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T" ii«, ?«>.«"W
HARVARD
COLLEGE
LIBRARY
THE GIFT OF
M iss Ellen Lang Wentworth
of Exeter, New Hampshire
3 2044 097 011 688
("
Elements of Algebra.
cL*j> . v t x jTcii ; xv s *'i%
BT
G. A. WENTWOETH, A.M.,
PBOFES80B OF MATHEMATICS IB PHILLIPS EXETER ACADEMY.
BOSTON:
PUBLISHED BY GINN & COMPANY.
1886.
MRV/.-' rvs.;r LiuRAfTY
r - . ■ Of
MISS £LLEN L WENWuhTrf
WENTWORTH'S
SERIES OF MATHEMATICS.
Primary School Arithmetic.
Grammar School Arithmetic.
Practical Arithmetic.
Practical Arithmetic (Abridged Edition).
Exercises in Arithmetic.
Shorter Course in Algebra.
Elements of Algebra.
Complete Algebra.
University Algebra.
Exercises in Algebra.
Plane Geometry.
Plane and Solid Geometry.
Exercises in Geometry.
PI. and Sol. Geom. and PI. Trigonometry.
Plane Trigonometry and Tables.
PI. and Sph. Trig., Surveying, and Tables.
Trigonometry, Surveying, and Navigation.
Log. and Trig. Tables [Seven).
Log. and Trig. Tables [Complete Edition).
Speoial Circular and Terms on application.
COPYRIGHT, 1881, BY Q. A. WENTWORTH.
J. 8. CU8HINQ & CO., PRINTERS, BOSTON.
CHAPTER V.
Simple Equations.
106. An equation is a statement that two expressions are
equal. Thus, 4 x - 12 = 8.
107. Every equation consists of two parts, called the
first and second sides, or members, of the equation.
108. An identical equation is one in which the two sides
are equal, whatever numbers the letters stand for. Thus,
(z + b)(x-b) = a*-V.
109. An equation of condition is one which is true only
when the letters stand for particular values. Thus, x + 5
= 8 is true only when x = 3.
110. A letter to which a particular value must be given
in order that the statement contained in an equation may
be true is called an unknown quantity.
111. The value of the unknown quantity is the number
which substituted for it will satisfy the equation, and is
called a root of the equation.
112. To solve an equation is to find the value of the
unknown quantity.
113. A simple equation is one which contains only the
first power of the unknown quantity, and is also called an
equation of the first degree.
60 ALGEBRA.
114. If equal changes be made in both sides of an equa-
tion, the results will be equal. § 43.
(1) To find the value of x in x + b = a.
c + b = a;
Subtract b from each side, x+ b— 6 = a— b;
Cancel + 6 — 6, x = a—b.
(2) To find the value of x in x — b = a.
x—-b = a;
Subtract — 6 from each side, x—b + b = a + b ;
Cancel — 6 + i, x = a + b.
The result in each case is the same as if b were trans-
posed to the other side of the equation with its sign
changed. Therefore, •
115. Any term may be transposed from one side of an
equation to the other provided its s\gn be changed.
For, in this transposition, the same number is subtracted
from each side of the equation.
116. The signs of all the terms on each side of an equa-
tion may be changed ; for, this is in effect transposing every
term.
117. When the known and unknown quantities of an
equation are connected by the sign + or — , they may be
separated by transposing the known quantities to one side
and the unknown to the other.
118. Hence, to solve an equation with one unknown
quantity,
Transpose all the terms involving the unknown quantity
to the left side, and all the other terms to the right side:
SIMPLE EQUATIONS. 61
combine the like terms, and divide both sides by the coefficient
of the unknown quantity.
119. To verify the result, substitute the value of the
unknown quantity in the original equation.
Exercise XXVIII.
Find the value of * in
1. 5*- 1-19. a 16* -11 = 7* + 70.
2. 3* + 6 = 12. 9. 24* -49 = 19*- 14.
3. 24*=7* + 34. 10. 3* + 23 = 78 — 2*.
4. 8* -29 = 26 -3*. 11. 26 — 8* = 80 — 14*.
5. 12 — 5* = 19-12*. 12. 13-3* = 5* — 3.
a 3* + 6 — 2* = 7*. 13. 3*-22 = 7* + 6.
7. 5* + 50 = 4* + 56. 14. 8 + 4* = 12*-16.
15. 5* -(3*— 7) = 4* — (6* — 35).
16. 6*- 2 (9 -4*) + 3 (5* -7) = 10*- (4 + 16* + 35).
17. 9*-3(5*-6) + 30 = 0.
ia *-7(4*-ll) = 14(*-5)-19(8-*)-61.
19. (*+7)(*-3) = (*-5)(*-15).
20. (*-8)(* + 12) = (* + l)(*-6).
21. (x - 2) (7 - *) + (* - 5) (* + 3) - 2 (* - 1) + 12 = 0.
22. (2*-7)(* + 5) = (9-2*)(4-*) + 229.
23. 14-*-5(*-3)(* + 2) + (5-*)(4-5*)=45*-76.
24. + 5)* -(4-*)* = 21*.
25. 5(*-2)* + 7(s-3) 2 = (3*-7)(4*-19) + 42.
62 ALGEBRA.
Exercise XXIX.
PROBLEMS.
1. Find a number such that when 12 is added to its
double the sum shall be 28.
Let x = the number.
Then 2x = its double,
and 2x + 12 = double the number increased by 12.
Bat 28 = double the number increased by 12.
.\2a?+12=28.
2s =28 -12,
2x = 16,
x= 8.
2. A farmer had two flocks of sheep, each containing the
same number. He sold 21 sheep from one flock and
70 from the other, and then found that he had left in
one flock twice as many as in the other. How many
had he in each ?
Let x = number of sheep in each flock.
Then x — 21 = number of sheep left in one flock,
and x — 70 = number of sheep left in the other.
.'. ar-21 =2(38-70),
a?-21=2a?-140.
a;-2x = -140 + 21,
- x = -119,
x = 119.
3. A and B had equal sums of money ; B gave A $5, and
then 3 times A's money was equal to 11 times B'a
money. What had each at first ?
Let x = number of dollars each had.
Then x + 5 = number of dollars A had after receiving $ 5
from B,
and 9 — 5 = number of dollars B had after giving A $5.
SIMPLE EQUATIONS. 63
.\3(s + 5) = 11(3-5),
33+15 = 113-55.
33-ll3 = -55-15.
-83 = -70,
3=8|.
Therefore, each had $8.75.
4. Find a number whose treble exceeds 50 by as much as
its double falls short of 40.
Let x = the number.
Then 33 = its treble,
and 33 — 50 = the excess of its treble over 50 ;
also, 40 — 23 = the number its double lacks of 40.
.-.33 — 50 = 40-23;
33+23 = 40 + 50.
53=90,
3=18.
5. What two numbers are those whose difference is 14,
and whose sum is 48 ?
Let 3 = the larger number.
Then 48 — x = the smaller number,
and 3 — (48 — x) = the difference of the numbers.
But 14 = the difference of the numbers.
...a;-(48-3) = 14.
3 — 48 + 3 = 14 ;
23 = 62;
3=31.
Therefore, the two numbers are 31 and 17.
6. To the double of a certain number I add 14, and obtain
as a result 154. What is the number ?
7. To four times a certain number I add 16, and obtain as
a result 188. What is the number ?
8. By adding 46 to a certain number, I obtain as a result
a number three times as large as the original number.
Find the original number.
64 ALGEBRA.
9. One number is three times as large as another. If I
take the smaller from 16 and the greater from 30,
the remainders are equal. What are the numbers ?
10. Divide the number 92 into four parts, such that the
first exceeds the second by 10, the third by 18, and
the fourth by 24.
11. The sum of two numbers is 20 ; and if three times the
smaller number be added to five times the greater,
the sum is 84. What are the numbers ?
12. The joint ages of a father and son are 80 years. If the
age of the son were doubled, he would be 10 years
older than his father. What is the age of each ?
13. A man has 6 sons, each 4 years older than the next
younger. The eldest is three times as old as the
youngest. What is the age of each ?
14. Add $24 to a certain sum and the amount will be as
much above $80 as the sum is below $80. What is
the sum?
15 Thirty yards of cloth and 40 yards of silk together cost
$ 330 ; and the silk cost twice as much a yard as the
cloth. How much does each cost a yard ?
16. Find the number whose double increased by 24 exceeds
80 by as much as the number itself is less than 100.
17. The sum of $500 is divided among A, B, 0, and D. A
and B have together $280, A and C $260, and A and
D $ 220. How much does each receive ?
18. In a company of 266 persons composed of men, women,
and children, there are twice as many men as women,
and twice as many women as children. How many
are there of each ?
SIMPLE EQUATIONS. 65
19. Find two numbers differing by 8, such that four times
the less may exceed twice the greater by 10.
20. A is 58 years older than B, and A's age is as much
above 60 as B's age is below 50. Find the age of
each.
21. A man leaves his property, amounting to $ 7500, to be
divided among his wife, his two sons, and three daugh-
ters, as follows : a son is to have twice as much as a
daughter, and the wife $500 more than all the chil-
dren together. How much was the share of each ?
22. A vessel containing some water was filled by pouring
in 42 gallons, and there was then in the vessel seven
times as much as at first. How much did the vessel
hold?
23. A has $72 and B has $52. B gives A a certain sum ;
then A has three times as much as B. How much
did A receive from B?
24. Divide 90 into two such parts that four times one part
may be equal to ixve times the other.
25. Divide 60 into two such parts that one part exceeds
the other by 24.
26. Divide 84 into two such parts that one part may be less
than the other by 36.
Note I. When we have to compare the ages of two persons at a
given time, and also a number of years after or before the given
time, we must remember that both persons will be so many years
older or younger.
Thus, if x represent A's age, and 2x B's age, at the present time,
A's age five years ago will be represented by a? — 5; and B's by
2x — 5. A's age five years hence will be represented by x + 5 ; and
3*s age by 2a? + 5.
66 ALGEBRA.
27. A is twice as old as B, and 22 years ago he was three
times as old as B. What is A's age ?
28. A father is 30 and his son 6 years old. In how many
years will the father be just twice as old as the son ?
29. A is twice as old as B, and 20 years since he was three
times as old. What is B's age ?
30. A is three times as old as B, and 19 years hence he
will be only twice as old as B. What is the age
of each ?
31. A man has three nephews ; his age is 50, and the joint
ages of the nephews is 42. How long will it be be-
fore the joint ages of the nephews will be equal to
N that of the uncle ?
Note II. In problems involving quantities of the same kind
expressed in different units, we must be careful to reduce all the quan-
tities to the same unit.
Thus, if x denote a number of inches, all the quantities of the same
kind involved in the problem must be reduced to inches.
32. A sum of money consists of dollars and twenty-five-cent
pieces, and amounts to $20. The number of coins is
50. How many are there of each sort ?
33. A person bought 30 pounds of sugar of two different
kinds, and paid for the whole $2.94. The better
kind cost 10 cents a pound and the poorer kind 7
cents a pound. How many pounds were there of
each kind ?
34. A workman was hired for 40 days, at $ 1 for every day
he worked, but with the condition that for every
day he did not work he was to pay 45 cents for his
board. • At the end of the time he received $22.60.
How many days did he work ?
SIMPLE EQUATIONS.
35. A wine merchant has two kinds of wine ; one worth 50
cents a quart, and the other 75 cents a quart. From
these he wishes to make a mixture of 100 gallons,
worth $2.40 a gallon. How many gallons must he
take of each kind.
36. A gentleman gave some children 10 cents each, and
had a dollar left. He found that he would have re-
quired one dollar more to enable him to give them
15 cents each. How many children were there ?
37. Two casks contain equal quantities of vinegar; from
the first cask 34 quarts are drawn, from the second,
20 gallons ; the quantity remaining in one vessel is
now twice that in the other. How much did each
cask contain at first ?
38. A gentleman hired a man for 12 months, at the wages
of $90 and a suit of clothes. At the end of 7 months
the man quits his service and receives $33.75 and
the suit of clothes. What was the price of the suit
of clothes ?
39. A man has three times as many quarters as half-dollars,
four times as many dimes as quarters, and twice as
many half-dimes as dimes. The whole sum is $ 7.30.
How many coins has he altogether ?
40. A person paid a bill of $15.25 with quarters and half-
dollars, and gave 51 pieces of money altogether.
How many of each kind were there ?
41. A bill of 100 pounds was paid with guineas (21 shil-
lings) and half-crowns (2} shillings), and 48 more
half-crowns than guineas were used. How many of
each were paid ?
CHAPTER X.
Problems.
Exercise LXVII.
Ex. Find the number the sum of whose third and fourth
parts is equal to 12.
Let x = the number.
Then ? = the third part of the number,
and - = the fourth part of the number,
4
.*. - + - = the sum of the two parts.
3 4 r
But 12 = the sum of the two parts,
••• H= 12 -
Multiply both sides by 12 :
4a; + 3a; =144,
7a; =144,
.-. ar = 20f
1. Find the number whose third and fourth parts together
make 14.
2. Find the number whose third part exceeds its fourth
part by 14.
3. The half, fourth, and fifth of a certain number are
together equal to 76 ; find the number.
4. Find the number whose double exceeds its half by 12.
5. Divide 60 into two such parts that a seventh of one
part may be equal to an eighth of the other.
138 ALGEBRA.
6. Divide 50 into two such parts that a fourth of one part
increased by five-sixths of the other part may be
equal to 40.
7. Divide 100 into two such parts that a fourth of one
part diminished by a third of the other part may be
equal to 11.
8. The sum of the fourth, fifth, and sixth parts of a cer-
tain number exceeds the half of the number by 112.
What is the number ?
9. The sum of two numbers is 5760, and their difference is
equal to one-third of the greater. What are the
numbers ?
10. Divide 45 into two such parts that the first part
divided by 2 shall be equal to the second part mul-
tiplied by 2.
11. Find a number such that the sum of its fifth and its
seventh parts shall exceed the difference of its fourth
and its seventh parts by X)9.
12. In a mixture of wine and water, the wine was 25 gal-
lons more than half of the mixture, and the water
5 gallons less than one-third of the mixture. How
many gallons were there of each ?
13. In a certain weight of gunpowder the saltpetre was
6 pounds more than half of the weight, the sulphur
5 pounds less than the third, and the charcoal 3
pounds less than the fourth of the weight. How
many pounds were there of each ?
14. Divide 46 into two parts such that if one part be
divided by 7, and the other by 3, the sum of the
quotients shall be 10.
PROBLEMS. 139
15. A house and garden cost $850, and five times the price
of the house was equal to twelve times the price of
the garden. What is the price of each ?
16. A man leaves the half of his property to his wife, a sixth
to each of his two children, a twelfth to his brother,
and the remainder, amounting to $600, to his sister.
What was the amount of his property ?
17. The sum of two numbers is a and their difference is b ;
find the numbers.
18. Find two numbers of which the sum is 70, such that
the first divided by the second gives 2 as a quotient
and 1 as a remainder.
19. Find two numbers of which the difference is 25, such
that the second divided by the first gives 4 as a quo-
tient and 4 as a remainder.
20. Divide the number 208 into two parts such that the
sum of the fourth of the greater and the third of the
smaller is less by 4 than four times the difference of
the two parts.
21. Find four consecutive numbers whose sum is 82.
Note I. It is to be remembered that if x represent a person's age
at the present time, his age a years ago will be represented by x — a,
and a years hence by x + a.
Ex. In eight years a boy will be three times as old as he
was eight years ago. How old is he ?
Let x = the number of years of his age.
Then x — 8 = the number of years of his age eight years ago,
and x + 8 — the number of years of his age eight years hence,
.*. x + 8 = 3 (x - 8),
x + S =3a>-24,
a;_3x = -24-8 t
-2s = -32,
x=16.
140 ALGEBRA.
22. A is 72 years old, and B's age is two-thirds of A's.
How long is it since A was five times as old as B ?
23. A mother is 70 years old, her daughter is half that age.
How long is it since the mother was three and one-
third times as old as the daughter ?
24. A father is three times as old as the son; four years
ago the father was four times as old as the son
then was. What is the age of each ?
25. A is twice as old as B, and seven years ago their united
ages amounted to as many years as now represent
the age of A. Find the ages of A and B.
26. The sum of the ages of a father and son is half what it
will be in 25 years ; the difference is one-third what
the sum will be in 20 years. What is the age of each ?
Note II. If A can do a piece of work in x days, the part of the
work that he can do in one day will be represented by J. Thus, if he
can do the work in 5 days, in 1 day he can do \ of the work.
Ex. A can do a piece of work in 5 days, and B can do it
in 4 days. How long will it take A and B together
to do the work ?
Let x = the number of days it will take A and B together.
Then J = the part they can do in one day.
' Now, \ = the part A can do in one day,
and ' \ = the part B can do in one day.
.'. \ + \ = the part A and B can do in one day.
-•• i + i-i,
4a + 5;c=:20,
9* = 20,
3 = 2f
Therefore they will do the work in 2$ days.
27. A can do a piece of work in 5 days, B in 6 days, and
C in 7 J days ; in what time will they do it, all work-
ing together ?
PBOBLEMS. 141
28. A can do a piece of work in 21 days, B in 3} days, and
C in 3} days ; in what time will they do it, all work-
ing together ?
29. Two men who can separately do a piece of work in 15
days and 16 days, can, with the help of another, do
it in 6 days. How long would it take the third man
to do it alone ?
30. A can do half as much work as B, B can do half as
much as C, and together they can complete a piece
of work in 24 days. In what time can each alone
complete the work ?
SL A does -J of a piece of work in 10 days, when B comes
to help him, and they finish the work in 3 days
more. How long would it have taken B alone to do
the whole work ?
32. A and B together can reap a field in 12 hours, A and
in 16 hours, and A by himself in 20 hours. In
what time can B and C together reap it? In what
time can A, B, and G together reap it ?
33. A and B together can do a piece of work in 12 days,
A and C in 15 days, B and C in 20 days. In what
time can they do it, all working together ?
Note III. If a pipe can fill a vessel in z hours, the part of the
vessel filled by it in one hour will be represented by J. Thus, if a
pipe will fill a vessel in 3 hours, in 1 hour it will fill J of the vessel.
34. A tank can be filled by two pipes in 24 minutes and 30
minutes respectively, and emptied by a third in 20
minutes. In what time will it be filled if all three
are running together ?
35. A tank can be filled in 15 minutes by two pipes, A and
B, running together. After A has been running by
142 ALGEBRA.
itself for 5 minutes, B is also turned on, and the tank
is filled in 13 minutes more. In what time may it
be filled by each pipe separately ?
36. A cistern could be filled by two pipes in 6 hours and 8
hours respectively, and could be emptied by a third
in 12 hours. In what time would the cistern be
filled if the pipes were all running together ?
37. A tank can be filled by three pipes in 1 hour and 20
minutes, 3 hours and 20 minutes, and 5 hours, re-
spectively. In what time will the tank be filled
when all three pipes are running together ?
38. If three pipes can fill a cistern in a, b, and c minutes,
respectively, in what time will it be filled by all
three running together ?
39. The capacity of a cistern is 755J gallons. The cistern
has three pipes, of which the first lets in 12 gallons
in 3 \ minutes, the second 15-J- gallons in 2J- minutes,
the third 17 gallons in 3 minutes. In what time
will the cistern be filled by the three pipes running
together ?
Note IV. In questions involving distance, time, and rate:
Distance = Time
Rate
Thus, if a man travels 40 miles at the rate of 4 miles an hour,
40
-— = number of hours required.
Ex. A courier who goes at the rate of 3L£ miles in 5 hours,
is followed, after 8 hours, by another who goes at the
rate of 22£ miles in 3 hours. In how many hours
will the second overtake the first ?
Since the first goes 31 J miles in 5 hours, his rate per hour is 6^
miles.
PROBLEMS. 143
Since the second goes 221 miles in 3 hours, his rate per hour is 7}
miles.
Let x — the number of hours the first is travelling.
Then x — 8 =» the number of hours the second is travelling.
Then 6^ x = the number of miles the first travels ;
(x — 8) 7} = the number of miles the second travels.
They both travel the same distance,
/. 6A*~ (3-8)7J.
The solution of which gives 42 hours.
40. A sets out and travels at the rate of 7 miles in 5 hours.
Eight hours afterwards, B sets out from the same
place and travels in the same direction, at the rate
of 5 miles in 3 hours. In how many hours will B
overtake A ?
4L A person walks to the top of a mountain at the rate
of 2£ miles an hour, and down the same way at the
rate of 3£ miles an hour, and is out 5 hours. How
far is it to the top of the mountain ?
42. A person has a hours at his disposal. How far may he
ride in a coach which travels b miles an hour, so as
to return home in time, walking back at the rate of c
miles an hour?
43. The distance between London and Edinburgh is 360
miles. One traveller starts from Edinburgh and
travels at the rate of 10 miles an hour ; another
starts at the same time from London, and travels at
the rate of 8 miles an hour. How far from London
will they meet ?
44. Two persons set out from the same place in opposite
directions. The rate of one of them per hour is a
mile less than double that of the other, and in 4
hours they are 32 miles apart. Determine their
rates.
144
ALGEBRA.
45. In going a certain distance, a train travelling 35 miles
an hour takes 2 hours less than one travelling 25
miles an hour. Determine the distance.
Note V. In problems relating to clocks, it is to be observed that
the minute-hand moves twelve times as fast as the hour-hand.
Ex. Find the time between two and three o'clock when the
hands of a clock are :
I. Together.
II. At right angles to each other.
III. Opposite to each other.
Fig. 1.
Fig. 2.
Fig. 8.
I. Let Clfand CAT (Fig. 1) denote the positions of the hour and
minute hands at 2 o'clock, and CB the position of both hands vnen
together.
Then arc HB = one-twelfth of arc MB.
Let x = number of minute-spaces in arc MB.
Then — =» number of minute-spaces in arc HB,
and 10 = number of minute-spaces in arc MH.
Now arc MB = arc MH + arc HB.
That is, x =
' 10+ f 2 -
The solution of this equation gives x =» 10ff .
Hence, the time is 10-J-^- minutes past 2 o'clock.
II. Let CB and CD (Fig. 2) denote the positions of the hour and
minute hands when at right angles to each other.
PROBLEMS. 145
Let x — number of minute-spaces in arc MHBD.
Then — — number of minute-spaces in arc HB,
and 10 =» number of minute-spaces in arc MH.
15 — number of minute-spaces in arc BD.
Now arc MHBD = arcs MH+ HB + BD.
Thatis, a; = 10 + ^ + 15.
The solution of this equation gives x — 27^.
Hence, the time is 27-^ minutes past 2 o'clock.
III. Let CB and CD (Fig. 3) denote the positions of the hour and
minute hands when opposite to each other.
Let x = number of minute-spaces in arc MHBD.
Then — = number of minute-spaces in arc JIB,
12 r
and 10 = number of minute-spaces in arc MH,
30 = number of minute-spaces in arc BD,
Now arc MHBD = arcs MH+ HB + BD.
Thatis, 3-10+ — + 30.
The solution of this equation gives x = 43^.
Hence, the time is 43^ minutes past 2 o'clock.
46. At what time are the hands of a watch together :
I. Between 3 and 4 ?
II. Between 6 and 7?
III. Between 9 and 10?
47. At what time are the hands of a watch at right angles :
I. Between 3 and 4 ?
II. Between 4 and 5 ?
III. Between 7 and 8?
48. At what time are the hands of a watch opposite to
each other :
I. Between 1 and 2?
II. Between 4 and 5 ?
III. Between 8 and 9?
146 ALGEBRA.
49. It is between 2 and 3 o'clock ; but a person looking at
his watch and mistaking the hour-hand for the
minute hand, fancies that the time of day is 55
minutes earlier than it really is. What is the true
time?
Note VI. It is to be observed that if a represent the number of
feet in the length of a step or leap, and x the number of steps or leaps
taken, then ax will represent the number of feet in the distance
made.
Ex. A hare takes 4 leaps to a greyhound's 3 ; but 2 of the
greyhound's leaps are equivalent to 3 of the hare's.
The hare has a start of 50 leaps. How many leaps
must the greyhound take to catch the hare ?
Let 3aj = the number of leaps taken by the greyhound.
Then 4 a? = the number of leaps of the hare in the same time.
Also, let a denote the number of feet in one leap of the hare.
Then -£ will denote the number of feet in one leap of the grey-
hound.
That is, 3 as X -~ -= the whole distance,
and (50 + 4x) a — the whole distance,
.-. ?|*-(50 + 4*)a.
Divide by a, 25-50 + 4*.
z
9x=100 + 8* f
*=100,
.-. 3* -300.
Thus the greyhound must take 300 leaps.
50. A hare takes 6 leaps to a dog's 5, and 7 of the dog's
leaps are equivalent to 9 of the hare's. The hare
has a start of 50 of her own leaps. How many leaps
will the hare take before she is caught ?
PROBLEMS. 147
51. A greyhound makes 3 leaps while a hare makes 4 ; but
2 of the greyhound's leaps are equivalent to 3 of the
hare's. The hare has a start of 50 of the greyhound's
leaps. How many leaps does each take before the
hare is caught ?
52. A greyhound makes two leaps while a hare makes 3 ;
but 1 leap of the greyhound is equivalent to 2 of the
hare's. The hare has a start of 80 of her own leaps.
How many leaps will the hare take before she is
caught ?
Note VII. It is to be observed that if the number of units in the
breadth and length of a rectangle be represented by x and x + a,
respectively, then x (x + a) will represent the number of surface units
in the rectangle, the unit of surface having the same name as the
linear unit in which the sides of the rectangle are expressed.
53. A rectangle whose length is 5 feet more than its breadth
would have its area increased by 22 feet if its length
and breadth were each made a foot more. Find its
dimensions.
54. A rectangle has its length and breadth respectively 5
feet longer and 3 feet shorter than the side of the
equivalent square. Find its area.
55. The length of a rectangle is an inch less than double
its breadth ; and when a strip 3 inches wide is cut
off all round, the area is diminished by 210 inches.
Find the size of the rectangle at first.
66. The length of a floor exceeds the breadth by 4 feet ; if
each dimension were increased by 1 foot, the area
of the room would be increased by 27 square feet.
Find its dimensions.
Note VIII. It is to be observed that if b pounds of metal lose a
pounds when weighed in water, 1 pound will lose \ of a pounds,
or | of a pound.
148 ALGEBRA.
57. A mass of tin and lead weighing 180 pounds loses 21
pounds when weighed in water ; and it is known that
37 pounds of tin lose 5 pounds, and 23 pounds of
lead lose 2 pounds, when weighed in water. How
'many pounds of tin and of lead in the mass ?
58. If 19 pounds of gold lose 1 pound, and 10 pounds of
silver lose 1 pound, when weighed in water, find the
amount of each in a mass of gold and silver weighing
106 pounds in air and 99 pounds in water.
59. Fifteen sovereigns should weigh 77 pennyweights ; but
a parcel of light sovereigns, having been weighed and
counted, was found to contain 9 more than was sup-
posed from the weight ; and it appeared that 21 of
these coins weighed the same as 20 true sovereigns.
How many were there altogether ?
GO. There are two silver cups, and one cover for both. The
first weighs 12 ounces, and with the cover weighs
twice as much as the other without it ; but the sec-
ond with the cover weighs one-third more than the
first without it. Find the weight of the cover.
61. A man wishes to enclose a circular piece of ground with
palisades, and finds that if he sets them a foot apart
he will have too few by 150 ; but if he sets them a
yard apart he will have too many by 70. What is the
circuit of the piece of ground ?
62. A horse was sold at a loss for $200 ; but if it had been
sold for $250, the gain would have been three-fourths
of the loss when sold for $200. Find the value of
the horse.
63. A and B shoot by turns at a target. A puts 7 bullets
out of 12, and B 9 out of 12, into the centre. Be-
tween them they put in 32 bullets. How many shots
did each fire?
PROBLEMS. 149
64 A boy buys a number of apples at the rate of 5 for 2
pence. He sells half of them at 2 a penny and the
rest at 3 a penny, and clears a penny by the tran-
saction. How many does he buy ?
65. A person bought a piece of land for $ 6750, of which
he kept -J for himself. At the cost of $250 he made
a road which took ^ of the remainder, and then sold
the rest at 12J- cents a square yard more than double
the price it cost him, thus clearing his outlay and
$500 besides. How much land did he buy, and
what was the cost-price per yard ?
66. A boy who runs at the rate of 12 yards per second
starts 20 yards behind another whose rate is 10^
yards per second. How soon will the first boy be
10 yards ahead of the second ?
67. A merchant adds yearly to his capital one-third of it,
but takes from it, at the end of each year, $5000 for
expenses. At the end of the third year, after de-
ducting the last $5000, he has twice his original
capital. How much had he at first ?
68. A shepherd lost a number of sheep equal to one-fourth
of his flock and one-fourth of a sheep ; then, he lost a
number equal to one-third of what he had left and
one-third of a sheep ; finally, he lost a number equal
to one-half of what now remained and one-half a
sheep, after which he had but 25 sheep left. How
many had he at first?
69. A trader maintained himself for three years at an ex-
pense of $ 250 a year ; and each year increased that
part of his stock which was not so expended by one-
third of it. At the end of the third year his original
stock was doubled. What was his original stock ?
150 ALGEBRA.
70. A cask contains 12 gallons of wine and 18 gallons of
water; another cask contains 9 gallons of wine and
3 gallons of water. How many gallons must be
drawn from each cask to produce a mixture contain-
ing 7 gallons of wine and 7 gallons of water ?
71. The members of a club subscribe each as many dollars
as there are members. If there had been 12 more
members, the subscription from each would have
been $10 less, to amount to the same sum. How
many members were there ?
72. A number of troops being formed into a solid square,
it was found there were 60 men over; but when
formed in a column with 5 men more in front than
before, and 3 men less in depth, there was lacking
one man to complete it. Find the number of troops.
73. An officer can form the men of his regiment into a hol-
low square twelve deep. The number of men in the
regiment is 1296. Find the number of men in the
front of the hollow square.
74. A person starts from P and walks towards Q at the
rate of 3 miles an hour; 20 minutes later another
person starts from Q and walks towards P at the
rate of 4 miles an hour. The distance from P to Q
is 20 miles. How far from P will they meet ?
75. A person engaged to work a days qn these conditions :
for each day he worked he was to receive b cents,
and for each day he was idle he was to forfeit c cents.
At the end of a days he received d cents. How
many days was he idle ?
76. A banker has two kinds of coins : it takes a pieces of
the first to make a dollar, and b pieces of the second
to make a dollar. A person wishes to obtain c pieces
for a dollar. How many pieces of each kind must
the banker give him ?
168 ALGEBRA.
7. If B give A $25 they will have equal sums of money;
but if A give B $22, B's money will be double that
of A's. How much has each ?
a A farmer sold to one person 30 bushels of wheat and
40 bushels of barley for $67.50; to another person
he sold 50 bushels of wheat and 30 bushels of barley
for $85. What was the price of the wheat and of
the barley per bushel ?
9. If A give B $5 he will then have $6 less than B ; but
if he receive $5 from B, three times his money will
be $20 more than four times B's. How much has
each?
10. The cost of 12 horses and 14 cows is $ 1900 ; the cost
of 5 horses and 3 cows is $650. What is the cost of
a horse and a cow respectively ?
Note I. A fraction of which the terms are unknown may he rep-
resented hy -•
Ex. A certain fraction becomes equal to \ if 3 be added to
its numerator, and equal to \ if 3 be added to its de-
nominator. Determine the fraction.
Let £ =x the required fraction.
„ , q
By the conditions = J,
V
and = \ .
y + 3
From the solution of these equations it is found that
s = 6,
y = 18.
Therefore the fraction = ^.
VL A certain fraction becomes equal to 2 when 7 is added
to its numerator, and equal to 1 when 1 is subtracted
from its denominator. Determine the fraction.
PROBLEMS. 169
12. A certain fraction becomes equal to -J- when 7 is added
to its denominator, and equal to 2 when 13 is added
to its numerator. Determine the fraction.
13. A certain fraction becomes equal to •$• when the denom-
inator is increased by 4, and equal to fj- when the
numerator is diminished by 15. Determine the frac-
tion.
14. A certain fraction becomes equal to ■§■ if 7 be added to
the numerator, and equal to f if 7 be subtracted from
the denominator. Determine the fraction.
15. Find two fractions with numerators 2 and 5 respectively,
whose sum is 1-J-, and if their denominators are inter-
changed their sum is 2.
16. A fraction which is equal to -| is increased to -fc when
a certain number is added to both its numerator and
denominator, and is diminished to % when one more
than the same number is subtracted from each. De-
termine the fraction.
Note II. A number consisting of two digits which are unknown
may be represented by 10 x + y, in which x and y represent the digits
of the number. Likewise, a number consisting of three digits which
are unknown may be represented by 100 x + lOy + z, in which x, y,
and z represent the digits of the number.
For example, consider any number expressed by three digits, as
364. The expression 364 means 300 + 60 + 4 ; or, 100 times 3 + 10
times 6+4.
Ex. The sum of the two digits of a number is 8, and if 36
be added to the number the digits will be inter-
changed. What is the number ?
Let x = the digit in the tens* .place,
and y = the digit in the units' place.
Then 10 x + y = the number.
By the conditions, x + y = 8, (1)
and 10a: + y + 36-10y + *. (2)
170 ALGEBRA.
From (2), 9* - 9y = - 36.
Divide by 9, x — y = — 4.
Add (1) and (3), 2x = 4,
.-.re = 2.
Subtract (3) from (1), 2y=12,
.\y = 6.
Hence, the number is 26.
17. The sum of the two digits of a number is 10, and if 54
be added to the number the digits will be inter-
changed. What is the number ?
18. The sum of the two digits of a number is 6, and if the
number be divided by the sum of the digits the quo-
tient is 4. What is the number ?
. 10. A certain number is expressed by two digits, of which
the first is the greater. If the number be divided by
the sum of its digits the quotient is 7 ; if the digits
be interchanged, and the resulting number diminished
by 12 be divided by the difference between the two
digits, the quotient is 9. What is the number ?
20. If a certain number be divided by the sum of its two
digits the quotient is 6 and the remainder 3 ; if the
digits be interchanged, and the resulting number be
divided by the sum of the digits, the quotient is 4
and the remainder 9. What is the number ?
21. If a certain number be divided by the sum of its two
digits diminished by 2, the quotient is 5 and the re-
mainder 1 ; if the digits be interchanged, and the
resulting number be divided by the sum of the digits
increased by 2, the quotient is 5 and the remainder
8. Find the number.
22. The first of the two digits of a number is, when doubled,
3 more than the second, and the number itself is less
by 6 than five times the sum of the digits. What
is the number ?
PROBLEMS. 171
23. A number is expressed by three digits, of which the first
and last are alike. By interchanging the digits in
the units' and tens' places the number is increased by
54 ; but if the digits in the tens' and hundreds' places
are interchanged, 9 must be added to four times the
resulting number to make it equal to the original
number. What is the number ?
24. A number is expressed by three digits. The sum of the
digits is 21 ; the sum of the first and second exceeds
the third by 3 ; and if 198 be added to the number,
the digits in the units' and hundreds' places will be
interchanged. Find the number.
25. A number is expressed by three digits. The sum of
the digits is 9 ; the number is equal to forty-two
times the sum of the first and second digits ; and the
third digit is twice the sum of the other two. Find
the number.
26. A certain number, expressed by three digits, is equal to
forty-eight times the sum of its digits. If 198 be
subtracted from the number, the digits in the units'
and hundreds' places will be interchanged ; and the
sum of the extreme digits is equal to twice the mid-
dle digit. Find the number.
Note III. If a boat move at the rate of x miles an hour in still
water, and if it be on a stream that runs at the rate o£ y miles an
hour, then
x + y represents its rate down the stream^
x — y represents its rate up the stream.
27. A waterman rows 30 miles and back in 12 hours. He
finds that he can row 5 miles with the stream in the
same time as 3 against it. Find the time he was
rowing up and down respectively.
172 ALGEBRA.
28. A crew, which can pull at the rate of 12 miles an hour
down the stream, finds that it takes twice as long to
come up the river as to go down. At what rate does
the stream flow ?
29. A man sculls down a stream, which runs at the rate of
4 miles an hour, for a certain distance in 1 hour and
40 minutes. In returning it takes him 4 hours and
15 minutes to arrive at a point 3 miles short of his
starting-place. Find the distance he pulled down
the stream and the rate of his pulling.
30. A person rows down a stream a distance of 20 miles
and back again in 10 hours. He finds he can row
2 miles against the stream in the same time he can
row 3 miles with it. Find the time of his rowing
down and of his rowing up the stream ; and also the
rate of the stream.
Note IV. When commodities are mixed, it is to be observed that
the quantity of the mixture = the quantity of the ingredients ; the
cost of the mixture = the cost of the ingredients.
Ex. A wine-merchant has two kinds of wine which coct
72 cents and 40 cents a quart respectively. How
much of each must he take to make a mixture of 50
quarts worth 60 cents a quart ?
Let x = required number of quarts worth 72 cents a
quart,
and y <=> required number of quarts worth 40 cents a
quart.
Then, 72 a; = cost in cents of the first kind,
40 y = cost in cents of the second kind of wine,
and 3000 = cost in cents of the mixture.
.-. x + y « 50,
72a? + 40y«3000.
From which equations the values of x and y may be found.
PROBLEMS 173
31. A grocer mixed tea that cost him 42 cents a pound
with tea that cost him 54 cents a pound. He had 30
pounds of the mixture, and by selling it at the rate
of 60 cents a pound, he gamed as much as 10 pounds
of the cheaper tea cost him. How many pounds of
each did he put into the mixture ?
32. A grocer mixes tea that cost him 90 cents a pound
with tea that cost him 28 cents a pound. The cost
of the mixture is $61.20. He sells the mixture at
50 cents a pound, and gains $3.80. How many
pounds of each did he put into the mixture ?
33. A farmer has 28 bushels of barley worth 84 cents a
bushel. With his barley he wishes to mix rye worth
$1.08 a bushel, and wheat worth $1.44 a bushel, so
that the mixture may be 100 bushels, and be worth
$1.20 a bushel. How many bushels of rye and of
wheat must he take ?
Note V. It is to be remembered that if a person can do a piece
of work in x days, the part of the work he can do in one day will be
represented by J.
Ex. A and B together can do a piece of work in 48 days ;
A and C together can do it in 30 days ; B and to-
gether can do it in 26-| days. How long will it take
each to do the work ?
Let x = the number of days it will tafce A alone to do the work,
y = the number of days it will take B alone to do the work,
and z = the number of days it will take C alone to do the work.
Then, _, -, -, respectively, will denote the part each can do
x v z j
9 ma day,
and - + - will denote the part A and B together can do in a day,
x y
but — - will denote the part A and B together can do in a day.
174
ALGEBRA.
Therefore,
1 + 1-Jl
x y 48
Likewise,
111
x + z = 30
and
11 1 3
y z = 26J~80
Add (1), (2), and (3),
2 2 2 11
x + y + z = 120
Multiply (1) by 2,
2 2 1
* + y "24
Subtract (5) from (4),
2 1
z 20
.-. 2 = 40.
0)
(2)
(3)
(4)
(5)
Subtract the double of (2) from (4), ? - —
y 40
.-. y = 80.
Subtract the double of (3) from (4), - = JL
.-. x=120.
34. A and B together earn $40 in 6 days ; A and C to-
gether earn $54 in 9 days; B and C together earn
$80 in 15 days. What does each earn a day ?
35. A cistern has three pipes, A, B, and C. A and B will
fill it in 1 hour and 10 minutes ; A and C in 1 hour
and 24 minutes ; B and C in 2 hours and 20 minutes.
How long will it take each to fill it ?
36. A warehouse will hold 24 boxes and 20 bales ; 6 boxes
and 14 bales will fill half of it. How many of each
alone will it hold ?
37. Two workmen together complete some work in 20 days ;
but if the first had worked twice as fast, and the sec-
ond half as fast, they would have finished it in 15 days.
How long would it take each alone to do the work ?
38L A purse holds 19 crowns and 6 guineas; 4 crowns and
5 guineas fill £J of it. How many of each alone will
it hold?
PROBLEMS. 175
39. A piece of work can be completed by A, B, and C to-
gether in 10 days ; by A and B together in 12 days ;
by B and C, if B work 15 days and 30 days. How
long will it take each' alone to do the work ?
40. A cistern has three pipes, A, B, and C. A and B will
fill it in a minutes ; A and C in b minutes ; B and
in c minutes. How long will it take each alone to
fill it?
Note VI. In considering the rate of increase or decrease in quan-
tities, it is usual to take 100 as a common standard of reference, so
that the increase or decrease is calculated for every 100, and there-
fore called per cent. '
It is to be observed that the representative of the number result-
ing after an increase has taken place is 100 + increase per cent ; and
after a decrease, 100 — decrease per cent.
Interest depends upon the time for which the money is lent, as
well as upon the rate per cent charged ; the rate per cent charged
being the rate per cent on the principal for one year. Hence,
a . , . , , Principal X Kate X Time
Simple interest = * »
r 100
where Time means number of years or fraction of a year.
Amount = Principal + Interest.
In questions relating to stocks, 100 is taken as the representative
of the stock, the price represents its market value, and the per cent
represents the interest which the stock bears. Thus, if six per cent
stocks are quoted at 108, the meaning is, that the price of $100 of
the stock is $108, and that the interest derived from $100 of the
stock will be jfa of $ 100, that is, $6 a year. The rate of interest on
the money invested will be \%% of 6 per cent.
41. A man has $ 10,000 invested. For a part of this sum he
receives 5 per cent interest, and for the rest 4 per
cent ; the income from his 5 per cent investment is
$50 more than from his 4 per cent. How much has
he in each investment ?
176 ALGEBRA.
42. A sum of money, at simple interest, amounted in 6
years to $26,000, and in 10 years to $30,000. Find
the sum and the rate of interest.
43. A sum of money, at simple interest, amounted in 10
months to $26,250, and in 18 months to $27,250.
Find the sum and the rate of interest.
44. A sum of money, at simple interest, amounted in m
years to a dollars, and in n years to 6. dollars. Find
the sum and the rate of interest.
45. A sum of money, at simple interest, amounted in a
months to c dollars, and in b months to d dollars.
Find the sum and the rate of interest.
46. A person has a certain capital invested at a certain rate
per cent. Another person has $ 1000 more capital,
and his capital invested at one per cent better than
the first, and receives an income $ 80 greater. A third
person has $1500 more capital, and his capital in-
vested at two per cent better than the first, and re-
ceives an income $ 150 greater. Find the capital of
each, and the rate at which it is invested.
47. A person has $12,750 to invest. He can buy three per
cent bonds at 81, and five per cents at 120. Find
the amount of money he must invest in each in order
to have the same income from each investment.
48. A and B each invested $1500 in bonds; A in three
per cents and B in four per cents. The bonds were
bought at such prices that B received $5 interest
more than A. After both classes of bonds rose 10
points, they sold out, and A received $50 more than
B.. What price was paid for each class of bonds?
PROBLEMS. 177
49. A person invests $10,000 in three per cent bonds,
$16,500 in three and one-half per cents, and has an
income from both investments of $1056.25. If his
investments had been $2750 more in the three per
cents, and less in the three and one-half per cents,
his income would have been 62£ cents greater.
What price was paid for each class of bonds ?
60. The sum of $2500 was divided into two unequal parts
and invested, the smaller part at two per cent more
than the larger. The rate of interest on the larger
sum was afterwards increased by 1, and that of the
smaller sum diminished by 1 ; and thus the interest
of the whole was increased by one-fourth of its value.
If the interest of the larger sum had been so in-
creased, and no change been made in the interest of
the smaller sum, the interest of the whole would have
been increased one-third of its value. Find the sums
invested, and the rate per cent of each.
Note VII. If x represent the number of linear units in the length,
and y in the width, of a rectangle, xy will represent the number of
its units of surface ; the surface unit having the same name as the
linear unit of its sides.
51. If the sides of a rectangular field were each increased
by 2 yards, the area would be increased by 220
square yards; if the length were increased and the
breadth were diminished each by 5 yards, the area
would be diminished by 185 square yards. What is
its area ?
52. If a given rectangular floor had been 3 feet longer and
2 feet broader it would have contained 64 square feet
more ; but if it had been 2 feet longer and 3 feet
broader it would have contained 68 square feet more.
Find the length and breadth of the floor.
178 ALGEBRA.
53. In a certain rectangular garden there is a strawberry-
bed whose sides are one-third of the lengths of the
corresponding sides of the garden. The perimeter of
the garden exceeds that of the bed by 200 yards;
and if the greater side of the garden be increased by
3, and the other by 5 yards, the garden will be en-
larged by 645 square yards. Find the length and
breadth of the garden.
Note VIII. Care must be taken to express the conditions of a
problem with reference to the same principal unit.
Ex. In a mile race A gives B a start of 20 yards and beats
him by 30 seconds. At the second 'trial A gives B a
start of 32 seconds and beats him by 9-j^ yards.
Find the rate per hour at which eatjh runs.
Let x = number of yards A runs a second,
and y = number of yards B runs a second.
Since there are 1760 yards in a mile,
1 7fiO
= number of seconds it takes A to run a
* mile,
1740 1750A-
and — -££ = number of seconds B was running in the
^ y first and second trials, respectively.
Hence, !2!2_i^ = 30l
and 1W&-1W2-83.
y x
The solution of these equations gives x = 5ff and y = 5^.
That is, A runs — 111, or -— -, of a mile in one second ;
1760 300
and in one hour, or 3600 seconds, runs 12 miles.
Likewise, B runs 10^ T miles in one hour.
64. In a mile race A gives B a start of 100 yards and beats
him by 15 seconds. In the second trial A gives B a
start of 45 seconds and is beaten by 22 yards. Find
the rate of each in miles per hour.
PEOBLEMS. 179
65. In a mile race A gives B a start of 44 yards and beats
him by 51 seconds. In the second trial A gives B a
start of 1 minute and 15 seconds and is beaten by 88
yards. Find the rate of each in miles per hour.
66. The time which an express-train takes to go 120 miles
is -jBj- of the time taken by an accommodation-train.
The slower train loses as much time in stopping at
different stations as it would take to travel 20 miles
without stopping ; the express-train loses only half
as much time by stopping as the accommodation-
train, and travels 15 miles an hour faster. Find the
rate of each train in miles per hour.
67. A train moves from P towards Q, and an hour later a
second train starts from Q and moves towards P
at a rate of 10 miles an hour more than the first
train; the trains meet half-way between P and Q.
If the train from P had started an hour after the
train from Q its rate must have been increased by 28
miles in order that the trains should meet at the
half-way point. Find the distance from P to Q.
68. A passenger-train, after travelling an hour, meets with
an accident which detains it one-half an hour ; after
which it proceeds at four-fifths of its usual rate, and
arrives an hour and a quarter late. If the accident
had happened 30 miles farther on, the train would
have been only an hour late. Determine the usual
rate of the train.
59. A passenger-train after travelling an hour is detained
15 minutes ; after which it proceeds at three-fourths
of its former rate, and arrives 24 minutes late. If
the detention had taken place 5 miles farther on, the
train would have been only 21 minutes late. Deter-
mine the usual rate of the train.
180 ALGEBRA.
00. A man bought 10 oxen, 120 sheep, and 46 lambs. The
cost of 3 sheep was equal to that of 5 lambs ; an ox,
a sheep, and a lamb together cost a number of dol-
lars less by 57 than the whole number of animals
bought; and the whole sum spent was $2341.50.
Find the price of an ox, a sheep, and a lamb, respec-
tively.
6L A farmer sold 100 head of stock, consisting of horses,
oxen, and sheep, so that the whole realized $11.75 a
head ; while a horse, an ox, and a sheep were sold
for $110, $62.50, and $7.50, respectively. Had he
sold one-fourth of the number of oxen that he did,
and 25 more sheep, he would have received the same
sum. Find the number of horses, oxen, and sheep,
respectively, which were sold.
62. A, B, and C together subscribed $100. If A's sub-
scription had been one-tenth less, and B's one-tenth
more, C's must have been increased by $ 2 to make
up the sum ; but if A's had been one-eighth more,
and B's one-eighth less, C's subscription would have
been $ 17.50. What did each subscribe ?
63. A gives to B and C as much as each of them has ; B
gives to A and as much as each of them then has ;
and C gives to A and B as much as each of them
then has. In the end each of them has $6. How
much had each at first ?
64. A pays to B and C as much as each of them has ; B
pays to A and C one-half as much as each of them then
has ; and C pays to A and B one-third of what each of
them then has. In the end A finds that he has
$1.50, B $4.16|, C $.58£. How much had each at
first?
QUADRATIC EQUATIONS. 215
(1) The sum of the squares of two consecutive numbers is
481. Find the numbers.
Let
x = one number,
and
x + 1 = the other.
Then
3 s + (x + l) s = 481,
or
2x* + 2x + 1 = 481.
The solution of which gives, x = 15, or — 16.
The positive root 15 gives for the numbers, 15 and 16.
The negative root — » 16 is inapplicable to the problem, as consecu-
tive numbers are understood to be integers which follow one another
in the common scale, 1, 2, 3, 4
(2) What is the price of eggs per dozen when 2 more in a
shilling's worth lowers the price 1 penny per dozen ?
Let x = number of eggs for a shilling.
Then, - » cost of 1 egg in shillings,
12
and — = cost of 1 dozen in shillings.
x
But, if x + 2 = number of eggs for a shilling,
12
— cost of 1 dozen in shillings.
x + 2
12 12 1
.*. = — (1 penny being ^ of a shilling).
x x + 2 12
The solution of which gives x = 16, or — 18.
And, if 16 eggs cost a shilling, 1 dozen will cost J| of a shilling,
or 9 pence.
Therefore, the price of the eggs is 9 pence per dozen.
If the problem be changed so as to read : What is the
price of eggs per dozen when two less in a shilling's worth
raises the price 1 penny per dozen ? the algebraic statement
Will be 12 12_1
x—2 x = 12*
The solution of which gives x = 18, or — 16.
Hence, the number 18, which had a negative sign and was inappli-
cable in the original problem, is here the true result.
216 ALGEBRA.
Exercise XC.
1. The sum of the squares of three consecutive numbers is
j 365. Find the numbers.
2. Three times the product of two consecutive numbers
exceeds four times their sum by 8. Find the numbers.
3. The product of three consecutive numbers is equal to
three times the middle number. Find the numbers.
4. A boy bought a number of apples for 16 cents. Had
he bought 4 more for the same money he would have
paid J of a cent less for each apple. How many did
he buy ?
5. For building 108 rods of stone-wall, 6 days less would
have been required if 3 rods more a day had been
built. How many rods a day were built ?
6. A merchant bought some pieces of silk for $900. Had
he bought 3 pieces more for the same money he would
have paid $ 15 less for each piece. How many did
he buy ?
7. A merchant bought some pieces of cloth for $168.75.
He sold the cloth for $ 12 a piece and gained as much
as 1 piece cost him. How much did he pay for each
piece ?
8. Find the price of eggs per score when 10 more in 62}
cents' worth lowers the price 311 cents per hundred.
9. The area of a square nfay be doubled by increasing its
length by 6 inches and its breadth by 4 inches. De-
termine its side.
10. The length of a rectangular field exceeds the breadth
by 1 yard, and the area is 3 acres. Determine its
dimensions.
QUADRATIC EQUATIONS. 217
11. There are three lines of which two are each $ of the
third, and the sum of the squares described on them
is equal to a square yard.* Determine the lengths of
the lines in inches.
12. A grass plot 9 yards long and 6 yards broad has a path
round it. The area of the path is equal to that of
the plot. Determine the width of the path.
13. Find the radius of a circle the area of which would be
doubled by increasing its radius by 1 inch.
14. Divide a line 20 inches long into two parts so that the
rectangle contained by the whole and one part may
be equal to the square on the other part.
15. A can do some work in 9 hours less time than B can do
it, and together they can do it in 20 hours. How
long will it take each alone to do it ?
16. A vessel which has two pipes can be filled in 2 hours
less time by one than by the other, and by both to-
gether in 2 hours 55 minutes. How long will it take
each pipe alone to fill the vessel ?
17. A vessel which has two pipes can be filled in 2 hours
less time by one than by the other, and by both to-
gether in 1 hour 52 minutes 30 seconds. How long
will it take each pipe alone to fill the vessel ?
ia An iron bar weighs 36 pounds. If it had been 1 foot
longer each foot would have weighed i a pound less.
Find the length and the weight per foot.
19. A number is expressed by two digits, one of which is the
square of the other, and when 54 is added its digits
are interchanged. Find the number.
20. Divide 35 into two parts so that the sum of the two
fractions formed by dividing each part by the other
may be 2^.
218 ALGEBRA.
21. A boat's crew row 3} miles down a river and back
again in 1 hour 40 minutes. If the current of the
river ii 2 miles per hour, determine their rate of row-
ing in still water.
22. A detachment from an army was marching in regular
column with 5 men more in depth than in front. On
approaching the enemy the front was increased by
845 men, and the whole was thus drawn up in 5 lines.
Find the number of men.
23. A jockey sold a horse for $ 144, and gained as much per
cent as the horse cost. What did the horse cost ?
24. A merchant expended a certain sum of money in goods,
which he sold again for $ 24, and lost as much per
cent as the goods cost him. How much did he pay
for the goods ?
25. A broker bought a number of bank shares ($100 each),
when they were at a certain per cent discount, for
$ 7500 ; and afterwards when they were at the same
per cent premium, sold all but 60 for $5000. How
many shares did he buy, and at what price ?
26. The thickness of a rectangular solid is $ of its width,
and its length is equal to the sum of its width and
thickness ; also, the number of cubic yards in its vol-
ume added to the number of linear yards in its edges
is -| of the number of square yards in its surface.
Determine its dimensions.
27. If a carriage-wheel 16} feet jound took 1 se # cond more
to revolve, the rate of the carriage per hour would be
1-J- miles less. At what rate is the carriage travelling ?
CHAPTER XV.
Simultaneous Quadratic Equations.
237. Quadratic equations involving two unknown quan-
tities require different methods for their solution, according
to the form of the equations.
238. Case I. When from one of the equations the value
of one of the unknown quantities can be found in terms of
the other, and this value substituted in the other equation.
Ex. Solve: 3*>- 2^ = 51 (1)
x-y = 2 J (2)
Transpose * in (2), y = x — 2.
Substitute in (1), 3 x* - 2x (x - 2) - 5.
The solution of which gives x = 1 or — 5.
.\y = -lor-7.
Special methods often give more elegant solutions of examples than
the general method by substitution.
I. When equations have the form, x ± y = a, and xy = b; x* ± y*
=» a, and xy =*b; or, x±y = a, and x* + y* = b.
a)
Dive: x + y =
= 401
)0 J
(1)
xy =
= 3(
(2)
Square (1),
**-
f 2xy+y* =
= 1600.
(3)
Multiply (2) by 4,
4*1/
= 1200.
= 400.
(4)
Subtract (4) from (3),
X*-
-2xy+y* =
Extract root of each side.
X-
-y-
= ±20.
(»)
Add (1) and (6),
2s =
\x =
= 60 or
= 30 or
20,
10.
Subtract (6) from (1),
2y =
•••y=
= 20 or
-10 or
60,
30.
220 ALGEBRA.
Square (1), a? -2^ + ^ = 16. (3)
Subtract (2) from (3), - 2 xy = - 24. (4)
Subtract (4) from (2), a? + 2 ay + y* = 64.
Extract the root, x + y = ± 8. (5)
By combining (5) a,nd (1), x = 6 or — 2.
y = 2 or -6.
(3) Solve:
ar + y 20
1 , 1 _ 41
x* y* 400 >
a* xy
t(3),
1 _ 81
y* = *400'
2 40
27 "400*
(1)
(2)
8 q uare<l), ? + £ + £-& < 3 >
Subtract (2) from (3), 1 - *j_ (4)
zy M00
Subtract (4) from (2), I - ^ + - 1 - = — .
' a* ^ zy y 8 400
Extract iiie root, I - I = ± - . (5)
x y 20 w
By combining (1) and (5), x = 4 or 5.
y =» 5 or 4.
II. When one equation may be simplified by dividing it by the other.
. x+y=1 i
(1)
(2)
Divide (1) by (2), x> - xy + y* = 13.
(3)
Square (2), a? +2 ay + y* = 49.
(4)
Subtract (3) &om (4), 3 xy = 36.
Divide by — 3, — xy = — 12.
(5)
Add (5) and (3), s*-2*y + y* = l.
Extract the root, x— y = ±l.
(6)
By combining (6) and (2), x =■ 4 or 3.
y-3or4.
SIMULTANEOUS QUADRATIC EQUATIONS.
221
Solve :
L a; + y = 13)
sy = 36 J
2. ar + y = 29j
xy = 100 J
3. a;-y=191
ary = 66 J
4. a: — y = 45 1
a^ = 250 J
5. x-'y = \0 \
^ 2 + y 2 =178JU
Exercise XCI.
11. # + y = 49 1
^ + 2^=1681/
12. z 8 + 3/ 8 =34n
a; + y = ll J
13. 2^ + ^=10081
x + y = l2 i
14. ^-y , = 98')
ir — y = 2 J
15. s*- 2 / 8 = 279)
X -y=S J
6. a; — y = 14 1
^ + ^ = 436J
7. a; + y = 12 1
^ + ^=104 J
16. x — 3y=ll
*y+y*= 5 J
17. 4y = 5a; + l \
2a;y = 33-a*J
a: y 4
a? V 16 J
9. M=5
a: y
10. 7**-8ay = 1591
5*+2y = 7 J
ia 1-1 = 3
a; y
1-1 = 21
19. 1_1 = 2}.
x y
1-1-8*
20. ^-2ay-y*=ll
* + y = 2 J
222 ALGEBRA.
)
239. Case II. When each of the two equations is homo-
gcneou8 and of the second degree.
Ex. Solve: 2^-4^ + 3^=171 (1)
y»-*»=16 J (2)
9 Let y = vx, and substitute vx for y in both equations.
From (1), 2v*x* - 4vx* + 3x* - 17,
2v» - 4v + 3*
From (2), v*3»
- a* = 16,
■.*- 16 .
Equate the values of x 9 ,
17 16
2v*
-4v + 3 v*-l'
Z2v*-(
34v + 48 =17^-17,
15v*-64i;=-65.
The solution gives,
„ 5 ^ 13
V= 3° r 5-
Substitute the value of v in
v»-l
then,
a-=9 or 25,
9
5
/. x = ± 3 or ± - ,
3
and
y =. vx = ± 5 or ± — .
c, , Exercise XCII.
Solve :
1.
z 2 + :ry + 2y 3 =74 |
2* 2 + 2:ry + y 2 =73J
4. £ 2 -4 2 / 2 -9 =
^ + 2 2 / 2 -3 =
0/
2.
3^ + 82/ 2 =14 /
5. of — xy — 35 =
*y + y"-18 =
0/
3.
y"-2ay = -15 /
2^ — ^ + 3^ =
= 441
= 16)
SIMULTANEOUS QUADRATIC EQUATIONS. 223
a? + xy-\b = 0\ 9. 2* 2 + 3zy + if= 701
xy — y 2 — 2 = J 6:^ + ^-—^ =
a 2 — #y + y 2 = 7 1 10. a? — xy — y* = b
3s 2 +13zy + 8y 2 =162J 2^ + 3^ + ^ = 28.
240. Case III. When the two equations are symrnetrical
with respect to # and y ; that is, when they have x and y
similarly inyolved in them. #
Thus, the expressions 2 a? + 3 afy* -\-2y 3 , 2xy — Sx — 3y + 1 , a? 4 —
3 s*y — 3 ay* + y 4 are symmetrical expressions.
(1) Solve: s' + ^lSsyl CD
' * + y = 12 J (2)
Put w + v for a;, and u — v for y, in (1) and (2).
(1) becomes (u + vf + (u — vf = 18 (u + v) (u — v) t
or u 8 + 3uv» = 9 (u* - v*). (3)
(2) becomes (w + v) + (w — t>) = 12,
or 2w = 12,
.*. u - 6.
Substitute 6 for u in (3).
(3) becomes
whence,
216 +
18v* = 9 (36 -
.*. v = ± 2,
-A
.*. a; = it + v =
= 6±2 =
■ 8 or 4,
y-«
i-v = 6*2 =
= 4 or 8.
* 4 + 2/
= 8 )
' = 706/
(1)
(2)
and
[2) Solve:
Put u + v for a?, and u — v for y, in (1) and (2).
(1) becomes (u + v) + (u — v) = 8,
/. u = 4.
(2) becomes u 4 + 6wV + v 4 = 353. (3)
Substitute 4 for u in (3),
256 + 96^ + ^ = 353,
or, v 4 + 96 1? = 97. (4)
The solution of (4) gives v = ± 1 or ± V— 97.
Taking the possible values of v t x = 5 or 3, and y = 3 or 5.
224 ALGEBRA.
Solve • Exercise XCIII.
1. 4zy = 96-*yi ± 4(s + y) = 3*y \
x + y = 6 J x + y + x * + y*= = 26)
2. a* + y*=18 — x-yl 5. 4^ + ^ + 4^ = 58 1
*y = 6 J bx* + by 2 = 65 J
3. 2(x* + y>) = 5xyl 6. xy(x + y) = S0)
4 (# — y)=xy i a 3 + ^* = 35 J
24L The preceding cases are general methods for the solution of
equations which belong to the kinds referred to ; often, however, in
the solution of these and other kinds of simultaneous equations in-
volving quadratics, a little ingenuity will suggest some step by which
the roots may easily be found.
o , . Exercise XCIV.
1. x-y=7 \ 8. x~y = l 1
x* + xy + y*=13) ^ + ^ = 8}/
2. ^ + ^=351 9. x* + ±xy = Z )
3y-yS = 6 J 4:xy + y* = 2i)
3. a^-12 = 0| 10. x* — xy + y*=4:8)
x-2y=5) x — y — 8 = J
4. sy_7 = 1 11. x* + Sxy + y*=l 1
^ + 3^ = 50/ Sx* + xy + 3y* = 13)
5. 2*--5y = 9 1 12. x*-2xy + 3y*=l%)
x* — xy + y a =l ) z* + zy—y* = i. J
& x — y = 9 1 13. x + y = a 1
^ + 8 = 0} 4^-a 2 = -46 2 /
7. hx — 7y = ^ 14 «-y = l ]
5^_1^ = 4 __ V ? + ? = 2i
4 J y a? J
SIMULTANEOUS QUADRATIC EQUATIONS.
225
15. z* + 9*y = 34Cn
16. x + y = 6 I
> = 72J
2a x*-y s = a 3 )
x— y = a J
27. x* — xy = a* + b*
^4-y3 = 7 2 / xy-y i = 2ab
17. 3zy + 2* + y = 485) 28. ^-y* = 4a5\
}
3:r-2y = J
18. a: — y = l \
s»-y» = 19J
19. a» + y» = 2728 \
x'-xy + y 2 = 124J
20. a; + 3/ = a 1
^ + ^ = 5 2 i
21. x*-y* =
Srf-lxy+by*
x + y , g-y_10
-.}
22.
z* + y» = 45
23.1 + 1 =
x y
1
17
x + 1 y+1 12
24. x z -xy + y i = 7 \
*H*y+y 4 = 133 J
25. 3 + 2/ = 4 1
a- l + y* = 82J
xy = a* — b* )
29. a;y = 1
s» + y» = 16J
3a a^ + 3y + ^ = 37 1
3 4 + x 2 y 2 + y 4 --=48lJ
31. 3* = ax + 5y 1
y* = ay + far J
32. s-y-2=0 |
15(3 i -y 2 )=163yJ
33. x +y | g -y_ 89 }
*— y *+y 40 J.
6*==20y + 9 J
34. 5 + 5=1
re y
35. s» + y»=7 + *y )
aJ + yS = 6icy-l/
36. ^-^ = 3093"!
x-y=Z J
37. |(*-l)-i(*+l)(y-l) = -lU
• i(y + 2) = i(* + 2) J
38. 10x> + 15xy = Sab- 2a' \
\0y l +\bxy = Zab-2b i i
22G ALGEBRA.
Exercise XCV.
1. If the length and breadth of a rectangle were each in-
creased by 1, the area would be 48 ; if they were each
diminished by 1, the area would ,be 24. Find the
length and breadth.
2. The sum of the squares of the two digits of a number is
25, and the product of the digits is 12. Find the
number.
3. The sum, the product, and the difference of the squares,
of two numbers are all equal. Find the numbers.
Note. Represent the numbers by x + y and x— y, respectively.
4. The difference of two numbers is f of the greater, and
the sum of their squares" is 356. What are the num-
bers?
5. The numerator and denominator of one fraction are each
greater by 1 than those of another, and the sum of
the two fractions is 1-^j- ; if the numerators were in-
terchanged the sum of the fractions would be 1-J-.
Find the fractions.
6. A man starts from the foot of a mountain to walk to its
summit. His rate of walking during the second half
of the distance is \ mile per hour less than his rate
during the first half, and he reaches the summit in
5} hours. He descends in 3f hours, by walking 1
mile more per hour than during the first half of the
ascent. Find the distance to the top and the rates
of walking.
Note. Let 2 »«= the distance, and y miles per hour = the rate at first.
Then - + — 5* hours, and -^- - 3f hours.
y y-i y+ l
SIMULTANEOUS QUADRATIC EQUATIONS. 227
7. The sum of two numbers which are formed by the same
two digits in reverse order is -f £ of their difference ;
and the difference of the squares of the numbers is
3960. Determine the numbers.
8. The hypotenuse of a right triangle is 20, and the area
of the triangle is 96. Determine the sides.
Note. The square on the hypotenuse = sum of the squares on the
sides ; and the area of a right triangle = J product of Bides.
9. Two boys run in opposite directions round a -rectangular
field the area of which is an acre ; they start from
one corner and meet 13 yards from the opposite cor-
ner ; and the rate of one is J- of the rate of the other.
Determine the dimensions of the field.
10. A, in running a race with B, to a post and back, met
him 10 yards from the post. To make it a dead heat,
B must have increased his rate from this point 41-f-
yards per minute ; and if, without changing his pace,
he had turned back on meeting A, he would have
come 4 seconds after him. How far was it to the
post?
11. The fore wheel of a carriage turns in a mile 132 times
more than the hind wheel ; but if the circumferences
were each increased by 2 feet, it would turn only 88
times more. Find the circumference of each.
12. A person has $6500, which he divides into two parts
and loans at different rates of interest, so that the two
parts produce equal returns. If the first part had
been loaned at the second rate of interest, it would
have produced $ 180 ; and if the second part had been
loaned at the first rate of interest, it would have pro-
duced $ 245. Find the rates of interest.
CHAPTER XIX.
Logarithms.
292. In the common system of notation the expression
of numbers is founded on their relation to ten.
Thus, 3854 indicates that this number contains 10 3 three times, 10*
eight times, 10 five times, and four unite.
293. In this system a number is represented by a series
of different powers of 10, the exponent of each power being
integral. But, by employing fractional exponents, any num-
ber may be represented (approximately) as a single power
of 10.
294. When numbers are referred in this way to 10, the
exponents of the powers corresponding to them are called
their logarithms to the base 10.
For brevity the word "logarithm" is written log.
From § 255 it appears that :
io»= i, io- 1 (=^) =1,
10* = 10, 10- , (= T * T ) =01,
10 s = 100, 10-»(= TT ^) = .001,
and so on. Hence,
log 1=0, log.l =-1,
log 10 = 1, log .01 =-2,
log 100 = 2, log .001 =-3,
and so on.
COLS ALGEBRA.
V
It is evident that the logarithms of all numbers between
1 and 10 will be -f- a fraction,
10 and 100 will be 1 + a fraction,
100 and 1000 will be 2 + a fraction,
1 and .1 will be — 1 + a fraction,
.1 and .01 will be — 2 + a fraction,
.01 and .001 will be — 3 + a fraction.
295. The fractional part of a logarithm cannot be ex-
pressed exactly either by common or by decimal fractions ;
but decimals may be obtained for these fractional parts,
true to as many places as may be desired.
If, for instance, the logarithm of 2 be required ; log 2 may be sup-
posed to be £.
Then 10* = 2 ; or, by raising both sides to the third power, 10 =■ 8,
a result which shows that J is too large. a
Suppose, then, log 2 = -j^. Then 10" = 2, or by raising both sides
to the tenth power, 10 3 = 2 10 . That is, 1000 = 1024, a result which
shows that -fa is too small.
Since £ is too large and ^ too small, log 2 lies between J and ^ ;
that is, between .33333 and .30000.
In supposing log 2 to be J, the error of the result is if^A = ^ = .2.
In supposing log 2 to be ^, the error of the result is XQO i°oo J o 02i B
f$fo = — .024 ; log 2, therefore, is nearer to ^ than to J.
The difference between the errors is .2 — (— .024) = .224, and the
difference between the supposed logarithms is .33333 — .3 = .03333.
The last error, therefore, in the supposed logarithm may be consid-
ered to be approximately -£fc of .03333 = .0035 nearly, and this added
to .3000 gives .3035, a result a little too large.
By shorter methods of higher mathematics, the logarithm of 2 is
known to be 0.3010300, true to the seventh place.
296. The logarithm of a number consists of two parts, an
integral part and a fractional part.
Thus, log 2 =» 0.30103, in which the integral part is 0, and the frac-
tional part is .30103 ; log 20= 1.30103, in which the integral part is 1,
and the fractional part is .30103.'
LOGAEITHMS. 263
297. The integral part of a logarithm is called the char-
acteristic ; and the fractional part is called the mantissa.
298. The mantissa is always made positive. Hence, in
the case of numbers less than 1 whose logarithms are nega-
tive, the logarithm is made to consist of a negative charac-
teristic and a positive mantissa.
299. When a logarithm consists of a negative character-
istic and a positive mantissa, it is usual to write the minus
sign over the characteristic, or else to add 10 to the charac-
teristic and to indicate the subtraction of 10 from the
resulting logarithm.
Thus, log .2 = 1.30103, and this may be written 9.30103 - 10.
300. The characteristic of a logarithm of an integral num-
ber, or of a mixed number, is one less than the number of
integral digits.
Thus, from I 294, log 1 = 0, log 10 = 1, log 100 = 2. Hence, the
logarithms of all numbers from 1 to 10 (that is, of all numbers con-
sisting of one integral digit), will have for characteristic ; and the
logarithms of all numbers from 10 to 100 (that is, of all numbers con-
sisting of two integral digits), will have 1 for characteristic ; and so
on, the characteristic increasing by 1 for each increase in the number
of digits, and therefore always being 1 less than that number.
301. The characteristic of a logarithm of a decimal frac-
tion is negative, and is equal to the number of the place occu-
pied by the first significant figure of the ^decimal.
Thus, from \ 294, log .1 - - 1, log .01 = -2, log .001 = -3. Hence,
the logarithms of all numbers from .1 to 1 will have — 1 for a charac-
teristic (the mantissa being plus) ; the logarithms of all numbers from
.01 to .1 will have — 2 for a characteristic ; the logarithms of all num-
bers from .01 to .001 will have — 3 for a characteristic ; and so on,
the characteristic always being negative and equal to the number of the
place occupied by the first significant figure of the decimal.
264 ALGEBRA.
302. The mantissa of a logarithm of any integral num-
ber or decimal fraction depends only upon the digits of the
number, and is unchanged so long as the sequence of the
digits remains the same.
For, changing the position of the decimal point in a number ii
equivalent to multiplying or dividing the number by a power of 10.
Its logarithm, therefore, will be increased or diminished by the expo-
nent of that power of 10 ; and, since this exponent is integral, the
mantissa of the logarithm will be unaffected.
Thus, if 27196 = 10 4 - 434 *,
then 2719.6 = 10 3434 *,
27.196 = 10 1 - 4345 ,
2.7196 - 10 04345 ,
.27196 = 10»-« 4 *- ,o f
.0027196 - 10 7434 *- 10 .
803. The advantage of using the number 10 ae the base
of a system of logarithms consists in the fact tnat the man-
tissa depends only on the sequence of digits* and the charac-
teristic on the position of the decimal point.
304. As logarithms are simply exponents (§ 294), therefore,
The logarithm of a product is the sum of the logarithms
of the factors.
Thus, log 20 _ log (2 X 10) = log 2 + log 10
- 0.3010 + 1.0000 = 1.3010 ; '
log 2000 = log (2 x 1000) = log 2 + log 1000,
- 0.3010 + 3.0000 = 3.3010 ;
log .2 - log (2 X .1) = log 2 + log .1,
- 0.3010 + 9.0000 - 10 - 9.3010 - 10 ;
log .02 = log (2 X .01) - log 2 + log .01,
= 0.3010 + 8.0000 - 10 = 8.3010 - 10.
Exercise CIX.
Given : log 2 = 0.3010 ; log 3 = 0.4771 ; log 5 = 0.6990 ;
log 7 = 0.8451.
Find the logarithms of the following numbers by resolv-
LOGARITHMS. 265
ing the numbers into factors, and taking the sum of the
logarithms of the factors :
1. log 6. 9. log 25. 17. log .021. 25. log 2.1.
2. log 15. 10. log 30. ia log .35. 26. log 16.
a log 21. 1L log 42. 19. log .0035. 27. log .056.
4. log 14. 12. log 420. 20. log .004. 28. log .63.
6. log 35. 13. log 12. 21. log .05. 29. log 1.75.
6. log 9. 14. log 60. 22. log 12.5. 30. log 105.
7. log 8. 15. log 75. 23. log 1.25. 3L log .0105.
a log 49. 16. log 7.5. 24. log 37.5. 32. log 1.05.
305. As logarithms are simply exponents (§ 294), there-
fore,
The logarithm of a power of a number is equal to the
logarithm of the number multiplied by the exponent of the
power.
Thus, log5 7 = 7 X log 5= 7x0.6990 = 4.8930.
log 3 11 = 11 x log 3 - 11 x 0.4771 - 5.2481.
306. As logarithms are simply exponents (§ 294), there-
fore, when roots are expressed by fractional indices,
The logarithm of a root of a number is equal to the loga-
rithm of the number multiplied by the index of the root.
Thus, log 2* = i of log 2 - J X 0.3010 - 0.0753.
log .002* = J of (7.3010 - 10).'
The expression i of (7.3010 - 10)
may be put in the form of J of (27.3010 - 30) which = 9.1003 - 10 ;
for, since 20 - 20 = 0, the addition of 20 to the 7, and of - 20 to the
— 10, produces no change in the value of the logarithm.
307. In simplifying the logarithm of a root the equal pos-
itive and negative numbers to be added to the logarithm must
be such that the resulting negative number, when divided by
the index of the root, shall give a quotient of — 10.
266 ALGEBRA.
16. 7f
a. 5i
17. 5*.
22. 2V.
18. 3A.
23. 5i
19. 7*.
24. 7V.
20. 3i
25. 21*.
Exercise CX.
Given : log 2 = 0.3010 ; log 3 = 0.4771 ; log 5 = 0.6990 ;
log 7 = 0.8451.
Find logarithms of the following :
1. 2 s . 6. 5 5 . 11. 5*.
2. 5 8 . 7. 2*. 12. 7tV.
3. 7 4 . a 5*. 13. 2*.
4. 3 8 . 9. 3*. 14. 5*.
5. 7 s . 10. 7*. 15. 3*.
308. Since logarithms are simply exponents (§ 294), there-
fore,
The logarithm of a quotient is the logarithm of the divi-
dend minus the logarithm of the divisor.
Thus, log f = log 3 - log 2 = 0.4771 - 0.3010 = 0.1761.
log } = log 2 - log 3 = 0.3010 - 0.4771 = - 0.1761.
To avoid the negative logarithm — 0.1761, we subtract the entire
logarithm 0.1761 from 10, and then indicate the subtraction of 10
from the result.
Thus, - 0.1761 - 9.8239 - 10.
Hence, log f = 9.8239 - 10.
309. The remainder obtained by subtracting the loga-
rithm of a number from 10 is called the oologarithm of the
number, or arithmetical complement of the logarithm of the
number.
Cologarithm is usually denoted by colog, and is most
easily found by beginning with the characteristic of the loga-
rithm and subtracting each figure from 9 down to the last
significant figure, and subtracting that figure from 10.
Thus, log 7 = 0.8451 ; and colog 7 = 9.1549. Colog 7 is readily
found by subtracting, mentally, from 9, 8 from 9, 4 from 9, 5 from
9, 1 from 10, and writing the resulting figure at each step.
LOGARITHMS. 267
310. Since colog 7 = 9.1549,
and log f = log 1 - log 7 = - 0.8451 = 9.1549 - 10,
it is evident that,
If 10 be subtracted from the cologarithm of a number, the
result is the logarithm of the reciprocal of that, number.
311. Since log £ = log 7 — log 5,
= 0.8451 - 0.6990 = 0.1461,
and log 7 + colog 5 - 10 = 0.8451 + 9.3010 - 10,
= 0.1461,
it is evident that,
The addition of a cologarithm — 10 is equivalent to the
subtraction of a logarithm.
The steps that lead to this result are :
l = 7xi
therefore, log J = log (7 X i) - log 7 + log £. \ 304.
But log \ = colog 5-10. \ 309.
Hence, log \ — log 7 + colog 5 — 10.
Therefore,
312. The logarithm of a quotient may be found by add-
ing together the logarithm of the dividend and the cohga-
rithm of the divisor, and subtracting 10 from the result.
In finding a cologarithm when the characteristic of the logarithm
is a negative number, it must be observed that the subtraction of a
negative number is equivalent to the addition of an equal positive
number.
Thus, log -^- = log 5 + colog .002 - 10,
= 0.6990 + 12.6990-10,
= 3.3980.
Here log .002 =» 3.3010, and in subtracting — 3 from 9 the result ifl
the same as adding + 3 to 9.
o
Again, log — =* log 2 + colog .07 — 10,
-0.3010 + 11.1549-10,
» 1.4559.
268 ALGEBRA.
Also,
log -W.
= 8.8451 -
-10 + 9.0970-
-10,
Here,
Hence,
log2» =
colog 2 s =
= 17.9421-20,
= 7.9421 - 10.
=» 3 log 2 = 3 x
= 10-0.9030 =
0.3010 «
. 9.0970.
: 0.9030.
Exercise CXI.
Given : log 2 = 0.3010 ; log 3 = 0.4771 ; log 5 = 0.6990 ;
log 7 = 0.8451.
Find logarithms for the following quotients :
1. I 7. I 13. $*. 19. M. 25. ™
5 3 3 .003 3 8
2. ?. 8. I 14. ^. 20. 427. ,26. *
7 2 2 .02 .02»
3. 2. 9. |. 15. 4- 7 . 21. i» «r. 3L
5 3 5 .007 .02 s
4. | 10. I 16. A. 22. ™. 28. **
7 2 .07 .07 .003 s
5. 5. 11. I 17. JL. 23. & 29. =°°?.
7 2 .007 7 7 s
6 . 7 . 12> 7 18. m 24. =2927 30. *
5 .5 7 .2 .005*
313. A table of four-place logarithms is here given, which
contains logarithms of all numbers under 1000, the decimal
point and characteristic being omitted. The logarithms of
single digits 1, 8, etc., will be found at 10, 80, etc.
Tables containing logarithms of more places can be pro-
cured, but this table will serve for many practical uses, and
will enable the student to use tables of six-place, seven-
place, and ten-place logarithms, in work that requires
greater accuracy.
LOGARITHMS. 269
314. In working with a four-place table, the numbers
corresponding to the logarithms, that is, the antilogarithms,
as they are called, may be carried to four significant digits.
To Find the Logarithm of a Number in this Table.
316. Suppose it is required to find the logarithm of 65.7.
In the column headed "N" look for the first two significant
figures, and at the top of the table for the third significant
figure. In the line with 65, and in the column headed 7,
is seen 8176. To this number prefix the characteristic and
insert the decimal point. Thus,
log 65.7 = 1.8176.
Suppose it is required to find the logarithm of 20347.
In the line with 20, and in the column headed 3, is seen
3075 ; also in the line with 20, and in the 4 column, is seen
3096, and the difference between these two is 21. The dif-
ference between 20300 and 20400 is 100, and the difference
between 20300 and 20347 is 47. Hence, -^ of 21 = 10,
nearly, must be added to 3075. That is,
log 20347 =4.3085.
Suppose it is required to find the logarithm of .0005076.
In the line with 50, and in the 7 column, is seen 7050 ; in
the 8 column, 7059 : the difference is 9. The difference be-
tween 5070 and 5080 is 10, and the difference between 5070
and 5076 is 6. Hence, -& of 9 = 5 must be added to 7050.
Thftt 1S ' log .0005076 = 6.7055 - 10.
To Find a Number when its Logarithm is Given.
316. Suppose it is required to find the number of which
the logarithm is 1.9736.
Look for 9736 in the table. In the column headed " N,"
and in the line with 9736, is seen 94, and at the head of
270
ALGEBRA.
N
1
2
3
4
5
6
7
8
9
10
11
12
13
14
0000
0414
0792
1139
1461
0043
0453
0828
1173
1492
0086
0492
0864
1206
1523
0128
0531
0899
1239
1553
0170
0569
0934
1271
1584
0212
0607
0969
1303
1614
0253
0645
1004
1335
1644
0294
0682
1038
1367
1673
0334
0719
1072
1399
1703
0374
0755
1106
1430
1732
15
16
17
18
19
1761
2041
2304
2553
2788
1790
2068
2330
2577
2810
1818
2095
2355
2601
2833
1847
2122
2380
2625
2856
1875
2148
2405
2648
2878
1903
2175
2430
2672
2900
1931
2201
2455
2695
2923
1959
2227
2480
2718
2945
1987
2253
2504
2742
2967
2014
2279
2529
2765
2989
20
21
22
23
24
3010
3222
3424
3617
3802
3032
3243
3444
3636
3820
3054
3263
3464
3655
3838
3075
3284
3483
3674
3856
3096
3304
3502
3692
3874
3118
3324
3522
3711
3892
3139
3345
3541
3729
3909
3160
3365
3560
3747
3927
3181
3385
3579
3766
3945
3201
3404
3598
3784
3962
25
26
27
28
29
3979
4150
4314
4472
4624
3997
4166
4330
4487
4639
4014
4183
4346
4502
4854
4031
4200
4362
4518
4669
4048
4216
4378
4533
4683
4065
4232
4393
4548
4698
4082
4249
4409
4564
4713
4099
4265
4425
4579
4728
4116
4281
4440
4594
4742
4133
4298
4456
4609
4757
30
31
32
33
34
4771
4914
5051
5185
5315
4786
4928
5065
5198
5328
4800
4942
5079
5211
5340
4814
4955
5092
5224
5353
4829
4969
5105
5237
5366
4843
4983
5119
5250
5378
4857
4997
5132
5263
5391
4871
5011
5145
5276
5403
4886
5024
5159
5289
5416
4900
5038
5172
5302
5428
35
36
37
38
39
5441
5563
5682
5798
5911
5453
5575
5694
5809
5922
5465
5587
5705
5821
5933
5478
5599
5717
5832
5944
5490
5611
5729
5843
5955
5502
5623
5740
5855
5966
5514
5635
5752
5866
5977
5527
5647
5763
5877
5988
5539
5658
5775
5888
5999
5551
5670
5786
5899
6010
40
41
42
43
44
45
46
47
48
49
5<T
51
52
53
54
6021
6128
6232
6335
6435
6031
6138
6243
6345
6444
6012
6149
6253
6355
6454
6053
6160
6263
6365
6464
6064
6170
6274
6375
6474
6075
6180
6284
6385
6484
6085
6191
6294
6395
6493
6096
6201
6304
6405
6503
6107
6212
6314
6415
6513
6117
6222
6325
6425
6522
6618
6712
6803
6893
6981
6532
6628
6721
6812
6902
6542
6037
6730
6821
6911
6551
6646
6739
6830
6920
6561
6656
6749
6839
6928
6571
6665
6758
6848
6937
6580
6675
6767
6857
6946
6590
6684
6776
6866
6955
6599
6693
6785
6875
6964
6609
6702
6794
6884
6972
6990
7076
7160
7243
7324
6998
7084
7168
7251
7332
7007
7093
7177
7259
7340
7016
7101
7185
7267
7348
7024
7110
7193
7275
7356
7033
7118
7202
7284
7364
7042
7126
7210
7292
7372
7050
7135
7218
7300
7380
7059
7143
7226
7308
7388
7067
7152
7235
7316
7396
LOGARITHMS.
271
N
1
2
3
4
5
6
7
8
9
55
56
57
58
59
7404
7482
7559
7634
7709
7412
7490
7566
7642
7716
7419
7497
7574
7649
7723
7427
7505
7582
7657
7731
7435
7513
7589
7664
7738
7443
7520
7597
7672
7745
7451
7528
7604
7679
7752
7459
7536
7612
7686
7760
74GG
7543
7619
7694
7767
7474
7551
7627
7701
7774
60
61
62
63
64
65
66
67
68
69
7782
7853
7924
7993
8062
7789
7860
7931
8000
8069
7796
7868
7938
8007
8075
7803
7875
7945
8014
8082
7810
7882
7952
8021
8089
7818
7889
7959
8028
8096
7825
7896
7966
8035
8102
7832
7903
7973
8041
8109
7839
7910
7980
8048
8116
7846
7917
7987
8055
8122
8129
8195
8261
8325
8388
8136
8202
8267
8331
8395
8142
8209
8274
8338
8401
8149
8215
8280
8344
8407
8156
8222
8287
8351
8414
8162
8228
8293
8357
8420
8169
8235
8299
8363
8426
8176
8241
8306
8370
8432
8182
8248
8312
8376
8439
8189
8254
8319
8382
8445
70
71
72
73
74
8451
8513
8573
8633
8692
8457
8519
8579
8639
8698
8463
8525
8585
8645
8704
8470
8531
8591
8651
8710
8476
8537
8597
8657
8716
8482
8543
8603
8663
8722
8488
8549
8609
$669
8727
8494
8555
8615
8675
8733
8500
8561
8621
8681
8739
8506
8567
8627
8686
8745
75
76
77
78
79
8751
8808
8865
8921
8976
8756
8814
8871
8927
8982
8762
8820
8876
8932
8987
8768
8825
8882
8938
8993
8774
8831
8887
8943
8998
8779
8837
8893
8949
9004
8785
8842
8899
8954
9009
8791
8848
8904
8960
9015
8797
8854
8910
8965
9020
8802
8859
8915
8971
9025
80
81
82
83
84
9031
9085
9138
9191
9243
9036
9090
9143
9196
9248
9042
9096
9149
9201
9253
9301
9355
9405
9155
9504
9047
9101
9154
9206
9258
9053
9106
9159
9212
92C3
9058
9112
9165
9217
9269
9063
9117
9170
9222
9274
9069
9122
9175
9227
9279
9074
9128
9180
9232
9284
9079
9133
9186
9238
9289
85
86
87
88
89
9291
9345
9395
9445
9494
9299
9350
9400
9450
9499
9309
9360
9410
9460
9509
9315
9365
9415
9465
9513
9320
9370
9420
9469
9518
9566
9614
9661
9708
9754
9325
9375
9425
9474
9523
9330
9380
9430
9479
9528
9335
9385
9435
9484
9533
9340
9390
9440
9489
9538
00
91
92
93
94
9542
9590
9638
9685
9731
9547
9595
9643
9689
9736
9552
9600
9647
9694
9741
9557
9605
9652
9699
9745
9562
9609
9657
9703
9750
9571
9619
9666
9713
9759
9576
9624
9671
9717
9763
9581
9628
9675
9722
9768
9586
9633
9680
9727
9773
95
96
97
98
99
9777
9823
9868
9912
9956
9782
9827
9872
9917
9961
9786
9832
9877
9921
9965
9791
9836
9881
9926
9969
9795
9841
9886
9930
9974
9800
9845
9890
9934
9978
9805
9850
9894
9939
9983
9809
9854
9899
9943
9987
9814
9859
9903
9948
9991
9818
9863
9908
9952
9996
272 ALQEBEA.
the column in which 9736 stands is seen 1. Therefore,
write 941, and insert the decimal point as the characteristic
directs. That is, the number required is 94.1.
Suppose it is required to find the number of which the
logarithm is 3.7936.
Look for 7936 in the table. It cannot be found, but the
two adjacent mantissas between which it lies are seen to be
7931 and 7938 ; their difference is 7, and the difference be-
tween 7931 and 7936 is 5. Therefore, f of the difference
between the numbers corresponding to the mantissas, 7931
and 7938, must be added to the number corresponding to
the mantissa 7931.
The number corresponding to the mantissa 7938 is 6220.
The number corresponding to the mantissa 7931 is 6210.
The difference between these numbers is 10,
and 6210 + £ of 10 = 6217.
Therefore, the number required is 6217.
Suppose it is required to find the number of which the
logarithm is 7.3882 - 10.
Look for 3882 in the table. It cannot be found, but the
two adjacent mantissas between which it lies are seen to be
3874 and 3892 ; their difference is 18, and the difference
between 3874 and 3882 is 8. Therefore, ■& of the differ-
ence between the numbers corresponding to the mantissas,
3874 and 3892, must be added to the number corresponding
to the mantissa 3874.
The number corresponding to the mantissa 3892 is 2450.
The number corresponding to the mantissa 3874 is 2440.
The difference between these numbers is 10,
and 2440 + £ of 10 = 2444.
Therefore, the number required is .002444.
LOGARITHMS. 273
Exercise CXII.
Find logarithms of the following numbers :
1. 60. 6. 3780. lL 70633. l& 877.08,
% 101. 7. 54327. 12. 12028. 17. 73.896,
3. 999. a 90801. 13. 0.00987. IS. 7.0699.
4. 9901. 9. 10001. 14. 0.87701. 19. 0.0897.
5. 5406. ia 10010. 15. 1.0001. 20. 99.778,
Find antilogarithms to the following logarithms :
21. 4.2488. 2& 4.7317. 29. 9.0410-10.
22. 3.6330. » 1.9730. 30. 9.8420-10.
23. 2.5310. 27. 9.8800-10. 31. 7.0216-10.
21 1.9484. 28. 0.2787. 32. 8.6580-10.
Ex. Find the product of $08.4 X .05392 X 2.117.
log 908.4 = 2.9583
log .05392 = 8.7318 -10
log 2.117 - 0.3257
2:0158= log 103.7. An*.
Find by logarithms the following products :
33. 948.76x0.043875. 35. 830.75x0.0003769.
34. 3.4097x0.0087634. 36. 8.4395x0.98274.
317. When any of the factors are negative, find their log-
arithms without regard to the signs; write the letter n
after the logarithm that corresponds to a negative number.
If the number of logarithms so marked be odd, the product
is negative; if even, the product is positive.
274
ALGEBRA.
Find the products of :
37. 7564 x (-0.003764).
3a 3.7648 X (-0.083497).
Ex. Find the quotient of
39. -5.840359x(-0.00178).
40. -8945.07x73.846.
8.3709 X 834.637
log 8.3709 .
log 834.637
colog 7308.946
Find the quotients of :
70654
41.
43.
44.
45.
54013'
58706
93078*
8.32165
0.07891'
65039
90761'
7.652
-0.06875*
46.
47.
4a
49.
7308.946
= 0.9227
■ 2.9215
. 6.1362-10
9.9804 -10 -log .9558. An*.
0.07654
83.947 X 0.8395
7564 x 0.07643
8093 X 0.09817'
89 x 753 x 0.0097
36709 X 0.08497 '
413 x 8.17 X 3182
915 X 728 X 2.315'
212 x(- 6.12) x(- 2008)
365 X(- 531) X 2.576 '
Ex. Find the cube of .0497.
log .0497 = 8.6964 -10
6.0892 -10 -log .0001228. Am.
Find by logarithms :
51. 6.05 s . 55. 0.78765 8 .
52. 1.051 7 . 56. 0.691 9 .
53. 1.1768*. 57. (£f) u .
54. 1.3178 10 . 5a m) 7 .
59. (lGf)*
eo. my.
«• (Mi) 6 -
62. (7 T V) 0JB .
63. (3tf )*»
64. (W*.
65. (8|) 2 - 8 .
66. (534.) - 37 *.
LOGARITHMS. 275
Ex. Find the fourth root of 0.00862.
log 0.00862= 7.9355-10
30. -30
4)37.9355-40
9.4839 - 10 - log .3047. Am.
Find by logarithms :
67. 7*. 7a 8379*. 7a 0.17643*. 7a (tA^t)*
68. 11*. 71. 906.80*. 74. 2.5637* 77. (9££)*.
69. 783*. 72. 8.1904*. 75. (flft)*. 78. (lift)*.
Find by logarithms the values of:
D.0075433* X 78.343 X 8172.4* X 0.00052
79.
80.
81.
82.
83.
81
85.
sa
87.
64285.* X 154.27 4 X 0.001 X 586.79*
15.832* X 5793.6* X 0.78426
\0.000327* X 768.94* X 3015.3x0.007*'
J 7.1895 X 4764.2* X 0.00326*
\0.00048953 X 457 3 X 5764.4*'
« 13.1 416x4771.21x2.7183*
A30.103 4 X 0.4343* x 69.897 4 '
7 f 0.03271* X 53.429 X 0.77542 8
\ 32.769 X 0.000371 4
J 732.056* x 0.0003572* x 89793
\42.2798 8 x 3.4574 X 0.0026518*'
a / 7932x 0.00657 X 0.80464
\ 0.03274 X 0.6428 '
s / 7.1206 x V0.13274 X .057389
\ V0.43468 X 17.385 X V0.0096372*
f 3.075526* x 5771.2* X 0.0036984* x 7.74 ]*
X 72258 x 327.93 8 x 86.97* J '
276 ALGEBRA.
318. Since any positive number other than 1 may be
taken as the base of a system of logarithms, the following
general proofs to the base a should be noticed.
I. The logarithm of the product of two or more numbers
is equal to the sum of the logarithms of the numbers.
For, let m and n be two numbers, and x and y their logarithms.
Then, by the definition of a logarithm, m = a* and n = a*.
Hence, mxn = a*xa?=* a?+*.
.\ log (m X n) — x + y,
= log m + log w.
In like manner, the proposition may be extended to any number
of factors.
II. The logarithm of a quotient is equal to the logarithm
of the dividend minus the logarithm of the divisor.
For, let m and n be two numbers, and x and y their logarithms.
Then m = a* and n = <&.
Hence, m-f-7i = a*-^-ay = a*-».
.'. log (m -5- n) = x — y t
= log m — log n.
From this it follows that log — = log 1 — log m.
tn
But, since log 1 — 0, log — = — log m.
771
III. The logarithm of a power of a number is equal to
the logarithm of the number multiplied by the exponent of
the power.
For, let x be the logarithm of m.
Then m = a*,
and mP = (a*)P = a".
/. log m* = px,
«* p log 771.
LOGARITHMS. 277
IV. The logarithm of the root of a number is equal to the
logarithm of the number divided by the index of the root
For, let x be the logarithm of m.
Then m = d*
and rr} = (a*f = a?.
.■.logm>«*--l2&«
319. An exponential equation, that is, an equation in
which the exponent is the unknown quantity, is easily
solved by logarithms.
For, let aF = m.
Then log a* = log m,
.*. x log a = log m,
log m
log a
Ex. Find the value of x in 81* = 10.
81* -10,
T _loglO
log 81'
.*. log x = log log 10 + colog log 81,
= + 9.7193-10,
.\*-0.5&.
k
320. Logarithms of numbers to any base a may be con-
verted into logarithms to any other base b by dividing the
computed logarithms bj the logarithm of b to the base a.
For, let log m = y to the base 6,
and log b = x to the base a.
Then m = &*, and 6 = a*
.-. m = (a*)* = a«*.
/. log m (to base a) =» xy =» log 6 (to base a) x log m (to base b),
■•■ log m (to base *) - ^ ?, (to *" «)
' log 6 (to base a)
This is usually written, log* m = °^* 7 ? .
log«o
CO
FIVE-PLACE
LOGARITHMIC AND TRIGONOMETRIC
TABLES.
ARRANGED BY
G. A. WENTWORTH, A.M.,
AND
G. A, HILL, A.M.
o*«o
BOSTON:
PUBLISHED BY GINN & COMPANY.
1885.
Entered according to Act of Congress, in the year 1882, by
G. A. WENTWORTH AND G. A. HILL,
In the office of the Librarian of Congress at Washington.
Ginn, Heath, A, Co., Printers :
j. s. cushing, supt., 16 hawley street,
Boston.
INTRODUCTION.
<>5*$o
3. If the natural numbers are regarded as powers of ten, the ex-
ponents of the powers are the Common or Briggs Logarithms of the
numbers. If A and B denote natural numbers, a and b their loga-
rithms, then 10 a = A, 10* = B ; or, written in logarithmic form,
log^. = a, logJ3=6.
2. The logarithm of a product is found by adding the logarithms
of its factors.
For, AXB =10«Xl0* = 10«+».
Therefore, log (A X B) = a + b = log A + log B.
3. The logarithm of a quotient is found by subtracting the logarithm
of the divisor from that of the dividend.
A
Therefore, log ^ = a — b = log A — log B.
B
4. The logarithm of a power of a number is found by multiplying
the logarithm of the number by the exponent of the power.
For, A n = (10«) tt = 10«».
Therefore, log A n = an = n log A.
& The logarithm of the root of a number is found by dividing the
logarithm of the number by the index of the root.
For, VZ=Vld* = l(F.
n . a log A
Therefore, log VA = - = — — •
6. The logarithms of 1, 10, 100, etc., and of 0.1, 0.01, 0.001, etc.,
are integral numbers. The logarithms of all other numbers are frac-
tions.
IV LOGABITHMS.
For, 10° = 1, hence log 1 = ; 10" 1 = 0.1, hence log 0.1 = — 1
10* = 10, hence log 10 = 1; 10-*= 0.01, hence log 0.01 = — 2
10 s = 100, hence log 100 = 2 ; 10-* = 0.001, hence log 0.001 = -3
10 8 = 1000, hence log 1000 = 3 ; and so on.
If the number is between 1 and 10, the logarithm is between and 1.
If the number is between 10 and 100, the logarithm is between 1 and 2.
If the number is between 100 and 1000, the logarithm is between 2 and 3.
If the number is between 1 and 0.1, the logarithm is between and —1.
If the number is between 0.1 and 0.01, the logarithm is between —1 and —2.
If the number is between 0.01 and 0.001, the logarithm is between —2 and —3.
And so on.
7. If the number is less than 1, the logarithm is negative (§ 6),
but is written in such a form that the fractional part is always positive.
For the number may be regarded as the product of two factors, one of
which lies between 1 and 10, and the other is a negative power of 10 ; the
logarithm will then take the form of a difference whose minuend is a positive
proper fraction, and whose subtrahend is a positive integral number.
Thus, 0.48 = 4.8X0.1.
Therefore (§ 2) , log 0.48 = log 4.8 + log 0. 1 = 0.68124 - 1. (Page 1.)
Again, 0.0007 = 7X0.0001.
Therefore, log 0.0007 = log 7 + log 0.0001 = 0.84510 - 4.
& Every logarithm, therefore, consists of two parts: a positive
or negative integral number, which is called the Characteristic, and
& positive proper fraction, which is called the Mantissa.
Thus, in the logarithm 3.52179, the integral number 3 is the characteristic,
and the fraction .52179 the mantissa. In the logarithm 0.78254 — 2, the inte-
gral number — 2 is the characteristic, and the fraction .78254 is the mantissa.
9. If the logarithm is negative, it is customary to change the form
of the difference so that the subtrahend shall be 10 or a multiple of 10.
This is done by adding to both minuend and subtrahend a number
which will increase the subtrahend to 10 or a multiple of 10.
Thus, the logarithm 0.78254—2 is changed to 8.78254 — 10 by adding 8 to
both minuend and subtrahend. The logarithm 0.92737 — 13 is changed to
7.92787—20 by adding 7 to both minuend and subtrahend.
10. The following rules are derived from § 6 : —
If the number is greater than 1, make the characteristic of the
logarithm one unit less than the number of figures on the left of the
decimal point.
If the number is less than 1 , make the characteristic of the loga-
rithm negative, and one unit more than the number of zeros between
the decimal point and the first significant figure of the given number.
INTRODUCTION.
If the characteristic of a given logarithm is positive, make the
number of figures in the integral part of the corresponding number
one more than the number of units in the characteristic.
If the characteristic is negative, make the number of zeros between
the decimal point and the first significant figure of the corresponding
number one less than the number of units in the characteristic.
Thus, the characteristic of log 7849.27 = 3 ;
the characteristic of log 0.037 = — 2 = 8.00000 — 10.
If the characteristic is 4, the corresponding number has five figures in its
integral part. If the characteristic is — 3, that is, 7.00000 — 10, the corre-
sponding fraction has two zeros between the decimal point and the first
significant figure.
U. The logarithms of numbers that can be derived from one
another by multiplication or division by an integral power of 10 have
the same mantissa.
For, multiplying or dividing a number by an integral power of 10 will
increase or diminish its logarithm by the exponent of that power of 10 ; and
since this exponent is an integer, the mantissa of the logarithm will be
unaffected.
Thus, log 4.6021 =0.66296. (Page 9.)
log 460.21 = log (4.6021 X 10») = log 4.6021 + log 10*
= 0.66296 + 2 = 2.66296.
log 460210 = log (4.6021 X 10*) = log 4.6021 + log 10»
= 0.66296 + 5 = 5.66296.
log 0.046021 = log (4.6021 ■+■ 10*) = log 4.6021 -log 10 1
= 0.66296 -2 = 8.66296 - 10.
TABLE I.
12. In this table (pp. 1-19) the vertical columns headed N contain
the numbers, and the other columns the logarithms. On page 1 both
the characteristic and the mantissa are printed. On pages 2-19 the
mantissa only is printed.
The fractional part of a logarithm can be expressed only approxi-
mately, and in a five-place table all figures that follow the fifth are
rejected. Whenever the sixth figure is 5, or more, the fifth figure is
increased by 1. The figure 5. is written when the value of the figure
in the place in which it stands, together with the succeeding figures, is
more than 4£, but less than 5.
Thus, if the mantissa of a logarithm written to seven places is 5328732,
it is written in this table (a five-place table) 53287. If it is 5328751, it is
written 53288. If it is 5328461 or 5328499, it is written in this table 53285.
Again, if the mantissa is 5324981, it is written 53250; and if it is 4999967,
it is written 50000.
VI LOGAEUHMS.
This distinction between 5 and £, in case it is desired to curtail still
further the mantissas of logarithms, removes all doubt whether a 5 in
the last given place, or in the last but one followed by a zero, should
be simply rejected, or whether the rejection should lead us to increase
the preceding figure by one unit.
Thus, the mantissa 1392ft when reduced to four' places should be 1392;
but 13925 should be 1393.
To Find the Logarithm op a Given Number.
13. If the given number consists of one or two significant figures,
the logarithm is given on page 1. If zeros follow the significant
figures, or if the number is a proper decimal fraction, the character-
istic must be determined by § 10.
14. If the given number has three significant figures, it will be
found in the column headed N (pp. 2-19), and the mantissa of its
logarithm in the next column to the right, and on the same line.
Thus,
Page 2. log 145 = 2.16137, log 14600 = 4.16137.
Page 14. log 716 = 2.85491, log 0.716 = 9.85491 - 10.
15. If the given number has four significant figures, the first three
will be found in the column headed N, and the fourth at the top of
the page in the line containing the figures l, 2, 3, etc. The mantissa
will be found in the column headed by the fourth figure, and on the
same line with the first three figures. Thus,
Page 16. log 7682 =3.88547, log 76.85 =1.88564.
Page 18. log 93280 = 4.96979, log 0.9468 = 9.97626 - 10.
16. If the given number has five or more significant figures, a
process called interpolation is required.
Interpolation is based on the assumption that between two con-
secutive mantissas of the table the change in the mantissa is directly
proportional to the change in the number.
Required the logarithm of 34237.
The required mantissa is (§ 11) the same as the mantissa for 3423.7 ; there-
fore it will be found by adding to the mantissa of 3423 seven-tenths of the
difference between the mantissas for 3423 and 3424.
The mantissa for 3423 is 53441.
The difference between the mantissas for 3423 and 3424 is 12.
Hence, the mantissa for 3423.7 is 63441 + (0.7 X 12) = 63449.
Therefore, the required logarithm of 34237 is 4.53449.
INTBODUCTION.
VU
Required the logarithm of 0.0015764.
The required mantissa is the same as the mantissa for 1576.4; therefore
It will be found by adding to the mantissa for 1576 four-tenths of the difference
between the mantissas for 1576 and 1577.
The mantissa for 1576 is 19756.
The difference between the mantissas for 1576 and 1577 is 27.
Hence, the mantissa for 1576.4 is 19756 + (0.4 X 27) = 19767.
Therefore, the required logarithm of 0.0015764 is 7.19767-10.
Required the logarithm of 32.6708.
The required mantissa is the same as the mantissa for 3267.08 ; therefore
it will be found by adding to the mantissa for 3267 eight-hundredths of the
difference between the mantissas for 3267 and 3268.
The mantissa for 3267 is 51415.
The difference between the mantissas for 3267 and 3268 is 13.
Hence, the mantissa for 3267.08 is 51415 + (0.08 X 13) = 61416.
Therefore, the required logarithm of 32.6708 is 1.51416.
17. When the fraction of a unit in the part to be added to the
mantissa for four figures is less than 0.5 it is to be neglected ; when
it is 0.5 or more than 0.5 it is to be taken as one unit.
Thus, in the first example, the part to be added to the mantissa for 3423
Is 8.4, and the .4 is rejected. In the second example, the part to be added to
the mantissa for 1576 is 10.8, and 11 is added.
%
To Find the Number Corresponding to a Given Logarithm.
18. If the given mantissa can be found in the table, the first three
figures of the required number will be found in the same line with the
mantissa in the column headed N, and the fourth figure at the top of
the column containing the mantissa.
The position of the decimal point is determined by the character-
istic (§ 10).
Find the number corresponding to the logarithm 0.92002.
Page 16. The number for the mantissa 92002 is 8318.
Therefore, the required number is 8.318.
Find the number corresponding to the logarithm 6.09167.
Page 2. The number for the mantissa 09167 is 1236.
Therefore, the required number is 1235000.
Find the number corresponding to the logarithm 7.50325
Page 6. The number for the mantissa 50325 is 3186.
Therefore, the required number is 0.003186.
10.
VU1 LOGARITHMS.
— ^ — , — m
19. If the given mantissa cannot be found in the table, find in the
table the two adjacent mantissas between which the given mantissa
lies, and the four figures corresponding to the smaller of these two
mantissas will be the first four significant figures of the required
number. If more than four figures are desired, they may be found by
interpolation, as in the following examples :
Find the number corresponding to the logarithm 1.48762.
Here the two adjacent mantissas of the table, between which the given
mantissa 48762 lies, are found to be (page 6) 48756 and 48770. The corre-
sponding numbers are 3073 and 3074. The smaller of these, 3073, contains
the first four significant figures of the required number.
The difference between the two adjacent mantissas is 14, and the difference
between the corresponding numbers is 1.
The difference between the smaller of the two adjacent mantissas, 48756,
and the given mantissa, 48762, is 6. Therefore, the number to be annexed to
3073 is ^ of 1 = 0.428, and the fifth significant figure of the required number
is 4.
Hence, the required number is 30.734.
Find the number corresponding to the logarithm 7.82326 — 10.
The two adjacent mantissas between which 82326 lies are (page 13) 82321
and 82328. The number corresponding to the mantissa 82321 is 6656.
The difference between the two adjacent mantissas is 7, and the difference
between the corresponding numbers is 1.
The difference between the smaller mantissa, 82321, and the given mantissa,
82326, is 5. Therefore, me number to be annexed to 6656 is f of 1 = 0.7, and
the fifth significant figure of the required number is 7.
Hence, the required number is 0.0066567.
In using a five-place table the numbers corresponding to mantissas
may be carried to five significant figures, and in the first part of the
table to six figures.*
20. The logarithm of the reciprocal of a number is called the
Oologarithm of the number.
If A denote any number, then
colog J. = logi = log 1 — log^l (§ 3) =r —log A.
Hence, the cologarithm of a number is equal to the logarithm of
the number with the minus sign prefixed, which sign affects the entire
logarithm, both characteristic and mantissa.
* In most tables of logarithms proportional parts are given as an aid to
interpolation; but, after a little practice, the operation can be performed
nearly as rapidly without them. Their omission allows a page with larger-
faced type and more open spacing, and consequently less trying to the eyes.
INTRODUCTION. IX
In order to avoid a negative mantissa in the cologarithm, it is
customary to substitute for — log A its equivalent
(lO--log^i)-lO.
Hence, the cologarithm of a number is found by subtracting the
logarithm of the number from 10, and then annexing — 10 to the
remainder.
The best way to perform the subtraction is to begin on the left and
subtract each figure of log A from 9 until we reach the last significant
figure, which must be subtracted from 10.
If log A is greater in absolute value than 10 and less than 20, then
in order to avoid a negative mantissa, it is necessary to write — log J.
in the form
(20-log^i)-20.
So that, in this case, colog A is found by subtracting log-4 from 20,
and then annexing — 20 to the remainder.
Find the cologarithm of 4007;
10 -10
Page 8. log 4007= 3.60282
colog 4007= 6.39718-10
Find the cologarithm of 103992000000.
20 -20
Page 2. log 103992000000 = 11 .01700
colog 103992000000 = 8.98300 - 20
If the characteristic of log A is negative, then the subtrahend, —10
or — 20, will vanish in finding the value of colog A.
Find the cologarithm of 0.004007.
10 -10
log 0.004007= 7.60282-10
colog 0.004007= 2.39718
With practice, the cologarithm of a number can be taken from the
table as rapidly as the logarithm itself.
By using cologarithms the inconvenience of subtracting the loga-
rithm of a divisor is avoided. For dividing by a number is
equivalent to multiplying by its reciprocal. Hence, instead of
subtracting the logarithm of a divisor its cologarithm may be added.
LOGABITHMS.
Computation by Logarithms.
2L (1) Find the value of x, if x = 72214 x 0.08203.
Page 14. log 72214 = 4.85862
Page 16. log 0.08203 =8.91897—10
By §2.
Page 11.
logx
x
= 8.77269
= 5923.63
(2) Find the value of a?, if x = 5250 -*- 23487.
Page 10. log 5250 = 8.72016
Page 4. colog 23487 = 5.62917 - 10
Page 4.
logx
.-. x
= 9.34933 — 10 = log 0.22353
= 0.22353
(3) FmdthevaIueofs,ifs=: 7.56x4667x567
Page 15.
Page 9.
Page 11.
Page 17.
Page 6.
Page 4.
Page 5.
log 7.66
log 4667
log 667
colog 899.1
colog 0.00337
colog 23435
logx
.-. x
899.1 x 0.00337 X 23435
= 0.87862
= 3.66904
= 2.76358
= 7.04619-10
= 2.47237
= 5.63013-10
= 2.44983 = log 281.78
= 281.73
(4) Find the cube of 376.
. Page 7. log 376
Multiply by 3 (§4),
Page 10. log 376 3
.-. 376 3
= 2.67619
3
= 7.72557 = log 53168600
= 53158600
(5) Find the square of 0.003278.
Page 6. log 0.003278= 7.51561-10
Page 2.
log 0.003278* = 16.03122 - 20 = log 0.000010745
.-. 0.003278«= 0.000010745
(6) Find the square root of 8322.
Page 16. log 8322 = 3.92023
Divide by 2 (§ 5) , 2 ) 3.92023
log V8322 = 1.96012 = log 91.226
.-. V8322 =91.226
If the given number is a proper fraction, its logarithm will have as
a subtrahend 10 or a multiple of 10. In this case, before dividing the
logarithm by the index of the root, both the subtrahend and the num-
INTRODUCTION. xi
ber preceding the mantissa should be increased by such a number as
will make the subtrahend, when divided by the index of the root,
10 or a multiple of 10.
(7) Find the square root of 0.000043641.
Page 8. log 0.000043641 = 5.63989-10
10 -10
Divide by 2 (§ 5), 2 )15.63989 -"JO
Page 13. log V0.00004364i = 7.81995 - 10 = log 0.0066062
.-. V0.000043641 = 0.0066062
(8) Find the sixth root of 0.076553.
Page 15. log 0.076553 = 8.88397-10
50 -50
Divide by 6 (§ 5), 6 )58.88397-60
Page 13 log v U076553 = 9.8H00 - 10 = log 0.66163
:•. vU076553 = 0.65163
TABLE II.
22. This table (page 20) contains the value of the number ir, its
most useful combinations, and their logarithms.
Find the length of an arc of 47° 32' 57" in a unit circle.
47° 32' 67" =171177"
log 171177 = 6.23344
log-i- =4.68567-10
a"
log arc 47° 32' 67" = 9.91901 - 10 = log 0.82994
. \ length of arc = 0.82994
Find the angle if the length of its arc in a unit circle = 0.54936.
log 0.64936 = 9.73986 - 10
colog-i-=loga" =6.31443
a"
log angle = 6.05429 = log 113316
.-. angle = 113316"= 31° 28' 36"
23. The relations between arcs and angles given in Table II. are
readily deduced from the circular measure of an angle.
In Circular Measure an angle is defined by the equation
, arc
angle =——,
radius
in which the word arc denotes the length of the arc corresponding to
the angle, when both arc and radius are expressed in terms of the
same linear unit.
XU LOGARITHMS.
Since the arc and radius for a given angle in different circles vary
in the same ratio, the value of the angle given by this equation is
independent of the value of the radius. If the radius is unity, the
equation denning the angle becomes
angle = arc.
That is, in circular measure an angle is measured by the length of its
arc in a unit circle. Therefore,
If the arc = circumference, the angle = 2 tc.
If the arc == semicircumference, the angle = v.
If the arc = quadrant, the angle = %ir.
If the arc = radius (= 1), the angle = 1 ;
that is, in circular measure the angular unit is the angle whose arc is
equal in length to the radius of the circle.
Since 180° in common measure equals v units in circular measure,
therefore
1° in common measure = -^- units in circular measure ;
180°
1 unit in circular measure = in common measure.
IT
By means of these two equations, the value of an angle expressed
in one measure may be changed to its value in the other measure.
Thus, the angle whose arc is equal to the radius is an angle of
180°
1 unit in circular measure, and is equal to , or 57° 17' 45", very
nearly.
TABLE III.
24. This table (pp. 21-49) contains the logarithms of the trigo-
nometric functions of angles. In order to avoid negative character-
istics, the characteristic of every logarithm is printed 10 too large.
Therefore, — 10 is to be annexed to each logarithm.
On pages 28-49 the characteristic remains the same throughout
each column, and is printed at the top and the bottom of the column.
But on page 30 the characteristic changes one unit in value at the
places marked with bars. Above these bars the proper characteristic
is printed at the top, and below them at the bottom, of the column.
25. On pages 28-49 the log sin, log tan, log cot, and log cos, ot 1°
to 89°, are given to every minute. Conversely, this part of the table
gives the value of the angle to the nearest minute when log sin, log tan,
log cot, or log cos is known, provided log sin or log cos lies between
8.23822 and 9.99992, and log tan or log cot lies between 8.23829 and
11.76171.
INTRODUCTION. Xlll
If the exact value of the given logarithm of a function is not found
in the table, the value nearest to it is to be taken, unless interpolation
is employed as explained in § 26.
If the angle is less than 45°, the number of degrees is printed at
the top of the page, and the number of minutes in the column to the
left of the columns containing the logarithm. If the angle is greater
than 45°, the number of degrees is printed at the bottom of the page,
and the number of minutes in the column to the right of the columns
containing the logarithms.
If the angle is less than 45°, the names of its functions are printed
at the top of the page ; if greater than 45°, at the bottom of the
page. Thus,
Page 88. log sin 21° 87' = 9.56681 — 10.
Page 45. log cot 86° 53' = 10.12473 - 10 = 0.12473.
Page 87. log cos 69° 14' = 9.64969-10.
Page 49. log tan 46° 69' = 10.01491 - 10 = 0.01491.
Page 48. If log cos = 9.87468 — 10, angle = 41° 28'.
Page 84. If log cot = 9.39363 — 10, angle = 76° 6'.
If log sin = 9.47760 — 10, the nearest log sin in the table is 9.47774 — 10
(page 36), and the angle corresponding to this value is 17° 29'.
If log tan = 0.76520= 10.76520 — 10, the nearest log tan in the table is
10.76490 — 10 (page 82), and the angle corresponding to this value is 80° 15'.
26. If it is desired to obtain the logarithms of the functions of
angles that contain seconds, or to obtain the value of the angle in
degrees, minutes, and seconds, from the logarithms of its functions,
interpolation must be employed. Here it must be remembered that,
The difference between two consecutive angles in the table is 60".
Log sin and log tan increase as the angle increases; log cos and
log cot diminish as the angle increases.
Find log tan 70° 46' 8".
Page 87. log tan 70° 46' = 0.46781.
The difference between the mantissas of log tan 70° 46 f and log tan 70° 47'
Is 41, and & of 41 = 6.
As the function is increasing, the 5 must be added to the figure in the fifth
place of the mantissa 45731 ; and
Therefore log tan 70° 46' 8" = 0.45736.
•Find log cos 47° 35' 4".
Page 48. log cos 47° 85' = 9.82899 - 10.
The difference between this mantissa and the mantissas of the next log cos
Is 14, and ^ of 14 = 1.
As the function Is decreasing, the 1 must be subtracted from the figure in
the fifth place of the mantissa 82899 ; and
Therefore log cos 47° 35' 4" = 9.82898 - 10.
XIV LOGARITHMS.
Find the angle for which log sin = 9.45359 — 10.
Page 35. The mantissa of the nearest smaller log sin in the table is 45834.
The angle corresponding to this value is 16° 30'.
The difference between 45334 and the given mantissa, 45359, is 25.
The difference between 45334 and the next following mantissa, 45377, is
43, and}} of 60" = 35".
As the function is increasing, the 35" must be added to 16° 30'; and the
required angle is 16° 30' 35".
Find the angle for which log cot = 0.73478.
Page 32. The mantissa of the nearest smaller log cot in the table is 73415.
The angle corresponding to this value is 10° 27'.
The difference between 73415 and the given mantissa is 63.
The difference between 73415 and next following mantissa is 71, and
ft of 60" = 53".
As the function is decreasing, the 53" must be subtracted from 10° 27' ;
and the required angle is 10° 26' 7".
27. If log sec or log esc of an angle is desired, it may be. found
from the table by the formulas,
sec-4 = 7 ; hence, log sec -4 = colog cos A.
cosA
esc A == — — - ; hence, log esc A = colog sin A.
smA
Page 31. log sec 8° 28' = colog cos 8° 28' =0.00476.
Page 42. log esc 69° 36' 44" = colog sin 59° 86' 44" = 0.06418.
2& If a given angle is between 0° and 1°, or between 89° and 90° ;
or, conversely, if a. given log sin or log cos does not lie between the
limits 8.23822 and 9.99992 in the table; or, if a given log tan or
log cot does not lie between the limits 8.23829 and 11.76171 in the
table ; then pages 21-24 of Table IEL must be used.
On page 21, log sin of angles between 0° and 0° 3', or log cos of the
complementary angles between 89° 57' and 90°, are given to every
second; for the angles between 0° and 0°3', log tan = log sin, and
log cos = 0.00000 ; for the angles between 89° 57' and 90°, log cot =
log cos, and log sin = 0.00000.
On pages 22-24, log sin, log tan, and log cos of angles between 0°
and 1°, or log cos, log cot, and log sin of the complementary angles
between 89° and 90°, are given to every 10".
Whenever log tan or log cot is not given, they may be found by the
formulas,
log tan = colog cot. log cot = colog tan.
Conversely, if a given log tan or log cot is not contained in the
table, then the colog must be found; this will be the log cot or
log tan, as the case may be, and will be contained in the table.
IOTBODTTCTION. XV
On pages 25-27 the logarithms of the functions of angles between
1° and 2°, or between 88° and 90°, are given in the manner employed
on pages 22-24. These pages should be used if the angle lies between
these limits, ancr if not only degrees and minutes, but degrees, min-
utes, and multiples of 10" are given or required.
When the angle is between 0° and 2°, or 88° and 90°, and a greater
degree of accuracy is desired than that given by the table, interpo-
lation may be employed ; but for these angles interpolation does not
always give true results, and it is better to use Table IV.
Find log tan 0° 2' 47", and log cos 89° 37' 20".
Page 21. log tan 0° 2' 47" = log sin 0° 2' 47" = 6.90829-10.
Page 23. log cos 89° 37' 20" = 7.81911-10.
Find log cot 0° 2' 15".
10 -10
Page 21. log tan 0° 2' 16" = 6.81591-10
Therefore, log cot 0° 2' 15" = 3.18409
Find log tan 89° 38' 30".
10 -10
Page 23. log cot 89° 38' 80" = 7.79617 - 10
Therefore, log tan 89° 88' 80"= 2.20383
Find the angle for which log tan = 6.92090 — 10.
Page 21. The nearest log tan Is 6.92110 — 10.
The corresponding angle for which is 0° 2' 52".
Find the angle for which log cos = 7.70240 — 10.
Page 22. The nearest log cos is 7.70261 — 10.
The corresponding angle for which is 89° 42' 40".
Find the angle for which log cot = 2.37368.
This log cot Is not contained in the table.
The cologcot = 7.62632 — 10 = log tan.
The log tan In the table nearest to this is (page 22) 7.62510 — 10, and the
angle corresponding to this value of log tan is 0° 14' 30".
29. If an angle x is between 90° and 360°, it follows, from formu-
las established in Trigonometry, that,
between 90° and 180°, between 180° and 270°,
log sin x = log sin (180° — x) , log sin x = log sin (x — 180°) n ,
log cos x = log cos (180° — «) n , log cos x = log cos (x — 180°) n ,
log tan x = log tan (180° — a?) n , log tan x = log tan (x — 180°) ,
log cot x = log cot (180° — x) n ; log cot x = log cot (x — 180°) ;
XVI LOGABITHMS.
between 270° and 360°,
log sin x = log sin (360° — x) n ,
log cos x = log cos (360° — x) ,
log tana? = logtan (360° — x) n , •
log cot x = log cot (360° — x) n .
The letter n is placed (according to custom) after the logarithms
of those functions which are negative in value.
The above formulas show, without further explanation, how to find
by means of Table HE. the logarithms of the functions of any angle
between 90° and 360°.
Thus, log sin 137° 46' 22" = log sin 42° 14' 38" = 9.82756 -10.
log cos 137° 46' 22" = log* cos 42° 14' 38" = 9.86940» - 10.
logtan 137° 46' 22" = log* tan 42° 14' 38" = 9.95815*- 10.
log cot 137° 46' 22" = log* cot 42° 14' 38" = 0.04185 n .
log sin 209° 32* 50" = log n sin 29° 32' 50'' = 9.69297* - 10.
Iogcos330°27'10" = logcos 29° 32' 50" = 9.93949 -10.
Conversely, to a given logarithm of a trigonometric function there
correspond between 0° and 360° four angles, one angle in each quad-
rant, and so related that if z denote the acute angle, the other three
angles are 180°— a?, 180° +x, and 360°— x.
If besides the given logarithm it is known whether the function is
positive or negative, the ambiguity is confined to two quadrants,
therefore to two angles.
Thus, if the log tan = 9.47451 - 10, the angles are 16° 36' 17" in Quadrant I.
and 196° 36' 17" in Quadrant III. ; but if the logtan = 9.47451*- 10, the angles
are 163° 23' 43" in Quadrant II. and 343° 23' 43" in Quadrant IV.
To remove all ambiguity, further conditions are required, or a
knowledge of the special circumstances connected with the problem in
question.
TABLE IV.
30. This table (page 50) must be used when great accuracy is
desired in working with angles between 0° and 2°, or between 88°
and 90°.
The values of S and T are such that when the angle a is expressed
in seconds,
S = log sin a — log a",
T = log tana — log a".
Hence follow the formulas given on page 50.
The values of S and T are printed with the characteristic 10 too
large, and in using them — 10 must always be annexed.
INTRODUCTION.
XV11
Find log sin 0° 58' 17".
0° 58' 17" = 3497"
log 3497 =3.54370
S = 4.68666 -10
log sin 0° 68' 17" = 8.22925 - 10
Find log tan 0° 52' 47.5".
0° 52' 47.6" = 3167.5"
log 3167.6 = 3.50072
T = 4.68661 -10
Find log cos 88° 26' 41. 2".
90° -88° 26' 41.2" = 1°33' 18.8"
= 6598.8"
log 5598.8 = 3.74809
8 = 4.68662-10
log tan 0° 52' 47.6" = 8.18633 - 10
log cos 88° 26' 41.2" = 8.43361 - 10
Find log tan 89° 54' 37.362".
90° — 89° 54' 37.362" = 0° 5' 22.638'
= 322.638"
log 322.638 = 2.60871
T = 4.68558 -10
log cot 89° 54' 37.362" = 7. 19429 - 10
log tan 89° 64' 87.862" = 2.80571
Find the angle, if log sin = 6.72306-10.
6.72306 - 10
= 4.68567 — 10
8 =
Subtract,
Find the angle for which
cologcot:
T:
Subtract,
2.03749
109.015"
= log 109.015
= 0° 1' 49.015"
log cot = 1.67604.
= 8.82396 — 10
= 4.68664-10
3.68832 = log 4348.3
4348.3" = 1° 12' 28.3"
Find the angle for which log tan = 1.55407.
colog tan = 8.44693 — 10
T = 4.68569 - 10
Subtract, 3.76024 = log 5757.6
5757.6" = 1° 85' 57.6",
and 90° - 1° 85' 57.6" = 88° 24' 2.4".
Therefore, the angle required is 88° 24' 2.4".
TABLE V.
33. Table V. (pp. 51-53), contains the natural sines, cosines,
tangents, and cotangents of angles from 0° to 90°, at intervals of 10'.
If greater accuracy is desired it may be obtained by interpolation.
TABLE VI.
32. This table (p. 54) , containing the circumferences and areas
of circles, does not require explanation.
Note. In preparing the preceding explanations, free use has been made
of the Logarithmic Tables by F. G. Gauss, from which, also, Tables II. and
VI. have been taken.
XV111
LOGABITHMB.
TABLE VII.
33. This table (pp. 55-60) gives the latitude and departure to
three places of decimals for distances from 1 to 10, corresponding
to bearings from 0° to 90° at intervals of 15'.
If the bearing does not exceed 45° it is found in the left-hand
column, and the designations of the columns under "Distance" are
taken from the top of the page ; but if the bearing exceeds 45°, it is
found in the rigkt-h&nd column, and the designations of the columns
under "Distance" are taken from the bottom of the page.
The method of using the table will be made plain by the following
examples : —
(1) Let it be required to find the latitude and departure of the
course N. 35° 15' E. 6 chains.
On p. 60, left-hand column, look for 35° 16' ; opposite this bearing, in the
vertical column headed "Distance 6," are found 4.900 and 3.463 under the
headings "Latitude" and " Departure" respectively. Hence, latitude or
northing = 4.900 chains, and departure or easting = 3.463 chains.
(2) Let it be required to find the latitude and departure of the
course S. 87° W. 2 chains.
As the bearing exceeds 46°, we look in the right-hand column of p. 65, and
opposite 87° in the column marked " Distance 2 " we find (taking the designa-
tions of the columns from the bottom of the page) latitude = .105 chains,
and departure = 1.997 chains. Hence, latitude or southing = .105 chains, and
departure or westing = 1.997 chains.
(3) Let it be required to find the latitude and departure of the
course N. 15° 45' W. 27.36 chains.
In this case we find the required numbers for each figure of the distance
separately, arranging the work as in the following table. In practice, only
the last columns under " Latitude " and " Departure " are written.
Distance.
Latitude.
Departure.
20 = 2 X 10
7
0.3 =8-*- 10
0.06 = 6 -r- 100
1.925X10 =19.25
6.737
2.887 -f- 10 = 0.289
6.775 + 100= 0.058
0.643X10 =5.43
1.90
0.814-*- 10 =0.081
1.628 -r- 100 = 0.016
27.86
26.334
7.427
Hence, latitude = 26.334 chains, and departure = 7.427 chains.
TABLE I
■
THE
COMMON OE
BEIGGS
LOGAEITHMS
OF THE
NATUEAL NUMBEES
From 1 to 10000.
—
1-100
17
log
?
log
17
log
N
log
N
log
1
0.00000
21
1.32 222
41
1.61278
61
1.78533
81
1.90849
2
0.30103
22
1.34 242
42
1.62 325
62
1.79 239
82
1.91381
3
0.47 712
23
1.36173
43
1.63 347
63
1.79934
83
1.91908
4
0.60206
24
1.38021
44
1.64345
64
1.80618
84
1.92428
6
0.69897
25
1.39 794
45
1.65 321
65
1. 81 291
85
1.92942
6
0.77815
•*
1.41497
46
1.66 276
66
1. 81 954
86
1.93 450
7
0.84 510
27
1.43136
47
1.67 210
67
1.82607
87
1.93952
8
0.90309
28
1.44 716
48
1.68124
68
1. 83 251
88
1.94448
9
0.95 424
29
1.46240
49
1.69020
69
1.83 885
89
1.94939
10
1.00000
30
1.47 712
50
1.69897
70
1.84 510
90
1.95 424
11
1.04139
31
1.49136
61
1.70757
71
1. 85 126
91
1.95 904
12
1.07918
i 82
1.50515
62
1.71600
72
1. 85 733
92
1.96379
13
1. 11 394
33
1.51851
63
1.72 428
73
1.86332
93
1.96848
14
1.14613
34
1.53148
64
1. 73 239
74
1.86 923
94
1.97313
15
1.17609
35
1.54407
65
1.74036
75
1. 87 506
95
1.97 772
16
1.20412
36
1.55 630
66
1.74819
76
1.88 081
96
1.98 227
17
1.23 045
37
1.56820
67
1. 75 587
77
1.88649
97
1.98677
18
1.25 527
83
1.57978
68
1.76343
78
1.89209
98
1.99123
19
1.27875
39
1.59106
59
1.77085
79
1.89 763
99
1.99564
20
1.30103
40
1.60206
60
1.77815
80
1.90309
100
2.00000
K
log
N
log
IT
log
IT
log
IT
log
1-100
2
100-150
N
1
2
3
4
5
6
7
8
100
00000
00043
00087
00130
00173
00217
00260
00303
00346 00389
101
00432
00475
00 518
00 561
00604
00647 00689 00732 00775 00817
102
00860
00903
00945
00988
01030
01072 01115 01157 01199 01242
103
01284
01326
01368
01410
01452
01494
01536
01578
01620 01662
104
01703
01745
01787
01828
01870
01912 01953 01995 02 036 02 078
105
02119
02160
02 202
02 243
02 284
02 325
02366
02 407 02 449 02 490
106
02 531
02 572
02 612
02 653
02 694
02 735 02 776 02 816 02 857 02 898
107
02938
02979
03 019
03 060
03100
03141
03181
03 222
03 262 03 302
108
03 342
03 383
03 423
03 463
03 503
03 543
03 583
03 623
03 663 03 703
109
03 743
03 782
03 822
03 862
03 902
03 941
03 981
04 021
04060 04100
uo
04139
04179
04 218
04 258
04 297
04336 04376 04415 04454 04493
111
04 532
04 571
04610 04650 04689
04 727
04 766
04805
04844 04883
112
04 922
04961
04999
05 038
05 077
05115
05154
05192
05 231 05 269
113
05 308 05 346 05 385 05 423 05 461
05 500 05 538 05 576 05 614 05 652
114
05 690 05 729 05 767 05 805 05 843
05 881
05 918
05 956
05 994 06032
115
06070
06108
06145
06183
06 221
06258
06296
06333
06371 06408
116
06446
06 483
06 521
06 558
06 595
06633
06 670
06 707
06 744 06 781
117
06819
06856
06 893
06930
06967
07004
07041
07 078 07115 07151
118
07188 07 225 07262 07 298 07 335
07 372
07408
07 445
07482 07 518
119
07 555 07 591 07628 07 664 07 700
07 737
07 773
07809
07 846 07 882
120
07 918
07954
07990
08 027
08063
08099 08135 08171
08 207 08 243
121
08 279
08314
08 350
08 386
08422
08458
08814
08493
08 529 08 565 08600
122
08 636
08672
08 707
08 743
08 778
08 849
08 884
08 920 08955
123
08991
09026
09061
09096
09132
09167
09 202
09237
09272 09307
124
09 342
09377
09412
09447
09482
09 517
09552
09587
09621 09656
125
09691
09726 09760 09 795
09830
09864
09 899
09934
09968 10003
126
10037
10072
10106
10140
10175
10209
10243
10278
10312 10346
127
10380
10415
10449
10483
10517
10551
10 585
10619
10 653 10687
128
10 721
10 755
10789
10 823
10857
10890
10924
10958
10992 11025
129
11059
11093
11126
11160
11193
11227
11261
11294
11327 11361
130
11394
11428
11461
11494
11528
11561
11594
11628
11661 11694
131
11727
11760
11793
11826
11860
11893
11926
11959
11992 12024
132
12057
12 090
12123
12156
12189
12 222
12 254
12 287
12 320 12 352
133
12385
12 418
12 450
12 483
12 516
12 548
12 581
12 613
12 646 12 678
134
12 710
12 743
12 775
12 808
12 840
12 872
12905
12f37
12 969 13 001
135
13 033
13 066
13 098
13130
13162
13194
13 226
13 258
13 290 13 322
136
13 354
13 386
13 418
13 450
13 481
13 513
13 545
13 577
13609 13640
137
13 672
13 704
13 735
13 767
13 799
13 830
13 862
13 893
13 925 13 956
138
13 988
14 019
14 051
14 082
14114
14145
14176
14 208
14239 14270
139
14301
14333
14364
14 395
14 426
14 457
14489
14 520
14 551 14 582
140
14 613
14644
14675
14 706
14 737
14 768
14 799
14829
14 860 14891
141
14922
14 953
14 983
15 014
15 045
15 076
15106
15137
15168 15198
142
15 229
15 259
15 290
15 320
15 351
15 381
15 412
15 442
15473 15 503
143
15 534
15 564
15 594
15 625
15 655
15 685
15 715
15 746
15 776 15 806
144
15 836
15 866
15 897
15 927
15 957
15 987
16017
16047
16077 16107
145
16137
16167
16197
16227
16256
16 286
16316
16346
16376 16406
146
16435
16465
16495
16524
16 554
16 584
16613
16643
16673 16 702
147
16 732
16 761
16 791
16 820
16850
16879
16909
16938
16967-16997
148
17026
17056
17 085
17114
17143
17173
17 202
17 231
17 260 17289
149
17319
17 348
17377
17406
17 435
17464
17493
17 522
17 551 17580
150
17 609
17638
17667
17696
17 725
17 754
17 782
17 811
17840 17869
N
1
2
3
4
5
6
7
8 9 J
100-150
150-200
3
N
1
2
3
4
5 6
7 8 9
150
17609
17638
17 667
17 696
17 725
17 754 17 782
17 811 17840 17 869
161
17 898
17926
17 955
17984
18013
18041 18070
18099 18127 18156
152
18184
18213
18 241
18 270
18298
18327 18355
18 384 18412 18441
153
18469
18498
18 526
18 554
18 583
18611 18639
18667 18 696 18 724
154
18 752
18 780
18808
18837
18865
18 893 18921
18949 18977 19005
155
19033
19061
19089
19 117
19145
19173 19 201
19229 19257 19 285
156
19312
19340
19368
193%
19424
19451 19479
19 507 19 535 19 562
157
19 590
19618
19645
19673
19 700
19 728 19 756
19 783 19811 19838
158
19 866
19 893
19921
19948
19976
20003 20030
20058 20085 20112
169
20140
20167
20194
20222
20249
20276 20303
20330 20358 20385
160
20412
20439
20466
20493
20520
20 548 20575 20602 20629 20656
161
20683
20 710
20 737
20 763
20 790
20 817 20844
20871 20 898 20925
162
20952
20978
21005
21032
21059
21085 21112
21139 21165 21192
163
21219
21245
21272
21299
21325
21352 21378 21405 21431 21458
164
21484
21511
21537 21564
21590
21617 21643
21669 21696 21722
165
21748 21775 21801
21827
21854
21880 21906
21932 21958 21985
166
22 011
22 037
22 063
22089
22115
22141 22167
22194 22 220 22 246
167
22 272
22298
22 324
22 350
22376
22401 22427
22 453 22 479 22 505
168
22 531
22 557
22 583
22 608
22 634
22660 22686
22 712 22 737 22 763
169
22 789
22 814
22 840
22 866
22 891
22917 22 943
22968 22994 23019
170
23 045
23 070
23 096
23121
23147
23172 23198
23 223 23 249 23 274
171
23300
23 325
23 350
23 376
23 401
23 426 23 452
23 477 23 502 23 528
172
23553
23 578
23 603
23 629
23 654
23 679 23 704
23 729 23 754 23 779
173
23 805
23 830
23 855
23 880
23 905
23 930 23 955
23980 24005 24030
174
24 055
24080 24105
24130
24155
24180 24 204
24229 24 254 24 279
175
24304
24329
24 353
24378
24403
24428 24452
24 477 24 502 24 527
176
24 551
24 576
24601
24625
24 650
24 674 24 699
24 724 24 748 24 773
177
24797
24822
24 846
24871
24895
24920 24944
24 969 24993 25 018
178
25 042
25 066
25 091
25115
25139
25164 25188
25 212 25 237 25 261
179
25 285
25 310
25 334
25 358
25 382
25 406 25 431
25 455 25 479 25 503
180
25 527
25 551
25 575
25 600
25 624
25 648 25 672
25 696 25 720 25 744
181
25 768
25 792 25 816 25 840
25 864
25 888 25 912
25 935 25 959 25 983
182
26007
26031
26055
26079
26102
26126 26150
26174 26198 26 221
183
26245
26269
26293
26316
26340
26364 26387
26411 26435 26458
184
26482
26505
26529
26 553
26576
26600 26623
26647 26670 26 694
185
26 717 26 741
26 764
26 788 26811
26834 26858
26881 26905 26928
186
26951
26975
26998
27021 27045
27068 27091
27114 27138 27161
187
27184
27 207
27 231
27254
27277
27 300 27323
27346 27370 27393
188
27 416
27439
27462
27485
27508
27 531 27 554
27 577 27600 27 623
189
27646
27669
27692
27 715
27 738
27 761 27 784
27807 27 830 27 852
190
27875
27898
27921
27944
27967
27989 28012
28035 28058 28081
191
28103
28126
28149
28171
28194
28 217 28240 28262 28285 28 307
192
28330
28353
28375
28 398
28421
28443 28466
28488 28 511 28533
193
28 556 28578
28601
28 623
28646
28668 28691
28 713 28 735 28 758
194
28 780
28803
28825
28847
28870
28 892 28914
28937 28959 28981
195
29003
29026
29048
29070
29092
29115 29137
29159 29181 29 203
196
29226
29 248
29270 29292
29314
29336 29 358
29 3S0 29403 29425
197
29447
29469
29491
29 513
29535
29 557 29 579 29601 29623 29645
198
29667
29688
29 710
29 732
29754
29 776 29 798
29820 29 842 29863
199
29885
29907
29929
29951
29973
29994 30016
30038 30060 30081
200
30103
30125
30146
30168
30190
30 211 30233
30255 30276 30298
N
1
2
3
4
5 6
7 8 9
150-200
4
200-250
N
12 8 4
5 6 7 8
200
30103 30125 30146 30168 30190
30211 30233 30255 30276 30 298
201
30320 30341 30363 30384 30406
30428 30449 30471 30492 30 514
202
30 535 30557 30578 30600 30621
30643 30664 30685 30 707 30 728
203
30 750 30 771 30 792 30814 30835
30856 30878 30899 30920 30942
204
30963 30984 31006 31027 31048
31069 31091 31112 31133 31154
205
31175 31197 31218 31239 31260
31281 31302 31323 31345 31366
206
31387 31408 31429 31450 31471
31492 31513 31534 31555 31576
207
31 597 31 618 31 639 31 660 31 681
31 702 31 723 31 744 31 765 31 785
208
31806 31827 31848 31869 31890
31911 31931 31952 31973 31994
209
32 015 32035 32 056 32077 32098
32118 32139 32160 32181 32 201
210
32 222 32 243 32263 32 284 32 305
32325 32346 32 366 32387 32 408
211
32428 32449 32 469 32 490 32 510
32 531 32 552 32 572 32 593 32 613
212
32 634 32 654 32675 32 695 32 715
32 736 32 756 32 777 32 797 32 818
213
32 838 32 858 32 879 32 899 32 919
32940 32960 32980 33001 33021
214
33041 33062 33082 33102 33122
33143 33163 33183 33203 33 224
215
33244 33264 33284 33304 33325
33345 33 365 33 385 33405 33 425
216
33445 33465 33486 33 506 33526
33 546 33 566 33 586 33 606 33 626
217
33 646 33666 33686 33 706 33 726
33 746 33 766 33 786 33806 33826
218
33846 33866 33885 33905 33925
33945 33965 33985 34005 34025
219
34044 34064 34084 34104 34124
34143 34163 34183 34203 34 223
220
34 242 34262 34 282 34301 34 321
34341 34361 34380 34400 34420
221
34439 34459 34479 34498 34 518
34 537 34 557 34 577 34 5% 34 616
222
34635 34655 34674 34694 34 713
34 733 34 753 34 772 34 792 34 811
223
34830 34850 34869 34889 34908
34928 34947 34967 34986 35 005
224
35025 35044 35064 35083 35102
35 122 35 141 35 160 35 180 35 199
225
35 218 35 238 35 257 35 276 35 295
35315 35334 35 353 35 372 35 392
226
35411 35430 35 449 35 468 35 488
35 507 35 526 35545 35 564 35 583
227
35 603 35 622 35 641 35 660 35 679
35 698 35 717 35 736 35 755 35 774
228
35 793 35 813 35 832 35 851 35 870
35889 35908 35927 35946 35 965
229
35 984 36003 36021 36040 36059
36078 36097 36116 36135 36154
230
36173 36192 36211 36229 36248
36267 36286 36305 36324 36342
231
36361 36380 36399 36418 36436
36455 36474 36493 36511 36 530
232
36549 36 568 36586 36605 36624
36642 36661 36680 36698 36 717
233
36 736 36 754 36773 36 791 36810
36829 36847 36866 36884 36903
234
36922 36940 36959 36977 36996
37014 37033 37051 37070 37 088
235
37107 37125 37144 37162 37181
37199 37218 37236 37 254 37 273
236
37291 37310 37328 37346 37365
37383 37401 37420 37438 37457
237
37475 37493 37511 37530 37 548
37566 37585 37603 37621 37639
238
37658 37676 37694 37 712 37731
37 749 37 767 37 785 37 803 37 822
239
37840 37858 37876 37894 37912
37931 37949 37967 37985 38 003
240
38021 38039 38057 38075 38093
38112 38130 38148 38166 38184
241
38202 38220 38 238 38 256 38274
38292 38310 38328 38346 38364
242
38382 38399 38417 38435 38453
38471 38489 38507 38 525 38 543
243
38 561 38 578 38 596 38 614 38632
38650 38668 38686 38 703 38 721
244
38 739 38 757 38 775 38 792 38 810
38828 38846 38863 38881 38899
245
38917 38934 38952 38970 38987
39005 39023 39041 39058 39076
246
39094 39111 39129 39146 39164
39182 39199 39217 39235 39252
247
39 270 39 287 39305 39322 39340
39358 39375 39393 39410 39428
248
39445 39463 39480 39498 39 515
39 533 39550 39 568 39 585 39602
249
39620 39637 39655 39672 39690
39707 39 724 39742 39 759 39 777
250
39794 39811 39829 39846 39863
39881 39898 39915 39933 39950
N
12 8 4
5 6 7 8
200-250
250-300 6
N
12 3 4
5 6 7 8 9
250
39 794 39 811 39829 39846 39863
39881 39898 39915 39933 39950
251
39967 39985 40002 40019 40037
40054 40071 40088 40106 40123
252
40140 40157 40175 40192 40209
40226 40243 40261 40278 40295
253
40312 40329 40346 40364 40381
40398 40415 40432 40449 40466
254
40483 40500 40518 40535 40552
40 569 40586 40603 40620 40637
255
40654 40671 40688 40705 40 722
40739 40756 40773 40790 40807
266
40824 40841 40 858 40875 40892
40909 40926 40943 40960 40976
257
40993 41010 41027 41044 41061
41 078 41 095 41 111 41 128 41 145
258
41 162 41 179 41 196 41 212 41 229
41246 41263 41280 41296 41313
259
41330 41347 41363 41380 41397
41414 41430 41447 41464 41481
260
41497 41514 41531 41547 41564
41581 41597 41614 41631 41647
261
41664 41681 41697 41714 41731
41 747 41 764 41 780 41 797 41 814
262
41830 41847 41863 41880 41896
41913 41929 41946 41963 41979
263
41996 42012 42029 42045 42062
42078 42095 42111 42127 42144
264
42160 42177 42193 42 210 42 226
42243 42259 42 275 42 292 42 308
265
42 325 42341 42 357 42374 42 390
42406 42423 42439 42455 42472
266
42488 42 504 42 521 42 537 42 553
42 570 42 586 42 602 42 619 42635
267
42 651 42 667 42 684 42 700 42 716
42 732 42 749 42 765 42 781 42 797
268
42 813 42 830 42 846 42 862 42 878
42 894 42911 42927 42943 42 959
269
42975 42991 43008 43024 43040
43056 43 072 43088 43104 43120
270
43 136 43 152 43 169 43 185 43 201
43 217 43 233 43 249 43 265 43 281
271
43 297 43 313 43 329 43345 43 361
43 377 43393 43409 43425 43441
272
43 457 43 473 43 489 43 505 43 521
43 537 43 553 43 569 43 584 43 600
273
43 616 43 632 43 648 43 664 43 680
43 696 43 712 43 727 43 743 43 759
274
43 775 43 791 43807 43823 43838
43854 43870 43886 43902 43917
275
43 933 43949 43 965 43981 43996
44012 44028 44044 44059 44075
276
44091 44107 44122 44138 44154
44170 44185 44201 44 217 44 232
277
44 248 44 264 44279 44 295 44 311
44 326 44342 44358 44373 44389
278
44404 44420 44436 44451 44467
44483 44498 44 514 44 529 44 545
279
44 560 44 576 44 592 44607 44623
44638 44654 44669 44685 44 700
280
44 716 44 731 44 747 44 762 44 778
44 793 44809 44824 44840 44855
281
44871 44886 44902 44917 44932
44948 44963 44979 44994 45010
282
45 025 45 040 45 056 45 071 45 086
45 102 45 117 45 133 45 148 45 163
283
45 179 45 194 45 209 45 225 45 240
45 255 45 271 45 286 45301 45 317
284
45 332 45 347 45 362 45 378 45 393
45408 45423 45 439 45 454 45 469
285
45 484 45 500 45 515 45 530 45 545
45 561 45 576 45 591 45 606 45 621
286
45 637 45 652 45 667 45 682 45 697
45 712 45 728 45 743 45 758 45 773
287
45 788 45803 45 818 45 834 45 849
45 864 45 879 45 894 45909 45 924
288
45 939 45954 45 969 45984 46000
46015 46030 46045 46060 46075
289
46090 46105 46120 46135 46150
46165 46180 46195 46210 46225
290
46240 46255 46270 46285 46300
46315 46330 46345 46359 46374
291
46389 46404 46419 46434 46449
46464 46479 46494 46 509 46 523
292
46538 46553 46568 46583 46 598
46613 46627 46642 46657 46672
293
46687 46702 46 716 46 731 46 746
46761 46776 46 790 46805 46820
294
46835 46850 46864 46879 46894
46909 46923 46938 46953 46967
295
46982 46997 47012 47026 47 041
47056 47070 47085 47100 47114
296
47129 47144 47159 47173 47188
47 202 47 217 47232 47246 47 261
297
47276 47 290 47305 47319 47334
47349 47363 47378 47 392 47407
298
47422 47436 47451 47465 47480
47494 47509 47 524 47 538 47 553
299
47 567 47 582 47596 47611 47625
47640 47654 47669 47683 47 698
300
47712 47 727 47741 47756 47 770
47 784 47 799 47813 47828 47842
N
12 3 4
5 6 7 8 9
250-2
too
6
300-350
N *
12 3
4
5 6 7 8 9
300
47 712
47 727 47 741 47 756 47 770
47 784 47 799 47813 47 828 47842
301
47 857
47 871 47 885 47900
47 914
47929 47943 47958 47972 47986
302
48001
48015 48029 48 044
4S058
48073 48087 48101 48116 46130
303
48144
48159 48173 48187
48 202
48 216 48 230 48 244 48 259 48273
304
48287
48302 48 316 48330
48344
4S359 48373 48387 48401 48416
305
48430 48444 48458 48473
48487
48 501 48 515 48 530 48 544 48 558
306
48572
48 586 48601 48615
48 629
48643 48657 48671 48686 48 700
307
48 714
48 728 48 742 48 756 48 770
48 785 48 799 48 813 48 827 48 841
308
48 855
48869 48883 48897
48 911
48926 48940 48954 48968 48982
309
48996 49010 49024 49038 49052
49066 49080 49094 49108 49122
310
49136
49150 49164 49178
49192
49206 49220 49234 49248 49262
311
49276 49 290 49304 49318
49 332
49346 49360 49374 49388 49402
312
49415
49429 49443 49457
49 471
49485 49499 49 513 49 527 49 541
313
49554
49 568 49582 49 5%
49 610
49624 49638 49651 49665 49679
314
49693
49707 49721 49 734
49 748
49762 49776 49790 49803 49817
315
49831 49845 49859 49872 49886
49900 49914 49927 49941 49955
316
49 969
49982 49996 50010
50024
50037 50051 50065 50079 50092
317
50106
50120 50133 50147
50161
50174 50188 50202 50 215 50229
318
50243
50256 50270 50 284
50 297
50311 50325 50338 50352 50365
319
50379
50393 50406 50420
50433
50447 50461 50474 50488 50501
320
50 515
50 529 50 542 50556
50 569
50583 50596 50610 50 623 50637
321
50651
50664 50678 50691
50 705
50 718 50 732 50 745 50 759 50 772
322
50 786
50 799 50 813 50 826
50 840
50853 50866 50880 50893 50907
323
50920
50934 50947 50961
50974
50987 510Q1 51014 51028 51041
324
51055
51068 51081 51095 51108
51121 51135 51148 51162 51175
326
51188
51202 51215 51228
51242
51255 51268 51282 51295 51308
326
51322
51335 51348 51362
51375
51388 51402 51415 51428 51441
327
51455
51468 51481 51495
51508
51521 51534 51548 51561 51574
328
51587
51601 51614 51627
51640
51654 51667 51(580 51693 51706
329
51720
51733 51746 51759 51772
51 786 51 799 51 812 51 825 51 838
330
51851
51865 51878 51891
51904
51917 51930 51943 51957 51970
331
51983
51996 52 009 52 022
52 035
52048 52 061 52 075 52 088 52101
332
52114
52127 52140 52153
52166
52179 52192 52 205 52 218 52 231
333
52 244
52 257 52 270 52 284
52 297
52310 52323 52 336 52 349 52 362
334
52 375
52 388 52401 52414
52427
52440 52 453 52466 52 479 52492
335
52504
52517 52 530 52 543
52 556
52 569 52 582 52 595 52 608 52 621
336
52 634
52 647 52 660 52 673
52 686
52699 52 711 52 724 52 737 52 750
337
52 763
52 776 52 789 52 802
52 815
52827 52840 52853 52866 52879
338
52 892 52905 52 917 52 930
52 943
52956 52969 52982 52994 53 007
339
53020
53033 53 046 53 058
53 071
53084 53 097 53110 53122 5313S
340
53148
53161 53173 53186
53199
53 212 53 224 53 237 53 250 53 263
341
53 275
53 288 53 301 53 314
53 326
53 339 53 352 53 364 53 377 53 39Q
342
53 403
53415 53 428 53441
53 453
53 466 53 479 53491 53 504 53 517
343
53 529
53 542 53 555 53 567
53 580
53 593 53 605 53 618 53 631 53 643
344
53 656
53 668 53 681 53 694
53 706
53 719 53 732 53 744 53 757 53 769
345
53 782
53 794 53 807 53 820
53 832
53 845 53 857 53 870 53 882 53 895
346
53 908
53 920 53 933 53 945
53 958
53 970 53 983 53 995 54 008 54 020
347
54 033
54045 54058 54070
54083
54095 54108 54120 54133 54145
348
54158
54170 54183 54195
54 208
54 220 54 233 54 245 54258 5427Q
349
54 283
54 295 54 307 54 320
54 332
54345 54357 54370 54 382 54394
350
54407
54 419 54432 54 444
54456
54469 54481 54 494 54 506 54518
N
12 3
4
5 6 7 8 9
300-350
360-4
too
7
N
1
2
3
4
5
6
7
8
350
54407
54 419
54 432
54444
54456
54469
54 481
54494 54 506
54 518
351
54 531
34543
54 555
54 568
54 580
54 593
54605
54 617
54630
54 642
352
54654
54 667
54 679
54 691
54 704
54 716
54 728
54 741
54 753
54 765
353
54 777
54 790
54 802
54 814
54 827
54 839
54 851
54 864
54 876
54888
354
54900
54 913
54 925
54937
54 949
54962
54 974
54986
54 998
55 011
355
55 023
55 035
55 047
55 060
55 072
55 084
55 096
55108
55121
55133
356
55 145
55157
55169
55182
55194
55 206
55 218
55 230
55 242
55 255
357
55 267
55 279
55 291
55 303
55 315
55 328
55 340
55 352
55 364
55 376
358
55 388
55 400
55 413
55 425
55 437
55 449
55 461
55 473
55 485
55 497
359
55 509
55 522
55 534
55 546
55 558
55 570
55 582
55 594
55 606
55 618
360
55 630
55 642
55 654
55 666
55 678
55 691
55 703
55 715
55 727
55 739
361
55 751
55 763
55 775
55 787
55 799
55 811
55 823
55 835
55 847
55 859
362
55 871
55 883
55 895
55 907
55 919
55 931
55943
55 955
55 967
55 979
363
55 991
56003
56015
56027
56038
56050
56062
56074
56086
56098
364
56110 56122
56134
56146
56158
56170
56182
56194
56205
56217
365
56229
56241
56 253
56265
56277
56289
56301
56 312
56324
56336
366
56348
56360
56372
56384
56396
56407
56419
56431
56443
56455
367
56467
56478
56490
56 502
56 514
56 526
56 538
56 549
56561
56573
368
56585
56 597
56608
56620
56632
56644
56656
56667
56679
56691
369
56 703
56 714
56 726
56 738
56 750
56 761
56 773
56 785
56 797
56808
370
56820
56832
56844
56855
56867
56879
56891
56902
56914
56926
371
56937
56949
56961
56972
56984
56996
57008
57019
57031
57043
372
57054
57066
57 078
57 089
57101
57113
57124
57136
57148
57159
373
57171
57183
57194
57 206
57 217
57 229
57241
57 252
57264
57276
374
57287
57 299
57310
57322
57334
57345
57357
57368
57380
57392
375
57403
57 415
57426
57438
57449
57461
57473
57484
57496 57 507
376
57 519
57 530
57 542
57 553
57 565
57 576
57 588
57600
57611
57623
377
57634
57646
57657
57669
57 680
57692
57 703
57 715
57 726
57 738
378
57 749
57 761
57 772
57 784
57 795
57807
57818
57 830
57841
57852
379
57864
57 875
57887
57898
57910
57921
57933
57944
57955
57967
380
57978
57 990
58001
58013
58024
58035
58047
58058
58070
58081
381
58092
58104
58115
58127
58138
58149
58161
58172
58184
58195
382
58 206
58 218
58229
58 240
58252
58263
58274
58286
58297
58309
383
58 320
58 331
58 343
58354
5^365
58478
58377
58388
58399
58410
58422
384
58433
58444
58456
58467
58490
58501
58 512
58524
58 535
385
58 546
58 557
58 569
58 580
58 591
58602
58614
58625
58 636
58 647
386
58659
58670
58681
58692
58 704
58 715
58 726
58 737
58 749
58 760
387
58 771
58 782
58 794
58805
58 816
58827
58838 58850 58861
58872
388
58 883
58 894
58906
58917
58928
58939
58950
58961
58973
58984
389
58995
59006
59017
59028
59040
59051
59062
59073
59084
59095
390
59106
59 118
59129
59140
59151
59162
59173
59184
59195
59207'
391
59 218
59 229
59240
59 251
59 262
59273
59 284
59295
59306
59318
392
59329
59340
59351
59 362
59373
59384
59395
59406
59417
59428
393
59439
59450
59 461
59472
59483
59494
59506
59517
59 528
59 539
394
59550 59 561
59572
59 583
59 594
59605
59616
59627
59638
59649
895
59660
59671
59682
59693
59 704
59 715
59 726
59 737
59 748
59 759
896
59 770
59 780
59 791
59 802
59813
59824
59835
59846
59857
59868
397
59879
59890
59901
59912
59 923
59934
59945
59956
59966
59977
898
59988
59999 60 010
60021
60032
60043
60054
60065
60076
60086
399
60097
60108
60119
60130
60141
60152
60163
60173
60184
60195
400
60206
60217
60228
60 239
60 249
60260
60 271
60282
60293
60304
N
1
2
3
4
5
6
7
8
9
350-4
LOO
400-460
N
12 8
4
5 6 7 8
400
60206 60217 60228 60239
60 249
60260 60271 60282 60293
60304
401
60314 60325 60336 60347
60 358
60369 60379 60390 60401
60412
402
60423 60433 60444 60455
60466
60477 60487 60498 60509
60 520
403
60 531 60 541 60 552 60563
60 574
60 584 60595 60606 60 617
60627
404
60638 60649 60660 60670
60681
60692 60703 60713 60724 60735
405
60746 60756 60767 60778 60788
60 799 60810 60821 60831
60842
406
60853 60863 60874 60885
60 895
60906 60917 60927 60938
60949
407
60959 60970 60 981 60991
61002
61013 61023 61034 61045
61055
408
61066 61077 61087 61098
61109
61 119 61 130 61 140 61 151
61162
409
61 172 61 183 61 194 61 204 61 215
61225 61236 61247 61257
61268
410
61278 61289 61300 61310
61321
61331 61342 61352 61363
61374
411
61384 61395 61405 61416
61426
61437 61448 61458 61469
61479
412
61490 61500 61511 61521
61532
61542 61553 61563 61574
61584
413
61595 61606 61616 61627
61637
61648 61658 61669 61679
61690
414
61700 61711 61721 61731
61742
61752 61763 61773 61784
61794
415
61805 61815 61826 61836
61847
61857 61868 61878 61888
61899
416
61909 61920 61930 61941
61951
61962 61972 61982 61993
62003
417
62 014 62 024 62 034 62 045
62 055
62066 62076 62086 62097
62107
418
62118 62128 62138 62149
62159
62170 62180 62190 62 201
62 211
419
62 221 62 232 62 242 62 252
62 263
62273 62284 62294 62304
62315
420
62 325 62 335 62 346 62 356
62 366
62 377 62387 62 397 62 408
62418
421
62 428 62439 62449 62 459
62 469
62 480 62 490 62 500 62 511
62 521
422
62 531 62 542 62 552 62 562
62 572
62 583 62 593 62 603 62 613
62 624
423
62634 62644 62655 62665
62 675
62685 62696 62 706 62 716
62 726
424
62 737 62 747 62 757 62 767
62 778
62 788 62 798 62808 62818
62 829
425
62 839 62 849 62 859 62 870
62 880
62890 62900 62910 62921
62 931
426
62941 62 951 62961 62 972
62 982
62992 63002 63012 63022
63 033
427
63043 63053 63063 63073
63 083
63094 63104 63114 63124
63134
428
63 144 63 155 63 165 63 175 63 185
63195 63205 63215 63225
63 236
429
63 246 63256 63266 63 276 63 286
632% 63306 63317 63327
63 337
480
63 347 63 357 63 367 63 377 63 387
63397 63407 63417 63428
63438
431
63448 63458 63468 63478 63488
63498 63 508 63 518 63 528
63 538
432
63 548 63 558 63 568 63 579 63 589
63599 63609 63619 63629 63639
433
63649 63659 63669 63679
63 689,
63699 63 709 63 719 63 729
63 739
434
63 749 63 759 63 769 63 779
63 789
63799 63809 63819 63829
63 839
435
63849 63859 63869 63879
63 889
63899 63909 63919 63929
63 939
436
63949 63 959 63969 63979
63 988
63998 64008 64018 64028
64038
437
64048 64058 64068 64078
64 088
64098 64108 64118 64128
64137
438
64147 64157 64167 64177
64187
64197 64207 64 217 64 227
64 237
439
64 246 64256 64266 64 276
64 286
64 296 64306 64316 64326 64 335
440
64 345 64 355 64365 64 375 64385
64395 64404 64414 64424
64 434
441
64444 64454 64464 64473
64 483
64493 64 503 64 513 64 523
$4 532
442
64 542 64 552 64562 64 572
64 582
64591 64601 64611 64621
64631
443
64640 64650 64660 64670 64680
64689 64699 64 709 64 719
64 729
444
64 738 64 748 64 758 64 768
64 777
64 787 64 797 64807 64816
64826
445
64836 64846 64856 64865
64 875
64885 64895 64904 64914
64924
446
64933 64943 64953 64963
64972
64982 64992 65 002 65011
65 021
447
65 031 65 040 65 050 65 060
65 070
65079 65 089 65 099 65108
65118
448
65128 65137 65147 65157
65167
65 176 65 186 65 196 65 205
65 215
449
65 225 65 234 65 244 65 254 65 263
65 273 65 283 65 292 65 302
65 312
450
65 321 65 331 65 341 65 350
65 360
65369 65379 65389 65398
65408
N
12 8
4
5 6 7 8
9
400-450
450-500
9
V
1
2
3
4
5 6 7 8
450
65 321
65 331
65 341
65 350
65 360
65 369 65 379 65 389 65 398
65 408
451
65 418
65 427
65 437
65 447
65 456
65 466 65 475 65 485 65 495
65 504
452
65 514
65 523
65 533
65 543
65 552
65 562 65 571 65 581 65 591
65 600
453
65 610
65 619
65 629
65 639
65 648
65 658 65 667 65 677 65 686
65 696
454
65 706 65 715 65 725 65 734 65 744
65 753 65 763 65 772 65 782
65 792
455
65 801
65 811
65 820
65 830
65 839
65 849 65 858 65 868 65 877
65 887
456
65 896
65 906
65 916
65925
65 935
65 944 65 954 65 963 65 973
65 982
457
65 992
66001
66011
66020
66030
66039 66049 66058 66068
66077
458
66087 66096 66106 66115 66124
66134 66143 66153 66162
66172
459
66181
66191
66200
66210
66 219
66229 66238 66247 66257
66266
460
66276
66285
66 295
66304
66 314
66323 66332 66342 66351
66361
461
66370 66380
66389
66398
66408
66417 66427 66436 66445
66455
462
66464
66474
66483
66492
66502
66 511 66 521 66 530 66 539
66 549
463
66 558
66 567
66 577
66 586
66596
66605 66614 66624 66633
66642
464
66652
66661
66671
66680
66689
66699 66 708 66 717 66 727
66 736
465
66 745
66 755
66 764
66 773
66 783
66792 66801 66811 66820
66829
466
66839
66848
66857
66867
66876
66885 66894 66904 66913
66922
467
66932
66941
66950 66960
66969
66978 66987 66997 67006
67015
468
67025
67034
67043
67052
67062
67071 67080 67 089 67099
67108
469
67117
67127
67136
67145
67154
67164 67173 67182 67191
67 201
470
67 210
67 219
67 228
67 237
67 247
67256 67265 67274 67 284
67 293
471
67302
67311
67321
67 330
67339
67348 67357 67367 67376
67385
472
67394
67403
67413
67422
67 431
67440 67449 67459 67468
67477
473
67 486
67495
67 504
67 514
67 523
67 532 67 541 67 550 67 560
67 569
474
67 578
67 587
67 596
67605
67614
67624 67 633 67 642 67651
67 660
475
67669
67679
67688
67697
67 706
67 715 67724 67 733 67 742
67 752
476
67 761
67 770 67 779
67 788
67 797
67806 67815 67 825 67 834
67843
477
67 852
67861
67870
67 879
67 888
67897 67906 67916 67925 67934
478
67943
67 952
67 961
67970
67979
67988 67997 68006 68015
68024
479
68034
68043
68052
68061
68070
68079 68088 68097 68106
68115
480
68124
68133
68142
68151
68160
68169 68178 68187 68196
68 205
481
68 215 68 224 68 233
68 242
68 251
68260 68269 68278 68287
68 296
482
68 305
68 314
68323
68332
68341
68350 68 359 68368 68377
68386
483
68 395
68404
68413
68422
68431
68440 68449 68458 68467
68476
484
68 485
68494 68502
68 511
68 520
68 529 68 538 68 547 68 556
68 565
485
68 574
68 583
68 592
68601
68610
68619 68628 68637 68646 68655
486
68 664
68 673
68 681
68690
68 699
68 708 68 717 68 726 68 735
68 744
487
68 753
68 762
68 771
68 780
68 789
68 797 68806 68815 68824
68 833
488
68842
68851
68860
68 869
68 878
68886 68895 68904 68913
68 922
489
68931
68940
68949
68958
68966
68975 68984 68993 69002
69011
490
69020
69028
69037
69046
69055
69064 69073 69082 69090
69099
491
69108
69117 69126 69135
69144
69152 69161 69170 69179
69188
492
69197
69205
69214
69 223
69232
69 241 69 249 69258 69267
69276
493
69285
69 294
69302
69311
69320
69329 69338 69346 69355
69364
494
69 373
69381
69390 69399
69408
69417 69425 69434 69443
69452
495
69461
69469
69478 69487
694%
69 504 69 513 69 522 69 531
69 539
496
69 548
69 557 69 566 69 574
69 583
69 592 69601 69 609 69 618
69627
497
69 636
69644
69653
69662
69 671
69679 69688 69 697 69 705
69 714
498
69 723
69 732
69 740 69 749
69 758
69 767 69 775 69 784 69 793
69801
499
69 810 69 819 69827 69836 69845
69854 69862 69 871 69880
69 888
500
69897
69906
69914
69923
69932
69940 69949 69958 69966 69975
N
O
1
2
3
4
5 6 7 8
450-500
10
500-550
V
1
2
3
4
5
6 7 8
500
69897 69906
69914
69923
69932
69940
69949 69958 69966
69 975
501
69984 69992
70001
70010
70018
70027
70036 70044 70053
70062
502
70070 70079
70088
70096 70105
70114
70122 70131 70140
70148
503
70157 70165
70174
70183
70191
70200
70209 70217 70 226
70234
504
70 243 70252
70260
70269
70278
70 286
70295 70303 70 312
70321
605
70329 70338
70346
70355
70364
70372
70381 70 389 70398
70406
506
70415 70424
70432
70441
70449
70458
70467 70475 70484
70492
507
70501 70 509
70 518
70 526
70 535
70 544
70 552 70 561 70 569
70 578
508
70 586 70 595
70603
70612
70 621
70629
70638 70 646 70 655
70663
509
70672 70680
70689
70697
70 706
70 714
70 723 70 731 70 740
70 749
510
70 757 70 766
70 774
70 783
70 791
70800
70808 70817 70 825
70834
511
70842 70851
70859
70 868
70876
70885
70893 70902 70910
70919
512
70927 70935
70944
70952
70 961
70969
70978 70986 70995
71003
513
71012 71020
71029
71037
71046
71054
71063 71071 71079
71088
614
71 096 71 105 71 113 71 122
71130
71139
71 147 71 155 71 164
71172
515
71181 71189
71198
71206
71214
71223
71231 71240 71248
71257
516
71265 71273
71282
71290
71299
71307
71315 71324 71332
71341
517
71349 71357
71366 7*374
71383
71391
71399 71408 71416 71425
518
71433 71441
71450 71458 71466
71475
71483 71492 71500
71508
519
71517 71525
71533
71542
71550
71559
71567 71575 71584
71592
520
71600 71609
71617
71625
71634
71642
71650 71659 71667
71675
521
71684 71692
71700
71709
71717
71725
71734 71742 71750
71759
622
71767 71775
71784
71792
71800
71809
71817 71825 71834
71842
523
71850 71858
71867
71875
71883
71892
71900 71908 71917
71925
524
71933 71941
71950 71958
71966
71975
71983 71991 71999
72 008
526
72 016 72 024
72 032
72041
72 049
72 057
72066 72074 72082
72 090
526
72 099 72107
72115
72123
72132
72140
72148 72156 72 165
72173
527
72181 72189
72198
72 206
72 214
72 222
72 230 72 239 72 247
72 255
628
72 263 72 272
72 280
72 288
72 296
72 304
72 313 72 321 72 329
72 337
629
72 346 72354
72 362
72370
72 378
72 387
72 395 72403 72411
72419
530
72428 72436
72444
72452
72460
72469
72477 72 485 72493
72 501
531
72 509 '72 518
72 526
72 534
72 542
72 550
72 558 72 567 72 575
72 583
632
72 591 72 599
72 607
72616
72 624
72 632
72640 72 648 72 656 72 665
633
72 673 72681
72 689
72 697
72 705
72 713
72 722 72 730 72 738
72 746
534
72 754 72 762
72 770
72 779 *72 787
72 795
72 803 72 811 72 819
72 827
535
72835 72 843
72 852
72 860
72 868
72876
72 884 72 892 72 900
72 908
536
72916 72 925
72 933
72 941
72 949
72 957
72 965 72 973 72 981
72 989
537
72997 73 006
73 014
73 022
73 030
73 038
73 046 73 054 73 062
73 070
538
73 078 73 086
73 094
73102
73111
73119
73127 73135 73143
73151
539
73159 73167
73175
73183
73191
73199
73 207 73 215 73 223
73 231
540
73 239 73 247
73 255
73 263
73 272
73 280
73 288 73 296 73 304
73 312
541
73 320 73 328
73 336
73 344
73 352
73 360
73 368 73 376 73 384
73 392
542
73400 73408
73 416
73 424
73 432
73 440
73 448 73 456 73 464
73 472
543
73480 73488
73 496
73 504
73 512
73 520
73 528 73 536 73 544
73 552
544
73 560 73 568
73 576
73 584
73 592
73 600
73 608 73616 73 624
73 632
545
73640 73648
73 656
73 664
73 672
73 679
73 687 73 695 73 703
73 711
646
73 719 73 727
73 735
73 743
73 751
73 759
73 767 73 775 73 783
73 791
647
73 799 73 807 73 815
73 823
73 830
73 838
73 846 73 854 73 862
73 870
548
73 878 73 886
73 894
73 902
73 910
73 918
73926 73 933 73 941
73 949
649
73 957 73 965
73 973
73 981
73 989
73 997
74005 74013 74020
74 028
550
74 036 74044
74052
74060
74068
74076
74084 74092 74099
74107
N
1
2
3
4
5
6 7 8
500-550
550-600
ii
N
12
3
4
5
6
7
8
550
74036 74044 74052
74060
74068
74076 74084
74092
74099
74107
551
74115 74123 74131
74139
74147
74155
74162
74170
74178
74186
652
74194 74 202 74 210
74 218
74 225
74 233
74 241
74 249
74257
74265
553
74 273 74 280 74 288
74 2%
74 304'
74312
74 320
74 327
74335
74 343
554
74351 74359 74367
74374
74382
74 390 74398
74406
74414
74421
555
74429 74437 74445
74453
74461
74468
74 476
74484
74492
74500
556
74 507 74 515 74 523
74 531
74 539
74 547
74 554
74 562
74 570
74 578
657
74 586 74 593 74601
74 609
74617
74624
74 632
74640
74648
74656
558
74663 74 671 74679
74 687 74695
. 74 702
74 710
74 718
74 726
74 733
659
74 741 74 749 74 757
74 764
74 772
74 780
74 788
74 796
74803
74 811
560
74 819 74 827 74834 74 842 74850
74858
74 865
74 873
74 881
74 889
561
74896 74 904 74912
74920
74927
74935 74 943
74 950
74 958
74 966
662
74974 74981 74989 74997 75005
75 012
75 020
75 028
75035
75 043
563
75 051 75 059 75 066
75 074
75 082
75 089 75 097 75105
75113
75120
564
75128 75136 75143
75151
75159
75166.
75174
75182
75189
75197
565
75 205 75 213 75 220
75 228
75 236
75 243
75 251
75 259
75 266
75 274
566
75 282 75 289 75 297
75 305
75 312
75 320
75 328
75 335
75343
75 351
567
75 358 75 366 75 374
75 381
75 389
75 397
75 404
75 412
75 420
75 427
568
75 435 75 442 75 450
75 458
75 465
75 473
75 481
75 488
75 496
75 504
569
75 511 75 519 75 526
75 534
75 542
75 549 75 557 75 565 75 572 75 580
570
75 587 75 595 75 603
75 610
75 618
75 626
75 633
75 641
75 648
75 656
571
75 664 75 671 75 679
75 686
75 694
75 702
75 709
75 717
75 724
75 732
572
75 740 75 747 75 755
75 762
75 770
75 778
75 785
75 793
75 800
75 808
573
75 815 75 823 75 831
75 838
75 846
75 853
75 861
75 868
75 876
75 884
574
75 891 75 899 75 906
75 914
75 921
75 929
75 937
75 944
75 952
75 959
576
75 967 75 974 75 982
75 989
75 997
76005
76012
76020
76027
76035
576
76042 76050 76057 76065 76072
76080
76087
76095
76103
76110
577
76118 76125 76133
76140
76148
76155
76163
76170
76178
76185
578
76193 76 200 76208
76215
76 223
76230
76238
76245
76253
76 260
679
76268 76275 76283
76290
76298
76305
76313
76320
76328
76335
580
76343 76350 76358
76365
76373
76380
76388
76395
76403
76410
581
76418 76425 76433
76440
76448
76455
76462
76470
76477
76485
582
76492 76500 76 507 76 515
76 522
76530
76 537
76545 76 552 76559
583
76 567 76 574 76582
76 589
76597
76604
76612
76619
76626
76634
684
76641 76649 76656
76664
76671
76678
76686
76693
76 701
76 708
585
76 716 76 723 76 730
76 738
76 745
76753 76760 76768 76775 76782
586
76 790 76 797 76805
76812
76819
76827
76834
76842
76849
76856
587
76864 76871 76879
76886
76893
76901
76908
76916
76923
76 930
588
76938 76945 76953
76960
76967
76975
76982
76989
76997
77004
589
77012 77019 77026
77034
77041
77048
77056
77063
77070
77078
590
77085 77093 77100
77107
77115
77122
77129
77137
77144
77151
591
77159 77166 77173
77181
77188
77195
77 203
77210
77217
77 225
692
77 232 77240 77247
77 254
77262
77269
77 276
77283
77291
77298
593
77 305 77313 77320
77327
77335
77342
77349
77357
77364
77 371
594
77379 77386 77393
77401
77408
77415
77422
77430
77437
77444
595
77452 77459 77466
77474
77481
77488
77495
77 503
77510
77 517
596
77 525 77532 77539
77 546
77 554
77561
77 568
77 576
77 583
77 590
597
.77597 77605 77 612
77 619
77627
77 634
77641
77648
77 656
77663
598
77 670 77677 77685
77692
77 699
77 706
77 714
77 721
77 728
77 735
599
77 743 77 750 77757
77 764
77 772
77 779
77786
77 793
77801
77 808
600
77815 77822 77 830
77837
77 844
77851
77 859
77866
77873
77880
N
12
3
4
5
6
7
8
9
550 -e
100
12
600-650
N
1 2
3
4
5
6
7
8
eoo
77 815
77822 77 830
77837
77 844
77851
77 859
77866
77873
77880
601
77887 77895 77902
77909
77916
77924
77 931
77938
77945
77952
602
77 960
77967 77974
77981
77988
77996
78 003
78010
78017
78025
603
78032
78039 78046
78053
78061
78068 78 075
78082
78 089
78 097
604
78104
78111 78118
78125
78132
78140
78147
78154
78161
78168
605
78176
78183 78190
78197
78204
78 211
78 219
78226 78233
78 240
606
78 247
78254 78262
78 269
78276
78283
78 290
78 297
78305
78312
607
78319
78326 78333
78 340
78347
78355 78362 78369 78376 78 383
608
78390
78398 78405
78412
78419
78426 78 433
78440
78447
78 455
609
78462
78469 78476
78483
78490
78497
78 504
78 512
78 519
78 526
610
78 533
78 540 78547
78 554
78 561
78 569
78 576
78 583
78 590
78 597
611
78604
78611 78618
78 625
78633
78640
78647
78654
78661
78668
612
78675
78682 78 689
78696
78 704
78 711
78 718 78 725
78 732
78 739
613
78 746
78 753 78 760
78 767
78 774
78 781
78 789
78 796
78803
78 810
614
78 817
78824 78831
78838
78845
78 852
78 859
78866
78873
78 880
615
78 888
78895 78902
78909
78916
78923
78 930
78937
78944
78951
616
78 958
78965 78972
78979
78986
78993
79000
79007
79014
79021
617
79029
79036 79043
79050
79057
79064
79071
79078 79085
79092
618
79099
79106 79113
79120
79127
79134
79141
79148
79155
79162
619
79169
79176 79183
79190
79197
79204
79211
79218
79 225
79 232
620
79 239
79246 79253
79 260
79267
79 274
79281
79 288
79295
79 302
621
79309
79316 79323
79330
79337
79 344
79351
79358
79365
79372
622
79379
79386 79 393
79400
79407
79414
79421
79428
79435
79 442
623
79449
79456 79463
79470
79477
79484
79491
79498 79 505
79 511
624
79 518
79 525 79532
79 539
79 546
79 553
79 560
79 567
79 574
79 581
625
79588 7959i 79602 79609 79616
79623
79630
79637 79644
79650
626
79657
79664 79671
79678
79685
79692
79699
79 706
79 713
79 720
627
79 727
79 734 79 741
79 748
79 754
79 761
79 768
79 775
79 782
79 789
628
79 796
79803 79810
79 817
79 824
79831
79837
79844
79851
79 858
629
79865
79872 79879
79886
79893
79900
79906
79913
79-920
79927
630
79934
79941 79948 79955
79962
79969
79975
79982
79989
79996
631
80003
80010 80017
80024
80030
80037
80044
80051
80058
80065
632
80072
80079 80085
80092
80099
80106
80113
80120
80127
80134
633
80140
80147 80154
80161
80168
80175
80182
80188
80195
80202
634
80209
80216 80223
80229
80236
80243
80 250
80257
80 264
80271
635
80277
80284 80291
80 298 80305
80312
80318
80325
80332
80339
636
80346
80353 80359
80366
80373
80380
80387
80393
80400
80407
637
80414
80421 80428
80434
80441
80448 80455
80 462
80468
80475
638
80482
80489 80496
80 502
80 509
80 516
80 523
80 530
80 536
80 543
639
80550
80557 80564
80570
80577
80 584
80 591
80598
80604
80611
640
80618 80625 80632
80 638
80645
80652
80659
80665
80672
80679
641
80686
80693 80699
80 706
80 713
80 720
80 726
80 733
80 740
80 747
642
80 754
80 760 80 767
80 774
80 781
80 787
80 794
80 801
80 808
80 814
643
80821
80828 80835
80841
80 848
80855
80862
80868
80875
80882
644
80 889
80895 80902
80909
80916
80922
80929
80936
80943
80949
645
80956
80963 80969 80976 80983
80990
80996
81003
81010
81017
646
81023
81030 81037
81043
81050
81057
81064
81070
81077
81084
647
81090
81097 81104
81111
81117
81124
81131
81137
81144
81151
648
81158
81164 81171
81178
81184
81191
81198
81204
81211
81218
649
81224
81231 81238 81245
81251
81258 81265
81271
81278
81285
650
81291
81298 81305
81311
81318
81325
81331
81338 81345
81351
N
1 2
/
3
4
5
6
7
8
9
600-650
650-700
13
N
O 1
2 8
4
5 6 7 8
650
81291 81298 81305 81311
81318
81325 81331 81338 81345
81351
651
81358 81365
81371 81378
81385
81391 81398 81405 81411
81418
652
81425 81431
81438 81445
81451
81458 81465 81471 81478 81485
653
81491 81498
81505 81511
81518
81525 81531 81538 81544
81551
654
81558 81564
81571 81578
81584
81591 81598 81604 81611
81617
656
81624 81631
81637 81644 81651
81657 81664 81671 81677
81684
656
81690 81697
81704 81710
81717
81723 81730 81737 81743
81750
657
81757 81763
81770 81776
81783
81790 81796 81803 81809
81816
668
81823 81829
81836 81842
81849
81856 81862 81869 81875
81882
669
81889 81895
81902 81908
81915
81921 81928 81935 81941
81948
660
81954 81961
81968 81974
81981
81987 81994 82000 82 007
82014
661
82 020 82 027
82 033 82040
82046
82053 82 060 82066 82073
82079
662
82 086 82 092
82099 82105
82112
82119 82125 82132 82138
82145
663
82151 82158
82164 82171
82178
82184 82191 82197 82 204
82 210
664
82217 82223
82 230 82 236
82 243
82 249 82 256 82263 82 269
82 276
665
82 282 82 289 82 295 82 302
82 308
82315 82321 82328 82 334
82 341
666
82 347 82 354
82 360 82 367
82 373
82 380 82387 82 393 82 400
82406
667
82 413 82 419
82426 82 432
82439
82445 82452 82458 82 465
82471
668
82 478 82484
82491 82 497
82 504
82 510 82 517 82 523 82 530
82 536
669
82 543 82 549
82 556 82 562
82 569
82 575 82 582 82 588 82 595
82 601
670
82607 82 614
82 620 82 627
82 633
82640 82646 82653 82 659
82666
671
82 672 82 679
82 685 82692
82 698
82 705 82 711 82 718 82 724 82 730
672
82 737 82 743
82 750 82 756 82 763
82 769 82 776 82 782 82 789
82 795
673
82802 82808
82 814 82 821
82 827
82 834 82 840 82 847 82 853
82 860
674
82866 82872
82 879 82 885
82 892
82 898 82905 82 911 82 918
82924
676
82 930 82 937
82943 82 950 82956
82963 82 969 82 975 82 982
82 988
676
82995 83 001
83008 83014
83 020
83027 83033 83040 83046
83 052
677
83 059 83 065
83072 83078 83085
83091 83097 83104 83 110
83117
678
83123 83129
83136 83 142
83149
83155 83161 83168 83174
83181
679
83187 83193
83 200 83 206
83 213
83 219 83 225 83 232 83 238
83 245
680
83 251 83 257
83 264 83 270
83 276
83283 83 289 83 296 83302
83 308
681
83 315 83 321
83 327 83 334
83 340
83 347 83 353 83 359 83 366
83 372
682
83 378 83 385
83 391 83 398
83 404
83 410 83 417 83 423 83 429
83 436
683
83442 83448
83 455 83 461
83 467
83474 83480 83487 83493
83 499
684
83 506 83 512
83 518 83 525
83 531
83 537 83 544 83 550 83 556
83 563
685
83 569 83 575 83 582 83 588
83 594
83 601 83 607 83 613 83 620
83 626
686
83632 83639
83645 83 651
83 658
83664 83670 83677 83683
83 689
687
83696 83 702 83 708 83 715
83 721
83 727 83 734 83 740 83 746
83 753
688
83 759 83 765
83 771 83 778
83 784
83 790 83 797 83 803 83 809
83 816
689
83 822 83 828 83 835 83 841
83 847
83 853 83 860 83 866 83 872
83 879
690
83 885 83 891
83897 83904
83 910
83916 83923 83929 83935
83 942
691
83948 83 954
83960 83967
83 973
83979 83985 83992 83 998
84004
692
84011 84017
84023 84029
84 036
84042 84048 84055 84061
84 067
693
84073 84080
84 086 84092
84098
84105 84111 84117 84123
84130
694
84136 84142
84148 84155
84161
84167 84173 84180 84186
84192
695
84198 84205
84 211 84 217
84 223
84230 84 236 84 242 84 248 84 255
696
84 261 84 267
84 273 84 280
84 286
84292 84298 84 305 84 311
84 317
697
84323 84 330
84 336 84 342
84348
84354 84361 84367 84373
84379
698
84386 84 392
84 398 84404
84410
84417 84423 84429 84435
84442
699
84448 84454
84460 84466 84473
84479 84485 84491 84497
84 504
700
84 510 84 516 84 522 84 528 84 535
84 541 84547 84 553 84 559
84 566
N
O 1
2 8
4
6 6 7 8
650-700
14
700-750
N
12
3 4
5
6
7
8 9
700
84 510 84 516 84 522 84 528 84 535
84 541
84 547
84 553
84 559 84 566
701
84 572 84 578 84 584
84 590 84 597
84603
84 609
84 615
84 621 84 628
702
84634 84640 84646
84 652 84658
84 665
84 671
84 677
84683 84689
703
84696 84 702 84 708
84 714 84 720
84 726
84 733
84 739 84 745 84 751
704
84 757 84 763 84 770
84 776 84 782
84 788
84 794
84800
84 807 84 813
705
84819 84825 84 831
84837 84844
84850 84856 84862 84868 84874
706
84880 84887 84893
84899 84905
84 911
84917
84924
84 930 84936
707
84942 84948 84954
84960 84967
84973
84979 84985
84991 84 997
708
85 003 85 009 85 016
85 022 85 028
85 034
85 040
85 046
85 052 85 058
709
85 065 85 071 85 077
85 083 85 089
85 095
85101
85107
85114 85120
710
85126 85132 85138
85144 85150
85156
85163
85169 85175 85 181
711
85187 85193 85199
85 205 85 211
85 217
85 224
85 230
85 236 85 242
712
85 248 85 254 85 260
85 266 85 272
85 278 85 285 85 291
85 297 85 303
713
85 309 85 315 85 321
85 327 85 333
85 339
85 345
85 352
85 358 85 364
7H
85 370 85 376 85 382
85 388 85 394
85 400
85 406
85 412
85 418 85 425
715
85 431 85 437 85 443
85 449 85 455
85 461
85467
85 473
85 479 85 485
716
85 491 85 497 85 503
85 509 85 516
85 522
85 528
85 534
85 540 85 546
717
85 552 85 558 85 564
85 570 85 576
85 582
85 588
85 594
85 600 85 606
718
85 612 85 618 85 625
85 631 85 637
85 643
85 649 85 655
85 661 85 667
719
85 673 85 679 85 685 85 691 85 697
85 703
85 709
85 715
85 721 85 727
720
85 733 85 739 85 745
85 751 85 757
85 763
85 769 85 775
85 781 85 788
721
85 794 85 800 85 806
85 812 85 818
85 824
85 830
85 836
85 842 85 848
722
85 854 85 860 85 866
85 872 85 878
85 884
85 890
85 8%
85 902 85 908
723
85 914 85 920 85 926
85 932 85 938
85 944
85 950 85 956
85 962 85 968
724
85 974 85 980 85 986
85 992 85 998
86004
86010
86016
86022 86028
726
86034 86040 86046
86052 86058
86064
86070
86076
86082 86088
726
86094 86100 86106
86112 86118
86124
86130
86136
86141 86147
727
86153 86159 86165
86171 86177
86183
86189
86195
86201 86 207
728
86 213 86219 86225
86 231 86 237
86 243
86249
86255
86 261 86267
729
86 273 86 279 86 285
86291 86297
86303
86308
86 314
86320 86326
730
86332 86338 86344
86350 86356
86362
86368
86374
86380 86386
731
86392 86398 86404
86410 86415
86421
86427
86433
86439 86445
732
86451 86457 86463
86469 86475
86481
86487
86493
86499 86 504
733
86 510 86 516 86 522
86 528 86 534
86540
86 546
86 552
86 558 86 564
734
86 570 86 576 86 581
86 587 86 593
86 599
86605
86611
86617 86623
736
86629 86635 86641
86646 86652
86658
86664
86670
86676 86 682
736
86688 86694 86 700
86 705 86 711
86 717
86 723
86 729
86 735 86 741
737
86 747 86 753 86 759
86 764 86 770
86 776
86 782
86 788
86 794 86 800
738
86806 86 812 86 817
86 823 86829
86835
86 841
86 847
86 853 86 859
739
86864 86 870 86876
86 882 86 888
86 894
86900
86 906
86911 86917 ■
740
86 923 86929 86935
86941 86947
86953
86958
86964
86970 86 976
741
86982 86988 86994
86999 87005
87011
87017
87023
87 029 87 035
742
87040 87046 87052
87 058 87 064
87070
87075
87081
87087 -87 093
743
87099 87105 87111 87116 87122
87128
87 134
87140
87146 87151
744
87157 87163 87169
87175 87181
87186
87192
87198
87 204 87 210
745
87216 87 221 87 227
87 233 87239
87245
87251
87256 87262 87 268
746
87 274 87 280 87 286
87291 87 297
87303
87309 87 315 87320 87 326
747
87 332 87 338 87344
87 349 87355
87361
87367
87373
87379 87384
748
87390 87 396 87402
87408 87413
87419 87 425
87431
87437 87 442
749
87 448 87 454 87460
87466 87471
87477
87483
87489
87495 87 500
750
87 506 87 512 87518
87 523 87 529
87 535
87 541
87 547
87 552 87 558
N
12
3 4
5
6
7
8 9
700-750
750-800
15
N
12 3
4
5 6 7 8 9
750
87 506 87 512 87 518 87 523
87529
87535 87541 87 547 87 552 87 558
751
87564 87570 87 576 87581
87 587
87593 87599 87604 87610 87 616
752
87 622 87628 87 633 87639 87645
87651 87656 87662 87668 87674
753
87679 87685 87691 87697
87 703
87 708 87 714 87 720 87 726 87 731
754
87 737 87 743 87 749 87 754
87 760
87 766 87 772 87 777 87 783 87 789
755
87 795 87800 87806 87812
87818
87 823 87829 87835 87841 87846
756
87852 87 858 87864 87 869
87875
87 881 87887 87892 87 898 87904
757
87910 87915 87921 87 927
87933
87938 87944 87950 87955 87961
758
87967 87973 87978 87984
87990
87996 88001 88007 88013 88018
769
88024 88030 88036 88041
88047
88053 88058 88 064 88 070 88076
760
88081 88087 88093 88098
88104
88110 88116 88121 88127 88133
761
88138 88144 88150 88156 88161
88167 88173 88178 88184 88190
762
88195 88201 88207 88 213
88218
88 224 88 230 88235 88241 88247
763
88252 88258 88264 88270
88 275
88281 88 287 88292 88298 88304
764
88309 88315 88321 88326
88332
88338 88343 88349 88355 88360
765
88366 88372 88377 88383
88389
88395 88400 88406 88412 88417
766
88423 88 429 88434 88440
88446
88451 88457 88463 88468 88474
767
88480 88485 88491 88497
88 502
88 508 88513 88519 88525 88 530
768
88 536 88 542 88 547 88 553
88 559
88 564 88 570 88 576 88 581 88 587
769
88593 88598 88604 88610
88615
88621 88627 88632 88 638 88 643
770
88649 88655 88660 88666
88672
88677 88683 88689 88694 88700
771
88 705 88 711 88 717 88 722
88 728
88 734 88 739 88 745 88 750 88 756
772
88 762 88 767 88 773 88 779
88 784
88 790 88 795 88801 88 807 88 812
773
88 818 88 824 88829 88 835
88 840
88 846 88 852 88 857 88 863 88 868
774
88 874 88880 88885 88 891
88 897
88902 88908 88 913 88919 88 925
775
88930 88936 88941 88947
88953
88958 88964 88969 88975 88 981
776
88986 88992 88997 89003
89009
89014 89020 89025 89031 89037
777
89042 89048 89053 89059
89064
89 070 89076 89081 89087 89092
778
89098 89104 89109 89115
89120
89126 89131 89137 89143 89148
779
' 89154 89159 89165 89170
89176
89182 89187 89193 89198 89 204
780
89209 89215 89221 89226 89232
89237 89 243 89248 89 254 89 260
781
89 265 89271 89276 89282
89287
89293 89298 89304 89 310 89315
782
89321 89326 89332 89337
89343
89 348 89354 89360 89 365 89 371
783
89376 89 382 89387 89393
89 398
89404 89409 89415 89421 89426
784
89432 89437 89443 89448
89454
89459 89465 89470 89476 89481
785
89487 89492 89498 89 504
89 509
89 515 89520 89 526 89 531 89 537
786
89 542 89 548 89 553 89 559
89 564
89 570 89 575 89581 89 586 89 592
787
89597 89603 89609 89614
89620
89625 89631 89636 89642 89647
788
89653 89658 89664 89 669 89675
89680 89686 89691 89697 89702
789
89 708 89 713 89 719 89 724
89 730
89 735 89 741 89 746 89 752 89 757
790
89763 89768 89774 89 779 89785
89790 89796 89801 89807 89812
791
89 818 89823 89829 89 834
89840
89 845 89851 89856 89862 89 867
792
89873 89878 89883 89889
89894
89900 89905 89911 89916 89922
793
89927 89933 89938 89944
89949
89955 89960 89966 89971 89977
794
89982 89988 89993 39998 90004
90009 90015 90020 90026 90031
795
90037 90042 90048 90053
90059
90064 90069 90075 90080 90086
796
90091 90097 90102 90108
90113
90119 90124 90129 90135 90140
797
90146 90151 90157 90162
90168
90173 90179 90184 90189 90195
798
90200 90206 90211 90217
90222
90227 90233 90238 90244 90249
799
90255 90260 90266 90271 90276
90282 90287 90293 90298 90304
800
90309 90314 90320 90325
90331
90336 90342 90347 90352 90358
N
12 3
4
5 6 7 8 9
750-800
16
800-850
N
12 8 4
5 6 7 8
800
90309 90314 90320 90325 90331
90336 90342 90347 90352 90358
801
90363 90369 90374 90380 90385
90390 90396 90401 90407 90412
802
90417 90423 90428 90434 90439
90445 90450 90455 90461 90466
803
90472 90477 90482 90488 90493
90499 90504 90509 90 515 90 520
804
90526 90531 90536 80542 90547
90553 90558 90563 90569 90574
806
90580 90585 90590 90596 90601
90607 90612 90617 90623 90628
806
90634 90639 90644 90650 90655
90660 90666 90671 90677 90682
807
90687 90693 90698 90 703 90 709
90714 90720 90725 90730 90736
808
90741 90747 90752 90757 90763
90768 90773 90779 90784 90789
809
90795 90800 90806 90811 90816
90822 90827 90832 90838 90843
810
90849 90854 90859 90865 90870
90875 90881 90886 90891 90897
811
90902 90907 90913 90918 90924
90929 90934 90940 90945 90950
812
90956 90961 90966 90972 90977
90982 90988 90993 90998 91004
813
91009 91014 91020 91025 91030
91036 91041 91046 91052 91057
814
91062 91068 91073 91078 91084
91089 91094 91100 91105 91110
815
91116 91121 91126 91132 91137
91142 91148 91153 91158 91164
816
91169 91174 91180 91185 91190
91196 91201 91206 91212 91217
817
91222 91228 91233 91238 91243
91249 91254 91259 91265 91270
818
91275 91281 91286 91291 91297
91302 91307 91312 91318 91323
819
91328 91334 91339 91344 91350
91355 91360 91365 91371 91376
820
91381 91387 91392 91397 91403
91408 91413 91418 91424 91429
821
91434 91440 91445 91450 91455
91461 91466 91471 91477 91482
822
91487 91492 91498 91503 91508
91514 91519 91524 91529 91535
823
91540 91545 91551 91556 91561
91566 91572 91577 91582 91587
824
91593 91598 91603 91609 91614
91619 91624 91630 91635 91640
826
91 645 91 651 91 656 91 661 91 666
91672 91677 91682 91687 91693
826
91 698 91 703 91 709 91 714 91 719
91 724 91 730 91 735 91 740 91 745
827
91 751 91 756 91 761 91 766 91 772
91777 91782 91787 91793 91798
828
91803 91808 91814 91819 91824
91829 91834 91840 91845 91850
829
91855 91861 91866 91871 91876
VIS82 91887 91892 91897 91903
830
91908 91913 91918 91924 91929
91934 91939 91944 91950 91955
831
91960 91965 91971 91976 91981
91986 91991 91997 92002 92007
832
92012 92018 92023 92028 92 033
92038 92044 92049 92054 92059
833
92065 92070 92 075 92080 92085
92091 920% 92101 92106 92111
834
92117 92122 92127 92132 92137
92143 92148 92153 92158 92163
836
92169 92174 92179 92184 92189
92195 92200 92 205 92 210 92 215
836
92 221 92 226 92231 92 236 92 241
92 247 92 252 92 257 92 262 92 267
837
92273 92278 92283 92288 92293
92 298 92304 92 309 92 314 92319
838
92 324 92330 92 335 92 340 92 345
92 350 92 355 92 361 92 366 92371
839
92 376 92381 92387 92392 92 397
92402 92407 92412 92418 92423
840
92428 92433 92438 92 443 92449
92454 92 459 92 464 92469 92474
841
92480 92485 92490 92 495 92 500
92 505 92 511 92 516 92 521 92 526
842
92 531 92 536 92 542 92 547 92 552
92 557 92 562 92 567 92 572 92 578
843
92 583 92 588 92 593 92 598 92 603
92 609 92 614 92 619 92 624 92 629
844
92634 92639 92 645 92 650 92 655
92 660 92665 92 670 92 675 92681
845
92686 92691 92696 92701 92706
92 711 92 716 92 722 92 727 92 732
846
92 737 92 742 92 747 92 752 92 758
92 763 92 768 92 773 92 778 92 783
847
92788 92793 92799 92804 92809
92 814 92 819 92 824 92 829 92 834
848
92840 92845 92850 92855 92860
92 865 92870 92 875 92 881 92 886
849
92891 92896 92901 92906 92911
92916 92921 92927 92932 92937
850
92942 92947 92952 92957 92962
92967 92 973 92978 92 983 92 988
N
12 8 4
5 6 7 8
800-850
850-900
17
N
12 3 4
5 6
7 8 9
850
92 942 92947 92952 92957 92962
92967 92 973
92978 92983 92 988
861
92993 92998 93003 93008 93013
93 018 93 024
93 029 93 034 93 039
852
93044 93049 93054 93059 93064
93069 93075 93080 93085 93090
853
93 095 93100 93105 93110 93115
93120 93125
93131 93136 93141
854
93146 93151 93156 93161 93166
93171 93176
93 181 93 186 93 192
865
93197 93 202 93 207 93 212 93 217
93 222 93 227 93 232 93 237 93 242
856
93 247 93 252 93 258 93 263 93 268
93 273 93 278
93 283 93288 93293
857
93 298 93 303 93 308 93 313 93 318
93 323 93 328
93 334 93 339 93 344
858
93 349 93 354 93 359 93 364 93 369
93 374 93 379
93384 93389 93394
859
93 399 93404 93409 93414 93420
93425 93430 93435 93440 93445
860
93 450 93455 93 460 93 465 93 470
93 475 93 480
93 485 93490 93 495
861
93 500 93 505 93 510 93 515 93 520
93 526 93 531
93 536 93 541 93 546
862
93 551 93 556 93 561 93 566 93 571
93 576 93 581
93 586 93 591 93 596
863
93 601 93 606 93611 93 616 93 621
93 626 93 631
93636 93641 93646
864
93 651 93 656 93 661 93 666 93 671
93 676 93 682
93 687 93 692 93 697
866
93 702 93 707 93 712 93 717 93 722
93 727 93 732
93 737 93 742 93 747
866
93 752 93 757 93 762 93 767 93 772
93 777 93 7S2
93 787 93 792 93 797
867
93802 93 807 93 812 93 817 93 822
93 827 93 832
93837 93842 93847
868
93 852 93 857 93 862 93 867 93 872
93 877 93 882
93 887 93 892 93 897
869
93902 93907 93 912 93 917 93 922
93927 93932
93937 93942 93947
870
93 952 93 957 93 962 93 967 93 972
93 977 93 982
93987 93 992 93997
871
94002 94007 94012 94017 94022
94027 94032
94037 94042 94047
872
94052 94057 94062 94067 94072
94077 94082
94086 94091 94096
873
94101 94106 94111 94116 94121
94126 94131
94136 94141 94146
874
94151 94156 94161 94166 94171
94176 94181
94186 94191 94196
875
94201 94206 94211 94216 94 221
94 226 94 231
94236 94240 94245
876
94250 94255 94260 94 265 94 270
94 275 94280
94285 94 290 94 295
877
94300 94305 94310 94315 94 320
94325 94330 94 335 94340 94345
878
94349 94354 94359 94364 94369
94374 94379
94384 94389 94394
879
94399 94404 94409 94414 94419
94424 94429
94433 94438 94443
880
94448 94453 94458 94463 94468
94473 94478
94483 94488 94493
881
94498 94 503 94 507 94512 94 517
94522 94 527
94 532 94 537 94 542
882
94 547 94 552 94 557 94 562 94 567
94 571 94 576
94 581 94 586 94 591
883
94 596 94601 94606 94611 94616
94621 94626
94630 94635 94640
884
94645 94650 94655 94660 94665
94670 94675 94680 94685 94689
865
94694 94699 94 704 94 709 94 714
94 719 94 724
94 729 94 734 94 738
888
94 743 94 748 94 753 94 758 94 763
94 768 94 773
94 778 94 783 94 787
887
94 792 94 797 94802 94807 94812
94817 94 822
94827 94832 94836
868
94841 94846 94851 94856 94861
94866 94871
94876 94880 94 885
889
94890 94895 94900 94905 94910
94915 94919 94924 94929 94934
890
94939 94944 94949 94954 94959
94963 94968
94973 94978 94983
891
94988 94993 94998 95002 95007
95 012 95 017
95 022 95 027 95032
892
95 036 95 041 95046 95 051 95 056
95061 95 066
95 071 95 075 95 080
893
95 085 95 090 95095 95100 95105
95 109 95 114 95 119 95 124 95 129
894
95134 95139 95143 95148 95153
95158 95163 95168 95173 95177
895
95 182 95 187 95 192 95 197 95 202
95 207 95 211
95 216 95 221 95 226
896
95 231 95 236 95 240 95 245 95 250
95 255 95 260 95 265 95 270 95 274
897
95 279 95 284 95 289 95 294 95 299
95 303 95 308 95 313 95 318 95 323
898
95 328 95 332 95 337 95 342 95 347
95 352 95 357 95 361 95 366 95 371
899
95376 95 381 95 386 95 390 95 395
95400 95 405 95 410 95415 95 419
900
95424 95429 95434 95 439 95444
95448 95 453
95 458 95463 95 468
X
12 3 4
S 6
7 8 9
850-900
18
900-950
N
O 1 2
3 4
5
6 7 8
9
900
95 424 95 429 95 434
95 439 95 444
95 448
95 453 95 458 95 463
95 468
901
95 472 95 477 95 482
95 487 95 492
95 497
95 501 95 506 95 511
95 516
902
95 521 95 525 95 530
95 535 95 540
95 545
95 550 95 554 95 559 95 564
90S
95 569 95 574 95 578 95 583 95 588
95 593
95 598 95 602 95 607 95 612
904
95 617 95 622 95 626
95 631 95 636
95 641
95 646 95 650 95 655
95 660
905
95 665 95 670 95 674
95 679 95 684
95 689
95 694 95 698 95 703
95 708
906
95 713 95 718 95 722
95 727 95 732
95 737
95 742 95 746 95 751
95 756
907
95 761 95 766 95 770 95 775 95 780
95 785 95 789 95 794 95 799 95 804
908
95 809 95 813 95 818
95 823 95 828
95 832
95 837 95 842 95 847
95 852
- 909
95 856 95 861 95 866 95 871 95 875
95 880
95 885 95 890 95 895
95 899
910
95 904 95909 95 914
95 918 95 923
95 928
95 933 95 938 95 942
95 947
911
95 952 95 957 95 961
95 966 95 971
95 976
95 980 95 985 95 990 95 995
912
95 999 96004 96009
96014 96019
96023
96028 96033 96038
96042
913
96047 96052 96057
96061 96066
96071
96076 96080 96085
96090
914
96095 96099 96104
96109 96114
96118
96123 96128 96133
96137
916
96142 96147 96152
96156 96161
96166
96171 96175 96180
96185
916
96190 96194 96199
96204 96209
96213
96218 96 223 96 227
96232
917
96237 96242 96246
96 251 96 256
96 261
96265 96270 96 275
96280
918
96284 96 289 96294
96298 96303
96308
96313 96317 96322
96327
919
96 332 96336 96341
96346 96350
96 355
96360 96365 96369 96374
920
96379 96384 96388
96393 96398
96402
96407 96412 96417
96421
921
96426 96 431 96435
96440 96445
96450 96 454 96459 96464
96468
922
96473 96478 96483
96487 96492
96497
96501 96 506 96511
96 515
923
96 520 96 525 96 530
96 534 96 539
96 544
96548 96553 96 558
96562
924
96 567 96 572 96 577
96 581 96 586
96591
96595 96600 96605
96609
926
96614 96619 96624
96628 96633
96638
96642 96647 96652
96656
926
96661 96666 96670
96675 96680
96685
96689 96694 96699
96703
927
96 708 96 713 96 717
96 722 96 727
96 731
96 736 96 741 96 745
96 750
928
96 755 96 759 96 764 96 769 96 774
96 778
96 783 96 788 96 792
96 797
929
96802 96806 96811
96 816 96820
96825
96830 96834 96839
96 844
930
96848 96853 96858
96 862 96 867
96872
96876 96 881 96 886
96 890
931
96895 96900 96904 96909 96914
96918
96923 96928 96932
96937
932
96942 96946 96951
96956 96960
96965
96970 96974 96979
96984
933
96988 96993 96997
97002 97007
97011
97016 97021 97025
97030
934
97035 97039 97044 97049 97053
97058
97063 97 067 97 072
97 077
936
97081 97086 97090
97095 97100
97104
97109 97114 97118
97123
936
97128 97132 97137
97142 97146
97151
97155 97160 97165
97169
937
97174 97179 97183
97188 97192
97197
97 202 97 206 97 211
97 216
938
97 220 97 225 97 230
97 234 97 239
97243
97 248 97 253 97257
97262
939
97 267 97 271 97 276
97 280 97 285
97 290
97 294 97 299 973OT97308
940
97313 97317 97 322
97 327 97331
97 336
97340 97 345 97350 97354
941
97359 97364 97 368
97 373 97 377
97382
97387 97 391 97 3%
97400
942
97405 97410 97414
97419 97424
97 428 97433 97437 97442
97447
943
97451 97456 97460
97465 97470
97474
97479 97483 97488
97493
944
97497 97 502 97 506
97 511 97 516
97 520 97 525 97 529 97 534
97 539
945
97 543 97 548 97 552
97 557 97 562
97 566 97 571 97 575 97 580 97 585
946
97 589 97 594 97 598
97 603 97 607
97 612
97617 97621 97626
97630
947
97635 97640 97 644
97649 97 653
97 658
97663 97667 97672
97 676
948
97 681 97 685 97690
97695 97 699
97 704
97 708 97 713 97 717
97 722
949
97 727 97 731 97 736 97 740 97 745
97 749
97 754 97 759 97 763
97 768
950
97 772 97 777 97 782
97 786 97 791
97 795
97800 97804 97 809
97 813
N
12
3 4
5
6 7 8
• 1
900-950
950-1000 1 9
N
1
2
3
4
5 6 7 8 9
950
97 772
97 777
97 782
97 786
97 791
97 795 97 800 97 804 97 809 97 813
951
97 818
97 823
97 827
97 832
97 836
97 841 97 845 97 850 97 855 97 859
952
97 864
97 868
97 873
97 877
97 882
97886 97 891 97 896 97 900 97905
963
97 909
97 914
97 918
97 923
97 928
97 932 97937 97 941 97 946 97 950
954
97955
97 959 97964
97968
97973
97 978 97982 97 987 97 991 97996
956
98000 98005
98 009
98014
98019
98023 98028 98 032 98037 98 041
956
98 046 98 050 98055
98059 98064
98068 98073 98 078 98082 98 087
957
98091 98096 98100 98105
98109
98114 98118 98123 98127 98132
958
98137
98141
98146 98150 98155
98159 98164 98168 98173 98177
959
98182
98186
98191
98195
98 200
98 204 98 209 98214 98 218 98 223
960
98 227
98 232
98 236
98 241
98 245
98 250 98 254 98 259 98 263 98 268
981
98 272
98 277
98 281
98 286
98 290
98 295 98 299 98 304 98308 98 313
962
98318
98 322
98 327
98 331
98336
98340 98345 98349 98354 98 358
963
98363
98367
98 372
98 376
98 381
98385 98390 98 394 98399 98403
964
98408 98412
98417
98421
98426
9^430 98435 98439 98 444 98448
965
98453
98457
98462
98466
98471
98 475 98480 98484 98489 98493
966
98498
98 502
98 507
98 511
98 516
98 520 98 525 98 529 98 534 98 538
967
98 543
98 547
98 552
98 556
98 561
98 565 98 570 98 574 98 579 98 583
968
98 588
98 592
98 597
98 601
98605
98 610 98 614 98619 98 623 98 628
969
98632
98637
98641
98646
98650
98 655 98659 98 664 98668 98 673
970
98 677
98682
98 686
98691
9869*
98 700 98 704 98 709 98 713 98 717
971
98 722
98 726
98 731
98 735
98 740
98 744 98 749 98 753 98 758 98 762
972
98 767
98 771
98 776
98 780
98 784
98 789 98 793 98 798 98802 98 807
973
98 811
98 816
98 820 98 825
98 829
98 834 98 838 98843 98 847 98 851
974
98 856 98 860 98 865
98869
98 874
98878 98883 98887 98892 988%
975
98900 98905
98909 98914
98 918
98923 98927 98932 98 936 98 941
976
98 945
98949
98954
98958
98963
98 %7 98972 98976 98981 98 985
977
98989
98994
98998
99 003
99007
99012 99016 99021 99 025 99029
978
99034
99 038
99 043
99047
99 052
99056 99061 99065 99069 99074
979
99078
99083
99087 99092
99096
99100 99105 99109 99114 99118
980
99123
99127
99131
99136
99140
99145 99149 99154 99158 99162
981
99167
99171
99176 99180 99185
99189 99193 99198 99 202 99 207
982
99 211
99 216
99220
99 224
99 229
99 233 99 238 99 242 99247 99 251
983
99 255
99 260
99264
99 269
99 273
99277 99282 99286 99291 99295
984
99300
99 304
99 308
99 313
99317
99 322 99 326 99330 99335 99 339
985
99 344
99348
99 352
99 357
99 361
99 366 99370 99374 99379 99 383
986
99 388
99392
99 3%
99401
99405
99410 99414 99419 99423 99 427
987
99 432
99436
99441
99445
99 449
99454 99458 99463 99467 99471
988
99 476
99480
99484
99489
99493
99498 99502 99506 99511 99 515
989
99 520
99 524
99 528
99 533
99 537
99 542 99 546 99550 99 555 99 559
990
99 564
99 568
99 572
99 577
99 581
99 585 99 590 99 594 99 599 99 603
991
99607
99 612
99 616
99621
99 625
99629 99634 99638 99642 99647
992
99651
99656
99660
99664
99669
99673 99677 99682 99686 99691
993
99 695
99699
99 704
99 708
99 712
99 717 99 721 99 726 99 730 99 734
994
99 739 99 743
99 747
99 752
99 756
99 760 99 765 99 769 99 774 99 778
995
99 782
99 787 99 791
99 795
99 800
99 804 99 808 99 813 99 817 99 822
996
99826 99 830 99 835
99 839
99 843
99848 99852 99856 99861 99865
997
99870
99 874
99 878
99 883
99 887
99891 99896 99900 99904 99909
998
99913
99 917
99922
99926
99930
99 935 99939 99944 99948 99952
999
99957
99 961
99965
99970
99974
99978 99983 99 987 99 991 99 996
1000
00000 00004 00009 00013
00017
00022 00026 00030 00035 00039
N
1
2
3
4
5 6 7 8 9
950-1
000
20 TABLE IL-LOGAEITHMS OF CONSTAOTS.
CircumfereDce of the Circle in degrees = 360
Circumference of the Circle in minutes = 21 600
Circumference of the Circle in seconds = 1 296 000
If the radius r = 1, half the Circumference of the Circle is
tc = 3. 14 159 265 358 979 323 846 264 338 328
log
2.55 630250
4.33445 375
6.11260500
0.49 714987
Also:
log
2tt =
6.28318 531
0.79 817987
4tt =
12.56637061
1.09 920986
IT _
1
1.57079633
0.19611988
IT _
T"
1.04 719 755
0.02002 862
4*r__
8 ~
4.18879020
0.62 208 861
IT _
T"
0.78 539816
9.89508988-
-10
IT __
0.52 359878
9.71899862-
-10
i _
IT
0.31830989
9.50285 013-
-10
0.15915 494
9.20182 013-
-10
IT
0.95 492966
9.97997138-
-10
4 _
IT ~
1.27323954
0.10491012
8__
0.23873 241
9.37791139-
-10
4*r
7T» =
V* =
1 =
4-
f/ff
A/ 6 =
9.86960440
0.10132118
1.77245 385
0.56418958
0.97 720502
1.12 837917
1.46459189
0.68278406
2.14 502 940
0.62 035 049
0.80 599 598
log
0.99429975
9.00570025-10
0.24 857494
9.75142 506-10
9.98998 569-10
0.05 245 506
0.16571662
9.83 428338-10
0.33143325
9.79 263 713-10
9.90633 287-10
Arc a, whose length is equal to the radius r, is :
TOA
in degrees a° = ™ = 57. 29 577 951° .
in minutes a' = -^2? = 3 437. 74 677' . .
in seconds a" .
7T
648 000
= 206264.806*'..
Arc 2«, whose length is equal to twice the radius,2r,
360
in degrees 2o° . . . . = — . . .
o i 21600
m minutes 2a' — = — - — .
in seconds 2a" .
= 114. 59 155 903°
= 6875.49354'..
1296 000
If the radius r = 1, the length of the arc is :
1 7T
for 1 degree .
for 1 minute -±- = -
180
a'
for 1 second . .
for} degree....—.
for * minute .
2a'
for * second — — -.
* 2a"
10 800
= 648000*
IT
= 860
21600
..= 412 529.612"..
= 0.01745 329...
= 0.00029089...
= 0.00000485...
= 0.00872665...
= 0.00014 544...
: 0.00000242.
1296000
Sin 1" in the unit circle = 0. 00000485. . .
log
1.75 812263
3.53 627388
5.31442 513
2.05915 263
3.83 730388
5.61545 513
8.24187 737-10
6.46372 612-10
4.68557487-10
7.94084 737-10
6.16269612 — 10
4.38454487 — 10
4.68557487-10
21
TABLE III.
THE LOGARITHMS
OF THB
TRIGONOMETRIC FUNCTIONS :
Rom 0° to 0° 3', or 89° 67' to 90°, for every second ;
From 0° to 2°, or 88° to 90°, for every ten seconds |
Proml°t
Note. To
log sin
a 89°, for e
all the log
yery minute.
3 appended.
tan =« log sin
008-10.00000
arithms —
10 is to b<
log
log
0°
ft
0'
1'
2'
tt
tf
0'
1'
2'
ft
6.46 373
6. 76 476
60
30
6. 16 270
6. 63 982
6. 86 167
30
1
4.68 557
6.47 090
6. 76 836
59
31
6. 17 694
6. 64 462
6. 86 455
29
2
4.98 660
6. 47 797
6. 77 193
58
32
6. 19 072
6. 64 936
6. 86 742
28
3
5. 16 270
6. 48 492
6. 77 548
57
33
6.20 409
6.65 406
6. 87 027
27
4
5. 28 763
6.49 175
6.77 900
66
84
6. 21 705
6. 65 870
6. 87 310
26
5
5. 38 454
6.49 849
6. 78 248
55
85
6. 22 964
6.66 330
6. 87 591
25
6
5.46 373
6. 50 512
6. 78 595
54
36
6. 24 188
6. 66 785
6. 87 870
24
7
5. 53 067
6. 51 165
6. 78 938
53
87
6. 25 378
6.67 235
6. 88 147
23
8
5. 58 866
6. 51 808
6. 79 278
62
38
6. 26 536
6.67 680
6. 88 423
22
9
5.63 982
6. 52 442
6. 79 616
61
39
6.27 664
6.68 121
6. 88 697
21
10
5.68 557
6. 53 067
6. 79 952
50
40
6. 28 763
6.68 557
6. 88 969
20
11
5. 72 697
6. 53 683
6. 80 285
49
41
6. 29 836
6.68 990
6. 89 240
19
12
5. 76 476
6. 54 291
6. 80 615
48
42
6.30 882
6. 69 418
6.89 509
18
13
5. 79 952
6. 54 890
6.80 943
47
43
6.31904
6. 69 841
6.89 776
17
14
5.83 170
6. 55 481
6.81268
46
44
6. 32 903
6. 70 261
6.90 042
16
15
5. 86 167
6. 56 064
6.81591
45
45
6. 33 879
6. 70 676
6.90 306
15
16
5.88 969
6. 56 639
6. 81 911
44
46
6. 34 833
6. 71 088
6.90 568
14
17
5. 91 602
6. 57 207
6. 82 230
43
47
6. 35 767
6. 71 496
6. 90 829
13
18
5.94 085
6.57 767
6. 82 545
42
48
6.36 682
6.71900
6. 91 088
12
19
5.96 433
6. 58 320
6.82 859
41
49
6. 37 577
6. 72 300
6. 91 346
11
20
5.98 660
6. 58 866
6.83 170
40
50
6.38 454
6. 72 697
6.91602
10
21
6.00 779
6.59 406
6. 83 479
39
51
6. 39 315
6.73 090
6.91857
9
22
6.02 800
6. 59 939
6. 83 786
38
52
6. 40 158
6. 73 479
6. 92 110
8
23
6.04 730
6.60 465
6.84 091
37
53
6. 40 985
6. 73 865
6. 92 362
7
24
6.06 579
6.60 985
6.84 394
36
64
6. 41 797
6. 74 248
6. 92 612
6
25
6.08 351
6. 61 499
6.84 694
35
55
6. 42 594
6. 74 627
6. 92 861
5
26
6. 10 055
6. 62 007
6. 84 993
34
56
6.43 376
6. 75 003
6.93 109
4
27
6. 11 694
6. 62 509
6. 85 289
33
67
6. 44 145
6. 75 376
6. 93 355
3
28
6. 13 273
6.63 006
6. 85 584
32
68
6.44 900
6. 75 746
6.93 599
2
29
6. 14 797
6. 63 496
6. 85 876
31
59
6.45 643
6. 76 112
6.93 843
1
30
6. 16 270
6.63 982
6. 86 167
80
60
6.46 373
6. 76 476
6.94 085
ft
59'
68'
57'
tt
tt
59'
58'
57'
tf
log OOt = log 008
log sin = 10. 00 000
89 c
log cos
22
0°
r tr
log sin
log tan
log 001
ft f
t ft
log sin
log tan 1 log cob
ft f
_
__
10.00000
60
10
7.46 373
7.46373
10.00000
50
10
5.68 557
5.68557
10.00000
50
10
7.47090
7.47091
10.00000
50
20
5.98660
5.98660
10.00000
40
20
TAT 191
7. 47 797
10.00000
40
30
6.16 270
6.16270
10.00000
30
30
7.48491
7.48492
10.00000
30
40
6. 28 763
6.28 763
10.00000
20
40
7.49175
7. 49 176
10.00000
20
60
6.38454
6.38454
10.00000
10
60
7.49849
7.49 849
10.00000
10
1
6.46373
6.46373
10.00000
059
11
7.50 512
7.50 512
10.00000
49
10
6.53 067
6.53 067
10.00000
60
10
7. 51 165
7. 51 165
10.00000
50
20
6.58 866
6.58866
10.00000
40
20
7. 51 808
7.51809
10.00000
40
30
6.63 982
6.63 982
10.00000
30
30
7.52 442
7.52 443
10.00000
30
40
6.68 557
6.68 557
10.00000
20
40
7. 53 067
7.53 067
10.00000
20
50
6.72697
6.72 697
10.00000
10
50
7. 53 683
7.53 683
10.00000
10
2
6.76476
6.76476
10.00000
58
12
7.54 291
7. 54 291
10.00000
48
10
6.79952
6.79952
10.00000
50
10
7.54 890
7.54 890
10.00000
60
20
6. 83 170
6. 83 170
10.00000
40
20
7. 55 481
7. 55 481
10.00000
40
30
6. 86 167
6. 86 167
10.00000
30
30
7.56064
7.56 064
10.00000
30
40
6.88969
6.88969
10.00000
20
40
7.56639
7.56 639
10.00000
20
60
6. 91 602
6. 91 602
10.00000
10
50
7.57 206
7. 57 207
10.00000
10
3
6.94 085
6.94085
10.00000
57
13
7. 57 767
7. 57 767
10.00000
4:7
10
6.% 433
6.% 433
10.00000
50
10
7.58 320
7.58320
10.00000
50
20
6.98660
6.98661
10.00000
40
20
7.58 866
7.58 867
10.00000
40
30
7.00 779
7.00 779
10.00000
30
30
7.59406
7.59406
10.00000
30
40
7.02 800
7.02 800
10.00000
20
40
7.59939
7.59939
10.00000
20*
50
7.04 730
7.04 730
10.00000
10
60
7.60465
7.60466
10.00000
10
4
7.06 579
7.06 579
10.00000
56
14
7.60 985
7.60986
10.00000
46
10
7.08351
7.08352
10.00000
50
10
7. 61 499
7.61500
10.00000
50
20
7.10055
7.10055
10.00000
40
20
7.62 007
7.62008
10.00000
40
30
7.11694
7. 11 694
10.00000
30
30
7. 62 509
7.62 510
10.00000
30
40
7. 13 273
7. 13 273
10.00000
20
40
7.63 006
7.63 006
10.00000
20
50
7. 14 797
7. 14 797
10.00000
10
60
7. 63 496
7.63 497
10.00000
10
5
7.16270
7.16270
10.00000
055
15
7. 63 982
7.63 982
10.00000
45
10
7.17694
7.17 694
10.00000
50
10
7.64 461
7.64462
10.00000
50
20
7.19072
7.19073
10.00000
40
20
7.64 936
7.64937
10.00000
40
30
7.20409
7.20409
10.00000
30
30
7.65 406
7. 65 406
10.00000
30
40
7. 21 705
7. 21 705
10.00000
20
40
7. 65 870
7. 65 871
10.00000
20
50
7.22 964
7.22964
10.00000
10
50
7.66 330
7.66330
10.00000
10
6
7. 24 188
7. 24 188
10.00000
054
16
7.66 784
7.66 785
10.00000
44
10
7. 25 378
7. 25 378
10.00000
50
10
7. 67 235
7. 67 235
10.00000
60
20
7.26 536
7.26 536
10.00000
40
20
7.67 680
7.67680
10.00000
40
30
7.27664
7.27664
10.00000
30
30
7. 68 121
7. 68 121
10.00000
30
40
7.28 763
7. 28 764
10.00000
20
40
7.68 557
7. 68 558
9.99999
20
50
7.29836
7.29836
10.00000
10
50
7.68 989
7.68 990
9.99999
10
7
7.30882
7.30882
10.00000
053
17
7.69 417
7.69418
9.99999
43
10
7.31904
7.31904
10.00000
50
10
7.69 841
7.69842
9.99999
60
20
7.32 903
7.32903
10.00000
40
20
7.70 261
7.70 261
9.99999
40
30
7. 33 879
7. 33 879
10.00000
30
30
7.70 676
7.70677
9.99999
30
40
7. 34 833
7. 34 833
10.00000
20
40
7. 71 088
7. 71 088
9.99999
20
50
7. 35 767
7.35 767
10.00000
10
60
7. 71 496
7. 71 4%
9.99999
10
8
7.36682
7.36682
10.00000
52
18
7.71900
7.71900
9.99999
42
10
7.37 577
7.37 577
10.00000
50
10
7.72 300
7.72 301
9.99999
50
20
7.38454
7.38455
10.00000
40
20
7.72 697
7.72 697
9.99999
40
30
7.39 314
7.39 315
10.00000
30
30
7.73 090
7.73 090
9.99999
30
40
7.40158
7. 40 158
10.00000
20
40
7. 73 479
7.73 480
9.99999
20
60
7.40985
7.40985
10.00000
10
50
7. 73 865
7. 73 866
9.99999
10
9
7.41797
7. 41 797
10.00000
51
19
7.74 248
7.74 248
9.99999
41
10
7.42 594
7.42 594
10.00000
50
10
7.74627
7.74628
9.99999
60
20
7.43 376
7. 43 376
10.00000
40
20
7.75 003
7.75 004
9.99999
40
30
7. 44 145
7.44145
10.00000
30
30
7. 75 376
7.75377
9.99999
30
40
7.44 900
7.44 900
10.00000
20
40
7. 75 745
7. 75 746
9.99999
20
50
7. 45 643
7.45 643
10.00000
10
50
7. 76 112
7. 76 113
9.99 999
10
10
7.46373
7.46373
10.00000
050
20
7.76475
7.76476
9.99999
40
f ft
log OOB
log cot
log sin
ft t
t ft
log 001
log oot
log sin
ft t
89 c
0°
23
r rt
log sin
log tan
log 008
ff f
t ft
log Bin
log tan
log 008
ft t
20
7.76 475
7.76476
9.99999
040
30
7.94084
7.94086
9.99998
30
10
7.76 836
7.76 837
9.99 999
50
10
7.94325
7.94326
9.99998
50
20
7. 77 193
7. 77 194
9.99999
40
20
7.94 564
7.94 566
9.99998
40
30
7. 77 548
7. 77 549
9.99999
30
30
7.94 802
7.94 804
9.99998
30
40
7. 77 899
7.77 900
9.99999
20
40
7.95 039
7.95040
9.99998
20
60
7.78 248
7.78249
9.99999
10
50
7.95 274
7.95 276
9.99998
10
21
7. 78 594
7.78 595
9.99 999
039
31
7.95 508
7.95 510
9.99998
29
10
7.78938
7.78938
9.99999
60
10
7. 95 741
7.95 743
9.99998
60
20
7.79278
7.79 279
9.99999
40
20
7.95 973
7.95 974
9.99998
40
30
7.79616
7.79 617
9.99999
30
30
7.96203
7.96 205
9.99998
30
40
7.79952
7.79952
9.99 999
20
40
7.96432
7.96434
9.99998
20
50
7.80284
7.80 285
9.99999
10
50
7.96660
7.96662
9.99998
10
22
7.80615
7.80615
9.99 999
038
82
7.96 887
7.96889
9.99998
28
10
7.80942
7.80 943
9.99999
50
10
7.97113
7.97114
9.99998
50
20
7. 81 268
7. 81 269
9.99999
40
20
7. 97 337
7.97 339
9.99998
40
30
7. 81 591
7. 81 591
9.99 999
30
30
7. 97 560
7.97 562
9.99998
30
40
7. 81 911
7.81912
9.99 999
20
40
7.97 782
7. 97 784
9.99998
20
60
7. 82 229
7.82 230
9.99999
10
60
7.98003
7.98005
9.99998
10
23
7. 82 545
7. 82 546
9.99999
37
33
7.98 223
7. 98 225
9.99998
27
10
7. 82 859
7.82 860
9.99999
50
10
7.98442
7.98444
9.99998
60
20
7. 83 170
7. 83 171
9.99999
40
20
7.98 660
7.98662
9.^9998
40
30
7.83479
7. 83 480
9.99 999
30
30
7.98876
7.98878
9.99998
30
40
7. 83 786
7.83 787
9.99999
20
40
7.99092
7.99094
9.99998
20
60
7.84091
7.84092
9.99 999
10
60
7.99 306
7.99308
9.99998
10
240
7.84 393
7.84 394
9.99 999
36
840
7. 99 520
7.99 522
9.99998
26
10
7.84 694
7. 84 695
9.99 999
60
10
7.99 732
7.99734
9.99998
50
20
7.84992
7.84994
9.99999
40
20
7.99943
7.99946
9.99998
40
30
7.85 289
7. 85 290
9.99 999
30
30
8. 00 154
8.00156
9.99998
30
40
7. 85 583
7. 85 584
9.99999
20
40
8.00363
8.00365
9.99998
20
50
7. 85 876
7.85 877
9.99 999
10
60
8.00 571
8.00574
9.99998
10
25
7. 86 166
7. 86 167
9.99 999
35
85
8.00 779
8. 00 781
9.99998
25
10
7.86455
7.86456
9.99999
50
10
8.00985
8.00987
9.99998
60
20
7. 86 741
7.86 743
9.99999
40
20
8. 01 190
8. 01 193
9.99998
40
30
7.8702*
7.87 027
9.99999
30
30
8. 01 395
8.01397
9.99998
30
40
7.87 309
7.87310
9.99999
20
40
8. 01 598
8.01600
9.99998
20
60
7. 87 590
7. 87 591
9.99999
10
60
8.01801
8.01803
9.99998
10
26
7. 87 870
7.87871
9.99 999
34
86
8.02002
8.02 004
9.99998
24
10
7.88147
7. 88 148
9.99999
50
10
8.02 203
8. 02 205
9.99998
50
20
7.88423
7.88 424
9.99999
40
20
8.02 402
8.02 405
9.99998
40
30
7.88697
7.88 698
9.99 999
30
30
8.02 601
8.02 604
9.99998
30
40
7.88969
7.88 970
9.99999
20
40
8. 02 799
8. 02 801
9.99998
20
60
7.89 240
7. 89 241
9.99 999
10
60
8.029%
8.02 998
9.99998
10
27
7.89 509
7.89 510
9.99 999
033
87
8.03192
8. 03 194
9.99997
23
10
7.89 776
7. 89 777
9.99999
60
10
8. 03 387
8.03 390
9.99997
60
20
7.90041
7.90043
9.99 999
40
20
8. 03 581
8.03 584
9.99997
40
30
7.90305
7.90307
9.99999
30
30
8.03 775
8.03 777
9.99997
30
40
7.90568
7.90569
9.99999
20
40
8.03 967
8. 03 970
9.99997
20
50
7.90 829
7.90830
9.99999
10
50
8.04159
8.04162
9.99997
10
280
7.91088
7.91089
9.99999
32
38
8.04 350
8.04 353
9.99997
22
10
7.91346
7.91347
9.99999
50
10
8.04 540
8.04 543
9.99997
60
20
7. 91 602
7.91603
9.99999
40
20
8. 04 729
8. 04 732
9.99997
40
30
7. 91 857
7.91858
9.99999
30
30
8.04918
8.04921
9.99997
30
40
7.92110
7.92111
9.99998
20
40
8.05105
8.05 108
9.99997
20
50
7.92 362
7.92 363
9.99998
10
60
8.05 292
8.05 295
9.99997
10
29
7. 92 612
7.92 613
9.99998
31
39
8. 05 478
8. 05 481
9.99997
21
10
7.92 861
7.92 862
9.99998
60
10
8.05 663
8.05 666
9.99 997
60
20
7.93108
7. 93 110
9.99998
40
20
8.05 848
8.05 851
9.99997
40
30
7.93 354
7. 93 356
9.99 998
30
30
8.06031
8.06034
9.99 997
30
40
7.93 599
7.93 601
9.99998
20
40
8.06214
8.06 217
9.99997
20
60
7.93 842
7.93 844
9.99998
10
50
8.06396
8.06399
9.99 997
10
30
7.94084
7.94086
9.99998
30
400
8.06 578
8.06 581
9.99997
20
/ ft
log 008
log oot
log sin
ft 9
9 ft
log 008
log oot
log sin
tf t
89 c
24
0°
t ft
log iin
log tan
log cos
ft f
1 ft
log sin
log tan
log 008
ft t
40
8.06 578
8.06 581
9.99997
20
500
8.16268
8.16 273
9.99995
010
10
8. 06 758
8.06 761
9.99997
60
10
8.16413
8.16417
9.99995
50
20
8.06938
8.06941
9.99997
40
20
8. 16 557
8.16 561
9.99995
40
30
8.07117
8.07120
9.99997
30
30
8. 16 700
8.16 705
9.99995
30
40
8.07 295
8.07 299
9.99997
20
40
8.16843
8.16848
9.99 995
20
60
8.07473
8.07476
9.99997
10
60
8.16986
8.16991
9.99995
10
41
8.07 6^0
8.07653
9.99997
19
51
8. 17 128
8. 17 133
9.99995
9
10
8.07826
8.07829
9.99997
60
10
8.17270
8.17 273
9.99995
60
20
8.08002
8.08003
9.99997
40
20
8.17411
8.17416
9.99995
40
30
8.08176
8.08180
9.99997
30
30
8. 17 552
8. 17 557
9.99995
30
40
8.08350
8.08354
9.99997
20
40
8.17692
8.17 697
9.99995
20
60
8.08 524
8.08 527
9.99 997
10
60
8. 17 832
8.17837
9.99995
10
420
8.08696
8.08 700
9.99997
18
520
8.17 971
8.17976
9.99995
8
10
8.08 868
8.08 872
9.99 997
50
10
8. 18 110
8. 18 115
9.99995
50
20
8.09040
8.09043
9.99997
40
20
8.18249
8.18 254
9.99 995
40
30
8.09210
8.09214
9.99 997
30
30
8.18387
8.18 392
9.99995
30
40
8.09 380
8.09384
9.99 997
20
40
8.18 524
8. 18 530
9.99993
20
60
8.09 530
8.09 553
9.99997
10
50
8.18662
8.18 667
9.99995
10
43
8.09 718
8.09 722
9.99997
17
530
8. 18 798
8.18 804
9.99995
7
10
8.09886
8.09 890
9.99997
50
10
8.18933
8.18940
9.99995
50
*20
8.10054
8.10057
9.99997
40
20
8.19071
8.19076
9.99 995
40
30
8.10220
8.10224
9.99997
30
30
8.19 206
8. 19 212
9.99995
30
40
8.10386
8.10390
9.99997
20
40
8.19341
8.19347
9.99995
20
60
8.10 552
8.10 555
9.99996
10
60
8.19476
8.19481
9.99995
10
440
8. 10 717
8. 10 720
9.99 996
16
540
8.19610
8.19616
9.99995
6
10
8.10881
8.10884
9.99 996
60
10
8. 19 744
8.19 749
9.99 995
50
20
8.11044
8. 11 048
9.99996
40
20
8.19877
8.19 883
9.99995
40
30
8. 11 207
8.11211
9.99996
30
80
8.20010
8.20016
9.99995
30
40
8. 11 370
8. 11 373
9.99 996
20
40
8. 20 143
8. 20 149
9.99995
20
60
8. 11 531
8. 11 535
9.99996
10
60
8.20275
8.20 281
9.99994
10
450
8. 11 693
8. 11 6%
9.99996
15
550
8.20407
8.20413
9.99994
5
10
8. 11 853
8.11857
9.99996
50
10
8.20538
8.20 544
9.99994
50
20
8.12 013
8.12 017
9.99996
40
20
8.20669
8.20675
9.99994
40
30
8. 12 172
8. 12 176
9.99996
30
30
8.20 800
8.20806
9.99994
30
40
8.12331
8.12 335
9.99996
20
40
8.20930
8.20936
9.99994
20
60
8.12 489
8.12493
9.99 996
10
60
8.21060
8.21066
9.99 994
10
46
8.12 647
8.12 651
9.99996
14
560
8. 21 189
8. 21 195
9.99 994
4
10
8.12 804
8. 12 808
9.99 996
50
10
8. 21 319
8.21324
9.99994
50
20
8.12961
8.12 963
9.999%
40
20
8. 21 447
8.21453
9.99994
40
30
8. 13 117
8. 13 121
9.99 996
30
30
8. 21 576
8. 21 581
9.99994
30
40
8. 13 272
8. 13 276
9.99996
20
40
8. 21 703
8. 21 709
9.99944
20
50
8.13 427
8.13 431
9.99 996
10
50
8. 21 831
8. 21 837
9.99994
10
47
8. 13 581
8. 13 585
9.99996
13
57
8. 21 958
8.21964
9.99994
3
10
8. 13 735
8. 13 739
9.99996
50
10
8.22 085
8.22 091
9.99994
50
20
8.13 888
8. 13 892
9.99 996
40
20
8.22 211
8.22 217
9.99994
40
30
8.14041
8.14045
9.99996
30
30
8.22 337
8.22 343
9.99994
80
40
8.14193
8. 14 197
9.99 996
20
40
8.22 463
8.22 469
9.99 994
20
60
8.14 344
8.14348
9.99 9%
10
60
8. 22 588
8.22 593
9.99994
10
480
8.14495
8. 14 500
9.999%
12
580
8.22 713
8. 22 720
9.99994
2
10
8.14 646
8. 14 650
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50
10
8. 22 838
8.22 844
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20
8. 14 796
8.14 800
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8.22 968
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8.14945
8.14930
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30
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8. 23 086
8.23 092
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40
8.15 094
8.15 099
9.99996
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40
8. 23 210
8. 23 216
9.99994
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60
8. 15 243
8. 15 247
9.999%
10
60
8. 23 333
8. 23 339
9.99994
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8. 15 391
8. 15 395
9.99 996
11
590
8.23 456
8. 23 462
9.99994
1
10
8. 15 538
8. 15 543
9.999%
50
10
8. 23 578
8. 23 585
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20
8. 15 685
8. 15 690
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8. 23 700
8. 23 707
9.99 994
40
30
8. 15 832
8. 15 836
9.99996
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8. 23 822
8. 23 829
9.99993
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40
8. 15 978
8. 15 982
9.99995
20
40
8.23 944
8. 23 950
9.99 993
20
50
8. 16 123
8. 16 128
9.99 995
10
50
8.24065
8.24071
9.99 993
10
5O0
8.16268
8.16273
9.99995
10
600
8.24186
8. 24 192
9.99993
/ ft
log OOB
log oot
log sin
ft t
t ft
log 008
log oot
log sin
ft t
89 c
1°
25
t tf
log sin
log tan
log OOB
ft t
t tf
log sin
log tan
log 008
ft f
o
8.24186
8. 24 192
9.99 993
060
10
8.30879
8.30 888
9.99991
050
10
8.24306
8.24313
9.99993
60
10
8.30983
8.30992
9.99991
50
20
8.24426
8.24433
9.99993
40
20
8.31086
8. 31 095
9.99991
40
30
8.24546
8. 24 553
9.99993
30
30
8.31188
8.31198
9.99991
30
40
8.24 665
8.24672
9.99993
20
40
8. 31 291
8.31300
9.99991
20
50
8. 24 785
8.24 791
9.99993
10
50
8. 31 393
8. 31 403
9.99991
10
1
8.24903
8.24910
9.99993
089
11
8.31495
8. 31 505
9.99991
049
10
8. 25 022
8. 25 029
9.99993
50
10
8.31597
8.31606
9.99991
50
20
8.25140
8.25147
9.99993
40
20
8.31699
8. 31 708
9.99991
40
30
8. 25 258
8. 25 265
9.99993
30
30
8.31800
8.31809
9.99991
30
40
8.25 375
8.25 382
9.99993
20
40
8. 31 901
8.31911
9.99991
20
50
8. 25 493
8. 25 500
9.99993
10
60
8.32 002
8.32 012
9.99991
10
2
8.25 609
8. 25 616
9.99993
058
12
8. 32 103
8.32112
9.99990
048
10
8. 25 726
8. 25 733
9.99993
50
10
8. 32 203
8.32 213
9.99990
50
20
8.25 842
8.25 849
9.99993
40
20
8.32 303
8.32 313
9.99990
40
30
8.25 958
8.25 965
9.99993
30
30
8.32 403
8.32 413
9.99990
30
40
8.26074
8.26081
9.99993
20
40
8.32 503
8. 32 513
9.99990
20
50
8.26189
8.26196
9.99993
10
60
8.32 602
8.32 612
9.99990
10
3
8.26304
8.26312
9.99993
57
13
8. 32 702
8. 32 711
9.99990
47
10
8.26419
8.26426
9.99993
60
10
8.32 801
8.32 811
9.99990
50 .
20
8.26 533
8.26 541
9.99993
40
20
8. 32 899
8.32909
9.99990
40
30
8.26648
8.26655
9.99993
30
30
8.32998
8.33 008
9.99990
30
40
8.26 761
8.26769
9.99993
20
40
8.330%
8. 33 106
9.99990
20
60
8.26875
8.26882
9.99993
10
60
8.33195
8.33 205
9.99990
10
4
8.26988
8.26996
9.99992
056
14
8. 33 292
8.33 302
9.99 990
46
10
8.27101
8. 27 109
9.99992
50
10
8.33 390
8.33 400
9.99990
50
20
8. 27 214
8. 27 221
9.99992
40
20
8.33 488
8.33 498
9.99990
40
30
8.27326
8.27334
9.99992
30
30
8.33 585
8.33 595
9.99990
30
40
8.27438
8.27446
9.99992
20
40
8. 33 682
8.33 692
9.99990
20
60
8.27 550
8.27 558
9.99992
10
60
8. 33 779
8. 33 789
9.99990
10
5
8.27661
8.27669
9.99992
#55
15
8. 33 875
8. 33 886
9.99990
045
10
8. 27 773
8. 27 780
9.99992
50
10
8.33 972
8.33 982
9.99990
50
20
8.27 883
8.27891
9.99992
40
20
8.34068
8.34078
9.99990
40
30
8.27994
8.28002
9.99992
30
30
8.34164
8.34174
9.99990
30
40
8.28104
8. 28 112
9.99992
20
40
8.34 260
8.34 270
9.99989
20
60
8. 28 215
8.28 223
9.99992
10
50
8.34355
8.34 366
9.99989
10
6
8.28 324
8.28332
9.99992
54
16
8.34450
8.34461
9.99989
44
10
8.28 434
8.28442
9.99992
50
10
8.34 546
8.34556
9.99989
50
20
8.28 543
8.28 551
9.99 992
40
20
8.34640
8.34651
9.99989
40
30
8.28652
8.28660
9.99992
30
30
8.34 735
8. 34 746
9.99989
30
40
8. 28 761
8. 28 769
9.99992
20
40
8.34830
8.34840
9.99989
20
60
8.28869
8.28 877
9.99992
10
50
8.34924
8.34935
9.99989
10
7
8.28977
8.28986
9.99992
053
17
8. 35 018
8. 35 029
9.99989
43
10
8.29085
8.29094
9.99992
60
10
8.35112
8. 35 123
9.99989
50
20
8. 29 193
8. 29 201
9.99992
40
20
8.35 206
8. 35 217
9.99989
40
30
8.29300
8.29309
9.99992
30
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8.35 299
8.35 310
9.99989
30
40
8.29407
8.29416
9.99992
20
40
8.35 392
8. 35 403
9.99989
20
50
8.29514
8.29523
9.99992
10
50
8.35 485
8. 35 497
9.99989
10
8
8.29621
8.29 629
9.99 992
52
18
8.35 578
8.35 590
9.99989
42
10
8.29727
8.29 736
9.99991
60
10
8. 35 671
8.35 682
9.99989
60
20
8.29833
8.29842
9.99991
40
20
8. 35 764
8. 35 775
9.99989
40
30
8.29939
8.29947
9.99991
30
30
8. 35 856
8.35 867
9.99989
30
40
8.30044
8.30053
9.99991
20
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8.35 948
8. 35 959
9.99989
20
60
8.30150
8.30158
9.99991
10
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8.36040
8.36051
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10
9
8.30255
8.30263
9.99991
51
19
8. 36 131
8. 36 143
9.99989
41
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8.30359
8.30368
9.99991
50
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8.36 223
8.36235
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60
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8.30464
8.30473
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40
20
8.36314
8.36326
9.99988
40
30
8.30568
8.30 577
9.99991
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8.36405
8.36417
9.99988
30
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8.30672
8.30681
9.99991
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8.36 508
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20
60
8.30 776
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8.36 587
8.36599
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too
8.30879
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20
8.36678
8.36689
9.99988
40
t ft
log 001
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tf t
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ft t
88 c
f ft
log iia
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9 ft
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ft t
20
8.36678
8.36 689
9.99988
40
30
8. 41 792
8.41807
9.99985
30
10
8. 36 768
8. 36 780
9.99988
60
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8. 41 872
8.41887
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50
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8.36 858
8.36 870
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8.41952
8. 41 967
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8.36 948
8.36960
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30
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8.42032
8.42048
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30
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8.37038
8.37050
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20
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8. 42 112
8.42127
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20
60
8. 37 128
8. 37 140
9.99988
10
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8. 42 192
8.42 207
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8.37 217
8. 37 229
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39
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8. 42 272
8.42 287
9.99985
29
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8. 37 306
8.37 318
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60
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8.42 351
8.42 366
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60
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8.37 395
8.37 408
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8.42 430
8.42 446
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40
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8.37 484
8.37497
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30
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8.42 510
8. 42 525
9.99985
30
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8.37 573
8.37 585
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20
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8.42 589
8.42 604
9.99985
20
50
8.37 662
8.37674
9.99988
10
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8.42 667
8.42 683
9.99985
10
22
8. 37 750
8. 37 762
9.99988
38
32
8.42 746
8.42 762
9.99984
28
10
8. 37 838
8.37 850
9.99988
60
10
8.42 825
8.42 840
9.99 984
50
20
8.37 926
8.37938
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40
20
8.42 903
8.42 919
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40
30
8.38014
8.38026
9.99987
30
30
8.42 982
8.42 997
9.99984
30
40
8.38101
8.38114
9.99 987
20
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8.43 060
8.43 075
9.99 984
20
50
8. 38 189
8.38 202
9.99987
10
60
8.43138
8.43154
9.99984
10
23
8.38 276
8.38 289
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37
33
8.43 216
8. 43 232
9.99 984
27
10
8.38363
8.38376
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60
10
8. 43 293
8.43 309
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50
20
8.38450
8.38463
9.99987
40
20
8.43 371
8.43 387
9.99984
40
30
8.38 537
8. 38 550
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30
30
8. 43 448
8.43 464
9.99984
30
40
8.38624
8.38 636
9.99987
20
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8.43 526
8.43 542
9.99984
20
50
8.38 710
8. 38 723
9.99987
10
60
8.43 603
8.43 619
9.99984
10
24
8.38 796
8.38 809
9.99987
36
34
8. 43 680
8.43 696
9.99984
26
10
8.38882
8.38 895
9.99987
50
10
8.43 757
8.43 773
9.99984
50
20
8.38968
8.38981
9.99987
40
20
8.43 834
8.43 850
9.99984
40
30
8.39054
8.39 067
9.99987
30
80
8. 43 910
8.43 927
9.99984
30
40
8.39139
8. 39 153
9.99987
20
40
8. 43 987
8.44003
9.99984
20
60
8.39225
8.39 238
9.99987
10
50
8.44063
8.44080
9.99983
10
25
8.39310
8.39 323
9.99987
35
35
8.44139
8.44156
9.99983
25
10
8.39395
8.39408
9.99987
50
10
8.44 216
8. 44 232
9.99983
50
20
8.39480
8.39 493
9.99987
40
20
8.44 292
8.44 308
9.99983
40
30
8.39565
8. 39 578
9.99987
30
30
8.44 367
8.44334
9.99983
30
40
8.39649
8.39 663
9.99987
20
40
8.44443
8.44460
9.99983
20
60
8.39 734
8. 39 747
9.99986
10
50
8.44519
8.44 536
9.99983
10
26
8. 39 818
8.39 832
9.99986
34
36
8.44 594
8.44 611
9.99983
24
10
8.39902
8.39916
9.99 986
60
10
8.44669
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82 c
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25 927
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80 c
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28 641
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32 670
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29 087
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32 728
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29 591
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29 654
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35
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65 724
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36
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29 716
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65 664
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29 779
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33
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33
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29 841
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29 903
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31
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30
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30
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30 028
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30 090
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98 765
24
36
40152
41578
58 422
98 574
24
37
37 185
38 423
61577
98 762
23
37
40 200
41629
58 371
98 571
23
38
37 237
38 479
61521
98 759
22
38
40 249
41681
58 319
98 568
22
39
37 289
38 534
61466
98 756
21
39
40 297
41733
58 267
98 565
21
40
37 341
38 589
61411
98 753
20
40
40 346
41784
58 216
98 561
20
41
37 393
38 644
61356
98 750
19
41
40 394
41836
58164
98 558
19
42
37 445
38 699
61301
98 746
18
42
40 442
41887
58 113
98 555
18
43
37 497
38 754
61246
98 743
17
43
40 490
41939
58 061
98 551
17
44
37 549
38 808
61192
98 740
16
44
40 538
41990
58 010
98 548
16
45
37 600
38 863
61137
98 737
15
45
40 586
42 041
57 959
98 545
15
46
37 652
38 918
61082
98 734
14
46
40 634
42 093
57 907
98 541
14
47
37 703
38 972
61028
98 731
13
47
40 682
42 144
57 856
98 538
13
48
37 755
39 027
60 973
98 728
12
48
40 730
42 195
57 805
98 535
12
49
37 806
39 082
60 918
98 725
11
49
40 778
42 246
57 754
98 531
11
50
37 858
39 136
60 864
98 722
10
50
40 825
42 297
57 703
98 528
10
51
37 909
39 190
60 810
98 719
9
51
40 873
42 348
57 652
98 525
9
52
37 960
39 245
60 755
98 715
8
62
40 921
42 399
57 601
98 521
8
53
38 011
39 299
60 701
98 712
7
53
40 968
42 450
57 550
98 518
7
64
38 062
39 353
60 647
98 709
6
64
41016
42 501
57 499
98 515
6
55
38 113
39 407
60 593
98 706
5
65
41063
42 552
57 448
98 511
6
66
38 164
39 461
60 539
98 703
4
56
41111
42 603
57 397
98 508
4
67
38 215
39 515
60 485
98 700
3
57
41158
42 653
57 347
98 505
3
68
38 266
39 569
60 431
98 697
2
68
41205
42 704
57 296
98 501
2
69
38 317
39 623
60 377
98 694
1
69
41252
42 755
57 245
98 498
1
00
38 368
39 677
60 323
98 690
00
41300
42 805
57195
98 494
n
o
in
o
n
n
in
ft
r
u
log 008
V
log oot
XXJ
log tan
log sin
t
f
log 008
log oot
log tan
log sin
f
76°
75 c
16°
16°
35
f
log sin
log tan
ft
log GOt
1 A
log 008
t
f
log sin
A
log tan
A
logoot
1A
log oos
1 9
?
41300
42 805
±U
57 195
98 494
60
44 034
45 750
■J.U ■■■■
54 250
98 284
60
1
41347
42 856
57 144
98 491
59
1
44 078
45 797
54 203
98 281
59
2
41394
42 906
57 094
98 488
58
2
44 122
45 845
54 155
98 277
68
3
41441
42 957
57 043
98 484
57
3
44166
45 892
54 108
98 273
57
4
41488
43 007
56 993
98 481
66
4
44 210
45 940
54 060
98 270
66
6
41535
43 057
56 943
98 477
65
5
44 253
45 987
54 013
98 266
55
8
41582
43 108
56 892
98 474
64
6
44 297
46 035
53 965
98 262
54
7
41628
43 158
56 842
98 471
63
7
44 341
46 082
53 918
98 259
53
8
41675
43 208
56 792
98 467
62
8
44 385
46130
53 870
98 255
52
8
41722
43 258
56 742
98 464
61
9
44 428
46 177
53 823
98 251
61
10
41768
43 308
56 692
98 460
50
10
44 472
46 224
53 776
98 248
50
11
41815
43 358
56 642
98 457
49
11
44 516
46 271
53 729
98 244
49
12
41861
43 408
56 592
98 453
48
12
44 559
46 319
53 681
98 240
48
13
41908
43 458
56 542
98 450
47
13
44 602
46 366
53 634
98 237
47
14
41954
43 508
56 492
98 447
46
14
44 646
46 413
53 587
98 233
46
16
42 001
43 558
56 442
98 443
45
15
44 689
46 460
53 540
98 229
45
16
42 047
43 607
56 393
98 440
44
16
44 733
46 507
53 493
98 226
44
17
42 093
43 657
56 343
98 436
43
17
44 776
46 554
53 446
98 222
43
18
42 140
43 707
56 293
98 433
42
18
44 819
46 601
53 399
98 218
42
19
42 186
43 756
56 244
98 429
41
19
44 862
46 648
53 352
98 215
41
20
42 232
43 806
56 194
98 426
40
20
44 905
46 694
53 306
98 211
40
21
42 278
43 855
56145
98 422
39
21
44 948
46 741
53 259
98 207
39
22
42 324
43 905
56 095
98 419
38
22
44 992
46 788
53 212
98 204
38
23
42 370
43 954
56 046
98 415
37
23
45 035
46 835
53 165
98 200
37
24
42 416
44 004
55 9%
98 412
36
24
45 077
46 881
53 119
98 196
36
26
42 461
44 053
55 947
98 409
35
25
45 120
46 928
53 072
98 192
36
26
42 507
44 102
55 898
98 405
34
26
45 163
46 975
53 025
98 189
34
27
42 553
44 151
55 849
98 402
33
27
45 206
47 021
52 979
98185
33
28
42 599
44 201
55 799
98 398
32
28
45 249
47 068
52 932
98 181
32
29
42 644
44 250
55 750
98 395
31
29
45 292
47 114
52 886
98 177
31
30
42 690
44 299
55 701
98 391
30
30
45 334
47160
52 840
98174
30
31
42 735
44 348
55 652
98 388
29
31
45 377
47 207
52 793
98 170
29
32
42 781
44 397
55 603
98 384
28
32
45 419
47 253
52 747
98166
28
83
42 826
44 446
55 554
98 381
27
33
45 462
47 299
52 701
98 162
27
34
42 872
44 495
55 505
98 377
26
34
45 504
47 346
52 654
98 159
26
35
42 917
44 544
55 456
98 373
25
35
4*547
47 392
52 608
98 155
25
36
42 962
44 592
55 408
98 370
24
36
45 589
47 438
52 562
98 151
24
37
43 008
44 641
55 359
98 366
23
37
45 632
47 484
52 516
98 147
23
38
43 053
44 690
55 310
98 363
22
38
45 674
47 530
52 470
98144
22
39
43 098
44 738
55 262
98 359
21
39
45 716
47 576
52 424
98 140
21
40
43 143
44 787
55 213
98 356
20
40
45 758
47 622
52 378
98 136
20
41
43 188
44 836
55 164
98 352
19
41
45 801
47 668
52 332
98 132
19
42
43 233
44 884
55 116
98 349
18
42
45 843
47 714
52 286
98 129
18
43
43 278
44 933
55 067
98 345
17
43
45 885
47 760
52 240
98 125
17
44
43 323
44 981
55 019
98 342
16
44
45 927
47 806
52 194
98 121
16
45
43 367
45 029
54 971
98 338
15
45
45 969
47 852
52 148
98 117
15
46
43 412
45 078
54 922
98 334
14
46
46 011
47 897
52 103
98 113
14
47
43 457
45 126
54 874
98 331
13
47
46 053
47 943
52 057
98 110
13
48
43 502
45 174
54 826
98 327
12
48
46 095
47 989
52 011
98 106
12
49
43 546
45 222
54 778
98 324
11
49
46 136
48 035
51965
98 102
11
50
43 591
45 271
54 729
98 320
to
50
46 178
48 080
51920
98 098
10
51
43 635
45 319
54 681
98 317
9
51
46 220
48 126
51874
98 094
9
62
43 680
45 367
54 633
98 313
8
52
46 262
48171
51829
98 090
8
53
43 724
45 415
54 585
98 309
7
53
46 303
48 217
51783
98 087
7
64
43 769
45 463
54 537
98 306
6
64
46 345
48 262
51738
98 083
6
65
43 813
45 511
54 489
98 302
5
55
46 386
48 307
51693
98 079
5
66
43 857
45 559
54 441
98 299
4
56
46 428
48 353
51647
98 075
4
57
43 901
45 606
54 394
98 295
8
57
46 469
48 398
51602
98 071
3
68
43 946
45 654
54 346
98 291
2
58
46 511
48 443
51557
98 067
2
69
43 990
45 702
54 298
98 288
1
69
46 552
48 489
51511
98 063
1
60
44 034
45 750
54 250
■lft
98 284
A
60
46 594
o
48 534
51466
10
98 060
9
f
-tf —
log 008
V
logoot
Mi'
log tan
U
log sin
t
f
log COB
logoot
log tan
log sin
t
74 c
73 c
36
17°
18°
9
log sin
Q
log tan
Q
log cot
in
log 008
n
t
t
log sin
log tan
log oot
1 A
log cos
t
46 594
48 534
" " JLU-
51466
98 060
60
48 998
51178
10
48 822
97 821
60
1
46 635
48 579
51421
98 056
59
1
49 037
51221
48 779
97 817
59
2
46 676
48 624
51376
98 052
58
2
49 076
51264
48 736
97 812
58
3
46 717
48 669
51331
98 048
57
3
49 115
51306
48 694
97 808
57
4
46 758
48 714
51286
98 044
56
4
49 153
51349
48 651
97 804
56
5
46 800
48 759
51241
98 040
56
5
49192
51392
48 608
97 800
55
6
46 841
48 804
511%
98 036
54
6
49 231
51435
48 565
97 7%
54
7
46 882
48 849
51 151
98 032
53
7
49 269
51478
48 522
97 792
53
8
46 923
48 894
51 106
98 029
52
8
49 308
51520
48 480
97 788
62
9
46 964
48 939
51061
98 025
51
9
49 347
51563
48 437
97 784
51
10
47 005
48 984
51016
98 021
50
10
49 385
51606
48 394
97 779
50
11
47 045
49 029
50 971
98 017
49
11
49 424
51648
48 352
97 775
49
12
47 086
49 073
50 927
98 013
48
12
49 462
51691
48 309
97 771
48
13
47 127
49 118
50 882
98 009
47
13
49 500
51734
48 266
97 767
47
14
47168
49 163
50 837
98 005
46
14
49 539
51776
48 224
97 763
46
15
47 209
49 207
50 793
98 001
45
15
49 577
51819
48181
97 759
45
16
47 249
49 252
50 748
97 997
44
16
49 615
51861
48139
97 754
44
17
47 290
49 296
50 704
97 993
43
17
49 654
51903
48 097
97 750
43
18
47 330
49 341
50 659
97 989
42
18
49 692
51946
48 054
97 746
42
19
47 371
49 385
50 615
97 986
41
19
49 730
51988
48 012
97 742
41
20
47 411
49 430
50 570
97 982
40
20
49 768
52 031
47 969
97 738
40
21
47 452
49 474
50 526
97 978
39
21
49 806
52 073
47 927
97 734
39
22
47 492
49 519
50 481
97 974
38
22
49 844
52 115
47 885
97 729
38
23
47 533
49 563
50 437
97 970
37
23
49 882
52 157
47 843
97 725
37
24
47 573
49 607
50 393
97 966
36
24
49 920
52 200
47 800
97 721
36
25
47 613
49 652
50 348
97 %2
35
25
49 958
52 242
47 758
97 717
35
26
47 654
49 6%
50 304
97 958
34
26
49 9%
52 284
47 716
97 713
34
27
47 694
49 740
50 260
97 954
33
27
50 034
52 326
47 674
97 708
33
28
47 734
49 784
50 216
97 950
32
28
50 072
52 368
47 632
97 704
32
29
47 774
49 828
50172
97 946
31
29
50110
52 410
47 590
97 700
31
30
47 814
49 872
50128
97 942
30
30
50148
52 452
47 548
97 6%
30
31
47 854
49 916
50 084
97 938
29
31
50185
52 494
47 506
97 691
29
82
47 894
49 960
50 040
97 934
28
32
50 223
52 536
47 464
97 687
28
33
47 934
50 004
49 9%
97 930
27
33
50 261
52 578
47 422
97 683
27
34
47 974
50 048
49 952
97 926
26
34
50 298
52 620
47 380
97 679
26
35
48 014
50 092
49 908
97 922
26
35
50 336
52 661
47 339
97 674
25
36
48 054
50 136
49 864
97 918
24
36
50 374
52 703
47 297
97 670
24
37
48 094
50180
49 820
97 914
23
37
50 411
52 745
47 255
97 666
23
38
48 133
50 223
49 777
97 910
22
38
50 449
52 787
47 213
97 662
22
39
48 173
50 267
49 733
97 906
21
39
50 486
52 829
47171
97 657
21
40
48 213
50 311
49 689
97 902
20
40
50 523
52 870
47130
97 653
20
41
48 252
50 355
49 645
97 898
19
41
50 561
52 912
47 088
97 649
19
42
48 292
50 398
49 602
97 894
18
42
50 598
52 953
47 047
97 645
18
43
48 332
50 442
49 558
97 890
17
43
50 635
52 995
47 005
97 640
17
44
48 371
50 485
49 515
97 886
16
44
50 673
53 037
46 963
97 636
16
45
48 411
50 529
49 471
97 882
15
45
50 710
53 078
46 922
97 632
15
46
48 450
50 572
49 428
97 878
14
46
50 747
53 120
46 880
97 628
14
47
48 490
50 616
49 384
97 874
13
47
50 784
53 161
46 839
97 623
13
48
48 529
50 659
49 341
97 870
12
48
50 821
53 202
46 798
97 619
12
49
48 568
50 703
49 297
97 866
11
49
50 858
53 244
46 756
97 615
11
50
48 607
50 746
49 254
97 861
10
50
50 896
53 285
46 715
97 610
io
51
48 647
50 789
49 211
97 857
9
61
50 933
53 327
46 673
97 606
9
52
48 686
50 833
49 167
97 853
8
52
50 970
53 368
46 632
97 602
8
53
48 725
50 876
49 124
97 849
7
53
51007
53 409
46 591
97 597
7
54
48 764
50 919
49 081
97 845
6
54
51043
53 450
46 550
97 593
6
55
48 803
50 962
49 038
97 841
5
55
51080
53 492
46 508
97 589
5
56
48 842
51005,
48 995
97 837
4
56
51117
53 533
46 467
97 584
4
57
48 881
51048
48 952
97 833
3
57
51154
53 574
46 426
97 580
3
58
48 920
51092
48 908
97 829
2
68
51191
53 615
46 385
97 576
2
59
48 959
51135
48 865
97 825
1
59
51227
53 656
46 344
97 571
1
60
48 998
o
51178
9
48 822
10
97 821
n
60
51264
53 697
46 303
in
97 567
n
O
f
log 008
log oot
log tan
log sin
f
f
— y
log 008
w- -
log oot
■±u
log tan
y
log sin
',
72 c
71 c
19°
•
20°
37
t
log sin
o
log tan
log oot
10
log cos
ft
f
t
log sin
n
log tan
Q
log oot
10
log 008
n
f
51264
53 697
46 303
97 567
60
53 405
56 107
43 893
97 299
60
1
51301
53 738
46 262
97 563
59
1
.53 440
56 146
43 854
97 294
69
2
51338
53 779
46 221
97 558
58
2
53 475
56 185
43 815
97 289
68
3
51374
53 820
46 180
97 554
57
3
53 509
56 224
43 776
97 285
57
4
51411
53 861
46139
97 555
56
4
53 544
56 264
43 736
97 280
56
6
51447
53 902
46 098
97 545
65
6
53 578
56 303
43 697
97 276
55
6
51484
53 943
46 057
97 541
54
6
53 613
56 342
43 658
97 271
54
7
51520
53 984
46 016
97 536
53
7
53 647
56 381
43 619
97 266
53
8
51557
54 025
45 975
97 532
52
8
53 682
56 420
43 580
97 262
62
9
51593
54 065
45 935
97 528
51
9
53 716
56 459
43 541
97 257
51
10
51629
54 106
45 894
97 523
50
10
53 751
56 498
43 502
97 252
50
11
51666
54 147
45 853
97 519
49
11
53 785
56 537
43 463
97 248
49
12
51702
54 187
45 813
97 515
48
12
53 819
56 576
43 424
97 243
48
13
51738
54 228
45 772
97 510
47
13
53 854
56 615
43 385
97 238
47
14
51774
54 269
45 731
97 506
46
14
53 888
56 654
43 346
97 234
46
16
51811
54 309
45 691
97 501
45
16
53 922
56 693
43 307
97 229
46
16
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60 931
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66
62 486
66 735
33 265
95 751
4
57
62 513
66 768
33 232
95 745
3
58
62 541
66 801
33 199
95 739
2
69
62 568
66 834
33 166
95 733
1
60
62 595
66 867
33 133
95 728
n
n
in
n
■
log 008
u
log oot
log tan
log sin
-
66 c
65 c
40
25 c
26 c
O
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
81
82
33
84
35
36
87
38
89
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
65
66
57
58
69
60
log sin
0-
62 595
62 622
62 649
62 676
62 703
62 730
62 757
62 784
62 811
62 838
62 865
62 892
62 918
62 945
62 972
62 999
63 026
63 052
63 079
63 106
63 133
63 159
63 186
63 213
63 239
63 266
63 292
63 319
63 345
63 372
63 398
63 425
63 451
63 478
63 504
63 531
63 557
63 583
63 610
63 636
63 662
63 689
63 715
63 741
63 767
63 794
63 820
63 846
63 872
63 898
63 924
63 950
63 976
64 002
64 028
64 054
64 080
64 106
64 132
64 158
64 184
log OOB
log tan
—9 —
66 867
66 900
66 933
66 966
66 999
67 032
67 065
67 098
67 131
67 163
67196
67 229
67 262
67 295
67 327
67 360
67 393
67 426
67 458
67 491
67 524
67 556
67 589
67 622
67 654
67 687
67 719
67 752
67 785
67 817
67 850
67 882
67 915
67 947
67 980
68 012
68 044
68 077
68 109
68142
68 174
68 206
68 239
68 271
68 303
68 336
68 368
68 400
68 432
68 465
68 497
68 529
68 561
68 593
68 626
68 658
68 690
68 722
68 754
68 786
68 818
— 9 —
log oot
logoot
—10—
33 133
33 100
33 067
33 034
33 001
32 968
32 935
32 902
32 869
32 837
32 804
32 771
32 738
32 705
32 673
32 640
32 607
32 574
32 542
32 509
32 476
32 444
32 411
32 378
32 346
32 313
32 281
32 248
32 215
32 183
32 150
32 118
32 085
32 053
32 020
31988
31956
31923
31891
31858
31826
31794
31761
31729
31697
31664
31632
31600
31568
31535
31503
31471
31439
31407
31374
31342
31310
31278
31246
31214
31182
—10—
log tan
log«
95 728
95 722
95 716
95 710
95 704
95 698
95 692
95 686
95 680
95 674
95 668
95 663
95 657
95 651
95 645
95 639
95 633
95 627
95 621
95 615
95 609
95 603
95 597
95 591
95 585
95 579
95 573
95 567
95 561
95 555
95 549
95 543
95 537
95 531
95 525
95 519
95 513
95 507
95 500
95 494
95 488
95 482
95 476
95 470
95 464
95 458
95 452
95 446
95 440
95 434
95 427
95 421
*L415
95 409
95 403
95 397
95 391
95 384
95 378
95 372
95 366
log sin
60
69
68
57
56
56
54
53
52
51
50
49
43
47
46
45
44
43
42
41
40
81
30
29
28
27
26
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
O
f
log gin
ft
log tan
ft
logoot
in
log 008
q
t
64184
68 818
lit—
31182
95 366
60
1
64 210
68 850
31150
95 360
59
2
64 236
68 882
31118
95 354
58
3
64 262
68 914
31086
95 348
57
4
64 288
68 946
31054
95 341
56
5
64 313
68 978
31022
95 335
56
6
64 339
69 010
30 990
95 329
54
7
64 365
69 042
30 958
95 323
63
8
64 391
69 074
30 926
95 317
52
9
64 417
69106
30 894
95 310
51
10
64 442
69138
30 862
95 304
50
11
64 468
69170
30 830
95 298
49
12
64 494
69 202
30 798
95 292
48
13
64 519
69 234
30 766
95 286
47
14
64 545
69 266
30 734
95 279
46
16
64 571
69 298
30 702
95 273
45
16
64 5%
69 329
30 671
95 267
44
17
64 622
69 361
30 639
95 261
43
18
64 647
69 393
30 607
95 254
42
19
64 673
69 425
30 575
95 248
41
20
64 698
69 457
30 543
95 242
40
21
64 724
69 488
30 512
95 236
39
22
64 749
69 520
30 480
95 229
38
23
64 775
69 552
30 448
95 223
87
24
64 800
69 584
30 416
95 217
38
26
64 826
69 615
30 385
95 211
36
26
64 851
69 647
30 353
95 204
34
27
64 877
69 679
30 321
95 198
38
28
64 902
69 710
30 290
95 192
82
29
64 927
69 742
30 258
95 185
31
30
64 953
69 774
30 226
95 179
30
31
64 978
69 805
30195
95 173
29
32
65 003
69 837
30163
95 167
28
33
65 029
69 868
30 132
95 160
27
34
65 054
69 900
30100
95 154
26
36
65 079
69 932
30 068
95 148
25
36
65 104
69 963
30 037
95 141
24
37
65 130
69 995
30 005
95135
23
38
65 155
70 026
29 974
95 129
22
39
65 180
70 058
29 942
95 122
21
40
65 205
70 089
29 911
95 116
20
41
65 230
70121
29 879
95 110
19
42
65 255
70152
29 848
95 103
18
43
65 281
70 184
29 816
95 097
17
44
65 306
70 215
29 785
95 090
16
45
65 331
70 247
29 753
95 084
15
46
65 356
70 278
29 722
95 078
14
47
65 381
70 309
29 691
95 071
IS
48
65 406
70 341
29 659
95 065
12
49
65 431
70 372
29628
95 059
11
50
65 456
70 404
29 596
95 052
10
51
65 481
70 435
29 565
95 046
9
62
65 506
70 466
29 534
95 039
8
53
65 531
70 498
29 502
95 033
7
54
65 556
70 529
29 471
95 027
6
65
65 580
70 560
29 440
95 020
5
56
65 605
70 592
29 408
95 014
4
57
65 630
70 623
29 377
95 007
3
58
65 655
70 654
29 346
95 001
2
59
65 680
70 685
29 315
94 995
1
60
65 705
70 717
29 283
94 988
o
n
in
'
log 008
logoot
All
log tan
u
log sin
f
64 c
63 c
27 c
f
log sin
Q
log tan
9
log oot
in
log 008
n
f
65 705
70 717
1U
29 283
94 988
60
1
65 729
70 748
29 252
94 982
69
2
65 754
70 779
29 221
94 975
58
3
65 779
70 810
29 190
94 969
57
4
65 804
70 841
29159
94 962
66
6
65 828
70 873
29127
94 956
65
6
65 853
70 904
29 096
94 949
54
7
65 878
70 935
29 065
94 943
53
8
65 902
70 966
29 034
94 936
52
9
65 927
70 997
29 003
94 930
51
io
65 952
71028
28 972
94 923
50
11
65 976
71059
28 941
94 917
49
12
66 001
71090
28 910
94 911
48
13
66 025
71121
28 879
94 904
47
14
66 050
71153
28 847
94 898
46
15
66 075
71184
28 816
94 891
45
16
66 099
71215
28 785
94 885
44
17
66124
71246
28 754
94 878
43
18
66148
71277
28 723
94 871
42
19
66173
71308
28 692
94 865
41
20
66197
71339
28 661
94 858
40
21
66 221
71370
28 630
94 852
39
22
66 246
71401
28 599
94 845
38
23
66 270
71431
28 569
94 839
37
24
66 295
71462
28 538
94 832
36
25
66 319
71493
28 507
94 826
35
26
66 343
71524
28 476
94 819
34
27
66 368
71555
28 445
94 813
33
28
66 392
71586
28 414
94 806
32
29
66 416
71617
28 383
94 799
31
30
66 441
71648
28 352
94 793
30
31
66 465
71679
28 321
94 786
29
32
66 489
71709
28 291
94 780
28
33
66 513
71740
28 260
94 773
27
34
66 537
71771
28 229
94 767
26
36
66 562
71802
28198
94 760
26
36
66 586
71833
28167
94 753
24
37
66 610
71863
28 137
94 747
23
88
66 634
71894
28 106
94 740
22
39
66 658
71925
28 075
94 734
21
40
66 682
71955
28 045
94 727
20
41
66 706
71986
28 014
94 720
19
42
66 731
72 017
27 983
94 714
18
43
66 755
72 048
27 952
94 707
17
44
66 779
72 078
27 922
94 700
16
45
66 803
72 109
27 891
94 694
15
46
66 827
72 140
27 860
94 687
14
47
66 851
72 170
27 830
94 680
13
48
66 875
72 201
27 799
94 674
12
49
66 899
72 231
2J769
94 667
11
50
66 922
72 262
27 738
94 660
10
51
66 946
72 293
27 707
94 654
9
52
66 970
72 323
27 677
94 647
8
53
66 994
72 354
27 646
94 640
7
64
67 018
72 384
27 616
94 634
6
66
67 042
72 415
27 585
94 627
6
56
67 066
72 445
27 555
94 620
4
67
67 090
72 476
27 524
94 614
3
58
67113
72 506
27 494
94 607
2
69
67137
72 537
27 463
94 600
1
60
67161
72 567
27 433
94 593
Q
— 9 —
log oot
in
n
f
a
log 008
log tan
log ain
f
28°
41
f
log sin
log tan
log oot
log 008
t
Q
9
10
q
67161
72 567
27 433
94 593
60
1
67185
72 598
27 402
94 587
59
2
67 208
72 628
27 372
94 580
58
8
67 232
72 659
27 341
94 573
57
4
67 256
72 689
27 311
94 567
56
5
67 280
72 720
27 280
94 560
56
6
67 303
72 750
27 250
94 553
54
7
67 327
72 780
27 220
94 546
53
8
67 350
72 811
27189
94 540
52
9
67 374
72 841
27159
94 533
51
10
67 398
72 872
27128
94 526
60
11
67 421
72 902
27 098
94 519
49
12
67 445
72 932
27 068
94 513
48
13
67 468
72 963
27 037
94 506
47
14
67 492
72 993
27 007
94 499
46
15
67 515
73 023
26 977
94 492
45
16
67 539
73 054
26 946
94 485
44
17
67 562
73 084
26 916
94 479
43
18
67 586
73 114
26 886
94 472
42
19
67 609
73 144
26 856
94 465
41
20
67 633
73 175
26 825
94 458
40
21
67 656
73 205
26 795
94 451
39
22
67 680
73 235
26 765
94 445
38
28
67 703
73 265
26 735
94 438
37
24
67 726
73 295
26 705
94 431
36
25
67 750
73 326
26 674
94 424
35
26
67 773
73 356
26 644
94 417
34
27
67 796
73 386
26 614
94 410
33
28
67 820
73 416
26 584
94 404
32
29
67 843
73 446
26 554
94 397
81
80
67 866
73 476
26 524
94 390
30
31
67 890
73 507
26 493
94 383
29
32
67 913
73 537
26 463
94 376
28
33
67 936
73 567
26 433
94 369
27
34
67 959
73 597
26 403
94 362
26
35
67 982
73 627
26 373
94 355
25
36
68 006
73 657
26 343
94 349
24
37
68 029
73 687
26 313
94 342
23
38
68 052
73 717
26 283
94 335
22
39
68 075
73 747
26 253
94 328
21
40
68 098
73 777
26 223
94 321
20
41
68121
73 807
26193
94 314
19
42
68144
73 837
26163
94 307
18
43
68167
73 867
26133
94 300
17
44
68190
73 897
26103
94 293
16
45
68 213
73 927
26 073
94 286
15
46
68 237
73 957
26 043
94 279
14
47
68 260
73 987
26 013
94 273
13
48
68 283
74 017
25 983
94 266
12
49
68 305
74 047
25 953
94 259
11
SO
68 328
74 077
25 923
94 252
10
51
68 351
74 107
25 893
94 245
9
52
68 374
74 137
25 863
94 238
8
53
68 397
74166
25 834
94 231
7
54
68 420
74 196
25 804
94 224
6
65
68 443
74 226
25 774
94 217
5
56
68 466
74 256
25 744
94 210
4
57
68 489
74 286
25 714
94 203
3
58
68 512
74 316
25 684
94196
2
59
68 534
74 345
25 655
94 189
1
60
68 557
74 375
25 625
94182
n
n
in
n
f
V
log 008
9
log oot
log tan
V
log sin
t
62 c
61 c
42
29°
1-
log sin
log tan
logoot
in
log 008
f
68 557
74 375
JLU -- ■
25 625
94 182
60
1
68 580
74 405
25 595
94 175
59
2
68 603
74 435
25 565
94168
68
3
68 625
74 465
25 535
94 161
67
4
68 648
74 494
25 506
94 154
56
6
68 671
74 524
25 476
94 147
55
6
68 69*
74 554
25 446
94 140
54
7
68 716
74 583
25 417
94133
53
8
68 739
74 613
25 387
94126
62
68 762
74 643
25 357
94 119
61
10
68 784
74 673
25 327
94 112
60
11
68 807
74 702
25 298
94 105
49
12
68 829
74 732
25 268
94 098
48
13
68 852
74 762
25 238
94 090
47
14
68 875
74 791
25 209
94 083
46
15
68 897
74 821
25 179
94 076
45
16
68 920
74 851
25 149
94 069
44
17
68 942
74 880
25 120
94 062
43
18
68 965
74 910
25 090
94 055
42
19
68 987
74 939
25 061
94 048
41
20
69 010
74 969
25 031
94 041
40
21
69 032
74 998
25 002
94 034
39
22
69 055
75 028
24 972
94 027
38
23
69 077
75 058
24 942
94 020
37
24
69 100
75 087
24 913
94 012
36
25
69122
75 117
24 883
94 005
36
26
69 144
75 146
24 854
93 998
84
27
69 167
75 176
24 824
93 991
33
28
69 189
75 205
24 795
93 984
32
29
69 212
75 235
24 765
93 977
81
30
69 234
75 264
24 736
93 970
30
31
69 256
75 294
24 706
93 963
29
32
69 279
75 323
24 677
93 955
28
33
69 301
75 353
24 647
93 948
27
34
69 323
75 382
24 618
93 941
26
35
69 345
75 411
24 589
93 934
26
36
69 368
75 441
24 559
93 927
24
37
69 390
75 470
24 530
93 920
23
38
69 412
75 500
24 500
93 912
22
39
69 434
75 529
24 471
93 905
21
40
69 456
75 558
24 442
93 898
20
41
69 479
75 588
24 412
93 891
19
42
69 501
75617
24 383
93 884
18
43
69 523
75 647
24 353
93 876
17
44
69 545
75 676
24 324
93 869
16
45
69 567
75 705
24 295
93 862
16
46
69 589
75 735
24 265
93 855
14
47
69 611
75 764
24 236
93 847
13
48
69 633
75 793
24 207
93 840
12
49
69 655
75 822
24 178
93 833
11
50
69 677
75 852
24 148
93 826
10
51
69 699
75 881
24 119
93 819
9
52
69 721
75 910
24 090
93 811
8
63
69 743
75 939
24 061
93 804
7
54
69 765
75 969
24 031
93 797
6
55
69 787
75 998
24 002
93 789
6
66
69 809
76 027
23 973
93 782
4
57
69 831
76 056
23 944
93 775
3
58
69 853
76 086
23 914
93 768
2
59
69 875
76 115
23 885
93 760
1
60
69 897
76144
23 856
93 753
o
Q
10
fl
f
log 008
log oot
log tan
log sin
f
30 c
f
log sin
q
log tan
g
logoot
in
log GOB
t
69 897
76144
1U
23 856
93 753
60
1
69 919
76173
23 827
93 746
69
2
69 941
76 202
23 798
93 738
68
8
69 963
76 231
23 769
93 731
67
4
69 984
76 261
23 739
93 724
56
6
70 006
76 290
23 710
93 717
65
6
70 028
76 319
23 681
93 709
64
7
70 050
76 348
28 652
93 702
63
8
70 072
76 377
23 623
93 695
62
9
70 093
76 406
23 594
93 687
61
10
70115
76 435
23 565
93 680
50
11
70137
76 464
23 536
93 673
49
12
70 159
76 493
23 507
93 665
48
18
70180
76 522
23 478
93 658
47
14
70 202
76 551
23 449
93 650
46
16
70 224
76 580
23 420
93 643
46
16
70 245
76 609
23 391
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77 250
86 630
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54
53 c
46
37°
t
log sin
Q
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Q
log cot
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77 946
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12 289
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60
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77 963
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69
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77 980
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68
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77 997
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66
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65
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64
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53
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26
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25
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24
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23
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22
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78 592
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21
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78 609
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20
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78 625
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89 840
19
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78 642
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18
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78 658
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17
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16
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78 691
88 890
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15
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88 916
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89 791
14
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13
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12
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10 980
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89 099
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7
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6
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10 849
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5
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78 869
89 177
10 823
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4
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78 886
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3
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78 902
89 229
10 771
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2
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78 918
89 255
10 745
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1
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78 934
89 281
10 719
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n
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log oot
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f
38 c
f
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log tan
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78 934
89 281
10 719
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60
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59
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68
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52
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51
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47
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45
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44
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43
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42
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41
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40
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37
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33
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32
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31
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5
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52 c
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39'
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79 887
90 837
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60
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6
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5
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4
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3
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2
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80 792
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88 436
1
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80 807
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07 619
88 425
n
n
in
n
f
log 001
log cot
log tan
log sin
9
40°
47
f
log sin
log tan
logoot
log 008
f
Q
Q
10
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80 807
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07 619
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60
1
80 822
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69
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68
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80 852
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57
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80 867
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56
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80 882
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55
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63
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51
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50
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48
13
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47
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46
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44
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40
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3
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2
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1
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in
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logOOB
a
logoot
XII
log tan
log sin
f
50 c
49°
48
4
1°
42°
?
log sin.
9
log tan
log oot
10
logoos
Q
f
f
log sin
log tan
log oot
logoos
9
o
816£t
93 916
06 084
87 778
60
82 551
V
95 444
10
04 556
87107
60
1
81709
93 942
06 058
87 767
59
1
82 565
95 469
04 531
87 096
59
2
81723
93 967
06 033
87 756
58
2
82 579
95 495
04 505
87 085
58
3
81738
93 993
06 007
87 745
57
3
82 593
95 520
04 480
87 073
57
4
81752
94 018
05 982
87 734
56
4
82 607
95 545
04 455
87 062
56
5
81767
94 044
05 956
87 723
56
6
82 621
95 571
04 429
87 050
56
6
81781
94 069
05 931
87 712
64
6
82 635
95 596
04 404
87 039
54
7
81796
94 095
05 905
87 701
53
7
82 649
95 622
04 378
87 028
53
8
81810
94 120
05 880
87 690
52
8
82 663
95 647
04 353
87 016
52
9
81825
94146
05 854
87 679
51
9
82 677
95 672
04 328
87 005
51
10
81839
94 171
05 829
87 668
50
10
82 691
95 698
04 302
86 993
50
11
81854
94 197
05 803
87 657
49
11
82 705
95 723
04 277
86 982
49
12
81868
94 222
05 778
87 646
48
12
82 719
95 748
04 252
86 970
48
13
81882
94 248
05 752
87 635
47
13
82 733
95 774
04 226
86 959
47
14
81897
94 273
05 727
87 624
46
14
82 747
95 799
04 201
86 947
46
15
81911
94 299
05 701
87 613
46
15
82 761
95 825
04175
86 936
46
16
81926
94 324
05 676
87 601
44
16
82 775
95 850
04 150
86 924
44
17
81940
94 350
05 650
87 590
43
17
82 788
95 875
04 125
86 913
43
18
81955
94 375
05 625
87 579
42
18
82 802
95 901
04 099
86 902
42
19
81969
94 401
05 599
87 568
41
19
82 816
95 926
04 074
86 890
41
20
81983
94 426
05 574
87 557
40
20
82 830
95 952
04 048
86 879
40
21
81998
94 452
05 548
87 546
39
21
82 844
95 977
04 023
86 867
39
22
82 012
94 477
05 523
87 535
38
22
82 858
96 002
03 998
86 855
38
23
82 026
94 503
05 497
87 524
37
23
82 872
96 028
03 972
86 844
37
24
82 041
94 528
05 472
87 513
36
24
82 885
96 053
03 947
86 832
36
25
82 055
94 554
05 446
87 501
35
25
82 899
96 078
03 922
86 821
35
26
82 069
94 579
05 421
87 490
34
26
82 913
96 104
03 896
86 809
34
27
82 084
94 604
05 396
87 479
33
27
82 927
96129
03 871
86 798
33
28
82 098
94 630
05 370
87 468
32
28
82 941
96155
03 845
86 786
32
29
82 112
94 655
05 345
87 457
31
29
82 955
96180
03 820
86 775
31
30
82 126
94 681
05 319
87 446
30
30
82 968
96 205
03 795
86 763
30
31
82 141
94 706
05 294
87 434
29
31
82 982
96 231
03 769
86 752
29
32
82 155
94 732
05 268
87 423
28
32
82 996
96 256
03 744
86 740
28
33
82 169
94 757
05 243
87 412
27
33
83 010
96 281
03 719
86 728
27
34
82 184
94 783
05 217
87 401
26
34
83 023
96 307
03 693
86 717
26
35
82 198
94 808
05 192
87 390
25
35
83 037
96 332
03 668
86 705
26
36
82 212
94 834
05 166
87 378
24
36
83 051
96 357
03 643
86 694
24
37
82 226
94 859
05 141
87 367
23
37
83 065
96 383
03 617
86 682
23
38
82 240
94 884
05 116
87 356
22
38
83 078
96 408
03 592
86 670
22
39
82 255
94 910
05 090
87 345
21
39
83 092
96 433
03 567
86 659
21
40
82 269
94 935
05 065
87 334
20
40
83 106
96 459
03 541
86 647
20
41
82 283
94 961
05 039
87 322
19
41
83 120
96 484
03 516
86 635
19
42
82 297
94 986
05 014
87 311
18
42
83 133
96 510
03 490
86 624
18
43
82 311
95 012
04 988
87 300
17
43
83 147
96 535
03 465
86 612
17
44
82 326
95 037
04 963
87 288
16
44
83 161
96 560
03 440
86 600
16
45
82 340
95 062
04 938
87 277
15
45
83 174
96 586
03 414
86 589
15
46
82 354
95 088
04 912
87 266
14
46
83 188
96 611
03 389
86 577
14
47
82 368
95 113
04 887
87 255
13
47
83 202
96 636
03 364
86 565
13
48
82 382
95 139
04 861
87 243
12
48
83 215
96 662
03 338
86 554
12
49
82 3%
95 164
04 836
87 232
11
49
83 229
96 687
03 313
86 542
11
50
82 410
95 190
04 810
87 221
10
50
83 242
96 712
03 288
86 530
10
51
82 424
95 215
04 785
87 209
9
51
83 256
96 738
03 262
86 518
9
52
82 439
95 240
04 760
87 198
8
52
83 270
96 763
03 237
86 507
8
63
82 453
95 266
04 734
87187
7
53
83 283
96 788
03 212
86 495
7
64
82 467
95 291
04 709
87175
6
54
83 297
96 814
03 186
86 483
6
65
82 481
95 317
04 683
87 164
6
55
83 310
96 839
03 161
86 472
6
66
82 495
95 342
04 658
87 153
4
56
83 324
96 864
03 136
86 460
4
67
82 509
95 368
04 632
87141
3
57
83 338
96 890
03 110
86 448
3
58
82 523
95 393
04 607
87130
2
58
83 351
96 915
03 085
86 436
2
69
82 537
95 418
04 582
87119
1
59
83 365
96 940
03 060
86 425
1
60
82 551
q
95 444
q
04 556
10
87107
o
60
83 378
96 966
03 034
86 413
n
f
log 008
log oot
log tan
log Bin
f
f
y
log 008
V
log oot
1U
log tan
8
logsin
f
48 c
47°
43°
44°
49
f
log sin
tygtan
9
log oot
10
log 008
f
f
log sin
log tan
Q
log oot
10
log 008
n
9
83 378
96 966
03 034
86 413
60
84 177
98 484
01516
85 693
60
1
83 392
96 991
03 009
86 401
59
1
84190
98 509
01491
85 681
59
2
83 405
97 016
02 984
86 389
68
2
84 203
98 534
01466
85 669
58
3
83 419
97 042
02 958
86 377
57
3
84 216
98 560
01440
85 657
67
4
83 432
97 067
02 933
86 366
56
4
84 229
98 585
01415
85 645
66
5
83 446
97 092
02 908
86 354
55
5
84 242
98 610
01390
85 632
55
6
83 459
97 118
02 882
86 342
64
6
84 255
98 635
01365
85 620
64
7
83 473
97143
02 857
86 330
63
7
84 269
98 661
01339
85 608
53
8
83 486
97168
02 832
86 318
52
8
84 282
98 686
01314
85 5%
62
9
83 500
97 193
02 807
86 306
51
9
84 295
98 711
01289
85 583
51
10
83 513
97 219
02 781
86 295
60
10
84 308
98 737
01263
85 571
50
11
83 527
97 244
02 756
86 283
49
11
84 321
98 762
01238
85 559
49
12
83 540
97 269
02 731
86 271
48
12
84 334
98 787
01213
85 547
48
13
83 554
97 295
02 705
86 259
47
13
84 347
98 812
01188
85 534
47
14
83 567
97 320
02 680
86 247
46
14
84 360
98 838
01 162
85 522
46
15
83 581
97 345
02 655
86 235
45
16
84 373
98 863
01137
85 510
45
16
83 594
97 371
02 629
86 223
44
16
84 385
98 888
01112
85 497
44
17
83 608
97 3%
02 604
86 211
43
17
84 398
98 913
01087
85 485
43
18
83 621
97 421
02 579
86 200
42
18
84 411
98 939
01061
85 473
42
19
83 634
97 447
02 553
86 188
41
19
84 424
98 964
01036
85 460
41
20
83 648
97 472
02 528
86 176
40
20
84 437
98 989
01011
85 448
40
21
83 661
97 497
02 503
86 164
39
21
84 450
99 015
00 985
85 436
39
22
83 674
97 523
02 477
86 152
38
22
84 463
99 040
00 960
85 423
38
23
83 688
97 548
02 452
86 140
37
23
84 476
99 065
00 935
85 411
37
24
83 701
97 573
02 427
86128
36
24
84 489
99 090
00 910
85 399
36
25
83 715
97 598
02 402
86 116
35
25
84 502
99 116
00 884
85 386
35
26
83 728
97 624
02 376
86 104
84
26
84 515
99141
00 859
85 374
34
27
83 741
97 649
02 351
86 092
33
27
84 528
99 166
00 834
85 361
33
28
83 755
97 674
02 326
86 080
32
28
84 540
99 191
00 809
85 349
32
29
83 768
97 700
02 300
86 068
31
29
84 553
99 217
00 783
85 337
31
30
83 781
97 725
02 275
86 056
30
30
84 566
99 242
00 758
85 324
30
31
83 795
97 750
02 250
86 044
29
31
84 579
99 267
00 733
85 312
29
32
83 808
97 776
02 224
86 032
28
32
84 592
99 293
00 707
85 299
28
33
83 821
97 801
02 199
86 020
27
33
84 605
99 318
00 682
85 287
27
34
83 834
97 826
02 174
86 008
26
34
84 618
99 343
00 657
85 274
26
35
83 848
97 851
02 149
85 996
26
36
84 630
99 368
00 632
85 262
25
36
83 861
97 877
02 123
85 984
24
36
84 643
99 394
00 606
85 250
24
37
83 874
97 902
02 098
85 972
23
37
84 656
99 419
00 581
85 237
23
38
83 887
97 927
02 073
85 960
22
38
84 669
99 444
00 556
85 225
22
39
83 901
97 953
02 047
85 948
21
39
84 682
99 469
00 531
85 212
21
40
83 914
97 978
02 022
85 936
20
40
84 694
99 495
00 505
85 200
20
41
83 927
98 003
01997
85 924
19
41
84 707
99 520
00 480
85 187
19
42
83 940
98 029
01971
85 912
18
42
84 720
99 545
00 455
85 175
18
43
83 954
98 054
01946
85 900
17
43
84 733
99 570
00 430
85 162
17
44
83 967
98 079
01921
85 888
16
44
84 745
99 5%
00 404
85 150
16
45
83 980
98 104
018%
85 876
16
46
84 758
99 621
00 379
85 137
15
46
83 993
98 130
01870
85 864
14
46
84 771
99 646
00 354
85 125
14
47
84 006
98 155
01845
85 851
13
47
84 784
99 672
00 328
85 112
13
48
84 020
98 180
01820
85 839
12
48
84 796
99 697
00 303
85 100
12
49
84 033
98 206
01794
85 827
11
49
84 809
99 722
00 278
85 087
11
50
84 046
98 231
01769
85 815
10
50
84 822
99 747
00 253
85 074
10
51
84 059
98 256
01744
85 803
9
61
84 835
99 773
00 227
85 062
9
52
84 072
98 281
01719
85 791
8
52
84 847
99 798
00 202
85 049
8
63
84 085
98 307
01693
85 779
7
53
84 860
99 823
00177
85 037
7
64
84 098
98 332
01668
85 766
6
64
84 873
99 848
00 152
85 024
6
65
84 112
98 357
01643
85 754
5
55
84 885
99 874
00 126
85 012
5
66
84 125
98 383
01617
85 742
4
56
84 898
99 899
00 101
84 999
4
67
84 138
98 408
01592
85 730
3
57
84 911
99 924
00 076
84 986
8
68
84 151
98 433
01567
85 718
2
58
84 923
99 949
00 051
84 974
2
69
84 164
98 458
01542
85 706
1
59
84 936
99 975
00 025
84 %1
1
60
84177
9-
98 484
o
01516
10
85 693
a
60
84 949
o
00 000
n
00 000
in
84 949
n
f
log C08
log cot
log tan
log sin
t
f
y
log 008
log cot
— iu- ■
log tan
— y —
log sin
t
46 c
45 c
50
TABLE IV.
Fob Determining with Greater Accuracy than can be done bt
means of Table III. :
1. log sin, log tan, and log cot, when the angle is between 0° and 2° ;
2. log cos, log tan, and log cot, when the angle is between 88° and 90° ;
8. The value of the angle when the logarithm of the function does not
lie between the limits 8. 64 684 and 11. 45 816.
FORMULAS FOR THE USE OF THE NUMBERS S AND T.
I. When the angle a is between 0° and 2° :
log sin o = log a" + 8.
log a" = log sin a — S,
log tana = log a" + T.
= log tana— T,
log cot a = coiog tan a. = colog cot a—T.
II. When the angle a is between 88° and 90° :
log cos a = log (90°-a)" + S.
log (90°-a)" = log cos a— &,
log cot a = log (90°— a)" + T.
= log cot a— T,
log tan a = colog cot a.
= colog tana— T f
and o = 90°-(90°-o).
^LTJES OF S AKD T.
•"
S
log sin a
•"
T
log tan a
a
T
log tan a
__.
,
_
5146
8.39713
4.68 557
4.68557
4.68567
2409
4.68556
8.06740
200
4.68558
6.98660
5424
4.68568
8.41999
3 417
4.68 555
8.21920
1726
4.68 559
7.92263
5689
4.68569
8.44072
3823
4.68555
8.26795
2432
4.68560
8.07156
5 941
4.68570
8.45955
4190
4.68554
8.30776
2976
4.68561
8.15924
6184
4.68571
8.47697
4840
4.68553
8.37038
3 434
4.68562
8.22142
6417
4.68 572
8.49305
5 414
4.68552
8.41904
3 838
4.68 563
8.26973
6642
4.68573
8.50802
5 932
4.68551
8.45 872
4204
4.68564
8.30930
6859
4.68574
8.52 200
6408
4.68550
8.49223
4540
4.68565
8.34270
7070
4.68575
8.53 516
6633
4.68 550
8.50 721
4699
4.68565
8.35 766
7173
4.68575
8.54145
6851
4.68549
8.52125
4 853
4.68566
8.37167
7274
8.54 753
7267
8.54 684
5146
8.39713
a"
S
log sin a
a"
T
log tan a
a
T
log tana
51
TABLE V.
— •—
THE iNATUKAL VALUES
OF
SINES, COSINES, TANGENTS, AND COTANGENTS,
JN THE UNIT CIRCLE.
0°— 8°
o f
•in
tan
oot
001
f o
O
0.0000
0.0000
infinite
1.0000
90
10
0.0029
0.0029
343.7737
1.0000
50
20
0.0058
0.0058
171.8854
1.0000
40
SO
0.0087
0.0087
114.5887
1.0000
30
40
0.0116
0.0116
85.9398
0.9999
20
50
0.0145
0.0145
68.7501
0.9999
10
1
0.0175
0.0175
57.2900
0.9998
89
o f
oot
oot
tan
sin
f o
o /
sin
ten
oot
001
f o
o /
sin
tan
oot
008
t o
1
0.0175
0.0175
57.2900
0.9998
89
5
0.0872
0.0875
11.4301
0.9962
085
10
0.0204
0.0204
49.1039
0.9998
60
10
0.0901
0.0904
11.0594
0.9959
60
20
0.0233
0.0233
42.9641
0.9997
40
20
0.0929
0.0934
10.7119
0.9957
40
SO
0.0262
0.0262
38.1885
0.9997
80
80
0.0958
0.0963
10.3854
0.9954
80
40
0.0291
0.0291
34.3678
0.9996
20
40
0.0987
0.0992
10.0780
0.9951
20
50
0.0320
0.0320
31.2416
0.9995
10
60
0.1016
0.1022
9.7882
0.9948
10
2
0.0349
0.0349
28.6363
0.9994
88
6
0.1045
0.1051
9.5144
0.9945
84
10
0.0378
0.0378
26.4316
0.9993
50
10
0.1074
0.1080
9.2553
0.9942
60
20
0.0407
0.0407
24.5418
0.9992
40
20
0.1103
0.1110
9.0098
0.9939
40
SO
0.0436
0.0437
22.9038
0.9990
80
30
0.1132
0.1139
8.7769
0.9936
30
40
0.0465
0.0466
21.4704
0.9989
20
40
0.1161
0.1169
8.5555
0.9932
20
60
0.0494
0.0495
20.2056
0.9988
10
50
0.1190
0.1198
8.3450
0.9929
10
8
0.0523
0.0524
19.0811
0.9986
87
7
0.1219
0.1228
8.1443
0.9925
083
10
0.0552
0.0553
18.0750
0.9985
60
10
0.1248
0.1257
7.9530
0.9922
60
20
0.0581
0.0582
17.1693
0.9983
40
20
0.1276
0.1287
7.7704
0.9918
40
80
0.0610
0.0612
16.3499
0.9981
80
30
0.1305
0.1317
7.5958
0.9914
80
40
0.0640
0.0641
15.6048
0.9980
20
40
0.1334
0.1346
7.4287
0.9911
20
60
0.0669
0.0670
14.9244
0.9978
10
60
0.1363
0.1376
7.2687
0.9907
10
4
0.0698
0.0699
14.3007
0.9976
86
8
0.1392
0.1405
0.1435
7.1154
0.9903
82
10
0.0727
0.0729
13.7267
0.9974
60
10
0.1421
6.9682
0.9899
50
20
0.0756
0.0758
13.1969
0.9971
40
20
0.1449
0.1465
6.8269
0.9894
40
SO
0.0785
0.0787
12.7062
0.9969
30
80
0.1478
0.1495
6.6912
0.9890
80
40
0.0814
0.0816
12.2505
0.9967
20
40
0.1507
0.1524
6.5606
0.9886
20
60
0.0843
0.0846
11.8262
0.9964
10
50
0.1536
0.1554
6.4348
0.9881
10
5
0.0872
0.0875
11.4301
0.9962
085
0.1564
0.1584
6.3138
0.9877
81
o t
008
oot
tan
Bin
f o
o /
008
cot
tan
sin
f o
81°-89 c
52
9°-
■26°
o f
sin
tan
cot
008
f o
o f
sin
tan
oot
008
t o
9
0.1564
0.1584
6.3138
0.9877
81
18
0.3090
0.3249
3.0777
0.9511
72
10
0.1593
0.1614
6.1970
0.9872
60
10
0.3118
03281
3.0475
0.9502
50
20
0.1622
0.1644
6.0844
0.9868
40
20
03145
03314
10178
0.9492
40
30
0.1650
0.1673
5.9758
0.9863
30
30
03173
0.3346
2.9887
0.9483
30
40
0.1679
0.1703
5.8708
0.9858
20
40
03201
03378
2.9600
0.9474
20
60
0.1708
0.1733
5.7694
0.9853
10
50
03228
03411
2.9319
0.9465
10
10 o
0.1736
0.1763
5.6713
0.9848
80
19
03256
0.3443
2.9042
0.9455
71
10
0.1765
0.1793
5.5764
0.9843
60
10
0.3283
0.3476
2.8770
0.9446
50
20
0.1794
0.1823
5.4845
0.9838
40
20
03311
03508
2.8502
0.9436
40
30
0.1822
0.1853
53955
0.9833
30
30
03338
03541
218239
0.9426
30
40
0.1851
0.1883
5.3093
0.9827
20
40
0.3365
03574
2.7980
0.9417
20
50
0.1880
0.1914
5.2257
0.9822
10
60
0.3393
03607
2.7725
0.9407
10
n o
0.1908
0.1944
5.1446
0.9816
79
20
0.3420
0.3640
2.7475
0.9397
70
10
0.1937
0.1974
5.0658
0.9811
50
10
03448
03673
2.7228
0.9387
50
20
IXX1965
0.2004
4.9894
0.9805
40
20
0.3475
0.3706
2.6985
0.9377
40
30
0.1994
0.2035
4.9152
0.9799
30
30
0.3502
03739
2.6746
0.9367
30
40
0.2022
0.2065
4.8430
0.9793
20
40
03529
0.3772
2.6511
0.9356
20
50
0.2051
0.2095
4.7729
0.9787
10
50
0.3557
03805
2.6279
0.9346
10
12
0.2079
0.2126
4.7046
0.9781
78
21
0.3584
03839
2.6051
0.9336
069
10
0.2108
0.2156
4.6382
0.9775
60
10
03611
0.3872
2.5826
0.9325
60
20
0.2136
0.2186
4.5736
0.9769
40
20
0.3638
0.3906
2.5605
0.9315
40
30
0.2164
0.2217
4.5107
0.9763
30
30
03665
03939
2.5386
0.9304
30
40
0.2193
0.2247
4.4494
0.9757
20
40
03692
0.3973
2.5172
0.9293
20
50
0.2221
0.2278
4.3897
0.9750
10
50
0.3719
0.4006
2.4960
0.9283
10
13
0.2250
0.2309
43315
0.9744
77
220
0.3746
0.4040
2.4751
0.9272
68
10
0.2278
0.2339
4.2747
0.9737
50
10
0.3773
0:4074
2.4545
0.9261
60
20
0.2306
0.2370
4.2193
0.9730
40
20
0.3800
0.4108
2.4342
0.9250
40
30
0.2334
0.2401
4.1653
0.9724
30
30
03827
0.4142
2.4142
0.9239
30
40
0.2363
0.2432
4.1126
0.9717
20
40
03854
0.4176
2.3945
0.9228
20
50
0.2391
0.2462
4.0611
0.9710
10
50
03881
0.4210
23750
0.9216
10
14
0.2419
0.2493
4.0108
0.9703
76
23
03907
0.4245
2.3559
0.9205
67
10
0.2447
0.2524
3.9617
0.9696
60
10
0.3934
0.4279
2.3369
0.9194
60
20
0.2476
0.2555
3.9136
0.9689
40
20
0.3961
0.4314
2.3183
0.9182
40
30
0.2504
0.2586
3.8667
0.9681
30
30
0.3987
0.4348
2.2998
0.9171
30
40
0.2532
0.2617
3.8208
0.9674
20
40
0.4014
0.4383
2.2817
0.9159
20
50
0.2560
0.2648
3.7760
0.9667
10
50
0.4041
0.4417
2.2637
0.9147
10
15
0.2588
0.2679
3.7321
0.9659
75
240
0.4067
0.4452
2.2460
0.9135
o ee
10
0.2616
Q.2711
0.2742
3.6891
0.9652
50 *
10
0.4094
0.4487
2.2286
0.9124
60
20
0.2644
3.6470
0.9644
40
20
0.4120
0.4522
2.2113
0.9112
40
30
0.2672
0.2773
3.6059
0.9636
30
30
0.4147
0.4557
2.1943
0.9100
30
40
0.2700
0.2805
3.5656
0.9628
20
40
0.4173
0.4592
2.1775
0.9088
20
50
0.2728
0.2836
3.5261
0.9621
10
50
0.4200
0.4628
2.1609
0.9075
10
16
0.2756
0.2867
3.4874
0.9613
74
25
0.4226
0.4663
2.1445
0.9063
65
10
0.2784
0.2899
3.4495
0.9605
50
10
0.4253
0.4699
2.1283
0.9051
50
20
0.2812
0.2931
3.4124
0.9596
40
20
0.4279
0.4734
2.1123
0.9038
40
30
0.2840
0.2962
3.3759
0.9588
30
30
0.4305
0.4770
2.0965
0.9026
30
40
0.2868
0.2994
3.3402
0.9580
20
40
0.4331
0.4806
2.0809
0.9013
20
60
0.2896
03026
3.3052
0.9572
10
60
0.4358
0.4841
2.0655
0.9001
10
17
0.2924
0.3057
3.270*
0.9563
73
26
0.4384
0.4877
2.0503
0.8988
64
10
0.2952
0.3089
3.2371
0.9555
50
10
0.4410
0.4913
2.0353
0.8975
60
20
0.2979
0.3121
3.2041
0.9546
40
20
0.4436
0.4950
2.0204
0.8962
40
30
0.3007
0.3153
3.1716
0.9537
30
80
0.4462
0.4986
2.0057
0.8949
30
40
0.3035
0.3185
3.1397
0.9528
20
40
0.4488
0.5022
1.9912
0.8936
20
60
0.3062
0.3217
3.1084
0.9520
10
50
0.4514
0.5059
1.9768
0.8923
10
18
0.3090
0.3249
3.0777
0.9511
72
27
0.4540
0.5095
1.9626
0.8910
63
o f
008
oot
tan
sin
f o
o t
008
oot
tan
sin
f o
63°-80 c
27°-
-44
53
o f
sin
tan
cot
008
f o
o /
sin
tan
oot
oos
t o
27
0.4540
0.5095
1.9626
0.8910
63
36
0.5878
0.7265
1.3764
0.8090
54
10
0.4566
0.5132
1.9486
0.8897
50
10
0.5901
0.7310
1.3680
0.8073
50
20
0.4592
0.5169
1.9347
0.8884
40
20
0.5925
0.7355
1.3597
0.8056
40
30
0.4617
0.5206
1.9210
0.8870
30
30
0.5948
0.7400
1.3514
0.8039
30
40
0.4643
0.5243
1.9074
0.8857
20
40
0.5972
0.7445
1.3432
0.8021
20
50
0.4669
0.5280
1.8940
0.8843
10
50
0.5995
0.7490
1.3351
0.8004
10
28
0.4695
0.5317
1.8807
0.8829
62
37
0.6018
0.7536
1.3270
0.7986
053
10
0.4720
0.5354
1.8676
0.8816
50
10
0.6041
0.7581
1.3190
0.7969
60
20
0.4746
0.5392
1.8546
0.8802
40
20
0.6065
0.7627
1.3111
0.7951
40
30
0.4772
0.5430
1.8418
0.8788
30
30
0.6088
0.7673
1.3032
0.7934
30
40
0.4797
0.5467
1.8291
0.8774
20
40
0.6111
0.7720
1.2954
0.7916
20
60
0.4823
0.5505
1.8165
0.8760
10
50
0.6134
0.7766
1.2876
0.7898
10
290
0.4848
0.5543
1.8040
0.8746
61
38
0.6157
0.7813
1.2799
0.7880
52
10
0.4874
0.5581
1.7917
0.8732
50
10
0.6180
0.7860
1.2723
0.7862
50
20
0.4899
0.5619
1.7796
0.8718
40
20
0.6202
0.7907
1.2647
0.7844
40
30
0.4924
0.5658
1.7675
0.8704
30
30
0.6225
0.7954
1.2572
0.7826
30
40
0.4950
0.5696
1.7556
0.8689
20
40
0.6248
0.8002
1.2497
0.7808
20
60
0.4975
0.5735
1.7437
0.8675
10
50
0.6271
0.8050
1.2423
0.7790
10
300
0.5000
0.5774
1.7321
0.8660
60
39
0.6293
0.8098
1.2349
0.7771
51
10
0.5025
0.5812
1.7205
0.8646
50
10
0.6316
0.8146
1.2276
0.7753
50
20
0.5050
0.5851
1.7090
0.8631
40
20
0.6338
0.8195
1.2203
0.7735
40
30
0.5075
0.5890
1.6977
0.8616
30
30
0.6361
0.8243
1.2131
0.7716
30
40
0.5100
0.5930
1.6864
0.8601
20
40
0.6383
0.8292
1.2059
0.7698
20
50
0.5125
0.5969
1.6753
0.8587
10
50
0.6406
0.8342
1.1988
0.7679
10
31
0.5150
0.6000
1.6643
0.8572
59
40
0.6428
0.8391
1.1918
0.7660
50
10
0.5175
0.6048
1.6534
0.8557
50
10
0.6450
0.8441
1.1847
0.7642
50
20
0.5200
0.6088
1.6426
0.8542
40
20
0.6472
0.8491
1.177S
0.7623
40
30
0.5225
0.6128
1.6319
0.8526
30
30
0.6494
0.8541
1.1708
0.7604
30
40
0.5250
0.6168
1.6212
0.8511
20
40
0.6517
0.8591
1.1640
0.7585
20
50
0.5275
0.6208
1.6107
0.8496
10
60
0.6539
0.8642
1.1571
0.7566
10
32
0.5299
0.6249
1.6003
0.S480
58
41
0.6561
0.8693
1.1504
0.7547
49
10
0.5324
0.6289
1.5900
0.8465
50
10
0.6583
0.8744
1.1436
0.7528
50
20
0.5348
0.6330
1.5798
0.8450
40
20
0.6604
0.8796
1.1369
0.7509
40
30
0.5373
0.6371
1.5697
0.8434
30
30
0.6626
0.8847
1.1303
0.7490
30
40
0.5398
0.6412
1.5597
0.8418
20
40
0.6648
0.8899
1.1237
0.7470
20
60
0.5422
0.6453
1.5497
0.8403
10
50
0.6670
0.8952
1.1171
0.7451
10
33
0.5446
0.6494
1.5399
0.8387
57
42
0.6691
0.9004
1.1106
0.7431
48
10
0.5471
0.6536
1.5301
0.8371
50
10
0.6713
0.9057
1.1041
0.7412
50
20
0.5495
0.6577
1.5204
0.8355
40
20
0.6734
0.9110
1.0977
0.7392
40
30
0.5519
0.6619
1.5108
0.8339
30
30
0.6756
0.9163
1.0913
0.7373
30
40
0.5544
0.6661
1.5013
0.8323
20
40
0.6777
0.9217
1.0850
0.7353
20
50
0.5568
0.6703
1.4919
0.8307
10
50
0.6799
0.9271
1.0786
0.7333
10
34
0.5592
0.6745
1.4826
0.8290
56
43
0.i6820
0.9325
1.0724
0.7314
47
10
0.5616
0.6787
1.4733
0.8274
60
10
0.6841
0.9380
1.0661
0.7294
50
20
0.5640
0.6830
1.4641
0.8258
40
20
0.6862
0.9435
1.0599
0.7274
40
30
0.5664
0.6873
1.4550
0.8241
30
30
0.6884
0.9490
1.0538
0.7254
30
40
0.5688
0.6916
1.4460
0.8225
20
40
0.6905
0.9545
1.0477
0.7234
20
60
0.5712
0.6959
1.4370
0.8208
10
50
0.6926
0.9601
1.0416
0.7214
10
35
0.5736
0.7002
1.4281
0.8192
055
44
0.6947
0.9657
1.0355
0.7193
46
10
0.5760
0.7046
1.4193
0.8175
50
10
0.6967
0.9713
1.0295
0.7173
60
20
0.5783
0.7089
1.4106
0.8158
40
20
0.6988
0.9770
1.0235
0.7153
40
30
0.5807
0.7133
1.4019
0.8141
30
30
0.7009
0.9827
1.0176
0.7133
30
40
0.5831
0.7177
1.3934
0.8124
20
40
0.7030
0.9884
1.0117
0.7112
20
50
0.5854
0.7221
1.3848
0.8107
10
50
0.7050
0.9942
1.0058
0.7092
10
36
0.5878
0.7265
1.3764
0.8090
054
45
0.7071
1.0000
1.0000
0.7071
45
o t
008
oot
tan
sin
t o
o f
008
cot
tan
sin
t o
45°-62 c
54 '
TABLE VL-
-CIRCUMFERENCES AND AREAS OP CIRCLES.
If y= the radius of the circle, the circumference = 2irJT.
It N= the radius of the circle, the area
= rN*.
If ^T= the circumference of the circle, the radius = ~N.
9 2»
If N= the circumference of the circle, the area = j- if*.
H
2irJV
*N*
2t
4*
H
%vN
*ir*
2»
4t
0.00
0.0
0.000
0.00
60
314. 16
7 854
7.96
198.94
1
6.28
3.1
0.159
0.08
61
320.44
8171
8.12
206.98
2
12.57
12.6
0.318
0.32
52
326. 73
8495
8.28
215. 18
3
18.85
28.3
0.477
0.72
53
333.01
8 825
8.44
223. 53
4
25.13
50.3
0.637
1.27
64
339.29
9161
8.59
232.05
6
31.42
78.5
0.796
1.99
65
345. 58
9503
8.75
240.72
6
37.70
113.1
0.955
2.86
56
351.86
9852
8.91
249. 55
7
43.98
153.9
1.114
3.90
67
358. 14
10207
9.07
258. 55
8
50.27
201.1
1.273
5.09
58
364.42
10 568
9.23
267.70
9
56.55
254.5
1.432
6.45
59
370. 71
10936
9.39
277. 01
10
62.83
314.2
1.592
7.96
00
376.99
11310
9.55
286.48
11
69.12
380.1
1.751
9.63
61
, 383. 27
11690
9.71
2%. 11
12
75.40
452.4
1.910
11.46
62
389.56
12 076
9.87
305.90
13
81.68
530.9
2.069
13.45
63
395.84
12 469
10.03
315.84
14
87.%
615.8
2.228
15.60
64
402.12
12 868
10.19
325.95
16
94.25
706.9
2.387
17.90
65
408.41
13 273
10.35
336. 21
16
100. 53
804.2
2.546
20.37
66
414.69
13 685
10.50
346.64
17
106..81
907.9
2.706
23.00
67
420.97
14103
10.66
357. 22
18
113. 10
1017.9
2.865
25.78
68
427.26
14 527
10.82
367.97
19
119.38
1 134. 1
3.024
28.73
69
433. 54
14957
10.98
378. 87
20
'125.66
1 256. 6
3.183
31.83
70
439.82
15 394
11.14
389.93
21
131.95
1385.4
3.342
35.09
71
446.11
15 837
11.30
401.15
22
138. 23
1 520. 5
3.501
38.52
72
452.39
16 286
11.46
412.53
23
144.51
1661.9
3.661
42.10
73
458.67
16 742
11.62
424.07
24
150.80
1809.6
3.820
45.84
74
464.%
17 203
11.78
435. 77
26
157.08
1963.5
3.979
49.74
76
471. 24
17671
11.94
447.62
26
163.36
2 123. 7
4.138
53.79
76
477. 52
18146
12.10
459.64
27
169.65
2290.2
4.297
58.01
77
483.81
18627
12.25
471. 81
28
175. 93
2463.0
4.456
62.39
78
490.09
19113
12.41
484.15
29
182.21
2 642.1
4.615
66.92
79
4%. 37
19607
12.57
4%. 64
30
188.50
2827.4
4.775
71.62
80
502.65
20106
12.73
509.30
31
194.78
3 019.1
4.934
76.47
81
508.94
20612
12.89
522. 11
32
201.06
♦ 3 217.0
5.093
81.49
82
515.22
21124
13.05
535.08
33
207. 35
3 421.2
5.252
86.66
83
521. 50
21642
13.21
548. 21
34
213.63
3 631.7
5.411
91.99
84
527. 79
22167
13.37
561. 50
35
219.91
3848.5
5.570
97.48
86
534.07
22 698
13.53
574.95
36
226. 19
4071.5
5.730
103. 13
86
540.35
23 235
13.69
588. 55
37
232.48
4300.8
5.889
108.94
87
546.64
23 779
13.85
602.32
38
238.76
4 536.5
6.048
114.91
88
552. 92
24328
14.01
616. 25
39
245.04
4 778.4
6.207
121.04
89
559.20
24 885
14.16
630.33
40
251. 33
5 026.5
6.366
127.32
90
565.49
25 447
14.32
644.58
41
257. 61
5 281.0
6.525
133. 77
91
571. 77
26016
14.48
658.98
42
263.89
5 541.8
6.685
140.37
92
578.05
26 590
14.64
673. 5+
43
270. 18
5808.8
6.844
147. 14
93
584.34
27172
14.80
688.27
44
276.46
6082.1
7.003
154.06
94
590.62
27 759
14.%
703. 15
46
282. 74
6361.7
7.162
161. 14
96
5%. 90
28353
15.12
718. 19
46
289.03
6647.6
7.321
168. 39
96
603.19
28 953
15.28
733. 39
47
295. 31
6939.8
7.480
175. 79
97
609.47
29 559
15.44
748.74
48
301. 59
7 238.2
7.639
183.35
98
615. 75
30172
15.60
764.26
49
307.88
7 543.0
7.799
191.07
99
622.04
30 791
15.76
779.94
50
314. 16
7 854.0
7.958
198.94
100
628.32
31416
15.92
795. 77
H
2vN
*ir*
2x
4ir
H
2vir
*&
i'
±-
A TABLE OF THE ANGLES
Which every Point and Quarter Point of the Compass make* with the Meridian.
North.
N. by E.
N. by W.
Point*
a
o I II
2 48 45
6 37 80
8 26 15
11 16
Points,
l" *
South.
S. by E.
S. by W.
N.N.E.
N.N.W.
i-%
14 3 45
16 52 30
10 41 15
22 30
l-i y
1-1
2
S.S.E.
S.S.W.
N.E.byN.
N.W.byN.
3
*
26 18 45
28 7 30
30 56 15
33 45
2-%
2-
2-
8
S.E.by S.
S.W. by S.
N.E.
N.W.
3-ly
3-*
4
36 33 45
39 22 30
42 11 15
45
8-V 4
S.E.
S.W.
N.E. by E.
N.W. by W.
ti
47 48 45
60 37 30
53 26 15
56 15
n
S.E.byE.
SW.byW.
E.N.E.
W.N.W.
E. by N.
W. by N.
b-h
6
~6-\
6-}
6-1
7
50 8 45
61 52 30
64 41 15
67 30
70 18 45
73 7 30
76 56 15
78 45
6
E.S.E.
W.S.W.
8:*
6-»4
E. by S.
W. by S.
East.
"West.
7-i>
7-1
8
81 33 45
84 22 30
87 11 15
90
n
East. "West.
56
TABLE VH
-TI
IAV1
SRSE TABLE.
Bearing.
Distance 1.
Distance 2.
Distance 3.
Distance 4.
Distance 5.
Bearing.
o f
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
o t
15
1.000
0.004
2.000
0.009
3.000
0.013
4.000
0.017
5.000
0.022
89 45
SO
1.000
0.009
2.000
0.017
3.000
0.026
4.000
0.035
5.000
0.044
30
45
1.000
0.013
2.000
0.026
3.000
0.039
4.000
0.052
5.000
0.065
15
1 o
1.000
0.017
2.000
0.035
3.000
0.052
3.999
0.070
4.999
0.087
89
15
1.000
0.022
2.000
0.044
2.999
0.065
3.999
0.087
4.999
0.109
45
80
1.000
0.026
1.999
0.052
2.999
0.079
3.999
0.105
4.998
0.131
30
45
1.000
0.031
1.999
0.061
2.999
0.092
3.998
0.122
4.998
0.153
15
2
0.999
0.035
1.999
0.070
2.998
0.105
3.998
0.140
4.997
0.174
88
15
0.999
0.039
1.998
0.079
2.998
0.118
3.997
0.157
4.9%
0.1%
45
80
0.999
0.044
1.998
0.087
2.997
0.131
3.9%
0.174
4.995
0.218
30
45
0.999
0.048
1.998
0.096
2.997
0.144
3.995
0.192
4.994
0.240
15
3
0.999
0.052
1.997
0.105
2.9%
0.157
3.995
0.209
4.993
0.262
87
16
0.998
0.057
1.997
0.113
2.995
0.170
3.994
0.227
4.992
0.283
46
SO
0.998
0.061
1.996
0.122
2.994
0.183
3.993
0.244
4.991
0305
30
46
0.998
0.065
1.996
0.131
2.994
0.1%
3.991
0.262
4.989
0327
15
4
0.998
0.070
1.995
0.140
2.993
0.209
3.990
0.279
4.988
0349
86
15
0.997
0.074
1.995
0.148
2.992
0.222
3.989
0.2%
4.986
0.371
46
30
0.997
0.078
1.994
0.157
2.991
0.235
3.988
0314
4.985
0.392
30
45
0.997
0.083
1.993
0.166
2.990
0.248
3.986
0.331
4.983
0.414
15
5
0.996
0.087
1.992
0.174
2.989
0.261
3.985
0.349
4.981
0.436
85
15
0.9%
0.092
1.992
0.183
2.987
0.275
3.983
0.366
4.979
0.458
45
30
0.995
0.096
1.991
0.192
2.986
0.288
3.982
0.383
4.977
0.479
30
45
0.995
0.100
1.990
0.200
2.985
0.301
3.980
0.401
4.975
0.501
15
6
0.995
0.105
1.989
0.209
2.984
0.314
3.978
0.418
4.973
0.523
84
15
0.994
0.109
1.988
0.218
2.982
0327
3.976
0.435
4.970
0.544
45
30
0.994
0.113
1.987
0.226
2.981
0.340
3.974
0.453
4.968
566
30
45
0.993
0.118
1.986
0.235
2.979
0.353
3.972
0.470
4.%5
0.588
15
7
0.993
0.122
1.985
0.244
2.978
0.366
3.970
0.487
4.963
0.609
83
16
0.992
0.126
1.984
0.252
2.976
0.379
3.968
0.505
4.960
0.631
45
30
0.991
0.131
1.983
0.261
2.974
0.392
3.966
0.522
4.957
0.653
30
45
0.991
0.135
1.982
0.270
2.973
0.405
3.963
0.539
4.954
0.674
15
8
0.990
0.139
1.981
0.278
2.971
0.418
3.%1
0.557
4.951
0.6%
82
15
0.990
0.143
1.979
0.287
2.%9
0.430
3.959
0.574
4.948
0.717
45
30
0.989
0.148
1.978
0.2%
2.%7
0.443
3.956
0.591
4.945
0.739
30
45
0.988
0.152
1.977
0.304
2.%5
0.456
3.953
0.608
4.942
0.761
15
9
0.988
0.156
1.975
0.313
2.963
0.469
3.951
0.626
4.938
0.782
81
15
0.987
0.161
1.974
0.321
2.%1
0.482
3.948
0.643
4.935
0.804
45
30
0.986
0.165
1.973
0.330
2.959
0.495
3.945
0.660
4.931
0.825
30
45
0.986
0.169
1.971
0.339
2.957
0.508
3.942
0.677
4.928
0.847
15
10
0.985
0.174
1.970
0.347
2.954
0.521
3.939
0.695
4.924
0.868
80
15
0.984
0.178
1.968
0.356
2.952
0.534
3.936
0.712
4.920
0.890
45
30
0.983
0.182
1.967
0.364
2.950
0.547
3.933
0.729
4.916
0.911
30
45
0.982
0.187
1.965
0.373
2.947
0.560
3.930
0.746
4.912
0.933
15
11
0.982
0.191
1.963
0.382
2.945
0.572
3.927
0.763
4.908
0.954
79
15
0.981
0.195
1.962
0.390
2.942
0.585
3.923
0.780
4.904
0.975
46
30
0.980
0.199
1.960
0.399
2.940
0.598
3.920
0.797
4.900
0.997
30
46
0.979
0.204
1.958
0.407
2.937
0.611
3.916
0.815
4.895
1.018
15
12
0.978
0.208
1.956
0.416
2.934
0.624
3.913
0.832
4.891
1.040
78
15
0.977
0.212
1.954
0.424
2.932
0.637
3.909
0.849
4.886
1.061
45
30
0.976
0.216
1.953
0.433
2.929
0.649
3.905
0.866
4.881
1.082
30
46
0.975
0.221
1.951
0.441
2.926
0.662
3.901
0.883
4.877
1.103
15
13
0.974
0.225
1.949
0.450
2.923
0.675
• 3.897
0.900
4.872
1.125
77
15
0.973
0.229
1.947
0.458
2.920
0.688
3.894
0.917
4.867
1.146
45
30
0.972
0.233
1.945
0.467
2.917
0.700
3.889
0.934
4.862
1.167
30
46
0.971
0.238
1.943
0.475
2.914
0.713
3.885
0.951
4.857
1.188
15
14
0.970
0.242
1.941
0.484
2.911
0.726
3.881
0.968
4.851
1.210
76
15
0.969
0.246
1.938
0.492
2.908
0.738
3.877
0.985
4.846
1.231
46
30
0.968
0.250
1.936
0.501
2.904
0.751
3.873
1.002
4.841
1.252
30
45
0.967
0.255
1.934
0.509
2.901
0.764
3.868
1.018
4.835
1.273
15
15
0.966
0.259
1.932
0.518
2.898
0.776
3.864
1.035
4.830
1.294
75
O 9
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
o f
Bearing.
Distance 1.
Distance 2.
Distance 3.
Distance 4.
Distance 5.
Bearing.
76°- 90°
0°-
15°
57
Bearing.
Distance 6.
Distance 7.
Distance 8.
Distance 9.
Distance 10.
Bearing.
o t
Lat,
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
o f
Ol5
6.000
0.026
7.000
0.031
8.000
0.035
9.000
0.039
10.000
0.044
89 45
30
6.000
0.052
7.000
0.061
8.000
0.070
9.000
0.079
10.000
0.087
30
45
5.999
0.079
6.999
0.092
7.999
0.105
8.999
0.118
9.999
0.131
15
1 o
5.999
0.105
6.999
0.122
7.999
0.140
8.999
0.157
9.999
0.175
89
15
5.999
0.131
6.998
0.153
7.998
0.175
8.998
0.1%
9.998
0.218
45
80
5.998
0.157
6.998
0.183
7.997
0.209
8.997
0.236
9.997
0.262
80
45
5.997
0.183
6.997
0.214
7.996
0.244
8.996
0.275
9.995
0.305
15
2
5.996
0.209
6.996
0.244
7.995
0.279
8.995
0.314
9.994
0.349
88
15
5.995
0.236
6.995
0.275
7.994
0.314
8.993
0.353
9.992
0.393
45
80
5.994
0.262
6.993
0.305
7.992
0.349
8.991
0.393
9.991
0.436
80
45
5.993
0.288
6.992
0.336
7.991
0.384
8.990
0.432
9.989
0.480
15
3
5.992
0.314
6.990
0.366
7.989
0.419
8.988
0.471
9.986
0.523
87
16
5.990
0.340
6.989
0397
7.987
0.454
8.986
0.510
9.984
0.567
45
80
5.989
0.366
6.987
0.427
7.985
0.488
8.983
0549
9.981
0.611
80
45
5.987
0.392
6.985
0.458
7.983
0.523
8.981
0.589
9.979
0.654
15
4
5.985
0.419
6.983
0.488
7.9?1
0.558
8.978
0.628
9.976
0.698
86
16
5.984
0.445
6.981
0.519
7.978
0.593
8.975
0.667
9.973
0.741
45
80
5.982
0.471,
6.978
0.549
7.975
0.628
8.972
0.706
9.969
0.785
30
46
5.979
0.497
6.976
0.580
7.973
0.662
8.%9
0.745
9.966
0.828
15
5
5.977
0.523
6.973
0.610
7.970
0.697
8.966
0.784
9.%2
0.872
85
15
5.975
0.549
6.971
0.641
7.966
0.732
8.%2
0.824
9.958
0.915
46
80
5.972
0.575
6.968
0.671
7.963
0.767
8.959
0.863
9.954
0.959
30
45
5.970
0.601
6.965
0.701
7.960
0.802
8.955
0.902
9.950
1.002
15
6
5.967
0.627
6.962
0.732
7.956
0.836
8.951
0.941
9.945
1.045
84
15
5.964
0.653
6.958
0.762
7.952
0.871
8.947
0.980
9.941
1.089
45
30
5.961
0.679
6.955
0.792
7.949
0.906
8.942
1.019
9.936
1.132
30
45
5.958
0.705
6.951
0.823
7.945
0.940
8.938
1.058
9.931
1.175
15
7
5.955
0.731
6.948
0.853
7.940
0.975
8.933
1.097
9.926
1.219
83
15
5.952
0.757
6.944
0.883
7.936
1.010
8.928
1.136
9.920
1.262
46
30
5.949
0.783
6.940
0.914
7.932
1.044
8.923
1.175
9.914
1.305
30
45
5.945
0.809
6.936
0.944
7.927
1.079
8.918
1.214
9.909
1.349
15
8
5.942
0.835
6.932
0.974
7.922
1.113
8.912
1.253
9.903
1.392
82
15
5.938
0.861
6.928
1.004
7.917
1.148
8.907
1.291
9.897
1.435
45
80
5.934
0.887
6.923
1.035
7.912
1.182
8.901
1.330
9.890
1.478
30
45
5.930
0.913
6.919
1.065
7.907
1.217
8.895
1.369
9.884
1.521
15
9
5.926
0.939
6.914
1.095
7.902
1.251
8.889
1.408
9.877
1.564
81
15
5.922
0.964
6.909
1.125
7.8%
1.286
8.883
1.447
9.870
1.607
46
30
5.918
0.990
6.904
1.155
7.890
1320
8.877
1.485
9.863
1.651
30
45
5.913
1.016
6.899
1.185
7.884
1.355
8.870
1.524
9.856
1.694
15
10
5.909
1.042
6.894
1.216
7.878
1.389
8.863
1.563
9.848
1.737
80
15
5.904
1.068
6.888
1.246
7.872
1.424
8.856
1.601
9.840
1.779
45
30
5.900
1.093
6.883
1.276
7.866
1.458
8.849
1.640
9.833
1.822
30
45
5.895
1.119
6.877
1.306
7.860
1.492
8.842
1.679
9.825
1.865
15
11
5.890
1.145
6.871
1.336
7.853
1.526
8.835
1.717
9.816
1.908
79
15 v
5.885
1.171
6.866
1.366
7.846
1.561
8.827
1.756
9.808
1.951
45
30
5.880
1.196
6.859
1.3%
7.839
1.595
8.819
1.794
9.799
1.994
30
45
5.874
1.222
6.853
1.425
7.832
1.629
8.811
1.833
9.791
2.036
15
12
5.869
1.247
6.847
1.455
7.825
1.663
8.803
1.871
9.782
2.079
78
15
5.863
1.273
6.841
1.485
7.818
1.697
8.795
1.910
9.772
2.122
45
30
5.858
1.299
6.834
1.515
7.810
1.732
8.787
1.948
9.763
2.164
30
45
5.852
1.324
6.827
1.545
7.803
1.766
8.778
1.986
9.753
2.207
15
13
5.846
1.350
6.821
1.575
7.795
1.800
8.769
2.025
9.744
2.250
77
15
5.840
1.375
6.814
1.604
7.787
1.834
8.760
2.063
9.734
2.292
45
30
5.834
1.401
6.807
1.634
7.779
1.868
8.751
2.101
9.724
2.335
30
45
5.828
1.426
6.799
1.664
7.771
1.902
8.742
2.139
9.713
2.377
15
14
5.822
1.452
6.792
1.693
7.762
1.935
8.733
2.177
9.703
2.419
76
15
5.815
1.477
6.785
1.723
7.754
l.%9
8.723
2.215
9.692
2.462
45
30
5.809
1.502
6.777
1.753
7.745
2.003
8.713
2.253
9.682
2.504
30
45
5.802
1.528
6.769
1.782
7.736
2.037
8.703
2.291
9.671
2.546
15
15
5.7%
1.553
6.761
1.812
7.727
2.071
8.693
2.329
9.659
2.588
75
o t
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
o f
Bearing.
Distance 6.
Distance 7.
Distance 8.
Distance 9.
Distance 10.
Bearing.
75°- 90°
58
16°-
30 c
>
Bearing.
Distance 1.
Distance 2.
Distance 3.
Distance 4.
Distance 5.
Bearing.
o f
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
o f
1515
0.965
0.263
1.930
0.526
2.894
0.789
3.859
1.052
4.824
1.315
74 46
30
0.964
0.267
1.927
0.534
2.891
0.802
3.855
1.069
4.818
1.336
30
45
0.962
0.271
1.925
0.543
2.887
0.814
3.850
1.086
4.812
1.357
16
16
0.961
0.276
1.923
0.551
2.884
0.827
3.845
1.103
4.806
1378
74
15
0.960
0.280
1.920
0.560
2.880
0.839
3.840
1.119
4.800
1.399
46
30
0.959
0.284
1.918
0.568
2.876
0.852
3.835
1.136
4.794
1.420
30
46
0.958
0.288
1.915
0.576
2.873
0.865
3.830
1.153
4.788
1.441
16
17
0.956
0.292
1.913
0.585
2.869
0.877
3.825
1.169
4.782
1.462
73
15
0.955
0.297
1.910
0.593
2.865
0.890
3.820
1.186
4.775
1.483
45
30
0.954
0.301
1.907
0.601
2.861
0.902
3.815
1.203
4.769
1.504
30
45
0.952
0305
1.905
0.610
2.857
0.915
3.810
1.220
4.762
1.524
16
18
0.951
0.309
1.902
0.618
2.853
0.927
3.804
1.236
4.755
1.545
72
15
0.950
0.313
1.899
0.626
2.849
0.939
3.799
1.253
4.748
1.566
46
30
0.948
0.317
1.897
0.635
2.845
0.952
3.793
1.269
4.742
1.587
30
46
0.947
0.321
1.894
0.643
2.841
0.964
3.788
1.286
4.735
1.607
16
19
0.946
0.326
1.891
0.651
2.837
0.977
3.782
1.302
4.728
1.628
71
16
0.944
0.330
1.888
0.659
2.832
0.989
3.776
1319
4.720
1.648
46
30
0.943
0.334
1.885
0.668
2.828
1.001
3.771
1335
4.713
1.669
30
45
0.941
0.338
1.882
0.676
2.824
1.014
3.765
1352
4.706
1.690
16
20
0.940
0.342
1.879
0.684
2.819
1.026
3.759
1368
4.698
1.710
70
16
0.938
0346
1.876
0.692
2.815
1.038
3.753
1384
4.691
1.731
46
30
0.937
0.350
1.873
0.700
2.810
1.051
3.747
1.401
4.683
1.751
30
45
0.935
0.354
1.870
0.709
2.805
1.063
3.741
1.417
4.676
1.771
15
21
0.934
0.358
1.867
0.717
2.801
1.075
3.734
1.433
4.668
1.792
69
15
0.932
0362
1.864
0.725
2.796
1.087
3.728
1.450
4.660
1.812
46
30
0.930
0.367
1.861
0.733
2.791
1.100
3.722
1.466
4.652
1.833
30
45
0.929
0.371
1.858
0.741
2.786
1.112
3.715
1.482
4.644
1.853
16
22
0.927
0.375
1.854
0.749
2.782
1.124
3.709
1.498
4.636
1.873
68
15
0.926
0.379
1.851
0.757
2.777
1.136
3.702
1.515
4.628
1.893
46
30
0.924
0.383
1.848
0.765
2.772
1.148
3.696
1.531
4.619
1.913
30
45
0.922
0.387
1.844
0.773
2.767
1.160
3.689
1.547
4.611
1.934
15
23
0.921
0.391
1.841
0.781
2.762
1.172
3.682
1.563
4.603
1.954
67
15
0.919
0395
1.838
0.789
2.756
1.184
3.675
1.579
4.594
1.974
46
30
0.917
0399
1.834
0.797
2.751
1.196
3.668
1.595
4.585
1.994
30
45
0.915
0.403
1.831
0.805
2.746
1.208
3.661
1.611
4.577
2.014
16
24
0.914
0.407
1.827
0.813
2.741
1.220
3.654
1.627
4.568
2.034
66
15
0.912
0.411
1.824
0.821
2.735
1.232
3.647
1.643
4.559
2.054
46
30
0.910
0.415
1.820
0.829
2.730
1.244
3.640
1.659
4.550
2.073
30
46
0.908
0.419
1.816
0.837
2.724
1.256
3.633
1.675
4.541
2.093
16
25
0.906
0.423
1.813
0.845
2.719
1.268
3.625
1.690
4.532
2.113
65
15
0.904
0.427
1.809
0.853
2.713
1.280
3.618
1.706
4.522
2.133
46
30
0.903
0.431
1.805
0.861
2.708
1.292
3.610
1.722
4.513
2.153
30
45
0.901
0.434
1.801
0.869
2.702
1303
3.603
1.738
4.503
2.172
16
26
0.899
0.438
1.798
0.877
2.6%
1.315
3.595
1.753
4.494
2.192
64
15
0.897
0.442
1.794
0.885
2.691
1327
3.587
1.769
4.484
2.211
46
30
0.895
0.446
1.790
0.892
2.685
1339
3.580
1.785
4.475
2.231
30
45
0.893
0.450
1.786
0.900
2.679
1.350
3.572
1.800
4.465
2.250
15
27
0.891
0.454
1.782
0.908
2.673
1362
3.564
1.816
4.455
2.270
63
15
0.889
0.458
1.778
0.916
2.667
1374
3.556
1.831
4.445
2.289
45
30
0.887
0.462
1.774
0.923
2.661
1385
3.548
1.847
4.435
2309
30
45
0.885
0.466
1.770
0.931
2.655
1.397
3.540
1.862
4.425
2.328
15
28
0.883
0.469
1.766
0.939
2.649
1.408
3.532
1.878
4.415
2347
62
15
0.881
0.473
1.762
0.947
2.643
1.420
3.524
1.893
4.404
2367
45
30
0.879
0.477
1.758
0.954
2.636
1.431
3.515
1.909
4.394
2386
30
45
0.877
0.481
1.753
0.962
2.630
1.443
3.507
1.924
4384
2.405
15
28
0.875
0.485
1.749
0.970
2.624
1.454
3.498
1.939
4.373
2.424
61
15
0.872
0.489
1.745
0.977
2.617
1.466
3.490
1.954
4.362
2.443
45
30
0.870
0.492
1.741
0.985
2.611
1.477
3.481
1.970
4.352
2.462
30
45
0.868
0.496
1.736
0.992
2.605
1.489
3.473
1.985
4.341
2.481
15
30
0.866
0.500
1.732
1.000
2.598
1.500
3.464
2.000
4330
2.500
60
o f
Dep.
Lat.
Dep. Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
o f
Bearing.
Distance 1.
Distance 2.
Distance 3.
Distance 4.
Distance 5.
Bearing.
60°- 75 c
15°-
-30°
59
Bearing.
Distance 6.
Distance 7.
Distance 8.
Distance 9.
Distance 10.
Bearing.
o f
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
o t
1515
5.789
1.578
6.754
1.841
7.718
2.104
8.683
2.367
9.648
2.630
74 46
30
5.782
1.603
6.745
1.871
7.709
2.138
8.673
2.405
9.636
2.672
30
45
5.775
1.629
6.737
1.900
7.700
2.172
8.662
2.443
9.625
2.714
16
16
5.768
1.654
6.729
1.929
7.690
2.205
8.651
2.481
9.613
2.756
74
15
5.760
1.679
6.720
1.959
7.680
2.239
8.640
2.518
9.601
2.798
45
30
5.753
1.704
6.712
1.988
7.671
2.272
8.629
2.556
9.588
2.840
30
45
5.745
1.729
6.703
2.017
7.661
2306
8.618
2.594
9.576
2.882
15
17
5.738
1.754
6.694
2.047
7.650
2339
8.607
2.631
9.563
2.924
73
15
5.730
1.779
6.685
2.076
7.640
2372
8.595
2.669
9.550
2.965
46
30
5.722
1.804
6.676
2.105
7.630
2.406
8.583
2.706
9.537
3.007
30
45
5.714
1.829
6.667
2.134
7.619
2.439
8.572
2.744
9.524
3.049
15
18
5.706
1.854
6.657
2.163
7.608
2.472
8.560
2.781
9.511
3.090
72
15
5.698
1.879
6.648
2.192
7.598
2.505
8.547
2.818
9.497
3.132
46
30
5.690
1.904
6.638
2.221
7.587
2.538
8.535
2.856
9.483
3.173
30
45
5.682
1.929
6.629
2.250
7.575
2.572
8.522
2.893
9.469
3.214
15
19
5.673
1.953
6.619
2.279
7.564
2.605
8.510
2.930
9.455
3.256
71
16
5.665
1.978
6.609
2308
7.553
2.638
8.497
2.967
9.441
3.297
46
30
5.656
2.003
6.598
2.337
7.541
2.670
8.484
3.004
9.426
3338
30
45
5.647
2.028
6.588
2.365
7.529
2.703
8.471
3.041
9.412
3.379
15
20
5.638
2.052
6.578
2.394
7.518
2.736
8.457
3.078
9397
3.420
70
15
5.629
2.077
6.567
2.423
7.506
2.769
8.444
3.115
9.382
3.461
45
30
5.620
2.101
6.557
2.451
7.493
2.802
8.430
3.152
9.367
3.502
30
45
5.611
2.126
6.546
2.480
7.481
2.834
8.416
3.189
9.351
3.543
16
21
5.601
2.150
6.535
2.509
7.469
2.867
8.402
3.225
9336
3.584
69
15
5.592
2.175
6.524
2.537
7.456
2.900
8.388
3.262
9.320
3.624
45
30
5.582
2.199
6.513
2.566
7.443
2.932
8.374
3.299
9304
3.665
30
46 *
5.573
2.223
6.502
2.594
7.430
2.964
8.359
3335
9.288
3.706
15
22 0*
5.563
2.248
6.490
2.622
7.417
2.997
8345
3371
9.272
3.746
68
15
5.553
2.272
6.479
2.651
7.404
3.029
8330
3.408
9.255
3.787
45
30
5.543
2.296
6.467
2.679
7.391
3.061
8.315
3.444
9.239
3.827
30
45
5.533
2320
6.455
2.707
7378
3.094
8.300
3.480
9.222
3.867
15
23
5.523
2344
6.444
2.735
7364
3.126
8.285
3.517
9.205
3.907
67
16
5.513
2368
6.432
2.763
7350
3.158
8.269
3.553
9.188
3.947
46
30
5.502
2392
6.419
2.791
7336
3.190
8.254
3.589
9.171
3.988
30
46
5.492
2.416
6.407
2.819
7322
3.222
8.238
3.625
9.153
4.028
15
24
5.481
2.440
6395
2.847
7308
3.254
8.222
3.661
9.136
4.067
66
15
5.471
2.464
6382
2.875
7.294
3.286
8.206
3.696
9.118
4.107
45
80
5.460
2.488
6370
2.903
7.280
3318
8.190
3.732
9.100
4.147
30
45
5.449
2.512
6.357
2.931
7.265
3349
8.173
3.768
9.081
4.187
15
25
5.438
2.536
6.344
2.958
7.250
3381
8.157
3.804
9.063
4.226
65
16
5.427
2.559
6331
2.986
7.236
3.413
8.140
3.839
9.045
4.266
46
30
5.416
2.583
6318
3.014
7.221
3.444
8.123
3.875
9.026
4.305
30
45
5.404
2.607
6305
3.041
7.206
3.476
8.106
3.910
9.007
4.345
15
26
5.393
2.630
6.292
3.069
7.190
3.507
8.089
3.945
8.988
4384
64
15
5.381
2.654
6.278
3.096
7.175
3.538
8.072
3.981
8.969
4.423
45
30
5370
2.677
6.265
3.123
7.160
3.570
8.054
4.016
8.949
4.462
30
45
5.358
2.701
6.251
3.151
7.144
3.601
8.037
4.051
8.930
4.501
15
27
5346
2.724
6.237
3.178
7.128
3.632
8.019
4.086
8.910
4.540
63
15
5.334
2.747
6.223
3.205
7.112
3.663
8.001
4.121
8.890
4.579
45
30
5322
2.770
6.209
3.232
7.096
3.694
7.983
4.156
8.870
4.618
30
45
5310
2.794
6.195
3.259
7.080
3.725
7.965
4.190
8.850
4.656
15
28-0
5.298
2.817
6.181
3.286
7.064
3.756
7.947
4.225
8.829
4.695
62
16
5.285
2.840
6.166
3313
7.047
3.787
7.928
4.260
8.809
4.733
45
30
5.273
2.863
6.152
3340
7.031
3.817
7.909
4.294
8.788
4.772
30
45
5.260
2.886
6.137
3.367
7.014
3.848
7.891
4329
8.767
4.810
15
29
5.248
2.909
6.122
3394
6.997
3.878
7.872
4363
8.746
4.848
61
15
5.235
2.932
6.107
3.420
6.980
3.909
7.852
4398
8.725
4.886
45
30
5.222
2.955
6.093
3.447
6.963
3.939
7.833
4.432
8.704
4.924
30
45
5.209
2.977
6.077
3.474
6.946
3.970
7.814
4.466
8.682
4.962
15
30
5.196
3.000
6.062
3.500
6.928
4.000
7.794
4.500
8.660
5.000
60
o f
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
.Lat.
Dep.
Lat.
o ,
Bearing.
Distance 6.
Distance 7.
Distance 8.
Distance 9.
Distance 10.
1 Bearing.
60°- 75 c
60
30°-
-46°
Bearing.
Distance 1.
Distance 2.
Distance 3.
Distance 4.
Distance 5.
Bearing.
o ;
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
o ;
8015
0.864
0.504
1.728
1.008
2.592
1.511
3.455
2.015
4.319
2.519
59 46
30
0.862
0.508
1.723
1.015
2.585
1.523
3.447
2.030
4.308
2.538
30
45
0.859
0.511
1.719
1.023
2.578
1.534
3.438
2.045
4.297
2.556
16
81
0.857
0.515
1.714
1.030
2.572
1.545
3.429
2.060
4.286
2.575
59
15
0.855
0.519
1.710
1.038
2.565
1.556
3.420
2.075
4.275
2.594
46
30
0.853
0.522
1.705
1.045
2.558
1.567
3.411
2.090
4.263
2.612
30
45
0.850
0.526
1.701
1.052
2.551
1.579
3.401
2.105
4.252
2.631
16
82
0.848
0.530
1.696
1.060
2.544
1.590
3.392
2.120
4.240
2.650
58
15
0.846
0.534
1.691
1.067
2.537
1.601
3383
2.134
4.229
2.668
45
30
0.843
0.537
1.687
1.075
2.530
1.612
3374
2.149
4.217
2.686
30
45
0.841
0.541
1.682
1.082
2.523
1.623
3364
2.164
4.205
2.705
16
88
0.839
0.545
1.677
1.089
2.516
1.634
3.355
2.179
4.193
2.723
57
15
0.836
0.548
1.673
1.097
2.509
1.645
3345
2.193
4.181
2.741
46
30
0.834
0.552
1.668
1.104
2.502
1.656
3336
2.208
4.169
2.760
30
45
0.831
0.556
1.663
1.111
2.494
1.667
3326
2.222
4.157
2.778
15
84
0.829
0.559
1.658
1.118
2.487
1.678
3316
2.237
4.145
2.796
56
15
0.827
0.563
1.653
1.126
2.480
1.688
3306
2.251
4.133
2.814
45
30
0.824
0.566
1.648
1.133
2.472
1.699
3.297
2.266
4.121
2.832
30
46
0.822
0.570
1.643
1.140
2.465
1.710
3.287
2.280
4.108
2.850
15
85
0.819
0.574
1.638
1.147
2.457
1.721
3.277
2.294
4.096
2.868
55
15
0.817
0.577
1.633
1.154
2.450
1.731
3.267
2309
4.083
2.886
45
30
0.814
0.581
1.628
1.161
2.442
1.742
3.257
2.323
4.071
2.904
30
45
0.812
0.584
1.623
1.168
2.435
1.753
3.246
2337
4.058
2.921
16
86
0.809
0.588
1.618
1.176
2.427
1.763
3.236
2.351
4.045
2.939
54
15
0.806
0.591
1.613
1.183
2.419
1.774
3.226
2365
4.032
2.957
45
30
0.804
0.595
1.608
1.190
2.412
1.784
3.215
2379
4.019
2.974
30
46
0.801
0.598
1.603
1.197
2.404
1.795
3.205
2393
4.006
2.992
15
87
0.799
0.602
1.597
1.204
2.396
1.805
3.195
2.407
3.993
3.009
53
15
0.796
0.605
1.592
1.211
2.388
1.816
3.184
2.421
3.980
3.026
45
SO
0.793
0.609
1.587
1.218
2.380
1.826
3.173
2.435
3.967
3.044
30
45
0.791
0.612
1.581
1.224
2.372
1.837
3.163
2.449
3.953
3.061
15
88
0.788
0.616
1.576
1.231
2.364
1.847
3.152
2.463
3.940
3.078
52
16
0.785
0.619
1.571
1.238
2.356
1.857
3.141
2.476
3.927
3.095
45
30
0.783
0.623
1.565
1.245
2.348
1.868
3.130
2.490
3.913
3.113
30
45
0.780
0.626
1.560
1.252
2.340
1.878
3.120
2.504
3.899
3.130
15
89
0.777
0.629
1.554
1.259
2.331
1.888
3.109
2.517
3.886
3.147
51
15
0.774
0.633
1.549
1.265
2.323
1.898
3.098
2.531
3.872
3.164
46
30
0.772
0.636
1.543
1.272
2.315
1.908
3.086
2.544
3.858
3.180
30
45
0.769
0.639
1.538
1.279
2307
1.918
3.075
2.558
3.844
3.197
15
40
0.766
0.643
1.-532
1.286
2.298
1.928
3.064
2.571
3.830
3.214
50
16
0.763
0.646
1.526
1.292
2.290
1.938
3.053
2.584
3.816
3.231
46
30
0.760
0.649
1.521
1.299
2.281
1.948
3.042
2.598
3.802
3.247
30
45
0.758
0.653
1.515
1.306
2.273
1.958
3.030
2.611
3.788
3.264
15
41
0.755
0.656
1.509
1.312
2.264
1.968
3.019
2.624
3.774
3.280
49
15
0.752
0.659
1.504
1.319
2.256
1.978
3.007
2.637
3.759
3.297
45
30
0.749
0.663
1.498
1.325
2.247
1.988
2.996
2.650
3.745
3313
30
45
0.746
0.666
1.492
1.332
2.238
1.998
2.984
2.664
3.730
3.329
15
42
0.743
0.669
1.486
1.338
2.229
2.007
2.973
2.677
3.716
3346
48
15
0.740
0.672
1.480
1.345
2.221
2.017
2.961
2.689
3.701
3.362
46
30
0.737
0.676
1.475
1.351
2.212
2.027
2.949
2.702
3.686
3.378
30
45
0.734
0.679
1.469
1.358
2.203
2.036
2.937
2.715
3.672
3.394
15
48
0.731
0.682
1.463
1364
2.194
2.046
2.925
2.728
3.657
3.410
47
15
0.728
0.685
1.457
1.370
2.185
2.056
2.913
2.741
3.642
3.426
45
30
0.725
0.688
1.451
1.377
2.176
2.065
2.901
2.753
3.627
3.442
30
46
0.722
0.692
1.445
1.383
2.167
2.075
2.889
2.766
3.612
3.458
15
44
0.719
0.695
1.439
1.389
2.158
2.084
2.877
2.779
3.597
3.473
46
15
0.716
0.698
1.433
1396
2.149
2.093
2.865
2.791
3.582
3.489
45
30
0.713
0.701
1.427
1.402
2.140
2.103
2.853
2.804
3.566
3.505
30
45
0.710
0.704
1.420
1.408
2.131
2.112
2.841
2.816
3.551
3.520
16
45
0.707
0.707
1.414
1.414
2.121
2.121
2.828
2.828
3.536
3.536
45
o t
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
O f
Bearing.
Distance 1.
Distance 2.
Distance 3.
Distance 4*
Distance 5.
Bearing.
46°- 60 c
30°-
-45
O
61
Bearing.
Distance 6.
Distance 7.
Distance 8.
Distance 9.
Distance 10.
Bearing.
o t
Lat.
Bep.
Lat.
Bep.
Lat.
Bep.
Lat.
Bep.
Lat.
Bep.
o f
8015
5.183
3.023
6.047
3.526
6.911
4.030
7.775
4.534
8.638
5.038
59 45
30
5.170
3.045
6.031
3.553
6.893
4.060
7.755
4.568
8.616
5.075
30
45
5.156
3.068
6.016
3.579
6.875
4.090
7.735
4.602
8.594
5.113
15
81
5.143
3.090
6.000
3.605
6.857
4.120
7.715
4.635
8.572
5.150
59
15
5.129
3.113
5.984
3.631
6.839
4.150
7.694
4.669
8.549
5.188
45
30
5.116
3.135
5.968
3.657
6.821
4.180
7.674
4.702
8.526
5.225
30
45
5.102
3.157
5.952
3.683
6.803
4.210
7.653
4.736
8.504
5.262
15
82
5.088
3.180
5.936
3.709
6.784
4.239
7.632
4.769
8.481
5.299
58
15
5.074
3.202
5.920
3.735
6.766
4.269
7.612
4.802
8.457
5336
46
30
5.060
3.224
5.904
3.761
6.747
4.298
7.591
4.836
8.434
5373
30
45
5.046
3.246
5.887
3.787
6.728
4328
7.569
4.869
8.410
5.410
15
88
5.032
3.268
5.871
3.812
6.709
4.357
7.548
4.902
8.387
5.446
57
15
5.018
3.290
5.854
3.838
6.690
4386
7.527
4.935
8.363
5.483
46
30
5.003
3312
5.837
3.864
6.671
4.416
7.505
4.%7
8.339
5.519
80
45
4.989
3.333
5.820
3.889
6.652
4.445
7.483
5.000
8.315
5.556
16
84
4.974
3.355
5.803
3.914
6.632
4.474
7.461
5.033
8.290
5.592
56
15
4.960
3377
5.786
3.940
6.613
4.502
7.439
5.065
8.266
5.628
46
30
4.945
3.398
5.769
3.%5
6.593
4.531
7.417
5.098
8.241
5.664
80
46
4.930
3.420
5.752
3.990
6.573
4.560
7395
5.130
8.217
5.700
16
85
4.915
3.441
5.734
4.015
6.553
4.589
7.372
5.162
8.192
5.736
55
15
4.900
3.463
5.716
4.040
6.533
4.617
7.350
5.194
8.166
5.772
45
30
4.885
3.484
5.699
4.065
6.513
4.646
7.327
5.226
8.141
5.807
80
45
4.869
3.505
5.681
4.090
6.493
4.674
7.304
5.258
8.116
5.843
16
ae o
4.854
3.527
5.663
4.115
6472
4.702
7.281
5.290
8.090
5.878
54
16
4.839
3.548
5.645
4.139
6.452
4.730
7.258
5.322
8.064
5.913
45
80
4.823
3.569
5.627
4.164
6.431
4.759
7.235
5353
8.039
5.948
80
46
4.808
3.590
5.609
4.188
6.410
4.787
7.211
5385
8.013
5.983
15
87
4.792
3.611
5.590
4.213
6.389
4.815
7.188
5.416
7.986
6.018
58
15
4.776
3.632
5.572
4.237
6.368
4.842
7.164
5.448
7.960
6.053
46
30
4.760
3.653
5.554
4.261
6347
4.870
7.140
5.479
7.934
6.088
80
45
4.744
3.673
5.535
4.286
6.326-
4.898
7.116
5.510
7.907
6.122
15
88
4.728
3.694
5.516
4310
6.304
4.925
7.092
5.541
7.880
6.157
52
16
4.712
3.715
5.497
4334
6.283
4.953
7.068
5.572
7.853
6.191
46
30
4.6%
3.735
5.478
4.358
6.261
4.980
7.043
5.603
7.826
6.225
80
45
4.679
3.756
5.459
4.381
6.239
5.007
7.019
5.633
7.799
6.259
16
89
4.663
3.776
5.440
4.405
6.217
5.035
6.994
5.664
7.772
6.293
51
15
4.646
3.7%
5.421
4.429
6.195
5.062
6.970
5.694
7.744
6.327
46
80
4.630
3.816
5.401
4.453
6.173
5.089
6.945
5.725
7.716
6361
30
45
4.613
3.837
5382
4.476
6.151
5.116
6.920
5.755
7.688
6.394
15
40
4.5%
3.857
5.362
4.500
6.128
5.142
6.894
5.785
7.660
6.428
50
15
4.579
3.877
5343
4.523
6.106
5.169
6.869
5.815
7.632
6.461
45
30
4.562
3.897
5323
4.546
6.083
5.1%
6.844
5.845
7.604
6.495
30
45
4.545
3.917
5303
4.569
6.061
5.222
6.818
5.875
7.576
6.528
15
41
4.528
3.936
5.283
4.592
6.038
5.248
6.792
5.905
7.547
6.561
49
15
4.511
3.956
5.263
4.615
6.015
5.275
6.767
5.934
7.518
6.594
45
80
4.494
3.976
5.243
4.638
5.992
5301
6.741
5.964
7.490
6.626
30
45
4.476
3.995
5.222
4.661
5.968
5327
6.715
5.993
7.461
6.659
15
42
4.459
4.015
5.202
4.684
5.945
5353
6.688
6.022
7.431
6.691
48
16
4.441
4.034
5.182
4.707
5.922
5379
6.662
6.051
7.402
6.724
46
80
4.424
4.054
5.161
4.729
5.898
5.405
6.635
6.080
7.373
6.756
30
46
4.406
4.073
5.140
4.752
5.875
5.430
6.609
6.109
7.343
6.788
15
48
4.388
4.092
5.119
4.774
5.851
5.456
6.582
6.138
7.314
6.820
47
16
4370
4.111
5.099
4.7%
5.827
5.481
6.555
6.167
7.284
6.852
46
30
4352
4.130
5.078
4.818
5.803
5.507
6.528
6.195
7.254
6.884
30
46
4334
4.149
5.057
4.841
5.779
5.532
6.501
6.224
7.224
6.915
15
44
4316
4.168
5.035
4.863
5.755
5.557
6.474
6.252
7.193
6.947
46
15
4.298
4.187
5.014
4.885
5.730
5.582
6.447
6.280
7.163
6.978
45
80
4.280
4.206
4.993
4.906
5.706
5.607
6.419
6.308
7.133
7.009
30
46
4.261
4.224
4.971
4.928
5.681
5.632
6.392
6.336
7.102
7.040
15
45
4.243
4.243
4.950
4.950
5.657
5.657
6.364
6.364
7.071
7.071
45
o f
Dep.
Lat.
Bep.
Lat.
Bep.
Lat.
Bep.
Lat.
Bep.
Lat.
o f
Bearing.
Distance 6.
Distance 7.
Distance 8.
Distance 9.
Distance 10.
Bearing.
45°- 60 c
WENTWORTH'S
MATHEMATICS.
In four years the Algebra has been adopted by
157 Colleges
and 932 High Schools and Academies.
In eight years the Geometry has been adopted by
243 Colleges
and 1460 High Schools and Academies.
These are leading institutions. A few of
the representative Public Schools are :
Maine : Auburn, Augusta, Bangor, 'Bath, Belfast, Calais, Eastport,
Ellsworth, Gardiner, Lewiston, Portland, Saco, Skowhegan.
. New Hampshire : Claremont, Concord, Dover, Exeter, Keene,
Manchester.
Vermont : Brattleboro', Burlington, Montpelier, Rutland, Spring-
field, St. Albans, Vergennes.
Massachusetts : Attleboro', Gloucester, Haverhill, Holyoke, Law-
rence, Lowell, Maiden, Natick, New Bedford, Newburyport,
Newton, Salem, Taunton, Waltham, Worcester.
Rhode Island : Pawtucket, Providence.
Connecticut : Bridgeport, Danbury, Hartford, Meriden, Middle-
town, New Britain, New London, Waterbury, Willimantic.
New York : Albany, Auburn, Binghamton, Brooklyn, Cohoes -
Dunkirk, Flushing, Hornellsville, Ithaca, Jamestown, Kings-
ton, Newburg, New York, Ogdensburg, Oswego, Poughkeepsie,
Rome, Saratoga, Syracuse, Troy, Utica, Watertown.
New Jersey : Jersey City, Hoboken, Montclair, Orange, Paterson,
Somerville.
Pennsylvania: Ashland, Bradford, Columbia, Danville, Erie,
Franklin, Johnstown, New Castle, Oil City, Philadelphia,
Pittsburg, Pottsville, Heading, Scranton, Warren, West
Pittston.
District of Colombia: Washington.
Virginia: Gordonsville, Lexington, Norfolk, Petersburg, Ports-
mouth, Richmond, Staunton.
North Carolina : Durham, Fayetteville, New Berne.
Georgia : Atlanta, Augusta, Columbus, Rome.
Alabama : Birmingham, Eufaula, Mobile, Tuscaloosa.
Texas : Brenham, Dallas, Galveston, San Antonio.
Ohio : Akron, Alliance, Bellaire, Canton, Cleveland, Gallon, Lan-
caster, Lima, Marietta, Mount Vernon, Oberlin, Sidney, Troy,
Wooster, Xenia.
Indiana: Greencastle, Indianapolis, Logansport, New Albany,
Richmond, South Bend, Vincennes.
Illinois : Aurora, Bloomington, Cairo, Chicago, Danville, Decatur,
Englewood, Galesburg, Hyde Park, Jacksonville, Knoxville,
Lincoln, Rock Island, Springfield.
Tennessee : Knoxville, Memphis, Nashville, Shelbyyille.
Kentucky : Hopkinsville, Louisville, Owensboro', Princeton.
Michigan:- Adrian, Bay City, St. Clair, Ypsilanti.
"Wisconsin: Eau Claire, Fond du Lac, Green Bay, Madison,
Oshkosh, Racine.
Minnesota : Duluth, Faribault, Mankato, Minneapolis, Red Wing,
Rochester, St. Paul, Stillwater, Winona.
Missouri : Kansas City, St. Joseph, Trenton.
Iowa : Creston, Burlington, Cedar Rapids, Clinton, Davenport,
Des Moines, Dubuque, Grinnell, Iowa City, Keokuk, Ottumwa.
Kansas : Concordia, Lawrence, Olathe, Winneld.
Nebraska : Grand Island, Niobrara, Omaha, Seward.
Dakota : Fargo, Grand Forks, Scotland.
California : Los Angeles, Oakland, Sacramento, San Francisco.
FACTS LIKE THESE
seem to be a practical endorsement of Wentworth's
Series of Mathematics, and to afford a
STRONG PRESUMPTION
in favor of two new books, now ready, which have
been prepared to meet the wants of the best public
and private schools :
^ABeed's PRIMARY ARITHMETIC, T^rtEdh^So'ota.
*+mm GRAMMA R SCHOOL A RITHMETIC, »*■
Descriptive Circulars on application. Copies for examina-
tion on receipt of Introduction price. Favorable terms for
introduction. Correspondence Invited.
CINN & COMPANY, Publishers,
Boston, New York, and Chicago.
The Teachers' Edition
OF
FIRST STEPS IN NUMBER
Part I. 1st Year. (Nos. 1 to 9 inclusive), 216 pp., boards, 30 cts.
Part II. 2d Year. (Nos. 10 to 20 inclusive), 116 pp., boards, 30 cts.
Part III. 3d Year. (Nos. from 21), 156 pp., boards, 30 cts.
Complete, 480 pp., cloth, 90 cts.
The Teachers' Edition takes up the work with the
number three, and proceeds step by step, following the law
of dependence and simplicity. Each of the smaller numbers
is presented in succession under the four heads : —
1. Perception of the number.
2. Analysis of the number.
3. Drill upon facts discovered by analysis.
4. Comparison with smaller numbers.
The book abounds in excellent little problems about articles
children buy ; about things they see and do ; about facts in
nature, as the number of toes a cat has, the number of wings
a butterfly has, the number of legs a fly has ; about numbers
applied arbitrarily, as the days in a week, the things in a
dozen, the things in a score, the sheets in a quire, gills in a
pint, quarts in a gallon, in a peck, in a bushel, inches in a foot,
feet in a yard ; — problems exactly suited to children, and de-
signed to make them think, to cultivate the reasoning faculty,
to awaken interest, to impress facts, and to put knowledge in
a form for use. The office of the examples is not simply to
FIBST STEPS IN NLMBER. 13
test for facts, but also to give the very best drill in the correct
understanding and use of simple language. The questions are
not puzzling, and, besides, directions are given the teachers to
illustrate with objects every question that the child does not
readily understand.
Notation and numeration are taught step by step, as occa-
sion requires ; and the principles are acquired gradually, with-
out special effort on the part of the pupil.
No other subject gives so many opportunities for mental
activity to children just beginning school life as simple num-
ber-work. In no other school work is it possible to lead them
to do, to talk, to think, as in simple questions of arithmetic.
Every lesson makes a special demand upon their powers of
observation and of quick response. Every lesson, rightly con-
ducted, gives them the keenest pleasure, and awakens the
greatest pride in success. With the aid of the little problems
and devices given in the book, it is hoped that teachers of
ordinary tact and interest in their work will change the gen-
erally dull and wearisome number- tasks into bright and
inspiring lessons.
The last chapters of the book relate to work that does not
properly belong to the primary grades; but inasmuch as a
great many pupils do not advance beyond the primary schools,
it is expected that teachers will make an effort to give such
pupils sufficient instruction in Percentage and Interest to
enable them to apply the principles of these subjects to the
affairs of e very-day life.
The chapter on the number Eight is here given to exhibit
the method of treating each of the smaller numbers.
The Pupils' Edition
* OF
FIRST STEPS IN NUMBER
160 pages, boards.
Introduction price 30 cts.
Allowance for old book ... 12 cts.
No text-book is calculated to be of much help to the child
during the early part of his school life. He needs none in
recitation, and the work which he does by himself is repre-
sented by figures, so that a book which contains a great deal
of number-work expressed in figures, and in the last part
some problems with blanks for figures, — which figures he is
to supply and then solve the problems, — is the only kind of
a book suitable for his use. The Pupils' Edition is just such
a book. It contains a great many number-lessons expressed
in figures. References to pages in the Teachers' Edition are
given at the beginning of each lesson, so that the teacher may
see at a glance at what stage of the children's progress the
lesson-work should be required of her pupils. It will be
observed that the Pupils' Edition is not designed to be taken
up until the first nine numbers have been mastered. The
Pupils' Edition is indispensable to the teacher in the matter
of saving time, and indispensable to the pupil for systematic
practice work. A few specimen lessons are here appended.
WENTWORTH'S
Grammar School Arithmetic!
Introduction price 75 cts.
Allowance for old book . . . 30 cts.
The Grammar School Arithmetic is intended to follow
the Primary Arithmetic, making with that a two-book series
for Common Schools. It is designed to give pupils of the
grammar-school age an intelligent knowledge of the subject
and a moderate power of independent thought.
Whether Arithmetic is studied for mental discipline or for
practical mastery over the every-day problems of common
life, mechanical processes and routine methods are of no
value.
Pupils can be trained to logical habits of mind and stim-
ulated to a high degree of intellectual energy by solving
problems adapted to their capacities. They become practical
arithmeticians, not by learning special business forms, but
by founding their knowledge on reasoning which they fully
comprehend, and by being so thoroughly exercised in logical
analysis that they are independent of arbitrary rules.
This Arithmetic contains a great number of oral and
written problems, well-graded and progressive, made up for
youths from ten to fourteen years of age. Definitions and
explanations are made as brief and simple as possible. It is
16 GRAMMAR SCHOOL ARITHMETIC.
not intended that definitions shall be committed to memory,
but that they shall be simply discussed by teacher and pupils.
Every teacher, of course, will be at liberty to give better
definitions, and to make a better presentation of methods,
than those exhibited in the book.
In short, the chief object in view will be gained if pupils
are trained to solve the problems by neat and intelligent
methods, and are kept free from set rules and formulas.
A great many number-problems are given in the first
pages of the book, so that the necessary facility and accu-
racy in computing under the four fundamental rules may
be acquired, as want of accuracy and rapidity in mere cal-
culations distracts the attention which should be given to the
investigation and correct statement of clothed exercises.
The last three chapters are a short chapter on the Metric
System, a chapter on Mensuration, and a chapter of Miscel-
laneous Problems. The Metric System is treated here be-
cause the great majority of grammar school pupils have no
time for the subject, while those who have can as well learn
the system at this stage of their progress as earlier. The
chapter on Mensuration is suited to the ability of beginners.
The intention is not to give a system of Geometry, but to
render familiar those notions of Geometry that are indispen-
sable for practical purposes. This chapter has been illus-
trated and enforced by numerous practical examples.
Wentworth's Mathematical Series.
Albro B. Chase, Prin. of High School, Portland, Me. : The Algebra suc-
ceeds better than any work I have tried. I like the Geometry equally well.
A. W. Bachelor, Prin. of High School, Manchester, N.IL: Went-
worth's Algebra has been our regular class-book, and we have never found
its equal. We esteem the Geometry as highly as the Algebra.
C. H. Pease, Teacher of Math., Burlington ( Vt.) High School: I think
your Algebra and Geometry the best I ever saw.
A. L. Hardy, Instructor in St. Johnsbury ( Vt.) Acad. : The more I
use the Geometry, and the more I compare it with others, the better I like it.
C. P. P. Bancroft, Prin. Phillips Acad., Andover, Mass. : The Geome-
try seems to me very clear and compact.
Frank O. Carpenter, Junior Master in English High School, Boston,
Mass. : Your Geometry is clear, concise, and admirably arranged.
D. B. Hagrar, Prin. Salem {Mass.) Normal School : The longer we use
the Geometry, the better we like it.
Arthur A. Brooks, Worcester (Mass.) High School: I consider it by
far the best text-book in Geometry with which I am acquainted.
W. H. H. Phillips, Prof, of Math., Wesleyan Acad., Wilbraham, Mass. :
. I have used the Algebra, and am very much pleased with it.
Prof. N. F. Davis, Brown Univ., Providence, R.I.: For beginners the
Geometry is the clearest and best with which I am acquainted.
Alexander Bevan, Teacher of Math., Providence (R.I.) High School :
I know of nothing better than the oest. Westworth's is the best.
B. H. Wilson, Prin. Middletown (Conn.) High School : The Algebra and
Geometry give entire satisfaction.
John D. Whitney, Prof, of Math., St. Francis Xavier Coll., New York
City : I consider Wentworth's Algebra and Geometry as undoubtedly the
best text-books published in this country on these subjects.
Charles W. Cole, Supt. Schools, Albany, N. Y. : Wentworth's Algebra
and Geometry, now in use in the High School, have contributed to the best
results ever attained in the history of the school.
H. F. Towle, Prof of Math, in Brooklyn (N. Y.) Central School :
Wentworth's Geometry has no equal among the text-books of the day.
George H. Barton, Supt. of Schools, Jersey City, N. J. : I consider
them superior to any I have ever used or examined.
H. E. Dawson, Prof of Math., Newark (N.J.) High School: I consider
Wentworth's series superior to all.
Selden J. Coffin, Prof, of Math, and Astron., Lafayette College, Pa. :
The Geometry has been used with increasing satisfaction and success. The
Algebra is a fine book, well prepared, and fulfils the aim of the author.
D. M. Sensenigr, Prof, of Math., State Normal School, West Chester,
Pa. : It has my heartiest approval in all its characteristic features.
W. W. Birdsall, Prin. Boys* High School, Wilmington, Del.: The
Algebra and Geometry are the best books on the subject for high-school use.
Miss M. D. Browne, Math. Teacher, High School, Washington, D.C. :
We have found Algebra and Geometry both admirably adapted to our work.
Wm. A. Obenchain, Pres., and Prof of Math., Ogden Coll., Bowling
Green, Ky.: The best text-books on these subjects I have ever found.
U WENT WORTH'S MATHEMATICAL SERIES.
Jas. Dinwiddle, Prof, of Math., Univ. of Tenn., Knoxville: Went-
worth's Geometry is the best book for class-room work that I know.
Wentworth's Algebra is also a most excellent text-book.
D. B. Johnson, Supt. of Schools, Columbia, S.C.: We consider the
Algebra and Geometry the best publications of their kind.
S. P. Sanford, Prof, of Math., Mercer Univ., Macon, Ga. : A pupil
who could not learn Geometry from Wentworth could not learn it at all.
T. W. Palmer, Prof, of Math., Univ. of Alabama, Tuscaloosa, Ala.: I
have never seen text-books so clear, concise, and every way desirable.
C. H. Churchill, Prof, of Math., Oberlin Coll., 0. : We are more and
more pleased with the Geometry.
Geo. W. Smith, Teacher of Math., Woodward High School, Cin., O. :
I consider the Algebra by far the best yet written by any American author.
B. L. Harris, Prof, of Math.. Central High School, Cleveland, 0. :
Wentworth's Geometry carries with itself its own recommendation. Went-
worth's Algebra is the best work for a high school that I have ever used.
M. B. Bogarte, Teacher of Math., Normal School, Valparaiso, Ind. :
I hardly see how Wentworth's Geometry could be improved.
F. R. Feitshans, Supt. Sch., Springfield, III. : We think the Algebra is
the best book on the subject that has yet appeared in this country.
R. M. Putnam, Supt. of Schools, Ypsilanti, Mich. : I have used Went-
worth's Practical Arithmetic with one of my own classes, and am much
pleased with results. It grows in favor with us every day.
Wm. H. Beach, Prin. Madison, Wis., High School: For systematic
arrangement, for clearness and accuracy of statement, and for neatness
and finish of mechanical work, I consider them unexcelled.
B. F. Wright, Supt. Sch., St. Paul., Minn. : The Algebra and Geometry
seem better suited to our work than any others I have seen.
W. M. West, Supt. of Schools, Faribault, Minn. : I believe Went-
worth's Geometry superior to any other book published on the subject.
Hiram L. Peet, Prin. of High School, Dubuque, la. : The Algebra,
Geometry, and Trigonometry are alike excellent.
Joseph Ficklin, Prof, of Math., Univ. of Missouri, Columbia: Profes-
sor Wentworth is evidently a practical teacher.
R. R. Steele, Supt. of Kirksville {Mo.) Public Schools: It is the best
Algebra it has been my fortune to meet.
D. C. Tillotson. Supt. of Schools, Topeka, Kan. : I am satisfied that
Wentworth's Plane Geometry is superior to any work of the kind published.
E. S. Lewis, Prof, of Math., Little Bock (Ark.) Univ.: Wentworth's
Algebra and Geometry embody the best principles of modern teaching.
H. P. Lewis. Prin. High School, Omaha, Neb. : The teacher of algebra
reports better success than with any other text which she has used.
C. J. A. Porter, Sackett School, Oakland, Cal. : I consider Wentworth's
mathematical works as the very best for general educational purposes.
A circular containing full critical endorsements from 518
leading teachers sent postjmid on application.
GINN & COMPANY, Publishers.
GlNN & COMPANY'S BOOKS. 1
ELEMENTARY ENGLISH : Classics for Children: Water-Babies,
Robinson Crusoe, Swiss Family Robinson, Greek Heroes, Tales
from Shakespeare, Lady of the Lake, Tales of a Grandfather,
Quentin Durward, Sketch Book, Merchant of Venice, Stories of
the Old World; Stickney's Primer; Turner's Primer and First
Reader, Stories ; Whitney & Knox Language Series: Elementary
Lessons (Pupils 1 edit., Teachers' edit.), Grammar; Hazen's Speller.
HIGHER ENGLISH: Arnold's Literature; Bancroft's Composi-
tion; Fulton & Trueblood's Choice Readings; Hudson's Har-
vard and School Shakespeares ; Sprague's Irving, Milton.
ANGLO-SAXON : Carpenter's Grammar, English of XIV. Century ;
Sievers' Grammar ; Beowulf; Exodus & Daniel; Be6wulf (Trans.).
LATIN : AUen & Greenough's Grammar, Caesar, Sallust, Cicero,
Ovid; Allen's New Method, Composition, Tacitus; Blackburn's
Grammar and Exercises; Greenough's Virgil; Leighton's Les-
sons; Shumway's Synonymes; Tetlow's Lessons; White's Lexi-
cons ; Whiton's Six Weeks' Preparation ; Classical Atlas.
GREEK: College Series of Greek Authors; Flagg's Demos-
thenes, Seven against Thebes; Goodwin's Grammar, Reader,
Moods and Tenses, Xenophon and Herodotus ; Goodwin & White's
Anabasis; Leighton's Lessons; Liddell & Scott's Lexicons;
White's Lessons, (EdipusTyranuus; Lysias; Pindar; Lyric Poets.
MATHEMATICS: Byerly'8 Calculus ; Hill's Geometry for Begin-
ners; Taylor's Calculus; Wentworth's Arithmetic, Algebras,
Geometries, Trigonometries, Surveying, Navigation ; Wentworth
& Hill's Practical Arithmetic, Exercises in Arithmetic, Algebra,
and Geometry; Wentworth & Reed's First Steps in Number
( Pupils' edit., Teachers' edit.) ; Wheeler, Trig, and Tables.
MUSIC: Mason's First, Second, Third, Intermediate, and Indepen-
dent Headers, Charts, Hymn and Tune Books; Sharland's Fourth
Headers ; Eichberg's High School Readers.
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& Heath's Classical Atlas ; Hall's Our World, No. 1 and No. 2 ;
Johnston's Wall Maps ; Joslin's Globes.
SCIENCE : Gage's Physics.
PHILOSOPHY: Ladd's Lotze's Outlines; Seelye's Hickok's Moral
and Mental Science.
MODERN LANGUAGES: Knapp's Spanish Grammar and Read-
ings, French Readings.
SANSKRIT : Perry's Primer ; Lanman's Reader ; Whitney's
Grammar.
AND MANY OTHERS.
FULL DESCRIPTIVE CATALOGUE FREE.
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