99,108.98
0.11
a. 99
9,79
2033
35.19
53.77
76.29
I0!.7g
l,3a09 167.37
206.59
847.78;t93.78
0.1!
3.07
80.74
35.46
;4i!
76.70
[03-80
133.63 167.98
806.26
248.46
294.56
3.15
8095
35.74
5446
77.10
103-67
134^17,190.58
806.9%
84ftl9
896.38
0.16
3.83
10.84
!1.)7
36.02
54.81
77.51
104.13
1S1.7] 169.19
807.60
849.93
!9S.le
0.18
asg
lU.39
81.38
36.30
55.16
77.9!
101.63
136.85 169.80
808 M
8U.B7
t99S9
0.10
3.411
10.54
81.60
36.59
55.50
7«.34
105.10
136.79 170.41
808.93
85U1
897.79
0.8!
3,4^
10 6!
36.6?
55.K6
78.73
1 05.58
136.34171.02
209.68
taJs
80840
0.!4
3.S8
8!.03
37.16
56.80
79.17
106.06
136.68171.63
210.30
861.89
(99.40
0.26
ll!o(
!!.a5
31.41
5S,5S
79,58
106.55
137.13 172.84
!lO-8e
263.S3
aoD.it
0.!9
3.76
11,15
!8.48
17.72
80 00
107.03
137.98' 172.86
211.66
25438
SOLDI
0.31
3.85
11.31
82.70
38.01
37.25
80.48
107.51
138.53,173.47
218.31
850.18
301^
0.34
a94
11*7
88,92
18.30
67.61
S0.84
108.00
139.08 174.09
!13.l»
tsasi
S0(.06
te
0.37
4.03
11.63
83.14
38-69
57.96
BLaS
108.48
139.63 174.70
213.ID
268.68
S03.46
iT
0.10
4.13
11.79
23.37
38.88
53.32
81.68
108.97
140.18 176.3!
814.88
861.87
S0U7
se
0.43
4.*!
11.95
23.60
39.17
68.63
8!.lfl
109.46
140.71176,94
2IA0T
158.18
805.09
89
0.46
4.3S
l!.ll
23.8!
39.4;
59.03
82.33
109.95
141,!9 176.66
81S.T4
259,87
J0&90
30
0.»9
4.4!
18.87
24.05
39.76
141.83177.18
81(U4
859.62
»6.78
31
0.62
4.5!
l!.44
84.88
39.76
43^38
110 93
142.40177.80
tlT.lt
HOST
301.51
38
a56
4.62
l!.60
84.51
40.35
60.15
83.81
111.4!
148.96,178.4!
tnAi
861.11
308.36
33
O.fiD
4.7!
1?.77
10.65
60.48
84.83
111.91
143.52179,05
818.60
861.88
309.1S
3i
0.63
4.8!
l!.91
24.93
ta93
60.84
84.6B
112,41
141.08 179.68
219.19
888.64
ilO.00
35
0.67
4 91!
13.10
S5.21
41.85
61.21
85.09
118.90
144.64180.30
819.89
263.39
3ia8!
36
0.71
5.03
[3.8!
25.45
41.55
85.52
113.40
1 15.20, 180.M
880.68
264.13
Ut.65
37
075
5.13
13.44
!5.68
41.85
61.91
113.90
116.77 [81.56
281.27
864^91
318.47
38
0.J9
S.!4
ia6!
!5.92
48.15
68.31
B6,39
114.40' 146.33' 188. 19
221.97
266.67
313.30
3!)
0.S3
3.35
13.7B
26.16
48 15
62.68
86.8!
114.eo| 146.90' 182.82
828.66
2S6.13
314.13
5.45
86.10
48.TO
tia.OS
87.26
I16.40|147.46,183.U
823.36
267.20
311.95
41
0.B!
5.56
14.14
!6.64
13.06
63.4!
87.70
115.90148.03184.09
824.06
!e7.96
313.78
4a
a96
5.B7
14.31
36.88
43.37
63.79
83.13
116.10 148.60.184.7!
!8t76
268.72
316.61
43
1.01
5.79
14.49
87.12
43.68
64.16
88.57
116.91 119.17 18S.3S
825.46
269.19
317.44
41
06
5.90
14.67
27.37
43.99
64.54
69.01
117.41119.74185,99
2!ei.l(,
270-26
3l8a'J
10
1485
!7.Gi
41.30
04.91
J9,16
117.92 lSO,3l!l86.63
886.86
271.03
319.11
46
S.I3
15.03
27.86
14.61
65.89
J9.90
118.43:i5i).88' 187.27
887.37
271.79
319 94
!0
6.24
1*.J1
28.I0|44.92
6S.67
sasi
1 18.94151.46 1B7.9I
828. !7
87!.67
320.78
48
!6
e.36
15.39
66.05
90.79
1 19.15152 OS'l 88.55
888.98
!73 3«
3814!
49
31
a.4S
15.S8
28.S0|4fi.65
66.43
^1.83
M9.M6LS2.eilS9.l9
229-69
!74.li
381 4i
60
1
14.Tfi
66.81
J80.47 153.19 189.83
230:45
274.89
383.29
51
4!
B.7!
15.95
!9.10
46.18
B7.19
98.13
120.98153.77,190.47
23I.U
273.66
3£f.I3
5S
48
6.84
16.14
29.36
46.50
67.58
38.57
I21..'i0 15435 I9I.1!
!31.81
!7«.43
384-97
53
63
6.96
16.3!
!9.6I
16.82
67.96
93.0!
182.01 154.93191.76
238,53
277.81
385.88
54
59
7.09
16 3
29.86
47.14
6^35
93.47
182.63155.51198.41
233.21
277-99
380.68
55
66
7.!1
16.70
30.1!
17.46
68.73
93.93
123,05156.09:193.06
833.93
278.77
327.iO
56
;i
7.34
16.89
30.38
69.12
94,38
183.67 158.68
193.71
234.67
879.53
S88.35
fiT
TT
17.09
30.64
18.11
69.51
B4.83
121.09137.26
194^6
836.38
18033
388.20
66
83
7.59
ir!8
30.89
48.43
69.90
05.29
184,61 1, ;7.85
196.08
836.10
281.11
330.01
59
90
7.78
1J.48
31.15
48.76
70.29
93.73
185.13 15^.43
lftS.67
8S6.B8
I8].89
■i30.t-9
'ais'
/
•* /
aoTJamaoS-
■OM
OM
0.07
0.09 1 aio 1 oTr
OTT^
MS
' ■' /
a.vi o.0i 0.06
ao9
ai8
0.14
an a.ta\ o.«
0.85
0.88
aao
0L33
■: h
0116.0810.10
ai4
ai6
aSl\0.l6\ 0.3u\a.3A\O.W\ll.*\\(v4*
q.49
JLk
€gfo.oa 10.13
0.18
0.24
O.SB 1 0.34 \ 0.40 \ 0.4a \ O-tiO \ «.V. \ n.Ml \ (vw \
r. XXVIII
TABLE XXIX
— Reduction to either Solsriec. 95
PiKT II.
Obl
quUT_
fthe Ecliptic 83- 87
40''.
Arg. F
edut
Diff.^^E
A^.
Rtduc
Diir.
WSr
Arg.
Rcduc
Diff
M^\^.
+
00
~7m
D.08 O.0ft00
lo'
1 IlJl
8,41
11.0661
80
4 46.83
180
0.8606
100
ao«
1 14!lS
t.45
0.0673
10
4 61.63
0.8660
o.no
!0
o-os
1 16.57
8.49
0O695
80
4 56 45
0.8694
so
0.0O
30
0.18
30
1 19.06
8.53
a0717
30
5 1.35
n.873B
1
0.00
40
0.3*
40
8.57
40
5 6.87
i.9r
0.8783
1 so
00
51
0.50
0.88
0.0005
50
I 84.16
8.61
0.0763
M
5 11.83
5 01
0.888S
11 8973"
«
aoo
1 00
Mi
0.0007
t 46.77
8.65
1).0JH7
5 18.83
2 SO
O.O0
0.98
0,890.0009
1(
1 89.48,8 69
O.OBIl
U
5 81.87
0.8919
3
0,00
1.87
D.34 0.O011
80
1 38.118.78
0.0836
80
5 86.34
a8965
3 30
0.00
1.H1
0,38 0.0014
30
1 34.83 8.77
00860
30
5 31.46
5,16
3018
*
0.00
400
1.99
0.48 OOIS
40
1 37.60 8.81
40
5 36.68
S.80
0.3059
4 30
ooi
50,0
8.41
0.46 0.0088
50
1 4011
8.B5
aosii
£0
6 41.88
i,8
3106
S
ooT
8 00
8.M7
o:mWo-86
I 13.86
a0937
!8
5 17.06
i.i>
10
aoi
10
13T
0.5*0.0031
10
1 46.15
8.9S
0.0963
10
6 68.34
0.3808
10
0-01
*00
3.&1
0.57 003fi
sol
1 49.08
8.B6
a0990
80
0.385O
30
0.01
300
4.48
68,0.0011
30
I 58.0113.01
t 55 05 3.05
0.1017
30
6 3.01
s,to
a38n9
400
6 8.41
5.44
0.3349
40
0.01
_^9
a. 76
o!b9 0.0U61
50
1 58 l0|iO9
0.1O78
SI)
6 13.85
6.48
0.33sa
6
0.01
3 0,0
i3~0
8 1.19,3.18
0.1100
i3 "O
6 19.33
11.3419
10
0.01
100
;.19
j!7eo.oouH
10
8 4.31
3.17
aiisn
10
6 81 B4
a3i9a
to
0.01
2o'o
7.97|0.ai 100073
80
8 7.48
3.81
aU57
80
6 30.40
5.6(
assiB
30
0.0*
3o;o
8.TBJ0.efi|0.0O8O
30
8 10.69
3.85
0.11B6
30
6 36.00
5.61
3600
*0
0.0!
400
9.S4
Q.SaO.OOHB
40
8 laBl
3.tB
0.1816
40
6 41.64
6.67
a3«fii
60
aoi
fio|o
10.53
0.940,OO9fl
8 17.88
S.33
0.1846
5i
6 17.31
5.78
a3703
7
aoi"
t 0,0
11.47
D.B7 0.0101
8 80.55
3.3b
a 187b
il
rrs
a3i*5
10
0.02
100
18.14
i.oeo.0113
10
8 8391
3.41
a 1307
10
6 58.TB
5.80
aSBOT
»
O.03
800
13. 4e
1.06 0.0188
80
8 87.38
3.45
a 1338
SO
T 4,56
5.81
0.3IMO
30
0.03
300
11.58
I.O»|0.0138
30
8 30,77
3.48
a 1369
30
7 10.48
5.87
0,391.1
to
0.03
400
15.S1
1.11
0.O148
40
8 31.85
3.53
a 1401
7 16.8S
6,98
a39fl7
SO
o.ot
500
1B.T5
1.18
0.0158
a 37.7M
3-56
a 1433
50
7 88.8!
S.95
a4(l2l
fl
OOi
5 00
17.93
1.81
aoioa
8 41.31
3.61
0.1465
aTlJ
J tB.16
0.407S
10
0.01
100
19.14
1.86
o.oni
10
8 41.95
3.64
0.1498
10
7 34.16
e!o3
a4iso
!0
0.05
iOD
80.40
1.89
0.O185
80
8 48,59
3.69 0-1531
80
7 40.1B
6,08
04185
30
0.O5
30 £1.69
1.3+
aoi 97
SO
8 58.8H
a78 0.1564
J 46.87
)i.ll
a,4M»
40
0.05
400 £3.03
1.37
40
8 56.00
3.77 0.1598
10
1 58.38
.-,1,5
0.4C9B
50
0.08
500
84.40
1.48
0.0881
8 59.77
3.B0
a 1 838
50
7 5ft53
6.80
a4358
9
0.06
tJ 00
85.88
i.4i
0.0831
16
3 3.57
115
U.lb67
86
8 4.7;i
6.83
alios
10
0.07
10
87.87
0.0817
10
3 7 48
3.88
0.1708
8 ia96
6.88
a4465
20
0.07
800
88.7b
1.51
ao8fii
80
3 11.30
3.98
0.1737
80
8 17.84
G.31
aifiis
30
0.O8
30 30.30
1.57
30
3 15.88
3.97
a 1773
30
8 83.55
6.35
a457S
*0
400
31.37
1.68
0,0889
3 19 19
1.0(
aiB09
HI
8 89.90
S.10
a 4637
BO
o.on
50
33.49
1.65
0.0303
3 83.19
a 1815
SO
8 36.30
a4896
Ivus"
7
35.11
0.U31H
mi
3 87.84
4^
0.1888
8 48.73
6147
04754
10
O.10
10
36.83
1.73
0.0333
10
3 31.38
4.18
8 4a80
6.58
ai8i8
80
0.11
38.56
1.76
0.0319
80
3 35.41
4.ir
ai937
80
8 55,78
6.55
4873
30
30
40.34
1.81
0.0366
3 39.60
4.81
a 1991
30
9 8.S7
6.59
0,4938
40
U.1Z
40
43. T 5
1.83
a038l
3 43.81
48.
0.8033
10
9 B.8n
6.63
a4^9i
5i)
0.13
50
44-00
1.S9
a0399
SO
3 4*06
4.88
0.8071
30
fl 15.411
6.S«
05O53
0.11
8
T
45.8a
r9i
0.01l*>
lit
3 58.33
D.8110
9 88.17
0.5 JU
0.1. i
ll>
4T.fl3
1.97
0.O131
10
3 56.66
43ta
D.8149
K
9 88.88
a5175
SO
0.15
EO
n
49.80
8.01
0.0458
to
1 1.08
410
aaiya
80
9 35.63
IT.
a5837
0.5299
30
0.1l>
8.05
1.M
0.8889
3(
9 48-48
6.83
40
1
53.8(1
8.09
1 9.86
1.48
0.88S9
9 19.85
6.87
0.538t
50
O.IS
50
55.95
8.13
0.0508
1 14.31
4 53
4 5b
aasio
.50
9 66.18
-i-91
0.5*«1
0.19
y
¥- -
5H.0M
i.\n
aosa7
1 1B,H7
0.8351
iO
10 :!,o.i
9.95
0-5187
10
0.81
10
1
0.!b
8.81
a0547
10
1 83.43
4.60
0.8393
10 9.98
6.99
0.5.160
80
o.st
80
I
8.47
8.85
0.0567
1 28,03
1.61
a8135
111
10 18-97
f.Ot
0.56tl
30
0.13
30
i
4.78
8.89
a 058 7
1 38.67
1.68
aa477
30
10 ai.nL
7 07
a507B
40
0.24
1
8.33
a 0608
1 37.35
1.78
a8fl80
40
10 31.QS
I.U
laMifi.
50
50
1
S.34;i.37[a0689
il.fl/ \omii
4 48.01
*.lB\<l,?.i&'i
k
\M ■.»..vs}ft.\r!ti.w«».>
13
£ii
10 01
30 <
4 *li.83\ \i.\mv
nvi 'v■^•^*, »■«*>
^
»6
TABLE XXX.
TABLE xxxr. 1
To change mean Solar
nto Sidereal Time.
To cliatiRe Sidereal ini
mean Solar Time. |
as
Add
SalK
Add
!^
'.ftSS-
mn.
KiMnd
[^
s!^^
^
^fSE^
] s'ssls.ie
1
0.164
"
0.003
I
a 3 55.903
,
0.rfil
I
0,()b3
t
7 .53.118
s
O.S29
2
0.006
8
7 Sl.ei6
0.38B
2
aoo5
S
f) 11 49,6R8
3
3
o.ooa
3
II 47.781
3
0.491
3
0008
«
D IS ifi.234
0.658
aoii
4
1) 15 43.6,32
4
0.655
4
011
s
19 *?.THO
S
(1828
s
OOlt
5
1) 19 39,540
5
OSlfl
S
0.014
S3 39.S;i6
0.9Bfi
0.O17
6
It 23 35,448
0.983
0.016
T
*J 35.HBS
7
1.150
0.OI9
7
87 S1.3Sfl
7
1.147
7
0.019
8
HI "igttS
8
1.315
s
0.038
S
31 27.8'!t
8
1.311
8
0.088
9
3,S !9.00t
9
1.479
9
0.085
I) 35 83 172
9
1.471
9
0.025
10
^S S.iflfifl
[0
1.643
10
0.027
10
1) 39 19,090
10
1,636
10
027
II
« 8j lie
-tT 1S.fi7«
TT
12
i.fi07
1.97S
18
0.030
a033
11
18
13 14.988
+7 1089'
12
l.BOi
1.966
18
030
0.032
13
51 l.i,2gH
13
8-136
13
0.036
13
51 6.901
13
2,130
13
0.035
li
55 11.78t
2.300
14
038
55 2.718
14
2.891
0039
\a
5W a^wi
[5
2.464
IS
041
IS
58 59.680
IS
2.IS7
IS
0.011
1 3 *.Hflli
TT
S.629
"ie"
ooTT,
16
TT
a.<i^i
la-
O.0«
n
1 J 1.153
17
S.fl93
17
0.047
17
1 6 50.436
17
2.185
ir
aoi6
IS
1 10 ssom
18
3.057
19
aoso
18
1 10 46.3 tl
18
8.949
19
aoi9
19
1 14 !HMi
10
3.221
19
O.0S3
I 14 42.258
3,113
19
0.058
»
I 18 SLIM
80
3 28fi
80
20
1 IB 3H.lfiO
80
.3.277
80
~ir~
1 *i *J.(*7fi
Dl
3.450
TT
0.059"
81
1 28 ■.l\.OW
21
.1110
*ar
0.057
*2
1 *6 4V23S
S!
3.614
88
0.01)1
88
1 2fi 89.976
88
,3.'in4
22
O060
n
1 30 4n.T»-
!3
3.77(1
83
O.rlfil
23
1 30 25.8^1
81
3,768
83
U.063
n
1 34 37 341
!+
3.913
8*
ome
21
1 34 2I.7(>8
84
3 938
81
0.066
!5
1 39 33.900
!5
4 108
85
0.069
85
1 39 17.700
85
4.0S6
8.1
a 068
e<i
1 42 30 45fi
4.*72
"is"
26
1 48 13.608
86
0071
1 48 g7.oia
27
4.4^6
27
0075
87
1 46 9.516
27
27
071
gB
I so g3.5«a
28
89
077
29
1 SO 5.421
4.587
29
0.070
S9
1 54 *(I.I8t
S9
4.TKt
89
080
1 54 1.338
29
4.751
89
O079
30
1 5H 16.(W0
■to
4,!)2fl
30
O0B8
30
I .57 57 8*1
30
4 111.1
0.098
31
ir
5.0!!^
IT
oims
al
i 1 53 lib
31
S079
0.08 S
3i
i 6 9.79?
32
32
38
8 5 4!l-051i
32
5.212
32
0.0H7
33
* 10 6.348
5.421
33
0O91
33
2 41S64
33
5.406
33
34
« 14 2.B04
St
5.595
34
0.0? 1
2 13 40878
34
5.570
31
0.093
3J
* 17 59.46(
35
5750
35
097
S3
8 17 36.780
35
5734
O0it6
EoLHn.
ItT
36
fld. Hr.
4.899
"3~
0.IW8
9.8585
37
6,078
3T
1
9.829
37
6.062
37
0.101
19.713
38
6.242
38
106
2
19.659
38
6.885
0.104
29.5^9
6.407
39
0-108
3
89.199
S.r>99
39
0.106
39.4S6
10
6.571
40
0.11 1
4
39,319
40
6.553
40
0.109
49.?»2
41
6.735
41
5
19.147
41
6.717
41
59.1 3D
42
6.900
42
0-IIG
6
59977
48
6.891
48
0.115
1 8.995
43
7.004
43
0.119
7
I 8.806
43
7.044
aii7
1 18.65!
7.22H
44
0.138
9
1 19.(i:l6
44
7.808
41
0.120
1 88-708
45
7.393
45
0-125
9
I 88.165
45
7.378
45
0.123
1 3B.fiS5
46
7.557
46
128
10
1 39.295
*li
7.436
46
0.126
1 4B.4S1
47
7.7*8
47
0.131
11
I 18.184,
47
7.6U9
17
0.188
1 5S.STS
48
7.886
49
0-133
18
1 57.954
49
7.864
0.131
i 6.1^
4»
0.136
13
2 7.793
49
B.IJJJ
49
0.131
i 17.991
SO
8.814
SO
139
14
8 17.613
50
9.191
50
0.137
S 87.847
51
8.379
51
0141
15
2 27.442
TT
9.355
% 37.704
S2
a543
0.14t
16
a 37.278
58
a 118
i 47.5(i0
53
9.707
53
0.147
17
S 17.101
9.(193
S3
aivs
! 57.417
9.972
54
ai5o
18
8 56.931
8.646
54
147
3 7.S73
55
9.036
55
0IS2
19
3 6.760
55
9 010
0.150
3 17.130
9.800
56
0.155
80
3 16.590
50"
a 153
3 tB.6S1
57
9.364
57
21
3 26.419
57
9338
57
0,156
3 36.844
5S
9.588
58
0.159
22
3 56.819
59
9,602
49
0.158
Kl
3 46.700
9 698
59
0.1H2
83
3 46078
59
9.666
59
0.161
it
3 M.65e
n.aw
60
aiG4
14
3 55.909
60
9.689
60
0.161
/
T^
^1ein.rbeu>edu
.h<..(h.B»n
iKisht
__
fcttnaum iAiik,Vn
«^«nilTlmc ' 1
TABLE XXXII.
To ooinert Mean Time inlo Parts of the Eqa«tor.
TAB.XXXm.lP7
Lengths of Circular
■nme.
EquBlor.
Time.
Equator.
Time.
Part, of .he
An..
15 t 87.817
SO t 55.694
U T t3.MI
60 9 si.nefl
7.5 18 19.835
15 8.464
45 7 398
1 9HS6
1 15 18..381
8
3
i
15 041
30,088
45.183
1 a 16*
1 15.805
8
3
4
0,01745388
0.03*90659
0.0583JS83
O,06»fll..ll7
0.08786646
90 U ♦7.0SI
lOS IT lt.988
180 19 *8.T75
135 88 10.688
150 H 3S.iS9
1 30 14^78.^
e iftTr.^
8 15 82.177
8 30 84,611
7
B
9
10
1 3.-U246
1 45,887
2 0,388
8 1.4.369
i 30.411
9
10
O.IU*7r97S
0.18817: 05
133(i8631
O.15TO7f)03
0,17tVti93
165 tr H.3I0
ISO 89 3*.I63
195 38 8.010
810 Si 89.857
8(4 3fl 57.703
2 45 27AII.',
3 89..509
3 15 38033
3 30 31.197
3 45 HSMi
13
11
TS
8 45.45*
3 04D3
3 15.534
3 3a57J
3 45.616
50
0,3lW6rB5
0.58:U9K78
11.69813170
a87?6(i463
1.0*719755
Hi) 39 85.550
855 41 53.397
870 44 81.84*
BUS 46 49.091
300 49 16.93H
4 39.4516
4 15 11.H911
4 30 41.354
* 45 4li.ftl8
5 49.888
17
18
ID
80
4 0,657
4 15.608
4 Sa73!)
* 45.780
5 a88]
90
luo
no
1.3!lb'86'l!0
1 57079fi33
1,71,538986
1 919S6818
330 54 18.631
3-15 56 40.478
360 59 8.385
S 15 51.T4S
S 30 .14.810
5 45 56,674
6 59.139
6 16 1.603
81
83
84
85
S M0,903
5 45,94*
6 aS85
6 16,087
180
130
U'l
1,^0
8 09139510
8.^6898803
1.4*3*61195
8.ftl79M8B
8.7 985^680
of Meui
Time.
PWtotihe
Eqouor.
30
31 4.067
6 *« 6.531
T 1 8.995
1 16 1I.4S9
7 31 13.983
86
87
SH
89
30
6 3I,0UB
7 1.150
7 16.191
7 31.233
170
180
810
810
!70
8.!1670597S
3,11159865
3.6KU9I13
4.1887*80
0.1
at
as
0.*
l.M*
300S
4.5li
6.016
3t
33
34
35
6 1 18.851
8 10 8i.3IS
8 31 83.779
8 *a 86.?**
31
38
33
3*
34
7 46.873
8 1.314
8 18.355
8 31.396
H 46,437
8
3
4
0.00058178
00OST866
0.O0I163SS
O.'KIUSllt
OS
D.G
0.1
o.«
0,9
9.08fi
10.689
18.033
13.537
37
38
39
40
9 16 3IJ78
9 31 33,636
9 46 36.100
10 1 38.565
37
3M
39
in
S I.4T8
U 16,519
9 31.560
B 46-601
10 1.643
8
»
10
aillll 71533
0,01)803688
0.00 .'387 11
O.OII8G1799
0-01(890888
aoi
aos
0.03
00*
0.1S0
0.301
0.451
aeot
41
43
44
45
10 16 41.U89
10 31 43,193
10 46 4S95T
11 1 48.481
11 16 50 8H5
48
43
1*
45
10 31.785
10 46.766
11 1.807
11 16.848
40
50
o.o;)a78665
0.0116355:1
ao 1*51111
0.0171.4389
0.0S
O.0S
001
ao9
758
0.903
1.053
1.803
1.35*
46
50
11 31 53,349
H 46 55.813
18 1 sasiT
18 17 0.711
18 38 3.806
47
48
*9
SO
U 46.930
18 1.971
18 17.018
18 38.053
0^)0000185
0,000110970
0.00001939
0.00008184
ftOOI
aoo8
O.0O3
aoot
aois
ao3o
ao45
aofio
51
SI
55
18 47 5,'.70
13 S 8,131
13 17 10..598
13 38 13.068
13 47 15.586
51
58
54
55
13 8.135
13 17.176
13 32.817
13^47.f59
0.0000*909
aO00033S*
0.110003879
0.00(101363
0.(10(1
0.007
O.008
aoo9
0.075
OOflO
&i05
a 180
a 135
56
ST
58
59
60
1* 8 17.990
14 17 80.*5l
1* 38 8i.9l8
1* 47 85.388
15 8 27 84-7
(W
11 a.3m.
11 17.311
14 38..388
\5 1.Vf.\
■ 80 ■
30
10
0.00009696
0.0OJI4VU
o.oiom^ffK
S
gs
TABLE XXXJV.
Annual PrecesBion of a SWr in R, A. in Time.
Argumenl, H- A. of the Siar in Time.
— +
— +
— +
— +
— +
(>>•
IgK
It 13^
gh li),
4" le"
5" 17"
N.
+
+ —
+ —
+ —
+ —
+ —
N.
!■. 1'
P. P
P.P
P.P
+
n
0.000
0.316
+
0,945
+
1.IS7
1.891
Id
O.05fl
fi
0.108
5
aTi8
0.9S5
a
1.185
1.305
•W
Vi
VI)
0.117
a*ST
H
0.766
I.0S4
4
tail
1.318
40
17
MO
0.17*
IH
O.SII
14
0,813
1.060
li
1.835
■*
1.385
SO
S-S
4fl
0.83!
??
O.50S
Ifl
ftBi9
1,095
H
1.356
St
1.331
m
M
fl"
0.889
91
0.61T
a903
1.187
11
1.S74
3
1.335
m
ss
b(J
0.34«
Si
0.6bS
SS
0,945
81
1. 157
13
1.891
4
1.33B
N.
+ —
n.
II
Z3i>
lOh ggli
Si. 2V
8" id"
7" 10"
6" 18^ , 1
s.
+
— +
— +
— +
— +
- + M-
Mulilp
j-thenu
nber Ibund froni Ihe Table, vith iu proper eij^, bf the natoral Ungent of the 1
Slai'6 dec
> wbich add ihe oiuiiant quantity S'.OeS for tAe annual precisaiao, =^aia the |
Sjnopaij.
1
TABLE XXXV.
Argumcm, R. A. of the Star in Time,
+
— +
— +
- +
— +
0"
[2b
Ji' 13"
gl> I4h
3h 15"
4" IS"
Sl> 17''
P.P
P.P
F.P.| ..■
+
rt
0.349
0.6T5
+
0.954
1.I6S
+ ] 1.304
61)
d
10
0.106
0.735
3
1.197
1.318
40
ISf
11
0.4fla
0.774
1.034
1.882
8
40
HIJ
OlTfi
1«
0.510
0,321
ll>
1,071
1.247
8
1.339
30
K4.
ill
0.83*
y«
0.571
0.8Sa
1.106
1.869
»
1.345
80
JUI
.'id
298
91
0.6S3
0.BI8
IT
1.138
1.887
4
1.349
10
3B
<Hi
0.319
33
0,675
2ti 1 0.951
*1
1.168
13
1.304
5
I.3S0
II
lob JSh
y 111!
tiX gO"
7- 19»
—
+
— +
— +
- +
- +
Tlic number from Uie Table = p, and p x mc dec. = Ii.
TABLE XXSVI.
Argument, B. A. of (he Son in Time.
1
4-
— +
- +
0"
la"
P 13*
S" .4^
3^ 15"
4S Iflh
5" nh
P.P. ..
P, P,
p. p
1.196
1.073
a8J6
afli9
0.381
60
1)
Ir)
1.83T
X
1,181
S
1, 045
*
0.B37
5
0.578
a86B
50
a
in
i.at*
i
1,164
7
1015
0.70fi
10
0.583
II
0.815
40
m
1.2i8
K
1.144
III
0.B93
11
0.751
0.474
111
ill
l.??0
M
1.1 83
U
0.919
IH
88
aiori
80
,50
1309
10
1.0911
a913
»?
tlh
0.373
87
0,054
10
I
nil
I.IDfi
13
1.073
SO
U.976
se
(>.iit9
0.381
38
0,000
33"
ion iii-
9b gin 1 B" au"
7^ in-
6l> Itt" 1
+
+ -
+ - 1 + -
+ -
+ - 1
'
Tht nnmber from the Tsblc ;= ;, am] g x ue. dec — r.
TABLE XXXVII. 99
Argumfiit, B. A. of the Sut in Time.
+ —
M.
OK IJIi
!« 131-
g1 11*
3" 15"
1" le"
N.
- +
— +
— +
— +
— +
N.
P.I',
P.P.
P. H,
n
O.fiW
0.631
0.557
0,155
0,398
0.166
irtr
m
0.613
in
V
0.519
0.135
3
0.99T
3
0.139
50
1
0.611
?
101
1
0.5!7
0.1(3
.1
0.979
(i
0.119
in
3
-■91
0.510
0.399
H
0.916
H
O.081
30
a
1(1
o.ea3
iK:i
H
0.193
0369
10
0.880
11
0.056
9(1
e
•■511
0-6S!8
^
(1
Ml
10
13
0.193
0098
10
3
60
0.991
7
557
IS!
0.455
U
0.328
18
0166
17
+ —
+ —
+ —
+ —
+ —
+ —
N.
11" 83"
10" 22"
ph 2ib
gn ao"
S.
— +
— +
— +
— +
— +
— +
&
Tlie nunibu &om the Table ^wee i, nod » x tang, dec. = d.
TABLE XXXVIU.
Annual PrccesBion of n Star in N. P. D.
ArgumcDt, R. A. af the Sue in Time.
S.
— +
— +
— +
— +
s.
0" 13"
1" las
41. IfiH
5" ill-
N.
+ —
+ —
+ —
+ —
+ —
N.
P. P
P. F
P. P
90.011
ifl.aei
17.3J9
11.173
10.099
5.188
60
13
III
90.03S
.■),'!
1
ll'i
,W
16.901
13.5*8,
HI
B.855
8T
50
ST
m
19.9fi8
71
835
110
16.419
110
19.881
lfi>i
B.4T1
174
3.181
m
40
m
19.872
inrj
1:
AIM
ir.i
15.909
ii|0
la.903
VM
1.670
?fil
3.SI6
10
53
in
19.739
Ul
lea
?19
15.354
fHO
11.497
3?1
6.855
31K
CG
m
19.569
ITK
.779
971
11.778
350
10.7fi9
405
6.09T
08T1
10
80
liO
2[S
1
.3iB
339
11.173
190
1(1.019
1^0
5.188
5^9
oono
N.
N.
1,1. aah
10* n^
Bb a in
8" 90i>
jU 191.
ft" 18"
S.
+ —
+ —
+ —
+ —
+ —
S.
The number from d.e Table = n'.
TABLE XXXIS.
Aberration in N. P. D. to fiad p.
■^-
-..
Argument, B. A. of the Stat in Time.
121.
3» ISl-
if Iti"
51' IT"
'f.V
— +
— +
— +
— +
— +
— +
P. P
P. 1'
P. P
P. 1'
30.S3S
19.5f!5
11.338
io.lsa
5.813
(id
15
11)
S{).a36
31
19.318
hi
17,083
7(1
13.681
Hi
B.353
HM
4.381
50
94.
ao.l7H
«H
19.033
KIH
16.398
IKI
13.01B
161
B.5H0
176
3.517
10
3G
3rt
ao.o8«
lOK
18.713
Vi*
16.069
*U\
18.330
HH
7.751
?fil
'^n
VI
19.917
i3n
18.357
S(lti
15.516
9m
11.618
3?M
6.988
31?
60
,'.!)
i9-T7S
170
IT.Sfle
970
11.931
350
10.881
iin
6.091
WD
0.883
in
72
(!0
ip.sim
tai
17.511
391
11.39?
10.198
199
5,913
6-iH
o.mn
+ —
+ —
+ —
llh jsli
J oh sgi.
a<' sih
H" 90'>
7" IS"
6" 18"
Ml
Iiiply the number found in tbe Table by the natural jine of the SUi'i decllnataii -, Mi^-i.5iEKl.i.
will give 6'. '
1
100 TABLE XL.
Abcnstion ID N. F. D. to And 7'.
Ai^iuncnt, R. A. of the SUr in Time.
tf> 18"
8" 14"
3» 15" 1
+ - 1
+ -
+ -
* - 1
+ -
+ -
P P
'.p.
P.P.
m.
0,(100
4.aoB
+
9.800
+
13.138
+
16.090
+
17,947
HI
O.HIO
!■••
S.5S9
1(4
9.99*
Ml
13.700
30
16.4eO
\a
19.140
50
l.'^l
B.3i5
IVH
10.657
NX)
14-833
16.839
81)
ia89B
40
ff.'i
7.110
iifi-
11.311
150
14-740
17.166
30
la.480
30
■ion
7.85S
f.W
11.943
for)
15.880
180
17.460
17 fi
■iw
18.558
■f-W
IS 660
l,W1
17.780
,w
1&««
480
4WI)9
4S0
9.800
3H4
13.138
300
16.090
IHO
IT. 347
60
18.580
1)
11" 23" 1
10" !g"
S" 21"
6" 80" 1
7" 19»'
6" 19"
Tbe number liom this Table, nrntdpUed by the DMunl tin< of the Stu'i decIiaUioD, giva ■ pro-
duct, to which r' being added, ilw remit will be c'.
TABLE XLI.
Argumenl, Deelinntim of the Sl«i.
Dee. North — Soolh +
to.
D.
No. 1 I..
No.
11.
No.
D.
No
D.
No.
i-
No.
I).
No.
flfifi
in
t".4^
?n
7.fi7S
fi.9B.S
40
,179
■iO
1.1 B
4.033
70
8.759
90
.401
i
im/i
li
■i
11
"1
T.530
1
i.tliH
111
.(IM7
51
,S07
61
3.910
8.686
HI
.as»
it
oni
I?
1
^1
9?
7.479
■?
IHV)
48
1'l+
4%
fi8
3.797
■?
8.493
Hi(
.m
ft
mn
n
ft
7.485
;i
4'1
1(111
,W
fiS
XG68
1
a.359
m
•mo
4
.046
»\
7.369
.803
,54
3536
8.823
M4
.»!4S
u
f.R
5.703
M
4fiy
«.■.
3.409
8.088
Hi
.7<I3
M
ia
7.850
w
K.,Wh
4«
5,«II3
.W
4.51
MK
3.881
■6
1.B51
Wi
L,«ll
T
(KIA
17
Tl
il
7.1S7
!7
H,44V
47
iWll
57
4 SI
67
J.158
■7
1.814
MT
t*
.!IHJ
IH
m
VH
7.188
(l.MM
M
.,;m
5fl
4i'7
3.088
'H
1.677
MM
l.tm
9
.SSI
IU
/
at
i9
J.055
39
G.868
49
5.898
50
4.15
69
8.891
79
I.S39
99
0,141
The number froni this Table i> r'.
TABLE XLII.
Luniir Nutation in B. A. to find *' = d'.
Argument, R. A. of the Star in Time.
— +
— +
— +
— +
— +
S.
1" 13^
8" 14*
31 15"
4" 16"
6" IT"
N.
+ —
+ —
+ —
+ —
+ —
N.
e.i
P. P
P. P
■+
41
0,000
e.49T
+
4.984
+
6.983
+
B.35S
9.319
m
0.481
V
8.901
3<
S.18S
a
7.113
n
fl,.5fif
4
0.419
.11)
»
an
0.S41
3.30C
(if
5.534
58
7.391
:u
B.T41
f)
9.501
IM
Ho
ISf
3.603
101
5.B74
71
7.655
51
S.9H
14
111
ia
[«(
4.0:7
VM
6.808
10'
7.D03
fif
9.06«
*.038
i-IK
4.455
ni
6.5IB
ru
8.137
(iU
8.4!JT
i4U
4.884
804
6.883
156
9.355
108
9.319
87
9,646
N.
+ —
11" 83"
101. agh
9* 81"
B" 80"
7" 19"
— +
- +
— +
— +■
— +
— +
s.
TABLES
101
FOB COKPUTINO THE NOTATION OF A STAB Ih
lUOHT ASCENSION
AND DECLINATION.
TABLE XLIU.
TABLE XLIV.
TABLE XLV.
OH
oa
EQU AXIOM OF EQDIHOXEE
■r«LE..0FKnTAT.OH.
TABLE n. OF KCTATIOM.
.HBIOHTABCB«B10».
ABaOMEST.
ASGCHSNT.
For the NuBlion in R. A.
For the NulBtioti in R. A.
AmOUMEMT.
R. A. S««— Lon. Moon'. Node
tlA. Stu— Lon. Moon'a Node.
Porlb« NuUtion io Dedin.
Far the Nutation in Deciin.
[LA. Sou + 3 tigat — Lon.
H, A. Slai + 3 signs + Lon.
Node.
Moon's Node.
Moon's Node.
s. S.
s. a.
S. B.
a. s.
5. ^
VI
VII
IVIll
VI
VII
IVIIl
} Vj
vn
IVIIl
— +
— +
— +
— +
— +
^ +
30
— +
-~. +
— . +
8.77
7.60
4.39
so
o'
1,29
1.11
0.64
0,00
8.62
11.93
30
1
ft77
7.68
4.35
59
1
1,88
l.IO
0.62
39
0,30
8.88
15.08
29
2
8.77
7.44
4.18
28
2
1.88
I.OM
0.60
28
2
0.6
9.14
15.83
88
3
are
7.36
3.98
3
1.28
1.08
0.58
87
3
0.90
9-39
15.36
27
4
aia
7.27
3.84
4
1.88
1.07
0.56
26
4
1.80
9.64
15.50
88
fi
8.74
l.\%
3.71
5
1.28
1.06
0.54
as
1.50
S.B9
15.63
25
6
8.72
7.10
3.57
1.28
1.04
0.52
u
1.80
10,14
15.75
84
T
8.71
7,00
3,43
7
1.88
1.03
0.50
23
7
8.10
10.39
15.87
23
8
8.G9
Cfll
3.89
8
1.87
1.01
0.4B
28
B
8.40
ia6i
15.99
2!
8.66
6.9S
3.14
S
1.87
1.00
0.40
21
_fl^
8.70
10,85
16.10
81
It)
M.G4
G.72
3.00
10
a9H
~0.U
80
8.U9
TLcS
16.80
80
9.61
6.62
19
11
1.86
0.97
0.42
Ifi
11
3.29
11.31
16,30
19
It
8.58
«.6*
2.71
12
1.26
0.95
0.40
IS
3.59
11.54
16.40
18
L-i
9-S5
6.41
2.56
13
1.85
O.Sli
0.38 : JT
13
3bB
11.76
16.49
17
14
8.fil
6.31
2.42
11
1.26
0.92
0.35
H
II. 98
16.58
16
IS
8.47
6.21
2.27
16
1.24
0,91
0.33
15
4.46
l*.!!!
16.66
15
16
B.4S
6.09
2.12
16
1.84
0.89
0.31
4.75
12.40
16.73
14
9.39
5.98
1.97
17
1.83
0.96
0.89
U
18,61
16.88
13
18
B.34
5.87
1-88
IS
1.28
0,86| 0,87
18
&.Si
18.81
16.87
18
1»
fl.2fl
5.76
1.67
19
1.88
0.8 1! 0.26
19
S.61
13,01
I6.a3
11
80
1.58
1.81
0.8*
80
5,90
16.98
10
81
8.19
5.52
1.37
$1
1.80
0.81
0.80
81
6.18
13.40
17.03
g
22
8.13
5.39
1.88
88
I.IB
079
0.18
2S
6.46
13.59
17.0S
8
83
8,07
5.88
1.07
83
1.16
0.77
ai6
83
6.74
13.77
17.12
7
84
8.01
6.16
0.98
24.
U7
0,76
0.13
84
7.01
13.95
n.ii
6
7.94
5.03
0.75
!5
0,74
ail
25
7.29
14.13
1J.18
5
5fi
7.88
4.90
0.61
US
0.78
0,09
26
7.56
14.30
17.20
4
87
7.81
4.78
0.46
87
1.14
0O7
27
7.83
11-46
17.22
4
2B
7.74
4.6.
0.31
28
1.13
0.68
0.04
88
8.10
14.68
17.23
2
29
7.67
4.52
0.15
89
1.12
0.66
0.02
5S
8.3B
14.78
17 2*
30
7.60
4.39
0.00
30
1.11
0.61
o.on
30
8.62
U.93^
17.24
1-
— +
— +
— +
^^T
+ —
+ —
+ —
S. S
S-. S
s. s.
S. 8
S. F
a. &
8. S
8. S.
S. S.
XIV
XIV
IX 11
XIV
XIV
IX 11
XI V
X IV
IXIII
TaJiTid the NulBlian of a Star tij Righ
Ascc«i.imi.
from Tuble£XLlII.XI.lV.
answering to their proper Drgumenu, odd the Iob. taog
nt of Ihe Siar'a declination )
Ihe Slim Kill be the \og. of pait (irst of nutation, end if Ih
e declimtion is eoulli, changa
the Bign— to which opply the equation from Table XLV
.an»irerinB "> lli« iongilude
e nutstion in riglit ascension.
To find Ihe NtOation of a Star in Ds
Utiatioa,
Incieaae the arguments of Tahlei XLlll. and XLIV.
eacb bi Ae *vtte m^^o*, i-a*. '
i\\ \« vtie nMVi'CvQtv vi^ it-tW^i-
iie(.Qi\4 ct^\iaVv(«\-
TABLB XL VI.
R.A. Star — Lon. Sun,
; 19-39 It
. 19.3711
. 19.34 1^
■ J9.!) \l
I \9AEU
H13.9; i
■el3.T3 t
iT13.« i
.7l3.g4 i
5 10.89 t
8 10.00 (
i fl.Tl (
VXIV
IX III
TABLE XLVII.
For the Aberration in R.A
H.A. Star+8un"a Lon,
Par the Aberration in Dec
R.Asccn. Sur+Snigns+Siu
Dec].
k S. S.
i 0.10
1 0,19
i 0.17
j 0-16
XI VX
TABLE XLVII L
Abgument.
r psrt id of Aber, in peel.
Sun's Lon-+S[Dr'> DmU
or part 3d Aber. in Deci.
Sun's Lon.+Star'6 Decl.
VIU
Jih.
Tafind Itit Aierratiou if a Star in Right Airention.
Vo ihe log. nf the sum or difference of the equations from Tables XLVI. XLVII.
wering to their itrguoieota, add the log. secant of the Star's declinalion, the sun
be the log. of the aberration In right Bseension.
Tojiud the Aberration qfa Slur in DeclinaSon.
,nd the sum or difference of the equations answering lo Ihe former argumenla, in
rsed by 111 sifina, to the log. of which, add the log- sine of the Star's declinalion
>um will be the log of part Ist of the abetralion. Take parts second and third o
aiion from Table XLVIII. which, applied W the tormet, *i\\©>iev"n? a^Knadoi
'Inmioa. If ifte Star's dedinntioo is south, clmnse ftio sign at ^MV^ t& Mife'^^\.\
^ IS-
TABLE LIL
TABLE XLIX.
Lun. Nut. uf ibi Equinoxes
in B.A.
««■■ ubiu^itj ortb.
EcHpUf.
V«r|" - '•
TABLE Lin.
!Zl
— +
— +
1-
s
lB00l83 ST fiS.84
1810.83 27 5i.B3
IBSO'SS ST 47.41
0.00
0.55
0.97
lo
iol Nut. o( (he KquinoE
isinR
A.
18*);S3 ST t3.0<
8
0.04
0.S9
0.98
88
Sun's
1 V
I VI
Sun
4
0.08
0.68
1,00
86
0.U13
Long.
IX 11
+ +
X IV
+ +
L^g
6
8
10
12
0.11
IJ.1S
U,19
0.83
0.69
0.71
0.7S
1.01
1.02
1,04
1.05
28
80
18
MunlhljUiolnutlDO.
t.-. 1
Feb. I
O.Oi
0.00
0.06
30
14
0.26
0.78
16
March 1
0.07
5
008
0.07
85
16
0.30
D.81
L07
14
ipril 1
0.11
10
0.03
0-07
20
18
0.34
0.83
1.08
18
May 1
o.u
IS
oot
0.07
IS
80
0.37
0.86
1.09
10
lun> 1
CIS
SO
005
0.07
10
ss
1.09
rui; 1
0.82
25
o.oe
0.0g
24
[1.44
D.eo
1.10
6
August 1
Sepl. 1
0.S.5
0.29
30
0.0ti
0.06
86
28
a. 48
0.92
0.95
1 10
1.10
i
2
3«. 1
0.33
IIVII
30
d's6
0.97
l.IO
Nov. 1
0.3S
XI \
xTv
ivir
n:
Dec. 1
0.40
+ +
+ —
+ —
+ —
TvlBLE L.
TABLE LI.
SoIbt Equation of the Obliquity.
Long-
Kqualioti
Dnyjof the31ontVi.
[=9"
Lunsr i-quBtimi uf ttic
Obliciuiry.
640 cmin I*ng. Moon's Node.]
■ . •
„
Lon«.
VI
I VII
IIVIII
Long.
VI
+0.43
Mar. 21
Sept. 23
Mnon'e
DIooa'B
s
10
0.43
0.41
26
31
88
Oct. 3
Node.
Node.
15
0.37
April a
9
9.65
8.34
482
30
80
0.33
10
14
0.64
8.25
4 67
80
S5
0.29
li
19
9.63
8.16
4.53
23
1 VII
0.22
20
21
9.68
8.08
4 37
ST
O.lfi
25
29
i
9.61
7.94
428
86
10
+0.08
May 1
Nqv. 3
s
9.eo
T.89
25
15
0.00
6
8
6
9.58
7.79
3.92
84
to
— 0.0rt
n
13
7
9.56
7.69
3T6
83
S5
o.ga
IH
18
9-54
fl.52
7.S9
7.49
3.61
3.45
82
81
11 Vlll
o.aa
21
8S
87
2J
10
B.60
1.38
339
20
10
0.3T
lune I
Dec. 2
U
9.46
7.87
3.11
19
IS
0.11
6
7
12
9.42
7.16
2.98
18
20
0.43
18
13
9.39
7.01
2.88
17
S5
0.43
16
17
14
9.35
6.93
2.66
18
III IX
0.43
is
S8
9.31
6.S1
8.19
15
0.41
27
27
le
9.26
6.69
2.33
11
10
l.^>
0.37
0.33
luly 2
Jan. 1
G
IT
18
9.21
9.16
6.57
6.15
2.17
8.00
13
18
so
0.S9
13
g.ii
638
1.81
i5
0.22
18
Ifi
20
9.06
8.99
8.9S
6.19
151
1.34
10
IV X
5
0.15
—11.08
23
20
21
g2
e.06
5.93
9
8
10
0.00
Aug. 3
30
23
8.87
5.80
1.17
7
15
+0.08
Feb. 4
21
8.80
5.66
I.Ol
6
to
O.IS
13
8S
8.73
5.53
0.81
5
S5
0.82
Ifi
14
2«
8.65
5-38
0.ri7
4
V XI
0.S9
23
19
87
8.57
5.25
0,50
3
5
0.3!)
SS
24
88
8.50
5.11
0.33
2
10
0.38
13
Vlar. 1
29
8.43
4.96
0.17
I
20
0.1.1
0.43
8
U
30
8.3*
4.88
0.00
WwW WA'S.wA
2.5
0.43
18
16
1'/ r> XII 1 o.*3
83
SI
m4 TABLt; LIU. Kight ABcensious uiid ilecliuiHiOD. of Stnrs tor lli2«. J
Cha-
Pr. Name.
Mug.
Declination. *y"^'
Conatellftlion*
Ascensiun-
Var.
Orac Minotis
Pole Star
59 a.6
-i- 15.00
flB 33 89 N -f- 19.45
etidnni
Achertiat
1 31 18.0
-1- a-ai
58 19 3 s — ia.6a
AniBTis
■y a3(i
32 38 42 N -t- 17.40
T.otti
4 aa 3-7
+ :ii3
16 9 31 N ■(- 7.93
Auriga:
Capella
* 4 0.0
+ 4.41
15 48 45 n'-I- 4.Sd
dri-nb
aigel
Bellaliix
"T"
5 (i lli.l]
+ 8.8M
8 84 ae S— 4.7 J
OrioniB
5 15 51.S
+ 3.28
6 U 11 JJ
+ 4.01
Orionis
Ueteigaese
5 45 51-9
+ 3.35
7 a? 2N
+ 1,36
Navis
CanqpuB
6 20 &3
-t- 1.33
58 36 18 S
■f 1,88
Car« MajoriB
Siriua
6 3T 31.1
+ 3.64
16 39 13 B
+ 4.41
T 83 36.9
+ 3.«S
38 15 26 N
— 7.18
Canis Minoris
T So 18.0
+ 3.17
5 39 33 N
— 9.83
GlMIMOiWK
Pollux
7 3V 47-0
+ 3.69
88 36 3 .\
_ 8.08
Navii
U 11 30.0
+ a75
69 42 S
+ 14.85
Lbonh
BeRUlus
9 69 12-4
+ 181
18 48 18 S
—17.23
Uri» Majorta
Dub he
53 a-5
4- 183
98 40 40 M
—19.36
Cruel.
2 IT 7.7
+ 185
83 7 36 S
+ 20.02
VlUGIHIt
Spi™
3 16 as
+ au
10 15 35 S
-1- 18.94
Centauri
3 51 46.9
+ 4.13
t9 38 11 S
■t- 17.88
Drnconis
3 59 44.8
+ 1,81.
85 la 2 N
_1T-4S
BlIDlil
* 7 49-3
+ 3,73
30 4 50 N
—18.97
Ciouuri
14 28 36,6
+ 4.45
60 8 8 S
+ 10.18
i Libra!
ZubcneEch
* 41 23.9
+ 3.30
15 19 13 S
■(-15.35
S 18 sa.6
+ 3.86
26 2 87 S
-I- 9.59
DracociiB
Ra
.ab,n
T Sa 37.1
+ 1.3>*
51 30 4C N
— 0.67
ra
8 31 7.8
+ 2.03
38 37 41 N
■t- 3.02
A«DIL«
Allflir
9 48 33.«
■^ 8.93
8 85 15 N
+ 9.06
Aquaril
*: 56 57-0
+ 109
I 9 fi S
-17.27
Gruis
al SG 57.9
+ 185
47 48 8 S
— 17.15
Pia. Auaik.
Fomalbaut
aa 49 7.7
+ 3.31
30 31 84 S
_1B-H6
PEd.il
Marcah
33 5(i ia.i
-1- a. 98
14 1(1 53 N
+ 19,33
■ I-AKLISLIV— U«c.n,alNun,
enforeuc IJny in the Year. 1
I
Months.
.TST "
brcti
Aprtt M£J^
June. 1 July.
aTIF
-^^
0.747
Nov. 1 D«.
0.833 0,914
va.ow
U.085
11)3
0.3Hi,0.3!9
oliTilawfi
0.58
0.686
ao.003
0.088
16+
0.849
0.331
0.41
0.58
0.668
0.750
0.835
0.917
J l}.00li
0.091
.IfiT
o-asa
0.334
0-41
0.58
6.671
0.753
0,833
0,930
4 0.'K)S
0.093
• ITO
o.as5
0.337
0.43
0.504
0.58
0.673
0,755
0.840
0.928
5 0.011
0.09fi
,173
0.a58
0.340
0-43
0.5O7
0.59
0,675
OTUTH
a7.5e
0.76U
a843
0.845
agas
0,988
tj u.ai4
•0.099
.175
o.a60
0-318
or
11.59
7 O.OtJ
U.10Z
.IIB
o.as3
0.345
0.43<
0.518
0.59
0,681
0,763
asiB
0.931
g 0.')19
0.104
.161
U.a66
0.31M
0.43,
0.515
O.(i0
0.684!O.T66
0.S51
0,933
» 0.022
Q.107
0.269
assi
0.43
0-518
0.60
a6M70.761
0,t(,M
0,938
.' .
9-035
0,1 OB
).1»6
o.an
o.n5:
0.43
t.fiO
0,689]a77a
0.HS6
0,9:!9
MH
t,lB9
aa7i
O.OU
0.6y2|a775
9L1U
(.19S
0.877
0.359
0-44
0-526
0.61
a695
a777
0.862 0,914
B.in
).1S5
0.380
0.368
0.44
0.58!
a698
a7?o
0.8650.947
d;i80
).19I
0.!Ba
0.;i64
0-4.1)
0.531
0.61
0.701
0.78S
U.867|0.950
3 0.039
0.183
(.300
0.385
a3fi7
0-45
0.S3-1
0,61
0.703
0.7RS
n.sTO
0.953
;
6 U.OtI
STfi
).g03
0.888
0.370
oM
0537
o.ua
|».70(.
O^TflW
loTS
0.955
0.1 i9
.306
o.gai
0.;iT3
0.45
0.64O
0.6a
0.JO9
0.791
0.958
H0.O16
0.131
.308
0.39:i
0-373
a.m
t).S4a
0.6a
0.711
0-793
0.878
g.981
il B.(11B
0.13 1
».811
0.896
o.3rtt
0-46
a54i
0.T14
0.796
[1.863
0.9U
(t J.OSg
0.13T
.iu
0.899
0.3S
(i-4e
0.54P
0.63.
0.7IT
0-7 9S
0-884
(1.SS6
"10.056
OillU
,81;
0.303
0.38;
(1-46
0551
0,63
[).78U
0.802
1887
iJM
a o.oa;
0.1(8
-aifl
0.304
0.3Sfi
0-17
0-722
0.804
0.89O
9.6 tl
3 rj.OSU
0.1tj
.332
0.307
0.3S9
0.47
0.i5'
0,6-1-
0.73S
0.1107
a8W4
fl.97*
;
tO.OG3
0,U8
.as5
0.309
o.wa
0.17
0,559
0,6 H
0.728
0.810
0.895
0.971
S ).09(
0.i51
.aa
u.3ia
0.395
11-47
0.562
0.64
0.73!
0.BI3
0.8B8
a980
; i
Wy.tJ(fts
0.315
0.:ia7
ij:^
0.664
oef
0.7^
0.815
:i:9ob
ol^
/ gjja-onlo.isshi'ia
0.31 S
0.4O0
0.485\o.5fn
0.65
0.736
0.818
0.903
0.985
1 ssao74kf.is»h.tm
o^ao
0.40
3.4«l\o.510\ft.655.Vj.l3a|iai.\^.Wit^.ftf»
29 0.O77p.l6m.23S
0.333
0.40
Mo.ojs! /a84I
0.38fi
0.40
olSol?£33*W^^^^
'•"la
'^'iSI
/"
241
0.41 1| \0.6l!*V>.
fi^\
\0.
^^\
-
^
^\_
■■ -^
■ ■ ' \.fa
Sub's lIiKhl Ascension lor ever
Ui, in the \
'^nm.
— in^
[lBy».
JmuEiry.
February.
' Alan:li.
■ ApriL
' Majf.
J
\% M S
20 55 36
82 49 40
43 11
2 34 81
4 37 12
2
IR i8 SO
21 41
28 53 81
46 49
2 38 16
4 41 19
3
IS 5g Sd
81 4 Vi
88 57 T
50 27
8 42 6
4 15 24
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