log tan (is = 80 45') = 10.788185
+ log tan [(is-ia) = 31 45'] = 9.791563
+ log tan [tt*~J&) = 25 45'] = 9.683356
+ log tan [(is-ic) = 23 15'] = 9.633098
Rejecting 10 = 9.948101 = log tan *K . /. K =166 20' 20*
* The + is always taken ; otherwise, K being > 90, K would be > 360 which is impos-
sible. (PART HI., 256).
110 SPHRIIICAL TUIGONOMETIIY.
Whence area = 166 c f n ' . & (4000) 2 = - . i* (4000) 2 = 46,450,440,
2. Given a = 70 U' 20", b = 49 24' 10", and c = 38 46' 10", to
find the area of a spherical triangle on a sphere whose diameter is 8
feet. Ans. 5.1, nearly.
. Prob. Having two sides and their included angle given
in a spherical triangle, to find the area.
SOLUTION. Compute the other two angles by Napier's Analogies, and find
the area from the angles. [The formula cot ^K = - r - - gives
the spherical excess in terms of two sides and their included angle ; but it is of
no practical value for finding the area, as it is not adapted to logarithmic compu-
tation. For the manner of producing it and several other forms for K, see
Todhunter's Spherical Trigonometry, (103)].
[NOTE. The three following problems are given merely to indicate to the
student some departments of investigation in which Spherical Trigonometry is
of essential service. The two sciences to which this branch of Pure Mathe-
matics is indispensable, are Geodesy, or the mathematical measurement of the
earth, and Astronomy.]
1. To find the shortest distance on the earth's surface be-
tween two points whose latitudes and longi-
tudes are known.
SUG'S. The shortest distance on the surface be-
tween two points is the arc of a great circle joining
the points. Hence, the Problem is : Given two sides
(the co-latitudes) and the included angle (the differ-
ence in longitude), to find the third side.
FIG. 7i. Ex. 1. Berlin is situated in Lat. 52 31'
13" N., Lon. 13 23' 52" E., and Alexandria, Egypt, in Lat. 31 13'
N"., Lon. 29 55' E. What is the shortest distance in miles on the
earth's surface between them, the earth being considered a sphere
whose radius is 3962 miles ?
Ans. 1691.96 miles.
Ex. 2. A ship starts from Valparaiso, Chili, Lat.33 02' S., Lon. 71
43' W., and sails on the arc of a great circle in a northwesterly direc-
tion 3840 miles, when her longitude is found to be 120 W. What
is her latitude ? Am. 51' W" S.
Ex. 3. A ship starts from Rio Janeiro, Brazil, Lat. 22 54' S., Lon.
42 45' W., and sails in a northeasterly direction on the arc of a great
circle 5624.4 miles, when her latitude is found to be 50 N. What is
her longitude ? Ans. 2 01' 18" W.
I?r6b. 2. To find the time of day from the altitude of the sun.
SUG'S. Let NESQ represent the projection of the concave of the heavens
upon the plane of the meridian of observation.
The equator of this concave sphere is simply the
intersection of the plane of the earth's equator with
this imaginary concave sphere, and its axis is the
prolongation of the earth's axis, the poles being
the points N and S where the axis pierces the
imaginary concave. EQ is the projection of this
celestial equator, NS the axis or the projection of
a great circle perpendicular to the meridian of the
observer (NESQ) and to the equator, HO the projec-
tion of the horizon, and ZZ' the projection of the
prime vertical (that is, a great circle of the heavens
passing through the zenith of the observer and the east and west points in his
Now let S' be the place of the sun at the time of observation. RS', the sun's
declination, is known from the almanac ; LS', the sun's altitude, is measured
with the sextant (or other instrument) ; and EZ is the latitude of the observer.
Hence, in the spherical triangle ZNS' we know the three sides, viz., NS' = the
co-declination of the sun, ZS' = the co-altitude of the sun, and ZN the co-
latitude of the observer. We may therefore compute the angle ZNS', which
reduced to time gives the time before or after noon as the case may be.
Ex. 1. On April 21st the master of a ship at sea in latitude 39
06' 20" N., observed the altitude of the sun's centre at a certain time
in the forenoon and found it to be 30 10' 40", and looking in the
almanac found the sun's decimation at that time to be 12 03' 10"
N. What was the time of day ?
90 - 30 10' 40" =
90 - 12 03' 10" =
90 39 06' 20" =
59 49' 20"
77 56' 50"
50 53' 40"
2)188 39' 50"
94 19' 55"
a. c. log sin 94 19' 55" = 0.001243
a. c. log sin 34 30' 35" = 0.24676~>
log sin 16 23' 05" = 9.450:381
log sin 43 26' 15" = 9 8:1 7:] 12
log tan 30 22' 03".3 = 9.767850
112 SPHK1UCAL TRIGONOMETRY.
Therefore | the hour angle NNS' = 30 22' 08" .8, and the hour angle is 60 d
44' 07". This reduced to time at 4 minutes to a degree, gives 4 h. 2 m. 56 s. be-
fore noon, or 7 h. 57 m. 4 s. A. M.
Ex. 2. In latitude 40 21' N., when the declination of the sun is
3 20' S., and its altitude 36 12', what is the time of day ?
Ans. 9 h. 42 m. 40 s. A. M.
Ex. 3. In latitude 21 02' S., when the sun's declination was 18
32' N., and the altitude in the afternoon 40 08', what was the time
of day ? Ans. 2 h. 3m. 57" P. M.
Prob. 3. To find the time of sunrising and sunsetting at any
given place on a given day.
SUG'S. The projection being the same as before, let M'RS'M represent the ap*
parent diurnal path of the sun. Since S'M is described in 6 hours, the time taken
to describe RS' is the time before 6 o'clock, at
which the sun rises, i. e. t passes the horizon HO.
But the time requisite to describe RS', is the same
part of 24 hours (360 angular measure) that the
angle CNL (= arc CL) is of 360V Hence, the arc
CL, in time, is the time before 6 o'clock at which
the sun rises. In a similar manner, C/, in time,
is seen to be the time after 6 o'clock when the sun
is south of the equator. The solution of the prob-
lem, therefore, consists in finding CL. Now, in
the triangle RLC, right angled at L, LR the
sun's declination at the time, and angle RCL = ECH = the co-latitude of the
place.* From these data CL is readily found.
Ex.l. Kequired the time of sunrise at latitude 42 33' N., when
the sun's declination is 13 28' N.
cot 47 27' = 9.962813
tan 13 28' = 9.379239
sin 12 41' 52" = 9.342052
(12 41' 52") x 4 gives the time before 6 o'clock as 50' 47". .'. The sun rises at
5 h. 09 m. 13 s.
* This may be seen thus : Suppose a person to start from the equator at Hand travel
north. When he is at , the south point of his horizon (H) is at S ! and for every degree lie
goes north, the south pole (S) sinks a degree below his horizon. Hence, HCS = his latitude,
and ECH = co-latitude.
Ex. 2. Required the time of sunrise at latitude 57 02' 54" K,
when the sun's declination is 23 28' N.
Sun rises at 3 h. 11 m. 49 s.
Ex. 3. How long is the sun above the horizon in latitude 58 12'
., when the sun's declination is 18 41' S., that is about January
Both ? Ans. 7 h. 35 m. 36 s.
Ex. 4. What is the length of the longest day at Ann Arbor, Mich.,
Lai 42 16' 48".3, the sun's greatest declination being 23 27' ?
Ans. 15 h. 05 m. 50 s.
[NOTE. In such problems as the foregoing, several small corrections have to
be made in order to entire accuracy, such, for example, as that for refraction in
taking the altitude, and for the time required for the sun's disk to pass the hori-
zon. But they would be out of place here.]
TABLE OF LOGARITHMS.
[NOTE. If the student understands the nature and use of logarithms so as to
be able to use the common tables, it will not be necessary that he should read
this introduction. Otherwise a careful study of it will be needful before reading
Section 4 of the Plane Trigonometry.]
1. A Logarithm is the exponent by which a fixed number ia
to be affected in order to produce any required number. The fixed
number is called the Base of the System.
ILL. Let the Base be 3 : then tho logarithm of 9 is 2 ; of 27, 3 ; of 81, 4 ; of
19623, 9 ; for 3 a = 9 ; 3 s = 27 ; 3 4 = 81 ; and 3' = 19683. Again, if 64 is the
base, the logarithm of 8 is |, or .5, since 64 , or 64'* = 8 ; i. ., |, or .5 is the
exponent by which 64, the base, is to be affected in order to produce the num-
ber 8. So also, 64 being the base, $, or .333 + , is the logarithm of 4, since 64 , or
54.333 + _ 4. ^ ^ or 333 +> jg tlie exponent by which 64, the base, is to be
affected in order to produce the number 4. Once more, since 64 , or 64 666 + =
16, |, or .666 +,is the logarithm of 16, if the base is 64. Finally, 64 ' or
64 6 = |, or .125 ; hence , or .5, is the logarithm of , or .125, when the
base is 64. In like manner, with the same base, , or .333 +, is the loga-
rithm of i, or .25.
1. If 2 is the base what is the logarithm of 4 ? of 8 ? of 32 ? of
128? of 1024?
SOLUTION. 7 is the logarithm of 128, if 2 is the base, since 7 is the exponent
by which 2 is to be affected in order to produce the number 128.
2. If 5 is the base, what is the logarithm of 625 ? of 15625 ? of
125? of 25?
2 INTRODUCTION TO THE TABLE OF LOGARITHMS.
3. If 10 is the base, what is the logarithm of 100 ? of 1000 ? ol
10,000? of 10,000,000 ?
4. If 2 is the base, what is the logarithm of J, or .25 ? of -J, or .125 ?
of -gV, or .03125 ? Ans. to the last, - 5.
5. If 8 is the base, of what number is f , or .666 -f- the logarithm ?
of what number is , or 1.333 +, the logarithm ? of what number is
2 the logarithm ? of what number is 2J, or 2.333 + ? of what num-
ber 3f, or 3.666 + ? Ans. to the last. 2048.
SCH. Since any number with for its exponent is 1, the logarithm of 1 is
in all systems. Thus 10* = 1, whence is the logarithm of 1, in a system in
^ which the base is 10.
2. A System of Logarithms is a scheme by which all num-
bers can be represented, either exactly or approximately, by expo-
nents by which a fixed number (the base) can be affected.
3. There are Two Systems of Logarithms in common use, called,
respectively, the Briggean or Common System, and the Napierian
or Hyperbolic System. The base of the former is 10, and of the
latter 2.71828 +. In speaking of logarithms of numbers, the com-
mon logarithm is always signified, if no specification is made.
4. The logarithms of all numbers, except the exact powers of the
base, indicate a power of a root, and are consequently fractional and
usually only approximations. It is customary to write them in the
form of decimal fractions. The integral part is called the Char-
acteristic, and the fractional part the Mantissa. The charac-
teristic can always be told by a simple inspection of the number
itself; hence only the mantissa is commonly given in the table.
5. Prop. The characteristic of the common logarithm of any
number greater than unity, is one less than the number of integral
figures in the given number.
' ILL. The logarithm of 4685 is more than 3, because 10* = 1000, and less than
4, because 10* = 10,000 ; hence it is 3 + a fraction. The same method may be
pursued to determine the characteristic of the logarithm of any other number
greater than unity, and the truth of the proposition be observed. Thus the
logarithm of 25645.827 is 4, since the number lies between the 4th and 5th
powers of the base, 10.
6. JProp. TJie mantissa of a decimal fraction, or of a mixed
number, is the same as the mantissa of the number considered as
INTRODUCTION TO THE TABI^E Oft LOGARITHMS. 3
DEM. Suppose it is known log 2845672 = 6.454185. This means thai
10'-" - 2845672. Dividing by 10 successively, we have
10 5 -" 4186 = 284567.2, or log 284567.2* = 5.454185,
10 4. 4nm _ 28456.72, or log 28456.72 = 4.454185,
10 s '* 64186 = 2845.672, or log 2845.672 = 3.454185,
10 a.4.4i _ 284.5672, or log 284.5672 = 2.454185,
10 i.4.4i _ 28.45672, or log 28.45672 = 1.454185,
10 o.464i 8 6 _ 2.845672, or log 2.845672 = 0.454185.
Now if we continue the operation of division, only writing 0.454185 - 1.
1.454185, meaning by this that the characteristic is negative and the mantissa
positive, and the subtraction not performed, we have
lO" 464186 = 0.2845672, or log 0.2845672 = T.454185,
10 T44i _ 0.02845672, or log 0.02845672 = &454185,
10 57 " 4185 = 0.002845672, or log 0.002845672 = 3.454185,
etc., etc. Q. E. D.
7. COR. The characteristic of a number consisting entirely of a
decimal fraction, is negative, and one more than the number of 0' $
immediately following the decimal point, as appears from the last
demonstration, or from the fact that 1 -1 = -^ ='.1 ; 10~* = -^ =
.01 ; 10- 8 = TTjVrr = .001 ; etc., etc.
8. One of the most important uses of logarithms is to facilitate
the multiplication, division, involution, and extraction of roots of
large numbers. These processes are performed upon the following
9. Prop. 1. The sum of the logarithms of two numbers is thb
logarithm of their product.
DEM. Let a be the base of the system. Let m and n be any two numbers
whose logarithms are x and y respectively. Then by definition a x = m, and
a' J = n. Multiplying these equations together we have a x + = mn. Whence
x + y is the logarithm of mn. Q. E. D.
JLO. Prop. 2. The logarithm of the quotient of two numbers is
the logarithm of the dividend minu$ the logarithm of the divisor.
DEM Let a be the base of the system, and m and n any two numbers whose
logarithms an?, respectively, x and y. Then by definition we have a x = m, and
a> n. Dividing, we have rf- = . Whence x y is the logarithm of
JJ. E. D.
* This is the common abbreviation indicating the logarithm of a number, and should iw
read "logarithm of 284567.2," not "log 2845G7.2,'' which is grossly inelegant.
4 INTRODUCTION TO THE TABLE OF LOGARITHMS.
11. Prop. 3. The logarithm of a power of a number is tht
logarithm of the number multiplied by the index of the power.
DEM. Let a be the base, and x the logarithm of m. Then of ~ m ; and raising
both to any power, as the 2th, we have a xz = m z . Whence xz is the logarithm
of the 2th power of m. Q. B. D.
12. Prop. 4. The logarithm of any root of a number is the
logarithm of the number divided by the number expressing the degree
of the root.
DEM. Let a be the base, and x the logarithm of m. Then a x = m. Extract-
ing the 2th root we have 0?= 3/m. Whence - is the logarithm of \/m. Q. E. D.
TABLE OF LOGARITHMS.
13. In order to apply the above principles practically, we need
what is called a Table of Logarithms. That is, a table from which
we can readily obtain the logarithm of any number, or the number
corresponding to any logarithm. Such a table is given on pages 11
to 28. For methods of computing it, the student is referred to
algebra. Its nature and manner of use will be learned from the
two following problems :
14. Prob. To find the logarithm of a number from the table.
SOLUTION. The logarithm of any number between 1 and 100 is seen directly
from the table on page 11. The column marked N contains the numbers, and the
adjacent column to the right gives the logarithm of the corresponding number
to 6 places of fractions. Thus, log 7 = 0.845098 ; log 33 = 1.518514.
The mantissa of the logarithm of any number expressed with three figures is
found in the column headed 0, on some one of the pages from 12 to 26 inclusive.
The given number being found in the column marked N, the mantissa of it?
logarithm stands opposite. The characteristic can be determined by (5), (6*), (7)
When but four figures are found opposite the number in the column, the two
left hand figures of the mantissa are the same as in the next mantissa above,
which lias six. Thus, log 443 = 2.646404.
To find the logarithm of a number consisting of four figures. Let it be required
to find the logarithm of 2936. Looking for 293 (the first three figures) in the
column of numbers, and then passing to the right until reaching the column
headed 6, the fourth figure, we find 7756, to which prefixing the figures 46,
which belong to all the logarithms following them till some others are indicated,
we have for the mantissa of the logarithm of 2936, .467756. But, as 3 is the
INTRODUCTION TO THE TABLE OF LOGARITHMS. 6
logarithm of 1000, and 4 of 10,000, log 2936 is 3 and this decimal, or log 2936 =
To find the logarithm of a number consisting of more than four figures. Let it
be required to find the logarithm of 2845672. Finding the decimal part of
logarithm of the first four figures 2845, as before, we find it to be .454082. Now
the logarithm of 2346 is 153 (million ths, really) more than that of 2845, as found
in the right-hand column, marked D. Hence, assuming that if an increase of the
number by 1000 makes an increase in its logarithm of 153, an increase of 672 hi
the number will make an increase in the logarithm of -$f a o, or .672 of 153, or
103, omitting lower orders, and adding this to .454082, we have .454185 as the
mantissa of log 2845672. The integral part is 6, since 2845672 lies between the
6th and 7th powers of 10. Hence, log 2845672 = 6.454185. Q. E. D.
Sen. 1. If in seeking the logarithm of any number, any of the heavy dots
noticed in the table are passed, their places are to be filled with O's, and the first
two figures of the decimal of the logarithm taken from the column in the line
below. Thus, log 3166 is 3.500511. This arrangement of the table is merely
a convenient method of saving space.
Sen. 2. The column marked D is called the column of Tabular Differences ;
and any number in it is the difference between the mantissas found in columns
4 and 5, which is usually the same as between any two consecutive logarithms
in the same horizontal line. The assumption made in using this difference, viz.,
that the logarithms increase in the same ratio as the numbers, is only approxi-
mately true, but still is accurate enough for ordinary use.
1. Find the logarithm of 2200. ....... Logarithm, 3.342423.
2. Find the logarithm of 24.36 Logarithm, 1.386677.
3. Find the logarithm of 2.698 Logarithm, 0.431042.
4. Find the logarithm of 3585.9 Logarithm, 3.554598.
5. Find the logarithm of 42.6634 Logarithm, 1.630056.
6. Find the logarithm of 331.957 Logarithm, 2.521082.
7. Find the logarithm of 2519.38 Logarithm, 3.401294.
8. Find the logarithm of .538329 Logarithm, 1.731047.
9. Find the logarithm of .087346 Logarithm, 2.941243.
10. Find the logarithm of .007389 Logarithm, 3.868586,
15. Sen. 3. It will be observed that the tabular difference varies rapidly at
the beginning of the table ; hence, for numbers between 10000 and 11000 it ia
better to use the last two pages of the table.
16. frob 2. To find a number corresponding to a given
6 INTRODUCTION TO THE TABLE OF LOGARITHMS.
SOLUTION. Let it be required to find the number corresponding to the
logarithm 5.515264. Looking in the table for the next less mantissa, we find
.515211, the number corresponding to which is 3275 (no account now being
taken as to whether it is integral, fractional, or mixed ; as in any case the figures
vvill be the same). Now from the tabular difference, in column D, we find that
an increase of 133 (millionths, really) upon this logarithm (.515211), would make
an increase of 1 in the number, making it 3276. But the given logarithm is
only 53 greater than this, hence it is assumed (th Aigh only approximately
correct) that the increase of the number is -ffe of 1, or 53 -f- 133 = .3984 + .
This added (the figures annexed) to 3275, gives 32753984 + . The characteristic,
being 5, indicates that the number lies between the 5th and 6th powers jf 10,
and hence has 6 integral places. /. 5.515264 = log. 327539.84 + . Q. E. D.
1. Find the number whose logarithm is 1.240050.
2. Find the number whose logarithm is 2.431203.
3. Find the number whose logarithm is 3.503780.
4. Find the number whose logarithm is 0.138934.
5. Find the number whose logarithm is 1.368730.
_ Number, .233738.
6. Find the number whose logarithm is 2.448375.
_ Number, .028078.
7. Find the number whose logarithm is 3.538630.
8. Find the number whose logarithm is .843970.
_ Number, 6.98184.
9. Find the number whose logarithm is 1.867372.
10. Find the number whose logarithm is .003985.
_ Number, 1.00921.
11. Find the number whose logarithm is .723460.
1. Find, by means of logarithms, the product of 57.98 by 18.
SOLUTION. As the logarithm of the product equals the sum of the logarithms
oi the factors (9), we find the logarithms of 57.98, and 18 from the table, and
INTRODUCTION TO THE TABLE OF LOGARITHMS. 7
adding them together, find the number corresponding to the sum. The latter
number is the proJuct required. Thus,
log 57.98 = 1.763278
log 18 = 1.225273
3.018551 = log 1043.64.
2. Multiply 23.14 by 5.062. Prod. 117.1347.
3. Multiply 0.00563 by 17. Prod. 0.09571.
4. Multiply 397.65 by 43.78. Prod. 17409.117.
5. Multiply 0.3854 by 0.0576. Prod. 0.022199.
6. Find the continued product of 3.902, 597.16,
and 0.0314728. Prod. 73.3354.
7. Multiply 832403 by 30243. Prod. 25174363929 *
8. Multiply 9703407 by 90807. Prod. 881137279449.
9. Multiply 3.47 by 9.83. Prod. 34.1101.
10. Multiply 12.763 by 10.976. Prod. 140.086688 *
[NOTE. The examples in division below will offer additional exercise, i(
1. Divide 24163 by 4567.
SOLUTION. Since the logarithm of the quotient equals the logarithm of th
dividend minus the logarithm of the divisor, we have the following operation
log 24163 = 4.383151
log 4567 = 3.659631
0.723520 = log 5.29078,
which number is the quotient.
2. Divide 56.4 by 0.00015.
OPERATION, log 56.4 = 1751279
log 0.00015 = 4.176091
Difference of log's = 5.575188, .'. The quotient is 876000.
SUG. Observe that only the characteristic of the logarithm is negative, and
that in subtracting we are to regard the nature of the logarithmic numbers w
positive or negative.
3. Divide 461.02876 by 21.4.
4. Divide 25.49052 by 2.46.
5. Divide 17610.8248 by 37.6.
6. Divide .00144 by 1.2.
7. Divide .0000025 by .005.
* Ordinary 6 or 7 place logarithms will not <nve these products correct. Why ?
8 INTRODUCTION TO THE TABLE OF LOGARITHMS.
8. Divide 43.2 by .24. Ans. 180
9. Divide 59.74514 by 1.36. Ans. 43.93025.
10. Divide .0001728 by 2.4. Ans. .000072.
[NOTE. The examples in multiplication given above will afford additional
xercise, if necessary.]
17. Sen. Arithmetical Complement. The arithmetical complement
of a number is simply the remainder after subtracting the number from some
particular fixed number. Thus, the a. c. of 5 with reference to 9 is 4 ; of 3, 6; of
7, 2; etc. The a. c. of 7 with reference to 10 is 3 ; of 4, 6 ; of 2, 8 ; etc. When
required to subtract one number from another, we may, if we choose, add its a. c.
and then subtract the number with reference to which the a. c. is taken. This
process will give the true difference. Thus, if we are to subtract 6 from 9, we
may add to 9 what 6 lacks of being 10 (10 6 = 4, the a. c. of 6 with reference
to 10) and then subtract 10. 9 6 = 9 + 4 10. A few such questions as the
following will render this simple process familiar. What number must I add to
576, in order that I may subtract 100 from the sum, and get the same remainder
as if I had subtracted 58 in the first instance ? Again, if I wish to take 37 from
160, what must I add to the latter, in order that I may subtract 40 from the
result, and get the difference sought ?
This principle is sometimes used in computing by means of logarithms. It is
especially convenient when there are several multipliers and several divisors
involved in the same computation. An example or two will make the process
familiar. The complements of logarithms are usually taken with reference to
10. If the logarithm exceeds 10, 20 may be used, etc.
18. Required the result of the following combinations : 346 X 27.8
+ 1156 X 3426 -7- 2.004 X 27 -r- 11.05.
OPERATION. log 346 = 2.539076
log 27.8 = 1.444045
a. c. log 1156 = 6.937042
log 3426 = 3.534787
a.c. log 2.004 = 9.698102
log 27 = 1.431364
a.c. log 11.05 = 8.956638
Rejecting 30.000000 as three complements arc
used. 4.541054 is the logarithm of the re
quired result. /. As 4.541054 = log 34757.92, the latter is the result sought
[NOTE. The preceding examples can be used to familiarize this principle if