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f 




TABLE BOOK AND TEST PROBLEMS 



IN 



MATHEMATICS 



s BY 



J. K: ELLWOOD, A.M. 

Principal of thb Ck>LFAx School, Pittsburgh, Psnn. 



-»o2o<o*- 






NEW YORK • : • CINCINNATI • : • CHICAGO 

AMERICAN BOOK COMPANY 



COPTBIOHT, 1802, 

By AMERIOAN BOOK COMPANY, 



f 

% 



printed bie 

TRnillfam fvtoon 

Hew l^tft, "QU 0. &• 



6u^ 



-. CONTENTS. 

\ 

T 

Part Paos 

^ I. THEOREMS, RULES, AND FORMULAS 9 

Table of Logarithms of Niunbers from 1 to 10,000 ... 27 
^, Table of Logarithmic Sines, Cosines, Tangents, and Co- 

tangents . . . . ^ 43 

Table of Natural Sines, Cosines, Tangents, and Cotangents 89 

n. TEST PROBLEMS Ill 

ArithmeticaX Problems Ill 

Denominate Numbers .• . . Ill 

Least Common Multiple and Greatest Common Divisor . . Ill 

Partnership. . 112 

Proportion 112 

Profit and Loss 113 

Stocks and Bonds < 116 

Interest 116 

Discount and Present Worth 117 

Involution and Evolution 117 

Alligation 118 

Annuities 118 

"Age" Problems 118 

« Time " Problems 119 

General Analysis 120 

Mensuration 122 

The Rectangle . 122 

The Triangle 123 

The Circle 126 

Pyramids and Cones 126 

3 



4 CONTENTS. 

Pabt Pagb 

Similar Solids 126 

Cubes and Spheres . . . .  127 

Miscellaneous Problems 127 

Algebraic Problems 130 

Factoring 130 

Fractions 130 

Simple Equations 131 

Radicals 133 

Quadratic Equations 133 

Special Expedients 137 

Simultaneous Equations 137 

Reciprocal or Recurring Equations 139 

Higher Equations 140 

Miscellaneous Problems 144 

Applications of Algebra 144 

Geometrical Problems, etc 146 

Trigonometrical Problems 148 

Problems involving Calculus 149 

Promiscuous Problems 149 

Problems with Curious Besults 151 

Digits 151 

** One Cent'* .161 

Involution of Imaginary Quantities 151 

The Zero Factor 161 

Something to investigate 151 

The Proposition of Archimedes 152 

1888 152 

Summation by Subtraction 162 

Series 163 

III. SOLUTIONS 156 



PREFACE. 



In nearly every class in mathematics there is to be found a larger 
or smaller number of pupils who early develop more than that 
average proficiency for which the problems in the regular text-books 
are graded and adapted. These problems do not afford sufficiently 
stimulating exercise to such pupils, whose development is often 
arrested or retarded in consequence. 

ThLs volume of " Test Problems " has been prepared for the use 
of such apt or advanced pupils, and for the convenience of teachers 
in examining advanced classes. It will also afford valuable supple- 
mental work in every school. It contains a collection of rather 
difficult problems in the various branches of elementary mathematics. 
While none of the problems involve higher mathematics, their solu- 
tion requires close reasoning and a thorough knowledge of ele- 
mentary principles. It is believed they will afford the drill needed 
by advanced classes or by pupils of rather more than average apti- 
tude in mathematics. 

The problems have been gathered from many sources. A few of 
them may occur in the regular text-books ; many have appeared in 
the columns of mathematical or educational journals; while still 
others have been supplied to the author by mathematicians, or are 
original with himself. 

For convenient reference, there has been embodied in Part I. a 
collection of rules of mensuration, important theorems, trigonomet- 
rical formulas, and tables of logarithms and of natural sines, cosines, 
tangents, etc., which it is believed will commend itself to all. In Part 
II. is embraced the statement of the test problems, which have been 
classified as closely as seemed possible. In Part III. are given the 
solutions of all problems stated in Part II. 

The solutions are not all original. Most of those that are not, 
however, are credited to their authors ; but some of them have been 
picked up " by the wayside," and their authors cannot be given. In 



6 PREFACE, 

nearly all solutions the aim has been to make every step clear rather 
than to present a brief operation ; and it is believed that any student 
with a fair knowledge of any of the branches of mathematics under 
which a given problem falls will be able to follow its solution intelli- 
gently and easily. The arithmetical and algebraic solutions are 
designed especially to aid the pupils and teachei*s of our public 
schools; while the chapters on ** Special Expedients'' and "Miscel- 
laneous Solutions " contain much that may be studied with profit by 
more advanced scholars, teachers, superintendents, etc., as they include 
solutions by some of the best mathematicians of the country. In 
solving equations, it has not been deemed necessary to give the values 
of all the unknown quantities ; yet in some instances they have been 
given, and in the others they are readily obtainable. 

The classification of the solutions is necessarily imperfect, owing to 
their promiscuous character, and to the fact that frequently two or 
more principles are almost equally involved in the solution of a 
problem, on account of which the solution might as legitimately be 
placed in another class as in the one to which it has been assigned. 
Especially is it difficult to classify " Special Expedients," or ** Artifices," 
as they are restricted and special, and as in their use the student must 
depend solely upon his ingenuity. Owing to these difficulties, the 
classification throughout has been based upon the topic or head under 
which a solution chiefly falls, or, in other words, upon the leading 
principle or operation involved. 

For fine solutions received, I desire to express my thanks to Dr. 
I. J. Wireback and Mr. L. B. Fillman of St. Petersburgh, Penn., and 
to Professor B. F. Burleson of Oneida Castle, N.Y., who has also 
rendered valuable assistance in reading proof, in removing imper- 
fections, and in supplying many of the problems in " Series," together 
with their admirable solutions. Special acknowledgment is due to 
Mr. Russell Hinman of the American Book Company, New York, to 
whom I am indebted for many excellent suggestions and uniform 
courtesy. 

tJ. JV« Jii* 



INDEX TO RULES AND THEOREMS, 



No. Page 
Beams. See Tlmbera. 

Basbel, cubic contents of ... 17 12 

Circle, area of 1 9 

area of a sector of 2 

area of a segment of .... 8 9 

cbord of an are and half are of 4 9 

circumference of 6 10 

and equivalent square .... 6 10 

Circles compared 7 10 

inscribed 01 20 

Cone. Bee Pyramid. 

Cosines, table of natural .... - 80 

table of logarithmic - 43 

Cotangents, table of natural . . - 09 

table of logarithmic > 43 

Cube, diagonal of 8 10 

Cubes, inscribed 10 

Cycloid, area of 10 11 

Cycloidal curve, length of ... 11 12 
Cylinder. See Prism. 
Difference and sum of two quan- 
tities, relaUons of 17 13 

Earth, radius of 17 13 

Ellipse, area of 12 12 

circumference of ...... 13 12 

Ellipsoid, surface of 14 12 

Oallon, cubical contents of. . . 17 12 

Helix, length of 15 12 

Kilogram, value of French, in 

pounds 17 12 

Liter, value of French, in cu. in. 17 12 

Logarithms, table of .... . - 27 

Lune, area of . . 16 12 

Meter, value of French, in feet . 17 12 

Miscellaneous formulas .... 17 12 

trigonometrical formulas . . 73 24 

w, value of . 17 12 

value of Log 17 12 

Parabola, area of 18 14 

area of a segment of .... 19 14 



No. ftfs 

Paraboloid, volume of .... 20 14 
Pendulum, length of seeonds, at 
New York, London, and 

Paris 17 12 

Polygon, area of a regular ... 81 • 14 

surface of a spherical .... 22 14 

Prism, surface of 28 14 

volume of 24 14 

Prismoid, volume of 25 15 

Pyramid or cone, center of grav- 

ity of a trianguhir 26 15 

volume of 27 15 

volume of a frustum of ... 28 15 
Ring, length of axis of an ellipti- 
cal 29 15 

volume of a cylindrical ... 30 15 

Sines, table of natural .... - 89 

table of logarithmic .... - 48 

Sphere, surface of 81 15 

surface of a segment of ... 82 15 

volume of 83 15 

volume of a segment of ... 34 15 

Spheres, inscribed 9 10 

Spherical sector, volume of. . . 85 16 

Spheroid, surface of 86 16 

surface of a frustum of ... 87 16 

surface of a segment of ... 88 16 

volume of 89 16 

volume of a frustum of . . 40, 41 16 
volume of a segment of . . 42, 48 17 
Spindle, surface of a circular . . 44 17 
surface of a cycloidal .... 45 17 
volume of a circular .... 46 17 
volume of a segment of a cir- 
cular . 47 17 

volume of a zone of a circular . 48 18 

volume of a cycloidal .... 49 18 

Tolumeof an elliptical. ... 50 18 
volume of middle frustum of an 

eUipUcal 51 18 

7 



8 



INDEX TO RULES AND THEOREMS. 



No. Page 
Bpiodle {continued), 

volume of Begment of an ellip- 
tical N 62 18 

volume of a parabolic .... 53 18 
volume of middle firuatum of 

a parabolic . M 18 

volume of segment of a para- 
bolic 65 18 

volume of an hyperbolic ... 60 19 
volume of middle frustum of 

an hyperbolic -.67 19 

volume of segment of an hyper- 
bolic 68 19 

Spiral line, length of a plane . . 69 19 
Square, circle and equivalent . . 6 10 
equilateral triangle and equiv- 
alent 68 22 

Square of any number . . . . flO 19 
Squares, inscribed . . . . 61, 62 20, 21 
Sum and difference of two quan- 
tities, relations of 17 13 

Tangents, table of natural ... - 99 



No. Rife 
Tangents {continued). 

table of logarithmic - 43 

Timbers, strength of 68 21 

Trapezium, area of 64 22 

Trapezoid, area of 66 22 

center of gravity of 66 22 

Triangle, area of 67 22 

area of equilateral, equivalent 

to a given square 68 22 

center of gravity of ..... 09 22 

Triangles compared 70 22 

formulas for oblique .... 71 23 

formulas for right-angled. . . 72 28 
Trigonometrical formulas, mis- 
cellaneous 73 24 

Ungulas, curved snrfaoe and vol- 

umeof 74 26 

Water, weight of cubic foot of . 17 12 

Wedge, volume of 76 26 

Zone, area of a circular .... 76 26 

surface of a spherical .... 77 26 

volume of a spherical .... 78 26 



TABLE BOOK AND TEST PROBLEMS 



IK 



MATHEMATICS. 



>J©«o^ 



Part L 
theorems, rules, and formulas. 

1. Area of a Circle. — Multiply the square of the radius by 
v, or 3.1416, or use the formula ttt*. 

2. Area of a Sector of a Circle. — Multiply the arc by half 
the radius ; or use this proportion, Area of circle : area of sec- 
tor : : 360 : number of degrees in arc. 

3. Area of a Segment of a Circle. — Find the area of a sector 
having the same arc, and the area of the triangle formed by 
chord of segment and radii of sector. The difference of these 
is the area of a segment less, and their sum that of one greater, 
than a semicircle. Or {approximate rule when segment is less 
than semicircle) to two thirds of the product of height of seg- 
ment by chord, add cube of height divided by twice chord. 

• 4. The Chord of an Arc. — Let (7= the chord of an arc, 
c = the chord of half an arc, v sin == versed sine, d = diam- 
eter. Then 

\(Sc — C) = length of arc nearly. 

VC'^ + 4vsin^ X 10 V sin* , o ^ ^x. £ 
^ — J — -f 2 c = length of arc. 

9 



10 TABLE BOOK AND TEST PROBLEMS. 

2c X 10 V sin 



60d-27vsiii 



'■\-2c= length of arc. 



2V?--vsm^=C. 
V(P-(d-v8inx2)*=C. 
|((72 + 4vsm2)i = c. 
Vd X V sin = c, 
c* -h V sin = d, 
c* -5- d = V sin. 



^(d-Vd^-C^) = vsin. 
When V sin is greater than a radius, 



^(d+V<f'-C2) = vsin. 

6. Circumference of a Circle. — Multiply the product of the 
radius and 3.1416 by 2, or use the formula 2 tit. 

6. Circle and Equivalent Square. — Diameter of circle multi- 
plied by .8862 is equal to the side of an equal square. 

7. Circles are to each other as the squares on their radii. 

8. Oiven the Side of a Cube, t o find its Diagonal. — The 

diagonal of the side is V2 x side^ This is one leg, and the 
edge of the cube is another leg, of a right-angled triangle 
whose hypothenuse is the diagonal of the cube. Hence the 
cube's diagonal = V3 x side* = side x V3. Therefore, to find 
the diagonal of a cube from its side, multiply the side by VS. 

9. Inscribed Cubes and Spheres. — A cube is inscribed in a 
sphere, a sphere in this cube, a cube in this sphere, and so on. 
Find the ratio of the first sphere to the fifth. 

The diameter of the first sphere is readily seen to be the 
diagonal of the first cube, and the edge of the first cube to be 
the diameter of the second sphere, and so on. If a = the edge 
of the cube, then a' + a' + a' = 3a' = square of diagonal, and 
aV3 = diagonal. Hence, to find the edge of a cube, divide the 
diagonal by VS. 



THEOREMS, RULES, AND FORMULAS. 11 

Let B = radius of first sphere. Then 2i? = its diameter = 
diagonal of first cube. 2 i2 -f- V3 = — 7= = edge of first cube = 

^ 2R 

diameter of second sphere = diagonal of second cube. 5- 

2E ^ 

V3 = -— = edge of second cube = diameter of third sphere 

o 

= diagonal of third cube, and so on ad infinitum^ 

We now observe that the first three diameters are 2B, 
— —, -—-, the diagonals the same, and the edges — -, — ^, 

— — , each forming part of an infinite decreasing series, whose 
3V3 

ratio is — • 

Having given the diameter of the first sphere, we may find 
any diameter by the rule or formula for finding the last term 
of a geometrical progression. Let R = radius, 2R = diameter 
of the first sphere. Then, in the problem. 



'-'H^^= 



2B 



9 

o p 

-— - is the fifth term of the series, or the diameter of the fifth 

9 
sphere. By similar solids, we have, First sphere : fifth sphere 

: : 2 JP : /^— Y; that is, as 22? : |^. Hence the first sphere 
V 9 / 729 

is 729 times the fifth. 

2 72 2 R 

As the diameters form the following series, 2Ry — -, -— -, 

V3 ^ 
, fit A±L^ zJi etc., we observe that any desired diam- 
3V3 9 9V3 27 

eter may be found by dividing the first diameter by ( V3)""^. 
For example, the seventh diameter = - — — = — - 

10. Area of a Cycloid. — Multiply the area of the generat- 
ing circle by 3. 



12 TABLE BOOK AND TEST PROBLEMS. 

11. Length of a Cydoidal Curve. — Multiply the diameter 
of the generating circle by 4. 

12. Area of an Ellipse. — Multiply the product of the diam- 
eters by Jtt. 

13. Circumference of an Ellipse. — Let D and d represent 
the long and short diameters. Then the circumference equals 

{approximaJte), 



■V 



D^^d' {D-d)' 
2 8.8 



14. Convex Surface of an Ellipsoid. — To four times the 
square of the height add the square of the base. Multiply 
the square root of half the sum by 3.1416, and this product by 
the radius of the base. 

15. Length of a Helix. — Square the circumference de- 
scribed by the generating point, add the square of the distance 
advanced in one revolution, extract the square root of the sum, 
and multiply by the number of revolutions. 

16. Area of a Lnne or Crescent. — Take the difference of the 
areas of the two segments formed by the arcs of the lune and 
its chord. 

17. Miscellaneous Formulas. 

IT = 3.14159 26535 89793 23846 26433 83280. 
Log ir = 0.49714 98726 94133 85435 12682 88291. 
United SUtes standard gallon = 231 cu. in. = 0.133681 cu. ft. 

United States standard bushel = 2150.42 cu. in. = 1.244456 cu. ft. 
British imperial gallon = 277.25678 cu. in. = 0.160449 cu. ft. 

French meter = 3.28083 ft. 
French liter = 61.02327 cu. in. 
French kilogram = 2.20462 lbs. Avoirdupois. 
Weight of cubic foot of water (maximum density 39.101°, barometer 

30 in., thermometer 39.83° F.) = 62.379 lbs. Avoirdupois. 
Weight of cubic foot of water (maximum density 39.101°, barometer 

30 in., thermometer 62° F.) = 62.321 lbs. Avoirdupois. 
Length of seconds pendulum at New York = 39.10120 in. 
Length of seconds pendulum at London = 39.13908 in. 
Length of seconds pendulum at Paris = 39.12843 in. 



TUE0BEM8, RULES, AND FORMULAS. 18 

Equatorial radius of earth according to Clarke = 20026061.779 ft. 

Polar radius of earth according to Clarke = 20855120.864 ft. 

Mean radius of earth = 20890501.316 ft. 

Calling the earth a sphere with the above mean radius, 100 feet on 
the surface subtends an angle of .987355124 of a second at the 
center of the earth. 

Let 8 = the sum, d = the difference, p = the product, of two 
numbers A and B, of which A is the larger number. Then we 
have the following formulas : — 

One half the sum of two quantities plus one half their differ- 
ence is equal to the larger quantity, or ^8 + ^d = A, 

One half the sum of two quantities minus one half their 
difference is equal to the smaller quantity, or ^s — ^d = 3. 

The square root of the difference between the square of the 
sum of two quantities and four times their product is equal to 
the difference of the quantities, or Vs^ — 4p = ^4 — J5. 

The square root of the sum of the square of the difference of 
two quantities and four times their product is equal to the 
sum of the quantities, or Vd^ + 4p = ^ + jB. 

The difference of two quantities divided by one less than 
the quotient arising from dividing the larger by the smaller 

is equal to the smaller quantity, or d-i-(—^l] = B, 

The square of the sum of two quantities minus the sum of 
their squares is equal to twice their product, or s^ — {A^ -f JB®) 

The square root of the quotient obtained by dividing the 
product of two quantities by the quotient of the larger by the 

smaller is equal to the smaller quantity, or -Wp -i- — = 5. 

The difference of the squares of two quantities divided by 
the sum of the quantities is equal to the difference of the 
quantities, or (A^ — B^) -s- s = d. 



14 TABLE BOOK AND TEST PROBLEMS, 

The square of the sum of two quantities is equal to the 
square of the first, plus the square of the second, plus twice 
the product of the two, or »^ = A^ + JB^'\- 2p. 

The square of the difference of two quantities is equal to the 
square of the first, plus the square of the second, minus twice 
the product of the two, or cP= A^ -{- B^ — 2 p. 

The prodtLct'of the sum and difference of two quantities is 
equal to the difference of their squares, or sd = A^ — J3®. 

The chord of an angle is equal to twice the sine of half the 
angle. 

18. Area of a Parabola. — Take two thirds of the product of 
the base by the height.* 

19. Area of a Segment of a Parabola. — Multiply the differ- 
ence of the cubes of the two ends of the segment by twice, its 
height, and divide the product by three times the difference 
of the squares of the ends. 

20. Volume of a Paraboloid. — Multiply the product of the 
height and the square of the radius by tt, and divide the result 
by 2. 

21. Area of a Eeg^ar Polygon. — Use the formula 

180° 
ia^n cot (a = side, n = number of sides). 



?i 



22. Surface of a Spherical Polygon. — Use the formula 

trr^ X — ~^^ ~^Q ^ (r = radius of sphere, S = sum of an- 

180 

gles, 71 = number of sides). 

23. Surface of a Prism or Cylinder. — Multiply the perim- 
eter by the height, and add the areas of the two ends. 

24. Volume of a Prism or Cylinder. — Multiply the area of 
the base by the height. 

* The area of a circular segment on railroad curves, where the chord is very long in 
proportion to the height, may be found with great accuracy by the above formula. 



THSOBEMS, RULES, AND FORMULAS. 15 

26. yolume of a Prismoid. — Use the Prismoidal Formula : 
Add the areas of the two bases to four times the area of a 
middle section parallel to them, and multiply the sum by one 
sixth of the perpendicular height. 

26. The Center of Oravity of a Triangular Pyramid is in the 
line joining the vertex and the center of gravity of the base, 
at one fourth the distance from the base to the vertex. 

27. Volume of a Pyramid or Cone. — Multiply the area of the 
base by one third of the altitude. 

28. Volume of a Frustum of a Pyramid or Cone. — Multiply 
the areas of the two bases together, and extract the square 
root of the product. To this root add the two areas, and mul- 
tiply the sum by one third of the altitude. 

29. Length of Axis of an Elliptical Bing. — Square the 
diameters- of the axes of the ring, and multiply the square 
root of half t^eir sum by 3.1416. 



30. Volume of a Cylindrical Bing. — Multiply the sum of 
the thickness and inner diameter by the square of the thick- 
ness, and that product by 2.4674. 

31. Surface of a Sphere. — Multiply the diameter by the cir- 
cumference. 

32. Surface of a Segment of a Sphere. — Multiply the height 
by the circumference of the sphere, and add the area of the 
base. 

33. Volume of a Sphere. — Multiply the cube of the diameter 
by .6236. 

34. Volume of a Segment of a Sphere. — Add the square of 
the height to three times the square of the radius of the base, 
and multiply the sum by the product of the height by .5236 ; 
or subtract twice the height of the segment from three times 
the diameter of the sphere, and multiply the remainder by the 
product of the square of the height by .5236. 



16 TABLE BOOK AND TEST PROBLEMS. 

35. Yolume of a Spherical Sector. — Multiply one third of 
the radius of the sphere by the external surface of the zone^ 
which is the base of the sector. 

36. Surface of a Spheroid.'*^ — Multiply the square root of 
half the sum of the squares of the diameters by 3.1416; and 
this product by the conjugate diameter if prolate, and by the 
transverse if oblate. 

37. Convex Surface of a Frustum of a Spheroid. — Proceed 
as by the following rule to obtain proportionate height of 
frustum. Then multiply this height by 3.1416, and this prod- 
uct by the diameter parallel to the base of the frustum. 

38. Convex Surface of a Segment of a Spheroid. — Add the 

squares of the diameters, and take the square root of half 
the sum. Then as the diameter from which the segment is 
cut is to this root, so is the height of the segment to the pro- 
portionate height required. Multiply the other diameter by 
3.1416, and this by the proportionate height of the segment. 

39. Volume of a Spheroid. — Multiply the square of the 
revolving axis by the fixed axis, and this product by .5236. 

40. Volume of the Middle Frustum of a Spheroid (when 
ends are circular) . — Add the square of the diameter of either 
end to twice the square of the revolving axis, and multiply 
the sum by the product of the length of frustum by .2618. 

41. Volume of the Middle Frustum of a Spheroid (when 
ends are elliptical). — Add the product of the transverse and 
conjugate diameters of either end to twice the product of the 
transverse and conjugate diameters of the middle section, and 
multiply the sum by the product of the length of the frustum 
by .2618. 



* A spheroid is a Bolid generated by the revolution of a aemi-ellipse about one of its 
diameters. 



THEOREMS, RULES, AND FORMULAS. 17 

42. Volume of a Segment of a Spheroid (when base is eirou- 

lar). — Take the diiference between three times the fixed axis 
and twice the height of the segment, and multiply the remain- 
der by the sqaare of the height of the segment, and this 
product by .5236. Then the square of fixed axis : square of 
revolving axis : : last product : volume. 

43. Yolnme of a Segment of a Spheroid (when the base is 
elliptical or perpendicular to the revolving axis). — Take the 
difference between three times the fixed axis and twice the 
height of the segment, and multiply the remainder by the square 
of the height of the segment, and this product by .5236. Then 
the fixed axis : revolving axis : : last product : volume of seg- 
ment. 

44. Convex Surface of a Circular Spindle. — Multiply the 
radius of the revolving arc by the length of the spindle. Mul- 
tiply the arc by the distance between center of spindle and 
center of revolving arc. Subtract this product from the 
former, and multiply the remainder by 2 tt. 

NoTB. — This rule gives also the surface of a zone, segment, or frus- 
tum of a spindle. 

45. Convex Surface of a Cyoloidal Spindle. — Multiply the 
area of the generating circle by ^ . 

46. Volume of a Circular Spindle. — Multiply half the area 
of the revolving segment by the central distance. Subtract 
the product from one third the cube of half the length, and 
multiply the remainder by 4ir. 

47. Volume of a Segment of a Circular Spindle. — From half 
the length of the spindle take the length of the segment. 
Find the volume of a middle frustum whose length is twice 
this difference. Then from the volume of the whole spindle 
take the volume of the middle frustum, and divide the remain- 
der by 2. 

BLLWOOD^S TEST PROB. — 2. 



18 TABLE BOOK AND TEST PROBLEMS. 

48. Yolune of a Zone or Fnutnm of a Circular Spindle. — 
Subtract one third of the square of half the length of the 
frustum from the square of half the length of the whole spin* 
dle^ and multiply the remainder by half the length of the 
frustum. From this product subtract the product of the cen- 
tral distance by the revolving area which generates the frus* 
tum, and multiply the remainder by 2ir. 

49. Volume of a Cycloidal Spindle. — Square twice the 
diameter of the- generating circle, multiply by 3.927 times the 
circumference, and divide the product by 8. 

50. Volume of an Elliptic Spindle. — Add the square of its 
diameter to the square of twice the diameter at one fourth of 
its length, and multiply the sum by the product of the length 
by .1309 (or^ijTr). 

61. Volume of the Middle Frustum of an Elliptic Spin- 
dle. — Add the squares of the greatest and least diameters to 
the square of twice the diameter midway between the two, 
and multiply the sum by the product of the length by .1309. 

52. Volume of a Segment of an Elliptic Spindle. — To the 

square of the diameter of the base of the segment add the 
square of twice the diameter midway between the base and 
the vertex, and multiply the sum by the product of the length 
of the segment by .1309. 

53. Volume of a Parabolic Spindle. — Multiply the square 
of the diameter by the length, and the product by ^v. 

54. Volume of the Middle Frustum of a Parabolic Spin- 
dle. — To eight times the square of the greatest diameter add 
three times the square of the least diameter and four times 
the product of these diameters, and multiply the sum by the 
product of the length by -^w, 

65. Volume of a Segment of a Parabolic Spindle. — To the 

square of the diameter of the base of the segment add the 



THEOREMS, BULES, AND FORMULAS. 19 

square of twice the diameter midway between the base and 
vertex, and multiply the sum by the product of the height of 
the segment by ^^ tt. 

56. Volume of an Hyperbolic Spindle. — Add the square of 
the diameter to the square of twice the diameter at one fourth 
its length, and multiply the sum by the product of the length 

57. Volume of the Middle Frustum of an Hyperbolic Spin- 
dle. — Add the squares of the greatest and least diameters to 
the square of twice the diameter midway between the two, 
and multiply the sum by the product of the length by ^v. 

58. Volume of a Segment of an Hyperbolic Spindle. — To 

the square of the diameter of the base of the segment add the 
square of twice the diameter midway between the base and 
vertex, and multiply the sum by the product of the length of 
the segment by ^jir. 

59. Length of a Plane Spiral Line. — Multiply half the sum 
of the greater and less diameters by 3.1416, and again by the 
number of revolutions ; or multiply the number of revolutions 
by the mean length of the circumferences. 

60. Square of any Number (Novel Method). — The follow- 
ing method of squaring any number was first brought to our 
notice by Mr. A. L. Foote of Merrick, N.Y., who says he has 
used it in his own practice for upwards of thirty years, but 
does not claim anything original in its use, it being merely an 
application of the well-known principle that the square of any 
polynomial is equal to the sum of the squares of its several 
terms, plus twice the product of every two terms of the poly- 
nomial. For example : — 



{a + b-^-c + ay 



b^c + dy 

= a* -f 6« -f (? + d* -h 2a& -f 26c 4- 2cd -f 2ac -f 26d 4- 2ad. 



20 



TABLE BOOK AND TEST PROBLEMS. 



Take the number 4567, and denoting 4 by a, 6 by b, 6 by c, 
and 7 by d, we have the following arrangement : 



a 
4 

16 




b 
5 

25 


c 
6 

36 


d 

7 

49 




2ab 


2 be 


2cd 






40 


60 

2ac 
48 

2 ad 


84 

2bd 
70 





56 

Sum = 20857489 = (4567)^ The placing and addition of 
these numbers are better shown by the following : — 

16253649 

406084 

4870 

56 

20857489 

This shows us that a^ = 16,000,000, not 16, and that 
6^ = 250,000, not 25 merely. Any number whatever may be 
squared by this method. 

61. Inscribed Squares and Circles. — Let a square be in- 
scribed in a circle, a circle in this square, a square in this 

circle, and so on. It will be 
observed from the figure that 
the diameter of the first circle 
is also the diagonal of the first 
square, and that the side of 
the first square is also equal 
to the diameter of the second 
circle. This law holds good 
ad infinitum. 

Let the side of any square, 
as AC, = a. Then 




THEOREMS, RULES, AJ^D FORMULAS. 21 

and AD = a V2 = the diagonal. Hence, to find the side of a 
square, the diagonal being given, divide the diagonal by V2. 
Let D = diameter of the first circle. It is also the diagonal of 

the first square. Then — - = side of the first square = diame- 
ter of second circle. It is also the diagonal of the second 
square : hence s- V2 = ~^ = side of the second square = di- 
ameter of third circle = diagonal of third square. Then 

— - -5- V2 = = side of third square = diameter of fourth 

2 2 V2 

circle = diagonal of fourth square, and so on. 

Reviewing this, we observe that the diameters and sides 
form two decreasing series, having the same ratio, but dif- 
fering in first terms. The diameters (and diagonals) are 

Z>, , — , -f etc. The sides are — -, — , , — , etc. 

V2 2 2-V2 V2 2 2V2 4 

The ratio is 



V2 

62. The Diameter of a Circle multiplied by .707107 is equal to 
the side of an inscribed square. 

63. Strength of Timbers or Beams. — A beam twice as wide 
as another is twice as strong ; one twice as deep is four times 
as strong ; while one twice as long is only half as strong. 

A beam loose at both ends and loaded in the middle will 
bear only two thirds as much as if both ends were firmly fixed. 

A beam will bear twice as much weight uniformly distrib- 
uted over its whole length as when the entire weight is placed 
in the middle. 

When a beam is fixed at one end and loaded at the end pro- 
jecting, it will bear but one fourth the weight it will support 
when fixed at both ends and loaded in the middle. 

The strength of rectangular beams varies as the breadth 
multiplied by the square of the depth. Hence a beam with 
its narrow side upward is as much stronger than with its broad 
Bide upward as the depth exceeds the breadth. 



22 TABLE BOOK AND TEST PROBLEMS. 

A triangular beam is twice as strong when resting ou a side 
as when resting on the opposite edge. 

An inch square bar will support a greater weight than an 
inch round one. 

A piece of oak one foot long and one inch square, when sup- 
ported at both ends, will sustain a weight of 600 pounds. A 
similar bar of iron will sustain a weight of 2190 pounds. The 
oak weighs half a pound, and the iron 3 pounds. 

A beam 2 by 8, placed on its edge, is four times as strong 
as one 2 by 4. Placed on their broad sides, the former is only 
twice as strong as the latter. 

64. Area of a Trapezium. — Multiply the diagonal by half 
the sum of the two perpendiculars falling upon it from the 
opposite angles. 

66. Area of a Trapezoid. — Multiply half the sum of the 

parallel sides by the perpendicular distance between them. 

66. The Center of Gravity of a Trapezoid' is on the line 
which bisects the parallel bases, and divides it in the ratio of 
twice the longer plus the shorter to twice the shorter plus the 
longer. 

67. Area of a Triangle. — Multiply the base by half the 
altitude, or use the formula. 

Area = Vs(s — a) (s — b) {s — o), 

in which s = half the sum of the sides a, b, and c. When the 
triangle is equilateral, the formula becomes. 

Area = -J^a^VS. 

68. Equilateral Triangle and Equivalent Square. — The side 
of a square multiplied by 1.52 is equivalent to the side of an 
equilateral triangle of equal area. 

69. The Center of Gravity of a Triangle is one third the 
distance from the middle of a side to the opposite angle. 

70. Similar Triangles are to each other as the squares on 
their homologous sides. 



THEOREMS, RULES, AND FORMULAS. 



28 



71. Formulas for Obliqno Triangles. 




Given, 


Sought, 


A, B, a 


b 


A, a, h 


B 


a, b, G 


A'-B 


a, b, c 


A 


A, J5, C, a 


Area 


A, 6, c 


Area 


a, 6, c 


Area 



Formulas, 



6 = 



amxiB 



sin 5 = 



sin^ 
beXnA 



a 



tan] (^-5) = (q-&)tan*(^-|.BX 



^ be 



cos 



6c 



Area = 



a* sin B sin C 



2sin^ 
Area = J 6c sin A, 



« = J (a + 6 + c), Area = V» (« — a) (« — 6) (» — c). 



72. Formulas for Bight Triangles. — Let ^ be any acute 
angle, and let a perpendicular BC be drawn from any point 
in one side to the other side. Then, if the sides of the right 
triangle thus formed are denoted by letters, as in the accompa- 
nying figure, we shall have these six formulas : — 



1. sin^ = -. 

c 

2. cos -4 = - 

c 

3. tan^ = ^. 



4, cosec -4 = — 

a 

5. sec A = -• 



6. cot^ =-• 

a 




24 



TABLE BOOK AND TEBT PB0BLJBM8. 



Given. 


Sought. 


a, c 


A,B,b 


a, h 


A,B,c 


A, a 


B,b,c 


A,b 


B,a,c 


A, c 


B,a,b 



Formulas. 



a 

— » 

c 



sin J. 

tan^ 

5 = 9(F-^, 



a 

— » 

b 



5 = 90°-^, 



a 

— » 

c 



cos£ 

COtB 

6 = acot4» 



a 

— » 

6 



a = & tan A, 



b= y/(c-\-aXc — <0' 



c = 
c = 

c = 
b = 



Va2 + 62. 

a 

sin^ 

6 
cos^ 

ccos^. 




73. Miscellaneous Trigonometrical Formulas. 

sin' A 4- cos' As=l. b 

sin (-4 ± 5) = sin A cos J5 ± sin J5 cos A* 

a 

COS (-4 ± ^) = cos A cos -B T sin -4 sin J5. 

sin 2A = 2 sin -4 cos A. 

cos 2 -4 = cos' u4 — sin' ^ = 1 — 2 sin' A = 2 cos' -4 — 1. 

sin' -4 = ^ — ^ cos 2 A cos' -4 = ^ + J cos 2 -4. 

sin il + sin J? = 2 sin ^ (^ + 5) cos ^(A — B). 

sin^ - sin ^ = 2 cos ^(A-^B) sin|(.4 - B). 

cos ^ + cos 5 = 2cos ^ (^ -f B) cos |(^ — B). 

cos B — cos ^ = 2 sin ^ (-4 -f -B) sin -J- (-4 — 5). 

sin' A — sin' 5 = cos' B — cos' -4 = sin (^ -f B) sin (A-^ B). 

cos' J. — sin' B = cos (^ -f B) cos (-^1 — B). 



tan^ = 



sin^ 

 • 

cos -4 



cot^ = 



cos -4 
sin^ 



THEOREMS, RULES, AND FORMULAS. 



25 



tan {A±B)z= 



tan A ± tan B 



tan A ± tan B 

   • 

1 qp tan A tan JB 

^ sin (^ ± B) 
cos -4 cos J5 

^ sin {A ± B) 
sin A sin ^ 

sin ^ 4- sin ^ _ tan ^ (A-^B) 

i3,n^{A-B) 



cot ^ ± cot B = 



sin -4 — sin B 

sin ^ -f sin JB 
cos -4 -h cos J5 



= tan^(^-f B). 



sin A 4- sin B 
cos B — cos -4 

sin ^ — sin jB 
cos A -h cos B 

sin -4 -— sin 5 



= cot^(^-5). 



= tani(^-2?). 



cos B — cos -4 

sin^ 



coti(A-{'B). 



tan ^ -4 = 



cot ^ -4 = 



1 -h cos ^ 
sin -4 

-■-■■  -    • 

1 — cos ^ 



74. Curved Surface and Volume of Cylindrical Ungulas. 

(a) When section is parallel to axis of cylinder, 
Curved surface = height x length of arc of one end. 
Volume = area of base x height of cylinder. 

(b) When section passes obliquely through opposite sides 
of cylinder, 

Curved surface = circumference of base of cylinder x 

half sum of greatest and least heights 
of ungula. 

Volume = area base of cylinder x half sum of 

greatest and least lengths of ungula. 

(c) When section passes through base and one side, and 
base of ungula does not exceed a semicircle, 



Curved surface = -^ — (sine of half arc of base x di 



lam- 



vsin 



eter of cylinder — length of arc x 
cosine). 



Volume 



height 
p'sin 
area of base x cosine of half arc). 



= "^^^^^ a sin» of half arc of base - 
vsin 



26 TABLE BOOK AND TEST PROBLEMS. 

(d) When base exceeds a seraicircle, 

Curved surface = — 5 — [sine of half arc of base x diam- 

vsm 

eter of cylinder + length of arc x 

(v sin — sine of base)]. 

Volume = -?i? — (I sin' of half arc of base -f area 

vsin 

of base x cosine). 

(e) When section passes obliquely through both ends of 
cylinder, find the surface as follows : — 

Conceive the section to be continued till it meets the side of 
cylinder produced. Then as the difference of versed sines 
of arcs of two ends of ungula is to v sin of arc of less end, so 
is height of cylinder to part of side produced. 

The surface of each of the ungulas thus found may be ascer- 
tained by preceding rules, and their difference will be the 
curved surface. Find the volume of each ungula by preceding 
rules, and take their difference for volume. 

75. Volume of a Wedge. — Add twice the length of the base 
to the length of the edge, and multiply the sum by one sixth 
of the product of the height of wedge and breadth of base. 

76. Area of a Circnlar Zone. — Subtract the areas of the 
segments from the area of the circle. 

77. Surface of a Spherical Zone. — Multiply height by cir- 
cumference of the sphere, and add the area of the two ends. 

78. Volume of a Spherical Zone. — Add one third of the 
square of height of zone to sum of the squares of the radii of 
ends, and multiply the sum by product of height by 1.5708. 



TABLE 



OP 



LOGARITHMS OF NUMBERS 



PROM 



1 TO 10,000 



%^>^k^^m^ i^^^^%^*%^^^^t^\^ A ^ "^^ 



' N. 


L.« 


N. 


Log. 


N. 


Lof. 


N. 


Lof. 


1 


0.000000 


36 


1.414973 


51 


1.707570 


76 


1.880814 


. s 


0.301U30 


37 


1.431364 


53 


1.716003 


77 


1.886491 


3 


0.452121 
0.1153000 


38 


1.447158 


53 


1.734376 


78 


1.892095 


4 


39 


1.463398 


54 


1.7321)94 


79 


1.897637 


5 


o.ddesTo 


30 


1.4T7121 


55 


1.740.163 


80 


1.903090 


6 


0.778151 


31 


1.491363 


56 


1.748188 


81 


1.908485 


7 


0.845098 


33 


1.505150 


57 


1.755875 


83 


1.913814 


8 


0.903090 


33 


l.5ia'>14 


58 


1.763428 


83 


1.919078 


9 


0.954343 


34 


1.531479 


59 


1.770&53 


84 


1.934379 


10 


1.000000 


35 


1.544068 


60 


1.778151 


85 


1.939419 


11 


1.041393 


36 


1.5.S6303 


61 


1.785330 


86 


1.93^498 


13 


1.079181 


37 


1.568303 


63 


1.792392 


87 


1.939519 


13 


1.113943 


38 


1.579784 


63 


1.799341 


88 


1.944483 


14 


1.146138 


39 


1.591065 


64 


].80<>180 


89 


1.949390 


15 


1.176091 


40 


1.603060 


65 


1.812913 


90 


1.954343 


16 


1.304130 


41 


1.613784 


66 


1.819544 


91 


1.959041 


17 


1.330449 


43 


1.633349 


67 


1.826075 


93 


1.963788 


18 


1.355273 


43 


1.633468 


68 


1.832509 


93 


1.968483 


19 


1.378754 


44 


1.G43453 


69 


1.838849 


04 


1.973138 


30 


1.301030 


45 


1.653313 


70 


1.845098 


95 


1.977734 


31 


1.323319 


46 


1.6637.')8 


71 


1.851358 


96 


1.983371 


n 


1.343433 


47 


1.672098 


72 


1.8573a3 


97 


1.986773 


33 


1.361728 


48 


1.681341 


73 


1.863333* 


98 


1.991336 


34 


1.380311 


49 


1.6g019() 


74 


1.869333 


99 


1.995635 


35 


1.397940 


50 


1.698970 


75 


1.875061 


100 


3.000000 



N. B. In the following table, in the lust nine columns of each page, 
where the first or leading figures change from 9*8 to O's, points or dots 
are introduced instead of the O's through the rest of the line, to catch 
the eye, and to indicate that from thence the annexed first two figures 
of the TiOgarithm in the second column stand in the next lower line. 

27 



2S 



A TABLE OF LOGARITHMS FROM 1 TO 10,000. 



N. 1 


1 


1 1 


2 1 


3 


4 


5 


6 


1 7 


8 


» 


1 D. 


100 


000000 


0434 


0868 


1301 


1734 


2166 


2506 


3029 


3461 


3891 


432 


101 


4321 


4751 


5181 


5609 


6038 


6466 


6894 


7321 


7748 


8174 


438 


lOS 


8600 


9026 


9451 


9876 


.300 


.724 


1147 


1570 


1993 


2415 


434 


103 


012837 


3250 


3680 


4100 


4521 


4949 


5360 


5779 


6197 


6616 


419 


104 


7033 


7451 


7868 


8284 


8700 


9116 


9532 


9947 


.361 


.775 


416 


105 


021189 


1603 


2016 


3428 


2841 


3252 


3664 


4075 


4486 


4896 


413 


106 


5306 


5715 


6125 


6533 


6942 


7350 


7757 


8164 


8571 


8978 


408 


107 


9384 


9789 


.195 


.600 


1004 


1406 


1812 


2216 


3619 


3021 


404 


108 


033424 


3H26 


4227 


4628 


5029 


5430 


5830 


(mo 


6629 


7028 


400 


109 


7426 


7825 


8223 


8620 


9017 


9414 


9811 


.207 


.602 


.996 


396 


110 


041393 


1787 


2182 


2576 


2969 


3362 


3755 


4148 


4540 


4932 


393 


111 


5323 


5714 


6105 


6495 


6885 


7275 


7664 


8053 


8443 


8830 


389 


112 


9218 


9606 


9993 


.380 


.766 


1153 


1538 


1934 


2309 


3604 


386 


113 


053078 


3463 


3846 


4230 


4613 


4996 


5378 


5760 


6142 


6524 


382 


114 


6905 


7286 


7666 


8046 


8426 


6805 


9185 


9563 


9942 


.320 


379 


115 


060696 


1075 


1452 


1829 


2206 


2582 


2956 


3333 


3709 


4063 


376 


116 


4458 


4832 


5206 


5580 


5953 


6326 


6699 


7071 


7443 


7815 


372 


117 


8186 


8557 


8928 


9298 


9668 


. .Jo 


.407 


.776 


1145 


1514 


369 


118 


071882 


2250 


2617 


2965 


3352 


3718 


4085 


4451 


4816 


5182 


366 


119 


5547 


5012 


6276 


6640 


7004 


7368 


7731 


8094 


8457 


8819 


363 


190 


079181 


9543 


9904 


.266 


•626 


.987 


1347 


1707 


2067 


2436 


360 


121 


062785 


3144 


3503 


3661 


4219 


4576 


4934 


5291 


5647 


6004 


357 


122 


6360 


6716 


7071 


7426 


7761 


8136 


8490 


8845 


9198 


9552 


355 


123 


9905 


.258 


.611 


.963 


1315 


1667 


2018 


2370 


2T21 


3071 


3511 


124 


093422 


3772 


4122 


4471 


4820 


5169 


5518 


5866 


6215 


6562 


349, 


125 


6910 


7257 


7604 


7951 


8298 


8644 


8990 


0335 


9681 


..36 


346 


126 


100371 


0715 


1059 


1403 


1747 


2091 


2434 


2777 


3119 


3462 


343 1 


127 


3804 


4146 


4487 


4826 


5169 


5510 


5851 


6191 


6531 


6871 


340 


128 


7210 


7549 


7888 


8227 


8565 


8903 


9241 


9579 


9916 


.253 


338 


129 


110590 


0926 


1263 


1509 


1934 


2270 


2605 


2840 


3275 


3600 


335 


130 


113943 


4277 


4611 


4944 


5278 


5611 


5943 


6276 


6606 


6940 


333 


131 


7271 


7603 


7934 


8265 


8595 


8926 


9256 


9586 


9915 


.245 


330 


132 


120574 


0903 


1231 


1560 


1868 


2216 


2544 


2671 


3198 


3535 


328 


133 


3852 


4178 


4504 


4830 


5156 


5481 


5806 


6131 


6456 


6781 


335 


134 


7105 


7429 


7753 


6076. 


8399. 


8722 


9045 


9368 


9600 


..12 


323 


135 


130334 


0655 


0977 


1296 


1939 


2260 


2580 


2900 


3219 


331 


136 


3539 


3858 


4177 


4496 


4614 


5133 


5451 


5769 


6086 


6403 


318 


137 


6721 


7037 


7354 


7671 


7967 


6303 


6618 


8934 


9249 


9564 


315. 


138 


9679 


.194 


.506 


.822 


1136 


1450 


1763 


2076 


2389 


2702 


314 


139 


143015 


3327 


3639 


3951 


4263 


4574 


4885 


5196 


5507 


5818 


311 


140 


146128 


6438 


6748 


7058 


7367 


7676 


7965 


8294 


8603 


8911 


309 


141 


9219 


9527 


9835 


.142 


.449 


.756 


1063 


1370 


1676 


1982 


307 


142 


152288 


2594 


2900 


3205 


3510 


3815 


4120 


4424 


4728 


5032 


305 


143 


5336 


5640 


5943 


6246 


6549 


68S2 


7154 


7457 


7759 


8061 


303 


144 


8362 


8664 


8965 


9266 


9567 


9868 


.168 


.469 


.769 


1068 


301 


145 


161368 


1667 


1967 


2266 


2564 


2863 


3161 


3460 


3758 


4055 


299 


146 


4353 


4650 


4947 


5244 


5541 


5838 


6134 


6430 


6726 


7022 


297 


147 


7317 


7613 


7908 


8203 


8497 


8792 


9066 


9380 


9674 


99uO 


295 


148 


170262 


0555 


0648 


1141 


1434 


1726 


2019 


2311 


2603 


2895 


293 


149 


3186 


3478 


3769 


4060 


4351 


4641 


4932 


5222 


5512 


5802 


291 


150 


176091 


6381 


C670 


6959 


7248 


7536 


7825 


8113 


8401 


8689 


289 


151 


8977 


9264 


9552 


9839 


.126 


.413 


.699 


.985 


1272 


1558 


287 


152 


181844 


2129 


2415 


2700 


2965 


3270 


3o55 


3839 


4129 


4407 


285 


153 


4691 


4975 


5259 


5542 


5825 


61U8 


6301 


6674 


6956 


7239 


283 


154 


7521 


7803 


8084 


8366 


8647 


8928 


9209 


9490 


9771 


..51 


281 


155 


190332 


0612 


0892 


1171 


1451 


1730 


2010 


2289 


2567 


2846 


279 


156 


3125 


3403 


3681 


3959 


4237 


4514 


4792 


5069 


5346 


5623 


278 


157 


5899 


6176 


6453 


6720 


7005 


7281 


7556 


7632 


8107 


8382 


376 


158 


8657 


8932 


9206 


9481 


9755 


..29 


.303 


.577 


.850 


1124 


374 


159 


201397 1670 


1943 


2216 


2488 


2761 


3033 


3305 


3577 


3848 


373 


N. 1 


1 1 


2 1 


3 1 


4 1 5 1 


6 1 


7 1 8 1 1 D. 1 



1 TiBLi or LOOiRiTHira raoM 1 to 10,000. 



sa4i^ 




Im 


-^r 




6856 


7096 


73GS 


7634 


7S04 
.586 


"JIm 


5109 


8010 


asai 




iSOlCB 


MM 


3836 


0892 
W90 


3^M 


s3oe 






6084 














asMie 




0960 






S598 


329) 


3a04 


8285 


M37 






SM8 




0049 










1M6 


S5U 


5TM 


ms 


318S 

earn 


«30 
0499 


TO73 


K19 




B7U9 




S40430 










S8S3 


30M 






3822 


**T6TO 


7ei8 


S7S5 


SB9S 


8^ 


^I 


03 LO 


8WS 


31b3 


3399 






IH90 




5761 


9513 


1406 




7675 




871842 


iwr4 










1389 


4620 


4830 




SMS 


fiGB3 


8951 


7151 




SWIM 
%1033 


8962 


^ 


9439 


9667 


3MJ 


35OT 

5782 


MU7 


623'2 


0456 


TBCrS 








era6 


S«CO» 


W78 


2aw 


0702 




14«ll 


+«r7 










8884 




7323 








B289 


9501 


9ra5 


1 


7710 


B 


1681 
38+4 

5990 


less 

4059 
.481 


"™ 


407B 


'S 


lo8» 


4ml 


SBTO 


6180 






6809 


e063 


BS72 






8098 


SD146 




0S«3 




09T7 




8438 








IS^.- 


4488 








8330 




ffre? 


fsiai 


9194 


MM 


«640 
463$ 


ii 


s 


litis 


is 


6MI> 


88M 


7«60 
90M 


7a«o 




06^ 




1039 


1*31 



1388 




UW 


iiS 






0082 


9938 


7630 


2334 


3488 


274S 


72« 




™^ 


























B9W 


7W 












1916 


^ 


m^ 


Im 


6937 








1W 


m» 


6233 






S! 


7130 


^ 


7518 


»«7 


^ 
















^ 






32.14 




aaa 








1S98 




2013 












aliH 








1S32 


40^ 












17 


*M6 


1830 







» 


— ^— 


",-( g 


3|4|S|«|1|8|e 


B. 


aio 


"j«4a 












3600 








1ST 




4398 


45ea 










5570 








IW 


ase 


«353 


SMS 




floas 


71» 












m 


ase 


B30S 


8UD 


eeei 




goes 












I»4 


sat 


3B0S4S 


OMIt 


0636 


oeis 






1410 






1980 


IBS 


aas 










MM 


3147 






3724 


3Blfl 


l«3 


«e 


4iae 












asw 






»34 


101 












8790 






















esoe 


8696 












IVO 


aaa 


»8K 




.ai5 


.404 


.SS3 






1101 






180 


S3D 






S105 


aaai 


S482 


8671 




3048 


3836 


3434 








3800 






4363 






4926 


SU3 




88 


^ 


WM 




7^ 


^ 




8287 


wra 




BHt3 
8845 


7109 
0030 


86 






MOl 






was 




.S8 


.313 


.6118 


.883 


89 






lffi3 




lffi£t 


1806 






S360 






84 


S36 


» 


3DM 


nso 


3464 


»47 












184 




4748 








5481 


seo4 


se46 




WIS 


8304 


89 






eru 


BB43 






7488 


7670 


7899 


8034 


me 


88 
















94S7 




8840 






3W 


3802 1 


039S 
3W5 


0S73 


s 


0B34 
4533 


Sftl7 


3W7 


3B7T 


345S 


3636 
5488 


170 














etM 
























8STB 


B45fl 






K«] 




















.4U5 








MB 


























firar 














3U'ie 








318 




















6ira 








«374 


6548 




8896 










77Wi 




230 


SBTMO 


6114 


^ 


?tS-3 


BS34 


8808 


8B81 


8|M 


0328 


^ 


J" 


^ 


3121 


^ 


sIm 


M35 


^ 


S?l 


«« 


43S0 


4492 


9949 
48t.'3 


in 














sfos 




B(Kt» 












8710 






7!H1 


7391 


75l;l 










tsfl 
















































as8 


*itex 
















asiG4 




168 


i» 


3300 




303S 


3ai3 










4639 






KM 












ssoa 






630B 








eeti 






















sea 
























sea 


MSC 




.see 


















S«4 


mm 




1933 










2754 


8S18 






S(B 


ado 




as74 






40ES 




4302 




4718 


164 


9«6 


46% 




saoe 


5371 


5534 






acs3 


6186 






ser 




Iw 






71S1 


7384 


7486 






9591 


ioa 






















18U3 




CT) 


4313W 


^ 


1684 


18« 


i 


ml 


E 


S488 


i»49 
Ml 7 


8800 


158 


«76 


mi 


MM 


wm's 


e8!M 


s 


iw5 


iSI 


ioM 


.594 


8^ 


1S8 












3106 
















4a4i 
















S803 






S79 




57«0 


MIS 


6071 


62J0 


63*! 


6537 


6083 




7003 


151 



A TABLB OF LOGARITHMS PROM 1 TO 10,000. 



31 



H. I 0(ltg|3|4i5|6}7|8|9| D. 



S80 
961 
382 
S83 

984 
985 
S86 
987 

988 



990 
90] 



903 
994 
995 



907 
998 



300 

301 

302 

303 

304 

305 

306^ 

307 

308 

303 

310 
311 
319 
313 
314 
315 
316 
317 
318 
319 

390 
391 
399 
393 
394 
395 
396 

:«7 

398 
399 

330 

331 

339 

333 

334 

335 

330 

337^ 

338 

330 



447158 
8706 

450949 
1786 
3318 
4845 
6366 
7889 
9399 

400896 

4IS9386 
3893 
5383 
66G8 
8347 
9699 

471999 
9756 
4916 
5671 

477191 
8566 

480007 
1443 
9874 
4300 
5721 
7138 
8551 
9958 

491369 
9760 
4155 
5544 
6930 
8311 
9687 

501059 
9497 
3791 

505150 
6505 
7856 
9903 

510545 
1883 
3918 
4548 
5874 
7196 

518514 
9698 

591138 
9444 
3746 
5045 
6339 
7to(t 
8917 

S30900 



7313 
8861 
0403 
1940 
3471 
4997 
6518 
8033 
9543 
1048 

9548 
4049 
5539 
7016 
8495 
9969 
1438 
9903 
4309 
5816 

7266 
8711 
0151 
158i> 
3016 
4449 
5863 
7980 
8699 
..99 

1509 
9900 
4994 
5683 
7068 
8448 
9894 
1196 
9564 
3997 

5988 
6640 
7991 
9337 
0679 
9017 
3351 
4681 
6006 
7398 

8646 
9959 
1969 
9575 
3876 
5174 
6469 
7759 
9045 
0988 



74G8 
9015 
0557 
9093 
3624 
5150 
6670 
8184 
9U94 
11U6 

9697 
4191 
5680 
7164 
8643 
.116 
1585 
3049 
4508 
5969 

7411 
8855 
0994 
1729 
3159 
4585 
6005 
7421 
8833 
.939 

1649 
3040 
4433 
5622 
7206 
8586 
99G9 
1333 
9700 
4063 

5491 
6776 
8126 
9471 
0813 
9151 
3484 
4813 
6139 
7460 

8777 
..90 
1400 
9705 
4006 
5304 
6598 
7888 
9174 
0456 



7C93 
9170 
0711 
9247 
3777 
5309 
6821 
8336 
9845 
1348 

9847 
4340 
5899 
7312 
8790 
.963 
1739 
3195 
4653 
6107 

7555 
691)9 
0438 
1879 
3309 
4727 
6147 
7563 
8974 
.380 

1789 
3179 
4579 
5060 
7344 
8724 
..99 
1470 
2837 
4199 

5557 
6911 
8960 
9606 
0947 
9284 
3617 
4946 
6971 
7599 

8909 
.991 
1530 
9835 
4136 
5434 
6727 
8016 
9309 
0584 



7778 
9324 
0665 
9400 
3930 
5454 
6973 
8487 
9995 
1499 

9997 
4490 
5977 
7460 
8938 
.410 
1878 
3341 
4'; 99 
6259 

7700 
9143 
0589 
9016 
3445 
4869 
6289 
7704 
9114 
.590 

1929 
3319 
4711 
6U99 
74»3 
8862 
.936 
1607 
9973 
4336 

5693 
7046 
8395 
9740 

loei 

9418 
3750 
5079 
6403 
7794 

9040 
.353 
1661 
9966 
4966 
5563 
6856 
8145 
9430 
0719 



7933 


8U88 


8242 


9478 


9633 


9787 


1018 


1172 


1326 


9553 


9706 


9859 


4082 


4235 


4387 


5606 


5758 


5910 


7195 


7276 


7428 


8638 


8789 


8940 


.146 


.996 


.447 


1649 


1799 


1948 


3146 


3996 


3445 


4639 


4788 


4936 


6196 


6974 


6423 


7608 


7756 


7904 


9085 


9933 


9380 


.557 


.704 


.851 


9025 


9171 


9318 


3487 


3633 


3779 


4944 


5090 


5235 


6397 


6549 


6687 


7844 


7989 


8133 


9287 


9431 


9575 


0725 


08C9 


1012 


9159 


9302 


9445 


3587 


3730 


3872 


5011 


5153 


5295 


6430 


6572 


0714 


7845 


7986 


8127 


9955 


9396 


9537 


.661 


.801 


.941 


9069 


9901 


9341 


3458 


3597 


3737 


4850 


4989 


5128 


6238 


6376 


6515 


762J 


7759 


7697 


891)9 


9137 


9275 


.374 


.511 


.648 


1744 


1880 


9017 


3109 


3246 


3382 


4471 


4607 


4743 


5828 


5964 


6099 


7181 


7316 


7451 


8530 


8064 


8799 


9874 


...9 


.143 


1215 


1349 


1482 


2551 


9684 


9818 


3883 


4016 


4149 


5211 


5344 


5476 


6535 


6668 


6800 


7855 


7987 


6119 


9171 


9303 


0434 


.484 


.615 


.745 


1792 


1999 


9053 


3096 


3936 


3356 


4396 


4526 


4656 


5693 


5892 


5951 


6965 


7114 


7943 


8974 


8402 


8531 


9559 


9687 


9615 


0640 


0968 


1096 



8397 
9941 
1479 
3019 
4540 
6069 
7579 
9091 
.597 
9098 

3594 
5085 
6571 
8C59 
0597 
.998 
3464 
3925 
5381 
6839 

8978 
9719 
1156 
9588 
4015 
5437 
6655 
8269 
9677 
1081 

9481 
3876 
5967 
G653 
8035 
9412 
.785 
9154 
3518 
4878 

6934 
7586 
8934 
.277 
1616 
9951 
4982 
5609 
6932 
8251 

9566 
.876 
21U3 
3486 
4785 
6081 
7372 
8660 
9943 
1993 



8559 
..95 
1633 
31G5 
4692 
6214 
7731 
9949 
.748 
9248 

3744 

5934 
6719 
8200 
9675 
1145 
9610 
4071 
5596 
6976 

8492 
9863 
1299 
2731 
4157 
5579 
6097 
8410 
9818 
1222 

2691 
4015 
5406 
6791 
8173 
9550 
.922 
921)1 
3(5J5 
5014 

6370 
7721 
9068 
.411 
1750 
3084 
4414 
5741 
7064 
8382 

9697 
1007 
9314 
3616 
4915 
6910 
7501 
8788 
..79 
13S1 



155 
154 
154 
153 
153 
152 
152 
151 
151 
150 

150 

149 1 

149 

148 

148 

147 

146 

146 

146 

145 

145 
144 
144 
143 
143 
142 
149 
141 
141 
140 

140 
139 
139 
139 
138 
138 
137 
137 
136 
136 

136 
135 
135 
134 
134 
133 
133 
133 
139 
139 

131 

131 
131 
130 
130 
199 
190 
190 
198 
198 



N.| 0|1|9|3|4|5|6|7|8|0 | D. 



32 




ATWL.0 




OH I 


TO 10,000. 






X. 


1 1 1 S 1 3 1 4 1 a 1 1 T 1 8 1 9 1 D. II 






ItOT 


1734 


IBflS 






9345 




3500 


Ml; 


198 








3039 


3i3e 








3645 




3899 










4989 


4407 




4661 




4BJ4 
















aff7* 




NIT 


6053 


6190 




0439 


1» 










9937 




7189 


T31S 


7441 




7«93 


196 




T819 














S699 


IXBi 


8951 


I3e 




wm 


taa 










S629 






.904 


195 




Mm9 




os« 




0830 












135 


Ma 










S079 














MS 


S825 








3323 






3696 








330 


jMoas 


JIM 


MM 


MT8 


^ 


4688 


48 IS 
6049 


om 


6996 


5183 


a 


3sa 




eaeit 


6T8J 


9913 


7036 




7i8J 


740S 




7653 




Xi3 










MS7 


8389 


8513 


as35 






1S3 




VCKU 


91SS 






9491 


9«16 






9984 


.106 








03£1 




0S9S 






9969 








m 










ISIO 


11)33 


weo 




9303 


3433 




m 










3033 


31iS 


aj76 


3398 




3640 




191 


3sa 




M04 




4MT 


43S8 




4610 


4731 


4852 


4973 




3» 














SftJO 




6061 


6189 




360 


"?^ 


T6i7 


™ 


^ 


1988 


81^ 


^ 


Sll 


^9 


7387 


130 


302 






8948 






9JU8 


912a 


9548 








363 














.984 




























S065 








S3»3 












iw 










36T 


Ji81 


3«d0 


«03 


^I 


Sisa 


S257 




5494 


tefi 


1™ 


111 


368 




sftsa 










65M 


6673 




69l» 




369 


H^ 


6319 


S43S 


MM 


8^1 


sm 


8905 


9023 


9140 


9157 


117 




DIM 








9843 


9959 




.193 


■3U9 








SJDMJ 






oajj 


1010 


jiaa 


1243 


13S9 








373 






I94a 


4058 




)«»i 


9407 


9523 


3639 






sn 


«31 


SMe 




3£M 


4494 


3451 

4610 






33U0 


3915 
5072 


lit 


jra 


5180 


S303 


S419 


S534 


5659 


5763 


5880 


^ 


6111 




115 






reuT 














8410 


8sa 




17» 










9097 






9441 




96S1 






Sr97S4 


989d 




.IM 




.355 










n 
































S391 


9404 




9631 






W!-2 


3US5 


11 


i 


m 


Si 






3B9S 


4896 


w» 


0^ 


5935 




11 




S5S 








703T 


7149 




7374 








387 






T9JS 


8947 


8160 


8fl2 


8384 


8196 












BIW 








9391 








9838 






9950 








.396 






















1399 


1510 


























9732 


9843 


9954 


3964 






3X1 




















4982 




3S3 


4393 
















5276 


5386 




3»4 


urn 


M08 




5KT 


















■SBT 


BTOT 




S9M 


7037 


7148 


7956 








no 




709$ 


TBOS 








8*43 


8353 










W7 
























age 


ees3 


9991 




















3M 


mm 


im 






14IB 


1317 


16SS 


1734 


1843 




109 


1 



A TABLJE OF LOOARITiUai FROM 1 TO 10,000. 



S3 



H. 


1 1 1 3 1 3 


1 * 1 


5 i 6 1 7 1 8 1 9 


rri 


400 


•08060 


8169 


8277 


8386 


3494 


3803 


8711 


8819 


8933 


3036 


108 1 


401 


3144 


3353 


3381 


3469 


3577 


3688 


3794 


9S0r2 


4010 


4118 


108 




403 


^% 


4334 


4443 


4550 


4658 


4766 


4874 


4983 


5089 


5197 


106 




403 


5305 


5413 


5531 


5638 


5736 


5844 


5951 


6059 


6166 


6374 


106 




404 


6381 


6489 


6598 


6704 


6811 


6919 


7038 


7133 


7241 


7348 


107 




405 


7455 


7563 


7669 


7777 


7884 


7991 


8098 


8205 


8313 


8419 


107 




406 


8536 


8833 


6743 


8847 


8954 


9061 


9167 


9274 


9381 


9488 


107 




407 


9594 


9701 


9808 


9914 


..31 


.138 


.334 


.341 


.447 


.554 


107 




406 


610660 


0787 


0673 


0979 


1088 


11^2 


1396 


1405 


1511 


1617 


106 




400 


1733 


1839 


1936 


9043 


3148 


3354 


8360 


3466 


3573 


3678 


106 




410 


613784 


8890 


8993 


3103 


3907 


3313 


3410 


3535 


3630 


3736 


106 




411 


3643 


3947 


4053 


4159 


4264 


4370 


4475 


4581 


4680 


4793 


100 




413 


4897 


5003 


5108 


5313 


5319 


5434 


5530 


5634 


5740 


5845 


105 




413 


5950 


6055 


6160 


6365 


6370 


6478 


6581 


6688 


6700 


6895 


105 




414 


7000 


7105 


7310 


7315 


742D 


7535 


7639 


7734 


7839 


7943 


105 




415 


8048 


8153 


6357 


83S3 


6486 


8571 


8676 


8780 




8089 


105 




416 


9093 


9193 


9302 


9401 


0511 


9815 


9719 


9834 


9^28 


• •33 


104 




417 


080136 


0340 


0344 


0448 


0553 


0858 


0760 


0864 


0068 


1073 


104 




418 


1176 


1383 


1384 


1488 


1592 


1895 


1799 


1903 


3007 


8110 


104 




410 


8214 


3316 


9431 


3535 


9628 


3733 


8835 


8939 


3043 


3146 


104 




490 


033349 


3353 


3458 


3559 


3663 


3766 


3860 


3973 


4076 


4179 


103 




431 


4383 


4385 


4483 


4591 


4605 


4796 


4901 


5004 


5107 


5310 


103 




433 


5313 


5415 


5518 


5631 


5734 


5827 


5939 


60^ 


6135 


0338 


103 




433 


6340 


6U3 


6546 


6648 


6751 


6853 


6056 


7058 


7161 


7363 


103 




434 


7366 


7468 


7571 


7673 


7775 


7878 


7980 


8083 


8185 


8287 


103 




435 


8389 


8491 


8593 


8695 


8797 


8903 


0003 


0104 


920G 


9308 


103 




436 


9410 


9513 


9S13 


9715 


0617 


9919 


..31 


.133 


.234 


.320 


102 




4S7 


630438 


0520 


0631 


0733 


0635 


0933 


1038 


1139 


1311 


1343 


103 




486 


1444 


1545 


1647 


1748 


1849 


1931 


3053 


3153 


3353 


3358 


101 




430 


3457 


3559 


3660 


3761 


8838 


3933 


3064 


3165 


3266 


3367 


101 




430 


633468 


3560 


3670 


3771 


3873 


3973 


4074 


4175 


4376 


4376 


100 




431 


4477 


4578 


4679 


4779 


4883 


4981 


5081 


5182 


5883 


5383 


100 




433 


5481 


5584 


5685 


5785 


5888 


5986 


6087 


6187 


6337 


6388 


100 




433 


6488 


6583 


6088 


6789 


6889 


6989 


7069 


7189 


7390 


7393 


100 




434 


7490 


7590 


7690 


7790 


7893 


7990 


8090 


8190 


8393 


8389 


99 




435 


8489 


8599 


8689 


8789 


CXX90 


8988 


9088 


9188 


9387 


9387 


09 




436 


M86 


0586 


9688 


9785 


9685 


9964 


..84 


.183 


.883 


.382 


09 




437 


640481 


0581 


0680 


0779 


0679 


0978 


1077 


1177 


1378 


1375 


99 




438 


1474 


1573 


1673 


1771 


1871 


1970 


3069 


3188 


32G7 


3366 


99 




430 


3465 


3563 


9633 


3761 


8860 


3959 


3058 


3156 


3255 


3354 


99 




440 


643453 


3551 


3650 


3749 


3847 


3946 


4044 


4143 


4343 


4340 


06 




441 


4439 


4537 


4636 


4734 


4832 


4931 


5039 


5127 


5228 


5334 


98 




443 


54^ 


5531 


5619 


5717 


5815 


5913 


6011 


6110 


6308 


6336 


96 




443 


6404 


6533 


6600 


6608 


6796 


6894 


6993 


7089 


7187 


7365 


98 




444 


7383 


7481 


7579 


7676 


7774 


7873 


7969 


8067 


8165 


8263 


06 




445 


8360 


8158 


8555 


8853 


8750 


8848 


8945 


9043 


9140 


9337 


97 




446 


9335 


0433 


9530 


9^7 


9734 


9621 


9919 


..18 


.113 


.310 


97 




447 


650303 


0405 


0503 


0599 


0698 


0793 


0690 


0987 


1084 


1181 


97 




448 


1378 


1375 


1473 


1589 


1688 


1762 


1859 


1956 


3053 


3150 


97 




|449 


3346 


3343 


3440 


3538 


3633 


3730 


3836 


3933 


3019 


3116 


97 




J450 


653313 


3300 


3405 


xxm 


3596 


3695 


3791 


aXXJO 


3984 


4060 


96 




451 


4177 


4373 


4360 


4485 


4563 


4658 


4754 


4850 


4946 


5043 


96 




453 


5138 


5335 


5331 


5437 


5533 


5619 


5715 


5810 


5906 


6003 


96 




453 


6006 


6194 


saso 


6386 


6483 


6577 


6673 


6769 


6864 


6060 


96 




'454 


7058 


7158 


7347 


7343 


7438 


7534 


7639 


7725 


7830 


7016 


96 




,455 


8011 


8107 


8303 


8398 


8393 


8488 


8584 


8679 


8774 


8870 


95 




456 


8965 


0030 


9153 


(»50 


9346 


9441 


0536 


9631 


9736 


0621 


95 




457 


9916 


..11 


.106 


.301 


.396 


.391 


.488 


.581 


.676 


.771 


95 




458 


660865 


0930 


1055 


1150 


1245 


1339 


1434 


1529 


1623 


1718 


95 




l4S0 


1813 


1907 


3003 


3038 


3191 


3286 


3380 


3475 


3569 


8663 


95 




H 1 


1 1 1 3 1 3 1 


4 1 


5 1 6 1 7 1 8 1 9 i 


D. 





BLLWOOD*S TEST PBOB. — 3. 



34 




A TABLE or UMUurma pmoM 1 


TO 10,000. 






H. 


1 1 1 


1 « 


1 3 1 


4 1 


5 


1 • 1 


7 


8 1 


9 1 D.| 


409 


•B73B 


9B9B 


3947 


30U 


31% 


S30 


3394 


3418 


3519 


zm 94I 


461 


3701 


3795 


3889 


3083 


4078 


4173 


«66 


4300 


4434 


4548 
5487 


94 


408 


4649 


4736 


4830 


4924 


5018 


5113 


5206 


Si99 


5393 


»4 


40 


5581 


5675 


5760 


adOBE 


505> 


6050 


6143 


623T 


6331 


64^ 


94 


464 


6518 


6613 


6705 


6799 


68»i 


6866 


707» 


7173 


7266 


7360 


94 


465 


7453 


7546 


7649 


7733 


?82'> 


7933 


8313 


8106 


8199 


8293 


93 


406 


8383 


8479 


8S73 


8663 


8759 


8652 


8945 


9038 


9131 


9824 


SS 


467 


9317 


9410 


8503 


9306 


9689 


8788 


9873 


9067 


• •60 


153 


93 


466 


C7IB46 


0339 


0431 


0524 


0617 


0710 


0802 


0893 


0989 


ueo 


93 


469 


1173 


1335 


1338 


1431 


1543 


1636 


1728 


1821 


UlA 


9005 


93 


470 


679098 


919) 


S83 


S375 


»4S7 


9363 


9RS3 


3744 


9836 


9999 


«3 


471 


30S1 


3113 


39.15 


3297 


3380 


3482 


3574 


3666 


3756 


3B30 


98 


4» 


3912 


4034 


4126 


4318 


4310 


4412 


4494 


4566 


4577 


4709 


98 


473 


4861 


4053 


5043 


5137 


52i8 


5320 


5413 


5503 


sae 


5037 


•3 


474 


5778 


5670 


5»S 


6053 


6145 


6336 


6328 


6419 


6511 


6003 


99 


475 


6691 


6783 


6876 


6068 


7039 


7151 


7943 


7333 


74M 


7516 




476 


7607 


7606 


7781 


7»jl 


7973 


8363 


8154 


8345 


8336 


8«7 




477 


8518 


860) 


8700 


8791 


m^ 


8973 


9064 


9155 


9346 


9337 




478 


9438 


9519 


OiilO 


970G 


9791 


9883 


9973 


..63 


.154 


M5 




479 


680336 


043S 


0517 


0607 


0688 


OTflS 


nsTQ 


OBTD 


1060 


1151 


90 


480 


681241 


13S 


1423 


1513 


1603 


1693 


1784 


1874 


1964 


9055 


481 


9145 


3235 


3323 


241t 


3536 


3396 


9R8i 


vm 


385>7 


3957 


90 


4a2 


3017 


3137 


3237 


3317 


3407 


3497 


35^ 


3R77 


3767 


3837 


90 


483 


3917 


4037 


4127 


4217 


4307 


4396 


4486 


4576 


4666 


4756 


90 


4<J4 


4815 


4835 


5025 


5114 


5304 


£34 


5383 


5473 


53(3 


5653 


90 


483 


5742 


5831 


5021 


6010 


6103 


61S9 


6379 


6»« 


6458 


6547 


89 


486 


6636 


6726 


6815 


6004 


69M 


7IH3 


7172 


73(il 


TiSl 


7449 


80 


487 


7S9 


7618 


ttiJl 


7796 


7886 


7975 


8«U 


8153 


8242 


8131 


89 


488 


8190 


850) 


8338 


8387 


8776 


88 5 


8933 


»343 


9131 


«£» 


89 


480 


93J9 


9388 


9486 


9375 


9664 


9733 


9611 


9333 


• •19 


.107 


89 


493 


680196 


l»85 


0373 


04» 


0653 


f/B^ 


0738 


0«I6 


C9QS 


0803 


89 


401 


1061 


117U 


1358 


1347 


1433 


1534 


1612 


171)0 


ITSil 


1877 


88 


4» 


1965 


S053 


3143 


3330 


3318 


34)6 


3494 


S563 


»i7l 


9759 


88 


483 


9847 


9935 


3J33 


3111 


3I9J 




3375 


3403 


3331 


3639 


88 


404 


37^7 


3815 


3903 


3991 


40*8 


4166 


4351 


4342 


4433 


4517 


88 


495 


4605 


4603 


4781 


48Gd 


4336 


5044 


5111 


5219 


5307 


5394 


88 


496 


5483 


5560 


5657 


5744 


5^02 


5919 


6037 


6094 


6182 


0369 


87 


497 


6356 


6444 


6531 


6618 


6706 


6793 


6880 


flow 


7»J55 


7143 


87 


488 


7229 


7317 


7404 


7491 


7578 


TfiflS 


7759 


7K» 


7923 


8014 


87 


499 


8101 


8188 


8375 


83GS 


8449 


8535 


8623 


8700 


8795 


8833 


87 


500 


08970 


9057 


9144 


0831 


9317 


9404 


9491 


9578 


or^ 


9751 


87 


501 


0838 


9924 


..11 


. •06 


.184 


.971 


.338 


.444 


.331 


.617 


87 


50S 


700704 


0790 


0877 


0963 


1033 


1136 


1323 


1309 


131»3 


1482 


86 


503 


1566 


1654 


1741 


1827 


1913 


1999 


9066 


9173 


3238 


3344 


86 


504 


9131 


3517 


9603 


3>89 


3773 


38U 


S947 


3033 


3119 


3303 


86 


506 


aaoi 


3377 


34^*3 


3519 


3633 


3721 


3807 


3893 


3979 


4065 


86 


506 


4151 


4236 


4322 


4408 


4494 


4579 


4665 


4751 


4837 


4922 


88 


507 


50J8 


5094 


5179 


58G5 


535J 


5436 


5522 


5607 


5693 


5778 


86 


503 


5864 


5949 


6035 


6120 


62)6 


6391 


6376 


64^ 


6547 


6633 


85 


509 


6718 


6803 


68H8 


6974 


7039 


7144 


7339 


7315 


7400 


7485 


85 


510 


707570 


7655 


7740 


7%6 


7911 


7996 


8061 


8166 


8351 


8336 


85 


511 


8431 


8506 


8591 


8676 


87G1 


H846 


89 U 


9U15 


9100 


9183 


85 


513 


9S70 


9355 


9440 


9324 


9609 


9634 


9779 


9663 


9948 


..33 


85 


513 


710117 


0903 


0887 


0371 


0436 


0543 


0335 


0710 


0794 


087^) 


85 


514 


0963 


1048 


1133 


1317 


1301 


1383 


1470 


1554 


1639 


1723 


81 


515 


1807 


1893 


1976 


9060 


3144 


3-229 


S3 13 


3397 


9481 


3566 


84 


516 


9650 


3734 


3818 


3009 


9966 


3070 


3154 


3338 


3323 


3407 


84 


517 


3491 


3575 


3fV50 


3743 


3826 


3010 


3994 


4078 


4163 


4246 


84 


518 


4330 


4414 


4497 


4581 


4665 


4749 


4833 


4916 


5000 


5084 


84 


|519 


»«7 


5851 


5335 


5418 


5503 


5586 


5660 


5753 


5836 


soao 


84 


H.| 


1 1 1 


3 


3 1 


4 1 


5 1 


6 1 


7 1 


8 1 


» Itt 1 



A TABLE OP LOGARITHMS PROM 1 TO 10,000. 



35 



N. 


1 


1 1 


1 2 1 


3 1 


4 


5 


1 • 1 


^ 


8 


9 


Itt 


a» 


716003 


6087 


6170 


6354. 


6337 


6421 


6504 


6588 


6671 


6754 


83 


531 


6838 


6931 


7004 


7088 


7171 


7954 


7338 


7431 


7504 


7587 


83 


533 


7671 


7754 


7837 


7920 


8003 


8086 


8160 


8253 


831)6 


8419 


83 


533 


8503 


8585 


8868 


8751 


8834 


8917 


0000 


9063 


9105 


0348 


83 


534 


0331 


9414 


0497 


0580 


9883 


0745 


9828 


0911 


0004 


..77 


83 


535 


730150 


0343 


tms 


0407 


0493 


0573 


0035 


0738 


0621 


0903 


83 


535 


0966 


1088 


1151 


1233 


1310 


1398 


1481 


1563 


1846 


1738 


82 


537 


1811 


1893 


1975 


3058 


9140 


9222 


9305 


9387 


3460 


9553 


89 


538 


3834 


2718 


9796 


9881 


9983 


3045 


3127 


3200 


3391 


3374 


89 


589 


3456 


3538 


3630 


3703 


3784 


3866 


3048 


4030 


4113 


4194 


83 


530 


734976 


4358 


4440 


4523 


4004 


4885 


4707 


4849 


4931 


5013 


89 


531 


5095 


5176 


53.^ 


5340 


5432 


5303 


5535 


5667 


5748 


5630 


83 


533 


5013 


5093 


0075 


6156 


6938 


6330 


6401 


6433 


6584 


6646 


89 


533 


67»7 


6809 


6890 


6973 


7053 


7134 


7918 


7397 


7379 


7480 


81 


534 


7541 


7833 


7704 


7785 


7886 


7948 


8329 


8110 


8191 


8373 


81 


535 


8354 


8435 


8516 


8597 


8678 


8750 


8841 


80-^ 


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81 


536 


9165 


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0733 


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0693 


81 


537 


9974 


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.136 


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.378 


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.031 


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81 


538 


730783 


0863 


0944 


1024 


1105 


1188 


1286 


1347 


1428 


1508 


81 


539 


1589 


1069 


1750 


1830 


1911 


1991 


9079 


3153 


9233 


9313 


81 


540 


733394 


9474 


9555 


98.15 


9715 


3798 


9876 


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3037 


3117 


80 


541 


3197 


3278 


3358 


3438 


3518 


3598 


3679 


3750 


3839 


3919 


80 


543 


3999 


4079 


4160 


4310 


4320 


4400 


4480 


4560 


4640 


4730 


80 


543 


4800 


4880 


4960 


5040 


5120 


5200 


5270 


5350 


5439 


5519 


80 


544 


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5750 


5838 


5918 


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6078 


6157 


6937 


6317 


80 


545 


6397 


6476 


6558 


6835 


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6795 


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7034 


7113 


80 


546 


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7273 


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7431 


7511 


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7G70 


7749 


7839 


7936 


79 


547 


7987 


8067 


8148 


8225 


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8632 


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70 


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138d 


1487 


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1733 


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70 


553 


1939 


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9098 


9175 


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9111 


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79 


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3735 


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3118 


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3275 


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78 


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3388 


3687 


3745 


3823 


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3960 


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4136 


4915 


78 


555 


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4449 


4523 


4808 


4684 


4763 


4843 


4919 


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78 


556 


5075 


5153 


5231 


5309 


5387 


5485 


5543 


5631 


5693 


5777 


78 


557 


5855 


5933 


6011 


6089 


6167 


6245 


6323 


6401 


6479 


6558 


78 


558 


6834 


6713 


6790 


6868 


6945 


7023 


7101 


7179 


7236 


7334 


78 


550 


7413 


7489 


7567 


7645 


7722 


7800 


7878 


7935 


8033 


8110 


78 


560 


748188 


8366 


8343 


8421 


8198 


8576 


8653 


8731 


8808 


8885 


77 


561 


8963 


0040 


0118 


0195 


9272 


0350 


0427 


0504 


9382 


0659 


77 


503 


0736 


0814 


0691 


9968 


..45 


.123 


.900 


.977 


.351 


.431 


77 


563 


750508 


0586 


0663 


0740 


0817 


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0071 


1048 


1125 


1203 


77 


564 


1379 


1356 


1433 


1510 


1587 


1604 


1741 


1818 


1835 


1973 


77 


565 


9048 


9125 


!»93 


»79 


9356 


9433 


9509 


9388 


9883 


9740 


77 


566 


9816 


S893 


9970 


3047 


3123 


3300 


3277 


3333 


3130 


3506 


77 


567 


3583 


3660 


3736 


3813 


3839 


3986 


4043 


4119 


4195 


4272 


77 


568 


4348 


4425 


4501 


4578 


4354 


4730 


4807 


4383 


4960 


5036 


76 


560 


5119 


5189 


5965 


5341 


5417 


5494 


5570 


5046 


5722 


5799 


70 


570 


755875 


5051 


6QS7 


6103 


618} 


0358 


6332 


6406 


6484 


6560 


76 


571 


6636 


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6788 


6864 


6940 


7016 


7392 


7168 


7244 


7320 


76 


573 


7396 


7472 


7548 


7034 


7700 


7775 


7851 


7927 


8003 


8079 


76 


573 


8155 


8330 


8306 


8382 


8458 


8333 


8609 


8083 


8761 


8836 


76 


574 


8913 


8988 


0083 


9139 


0214 


0230 


0368 


9441 


9517 


9592 


76 


1575 


9668 


9743 


0S19 


9691 


0370 


..45 


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.198 


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.347 


75 


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760433 


0493 


0573 


0849 


0724 


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0875 


0950 


1035 


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75 


577 


1176 


1251 


1338 


1402 


1477 


1552 


1827 


1702 


1778 


1853 


75 


576 


1938 


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9078 


2153 


2^8 


3303 


3378 


3453 


9S29 


9604 


75 


570 


9679 


9754 


3839 


9904 


2978 


3053 


3128 


3233 


3378 


3353 


75 


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1 


1 1 


8 1 


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8 


1 9 ID. II 



36 




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BIO 


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7304 


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TOS7 


vn 


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8040 


8190 




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74 




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srso 


seoo 


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tm 




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9968 


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74 


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IML 


1734 


i»oe 


1881 


losa 


MSB 


1I0« 




3348 




592 


xm 




HW 




(615 


asea 






MDfl 






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3540 














fOOfl 


4079 






4208 






73 








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5100 






686 






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7M4 


7717 


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8079 




SM 


rmiii 


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won 


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73 






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73 
















1468 


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11184 




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73 


















3080 




















4261 


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4760 


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7177 


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6609 
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6680 
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7531 


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7744 


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1059 


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70 




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SSK 


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3073 


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70 




3UM 




323! 


3301 


3371 








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70 

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80 


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1078 


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3568 


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09 


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08 
















3887 






















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4016 




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0044 




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* Ti«LB or 


l^.mD4.R. 


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o 10.000. 


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BDeim 1 oite 1 631B ] 0384 


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8211 


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Biom 


0300 


0387 








9636 


0703 


OTTO 


9837 






HNM 








1173 


1340 




1374 








MS 






1709 


1776 




1910 












sw 








344S 






3648 






9847 








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3047 












344t 


3114 






3U1 


3MS 












4049 








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4514 


4S8) 


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4889 








SMS 












SH 




M44 




4777 




3910 








617S 








8308 


8374 














8838 




ssa 


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7367 


7433 


7499 




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7898 




7839 


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8038 


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81 


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7841 


7904 


7967 








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1948 


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Jill 


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iiraikV -B 


'I"- II 



38 




A TABI«B OP LOOARITHMt PKOM 1 


TO 10,000. 






1 *• 





1 


8 1 3 


4 1 5 1 6 1 


7 


1 8 


» 1 D. 1 


TOO 


845UIH 


5160 


5223 


5334 


5346 


5406 


5470 


5532 


5994 


5056 


68 


701 


5718 


5730 


5843 


5iM)l 


5056 


6028 


6090 


6151 


0313 


6375 


63 


702 


6337 


6399 


6411 


6523 


6585 


66(8 


6706 


6770 


6833 


6694 


68 


703 


<053 


7017 


7079 


7141 


73 J3 


7384 


7326 


7388 


7449 


7511 


63 


704 


7573 


7634 


760'» 


7753 


7819 


7881 


7943 


8004 


8066 


6138 


08 


705 


8189 


8251 


8312 


8374 


8415 


8497 


8559 


8620 


8683 


8743 


68 


706 


8805 


88:H) 


8J23 


8939 


9051 


0112 


9174 


9235 


9397 


9356 


61 


707 


9419 


9431 


9542 


9004 


90t>5 


9738 


9788 


9349 


9911 


9973 


61 


706 


850033 


0JJ5 


015!) 


0317 


0279 


0340 


04Q1 


0(82 


0521 


0385 


61 


700 


0646 


0707 


0769 


0630 


0891 


0953 


1014 


1075 


11J6 


1197 


61 


710 


851258 


1330 


1381 


1443 


1503 


1564 


1625 


1686 


1747 


1809 


61 


711 


187J 


1931 


19:)2 


8053 


8114 


8175 


3-236 


8-297 


3358 


3419 


61 


71-1 


8480 


8541 


S6>2 


3683 


8724 


3785 


3646 


3997 


8J66 


30-29 


61 


7ia 


3090 


3150 


3211 


3273 


3333 


3394 


»455 


3516 


3577 


3637 


61 


711 


36J6 


3750 


38-23 


3881 


3041 


4003 


4003 


41-24 


4185 


4345 


61 


715 


4)C6 


4367 


4428 


4488 


4549 


4619 


4670 


4731 


4793 


4852 


61 


716 


4913 


4374 


50J4 


5905 


5156 


5216 


5377 


5337 


5393 


5459 


61 


717 


5519 


5580 


5640 


5701 


5751 


5832 


5833 


5943 


6003 


6064 


61 


718 


6114 


6185 


6245 


6300 


6336 


6437 


6487 


6548 


6606 


6668 


60 


713 


6739 


6789 


6850 


0910 


6970 


7031 


7091 


7153 


7313 


7373 


60 


720 


8S7333 


7393 


7453 


7513 


7574 


7834 


7634 


7755 


7815 


7875 


60 


7«1 


7935 


79J5 


8058 


8116 


8176 


8238 


8397 


8357 


8417 


8477 


60 


721 


8537 


8597 


8357 


8718 


8n8 


8833 


ooJH 


8953 


9018 


9078 


60 


723 


9133 


9193 


0358 


9318 


0379 


9439 


9499 


9559 


0619 


9679 


60 


7^ 


9739 


9799 


9359 


9918 


9078 


..33 


• .Uo 


.158 


.318 


.278 


60 


725 


860334 


0393 


0453 


0518 


0578 


0637 


0897 


0757 


0617 


0877 


00 


735 


0937 


0090 


1058 


1116 


1176 


1336 


1295 


1355 


1415 


1475 


60 


727 


1534 


1591 


1851 


1714 


1773 


1833 


1893 


1952 


3013 


3073 


60 


7V8 


8131 


3191 


3351 


3310 


3379 


3439 


2489 


3549 


36 J8 


2668 


60 


739 


3733 


3737 


2»n 


3908 


8936 


3025 


3035 


3144 


3294 


3363 


00 


730 


863323 


3383 


3443 


3501 


3561 


3620 


3T80 


3739 


3799 


3858 


90 


731 


3917 


3977 


4036 


4090 


4155 


4314 


4274 


4333 


4392 


4452 


50 


732 


4511 


4570 


4639 


4689 


4748 


4808 


4887 


4928 


4985 


5045 


90 


733 


5104 


5163 


5^2 


5282 


5341 


5400 


5459 


5519 


5578 


5637 


99 


734 


5696 


5755 


5314 


5874 


5933 


5992 


6051 


6110 


6169 


63-28 


90 


735 


6287 


6346 


6405 


6415 


6524 


6583 


6643 


6701 


6780 


6619 


90 


736 


6878 


6937 


699> 


7055 


7114 


7173 


7232 


73iil 


7359 


7409 


99 


737 


7467 


7523 


7585 


7644 


7703 


7783 


7821 


788J 


7939 


7998 


99 


738 


8056 


8115 


8174 


8333 


8392 


8350 


8409 


8408 


8527 


8588 


90 


739 


8644 


8703 


8762 


8821 


6879 


8938 


8997 


9058 


9114 


9173 


90 


740 


860332 


0299 


9349 


9408 


0466 


0535 


9584 


9642 


9701 


9760 


99 


741 


9818 


9877 


9935 


0994 


..53 


.111 


.170 


.223 


.337 


.345 


90 


743 


870401 


04G2 


0521 


0579 


0633 


0893 


0755 


0313 


0872 


0030 


58 


743 


0989 


1047 


1106 


1184 


1223 


1381 


1339 


1393 


1456 


1515 


98 


744 


1573 


1631 


1890 


1748 


18)6 


18.55 


19i3 


1981 


3040 


3098 


98 


' 745 


3150 


3315 


3273 


3331 


3389 


3448 


2503 


3564 


3822 


3681 


58 


740 


8739 


3797 


3355 


3913 


3973 


3J39 


3083 


3148 


3-204 


3363 


58 


747 


3321 


3379 


3437 


3495 


3553 


3611 


38!'.9 


37-27 


3785 




56 


743 


3902 


39vl0 


4018 


4078 


4134 


4192 


4350 


4308 


4386 


4434 


58 


749 


4482 


4540 


4598 


4858 


4714 


4772 


4830 


4888 


4045 


5003 


58 


750 


875061 


5119 


5177 


5335 


5393 


5351 


5409 


5486 


5534 


5583 


58 


751 


5640 


5898 


5756 


5813 


5871 


5929 


5987 


6045 


6102 


6160 


58 


753 


0218 


6276 


6333 


6391 


6449 


6507 


6564 


6633 


6663 


6737 


58 


753 


6795 


6833 


6910 


6988 


7028 


7083 


7141 


7199 


7356 


7314 


58 


754 


7371 


7429 


7487 


7544 


7602 


7659 


7717 


7774 


7833 


7889 


68 


755 


7947 


8001 


8063 


8119 


8177 


8-234 


8292 


8349 


8407 


8464 


57 


756 


8532 


8579 


8837 


Si^)! 


8753 


8809 


8368 


8924 


8981 


9030 


57 


757 


0996 


0153 


(»11 


9288 


9325 


9383 


9440 


0497 


9555 


9612 


57 


758 


9669 


9730 


0784 


9841 


9896 


9958 


..13 


..70 


.137 


.185 


57 


750 


880243 


0299 


0356 


0413 


0471 


0538 


0585 


0643 


0699 


0756 


57 


N. 


1 


1 1 


3 1 3 1 


4 1 5 1 6 


7 


1 8 


1 9 


1 D- 







A TABLE OF 


LOOARITBMS FROM 1 TO 10,000. 




39 


N. 


1 


1 


2 


3 


4 


5 1 6 


7 


1 8 


9 1 D. ;| 


7«0 


flRoeu 


0671 


0928 


0985 


1043 


1099 


1156 


1313 


1371 


132R 


57 


7B1 


1385 


1442 


1499 


1556 


1613 


1670 


1727 


1784 


1841 


1898 


57 


7«2 


1955 


3012 


2069 


3136 


3183 


2340 


2-297 


2354 


3411 


3468 


57 


1 763 


3525 


2581 


3638 


3695 


3753 


3609 


2866 


3923 


3980 


3037 


57 


764 


3093 


3150 


3207 


3364 


:{321 


33T7 


3434 


3491 


3546 


3605 


57 


765 


3661 


3718 


3775 


3832 


3888 


3945 


4002 


4059 


4115 


4172 


57 


766 


4329 


4385 


4342 


4399 


4455 


4513 


4569 


4625 


4H82 


4739 


57 


767 


4795 


4852 


4909 


4985 


5022 


5078 


5135 


5192 


5248 


5305 


57 


7G8 


5361 


5418 


5474 


5531 


5587 


5644 


5700 


5757 


5813 


5870 


57 


760 


5928 


5963 


6030 


6096 


6152 


6209 


6365 


6321 


6378 


6434 


56 


770 


880491 


6547 


6604 


6660 


6716 


67rj 


6829 


6885 


6!U3 


6096 


56 


771 


7054 


7111 


7167 


7223 


7280 


7336 


7392 


7449 


7505 


7561 


56 


772 


7617 


7674 


7730 


7788 


7842 


7898 


7955 


6011 


8067 


6133 


56 


773 


8179 


8236 


8292 


8^48 


8404 


8460 


8516 


8573 


6639 


6685 


56 


774 


8741 


8797 


8853 


8909 


8985 


9021 


9077 


9134 


9190 


9246 


56 


775 


9302 


9358 


9414 


9470 


9a28 


9582 


9638 


9694 


9750 


9606 


56 


776 


9863 


9D1S 


9974 


..30 


..83 


.141 


.197 


.353 


.309 


.365 


56 


77? 


800421 


0477 


0533 


0589 


0645 


0703 


0756 


0813 


m\8 


0924 


56 


778 


093.1 


1035 


1091 


1147 


1-203 


1259 


1314 


1370 


1426 


1482 


56 


779 


1537 


1593 


1649 


1705 


1760 


1816 


1872 


1928 


1983 


2939 


56 


780 


892095 


2150 


2206 


2362 


2317 


2373 


2429 


2484 


3540 


3595 


5S 


781 


2351 


2707 


2762 


2818 


8873 


2923 


2985 


3040 


3096 


3151 


.56 


78i 


3207 


32:;2 


3318 


3373 


3439 


3484 


3540 


3595 


3651 


3706 


56 


783 


3762 


3817 


3873 


3928 


3984 


4039 


4094 


4150 


4205 


4261 


55 


784 


4316 


4371 


44-27 


4482 


4538 


4533 


4648 


4704 


4759 


4814 


55 


785 


4870 


4925 


4980 


5038 


5091 


5141) 


5-201 


5257 


5312 


5367 


55 


786 


5423 


5478 


5533 


5583 


5644 


5699 


5754 


5809 


5864 


5929 


55 


787 


5975 


6O30 


6085 


6140 


6195 


6-251 


6336 


6361 


0416 


6471 


55 


788 


6526 


6581 


6636 


6692 


6747 


6802 


6857 


6912 


6967 


7022 


55 


789 


7077 


7133 


7187 


7242 


7397 


735S 


7407 


7462 


7517 


7572 


55 


790 


897627 


7682 


7737 


7792 


7847 


7932 


7957 


6012 


6087 


61SS 


55 


791 


8176 


8231 


8286 


8341 


8393 


8451 


8536 


8561 


8615 


6670 


55 


792 


8725 


878;) 


8835 


as90 


8344 


8933 


9054 


9109 


9164 


9218 


55 


793 


0273 


9328 


9383 


9437 


0492 


9547 


9802 


0656 


9711 


9766 


55 


794 


9821 


9375 


9930 


9935 


..39 


 9 9^ 


.149 


.203 


.258 


.312 


55 


795 


900367 


W22 


0476 


0531 


0585 


0640 


0695 


0749 


0604 


0859 


55 


793 


0913 


0918 


1022 


1077 


1131 


118') 


1-240 


1295 


1349 


1404 


55 


797 


1458 


1513 


1587 


1622 


1676 


1731 


1785 


1840 


1894 


1948 


54 


793 


30U3 


2057 


2112 


2168 


3231 


2275 


2329 


2384 


2438 


3492 


54 


799 


2547 


3601 


2655 


2710 


3764 


2818 


3873 


2927 


3981 


3036 


54 


800 


903090 


3144 


3199 


3253 


3307 


3361 


3416 


3470 


3524 


3578 


54 


, 801 


3633 


3687 


3741 


3795 


3849 


3904 


3953 


4012 


4066 


4120 


54 


802 


4174 


^29 


4283 


4:«7 


4391 


4445 


4499 


4553 


4607 


4601 


54 


803 


4716 


4770 


4824 


4878 


4933 


4988 


5040 


5094 


5148 


5202 


54 


804 


£256 


5310 


5364 


5418 


5473 


5&28 


5580 


5634 


56d3 


5742 


54 


805 


5796 


5850 


5934 


5958 


6012 


6066 


6119 


6173 


6-227 


6281 


54 


806 


6335 


6389 


6443 


6497 


6551 


6604 


6658 


6712 


6766 


6820 


54 


807 


6874 


6927 


6981 


7035 


7083 


7143 


7198 


7-250 


7304 


7358 


54 


808 


7411 


7465 


751D 


7573 


7028 


7680 


7734 


7787 


7841 


7895 


54 


809 


7949 


8003 


8J58 


8110 


8183 


8217 


8370 


8324 


8378 


8431 


54 


810 


G08485 


8539 


8592 


8846 


8899 


8753 


8807 


8880 


6914 


6967 


54 


811 


9021 


9074 


9128 


9181 


9235 


9283 


9342 


9396 


9449 


0503 


54 


813 


9556 


9610 


9863 


9718 


9770 


9823 


9877 


9930 


9984 


..37 


53 


813 


010091 


0144 


0197 


0251 


0304 


0358 


0411 


0464 


0518 


0571 


53 


814 


06^ 


0678 


0731 


0784 


0838 


0831 


0944 


0998 


1051 


1104 


53 


815 


1158 


1311 


1284 


1317 


1371 


14-24 


14T7 


1530 


1584 


1637 


53 


816 


1690 


1743 


179? 


1J50 


1903 


1956 


2039 


3063 


3116 


2169 


53 


817 


3322 


2275 


2328 


2381 


3435 


2488 


2541 


8594 


3647 


2700 


53 


818 


2753 


3806 


3859 


3913 


398* 


3019 


3072 


3125 


3178 


3231 


53 


819 


3384 


3337 


3390 


:)443 


3496 


3549 


3602 


3655 


3708 


3761 


53 


N. 


1 1 


1 1 


3 


3 


4 


5 1 6 


1 7 


1 8 


1 » 1 1> 1 



40 




1 TIBLI 




TO 10,000. 






H. 


a 1 1 1 a 


3 


4 1 i 


8 1 7 


a 


D.II 


»r 


«l3gl4 






■^3 


Tom 


WTO 


4l3B 


"iTsi 


ImT 


-4S0- 


33 


ni 


4343 


43B6 




4VB 


4US 


4608 


4060 




4766 


481S 










4Bn 


»30 


MBS 


S130 




5*41 


5394 




53 




9400 


»S3 


SMU 


SH8 


acii 


ue4 






SB29 






at 


5MT 


aeo 












ftue 








8U 


«4M 


C30T 


nsg 




0064 














BM 


XSSO 




TMU 




TIOO 


7143 


7393 


7348 


7400 






tan 


IMM 






Tim 




7768 


TWO 


7OT3 


1K» 


7978 


sa 


ffie 




80B3 


81» 




8M0 










B303 


33 


89S 


8US 


8*07 


tnw 
















33 


830 


"wM 


B130 


0183 


^ 


MID 


^ 


JW^ 


9444 


9490 
..10 


9540 


» 


83a 


W)1S3 










0384 








aw3 


sa 


833 


064S 


OOOT 


0748 


0801 


0833 


0006 


oose 








91 




lite 


1318 


K70 


l»3 


1374 


1490 




1330 


isn 


1(34 


H 








17W 


1843 


1804 


1940 




WW 






91 




ffioe 




331(1 




!M14 


34<I0 


3318 


»7« 




HI* 


9S 










aesi 


»S3 


»«5 


3037 




3140 


3ID3 


51 






we 




3300 




3303 




3007 


3038 


3710 






MOT 


3814 


386S 




3900 












99 


8M 


iwmi 














4641 


4603 












4H99 




SOOJ 


3054 












8«t 




53G4 














5723 


S770 




S43 




M78 


»31 


5963 










0340 






§44 


0343 


63M 
















6803 








6«oe 














73C8 






Mfl 




74« 
















7839 




s 


8390 


nss 


TWO 

B4ee 


isS 


emi 


HBSa 


1™ 


W4S 
8754 


8803 


883 


SI 




eaoe 


»i» 


WIO 


9061 








9900 


9317 


9308 






smie 




RRI 








979S 




0897 


9870 






■030 












.230 






.389 




US 


030440 














OTOfl 








saa 


DStfl 


IDOO 










ISM 


1305 










14S» 


1500 










1763 




1B65 




5 


8U 


i»ee 


901- 








wso 


9971 


9322 


9373 




S 


SMI 




8534 




2626 






977S 




S879 




s 


857 




3031 


30W 








3983 




3380 




5 


8» 








3039 




JT40 






3^3 




51 


8W 




4044 








4S46 


4906 










8W 


B344ge 




4500 


4(130 










4009 


4053 








MM 


3104 




K»a 


5SM 


5308 




5406 






§sa 


UOT 






36.'« 




1759 


5809 




39 D 


SD60 




en 




HIOl 


6111 






6863 


0313 










864 


UI4 


















0900 




eiu 


TOM 


TOOO 




716T 
















am 








70ns 


















8010 


8000 


llsn 


WTO 


BTM 


mo 


Pm 


8870 


8490 


8470 


50 


sea 










ffSO 




9390 


9369 






30 


ffTO 


B30SI9 


osss 
oosa 


o?!e 


9008 
01B8 


(el 


B7B9 
0367 


981B 


0869 


wn 


9908 


50 






OHO 




0660 






0815 


neos 


0915 










1004 




















8T4 


ISII 






lOBO 








1890 


1900 






8T5 


me 


n» 






aso7 




9300 










878 


3S04 


KM 












9851 


9901 




90 




3000 




3on 










3340 


3300 


344S 


M 


878 


34DS 




3KI3 


3S4J 








3811 


3800 




49 


S7« 




«03e 


4aee 




4188 


4336 




«0S 


43S4 


M33 


49 


IT. 


0~ 


1 


s 


""3~ 


* 


* 


* 


' 


8 


9 


"s: 







A TABLE OP LOaARlTBMS PROM t ' 


ro 10,000. 




41 


N. 


1 1 1 


1 2 


1 3 


4 


1 5 


1 6 1 7 


1 8 


1 9 1 B. 1 


880 


944483 


4572 


4581 


4631 


4680 


4729 


4779 


4828 


4877 


40S7 


49 


881 


4976 


5025 


5074 


5124 


5173 


5222 


5873 


5321 


5370 


5419 


49 


882 


5469 


5518 


5567 


5616 


5665 


5715 


5764 


5813 


5862 


5912 


49 


883 


3981 


6010 


6059 


6106 


6157 


6207 


6256 


6305 


6354 


6403 


49 


884 


64^ 


6501 


6551 


6600 


6649 


6696 


6747 


6798 


6845 


6694 


49 


885 


6943 


6992 


7041 


7090 


7140 


7189 


7238 


7287 


7336 


7385 


49 


886 


7434 


7483 


7532 


7581 


7630 


7679 


7728 


7777 


7826 


7875 


49 


887 


7934 


7973 


8022 


8070 


8119 


8168 


8217 


fTOaO 


8:115 


8364 


40 


888 


8113 


8462 


8511 


8560 


8609 


8357 


8706 


8755 


8604 


8853 


40 


889 


891M 


8951 


8999 


9048 


9097 


9146 


9195 


9244 


9292 


9341 


49 


800 


949393 


9439 


9488 


9536 


9585 


9634 


9683 


9731 


9780 


9H29 


49 


891 


9878 


9926 


9975 


..24 


..73 


.121 


.170 


.219 


.267 


.316 


49 


892 


959365 


0414 


0462 


0511 


0560 


0606 


0657 


07 J6 


0754 


0803 


49 


893 


0651 


0900 


0949 


0997 


1046 


1095 


1143 


1192 


1240 


1269 


49 


894 


1338 


1386 


1435 


1483 


l.'n2 


1580 


16-29 


1677 


1726 


1775 


49 


895 


1823 


1872 


1920 


1969 


2017 


2066 


2114 


2163 


2211 


2260 


48 


896 


3308 


2356 


2405 


2453 


2502 


2550 


2599 


2647 


2696 


2744 


46 


897 


2792 


2841 


2889 


2938 


2966 


3034 


3063 


3131 


3180 


3228 


46 


898 


327d 


3325 


3373 


3421 


3470 


3518 


3566 


3615 


3663 


3711 


48 


890 


3760 


3806 


3356 


3905 


3933 


4001 


4043 


4098 


4146 


4194 


48 


900 


954243 


4291 


4339 


43OT 


4435 


*4484 


4532 


4580 


4028 


4677 


48 


901 


4725 


4773 


4821 


4860 


4918 


4966 


5014 


5062 


5110 


5158 


48 


9G3 


5207 


5255 


5303 


5331 


5399 


5447 


5495 


5543 


5592 


5640 


48 


903 


5688 


5736 


5784 


.5832 


5880 


5928 


5976 


60^ 


6072 


6120 


48 


901 


6168 


6216 


6265 


i;3]3 


6361 


6409 


6457 


6505 


Kna 


6601 


48 


905 


6619 


6697 


6745 


6793 


6840 


^HU 


6936 


60&4 


7032 


7060 


48 


906 


7128 


7176 


7221 


7272 


7320 


'm68 


7416 


7464 


7512 


7550 


48 


907 


7607 


7655 


770J 


7751 


7r9J 


7847 


7894 


7942 


7990 


8038 


48 


gos 


8066 


8134 


8181 


8229 


8277 


8325 


8373 


8421 


84(38 


8516 


48 


909 


8534 


8612 


8650 


8707 


8755 


8833 


8853 


8898 


8946 


annt 
own 


48 


910 


959041 


9089 


9137 


9185 


9232 


9280 


9328 


9375 


9423 


9471 


48 


911 


9518 


9566 


9614 


9661 


9709 


9757 


9604 


9852 


990O 


9947 


48 


912 


9995 


..42 


..93 


.138 


.185 


.233 


.280 


.328 


.376 


.423 


48 


913 


930471 


0518 


0566 


0613 


0861 


0709 


0756 


06m 


0651 


0699 


48 


914 


0946 


0994 


1041 


1080 


1136 


1184 


1231 


1279 


1326 


1374 


47 


915 


1421 


1469 


1516 


1563 


1611 


1658 


1706 


1753 


1801 


1848 


47 


916 


1893 


1943 


1990 


2038 


2085 


2132 


2180 


2-227 


2275 


2322 


47 


917 


2i69 


»417 


2464 


2511 


2559 


2006 


2653 


2701 


2748 


2795 


47 


918 


2843 


2890 


2937 


29a3 


3032 


3079 


3126 


3174 


3221 


3268 


47 


919 


3316 


3363 


3410 


3457 


3504 


3552 


3599 


3646 


3693 


3741 


47 


990 


063788 


3835 


3882 


3929 


3977 


4024 


4071 


4118 


41&5 


4212 


47 


021 


4260 


4307 


4354 


4401 


4448 


4495 


4542 


4393 


43:r7 


4684 


47 


022 


4731 


4778 


4825 


4872 


4919 


4966 


5013 


5061 


5103 


5155 


47 


923 


5202 


5249 


5206 


5343 


5393 


5437 


5484 


5531 


5578 


5625 


47 


024 


5672 


5719 


5706 


5813 


5860 


5307 


5954 


6001 


6048 


6095 


47 


925 


6142 


6189 


6236 


62K:{ 


6321 


6376 


6423 


6470 


6517 


6564 


47 


926 


6611 


6658 


6705 


6752 


6793 


6845 


6892 


6939 


6986 


7033 


47 


927 


7063 


7127 


7173 


7220 


7267 


7314 


7361 


7408 


7454 


7501 


47 


928 


7548 


7595 


7642 


7688 


7735 


7782 


7829 


7875 


79-22 


7969 


47 


929 


8016 


8063 


8109 


8156 


8203 


8249 


8296 


8343 


8300 


8436 


47 


930 


968483 


8530 


8576 


8623 


8670 


8716 


8763 


8810 


8656 


8903 


47 


931 


8950 


8996 


9043 


9093 


9136 


9183 


9229 


9276 


9323 


0369 


47 


932 


9416 


9463 


9509 


9556 


9602 


9649 


9695 


9742 


9789 


9835 


47 


933 


9882 


9928 


9975 


..21 


..68 


.114 


.161 


.207 


.254 


.300 


47 


034 


970347 


0393 


0440 


0486 


05.T1 


0579 


0626 


0672 


0719 


0765 


46 


035 


0612 


0856 


092)4 


0951 


0997 


1044 


1090 


1137 


1183 


1229 


46 


936 


1276 


1322 


1369 


1415 


1461 


1508 


1554 


1601 


1647 


1693 


46 


937 


1740 


1786 


1832 


1879 


1925 


1971 


2018 


2064 


2110 


2157 


46 


938 


2203 


2249 


2295 


2342 


2388 


2434 


2481 


2527 


2573 


2619 


46 


939 


2666 


2712 


2758 


2804 


2851 


2897 


2943 


2989 


3035 


3062 


46 


N. 


1 1 1 


2 


1 3 


4 


1 5 


1 6 1 7 


1 8 


1 » Il>-|| 



42 



A TABLE OF LOGARITHMS FROM 1 TO 10,000. 



N. 1 





1 1 


1 2 


1 3 


1 4 


1 5 


1 6 


7 


8 1 


9 


1 ». 


MO 


973138 


3174 


3330 


3366 


3313 


3353 


3403 


»451 


3497 


3343 


46 


Ml 


3593 


3'i36 


3533 


3723 


3774 


3330 


3836 


3913 


3959 


40i)5 


46 


943 


4051 


4097 


4143 


4183 


4335 


4331 


4337 


4374 


4430 


4466 


46 


943 


4512 


4358 


4604 


4530 


4696 


4743 


4783 


4834 


4830 


4936 


46 


944 


4073 


5018 


5334 


5110 


5156 


53>3 


.5248 


5394 


3340 


5386 


46 


945 


5432 


5478 


5331 


5370 


5616 


5563 


5707 


5733 


5799 


5845 


46 


946 


5801 


5937 


5aS3 


6030 


6075 


6121 


6167 


6213 


6358 


6304 


46 


947 


6350 


6396 


6443 


6488 


6333 


6579 


66-25 


6671 


6717 


6763 


46 


948 


6808 


6d.>4 


6300 


6346 


6393 


70:i7 


7083 


7139 


7175 


7230 


46 


949 


7366 


7313 


7358 


7403 


7449 


7433 


7541 


7586 


7633 


7678 


46 


953 


977724 


7769 


7815 


7861 


7306 


7933 


7938 


8043 


8069 


8135 


4« 


951 


8181 


8226 


8273 


8317 


8363 


8409 


8454 


8330 


8546 


8591 


46 


953 


8637 


8683 


8?28 


8774 


8819 


8865 


83 J 1 


8956 


9003 


9047 


46 


953 


9393 


9138 


9184 


9330 


9275 


9331 


9366 


9113 


9457 


9503 


46 


954 


9348 


9594 


9339 


9383 


9730 


9776 


9621 


9667 


9i)13 


9058 


46 


955 


960033 


0343 


0094 


0140 


0183 


0231 


0276 


03-23 


0367 


0413 


45 


956 


(M58 


0503 


0349 


0394 


0640 


0685 


0730 


0776 


0631 


0667 


45 


957 


0912 


0937 


1003 


1)48 


1093 


1139 


1184 


!-£» 


1375 


13-20 


45 


958 


1363 


1411 


1456 


1301 


1547 


1592 


1637 


1683 


17-28 


1773 


45 


959 


1819 


1864 


1909 


1354 


2000 


8045 


3300 


3135 


3181 


3336 


45 


963 


962271 


3316 


3363 


3407 


34i3 


3497 


3513 


3388 


3633 


3678 


45 


961 


2723 


3769 


3814 


3339 


3304 


3049 


3994 


3040 


3085 


3130 


45 


962 


3173 


323) 


33:'»3 


3310 


3356 


3401 


3446 


3431 


3536 


3581 


45 


963 


3636 


3671 


3716 


37^2 


3307 


3833 


3897 


3943 


3987 


4033 


45 


964 


4077 


413i 


4167 


4212 


4357 


4.11)3 


4347 


4.392 


4437 


4433 


45 


965 


4527 


4572 


4617 


4663 


4707 


4752 


4797 


4843 


4887 


4333 


45 


966 


4977 


5032 


5U67 


5113 


5157 


5303 


5347 


5232 


5337 


5383 


45 


967 


5426 


5471 


5516 


5561 


5606 


5;i5i 


5696 


5741 


5786 


5830 


45 


968 


5875 


5030 


5965 


6010 


6033 


6100 


6144 


6189 


6234 


6379 


45 


969 


6334 


6360 


6413 


6458 


6503 


6518 


6503 


6637 


66R2 


6727 


45 


970 


983772 


ten 


6861 


6906 


6951 


6996 


7040 


7085 


7130 


7175 


45 


971 


"^19 


7264 


7309 


7353 


7398 


7443 


7488 


7333 


7577 


7633 


45 


972 


7666 


7711 


7756 


7800 


7845 


7890 


7934 


7979 


8334 


8068 


45 


973 


8113 


8157 


83 J3 


8-247 


8-291 


8336 


8381 


84-25 


8470 


8514 


45 


974 


a550 


8604 


8648 


8693 


8737 


8783 


88-26 


8871 


8916 


8960 


45 


975 


9003 


9049 


9091 


9138 


9183 


92-27 


9373 


9316 


9361 


9405 


45 


976 


9453 


9404 


9539 


9533 


9:38 


9673 


9717 


9761 


9606 


9830 


44 


977 


9833 


9930 


9333 


..38 


..73 


.117 


.161 


.306 


.350 


.394 


44 


978 


9933:)9 


0333 


0438 


0472 


0516 


0561 


0605 


0630 


0694 


0738 


44 


979 


0783 


0827 


0671 


0916 


0960 


1004 


1049 


1093 


1137 


1183 


44 


960 


991326 


1-270 


1315 


1339 


1403 


1448 


1493 


1536 


1580 


1635 


44 


981 


1669 


1713 


1738 


1803 


1846 


1890 


1935 


1979 


3023 


3067 


44 


9R-2 


3111 


3156 


3300 


3344 


3338 


33:« 


3377 


3421 


3165 


2509 


44 


083 


3534 


3598 


3G43 


3686 


3730 


3774 


3819 


38ii3 


3307 


3951 


44 


934 


3395 


3039 


3083 


3127 


3172 


3316 


3260 


3304 


3348 


3393 


44 


985 


3436 


3480 


3324 


35r)8 


3013 


3637 


3701 


3745 


3789 


3833 


44 


986 


3877 


3931 


3965 


4009 


4033 


4iI97 


4141 


4133 


4330 


4273 


44 


987 


4317 


4361 


4405 


4449 


4493 


4537 


4381 


4633 


4669 


4713 


44 


988 


4757 


4831 


4843 


4889 


4933 


4977 


5031 


5)65 


5106 


5133 


44 


939 


5196 


5340 


5284 


5338 


5373 


5416 


5460 


5504 


5547 


5591 


44 


090 


995635 


5679 


57-23 


5767 


5811 


5834 


5896 


5042 


5086 


6030 


44 


991 


6374 


6117 


6161 


6-205 


6349 


6-293 


63:t7 


6.130 


6434 


6468 


44 


903 


6513 


6555 


6393 


6643 


6687 


6731 


6774 


6818 


6883 


C906 


44 


993 


6919 


6993 


7037 


7080 


7134 


7168 


7313 


7-255 


7390 


7343 


44 


994 


7386 


7430 


7474 


7317 


7561 


7605 


7648 


7692 


77:w 


7779 


44 


905 


7833 


7867 


7910 


7034 


7908 


8041 


8085 


81-20 


8173 


8316 


44 


^6 


8259 


8303 


8347 


8390 


8434 


8177 


8531 


8364 


8608 


8653 


44 


997 


8693 


8739 


8733 


8326 


8869 


8913 


8956 


9000 


9043 


9087 


44 


998 


0131 


9174 


9213 


0-261 


9305 


9348 


9392 


9435 


0479 


9533 


44 


999 


9365 


9609 


9632 


9(>06 


9739 


9783 


9826 


9870 


0913 


9957 


43 


N. 1 


1 1 1 3 


3 


1 4 


1 5 1 6 1 7 1 8 1 9 1 D. 1 



TABLE 



or 



LOGARITHMIC SINES, COSINES, 
TANGENTS, AND COTANGENTS 



roB 



EVERY DEGEEE AND MINUTE 



OF THE QUADRANT. 



NoTB. — The minutes in the left-hand column of each page, increas- 
ing downwards, belong to the degrees at the top ; and those increasing 
upwards, in the right-hand column, belong to the degrees below. 



43 



44 




(0 Degree.) a table 


OP LOGARITHMIC 






M. 1 


Sine 1 


D 1 


Cosine. 


D-l 


Tang. 


D. 1 


Cotang. 







O.'VXXKW 




10.000000 




0.000000 




Infinite. 


60 


1 


6.t63736 


501717 


000000 


00 


6.463726 


501717 


13.536274 


59 


3 


764758 


393485 


000000 


00 


764756 


393483 


335344 


58 


3 


940847 


306331 


ouoooo 


00 


940647 


308231 


059153 


57 


4 


7.065786 


161517 


000000 


00 


7.065786 


161517 


13.934314 


56 


5 


163606 


131968 


000000 


00 


163696 


131969 


837304 


55 


6 


341877 


111575 


9.999999 


01 


341878 


111578 


758123 


54 


7 


308834 


96653 


vaifsWf 


01 


306825 


9;)i53 


691175 


53 


8 


366816 


85354 


jfaWrilv 


01 


366817 


85254 


633 183 


53 


9 


417968 


76383 


WamfaV 


01 


417970 


76363" 


^ 582030 


51 


10 


463735 


6R9A8 


OQOQQ0 


01 


463727 


63938 


536273 


50 


11 


7.505118 


03981 


9.W0VO6 


01 


7.505130 


62981 


18.494880 


49 


13 


543906 


57936 


999997 


01 


543909 


57933 


457091 


48 


13 


577668 


53641 


OQQQQ7 


01 


577672 


53642 


432338 


47 


14 


600653 


49938 


999996 


01 


609657 


49939 


390143 


46 


15 


639816 


46714 


999996 


01 


8198-30 


46715 


360180 


45 


16 


667845 


43881 


999995 


01 


667849 


43882 


3:13151 


44 


17 


694173 


41373 


999995 


01 


69(179 


41373 


305821 


43 


18 


718997 


39135 


vinfinn 


01 


719003 


39136 


360997 


43 


19 


743477 


37137 


990993 


01 


743484 


37138 


357516 


41 


SO 


764754 


35315 


909993 


01 


764761 


35136 


335339 


40 


31 


7.785943 


33673 


"'WaVai 


01 


7.785951 


33673 


13.314049 


39 


33 


806146 


33175 


999991 


01 


806155 


33176 


193845 


38 


33 


835451 


30605 


999990 


01 


825460 


30606 


174540 


37 


34 


SA3SU 


39547 


999989 


02 


843944 


39549 


1.56056 


36 


35 


861663 


383R8 


999988 


03 


861674 


38393 


138336 


35 


36 


878695 


37317 


VifrnfaOO 


03 


878708 


27318 


131393 


34 1 


37 


895085 


36323 


999937 


03 


895099 


36325 


104901 


33 


38 


910679 


35399 


999988 


02 


910394 


35401 


089106 


33. 


39 


936119 


34538 


999985 


03 


936134 


34540 


073866 


31 


30 


940643 


33733 


999983 


03 


040658 


33735 


059143 


30 


31 


7.955083 


23980 


9.999932 


03 


7.955100 


ies2981 


13.044900 


29 


33 


968870 


3^73 


999381 


03 


9o8889 


32275 


031111 


38 


33 


982233 


31608 


999980 


03 


982253 


21610 


017747 


27 


34 


995198 


30981 


999979 


03 


995219 


30983 


004781 


36 


35 


8.007787, 


30390 


999977 


03 


8.007809 


30392 


11.992191 


35 


36 


020021 


19831 


999976 


02 


020045 


19833 


979955 


34 


37 


031919 


19302 


999975 


02 


031915 


19305 


968055 


23 


38 


043501 


18801 


999973 


03 


043537 


18803 


956473 


23 


39 


054781 


18335 


999972 


02 


054809 


18327 


945191 


31 


40 


065776 


17873 


999971 


02 


065806 


17874 


934194 


30 


41 


8.076500 


17441 


9.999969 


(K2 


8.076531 


17444 


11.923469 


19 


43 


066965 


17031 


OQQQ'M 
VSWSfvJO 


03 


086997 


nou 


913003 


18 


43 


097183 


i&m 


999906 


02 


097217 


16642 


902783 


17 


44 


107167 


16365 


999964 


03 


107302 


16268 


892797 


16 


45 


116926 


15908 


999963 


03 


116963 


15910 


883037 


15 


46 


126471 


15566 


999961 


03 


136510 


15i>68 


873490 


14 


47 


135810 


15238 


999959 


03 


135851 


15241 


864149 


13 


48 


144953 


14924 


909958 


03 


144996 


14927 


85.5004 


13 


49 


153907 


14622 


999956 


03 


153952 


14627 


846048 


11 


50 


162t}81 


14333 


999U54 


03 


162727 


14336 


837373 


10 


51 


8.171380 


14054 


9.999952 


03 


8.171328 


14057 


11.838673 


9 


53 


J79713 


13786 


999950 


03 


179763 


13790 


830337 


8 


53 


187965 


13529 


999948 


03 


188036 


13532 


811964 


7 


54 


196103 


13280 


999946 


03 


196156 


13284 


803844 


6 


55 


304070 


13041 


999944 


03 


304126 


13044 


795874 


5 


50 


311895 


12810 


999943 


04 


311953 


12814 


788047 


4 


57 


319581 


13587 


999940 


04 


319641 


12590 


780350 


3 


58 


337134 


1337-i 


999938 


04 


5^27195 


13376 


7T2805 


3 


59 


334557 


13164 


999936 


04 


334631 


13168 


765379 


1 


60 

1 1 


341855 


11963 


999934 


04 


341931 


11967 


758079 





Oadne 




1 Sine 


1 


1 Cotang. 


1 


1 Tang. 


IBt 



89 



" 




SIMBS 


AND TANOKNTB. (1 Dej 


pnee.) 




48 


» 




Hm 1 


». 1 


CMlne 1 D. 1 


Ttog. 


D. 


1 OolMf . 1 1 







8.341855 


11963 


9.900934 


04 


8.341031 


11067 


11.758079 


60 




1 


249033 


11768 


999933 


04 


340103 


11773 


750898 


50 




3 


856004 


11560 


909929 


04 


256165 


11584 


743835 


58 




3 


263043 


11366 


lllWv«7 


04 


363115 


11403 


736885 


57 




4 


360681 


11331 


909935 


04 


960056 


11235 


730044 


56 




5 


276U14 


11050 


909932 


04 


37G691 


11054 


723309 


55 




6 


383343 


10883 


90M^20 


04 


383333 


10887 


7166T7 


54 




7 


3811773 


10731 


999918 


04 


389656 


10736 


710144 


53 




8 


396307 


10565 


999915 


04 


396393 


10570 


70371)8 


58 




9 


303546 


10413 


999913 


04 


302634 


10418 


097366 


51 




10 


306794 


10366 


999910 


04 


308884 


10870 


601116 


50 




11 


8.314954 


10123 


0.990907 


04 


8.315046 


10136 


11.684964 


48 




IS 


331037 


9963 


999905 


04 


331132 


9967 


678878 


48 




13 


387016 


9847 


990903 


04 


337114 


9851 


673886 


47 




14 


332934 


9714 


009699 


05 


333025 


9719 


666875 


46 




15 


338753 


9586 


999897 


05 


338856 


9590 


661144 


45 




16 


344504 


9460 


ntmanA 
VWODn 


05 


344610 


9465 


655390 


44 




17 


350181 


0338 


999691 


05 


350280 


9343 


649711 


43 




18 


355783 


9319 


999888 


05 


355895 


9294 


641105 


42 




19 


361315 


9103 


999685 


05 


361430 


9108 


638570 


41 




90 


366rr7 


8990 


909883 


05 


366895 


8995 


633105 


40 




21 


8.372171 


8880 


9.999870 


05 


8.373398 


8885 


11.637708 


39 




& 


377499 


8773 


90;)876 


05 


377633 


8777 


6S3378 


38 




S3 


383763 


8667 


999873 


05 


382880 


8672 


617111 


37 




S4 


387963 


8564 


909670 


05 


388092 


8570 


611908 


36 




35 


393101 


8464 


999867 


05 


393334 


8470 


606766 


35 




36 


398179 


8366 


990864 


05 


398315 


8371 


601685 


34 




27 


403199 


8371 


099861 


05 


40333d 


82:6 


5966G3 


33 




38 


406161 


8177 


999858 


05 


4083(M 


8188 


591696 


32 




39 


413068 


8086 


999854 


05 


413213 


8091 


586787 


31 




30 


417919 


7996 


999851 


06 


418068 


8009 


581033 


30 




91 


8.433717 


7909 


9.999848 


06 


8.422860 


7914 


11.577131 


29 




33 


437463 


7833 


909844 


06 


437618 


7830 


573383 


28 




33 


433156 


7740 


999641 


06 


433315 


TJ45 


567685 


27 




34 


436800 


7057 


999838 


06 


436068 


7663 


563018 


90 




35 


441394 


7577 


999834 


06 


441560 


7583 


558440 


25 




36 


445041 


7499 


909H31 


06 


446110 


7505 


553800 


24 




37 


450440 


7433 


990827 


06 


450613 


7498 


549387 


23 




38 


454893 


7346 


999833 


06 


455070 


7353 


544930 


22 




39 


459301 


7273 


999830 


06 


450481 


7279 


540519 


31 




40 


463665 


730O 


999616 


06 


463849 


7306 


536151 


30 




41 


8.467085 


7128 


9.999613 


06 


8.468173 


7135 


11.531888 


19 




43 


473363 


7060 


999809 


06 


479454 


7066 


587546 


18 




43 


476498 


6001 


999605 


06 


476603 


6098 


583307 


17 




44 


480693 


6934 


999801 


06 


48U893 


6831 


519108 


16 




45 


484848 


6850 


999797 


07 


485050 


6865 


5149S0 


15 




46 


488963 


6794 


999793 


07 


480170 


6801 


510630 


14 




47 


493040 


6731 


999790 


07 


493350 


6738 


506750 


13 




48 


497078 


6669 


999786 


07 


497393 


6676 


502707 


13 




49 


501060 


6608 


9907R3 


07 


501398 


6615 


498703 


LI 




5U 


505045 


6548 


999778 


07 


505267 


6555 


494733 


10 




51 


8.508974 


6489 


9.999774 


07 


8.509300 


6406 


11.490600 


9 




S3 


513867 


6431 


999769 


07 


513008 


6438 


486003 


8 




53 


516726 


6375 


009765 


07 


516061 


6382 


483039 


7 




54 


530551 


6319 


999761 


07 


530790 


6336 


479210 


6 




55 


534343 


6364 


999757 


07 


534566 


6873 


475414 


5 




5tf 


588108 


6311 


999753 


07 


538340 


6818 


471651 


4 




57 


531898 


6158 


909748 


07 


533080 


6165 


467990 


3 




58 


535533 


6106 


999744 


07 


535779 


6113 


464221 


3 




iS9 


539186 


6055 


999740 


07 


539447 


6063 


460553 


1 




60 


540819 


0004 


999T35 


07 


543064 


6013 


456916 







1 


Cosine 


1 1 


fine 1 


1 Ootang. 


1 


1 Tang. i M. 





88 



46 




(2 Degrees.) a 


TABLE OP LOOARITBinC 




M. 


1 SiiM 


1 D 


Ocwlne. 1 


»• 1 


Ttog. 1 


D 


1 Coteng. 


1 





8.543819 


6004 


9.990735 


07 


8.543084 


601S 


11.456016 


60 


I 


546433 


5055 


999731 


07 


546631 


5963 


453309 


59 


s 


54D935 


5936 


099726 


07 


550368 


5914 


449732 


58 


1 3 


553539 


5858 


0991^ 


08 


553817 


OODO 


446183 


57 


4 


557054 


5811 


099717 


06 


557I36 


5819 


442664 


56 


• 5 


530540 


5765 


999713 


08 


560828 


5773 


439172 


55 


6 


563039 


5719 


993708 


08 


564391 


5727 


435700 


54 


7 


567431 


5674 


999704 


06 


567727 


5683 


433373 


53 


8 


5708J6 


5630 


999699 


08 


571137 


5638 


438863 


53 


9 


5742H 


5587 


993694 


08 


574530 


5395 


435480 


51 


10 


577506 


5544 


993689 


06 


577877 


5553 


433133 


50 


11 


8.58 893 


5303 


9.939683 


08 


8.581306 


5510 


11.418793 


49 


13 


584193 


5460 


999380 


06 


584514 


5168 


413486 


48 


13 


587460 


5119 


999^5 


06 


587795 


5137 


411205 


47 


14 


590731 


5379 


999370 


06 


591051 


5387 


436949 


40 


15 


593948 


5339 


993665 


08 


594383 


5347 


405717 


45 


16 


597153 


5330 


999660 


06 


597403 


5308 


4U3S08 


44 


17 


6.10333 


5361 


999355 


08 


600677 


5270 


39U323 


43 


18 


603489 


5333 


939650 


08 


603839 


5333 


396181 


43 


19 


606633 


5186 


939345 


09 


603978 


5194 


393022 


41 


20 


609734 


5149 


993340 


00 


610094 


5158 


389306 


40 


81 


8.613823 


5113 


9.999135 


00 


8.613189 


5131 


11.388811 


39 


22 


015891 


5J75 


993339 


03 


616363 


5065 


383738 


38 


23 


618937 


5041 


993634 


09 


619313 


5050 


380687 


37 


134 


021932 


5303 


999819 


09 


633343 


5015 


377657 


36 


25 


624965 


4973 


999614 


09 


635353 


4981 


374648 


35 


26 


627948 


4938 


933608 


09 


638340 


4947 


371660 


34 


27 


630911 


4934 


939603 


0^ 


631308 


4913 


368693 


33 


28 


633834 


4871 


999597 


00 


634256 


4883 


3T5744 


33 


29 


638776 


4839 


939593 


OJ 


637184 


4848 


362816 


31 


30 


639389 


4806 


933JM6 


09 


640093 


4816 


353937 


30 


31 


8.643563 


4775 


9.933581 


09 


8.643962 


4784 


11.357018 


29 


32 


645438 


47-!3 


993575 


09 


645833 


4753 


354147 


28 


33 


648374 


4712 


993570 


09 


648704 


47^ 


351396 


37 


ai 


9SUG2 


4883 


933564 


09 


651537 


4331 


348463 


23 


35 


653911 


4653 


933558 


10 


654353 


4861 


345648 


23 


36 


656703 


4632 


993'^ 


10 


657149 


4631 


342851 


24 


37 


659475 


4593 


999347 


10 


659938 


4602 


340072 


33 


38 


662333 


45S3 


999541 


10 


6S2689 


4573 


3T7311 


23 


39 


664968 


4535 


999335 


10 


665433 


4544 


334567 


21 


40 


667689 


4536 


999539 


19 


688J60 


4323 


331840 


20 


41 


8.670393 


4479 


9.999324 


10 


8.670870 


4488 


11.339130 


19 


43 


673083 


4451 


933518 


10 


673563 


4481 


3264.37 


18 


43 


675751 


4434 


999513 


10 


676339 


4434 


323761 


17 


44 


678405 


4*397 


999536 


13 


678903 


4417 


331100 


16 


45 


681043 


4370 


993500 


10 


681544 


4383 


318456 


15 


46 


6R3665 


4344 


933493 


10 


684173 


4354 


315826 


14 


47 


686373 


4318 


939487 


13 


686784 


43^8 


313316 


13 


48 


688863 


4232 


939481 


10 


689381 


4303 


310619 


13 


49 


691438 


4267 


939475 


10 


691963 


4377 


306037 


11 


50 


693998 


4242 


999489 


10 


694539 


4253 


305471 


10 


51 


8.696543 


4217 


9.999463 


11 


8.697061 


4338 


11.303919 


9 


53 


699073 


4192 


993453 


11 


699317 


4303 


300383 


8 


53 


701589 


4168 


999450 


11 


703139 


4179 


397861 


7 


54 


704090 


4144 


939443 


11 


704646 


4155 


395354 


6 


55 


706577 


4121 


• 939437 


11 


707140 


4133 


392860 


5 


56 


709049 


4097 


999431 


11 


709618 


4106 


39II3H3 


4 


57 


711507 


4074 


999434 


11 


713083 


4065 


387917 


3 


58 


713952 


4051 


999418 


11 


714534 


4002 


385465 


8 


59 


716383 


4039 


999411 


11 


716973 


4040 


3H:«K28 


] 1 


60 


718800 


4006 


999404 


11 


719396 


4317 


380604 


1 


L 


OofliDe 




Sine 


1 


Gotang. 


1 


1 Tang. 


1 M 



87 Degrees 







8INE8 


AND TANGENTS. (3 Degrees.) 




47 


M. 1 


Sine 1 


D. 1 


Cofllne 


D. 


1 Tang. 


1 D. 


1 Cotong. 1 1 





8.718800 


4006 


V.W94U4 


11 


8.719306 


4017 


11.380604 


00 1 


1 


T21204 


3984 


999398 


11 


721806 


3995 


878194 


59 


2 


733595 


39S2 


999391 


11 


724204 


3974 


375796 


58 


3 


725972 


3941 


999384 


11 


726588 


3953 


373413 


57 


4 


738337 


3919 


99srJ78 


11 


728959 


3030 


271041 


56 


5 


730688 


3898 


999371 


11 


731317 


3999 


368683 


55 


6 


733027 


3877 


099364 


12 


733663 


3889 


366337 


54 


7 


735354 


3857 


999357 


12 


735998 


3868 


264004 


53 


8 


737667 


3836 


9993.50 


13 


7:«317 


3848 


261683 52 1 


9 


739969 


3816 


999343 


12 


740626 


3827 


250374 51 1 


10 


742259 


3793 


999330 


13 


7429^ 


3807 


257078 


50 


11 


8.744536 


3776 


9.999329 


13 


8.745307 


3787 


11.254793 


49 


13 


746832 


3756 


991)332 


12 


747479 


3768 


253521 


48 


13 


749055 


3737 


999315 


13 


749740 


3749 


350260 


47 


14 


751297 


3717 


999308 


12 


751969 


3739 


3«8011 


46 


15 


753528 


3696 


999301 


13 


754327 


3710 


845773 


45 


16 


755747 


3679 


999294 


13 


750453 


3603 


343547 


44 


17 


757955 


3661 


999286 


13 


758668 


3673 


841332 


43 


18 


760151 


3642 


999279 


13 


76067S 


3655 


239136 


43 


19 


763337 


3624 


999272 


13 


763065 


3636 


836935 


41 


90 


764511 


3606 


999265 


13 


765346 


3618 


234754 


40 


31 


8.766675 


3588 


9.999257 


12 


8.787417 


3600 


11.832583 


39 


S3 


768828 


3570 


999250 


13 


769578 


3563 


330423 


38 


S3 


770970 


3553 


999242 


13 


T71T27 


3565 


238273 


37 


34 


773101 


3535 


999235 


13 


773866 


3518 


336134 


36 


35 


775223 


3518 


999227 


13 


775995 


353] 


334005 


35 


36 


V7TJ33 


3531 


999-230 


13 


778114 


3514 


321886 


34 


27 


779434 


3484 


999213 


13 


780222 


3497 


819778 


33 


38 


781524 


3467 


999335 


13 


782320 


3480 


317680 


38 


SB 


783605 


3451 


999197 


13 


784408 


3464 


315503 


31 


30 


785675 


3431 


999189 


13 


786486 


3447 


313514 


30 


31 


8.787736 


3418 


9.999181 


13 


8.788554 


3431 


11.311446 


SB 


33 


789787 


3402 


999174 


13 


790613 


3414 


309367 


S8 


33 


791828 


3386 


999166 


13 


793663 


3309 


307338 


S7 


34 


793859 


3370 


999138 


13 


794701 


3383 


305899 


96 


35 


795881 


3354 


999150 


13 


796731 


3368 


303269 


35 


36 


797894 


3339 


999143 


13 


796752 


3353 


301348 


34 


37 


799897 


3323 


999134 


13 


800763 


3337 


199337 


23 


38 


801892 


3306 


99913S 


13 


802765 


3322 


197335 


83 


39 


803876 


3393 


999118 


13 


804758 


3307 


195242 


81 


40 


805852 


3378 


999110 


13 


806742 


3202 


193356 


90 


41 


8.807819 


3363 


9.999102 


13 


8.a)8717 


3278 


11.191283 


19 


43 


809777 


3349 


999094 


14 


810663 


3262 


189317 


18 


43 


811726 


3334 


999066 


14 


812641 


3248 


187359 


17 


44 


813667 


3319 


990077 


14 


814589 


3333 


185411 


16 


45 


815599 


^05 


999069 


14 


816539 


3219 


183471 


15 


46 


817522 


3191 


999061 


14 


818461 


3305 


181539 


14 


47 


819436 


3177 


999053 


14 


^20384 


3191 


179616 


13 


48 


821343 


3163 


999044 


14 


822298 


3177 


177702 


13 


49 


823240 


3149 


999036 


14 


824205 


3163 


175795 


11 


50 


825130 


3135 


999027 


14 


836103 


3150 


173897 


10 


51 


8.837011 


3122 


9.999019 


14 


8.837993 


3136 


11.172006 


9 


53 


838884 


3108 


999010 


14 


829674 


3123 


170126 


8 


53 


830749 


3095 


099002 


14 


831748 


3110 


168352 


7 


54 


832607 


3082 


996993 


14 


833613 


3096 


166387 


6 


55 


834456 


3069 


998984 


14 


835471 


3063 


164529 


5 


56 


836297 


3056 


998976 


14 


837321 


3070 


162679 


4 


57 


838130 


3043 


9989fi7 


15 


839163 


3057 


160837 


3 


56 


830956 


3030 


998958 


15 


840998 


3045 


159002 


2 


59 


841774 


3017 


998950 


15 


842825 


3033 


157175 


1 


60 


843585 


3000 


9969 tl 


15 


a A AAA A 


3019 


155356 





^BB 


Gorina 




1 Biiie 


1 


1 Ootang. 


1 


1 Tang. |M. 



86Dcfrees 



48 




(4 Degrees.) a 


TABLE OF LOGARITHMIC 




M.I 


HIM 1 


D. 1 


CWm ( 


». 1 


Tug. 


D. 


1 Cotens i 1 





8.843S8S 


3005 


9.998941 


15 


a OAAAAA 


3019 


11.155356 


6U 


1 1 


845387 


2003 


908933 


15 


846455 


3007 


153545 


59 


3 


847183 


3980 


l.»08923 


15 


848260 


3995 


151740 


58 


3 


848971 


8067 


9D8914 


15 


850057 


3963 


149943 


57 


4 


850751 


8955 


998905 


15 


851846 


8970 


148154 


56 


5 


8525-25 


3943 


998896 


15 


853628 


8956 


146378 


55 


6 


854291 


3931 


998887 


15 


855403 


8946 


144397 


54 


7 


856049 


8919 


998878 


15 


857171 


8935 


142829 


53 


8 


857801 


3907 


998869 


15 


858932 


8933 


141068 


52 


9 


859546 


8896 


998860 


15 


860688 


3911 


139314 


51 


10 


861283 


8884 


998851 


15 


863433 


8900 


137567 


50 


11 


8.863014 


3873 


9.998841 


15 


8.864173 


XOOD 


11.135837 


49 


13 


864738 


88tl 


996833 


15 


885906 


9Bnni 


134094 


48 


13 


866455 


9850 


99333:1 


16 


867632 


3866 


139368 


47 


14 


868165 


8839 


999813 


10 


869351 


8854 


130640 


46 


U 


869888 


8828 


998804 


16 


871064 


8843 


188036 


45 


16 


871565 


8817 


996795 


16 


872770 


3833 


137330 


44 


17 


873355 


8806 


908785 


18 


874460 


3831 


135531 


43 


18 


874938 


8795 


906776 


16 


876162 


3811 


133636 


42 


19 


876615 


8786 


998766 


16 


877849 


3609 


133151 


41 


90 


87^85 


8773 


903757 


16 


879539 


3789 


130471 


40 


SI 


8.879949 


3783 


9.998747 


16 


8.881303 


S779 


11.118793 


30 


23 


8816U7 


3753 


996738 


16 


882869 


3768 


117131 


38 


S3 


883258 


3743 


998728 


16 


684530 


3758 


115470 


37 


 34 


884903 


8731 


998718 


16 


886185 


3747 


113815 


36 


. 35 


886543 


8731 


996708 


18 


837833 


«ri7 


113167 


35 


96 


888174 


8711 


996399 


16 


889476 


3737 


1105*34 


34 


87 


889891 


3700 


938889 


16 


891113 


3717 


106888 


33 


S8 


891431 


369!) 


098679 


16 


893742 


3707 


107356 


33 


SO 


893035 


8680 


998869 


17 


894366 


8607 


105634 


31 


90 


894643 


8870 


998659 


" 


895964 


8687 


101016 


30 


31 


6.806346 


9660 


9.996649 


17 


8.897596 


9677 


11.103404 


39 


39 


807843 


3C51 


998639 


17 


899203 


9667 


100797 


38 


33 


899432 


9641 


998189 


17 


000603 


8658 


099197 


37 


34 


901017 


3631 


908619 


17 


903398 


3648 


097603 


86 


35 


903596 


sum 


998600 


J7 


903987 


36:18 


096013 


85 


36 


904169 


9613 


006500 


17 


905570 


8089 


094430 


84 


37 


905736 


8603 


096589 


17 


907147 


3630 


092H53 


83 


38 


007397 


8503 


998578 


17 


906719 


9610 


091381 


23 


39 


908853 


8584 


998568 


17 


910385 


3601 


069715 


81 


40 


910404 


3575 


996556 


17 


911846 


3502 


088154 


90 


41 


8.911949 


3566 


9.996548 


17 


8.913401 


3563 


11.066599 


19 


43 


913488 


3556 


906537 


17 


914951 


3574 


065049 


18 


43 


915022 


8547 


996537 


17 


916495 


3565 


063505 


17 


44 


916550 


8538 


996516 


18 


918034 


8556 


061966 


16 


45 


918073 


3530 


906500 


18 


919568 


3547 


080433 


15 


46 


919591 


3530 


996495 


18 


031096 


3538 


078904 


14 


47 


931103 


3518 


996485 


18 


933619 


3530 


077381 


13 


48 


922610 


3503 


996474 


18 


934136 


3531 


075864 


13 


40 


924112 


8494 


998464 


18 


935649 


3512 


074351 


11 


50 


K2S609 


3486 


996453 


18 


937156 


3503 


073844 


10 


51 


8.927100 


8477 


9.908443 


18 


8.928658 


3495 


11.071343 


9 


S3 


928587 


8460 


996431 


18 


930155 


8486 


069845 


8 


53 


930068 


3460 


998421 


18 


ffll647 


8478 


068353 


7 


54 


931544 


3453 


998410 


18 


933134 


3470 


066866 


6 


55 


933015 


»I43 


998399 


18 


934616 


8461 


065384 


5 


56 


034481 


3435 


998388 


18 


936093 


3453 


063007 


4 


57 


035042 


3427 


998377 


18 


937565 


8445 


003435 


3 


58 


937398 


3419 


998366 


18 


939032 


3437 


060068 


3 


59 


938850 


3411 


908355 


18 


040494 


3430 


050506 


1 


60 


940296 


3403 


996344 


18 


941952 


8431 


058048 





1 


1 Ooiliie 


1 


1 Sine 


1 


1 Ooteng. 


J 


1 TMg. 1 M 1 



86 



SINES AND TANOENTS. (5 Degrees.) 



49 



M. 


1 Sine 


1 D. 


1 Codxie 


ID. 


1 TMig. 


1 ». 


1 cotuff. 1 i 





8.940396 


3403 


9.998344 


19 


8.941952 


3431 


11.058048 


60 


1 


941738 


8394 


998333 


19 


943404 


8413 


056596 


59 


8 


943174 


8387 


996322 


19 


944852 


8405 


055148 


58 


3 


944606 


2379 


998311 


19 


946395 


8397 


0537U5 


57 


4 


946034 


8371 


998300 


19 


947TJ4 


8390 


052266 


56 


5 


947456 


3363 


998289 


19 


949168 


8382 


050832 


55 


6 


948874 


3355 


998277 


19 


950597 


8374 


049403 


54 


7 


950287 


8348 


998266 


19 


952081 


3366 


047979 


531 


8 


951G96 


3340 


998255 


19 


953441 


8360 


046559 


52 


9 


953100 


3332 


998243 


19 


954856 


3351 


045144 


51 


10 


954499 


2325 


998232 


19 


956367 


8344 


043733 


50 


11 


8.955894 


8317 


9.998220 


19 


8.957674 


8337 


11.048390 


49 


IS 


957384 


8310 


998209 


19 


959075 


8330 


040995 


48 


13 


958670 


33(13 


998197 


19 


960473 


8323 


039537 


47 


14 


900053 


8395 


998186 


19 


961866 


8314 


038134 


46 


15 


961439 


2888 


996174 


19 


963255 


3307 


036745 


45 


16 


962801 


8890 


998163 


19 


964639 


8300 


035361 


44 


17 


964170 


2273 


908151 


19 


966019 


2393 


033981 


43 


16 


965534 


8366 


998139 


30 


967J94 


3366 


032606 


43 


19 


966893 


3259 


998128 


30 


968766 


8379 


031334 


41 


99 


968249 


8358 


998116 


30 


970133 


3371 


039667 


40 


21 


8.969600 


8244 


9.998104 


80 


8.971496 


8965 


11.028504 


39 


28 


970947 


2238 


998092 


80 


972855 


3357 


087145 


38 


23 


978289 


1^1 


998080 


80 


974909 


8351 


085791 


37 


34 


973098 


tetu 


996068 


30 


975560 


8344 


084440 


36 


25 


974963 


2217 


998056 


30 


976996 


3337 


023094 


35 


96 


976293 


8210 


998044 


30 


978248 


!&30 


(»1753 


34 


87 


977619 


2203 


998032 


30 


979586 


8383 


030414 


33 


2B 


978941 


8197 


996020 


80 


980921 


%ii7 


019079 


33 


29 


980259 


8190 


998008 


20 


932251 


3310 


017749 


31 


30 


981573 


8183 


997996 


80 


9835n 


«»)4 


016433 


30 


31 


8.982883 


3177 


9.997984 


80 


8.984899 


8197 


11.015101 


89 


33 


981189 


3170 


997973 


30 


06G217 


3191 


013783 


88 


33 


985491 


8163 


997959 


SO 


967532 


9184 


013468 


87 


34 


986789 


3157 


997947 


90 


988842 


9178 


011158 


86 


35 


988683 


8150 


997935 


91 


990149 


9171 


009851 


95 


36 


989374 


8144 


997922 


81 


991451 


9165 


006549 


84 


37 


990660 


8138 


997910 


21 


992750 


3158 


037250 


83 


38 


991943 


8131 


937897 


31 


994045 


8153 


005955 


82 


39 


993222 


8185 


997885 


81 


995337 


8146 


004863 


81 


40 


994497 


8119 


997873 


31 


996624 


3140 


003376 


90 


41 


8.995768 


8118 


9.997860 


31 


8.997908 


8134 


11.0030U8 


19 


43 


997036 


3106 


997847 


81 


999186 


8137 


000813 


18 


43 


998299 


8100 


997835 


81 


9.000465 


8131 


10.999535 


17 


44 


999560 


8094 


997832 


31 


001736 


3115 


998262 


16 


45 


9.000816 


8087 


997809 


31 


003007 


3109 


996993 


15 


46 


002069 


8082 


997797 


81 


004373 


8103 


995728 


14 


47 


003318 


2076 


997784 


21 


005534 


9097 


994466 


13 


48 


004563 


3070 


997771 


81 


006793 


9091 


993208 


18 


49 


005805 


8064 


997756 


81 


008047 


9085 


991953 


11 


50 


007044 


3058 


997745 


31 


009298 


9080 


990703 


10 


51 


9.008278 


3053 


9.997732 


81 


9.010546 


8074 


10.969454 


9 


58 


000510 


3046 


997719 


81 


011790 


8068 


988310 


8 


53 


0]0r37 


3040 


997706 


31 


013031 


3063 


966969 


7 


54 


011902 


8034 


997693 


23 


014268 


8056 


985732 


6 


55 


013182 


9039 


997680 


92 


015502 


8051 


984498 


5 


56 


014400 


8023 


997667 


82 


016732 


8045 


983268 


4 


57 


015613 


8017 


997654 


82 


017950 


8040 


982041 


3 


58 


016824 


8013 


997641 


22 


019183 


3033 


980817 


8 


59 


018031 


8006 


997028 


82 


020403 


9028 


979597 


1 


60 


019835 


9000 


997614 


23 


021690 


9083 


978380 







1 COflllM 


1 


1 Slna 


1 


1 Cotaog. 




1 T»ng. IM.J 








84 


Degit 


BW. 








ELLWO 


OD'8 TI 


:ST PROS. 


___ i 


i. 









50 



(6 Oegreei.) a table op LoaARitRinc 



M. 


1 Sine 


1 D. 


1 Cosine ! P 


1 Tang. 


1 D. 


1 Ootaag. 1 







9.019835 


1 8000 


9.907614 22 


9.031090 


2033 


10.978380 00 


1 


020435 


1995 


997601 22 


022834 


3017 


977166 50 1 


8 


021632 


1989 


997588 32 


024044 


2011 


975056 28 I 


3 


022825 


1964 


997574 


22 


025251 


2006 


974749 57 




4 


U24016 


1978 


997561 


22 


026455 


2000 


973545 1 50 




5 


02:;203 


1973 


997547 


22 


027655 


1995 


972345 i 55 




6 


026386 


1967 


997534 


23 


028852 


1990 


971148 


54 




7 


027567 


1962 


997520 


23 


030046 


1985 


969954 


53 




8 


028744 


1957 


997507 


23 


031237 


1979 


968763 


58 




9 


029918 


1951 


997493 


23 


033425 


1974 


967575 


51 




10 


031089 


1947 


997480 


23 


033609 


1969 


966391 


50 




11 


9.032257 


1941 


9.997466 


23 


9.034791 


1964 


10.965809 


49 




13 


033421 


1936 


997452 


23 


035969 


1958 


964031 


48 




13 


034582 


1930 


997439 


23 


037144 


1953 


963856 


47 




14 


035741 


1925 


997425 


23 


038316 


1948 


961684 


48 




15 


036896 


1920 


997411 


23 


039485 


1943 


960515 


45 




16 


038048 


1915 


997397 


23 


040651 


1938 


950949 


44 




17 


039197 


1910 


997383 


23 


041813 


1933 


958187 


43 




18 


040342 


1905 


997369 


23 


042973 


1938 


957037 


48 




19 


041485 


1899 


997355 


23 


044130 


1923 


955870 


41 




20 


042625 


1894 


997341 


23 


045284 


1918 


054716 


40 




21 


9.043762 


1889 


9.99rJ27 


24 


9.046434 


1913 


10.9S3566 


30 




22 


044895 


1884 


997313 


24 


047582 


1906 


952418 


38 




23 


046026 


1879 


997299 


21 


048727 


1903 


951273 


37 




24 


047154 


1875 


997385 


24 


049869 


1898 


950131 


30 




25 


048279 


1870 


997271 


24 


051006 


1893 


948992 


35 




26 


049400 


1865 


997257 


24 


052144 


1889 


947856 


34 




27 


050519 


1860 


997342 


24 


053277 


1884 


916733 


33 




28 


051635 


1855 


997238 


24 


054407 


1879 


945593 


33 




29 


052749 


1850 


997214 


24 


055535 


1874 


944465 


31 




30 


053859 


1845 


997199 


24 


056659 


1870 


943341 


30 




31 


9.054966 


1841 


9.997185 


24 


9.057781 


1865 


10.942219 


89 




38 


056071 


1836 


997170 


24 


058900 


1869 


941100 


38 




33 


057172 


1831 


997156 


24 


060016 


1855 


939964 


27 




34 


058271 


1827 


997141 


24 


061130 


1851 


938870 


26 




35 


059367 


1828 


997127 


24 


062240 


1846 


937760 


25 




36 


060460 


1817 


997112 


24 


063348 


1842 


936652 


24 




37 


061551 


1813 


997098 


24 


064453 


1837 


935547 


23 




38 


062639 


1808 


997083 


25 


065556 


1833 


934444 


23 




39 


063724 


18U4 


997068 


25 


066655 


1828 


933345 


21 




40 


064806 


1799 


997053 


25 


067752 


1824 


932248 


20 




41 


9.0G5885 


1794 


9.997039 


25 


9.068846 


1819 


10.931154 


19 


« 


42 


0G69<)2 


1790 


997024 


25 


069938 


1815 


930062 


18 




43 


068036 


1786 


997009 


25 


071027 


1810 


988973 


17 




44 


OC9107 


1781 


996994 


25 


072113 


1806 


927887 


16 




45 


070176 


1T77 


996979 


25 


073197 


1802 


936803 


15 




46 


071242 


1772 


996964 


25 


074278 


1797 


925733 


14 




47 


0712306 


1768 


996949 


25 


075356 


1793 


924644 


13 




48 


073366 


1763 


996934 


25 


076432 


1789 


923568 


13 




49 


074424 


1759 


996919 


25 


077505 


1784 


932495 


11 




50 


075480 


1755 


996904 


25 


078576 


J780 


931434 


10 




51 


9.076533 


1750 


9.99G889 


25 


9.079644 


1776 


10.(120356 


9 




52 


077583 


1746 


996874 


25 


080710 


.1772 


919290 


8 




53 


078631 


1742 


996858 


25 


081773 


1767 


918237 


7 




54 


079676 


1738 


996843 


25 


083833 


1763 


917167 


6 




55 


080719 


1733 


99682b 


25 


083891 


1759 


916109 


5 




56 


081759 


1729 


996812 


26 


084947 


1755 


915053 


4 




57 


082797 


1725 


996797 


26 


086000 


1751 


914000 


S 




58 


083832 


1721 


996782 


26 


087050 


1747 


913950 


3 




59 


084864 


1717 


996766 


26 


088098 


1743 


911908 


1 




60 


085894 


1713 


996751 


26 


069144 


1736 


910656 







1 


Cosine 1 


! 


Sine i 1 


Coteng. 1 


1 


Ttog. |ltj 









fliNca 


AND TANOBNT0. (7 Degrees.) 




51 


L 


M. 


1 «IM 


1 D- 


1 Cotlne 1 D. 


1 T«ng. 


1 D- 


1 ColMg. 1 1 







9.085894 


1713 


9.996751 


36 


9.089144 


1738 


10.910856 


60 




1 


086922 


1700 


996735 


36 


000187 


irj4 


009813 


50 




s 


087947 


1704 


996720 


36 


091228 


1730 


906773 


58 




3 


088970 


1700 


996704 


26 


092266 


1797 


9U77J4 


57 




4 


089990 


1696 


996688 


26 


093302 


1732 


906608 


56 




5 


091008 


1693 


996673 


28 


094336 


1719 


905664 


55 




6 


093024 


1688 


996657 


26 


095367 


1715 


904633 


54 




7 


093037 


1664 


996641 


26 


096395 


1711 


903605 


53 




6 


094047 


1680 


996635 


26 


007422 


1707 


902578 


58 




9 


095056 


1676 


996610 


26 


098446 


1703 


901554 


51 




10 


090062 


1673 


096504 


26 


099468 


1699 


900533 


50 




11 


0.097065 


1668 


9.996578 


27 


0.100487 


1695 


10.899513 


40 




IS 


098066 


1665 


996568 


27 


101504 


1691 


898496 


48 




13 


099065 


1661 


996546 


27 


109519 


1687 


'897481 


47 




14 


100063 


1657 


996530 


37 


103533 


1684 


896468 


46 




15 


101066 


1653 


996514 


37 


104543 


1680 


895458 


45 




16 


103048 


1640 


996498 


27 


105550 


1676 


891450 


44 




17 


103037 


1645 


996483 


37 


106556 


1673 


893444 


43 




18 


104QQ5 


1641 


996465 


37 


107550 


1660 


893441 


43 




19 


105010 


1638 


996449 


27 


108560 


1665 


801440 


41 




90 


105003 


1634 


996433 


27 


100550 


1661 


890441 


40 




SI 


9.106973 


1630 


9.996417 


27 


0.110556 


1658 


10.889444 


30 




29 


107951 


1037 


996400 


27 


111551 


1654 


888449 


38 




33 


108937 


1633 


006384 


27 


113543 


1650 


887457 


37 




34 


109901 


1619 


006368 


27 


113533 


1646 


886467 


36 




35 


110873 


1616 


096351 


27 


114521 


1643 


885479 


35 




36 


111843 


1613 


996335 


27 


115507 


1639 


884493 


34 




37 


113809 


1606 


996318 


27 


116491 


1636 


883500 


33 




38 


113T74 


1605 


996303 


28 


117478 


1633 


883538 


3i 




SO 


114';37 


1601 


990285 


28 


118453 


1629 


831548 


31 




30 


115698 


1507 


996969 


28 


110420 


1625 


880571 


30 




31 


9.116656 


1594 


9.996353 


28 


9.130404 


16S3 


10.879506 


39 




39 


117613 


1590 


996235 


28 


1213T7 


1618 


878883 


38 




33 


118567 


1587 


990219 


28 


122348 


1615 


877653 


37 




34 


119519 


1583 


990303 


28 


123317 


1611 


876683 


36 




35 


190409 


1580 


006185 


28 


124284 


1607 


875716 


35 




36 


131417 


1576 


996168 


88 


135349 


1604 


874751 


34 




37 


183363 


1573 


996151 


28 


136311 


1601 


873789 


33 




38 


123306 


1560 


996134 


28 


127173 


1507 


873R2H 


38 




30 


134348 


1566 


996117 


28 


128130 


1504 


871870 


31 




40 


1S5187 


1563 


996100 


28 


120067 


1501 


870913 


30 




41 


0.1S6135 


1550 


9.996083 


89 


0.130041 


1587 


10.860950 


19 




43 


137060 


1556 


996066 


29 


130994 


1584 


860006 


18 




43 


137063 


1553 


996040 


89 


131944 


1581 


868056 


17 




44 


118025 


1540 


996033 


39 


132893 


1577 


867107 


16 




45 


129854 


1545 


996015 


89 


133839 


1574 


866161 


15 




40 


130781 


1543 


995998 


89 


134784 


1571 


865316 


14 




47 


131706 


1530 


995080 


89 


135726 


1567 


864274 


13 




48 


138630 


1535 


995963 


89 


136667 


1564 


863333 


IS 




49 


133551 


1533 


995046 


89 


137605 


1561 


863305 


11 




50 


134470 


1539 


995038 


89 


138543 


1558 


861458 


10 




61 


9.135387 


1535 


9.995911 


89 


9.139476 


1555 


10.860534 


9 




53 


136303 


1523 


995894 


89 


140409 


1551 


850501 


8 




53 


137S16 


1519 


995876 


89 


141340 


1548 


^18660 


7 




54 


138128 


1516 


995859 


89 


142369 


1545 


857731 


6 




55 


139037 


1513 


995841 


89 


143196 


1543 


856804 


5 




56 


130044 


1500 


995833 


89 


144121 


1530 


855879 


4 




57 


140R50 


1506 


995806 


89 


145044 


1535 


854956 


3 




58 


141754 


1503 


095788 


29 


145966 


15.33 


854034 


8 




59 


149655 


1300 


095771 


29 


146885 


1539 


853115 


1 




00 


143555 


1496 


905753 


29 


147803 


1536 


853197 







1 C}(mIim 


1 


1 Sine 1 


1 CotlDg. 




1 Tug. |M. 





82 



52 



(B Degree*.) a tablb or looarithmic 



M. 


1 8iiM 


1 D 


CodM 


1 ». 


1 Ttog. 


1 »• 


1 Colug. 1 1 





9.143555 


1496 


9.995753 


30 


9.147803 


1590 


10.853197 


60, 


1 


144453 


1493 


905735 


30 


146718 


1533 


851363 


501 


S 


145349 


1490 


995717 


30 


149033 


1530 


650366 


58 


3 


146343 


1487 


995699 


30 


150544 


1517 


849456 


57 


4 


147136 


1484 


995681 


30 


151454 


1514 


848546 


56 


5 


148036 


1481 


995664 


30 


153:M>3 


1511 


647637 


55 


6 


148915 


1478 


995646 


30 


1533!I9 


15U8 


846731 


54 


7 


149803 


1475 


995rS8 


30 


154174 


1505 


845826 


53 


8 


150G86 


14718 


995C10 


30 


155077 


1503 


8441^23 


52 


9 


151509 


1469 


995591 


30 


155978 


1499 


844022 


51 


10 


153451 


1466 


995573 


30 


156877 


1490 


843133 


50 


11 


0.153330 


1463 


9 995555 


30 


0.15m75 


1403 


10.84ie235 


49 


IS 


1543:^8 


14^4) 


995537 


30 


158671 


1490 


641339 


4b 


13 


155083 


1457 


995510 


30 


150565 


1487 


640435 


47 


14 


155957 


1454 


995501 


31 


160457 


1464 


839543 


46 


15 


156830 


1451 


995483 


31 


161347 


1481 


836(153 


45 


16 


157T00 


1448 


995464 


31 


162336 


1479 


837764 


44 


17 


158569 


1445 


995446 


31 


163133 


1476 


836877 


43 


18 


159435 


1443 


995437 


31 


164006 


1473 


635093 


43 


19 


160301 


1439 


995409 


31 


164)993 


1470 


835106 


41 


30 


161164 


1436 


995390 


31 


165774 


1467 


634336 


40 


31 


9.163035 


1433 


9.995372 


31 


9.166654 


1464 


10.833346 


39 


22 


163885 


1430 


995353 


31 


167533 


1461 


633466 


38 


33 


163743 


1427 


905334 


31 


166409 


1458 


831501 


37 


34 


164600 


1424 


995316 


31 


169284 


1455 


630716 


36 


35 


165454 


1433 


995297 


31 


170157 


1453 


620843 


35 


36 


166307 


1419 


995378 


31 


171029 


1450 


838971 


34 


37 


167159 


1416 


995360 


31 


171899 


1447 


828101 


331 


38 


168008 


1413 


995241 


32 


172767 


1444 


tiUim 


32 


38 


168856 


1410 


995233 


33 


171)634 


1443 


8263G6 


31 


30 


169703 


1407 


995203 


33 


174490 


1430 


825501 


30 


31 


9.17054? 


1405 


9.995184 


33 


0.175363 


1436 


10.834638 


39 


33 


171389 


1403 


9951C5 


33 


176324 


1433 


633776 


88 


33 


172230 


1399 


995146 


33 


177084 


1431 


833916 


87 


34 


173070 


1396 


995127 


33 


17T943 


1426 


833056 


86 


35 


173908 


1394 


995106 


33 


178799 


1435 


831301 


35 


36 


174744 


1391 


995069 


33 


179655 


1433 


830345 


84 


37 


175578 


1388 


995070 


33 


1835G8 


1420 


819493 


83 


38 


176411 


1386 


995051 


33 


181360 


1417 


818640 


83 


39 


177242 


1383 


995033 


33 


183211 


1415 


617789 


81 


40 


178073 


1380 


995013 


33 


163050 


1413 


616041 


90 


41 


9.178900 


1377 


9.994903 


33 


0.183907 


1400 


10.8160B3 


19 


43 


179726 


1374 


994974 


33 


184758 


i4(nr 


815346 


16 


43 


180551 


1373 


994955 


33 


165597 


1404 


814403 


17 


44 


181374 


1369 


994935 


33 


166439 


1403 


813561 


16 


45 


182196 


1366 


994916 


33 


187280 


1399 


613730 


15 


46 


183016 


1364 


994896 


33 


188130 


1396 


811880 


14 


47 


183834 


1361 


994877 


33 


186956 


1393 


811043 


13 


46 


184651 


1359 


994857 


33 


189794 


1391 


810906 


13 


49 


185466 


1356 


994838 


33 


190629 


1369 


800371 


11 


50 


186380 


1353 


994818 


33 


191468 


1386 


806538 


10 


51 


9.187093 


1351 


9.994798 


33 


0.193394 


1364 


10.807706 


9 


53 


187903 


1348 


994779 


33 


103134 


1361 


806876 


8 


53 


188713 


1346 


994759 


33 


193953 


1379 


806047 


7 


54 


189519 


1343 


904730 


33 


194780 


1.376 


60SS20 


6 


55 


190325 


1341 


994719 


33 


195606 


1374 


6043D4 


5 


56 


191130 


1338 


994700 


33 


196430 


1371 


803570 


4 


57 


191933 


1336 


994680 


33 


197353 


1369 


803747 


3 


58 


192734 


1333 


994660 


33 


196074 


1366 


801036 


8 1 


59 


193534 


1330 


994640 


33 


108894 


1364 


801106 


1 


00 


194332 


1338 


994630 


33 


199713 


1361 


600387 





1 


Coane 




Sin* 


1 


1 Cotang. 


1 


1 T«g. 1 M. J 



81 







SINES 


AND TANGENTS. (9 DcgFCCS.) 




5 


M.| 


SiiM 


1 D- 


1 CkMlne 


ID. 


1 Tang. 1 


D. 1 


Cotang. 1 





9.194333 


1338 


9.994630 


33 


9.199713 


1361 


10.800387 


60 


1 


195139 


1396 


094600 


33 


300539 


1359 


799471 


59 


3 


195^25 


1333 


994580 


33 


301345 


1356 


798655 


58 


3 


196719 


1331 


994560 


34 


303159 


1354 


797841 


57 


4 


197511 


1318 


994540 


34 


803971 


1353 


797029 


56 


5 


198303 


1316 


994519 


34 


803782 


1340 


796318 


55 


6 


199091 


1313 


994499 


34 


304593 


1347 


795406 


54 


7 


199879 


1311 


994479 


34 


305400 


1345 


794600 


53 


8 


300666 


1306 


994459 


34 


3063U7 


1343 


793793 


53 


9 


301451 


1306 


•KI44<x} 


34 


307013 


1340 


793987 


51 


10 


303334 


1304 


994418 


34 


307817 


1338 


793183 


50 


11 


9.303317 


1301 


9.994397 


34 


9.308619 


1335 


10.791381 


49 


12 


303797 


1399 


9iM377 


34 


309130 


1333 


793580 


48 


13 


804577 


]396 


994357 


34 


810320 


1331 


789780 


47 


14 


305354 


1394 


994336 


34 


811018 


1338 


•/88983 


46 


15 


308131 


1393 


994316 


34 


311815 


1336 


788185 


45 


16 


8U0906 


1389 


994395 


34 


313611 


1324 


787389 


44 


17 


307679 


1387 


994374 


35 


813405 


1331 


786595 


43 


18 


806i53 


1385 


994354 


35 


314198 


1319 


785803 


43 


19 


3093^ 


1382 


994333 


35 


314989 


1317 


785011 


41 


90 


809993 


1380 


994313. 


35 


315780 


1315 


784320 


40 


81 


9.310760 


1378 


9.994191 


35 


9.316568 


1313 


10.783433 


39 


92 


811536 


1375 


994171 


35 


317356 


1310 


782644 


38 


S3 


818391 


1373 


994150 


35 


818143 


1308 


781858 


37 


S4 


813055 


1371 


994139 


35 


818936 


1305 


781074 


36 


35 


813R18 


1368 


9^108 


35 


319710 


1303 


780290 


35 


38 


814579 


1300 


994087 


35 


330493 


1301 


779508 


34 


87 


8i5:t:« 


1364 


994066 


35 


331373 


1299 


778728 


33 


88 


816097 


1361 


994045 


35 


333053 


1397 


777948 


33 


89 


816854 


1359 


994034 


35 


823830 


1394 


777170 


31 


30 


817609 


1357 


994003 


35 


833606 


1393 


776394 


30 


31 


9.318363 


1355 


0.993981 


35 


9.334383 


1390 


10.775618 


39 


33 


319116 


1353 


993900 


35 


3-25156 


1388 


774844 


38 


33 


819868 


1350 


993939 


35 


835^29 


1386 


774071 


27 


34 


^0618 


1348 


993918 


35 


326700 


1384 


773300 


36 


35 


831367 


1346 


9U38J0 


36 


837471 


1381 


772529 


35 


36 


833115 


1344 


993875 


36 


828239 


1379 


771761 


34 


37 


333861 


1343 


993854 


36 


839307 


1277 


770933 


33 


38 


833606 


1339 


993833 


36 


339773 


1375 


770227 


33 


39 


834349 


1337 


993811 


36 


830339 


1373 


769461 


31 


40 


835093 


1335 


993789 


36 


831303 


1371 


768698 


30 


41 


9.835833 


1333 


9.993768 


36 


9.332065 


1369 


10.767935 


19 


43 


836573 


1331 


993746 


36 


833826 


1367 


767174 


18 


43 


827311 


1328 


993735 


36 


233586 


1365 


766414 


17 


44 


338048 


1326 


993703 


36 


834345 


1263 


765655 


16 


45 


828784 


1334 


999681 


36 


835103 


1360 


764897 


15 


46 


329518 


1^3 


093680 


36 


835859 


1358 


764141 


14 


47 


8303S3 


1330 


993638 


36 


836614 


1356 


76338G 


13 


48 


830984 


1318 


903616 


36 


337308 


1354 


763832 


13 


49 


831714 


1316 


903594 


37 


838130 


1253 


761680 


11 


50 


833444 


1314 


993573 


37 


338873 


1350 


761128 


10 


51 


9.833178 


1313 


0.9935S0 


37 


9.339({33 


1348 


10.760378 


9 


S3 


833899 


1309 


993528 


37 


340371 


1316 


759629 


8 


53 


834635 


K07 


993500 


37 


841118 


1344 


758883 


7 


54 


835349 


1305 


093484 


37 


841865 


1343 


758135 


6 


55 


336073 


1303 


993463 


37 


843610 


1840 


757390 


5 


56 


836795 


1201 


991440 


37 


843354 


1338 


756646 


4 


57 


337515 


1199 


993418 


37 


844097 


1336 


755903 


3 


58 


83B335 


1197 


993396 


37 


844839 


1334 


755161 


3 


59 


838953 


1195 


993374 


37 


845579 


1333 


754431 


1 


60 


830670 


1193 


993351 


37 


846319 


1330 


753681 





.aaa 


1 OaHiw 1 


1 


SiiM 




1 Gotang. 


1 


1 Twg. 1 M. 



80 Degrees. 



54 




(10 Degrees.) a 


TABLE OF LOOARITHMIC 




M. 


1 8iiM 


1 I>. 


Coitoe 


D. 


1 TMlf . 


1 ». 


1 Coteag. 


1 





0.839670 


1193 


9.993351 


37 


0.346319 


1230 


10.753661 


60 


1 


840380 


1191 


993329 


37 


847057 


1338 


753943 


59 


8 


341101 


1189 


093307 


37 


847794 


1336 


758206 


58 


S 


341814 


1187 


993385 


37 


848530 


lim 


751470 


57 


4 


843536 


1185 


993^62 


37 


819264 


1833 


750736 


56 


5 


343337 


1183 


993340 


37 


349998 


1820 


750002 


55 


6 


843947 


1181 


993217 


38 


250730 


1318 


7492T0 


54 


7 


344656 


1179 


993195 


38 


351461 


1317 


748539 


53 


8 


345363 


1177 


993172 


38 


852191 


1315 


747809 


52 


9 


846069 


1175 


993149 


38 


352920 


1313 


747080 


51 


10 


846T75 


1173 


993127 


38 


853648 


1211 


746352 


50 


11 


9.847478 


1171 


9.993104 


38 


0.354374 


1800 


10.745636 


40 


IS 


848181 


1160 


9939H1 


38 


355100 


1307 


744900 


48 


13 


848883 


1107 


993059 


38 


355824 


1805 


744176 


47 


14 


849383 


1165 


993036 


38 


856547 


1303 


743453 


46 


15 


850383 


1183 


993913 


33 


857269 


1301 


742731 


45 


16 


83(»80 


1161 


992990 


38 


857990 


1300 


742010 


44 


17 


851677 


1159 


992937 


38 


858710 


1198 


741390 


43 


18 


852373 


1158 


992944 


38 


859429 


1193 


740571 


42 


19 


8530:>7 


1156 


932921 


38 


860146 


1194 


739654 


41 


99 


353761 


1154 


992896 


38 


860663 


1192 


730137 


40 


31 


9.354451 


1153 


9.9(^2875 


38 


0.861578 


1190 


10.738483 


39 


22 


355141 


1150 


992852 


38 


3^292 


1189 


737706 


38 


£1 


855834 


1148 


992829 


39 


363005 


1187 


736095 


37 


U 


S5S52J 


1146 


992806 


39 


863717 


1185 


730283 


36 


25 


357211 


1144 


992783 


39 


864438 


1183 


735572 


35 


36 


857898 


1143 


«»759 


39 


365138 


1181 


734862 


34 


27 


858583 


1141 


992736 


39 


865847 


1179 


734153 


33 


38 


359368 


1139 


992713 


39 


866555 


1178 


733445 


32 


29 


3591)51 


1137 


9936!)0 


30 


367361 


1176 


732739 


31 


39 


860633 


1135 


992866 


39 


367967 


1174 


7^033 


30 


31 


9.361314 


1133 


9.992643 


39 


0.366671 


1172 


10.731389 


89 


32 


8619J4 


1131 


992619 


30 


369375 


1170 


730625 


38 


33 


363673 


1130 


092595 


39 


370077 


1160 


739923 


87 


34 


863351 


1128 


092372 


39 


870779 


1167 


739221 


36 


35 


864027 


1126 


992549 


30 


371479 


1165 


738521 


35 


36 


864703 


1134 


992325 


30 


373178 


1164 


727822 


84 


37 


865377 


11» 


092531 


39 


872876 


1162 


727124 


33 


38 


366051 


1130 


998473 


40 


373573 


1160 


728437 


^1 


39 


866723 


1119 


092454 


40 


874269 


1158 


725731 


31 


40 


367395 


1117 


9^130 


40 


874964 


1157 


735036 


90 


41 


0.368065 


1115 


0.992406 


40 


0.27.'i65A 


1155 


10.724343 


19 


42 


868734 


1113 


902382 


40 


376351 


1153 


723649 


18 


43 


369402 


1111 


992359 


40 


377043 


1151 


723957 


17 


44 


8700G9 


1110 


992335 


40 


377734 


1150 


722266 


16 


45 


870735 


1108 


093311 


40 


378434 


1148 


721576 


15 


46 


371400 


1106 


992237 


40 


379113 


1147 


72J887 


14 


47 


373064 


1105 


992263 


40 


379601 


1143 


730199 


13 


48 


872726 


1103 


992239 


40 


880488 


1143 


719512 


12 


49 


873388 


1101 


993314 


40 


881174 


1141 


718836 


11 


50 


374049 


1099 


992190 


49 


381858 


1140 


718143 


10 


51 


9.374708 


1098 


9.993166 


43 


0.382543 


1138 


10.717458 


9 


53 


875367 


1006 


992142 


40 


883225 


1136 


710775 


8 


53 


876024 


1094 


992117 


41 


383937 


1135 


716093 


7 


54 


876681 


1003 


933093 


41 


381583 


1133 


715413 


6 


55 


877337 


1091 


0920f^9 


41 


285268 


1131 


714733 


5 


56 


277091 


1089 


992044 


41 


885047 


1130 


714053 


4 


57 


378844 


1087 


992030 


41 


886624 


1128 


713376 


3 


58 


379297 


1066 


091996 


41 


387301 


1136 


713609 




50 


879948 


1064 


991971 


41 


287977 


1125 


713033 


1 


60 


880599 


1063 


991947 


41 


388652 


1133 


711348 





Lj 


Oodne | 


1 


Sioa 


1 


1 Ootang. 


1 


Ttaig. 


"ST 



79 







tIN£S 






55 


M. 


I Bine 


1 I>. 


1 Coehw 1 D. 


1 Tmg. 


1 ©. 


1 Cotang. 


1 





9.380599 


1083 


9.991947 


41 


0.888653 


1183 


10.711348 


60 


1 


3815M8 


1061 


991933 


41 


389336 


1128 


710674 


50 


3 


381897 


1079 


991897 


41 


889999 


1190 


710001 


58 


3 


383544 


1077 


991873 


41 


890671 


1118 


709339 


57 


4 


383190 


1076 


991848 


41 


391343 


1117 


7U8658 


56 


5 


28:HT6 


1074 


991833 


41 


893013 


1115 


707987 


55 


6 


28448U 


1078 


991799 


41 


893683 


1114 


707318 


54 


7 


885134 


1071 


991774 


43 


39.1350 


1118 


706650 


53 


8 


885766 


1069 


991749 


48 


394017 


1111 


705983 


53 


9 


886408 


1067 


991T24 


43 


894684 


1100 


705316 


51 


ID 


887048 


1066 


991699 


43 


395349 


1107 


704651 


50 


11 


0.887087 


10G4 


9.991674 


43 


9.396013 


1106 


10.703987 


49 


12 


888330 


1003 


991649 


43 


896677 


1104 


703333 


48 


13 


888964 


1061 


991034 


48 


897339 


1103 


709661 


47 


14 


88960J 


1059 


091599 


48 


898001 


1101 


701999 


46 


15 


89033G 


1058 


991574 


42 


898663 


1100 


701338 


45 


16 


890670 


1056 


091549 


42 


890383 


1096 


700678 


44 


J7 


891304 


1054 


991534 


42 


899980 


1096 


700030 


43 


18 


893137 


1053 


991498 


43 


300638 


1095 


699363 


43 


19 


3037G8 


1051 


091473 


42 


301395 


1093 


6087U9 


41 


90 


893399 


1050 


991448 


42 


301951 


1093 


698049 


40 


SI 


0.394089 


1048 


9.091433 


43 


9.309607 


1000 


10.607303 


39 


22 


S0SG58 


1016 


091397 


48 


303961 


1080 


606739 


38 


S3 


805386 


1045 


991373 


43 


303014 


1087 


606086 


37 


S4 


895Q13 


1043 


991346 


43 


304567 


1086 


605433 


36 


35 


896539 


1043 


991.131 


43 


305318 


lO&t 


604783 


35 


86 


897164 


1040 


991395 


43 


305869 


1083 


604131 


34 


87 


897788 


1039 


991370 


43 


306519 


1081 


693481 


33 


88 


398413 


1037 


991344 


43 


307168 


1080 


603833 


33 


89 


899034 


1036 


991318 


43 


3U7815 


1078 


608185 


31 


30 


899655 


1034 


9J1193 


43 


308463 


1077 


601537 


30 


31 


O.30O376 


1033 


9.991167 


43 


0.309109 


1075 


10.600901 


89 


33 


300895 


1031 


991141 


43 


309754 


1074 


690346 


88 


33 


301514 


1033 


991115 


43 


310308 


1073 


689603 


87 


34 


303133 


1038 


991090 


43 


311043 


1071 


688958 


86 


35 


308748 


103S 


991004 


43 


311685 


1070 


68H315 


85 


36 


303364 


1035 


991038 


43 


313337 


1068 


687673 


84 


37 


303979 


1033 


091013 


43 


31391)7 


1067 


687U33 


83 


38 


304593 


1033 


990986 


43 


3i:i608 


1065 


686393 


83 


39 


3U5307 


3030 


990960 


43 


314^17 


1064 


6&'>753 


31 


40 


305819 


1019 


990034 


44 


314885 


1063 


685115 


SO 


41 


0.306430 


1017 


0.990008 


44 


9.315533 


1061 


10.684477 


19 


48 


307041 


1016 


990883 


41 


316159 


1060 


683841 


18 


43 


397650 


1014 


990855 


44 


316795 


1058 


683305 


17 


44 


308359 


1013 


990839 


44 


317430 


1057 


683570 


16 


45 


308867 


1011 


990803 


44 


318064 


1055 


681936 


15 


46 


309474 


1010 


990777 


44 


318697 


1054 


681303 


14 


47 


310080 


1008 


990750 


44 


319339 


1053 


680671 


13 


48 


310685 


1007 


990734 


44 


319961 


. 1051 


680039 


13 


49 


311389 


1005 


990697 


44 


330593 


1050 


679408 


11 


50 


311893 


1004 


990671 


44 


331333 


1048 


678778 


10 


51 


0.318405 


1003 


0.090644 


44 


0.331851 


1047 


10.678149 





58 


313097 


1001 


990618 


44 


333479 


1045 


677531 


8 


53 


313698 


1000 


090591 


44 


333106 


1044 


676894 


7 


54 


314397 


098 


990565 


41 


333733 


1043 


676367 


6 


55 


314897 


097 


090538 


44 


334358 


1041 


675643 


5 


56 


315495 


096 


090511 


45 


334983 


1040 


675017 


4 


57 


316093 


094 


090485 


45 


395607 


1039 


674393 


3 


58 


316689 


093 


090458 


45 


396331 


1037 


673'<'69 


8 


50 


317384 


091 


990431 


45 


326853 


1036 


67:J147 


1 


60 


317879 


090 


090404 


45 


337475 


1035 


G73535 





1 1 


OOBllM 


1 


aizM 1 1 


OolaDg. 1 


1 


TMg. 1 


"m. 



7BPeiprwi. 



66 



(13 Degrees.) a tabus or logibitbhic 



M. 


1 fliiie 


B 


GoaiiM 


ID. 


1 Ttog. 


D. 


Coteng. 


60 





0.317879 


900 


9.99M04 


45 


9.337474 


1035 


10.679596 


1 


318473 


988 


990378 


45 


398095 


1033 


671905 


59 


1 8 


319066 


967 


900351 


45 


398715 


1039 


6719K5 


58 


3 


319658 


986 


990394 


45 


399334 


1030 


670666 


57 


4 


390349 


984 


990997 


45 


399953 


1099 


670047 


56 


5 


390840 


963 


990970 


45 


330570 


1098 


669430 


55 


6 


321430 


963 


990943 


45 


331187 


1096 


668813 


54 


7 


393019 


960 


990915 


45 


3318a3 


1095 


668197 


53 


8 


399fi07 


979 


990188 


45 


339418 


1094 


667589 


53 


9 


393194 


977 


990161 


45 


333033 


1093 


666967 


51 


10 


333780 


976 


990134 


45 


333646 


1091 


666354 


50 


11 


9.394366 


975 


9.990107 


46 


9.334950 


1090 


10.665741 


49 


13 


394950 


973 


990070 


46 


334871 


1019 


665199 


48 


13 


395534 


973 


990053 


46 


335489 


1017 


664518 


47 


14 


396117 


970 


990095 


46 


336093 


1016 


663907 


46 


15 


396700 


969 


989097 


46 


336709 


1015 


663998 


45 


16 


397981 


968 


069930 


46 


337311 


1013 


669680 


44 


17 


397869 


966 


069949 


46 


337919 


1019 


663061 


43 


18 


398449 


965 


989915 


46 


33R937 


1011 


661473 


49 


19 


399031 


964 


989887 


46 


339133 


1010 


600867 


41 


90 


399509 


969 


989860 


46 


339739 


1006 


660961 


40 


91 


9.330176 


961 


9.989839 


4» 


9.340344 


1007 


10.659656 


30 


S9 


330753 


960 


989604 


46 


340948 


1006 


650059 


38 


93 


331399 


958 


969777 


46 


341559 


1004 


658448 


37 


94 


331903 


957 


969749 


47 


349155 


1003 


657B45 


36 


95 


- 333478 


956 


969791 


47 


349757 


1009 


657343 


35 


96 


333051 


954 


969693 


47 


343358 


1000 


t>56649 


34 


97 


333694 


953 


9696()5 


47 


343958 


999 


656049 


33 


98 


334195 


953 


969637 


47 


344558 


WiO 


655449 


39 


99 


334766 


950 


969609 


47 


345157 


997 


654843 


31 


30 


335337 


949 


969569 


47 


345755 


996 


654945 


30 


31 


9.335006 


948 


9.969553 


47 


9.346353 


994 


10.653647 


99 


39 


336475 


946 


969595 


47 


346049 


993 


653051 


98 


33 


333043 


945 


969497 


47 


347545 


999 


659455 


97 


34 


337610 


944 


969469 


47 


348141 


991 


651859 


96 


35 


338176 


943 


969441 


47 


348735 


990 


651965 


95 


36 


338743 


941 


969413 


47 


349329 


968 


650671 


94 


37 


330306 


940 


969384 


47 


349999 


967 


650078 


93 


38 


339671 


939 


989356 


47 


350514 


966 


649486 


99 


39 


340434 


937 


989398 


47 


351106 


965 


648894 


91 


40 


340996 


936 


969300 


47 


351697 


983 


648303 


90 


41 


9.341558 


935 


9.969971 


47 


9.359987 


989 


10.647713 


19 


42 


3^119 


934 


969943 


47 


359676 


981 


647194 


18 


43 


349679 


939 


969914 


47 


353465 


960 


646535 


17 


44 


343939 


931 


969186 


47 


354053 


979 


645947 


16 


45 


343797 


930 


969157 


47 


354640 


977 


645360 


15 


46 


344355 


999 


9691S8 


48 


355937 


976 


644773 


14 


47 


344919 


997 


969100 


48 


355813 


975 


644187 


13 


48 


345469 


996. 


969071 


48 


356398 


974 


643603 


19 


49 


346094 


995 


969049 


48 


356963 


973 


643018 


11 


50 


346579 


094 


989014 


48 


357566 


971 


649434 


10 


51 


9.347134 


939 


9.988985 


48 


9.358149 


970 


10.641851 


9 


53 


347687 


921 


988956 


48 


358731 


969 


.641969 


8 


53 


348340 


9ao 


968997 


48 


359313 


968 


640687 


7 


54 


346793 


919 


968696 


48 


359693 


967 


640107 


6 


55 


340343 


917 


968869 


48 


360474 


966 


639596 


6 


56 


349893 


916 


988640 


48 


361053 


965 


63^1M7 


4 


57 


350443 


915 


068811 


49 


361639 


963 


63R368 


3 


58 


350993 


914 


988789 


49 


369910 


969 


637790 


9 


59 


351540 


913 


968753 


49 


369787 


961 


637913 


1 


09 




OU 


088794 


49 


363364 


960 


636636 





g^ 


OtMliie 1 


^^^^^^ 


1 8*iM» 1 


! 


OotftDg. 


BBS 


1 T»g. 


in. 



77 Dtgmm, 







■IMES 


AND TANOKNT8. (13 Degree*.] 


1 


5' 


r 


'm.| 




BilM 1 


D. 


1 Coilne 


1 D. 


1 TmW. 


1 ». 


1 Coteog. 1 




9.352088 


911 


9.088724 


49 


9.363364 


960 


10.636636 


60 




1 


352635 


910 


968695 


40 


363940 


050 


636060 


50 




s 


353181 


909 


988666 


49 


364515 


958 


635485 


56 




3 


353726 


906 


988636 


49 


365090 


957 


634910 


57 




4 


354271 


907 


988607 


49 


365664 


955 


634336 


56 




5 


354815 


905 


988578 


49 


366237 


954 


6:f3763 


55 




6 


355358 


904 


988518 


49 


366810 


953 


633190 


54 




7 


355901 


903 


988319 


49 


367382 


953 


632618 


53 




8 


356443 


902 


988489 


49 


367953 


951 


632047 


52 




9 


356984 


901 


988460 


49 


368534 


950 


631476 


51 




10 


357524 


899 


988430 


49 


360094 


949 


630906 


50 




11 


9.358064 


806 


9.988401 


49 


9.369663 


948 


10.63tK»7 


49 




13 


358603 


897 


988371 


49 


37U232 


946 


620768 


48 




13 


359141 


896 


gHH:m 


49 


370799 


945 


629201 


47 




14 


359678 


895 


9CSKn2 


50 


371367 


944 


028633 


46 




J5 


360215 


893 


988;M2 


50 


371933 


943 


628067 


45 




16 


360752 


892 




50 


372499 


943 


627301 


44 




1 17 


361287 


891 


98H223 


50 


373064 


941 


626936 


43 




18 


3S1822 


890 


988193 


50 


373629 


940 


626371 


43 




19 


362356 


889 


988163 


50 


374193 


939 


625807 


41 




SO 


362889 


888 


988133 


50 


374756 


938 


635244 


40 




21 


0.363422 


887 


9.988103 


50 


9.375319 


937 


10.<«4661 


ao 




23 


363954 


885 


988073 


50 


375881 


935 


624119 


as 




23 


364485 


884 


988043 


50 


376442 


934 


623558 


37 




24 


365016 


883 


988013 


50 


377003 


933 


<»2997 


36 




25 


365546 


882 


987983 


50 


377563 


933 


62sM37 


35 




26 


366075 


881 


987953 


50 


378122 


931 


621678 


34 




27 


366604 


880 


987922 


50 


378681 


930 


621319 


33 




28 


367131 


879 


987892 


50 


379239 


029 


620761 


as 




29 


367659 


877 


987862 


50 


379797 


928 


620203 


31 




30 


368185 


876 


987832 


51 


380354 


927 


619646 


30 




31 


9.368711 


875 


9.987801 


51 


9.380910 


920 


10.619000 


99 




32 


360236 


874 


987771 


51 


381466 


925 


618534 


28 




33 


369761 


873 


967740 


51 


382020 


924 


617080 


97 




34 


370285 


872 


987710 


51 


383575 


923 


617425 


96 




35 


370806 


871 


987679 


51 


383120 


923 


616871 


95 




36 


371330 


870 


987649 


51 


383682 


921 


616318 


94 




37 


371852 


869 


967618 


51 


384234 


920 


615766 


S3 




38 


372373 


867 


987588 


51 


384786 


019 


61^214 


93 




39 


372894 


866 


987557 


51 


383337 


018 


614663 


21 




40 


373414 


865 


987526 


51 


385888 


017 


614113 


90 




41 


9.373933 


864 


9.967496 


51 


9.386438 


915 


10.613563 


19 




42 


374452 


863 


987465 


51 


386987 


914 


613013 


18 




43 


374970 


862 


967434 


51 


387536 


913 


612464 


17 




44 


375487 


861 


987403 


52 


388084 


913 


611916 


16 




45 


376003 


860 


987372 


52 


388031 


911 


611369 


15 




40 


376519 


850 


987341 


52 


389178 


910 


610822 


14 




47 


377035 


8S8 


987310 


52 


389724 


009 


610276 


13 




48 


3n549 


857 


987279 


52 


390270 


9J8 


609730 


13 




49 


378003 


856 


987248 


52 


3908J5 


907 


609185 


11 




50 


378577 


854 


967217 


53 


391360 


906 


608640 


10 




51 


9.379089 


853 


9.967186 


58 


9.391903 


905 


10.606007 


9 




52 


379601 


853 


967155 


53 


393447 


904 


607553 


8 




53 


380113 


851 


967124 


52 




903 


607011 


7 




54 


380624 


850 


967092 


52 


383531 


903 


006469 


6 




55 


38u:h 


840 


987061 


52 


304073 


001 


605027 


5 




56 


381643 


848 


087030 


52 


394614 


000 


605386 


4 




57 


382152 


847 


986998 


52 


395154 


899 


604646 


3 




58 


382661 


846 


986967 


52 


395694 


896 


604306 


2 




50 


38:1168 


' 845 


986936 


53 


396233 


897 


603767 


1 




60 


383675 


844 


986004 


53 


306771 


896 


603229 







BB 


1 Codne 




1 Sine 




1 Gotang. 




Ttog. 1 M. 





TODagretf 



68 


( 


14 Degrees.) a 


T1BI.R OP LOGARITHMIC 


S991 


M.| Bine 


1 D 


COBbM 


1 ». 


1 Tw>g. 


1 D. 1 


Coteng. 


r^ 


d 


9.383675 


844 


9 986904 


52 


9.396771 


696 


10.603329 


60 


1 


384182 


843 


986873 


53 


397309 


896 


602691 


59 


2 


384G87 


642 


980841 


53 


397846 


895 


602154 


58 


3 


385192 


841 


98G809 


53 


. 398383 


894 


601617 


57 


; 4 


385697 


840 


986778 


53 


398919 


893 


601081 


50 


5 


38G201 


839 


985746 


53 


399455 


892 


600545 


55 


6 


386704 


838 


986714 


53 


399990 


691 


600010 


54 


7 


387207 


837 


986683 


53 


4005-24 


890 


599476 


53 


8 


387709 


836 


966651 


53 


401058 


889 


598943 


52 


9 


388210 


KU 


986619 


53 


401591 


888 


598409 


51 


10 


388711 


834 


986587 


53 


402124 


887 


597876 


50 


IJ 


9.389211 


833 


9 986555 


53 


9.402656 


886 


10.597344 


49 


12 


389711 


832 


9H(»23 


53 


403187 


685 


596813 


48 


13 


390210 


831 


986491 


53 


403718 


884 


596283 


47 


14 


390708 


830 


986459 


53 


404249 


883 


595751 


46 


15 


391206 


828 


966427 


53 


404778 


882 


595223 


45 


16 


391703 


827 


966395 


53 


405308 


881 


594692 


44 


17 


392199 


826 


986363 


54 


405836 


880 


504164 


43 


18 


392G95 


825 


986331 


54 


406364 


879 


593636 


42 


19 


393191 


824 


986299 


5( 


406892 


878 


593106 


41 


90 


393685 


823 


986266 


54 


407419 


877 


592581 


40 


21 


9.394179 


R22 


9966234 


54 


9.407945 


876 


10.592055 


39 


22 


394C73 


821 


966202 


54 


408471 


875 


591529 


38 


23 


395166 


820 


966169 


54 


408997 


874 


591003 


37 


24 


395658 


819 


966137 


54 


409521 


874 


590479 


36 


25 


396150 


818 


986104 


54 


410045 


873 


589955 


35 


26 


396641 


817 


988072 


54 


4105(>9 


872 


589431 


34 


27 


397132 


817 


986039 


5^1 


411092 


871 


588908 


33 


28 


397621 


816 


986007 


54 


411615 


870 


588385 


32 


29 


398111 


815 


985974 


54 


412137 


869 


587863 


31 


30 


398iJ00 


814 


965942 


54 


412658 


868 


587342 


30 


31 


0.399088 


813 


9.985909 


55 


9.413179 


867 


10.586821 


29 


32 


399575 


812 


965876 


55 


413699 


866 


586301 


28 


33 


400m)2 


81 1 


965843 


55 


414219 


865 


585781 


27 


34 


400549 


810 


965811 


55 


414738 


804 


585202 


26 


35 


401035 


8J9 


985778 


55 


415257 


864 


584743 


25 


36 


401520 


8J8 


985745 


55 


415775 


m3 


584225 


24 


37 


402005 


807 


965712 


-55 


416293 


862 


583707 


23 


38 


40^89 


806 


965679 


^55 


416810 


861 


583190 


22 


39 


402972 


805 


985646 


55 


417328 


860 


582074 


21 


40 


403455 


m 


985613 


55 


417842 


859 


582158 


20 


41 


9.403938 


803 


9.965580 


55 


9.418353 


858 


10.581642 


19 


42 


4()4420 


802 


985517 


55 


418873 


857 


581127 


18 


43 


404901 


801 


965514 


55 


419387 


856 


580613 


17 


44 


405382 


800 


985480 


55 


419901 


855 


580099 


16 


45 


405862 


799 


985447 


55 


420<n5 


855 


579585 


15 


46 


406341 


798 


985414 


56 


420927 


854 


579073 


14 


47 


406820 


797 


985380 


56 


421440 


853 


578560 


13 


48 


407299 


796 


985347 


56 


421952 


852 


578048 


12 


49 


4077VV 


795 


985314 


56 


432463 


851 


5775:n 


11 


50 


408254 


794 


965280 


56. 


423974 


850 


577026 


10 


51 


9.408731 


794 


9.965247 


56 


9.423484 


849 


10.576516 


9 


53 


409207 


793 


965213 


56 


423993 


648 


576007 


8 


53 


409682 


792 


965180 


56 


424503 


848 


575497 


7 


54 


410157 


791 


965146 


53 


425011 


847 


574989 


6 


55 


41063;2 


790 


965113 


56 


425519 


846 


574481 


5 


56 


411106 


789 


985079 


56 


426027 


845 


573973 


4 


57 


411579 


788 


985045 


5;] 


426534 


844 


5734(>6 


3 


58 


412052 


787 


985011 


56 


427041 


843 


572959 


2 


59 


412524 


786 


964978 


56 


427547 


843 


5724.53 


1 


60 


412996 


785 


984944 


56 


428052 


842 


571948 





1 


Codne 1 


1 


Sine 




Cotang. 


1 


1 Tang. 


1 M. 



Imm 



751)egreec 







•INE4 


1 AND TANGENTS. (15 Degrees.) 




6S 


» 


M. 


, Siiw 


1 D. 


Coabie 


1 D. 


1 TMlg. 


1 »• 


1 CoteBg. 


1 







9.412996 


785 


Q QAiQAi 


57 


9.428a'i2 


842 


10.571948 


00 


1 


413467 


784 


964910 


57 


428557 


841 


571443 


59 




9 


413938 


783 


9B4876 


57 


429062 


840 


570938 


58 




3 


414406 


783 


984842 


57 


429566 


839 


570434 


57 




4 


414878 


782 


984808 


57 


430070 


838 


569930 


56 




5 


415347 


781 


984T74 


57 


430573 


838 


569427 


55 




6 


415815 


780 


984740 


57 


431075 


837 


5G8925 


54 




7 


416283 


779 


964706 


57 


431577 


836 


568423 


53 




8 


416751 


778 


984672 


57 


432079 


835 


567921 


52 




9 


417217 


777 


984637 


57 


432580 


834 


567420 


51 




10 


417684 


776 


984603 


57 


433060 


833 


566920 


50 




11 


9.418150 


775 


9.984569 


57 


9.433580 


832 


10.566480 


40 




18 


418615 


774 


984535 


57 


434080 


838 


565920 


48 




18 


419079 


773 


984500 


57 


434570 


831 


565421 


47 




14 


419544 


773 


984466 


57 


435078 


830 


564922 


46 




15 


420007 


772 


984432 


58 


435576 


820 


564424 


45 




16 


420470 


771 


984397 


58 


436073 


828 


563927 


44 




17 


420933 


770 


984363 


58 


436570 


828 


563430 


43 




18 


421395 


769 


9843^^8 


58 


437067 


827 


562933 


42 




19 


421857 


768 


984294 


58 


437563 


826 


562437 


41 




SO 


422318 


767 


984259 


58 


438U59 


825 


561941 


40 




21 


9.422778 


767 


9.984!»4 


58 


9.438554 


884 


10.561446 


30 




S2 


423238 


766 


984190 


58 


439048 


823 


560952 


38 




23 


423607 


765 


984155 


58 


439543 


823 


560457 


37 




84 


424156 


764 


984120 


58 


410036 


828 


550964 


36 




25 


424615 


763 


984085 


58 


440529 


82i 


550471 


35 




86 


425073 


762 


984050 


58 


441022 


830 


558978 


34 




27 


425530 


761 


984015 


58 


441514 


810 


558486 


33 




98 


42S087 


760 


983981 


58 


442006 


819 


557994 


32 




29 


42C443 


760 


983016 


58 


442497 


818 


557503 


31 




30 


426899 


759 


983911 


58 


442968 


817 


557012 


30 




81 


9.427354 


758 


9.983875 


58 


9.443479 


816 


10.556521 


30 




92 


427809 


757 


98.3840 


50 


443968 


816 


556032 


36 




33 


428263 


756 


983805 


59 


444458 


815 


555542 


27 




34 


428717 


755 


983770 


50 


444947 


814 


555053 


36 




35 


429170 


754 


983735 


50 


445435 


813 


554565 


35 




36 


429623 


753 


983700 


50 


445923 


812 


554077 


24 




37 


430075 


752 


983664 


50 


446411 


812 


553589 


23 




38 


430527 


752 


983629 


50 


446898 


811 


553102 


22 




39 


430978 


751 


983594 


50 


447384 


810 


552616 


21 




40 


431429 


750 


983558 


59 


447870 


800 


552130 


80 




41 


9.431879 


749 


9.9R3S23 


50 


9.448356 


809 


10.551644 


19 




42 


432329 


749 


983487 


59 


448841 


808 


551159 


18 




43 


432778 


748 


983452 


50 


449326 


807 


550674 


17 




44 


433226 


747 


983416 


50 


449810 


806 


550190 


16 




45 


433675 


746 


983381 


50 


450894 


806 


549706 


15 




46 


434122 


745 


983345 


59 


450777 


805 


549223 


14 




47 


434569 


744 


983309 


59 


451260 


804 


54B740 


13 




48 


435016 


744 


983273 


60 


451743 


803 


548257 


12 




40 


435462 


743 


983238 


GO 


452225 


803 


54Tn5 


11 




50 


435908 


742 


983202 


60 


452706 


808 


547294 


10 




51 


0.4363.13 


741 


9.983166 


60 


0.453187 


801 


10.546813 







52 


436798 


740 


983130 


60 


453668 


800 


546332 


8 




53 


437242 


740 


983094 


60 


454148 


799 


545852 


7 




54 


437686 


739 


963058 


60 


454628 


799 


545372 


6 




55 


438129 


738 


963022 


60 


455107 


796 


544893 


5 




S6 


4S8572 


737 


962986 


60 


455586 


797 


544414 


4 




57 


439014 


736 


082950 


60 


456064 


796 


543936 


3 




58 


439456 


736 


982914 


GO 


456542 


796 


543458 


8 




50 


439897 


735 


982878 


60 


457019 


705 


542981 


1 




60 


440338 


734 


082842 


60 


457496 


794 


542501 







' 


Coifaw 1 




1 Sine 


1 


Cotang. 




1 TMIg. 


1"-, 





74I>ogreef 



60 


1 


[16 Degrees.) a 


TABLB OP LOOABITHMIC 




1L| Sine 


1 ». 


1 CoriiM 1 D 


1 TMIg. 


1 D 


CotaBg. 


1 





9.440338 


734 


9.9e!»43 


60 


9.457496 


794 


10.543504 


60 


1 


440778 


733 


983805 


60 


4579rj 


793 


548037 


50 


3 


441318 


738 


W2769 


61 


45R449 


793 


541551 


58 


3 


441658 


731 


982733 


61 


458925 


793 


541075 


57 


4 


443096 


731 


983696 


61 


459400 


791 


540600 


56 


5 


443535 


TM 


983660 


61 


459875 


790 


540135 


55 


6 


443973 


729 


983634 


61 


460349 


790 


539651 


54 


7 


443410 


738 


982587 


61 


460823 


789 


539177 


53 


8 


443847 


737 


9H3551 


61 


461297 


788 


536703 


58 


9 


444284 


737 


983514 


61 


461770 


788 


53H330 


51 


10 


444720 


736 


983477 


61 


468343 


787 


537758 


50 1 


11 


9.445155 


735 


9.083441 


61 


9.468714 


786 


10.537886 


*9 1 


12 


445590 


734 


983404 


61 


463186 


765 


536614 


48 


13 


446025 


733 


982367 


61 


463658 


785 


536343 


47 


14 


446459 


733 


983331 


61 


464139 


784 


535871 


46 


15 


446893 


723 


9H2394 


61 


464599 


783 


535401 


45 


16 


447326 


731 


9R2257 


61 


465060 


783 


534931 


44 


17 


447759 


730 


983230 


63 


465539 


783 


534461 


43 


18 


448191 


730 


983183 


63 


466006 


781 


533993 


48 


19 


448623 


719 


982146 


63 


466476 


780 


533524 


41 


90 


449054 


718 


962109 


63 


466945 


780 


533055 


40 


31 


9.449485 


717 


9.963073 


68 


9.467413 


779 


10.533587 


30 


33 


449915 


716 


983035 


68 


467880 


778 


533120 


38 


S3 


450345 


716 


961998 


63 


468347 


778 


531653 


37 


34 


450775 


715 


961961 


68 


468814 


777 


531186 


36 


35 


451204 


714 


981924 


63 


469380 


776 


530730 


35 


36 


451633 


713 


981886 


63 


469746 


775 


530354 


34 


37 


453060 


713 


981849 


63 


47U311 


775 


539789 


33 


38 


453488 


713 


961812 


63 


470676 


774 


529324 


38 


39 


453915 


711 


981774 


63 


471141 


773 


528859 


31 


30 


453343 


710 


961737 


63 


471605 


773 


S3K395 


30 


31 


9.453768 


710 


9.981699 


63 


9.473068 


778 


10.527938 


89 


33 


454194 


709 


981662 


63 


472533 


771 


5274C8 


88 


33 


454619 


706 


961625 


63 


473995 


771 


53 < 005 


87 


34 


455044 


707 


981587 


63 


473457 


770 


526543 


96 


35 


455469 


707 


981549 


63 


473919 


769 


526081 


85 


36 


455893 


706 


981512 


63 


474361 


760 


525619 


84 


37 


456316 


705 


981474 


63 


474843 


768 


585158 


83 


38 


456739 


704 


9H1436 


63 


475303 


767 


524697 


83 


39 


457163 


704 


961399 


63 


475763 


767 


584237 


81 


40 


457584- 


703 


961361 


63 


476833 


766 


583777 


90 


41 


9.458006 


703 


9.981323 


63 


9.476683 


765 


10.583317 


19 


43 


458437 


701 


981285 


63 


477143 


765 


532858 


18 


43 


458848 


701 


981247 


63 


477601 


764 


588399 


17 


44 


459268 


700 


961309 


63 


478050 


763 


581941 


16 


45 


459688 


699 


961171 


63 


478517 


763 


581483 


15 


46 


460106 


698 


981133 


64 


478075 


768 


521085 


14 


47 


460527 


698 


981095 


64 


479433 


761 


580568 


13 


48 


460946 


697 


961057 


64 


479689 


761 


590111 


19 


49 


461364 


696 


981019 


64 


480345 


760 


519655 


11 


50 


461782 


695 


980961 


64 


480801 


750 


519199 


10 


51 


9.463199 


695 


9.960943 


64 


9.481357 


759 


10.518743 


9 


52 


462616 


694 


960904 


64 


481713 


758 


5183RR 


8 


53 


463033 


093 


980666 


64 


482167 


757 


517833 


7 


54 


463448 


693 


980837 


64 


483621 


757 


517379 


6 


55 


463864 


603 


980789 


64 


483075 


756 


516935 


5 


56 


464279 


691 


980750 


64 


483539 


755 


516471 


4 


57 


464694 


690 


9H0713 


64 


483083 


755 


516018 


3 


58 


465108 


690 


960673 


64 


484435 


754 


515565 


8 


59 


465533 


680 


980635 


64 


484887 


753 


515113 


1 


60 


465935 


688 


980596 


64 


485339 


753 


514661 


' 


I 1 


Oorine i 


1 


Sine 1 1 


Ck>Cang. 1 


1 


«Mf. 1 


M.  



78 DegKM 



siifBa AND TANOENTS. (17 Degrees.) 



61 



M.| 



SIM 


D- 1 


9.465935 


688 


466348 


688 


466761 


687 


467173 


686 


467585 


685 


467996 


685 


468407 


684 


468817 


683 


469*227 


683 


469637 


683 


470046 


681 


9.470455 


680 


470863 


680 


471271 


679 


471679 


678 


473066 


678 


47SI» 


en 


473898 


676 


473304 


676 


473710 


675 


474115 


674 


9.474519 


674 


474933 


673 


475337 


672 


475730 


673 


476133 


671 


476538 


670 


476038 


669 


477340 


669 


4Tn41 


668 


478142 


667 


0.478543 


667 


478943 


606 


479343 


665 


479741 


6'J5 


480140 


664 


480639 


663 


480^7 


663 


481334 


663 


481731 


661 


482138 


661 


9.482533 


660 


483921 


659 


483316 


650 


483712 


658 


484107 


657 


484501 


657 


484833 


656 


485289 


653 


485382 


653 


48S075 


654 


9.486467 


633 


488860 


653 


487251 


653 


487643 


651 


488034 


651 


488434 


653 


4888J4 


650 


489204 


649 


489593 


648 


489962 


648 



I CkMine | D. 



Tang. 


D. 1 


9.485339 


ViS 


485791 


753 


486243 


751 


486693 


751 


487143 


750 


487593 


749 


488 J43 


749 


488493 


748 


488941 


747 


489390 


747 


489838 


746 


9.490386 


746 


490733 


745 


491180 


744 


491637 


744 


492073 


743 


493319 


743 


492965 


743 


493410 


741 


49J854 


740 


494299 


740 


9.494743 


740 


493186 


739 


493633 


TJ8 


496073 


737 


498515 


737 


496937 


736 


497399 


736 


497841 


735 


498282 


TJ4 


498733 


734 


9.499163 


733 


499303 


733 


500043 


733 


500481 


731 


503^30 


731 


501359 


730 


501797 


730 


503235 


739 


502672 


728 


503109 


728 


9.503546 


727 


503982 


727 


504418 


7-26 


504854 


7-25 


505289 


725 


505724 


7-24 


508159 


724 


538593 


723 


507027 


722 


507460 


722 


0.507893 


721 


506326 


721 


508759 


720 


509191 


719 


509322 


719 


510054 


718 


510483 


718 


510916 


717 


5U346 


716 


511776 


716 



I Ootaos. I 








1 
o 

3 

4 
5 
6 
7 
8 

10 

11 
13 
13 
14 
15 
16 
17 
18 
10 
90 

81 
82 
83 
84 

85 
96 
87 
88 
89 
30 

31 
33 
33 
34 
35 
36 
37 
38 
39 
40 

41 
43 
43 
44 

43 

46 

47 
48 
49 
50 

51 
52 
53 
54 

55 
56 

57 
58 
50 
00 



9.9S0396 


64 


980558 


64 


980519 


65 


980480 


65 


083442 


65 


980403 


65 


963364 


65 


963325 


65 


983288 


65 


980247 


65 


960208 


65 


9.960160 


65 


980130 


63 


960091 


65 


960052 


65 


080012 


65 


979973 


65 


979934 


66 


970695 


66 


079635 


66 


079816 


66 


9.979776 


66 


9T9737 


66 


979807 


66 


979358 


66 


979618 


6J 


979579 


66 


979339 


66 


979499 


6J 


979459 


6a 


979420 


66 


9.979383 


66 


979340 


66 


979330 


67 


979260 


67 


979220 


67 


079183 


67 


979140 


67 


979100 


67 


979339 


67 


979319 


67 


9.978979 


67 


978939 


67 


078898 


67 


078858 


67 


978817 


67 


978777 


67 


978736 


67 


978693 


68 


978353 


68 


978615 


68 


9.978574 


08 


978333 


68 


978493 


68 


978452 


68 


978411 


68 


978370 


68 


978329 


68 


978288 


68 


97aM7 


68 


978206 


68 



10.514601 
514200 
513758 
513307 
512857 
512407 
511957 
511308 
511050 
510310 
510162 

10.509714 
509267 
508820 
508373 
507927 
507481 
507035 
506590 
506146 
505701 

10.505257 
504814 
504370 
503927 
503485 
503043 
502601 
502159 
501718 
501278 

10.500637 
500397 
499958 
499519 
499060 
496341 
498303 
497765 
497338 
498891 

10.490454 
496018 
493582 
493146 
494711 
494276 
493841 
493407 
492973 
492340 

10.492107 
491874 
491241 
490609 
490378 
489940 
489515 
489084 
488654 
488224 



60 
59 
58 
57 
56 
55 
54 
53 
52 
51 
50 

49 
48 
47 
46 
45 
44 
43 
43 
41 
40 

39 
38 
37 
36 
35 
34 
33 
33 
31 
30 

39 
38 
27 
90 
85 
84 
83 
83 
31 
80 

19 
18 
17 
16 
15 
14 
13 
13 
11 
10 


8 
7 
6 
5 
4 
3 
8 
1 




I (^M&IM I 



I Sine I I Cotang. | 



t Tuig. I M. 



72 



62 



(18 Degrees.) a tablc op looasitbmic 



[. I Sine I D I Codne I D. ( Tang. | D. | Ooiang. j 




1 
3 
3 
4 
5 
6 
7 
8 
9 
10 

11 
13 
13 
14 
15 
16 
17 
18 
19 
30 

31 
33 
33 
34 
35 
36 
37 
38 
39 
30 

31 
33 
33 
34 
35 
36 
37 
38 
39 
40 

41 
43 
43 
44 
45 
46 
47 
48 
49 
50 

51 
53 
53 
54 
55 
56 
57 
58 
50 
60 



fUUotx* 
490371 
490759 
491147 
491.'^ 
491933 

V 49230? 
493695 
493081 
493466 
493851 

9.494336 
494621 
495005 
495388 
495773 
496154 
496537 
496919 
497301 
497683 

9.498064 
498444 
498825 
499304 
499584 
499963 
500343 
500721 
501099 
501476 

9.501854 
502231 
503607 
503984 
503360 
503735 
504110 
504485 
5048G0 
505234 

9.505608 
505981 
506354 
506727 
507099 
507471 
507843 
508214 
506585 
508956 

9.509336 

509n9<> 

510434 
510803 
511173 
511540 
511007 
513375 
513643 



648 
648 
647 
646 
646 
645 
644 
644 
643 
643 
643 

641 
641 
640 
639 
639 
638 
637 
637 
636 
636 

635 
634 
634 
633 
633 
632 
631 
631 
630 
629 

629 
628 
628 
627 
626 
626 
635 
625 
634 
633 

633 
623 
622 
621 
620 
620 
619 
619 
618 
618 

617 
616 
616 
615 
615 
614 
613 
613 
613 
613 



9.978306 


68 


978165 


68 


978134 


68 


978083 


69 


978043 


69 


978001 


69 


977959 


69 


977918 


63 


977877 


69 


977835 


69 


9T7794 


69 


9.977753 


69 


9Tnil 


69 


977669 


69 


977628 


69 


977586 


69 


977544 


70 


977503 


70 


977461 


70 


977419 


70 


977377 


70 


9.977335 


70 


977293 


70 


977351 


70 


977309 


70 


977167 


70 


977125 


70 


977083 


70 


977041 


70 


976999 


70 


970057 


70 


9.976914 


70 


976873 


71 


976830 


71 


976787 


71 


976745 


71 


976703 


71 


976660 


71 


976617 


71 


976574 


71 


976533 


71 


9.976489 


71 


976446 


71 


976404 


71 


976361 


71 


976318 


71 


976275 


71 


976233 


73 


076189 


73 


976146 


73 


976103 


73 


9.976060 


72 


076017 


73 


975974 


73 


97.1930 


73 


975887 


72 


975844 


73 


975800 


73 


975757 


73 


975714 


73 


975670 


73 



9.511776 
513306 
513635 
513064 
513493 
513931 
514349 
514777 
515304 
515631 
516057 

9.516484 
516910 
517335 
517761 
518185 
518610 
519034 
519458 
519883 
530305 

9.530738 
531151 
531573 
531995 
533417 
5£i838 
533259 
533680 
534100 
534530 



9.534939 


699 


535359 


698 


535T78 


698 


536197 


697 


536615 


697 


587033 


696 


537451 


696 


537868 


695 


528285 


695 


528702 


694 



9.529119 
529535 
5299.50 
530366 
530781 
531196 
531611 
53302'» 
532439 
532853 

9.533266 
533P79 
534092 
534504 
534916 
535328 
535739 
536150 
536561 
536972 



716 
716 
715 
714 
714 
713 
713 
713 
713 
711 
710 

710 
709 
700 
708 
708 
707 
706 
700 
705 
705 

704 
703 
703 
703 
703 
703 
701 
701 
700 
699 



693 
693 
693 
693 
691 
691 
690 
600 
689 
689 

688 
688 
687 
687 
686 
686 
685 
685 
684 
684 



10.488334 
487794 
487365 
486936 
486507 
486079 
485651 
485333 
484796 
484369 
483943 

10.483516 
483090 
483665 
483339 
481815 
481390 
480066 
480543 
480118 
479695 

10.479373 
478849 
478437 
478005 
4T7583 
477163 
476741 
476330 
475900 
475480 

10.475061 
474641 
474223 
473803 
473385 
473067 
473949 
473133 
471715 
471398 

10.470881 
470465 
470050 
469634 
469319 
468804 
468389 
467975 
467561 
467147 

10.466734 
466331 
465908 
465496 
465064 
464673 
464361 
463850 
463439 
463038 



60 
50 
58 
57 
56 
55 
54 
53 
53 
51 
50 I 

40 
48 
47 
46 
45 
44 
43 
43 
41 
40 

30 
38 
37 
36 
35 
34 
33 
33 
31 
30 

39 
38 
37 
96 
85 
34 
33 
S3 
31 
90 

19 
18 
17 
16 
15 
14 
13 
13 
11 
10 

9 
8 
7 
6 
5 
4 
3 
9 
1 




I Oodne I 



I Sine I I Ooteng. | 
71 Degree! 



I TMig. I M . 



liSj 







mifRB 


AMD TANOBNT8. (19 Degrees.^ 


1 


63 


M. 


1 Sine 


1 B. 


1 GoriM 


1 D. 


1 Tang 


1 » 


1 Cotwag. 


1 





9.513643 


612 


9.975670 


73 


9.536973 


684 


10.463028 


60 


1 


513009 


611 


975627 


73 


S3T3»2 


683 


462618 


59 


8 


513375 


611 


975583 


73 


537792 


683 


463208 


58 


3 


513741 


GIO 


975539 


73 


538202 


683 


461796 


57 


4 


514107 


609 


975496 


73 


538611 


683 


461389 


56 


5 


514472 


609 


975452 


73 


539030 


681 


460980 


55 


6 


514837 


008 


975408 


73 


539439 


681 


460571 


54 


7 


515302 


608 


975365 


73 


539837 


G80 


460163 


53 


8 


515506 


607 


975321 


73 


540345 


680 


459755 


53 


9 


515930 


007 


975277 


73 


540653 


679 


459347 


51 


10 


516394 


606 


975233 


73 


541061 


679 


458939 


50 


11 


9.516657 


605 


9.975189 


73 


9.541468 


678 


10.458532 


49 


12 


517030 


605 


975145 


73 


541875 


678 


458135 


48 


13 


517382 


604 


975101 


73 


542281 


677 


457719 


47 


14 


517745 


604 


975057 


73 


542688 


677 


457313 


46 


15 


518107 


603 


975013 


73 


543094 


676 


456906 


45 


IG 


518468 


603 


974969 


74 


543499 


676 


456501 


44 


17 


518829 


603 


974925 


74 


543905 


675 


456095 


43 


18 


519190 


601 


974880 


74 


544310 


675 


455690 


43 


19 


519551 


601 


974836 


74 


544715 


674 


455285 


41 


90 


519911 


600 


974792 


74 


545119 


674 


454881 


40 


31 


9.520271 


600 


9.974748 


74 


9.545524 


673 


10.454476 


39 


n 


520631 


599 


974703 


74 


545928 


673 


454073 


38 


33 


530990 


599 


974650 


74 


546331 


673 


453669 


37 


34 


531349 


o98 


974614 


74 


546735 


673 


453365 


36 


25 


531T07 


596 


974570 


74 


547138 


671 


453863 


35 


36 


532066 


597 


974525 


74 


547540 


671 


452460 


34 


37 


532424 


506 


9744S1 


74 


547943 


670 


452057 


33 


38 


533781 


596 


974436 


74 


548345 


670 


451655 


33 


29 


533138 


505 


974391 


74 


548747 


669 


451253 


31 


30 


533495 


595 


974347 


75 


549149 


669 


450651 


30 


31 


9.533853 


594 


9.974302 


75 


9.549550 


668 


10.450450 


39 


33 


524808 


594 


974257 


75 


549951 


668 


450049 


38 


33 


534564 


593 


974213 


75 


550353 


667 


449648 


37 


34 


534930 


593 


974167 


75 


550753 


C67 


449248 


26 


35 


525375 


593 


9741^ 


75 


551153 


666 


448848 


25 


36 


525630 


591 


974077 


75 


551553 


666 


448448 


24 


37 


535964 


591 


974033 


75 


551953 


665 


448048 


23 


38 


536339 


590 


973987 


75 


552351 


665 


447649 


22 


39 


536693 


590 


973943 


75 


552750 


665 


447250 


21 


40 


537046 


589 


973897 


75 


553149 


664 


446851 


20 


41 


9.527400 


589 


9.973853 


75 


9.553548 


664 


10.446453 


19 


43 


537753 


588 


973807 


75 


553946 


663 


446054 


18 


43 


528105 


588 


9rj761 


75 


554344 


663 


445656 


17 


44 


528458 


587 


973716 


76 


554741 


663 


445259 


16 


45 


528810 


587 


973671 


76 


555139 


663 


444861 


15 


46 


539161 


586 


973635 


76 


5555:» 


661 


444464 


14 


47 


539513 


586 


973580 


76 


555933 


6(}1 


444067 


13 


48 


5398.')4 


585 


973535 


76 


556329 


6('>0 


443671 


12 


49 


530213 


585 


973489 


76 


5.56725 


C60 


443275 


11 


50 


530565 


584 


973444 


76 


557121 


659 


442879 


10 


51 


9.53D9I5 


584 


9.973396 


76 


9.557517 


659 


10.443483 


9 


5fl 


53I2H5 


583 


973353 


76 


557913 


659 


442087 


8 


53 


531614 


582 


973307 


76 


558308 


C58 


441692 


7 


54 


531963 


582 


973261 


76 


558702 


6.58 


441298 


6 


55 


532312 


581 


973215 


76 


559097 


657 


440903 


5 


56 


532661 


581 


973169 


76 


559491 


657 


440509 


4 


57 


533009 


580 


973124 


76 


559885 


656 


440115 


3 


56 


533357 


580 


9T3078 


76 


560279 


€56 


439721 


3 


99 


533704 


579 


973033 


77 


560((73 


055 


439327 


1 


60 


534052 


578 


972066 


77 


561066 


655 


438934 





1 


Oosine 




Sine 


1 


Cotaog. 


1 


1 T*ng. 


|M. 



70 



64 


{ 


[20 Degrees.) a 


TABLE OP LOOARITHMIC 




riT 


1 Sine 


1 ». 


Cortne 


1 D. 


1 TMg. 


1 D- 


1 CoUog. 







9.534053 


578 


9.973986 


T7 


9.5(1066 


655 


10.4W914 


60 


1 


534399 


577 


973940 


77 


531459 


654 


438341 


50 


3 


534745 


577 


972891 


77 


561851 


654 


438149 


56 


3 


535092 


577 


973348 


77 


562344 


653 


437756 


S7 


4 


535138 


578 


972802 


77 


562636 


653 


437364 


56 


5 


535783 


576 


972755 


77 


563028 


633 


436973 


55 


6 


536139 


575 


972709 


77 


553419 


652 


43(1581 


54 


7 


536474 


574 


972663 


77 


563811 


653 


436189 


53 


8 


536818 


574 


972617 


T7 


5643 J3 


651 


435798 


53 


9 


537163 


573 


972570 


77 


564593 


651 


4354U8 


51 


10 


537537 


573 


973534 


77 


564933 


659 


435017 


50 


1] 


9.537851 


572 


9.973478 


77 


9.5653:3 


659 


10.434637 


49 


13 


538194 


573 


972431 


78 


56S763 


619 


434337 


48 


13 


838538 


571 


972335 


78 


566153 


640 


433847 


47 


14 


538880 


571 


972338 


78 


536343 


649 


433458 


46 


15 


539233 


570 


973391 


78 


566033 


618 


433068 


45 


16 


5395S5 


570 


972345 


78 


567330 


648 


433630 


44 


17 


539907 


569 


973198 


78 


567709 


647 


433391 


43 


18 


540349 


569 


973151 


78 


563098 


647 


4319U3 


43 


19 


540590 


568 


972105 


78 


568486 


646 


431514 


41 


30 


540031 


568 


973058 


78 


568873 


646 


431137 


40 


31 


9.541373 


567 


9.972011 


78 


9.569251 


645 


10.430739 


39 


^ 


541613 


567 


971934 


73 


569648 


645 


4.30352 


38 


23 


541953 


566 


971917 


78 


570035 


645 


439965 


37 


34 


54^293 


566 


971870 


78 


570433 


644 


439578 


38 


35 


542633 


565 


971823 


/8 


570809 


644 


439191 


35 


26 


542971 


565 


971776 


78 


571195 


643 


438805 


34 


37 


543310 


564 


971739 


79 


571581 


643 


438419 


33 


38 


543649 


564 


971683 


79 


571967 


643 


438033 


33 


30 


543937 


533 


971635 


79 


573353 


&13 


437648 


31 


30 


514325 


563 


971.W8 


79 


5T2738 


643 


437363 


30 


3J 


9.544663 


503 


9.071540 


79 


9.573123 


641 


10 438877 


39 


33 


545000 


552 


971493 


79 


573507 


641 


436493 


38 


33 


545338 


561 


971446 


79 


573802 


640 


436106 


37 


34 


545674 


561 


971398 


79 


574376 


640 


435734 


38 


35 


546011 


560 


971351 


79 


574660 


639 


435340 


35 


36 


546347 


560 


971333 


79 


575044 


639 


434956 


34 


37 


546683 


550 


971358 


79 


575427 


639 


434573 


33 


38 


547019 


5.59 


971-208 


79 


575810 


638 


434190 


33 


39 


547354 


558 


971161 


79 


57(5193 


638 


433807 


31 


40 


547689 


558 


971113 


79 


576576 


637 


433434 


30 


41 


9.548024 


557 


9.971068 


80 


9.576938 


637 


10 433041 


10 


43 


548359 


557 


971018 


8J 


577341 


636 


432650 


18 


43 


548693 


556 


970970 


80 


577733 


636 


433377 


17 


44 


549037 


556 


970932 


80 


578101 


636 


431896 


16 


45 


549360 


555 


970874 


80 


578486 


635 


431514 


15 


46 


549693 


555 


9708-27 


80 


578867 


635 


431133 


14 


47 


550036 


554 


970779 


80 


579248 


634 


430753 


13 


48 


550359 


551 


970731 


80 


579539 


634 


420371 


13 


49 


550692 


553 


970683 


89 


580000 


634 


419991 


U 


50 


551034 


553 


970335 


80 


580389 


633 


419611 


10 


51 


9.551356 


553 


9.970586 


80 


9.580769 


633 


10 419331 


9 


53 


551687 


553 


970538 


80 


581149 


633 


418851 


8 


53 


553018 


552 


970490 


80 


581538 


633 


418473 


7 


54 


553349 


551 


970443 


80 


581907 


633 


418093 


6 


55 


553680 


551 


970394 


80 


583386 


631 


417714 


5 


56 


553010 


550 


970345 


81 


58-2665 


631 


417335 


4 


57 


553341 


550 


970397 


81 


583043 


630 


416957 


3 


58 


553670 


549 


970349 


81 


583432 


630 


416578 


8 


59 


554000 


549 


970300 


81 


583800 


639 


416300 


1 


60 


554339 


548 


970153 


81 


584177 


639 


415833 





1 


Gofllne 


1 


Sine 1 




Gotang. 




Tanf. 


M. 







iOfSS 


AND TANGBMTB. (31 Degrees.) 


65 


M.| 8lM 


D. 


1 GoeiM 1 


D. 


1 Tfciig. 


1 D. 


1 Cotonf . 1 




• 


9.554339 


M8 


9.970158 


81 


9.584177 


630 


10.415833 


60 


1 


554658 


548 


970103 


81 


584555 


699 


415445 


59 


9 


564887 


547 


970055 


81 


584933 


638 


415068 


58 


3 


555315 


547 


970U06 


81 


565309 


638 


414601 


57 


4 


555643 


546 


969957 


81 


585686 


637 


414314 


56 


5 


555071 


546 


969909 


81 


586063 


637 


413938 


55 I 


6 


556399 


545 


069860 


81 


58S439 


637 


413561 


54 


7 


556636 


545 


9698U 


81 


586815 


638 


413185 


53 


8 


556053 


544 


969763 


81 


587190 


638 


413810 


58 


9 


557380 


544 


969714 


81 


567566 


633 


413434 


51 


10 


557606 


543 


969665 


81 


567941 


685 


413050 


50 


11 


9 557933 


543 


9.969616 


83 


0.588316 


635 


10.411681 


49 


13 


538358 


543 


969567 


83 


588691 


^4 


411309 


48 


13 


558583 


543 


969518 


83 


589066 


634 


410934 


47 


14 


558900 


543 


969469 


83 


589440 


633 


410560 


46 


15 


550334 


541 


969430 


83 


589614 


633 


410186 


45 


16 


550558 


541 


969370 


83 


590188 


633 


400813 


44 


17 


5S8883 


540 


960331 


83 


500563 


633 


400438 


43 


18 


560307 


540 


969373 


83 


500935 


633 


400065 


43 


19 


560531 


539 


969333 


83 


501308 


633 


408603 


41 


90 


560855 


539 


969173 


83 


591681 


631 


406310 


40 


81 


0.561178 


538 


9.969134 


83 


9.599054 


631 


10.407946 


39 


» 


561501 


538 


969075 


83 


503436 


630 


407574 


38 


S3 


561834 


537 


960035 


83 


593796 


630 


407903 


37 


34 


503146 


537 


968976 


83 


593170 


610 


406839 


36 


3S 


563468 


530 


968938 


83 


503543 


619 


406458 


35 


S8 


562790 


536 


968877 


83 


593014 


618 


406086 


34 


37 


563113 


536 


968837 


83 


504385 


618 


405715 


33 


38 


563433 


535 


968777 


83 


594656 


618 


405344 


33 


39 


563755 


535 


968738 


83 


595037 


617 


404973 


31 


30 


564075 


534 


988678 


83 


505308 


617 


404603 


30 


31 


0.564306 


534 


9.968038 


83 


9.595768 


617 


10.404339 


90 


33 


564716 


533 


968578 


83 


596138 


616 


403863 


88 


33 


565036 


533 


968588 


83 


596508 


616 


403493 


97 


34 


565356 


533 


968479 


83 


596878 


616 


403193 


90 


3S 


565676 


533 


968489 


83 


507347 


615 


403753 


95 


30 


565095 


531 


968379 


83 


507616 


615 


403384 


94 


37 


566314 


531 


9683S9 


83 


507085 


615 


403015 


93 


38 


566633 


531 


968378 


83 


506354 


614 


401646 


99 


30 


566931 


530 


968338 


84 


508733 


614 


401378 


91 


40 


567389 


530 


968178 


84 


509091 


613 


400009 


90 


41 


9.567587 


539 


9.968138 


84 


9.599459 


613 


10.400541 


19 


43 


567904 


539 


968078 


84 


509837 


613 


400173 


18 


43 


56R332 


538 


968037 


84 


600104 


613 


399806 


17 


44 


568539 


538 


967977 


84 


600563 


613 


399438 


16 


45 


56H56 
56IP3 


538 


967937 


84 


600939 


611 


399071 


15 


46 


527 


967876 


84 


601306 


61] 


396704 


14 


47 


569488 


537 


967896 


84 


601663 


611 


398338 


13 


48 


569604 


536 


967775 


84 


603030 


610 


307971 


18 


49 


570130 


536 


967735 


84 


603395 


610 


397605 


11 


50 


570435 


535 


967674 


84 


603761 


610 


397330 


10 


51 


9.570751 


535 


9.967634 


84 


0.603137 


600 


10.396873 





98 


571066 


534 


967573 


84 


603493 


609 


396.107 


8 


S3 


571380 


534 


967533 


85 


603858 


600 


396143 


7 


54 


571695 


533 


967471 


85 


604333 


606 


395777 


6 


55 


573009 


533 


967431 


85 


604588 


608 


395413 


5 


56 


573333 


533 


967370 


85 


604953 


607 


395047 


4 


57 


573636 


532 


967319 


85 


005317 


607 


304683 


3 


58 


573950 


533 


967368 


85 


605683 


607 


394318 


9 


50 


573363 


531 


967217 


85 


606046 


606 


393954 


1 1 


60 


573575 


531 


967166 


85 


606410 


606 


393590 


1 


1 


Oodne i 


1 


Sine 1 


1 


Ootang. 


1 


Tmg. |M. 1 



ELLWOOD^S TEST PBOB. — 6. 



66 



(23 Degrees.) a tabue op LOOiRtTHMic 



>L| Sbum I D. I CodM | D. | Tftng. | D. | Ooteng | 



^ 


0.573575 


591 


! 1 


573888 


590 


i s 


574900 


590 


3 


474519 


519 


4 


574894 


519 


i 5 


575136 


519 


6 


575447 


518 


7 


575758 


518 


6 


576069 


517 


9 


576379 


517 


10 


576689 


516 


11 


9.576990 


516 


13 


577309 


516 


13 


577618 


515 


14 


5T7997 


515 


15 


578936 


514 


16 


578545 


514 


17 


578853 


513 


18 


579109 


513 


19 


579470 


513 


90 


579777 


519 


91 


9.580085 


519 


99 


580389 


511 


93 


580699 


511 


94 


581005 


511 


95 


581319 


510 


96 


581618 


510 


97 


581924 


500 


98 


589229 


509 


90 


589535 


500 


1 ^ 


589840 


508 


31 


9.583145 


508 


39 


583449 


507 


33 


583754 


507 


34 


584058 


506 


35 


584361 


506 


36 


584665 


506 


37 


584968 


505 


38 


585972 


505 


30 


585574 


504 


40 


585877 


504 


41 


0.586179 


503 


49 


586489 


503 


43 


586783 


503 


44 


587065 


509 


45 


58rj86 


503 


46 


587688 


501 


47 


587980 


501 


48 


588989 


501 


49 


588590 


500 


50 


588890 


500 


51 


9.589190 


499 


59 


589489 


499 


53 


569789 


499 


54 


590088 


496 


55 


590387 


498 


56 


500686 


497 


57 


500964 


497 


58 


591989 


497 


50 


591580 


496 


60 

1 


581878 


496 



9.967166 
967115 
967064 
9G7013 
986961 
966010 
966859 
9668J6 
966756 
986705 
986653 

9.966603 
966550 
966499 
966447 
966395 
966344 
966993 
966240 
906188 
966136 

9.966065 
966033 
965961 
965938 
965876 
965824 
965772 
965790 
965668 
965615 

9.965563 
965511 
965458 
965406 
965353 
965301 
965348 
965105 
905143 
965090 

9.965037 
964964 
964931 
964879 
964896 
964773 
964719 
964666 
964613 
964360 

0.964507 
964454 
964400 
064347 
964204 
964240 
964187 
9G4133 
964060 
964036 



85 

85 
85 
85 
85 
85 
85 
85 
86 
83 
80 

86 
86 
84 
86 
88 
86 
86 
86 
86 
86 

87 
87 
87 
87 
87 
87 
87 
87 
87 
87 

87 
87 
87 
87 
88 
88 
88 
88 
88 
88 

88 
88 
88 
88 
88 
88 
88 
89 
89 
89 

80 
89 
89 
89 
89 
89 
89 
89 
89 
89 



9.606410 
606773 
607137 
607500 
607863 
606235 
60a588 
606950 
609313 
609674 
610036 

9.610397 
610750 
611190 
611480 
611841 
612201 
613561 
613921 
613381 
613641 

0.614000 
614359 
614718 
615077 
615435 
615793 
616151 
616509 
616867 
617324 

0.617582 
617939 
618295 
618653 
619006 
619364 
619731 
620076 
620433 
630787 

9.631143 
621497 
631853 
633207 
633561 
632915 
623269 
633693 
623976 
634330 

0.624683 
625036 
625388 
635741 
636093 
636445 
696797 
627149 
637501 
637853 



606 
606 
605 
605 
604 
604 
604 
603 
603 
603 
603 

603 
603 
601 
601 
601 
600 
600 
600 
599 
599 

598 
598 
598 
597 
597 
597 
596 
596 
596 
595 

595 
505 
504 
594 
594 
503 
593 
593 
593 
593 

503 

501 
591 
590 
590 
500 
589 
580 
589 
588 

588 
588 
587 
587 
587 
586 
586 
586 
585 
585 



10.303590 
303327 
392863 
392500 
392137 
391775 
391412 
391050 
390688 
390336 
389964 

10.389603 
389341 
388880 
388530 
388159 
387799 
387439 
387079 
386719 
386359 

10.386000 
385641 
385383 
384923 
384565 
384207 
383849 
383491 
383133 
382776 

10.383418 
383061 
381705 
381348 
380093 
380636 
380379 
379934 
379568 
379213 

10.378858 

378503 
378148 
r7793 
r439 

376731 
376377 
376034 
375670 

10.375317 
374964 
374613 
374359 
373907 
373555 
373903 
379851 
373499 
372148 



60 
50 
58 
57 
56 
55 
54 
53 
52 
51 
50 

49 
48 
47 
46 
45 
44 
43 
43 
41 
40 

39 
38 
37 
36 
35 
34 
33 
33 
31 
30 

30 
38 
37 
96 
35 
34 
23 
29 
31 
30 

10 
18 
17 
16 
15 
14 
13 
13 
11 
10 


8 
7 
6 
5 
4 
3 
3 
1 




[ I Codne \ | Sine \ \ Cotang. | | Tang. | M. 



67 







SINB8 


AND TANOBNTS. (23 Degrees.) 




67 


M-l 


aiiM 


D. 


CoriiM 


D. 


1 Ttag. 


1 D. 


1 CotaBf. 


1 





0.501878 


496 


9.964036 


80 


9.037853 


585 


10.373148 


60 . 


1 


502176 


495 


063073 


80 


028303 


585 


371797 


50 


s 


503473 


495 


963919 


80 


628554 


585 


37144G 


58 


3 


593770 


495 


963865 


90 


638905 


584 


371095 


57 


4 


503067 


494 


9i$3811 


90 


639355 


584 


370745 


56 


5 


593363 


4M 


963757 


90 


639606 


583 


370394 


55 


6 


503650 


493 


9!i3704 


90 


639936 


583 


370044 


54 


7 


503055 


493 


963650 


90 


630306 


583 


360604 


53 


8 


594351 


493 


063506 


90 


630656 


583 


369344 


58 


9 


591547 


493 


063543 


90 


631005 


583 


368905 


51 


10 


504842 


493 


963488 


90 


631355 


583 


368645 


50 


11 


0.505137 


491 


9.963434 


90 


0.631704 


583 


10.368396 


49 


13 


595432 


491 


963379 


90 


6»2»53 


581 


367947 


48 


13 


505737 


491 


963335 


90 


6^401 


581 


367509 


47 


14 


506031 


490 


963371 


90 


633750 


581 


367350 


46 


15 


506315 


400 


963317 


93 


633098 


580 


366003 


45 


16 


506609 


480 


963163 


9J 


6.13447 


580 


366553 


44 


17 


506003 


489 


963108 


91 


633795 


580 


306305 


43 


18 


597193 


489 


963054 


91 


634143 


579 


365857 


43 


19 


597490 


488 


963999 


91 


634490 


579 


365510 


41 


90 


597783 


488 


963945 


91 


634838 


579 


363163 


40 


31 


9.598075 


487 


9.963890 


91 


9.635185 


57^ 


10.364815 


39 


83 


598368 


487 


968R36 


91 


635533 


578 


364468 


38 


83 


598060 


487 


963781 


91 


635879 


578 


364131 


37 


94 


598953 


486 


903737 


91 


636396 


577 


363774 


36 


8S 


5093i4 


486 


963673 


91 


636573 


577 


363483 


35 


96 


599535 


485 


963617 


91 


036919 


577 


363061 


34 


87 


599^27 


485 


963563 


91 


637905 


577 


363735 


33 


38 


600118 


485 


963508 


91 


637611 


576 


3033R9 


33 


38 


600409 


484 


063453 


91 


637956 


576 


362044 


31 


30 


600700 


484 


902398 


92 


638303 


576 


361606 


30 


31 


0.600090 


484 


9.9^343 


92 


9.638647 


575 


10.311353 


39 


33 


601383 


483 


9^388 


92 


638993 


575 


361008 


38 


33 


601570 


483 


96^33 


92 


639337 


575 


360663 


37 


34 


601860 


483 


963178 


92 


639683 


574 


360318 


36 


3S 


603150 


482 


962133 


92 


640037 


574 


359973 


35 


36 


<«3139 


483 


963067 


92 


640371 


574 


359639 


34 


37 


603738 


481 


9^013 


92 


640716 


573 


359384 


33 


38 


603017 


481 


961957 


92 


641060 


573 


358040 


33 


30 


603305 


481 


9S1903 


92 


641404 


573 


358596 


31 


40 


603504 


480 


961846 


92 


641747 


573 


358253 


30 


41 


0.603882 


480 


9.961791 


92 


0.649091 


572 


10.357909 


10 


43 


604170 


479 


961735 


92 


643434 


573 


357566 


18 


43 


604457 


479 


961680 


92 


643T77 


573 


357333 


17 


44 


604745 


479 


961024 


93 


643130 


571 


356880 


16 


45 


605032 


478 


961569 


93 


643163 


571 


356537 


15 


46 


605319 


478 


961513 


93 


643836 


571 


350194 


14 


47 


605006 


478 


961458 


93 


644148 


570 


355853 


13 


48 


wisfm 


477 


9614(» 


93 


(144490 


570 


355510 


13 


49 


606179 


477 


961346 


93 


644833 


570 


355168 


11 


50 


606465 


476 


961390 


93 


645174 


569 


354836 


10 


51 


9.696751 


476 


9.961335 


93 


0.645516 


569 


10.354484 


9 


53 


607036 


476 


961179 


93 


645857 


5(i9 


351143 


8 


53 


607332 


475 


961133 


93 


646199 


569 


353801 


7 


M 


607607 


475 


961067 


9J 


646540 


566 


353460 


6 


55 


607892 


474 


961011 


93 


646881 


5G8 


353119 


5 


56 


606177 


474 


9S0955 


93 


617323 


566 


353778 


4 


1 S7 


608461 


474 


060899 


93 


647563 


567 


352438 


3 


58 


608745 


473 


960843 


94 


647903 


567 


353097 


3 


SO 


600039 


473 


960786 


94 


648343 


5r>7 


351757 


1 


60 


609313 


473 


960730 


94 


648583 


56(f 


351417 





i 1 


Chorine | 


1 


Sine 1 


1 


Cotong. 1 


1 


Tttg. 1 


M. 



66 



68 



(24 Degrees.) a tabu of LoaA«iTRMic 



M. 1 


Sin* 


1 D- 1 


CoaiM j 


D. 


Tu«. 


1 D. 


1 ColMlg. 


r 





9.609313 


473 


9.960730 


94 


9.048583 


500 


10.351417 


00 


1 


609597 


47t 


960674 


94 


648923 


506 


351077 


50 


2 


609680 


472 


96(Mil8 


94 


641^363 


560 


350TJ7 


58 


3 


610164 


472 


9H0561 


04 


049602 


560 


350306 


57 


4 


610447 


471 


96<»505 


94 


049942 


505 


35U058 


50 


5 


610729 


471 


960448 


94 


050361 


565 


349719 


55 


6 


611012 


470 


960^12 


94 


650630 


565 


349380 


54 


7 


611294 


470 


mm^ 


94 


650959 


564 


340041 


53 


8 


611576 


470 


960279 


94 


051297 


504 


648703 


53 


9 


611858 


469 


9602:^ 


94 


651636 


564 


348364 


51 


10 


612140 


400 


960165 


94 


651074 


503 


348026 


50 


11 


9.612421 


460 


9.960109 


95 


9.053313 


56S 


10.347688 


49 


IS 


612702 


468 


960052 


95 


053650 


563 


347350 


48 


13 


612983 


468 


950005 


i» 


fiS2088 


503 


347012 


47 


14 


613264 


467 


959038 


95 


6^3390 


568 


346674 


40 


15 


613545 


407 


959682 


95 


653003 


568 


340337 


45 


16 


613825 


407 


950825 


95 


6$4000 


503 


344iOOO 


44 


17 


614105 


466 


059768 


95 


054337 


561 


345663 


43 


18 


614385 


466 


959711 


95 


054074 


561 


345396 


43 


19 


614G65 


466 


950654 


95 


055011 


561 


341989 


41 


90 


614944 


465 


959596 


95 


055348 


501 


344652 


40 


31 


9.615223 


405 


9.9595.W 


95 


0.655GR4 


500 


10.344310 


30 


23 


615502 


465 


959482 


95 


056030 


560 


343980 


38 


23 


615781 


464 


959425 


95 


050356 


560 


343644 


37 


24 


616060 


464 


959368 


95 


056692 


550 


34.1306 


30 


25 


616338 


464 


9.59310 


96 


057028 


5.59 


343972 


35 


26 


616016 


463 


9.)9253 


96 


057364 


5.5P 


342636 


34 


27 


616804 


463 


950195 


96 


657699 


5.59 


342301 


33 


28 


617172 


462 


ft59l38 


96 


058034 


558 


341966 


33 


20 


617450 


462 


959081 


96 


058369 


558 


341631 


31 


30 


617727 


462 


959023 


96 


658704 


558 


341396 


30 


31 


9.618004 


461 


9.958965 


96 


9.650099 


558 


10.340061 


39 


32 


618281 


461 


958908 


96 


6593rj 


557 


340637 


36 


33 


618558 


461 


958850 


96 


659708 


557 


340292 


37 


34 


618834 


460 


958792 


96 


660042 


557 


330958 


90 


35 


619110 


460 


958734 


96 


660376 


557 


339624 


95 


36 


619.18G 


460 


958677 


96 


660710 


5.56 


339290 


34 


37 


619662 


450 


958619 


96 


061043 


550 


338957 


S3 


38 


619938 


459 


958561 


96 


661377 


550 


338623 


23 


30 


620213 


459 


958503 


97 


061710 


555 


338290 


31 


40 


620488 


458 


958445 


97 


062043 


555 


337957 


90 


41 


9.620703 


458 


9.058387 


97 


9.662.T76 


5.55 


10.337634 


19 


42 


621038 


457 


958329 


97 


662709 


554 


337291 


18 


43 


621313 


457 


958271 


97 


663042 


554 


336958 


17 


44 


621587 


457 


958213 


97 


663375 


554 


336625 


16 


45 


621861 


456 


958154 


97 


6GJ707 


5.54 


336293 


15 


46 


622135 


456 


958096 


97 


664039 


553 


335961 


14 


47 


622409 


456 


958038 


97 


664371 


553 


3:45029 


13 


48 


6226R2 


455 


957979 


97 


664703 


553 


335297 


19 


49 


622956 


455 


9.57921 


97 


665035 


553 


334965 


11 


50 


623229 


455 


957863 


97 


665366 


558 


334634 


10 


51 


9.623503 


454 


9.957B04 


97 


9.665607 


553 


10.334.103 


9 


52 


623774 


454 


957746 


98 


666029 


5.52 


33:«I71 


8 


53 


624047 


454 


957687 


98 


66^.360 


551 


333640 


7 


54 


624319 


453 


957628 


98 


666691 


551 


333300 


8 


55 


624591 


4.53 


957570 


93 


667021 


551 


332979 


5 


56 


624863 


453 


9.57511 


96 


667352 


551 


332848 


4 


57 


625135 


452 


957452 


98 


667682 


550 


3:i2318 


3 


58 


625406 


452 


957393 


98 


668013 


550 


331987 


3 


59 


625677 


452 


957335 


98 


668343 


5.50 


331657 


1 


60 


025948 


451 


957276 


98 


668672 


5.50 


331328 







1 Ootin* 


1 


1 Sine 


1 


1 Cotang. 


1 


1 T%ng. 


|M. 



Oft 









STNB8 AND TANGENTS. 


(25 Degrees.) 




69 







1 Bine | 


D. 


1 Corfne 1 


D. 


Tang. 


i »• 


i GoteDg. 1 || 




9.095948 


451 


9.957276 


96 


9.668673 


550 


10.331327 


60 




1 


096319 


451 


957217 


98 


669003 


549 


330998 


59 




8 


036490 


451 


957158 


96 


669332 


549 


330668 


58 




S 


096700 


450 


957099 


98 


669661 


549 


330339 


57 




4 


037030 


450 


957040 


98 


669991 


Dto 


330009 


56 




5 


037300 


450 


956981 


98 


670320 


548 


329680 


55 




6 


OS7570 


449 


950921 


99 


670649 


548 


329351 


54 




7 


037840 


449 


956803 


99 


670977 


548 


329023 


53 




8 


038100 


449 


956803 


99 


671306 


547 


328694 


52 




9 


eais:nH 


448 


956744 


99 


671634 


547 


328366 


51 




10 


03R647 


448 


ruuutOA 


99 


671963 


547 


328037 


50 




11 


9.038016 


447 


9.956625 


99 


9.672291 


547 


J0.32T709 


49 




IS 


039185 


447 


956566 


99 


679619 


546 


327381 


48 




13 


020453 


447 


956506 


99 


672047 


5^ 


327053 


47 




14 


^731 


446 


956447 


99 


673274 


546 


326726 


46 




15 


039989 


446 


956387 


99 


673602 


546 


326398 


45 




16 


630257 


446 


956397 


99 


673929 


M5 


326071 


44 




17 


<mRI34 


446 


954968 


99 


674257 


545 


325743 


43 




18. 


630793 


445 


956308 


100 


674584 


545 


325416 


42 




19 


631059 


445 


956148 


100 


674910 


544 


325090 


41 




» 


631336 


445 


956089 


100 


675237 


544 


324763 


40 




21 


9.631503 


444 


9.956039 


100 


9.675564 


544 


10.324436 


39 




» 


C318S9 


444 


955()69 


100 


675890 


544 


3241J0 


38 




23 


633135 


444 


955909 


100 


676216 


543 


323784 


37 




9i 


633393 


443 


955849 


100 


676543 


543 


323457 


36 




35 


633658 


443 


955789 


100 


676669 


543 


323131 


35 




96 


633023 


443 


955739 


100 


677194 


543 


332806 


34 




97 


633189 


448 


955669 


100 


677520 


542 


322480 


33 




38 


633454 


443 


955609 


100 


677846 


542 


3!£2154 


32 




39 


633719 


443 


955548 


100 


678171 


542 


321829 


31 




30 


633964 


441 


955488 


100 


67d496 


543 


321504 


30 




31 


9.634249 


441 


9.955438 


101 


9.678821 


541 


10.321179 


29 




SB 


634514 


440 


955368 


101 


679146 


541 


320854 


28 




33 


634778 


440 


955307 


101 


679471 


541 


320529 


27 




34 


635042 


440 


055347 


101 


679795 


541 


320205 


26 




35 


635300 


430 


955186 


101 


680120 


540 


319880 


25 




36 


635570 


439 


955196 


101 


680444 


540 


319556 


24 




37 


635834 


439 


955065 


101 


680768 


540 


319232 


23 




38 


636097 


438 


955005 


101 


681092 


540 


318908 


22 




39 


636360 


438 


054944 


101 


681416 


539 


318584 


21 




40 


636633 


438 


954883 


101 


681740 


539 


318260 


20 




41 


9.636886 


437 


9.954823 


101 


9.689063 


539 


10.317937 


19 




4St 


637148 


437 


954762 


101 


682387 


539 


317613 


18 




43 


637411 


437 


954701 


101 


683710 


538 


317290 


17 




44 


637673 


437 


954640 


101 


683033 


538 


316967 


16 




45 


637935 


436 


954579 


101 


683356 


538 


316644 


15 




46 


638197 


436 


954518 


108 


683679 


538 


316321 


14 




47 


638458 


436 


954457 


108 


684001 


537 


319999 


13 




48 


638730 


435 


954396 


109 


684324 


537 


315676 


12 




49 


638981 


435 


954335 


108 


684646 


537 


315354 


11 




50 


639242 


435 


954374 


103 


684968 


537 


315032 


10 




51 


9.639503 


434 


9.954213 


108 


9.685890 


536 


10.314710 


9 




58 


639764 


434 


954158 


103 


685673 


536 


314388 


8 




S3 


640024 


434 


954090 


108 


6a'i934 


536 


314066 


7 




54 


640984 


433 


954099 


108 


686255 


536 


313745 


6 




55 


640544 


433 


953968 


102 


686577 


535 


313423 


5 




50 


640804 


433 


953906 


wst 


686898 


535 


313102 


4 




57 


641064 


438 


953845 


108 


687219 


535 


312781 


3 




58 


641384 


438 


953783 


108 


687540 


535 


312460 


2 




50 


641584 


432 


953?^ 


103 


687861 


534 


312139 


1 




00 


641843 


431 


953660 


103 


688182 


534 


311818 









Cortne | 


1 


ffise 1 


\ 


Gotang. 1 


1 


Tang. |M.|| 



64 



70 


(96 Degiees.) a 


TABI.R OP LOGARITHMIC 




M. 1 


am 


1 I>. 


1 COfllB« 1 


D. 


1 Tang, 


D. 


1 CMaag. 


1 





9.641843 


431 


0.953660 


103 


0.688183 


534 


10.311818 


60 


1 


(M2101 


431 


953599 


103 


688503 


534 


311498 


50 


s 


642360 


431 


053537 


103 


688833 


534 


311177 


58 


3 


642618 


430 


053475 


103 


680143 


533 


310857 


57 


4 


642877 


430 


053413 


103 


680463 


533 


310537 


56 


5 


643135 


430 


053353 


103 


689783 


533 


310317 


55 


6 


643303 


430 


053300 


103 


600103 


533 


300897 


54 


7 


643650 


430 


053238 


103 


600433 


533 


309577 


53 


8 


643908 


430 


053166 


103 


600743 


533 


309358 


58 


9 


644165 


439 


053104 


103 


601063 


538 


308938 


51 


10 


644423 


438 


053048 


103 


601381 


538 


308619 


50 


11 


0.644680 


488 


9053960 


104 


9.091700 


531 


10.306300 


49 


13 


644936 


488 


058018 


104 


603010 


531 


307981 


48 


13 


645193 


487 


058855 


104 


603338 


531 


307663 


47 


14 


645450 


487 


058793 


104 


608656 


531 


307344 


46 


15 


645706 


437 


953731 


104 


6B8075 


531 


307035 


45 


16 


645063 


486 


953669 


104 


603893 


530 


306707 


44 


17 


646318 


486 


053606 


104 


603618 


530 


306368 


43 


18 


646474 


486 


052544 


104 


603030 


530 


306070 


48 


19 


64(i729 


435 


05^81 


104 


604848 


530 


305758 


41 


20 


646984 


435 


053410 


104 


604566 


589 


305434 


40 


31 


9.647240 


435 


0.952358 


104 


9.604883 


589 


10.305117 


30 


S3 


647494 


434 


053294 


104 


605801 


589 


304700 


38 


S3 


647749 


434 


99^31 


104 


005518 


589 


304483 


37 


SI 


648004 


434 


0521G8 


105 


093836 


539 


304164 


36 


85 


•48258 


434 


953106 


105 


606153 


588 


303847 


35 


86 


648513 


433 


058043 


105 


606470 


538 


303530 


34 


87 


648766 


483 


051080 


105 


606787 




303813 


33 


38 


649020 


433 


051017 


105 


607103 


588 


303807 


33 


89 


649274 


433 


051854 


105 


007480 


587 


303580 


31 


30 


649527 


428 


051791 


105 


607736 


587 


303364 


30 


31 


9.649781 


423 


0.051738 


105 


0.608053 


537 


10.301047 


80 


33 


650034 


^3 


051665 


105 


008369 


587 


301631 


88 


33 


650387 


431 


051603 


105 


608685 


586 


301315 


87 


34 


650539 


431 


051530 


105 


600001 


586 


300000 


86 


35 


6S0792 


431 


051476 


105 


600316 


586 


300664 


85 


36 


651044 


480 


051413 


105 


600633 


586 


300368 


34 


37 


651297 


480 


051340 


106 


600947 


536 


300053 


83 


38 


651549 


430 


051386 


106 


700863 


535 


800rj7 


88 


39 


651800 


419 


051323 


106 


700578 


535 


399483 


81 


40 


653053 


419 


051150 


106 


700803 


535 


899107 


80 


41 


0.658304 


419 


0.051006 


KM 


0.701806 


584 


10.808703 


19 


43 


658555 


418 


051033 


106 


701533 


534 


808477 


18 


43 


653806 


418 


050068 


106 


701837 


534 


888163 


V 


44 


653057 


418 


050005 


106 


703153 


534 


807848 


16 


45 


653306 


418 


050641 


106 


708466 


584 


807534 


15 


46 


653558 


417 


050778 


106 


708780 


sa 


807830 


14 


47 


653808 


417 


050714 


106 


703005 


533 


896005 


13 


48 


654059 


417 


050650 


106 


703400 


583 


806501 


13 


49 


654309 


416 


050586 


106 


703783 


533 


806377 


U 


50 


654558 


416 


050583 


107 


704036 


538 


885064 


10 


5] 


9.654808 


416 


0.050458 


107 


9.704350 


583 


10.805650 


9 


53 


655058 


416 


050304 


107 


704663 


588 


895337 


8 


53 


655307 


415 


050330 


107 


•7049n 


533 


895033 


7 


54 


655556 


415 


050866 


107 


705300 


533 


894710 


6 


55 


655805 


415 


050303 


107 


705603 


531 


S04307 


5 


56 


656054 


414 


050138 


107 


705016 


531 


804064 


4 


57 


656302 


414 


050074 


107 


706^8 


581 


803778 


3 


58 


656551 


414 


050010 


107 


706541 


531 


303450 


8 


50 


656799 


413 


040045 


107 


706654 


581 


803146 


1 


60 


657047 


413 


040881 


107 


707166 


580 


893834 







CoidQe 


1 


Sine 1 




1 Coteng. 


1 


1 TMlg. 


|M. 



6dDegre«k 



mas AND TAHOBHTt. (27 DegrtM.) 



71 



M. 


1 Siiw 


1 ». 


1 CkMiM 


1 0. 


1 Ttag. 


1 ». 


t Gotang. 


1 





9.657047 


413 


9.949881 


107 


9.707166 


520 


10.392834 


60 


1 


657395 


413 


949816 


107 


707478 


530 


302532 


50 


s 


657542 


412 


949752 


107 


707790 


530 


393310 


58 


3 


657790 


413 


949688 


108 


708102 


520 


391896 


57 


4 


658037 


413 


949623 


108 


708414 


519 


391586 


56 


5 


658384 


412 


940558 


108 


708726 


519 


391274 


55 


6 


658531 


411 


949(94 


108 


709037 


519 


390963 


54 


7 


658778 


411 


949429 


108 


709349 


519 


390651 


53 


8 


630025 


411 


949364 


108 


709660 


519 


390340 


53 


9 


659271 


410 


949300 


108 


709971 


518 


390029 


51 


10 


659517 


410 


949235 


106 


7102%S 


518 


389718 


50 


11 


9.659763 


410 


9.949170 


106 


9.710303 


518 


10.389407 


40 


13 


663009 


409 


949105 


106 


710904 


518 


3H9096 


48 


13 


660235 


409 


949049 


106 


711313 


818 


888785 


47 


14 


660501 


409 


948976 


106 


711525 


517 


888475 


46 


15 


060746 


40^ 


9489ia 


108 


711836 


5J7 


368164 


45 


16 


660991 


406 


948845 


106 


712146 


517 


867834 


44 


17 


661236 


408 


948780 


100 


71345S 


517 


387544 


43 


18 


661481 


408 


948715 


109 


712766 


516 


387234 


43 


19 


661726 


407 


948(150 


109 


713076 


516 


366924 


41 


90 


661970 


407 


948584 


109 


713386 


516 


386614 


40 


31 


9.662314 


407 


9.948519 


109 


9.713690 


516 


10.386304 


39 


S3 


663459 


407 


948454 


109 


714005 


516 


385095 


38 


33 


663703 


406 


948388 


109 


714314 


515 


985686 


37 


34 


663946 


406 


94(m3 


109 


714^4 


515 


385376 


36 


35 


663190 


406 


948257 


109 


714933 


515 


385067 


35 


96 


663433 


405 


048193 


109 


715242 


515 


384753 


34 


97 


663677 


405 


948126 


109 


715551 


514 


384449 


33 


'J8 


663920 


405 


948060 


109 


715860 


514 


3CJ4140 


33 


39 


664163 


405 


947995 


110 


716168 


514 


383833 


31 


30 


664406 


404 


947929 


110 


716477 


514 


383523 


30 


31 


9.664648 


404 


9.947863 


110 


9.716785 


514 


10.383315 


29 


33 


664891 


404 


947797 


110 


717093 


513 


382937 


88 


33 


665133 


403 


947731 


110 


717401 


513 


883599 


37 


34 


665373 


403 


947665 


110 


717709 


513 


882291 


36 


35 


665617 


403 


947609 


110 


718017 


513 


881983 


85 


36 


665839 


403 


047533 


110 


718323 


513 


381675 


34 


37 


666103 


403 


947467 


110 


718633 


512 


381367 


33 


38 


666342 


403 


947401 


110 


718940 


512 


381060 


32 


30 


666583 


403 


947335 


110 


719248 


513 


380752 


31 


40 


6H6H24 


401 


947269 


110 


719533 


513 


880445 


30 


41 


9.667065 


401 


9.947303 


110 


9.719662 


513 


10.380138 


19 


43 


6C7305 


401 


947133 


111 


730169 


511 


379631 


18 


43 


66734G 


40J 


947070 


111 


730476 


511 


879534 


17 


44 


667786 


400 


947094 


111 


790783 


511 


379217 


16 


43 


668027 


400 


9469J7 


111 


721089 


511 


378911 


15 


46 


668267 


400 


946871 


111 


731396 


511 


878604 


14 


47 


668506 


309 


946804 


111 


721702 


510 


878293 


13 


48 


668746 


399 


946738 


111 


733009 


510 


277991 


12 


49 


6689B6 


399 


»ld671 


111 


733315 


510 


377685 


11 


50 


669325 


399 


946604 


HI 


733021 


510 


377379 


10 


51 


9.669464 


398 


9.946538 


111 


9.722927 


510 


10.377073 


9 


52 


669703 


ftRffS 


946471 


111 


723232 


509 


376768 


8 


53 


669942 


JU8 


946404 


111 


733538 


509 


376462 


7 


54 


670181 


397 


046337 


111 


733844 


509 


876136 


6 


S3 


670419 


397 


946270 


112 


724149 


509 


873831 


5 


56 


670658 


397 


946203 


112 


724454 


509 


875546 


4 


57 


67069S 


397 


946136 


112 


734739 


508 


875241 


3 


58 


671134 


393 


946069 


112 


723063 


508 


374935 


3 


59 


671372 


396 


946033 


112 


733369 


508 


374631 


1 


60 


671609 


396 


945935 


113 


725674 


508 


374326 





1 


Cosine 1 


1 


Sine 1 


1 


Cotang. 1 


1 


TM»g. 1 


M. 








62 


Degre 


ef. 






■"^ 



72 



(28 Degraet.) a tabu op LOGAmiTBific 



 

1 
9 
3 

4 
5 
6 
7 
8 
9 
10 

11 
18 
13 
14 
15 
16 
17 
18 
10 
90 

31 
S3 
93 
94 

95 
96 
97 
9B 
90 
30 

31 
33 
33 
34 
35 
36 
37 
38 
39 
40 

41 
49 
43 
44 

45 
46 
47 
48 
49 
50 

51 
S3 
53 
54 
55 
56 
57 
58 
50 
00 



(Him 1 


D- 1 


9.671609 


396 


671847 


395 


679084 


395 


679331 


395 


679558 


395 


679795 


394 


673039 


394 


673968 


394 


673505 


394 


673741 


303 


6rJ977 


393 


9.674913 


3B3 


674448 


398 


674684 


399 


674919 


393 


675155 


399 


675390 


391 


675634 


391 


675859 


391 


676094 


391 


678328 


390 


9.676569 


390 


676796 


390 


677030 


390 


677964 


389 


677498 


389 


677731 


389 


677B64 


388 


678197 


388 


678430 


388 


678663 


388 


9.678895 


387 


679128 


387 


679360 


387 


679593 


387 


679824 


386 


680056 


386 




386 


680519 


385 


680750 


385 


680983 


385 


0.681813 


385 


681443 


384 


681674 


384 


681905 


384 


683135 


384 


683365 


383 


683595 


383 


682825 


383 


683055 


383 


683384 


383 


9.683514 


388 


663743 


383 


683973 


389 


684301 


381 


684430 


381 


684658 


381 


684887 


380 


685115 


380 


685343 


380 


685571 


380 



CoriM I D. t ^f • I ^- I Oo«»8- I 



9.945935 


113 


945868 


113 


945800 


119 


945733 


119 


945666 


119 


945508 


119 


945531 


119 


1M5464 


113 


945396 


113 


945338 


113 


945261 


113 


9.945193 


113 


945135 


113 


945058 


113 


944990 


113 


944932 


113 


944854 


113 


944786 


113 


944718 


113 


944650 


113 


944583 


114 


9.944514 


114 


944446 


114 


944377 


114 


944309 


114 


944241 


114 


944173 


114 


944104 


114 


944036 


114 


943967 


114 


943899 


114 


9.943830 


114 


943761 


114 


94.3693 


115 


943684 


115 


943555 


115 


943486 


115 


943417 


115 


943348 


115 


943879 


115 


943810 


115 


9.943141 


115 


943079 


115 


943003 


115 


949934 


115 


942864 


115 


949795 


116 


949796 


116 


943656 


116 


949587 


116 


943517 


116 


9.949448 


116 


949378 


116 


943308 


116 


043839 


116 


943169 


116 


943009 


116 


943039 


116 


941959 


116 


941889 


117 


941819 


117 



9.795674 
725079 
790364 
796588 
796803 
797197 
797501 
797805 
798109 
728413 
7SB716 

9.729000 
799393 



7S9099 
730933 
730535 
730838 
731141 
rJ14i4 
731746 

9.739048 
738351 
739653 
733955 
733857 
733558 
733860 
734163 
734463 
734764 

9.735066 
735367 
735668 
735009 
736809 
736570 
736871 
737171 
737471 
737771 

0.738071 
738371 
738671 
738971 
739871 
739570 
739870 
740169 
740468 
740767 

9.741066 
741365 
741664 
741063 
743961 
743559 
748858 
743156 
743454 
743758 



508 

506 
507 
SU7 
507 
507 
507 
506 
506 
506 
506 

506 



506 

505 
506 

505 
504 
504 
504 
504 

504 
503 
503 
503 
503 
503 
SO) 
508 
508 
508 

508 
508 
501 
501 
501 
501 
501 
500 
500 
500 

500 
500 

499 
409 
409 
499 
400 
499 
498 
408 

498 
498 
496 
497 
497 
497 
497 
497 
497 
496 



10.874396 
274081 
973716 
973419 
973108 
872803 
972499 
972195 
971891 
971588 
971984 

10.970980 
970677 
970374 
970071 
960767 
909465 
909162 
968859 
968556 
968854 



]0.1»«^ 
967649 
967347 
967045 
966743 
906149 
966140 
965838 
965537 
965836 

10.964034 
9B4633 
S64339 
964031 
963731 
963430 
963139 



10.961939 
961639 
961399 
961029 
960799 
900430 
960130 
959831 
950539 
990933 

10.958934 
958635 
958336 
958038 
957739 
957441 
957149 
956844 
956546 
950348 



60 
59 
58 
57 
56 
55 
54 
S3 
53 
51 
50 

40 
48 
47 
46 
45 
44 
43 
48 
41 
40 

39 
38 
37 
36 
35 
34 
33 
38 
31 
30 

90 
98 
97 
96 
95 
94 
93 
99 
91 
90 

19 
18 
17 
J6 
15 
14 
13 
19 
11 
10 

9 

8 
7 
6 
5 
4 
3 
9 
1 




I OoaiiM I 



I 



t Ooteng. I 



I Tung. I M. 



OlDegreef 







flntM AMD TAN0BNT8. (39 Degnes.] 




Z 


3 


M.I 


8lM 


1 ». 


1 OoriiM 


1 D- 


1 Twag. 


1 »• 


1 Oottuig. 


1 







9.685571 


380 


0.041810 


117 


0.743753 


406 


10.856848 


60 




1 


685799 


370 


041749 


117 


744050 


496 


855050 


50 




2 


686027 


379 


011679 


117 


744348 


496 


855653 


58 




3 


688354 


370 


041600 


117 


744645 


406 


855355 


57 




4 


686482 


370 


041539 


117 


744043 


406 


855A57 


56 




5 


686709 


378 


041400 


117 


745840 


406 


854760 


55 




6 


686936 


378 


041398 


117 


745538 


405 


854463 


54 


7 


6871G3 


378 


941328 


117 


745835 


405 


854165 


53 


8 


687389 


378 


941358 


117 


746132 


495 


353868 


53 


9 


687616 


377 


041187 


117 


746429 


495 


853571 


51 


10 


681843 


377 


041117 


117 


746736 


405 


853874 


50 


U 


9.688080 


377 


0.1M1046 


118 


0.747083 


404 


10.858977 


40 


13 


688205 


jn 


040075 


118 


747310 


«>4 


8»681 


48 


13 


688S81 


376 


040905 


118 


747616 


404 


SS8384 


47 


14 


688747 


376 


040834 


118 


747013 


4M 


8581187 


46 


15 


688073 


376 


040763 


118 


748200 


404 


851701 


45 


16 


680198 


376 


040603 


118 


748505 


403 


851495 


44 


17 


68M33 


375 


040622 


118 


748801 


403 


851199 


43 




18 


vowno 


375 


040551 


118 


749097 


493 


850003 


48 




19 


680873 


375 


040480 


118 


749393 


493 


850607 


41 




90 


600008 


375 


940409 


118 


740689 


403 


850311 


40 




91 


0.600383 


374 


0.040338 


118 


0.740965 


403 


10.850015 


30 




99 


600548 


374 


04<)867 


118 


750381 


402 


849719 


38 




93 


600773 


374 


040106 


118 


750576 


403 


840424 


37 




94 


600096 


374 


040185 


110 


750873 


403 


840138 


36 




95 


601220 


373 


040054 


110 


751167 


493 


8l8ai3 


35 




96 


601444 


373 


039082 


110 


751493 


4fa 


848.138 


34 




87 


601668 


373. 


030911 


110 


751757 


492 


848243 


33 




98 


601892 


373 


939840 


110 


752052 


491 


847048 


33 




90 


002115 


372 


03Sn'68 


110 


752347 


491 


847653 


31 




30 


602330 


373 


039607 


110 


758643 


491 


847358 


30 




31 


0.6(KiS02 


372 


0.030985 


no 


0.7.18037 


^1 


10.8470<>3 


30 




38 


«»785 


371 


030554 


119 


753231 


491 


846760 


38 




33 


693008 


371 


930482 


119 


753526 


401 


846474 


87 




34 


603231 


371 


039410 


119 


753820 


490 


846180 


36 




35 


603453 


371 


930339 


119 


754115 


490 


845885 


35 




36 


603676 


370 


9»»67 


120 


754400 


490 


845591 


34 




37 


603898 


370 


930105 


120 


754703 


499 


845297 


83 




38 


604120 


370 


930123 


120 


754007 


490 


845003 


22 




30 


604342 


370 


031)952 


190 


755201 


490 


344709 


31 




40 


604564 


300 


038080 


190 


755585 


480 


844415 


80 




41 


0.604786 


360 


0.938908 


120 


0.755878 


489 


10.844132 


19 




48 


695007 


360 


938836 


130 


756172 


489 


843828 


18 




43 


605329 


300 


938763 


130 


756465 


480 


343535 


17 




44 


605450 


368 


938601 


130 


756750 


480 


843811 


16 




45 


605671 


368 


038610 


130 


757052 


489 


843048 


15 




46 


606892 


368 


038547 


130 


757345 


488 


843655 


14 




47 


606113 


368 


638475 


120 


757638 


488 


843363 


13 




48 


696334 


367 


938402 


131 


757931 


488 


843060 


13 




49 


606554 


367 


938330 


131 


7582*4 


488 


841776 


11 




50 


606775 


367 


938258 


121 


758517 


488 


841483 


10 




51 


0.606005 


367 


9.038185 


131 


0.758810 


488 


10.841100 







58 


607815 


366 


938113 


121 


759102 


487 


840898 


8 




S3 


607435 


366 


938040 


121 


750305 


487 


840605 


7 




54 


607654 


366 


037067 


121 


750087 


487 


840313 


6 




55 


607874 


366 


037895 


131 


750979 


487 


840021 


5 




56 


698094 


365 


037822 


121 


700273 


487 


83U728 


4 




57 


608313 


365 


037749 


121 


760564 


487 


839436 


3 




58 


608532 


365 


937676 


121 


760856 


486 


839144 


3 




50 


808751 


365 


937H04 


131 


761148 


486 


338853 


1 




f^ 


608070 


364 


937531 


131 


761439 


486 


838561 







\L 


1 OlMllM 




fliiie 




Cotang. 1 




1 Ting. 


1 M. 





ao 



74 



(30 Degieet.) - * tiblk op looarithmic 



n? 


1 able 


1 1> 


1 CosbM 


1 D. 


1 Ttog. 


1 »• 


1 OotaBff. 


1 







9.608970 


364 


9.937531 


131 


9.761439 


486 


10.338561 


60 




1 


609189 


364 


937458 


133 


761731 


486 


838369 


50 




8 


600407 


364 


937385 


133 


769023 


486 


837977 


58 




3 


600626 


364 


937313 


133 


763314 


488 


837686 


57 




4 


600844 


303 


937238 


133 


763606 


465 


837394 


56 




5 


700062 


363 


937165 


133 


763897 


485 


837103 


55 




6 


70U880 


363 


937093 


133 


763188 


465 


836813 


54 




7 


709496 


363 


937010 


1^ 


763479 


485 


836.'i31 


53 




8 


700716 


363 


936946 


133 


763770 


485 


836230 


53 




9 


700933 


363 


936873 


123 


764061 


485 


235039 


51 




10 


701151 


363 


036799 


132 


764352 


484 


83S648 


50 




11 


0.701368 


302 


9.936735 


133 


9.764643 


484 


10.835357 


49 




IS 


701585 


368 


936652 


123 


764933 


484 


835067 


48 




13 


701802 


361 


936578 


123 


765324 


484 


834776 


47 




14 


709019 


361 


936505 


123 


765514 


484 


834486 


46 




15 


7029:M 


361 


936431 


123 


765805 


484 


834195 


45 




16 


70945S 


361 


936357 


123 


766095 


484 


833905 


44 




17 


702660 


360 


9.16884 


123 


766385 


483 


233615 


43 




18 


703885 


360 


936310 


123 


766G75 


483 


833325 


43 




19 


703101 


360 


936136 


123 


766965 


483 


833035 


41 




90 


703317 


360 


936063 


123 


767255 


483 


833745 


40 




SI 


9.703533 


359 


9.935988 


123 


9.767545 


483 


10.838455 


39 




2S 


703749 


359 


935914 


123 


767834 


483 


832166 


38 




S3 


7U3964 


350 


935840 


123 


768124 


483 


831876 


37 




34 


704179 


359 


935766 


124 


7C8413 


483 


831587 


36 




S5 


704395 


350 


935693 


134 


7C8703 


463 


831397 


35 




S6 


704610 


358 


935618 


124 


768993 


483 


231008 


34 




27 


704825 


358 


935543 


124 


769381 


483 


830719 


33 




S8 


705040 


358 


935469 


134 


769570 


488 


830430 


38 




29 


705254 


358 


935395 


124 


769600 


481 


830140 


31 




30 


7U5463 


357 


935330 


124 


770148 


481 


899653 


30 




31 


0.705683 


357 


9.935346 


134 


9.770437 


481 


10.829563 


89 




33 


.705898 


357 


935171 


134 


770736 


481 


SS9374 


88 




33 


706112 


357 


935097 


124 


771015 


481 


8S8985 


87 




34 


70632U 


356 


935033 


134 


771303 


481 


238697 


86 




35 


706530 


356 


034948 


134 


771503 


481 


238408 


85 




36 


706753 


356 


934873 


134 


771880 


480 


228120 


84 




37 


706967 


356 


934798 


J35 


773168 


480 


237832 


83 




38 


707180 


355 


934733 


135 


773457 


480 


227543 


88 




39 


70rJ93 


355 


934649 


lii5 


773745 


480 


237255 


81 




40 


707606 


355 


034574 


125 


773033 


480 


' 236967 


80 




41 


9.707819 


355 


9.934499 


135 


9.773331 


489 


10.226679 


19 




42 


706032 


354 


934424 


135 


7730<« 


479 


22039'4 


18 




43 


708245 


354 


934349 


135 


T73890 


479 


296104 


17 




44 


708458 


354 


934374 


135 


774184 


479 


235616 


16 




45 


708670 


354 


934199 


125 


774471 


479 


285580 


15 1 




46 


708882 


353 


934123 


135 


774759 


479 


825^1 


14 




47 


709094 


353 


934048 


125 


775046 


479 


824954 


13 




48 


709300 


353 


933i>73 


125 


775333 


479 


824667 


18 




49 


709518 


353 


933898 


126 


775621 


478 


824379 


11 




50 


709730 


353 


933832 


136 


775008 


478 


824092 


10 




51 


9.7U9941 


352 


9.933747 


136 


9.776195 


4:8 


10.833605 


9 




58 


710153 


353 


933G71 


126 


776483 


478 


233518 


8 




53 


710364 


353 


933596 


136 


776769 


478 


323331 


7 




54 


710575 


.353 


933520 


126 


777055 


478 


832945 


6 




55 


710786 


351 


933445 


130 


777342 


478 


332H58 


5 




56 


710997 


351 


933369 


136 


777G38 


477 


833:f73 


4 




57 


711208 


351 


933393 


136 


777915 


477 


832085 


3 




58 


711419 


351 


933317 


136 


778201 


477 


831799 


8 




59 


7 J 1629 


350 


933141 


12n 


778487 


477 


831513 


1 




60 


711839 


350 


933066 


126 


778774 


477 


831336 










j Oarine | 


I 


Sina 1 


1 


OotMig. 1 


1 


Tang. 


M. 





fiO 







•INBt 


AND TANORNTfl 


. (31 Degrees.) 




75 


Ml 


Sine 1 


». 


Coane | 


D. 


1 Ttaig. 


1 J). 


i Cocang. 







9.711830 


350 


9.933066 


126 


9.778774 


477 


10.321236 


60 


1 


713050 


350 


932990 


127 


779060 


477 


220940 


59 


3 


71^263 


350 


932914 


127 


779346 


476 


820654 


58 


3 


712409 


349 


9328:18 


127 


779632 


470 


830368 


57 


4 


713679 


319 


932762 


137 


779918 


476 


830083 


56 


5 


712880 


349 


932na'( 


127 


780203 


476 


819797 


55 


6 


713098 


349 


932G09 


127 


7804ft) 


476 


819511 


54 


7 


713308 


349 


932533 


127 


780775 


476 


819325 


53 


8 


713517 


348 


932457 


127 


781060 


476 


318940 


52 


9 


713728 


348 


932380 


127 


781346 


475 


318654 


51 


10 


713935 


348 


932304 


127 


781631 


475 


318369 


50 


11 


9.714144 


348 


9.KU828 


127 


9.781916 


475 


10.318084 


49 


13 


714352 


347 


933151 


127 


78-2301 


475 


317799 


48 


13 


714561 


347 


93S075 


128 


782486 


475 


217514 


47 


14 


714769 


347 


931998 


128 


782771 


475 


817229 


46 


15 


714978 


347 


931931 


128 


783vl56 


475 


816944 


45 


10 


715183 


347 


931&15 


128 


783341 


475 


216659 


44 


17 


715394 


346 


931768 


128 


783626 


474 


81G374 


43 


18 


715602 


346 


931691 


128 


783910 


474 


816090 


43 


19 


715809 


346 


931614 


128 


7S4195 


474 


315803 


41 


90 


716017 


346 


931537 


128 


78*479 


474 


215521 


40 


81 


9.716234 


345 


9.931460 


128 


9.784764 


474 


10.315336 


39 


SJ 


716432 


345 


9313ai 


1-28 


785048 


474 


214952 


38 


23 


716639 


345 


931306 


128 


785332 


473 


2146G8 


37 


34 


716846 


345 


931i^ 


129 


785616 


473 


814384 


36 


95 


717053 


345 


931152 


129 


785909 


473 


814100 


35 


96 


717259 


344 


931075 


1-29 


786184 


473 


313816 


34 


S7 


717466 


344 


930998 


1-29 


786468 


473 


3135:i3 


33 


28 


717673 


344 


930921 


129 


786752 


473 


813348 


33 


80 


717879 


344 


930&13 


129 


787033 


473 


813964 


31 


30 


718085 


343 


930766 


129 


787319 


473 


813681 


39 


31 


9.71^91 


343 


9.933688 


189 


9.787603 


478 


10.813397 


89 


33 


718497 


343 


930611 


129 


787886 


478 


313114 
811d30 


88 


33 


718703 


343 


930533 


129 


788170 


472 


87 


3t 


718999 


343 


930456 


1-29 


788453 


473 


811547 


96 


35 


719114 


342 


930378. 


129 


788736 


473 


811364 


85 


36 


719320 


342 


930300 


130 


789019 


473 


810961 


84 


37 


719525 


342 


930223 


130 


789303 


471 


810698 


S3 


38 


719730 


342 


930145 


130 


789585 


471 


810415 


83 


30 


7199:^5 


341 


930367 


130 


789868 


471 


310132 


31 


40 


790140 


341 


029989 


130 


790151 


471 


209849 


80 


41 


9.780345 


341 


9.939911 


130 


9.700433 


47J 


10.800567 


19 


42 


730549 


311 


989833 


130 


790716 


471 


809284 


18 


43 


780754 


340 


989755 


130 


790999 


471 


809001 


17 


44 


780958 


340 


1»0677 


130 


791281 


471 


808719 


16 


45 


721163 


310 


989599 


130 


791563 


470 


308437 


15 


46 


781366 


340 


939531 


130 


79184S 


470 


806154 


14 


47 


721570 


340 


^29443 


130 


792128 


470 


807U72 


13 


48 


721774 


339 


920364 


131 


792410 


470 


807590 


13 


40 


731978 


339 


929386 


131 


792692 


470 


807308 


11 


50 


722181 


339 


92$»07 


131 


792074 


470 


807026 


10 


91 


9.723385 


339 


9.939139 


131 


9.793356 


470 


10.806744 


9 


53 


73358H 


339 


929050 


131 


793538 


409 


306462 


8 


53 


723791 


338 


•928972 


131 


793819 


469 


306181 


7 


51 


732994 


338 


928893 


131 


794101 


469 


305699 


6 


55 


733197 


338 


928815 


131 


794383 


469 


305617 


5 


56 


7234U0 


338 


9i»738 


131 


791664 


469 


805336 


4 


57 


723603 


337 


928657 


131 


7949^5 


469 


805055 


3 


58 


rixm 


337 


928578 


131 


795227 


460 


204773 


2 


59 


724007 


337 


928499 


131 


795508 


468 


804492 


1 


60 


721210 


337 


928420 


131 


795789 


46« 


804211 





1 <;oiiiM 


1 


1 Sine 


1 


1 Cotang. 


1 


1 Tw*. 


|M. 



KJ 



76 


(33 Degrees.) a 


TABLB OP LOaARlTRMIC 




M. 


«IW 


D. 1 


GoiiM 


1 D. 


1 Ttaif . 


1 D. 


1 Cot«Dg. 1 ll 





9.724210 


337 


9.928420 


132 


9.795789 


468 


10.904211 


60 1 


1 


724412 


337 


928342 


139 


796070 


468 


903930 


59 


« 


7M614 


336 


928263 


132 


796351 


468 


903649 


58 


3 


724816 


336 


928183 


139 


796639 


408 


903368 


57 


4 


725017 


336 


928104 


139 


796913 


468 


903087 


56 


5 


725219 


330 


928025 


139 


797194 


466 


902806 


55 


6 


725420 


335 


927946 


139 


797475 


468 


902S35 


54 


7 


725622 


335 


927867 


139 


797755 


468 


902245 


S3 


8 


725823 


335 


927787 


139 


796036 


467 


901964 


52 


9 


726024 


335 


927708 


139 


798316 


467 


901684 


51 


10 


790225 


335 


927629 


139 


798596 


467 


901404 


50 


11 


9.796420 


334 


9.9S7549 


139 


9.798B77 


467 


10.901193 


40 


18 


796626 


334 


927470 


133 


799157 


467 


900843 


48 


13 


726H27 


334 


927390 


133 


799437 


467 


900563 


47 


14 


727027 


334 


927310 


133 


799717 


467 


2002K) 


46 


15 


727228 


334 


927931 


133 


799997 


466 


900003 


45 


16 


727428 


333 


927151 


133 


800277 


466 


199723 


44 


17 


727628 


333 


927071 


133 


800557 


466 


199443 


43 


18 


727828 


333 


926991 


133 


800836 


466 


199164 


49 


19 


728027 


333 


926911 


133 


801116 


466 


198884 


41 


90 


728227 


333 


926831 


133 


801396 


466 


198604 


40 


SI 


9.728427 


338 


9.928751 


133 


9.801675 


466 


10.196325 


30 


22 


728626 


332 


926671 


133 


801955 


466 


198045 


36 


33 


728825 


332 


926591 


133 


802234 


465 


lonee 


37 


SI 


729024 


332 


926511 


134 


802513 


465 


197487 


36 


25 


729223 


331 


926431 


134 


802799 


465 


197906 


35 


96 


729422 


331 


926351 


134 


803072 


465 


196928 


34 


27 


729621 


331 


926270 


134 


803351 


465 


196649 


33 


28 


•229820 


331 


926190 


134 


803630 


465 


196370 


39 


S9 


730018 


330 


926110 


134 


80390S 


465 


196092 


31 


30 


730216 


330 


926099 


134 


804187 


465 


195813 


30 


31 


9.730415 


330 


9.925949 


134 


9.804466 


464 


10.195534 


89 


32 


730613 


330 


925868 


134 


804745 


464 


195255 


88 


33 


730811 


330 


925788 


134 


805023 


464 


194977 


27 


34 


731009 


329 


925707 


134 


805302 


464 


194696 


96 


35 


731206 


329 


925626 


134 


805580 


464 


194420 


85 1 


36 


731404 


329 


925545 


135 


805059 


464 


194141 


84 


37 


731602 


329 


925465 


135 


806137 


464 


193863 


83 


38 


731799 


329 


925384 


135 


806415 


463 


193585 


82 


30 


731996 


328 


925303 


135 


806693 


463 


193307 


81 


40 


732193 


328 


925292 


135 


806971 


463 


193029 


90 


41 


9.732390 


328 


9.925141 


135 


9.807949 


463 


10.198751 


19 


42 


732.'i«7 


328 


92S060 


135 


807527 


463 


192473 


18 


43 


732784 


328 


924979 


135 


807805 


463 


199195 


17 


44 


732960 


327 


924897 


135 


808063 


463 


191917 


16 


45 


733177 


387 


924816 


135 


806361 


463 


191639 


15 


46 


733373 


327 


924735 


136 


8UH638 


462 


191369 


14 


47 


733569 


327 


924654 


136 


806916 


469 


191084 


13 


48 


rJ3765 


327 


924579 


136 


809193 


4m 


190807 


12 


49 


733961 


326 


924491 


136 


809471 


469 


190529 


11 


50 


734157 


326 


924409 


136 


809746 


463 


190252 


10 


51 


9.734353 


326 


9.924328 


136 


9.810025 


469 


10.189975 


9 


52 


734549 


396 


924246 


136 


810309 


462 


189698 


6 


53 


734744 


325 


924164 


136 


810580 


468 


189420 


7 


54 


734939 


325 


924063 


136 


810657 


462 


189143 


6 


55 


735135 


395 


924001 


136 


811134 


461 


188866 


5 


56 


735330 


325 


92:1919 


136 


811410 


461 


18Ki90 


4 


57 


735525 


325 


923837 


136 


811687 


461 


188313 


3 


58 


735719 


394 


923755 


137 


811964 


461 


188036 


2 


59 


735914 


394 


923673 


137 


819941 


461 


1877.59 


1 


60 


736109 


394 


923591 


137 


819517 


461 


187483 


6 




1 Corine 


1 


1 Bine 




1 OotBag. 


1 


1 »tog. |M. II 



67 







BINES 


AND TANGENTS 


. (33 Degrees.) 




77 


ll.j 


siiM 1 


D. 1 


CfMiiM 


I>. 1 


TMMf. 1 


» 1 


Cotaog 1 II 





9.736109 


334 


9.923591 


137 


9.813517 


461 


10.187488 


60 


1 


736303 


324 


923509 


137 


813794 


461 


187306 


50 


s 


736496 


334 


923437 


137 


813070 


461 


186930 


58 


3 


736693 


333 


923343 


137 


813347 


460 


186653 


57 


4 


736H86 


333 


923363 


137 


813633 


460 


180377 


56 


5 


737UeO 


333 


923181 


137 


813899 


460 


186101 


56 


« 


737274 


333 


933096 


137 


814175 


460 


185635 


54 


7 


737467 


333 


933016 


137 


814453 


460 


1&>548 


53 


8 


TJ7661 


333 


{^23933 


137 


814798 


460 


185273 


92 


9 


737855 


323 


922851 


137 


815004 


460 


184996 


51 


10 


738048 


333 


933768 


138 


815379 


460 


184731 


90 


11 


9.738341 


393 


9.933086 


238 


9.815555 


4.19 


10.181445 


40 


13 


738434 


333 


933603 


138 


815831 


450 


184160 


48 


13 


738637 


321 


933530 


138 


816107 


450 


183893 


47 


14 


736^0 


331 


933438 


138 


816383 


459 


183618 


46 


15 


739013 


331 


932355 


138 


816658 


459 


183343 


45 


16 


739306 


331 


922272 


138 


816933 


459 


183067 


44 


17 


7393U8 


3S1 


922180 


138 


817309 


450 


183791 


43 


18 


739500 


330 


922106 


138 


817484 


450 


182516 


43 


19 


739783 


3S0 


932023 


138 


817759 


459 


182241 


41 


ao 


739975 


330 


921940 


138 


818035 


458 


181965 


40 


21 


9.740167 


330 - 


9.921857 


139 


9.818310 


458 


10.181690 


39 


n 


740359 


330 


921774 


139 


818585 


458 


181415 


38 


23 


740550 


319 


921691 


139 


818860 


458 


181140 


37 


34 


746743 


319 


921607 


139 


819135 


458 


180865 


36 


35 


740934 


319 


931584 


139 


819410 


458 


180500 


35 


96 


741135 


319 


931441 


139 


819C84 


458 


180316 


34 


37 


741316 


319 


931357 


139 


819959 


458 


180041 


33 


38 


741508 


318 


921274 


139 


830334 


458 


179766 


33 


29 


741699 


318 


931190 


139 


890506 


457 


179493 


31 


ao 


741889 


318 


931107 


139 


830783 


457 


179217 


30 


31 


9.742080 


318 


9.931033 


139 


9.831057 


457 


10.178943 


SO 


32 


743971 


31'» 


930939 


140 


831333 


457 


178668 


98 


33 


743468 


317 


930856 


140 


831606 


457 


17a394 


37 


34 


749658 


317 


990772 


140 


831880 


457 


178190 


36 


35 


743843 


317 


930688 


140 


833154 


357 


177846 


35 


36 


743033 


317 


930604 


140 


833439 


457 


177571 


84 


37 


74:U33 


317 


930580 


140 


823703 


457 


177297 


33 


38 


743413 


316 


930436 


140 


832977 


459 


177023 


33 


39 


743603 


316 


920352 


140 


823250 


456 


176750 


31 


40 


743703 


316 


920868 


140 


823534 


456 


176476 


90 


41 


9.7430H3 


316 


9.999184 


140 


9.823798 


456 


10.178209 


19 


43 


744171 


316 


990099 


140 


834072 


456 


175028 


16 


43 


744.161 


315 


920015 


140 


824345 


456 


175655 


17 


44 


7445A0 


315 


919931 


141 


824619 


456 


175381 


16 


45 


744739 


315 


919646 


141 


834893 


456 


175107 


15 


46 


744928 


-315 


919763 


141 


825166 


456 


174834 


14 


47 


745117 


315 


919677 


141 


825439 


455 


174561 


13 


48 


745306 


314 


919.'W3 


141 


825713 


455 


174287 


IS 


49 


745494 


314 


919508 


141 


825986 


455 


174014 


11 


50 


745G83 


314 


919434 


141 


826259 


455 


173741 


10 


51 


0.745871 


314 


9.919339 


141 


9.826532 


455 


10.173468 


9 


92 


746059 


314 


9192.S4 


141 


836805 


455 


173195 


8 


93 


746348 


313 


919160 


141 


837078 


455 


172922 


7 


94 


746436 


313 


919U85 


141 


827351 


455 


172tM9 


6 


55 


746634 


313 


919000 


141 


837634 


455 


172376 


5 


50 


746813 


313 


918915 


142 


837897 


454 


172103 


4 


57 


746999 


313 


918830 


142 


828170 


454 


171H30 


3 


58 


747187 


313 


918745 


142 


828442 


454 


171558 


3 


50 


747374 


313 


918659 


142 


828715 


454 


171285 


1 


I 60 


7475(33 


313 


918574 


142 


828987 


454 


171013 





L 


1 GoMiie 


1 


1 Sine 


1 


1 Gotang. 


1 


1 T«g. 1 M. II 



66 



78 



(34 Degnea.) a table or looarithmic 



M. 


1 SlM 


1 ». 


1 GodiM 


1 D. 


1 Tutg. 


D. 


1 Cocang. 


1 1 





9.747563 


31S 


9.918374 


143 


9.828987 


454 


10.171013 


00 


1 


747749 


313 


918489 


143 


82!l3n0 


454 


170740 


50 


S 


7479:W 


313 


918404 


143 


82J5:« 


454 


110**18 


58 


3 


748123 


311 


918318 


143 


8208a> 


454 


170193 


57 


4 


748310 


311 


9182.-13 


142 


830077 


454 


169923 


56 


5 


748497 


311 


918147 


142 


83.k'M9 


453 


169651 


55 


6 


748683 


311 


918.)62 


143 


830031 


453 


160379 


54 


7 


748870 


311 


917976 


143 


830893 


433 


160107 


53 


9 


749053 


310 


917891 


143 


831163 


453 


168835 


52 


9 


749-243 


310 


917805 


143 


831437 


453 


168363 


51 


10 


749429 


310 


9ini9 


143 


83170)^ 


453 


168291 


50 


11 


9.749ni5 


310 


9.917034 


143 


9.831961 


453 


10.166010 


49 


IS 


7498J1 


310 


917348 


143 


&32253 


459 


167747 


48 


13 


749387 


309 


917403 


143 


832335 


453 


167475 


47 


14 


750172 


309 


917376 


143 


832796 


453 


167304 


46 


19 


75(»»58 


309 


917290 


143 


8:13068 


453 


166933 


45 


16 


7ja>4J 


3(» 


917204 


143 


833.i39 


453 


166661 


44 


17 


75'J7:i9 


309 


917118 


144 


833611 


4S3 


166389 


43 


Id 


750014 


306 


917033 


144 


833883 


453 


166118 


43 


19 


731099 


308 


916946 


144 


834154 


453 


165846 


41 


SO 


751384 


308 


916d39 


144 


834435 


453 


165575 


40 


SI 


9.751469 


306 


9.916773 


144 


9.834696 


-458 


10.165304 


39 


3S 


751054 


306 


916687 


144 


834967 


453 


165033 


38 


33 


75183U 


306 


916600 


144 


835238 


453 


164762 


37 


S4 


752023 


307 


916514 


144 


835599 


453 


164491 


38 


35 


752208 


307 


916427 


144 


835780 


451 


164320 


35 


36 


752392 


3J7 


916341 


144 


836351 


451 


163949 


34 


27 


752576 


307 


916254 


144 


830X» 


451 


163678 


33 


38 


752760 


307 


916167 


145 


836503 


451 


163107 


33 


39 


75i944 


306 


916081 


143 


831864 


451 


163136 


31 


30 


753128 


300 


915994 


145 


837134 


451 


162866 


30 


31 


9.753:)12 


306 


9.915937 


145 


9.8T7405 


451 


10.163505 


39 


33 


753495 


300 


915^20 


145 


837675 


45i 


163333 


38 


33 


753679 


30S 


9157:13 


145 


837946 


451 


163054 


37 


34 


7538G2 


3J5 


915646 


145 


838216 


451 


161784 


36 


35 


754046 


305 


915539 


145 


83H487 


450 


161513 


35 


36 


754229 


305 


915472 


145 


838757 


450 


161343 


34 


37 


754413 


305 


915:183 


143 


830037 


450 


1609rj 


33 


38 


7SI595 


305 


915297 


145 


839297 


450 


160703 


S3 


30 


754778 


304 


915210 


143 


839566 


450 


160433 


31 


40 


754960 


304 


915133 


146 


83X38 


450 


160162 


SO 


41 


9.755143 


304 


0.915035 


146 


9.840106 


450 


10.159893 


19 


48 


755326 


304 


914948 


146 


&10378 


450 


139623 


18 


43 


755308 


304 


914860 


146 


840647 


450 


l.'>9353 


17 


44 


755690 


304 


014773 


146 


840917 


449 


159083 


16 


45 


755872 


303 


914685 


146 


841187 


449 


158813 


15 


46 


756054 


303 


914506 


146 


»tl457 


449 


138513 


14 


47 


756336 


303 


914510 


146 


841796 


449 


158274 


13 


48 


756418 


303 


914423 


146 


841996 


449 


158004 


IS 


49 


756H00 


303 


914334 


146 


842266 


449 


157734 


11 


50 


756783 


303 


914246 


147 


842535 


449 


157405 


10 


51 


9.7509!hl 


303 


9.914158 


147 


9.843805 


449 


10.157195 





53 


757144 


303 


914070 


147 


843074 


449 


156926 


8 


53 


757326 


303 


013983 


147 


8i:«43 


449 


156657 


7 


54 


757W)7 


303 


913894 


147 


843613 


449 


150388 


6 


55 


757688 


301 


913830 


147 


843883 


448 


156118 


5 


56 


757869 


301 


913718 


147 


844151 


448 


155849 


4 


57 


758050 


301 


913630 


147 


844430 


448 


155580 


3 


58 


758330 


301 


913541 


147 


844689 


448 


155311 


8 ' 


50 


758411 


301 


913453 


147 


844938 


448 


155043 


1 


60 


758591 


301 


9133('>5 


147 


845227 


448 


154773 







Cotlne 




Sine 




1 Cotang. 


1 


1 Tang. 



65 







SINM 


AND TANGENTS. (35 Def^Tees.) 




71 


) 


M. 1 


fllii« 


». 


1 Codne 


1 D. 


1 Twig. 


1 D. 


1 CoteDg. 


r™ 







9.758391 


301 


9.913385 


147 


9.845227 


448 


10 134773 


60 




1 


758772 


300 


913270 


147 


843496 


448 


134304 


SB 




3 


758952 


300 


913187 


148 


845764 


448 


134236 


58 




3 


759132 


303 


913099 


148 


846033 


448 


133907 


57 




4 


759312 


300 


913010 


148 


843302 


448 


133698 


56 




5 


759492 


300 


9129^2 


148 


848570 


447 


13:)430 


55 




6 


759672 


299 


9128:i3 


148 


84^39 


447 


133161 


54 




7 


759652 


299 


912744 


148 


847107 


447 


13M93 


53 




8 


760031 


29;) 


9irKi5 


148 


847376 


447 


132824 


52 




9 


760211 


299 


912516 


J48 


847644 


447 


132336 


51 




10 


760390 


289 


912477 


148 


847913 


447 


132087 


50 




11 


9.760569 


398 


9.912388 


148 


9.848181 


447 


10.151819 


49 




12 


760748 


298 


912399 


149 


848449 


447 


151551 


48 




13 


760927 


29S 


912310 


149 


848717 


447 


151283 


47 




14 


761106 


398 


012121 


149 


848986 


447 


151014 


46 




13 


761285 


396 


913J31 


149 


849i54 


447 


ldtf746 


45 




16 


761464 


396 


911943 


149 


8495-:» 


447 


150478 


44 




17 


761642 


297 


911S33 


149 


649793 


446 


150210 


43 




18 


761821 


397 


911763 


149 


83Ja58 


446 


149942 


43 




19 


761999 


297 


911674 


149 


83)335 


446 


149675 


41 




30 


762177 


297 


911584 


149 


850593 


446 


149407 


40 




21 


9.763356 


297 


9.911495 


149 


9.850861 


446 


10.149139 


39 




23 


762534 


396 


911403 


149 


831129 


446 


148871 


38 




23 


762712 


296 


911315 


150 


851396 


446 


148601 


37 




24 


762889 


398 


911226 


130 


851664 


446 


148336 


36 




23 


763067 


396 


911136 


130 


651931 


446 


148069 


35 




28 


7e3245 


398 


911046 


150 


852199 


446 


147801 


34 




27 


763422 


390 


010936 


150 


85-3466 


446 


147534 


33 




28 


763600 


393 


910866 


150 


8.>3r33 


445 


147267 


33 




29 


703777 


293 


910776 


153 


853001 


445 


146999 


31 




30 


763954 


295 


910686 


150 


833268 


445 


146732 


30 




31 


9.764131 


295 


9.910596 


150 


9.853535 


445 


10.145465 


20 




32 


764306 


295 


910308 


150 


853803 


445 


140198 


28 




33 


764483 


294 


910415 


150 


834060 


445 


143931 


27 




34 


764662 


294 


9103-25 


151 


834336 


445 


145664 


26 




35 


764838 


294 


010235 


151 


854803 


445 


145397 


25 




36 


765J15 


294 


910144 


151 


854870 


445 


1451.10 


24 




37 


765191 


294 


OtOOM 


131 


855137 


445 


144863 


33 




38 


765387 


294 


939963 


151 


855404 


445 


144596 


» 




39 


765344 


293 


909373 


151 


853871 


444 


144329 


31 




40 


765720 


393 


909783 


151 


855938 


444 


1440^ 


30 




41 


9.765896 


293 


9.9^191 


151 


9.856204 


444 


10.143796 


10 




42 


766072 


393 


909 X)l 


151 


856471 


444 


143529 


18 




43 


766247 


293 


939310 


151 


856737 


444 


143363 


17 




44 


76642:) 


293 


• 909419 


151 


&57O04 


444 


142998 


16 




45 


765398 


392 


909326 


152 


857270 


444 


142730 


15 




46 


766774 


292 


90J237 


152 


837.537 


444 


142463 


14 




47 


76t»«9 


292 


909146 


152 


857833 


444 


142197 


13 




48 


767124 


292 


90:^035 


152 


858069 


444 


141931 


13 




49 


767300 


292 


903964 


152 


838336 


444 


141664 


11 




50 


767475 


291 


908873 


152 


858603 


443 


141398 


10 




51 


9.767649 


291 


9.908781 


152 


0.838868 


443 


10.141132 


9 




52 


767824 


391 


916690 


153 


859134 


443 


140886 


8 




53 


767909 


391 


dossm 


132 


8.50400 


443 


140600 


7 




54 


768173 


291 


903.507 


1.52 


8396G6 


443 


140334 


6 




55 


768348 


390 


9;)8416 


133 


859933 


443 


140068 


5 




56 


768333 


390 


906324 


133 


830198 


443 


130602 


4 




57 


768697 


390 


90R233 


133 


880464 


413 


139.536 


3 




58 


76R871 


390 


908141 


133 


860730 


443 


139270 


3 




5» 


760045 


290 


908349 


153 


880995 


443 


139005 


1 




60 


709319 


290 


007938 


133 


861201 


443 


138739 









1 Cofltne 


1 


Sine 




Cotang. 




T«Bg. 


|M. 





54 



80 



(96 Degrees.) a tablb op looaeithmic 



M.f Stale 


1 D 


I CoaliM 


I D. 


1 Twxig. 


1 D. 


1 CoOBg. 1 )| 





9.760319 


290 


9.907958 


153 


9.861261 


443 


10.138739 


60 


I 


760393 


289 


907866 


153 


861537 


443 


138473 


59 


« 


7G9566 


289 


907774 


153 


801792 


443 


138208 


58 


3 


769740 


289 


907682 


153 


863058 


443 


137943 


57 


4 


769913 


389 


907590 


153 


862333 


443 


137677 


56 


5 


770087 


289 


9U7498 


153 


863589 


443 


137411 


55 


6 


770360 


288 


907406 


153 


862854 


449 


137146 


54 


7 


770433 


288 


907314 


154 


863119 


443 


136881 


53 


8 


770606 


288 


907222 


154 


863385 


443 


136615 


53 


9 


770779 


288 


907129 


154 


863G50 


443 


136350 


51 


10 


770953 


288 


907097 


154 


863915 


443 


136085 


50 


11 


9.771125 


288 


9.906945 


154 


9.864180 


449 


10.135830 


49 


13 


771298 


287 


906858 


154 


8H4445 


443 


135555 


48 


13 


771470 


287 


006760 


154 


864710 


443 


135200 


47 


14 


771643 


287 


906667 


154 


864975 


441 


135035 


46 


1$ 


771815 


287 


006575 


154 


863340 


441 


134760 


45 


16 


771U87 


287 


906483 


154 


865505 


441 


134495 


44 


17 


772159 


287 


906389 


155 


865770 


441 


134330 


43 


18 


772331 


286 


906296 


155 


866035 


441 


133965 


43 


19 


772503 


286 


900304 


155 


860300 


441 


133700 


41 


90 


773675 


286 


906111 


155 


860564 


441 


133436 


40 


21 


9.772847 


286 


9.906018 


155 


9.866820 


441 


10.133171 


39 


33 


T730I8 


286 


905035 


155 


867094 


441 


133906 


38 


23 


T731QO 


286 


905833 


155 


867358 


441 


133643 


37 


34 


773361 


285 


005739 


155 


867623 


441 


132377 


36 


25 


773533 


285 


905645 


155 


807887 


441 


132113 


35 


26 


773704 


285 


905553 


155 


808152 


440 


131848 


34 


27 


773875 


285 


905450 


155 


868416 


440 


131584 


33 


28 


774046 


285 


005366 


156 


868680 


440 


131320 


33 


2D 


774217 


2H5 


905273 


156 


868945 


440 


131055 


31 


30 


774388 


284 


905179 


156 


869309 


440 


130791 


30 


31 


9.774558 


284 


9.935065 


156 


9.869473 


440 


10.130527 


29 


32 


774729 


284 


904993 


156 


809rJ7 


440 


130363 


28 


33 


774899 


284 


904898 


156 


870001 


440 


129999 


37 


31 


775JJ70 


284 


901804 


156 


870365 


440 


129735 


96 


35 


773240 


284 


904711 


156 


870.539 


440 


129471 


35 


36 


775410 


283 


904617 


156 


870793 


440 


129S07 


34 


37 


775580 


283 


904533 


156 


671057 


440 


1-28943 


33 


38 


775750 


283 


904430 


1.57 


871321 


440 


128679 


23 


39 


775930 


283 


904335 


157 


871585 


440 


128415 


31 


40 


776U00 


283 


004341 


157 


871849 


439 


128151 


90 


41 


9.776259 


28:< 


9.004147 


157 


9.873113 


439 


10.137888 


19 


43 


776429 


282 


904a'^ 


157 


872376 


439 


127634 


18 


43 


776598 


283 


903950 


157 


873640 


439 


137360 


17 


44 


7767C8 


283 


90.1864 


157 


872003 


439 


137097 


16 


45 


770937 


283 


903770 


157 


873167 


4.19 


126833 


15 


46 


777106 


283 


903676 


157 


873430 


439 


136570 


14 


47 


777275 


281 


903581 


157 


873604 


4:}9 


126306 


13 


48 


777444 


381 


9Q3487 


157 


873957 


4.19 


126043 


13 


49 


T77C13 


281 


903.103 


156 


8742a) 


439 


125780 


11 


50 


777781 


281 


903208 


158 


874484 


439 


125516 


10 


51 


9.777950 


281 


9.903303 


158 


9.874747 


439 


10.12SS53 


9 


53 


778119 


281 


9031U8 


158 


875010 


4.19 


124990 


8 


53 


778887 


280 


903014 


138 


875373 


438 


124727 


7 


54 


778455 


280 


902919 


158 


875536 


4.18 


124464 


• 


55 


778624 


280 


003834 


158 


875800 


438 


124300 


5 


56 


778792 


280 


902729 


las 


876063 


438 


133937 


4 


57 


778960 


280 


9036.34 


158 


87(>338 


438 


133674 


3 


58 


770128 


280 


9025.19 


l.-^ 


876589 


438 


133411 


9 


59 


TT9295 


279 


903444 


139 


876851 


438 


133149 


1 


60 


779463 


279 


903349 


159 


877114 


438 


133886 





1 


Godne | 


1 


8in« 




Ooteng. 




T»g. 1 M . 1 



63 







■INU 


4ND T«a«T. 


(37 DegKM.) 




81 


H. 


SilH 


n. 


C0<1H 


D. 


T-* 


». 


Couo,. 






B.TTIMO 


-m- 


D.IM314R 




9.S771H 


-oT" 


10.]3a« 


-to- 


I 






903^3 






438 


IHOJl 


rn 


i 


7^™ 




MI31S8- 


150 




438 




i» 


1 


T7g»sa 

7S0133 


379 


wwei 

001067 


1» 


877903 


418 

438 


j*^ 






T803Q0 


878 


001873 


ISO 


878438 


438 


laisTs 


Si 




7B0WT 








87S601 


438 


131300 


u 




780631 








87a9U 








e 
















M 






sra 






879178 


C7 






10 


TBim 


378 


W1301 




87S741 


437 


130309 


M 




I.TBtXl 


377 


3.0013B9 


m 


0.881001 


437 




49 


u 






IW1303 






417 






11 








Ha 


880S38 










I8I800 




ooioio 


lOtI 






119910 


40 


u 


M1KS6 


377 


000814 


i» 


e81U3 




1IS048 


43 




7S113S 


»n 


ooouis 


109 


881111 




118086 








370 








417 






la 




370 






8tll830 


417 




43 


n 


■mm 


370 




ISO 


883101 


43T 


117890 




» 


TSina 


no 


»M13 




88003 






40 


11 


s.; 11 


sm 


0.900037 




o.8eMS 


00 




39 


» 




no 






mxeu 








» 




375 






883148 




iioan 




3* 




8TS 


100047 


ISl 


883410 


lie 


116300 




M 






8B99S1 




88^73 


430 


116338 


3S 


M 






890811 




883914 


136 


116060 


14 








eW7S7 






438 


115804 




«. 




373 




161 


881417 




1 15343 


n 


ai 






B90J64 






436 






» 






bo;hs7 




881980 


41S 


113030 




11 










B.88S31S 


436 




39 


» 


'7*4770 


374 








IKI 


11407 




M 


784911 


371 




S-1 




116 


M33i 






78S10S 




BBBOTS 


m 


8860M 


4M 




30 


IS 


TBuao 




898981 


83 


8S0388 


438 




3J 


M 


laHn 






09 


aasMti 


4111 




34 




78S»T 
















M 


TBSIBl 


371 




169 










u 


7BSW5 




898*93 








I1MS7 




40 


78S08B 




898494 




8875M 






SO 


41 






9.»s)g7 






as 




9 


41 


'7M416 


373 














a 


TBOJTB 


37% 
















TWr4S 


373 


898101 




88^039 










TSOStM 




898006 




888900 








« 


7B708B 


373 


897908 




889190 








4T 




371 














48 


TO73BS 


371 










10118 




4* 


787*57 




anoii 








19057 




sa 






897SI0 




890»4 


431 




















19. 00533 




» 


'788D43 










434 


99373 




n 


788908 




897«« 




8909SA 


434 


09914 




u 


TSKirO 


370 


897133 




891317 










imssi 








89IS07 








X 




370 


890930 




891798 




08-09 






788890 
















K 


78W.8 










434 


07711 




K 


789iai 


370 


80«fl31 




89-lMfl 








SO 


780343 


301 


896S33 














°°*» 


___ 


ffi» 




C°u>» 




Tm». 


IT 



82 


( 


[38 Degrees.) a 


TABLB OP LOOARITRMTC 




M. 


1 8iM 


1 ». 


1 CotiM 


1 D. 


1 TMg. 


1 D. 


1 CotoBg. 1 1 





0.789342 


260 


9.896532 


164 


9.892810 


434 


10.107190 


60 


1 


789504 


260 


890433 


165 


89:1070 


434 


106930 


59 


9 


7B96G5 


260 


8963:U 


165 


893331 


434 


106660 


58 


3 


788827 


209 


896236 


165 


893591 


434 


106409 


57 


4 


789988 


260 


896137 


165 


893851 


4C4 


106149 


56 


5 


790149 


269 


896038 


165 


894111 


434 


105889 


53 


6 


790310 


268 


895939 


165 


894371 


434 


105620 


54 


7 


790471 


268 


895840 


165 


894632 


433 


105368 


53 


8 


790632 


268 


895741 


lf5 


894892 


433 


]a'>108 


52 


9 


790793 


268 


895641 


165 


895152 


433 


104848 


51 


10 


790954 


268 


895542 


165 


8954i2 


433 


104588 


50 


11 


0.791115 


S68 


9.895443 


166 


0.895672 


433 


10.10(328 


49 


12 


791275 


267 


895343 


166 


895932 


433 


104068 


48 


13 


791436 


2C7 


895344 


166 


896192 


433 


103808 


47 


14 


791506 


967 


895145 


166 


896452 


433 


103548 


46 


15 


791757 


267 


895045 


166 


896712 


433 


103288 


45 


16 


791917 


367 


894945 


166 


896971 


433 


103029 


44 


17 


792077 


267 


894846 


166 


897231 


433 


102769 


43 


18 


792237 


266 


894746 


166 


897491 


433 


102509 


42 


19 


792397 


266 


894646 


166 


897751 


433 


103249 


41 


99 


792557 


966 


894546 


166 


898010 


433 


101990 


40 


SI 


9.792716 


266 


9.894446 


167 


9.89»r70 


433 


10.101730 


30 


SS 


793876 


206 


894346 


107 


896330 


433 


101470 


38 


23 


793035 


966 


894i:46 


167 


8J8789 


433 


101211 


37 


24 


793195 


965 


894146 


107 


899049 


439 


100951 


36 


SS 


793351 


265 


894046 


167 


899308 


439 


100692 


35 


88 


7935H 


965 


893946 


167 


899568 


433 


100432 


34 


27 


793673 


265 


893846 


167 


899827 


439 


100173 


33 


28 


793R32 


265 


893745 


167 


900086 


439 


099914 


39 


29 


793991 


265 


893645 


167 


000346 


439 


099654 


31 


30 


794150 


264 


893544 


167 


900605 


439 


099395 


30 


31 


9.794308 


264 


9.893444 


168 


9.900864 


439 


10.099138 


99 


32 


794407 


204 


893343 


1''8 


901124 


439 


098876 


98 


33 


794628 


264 


893243 


im 


901383 


439 


098617 


97 


34 


794784 


264 


893142 


1G8 


901643 


439 


096358 


90 


35 


794942 


264 


893041 


168 


901901 


439 


098099 


95 


38 


795101 


264 


892940 


168 


902160 


439 


097810 


94 


37 


793259 


263 


892839 


168 


902419 


439 


097581 


93 


38 


795417 


263 


892739 


168 


902679 


439 


007321 


99 


30 


795575 


263 


892038 


168 


902938 


439 


097062 


91 


40 


795733 


263 


892536 


168 


903197 


431 


096803 


90 


41 


9.795891 


363 


9.892435 


169 


9.903455 


431 


10.096545 


19 


49 


79C049 


263 


892334 


169 


903714 


431 


096286 


18 


43 


796206 


263 


892233 


169 


903973 


431 


096037 


17 


44 


790364 


263 


892132 


169 


904332 


431 


095768 


18 


45 


796521 


263 


89:»30 


169 


904491 


431 


095509 


15 


46 


796679 


263 


891999 


169 


904750 


431 


095350 


14 


47 


796836 


268 


891827 


169 


905008 


431 


094993 


13 


48 


796993 


262 


891796 


169 


905267 


431 


094733 


19 


49 


797150 


261 


891624 


169 


905526 


431 


094474 


11 


50 


797307 


261 


891523 


170 


905784 


431 


094318 


10 


51 


9.797464 


261 


9.891421 


170 


9.906043 


431 


10.093957 


9 


52 


797621 


261 


891319 


170 


906302 


431 


093698 


8 


53 


797777 


961 


891217 


170 


906560 


431 


003440 


7 


54 


797934 


201 


891115 


170 


906819 


431 


093181 


6 


55 


798091 


261 


891013 


170 


907077 


431 


092923 


5 


56 


798247 


261 


890911 


170 


90rJ36 


431 


002664 


4 


57 


798403 


260 


890609 


170 


907594 


431 


092406 


3 


58 


798560 


260 


890707 


170 


907853 


431 


092148 


S 


59 


798716 


960 


890605 


170 


908111 


430 


001889 


1 


60 

1 


798872 


260 


890503 


170 


908369 


430 


091631 





Cosine i 


1 


8iD« 1 


1 


CotaDg. 1 


1 


Tug. |M. II 



61 







■INO 


ANDT*m 


TNT. 


(39 Deg™*..) 




83 


"m. 


n» 


''^^ 


OdM 1 D, 


T«g. 






m 




B.TywR 


MO 




m 








WO 


















wt 


sstrigB 




Qie88« 


430 










2U 


mhjs 


171 


909144 


430 








7994US 


2» 


B9W93 














TJW^l 


*5B 


S89990 




909(100 










7«ge.]6 








009918 










Tsoget 






171 














ssa 


seow 


171 


910435 


430 


C8940S 


S3 




801R-.9 


3M 


889570 




010893 


430 


089307 






eou«7 


398 


B8M;7 






430 


089040 


50 




«.eocK8i 


































1 












430 


088970 






B01O47 


S58 


8SU064 




911082 


430 


088018 




JS 


901901 










430 


087760 


45 










79 


































8S8G3I 


179 




439 




« 




B0181D 


ssr 


B88M8 






4'» 


ii8c;so 




9) 


















41 






0888341 


173 








30 


21 


SUt&BJ 


sss 




171 


014044 








13 


8»IM» 


BM 


888131 


173 


01430S 


4iS 






S4 


eosseo 


sse 


8880W 


173 


014500 








93 


eiB;« 


3ia 


88;nM 












» 


SHSBT 


330 














n 




310 


887718 


173 










98 


ni3!W4 


3M 




ira 




439 




33 








887510 


173 


915847 




084153 




» 


MJSIl 




ee74us 








083896 




w 


>(U3ltM 














U 


» 




33S 








439 




38 


XI 


803870 






74 




4» 


l)t)3]33 


SI 




».im 




88C9W 




917134 




083880 






804370 














95 


M 


80MW 










439 


0833S3 


»l 


ST 


SJ?^ 


354 










08W9S 




38 


IM 


880S7I 






4W 


081S37 


BS 


30 


80480) 


an 


88048* 




918430 


4SS 


081SHO 






8050» 




e8e3iis 




918677 




081333 


SO 




«-«)5ll)l 




».88ffiU7 




9018034 


4SS 


10.081080 




« 


805M3 










43S 






U 


eoHBS 


ssj 


S80U4T 


!* 


919448 


488 


080553 






BOSW? 


SSJ 


8f51M2 




BioTia 


498 












eesen 


71 


9I0D63 


498 












sBSTsa 






43S 






« 






R83sn 






49S 


079534 


« 




9S.1 




ITJ 




438 


079907 






eoMoe 


S33 


885418 




naomo 


489 






9U 




331 


88S3II 








078753 




SI 


o-eoeros 


9» 


g.aasau 












SB 












438 








BD701I 










438 


077983 






BOTiro 


as 


88488B 


178 


0-^4 


498 


077738 








, 333 




70 


9-S3ao 


438 






H 


imeis 




^5T2 






438 
498 


070950 
















438 


076700 








SSI 


881300 


70 




497 






60 


e08DG7 


2S1 


884J54 


177 


913813 


*a 




mTi 




OhIm 




»» 


Co<«.(. 




"is; 



84 



(40 Degree*,) a tamlk op ukiabitiuiic 



M. 


1 aiM 


1 D. 


i CodiM 1 D. 


1 Twif. 


r?" 


1 Cotang. 


1 





9.8oeJ67 


351 


0.684354 


177 


0.923813 


437 


10.076187 


00 


1 


608318 


351 


884148 


177 


024070 


487 


075030 


59 


3 


808368 


351 


884043 


177 


934327 


437 


075G73 


58 


3 


808519 


350 


883930 


177 


924563 


437 


075417 


57 


4 


608669 


350 


883829 


177 


924640 


437 


075160 


56 


5 


808819 


350 


683723 


177 


925096 


437 


074004 


55 


6 


808969 


350 


683617 


177 


925352 


437 


074648 


54 


7 


809119 


350 


883510 


177 


825609 


437 


074391 


53 


8 


809269 


350 


88:U04 


177 


925865 


437 


074135 


53 


9 


809419 


^9 


683297 


178 


926122 


437 


073878 


51 


10 


809569 


349 


8c(3191 


178 


936378 


437 


073(i23 


50 


11 


9 609716 


349 


9.883064 


1-/6 


0.936634 


437 


10.07330f 


48 


12 


609668 


349 


682977 


178 


93ti890 


427 


073110 


46 


]3 


810017 


349 


882871 


178 


lKrri47 


427 


073853 


47 


14 


610167 


349 


8:j2764 


173 


927403 


427 


073507 


46 


15 


810316 


5M8 


6c«557 


178 


927659 


437 


073341 


45 


16 


810465 


348 


aS3550 


17i 


027915 


427 


072085 


44 


17 


810614 


348 


882443 


178 


928171 


437 


071839 


43 


18 


810763 


248 


882336 


179 


928487 


427 


071573 


48 


19 


810913 


348 


K82229 


179 


928663 


437 


071317 


41 


90 


811061 


348 


882131 


179 


928940 


437 


071060 


40 


SI 


9 611310 


348 


9.682014 


179 


9.029196 


437 


10.070604 


38 


S2 


811358 


347 


881907 


179 


929453 


427 


070546 


38 


33 


8U507 


347 


681799 


179 


029708 


437 


070293 


37 


34 


8J1655 


347 


881693 


179 


929964 


436 


070036 


38 


35 


811804 


247 


881584 


179 


930220 


426 


069760 


35 


3G 


811952 


347 


881477 


179 


930475 


438 


009525 


34 


37 


812100 


247 


881369 


179 


930731 


436 


069369 


33 


38 


812248 


347 


881261 


180 


93)967 


430 


060013 


33 


39 


812396 


346 


881153 


180 


931243 


426 


068757 


31 


30 


812544 


346 


881046 


180 


031403 


436 


068301 


30 


31 


9 8126^ 


340 


9.880938 


160 


9.931755 


426 


10.068345 


98 


33 


81 -2840 


246 


88U830 


180 


932010 


426 


067990 


88 


33 


812938 


316 


880722 


180 


932266 


426 


067734 


37 


34 


813135 


216 


880G13 


180 


93-2522 


433 


067478 


96 


35 


813283 


246 


880505 


183 


9;J2776 


436 


067-233 


85 


36 


813430 


345 


880397 


183 


9:13033 


436 


066067 


84 


37 


813578 


245 


8802^9 


181 


933289 


426 


066711 


83 


38 


813725 


245 


880180 


181 


933545 


436 


066455 


33 


38 


813872 


345 


880072 


181 


933800 


436 


066200 


31 


40 


814019 


245 


879963 


181 


934050 


436 


065944 


30 


41 


9.814166 


345 


9.879855 


181 


9.934311 


438 


10.065680 


10 


43 


814313 


345 


879746 


181 


034567 


438 


065433 


18 


43 


814460 


344 


879637 


181 


034833 


438 


065177 


17 


44 


814607 


344 


879529 


181 


035078 


436 


064923 


18 


45 


814753 


344 


879 '20 


161 


035333 


436 


064667 


15 


46 


814900 


344 


879311 


181 


935589 


436 


064411 


14 


47 


815046 


344 


879202 


183 


935844 


436 


064156 


13 


48 


815193 


344 


679093 


183 


036100 


430 


063900 


12 


49 


815339 


244 


878984 


183 


936355 


438 


063845 


11 


SO 


815485 


343 


678875 


183 


036610 


4S» 


063390 


10 


51 


9.815631 


343 


0.878766 


183 


0.93686A 


435 


10.063134 


8 


52 


815778 


343 


878656 


183 


937121 


435 


002879 


8 


53 


815924 


343 


878547 


183 


037376 


435 


waxsu 


7 


54 


816069 


343 


878438 


183 


937632 


435 


063366 


6 


55 


816215 


343 


878328 


183 


937887 


435 


063113 


5 


50 


816361 


343 


878219 


183 


938142 


435 


061858 


4 


57 


816507 


343 


878109 


183 


936398 


435 


061603 


3 


• 58 


616652 


343 


877999 


183 


936853 


435 


061347 


8 


59 


810798 


343 


877890 


183 


936008 


435 


001093 


1 


60 


816943 


342 


877780 


183 


930163 


435 


060837 





1 


Co«lne 




Sine 1 


1 CotftDg. 


1 


1 Tuig. 


1 M. 1 








49 


Xhign 


M. 















MNU 






85 


M. 


1 81n« 


1 D 


1 Cosin« i D. 


1 Tang. 


1 D- 


1 Coung. 1 j] 







9.81G943 


842 


9.877780 


183 


9.939163 


425 


10.060837 


60 




1 


HlTtWH 


243 


877670 


183 


939418 


425 


060583 


59 




2 


8irJ33 


842 


8naoo 


183 


939G73 


425 


060327 


58 




3 


817379 


842 


877450 


183 


93991*8 


425 


060072 


57 




4 


817524 


241 


8n340 


183 


940183 


425 


059817 


56 




5 


817668 


341 


877230 


184 


940438 


425 


050562 


55 




6 


817813 


241 


877120 


184 


940694 


425 


059306 


54 




7 


817958 


241 


877010 


184 


940049 


425 


059051 


53 




8 


818103 


241 


87ft899 


184 


941204 


425 


058796 


52 




9 


818247 


241 


876789 


184 


941458 


425 


058542 


51 




16 


818393 


241 


876678 


184 


941714 


425 


058286 


50 




11 


9.818536 


240 


9.876568 


184 


9.941968 


435 


10.058033 


49 




18 


818C81 


240 


876457 


184 


942223 


435 


057777 


48 




13 


8I882S 


240 


876347 


184 


942478 


425 


057522 


47 




14 


8189G9 


240 


876236 


185 


942733 


425 


057267 


46 




15 


819113 


240 


876125 


185 


JM2988 


425 


057012 


45 




M 


819257 


340 


876014 


185 


94:^243 


435 


056757 


44 




n 


819401 


240 


875904 


185 


943498 


425 


056502 


43 




18 


819545 


339 


875793 


185 


943753 


425 


056248 


43 




19 


819689 


239 


875682 


185 


944007 


425 


055993 


41 




90 


819832 


339 


875571 


165 


944262 


425 


055738 


40 




21 


9.819976 


339 


9.875459 


185 


9.944517 


425 


10.055483 


39 




22 


820J20 


339 


875348 


185 


944771 


484 


055229 


38 




83 


8202G3 


839 


875237 


185 


945036 


424 


054974 


37 




84 


S20406 


839 


875186 


186 


945281 


434 


054719 


36 




85 


820550 


238 


875014 


186 


945535 


434 


054465 


35 




96 


820C93 


238 


874903 


186 


945790 


434 


054210 


34 




87 


880630 


238 


874791 


186 


946045 


424 


053955 


33 




88 


820979 


238 


874680 


186 


946299 


434 


053701 


33 




89 


831122 


238 


874568 


186 


946554 


424 


053446 


31 




30 


' 821S65 


838 


874456 


186 


946808 


424 


053192 


30 




31 


9.821407 


838 


9.874344 


186 


9.947063 


424 


10.052937 


39 




38 


821550 


238 


874233 


187 


947318 


434 


052682 


88 




33 


831693 


837 


874131 


187 


947572 


424 


052428 


87 




34 


831835 


837 


874009 


187 


947826 


434 


052174 


86 




35 


831977 


S37 


873896 


187 


948081 


494 


051919 


25 




36 


833139 


837 


873784 


187 


94R336 


434 


051.^ 


24 




37 


833263 


837 


873678 


187 


1M8590 


434 


051410 


23 




38 


823404 


837 


873560 


187 


1M8844 


434 


051156 


23 




39 


822546 


837 


873448 


187 


9^slC99 


4S4 


050901 


31 




40 


8396RH 


836 


873335 


187 


949353 


484 


050647 


30 




41 


8.832830 


830 


9.873233 


187 


9.949607 


434 


10.050393 


19 




48 


823973 


836 


873110 


188 


949662 


484 


050138 


18 




43 


823114 


836 


873998 


188 


950116 


424 


040884 


17 




44 


8232.15 


836 


879fi«> 


188 


950370 


434 


049630 


16 




45 


823397 


836 


874772 


188 


950625 


424 


049375 


15 




46 


fVSSKiQ 


236 


873659 


188 


950879 


434 


049121 


14 




47 


8£(r>80 


236 


878547 


188 


951133 


424 


048867 


13 




46 


823821 


335 


87»I34 


188 


951388 


434 


048612 


13 




48 


8Z)963 


835 


872331 


188 


951642 


434 


048358 


11 




50 


8S4104 


935 


873808 


188 


951896 


424 


048104 


10 




51 


9.834845 


835 


9.878095 


189 


9.052150 


434 


10.047850 







53 


884386 


835 


871981 


189 


953105 


434 


047595 


8 




53 


834527 


835 


871868 


189 


959f<59 


434 


047341 


7 




54 


834668 


834 


871755 


189 


952913 


434 


047087 


6 




55 


834808 


834 


871641 


V99 


95:tlG7 


433 


046833 


5 




56 


824949 


934 


871538 


189 


053431 


423 


046579 


4 




57 


88B090 


934 


871414 


189 


953675 


423 


046325 


3 




58 


885S30 


834 


871301 


189 


953939 


483 


046071 


8 




59 


885371 


934 


871187 


189 


954183 


433 


045817 


1 




60 


8B511 


934 


871073 


190 


954437 


483 


045563 







1 


CorfM 1 


1 


flfaM 1 1 


Ootaag. 1 


1 


Tug. 1 M. 1 



86 


(43 Degraes.) a 


TABLI OP LfMSABITHMIO 




~m!" 


Sin* 


1 D. 


1 GodM 


1 D. 


i TkDg. 


1 ». 




SB 





9.d3S511 


834 


9.871073 


190 


9.954437 


48S 


10.(MSM3 


In 


1 


825rt51 


833 


870960 


190 


954691 


483 


045.109 


» 


« 


835791 


833 


870846 


190 


954945 


433 


045055 


58 


3 


835031 


833 


870733 


190 


955300 


433 


044800 


57 


4 


836071 


833 


870G18 


190 


955454 


433 


(M4546 


56 


5 


820311 


833 


870504 


190 


955707 


433 


044293 


55 


fi 


836351 


833 


870390 


J90 


95.5961 


433 


044039 


54 


7 


836491 


833 


870376 


190 


«J5G215 


433 


043785 


53 


8 


836<)31 


833 


870161 


190 


951^469 


433 


013.'>3I 


53 


9 


836770 


833 


870047 


191 


95673:< 


433 


013377 


51 


10 


836910 


833 


869933 


191 


956977 


433 


043083 


50 


11 


9.S27040 


838 


9.800818 


191 


0.957331 


433 


10.043760 


48 


18 


837189 


838 


869704 


191 


057485 


483 


043515 


48 


13 


837338 


838 


860589 


191 


957739 


483 


043961 


47 


1 1^ 


837467 


838 


860474 


191 


B57993 


433 


043007 


46 


15 


837606 


838 


869360 


191 


958846 


433 


041754 


45 


16 


R2T745 


833 


809345 


191 


858500 


483 


041500 


44 


17 


837884 


331 


869130 


191 


958754 


483 


041346 


43 


18 


tt«033 


831 


869015 


193 


959008 


483 


040993 


^ 


19 


838163 


831 


868900 


193 


959363 


433 


040738 


41 


80 


838301 


831 


868785 


193 


959516 


433 


040484 


40 


81 


9.838439 


831 


9.868670 


193 


9.959760 


4-23 


10.040331 


39 


93 


838578 


831 


868555 


193 


930(h2:) 


4sa 


03il977 


38 


23 


8S8716 


331 


868440 


193 


960377 


433 


039733 


37 


84 


H38855 


830 


868334 


193 


960531 


433 


039469 


36 


85 


838993 


330 


868309 


193 


960784 


433 


QOMl^ 


35 


86 


839131 


830 


868093 


193 


961038 


433 


038963 


34 


87 


839309 


330 


807978 


193 


961391 


433 


038709 


33 


88 


839407 


830 


807863 


193 


961545 


433 


038455 


33 


80 


839545 


330 


«rr747 


193 


961799 


433 


038301 


31 


30 


839683 


830 


867631 


193 


963053 


433 


037948 


30 


31 


9.839831 


889 


9.8ir7515 


193 


9.968300 


433 


10.037604 


80 


33 


839959 


839 


867399 


193 


963560 


433 


037440 


88 


33 


830097 


839 


867983 


19B 


963813 


483 


037187 


87 


34 


830334 


839 


867167 


193 


0(»3067 


433 


036933 


86 


35 


830373 


839 


867051 


193 


963330 


483 


036680 


85 


36 


830509 


889 


866035 


194 


963574 


483 


036436 


94 


37 


830646 


839 


866819 


194 


963837 


433 


036173 


83 


38 


830784 


839 


866703 


194 


964081 


4sa 


035919 


88 


30 


830931 


338 


866586 


194 


964.135 


433 


035665 


81 


40 


831058 


388 


866470 


191 


964588 


438 


035413 


90 


41 


9.831195 


888 


9.866353 


194 


0.964848 


488 


10.035156 


19 


43 


831333 


388 


8668:17 


194 


965085 


438 


034005 


18 


43 


8314<}9 


338 


860130 


194 


965349 


482 


034651 


17 


44 


831606 


838 


866004 


195 


965603 


483 


(i34398 


16 


45 


831743 


838 


865887 


195 


965855 


433 


034145 


15 


46 


831879 


888 


865770 


195 


966109 


438 


0:i3891 


14 


47 


8330]5 


987 


865653 


195 


966363 


483 


ai3638 


13 


48 


833153 


837 


8655.16 


195 


966616 


438 


033384 


13 


49 


833388 


897 


865419 


195 


966869 


433 


033131 


11 


50 


838435 


887 


865303 


195 


967133 


438 


033877 


10 


51 


9. 8^961 


887 


9.8li5l85 


195 


0.967376 


438 


10.039684 


9 


58 


839697 


887 


865068 


195 


967639 


483 


038371 


8 


53 


833833 


887 


864950 


195 


967883 


438 


038117 


7 


54 


833969 


880 


864833 


196 


968136 


433 


031864 


6 


55 


833105 


996 


864716 


196 


968389 


488 


031611 


5 


56 


833341 


9H 


864598 


196 


968643 


488 


031357 


4 1 


57 


833377 


896 


864481 


196 


968896 


438 


031104 


3| 


58 


833513 


896 


864363 


196 


969149 


488 


030651 


8 


50 


833648 


896 


864345 


196 


969403 


488 


030597 


1 


60 


833783 


886 


864137 


196 


969656 


4^ 


030344 


«l 




1 OoriM 


1 


8iM 




CotMig. 


1 


1 Vteg. 


Ll! 



47 







«N£I 


AND TANOENTfl. (43 Degieet.j 




87 


M. 


1 8ln« 


1 D 


1 Corin* 1 D. 


1 T«f. 


! D. 


1 OoUDg. 


1 





9.8XS783 


336 


0.864137 


193 


9.909056 


433 


10.U30.;44 


00 


1 


833919 


335 


864010 


196 


0099:)9 


423 


090091 


50 


ft 


834054 


335 


863833 


197 


070103 


433 


Q398:m 


58 


3 


834189 


335 


803774 


197 


970416 


433 


039384 


57 


' 4 


8343-25 


235 


803656 


197 


9700(U» 


433 


033331 


56 


1 5 


834400 


235 


863538 


ltl7 


9709-23 


433 


039078 


55 


6 


834595 


335 


803419 


197 


971175 


423 


038835 


54 


7 


834730 


235 


863301 


197 


971429 


4^2 


038371 


53 


8 


8348G5 


333 


863183 


197 


071082 


433 


0-28318 


S3 


9 


834999 


3-24 


80.'W64 


197 


971935 


423 


036005 


51 


10 


833134 


SM 


8U3946 


198 


07-2188 


433 


037813 


50 


11 


9.83S360 


3-24 


9.863837 


108 


9.97-2441 


433 


10.037550 


49 


13 


833403 


ftU 


862709 


198 


97-2:;94 


433 


(K273M 


48 


13 


835338 


3-24 


803590 


198 


97-2918 


433 


037053 


47 


14 


835673 


334 


803471 


198 


973-201 


433 


0-3r>7iJ9 


46 


15 


835807 


334 


808353 


196 


973434 


433 


020546 


45 


16 


835IM1 


834 


803334 


198 


973707 


423 


030393 


44 


17 


airi075 


233 


863115 


196 


073900 


432 


030040 


43 


Id 


836i39 


333 


861996 


19d 


974313 


423 


025787 


43 


19 


83(»43 


333 


801877 


198 


9744% 


423 


0-25534 


41 


3U 


836477 


333 


861758 


1;I9 


074719 


4-22 


033-2rSl 


40 


31 


0.836611 


333 


9.mi638 


199 


9.074973 


433 


10.035037 


39 


32 


836745 


233 


861319 


199 


073220 


433 


024774 


38 


33 


836878 


333 


861400 


VJd 


973479 


433 


024521 


37 


34 


837013 


233 


861380 


199 


075733 


428 


0343G8 


30 


33 


837146 


333 


861161 


199 


075983 


433 


034015 


35 


36 


837379 


333 


861041 


199 


076-218 


4-23 


0337G3 


34 


37 


837413 


333 


800933 


19^ 


970491 


423 


033509 


33 


38 


837546 


333 


800893 


1!>9 


970744 


in 


0-23-256 


33 


89 


837679 


333 


800683 


303 


0769J7 


423 


033003 


31 


30 


837813 


333 


860503 


2M 


977330 


433 


032730 


90 


31 


9.837945 


833 


0.800443 


300 


0.077503 


4S3 


10.033497 


29 


33 


8%o:8 


221 


6.10333 


300 


«7n50 


433 


033344 


88 


33 


838-311 


331 


860303 


300 


0780)9 


423 


031991 


37 


34 


838344 


331 


800082 


2C0 


978303 


433 


031738 


86 


3S 


83W77 


fai 


839963 


»A 


978515 


433 


031485 


25 


36 


838610 


831 


839843 


300 


078708 


433 


031333 


34 


37 


83874-2 


221 


859731 


301 


079031 


4-23 


083979 


33 


38 


838875 


8-21 


859001 


301 


079274 


433 


(»07-26 


23 


39 


839007 


221 


899480 


301 


979527 


433 


030473 


21 


40 


839140 


£» 


850360 


301 


079780 


423 


030-230 


20 


41 


9.839372 


230 


9.859339 


301 


0.080033 


483 


10.019967 


19 


43 


839104 


2-20 


859119 


301 


980280 


433 


019714 


18 


43 


83B53a 


230 


83893S 


2)1 


960338 


433 


019403 


17 


44 


839068 


220 


858877 


201 


0d0791 


431 


019300 


16 


45 


&I9800 


2J0 


838750 


303 


981044 


431 


018936 


15 


46 


839333 


330 


858035 


303 


9813;)7 


431 


018703 


14 


47 


840004 


319 


858514 


80-2 


081330 


431 


018450 . 


13 


48 


84U19i 


819 


858393 


303 


961803 


431 


018197 


13 


49 


8403-28 


819 


958373 


Qifi 


0830:.6 


431 


017944 


11 


50 


840459 


3J9 


awi5i 


803 


983309 


431 


017091 


10 


51 


9.840591 


319 


0.8.18039 


313 


0.9835T3 


431 


10.017438 


9 


93 


8407-^ 


319 


a57fl06 


3.13 


983814 


431 


017IR0 


8 


53 


840854 


319 


857780 


803 


0830(J7 


431 


010933 


7 


51 


840985 


319 


at7005 


303 


983320 


431 


010080 


6 


55 


841116 


318 


857543 


803 


083373 


431 


010437 


5 


50 


841347 


318 


857433 


303 


963828 


431 


010174 


4 


57 


841378 


318 


857300 


303 


084079 


431 


015931 


3 


58 


841509 


318 


857178 


203 


984331 


431 


015069 


3 


50 


841010 


318 


a'>705fl 


303 


984584 


431 


015410 


1 


00 


811771 


318 


856^34 


303 


984837 


431 


015103 



M. 




1 OorilM 


1 


8lii« 1 


OoCang. 


i 


1 t"g. 1 



40 



88 



(44 Deften.) loo. sine* and TANaENTs. 



M. 1 


fiiiM 


D. 


1 CoiiiM 


D. 


1 Ttog. 


». 


1 Ooton» 1 







0.841771 


818 


0.856934 


303 


9.984837 


481 


10.015163 


00 




1 


841003 


318 


856813 


303 


985090 


431 


014910 


50 




8 


843033 


818 


856690 


304 


985343 


431 


014657 


56 




3 


843163 


317 


esnsefi 


804 


965596 


431 


014404 


57 




4 


843394 


817 


856446 


304 


965848 


431 


014153 


56 




5 


843434 


817 


856333 


304 


966101 


431 


013H99 


55 




6 


813555 


217 


856301 


304 


960354 


431 


013646 


54 




7 


843685 


317 


856078 


304 


966607 


431 


013393 


53 




8 


843R15 


317 


K{5056 


304 


966860 


431 


013140 


53 







813946 


817 




304 


987113 


431 


013888 


51 




10 


813076 


817 


855711 


305 


987365 


431 


0136:i'i 


50 




11 


0.843306 


816 


9.855568 


305 


9.087618 


481 


10.013383 


40 




13 


843336 


316 


855465 


305 


987871 


431 


013139 


48 




13 


843466 


316 


855343 


305 


088133 


431 


011877 


47 




14 


843505 


316 


855310 


305 


088376 


431 


011634 


46 




15 


843735 


316 


855006 


805 


968639 


431 


011371 


45 




16 


843855 


816 


854973 


305 




481 


011118 


44 




17 


843084 


316 


854850 


305 


969134 


431 


010866 


43 




18 


844114 


315 


854737 


806 


089387 


481 


010613 


43 




10 


844343 


315 


854603 


806 


969640 


431 


010360 


41 




00 


844373 


815 


854480 


306 


069803 


431 


010107 


40 




81 


0.844508 


815 


0.854356 


806 


0.900145 


431 


10.000855 


30 




83 


844631 


315 


854333 


306 


090396 


431 


009603 


38 




83 


844760 


3J5 


854100 


306 


900651 


48] 


009349 


37 




84 


844889 


315 


853986 


306 


090903 


431 


009097 


s 




85 


845018 


315 


853868 


306 


901156 


481 


008844 


35 




86 


845147 


315 


853738 


306 


901400 


431 


008591 


31 




87 


845376 


314 


853614 


307 


901663 


431 


006338 


sa 




38 


845405 


314 


853490 


307 


901914 


431 


008086 


33 




80 


845533 


314 


853:166 


307 


993167 


431 


007833 


31 




30 


845663 


314 


853343 


307 


093430 


431 


007580 


30 




31 


0.845790 


814 


9.853118 


907 


0.998073 


431 


10 007388 


30 




33 


845010 


814 


853904 


307 


903935 


431 


007075 


38 




33 


846047 


814 


858860 


307 


993178 


431 


006833 


37 




34 


846175 


814 


853745 


307 


993430 


431 


006570 


36 




35 


846304 


314 


858030 


807 


993683 


431 


006317 


35 




36 


846433 


313 


853496 


3U8 


993036 


431 


006064 


34 




37 


846560 


313 


853371 


308 


001180 


431 


005811 


33 




38 


846688 


313 


853347 


308 


994441 


431 


oavuig 


33 




30 


846816 


313 


853133 


308 


994694 


431 


005306 


31 




40 


846044 


313 


851097 


808 


904047 


431 


005053 


90 




41 


0.847071 


813 


0.851873 


908 


0.995199 


431 


10 004801 


10 




43 


847199 


313 


851747 


806 


995453 


431 


004548 


18 




43 


847337 


313 


851633 


908 


995705 


431 


004305 


n 




44 


847454 


313 


851497 


300 


905957 


431 


004043 


16 




45 


847583 


313 


851373 


309 


996310 


431 


003790 


15 




46 


847709 


313 


851346 


309 


996463 


431 


003537 


14 




47 


847836 


313 


851131 


309 


996715 


431 


INi;i3K5 


13 




48 


847964 


313 


850996 


309 


906968 


431 


003033 


13 




40 


848091 


313 


850870 


309 


997331 


431 


003779 


11 




50 


84^18 


313 


850745 


909 


997473 


431 


003537 


10 




51 


0.846345 


313 


0.850619 


909 


0.007796 


431 


10aMJ3374 







53 


848173 


311 


850403 


310 


997979 


431 


003031 


8 




53 


848500 


311 


850368 


310 


996331 


431 


001769 


7 




54 


848736 


311 


850343 


310 


998484 


431 


001516 


6 




55 


84Ra%3 


311 


850116 


310 


996737 


431 


001363 


5 




56 


848070 


311 


849990 


310 


998069 


431 


001011 


4 




57 


840106 


311 


849864 


810 


999343 


431 


000758 


3 




58 


849333 


311 


849738 


310 


999405 


431 


000505 


3 




50 


8403S0 


311 


849611 


810 


090748 


431 


000353 


1 




60 


840485 


311 


849485 


810 


10.000000 


431 


000000 







1 


Codne 




1 8kM 




Cotang. 


1 


1 TmDg. 1 M. 





45DegnM. 



TABLE 



or 



NATURAL SINES, COSINES, 
TANGENTS, AND COTANGENTS 



TOR 



EVEKY DEGEEE AND MINUTE 



OF THE QUADRANT. 



Note. — The minntes in the Ielt>hand column of each page, increaa- 
ing downwards, belong to the degrees at the top ; and those increasing 
upwards, in the right-hand column, belong to the degrees below. 



90 



NATURAL 81MBJI. 





Deg. 


1 Deg. 


2 Deg. 


3 Deg. 


4 Deg. 






Nat. 


N.Ck>. 


Nat. iN.Co- 


Nat. 


N.Uo 


Nut. 


N. Co- 


Nat. 


N.Co- 







Sine 


Sine 


S:ii«. Sine 


Sine 


Sine 


Sine 


Sine 


Sine 


fine 


M 

60 




OJOUO 


UniL 


01745 99J8o 


(r^90 


99939 


052:4 


998(3 


06976 


99756 




I 


00029 


00000 


01774 


99984 


03519 


99938 


05263 


99361 


07C05 


99754 


59 




2 


00053 


COOOO 


01803 


99984 


03548 


99937 


05292 


99860 


07034 


99752 


58 




3 


00087 


00000 


01832 


99983 


0:577 


99930 


05321 


99858 


07063 


99750 


57 




4 


00116 


00000 


018C2 


99983 


03606 


99U35 


0.350 


99857 


07092 


99748 


56 




5 


00145 


00000 


01891 


999S2 


G3G35 


99934 


05379 


99P55 


07121 


99746 


55 




6 


00175 


ooouo 


01920 


99982 


03664 


Qim.1 


0J4J8 


9.^.->4 


07150 


9U744 


54 




7 


00304 


00900 


01949 


99981 


03633 


99J32 


0.*437 


99352 


071:9 


99743 


53 




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89Dcg^ 


»8Deg. 1 


87 Deg. L 


86 Deg. 1 


85 Deg. 1 





NATURAL IINBI. 



91 



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5 Deg. 


6Deg. 


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8 Deg. 1 9 Deg. 


M 
60 


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10434 


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15615 


96773 


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1 

M 


N.CS 


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N.CS N.S. 


N.CS 


N.S. 


N.CS. 


N.S. 


^84Deg. 1 


83l>cg. J 


82Deg. 


81 Deg. 


80 Deg. 1 



92 



NATOSAI, (INn. 



M 




lODeg. 1 


11 Deg. 


13 Deg. 1 


13 Deg. 1 14 Deg. 


M 

60 


N.S. 


N.CS. 


N.S. 


N.CS. 


N.S. 


N.CS. 


N.S. 


N.CS. N.S. 


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173G5 


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97437 


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97430 


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30848 


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58 


3 


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98466 


19167 


96146 


30877 


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22581 


97417 


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57 


4 


17479 


98461 


19195 


98140 


20905 


97791 


32608 


97411 


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36 


5 


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55 


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17623 


96435 


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44 


17 


17852 


98394 


19566 


980ff7 


31375 


97711 


23977 


97335 84672 


96000 


43 


Id 


17880 


98389 


19595 


96061 


31303 


97705 


33005 


97318 , 84700 


96008 


42 


J9 


17009 


98383 


19833 


98056 


31331 


97698 


23033 


9TJ11 


84738 


96894 


41 


20 


17937 


98378 


19653 


98050 


31360 


97093 


23062 


97304 


84756 


96887 


40 


21 


17966 


08373 


19680 


98044 


31388 


97686 


23090 


97396, 


84784 


96680 


30 


32 


17995 


98368 


19700 


98039 


31417 


97680 


23118 


97291 


84813 


96873 


38 


23 


180^ 


98362 


19737 


98033 


31445 


97673 


23146 


97284< 


84841 


96866 


37 


24 


18053 


98357 


19766 


98037 


31474 


97667 


23175 


9TB78 


84869 


96858 


36 


25 


ISO&l 


98.^'i3 


19794 


98031 


31503 


97661 


23303 


97271 


848ir7 


96851 


35 


26 


18109 


98347 


19833 


98016 


21530 


97655 


23331 


97364 


84935 


96844 


34 


27 


18138 


98341 


19851 


98010 


215.59 


97648 


33360 


9T257 


84953 


96637 


33 


28 


18166 


9iSXK 


19880 


98004 


21587 
31616 


97643 
97636 


33388 

33:n6 


97351 
97-244 


249U2 
35010 


96829 

QAflOO 


32 
31 


39 


18195 


98331 


19908 


97998 


wxjcas 


30 


18224 


98335 


19937 


97993 


31644 


97630 


33345 


97237 


85038 


96815 


30 


31 


18352 


98330 


19965 


97967 


21672 


97023 


33373 


07830 


35066 


96807 


S9 


33 


18281 


96315 


19994 


97981 


31701 


97617 


23401 


97233 


35094 


96800 


88 


33 


18309 


96310 


30033 


9T975 


31^29 


97611 


23429 


97317 


85133 


96793 


27 


34 


18338 


96304 


30051 


97969 


31758 


97664 


23458 


97310 


85151 


96786 


86 


35 


18367 


08399 


30079 


97963 


21786 


97598 


23486 


97203 


85179 


96778 


85 


36 


18395 


96394 


20108 


97958 


21814 


97592 


23514 971961 


25307 


96771 


24 


37 


18424 


98288 


30136 


97953 


21843 


97585 


23542 


97189 


35335 


96764 


23 


38 


18453 


9838:) 


30165 


97946 


21871 


97579 


23571 


97183 


35363 


90756 


88 


39 


18481 


98277 


30193 


97940 


31899 


97573 


23599 


97176 


35391 


96749 


81 


40 


18509 


982T2 


30333 


97934 


21938 


97566 


23627 


97109 


35330 


96742 


20 


41 


1R538 


98387 


30350 


97938 


31956 


97560 


23656 


97103 


35348 


96734 


19 


42 


18567 


98361 


30379 


97933 


31985 


97553 


23684 


97155 


35376 


96737 


18 


43 


18595 


98356 


30307 


97916 


33013 


97547 


23712 


97148 


35404 


96719 


17 


44 


18834 


98350 


30336 


97910 


33041 


97541 


23740 


97141 


35433 


96712 


16 


45 


18653 


98345 


30364 


97905 


23070 


97534 


23769 


97134 


35460 


96705 


15 


46 


18681 


98340 


20303 


97899 


32098 


97528 


23797 


97127 


35488 


96607 


14 


47 


18710 


96334 


30431 


97893 


22126 


97521 


23825 


97130 


35516 


96690 


13 


48 


18738 


0833!) 


30450 


97887 


33155 


97515 


23853 


97113 


85545 


96683 


IS 


49 


18767 


98333 


30478 


97881 


33183 


97508 


25382 


97106 


35573 


96675 


11 


50 


18795 


98318 


30507 


97875 


^3313 


97502 


23910 


97100 


85601 


96667 


10 


51 


18834 


98313 


30535 


97869 


33340 


97496 


23938 


97093 


35639 


96660 


9 


52 


18853 


98207 


30563 


97863 


333()8 


97489 


23966 


97086 


35657 


966S3 


8 


53 


1H881 


98301 


20593 


97857 


33397 


97483 


23995 


97079 


25C85 


96645 


7 


54 


18910 


98196 


30630 


97851 


33.335 


97476 


24023 


97073 


S5713 


96638 


6 


55 


18938 


98190 


30649 


97845 


23353 


97470 


24051 


97065 


35741 


96630 


5 


56 


18967 


96185 


30677 


97839 


33383 


97463 


24079 


97058 


35769 


96633 


4 


57 


18995 


98179 


30706 


97833 


32410 


97457 


34106 


97051 


35798 


96615 


3 


58 


19034 


98174 


30734 


97827 


23438 


97450 


34136 


97044 


85826 


96606 


8 


50 
M 


19053 
N.CS. 


96168 

N.S. 


30763 


97831 


33467 


97444 


34164 


97037 


258S4 


96000 


1 


N.CS. 


N.S. 


N.CS. 


N.S. 


N.CS 


N.S. 


N CS 


NS 


M 


TJ Deg. 


78 Deg. ! 


77 Deg. 1 


76 Deg. 


75 Deg. 


- 



NATURAL SINE8. 



93 



M 



15 Deg. 


16 Deg. 


17 Deg. 


18 Deg. 


19 Deg. 


M 

60 


N.S. 


N.<». 


N.S. 


N.CS. 


N.S. 


N.Cd. 


N.S. 


N.CS. 


N.S. 


N.CS. 


35883 


96503 


37564 


96136 


29337 


93630 


30902 


95106 


32557 


94552 


1 


35910 


96585 


37592 


96118 


39365 


05632 


30929 


95097 


32584 


94542 


59 


8 


35938 


96578 


37620 


96110 


39393 


95613 


30937 


95088 


32612 


94533 


58 


3 


35066 


96570 


37648 


96103 


39331 


95605 


30985 


95079 


33639 


94523 


57 


4 


35094 


96563 


37676 


96094 


39348 


95596 


31012 


95070 


32667 


94514 


56 


5 


3fMY33 


96555 


37704 


96086 


29376 


95588 


31040 


95061 


39694 


94504 


55 


6 

7 


36050 


96547 
96540 


27731 
37759 


96078 
96070 


29404 
29433 


95579 
95571 


31068 
31005 


95053 
95043 


32722 
32749 


944i»5 
94485 


54 
53 


36079 


8 


38107 


96533 


37787 


96063 


294G0 


95563 


31123 


95033 


32777 


94476 


53 


9 


36135 


96534 


37815 


96054 


39487 


95554 


31151 


95034 


32804 


94466 


51 


10 


36163 


90517 


27843 


96046 


39515 


95545 


31178 


05015 


32832 


94457 


50 


U 


36191 


96509 


37871 


96037 


39543 


05536 


31306 


95006 


33859 


94447 


49 


13 


36319 


96503 


37899 


96039 


39571 


95528 


31333 


94997 


33887 


94438 


48 


13 


36247 


96494 


37937 


96021 


39599 


95519 


31261 


94988 


32914 


94438 


47 


' 14 


36375 


96486 


27955 


96013 


39626 


93311 


31289 


94979 


32942 


94418 


46 


15 


36303 


96479 


37983 


96005 


39654 


95502 


31316 


94970 


33969 


94409 


45 


16 


36331 


96471 


98011 


95997 


396ffi 


05493 


31344 


94961 


33997 


94399 


44 


17 
Id 


WllO 


96463 
96456 


3H039 
38067 


95989 
95U81 


99710 
39737 


95485 
95476 


31373 
31399 


94959 
94943 


33024 
33051 


94390 
94380 


43 
43 


36387 


19 


36415 


96448 


98095 


95973 


39765 


95467 


31427 


94933 


'33079 


94370 


41 


90 


36443 


96440 


28133 


95964 


39793 


95450 


31454 


94934 


3310b 


94361 


40 


31 


36471 


96433 


28150 


95956 


39631 


95450 


31488 


94915 


i 33134 


94351 


39 


33 


36500 


96435 


28178 


95948 


39849 


95441 


31510 


94906 


,33161 


04343 


38 


33 


36538 


96417 


38206 


95940 


39876 


95433 


31537 


94897 


'33189 


94333 


37 


34 


36556 


96410 


28334 


95931 


39904 


95434 


31565 


94^>88 


33316 943331 


36 


35 


36584 


96403 


3rt263 


95923 


39933 


95415 


31593 


94878 


33344 


94313 


35 


36 


36613 


96304 


2H3SI0 


95915 


39960 


95407 


31620 


94869 


33371 


94303 


34 


37 


36640 


96386 


28318 


95907 


29987 


95398 


31648 


94860 


33298 


94393 


33 


38 


IBDINXS 


96379 


28346 


95898 


30015 


95389 


31675 


94851 


3:)326 


94384 


33 


30 


360816 


96371 


28374 


9589U 


30043 


95380 


31703 


94843 


33353 


94374 


31 


30 


96734 


96363 


38403 


95883 


30071 


05373 


31730 


94833 


1 33381 


94364 


30 


31 


96759 


96355 


38439 


95874 


30098 


95363 


31758 


94893 


33406 


94354 


39 


33 


38780 


96347 


28457 


95865 


30126 


95354 


31786 


94814 


33436 


94345 


38 


33 


96808 


96340 


28485 


95857 


30154 


95345 


31813 


94805 


33463 


94335 


37 


34 


96636 


96333 


28513 


95849 


30183 


95337 


31841 


94795 


33490 


94335 


36 


35 


96864 


96334 


38541 


95841 


30309 


95338 


31868 


94786 


33518 


94315 


35 


36 


96803 


06316 


28569 


95833 


30337 


95319 


31896 


94777 


33545 


94306 


34 


37 


36930 


96308 


28597 


95834 


30965 


95310 


31933 


94768 


33573 


04196 


33 


38 


36948 


96301 


28635 


95816 


30393 


05301 


31951 


94758 


33600 


94186 


33 


30 


36076 


96393 


38653 


95807 


30320 


95393 


31979 


94749 


33627 


94176 


31 


40 


37004 


OSSdS 


38680 


95799 


30348 


95284 


:)3006 


94740 


33655 


i»4167 


30 


41 


37033 


96377 


38708 


95791 


30376 


95275 


:t2034 


94730 


3:1683 


94157 


19 


43 


37060 


9636U 


38736 


95783 


30403 


93266 


:^i06l 


94721 


33710 


im47 


18 


43 


37088 


96361 


28764 


95774 


30431 


95257 


:t2069 


94713 33737 


94137 


17 


44 


37116 


96353 


28793 


95766 


30459 


95248 


:i2ii6 


94703 33764 


94127 


16 


45 


37144 


96346 


38830 


95757 


30486 


95240 


:t3144 


94693 


33793 


94118 


15 


46 


37173 


96338 


38847 


95749 


30514 


05331 


33171 


94684 


33819 


94108 


14 


47 


37900 


90330 


3R875 


95740 


30543 


95322 


:i3199 


94674 


33846 


94098 


13 


48 


37338 


96333 


38U03 


95733 


30570 


05313 


32327 


9-1665 


33874 


94088 


13 


40 


37356 


96311 


28931 


95734 


30597 


95204 


:^2254 


94656 


33901 


94078 


11 


50 


37384 


96306 


38959 


95715 


30635 


95195 


:^2283 


94646 


33929 


94068 


10 


51 


37313 


96198 


28987 


95707 


30653 


95186 


32309 


94637 


33956 


94058 


9 


53 


37340 


9619U 


3U015 


9361)8 


30680 


95177 


32:137 


94637 


33983 


94049 


8 


53 


37:»8 


9618-i 


39043 


95090 


30708 


95168 


32364 


94618 


34011 


94039 


7 


54 


37306 


96174 


39(170 


95681 


30736 


95159 


32393 


94609 


34038 


94039 


6 


55 


37434 


96166 


39098 


95673 


30763 


95153 


32'! 19 


94599 1 34065 


94019 


5 


56 


37453 


96158 


29136 


95664 


30701 


95142 


32447 


94590 ! 34993 


94009 


4 


57 


37480 


96150 


29154 


95656 


30819 


95133 


32474 


04580 34120 


93999 


3 


58 


97508 


9614-2 


39182 


95647 


30646 


95124 


33502 


94571 34147 


93989 


2 


59 
M 


37536 


96134 


29209 


95639 


30874 


95115 


32529 


94561 1 34175 


93979 


1 
M 


N.C8. 


N.S. 


N.C3. 


N.S. 


N.CS 


N.S. 


N.OS 


N.S. ,N.CS 


N.S. 


741>eg. 


73Deg. 


72 Deg. 


71 Deg. 70 Deg. 



94 



NATUHAL BINR8. 



M 

U 


20 Deg. 


21 Deg. 


22 Deg. 


23 Deg. If 24 Deg. 


1 

M 

80 


N.S. 


N.C8 


rt.ri. N.OsJ 


N.S. 


N.C8. 


N.S. 


N.CS. iN.S. 


N.l». 


34-i02 


93969 


35t£<7 9J358 


374U1 


92718 


39(irj 


92050 


40674 


91355 


1 


34229 


93959 


35864 


93348 


37488 


92707 


39100 


92039 


40700 


91343 


59 


s 


34257 


i>3949 


35891 


93337 


37515 


92697 


39127 


92028 '40737 


91331 


58 


3 


34i84 


93939 


35918 


93327 


37542 


92(i86 


39153 


92016 40753 


91319 


57 


4 


34311 


93929 


35945 


93316 


37569 


92675 


39180 


92005 40780 


91307 


56 


5 


34339 


93919 


359r3 


93306 


37595 


92664 


39207 


91994 40606 


91395 


55 


6 


34366 


93909 


36000 


93395 


37622 


92653 


39234 


01983 


40833 


91983 


54 


7 


31393 


03899 


3li027 


93285 


37649 


92642 


39260 


91971 


40660 


91373 


53 


8 


34421 


93889 


36U54 


93274 


37676 


93631 


39287 


91959 


40886 


91360 


53 


9 


34448 


U3879 


36081 


93264 


37703 


92630 


39314 


91948 


40913 


91348 


51 


10 


34475 


93869 


36108 


93253 


37rj;j 


93609 


39341 


91936 


40939 


91336 


50 


11 


34503 


93859 


30135 


93243 


37757 


92598 


39367 


91935 


40966 


91324 


49 


12 


34530 


93849 


36163 


03232 


37784 


92587 


39394 


91914 


40902 


91313 


46 


13 


34557 


93839 


361}H) 


93222 


37811 


»576 


3J421 


91903 


41019 


91300 


47 


14 


34584 


93829 


30217 


9:^211 


37B38 


92565 


39448 


9J891 


41045 


91188 


46 


15 


34612 


93819 


36244 


9J291 


37865 


92554 


39474 


91879 , 


41073 


91178 


45 


10 


34630 


93809 


38271 


03100 


37893 


92543 


39501 


91888 


41096 


91164 


44 


17 


34666 


93799 


30298 


93180 


37919 


92532 


39528 


91856 


41135 


91153 


43 


Id 


34694 


93789 


33335 


93169 


37946 


93531 


39555 


91845 


41151 


91140 


43 


J9 


34721 


93779 


36352 


93159 


37973 


92510 


39581 


91833 


41178 


91128 


41 


30 


34748 


93769 


36379 


93148 


37999 


93499 


3;)60rt 


91832 


41204 


91116 


40 


31 


34775 


93759 


36406 


93137 


38ih» 


92488 


39635 


91810 


41331 


91104 


30 


22 


34803 


93748 


36434 


93127 


38053 


92477 


3J661 


91799 


41357 


91003 


38 


33 


34830 


93738 


36461 


93116 


38U60 


92466 


39688 


91787 


41284 


91080 


37 


24 


34857 


93728 


36488 


93106 


38107 


92455 


39715 


9m5 


41310 


91068 


36 


25 


34884 


93718 


3r>515 


93J95 


38134 


92444 


3d741 


91764 


41337 


91058 


35 


26 


»912 


93708 


33542 


9.IOd4 


38161 


92432 


39768 


91753 


41363 


91044 


34 


27 


34939 


93G9S 


3G5:J9 


93J74 


38188 


92421 


39795 


91741 


41390 


91033 


33 


23 


349GG 


9368S 


3S593 


93J63 


38215 


»2410 


:)9822 


91739 


41416 


91090 


32 


29 


34933 


93677 


33623 


93052 


38241 


923.09 


3J848 


91718 


41443 


91006 


31 


30 


35021 


93667 


36650 


93042 


38268 


^2388 


39875 


91708 

1 


41460 


90996 


30 


31 


35048 


03857 


388n 


93031 


38295 


92377 


39302 


01894 


41498 


90964 


30 


32 


35075 


93647 


33704 


93020 


38332 


93366 


39928 


01683 


41533 


90973 


38 


33 


35102 


03637 


36731 


93010 


38349 


92355 


39955 


91671 


41549 


90960 


87 


34 


35130 


93636 


36758 


92J99 


38376 


92343 


39962 


91660 


41575 


90948 


96 


35 


35157 


93616 


36785 


92988 


38103 


93333 


40008 


91648 


41603 


90936 


35 


30 


35183 


93606 


36812 


92978 


38430 


92331 


40035 


91636 


41838 


90B94 


34 


37 


35211 


93596 


35839 


92967 


38456 


92310 


40063 


01635 


41855 


90911 


33 


38 


35239 


93585 


363S7 


92956 


38483 


92299 


40088 


91613 


41881 


90699 


33 


39 


35266 


93575 


36894 


02945 


38510 


92387 


40115 


91601 


41707 


90667 


31 


40 


r>293 


93535 


36921 


92935 


3a=>37 


92276 


40141 


91500 


41734 


90675 


30 


41 


35320 


93555 


36348 


92924 


38564 


92265 


40168 


91578 


41760 


90663 


19 


42 


35347 


93544 


30975 


82913 


38591 


92354 


40195 


91568 : 


41787 


90651 


18 


43 


35375 


93534 


37002 


92902 


38617 


92243 


40221 


91555 


41613 


90639 


17 


44 


35402 


93524 


37029 


92316 


38644 


922:)1 


40248 


91543 


41840 


90836 


16 


45 


35429 


93514 


37056 


92881 
92870 


38671 


92330 


40275 


91531 


41868 


90614 


15 


46 


35456 


03593 


37003 


•RJoOo 


92309 


40301 


91510 


41893 


90608 


14 


47 


35484 


93493 


37110 


92859 


38725 


92198 


40328 


91506 


41919 


90790 


13 


48 


35511 


93483 


37137 


92849 


38752 


92186 


40355 


91496 1 


41945 


90778 


13 


49 


35538 


934?2 


37164 


928.18 


38778 


92175 


40381 


91484' 


41973 


90766 


11 


50 


35535 


03462 


37101 


92827 


38805 


92164 


40408 


91473: 


41996 


90753 


10 


51 


:»:>92 


03452 


37218 


92816 


38832 


92152 


40434 


91461! 


43034 


00741 


9 


52 


35619 


93441 


37245 


92805 


38859 


92141 


40461 


91449 


43051 


90780 


8 


53 


35S47 


93431 


37272 


92T94 


38886 


92130 


40488 


91437 ; 


43077 


90717 


7 


54 


35G74 


93430 


37299 


92784 


38913 


92119 


40514 


91435 


43104 


90704 


6 


55 


35701 


93410 


37328 


92773 


38939 


92107 


40541 


91414 


42130 


90093 


5 


56 


35728 


93400 


37353 


92762 


38966 


93096 


40567 


91403 


43156 


90880 


4 


57 


35755 


93389 


37380 


92751 


38993 


92085 


40594 


91390 


43183 


90868 


3 


58 


35782 


93379 


37407 


92740 


39020 


92073 


40631 


91378 


43309 


90655 


8 


59 
M 


35810 


93368 


37434 


92729 


39046 


92062 


40647 


91366 


43335 


90843 


1 
M 


N.Cd. 


N.S. 


N.OS. 


N.S. 


N.CS. 


N.S. 


N.08 


N.S. 


N.CS. 


N.S. 


69 Deg. 


68 Deg. { 


67 Deg. 


66 Deg. 


65 Deg. 



NATURAL SINKS. 



95 



U 


35 


Deg. 


36 Deg. 


37 Deg. 


38 Deg. 


II a» 


Deg. 


n 


N.8. |N.C8. 


N.S. N.CS 


S.8. 


N.US. 


N.S. 


N.CS. 


N.S. 


N.CS. M 1 
87462 6U 1 


|-2at69 


90631 


43837 89879 


45399 


tiUlUA 


46947 


88395 ,4&kil 


1 


42388 


90618 


43863 89H07 


45425 


89087 


46973 


88-281 '48506 


87448 


hO 


2 


4-2315 


90606 


43889 89851 


45451 


89074 


46999 


88367 


48533 


87434 


58 


3 


42:mi 


00594 


43916 


Bdnii 


45477 


8J061 


47U24 


88254 


48557 


87420 


57 


4 


4-2367 


90583 


43943 


89628 


455U3 


89048 


47050 


88340 '48583 


87406 


56 


5 


42»I4 


90569 


4391)8 


89816 


4.55-29 


89035 


47076 


88236 48j08 


87391 


55 


6 


42430 


00557 


43994 


8^803 


45554 


89021 


47101 


88213 48H34 


87377 


54 


7 


43446 


90545 


44030 


89790 


45580 


89008 


47127 


88199 


48659 


87363 


53 


8 


45M73 


90533 


44046 


897i/ 


45608 


88905 


47153 


KK185 


48684 


87349 


52 


9 


42409 


90530 


44073 


89764 


45633 


88981 


47J78 


88173 


48710 


87335 


51 


10 


43535 


90507 


44098 


89752 


45658 


88968 


47304 


88158 


48735 


87321 


fM 


11 


43553 


90495 


44134 


89739 


45684 


88955 


47329 


88144 


48761 


87306 


49 


13 


42578 


90483 


44151 


89726 


45710 


88942 


47255 


88130 


48786 


87293 


48 


13 


4-2804 


90470 


44177 


89713 


45738 


88928 


47281 


88117, 


48811 


87378 


47 


14 


4-2!.ni 


90458 


44303 


89700 


45763 


88915 


47308 


88103 


48837 


87264 


46 


15 


42657 


90446 


44339 


89687 


45787 


a^903 


4rJ32 


88069 


48862 


8/250 


45 


16 


43683 


00433 


44355 


89674 


45813 


OQQQQ 
OOCKIO 


47358 


88075 


48888 


87235 


44 


17 


43709 


90431 


44381 


8'J6Q3 


45839 


HH875 


47383 


88063 


48913 


87221 


43 


18 


42736 


90408 


44307 


89649 


45865 


F8863 


174U9 


88048 


48938 


87307 


42 


19 


43163 


90396 


41333 


89636 


45891 


OClO'tO 


474J4 


88034 1 


48964 


87193 


41 


90 


437H8 


003H3 


44350 


89633 


45017 


HHH35 


47460 


8:020 1 


48069 


67178 


40 


91 


42815 


90371 


44385 


89610 


45043 


88H23 


47486 


880061 


49314 


87164 


39 


93 


42iMl 


90358 


44411 


89597 


45968 


86808 


47511 


87993 ' 


40040 


87150 


38 


23 


43867 


90346 


44437 


89584 


45094 


89795 


47537 


87979 


49065 


<J7136 


37 


34 


43804 


90334 


44464 


89571 


46039 


88783 


47562 


87965 1 


49000 


87121 


36 


85 


42930 


90331 


44490 


89558 


46046 


88768 


47588 


87951 1 


49116 


87107 


35 


26 


42946 


9U309 


44516 


89545 


46072 


88755 


47614 


87937 


49141 


87U93 


34 


27 


42972 


90396 


44543 


89533 


46097 


88741 


47639 


87923 


49166 


87079 


33 


28 


43999 


90284 


445G8 


89519 


46133 


88738 


47665 


87909 


49192 


87064 


32 


29 


43035 


90371 


44594 


89506 


46149 


88715 


47690 


87896 


49217 


87U50 


31 


30 


43051 


90359 


44630 


89493 


40175 


887D1 


47716 


87883 


49343 


87036 


30 


31 


4.^77 


99346 


44640 


89489 


46201 


88688 


47741 


87868' 


40268 


87021 


29 


33 


43104 


90233 


44673 


89467 


46338 


88674 


47767 


87854 


49393 


87007 


28 


33 


43130 


00321 


44608 


89454 


46252 


88861 


47793 


87&10 


49318 


8G993 


27 


34 


43156 


90308 


44734 


89441 


46278 


88647 


47bl8 


87836] 


49344 


86978 


26 


35 


43182 


00196 


44750 


89438 


46304 


88634 


47844 


878131 


49369 


88964 


25 


36 


43309 


90183 


44776 


89415 


46330 


88630 


47869 


87798; 


49394 


86949 


24 


37 


43235 


90171 


44803 


89403 


4a^'» 


88607 


47895 


877841 


49419 


8G935 


23 


38 


43361 


90158 


44838 


89389 


46381 


88593 


47920 


87770 


49445 


86921 


22 


39 


43387 


00146 


44854 


89376 


46407 


88580 


47946 


87756' 


49470 


83906 


21 


40 


43313 


90133 


44880 


89363 


46433 


88566 


47971 


87743! 


4!M95 


86892 


30 


41 


4a340 


90130 


4400S 


89350 


46458 


88553 


47997 


87739; 


49531 


8C878 


19 


43 


43300 


90108 


44933 


89337 


46484 


88539 


48023 


87715 


49546 


868<i3 


18 


43 


43393 


09995 


44958 


89334 


46510 


885-36 


48048 


877011 


49571 


86849 


17 


44 


43418 


99083 


44984 


89311 


46536 


88513 


48073 


87687 


49506 


86834 


16 


45 


43445 


90070 


45010 


89396 


46561 


68499 


48099 


8T673 


49632 


86820 


15 


46 


43471 


g0057 


45036 


89C85 


46587 


88485 


48134 


87650 


49647 


86605 


14 


47 


43497 


O'KMS 


45062 


89273 


46613 


88473 


48150 


87045 


49673 


88791 


13 


48 


43523 


90032 


45088 


89259 


48639 


88458 


48175 


87G31 


49097 


8Gr<7 


12 


49 


4J549 


90019 


45114 


89245 


46664 


cxJ44u 


48201 


87017 


49T23 


88762 


11 


50 


43575 


90007 


45140 


89232 


46690 


88431 


48-226 


87603 


49748 


88748 


10 


51 


43803 


89994 


45166 


89219 


46716 


88417 


48252 


87589 


49773 


86733 


9 


53 


43638 


89981 


45193 


89206 


46743 


88404 


48277 


87575 


49798 


86719 


8 


53 


43854 


699n8 


45218 


89193 


46767 


88390 


48303 


87581 


49824 


86704 


7 


54 


43680 


89951 


45343 


89180 


48793 


88377 


48328 


87546 


49849 


86690 


6 


55 


43706 


89943 


45-280 


89167 


4S819 


88383 


48354 


87532 


49874 


86675 


5 


56 


43733 


89930 


45295 


89153 


48844 


88349 


48379 


87518 


49899 


86661 


4 


57 


43759 


89918 


45331 


8D140 


40870 


88336 


48405 


87504 


49934 


86646 


3 


58 


43785 


89905 


45347 


89127 


48896 


88322 


48430 


87490 


4995U 


86633 


2 


50 

m" 

1 


43811 


89892 


45373 


89114 

N.B. 


46931 


88308 


48456 


87476 


49075 


86617 


1 
M 


N.CS. 


N B. 


N.C9. 


N.CS. 


N.S. 


N.CS 


N.S. 


N.CS 


N H. 


64 Deg. 1 


63 Deg. 


63 Deg. 


61 Deg. 


GO Deg. 



9« 



NATDKAL anfM. 



M 




30 


Deg. 


31 Deg. 


33 Deg. 


33 Deg. f 34 Deg. 


M 

80 


N.8. |N.C8. 


N.S. N.CS. 


N.8. 


N.CU 


N.S. 


N.CS.'(N.8. 


N.CS 


50U00 


86603 


51504 85717 


52992 


84805 


54464 


63887 


55010 


82904 


1 


50025 


86588 


51529 


85703 


53017 


84789 


54488 


83851 


> 55943 


8S887 


50 


s 


50050 


86573 


51554 


85687 


53041 


84774 


54513 


83835 


55968 


82871 


58 


3 


50076 


89559 


51579 


85672 


53066 


84759 


54537 


63810 


55093 


82855 


57 


4 


50101 


86544 


51604 
51638 


85657 
85642 


53091 
53115 


84743 
84728 


54561 
54586 


83804 
83788 


56016 
56040 


83830 


56 


5 


50120 


86530 




55 


6 


50151 


86515 


51653 


85627 


53140 


84712 


54610 


83779 


58064 


88806 


54 


7 


50176 


86501 


51678 


85612 


53164 


84697 


54635 


63758 


56088 


83700 


53 


8 


50201 


88488 


51703 


85507 


53180 


84681 


54659 


63740 


58119 


83773 


58 


9 


50227 


88471 


51728 


855fl2 


53214 


84666 


54683 


83734 


56136 


82757 


51 


10 


50252 


86457 


51753 


85567 


53238 


84650 


54708 


83708 


58160 


83741 


50 


11 


50277 


86442 


51778 


85551 


53263 


84635 


54732 


83699 


56184 


62734 


49 


12 


50302 


86427 


51803 


85536 


532HR 


84619 


54756 


83676 


56906 


82708 


48 


13 


50327 


86413 


51828 


85521 


53312 


84604 


5478J 


83680 


56939 


63893 


47 


14 


50352 


80398 


51853 


85506 


53337 


84588 


54805 


83645 


56956 


83675 


46 


15 


50377 


86384 


51877 


85491 


53361 


84573 


54839 


83639 


50380 


89650 


45 


16 


50403 


86369 


51903 


8547B 


53386 


84557 


54854 


83613 


56305 


89643 


44 


17 


50428 


86354 


51927 


85461 


53411 


84543 


54878 


83507; 


58389 


82696 


43 


A 1 

16 


50453 


mMQ 


51952 


85440 


53435 


84526 


54902 


635811 


56353 


89610 


49 


19 


50478 


86325 


51977 


65431 


53460 


84511 


54937 


83565 


58377 


83503 


41 


90 


50503 


80310 


52003 


85416 


53484 


84495 


54051 


83549 


56401 


8-2577 


40 


21 


50528 


86295 


52026 


85401 


535U0 


84480 


54975 


63533, 


56435 


82561 


30 


22 


50553 


86381 


52051 


85385 


53534 


84464 


54999 


R3517I 


56449 


63544 


36 


23 


50578 


86266 


52076 


85370 


53558 


84443 


55084 


83501 


58473 


62538 


37 


24 


50603 


80251 


53101 


85aS5 


53583 


84433 


55048 
55073 


63485 


1^6497 


69511 


36 


25 


50628 


86237 


53126 


85340 


53607 


84417 


63460 


56591 


83495 


35 


26 


50654 


86232 


53151 


R5,>25 


53632 


84402 


55007 


63453 


56545 


82478 


34 


27 


50579 


86207 


52175 


85310 


53656 


84383 


55131 


83437 


58560 


89463 


33 


28 


50704 


86193 


52300 


85291 


53681 


84370 


55145 


83421 


56503 


83448 


33 


29 


50729 


86178 


52225 


85379 


53705 


84355 


55160 


83405 


56617 


82429 


31 


30 


50754 


86163 


52250 


85264 


537:iO 


84339 


55104 


83380 


56641 


89413 


30 


31 


50779 


86148 


52275 


85249 


53754 


84324 


55316 


83373 


56065 


62386 


20 


32 


50604 


86133 


53299 


85334 


53T79 


84308 


55342 


83356 


58689 


82380 


28 


33 


508-!9 


86119 


52324 


85318 


538(M 


84302 


55966 


63340, 


56713 


82363 


27 


34 


50854 


86104 


52349 


85203 


53828 


84277 


55391 


63324 


58736 


82347 


26 


35 


50879 


86089 


52374 


85188 


53853 


84961 


55315 


83308 


56780 


82330 


95 


36 


50904 


86074 


52399 


85173 


53877 


84345 


55.199 


63393 


56784 


82314 


94 


37 


50929 


86059 


53423 


85157 


53903 


84230 


.'>5363 


H3376; 


56808 


82297 


83 


38 


50954 


86045 


52448 


85142 


53926 


84214 


5.'>3fl8 


63960 


56839 


82381 


99 


39 


50979 


86030 


52473 


85127 


53951 


84198 


55413 


83944, 


58858 


89364 


91 


40 


51004 


86015 


52498 


85112 


53975 


84182 


55436 


83236! 


56860 


82248 


90 


41 


51029 


86000 


52522 


85096 


54O00 


84167 


5.'M60 


83319 


56904 


max 


19 


42 


51054 


85985 


53547 


85081 


54024 


84151 


55484 


831051 


56938 


69314 


18 


43 


51079 


85970 


52572 


85066 


54019 


84135 


55509 


83170 


56053 


82196 


17 


44 


51104 


85056 


53597 


85051 


54073 


84120 


.'15533 


83163 


56976 


82181 


16 


45 


51129 


85941 


52631 


a')035 


54097 


84104 


55557 


83147 


57000 


82165 


15 


46 


51154 


85036 


53646 


85030 


54123 


84088 


55581 


63131 


57094 


82148 


14 


47 


51179 


85911 


52671 


85005 


54146 


84073 


55605 


83115 


57047 


82132 


13 


48 


51304 


85893 


52({93 


84S)89 


54171 


84057 


55630 


830961 


57071 


82115 


19 


49 


51239 


85881 


52720 


84974 


54195 


84041 


55654 


63069: 


57005 


82098 


11 


50 


51254 


85886 


53745 


84059 


54220 


84025 


55678 


63066< 


57119 


89089 


10 


51 


51379 


85851 


.53770 


84943 


54244 


84009 


55703 


63050 


57143 


89065 





52 


51304 


a'i836 


52794 


84938 


54269 


83904 


55796 


83034 


57167 


69048 


8 


53 


51329 


85821 


5-2819 


84913 


54293 


83978 


55750 


83017 


57101 


89039 


7 


54 


51354 


85806 


5-2844 


84897 


54317 


83962 


55775 


63001 


57915 


62015 


6 


55 


51379 


85792 


52869 


84882 


54342 


83946 


55709 


82965 


57336 


81900 


5 


56 


51404 


85777 


53803 


84866 


54366 


83930 


55823 


82969 


57963 


81962 


4 


57 


51429 


85762 


5-2918 


84851 


54391 


83915 


55847 


83953 


57366 


81965 


3 


58 


51454 


85747 


J3943 


84836 


54415 


83899 


5,W71 


H399(3 


57310 


81949 


9 


50 
M 


51479 


85732 


52987 


84820 


54440 


83883 


55895 


83990 


57334 


81032 


1 
M 


N.CS. N.8. 


N.OS. N.S. 


N.CS. 


N.S. 


N.CS 


N.S. 


N.CS. 


^ B. 


59 Deg. 


58 Deg. 


57 Deg. 


56 Deg. 


55 Deg. 1 



J 



NATURAL SINES. 



97 





M 




35 


Deg. 


36 Deg. 


37 Deg. 


38 Deg. 1 39 Deg. 


 M 




N.s: 


N.C8 


N.s. 
58779 


N.CS 


N.S. 


N.CS. 


N.S. 


N.CS 
78801 


. N.S. 


N.CS 




5735ti 


81915 


80902 


6018S 


! 70864 


6156() 


62932 


77713 


60 




1 


57381 


81899 


58802 


80885 


60205 


79846 


61589 


78783 ; 62935 


77096 


59 




2 


57405 


81882 


58826 


80867 


602-28 


79829 


61612 


78763 62977 


77678 


58 




3 


57429 


818(>5 


58849 


80830 


60231 


79811 


61635 


78747 6:«)00 


77660 


57 




4 


57453 


81848 


58873 


80833 


60274 


79793 


61658 


78729 63022 


77041 


56 




5 


57477 


81832 


58896 


80816 


60298 


79776 


61681 


78711 1 63013 


77623 


55 




6 


57501 


81815 


58920 


80799 


60321 


79758 


61704 


78694 63068 


77605 


54 




7 


57524 


81798 


58943 


80782 


60344 


79741 


61728 


78676 1 63090 


7T586 


53 




8 


57548 


81782 


58il67 


80765 


60367 


79723 


61749 


78658 63113 


77568 


52 




9 


57572 


81765 


58990 


80748 


60390 


79706 


61772 


78640 63135 


77550 


51 




10 


57596 


81748 


59014 


80730 


60414 


79638 


61795 


78622 63158 


77531 


50 




11 


57619 


81731 


59037 


80713 


60437 


79671 


61818 


78604 6318;) 


77513 


49 




12 


57643 


81714 


59061 


8iJ696 


60460 


79653 


61841 


7858!J 1 63203 


77494 


48 




13 


57667 


81698 


59084 


80679 


60483 


79&15 


61864 


7^508 1 63225 


T7476 


47 




14 


57691 


81681 


59108 


&nm 


60306 


79618 


61887 


78550 63248 


77458 


46 




15 


57715 


81664 


59131 


8J644 


60529 


79600 


61909 


78532 ,63271 


77439 


45 




16 


57738 


81647 


59154 


8D627 


60553 


79583 


61932 


78514 


63293 


77421 


44 




17 


57762 


81631 


59178 


80610 


60576 


79535 


619.J3 


78496 


63316 


77402 


43 




18 


57780 


81614 


59-201 


80593 


60599 


79547 


61978 


78478 


63338 


77384 


42 




19 


57810 


81.'>»7 


392-25 


80576 


60622 


79530 


6-2001 


784rt0 


63361 


7ri6!i 


41 




20 


57833 


81580 


59-248 


80558 


60l>45 


79512 


62024 


78442 63383 


77347 


40 




21 


57837 


81.563 


59272 


80541 


60668 


79494 


6-2046 


78424 63406 


77329 


39 




22 


57881 


81546 


59295 


80524 


60691 


79477 


6-20r»9 


78405 63428 


77310 


38 




23 


57904 


81530 


59318 


80507 


60714 


79459 


l?2392 


78387 , 63451 


77292 


37 




21 


57928 


81513 


59342 


80489 


60738 


79441 


62113 


78369 ,63473 


772TJ 


36 




25 


57952 


81496 


5J.T65 


80472 


60761 


79424 


62138 


78351 63496 


77255 


35 




26 


57976 


81479 


59.189 


80455 


60T84 


79406 


62160 


78333 63518 


77236 


34 




27 


57999 


81462 


59412 


804:W 


6()8J7 


79.188 


62183 


78315 63540 


77218 


33 




28 


58023 


81445 


59436 


80420 


608:K) 


79.171 


622:Mi 


78297 i 63563 


77199 


32 




29 


58047 


81428 


59459 


80403 


00833 


79333 


622-29 


78279 


63383 


77181 


31 




30 


58070 


81412 


59482 


80386 


60876 


79335 


62231 


78261 


63608 


77162 


30 




31 


58094 


81395 


59506 


80368 


60899 


79318 


i?2274 


78-243 


63630 


77144 


29 




32 


58118 


81378 


5D52D 


80351 


60922 


79301) 


62297 


78-225 


63653 


77125 


28 




33 


58141 


81361 


59532 


803.14 


60945 


70282 


62.120 


78206 


63675 


77107 


27 




34 


58165 


81344 


59.376 


8.)3I6 


60968 


792:>4 


02342 


78188 


63698 


77088 


26 




35 


58189 


81327 


59399 


80299 


60991 


79-247 


62365 


78170 


63720 


77070 


25 




36 


58212 


81310 


.59622 


80282 


61015 


79-229 


02388 


73152 1 6.1742 


77051 


24 




37 


58236 


81293 


59r>4a 


80-2(H 


61038 


79211 


62411 


78134 63765 


77033 


23 




38 


58260 


81276 


59669 


80247 


61061 


79 J 93 


62433 


78116 6.1787 


77014 


22 




39 


58283 


81259 


59693 


80230 


61084 


79176 


62456 


78098 63810 


76996 


21 




40 


58307 


81242 


59716 


80212 


61107 


79158 


6-2479 


78079 638.12 


76977 


20 




41 


58330 


81225 


59TJ9 


80193 


61130 


79140 


62502 


78061 638.>4 


76959 


19 




42 


58354 


81208 


59763 


80178 


61153 


79122 


62324 


78043 638n 


76940 


18 




43 


58378 


81191 


5D786 


80160 


(Una 


79103 


62547 


78025 


6389!) 


?>921 


17 




44 


58401 


81174 


59809 


80143 


61199 


79087 


6-r»70 


78007 


63922 


7H9ai 


16 




45 


58425 


81157 


59832 


80125 


61222 


79069 


62592 


77988 


63944 


76884 


15 




46 


58449 


81140 


59856 


80108 


61245 


79051 


62615 


77970 


63066 


76866 


14 




47 


58472 


81123 


59879 


H;H)91 


61268 


79033 


626.18 


779.32 


63969 


76847 


13 




48 


58496 


81106 


59902 


8;M)73 


61291 


79015 


02660 


77934 


64011 


76828 


12 




49 


58519 


81089 


59926 


8iK)56 


61314 


78998 


62683 


77916 


64033 


76810 


11 




50 


58543 


81(*72 


59949 


80038 


613.17 


78980 


62706 


77897 


64056 


76791 


10 




51 


58567 


81035 


59972 


80021 


613()0 


78982 


62728 


77879 


64078 


76772 


9 




52 


58590 


81038 


59995 


80003 


61.183 


78944 


62751 


77861 


64100 


76754 


6 




53 


58614 


81021 


60019 


79986 


61406 


78926 


62774 


77843 


64123 


767.15 


7 




54 


ss^rr 


81004 


60042 


79908 


614-29 


78908 


6279fi 


77824 


64145 


76717 


6 




55 


58661 


80987 


60065 


79931 


61451 


78891 


02819 


77806 


64167 


76698 


5 




56 


58684 


80970 


60089 


7»934 


61474 


78873 


62842 


77788 


64190 


76679 


4 




57 


58708 


80953 


60112 


79916 


61497 


788.35 


62»;4 


77769 


64212 


76661 


3 




58 


58731 


80936 


60135 


79899 


61.5-20 


7a'^37 


6-2887 


77731 i 


64234 


76642 


2 




59 
M 


58755 


80919 


60158 


79881 


61343 


78819 


62909 


77733 


64256 


76623 


1 
M 




N CS. 


N.S. 


N. OS. 


N.S. 


N. CS. 


N.S. 


N.CS 


N.S. 1 


N CS 


N S. 




54 Deg. 


53 Deg. 1 


52 Deg. 


51 Deg. 


50 Deg. 1 



ellwood's test prob. — 7 



98 



NATURAL SINGS. 





40Deg. 


M 


N.S. 


N.CS. 





6427i» 


76M>4 


1 


64301 


76586 


3 


64323 


76567 


3 


64:M6 


76548 


4 


64368 


76530 


5 


64390 


76511 


6 


64412 


76492 


7 


64435 


76473 


8 


64457 


76455 


9 


64479 


76436 


10 


64501 


76417 


IJ 


64524 


76398 


12 


64546 


76380 


13 


64568 


76361 


14 


64590 


76342 


15 


64612 


76323 


16 


64635 


76304 


17 


64657 


76286 


18 


64679 


76267 


19 


64701 


76248 


2U 


64723 


76229 


21 


64740 


76210 


32 


64768 


76192 


33 


64790 


76irj 


24 


64812 


76154 


25 


64834 


761 35 


26 


64856 


76116 


37 


64878 


76097 


38 


649U1 


76078 


29 


64923 


76059 


30 


C4945 


76041 


31 


64967 


76022 


32 


64989 


76003 


33 


65011 


75984 


34 


65033 


75935 


35 


65055 


75946 


36 


«5077 
65099 


75927 


37 


75908 


38 


65122 


75889 


39 


65144 


75870 


40 


65166 


75851 


41 


65188 


75832 


43 


65210 


75813 


43 


65232 


75794 


44 


65254 


75775 


45 


65276 


75756 


46 


65398 


75T38 


47 


65320 


75719 


48 


65342 


75699 


49 


65364 


75680 


50 


65386 


75661 


51 


65108 


75642 


53 


65430 


75623 


53 


65452 


75604 


54 


65474 


75585 


55 


65496 


7556l> 


56 


65518 


75547 


57 


65540 


75.528 


58 


6')562 


75509 


50 


65584 


75490 


60 


65606 


75471 

N.S. 


M 


N.CS. 




49 1 


>cg. 1 



41 Deg. 



M.S. 1 


N.CS. 


656U6 


75471 


65628 


7545« 


65650 


75433 


65672 


75414 


65694 


75395 


65716 


75375 


65738 


75356 


65759 


75337 


65781 


75318 


65803 


75299 


65825 


75280 


65847 


75261 


65869 


75241 


65891 


75222 


6a013 


75203 


65035 


75184 


65956 


75165 


65978 


75146 


65000 


75126 


66022 


75107 


66044 


75088 


66066 


75069 


66088 


75050 


66109 


75030 


66131 


75011 


66153 


74992 


66175 


74973 


66197 


74953 


66218 


74934 


66240 


74915 


66363 


74896 


06384 


74876 


66306 


74857 


66327 


74838 


mm 


74818 


66371 


74799 


66393 


74780 


66414 


74760 


66436 


74741 


66458 


74722 


66480 


74703 


66501 


74683 


66523 


74664 


66545 


74644 


66566 


74625 


66588 


74606 


66610 


74586 


66632 


74567 


66653 


74.M8 


66675 


74528 


66697 


74509 


66718 


74489 


66740 


74470 


66762 


74451 


66783 


74431 


66805 


74412 


66827 


74392 


66848 


74373 


66870 


743,53 


66891 


74334 


66913 


74314 


N.CS. 


N.S. 



48 Dei;. 



42 Deg. 



N.S. 



66913 
66935 
66956 
615978 
6>99i) 
67021 
67043 
67064 
67086 
07107 
67129 
67151 
67172 
67194 
67215 
67237 

67258 
67280 
67301 
67323 
67344 
67366 
67387 
67409 
67430 
07453 
67473 
67495 
67516 
67538 
67559 

67580 
67602 
67633 
67645 
67666 
67688 
67709 
67TJ0 
677.53 
67773 
67795 
67816 
67837 
67859 
67880 

67901 
67923 
67944 
67965 
67987 
68008 
68029 
68051 

mm 

68093 
68115 
68136 
68157 
68179 
08200 



N.CS. 



74314 
74295 
74276 
74256 
74237 
74217 
741il8 
74178 
74159 
74139 
74130 
74100 
74080 
74061 
74J41 
74032 

74002 
73083 
73933 
73944 
73934 
73904 
73885 
73865 
73846 
73826 
73806 
73787 
73767 
73747 
73728 

73708 
7:i683 
736(;9 
73649 
73029 
73610 
73590 
73570 
73551 
73531 
73511 
73491 
73475 
73452 
73432 

73413 
73393 
73373 
73353 
73333 
73314 
73294 
73274 
732.54 
73234 
73215 
73195 
73175 
731.55 
73135 



N.CS. N.S 
47 Deg. 



43 Deg. 



N.S. 



682U0 
68221 
68342 
68264 
68285 
68306 
68327 
68349 
68370 
C'8391 
68412 
68433 
68455 
68476 
68497 
68518 

68539 
68561 
68582 
G8603 
68G34 
68645 
68666 
68688 
68709 
68730 
68751 
68772 
68793 
68814 
68835 

68857 
08878 
68899 
68920 
68941 
68962 
68983 
69004 
69t)25 
69046 
69067 
69088 
69109 
69130 
69151 

69172 
69193 
69214 
69235 
69256 
69277 
69298 
69319 
69340 
69361 
69382 
69403 
69434 
(i944.i 
69406 



N.CS. 



73135 
73116 
73096 
73«.76 
73056 
73U36 
73U16 
72996 
72976 
729.57 
72937 
72917 
728i«7 
72877 
72857 
72837 

72817 
72797 
72777 
72757 
72737 
72717 
72697 
72677 
72657 
7-2637 
72617 
72597 
725'. 7 
72557 
72537 

72517 
72497 
72477 
72457 
72437 
72417 
72397 
72377 
72357 
72337 
72317 
72297 
72277 
72257 
72236 

72216 
72196 
72176 
72156 
72136 
72116 
72095 
72075 
790.55 
72035 
79015 
71995 
71974 
71954 
71934 



N.CS N.S 
46 Deg. 



44 Deg. 



N.S. N.CS. M 



69466 
69487 
69508 
69529 
69549 
69570 
69591 
69612 
696.33 
6<)654 
69675 
69696 
6'J717 
69737 
69758 
69779 

G9809 
698-21 
69842 
69862 
69883 
69904 
69925 
69946 
69966 
69987 
70008 
70029 
70049 
70070 
70091 

70112 
70132 
70153 
70174 
70195 
70215 
7023(*) 
70257 
70277 
70298 
70319 
70339 
703(i0 
70381 
70401 

70423 
70443 
70463 
70484 
70505 
70525 
70546 
7a567 
70587 
70608 
70698 
70649 
70670 
70690 
70711 



71934 
71914 
71894 
71873 
71853 
71833 
71813 
71792 
71772 
71752 
71732 
71711 
71691 
71671 
71650 
71630 

71610 
71590 
71569 
71549 
71529 
71508 
71488 
71468 
71447 
71427 
71407 
71386 
71366 
71345 
71325 

71305 
71284 
71264 
71243 
71223 
71203 
71182 
71163 
71141 
71121 
I 71 100 
71080 
71059 
710.''9 
71019 

70908 
70978 
70957 
70937 
70i>16 
70896 
70875 
70855 
70834 
70813 
70793 
70772 
70752 
70731 
70711 



N CS.| N.S. 
45 Deg. 



60 
59 
58 
57 
56 
55 
54 
53 
52 
51 
50 
49 
48 
47 
46 
45 

44 
43 
43 
41 
40 
39 
38 
37 
36 
35 
34 
33 
33 
31 
30 

39 
38 
37 
36 
25 
34 
23 
33 
31 
30 
19 
18 
17 
16 
15 

14 

13 

IS 

11 

10 

9 

8 

7 

6 

5 

4 

3 

3 

] 





M 



NATURAL TANOEirrS. 



99 





Degrees. | 


1 Degree. | 2 Degrees. ] 


3 Degrees. 


M 
60 


M 


N.T.ID, 


N. Cot. 


N. Tan. 


N.Oot. 


N. Tho. 


N. U«)t 


N.Tan. N.Oot. 





ooouo 


OUOO.OU 


01746 


57.3900 


03493 


38.H3<» 


05341 19.0811 


1 


00039 


3437.75 


01775 


56.3506 


03531 


38.3991 


05370 


18.9755 


59 


s 


00058 


1718.87 


01804 


55.4415 


0.3550 


38.1604 


05399 


18.8711 


58 


3 


00387 


1145.92 


01833 


54.5613 


03579 


37.9373 


0533H 


1»».7C7j? 


57 


4 


OOllG 


839. 43() 


01863 


53.7086 


03609 


37.7117 


05357 


18.6656 


S6 


5 


00145 


687.549 


01891 


53.8831 


03638 


37.48911 


05387 


18.5645 


55 


6 


00175 


573.957 


01930 


53.0807 


03667 


37.3715 


05416 


18.4645 


54 


7 


00334 


491.106 


01949 


51.3033 


03696 


37.0566 


05445 


18.3655 


5:) 


8 


03333 


439.718 


01978 


50.5485 


03735 


36.8450 


05474 


18.2677 


53 


9 


00263 


381.971 


03007 


49.8157 


03754 


36.6367 


05503 


18.1706 


51 


10 


003U1 


343.774 


03336 


49.1039 


03783 


36.4316 


a5533 


18.0750 


50 


11 


00330 


313.521 


030136 


48.4131 


03813 


36.339ii 


05563 


17.9801 


49 


13 


00349 


386.478 


03095 


47.7395 


03843 


36.0307 


05591 


17.8863 


48 


13 


00378 


364.441 


03134 


47.0853 


03871 


35.8348 


05630 


17.7934 


47 


14 


00407 


345.553 


03153 


46.448J 


03930 


35.6418 


05649 


17.7015 


46 


IS 


00436 


339.182 


03183 


45.8394 


03939 


35.4517 


05678 


17.6106 


45 


Id 


00465 


314.858 


0»11 


45.3961 


03958 


25.9644 


05706 


17.5305 


44 


17 


00495 


303.319 


03340 


J4.6386 


03987 


35.0798 


05737 


17.4314 


43 


18 


005:^ 


193.984 


03369 


44.0661 


04016 


34.8978 


05766 


17.3432 


43 


19 


00553 


180.933 


03398 


43.5081 


04040 


34.7185 


05795 


17.3558 


41 


90 


005»3 


171.885 


03338 


43.0641 


04075 


34.5418 


05334 


17.1693 


40 


31 


00611 


163.700 


03357 


43.4335 


04104 


34.3675 


05854 


17.0837 


39 


» 


00640 


156.359 


03386 


41 9158 


04i:{3 


34.1957 


05833 


16.9990 


38 


33 


00669 


149.465 


03415 


41.410G 


04163 


34.0363 


05913 


16.9150 


37 


»l 


00698 


143.337 


03444 


40.9174 


04191 


33.8593 


05941 


16.8319 


36 


2S 


00737 


137.507 


03473 


40.4358 


04320 


33.694.) 


05970 


16.749(3 


35 


36 


00756 


133.319 


03503 


39.9655 


04250 


33.5)31 


05939 


16.6U81 


34 


37 


00785 


137.331 


03531 


39.5053 


04379 


33.3718 


0()02J 


16.5874 


33 


38 


00814 


133.774 


03530 


39.0568 


04308 


33.3137 


06058 


16.5075 


32 


80 


00844 


118.540 


03383 


38.6177 


04337 


33.0577 


06037 


16.4383 


31 


30 


00873 


114.589 


03619 


38.1885 


04306 


33.9J37 


06116 


16.3499 


30 


31 


00902 


110.893 


09648 


37.7686 


04395 


23.7518 


06145 


16.3733 


39 


33 


00931 


107.43') 


03677 


37.3579 


04414 


33.6020 


06175 


16.1953 


38 


33 


00960 


104.171 


03700 


36.9560 


01454 


23.4541 


00334 


16.1190 


37 


34 


00089 


101.107 


03735 


36.5337 


0148:) 


33.3081 


06333 


16.0435 


36 


35 


01018 


98.3179 


03764 


36.1776 


(M513 


32.1640 


063(32 


15.9087 


25 


36 


01047 


93.4895 


03793 


35.8006 


04.>11 


32.0217 


063i)l 


15.8945 


34 


37 


01076 


93.9J6'> 


0383J 


35.4313 


04570 


31.8dl3 


OSXil 


15.8211 


23 


38 


01105 


90.4633 


00851 


35.0695 


04599 


31.7426 


06350 


15.7483 


23 


39 


01135 


88.1433 


03881 


34.7151 


04,338 


31.6056 


06379 


15.6762 


31 


40 


01164 


83.9398 


03910 


34.31)78 


04(358 


31.4704 


06408 


15.604d 


30 


41 


01193 


83.8135 


03J39 


34.0373 


04687 


31.3369 


06437 


15.5343 


19 


43 


01333 


81.8470 


03938 


33.6935 


04716 


31.3349 


06467 


15.4638 


18 


43 


01351 


79.9434 


03937 


3J.3GG2 


04745 


31.0747 


Or>493 


15.3943 


n 


44 


0138J 


78.1363 


03:)3n 


33.0452 


04774 


33.9460 


06525 


15.3254 


16 


45 


01309 


76.3900 


03055 


3J.7303 


04803 


90.8188 


06554 


15.3571 


15 


46 


01338 


74.7393 


03084 


33.4313 


04333 


30.6033 


06584 


15.1893 


14 


47 


01367 


73.1390 


03114 


33.1181 


04863 


30.5691 


06313 


15.1332 


13 


48 


01396 


71.6151 


03143 


31.83(15 


04891 


30.4465 


06043 


15.0557 


13 


49 


01435 


70.1533 


03173 


31.5384 


U4930 


33.3353 


06671 


14.i)898 


11 


50 


01455 


68.7501 


03301 


31.3416 


04949 


30.3058 


06700 


14.9244 


10 


51 


01484 


67.4019 


03330 


33.959a 


04978 


30.0873 


06730 


14.859<) 


9 


53 


01513 


63.1055 


03359 


33.(i833 


0.-i037 


19.9732 


06759 


14.7951 


8 


53 


01543 


&t.8580 


03388 


33.4L:(i 


05037 


19.8546 


06788 


14.7317 


7 


54 


01571 


63.6567 


03317 


30. 1440 


05066 


19.7403 


06817 


14.6685 


6 


55 


01600 


63.4993 


03346 


39.8823 


05095 


19.6373 


06847 


14.6059 


5 


56 


01639 


61.3839 


03376 


39.6345 


05134 


19.51.>n 


0r»876 


14.5438 


4 


57 


0165H 


60.3058 


03405 


39.3711 


05153 


19.4051 


(K>!K)5 


14.4823 


3 


58 


01687 


5».965!l 


03434 


39.1330 


05183 


19.3959 


06934 


14.4313 


3 


50 


01716 


58.3613 


03463 


38.8771 


05313 


19.1S79 


06963 


14.31)07 


1 


60 
M 


01746 


57.3900 


03493 


38.6363 


05341 


19.0811 


06093 


14.3007 




M 


N.CoC. 


N.Tan. 


N.Cot 


N. Tan. 


N. OoL 


N. Tan. 


N.Cot 


N. Tan. 


89 Degrees. | 


88 Degrees. { 


87 Degrees. I 


8G Degrees. 







M 


JJ, L M 












31 SB 




m 
e K 


s; 






u 










a II 






11 


ss 






so 












48 












40 












44 




a n 


43 


18 


" ": 


41 


19 








t t 


40 






3> 




1 « 


38 


33 






H 












9B 




31 


tJ 




33 


aa 












30 


1 i 


30 


31 


« 'S 






'B 


sa 


33 




n 


34 




at 


3S 




as 


at 




14 






33 


38 




31 


3» 






4a 




10 








43 




IS 


43 












43 






4B 


a 


14 


47 
















11 






10 


SI 






S3 








> 






d ii 




S 


IS 










in 






98 




3 


S9 






80 












H 


1 


M 







NATURAL TANGENrS. 



101 



M 




8 Degrees, j 


9 Degrees. | 


10 Degrees. | 


11 Degrees. 


M 

60 


N.Tan. 


N. Cot 


N. Ta \. 


N. Cot. 


N. Tiin. 


N. Cot. 


N. Tan. 


N. Col. 


14054 


7.11537 


L'-dSS 


6.31375 


1763:i 


5.6713rt 


19438 


5.14455 


1 


14084 


7.10038 


158ti8 


6.30189 


17663 


5.(i6165 


liH68 


5.13658 


59 


3 


14113 


7.0854fi 


15898 


6.S9U07 


17693 


5.65205 


19498 


5.12862 


58 


3 


14143 


7.07U59 


15938 


6.37829 


1772.1 


5.6434B 


19529 


5.12069 


57 


4 


14173 


7.05579 


15958 


6.36655 


17753 


5.63-295 


19559 


5.11279 


56 


5 


14303 


7.04105 


15988 


0.2548H 


17783 


5.62344 


19589 


5.10490 


55 


6 


143:i3 


7.03637 


icon 


6.21321 


17813 


5.61397 


10)19 


5.09704 


54 


7 


14363 


7.01174 


16)47 


6.23160 


17843 


5.60452 


19049 


5.06921 


53 


8 


14391 


6.99718 


16077 


6.22.)0:^ 


17873 


5.59511 


19680 


5.08139 


52 


9 


14331 


6.9j3!)8 


16107 


6.20851 


179J3 


5.53573 


19710 


5.07360 


51 


10 


14351 


6.9()8J3 


16137 


6.19703 


17933 


5.57638 


19740 


5.06584 


50 


11 


14381 


6.95385 


16167 


6.18559 


17963 


5.5(»706 


19770 


5.05809 


49 


13 


14410 


6.939.J2 


16196 


6.17419 


17993 


5.55777 


19801 


5.05037 


48 


13 


14440 


6.935-25 


16226 


6.16383 


18;>23 


5.54851 


19831 


5.04-267 


47 


14 


14470 


6.91104 


16256 


6.15151 


18053 


5.53927 


19881 


5.03499 


46 


15 


14499 


6.89688 


16286 


6.14033 


18083 


5.53007 


19891 


5.02734 


45 


16 


14539 


6.88378 


16316 


6.13899 


16113 


5.53090 


19921 


5.01971 


44 


17 


14550 


6.86874 


16346 


6.11779 


18143 


5.51176 


19952 


5.01210 


43 


18 


14588 


6.85175 


16376 


6.10664 


18173 


5.50264 


19J82 


5.00451 


42 


19 


14018 


6.8408-2 


16405 


6.09552 


18203 


5.49:156 


3(M)12 


4.99695 


41 


30 


14648 


6.83394 


16435 


6.03444 


18233 


5.4815J 


2U042 


4.98940 


40 


31 


14678 


6.81312 


16465 


6.07340 


18263 


5.47548 


20073 


4.93188 


39 


33 


14707 


6.79938 


16495 


6.03340 


18293 


5.40648 


20103 


4.97438 


38 


33 


14737 


6.78364 


16525 


6.05143 


18323 


5.45751 


30133 


4.98690 


37 


34 


14767 


6.77199 


16555 


6.04051 


18353 


5.44857 


30104 


4.05945 


36 


35 


14796 


6.75838 


16585 


6.02902 


18383 


5.4:J90() 


30194 


4.95-201 


35 


36 


14836 


6.74483 


16615 


6.0l67o' 


18414 


5.43077 


20224 


4.94460 


34 


37 


14856 


6.73133 


16645 


6.00797 


18144 


5.42192 


30-254 


4.93721 


33 


38 


14886 


6.7178J 


16674 


5.99730 


18174 


5.41309 


33-285 


4.9-2984 


32 


39 


14915 


6.70450 


16704 


5.93646 


18504 


5.40429 


20315 


4.92-249 


31 


30 


14945 


6.69116 


16734 


5.97576 


18534 


5.39552 


20345 


4.91516 


30 


31 


14975 


6.67T87 


16764 


5.96510 


18564 


5.38677 


30376 


4.90785 


29 


33 


151)05 


6.06463 


16794 


5.95448 


18594 


5.37805 


3)4v)6 


4.90056 


28 


33 


15934 


6.65144 


16^4 


5.943J0 


18634 


5.3393i 


3.>4:)6 


4.89330 


37 


34 


15J61 


6.63831 


16854 


5.933J5 


18654 


5.38070 


30406 


4.88605 


38 


35 


15091 


6.63533 


16884 


5.92383 


18684 


5.35206 


20497 


4.878^ 


35 


36 


15134 


6.61319 


16914 


5.91335 


18714 


5.34345 


30527 


4.87162 


34 


37 


15153 


6.50931 


16944 


5.9J191 


18745 


5.33487 


30557 


4.86444 


33 


38 


15183 


6.58037 


16974 


5.89151 


18775 


5.33631 


30588 


4.85737 


32 


39 


15313 


6.57339 


17004 


5.88114 


18835 


5.31778 


30618 


4.85013 


21 


40 


15343 


6.56055 


17033 


5.87080 


183.15 


5.30928 


30648 


4.84300 


30 


41 


15373 


6.54777 


17063 


5.8(»51 


18865 


5.30080 


30679 


4.83590 


19 


43 


15303 


6.53503 


17093 


5.85034 


18895 


5.39235 


30709 


4.82882 


18 


43 


15333 


6.5S034 


17133 


5.84001 


18925 


5.383i)3 


30739 


4.82175 


17 


44 


15303 


6.50970 


17153 


5.82962 


li9.55 


5.37553 


20770 


4.81471 


16 


45 


15391 


6.49710 


17183 


5.81966 


18986 


5.36715 


30800 


4.80769 


15 


46 


15481 


6.48456 


17313 


5.80053 


19016 


5.35880 


30R30 


4.80068 


14 


47 


15451 


6.47336 


17343 


5.79944 


19046 


5.35048 


30881 


4.79370 


13 


48 


15181 


6.45961 


17373 


5.78938 


19076 


5.34318 


20891 


4.78673 


12 


49 


15511 


6.44730 


17303 


5.T7936 


19106 


5.33391 


20921 


4.77978 


11 


50 


15540 


6.43184 


17333 


5.76937 


19136 


5.33566 


30953 


4.7728G 


10 


51 


15570 


6.43353 


17363 


5.75941 


19166 


5.21744 


30982 


4.76595 


9 


53 


15600 


6.41036 


17393 


5.74949 


19197 


5.30925 


21013 


4.75900 


8 


53 


15630 


6.39304 


17433 


5.73960 


19327 


5.30107 


31043 


4.75219 


7 


54 


15)60 


6.38587 


n4S3 


5.73974 


19257 


5.19293 


31073 


4.74534 


6 


55 


15089 


6.37374 


17483 


5.71992 


1«»7 


5.18480 


31104 


4.73851 


5 


56 


15719 


6.36]6.> 


17513 


5.71013 


19317 


5.17671 


21134 


4.73170 


4 


57 


15749 


6.349<*)1 


17543 


5.70037 


19.347 


5.16863 


21164 


4.73490 


3 


58 


15779 


6.33761 


17573 


5.69064 


19378 


5.16058 


31195 


4.71813 


2 


50 


15809 


6.3'25ai; 


171:03 


5.680!M 


19408 


5.15256 


21225 


4.71137 


1 


60 
M 


15838 


6.31375 


17633 


5.671-28 


19433 


5.14455 


31256 


4.70463 



M 


N. Cot. 


N. Tan. 


N. Cot. 


N. Tan. 


N. Cot 


N. Tan. 


S. Col. 


N. Tan. 


81 Degrees. 


80 Degrees. 


79 Degrees. 1 


78 Degrees. 





12 Degrees. 


13 Degrees. 


14 Degree). 


ISDepa^l II 


 


N.IW 


n.a,t. 




N.Coi. 


N ■IVi, N 




N.-r^^ 


_2?llJL 












liSitTT 




3S711S 


7390S. Will 












34W14 1 


OOSBS 


WBU 




58 










4.3!»01 






SBesJ 


73331 


38 






*'.BMyii 






asiH« 3 


»SOi 


S6B8S 












mail 


4!9aM. 




mw 


30930 


7147« 






!1«M 




ana 


4.3U'J!IJ 


351101 S 


IMIIU3 




11040 








i.e^isa 


SK71 


4, MIT*! 


35118 3 




nam 


79810 












4.39159 


45HB 3 




S7913 


70188 
















BT13S 


91944 


»1SI 


S3 




xlo» 








MS" : 


geesi 


nOTII 








iiseo 










9ai«5 










2)390 






4^380]] 




99880 




1848: 


4B 






i'.«iai» 


3J15S 












48 














M7ia 


31301 


(1038 














i539« 3 


943^ 


?re33 








aiTia 






4.I4eSj 


3S39T 3 


037M 


?JS03 








«1743 


i.w«tr 


33SS 


4.94I3£ 


354% 3 


g3i7i 


31394 3 


I83M 














35439 3 


l3nQ 


n33« 3 






is 






^4 


4:mw 


3S490 3 






tH38 










23ino 






^839 


37388 
















355M 3 


91394 


31419 


H705 






sieo; 










90890 


«74SI 








3lift» 


i'.vmi 






3WH 3 


•9417 


31483 


08H 








4.IIMJ» 




lisie^ 








0461 






llSm 








gM!7e 3 


3UT 








39 






S38H 




|57m ! 










X 






33885 








sieoi 






!7 












mee 


snas 










t'.SBIi 




4.rrjw 




arwi 


31970 


tl40i 














HS31 3 


S7136 


«7T0I 


OOMR 


 




WIM 






4^l«530 


33883 3 




3773S 


0058) 












4.1S9B7 


S.W93 ; 




3T764 








a^ai 


4.4»tJ 








85744 




S»77i 




33 






W[nn 






S.1S8 




S937I 








lllSnOO 






35889 3 






HM 






«u^ 


4.4:9m 


















»153 


4.47374 


14I93 


4! 13350 


3SU4S 3 












stita 






4.IS15 






am 


sm 












4.iani 


38119 3 




B7981 


5TI57 






»MJ4 












BBOIS 


Man 
















sum 


MS37I 












BB303 3 




fsan 


50151 


19 


« 


>^30 


4:43™ 


Si 


4!o',Vi!l9 


SliSU 3 
aOSffil 3 

a:3u7 3 


?s 






JS 






4!411»3r. 






90338 3 


19837 


ssata 


Msr. 






«638 




34M1 






iKm 




Mm 






istm 


4.4(nir. 




4!inii39 




■M931 








4§ 


















13 




nrw 


4!3IUM 


U5t3 


4!lMfllG 


aMii 3 


TSMD 




owi 






nrai 




StSH 


4.oaio7 




775M 






19 




e0<ii 


liietai 


SICKS 


4.«U99 














!a84S 


4.3nM 


S4r«l 


4.05OTO 


3III4R S 




38433 


llrtfl 














(nr>:T 3 




sU** 








49903 








SmuiS 3 
















84778 




inciii 3 






soeon 




sn 


SSVA 








emni 3 






3031,^ 




31 


■s 


4!3«<7D 
4.343flll 


l£'l 


4!035}4 




7wls 


a85W 


4968- 


I 




aose 


4.3I^n 


nvs 






7*M4 


3SG43 










4.33148 


swo 


iloims 


38793 3 


rj»5 




48741 








Nrrinr 






N, Coi. N 

75 Deer 


eT; 






" 


77D 


eree.. 


74 U gr^cs. 



NATURAL TANSBNTS. 



103 



M 




16 Degrees. | 


17 Degrees. 


18 Degrees. | 


19 Degrees. 


M 
60 


N.Tan. 


W. Cot. 


N. Ta 1. 


S.Coi. 


N. Tan. 


N. Cot. 


\. Tan. 


N. Coi. 


28675 


3. 4874 J 


30i73 


3.27U85 


324J2 


3.077df 


344:)3 


2.90421 


1 


28706 


3.4&359 


3.M)U5 


3.26745 


3-2524 


3.07464 


3(465 


2.90147 


59 


2 


28738 


3.47977 


33837 


3.28408 


32556 


3.07180 


34498 


3. 0^873 


58 


3 


2876J 


3.4759ti 


30J69 


3.28067 


32588 


3.06857 


34530 


2.89600 


57 


4 


28800 


3.47216 


30700 


3.25729 


3-2621 


3.06654 


34563 


2.89327 


56 


5 


288:^^ 


3.48637 


30732 


3.25392 


32653 


3.08252 


34596 


2.89055 


55 


6 


28884 


3.4<H5d 


30764 


3.25055 


3-2685 


3.0595U 


31628 


2.08783 


54 


7 


28895 


3. 46080 


3079<> 


3.24719 


32717 


3.05649 


34661 


2.88511 


53 


8 


28927 


3.45703 


30828 


3.24383 


32749 


3.05349 


34693 


2.86240 


52 


9 


28i)58 


3.45327 


30880 


3.24049 


32762 


3.05049 


34726 


2.87970 


51 


10 


289JU 


3.44951 


30891 


3.2:1714 


32814 


3.04749 


34758 


2.87700 


50 


11 


2S021 


3.44576 


30923 


3.-23381 


^£2846 


3.04450 


34791 


2.87430 


49 


12 


29033 


3.44202 


30955 


3.23048 


32878 


3.04152 


34824 


2.87161 


46 


13 


29384 


3.43829 


30987 


3.22715 


32911 


3.03854 


34856 


2.86892 


47 


14 


291 iti 


3.43456 


31019 


3.2-2:W4 


32943 


3.03558 


34889 


2.8(i624 


4u 


15 


29147 


3.43084 


31051 


3.2-2053 


32975 


3.03260 


34922 


2.88358 


45 


M) 


3^179 


3.49713 


31083 


3.21722 


33007 


3.02963 


34954 


2.86069 


44 


17 


29210 


3.f2313 


31115 


3.21392 


33040 


3.02667 


34987 


2.85822 


43 


18 


29212 


3.41973 


31147 


3.21063 


:O072 


3.02372 


35019 


2.85555 


42 


19 


29274 


3.41604 


31178 


3.23734 


33104 


3.02077 


35052 


2.S.5289 


41 


30 


29J05 


3.412)6 


31210 


3.20406 


33i:i8 


3.01783 


3.7085 


2.85023 


40 


31 


29:i37 


3.408)9 


31242 


3.23079 


33189 


3.01489 


35117 


2.84758 


39 


32 


29368 


3.4U50-2 


31274 


3.1975-2 


:«-2ni 


3.0ll9(f 


35150 


2.84494 


38 


23 


29400 


3.40136 


31308 


3.19426 


.33233 


3.00903 


35183 


2.84229 


37 


24 


29132 


3.39771 


313:18 


3.19100 


3;«!i6 


3.00611 


35218 


2.83965 


36 


25 


39463 


3.39406 


31370 


3.18775 


33298 


3.00319 


35248 


2.83702 


35 


26 


29495 


3.39J42 


31403 


3. 18451 


33330 


3.00028 


35281 


2.83439 


34 


27 


29528 


3.38379 


31434 


3. 18127 


333t>3 


2.99738 


35314 


2.63176 


33 


38 


29558 


3.38317 


31466 


3.17834 


3:»95 


2.99447 


35346 


2.82914 


32 


29 


29590 


3.37955 


31498 


3.17481 


3:M27 


3.99158 


35370 


2.82653 


31 


30 


29621 


3.37594 


3153J 


3.17159 


33460 


3.98868 


35412 


2.R2391 


30 


31 


39653 


3.37234 


31562 


3.18838 


33492 


3.98560 


35445 


3.63130 


3i) 


32 


29885 


3.36875 


31594 


3.16)17 


33524 


3.98292 


354T7 


3.81870 


38 


33 


29716 


3.36516 


31826 


3.18197 


33557 


2.98004 


35510 


3.81610 


37 


34 


29748 


3.36158 


31658 


3.15877 


33589 


2.97717 


35543 


3.81350 


38 


35 


29780 


3.35800 


31690 


3.15558 


33621 


2.97430 


35576 


2.81091 


35 


36 


29611 


3.35443 


32722 


3.15240 


33654 


2.97144 


35608 


3.60833 


24 


37 


29843 


3.35087 


31754 


3.14922 


3388«} 


2.96858 


35641 


3.80574 


23 


38 


29875 


3.34732 


31786 


3.14605 


33718 


2.98573 


35(i74 


3.60318 


22 


39 


29936 


3.34377 


31818 


3.14288 


33751 


2.96288 


35707 


2.60059 


21 


40 


20938 


3.34023 


31850 


3.13972 


33783 


2.98004 


35740 


2.79803 


30 


41 


29070 


3.33670 


31882 


3.13856 


33816 


3.95720 


35772 


2.79S4S 


19 


43 


30001 


3.33317 


31014 


3.LT341 


33848 


2.95437 


35805 


2.79289 


18 


4;) 


30J33 


3.32905 


31946 


3.13027 


33881 


2.95155 


35838 


2.79033 


17 


44 


30J65 


3.32614 


31978 


3.12713 


33913 


2.?4872 


3.W71 


2.76778 


16 


45 


30097 


3.33264 


32010 


3.12400 


33945 


3.04590 


35904 


2.7a'>23 


15 


46 


30138 


3.31914 


32042 


3.12087 


33978 


3.94309 


35937 


2.76369 


14 


47 


30180 


3.31565 


32074 


3.11775 


34010 


2.94(F28 


35»fi9 


2.76C14 


13 


48 


30192 


3.31216 


.32106 


3.11484 


34043 


2.93748 


'Mnhti 


2.77761 


13 


; 49 


302-24 


3.30668 


32139 


3.11153 


34075 


2.931GH 


36035 


2.77507 


11 


50 


3i)255 


3.30321 


32171 


3.10842 


34108 


2.931p9 


36068 


2.77254 


10 


51 


30287 


3.30174 


32203 


3.10532 


34140 


2.92910 


38101 


2.77002 


9 


52 


30319 


3.29629 


32235 


3.10223 


34173 


2. 9^28.32 


38134 


2.76750 


8 


53 


3^351 


3.294^3 


32267 


3.09914 


34205 


2.92354 


36167 


2.76498 


7 


54 


30382 


3.2()130 


32299 


3.09606 


342,38 


2.92076 


36199 


2.76247 


6 


55 


30414 


3.28795 


3-2331 


3.09298 


34270 


2.91799 


imam 


2.75996 


5 


56 


30446 


3.28452 


32363 


3.08991 


34303 


2.91523 


36265 


2.75746 


4 


57 


30478 


3.28109 


32396 


3.08685 


34335 


2.91246 


3<>298 


3.75496 


3 


58 


30509 


3.27767 


32428 


3.08379 


34368 


2.90(J71 


36331 


2.75246 


3 


50 


30541 


3.27426 


32460 


3.08073 


34400 


2.90698! 


3(»tVI 


2.74997 


1 


60 
M 


30573 


3.27085 


32492 


3.07768 


34433. 2.90421 


2&:m 


2.74748 



M 


N. C.»t. 


N. Tan. 


N. Cot. 


N. Tan. 


N.Oot. N.Tan. 


S. Cot. 


N. Tan. 


73 Degrees. 


72 D^ 


Bgrees. 


71 Degrees. 


70 Degrees. 



104 



NATURAL TANGENTS. 



u 

u 


20 Decries. | 


21 Degrees. 


22 Degreea 


23 Degrees. 


M 

60 


N.Tan. 


N. Cot. 


N. Tan 


N.Coi. 
2.60509 


N.Tan. 


N. Cot. 


N.Tan. 


N.Cot. 


36397 


2.74748 


38386 


40403 


2.47509 


43447 


3.35SB5 


1 


36430 


2.74499 


38430 


2.60283 


40436 


2.47302 


42482 


3.35395 


59 


2 


36463 


2.74251 


38453 


2.60057 


4J»470 


2.47095 


43516 


3.35205 


58 


3 


36496 


2.74064 


38487 


2.59831 


40504 


2.46888 


43551 


2.35015 


57 


4 


36529 


2.73756 


38530 


2.59606 


40538 


2.46682 


43585 


2.34825 


56 


5 


36563 


2.73509 


38553 


2.59381 


40572 


2.46476 


42619 


2.:)4636 


55 


6 


36595 


2.73263 


38587 


2.59156 


40606 


2.46270 


42654 


2.34447 


54 


7 


36628 


2.73017 


38620 


2.58932 


40640 


2.46065 


42688 


2.34258 


53 


8 


36661 


2.72771 


38654 


2.58708 


40674 


2.45860 


42722 


2.34069 


53 


9 


36694 


2.72526 


38687 


2.58484 


40707 


2.45655 


42757 


2.33881 


51 


10 


36727 


2.72281 


38721 


2.58261 


40741 


2.45451 


42791 


2.33693 


50 


]] 


36769 


2.72036 


38754 


2.58038 


40r/5 


2.45246 


42826 


2.33505 


49 


12 


36793 


2.71792 


38787 


3.57815 


40809 


2.45043 


42860 


2.3.1317 


48 


13 


36828 


2.71548 


38H21 


2.57593 


40843 


2.44839 


42894 


2.33130 


47 


14 


36859 


2.71305 


388M 


2.57371 


40877 


2.44636 


42929 


2.32943 


46 


15 


36892 


2.71062 


38888 


2.57150 


40911 


2.44433 


42963 


2.32756 


45 


16 


36925 


2.70819 


38921 


2.56938 


40945 


2.44230 


42998 


2.32570 


44 


17 


36958 


2.70577 


38955 


2.56707 


4G979 


2.44027 


4.1032 


3.333ai 


43 


18 


36991 


2.70335 


38988 


2.56487 


41013 


2.4.3825 


43007 


3.32197 


43 


19 


37024 


2.70094 


39022 


2.56266 


41047 


2.43623 


43101 


3.33012 


41 


20 


37057 


2.69853 


39055 


2.56046 


41081 


2.43422 


43136 


2.31826 


40 


21 


37090 


2.69612 


39089 


2.55827 


41115 


2.43220 


43170 


2.31641 


39 


22 


37123 


2.C9371 


39122 


2.55608 


41149 


2.43019 


43205 


2.31456 


38 


23 


37157 


2.69131 


39156 


2.55389 


41183 


2.42819 


43239 


2.31271 


37 


24 


37190 


2.68892 


39190 


2.55170 


41317 


2.42618 


43274 


2.31086 


36 


25 


37223 


2.68653 


39223 


2.54952 


41351 


2.42418 


43308 


S.30902 


35 


26 


37256 


2.68414 


39257 


2.54734 


41285 


3.43218 


43343 


3.30718 


34 


27 


37289 


2.68175 


39290 


3.54516 


41319 


3.43019 


43378 


3.30534 


33 


28 


37322 


2.67937 


39324 


3.54299 


41353 


3.41819 


43413 


3.30351 


33 


29 


37355 


2.67700 


39357 


2.54082 


41387 


3.41620 


43447 


3.30167 


31 


30 


37388 


2.67462 


39391 


2.53865 


41421 


3.41431 


43481 


3.39964 


30 


31 


37422 


2.67225 


39425 


3.53648 


41455 


3.41323 


43516 


2.39601 


29 


32 


37455 


2.66989 


:^9458 


2.5:M32 


41490 


3.41025 


43550 


3.39619 


38 


33 


37488 


2.66752 


39492 


2.5.'«17 


41534 


2.40827 


43585 


2.39437 


37 


34 


37521 


2.66516 


39526 


2.53001 


41558 


2.40629 


43630 


3.39354 


36 


35 


37554 


2.66281 


39559 


2.52786 


41592 


2.40432 


43654 


3.89073 


35 


36 


37588 


2.66046 


39593 


2.52571 


41626 


2.40235 


43689 


3.26891 


34 


37 


37021 


2.65811 


39626 


2.52357 


41660 


2.40038 


43734 


2.28710 


33 


38 


37654 


2.65576 


39660 


2.52142 


41694 


2.39841 


43758 


2.28538 


33 


39 


37687 


2.65342 


39694 


2.51929 


41728 


2.39645 


43793 


3.28348 


31 


40 


37720 


2.65109 


39727 


2.51715 


41763 


2.39449 


43828 


3.38167 


30 


41 


37754 


2.64875 


39761 


2.51502 


41797 


2.39253 


43862 


3.37967 


19 


42 


37787 


2.64642 


39795 


2.51289 


41831 


2.39058 


43897 


3-27806 


18 


43 


37820 


2.64410 


39829 


2.51076 


41865 


2.38862 


43932 


3.37626 


17 


44 


37853 


2.64177 


39662 


2.50864 


41899 


2.38668 


4:^966 


3.27447 


16 


45 


37887 


2.63945 


39896 


3.50652 


41933 


2.38473 


44001 


3.S7S07 


15 


46 


37920 


2,63714 


39930 


3.50440 


41968 


2.38279 


44036 


3.87088 


14 


47 


37953 


2.63483 


39963 


3.50229 


42002 


2.38084 


44071 


3.36900 


13 


48 


37966 


2.63252 


39997 


2.50018 


420% 


2.37891 


44105 


3.26730 


13 


49 


38020 


2.63031 


40031 


2.49807 


42070 


3.37697 


44140 


3.365.12 


11 


50 


38053 


2.62791 


40065 


2.49597 


42105 


3.37504 


44175 


3.36374 


10 


51 


38066 


2.62561 


40098 


2.49386 


42139 


3.37311 


44310 


3.36196 


9 


52 


38120 


2.62332 


40133 


2.49177 


42173 


3.37118 


44344 


3.86018 


8 


53 


38153 


2.62103 


40166 


2.48967 


42207 


3.3()925 


44379 


2.25840 


7 


54 


38186 


2.61874 


40300 


2.48758 


42343 


2.36733 


44314 


2.35663 


6 


55 


38220 


2.61646 


40234 


2.48549 


43276 


2.36541 


44349 


3.5i5486 


5 


56 


38253 


2.61418 


40267 


2.48340 


42310 


3.36349 


44384 


3.35309 


4 


57 


38286 


2.61190 


40301 


2.4813-2 


42345 


3.36158 


44418 


2.35132 


3 


58 


38320 


2.60963 


40335 


2.47934 


42379 


3.35967 


44453 


2.24956 


3 


59 


38353 


2.60736 


40369 


3.47716 


42413 


3.35776 


44488 


2.34780 


1 


60 
M 

1 


38386 


2.60509 


40403 


3.47509 


42447 


3.35585 


44533 


3.34604 



M 


N Cot. 


N. Tan. 


N. Cot. 


N. Tan. 


N. Cot. 


N. Tan. 


N. Cot. 


N. Tan. 


69 Degrees. 


68 D( 


agrees. 


67 Degrees. I 


66 Degrees. 



NATURAL TANGENTS. 



105 



M 




24 De^preea. | 


25 Degrees. | 


26 Degrees. | 


27 Degrees. 


M 
60 


N.Tan. 


N. Cot 


N. Tan. 


N. Cot. 


N. T.m. 


iN. U<>t. 


N. Tan 


N. Cot. 


44533 


3.34li04 


4u631 


3.14451 


48773 


3.05000 


50953 


1.96261 


1 


44558 


3.34438 


46666 


3.14388 


48809 


2.04879 


50989 


1.96120 


59 


8 


44593 


3.34352 


46703 


3.14125 


48845 


3.04738 


51026 


1.95979 


58 


3 


44637 


3.34077 


467^*7 


2.13963 


48881 


3.04577 


51003 


1.95838 


57 


4 


44663 


3.3390*2 


46773 


2.1)801 


48917 


2.04426 


51099 


1.95698 


56 


5 


44697 


3.33727 


4^838 


2.13630 


489.13 


2.04276 


51136 


1.95557 


55 


6 


44733 


2.33553 


46843 


2.13477 


48989 


2.041-25 


51173 


1.95417 


54 


7 


44767 


3.3:«78 


46879 


3.l331«i 


4J036 


3.U3975 


51309 


1.953T7 


53 


8 


44803 


3.33304 


46914 


2.13J51 


490G3 


2.o:«-25 


51346 


1.95137 


53 


9 


44837 


3.3:hkio 


46950 


3.12993 


49098 


3.U3675 


51*283 


1.94997 


51 


10 


44873 


3.33857 


4^985 


3.1-28321 


49134 


3.035-26 


51319 


1.94858 


50 


11 


44907 


3.33683 


47031 


3.12671 


49170 


3.0.137(1 


51356 


1.94718 


49 


13 


44943 


3.33510 


47056 


2.13511 


49206 


3.033-27 


51393 


1.94579 


48 


13 


44977 


3.32337 


47093 


3.13350 


49243 


3.03078 


51430 


1.94440 


47 


14 


45013 


3.32164 


47138 


2.12190 


49278 


3.03929 


51467 


1.94301 


46 


15 


45047 


3.31992 


47163 


3.13030 


49315 


2.03780 


51503 


1.94163 


45 


16 


45033 


3.31819 


47199 


3.11871 


49351 


3.03631 


51540 


1.94033 


44 


17 


45117 


3.31647 


47234 


3.11711 


49387 


3.02483 


51577 


1.93885 


43 


18 


45158 


3.31475 


47370 


3.11552 


49133 


3.03335 


51614 


1.93746 


43 


19 


45187 


3.31304 


47305 


3.11393 


49459 


3.03187 


51651 


1.93808 


41 


ao 


45^^ 


3.31133 


47341 


3.11333 


49495 


3.03039 


51688 


1.93470 


40 


31 


45357 


3.30961 


47377 


3.11075 


49533 


3.01891 


51734 


1.93333 


39 


33 


45392 


3.30790 


47413 


3.10916 


49588 


3.01743 


51761 


1.93197 


38 


33 


45337 


3.30619 


47448 


3.10758 


40604 


3.01596 


51798 


1.93057 


37 


34 


45363 


3.30449 


47483 


3.10600 


49640 


8.01449 


51835 


1.9-2930 


36 


35 


45397 


3.30-278 


47519 


3.10441 


49677 


3.01303 


51873 


1.92782 


35 


36 


45433 


3.30108 


47555 


3.10384 


49713 


3.01155 


51909 


1.93645 


34 


37 


45467 


3.19338 


47590 


3.1013C 


49749 


3.01006 


51946 


1.93.508 


33 


38 


45503 


3.19769 


47636 


3.09969 


49786 


3.00863 


51983 


1.^2371 


33 


39 


45537 


3. 19599 


47662 


3.09811 


49823 


3.00715 


53030 


1.93235 


31 


30 


45573 


3.19430 


47698 


3.09654 


49858 


3.00569 


53057 


l.{»008 


30 


31 


45608 


3.193&] 


47733 


3.03498 


49894 


3.00433 


53094 


1.91963 


29 


33 


45643 


3.19;)93 


47769 


3.00341 


49931 


3.003n 


52131 


1.91825 


38 


33 


45378 


3.18J33 


47805 


3.0J1S4 


49967 


3.00131 


53168 


1.91690 


37 


34 


45713 


3.18755 


47840 


3.09038 


50004 


1.99986 


53-205 


1.91554 


36 


35 


45748 


3.18587 


47876 


3.08^ 


50040 


1.99841 


52243 


1.91418 


33 


36 


4i7d4 


3.18419 


47913 


3.05716 


50076 


1.99695 


5^79 


1.91282 


34 


37 


45819 


3.18351 


47948 


3.08580 


50113 


1.99550 


5-2316 


1.91148 


33 


38 


45854 


3.18084 


47984 


3.08405 


50149 


1.99406 


53353 


1.91017 


33 


39 


45889 


3.17916 


48019 


3.08250 


50185 


1.99-261 


53.190 


1.90876 


31 


40 


45924 


3.17749 


48J55 


3.08094 


50333 


1.99116 


5-2437 


1.90741 


30 


41 


45930 


3.17582 


48091 


3.07939 


50358 


1.98072 


52464 


1.90807 


19 


43 


45995 


3.17416 


48127 


3.07785 


50395 


1.988-28 


53501 


1.90473 


18 


43 


46030 


3.17349 


48163 


2.07630 


50331 


1.98684 


53538 


1.9J337 


17 


44 


46065 


3.17083 


48198 


3.07476 


50368 


1.98540 


53575 


1.90303 


16 


45 


46101 


3.16917 


«S34 


3.07331 


50404 


1.98396 


53613 


1.90060 


15 


46 


46136 


3.16751 


48370 


3.07167 


50441 


1.98353 


53650 


1.89035 


14 


47 


46171 


3.16585 


48306 


3.07014 


50477 


1.98110 


52687 


1.89801 


13 


48 


46306 


3.16430 


48343 


3.01860 


50514 


1.9796-1 


5-2734 


1.89667 


13 


49 


46343 


3.1U355 


48378 


3.06706 


50550 


1.9782:) 


52761 


1.89533 


11 


50 


46377 


3.16090 


48414 


3.06553 


50587 


1.9768(J 


52798 


1.89400 


10 


51 


46313 


3.15935 


48450 


3.0S400 


50633 


1.97538 


52836 


1.89266 


9 


5S 


46348 


3.15760 


48486 


3.06947 


50660 


1.97395 


52873 


1.89133 


8 


53 


46383 


3.15596 


48521 


3.06094 


50696 


1.97253 


52010 


1.89000 


7 


54 


4()418 


3.15432 


48557 


3.05942 


50733 


1.97111 


52947 


1.88867 


6 


55 


46454 


3.15368 


48593 


2.05789 


50769 


1.96969 


.52984 


1.88734 


5 


56 


46489 


3.15104 


48639 


3.05637 


50806 


1.96827 


5.3024 


1.88602 


4 


57 


46535 


3.14940 


48665 


2.05485 


50843 


1.98685 


53059 


1.88469 


3 


58 


46560 


3.147T7 


48701 


3.0533:) 


50879 


1.96544 


53096 


1.883.17 


3 


59 


46595 


3.14614 


48737 


3.05182 


50916 


1.9(>402 


53134 


1.88205 


1 


60 

: 


46631 


3.14451 


48773 


3.05030 


50953 


1.98361 


5:)171 
N. Cot. 


1.88073 



M 


N.CuL 


N.Tan. 


N. Cot. 


N. Tan. 


N.Cot. 


N. Tan. 


N. Tan. 


65 Degrees. 


64 Degrees. 


63 Degrees. 


1 62 Degrees. 



106 



NATURAL TANGENTS. 



M 

U 


28 Degrees. | 


29 Degrees. 


30DegTeea 


31 Degrees. 




N.Tan. 


N. Col' 


N.Tan 


N.Coi. 


N.Tan. 


N.Cot. 


N.Tan. 


N.Cot. 


M 


53171 


1.88073 


55431 


1.80405 


57735 


l.rJ205 


600S6 


1.66438 


60 


1 


53208 


1.87941 


55469 


1.80281 


57774 


1.73089 


60136 


1.66318! 59 II 


3 


53246 


1.87809 


55507 


1.80158 


57813 


1.73973 


60165 


1.66209 


58 


3 


53283 


1.87077 


55545 


1.8U034 


57851 


1.72857 


60205 


1.66099 


57 


4 


53320 


1.8754G 


55583 


1.79911 


57890 


1.72741 


60245 


1.65990 


56 


5 


53358 


1.87415 


55621 


1.79788 


57929 


1.72625 


60384 


1.65881 


55 


6 


53395 


1.87283 


55059 


1.79665 


57968 


1.72509 


60334 


1.65773 


54 


7 


53432 


1.87152 


55697 


1.79542 


58007 


1.72393 


60364 


1.65663 


53 


8 


53470 


1.87021 


55736 


1.79419 


58046 


1.72278; 


60403 


1.65554 


52 


9 


53507 


1.8G891 


55774 


1.79296 


58085 


1.72163 


60443 


1.65445 


51 


10 


53545 


1.86760 


55812 


1.79174 


58124 


1.72047| 


60483 


1.65337 


50 


1] 


53582 


1.86630 


55850 


1.79051 


58163 


1.71932 


60522 


1.65328 


49 


12 


53620 


1.86490 


55888 


1.78929 


58-201 


1.71817 


60562 


1 -65120 


4» 


13 


53657 


1.86369 


559-iO 


1.78807 


58340 


1.71702 


60602 


1.65011 


47 


14 


53694 


1.88239 


5(j964 


1.78685 


58279 


1.71588, 


60642 


1.649a3 


46 


15 


53732 


1.80109 


56003 


1.78563 


58318 


1.71473 


60681 


1.64795 


45 


16 


53769 


l.a'v979 


56041 


1.78441 


58357 


1.71358 


60721 


1.64687 


44 


17 


53807 


1.85850 


56079 


1.78319 


58396 


1.71244 


60761 


1.64579 


43 


18 


53844 


1.85730 


56117 


1.78198 


58435 


1.71129 


60801 


1.64471 


43 


19 


53882 


1.85591 


56156 


1.78077 


58474 


1.71015 


60841 


1.64363 


41 


20 


53920 


1.85462 


56194 


1.77955 


58513 


1.70901 


60881 


1.64356 


40 


21 


53957 


1.85333 


56232 


l.n834 


58553 


1.70787 


60921 


1.64148 


39 


32 


53995 


1.85204 


56270 


1.77713 


58591 


1.70G73 


60960 


1.64041 


38 


23 


54032 


1.85075 


56309 


1.77592 


58631 


1.70560 


61000 


1.63933 


37 


24 


54070 


1.84946 


56347 


1.77471 


56670 


1.70446 


61040 


1.63836 


36 


25 


54107 


1.84818 


56385 


1.77351 


58709 


1.70332 


61080 


1.63719 


35 


26 


54145 


1.84G89 


56424 


1.77230 


58748 


1.70219 


61130 


1.63613 


34 


27 


54183 


1.8456 J 


56462 


1.77110 


58787 


1.70106 


61100 


1.6:)505 


33 


28 


54220 


1.844ai 


56500 


1.76990 


58826 


1.69992 


61200 


1.63398 


33 


29 


54258 


1.84305 


56539 


1.76869 


58865 


1.69879 


61240 


1.63292 


31 


30 


54296 


1.84177 


56577 


1.76749 


58904 


1.G9766 


61280 


1.63185 


30 


31 


54333 


1.84049 


56616 


1.76639 


58944 


1.60653 


61330 


1.630TO 


39 


32 


54371 


1.83922 


56654 


1.76510 


58983 


1.69541 


61360 


1.62972 


38 


33 


54409 


1.83794 


56693 


1.76390 


59022 


1.69428 


61400 


1.62866 


37 


34 


54446 


1.83667 


56731 


1.76271 


59061 


1.63315 


61440 


1.63760 


36 


35 


54484 


1.83540 


56769 


1.76151 


50101 


1.C9203 


61480 


1.62C54 


35 


36 


54532 


1.83413 


56808 


1.76032 


59140 


1.69091 


61.530 


1.62548 


34 


37 


54560 


1.83286 


56846 


1.75913 


59179 


1.68979 


61561 


1.63442 


33 


38 


54597 


1.83150 


56885 


1.75794 


59218 


l.(>8866 


61601 


1.63336 


23 


39 


54635 


1.83033 


56923 


1.75675 


59258 


1.68754 


61641 


1.63330 


31 


40 


54673 


1.82906 


57963 


1.75556 


59297 


1.68643 


61681 


1.63135 


30 


41 


54711 


1.82780 


57000 


1.75437 


59336 


1.08531 


61731 


1.63019 


19 


42 


54748 


1.82654 


57039 


1.75319 


59376 


1 .68419 


61761 


1.61914 


18 


43 


54786 


1.82528 


57078 


1.75200 


59415 


1.68.108 


61801 


1.61808 


17 


44 


54834 


1.83402 


57116 


1.75082 


59454 


1.68196 


61842 


1.^1703 


16 


45 


54862 


l.«2?!76 


57155 


1.74964 


59494 


1.68085 


61883 


1.61598 


15 


46 


54900 


1.83150 


57193 


1.74846 


59533 


1.67974 


61933 


1.61493 


14 


47 


549:« 


1.82(^25 


57333 


1.74728 


595V3 


1.678C9 


61963 


1.61388 


13 


48 


54975 


1.81899 


57271 


1.74610 


59r>12 


1.67752 


62003 


1.61283 


13 


49 


55013 


1.81774 


57309 


1.74492 


59r5l 


1.67641 


62043 


1.61179 


11 


50 


55051 


1.81649 


57348 


1.74375 


59691 


1.67530 


62083 


1.61074 


10 


51 


55089 


1.81524 


57386 


1.74257 


59730 


1.67419 


62124 


1.60tf70 


9 


52 


55127 


1.81399 


57425 


1.74140 


59770 


1.67309 


62164 


1.60865 


8 


53 


55165 


1.81274 


57464 


1.74022 


59809 


1.67198 


62204 


1.60761 


7 


54 


55203 


1.81150 


57503 


1.73905 


59849 


1.67088 


62245 


1.60657 


6 


55 


55241 


1.81025 


57541 


1.73788 


59888 


1.66978 


62285 


1.60553 


5 


56 


55279 


1.80901 


57580 


1.73671 


59928 


1.6r8f.7 


62325 


1.60449 


4 


57 


55:«7 


1.80777 


57619 


1.73555 


59967 


1.66757 


623C6 


1.60345 


3 


58 


55355 


1.80653 


57657 


1.73438 


60007 


1.66647 


62406 


1.60341 


3 


59 


55393 


1.80529 


57G96 


1.73321 


60046 


1.66538 


63446 


1.60137 


1 


60 
M 

1 


55431 


1.80(05 


57735 


1.73205 
N. Tan. 
agrees. 


60086 


1.66428 


62487 


1.60033 




M 


N Cot. 


N. Tan 


N. Cot. 
60 D< 


N. Cot. 


N. Tan. 


N. Col. 


N. Tan. 


61 D< 


sgrees. 


59 D< 


igrees. 


58 D< 


sgrees. 





32Deg««. 


33 Degree 


35 Degree. 


— 


H 


H.Tu. 


P) Cot 


Htm 




N.Tui. 




If 




"Mlm 


I.MIU3J 


64641 


"iIm » 




T^is 


lo 




6*sn 


1.59936 


64983 




7U0d4 










t.508«! 


65623 


i.x '< 


761UT 


1.4S63I 








t.Bsraa 


65665 


.53 " 


TUISt 


1.43550 
















l.«lft 








:ja5i7 


ui^ 












e^73D 


.59411 




.M 1 










MTTO 


.5031 








1.41190 


63 


8 


69811 










1.43110 








^59105 




is:! s 


TOIll 


l.tVXti 








.JW6J 


6S3M 


.a II 


704S5 


1.41931 




11 


62933 


.S89Un 


65391 




70199 






■a 


6»73 


.58797 


63138 








48 












TOjgii 








SSOSS 






'.K S 


T0«» 


lillSSI 


18 




630BS 


.5849!) 


65563 




70tf73 


1.41497 




16 


63136 


.98388 


6U)61 






1.11469 








.5»186 


WiG46 






.41332 


43 








M0S8 


'.S3 a 


70804 


.lisj; 


43 


Ifi 


63S58 


'.iem\ 


657S9 


.53 3 












.saBI 










40 
















3S 


3St 




ijTTJS 


BSBH 


.51 9 


70979 




3S 






.17678 


S58» 


.51 7 


now 








e:nm 


.57515 




.51 6 


71066 


:4on 


36 




KSS03 










.40637 


15 


m 






B6021 


^51 1 




.40510 


34 


37 






66063 




71198 


.40454 


33 






.57179 


60195 


■*' 


71243 


.403S 






<am 


1.5706 






















T13J9 


iiom 


30 


31 


67746 


.56868 


66330 


1.S0 


71373 


.40169 


39 




63789 


.56707 


(m:3 


1.50 6 


71417 


.400i« 


38 




63836 










.39036 






63B71 




66356 


l.M 9 




.30630 


3t 


35 


63011 


is6466 


6639B 


1.50 a 




.39764 


«S 


X 


63953 


.563a 


66440 


1.S6 S 


T1593 


.3B6ra 


34 




63991 




B6483 




71637 


.3>»3 


33 


X 


61035 






ilso s 




.39507 


a 


3S 






66566 


1.50 8 






31 






'.Siaes 


66006 


1.50 8 




!39336 


« 






.55806 


66656 


1.50 e 


71813 


.30350 


19 


4S 


61199 


.55766 




1.49 8 


71B37 


.3916S 




42 


64(!4D 




66734 


1.19 9 


71901 


.30070 










S6r76 


1.49 H 




.38994 






6ti»3 


!S5«7 


86818 


1.49 9 






15 


V. 


61383 


1.65368 


BtSSD 




7»34 


1.38831 






ena* 


.SJM9 




l.4;i 6 


1.J078 


1.38738 




46 


64116 






1.40 J 








19 


6M8I 


'5S6Tt 


00086 






l!38308 


II 






.54973 


STOM 


1:49 3 




1.38481 


10 


St 




.S4SJ3 


67071 


1.19 4 


7»55 


I.3H399 






61616 






1.49 5 


TMtB 










]!54615 










7 








6719T 


1148 7 




l!38115 






61T34 




67*30 


1.4« 8 




1.38066 




X 


6i7rs 






1.43 9 




1.37978 






64817 






1.48 






3 










T,48 5 






9 










1.4H 




l!37723 1 11 


60 


81W1 


l!539Bfl 


67451 




7S654 


h^M 


M 


N Cat. 


tTTiiT 


s:coi 


iTt ^ 




N.T^-»-H 


57 Degree*. 


56"D^ 


Jiee, 


"M"D" 


ree«- 


_J 



108 






NATURAL 


TANGBNTS. 








M 



36 Degrees. | 


37 Degrees. 


38 Degrees. 


39 Degrees. 


M 

60 


N.Tan. 


N. Cot* 


N. Tan. 


N. Cot. 


N. Tan. 


N.Cot 


N.Tan 


N. Cou 


72654 


1.37638 


75355 


1.32704 


78129 


1.27994 


8li978 


1.23489 


1 


72699 


1.37554 


75401 


1.32624 


78175 


1.27917 


81027 


1.23416 


59 


2 


T2743 


1.37470 


75447 


1.32544 


78222 


1.27841 


81075 


1.23343 


58 


3 


72788 


1.37386 


75492 


1.32464 


78269 


1.27764 


81123 


1.2;K70 


57 


4 


72832 


1.37302 


755.T8 


1.32384 


78316 


1.27688 


81171 


1.23196 


56 


5 


72877 


1.37218 


75584 


1.32304 


78363 


1.27611 


81230 


1.3.3123 


55 


6 


72921 


1.37134 


75629 


1.32224 


78410 


1.27535 


812G8 


1.33050 


54 


7 


72966 


1.37050 


75675 


1.32144 


78457 


1.27458 


81316 


1.32977 


53 


8 


73010 


1.36967 


75721 


1.32064 


78504 


1.27382 


813(>4 


1.32904 


52 


9 


73055 


1.36883 


75767 


1.31984 


78551 


1. 27306 


81413 


1.22831 


51 


10 


73100 


1.36800 


75812 


1.31904 


78598 


1.27230 


8K61 


1.22758 


50 


11 


73144 


1.36716 


75858 


1.31825 


78645 


1.27153 


81510 


1.22685 


49 


13 


73189 


1.36633 


75904 


1.31745 


78693 


1.27077 


81558 


1.22612 


48 


13 


73234 


1.36549 


75950 


1.31660 


78739 


1.27001 


81606 


1.22539 


47 


14 


73278 


1.36466 


75996 


1.31586 


78786 


1.26925 


81655 


1.22467 


46 


15 


73323 


1.36383 


76042 


1.31507 


78834 


1.26849 


81703 


1.22394 


45 


16 


73368 


1.36300 


76088 


1.31427 


78881 


1.26774 


81752 


1.22321 


44 


17 


73413 


1.36217 


76134 


1.31348 


78938 


1.26698 


81800 


1.22249 


43 


18 


73457 


1.36133 


76180 


1.31269 


78975 


1.26622 


81849 


1.22176 


42 


19 


73502 


1.36051 


76226 


1.31190 


79022 


1.26540 


81898 


1.22104 


41 


20 


73547 


I.:i5g68 


76272 


1.3111U 


79070 


1.26471 


81946 


1.23031 


40 


2i 


73592 


1.35885 


76318 


1.31031 


79117 


1.26395 


81995 


1.21959 


39 


22 


73037 


1.35802 


76364 


1.3095'i 


79164 


1.26319 


82044 


1.21886 


38 


23 


73C81 


1.35719 


76410 


1.30873 


79212 


1.26244 


82092 


1.21814 


37 


34 


73726 


1.35637 


76456 


1.30795 


79259 


1.26169 


82141 


1.21743 


36 


25 


73771 


1.35554 


76502 


1.30716 


79306 


1.26093 


82190 


1 31670 


35 


26 


738 li> 


1.35472 


76548 


1.30637 


79354 


1.26018 


8223R 


1.31598 


34 


27 


738J6 


1.35389 


76594 


1.30558 


79401 


1.25943 


82287 


1.31536 


33 


28 


73906 


1.35307 


76640 


1.30480 


79449 


1.25867 


82336 


1.31454 


32 


29 


73951 


1.35234 


76686 


1.30401 


79496 


1.25792 


82385 


1.21382 


31 


30 


73996 


1.35142 


76733 


1.30323 


79544 


1.25717 


83434 


1.31310 


30 


31 


74041 


1.35060 


76779 


1.30244 


79591 


1.25642 


82483 


1.21338 


39 


32 


74086 


1.34978 


76825 


1.3016G 


79639 


1.2556: 


82531 


1.31166 


88 


33 


74131 


1.34896 


76871 


1.30087 


79686 


1.2549-2 


83580 


1.31094 


37 


34 


74170 


1.34814 


76918 


1.30009 


79734 


1.25417 


83639 


1.21023 


36 


35 


74221 


1.34732 


76964 


1.29931 


79781 


1.25343 


82678 


1.20951 


35 


36 


74267 


1.34650 


77010 


l.29H.'>3 


79829 


1.25368 


82727 


1.30879 


24 


37 


74312 


1.34568 


77057 


1.29775 


79877 


1.25193 


82776 


1.20808 


33 


38 


74357 


1.34487 


77103 


1.29696 


79924 


1.25118 


82825 


1.20736 


33 


39 


74402 


1.34405 


77149 


1.29618 


79972 


1.25044 


82874 


1.30665 


31 


40 


74447 


1.34323 


77196 


1.29541 


80020 


1.24969 


82923 


1.30593 


30 


41 


74492 


1.34242 


77242 


1.29463 


80067 


1.24895 


82972 


1.30522 


19 1 


43 


74538 


1.34160 


77289 


1.29385 


80115 


1.24820 


83022 


1.30451 


18 1 


43 


74583 


1.34079 


Tr335 


1.29307 


80163 


1.24746 


83071 


1.20379 


17 


44 


74628 


1.33998 


77382 


1.29229 


80211 


1.24672 


83120 


1.20308 


16 


45 


74674 


1.33916 


77428 


1.29152 


80258 


1.24597 


83169 


1.30337 


15 


46 


74719 


1.33835 


77475 


1.29074 


80306 


1.24.523 


83218 


1.30166 


14 


47 


74764 


1.33754 


77521 


1.28997 


80354 


1.24449 


83268 


1.30095 


13 


48 


74810 


1.33673 


77568 


1.28919 


80402 


1.243/5 


83317 


1.30034 


13 


49 


74855 


1.33592 


77615 


1.28842 


80450 


1.24301 


83366 


1.19953 


11 


50 


74900 


1.33511 


77661 


1.28764 


80498 


1.24227 


83415 


1.19882 


10 


51 


74946 


1.33430 


77708 


1.28687 


80546 


1.24153 


83465 


1.19811 


9 


52 


74991 


1. 3:^349 


77754 


1.28610 


80594 


1.24079 


83514 


1.19740 


8 


53 


75037 


1.33268 


77801 


1.28533 


80642 


1.24005 


83564 


1.19669 


7 


54 


750H2 


1.33187 


77848 


1.2845(; 


80690 


1.23931 


83613 


1.19599 


6 


55 


75128 


1.33107 


77895 


1.28379 


807^8 


1.23a58 


836(S 


1.19528 


5 


56 


75174 


1.33026 


77941 


1.28302 


80786 


1.23784 


8.3712 


1.19457 


4 


57 


75219 


1.32946 


77988 


1.28225 


80834 


1.23710 


83761 


1.19387 


3 


58 


75264 


1.32865 


77035 


1.28148 


80882 


1.23637 


83811 


1.19316 


3 


50 


75310 


1.32785 


77082 


1.28071 


80930 


1.23563 


83860 


1.19246 


1 


GO 
M 


75355 


1.32704 


77129 


1.27994 


80978 


1.23490 


83910 


1.19175 



M 


N. Cot. 


N. Tan. 


N. Cot. 


N. Tan. 


N.Cot. 


N.Tnn. 


N.Cot 


N. Tan. 


53 Df 


igreea. 


52 D< 


jgrees. 


51 D< 


agrees. 


50 D< 


egiees. 



M 


40 Degrees. 


j^ 


43 Degree.. 


~ 


N.-UO. 




n:ts:: 




1 


"SIT 

1M0U9 


lil8M4 




033D6 

93360 


r.OTiB 

!<no4i 


eo 

59 


1 


15'sB 






KMSK 


.*>9ei 

.00985 
.08861 


i 




»4i5S 


i 18084 










J 


B«»7 


.]98M 




g3l»4 
1B749 


ioMi; 


i 




MIJT 


'.ism 










la 


84SOT 


.18331 




SXKM 


looist 


48 


13 


atew 






ICMl 
















'!0S30J 






847M 


.latu 






.OftWI 






MTM 


. TSBIi 








43 










942J3 










'■ W 






:o60M 




so 




. 7T77 








W 




max 


. 7708 








30 




Bwje 


. 7638 






'.asBW 


38 






- rsw 




Hiin 




31 










us*» 


'0ST« 


30 


33 








SMM 


.0M8S 


35 


M 


8S3OT 


. rwi 






.05IW4 


















29 














S9 
















8HUB 






»1896 


!0J378 




31 


9S4» 






W9S3 


.115317 


n 




























»T 












'■taia 




3S 


asm 


1. 8741 




ftsra 


.05OU 




38 


RiJlO 






















ioMMU 














.01888 




1 


(W»a3 
ttnuM 


; 83!ie 






;04TIW 
.04T(IS 


21 


« 


snuM 


: *|^ 






lotsM 


le 






! liUSU 




lUTW 




IS 


«fi 


bUm? 


1.15987 
I.I.WIS 




9S78J 


-OMOl 


» 


« 








BWOT 




















s? 


dMJO 


i- s;js 

1. SilT 




MOM 




10 




BSMl 






MI90 






a 


8n.S7S 
















1. M4.1 

i! Kina 




9K233 


i039IX 
-IKITW 






an8n 






















;.niKij 




H 


BS»29 






MMg 






'U 


[TcS 


N. T»i>. 


i 


N. Cot, 


nTtm. 


"m 


_ 


49 Degree*. 




4S Degree.. 


_ 



Jl^*?^ 




« Degree* | 


^ 


t ^ 


':- 


W^ 


N 


Cul. 


owei 












wrm 






















)4es 


St 


- 4 


37 


»8liJ3 






06U63 


 S 


38 


^ 




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(i;o78 






sews 




nm 








B93a] 






071BS 


 « 


■la 






ma 


K 


 3 


40 


«II3i 




«j8ie 

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Sl^ 


1; j 


s 


K 




SS^ 


(I7S3 


1- 1 


w 


WOfiS 




mUh 


ansa 


1- n 


w 


119336 




»«i 








»B5M 




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B8U4I 














1. 3 




09834 


















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«0 


lOUOO 






9*t70 


1- 10 








nTc^. 


W: r 








45D 


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~45"D"e 


C" 


^. i 



Part IL 



TEST PROBLEMS. 



ABITHMETICAL PROBLEMS. 

DENOMINATE NUMBERS. 

1. A printer used 3 reams, 5 quires, 19 sheets of paper for 
printing half-sheet posters. How many did he print, allow* 
ing 1 quire to a ream for waste ? 

2. If a grocer's weights are one fourth of an ounce in a 
pound below the legal standard, how much does he gain fraud- 
ulently from the sale of 2 bags of Rio coffee, 116 pounds each, 
true weight, at 18f cents a pound ? 

3. Which is heavier, — a pound of gold, or a pound of lead ; 
an ounce of gold, or an ounce of lead ? 

4. A man starts from Philadelphia, and travels westward. 
When he stops, he ascertains that his watch is 6^ hours slow. 
In what longitude did he stop, and through how many degrees 
did he travel ? 

5. What is the weight of a 2-foot cube of gold, the specific 
gravity of gold being 19.36 ? 

LEAST COMMON MULTIPLE AND GREATEST COMMON 

DIVISOR. 

6. Find the L. C. M. of |, |, and ^^. 

7. Find the G. C. D. of |, f , f . 

8. Four bells toll at intervals of 3, 7, 12, and 14 minutes 
respectively. If they begin at the same time, how often will 
they toll together in 7 hours ? 

in 



112 TABLE BOOK AND TEST PROBLEMS. 

9. Four men make regular excursions into the country, 
between which each stays at home just one day. A is always 
absent 3 days, B 5 days, and C and D each 7 days. Provided 
they all start out together, how many days must elapse till 
they can sill be at home at one time ? 

10. A, B, and C start at noon from the same point to travel 
around a circle of 320 rods. A walks 8, B' 13, and C 24, rods 
per minute. A and B travel in one direction, and C in the 
opposite direction. How long before they all meet at the 
starting-point, no one varying his rate of travel ? 

PARTNERSHIP. 

11. A and B enagge in business as equal partners. On set- 
tlement it is found that A owes the firm $ 240, and that the 
firm owes B $260. How much should A give B to square 
the account ? 

12. Two partners, A and B, gain $249. A owns three 
fourths of the stock, lacking $ 10, and gains $ 175. Find the 
amount of their stock. 

13. A, B, and C, having 4 loaves, for which A paid 5 cents, 
B 8 cents, and C 11 cents, eat 3 loaves, and sell the fourth to 
D for 24 cents. Divide the 24 cents equitably. 

14. Jones hires a rig for $ 10 to go from Salem to Derry, a 
distance of 10 miles. At Tiffin, midway between the two 
places, he takes in Smith, who agrees to pay his proportionate 
share if Jones will take him to Derry and back again to Tifl&n. 
How much should Smith pay ? 

PROPORTION. 

16. If one third of 6 were 3, what would one half of 6 be ? 
If 3 were one third of 6, what would one half of 6 be ? 

16. How long will it take 18 men to empty a tank if 15 
men can empty it in 30 minutes, and 12 men can empty it in 
40 minutes, supposing water to be running in at a uniform 
rate? 



ARITHMETICAL PROBLEMS. 113 

17. A burden of 200 pounds suspended on a pole 4 feet in 
length, the point of suspension being 6 inches from the middle, 
is carried by two men, one at each end of the pole. How 
many pounds does each man carry ? 

18. A body weighs 1800 pounds at the surface of the earth. 
If the earth's diameter is 8000 miles, what will the body 
weigh 2000 miles above the surface ? 

19. A wins 9 games out of 12 when playing against B, and 
5 out of 8 when playing against C. How many games out of 
70 should B win when playing against C ? 

20. In a pair of scales a body weighed 31^ pounds in one 
scale, and only 20 pounds in the other. Required the true 
weight. 

21. If 20,000 cubic feet of air per minute pass through an 
air-course 5 feet square and 1000 feet in length, what volume 
will pass through the same air-way per minute if its length be 
increased to 4000 feet, the power remaining the same ? 

22. A town clock, whose pendulum is 15 feet long, loses 3 
hours per week. How much must the " bob " be raised that 
the clock may run true ? 

PROFIT AND LOSS. 

23. A pays B $ 190 for a $ 240 note due in 4 years. What 
amount does A gain, money being worth 6 per cent ? 

24. If my gain is 12^ per cent of my selling price, what is 
my rate per cent of gain ? 

26. Bought 3 watches for $90, and sold them at equal 
prices. On the first I gained 80 per cent, on the second 20 
per cent, and on the third I lost 10 per cent. Find the cost 
of each. 

26. Sold a cargo of sugar at an advance of 20 per cent. 
Had it cost $ 500 more, my gain would have been only 15 per 
cent. What was the cost of the cargo ? 
ellwood's test prob. — 8. 



114 TABLE BOOK AND TEST PROBLEMS. 

27. When goods are bought at 10, 10 and 5 per cent off, and 
sold at an advance of 50 per cent, at what per cent of the origi- 
nal price are they sold ? 

SJ8. A drover bought 20 cows and 30 oxen for $ 1500. He 
sold his cows for $21.60 each, his oxen for $37.50 each, and 
gained as much as one cow and one ox cost him. Find the 
price of one. 

29. I marked goods to gain 40 per cent ; but, my yard-stick 
being too long, I made only 20 per cent. Find the length of 
the yard-stick. 

30. A sells an article to B at a gain, and B to C at same 
rate of gain for $ 16. If B had sold for $ 10, his loss would 
have been half what he now gains. Find what A paid for the 
article. 

31. If I sell my sugar at a certain price per pound, I will 
lose $ 1 ; but if I increase the price 3 cents per pound, I will 
gain 50 cents. How many pounds have I ? 

32. A brewery is worth 4 per cent less than a tannery, and 
the tannery 16 per cent more than a boat. The owner of the 
boat has traded it for 75 per cent of the brewery, losing thus 
$ 103. What is the tannery worth ? 

33. I marked goods to gain 50 per cent ; but, by using an 
incorrect, yard-stick, I made only 20 per ceiit. Find the 
length of the yard-stick. 

34. A shop-keeper buys wool from Mr. Jones, weighing it 
on false scales, and thus defrauding him to the extent of 20 
per cent. The dealer, in selling the wool, weighs it on true 
scales. If his entire gain on a given quantity of wool is 50 
per cent, what per cent has been gained by honest trading ? 

36. A horse which I bought for 30 per cent less than his 
real worth, having been injured, was bought from me for 25 
per cent less than he cost, and by the transaction I lost $ 55 of 
his original value. What did I get for the horse ? 



ARITHMETICAL PROBLEMS. 115 

36. A grocer sells lard at a profit of 14^ per cent. In both 
buyiug and selling he uses a false balance, 13 pounds in one 
scale balancing 14 pounds in the other, and thus on a given 
quantity of lard increases his profits by $ 29. Find the cost 
of the " given quantity." 

37. If an article had cost me 20 per cent less, my rate of 
gain would have been 30 per cent more. Find the gain. 

38. A bought a lot for $ 100 on a credit of 6 months, and 
sold it at once for $200 cash. What did he gain, money 
being worth 6 per cent ? 

39. Sold a lot for $ 100 (cost price) on a credit of 6 months, 
and bought it back at once for $ 200 cash. What did I lose, 
money being worth 6 per cent ? 

STOCKS AND BONDS. 

40. My U. S. 5's yield 7 per cent. At what discount were 
they bought ? 

41. What sum must be invested in mining stock (par $ 10) 
at 20 per cent premium, \ per cent brokerage, if it pays 6 per 
cent semi-annual dividends, to yield $ 2000 yearly ? 

42. A man pays $21,200 for 5-20's when selling at 106. 
What is his annual income in currency, and per cent, gold 
being 112^ ? 

43. Which is the better investment, — U. S. 6's at 75, or 
U. S. 6's at 85 ? 

44. What rate per cent of income shall I receive if I buy 
U. S. 5'8 at a premium of 10 per cent, and receive payment at 

par in 15 years ? (From Fish's Complete Arithmetic.) 

45. I made $ 5000 by a speculation, and, wishing to invest 
it permanently, I bought $2000 6's of '81 at 117|, and in- 
vested the remainder in the new 4i^'s at 110^. What surplus 
remained after deducting brokerage, and what was my annual 
income ? 



116 TABLE BOOK AND TEST PROBLEMS. 

46. John Smith, through his broker, invested a certain sum 
of money in New York State 6's at 107^, and twice as much" 
in U. S. 5's of '81 at 98^, brokerage in each case \ per cent. 
The annual income from both is $ 3348. How much did he 

invest in each kind of stock ? (Fish's Compute Arithmetic, p. 840.) 

INTEREST. 

47. B's fortune added to two thirds of A's, which is to B's 
as 2 to 3, being on interest for 6 years at 8 per cent, amounts 
to f 8880. Find fortune of each. 

48. A borrows $ 100 for a year at 6 per cent interest, pay- 
ing the interest in advance. At the end of the year he pays 
$ 40 on account, and gives a new note for the balance, paying 
interest for a year in advance out of the $40. For what 
amount must the note be drawn, not reckoning days of grace ? 

49. A owes B $ 1000, but is able to raise only f 730, with 
which he proposes to pay part of the principal and the inter- 
est in advance on the remainder. For what sum must he give 
his note at simple interest for 2 years at 6 per cent ? 

50. The present value of a freehold estate of $100 per 
annum, subject to a payment of a certain sum at the end of 
every two years, is $1000, allowing 5 per cent compound 
interest. Find the biennial payment. 

51. What rate per cent does a bank make on its money by 
loaning it on 90-day paper ? 

52. A man agreed to pay $ 6000 for a store, the principal 
and interest to be paid in three equal annual payments. Find 
the yearly payment, interest being 6 per cent. 

53. A note bearing 6 per cent interest amounted to $ 325 
May 1, 1885. It would have amounted to $ 375 Aug. 1, 1886, 
if the rate had been 8 per cent. Find face and date of note. 



ABITHMETICAL FB0BLEM8. 117 

DISCOUNT AND PRESENT WORTH. 

64. Find the present worth of $1000 due in 60 days, 
money being worth 6 per cent. 

66. Bought a lot for $ 600, of which I paid $ 50 cash, $ 150 
in 9 months, f 200 in 1 year 9 months, and f 200 in 2 years 9 
months, interest at 8 i)er cent. What was the cash value, 
money being worth 6 per cent ? n 

66. Bought a horse for $ 500, and immediately sold him on 
a credit of 6 months. The First National Bank discounted 
the note I received ; and, having examined my money, I found 
I had gained 20 per cent on my purchase. What was the face 
of the note ? 

67. At what rate must a note, payable in 60 days, be dis- 
counted to produce 6 per cent interest ? 

68. What must be the face of a note that yields $2000 
when discounted at bank the day of its date, if drawn 30 days 
after date ? 

69. A note is payable in 30 days. At what rate must it be 
discounted to produce 6 per cent interest ? 

80. A bank by discounting a note at 6 per cent receives for 
its money a discount equivalent to 6J- per cent interest. How 
long must the note have been discounted before it was due ? 

81. A man bequeathed $9000 to his three sons, aged 13, 
15, and 17 years, in such a manner that the share of each, 
placed at compound interest at 6 per cent until he arrived at 
the age of 21 years, should amount to the same sum. Find 
the share of each. 

INVOLUTION AND EVOLUTION. 

62. Extract the square root of .625. 

83. Find the number when two thirds of its cube is 10 more 
than the cube of its two thirds. 



118 TABLE BOOK AND TE8T PROBLEMS. 

64. Two thirds of the square of three fourths of a number 
is 12 more than three fourths of the square of one half the 
number. What is the number ? 

65. What must be the width of a walk around a park 40 
rods square, to contain one fourth the area of the park ? 

ALLIGATION. 

66. Sold 50 dozen eggs for $ 8. How many dozens of each 
were there at 13, 14, 18, and 21 cents respectively ? 

67. What relative quantities of gold and silver, whose spe- 
cific gravities are 19|^ and lOJ, will make a compound whose 
specific gravity shall be 16.84 ? 

ANNUITIES. 

68. I rent a house for $300 a year, the rent to be paid 
monthly in advance. What amount of cash at the beginning 
of the year will pay one year's rent ? 

69. B bought a house for $6000 down, or equal install- 
ments of $ 1200 a year for 6 years. Which is the better for 

B, money being worth 6 per cent ? (From Brooks's Arithmetic.) 

70. A man borrows $ 1000 at 6 per cent per annum, to be 
repaid in five equal annual payments. How much must be 
paid each year, interest being added to principal at end of 
each year, and before the annual payment is deducted ? 

 

"AGE" PROBLEMS. 

71. My age now is one fourth of yours, but in 20 years I 
will be one half as old as you. How old are we ? 

72. Ten years ago, when Mrs. C. was married, she was one 
third as old as her husband, but now she is three sevenths as 
old. How old was each when they were married ? 



ARITHMETICAL PROBLEMS. 119 

73. Ten years ago Edwin was one third as old as his father, 
but 2 years hence he will be one half as old. What is the age 
of each ? 

74. Three times Jennie's age equals three eighths of Gertie's 
age. In how many years will Gertie be just twice as old as 
Jennie ? 

"TIME" PROBLEMS. 

76. What is the time if one fifth of the time past noon 
equals one third of the time to noon again ? 

76. Three fourths of the time past noon equals one half the 
time to midnight lacking 1 hour. What time is it ? 

77. It is between 4 and 5 o'clock, and the hands on the dial 
are exactly opposite. What time is it ? 

78. The hour hand of a clock moves 10 per cent too fast, 
and the minute hand 5 per cent too slow. In 20 minutes 
(true time) they will be together. How many minute spaces 
are they apart now ? 

79. At what times between 6 and 7 o'clock are the hour 
hand and minute hand 20 minutes apart ? 

80. At a certain time between 1 and 2 o'clock the minute 
hand was between 2 and 3. Within an hour the hands had 
changed places. What was the time when the hands were in 
the first position ? 

81. It is between 5 and 6 o'clock, and the minute hand has 
passed the 6 as far as the hour hand lacks having reached it. 
What time is it ? 

82. At a certain time between 2 and 3 o'clock the minute 
hand was between 3 and 4. Within an hour after, the hour 
hand and minute hand had exactly changed places. What was 
the time when the hands were in the first position ? 

83. A clock has three hands, — hour, minute, and second, — 
all turning on the same center. At 12 o'clock all are together, 



120 TABLE BOOK AND TEST PB0BLEM8. 

aad point at 12. How long will it be before each hand will be 
midway between the other two ? 

GENERAL ANALYSIS. 

84. If 48 pounds of sea-water contain 1^ pounds of salt, 
how much fresh water must be added to the 48 pounds so that 
40 pounds of the mixture shall contain one half a pound of 
salt ? 

86. Suppose A and B work together for 4 hours, after which 
A leaves, and B finishes in 3 hours 36 minutes ; but if B had 
left, A would have done the remainder in 4 hours 30 minutes. 
In what time could each alone have done the work ? 

86. A, B, and C together have $ 150. If B's money were 
taken from the sum of the other two, the remainder would be 
$ 60 ; and if C's were taken from the sum of the other two, 
the remainder would be one half of his (C's) money. Find 
how much each had. 

87. When a certain number of men had labored 12 days, 
and had the job one third completed, 20 more men were set to 
work, and the remainder of the job was finished in 14 days. 
How many men were employed at first ? 

88. A tree 33 feet high breaks off. If the broken part 
were a foot longer and the stump 4 feet shorter, the stump 
would be twice as long as the broken part. Find the length 
of both parts. 

89. A and B can do a piece of work in 15 days, B and C iu 
10 days, and A and C in 12 days. How long will it take all 
to do the work ? 

90. Bought 10 bushels potatoes and 20 bushels of apples 
for $ 11. At another time I bought 20 bushels potatoes and 
10 bushels of apples for $13. What price per bushel was 
paid? 



ARITHMETICAL PROBLEMS. 121 

91. Ay B, and each contemplated purchasing a warehouse. 
They agreed that if A and B bought it, A should pay two thirds 
of the price; but if B and C bought it, B should pay two 
thirds of the price. They finally agreed to buy it together, 
and it was found that A paid $12,000 more than C.*^ Find 
the cost. 

92. Two boats leave the same shore at the same time. It 
takes one 12 minutes to reach the opposite shore, and the 
other sails three times as fast. When will they first meet if 
the fast boat stops 2 minutes at the opposite shore ? 

93. How many tucks one fourth of an inch wide can be 
made in a strip of muslin a yard long, leaving one eighth of an 
inch between the edge of one tuck and stitching of the next ? 

94. I have two fifths of my money stolen. I then earn 
9 60, and spend two thirds of all I now have. My uncle then 
gives me $ 10, and after losing one sixth of what I have I find 
I have half as much as I had at first. How much had I 
at first ? 

95. A tree ^ feet high breaks off. If the broken part 
were 2 feet longer and the stump 8 feet shorter, the former 
would be half as long as the latter. Find the length of the 
stump. 

96. A man bought a cow, a horse, and an ox for $350. 
The horse cost 4 times as much as the ox lacking $ 40, and 
the ox cost twice as much as the cow lacking $ 12. Find the 
cost of each. 

97. I can pick 40 bushels of potatoes or dig 20 bushels in a 
day. How many bushels can I dig and pick in a week ? 

98. If a man can pasture 2 cows on an acre, and allows 
1 acre of roots to every 6 cows, how many cows can he keep on 
20 acres, and how much land has he in grass ? 

99. The head of a fish is 3 inches long ; the tail is as long 
as the head plus one fourth of the body ; and the body is as 
long as the head and tail. How long is the fish ? 



122 TABLE BOOK ANB TEST PROBLEMS, 

100. A watch and chain cost $ 100. Half of the cost of the 
watch is $20 more thaa the cost of the chain. Pind the cost 
of each. 

101. A beat B by 80 feet in a foot-race; but if A had 
walkea eight ninths of his rate, and B nine tenths of his, A 
would have beaten B by only 26 feet. Find the length of the 
course. 

102. Equal weights of gold and silver are in value as 20 to 
1, and equal volumes are in value as 1284 to 35. A certain 
volume is composed of equal weights of gold and silver. Find 
how many times more valuable it would be were it composed 
of gold alone. 

103. A and B engaged to reap a field for 90 shillings. A 
could reap it in 9 days, and they promised to complete it in 
6 days. They found, however, that they were obliged to call 
in C, an inferior workman, to assist them the last two days, 
in consequence of which B reeeived Ss, 9d. less than he other- 
wise would. In what time could B and C each reap the field ? 

(From Mathematical Magazine.) 

104. The product of two numbers is 117, and their quo- 

• 

tient is 1.4. Find the numbers. 

105. Two trains start at the same time, one from Pittsburgh 
to Altoona, the other from Altoona to Pittsburgh. If they 
arrive at destinations 1 hour and 4 hours after passing, what 
are their relative rates of running ? 

MENSURATION. 

The Rectangle, 

106. The perimeter of a rectangular field is 200 rods, and 
three times the width is twice the length. Find the area in 
acres. 

107. The sides of a rectangular room are as 4 to 3. If each 
side were 2 feet longer, the area would be 1344 square feet. 
Find the sides. 



ABITHMETICAL PROBLEMS. 123 

108. A rectangular trough is two thirds full of water. After 
35 gallons are taken out, it is three eighths full. What is the 
depth, the length being 10 feet and the width 2f feet ? 

109. Around a garden 80 feet square is a walk cont^ning 
one sixth as much area as the garden. Find the width of 
the walk. 

110. The largest square stick that I can cut from a 30-foot 
log contains 45 cubic feet. Find its diameter. 

111. Find the area of a square field whose diagonal is 8.284 
rods longer than its side. 

112. From one corner of a rectangular pyramid 6 by 8 feet 
it is 13 feet to the apex. Find the dimensions of a rectangular 
solid whose dimensions are as 2, 3, and 4^ that may be equiv- 
alent in volume. 

113. Two men run in opposite directions around a rectangu- 
lar field whose area is 1 acre. They start from one corner, 
and meet 13 yards from the opposite corner. The rates of 
running being as 5 to 6, find the dimensions of the field. 

114. The inside dimensions of two similar rectangular oil- 
cans are as 3, 7, and 11. The first holds 8 gallons, and the 
second requires 4 times as much tin as the other. Find the 
dimensions of the smaller, and the volume of the larger. 

The Triangle, 

116. A doorway is 8 feet high, and just wide enough to 
allow a circular saw 10 feet in diameter to pass through. 
Find the width. 

116. An isosceles triangle contains 6 acres and 12 perches, 
and its base is 72 perches. Find the sides. 

117. Find the area of a triangle whose sides are 9, 14, and 
5 rods. 

118. A man has a triangular field whose sides are 40, 50, 
and 60 rods respectively. A fence is built from the middle 



124 TABLE BOOK AND TE8T PROBLEMS. 

point of the side 50 rods long to the middle of the side 60 rods 
long. How long is the fence ? 

119. In a two-thirds pitch roof what is the length of the 
rafters if the building is 36 feet wide ? 

120. How long a rope will wind once around a cylinder 10 
feet long and 6 feet in diameter^ commencing at one end and 
going spirally around to the other ? 

121. A level lot of land 300 feet square has a wall 9 feet 
high surrounding it. What is the least height from the center 
of the ground that a man must stand who measures 5 feet to 
his sight, so that by looking over one comer he may see an 
object on the ground 20 feet distant from it ? 

(From Botsb's Problem$.) 

122. A triangular meadow whose sides each measure 100 
rods has a horse tied at one vertex. How long a rope will 
allow him to graze over half the meadow ? 

123. In a cubical room a line drawn from an upper corner 
to the middle of the floor is 24 feet. What is the size of the 
room ? 

124. Two trees stand on opposite sides of a stream 40 feet 
wide. The height of one tree is to the width of the stream as 
8 to 4, and the width of the stream is to the height of the other 
as 4 to 5. What is the distance between their tops ? 

125. A room is 12 feet wide, 16 feet long, and 8 feet high. 
A fly wishes to crawl from a lower corner to an opposite upper 
corner by the shortest route. How far must it travel ? 

126. How many feet of boards can be sawed from a log 
whose diameter is 2^2 feet, and length 16 feet, allowing one 
fourth of an inch as the width of saw ? 

127. A trapezoidal board, 12 feet long, is 16 inches wide at 
one end, and 8 inches at the other. How far from either end 
must it be cut transversely so that each part may contain one 
half of it ? 



ARITHMETICAL PROBLEMS. 125 

The Circle. 

128. A circular floor 30 feet in diameter is surrounded by 
a granary 4 feet wide and 3 feet high. Find the contents of 
the granary. 

129. If my plow cuts 18 inches wide, how many times must 
I plow around a circular quarter section to plow one half 
of it? 

130. In turning a sulky, whose wheels are 5 feet high and 
6 feet apart, so that the outer wheel is kept on the circumfer- 
ence of a certain circle, it is observed that the outer wheel 
makes two revolutions while the inner wheel makes one. 
Find the circumference of the inner track. 

131. If a man 6 feet high should walk once around the 
earth on the equator, how much farther would the top of his 
head move than his feet ? 

132. A horse is tied to one corner of a square 10-acre field. 
How long must a rope be to allow him to graze over 10 acres 
outside the field ? 

133. A, B, and C bought a grindstone 3 feet in diameter 
and 3 inches thick for $5. A paid f 2; B, f 1.75; and C, 
$1.25. How many inches must each grind off to get the 
worth of his money ? 

134. The length of the longest straight line that can be 
drawn on the surface of a circular race track is 20 rods. What 
is the area of the track, and what its width ? 

Pyramids aiid Cones. 

135. How many square yards of cloth will be required to 
make a conical tent 10 feet in diameter and 12 feet high ? 

136. Three men bought a conical sugar-loaf 20 inches high, 
and divided it into three equal solids by sections parallel with 
the base. Find the height of each section. 



126 TABLE BOOK AND TEST PROBLEMS. 

137. A conical vessel, whose altitude is 12 inches, contains 
13 gallons of water, and the area of the bottom is to that of 
the top as 5 to 3. Find the two dimensions. 

188. The diameters of the frustum of a cone are 40 and 20 
feet, and the altitude 50 feet. What length must be taken 
from the larger end to contain 12,000 cubic feet ? 

189. The volume of the frustum of a cone is 7050 cubic 
inches, its altitude 12 inches, and the diameter of its lowet 
base twice that of the upper base. Find the diameters of the 
two bases. 

140. A small bucket, one third full, is 8 inches deep, and 
its upper and lower diameters are respectively 7 and 6 inches. 
Find the volume of a ball, which, falling in, would cause a rise 
of 3 inches. 

141. Find the volume of the largest square pyramid that 
can be cut from a cone whose diameter is 10 feet, and altitude 
30 feet. 

Similar Solids. 

142. A bushel measure is 18^ inches wide and 8 inches deep. 
What are the dimensions of a similar measure containing 8 
bushels ? 

143. A conical wineglass 2 inches in diameter and 3 inches 
deep is one fourth full of water. Find depth of water. 

144. Goliath of Gath weighed 1015 pounds. What was his 
height if a man 5 feet 10 inches in height weigh 180 pounds ? 

145. There are three balls whose diameters are 3, 4, and 6 
inches respectively. What is the diameter of a ball contain- 
ing as much as the three ? 

146. A solid metal ball has a radius of 4 inches, and weighs 
8 pounds. What is the thickness of a spherical shell of the 
same metal, weighing 7f pounds, the external diameter of 
which is 10 inches ? 



ARITHMETICAL PROBLEMS. 127 

Cubes ajid Spheres. 

147. How many gallons of water in a hollow sphere, the 
diameter of which is 12 inches, and the crust 1 inch thick ? 

148. A cube immersed in a rectanglar reservoir 36 inches 
long and 16 inches wide raises the water 3 inches. What is 
the edge of the cube ? 

149. Find the diameter of a metal sphere which, dropped 
into a 6-inch cylindrical vessel, raises the water 4 inches. 

150. A cylindrical bucket 10 inches in diameter is one third 
full of water. A ball dropped into it raises the water to the 
brim. Find the depth of the bucket if the ball is just sub- 
merged. 

151. The greatest cube that can be inscribed in a given 
sphere has 10 inches for its diagonal. What are the contents 
of that portion of the sphere between its surface and the cube ? 

152. Four ladies own a ball of fine thread 3 inches in diam- 
eter. What portion of the diameter must each wind off in 
order to have equal shares of thread ? 

153. A conical wineglass which is brimful measures across 
the mouth 6 inches, and in depth 8 inches. What amount of 
water will run over if a sphere 4 inches in diameter be put 

into it ? (From Bpikobl's Live Questions.) 

MISCELLANEOUS PROBLEMS. 

154. In division of fractions why do we invert the divisor 
and multiply ? 

155. Find the value of 15 4- 9 -f- 3 - 2 x 3. 

156. A dry oak log is 20 feet long, 3 feet wide, and 2^ feet 
thick. What is its weight, the specific gravity of dry oak 
being .925 ? 



128 TABLE BOOK AND TEST PROBLEMS. 

167. The probability that Ulerich can lift 300 pounds is two 
thirds: the probability that Bierer can lift the same is five 
twelfths. If both try, what are the probabilities (1) that U. 
lifts it and B. fails ; (2) that U. fails and B. lifts it ; (3) that 
both lift it; (4) that neither lifts it ? 

168. The population of a State is 6,000,000. One seventieth 
die yearly. One sixtieth are born annually. What will be 
the population in 300 years ? 

169. A took 60, B 30, and C 10, eggs to market. They sold 
at the same price, and all received the same amount of money 
for their eggs. How was that possible ? 

160. Two trains, one 210 feet long, and the other 230, move 
on parallel tracks. When going in the same direction, they 
pass each other in 15 seconds; and when going in opposite 
directions, they pass in 3| seconds. Find the rates of the 
trains. 

161. What number divided by 11 leaves a remainder of 9, 
divided by 9 leaves a remainder of 6, by 7 leaves a remainder 
of 5, by 5 leaves a remainder of 4 ? 

162. From a cask of wine containing 100 gallons, 10 gallons 
are drawn, and the cask filled up with water ; 10 gallons are 
again drawn, and the cask filled. This process is repeated until 
100 gallons have been drawn from the cask. How much wine 
remains ? 

163. A postman delivered daily for a period of 42 days 4 
letters more than on the previous day. The aggregate delivery 
for the last 18 days was the same as that of the first 24 days. 
How many letters did he deliver altogether ? 

164. Three men, named Smith, Brown, and Hugus, with 
their sons Adam, Paul, and Dick, each own a piece of land in 
square form. Mr. Smith's piece is 23 rods longer on each side 
than Paul's, and Mr. Brown's piece is 11 rods longer on each 
side than Adam's. Each man has 63 square rods more than- 
his son. What are the full names of the boys ? 



ARITHMETICAL PROBLEMS. 129 

166. If 3 acres of grass, together with what grew on the 3 
acres while they were grazing, keep 12 oxen 4 weeks, and in 
the same manner 5 acres keep 15 oxen 6 weeks, how many 
oxen can in the same manner graze on 6 acres for 9 weeks ? 

166. Eequired the greatest possible number of hills of com 
that can be planted on a square acre, the hills to occupy only 
a mathematical point, and no two hills to be nearer than 3^ 
feet. 

167. How many trees can be planted on a small field 11 
rods square, no two trees to be nearer to each other than 1 
rod? 

168. Arrange the figures 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, in such a 
way, that, when added, the sum will be just lOO, 



BLLWOOD^S TBST PBOB. — 9. 



180 TABLE BOOK AND TEST PROBLEMS. 



ALGEBRAIC PROBLEMS. 
FACTORING. 

169. Factor — 16 ±40V«-25aj. 

170. Factor a^-a;-20. 

171. Factor aj»-2ar'-3a? + 6. 

172. Resolve x into 3 unequal factors. 

173. Factor a? - aV - aV + ajy*. 

174. Factor ar^H- 10 a? -39. 

175. Factor ic* - 6 oj - 14. 

176. Factor lOa/'-H-^- 20a. 

177. Factor a:' — y' and ^ + f. 



« + 



FRACTIONS. 

a 



M X 

178. Simplify — " 



a 

X 



1-5 



a 



179. What is the reciprocal of ^ — ^P? 

(a + 6)- 

180. Free (^""2/) of negative exponents. 



ALOEBBAIC PROBLEMS. 181 

SIMPLE EQUATIONS. 

181. Given x + y=a (1) 

x + z=b (2) 

y + z = c (3) 

182. Divide the fraction f into two parts, so that the 
numerators of the two parts taken together shall be equal to 
their denominators taken together. 

183. A person engaged to work a days on 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 ? 

184. A man can row with the current from A to B, a dis- 
tance of 42 miles, in 3 hours. When the current is two thirds 
as strong, it takes him 10^ hours to row from B to A. What 
was the velocity of the current in the first case ? 

185. A banker has two kinds of change. There must be a 
pieces of the first to make a crown, and b pieces of the second 
to make the same. ]^ow, a person wishes to have c pieces for 
a crown. How many pieces of each kind must the banker 
give him? 

186. A number is represented by 6 digits, of which the 
left-hand digit is 1. If the 1 be removed to units' place, 
the others remaining in the same order as before, the new 
number will be 3 times the original number. Find the number. 

187. The figure 7 is exactly midway between the hour and 
minute hands, and it is between 6 and 7 o'clock. What is the 
time? 

188. In an alloy of silver and copper, -- of the whole +p 

1 ^ 

ounces was silver, and - of the whole — q ounces was copper. 

n 

How many ounces of each were there ? 



182 TABLE BOOK AND TEST PROBLEMS. 

189. A and B ran a mile^ A giving B a start of 20 yards at 
the first heat, and beating him 30 seconds. At the second 
heat A gives B a start of 32 seconds, and beats him 9y\- yards. 
At what rate per hour does A run ? 

190. A certain article of consumption is subject to a duty 
of 6 cents per pound ; but, in consequence of a reduction of 
the duty, the consumption increased one half, but the revenue 
fell one third. What was the duty per pound after the reduc- 
tion? 

191. It is between 3 and 4 o'clock, and the hour hand is 
exactly opposite the minute hand. What time is it ? 

192. Find four numbers, such that the first plus half the 
rest, the second plus a third the rest, the third plus a fourth 
the rest, and the fourth plus a fifth of the rest, shall each be 
equal to a. 

193. A person has just 2 hours' spare time. How far may 
he ride in a stage which travels 12 miles an hour, so as to 
return home in time, walking back at the rate of 4 miles an 
hour? 

194. A vessel sailed with the wind and tide 60 miles, and 
returned with the wind and against the tide. She reached the 
same point in 12 hours, and the rates of sailing out and in 
were as 5 to 3. Eequired the time each way, and the strength 
of the wind and tide. 

195. Two years ago Mr. Jones was 5 times as old as his son 
John will be 2 years hence, and 3 years hence his age will 
be 15 times John's age 3 years ago. How old is each ? 

196. At 12 o'clock the three hands of a clock are exactly 
together, all moving on the same center. How long before 
each hand will be exactly midway between the other two ? 

197. 1^ ? f f ? iLj a, B, and C start at 

the same time from R to travel 40 miles to W, A walks at 
the rate of 1 mile an hour, B 2 miles an hour, while G rides 8 



ALGEBRAIC PROBLEMS. 188 

miles an hour. G rides to W, and then back until lie meets A, 
whom he picks up and carries a certain distance, then again 
rides back and picks up B, whom he carries just far enough to 
allow all three to reach W at the same time. Find the time 
of the trip. 



RADICALS. 



198. From (a — aj)V^^^^ subtract (a — «)\/^^ 

199. Extract the square root of (1 - a?)~* + 1. 

200. Prove V^ - V^ ^ J^ziVJ. 

•y/x —a V« + Va 



X 

X 



201. Find the square root of Jm' + ^nVm* — n*. 

QUADRATIC EQUATIONS. 

202. A tree a feet in height stands at the edge of a stream 
b feet in width. Where must this tree break so that the top 
may reach across the stream, while the broken parts remain in 
contact ? 

203. What is the diameter of a sphere which contains as 
many cubic inches as there are square inches in its surface ? 

201 Given x + y=za (1) 

xy=^b (2) 

to find X and y. 

206. Given Vx—Vy==2Vxy (1) 

« + y = 20 (2) 

to find X and y. 

206. If a number be increased by 6, and then the same num- 
ber be decreased by 6, the difference between the square roots 
of the results is 6. What is the number ? 

207. Given a^ + l-^O, 

or 

to find x. 



184 TABLE BOOK AND TEST PROBLEMS. 

208. From a cask containing 81 gallons of wine^ a man 
draws off a certain quantity, and then, filling the cask with 
water, draws off the same quantity again, and then there 
remain only 36 gallons of pure wine. How much wine did 
he draw off each time ? . (To be solved by pure quadratics.) 

209. A picture is 10 by 16 inches, and the surface of its 
frame is equal to the surface of the picture. Find the width 
of the frame. 

210. The number of rods around a square field is equal to 
the number of acres in it. Find the area. 

o- - ^ . VoS + Vw Va + Vn 

211. Given ^ /- = — 7= — ' 

■\/ax— -y/n Vn 

to find X, 

212. Given a^-{a-b + c)x = {b-a)c, 
to find X. 

213. An army, 25 miles from front to rear, moved forward 
just the length of itself. When it commenced to move, an 
ofl&cer started from the rear, rode to the front, and then back 
to the rear, which he reached just as the army halted. What 
distance did the officer travel ? 

214. The loudness of one bell is 3 times that of another. 
Now, supposing the strength of sound to be inversely as the 
square of the distance, at what place on the line of the two 
will the bells be equally well heard, the distance between them 
being a ? 

216. The intensity of two lights, A and B, is as 7 to 17, and 
their distance apart 132 feet. Assuming that the intensity 
varies inversely as the square of the distance, where in the 
line of the lights are the points of equal illumination ? 

216. What price are eggs per dozen, when to give 2 more 
for 12 cents would decrease the price per dozen 1 cent ? 

217. Given -^l-f- = ^1-- + 1, to find x. 



ALOEBBAIC PROBLEMS. 136 

218. A and B husk a field of corn for $ 20. When they 
husk 5 rows at a time, A husks 3 standing rows, while B husks 
the "down" row and 1 standing row. When they husk 6 
rows at a time, B husks 3 standing rows, while A husks the 
"down" row and the other 2 standing rows. How must 
they share the money ? Two elevenths of the rows are 
" down " rows : what is it worth to husk them ? 

219. Given a^ — 2aj*' + a?* = 6, to find x. 

220. There is a number consisting of two digits, which 
being multiplied by the left-hand digit gives the product 46 ; 
but if the sum of the digits be multiplied by the same digit, the 
product is only 10. What is the number ? 

221. Pind two numbers such that their sum, their product, 
and the difference of their squares, shall be all equal to each 
other. 

222. Given V^-V a-V^^ ^^^ to find cb. 

Va + '^d — Va' — ax 

(From Stoddabd Ain> Hbnkue's Algdtra,) 

223. The sum of three numbers in harmonical proportion is 
191, and the product of the first and third is 4032. Find the 
numbers. 

224. How large a trough can be dug out of a square stick of 
timber twice as long as wide and deep; the sides, end, and 
bottom to be 3 inches in thickness ; and when completed to 
contain exactly 11,772 solid inches of timber ? 

225. Find three numbers such that if the first be multiplied 
by the sum of the second and third, the second by the sum of 
the first and third, and the third by the sum of the first and 
second, the products shall be respectively 26, 50, and 56. 

226. A and B engaged to reap ^ field for 90 shillings. A 
could reap it in 9 days, and they promised to complete it in 
5 days. They found, however, that they were obliged to call 



186 TABLE BOOK AND TEST PROBLEMS. 

in C, an inferior workman, to assist them the last 2 days, in 
consequence of which B received 3«. Qd. less than he other- 
wise would. In what time could B and C each reap the field ? 

227. A railroad train, after traveling for 1 hour, meets 
with an accident which delays it 1 hour, after which it pro- 
ceeds at three fifths of its former rate, and arrives at the 
terminus 3 hours behind time. Had the accident occurred 50 
miles farther on, the train would have arrived 1 hour and 
20 minutes sooner. Required the length of the line, and the 

original rate of the train. (From Todhuntbb'b Algebra,) 

228. The sum of four numbers in geometrical progression 
is 15, and the sum of their squares 85. Find the numbers. 

229. The product of two numbers is jp, and the difference of 
their cubes is equal to m times the cube of their difference. 
Find the numbers. 



SPECIAL EXPEDIENTS. 187 



SPECIAL EXPEDIENTS. 
SIMULTANEOUS EQUATIONS. 

230. Given «* + y = H (1) 

x-\-f = l (2) 

to find X and y by quadratics. 

281. Given r^ -\-rs^^:=m^ (1) 

«« + ««4-i« = w« (2) 

<2 4_r«4.7^ = p« (3) 

to find r, s, and t 

282. Given x-i-y^lO (1) 

a;Vy = 12 (2) 

to find X. 

288. Given a?» + y'=34 (1) 

a:y = 16 (2) 

to find X and y. 

284. Given a;4-y = a* (1) 

Sy-x = y' (2) 

to find 0? and 3^. 

285. Given aj +y =4 (1) 

a^ + 2^ = 82 (2) 

to find X and y. 

286. Given « - ^ = 1 (1) 

aj»-y» = 19 (2) 

to find a; and y. 

287. Given « + y + «y(a + 3/) + «'2^ = 85 (1) 

xy + {x -{^yy + xy(x + y) = 97 (2) 

to find X and y. 



138 TABLE BOOK AND TEST PROBLEMS. 

«y-(« + y)=54 (2) 

to find X and y. 

239. Given 0*4-02^ = 104 (1) 

ic» + 3^« = 89 (2) 

to find a; and y. 

240. Given (a + y) = 70 (1) 

a? + f=^l^Z (2) 

to find X and ^. 

241. Given « + t/ = 5 (1) 

jc» + 2/« = 65 (2) 

to find X and y. 

' 242. Given ^^f^U (1) 

j»*-a;3^ = 10 (2) 

to find a; and y. 

243. Find two numbers whose product is equal to the differ- 
ence of their squares^ and the sum of whose squares is equal 
to the difference of their cubes. 

244. What two numbers are those whose difference multi- 
plied by the difference of their squares is 32, and whose sum 
multiplied by the sum of their squares is 272 ? 

245. Given 2(a? + y)«+l = (aj* + 3^)(ay + 0^ + 2/') (1) 

a + 2^ = 3 (2) 

to find X and y. 

246. Given a?* + 2/^ = 3 a: (1) 

x^ + y^^x (2) 

to find X and y. 



SPECIAL EXPEBISNTa. 189 



247. Given 

• 


xy = 7? — r^ 


(1) 




si? + y' = a?—Y 


(2) 


to find X and y. 






248. Given 


{x + y){l + xy)=:lSxy 


(1) 




(!r» + y«)(l + «y) = 208a!y 


(2) 


to find X and y. 






249. Given 


X — y = 2 


(1) 




ir»-y» = 242 


(2) 


to find X and y. 






250. Given 


2y» - 8-v^ + 2 Vi Vj'* - 4 V^ = 1* 


(1) 




V« + V8(»-Va!-4) = y + l 


(2) 


to find X and y. 






261. Given 


to* + 4to + « = 6 


(1) 




a» + 22 + w = 4 


(2) 



to find w and z. 

RECIPROCAL OR RECURRING EQUATIONS. 

262. Given {a^ + l){x' + l){x + l) = 30a^, 

to find X. 

253. Given 8aj8-16a?*- 25ar»-16a^ + 8 = 0, 
to find X. 



264. Given Jaj-1-Jl_ 1 = 5:111, 

M X -yl X X 

to find a;. 

266. Given 1 + ««= a(l + a)*, 

to find X. 



266. Given 2ajVl — aj* = a(l + aj*), 

to find 07. 



140 TABLE BOOK AND TEST PROBLEMS. 

267. Given 

(aj*+l) (aj»+l) (aj«+l) {x+1) =8aj»-|-10a?-8 aj^+lOar'+Sa?', 
to find all the real and imaginary values of x, 

HIGHER EQUATIONS. 

258. Given V?^=^ + V^^^* = V?'^^ + V^?^^, 
to find X, 

269. Given 2a?\/n^ = m(2-a«), 

to find X. 

260. Given 2aj \/nr^ = 2 - a?, 
to find X. 

261. Given V«-^=100, 
to find X, 

262. Given 8aj8 + 16a? = 9, 
to find X. 

263. Given «» + 2«« + aj=18, 
to find X. 



264. Given JIET^^^, 



to find 0?. 



266. Given x\x + Z)^2{Zx + ^), 

to find X. 

266. Given a?-6a^ + llaj = 6, 
to find X, 

267. Given a?*-6aj» + 5»' + 12aj = 60, 
to find X. 

268. Given aj» + 16aj = 128, 
to find X. 



SPECIAL EXPEDIENTS. 141 

269. Given a^-V«=14, 
to find X, 

270. Given 2x»-a» = l, 
to find X. 

271. Given aj»-6a + 4 = 0, 
to find X. 

272. Given iB»-3a' + 4 = 0, 
to find X, • 

278. Given Vac->^ = 100, 

to find X. 

274. Given 8iB*-20iB» + 20«» -60a; = 108, 

to find a;. 

276. Given a? + a?^SO, 

to find X. 

276. Given ?L lJa._2a-- = l, 

ar aj ^ x 

to find a;. 

277. Given J^±^^^ = x-2 
to find X. 

278. Given (l±^V%Z^=a, 

1+aJ* 1-a* 
to find a;. 

279. Given a;(V» + l)' = 32(aj + Va)-240, 
to find a;. 

280. Given ^^ --_!_ = §, 
to find X by quadratics. 



142 TABLS BOOK AND TS8T PROBLEMS. 

281. Given yja - ^ + yjo^ - ^ = A 
to find X. 

282. Given (aJ* + 1) y = (j(* + 1) »« (1) 

(,j/' + l)x = {3? + l)9y> (2) 

to find X and y. 

283. Given ^^+p^E^^^(j^\ 
to find X. 



284. Given a?*/"! + —jS^ - a? = 70, 



to find X. 



286. Given 


l+x» = a(l + aj)», 




to find X. 






286. Given 


iB8-6aj' + lla?-6 = 0, 




to find X. 






287. Given 
to find X, 


^ 12 + 8VS 
"^^ aj-5 ' 




288. Given 


a? + f = 72 


(1) 


to find X and j^. 


xy{x + y)z=4S 


(2) 


289. Given 


a:« + 2/8 = 35 


(1) 


to find X and 2^. 


x' + f^lS 


(2) 


290. Given 


(aj-y)(a^-y») = 160 


(1) 


to find X and y. 


(aj + y)(aj« + 3^) = 580 


(2) 


291. Given 


«y«(» + y — 2;) = a = 24 


(1) 




i»^(« — y + 2) = & = 72 


(2) 


to find X, y, and 


a^2;(— oj + y + 2) = c = 120 
2;. 


(3) 



SPECIAL EXPEDIENTS, 14S 

292. Solve the following equation by quadratics : — 

293. Solve a:8-3a^ + 4 = 0. 

294. Solve a?-6a? + 4 = 0. 

295. Solve «»-.8»' + 19»-12=:0. 



144 TABLE BOOK AND TEST PB0BLEM8. 



MISOELLANEOUS PROBLEMS. 

APPLICATIONS OF ALGEBRA. 

296. The perimeter of a rectangle is 140 feet. From one 
corner to the center is 25 feet. Find dimensions and area. 

287. In a right-angled triangle are given the difference 
between the base and perpendicular^ and also the difference 
between the base and hypothenuse, to find the sides. 

298. A room is 40 feet long and 13 feet wide. What is the 
length of the longest piece of carpet that can be laid on the 
floor of said room ? 

288. What is the area of a circular field in which the num- 
ber of acres is equal to the number of boards inclosing it; the 
fence being 5 boards high^ and each board 1 rod long ? 

300. I have a board whose surface contains 49f square feet. 
The board is 1^ inches thick, and I wish to make a cubical 
box of it. Eequired the length of one of its equal sides. 

901. Two poles stand on the same plane. From the top of 
the shorter to the foot of the longer is 40 feet, and from the 
top of the longer to the foot of the shorter is 60 feet. These 
two lines intersect 15 feet from the plane. Find the height 
of each pole. 

802. Draw a right line parallel to the base of a triangle, so 
that the parallel shall be equal to the difference of the lower 
segments. 

303. A pole standing perpendicularly on a hillside was 
broken by a storm a feet from the top, which touched the 
ground up the hill b feet from the foot of the pole. The mend- 
ing shortened it so that the broken part was exactly the length 
of the stump ; but it broke again at the same place, when it 



MISCELLANEOUS PROBLEMS. 145 

was observed that the top touched the ground c feet nearer the 
foot of the pole than before. Find original height of the pole. 

304L The hypothenuse of a right triangle is 35, and the side 
of the inscribed square is 12. Find the sides. 

305. A square farm contains as many acres as there are 
boards in the fence inclosing it. The fence is 7 boards high, 
and each board is half a rod long. How many acres are there 
in the farm ? 

808. A solid globe 1 foot in diameter has been blown into 
a hollow sphere one eighth of an inch thick. Find diameter 
of the hollow sphere. 

807. A circle is inscribed in an isosceles triangle whose 
base is twice its altitude. Show that the radius of the circle,- 
divided by the altitude of the triangle, is the V2 — 1. 

808. The area of a triangle is a rods, one side is b rods, and 
the other two sides are to each other as 2 to 3. Find them. 

809. Given AB to be a 
straight line, AC = 60, 
CD =20, the angle ECD 
to be the complement of 
the angle J., the angle 
EDB to be twice the angle 
A, and EB to be a perpen- 
dicular to AB, to find EB. 

^ 810. A tree 74 feet high, standing perpendicularly on a hill- 
side, was broken by the wind, but not severed ; and the top 
fell down the hill, striking the ground 34 feet from the root of 
the tree, the horizontal distance from the root to the fallen 
part being 18 feet. At what height did it break ? 

811. From a point within a square, three lines are drawn to 
three corners of the square. Find a side of the square. 

>f 812. The sides of an equilateral triangle are 200 feet. At 
each comer stands a pole, the height of the first being 30 
bllwood's test pbob. — 10. 




146 TABLE BOOK AND TEST PROBLEMS. 

feet, the second 40, and the third 50. At .what distance from 
the foot of each pole must a fourth pole be placed so that its 
top may just reach the top of each of the others ? How long 
is the fourth pole^ the triangle being a horizontal plane ? 

313. A circle is inscribed in a quadrant. Eequired the 
radius of the next greatest circle that can be drawn in the 
quadrant^ which shall be tangent exteriorly to this inscribed 
circle. 

GEOMETRICAL PROBLEMS, ETC. 

314. Construct the square root of any number N. 
316. Construct the square root of 46. 

316. Construct the square root of 19. 

317. Let a square be inscribed in a circle, a circle in this 
square, a square in this circle, and so on. In such a construc- 
tion find the ratio of the diameter of the first circle to that of 
the serenth. 

318. Eind the sum of the infinite series of diameters in 
the above. 

319. Demonstrate that the square inscribed in a semicircle 
is to the square inscribed in the entire circle as 2 to 6. 

320. Demonstrate that the square inscribed in a semicircle 
is to the square inscribed in a quadrant of the same circle as 
8 to 5. 

321. Demonstrate that the side of an equilateral triangle 
inscribed in a circle is to the radius as the V3 is to unity. 

322. Eequired the radius of the largest circle that can be 
inscribed in a triangle whose sides are respectively 11, 12, 
and 13 feet. 

323. Cut a board 10 feet long and 2 feet wide into four 
pieces which may be put together to form a square. 

324. A diameter of a circle is intersected by a chord. Show 
i.\2X the square of the radius is equal to the square of the diA- 



MISCELLANEOUS PROBLEMS, 147 

tance from the center of the circle to the point of intersection, 
plus the rectangle of the segments of the chord. 

325. Suppose a hemispherical vessel 3 feet in diameter were 
filled with water, and that ice would freeze uniformly 6 inches 
thick on the top and on the inner surface of the vessel. What 
would be the volume of ice thus formed ? 

326. If a circle be described touching the major axis of an 
ellipse in one focus, and passing through one extremity of the 
minor axis, half the major axis will be a mean proportional 
between half the minor axis and the diameter of this circle. 

327. The difference between the inscribed and circumscribed 
squares of a circle is 72 square feet. Find diameter of circle. 

328. Trisect the diagonal of a parallelogram. 

-^329. In the center of a rectangular lot 150 feet long and 80 
feet wide is a stake, to which a horse is tied by a rope 60 feet 
long. Over how much can the horse graze ? 

330. Three circular fields, each containing 80 acres, lie 
joining each other so as to inclose a small triangular piece 
of land. Find area of inclosed triangle. 

331. One acre of land lies between three equal circles drawn 
tangent to each other. Find the diameter of circles. 

332. Find the perimeter of a rhombus whose area is 216 
square inches, and one of whose diagonals is 24 inches. 

333. A ball 2 feet in diameter is in a corner. What is the 
diameter of the largest ball that can lie on the floor behind 
this one, both touching the same walls ? 

334. How high above the surface of the earth must a person 
be raised that he may see one third of the surface ? 

336. The three distances from a point within an equilateral 
triangle, to the angles, are a, h, and c. Determine the triangle. 

336. In a circle whose radius is 3, find the area of the part 
between parallel chords whose lengths are 4 and 5, both being 
on the same side of center. 



148 TABLE BOOK AND TEST PROBLEMS. 

837. By boring through the center of a sphere 4 inches in 
diameter, what volume will be bored away by an auger 2 inches 
in diameter ? 

388. Two circles touch each other, and also the base and arc 
of the semicircle in which they are inscribed. If their radii 
are a and b, what is the radius of the semicircle ? 

339. A hemispherical kettle of known and uniform thick- 
ness of shell is made from a given quantity of copper. Mnd 
an expression for the capacity of kettle. 

340. Demonstrate that the square described on the hypothe- 
nuse of a right triangle is equivalent to the sum of the squares 
on the other two sides. 

341. A conical glass 9 inches deep, and 6 inches wide at the 
top, is one third full of water. What is the radius of a ball 
that will just immerse ? 

TRIGONOMETRICAL PROBLEMS. 

342. At what latitude does a degree of longitude equal just 
half a degree at the equator ? 

343. What is the length of a chord cutting off one fifth of 
the area of a circle whose diameter is 10 feet ? 

344. From a point without a circular pond two tangents to 
its circumference are drawn, forming with each other an angle 
of 60**, and the length of each tangent is 18 rods. Find the 
diameter of the pond. 

345. A wheel 4 feet in diameter is sunk 1 foot in the mud. 
What portion of the area is in mud ? 

346. A ship, starting from the equator, sails due northeast 
1000 miles. Supposing the earth a sphere, find the ship's 
latitude. 

347. Find the angle of elevation, and velocity of projection, 
of a shell, so that it may pass through two points : the co6rdi- 



MISCELLANEOUS PROBLEMS. 149 

nates of the first being a?' = 1700 feet, y' = 10 feet ; and of the 
second, a" = 1800 feet, y" = 10 feet. (Olmstkd's coiuge PhUotcphy,) 

848. Upon how much surface can a horse graze when tied 
by a rope 100 feet long to a corner of a barn 25 feet square ? 

348. If a heavy sphere, whose diameter is 4 inches, be let 
fall into a conical glass full of water, whose diameter is 5 and 
altitude 6 inches, how much water will run over ? 

850. A horse is tied to a stake in the circumference of a 
circular 10-acre field. How long must the line be to allow him 
to graze over one acre inside the field ? 

851. Twenty acres are inclosed in circular form, and a 
stake is driven 10 feet from the circumference. How long a 
rope fastened to the stake will allow an animal to graze on 
one acre inside the fence ? 

PROBLEMS INVOLVING CALCULUS. 

852. Find the number whose nth root exceeds itself by the 
greatest possible quantity. 

858. The side of a square inscribed in a right triangle 
is 12. What are the sides, if the hypothenuse is the least 
possible ? 

854. How far above the center of a horizontal circle whose 
radius is 2 V2 rods, must a light be placed that the illumina- 
tion may be a maximum ? 

PROMISCUOUS PROBLEMS. 

855. Show that 9.45 = 9,^. 

J356. A flash of lightning was seen 8 seconds before the 
thunder was heard. How far away was the cloud ? 

857. How far will a nail head in the tire of a wheel move 
in driving 3.1416 rods ? 

858. A can dig a row of potatoes while B weeds 1. B can 
dig a row while A weeds 4, If both receive $ 6 a day, what is 
the share of each ? 



160 TABLE BOOK AND TEST PROBLEMS, 

309. A composition of gold and silver which weighs a 
pounds loses m pounds in water. Now, a pounds of gold lose 
n pounds in water, and a pounds of silver lose p pounds in 
water. How many pounds of gold in the composition ? 

360. A and B can dig a ditch in 12 days ; B and C, in 20 
days ; and A and C, in 15 days. How long would it take all 
of them to dig it ? How long each ? 

361. Three poles, each 50 feet long, were erected on a plain 
so that the upper ends met, and the lower ends were 60 feet 
apart. What length of plumb-line was required to reach from 
their point of meeting to the ground ? 

362. Bought $7000 worth of bonds due in 20 years, the 
interest on which was 7 per cent, payable semi-annually. They 
yielded me 8 per cent, payable semi-annually. What did I 
pay for them ? 

363. Given a?' - 6 a? = 7, 
to find X without completing the square. 

364. Given a?2 + 2a;-8 = 0, 
to find X without completing the square. 

365. How much longer is the fence around a rectangular 
10-acre field, whose breadth is one fourth the length, than 
the fence around a square field containing the same number of 
acres ? 

366. Satisfy the conditions in the equation 

ic2 4- aj = a square number. ^ 

367. A man pulls out the stumps in a field at the rate of 
25 cents apiece, and piles the stones at 10 cents a score. One 
stump occupies him as long as 40 stones. He works 3 days 
and earns $ 8, then goes on at the same rate of working, and 
finishes the job in 3f days more, earning altogether $ 20. How 
many stumps and stones were there in the field ? 



PBOBLEMS WITH CUBIOUS RESULTS. 161 



PROBLEMS WITH CUBIOUS RESULTa 

DIGITS. 

368. Multiply the number 12,345,679 by some number that 
will make the product contain but one of the nine digits. 

369. Arrange the nine digits so that their cube root can be 
extracted. 

"ONE CENT." 

870. Suppose one cent to have been placed at 6% compound 
interest at the commencement of the Christian era. What 
would the amount have been Jan. 1, 1882 ? 

(From fhe late B. B. Beitz, in Mathemaiieal Maff€tzine.) 

INVOLUTION OF IMAGINARY QUANTITIES. 

871. Show the absurdity of assuming that V— a* = a?. 



372. Find the values of (V--T)^ (V-T)«, (V^^)*. 
378. Why does mimis multiplied by minus give plus 9 

THE ZERO FACTOR. 

874. Prove the absurdity of the equation a^x. 

376. Prove the absurdity of the proportion —a: a:: a: —a, 
in which a is any finite quantity. 

SOMETHING TO INVESTIGATE. 

876. It is well known that any number of figures, multiplied 
by 3 or any of its multiples, and the digits of the product 
added until a single digit results, will give as a result either 
3, 6, or 9. Exemplify this. 



152 



TABLE BOOK AND TEST PROBLEMS. 



THE PROPOSITION OF ARCHIMEDES. 

877. "Archimedes, the celebrated philosopher and mechanic 
of Syracuse, once exclaimed, * Give me a place to stand, and I 
will move the world.' " Prove his inability to do this. 

(From Professor B. F. Buslxsok of Oneida Castle, N.T., in Nate» and Qneriet.) 



1888. 

378. Substitute numbers for these 
letters, so that when added in any one 
of the following ways the sum will be 
1888 : — 



a + 5 + c + d. 
d •^h-\-l +i>. 

a +/-f ^H-2>. 

Jc +1 +0 -\-p. 
a +b + o +p. 
m + i +d + h, 
m + i +k-{-o. 



a + e + i + m. 
p-^-o +n 4- ?ft. 
i +j +k -f-i. 

^ + ^ +i + ^• 

a-^b'\-e 4-/ 

t + i + «i 4- ». 
c -{- d -\- m -{- n. 

a-\-c+e -f^. 

d + h + b +f. 



a 


b 


c 


d 


e 


f 


9 


h 


• 

t 


J 


k 


I 


m 


n 





P 



^ + f+j 4- w. 

c + d + g-^-h, 

f-\-9+J+k. 
P'{'l +e + a. 
p + l^n-^-g. 



SUMMATION BY SUBTRACTION. 



379. Find the sum of 379, 8452, 31, and 60 by subtraction. 



8£BIE8. 153 



SERIES. 

380. Sum the series — 1 = 1 = 1 — etc. 

2.3.4 4.5.6 6.7.8 

381. Find the sum a of the series 

1+2+3+4 2+3+4+5 ^ n+(n+l) + (n+2) + (n+3) 
1.2.3.4 2.3.4.5 w(w+l)(n+2)(n+3) 

382. Find four numbers in proportion such that their sum 
shall be a, the sum of their squares 6, and the sum of their 
cubes c. 

383. The sum of the terms in an arithmetical series is 
8 = 1146, the sum of their squares is 6 = 126,746, and the sum 
of their cubes is c = 15,409,116. Find the first term a, the 
common difference c?, and the number of terms n. 

384. The sum of the terms in a geometrical series is « = 
2059, the sum of their squares is 6 = 953,317, and the sum of 
their cubes is c = 550,434,529. Find the first term a, the 
ratio r, and number of terms n. 

385. The sum of n terms of the series (2* . 4* - 1^ • 3*) + 
(6« . 8*- 5* . 7*) + (10* . 12«- 9* . 11«) + . . . etc., is « = 22,395,834,- 
549,559. Find the number of terms 7t. 

386. Find the nth term, and the sum of n terms, of the 
series 1, 19, 10, 14^, 12J, etc., in which each term after the 
second is an arithmetical mean between the preceding two. 

887. Find the nth term, and the continued product of n 
terms, of the series 4, 64, 16, 32, 16V2, etc., in which each 
term after the second is a geometrical mean between the pre- 
ceding two. 

388. Given - — ^^^"if ^ , = -. to find by series the 

l4.2aj-3a:* + 2aj' a 

value of X when a is greater than 1. 



1 

1 



154 TABLE BOOK AND TEST PROBLEMS. 

389. Sum to n terms the series whose general term is 
n*(3n — 2)iC""*, and find the numerical value of the same when 
x = 5f and n = all integral values from 1 to 100 ; also when 
X = .999, and n = all integral values from 1 to oo. 

390. A note of $ 400 at annual interest amounted to $ 441.50 
in 4 years. Required the rate. 

391. A person who enjoyed a perpetuity of $1000 per 
annum provided in his will that after his death it should 
descend to his son for 10 years, to his daughter for the next 
20 years, and to a hospital forever afterwards. What was the 
value of each bequest at the time of his decease, allowing com- 
pound interest at 6 per cent ? 

392. The number of balls in the rth = 13th course of a com- 
plete rectangular pile of cannon balls is & = 32, and the 
number in the qth = 7th course is c = 140. How many balls 
in the pile ? 

393. If the first pair of doves had produced not any at the 
end of their first year, 2 pairs at the end of their second, 3 
pairs at the end of their third, not any at the end of their 
fourth, 2 pairs at the end of their fifth, 3 pairs at the end of 
their sixth, and so on, repeating the numbers 0, 2, 3 in regular 
order; and if each pair of doves produced had bred in a simi- 
lar manner, and none had died, — how many pairs of doves 
would there have been at the end of the first century ? 

394. Sum the series ^^jtA' 4. ?1±A' + ?!±^ + etc. 



Part III. 



SOLUTIONS. 



KoTS. ~Tlie sohttion* mre iramberecl to correspond vitfa the probtemi in PartU., 

to wbich they refer. 

ARITHMETICAL SOLUTIONS. 

DENOMINATE NUMBERS. 

1. 3 reams = 1440 sheets ; 5 quires = 120 sheets ; the total = 1579 
sheets. Since there are 20 quires in a ream, and 1 quire to a ream is 
wasted, the waste = ^ of 1579 = 79-. 1679 - 79 = 1500, the number 
used. Each sheet makes 2 posters : hence 1500 x 2 = 3000, the number 
printed. 

2. For every pound he sells 15| ounces. 232 pounds or 3712 ounces 
-f- 15.75 = 235}}, the number of fraudulent pounds he sells. 235}| ~ 232 
= 3}] fraudulent pounds. 18| x 3|f = 69^ cents, the gain. 

8. The Troy pound, by which gold is weighed, contains 5700 grains; 
and the Ayoirdupois pound, which is used in weighing lead, contains 7000 
grains : hence the lead pound is the heavier. 

7000 grains -^ 16 = 437} grains = 1 ounce Avoirdupois. 

5760 grains -^ 12 = 480 grains = 1 ounce Troy. 

Hence the ounce of gold is the heavier. 

4. As his watch is too slow by 6} hours, he must be 15 x 6} = 97° 30' 
east of Philadelphia, or 97° 30' - 75° 10' = 22° 20' east longitude. He 
must therefore have travelled through 360° - 97° 30' = 262° 30'. 

6. Contents of cube = 2x2x2 = 8 cubic feet. 1000 x 8 = 8000, 
weight of same bulk of water. 8000 x 19.36 = 154,880 ounces = 9680 
pounds. 

Notes. — 1. The specific gravity of any aabstance is ita weight compared with the 
weight of an equal bnlk of water. 

2. Aaemne that a cnbic foot of water weighs 1000 oonces; and a cable foot of air at the 
earth's surface, about 1.22 ounces. 

1R6 



166 TABLE BOOK AND TEST PROBLEMS. 



LEAST COMMON MULTIPLE AND GREATEST COMMON 

DIVISOR. 

6. L.C.M. of 8,6, and7 = 106; G. CD. of 4, 6, and 10 = 2: 

.-. 105 -^ 2 = 62). Ans, 

7. G.C.D. of 2, 4, and8 = 2; L.C.M. of 3, 7, and 9 = 63: 

.-. the required G. C. D. = 2 h- 63 = A- 

8. The L. C. M. of 3, 7, 12, and 14 = 84 : hence they will toll together 
once in every 84 minutes ; and in 7 hours, or 420 minutes, they will toll 
together 6 times, not counting the first tolling. 

9. A makes a trip every 4 days, B every 6 days, and C and D every 8 
days. The L. C. M. of 4, 6, and 8 = 24 ; that is, they start out together 
every 24 days. But this includes the stay-at-home day : hence 24 — 1 = 23, 
the required number of days. 

10. B will overtake A once in every 320 -i- 6 = 64 minutes. A and C 
will meet once in every 320 -f- (8 + 24) = 10 minutes. Therefore all will be 
together in the L. C. M. of 64 and 10, which is 320 minutes, or 6 hours 
20 minutes. 

PARTNERSHIP. 

11. Being equal partners, they must share all indebtedness equally: 
hence A should give to B } of $240 + i of $260, or $260. 

12. Had A owned f of the stock, his gain would have been f of $ 249 
= $ 186.76. $ 186.76 - $ 176 = $ 11.75 = the gain on $ 10 of the stock : 
hence $1 of stock gains $1,176. To gain $249 requires $249 -i- 1.175 
= $211.91, the whole stock. 

18. This was a partnership, A owning ^j, B J, and C JJ. They have 
for division 3 loaves and 24 cents : hence A should have } of a loaf and 5 
cents ; B, 1 loaf and 8 cents ; C, -y^ of a loaf and 11 cents. But each ate 
1 loaf : hence B should get 8 cents. A, having eaten { of a loaf more 
than his share, must deduct the price thereof from his 6 cents. Since 
each loaf cost 6 cents, f of a loaf is worth f of 6, or 2} cents; and 5 — 2 J 
= 2} cents = A's share. The 2J cents go to C, who supplied the bread to 
A : hence C*s share = 11 -f 2J = 13} cents. 

14. First Solution, If Smith had been a partner all the way, he would 
have paid } of $ 10, or $6. But since he rides only half the distance, he 
should pay only } of $6, or $2 J. 



ABITBMETICAL SOLUTIONS. 157 

Second Solution, Jones rides 20 miles, and Smith rides 10 miles. 
Therefore, since Jones rides twice as far as Smith, he should pay twice 
as much. Hence Jones should pay f 6J, and Smith $3^. 



PROPORTION. 

16. If i of 6 were 3, f of 6 would be 9, and J of 6 would be 4}. If 8 
were J of 6, 9 would be } of 6, or 6 ; and 6 would therefore be 4 : hence 
i of 6 would be 2. The two statements are different. In the first case 
the condition increases our numbers 60 per cent, while in the second 
instance it decreases them 33i per cent. The problem might be written 
thus : (a) If 2 were 8, what would 3 be ? (6) If 3 were 2, what would 
8 be ? By proportion we have, 

2 : 3 : : 3 : X = 4}, Ans. to (a); 
and 3:2::3:x = 2, Ana. to (b). 

16. Suppose n = number of men that carry off the influx. As the 
number of men increases, the time of emptying decreases. Hence we have 
this proportion: 12 - n: 15 - n: : 30: 40. .-. w = 3. Then 12 - 3 (= 9) 
men empty the cistern (without the influx) in 40 minutes, or 1 man does 
it in 360 minutes. Hence 18 — 3 (= 16) men can empty it in 300 + 16 = 24 
minutes. 

17. In all such problems the weights borne are inversely as the dis- 
tances of the carrying points from the point of suspension. Hence, A 
being 1} feet, and B 2} feet, from the point of 8usi>ension, A carries 5 
pounds to B's 8. Therefore A carries 126 pounds, and B 76 pounds. 

18. From the earth^s center to the given height is 1} times the radius 
of the earth. Since gravitation varies inversely as the square of the dis- 
tance between the centers of gravity, we have the proportion 

(1})2 : 12 : : 1800 : x, the required weight. 
Whence x = 1800 -?- f = 800 pounds. 

19. Since A wins 9 while B wins 8, B^s skill is } of A^s ; and since A 
wins 5 to C's 3, C's skill is f of A's. Therefore B's skill is to C's as J to 
{, or 6 to 9. Hence, out of 70 games, B should win 26, and C 46. 

90. 20 : true weight : : true weight :SH; whence the square of the true 
weight = 20 X 31 J = 626, and the true weight = V625 = 26 pounds. 

81. The sectional area being unchanged, the volume and velocity of 
air currents are inversely proportional to the square roots of the lengths. 



158 TABLE BOOK AND TEST PBOBLEM8. 

Hence >/S00 : vlOOO :: 20,000 : (x), 

or 2: 1:: 20,000: (x); 

whence x = 10,000 cu. ft 

38. We assume the clock to be on the equator. The third law of the 
pendulum is this : The lengths are inversely proportional to the squares 
of the numbers of vibrations in a given time. The proportion is ex- 
pressed thus : Liliin^: iV^. The length of a second^s pendulum at the 
equator is 39 inches : hence we have the proportion 15 feet (= 180 inches) : 
39 inches : : 1 : x, whence x = ^^ ; that is, the 16-foot pendulum vibrates 
^^ of a time in 1 second. Jn a minute it vibrates ^^ x 60 = 13 times. 
Since it loses 3 hours, or 180 minutes, in a week, it loses 13 x 180 = 2340 
vibrations. In 1 hour it loses 2340 -r- 168 = 13.928 vibrations. It now 
vibrates 13 x 60 = 780 times in an hour, and to insure correct time it 
must vibrate 780 + 13.928 = 793.928 times. Hence this proportion : 
16: ^: : (793.928)2 : (780) «, whence I = 14.469 = length of pendulum after 
correction. Hence the <<bob*' must be raised 15 — 14.469 = .631 of a 
foot = 6.372 inches. 

PROFIT AND LOSS. 

23. The present worth of the note is $240 -f- 1.20 = $200. Hence A 
gains $200 — $190 = $10, and interest on the same for 4 years at 5 per 
cent, which is $2. 

94. Let 100 per cent = cost Then 4 — i = } of selling price = cost = 
100 per cent | = | of 100 per cent = 14 1, and { = 114f per cent = selling 
price. Then 114 J — 100 = 14f per cent = rate of gain. 

96. 180 per cent of first = 120 per cent of second = 90 per cent of 
third. Hence the third cost twice the first, and the second 1^ times the 
first. Then the Ist + twice the 1st + IJ times the 1st = 4J times the 
1st = $90, and the 1st = $20. $20 x 2 = $40 = 3d. $20 x li=$30 = 2d. 

26. First Solution, By the problem, f of cost = selling price. By 
conditions, the selling price is 115 per cent of real cost, plus 115 per cent 
of $ 600 ; or J J of cost -f $ 576 = selling price. Therefore $ 676 is the dif- 
ference between f of cost and jj of cost, or ^ of cost Hence the cost 
was $576 X 20 = $11,600. 

Second SoliUion. Difference in rate of gain = 20 per cent — 16 per 
cent = 6 per cent, which is $500+16 per cent of $600, or $676. Then 
1 per cent = | of $ 676 = $ 116, and 100 per cent = $ 11,500. 



ARITHMETICAL SOLUTIONS. 169 

97. 100 per cent less 10 per cent = 90 per cent. 
00 per cent less 10 per cent = 81 per cent. 
81 per cent less 6 per cent = 76.95 per cent. 

150 per cent of 76.95 = 115.425 per cent of the original cost. 

28. (^37.50 X 30) + ($21.60 x 20) = $1557 = selling price. $1557 - 
$1500 = $57 = gain = 1 ox and 1 cow. Then $57 x 20 = $1140 = 20 
oxen and 20 cows. Hence 10 oxen must have cost $ 1500 — $1140 = $360, 
and 1 ox cost $36. Then $57 - $36 = $21» cost of 1 cow. 

89. 120 per cent of real sales = 140 per cent of supposed sales. 

1 per cent of real sales = 1 J per cent of supposed sales. 

100 per cent of real sales = 116} per cent of supposed sales. 

Hence he sold 16} per cent = | more than he intended to sell, and the 
length of his yard-stick was 36 x 1| = 42 inches. 

80. $16 — $10 = $6 = 1} tunes B*s gain. Hence B gains $4, and the 
article cost him $12. His rate of gain is ^, or 33^ per cent. Then 
$ 12 = 133} per cent of what A paid. Hence A paid $ 12 -^ 1.33} = $9. 

81. $1 + $.50 = $1.50 = gain by the 3-cent increase. An increase of 
1 cent per pound would give a gain of $1.50^-3 = 50 cents. Hence 
50 -i- 1 = 50 = the number of pounds. 

82. Let 10$ per cent = the boat. Then the tannery = 116 per cent, and 
the breweiy = 116 per cent x .96 = 111.36 per cent. 75 per cent of the 
brewery = 83.52 per cent of the boat. 100 per cent — 83.52 per cent 
= 16.48 per cent, the loss. .*. 16.48 per cent of the boat = $ 103, 1 per 
cent = $6.25, and 100 per cent = $625. Tannery = $625 x 1.16 = $725. 

88. Marked price = 150 per cent = 120 per cent of the cost of real 
amount sold. 150 -h 120 = 1.25 = 1 per cent ; and 125 per cent = 100 per 
cent, or what is sold for a yard. Hence 125 — 100 = 25 per cent, the 
amount sold by mistake, = 9 inches. Then 36 + 9 = 45 inches, length of 
yard-dtick. 

84. Let 100 per cent = value of goods. Then 80 per cent = cost. 50 per 
cent of 80 per cent = 40 per cent of value = the entire gain. Hence 40 
per cent — 20 per cent = 20 per cent = the ** honest " gain. 

85. Let 100 per cent = original value. Then 70 per cent = cost, and 

52.5 per cent = selling price. 100 per cent — 5^.5 per cent = 47.5 per cent 

100 
of first value = $55. Hence original value was $55x -— = $115(i« 

($ 115^ X .70) X .75 = $60.78Jf . 



160 TABLE BOOK AND TE8T PROBLEMS. 

86. Let 100 per cent = the cost. Then IH} per cent = the selling price. 
For 14 pounds he pays the price of 13, or |J of 100 per cent = 92f per 
cent. He sells 18 pounds for the price of 14, or j J of 114 J per cent = 123 ^i^ 
per cent. The gain is 123^^ - 92} = 30|J per cent. The gain ought to 
have been 14 J per cent. Therefore 30Jf — 14f = 16|} per cent = gain by 
fraud = $29 ; and 100 per cent = $182, the required cost 

87. First Solution. Let 100 per cent = the cost. Then 80 per cent = 
supposed cost. Now, 30 per cent of 80 per cent = 24 per cent of 100 per 
cent. Hence, if the 20 per cent (supposed decrease) yield 24 per cent, 
100 per cent yields 120 per cent. Therefore 120 per cent — 100 per 
cent = 20 per cent = gain. 

Second Solution. ^ of selling price — yj^ of same = ^ of selling 
price = difference in rates = 30 per cent = ^j. Whence selling price = 
120 per cent of cost. .*. gain = 20 per cent. 

88. First Solution. $200 - $100 = $100 = part of gain. 3 per cent 
of $100 = $3, which is also gain, being interest. Hence $103 is the gain. 

Second Solution. He gains $100 + the interest of $200 for 6 months 
at 6 per cent, or $106. 

Third Solution. Must A pay interest for 6 months on $100 ? If not, 
the following is correct: $100 -s- 1.03 = $97,087, the present worth. 
$200 - $97,087 = $102,913, the gain. 

Fourth Solution. Let A set apart $100 of his cash totieet his pay- 
ment at the end of 6 months. The other $ 100 is his gain, and it is his 
only gain. 

89. If I get interest from purchaser, I receive $ 103 in 6 months, which 
is equivalent to $100 now. Hence $200 - $100 = $100, the loss. If I 
do not get interest, I receive $ 100 in 6 months, the present worth of which 
is $97,087. Hence I lose $200 - $97,087 = $102,913. 



STOCKS AND BONDS. 

40. Each bond yields $6 interest yearly. If $5 = 7 per cent of cost, 
100 per cent = ^^ of $5 = $71f. $100 - $71} = $28}, the discount. 
$28} H- 100 = 28} per cent, the rate of discount. 

41. 5 per cent semi-annually = 10 per cent yearly. Then $2000 = in- 
come = 10 per cent of the face value of the stock, and $20,000 = face = 
2000 shares. 1 share costs $10 x 1.20} = $12.0}, and 2000 shares coat 
12.0} X 2000 = $24,060. 



ARITHMETICAL SOLUTION 3. 161 

48. .1^21,200 -T- 106 = 200 shares. Each share yields $6 in gold yearly. 
Hence $6 x 200 = $1200, income in gold. $1200 xli = $1350, mcome 
in currency. $6 in gold = $6.76 in currency ; and $6.76-5- 106 = e^Vy 
per cent. 

48. Each U. S. 6 brings in a yearly interest of $5. But since the cost 
of the bond was 76 per cent, or $76 for a $100 bond, the $6 is the yearly 
interest of $76. Hence 6 -*- 76 = y^ = ^i V^^ cent = rate of income. 
Each U. S. 6 produces $6 yearly. But when bought at 86 per cent, the 
rate of income is 6 -j- 86 = T^j per cent. Hence the 6's are 7^ per cent 

— 6 J per cent = |^ of one per cent, the better investment. 

44. The interest on each bond for 16 years = $76, and the loss by 
selling at par = $10. Therefore the net gain is $66 in 16 years, or $4j^ 
in 1 year. If the investment, $110, yields $4 J, the rate is 4J -r- 110 = 
m per cent. 

45. 117f X 2000 = $2362.60 = cost of first kind of bonds. Then $6000 

- $2362.60 = $2647.60 = remainder. $2647.60 -^ llOJ = 23J bonds, and 
a surplus of $60.75. The interest on the $2000 6's = $120 annually. 
The interest on 23 J bonds at 4} per cent = $105.76. Then $120 + 
$105.76 = $225.75 = annual income. 

46. Cost of first kind of stock = 107} + } = 108 : hence the gain on $ 1 
= iSff = A ^^ ^^' ^^^* °^ second kind of stock = 98J -|- J = 99: hence 
gain on $1 = /^ of $1, and on $2 it is (g. Then the gain on every $3 
invested is ^^ -|- JJ = ^^ of $1. Consequently $3 were invested as often 
as ^^f is contained in $3348, or 21,384 times ; that is, the whole amount 
invested was $64,162, of which J, or $21,384, was invested in the first 
kind of stock, and J, or $42,768, in the other. 

INTEREST. 

47. A's fortune = § of B's. J of A's = J of B's. Then B's -|- J B's = 
?/ of B's = amount on interest. Amount of $ 1 for 6 years at 8 per cent 
= $1.48. $8880 -^ 1.48 = $6000 = amount on interest. Hence V^ of B's 
money = $6000, and B's = $4163}f f of $4153f} = $2769^% = A's. 

48. It is evident that the note must be drawn for $60 + the interest. 
It is also evident that 6 per cent of $60 -|- 6 per cent of the interest must 
be the interest. Hence $ 3.60 -|- 6 per cent of the interest = the inter- 
est, or $3.60 = 94 per cent of the interest. The interest, therefore, is 
100 (3.60 -f- 94) = $3,829, and the face of the note is $60 -|- $3,829 = 
$63,829. ^ 

ellwood's test prob. — 11. 



162 TABLE BOOK AND TEST PROBLEMS. 

i9. The interest for 2 years at 5 per cent rs ^ of the principal. Since 
the $730 includes ^ of the note, and also part paid of the debt, f 1000 
-- 9730 = 9270 = ^ of note. Hence the note must be given for 9270 
X y = 9300. 

50. In two years the holder of the estate receiyes two payments of 
9 100 each, one of which might have been pnt at interest for 1 year at 6 
per cent. Therefore at the end of the second year he receives 9206 ; and, 
letting 8 = the sum to be paid out biennially, 9205 —8= the net income, 
which is equal to the compound interest of 91000 for 2 years, or 9102.50. 
Hence 9205 ~ iS^= 9102.50, or 5= 9102.50. 

51. Interest of 91 for 03 days at 6 per cent = .0155. Proceeds of 91 
= 9 1 — -0155 = .0846. Hence in 03 days the bank makes .0156 on .0845, 
which is equivalent to about 0.2 x)er cent. 

None. — By the interest tables, the interest on $1 for 93 days at 6 per cent is 2 cents. 
Then the bank makes 2 cents on 98, or about 8 per cent. 

52. The compound amount of 9 1 for 3 years at 6 per cent is 9 1.191016, 
and for 96000 it is 91.101016 x 6000 = 97146.096. Then 91+ 91.06 + 
91.1236 = 93.1836 ; and the annual payment == 97146.096 -j- 93.1836 = 
92244.66. 

58. Let 100 per cent = the principal. The difference in time is 1|^ 
years. Interest on 100 per cent for IJ years at 8 per cent = 10 per cent. 
Then 960 — 10 per cent = interest on principal at 2 per cent for the 
required time, or 

925 — 6 per cent = same at 1 per cent (1) 

1 yearns interest = 6 per cent, and the entire interest is 9325 — 100 per 
cent, which divided by 1 year's interest gives the time from date of note 

to May 1, 1885. Hence ^^25- 100 percent ^ ^^^ number of years. At 

6 per cent 

1 per cent the interest on the principal for 1 year is 1 per cent, and for 

the required time it is 

9326 - 100 per cent ^ ^ ^^^ __ 93.25 - 1 per cent ^^^ 

6 per cent 6 per cent 

Now, equatmg (1) and (2), we have ^3-^5 - 1 per cent = ^25 - 5 per 

6 per cent 

cent. Multiplying both sides by 6 per cent, or yj^, we have 93.25 — 1 

X)er cent = 9 1.50 — .3 per cent. Hence 93.25 — 91-50 must be 1 per 

cent — .3 per cent, or .7 per cent = 9 1.75. Then 1 per cent = 91.75 x ^ 

= 92.60, and 100 per cent = 9250, the face of the note. The interest for 



ARITHMETICAL SOLUTIONS. 168 

1 year = $15: hence to produce $826 — $250 = $75 requires 75 -^ 15 
= 5 years. The date must therefore be 5 years prior to May 1, 1885, or 
May 1, 1880. 

DISCOUNT AND PRESENT WORTH. 

M. First Solution. By bank discount, the interest of $ 1000 for 68 
days = $ 10.50. Hence the present worth = $ 1000 — $ 10.50 = $ 080.50. 

Second Solution, By true discount, the amount of $ 1 for 60 days = 
$ 1.01. Then $ 1000 -^- $ 1.01 = $ 990.099+ = present worth. 

55. Amount of $ 150 for 9 months at 8 per cent =: $ 159 ; amount of 
$200 for 1 year 9 months at 8 i)er cent = $228 ; amount of $200 for 2 
years 9 months at 8 per cent = $244. The present worth of $159 at 6 
per cent = $152. 1 5 + ; the present worth of $ 228 at 6 per cent = $ 206.38 + ; 
the present worth of $244 at 6 per cent = $209.44+. Hence the cash 
value was $152,158 + $206,884 + $209,442 + $50 = $617,929. 

66. 120 per cent of $500 = $600, the money I received. Discount on 
$1 for 188 days = .0805. $1 - .0805 = .9695, proceeds of $1. $600 h- 
.9695 = $618|ff f, face of note. 

57. Amount of $1 for 68 days = $1.0105. .06 -h 1.0105 = 5.987+ per 
cent. 

68. Interest of $ 1 for 88 days at 6 per cent = .0055. $ 1 — .0055 = .9945, 
the proceeds of $ 1. Hence $2000 -s- .9945 = $2011.06, the face. 

Note. — Many banks ignore fractions of cents, and take 1 cent as the interest of $1 for 
83 days. Using tliis interest instead of $ .0055, we find the face of the note to be $2020.20. 
Borne banks coant also the day of disconnt, which makes 34 days. 

69. In bank discount the face of the note is equal to the proceeds plus 
the interest on the proceeds for the given time at the rate received at the 
bank. Hence the face value of $ 1 of proceeds for SS days at 6 per cent is 
$1.0055. Now, if $1 yield a certain sum at 6 i)er cent, $1.0055 must 
yield the same amount in the same time at .06 -^ 1.0055 = 5|jf} per cent. 

60. Reversing the process in the preceding solution, we find the face 
value of $1 of proceeds to be .065 -^ .06 = $ 1.088^. Hence the interest 
of $1 for the required time at 6} per cent = $1.08J - $1 = .08J ; for 1 
year it is .065, hence the required time is .08i -s- .065 = 1.28205+ years 
= 1 year 8 months 12 days. 

61. The shares were at interest 8, 6, and 4 years. The compound 
amount of $1 for 8 years at 6 per cent is $1.598848, and the present 
worth of $1 for 8 years is $1 -4- 1.593848 = .62741+ ; for 6 years it is 
$1 -^ 1.418519, or .70496 ; for 4 years it is $1 -f- 1,262477, or .79209. It 
is easily seen now that the shares must be to each other in the proportion 



164 



TABLE BOOK AND TEST PROBLEMS. 



of these present worths. Hence A, the youngest, gets ifi^y B ififi^^ 
and C iff^ii, of |9000. A, therefore, gets 1(2657.94+ ; B, $2086.47+ ; 
andC, i 3356. 59-. 

INVOLUTION AND EVOLUTION. 

62. When we point off decimals, we most count from the decimal 
point. Hence \/.^25 = .79+. 

63. } of a number cubed = ^j of the cube of the number. Then } of 
the cube, or ^f, — fj = 10, and j^ = 10 ; whence |f = 27, the cube of the 
number. The number, therefore, is V^ = 3. 

64. The square of } of a number = ^, of the square of the number, 
and ] of this = } of the square of the number. The square of } a number 
z=\ ot the square of the number, and } of this = ^ of the square of the 
number. By the problem, f of the square of the number is 12 more than 
^*g of the same. Hence i — A = A ®^ ^^® square of the number = 12, 
and 1} = 12 X -^ = 64, the square of the number. Then V64 = 8 = the 
required number. 



65. 40^ = 1600 ; | of 1600 

1600-400 

Vmo 

then 40 - 34.64 

Hence 5.36 h- 2 



400 = area of walk. 
1200 = area inside of the walk. 
34.64 = length of one side ; 

5.36 = twice the width of walk. 
2.68+ = width of walk. 



66. First Solution, 



16 









(«) 


(b) 


13 
14 


J 




2 






i 




5 


18 


i 




3 




21 

V 




i 


5 


2 

7 



Second Solution, 



21 







{^ (fe). 


i 




5 






i 




1 




i 




1 


i 




3 








8 


2 



ALLIGATION. 

5x3 + 7x5 = 50: hence multiply column (a) 
by 3, and column (6) by 5, which gives • 

6 at 13 = .78 
25 at 14 = 3.50 

9 at 18 = 1.62 
10 at 21 = 2.10 

50 $8.00 

Multiplying both columns by 5, we have 

25 at 13 

5 at 14 

5 at 18 

15 at 21 



ABITHMETICAL SOLUTIONS. 165 

Or multiplying colomn (a) by 2, and column (6) by 17, gives 

10 at 13 
17 at 14 

17 at 18 
6 at 21 

Or multiplying column (a) by 6, and column (6) by 1, gives 

.80 at 13 
1 at 14 
1 at 18 

18 at 21 

Or multiplying column (a) by 1, and column (&) by 21, gives 

5 at 13 
21 at 14 
21 at 18 

3 at 21 

Or multiplying column (a) by 3, and column (6) by 13, gives 

15 at 13 

13 at 14 

13 at 18 

at 21 

67. Since 1 cubic centimeter of water weighs 1 gram, the specific gravity 
of any substance is the weight of a cubic centimeter in grams. Then 1 
cubic centimeter of gold weighs 19.25 grams ; of silver, 10.5 grams ; and 
of the alloy, 16.84 grams. By alligation 

{19.25 1 6.34 volumes of gold. 
10.50 1 2.41 volumes of silver. 
Now, 19.25 X 6.34 = 122.045 grams of gold, 

and 10.5 x 2.41 = 25.305 grams of silver : 

hence the volumes of the relative quantities of gold and silver are as 634 
to 241 ; and the weights as 122,045 to 25,305, or as 3487 to 723. 

(By Dr. I. J. Wibbback.) 



166 TABLE BOOK AND TEST PB0BLEM8. 



ANNUITIES. 

68. $300 -t- 12 = 925.00, the monthly rent. 

12 y 13 

i— J — X 26 = $9.75, the mterest on payments. 
4 

$300 4- $9.76 = $ 309.75, the final value. 
$309.76 -f- 1.06 = $292.22. 

69. To give an income of $ 1200 a year requires $ 1200 h- .06 = $20,000. 
The compound interest of $ 1 for 6 years at 6 per cent is $0.418619, which 
multiplied by 20,000 gives $8370.38, the final value. $ 8370.38 -^ 1.418619, 
the compound amount of $1 for the given rate and time, = $6900.79, the 
present value. Hence $6000 — $6900.79 = $99.21, the amount B gains 
by installment plan. 

70. The annual payment must be an annuity, the amount of which in 

6 years will be 

1000 X (1.06)* (1) 

If no payments were made, there would be due at the end of 5 years the 
compound amount of $ 1000, or 

1000 X (1.06)6 (2) 

This amount is composed of five amounts, as follows: 1. The first pay- 
ment at compound interest for 4 years; 2. The second for 3 years; 
3. The third for 2 years ; 4. The fourth for 1 year ; 6. The fifth payment. 
The amount of an annuity of $ 1 for 6 years is 

(1.06)* 4- (1.06)» + (1.06)2 + (1.06)1 + (1.06)o. 

The sum of this geometrical progression is 

(1.06)5 _ 1 



.06 



(3) 



If $ 1 amounts to (3), it will require as many dollars to amount to (2) as 
(3) is contained times in (2), or 

1000 X (1.06)5 _ 1000 X .06 X (1.06)» _ ^^^^ ^^ 
(1.06)5 - 1 - (1.06)5 _ 1 - « 237.39. 

.06 



"AGE" PROBLEMS. 

71. J of yours + 20 = } (yours + 20), or J of yours + 20 = J yours + 
10 ; whence } of yours = 10, yours = 40, and mine = J of 40 = 10 years. 



ARITHMETICAL SOLUTION 8. 167 

72. Ten years ago twice her age was the difference between thehr ages, 
and her age was i the difference. Now his age is 2^ times hers, and 1| 
times her age is the difference, or her age is f the difference. As the 
difference always remains the same, 10 years must be } — i = i of the 
difference: hence the difference is 40 years, which is twice her age 10 
years ago. 

78. First Solution. Twice Edwin's age = the 4ifference between tiieir 
ages 10 years ago, and once his age = i the difference. But 12 years 
later Edwin^s age equals the difference. Hence 12 years = } the differ- 
ence ; and the difference is 24 years, which is twice Edwin's age 10 years 
ago. Edwin is now 12 + 10 = 22 years of age. 

Second Solution. Let F and E represent their ages 10 years ago. 
Then Ji^rr E, and JJ^^-f 12 = } (F+12), or } J'-f 24 ;= F-\- 12. Hence 
I F = 12, Edwin's age 10 years ago. 

74. This problem is found in Raub's •* Complete Arithmetic," p. 823, 
and is generally regarded as insolvable. Let G = Gertie's age now, and 
t7= Jennie's. By the conditions of the problem, G = SJ, After the 
required number of years, 2J+ twice the required number of years 
= G + the required number of years, or 2J -\- the required number of 
years ^ G or SJ, Hence the required number of years must be 6 J", or 
6 times Jennie's age now. That is, no matter what Jennie's age may be, 
when it shall have been increased by 6 times itself, it will be ^ of 
Gertie's. To illustrate : suppose Jennie's age is 4, Gertie's 82. Jennie's 
age added to 6 times itself = 28 years, which is ^ of Gertie's, the latter 
being 32 + 24, or 56 years. Illustrations might be multiplied ad infinitum. 

"TIME" PROBLEMS. 

75. If ^ = I, I of the time past noon — } of the time to noon ; and 
I + I, or }, of the time to noon = 24 hours. Hence the time to noon is 
9 hours, and the time is 3 a.m. 

76. f of time past = } of time to come — 1 hour. 
J of time past = J of time to come — J hour. 
J of time past = J of time to come — J hour. 

That is, the time past noon \b l\ hours less than } of the time to mid- 
night I of time to come + J of same — 1^ hours = 12 hours = time from 
noon to midnight. Hence f of time to midnight = 12 -f 1} = 13} hours, 
and } of time to midnight = } of 13} = 8 hours. Therefore the time is 
8 hours prior to midnight, or 4 p.m. 



168 TABLE BOOK AND TEST PROBLEMS. 

Tt. At 4 o'clock the hands are at 12 and 4. To overtake the hour 
hand, the other must gain 20 minute spaces ; and after passing the hour 
hand the other must gain 30 spaces; or 50 in all. To gain 1 space 
requires 1^ minutes, and to gain 50 requires 50 x 1^ = 54/^ minutes. 
Hence the time is 6if^ minutes past 4 o'clock. 

78. In a true minute the hour hand moves over H ^^ i^ = iVir ^^ ^ 
minute space. In the same time the minute hand moves over ^^ of a 
space. Hence the minute hand gains ^J — ^^ = }{} of a space in 1 true 
minute, and in 20 true minutes it will gain 20 x f §} = 17^ spaces, the 
number the hands are apart. 

79. They are 20 minutes apart twice, (a) At 6 o'clock the hands are 
30 minutes apart, and the minute hand must gain 10 minutes: hence 
10 X }f = lOJ^ = the number of minutes it must travel ; therefore the 
time is lOJJ minutes past 6. (6) At 32^ minutes past 6 o'clock the 
hands are together, and the minute hand must gain 20 minutes, to do 
which it must move over 20 x j} = 21^j spaces : hence the time is 
32^^ + 21y»x = 54^ minutes past 6. 

80. On a clock face there are 60 minute spaces. Let s denote the 
number of spaces between the two hands. If they exactly change places, 
the hour hand will move through s spaces, and the minute hand through 
60 — 8 spaces. But the minute hand goes over 12 spaces while the hour 
hand goes over 1 ; that is, their rates and distances are as 12 to 1. Hence 
12 : 1 : : 60 — « : 8, whence s = 4^y spaces. At 6^^ minutes past 1 the 
hands were together. Since that time the minute hand has gained 4^y 
spaces, which is {^ of the distance it has traveled. Therefore the minute 
hand has moved 4^j x |f = {ff spaces since 5^^^ minutes past 1. Then 
the time was 5^^ + iis = l^A°f minutes past 1 = 10 minutes 29^j*y 
seconds. 

81. First Solution, At 5 o'clock the minute hand is at 12, the other 
at 5. As the former moves 12 times as fast as the latter, the distance 
traveled by the latter in any given time is ^ of the gain of the former. 
In this instance, to overtake the hour hand, the other must gain 25 
minute spaces : hence the hands will be together at |^ x 25, or 27^ 
minutes past 5. The hour hand is then 2^^ spaces from 6. Half the 
gain from then until the required time is the distance each hand is from 
6. The gain is 11 times the distance traveled by the hour hand. Then 
5 J times that distance + that distance, or 6} times the same, is 2^ 
spaces; and A of |J = j®;^, "the distance" moved by the hour hand. 
2^^ — ^^ = 2^ spaces, the distance of hands from 6. Hence 30 + 2^^ 
= 32j^ minutes past five, the time. 



4ltlTBMETICAL SOLUTIONS. 169 

Second Solution. At half-past 5 the minute hand is at 6, and the other 
midway between 5 and 6. The minute hand moves 12 times as fast as 
the other : hence, by the conditions of the problem, the distance between 
each hand and 6 is 12 times the distance the hour hand moves after half- 
past 5 ; that is, the hour hand has traveled ^ of the distance to 6, or ^^ 
of 2} si)aces = /^ spaces ; and the minute hand ^^ of the distance to 12, 
or 3^^ of 30 spaces = 2^ spaces. The time, therefore, is 82 minutes 18^ 
seconds past 5. 

88. Since the minute hand moves 12 times as fast as the hour hand, 
and since in changing places both hands move just once around the dial, 
the distance between the hands at first is ^^ of the diaPs circumference 
(60 spaces), or 4^^ spaces. Since 2 o'clock the minute hand has gained 
11 times the distance the hour hand is past 2, or 10 + 4^ minute spaces. 
Then ^ of 14^ = l^V;, the number of spaces the hour hand is past 2 ; 
and 1^^^ + 4^j + 10 = 15f}f , the number of spaces the minute hand is 
past 12, or the time required is 15 minutes 56]^^ seconds past 2. 

• 

83. (a) It is evident that the hour hand will be the first to be midway 
between the others, and that it will be there inside of a minute. Suppose 
the hour hand moves 8 spaces : then the minute hand moves 12 «, and the 
second hand 720 s. For convenience denote the hands by H, M, and S, 
Now, from i7 to 3/ is 12a — « = lis. Since 720a is less than a minute, 
60 - 720a is the distance from S to 12. Then 60 - 720 a + « is the dis- 
tance from S to H, which, by condition, must be the same as from H 
to 3f. Hence 60 — 719a = 11 a, or 730 a = 60, a = 7^ = the part of a min- 
ute space the hour hand has passed over. ^^^ x 12 = |} = distance 
minute hand has gone. Then || of 60 seconds = 69^f seconds, the 
required time. 

(6) When S passes H, it will directly be midway between H and M. 
This will occur in less than 2 minutes after 12. Suppose H moves over 
a spaces : then Jf moves over 12 a, and S over 720 a. Then 

720a - 60= distance from 12 to jS' (1) 

From ^to ilfisl2a — a = lla spaces. Therefore from H to Sia^s, 

and 8 + ^8 = distance from 12 to /S^ (2) 

We have now 720 a — 60 = a -|- ^^ a, whence a = ^^^j. 

i^ij X 720 = -^iVkV = 60^^y seconds, 

the time till >S^ is midway between H and Jf. 

(c) When the second hand passes M, it will soon be far enough to 
leave M midway between itself and H, This will occur in a very few 



170 TABLE BOOK AND TEST FB0BLEM8. 

seconds. Let 8 be the number of spaces IT moves from 12. Then Jlfgoes 
12 < spaces ; and S, 7200. 

720« - 60 = distance from 12 to iS' (1> 

and 11 < = distance frcMn Hto M, After the first minute, S moyes over 

« + 11« + 11« = 23« = distance from 12 to 5 (2) 

Comparing (1) and (2), we see that 720 « — 60 = 23 8, or s = -ffj, which 
reduced to seconds gives 6l}tf seconds, the required time. 

GENERAL ANALYSIS. 

84. If 40 pounds contain i a pound of salt, to contain 1} pounds will 
require 40 x 3 = 120 pounds. Then 120 — 48 = 72 pounds = amount to 
be added. 

85. In the same time B does 4} h- 3f = 1}^ times as much work as A. 
A works 4 hours, and B in 4 hours does as much asAin4x 1\ = 6 hours. 
Hence what both did in 4 hours could have been done by A in 9 hours. 
Then 9 + 4^ = 13} hours, A's time ; and | of 13} = lOf hours, B*s time. 

88. A's + B's + C's = $ 150 (1) 

A's - B^s + C's = 850, by first condition. 
Hence 2 B's = ^ 100, and B*s = f 60. 

Taking B's from (1), we have 

A's + C's = ^ 100 (2) 

By the second condition, 

A's + B's - C's = J C's, 
or A's - C's = } C's - B's, 

or A's- JC's = -f50 (3) 

Comparing (2) and (3), we find that f A's = $ 100, or A's = 140. Then 
(2) becomes 840 -f C's = 8100, whence C's = 860. 

87. In 14 days the men first employed would do J x |f , or ^ of the 
work. Then J — -j^^ = ^, what 20 men do in 14 days, and in 1 day they 
do T^j of ^ = ^y of the work. 1 man does ^\^ of ^^^ = y^jts ^ ^ ^y> ^' 
all the work in 1008 days. He does \ of it in 336 days. Hence to do ^ 
of it in 12 days requires 336 4- 12 = 28 men, the number employed at first. 

88. Call the broken part B, and the stump S. Then 2 (jB + 1 foot) 
= /8'-4, or 2J? + 2 = /S'-4, or 25 + 6 = iS'. But B+S=SS feet. 
Using the value of S, we have 5 + 25 + 6 = 38, or 3 JB = 27, and B = d, 
the broken part. 2 5 + 6 = 24, the stump. 



^ ARITHMETICAL SOLUTIONS. 171 

89. In 1 day A and B can do ^ of the work. 
In 1 day B and C can do ^ of the work. 
In 1 day A and C can do ^ of the work. 

Adding these, we see that A + B + C can do } (^ + A + A) = J o^ 
the work in 1 day, and to do f will require | -r- 1 = 8 days. 

90. 10 + 20 cost $ 11.00 (1) 

20 + 10 cost $ 13.00 (2) 

30 + 30 cost $24.00 

or 10 + 10 cost I 8.00 

But 10 + 20 cost 1 11.00 

10 cost $ 3.00 

1 cost .30 

Substituting this value in (1) or (2), we find that P cost 50 cents. P 
represents a bushel of potatoes, and A a bushel of apples. 

91. We observe that A*s financial ability was twice B*8, and that B^s 

was twice C's. Hence A*s was 4 times C's. 4 times C's — C's = 3 times 

C*s = 9 12,000, and 

C^s share of cost = $ 4,000 

A's share of cost = 16,000 

B's share of cost = 8,000 

Total cost =$28,000 

92. Fast boat crosses in 4 minutes, and is ready to start back in 6 
minutes, when the slow boat is just in the middle of the river. They 
now travel towards each other, and the two together must travel half the 
width. Since the fast boat travels 3 times as fast as the other, it will 
travel f of the distance, or f of J = | of the width. To do this will re- 
quire I of 4 minutes =1} minutes. Hence 6 + 1J[ = 7} minutes = time 
from start till first meeting. 

98. Every «uck is three-ply, and contains ]^ x 3 = f of an inch of cloth. 
For every tuck there is a single ply J of an inch wide. Every tuck, there- 
fore, requires J + | = | of an inch of cloth. 1 yard = 36 inches, and 
36 -4- { = 41, the number of tucks. 

94. If f were stolen, j remained. J of (| + 860) = | + 840 = amount 
spent. ( i + 860) - (f + 840) = t + 820 = remainder. J + 820 + 8 10 
= amount after getting 810. J of (J + 830) = t^V + #5 = amount lost 
^ + 830 — (tV + 95) = i + 825 — remainder, which is half the original 
sum. i + 825 = i, or } : hence 825 = |, or |, of his money, and 876 
= what he had at first. 



172 TABLE BOOK AND TJEST PROBLEMS. 

95. By the conditions, 2 (broken part 4- 2) = stomp — 8, or (2 x broken 
part) 4- 4 = stump — 8. 2 x broken part = stump — 12. Then 66 — 12 
= 54 = 3 times broken part, and broken part = 18 feet. 66 — 18 = 48 
= stump. 

96. The cow cost the least. The cost of an ox = twice that of cow 

— $12. The cost of horse = 4 times this = 8 times cost of cow — $48 

— $40. Hence 11 times the cost of cow — $100 = $360. 11 times the 
cost of cow = $460, whence the cow cost $40^, ox $69^, and horse 
$239j3r. , 

97. I can pick twice as many as I can dig : therefore I must dig twice 
as long as I pick. Hence I must dig 4 days and pick 2. Then 20 x 4 
= 80 bushels, the required number. 

98. For pasture, 1 cow requires \ an acre, and for roots i of an acre; 
for both she requires ^ + ^ = § of an acre. 20 -r- j = 30, the number of 
cows. 30 H- 2 = 16, the number of acres in grass. 

99. The head and tail are 6 inches + i of the body. The body is the 
same. Hence 6 inches must be f of the body, and 8 inches the body. 
The tail is 3 inches -f 2 inches = 6 inches ; and 8 + 5 -f 3 = 16 inches, 
length of fish. 

100. The cost of the watch + } its cost = } of its cost, which is $20 
more than both cost, or $120. } = } of $120, or $40 ; and }, or cost of 
the watch, is $80. 

101. A, the more rapid walker, determines the time of the race. 
Therefore the second time was { of the first ; and f x ^^ = fj of B's first 
distance — his first distance = ^ of the same, or 54 feet. Then |J, or 
B's distance, was 54 x 80 = 4320 feet ; and 4320 + 80 = 4400 feet, the 
length of the course. 

102. By the conditions of the problem, equal weights of gold and 
silver are in volume as 1 to (^jf^ -i- 20) = \}\, Now, if gold were sub- 
stituted for silver in the given volume, which has 2 units of weight and 
21 units of value, there would be 1 -f f JJ = ^ff units of weight; 
and {^ X 20 = iff^ units of value. Then JL||1 -i. 21 = 2f J|, the re- 
quired number of times. 

108. The italicized clause may be differently interpreted. This solu- 
tion assumes that B received 3s. 9d. less than he would have received had 
he and A finished the work in the specified time, 6 days. For the other 
interpretation, see '* Algebraic Solutions.'' For f of the work A should 
receive ^ of the 00 shillings, or 50 shillings ; then B and C would receive 
90 - 60 = 40 shillings, of which B receives 40 - 3} = 36} shillmgs, for 6 



ARITHMETICAL SOLUTIONS. 



178 



days, or 7} shillings per day. Hence 90 -s- 7^ = 12^) days, the time in 
which B could do the work. 3| -i- 2 = 1} shillings, C*s wages for 1 day. 
Then 00 -4- 1} = 48 days, the time in which C could reap the field. 

104. 117 -i- 1.4 = 81, which is the square of the smaller number. Hence 
Vsl = 9, the smaller number ; and 9 x 1.4 = 13, the larger. 

105. Before meeting, the slower train travels the shorter distance; 
after meeting, it travels } the longer distance, while the faster train 
travels the shorter. Hence shorter distance : longer distance : : \ longer 
distance : shorter distance, or shorter distance squared = \ longer distance 
squared, or square of longer distance = 4 times square of shorter dis- 
tance. But the square of any number is equal to 4 times the square of its 
half. Hence shorter = } longer, and rates are as 1 to 2. 



or 



106. 

Hence 
Then 



107. 

and 

Hence 
and 

108. 



and 
109. 



MENSURATION. 
The Bectangle. 

J its perimeter = 100 rods. 

Length = 1} x width. 
2J X width = 100, 
width = 40. 
length = 60. 
Area = 40 x 60= 2400 square rods. 
2400 -f- 160 = 15 acres. 



1344 



(4x3) 

4X10: 
3X 10: 



H 

120 X 231 
10 X 2} X 144 
27,720 -i- 3960 



112, 

10 + = ratio of sides. 

40 = length, 

30 = width. 

^ = 36 gallons. 

36 X ^ = 120 = contents in gallons. 

27,720 = contents in cubic inches. 

3960, 

7 feet, the required length. 



80 X 80 = 6400 square feet = area of garden. 
J of 6400 = 1006J = area of the walk. 



Hence 



6400 + 1066f 

\/7466i 

86.4-80 

6.4^2 



7466 1 = area of square including both. 
86.4+ = side of square including both. 
6.4 = twice the width of walk. 
3.2 feet = width of walk. 



174 TABLE BOOK AND TEST PMOBLEMS. 

110. 45 -i- 80 = 1.5 = area of square end. 

VT5 = side of square. 
Then VTP + VTP = square of diagonal, 

and VS = diagonal = diameter of log. 

111. For conyenience put « = the side. Then the diagonal is V«*+«2= 
8y/2 = 1.4142 X 8. By the p]X)blem, 1.4142 s - 8, or .41429 = 8.284 rods. 
Hence 8.284 -i- .4142 = 20 rods, the side. The area is 20^ = 400 square 
rods, or 400 + 160 = 2^ acres. 

113. The diagonal of base = Vb* + O^ = 10. 

The altitude = VIS^ - 6* = 12. 

Volume = 6 xS X ^ = 102 cubic units. 

Let 2a;, 3 X, and 4x = the dimensions of the rectangular solid: then its 
volume = 24 «*, which, by condition, is equal to 192. Hence x* = 8, and 
X = 2 ; and the sides of the solid are 4, 6, and 8. 

118. Since their rates are as 5 to 6, their distances are as 5 to 6, and 
one runs 26 yards farther than the other: hence 26 x (6 + 6)= 286 = 
perimeter of field, and half the sum of the length and width = -i^|^. 
Half the difference of the same is VCCH*)^ — 4840 yards] = J^ yards. 
.'. ^t^ + V = 88 yards, the length ; and ifi - ^/^ = 55 yards, the width. 

114. 8 gallons = 1848 cubic inches. Assuming the dimensions of the 
smaller can to be 8, 7, and 11 inches, the contents would be 281 cubic 
inches. By similar solids, we have 231 : 1848 : : 3* : jc* (= 216), whence x, 
or end of larger box, is 6. Then the other sides are 14 and 22. Since the 
second can requires 4 times as much tin, it has 4 times as much surface ; 
and, by similar surfaces, its dimensions are Vi = 2 times the first Its 
contents are 2' = 8 times the smaller = 8 x 8 = 64 gallons. 

The Triangle. 

115. The saw is the hypothenuse of a right-angled triangle, whose per- 
pendicular is 8 feet. .-. 10^ - 8^ = 36 j and VSQ = 6, the width. 

116. The base x } the altitude, or } the base x altitude = area. But 
the area = 972 perehes, and ^ the base = 36 perehes. Hence 36 x the 
altitude = 972, whence the altitude is found to be 27. Now, either side 
is the hjrpothenuse of a right-angled triangle whose per pendicular is the 
altitude, or 27, and whose base is 36. Hence V362 + 27^ = 45, the length 
of the sides. 



ARITHMETICAL SOLUTIONS. 



176 



117. The sum of the sides is + 14 + 5 = 28 rods. The half smn is 
14. Sttbtxacting each side separately, we have 

14- 9 = 6. 
14-14 = 0. 
14- 6 = 9. 

Multiplying these three remainders and the half sum together, we have 

6x0x0x14 = 0. 

Extracting the square root of 0, we have Vo = 0, the required area ; 
that is, there is no area, and the problem is a *' catch/' The three given 
sides or lines do not form a triangle, since one is as long as the other two. 
They form one straight line only, or, if not, then two of the sides do not 
meet. 

118. Because it cuts the sides proportionally, it is parallel to the 40-foot 
side, and is therefore half as long as the base, or 20 rods long. 

119. In this problem the height of the pitch is ) of 36 = 24 feet. This 
is the perpendicular distance from the vertex of roof to the square, and is 
one leg of a right-angled triangle whose hypothenose is the length of a 
rafter. Half the width of the building is the other leg. Hence 

V24« + 182 = 30 = length of rafters. 
NoTB. — If the rafters project, add the projection to the length of rafters. 

190. The rope will form the hjrpothenuse of a right-angled triangle 
whose perpendicular is 10, and whose base is the circumference of the 
cylinder. The curcumference is 6 x x = 18.8496. Then 

the length of the rope = V[(18.8496)2 + (10)2] = 21.337 feet 

191. Let ED represent the comer of the garden waU, C the position of 
the object, B the center, and 

BF the height required. 
From one comer to the cen- 
ter of the lot is half the diag- 
onal = J V3002 + 3002 

= 212.13 - feet = BD. 

CDE and CSA are similar 

right triangles: hence CD: 

CB::DE:BA, or 20:(20 + 

212.13) : : 9 : BA ; whence BA = 104.46. Then BF = 104.45 - 6 = 99.46 

feet. 




176 TABLE BOOK AND TEST PROBLEMS. 



128. \ sum of sides = 160 : hence area = >/(160 x 60«)= 4330+ square 
rods. \ the area = 2165 rods, which is a sextant, or ^ of a circle, because 
each angle of the triangle is 60^, or ( of a circle. The entire circle = 2166 
X 6 = 12,990 = irra. 12,990 + 3.1416^ 4134.83 = »^. V4 134.83 = 64.3 = r, 
the length required. 

128. Assume a similar room whose side is unity, or 1. Its diagonal is 
V2 ; and from the middle or center of floor to one comer is \y/2. There- 
fore the distance from the center to an upper comer is 



V(J V2)2 + 1« = Vf = J Vtf. 

By similar triangles, then, we have }V6:24::l:a;, the required side, 
whence x = 19.696 feet. 

124. By the first condition, one tree is 80 feet high ; and by the second 
condition, the other is 60 feet high. One tree is therefore 30 feet higher 
than the other. This 30 feet is the perpendicular, and the width of the 
river is the base of a right-angled triangle whose hypothenuse is the dis- 
tance between the trees* tops. Then 30^ + 40^ = 2600 = square of the 
hypothenuse, which is 60 feet 

125. Since the shortest distance between two i)oints is a straight line, 
the fly must travel along the hypothenuse of a right-angled triangle. One 
leg of this triangle is the length of the room, and the other is the width 
+ the height Hence (16)3 ^ (12 + 8)* = 666 = the square of the hypoth- 
enuse. The fly must therefore travel V656 = 26.61 + feet 

NoTB. — This la clearly seen by setting up fonr small boards to represent the sides 
and ends of the room, then laying down the boards without displacing them, and drawing 
a line from the starting point of the fly to its destination. 

126. The diameter of the log will be the diagonal of the log when 
squared. Hence it will be the hypothenuse of an isosceles right-angled 
triangle. One leg of this triangle, or one side of the squared log, is found 
by taking the square root of J of the square of the diameter of the round 

log. In this case it is Vi(2\/2)2 = V4 = 2 feet = 24 inches. Therefore 
one board contains 16 x 2 = 32 feet To find the number of boards, 
divide 24 by 1 + J, which gives 19 -|- (if nothing were wasted by the width 
of saw, there would be 24 boards). Therefore 32 x 19 = 608 board feet. 

N0TX8. — 1. This method of finding the number of boards is not exactly correct, since 
there is one more board than saw-cut, provided there is no loss of a thin board at the last. 
Then, too, it is possible to cut some boards from the slabs cut off in squaring. We have 
considered only the squared log. 

2. Arithmetics give the following rule for finding the side of the Inscribed square, but 
do not explain: "Multiply the given diameter by .707106." In the given problem, 
2/2 X .707106 s 2, same result as obtained in a different way. 



ABITHMETICAL SOLUTIONS. 



177 



127. Let BCDE be the board. We have given BG = 4 inches, or 
I of a foot; £7=12 feet; and EF=S inches, or } of a foot; also 
JEI= i (16 — 8) = 4 inches, or | of a foot. Since the trapezoid decreases 




8 inches in 12 feet, it will decrease 16 inches (that is, come to a point) 
in 24 feet Hence AF=: 24 feet. Then 24 - 12 = ^G = 12 feet. The 
area of the board BCDE is 12 square feet. Area of EFHK (J of the 
board) = 3 square feet The area of the triangle EFA = J x 12 = 8 
square feet The area of EFA — EFHK =^ — Z=:b = 9.r%2k AHK, 
Then, by similar surfaces, we have EFA : KHA : : AF^ : AH^^ or 
8 : 5 : : 676 : (a; = 360). Hence AH = V360 = 18.97+ feet AH - AG 
= GH, or 18.97 - 12 = 6.97, the distance from narrow end. 12 - 6.97 
= 5.03 feet = distance from wider end. 



Tke Circle. 



128. Let AB be the diameter of the floor, and AD = BE = width of 
granary. Put JBC = 16 = r, and CE = 19 = iJ. Then xr* = area of inner 
circle, or floor ; and tB^ = area of outer 
circle, or floor and granary ; and 

xB^ — wr^ = area of granary 

= r(B^ - ra)= ir(361 - 226) 

= 136 T = 427.2676 square feet, 

which multiplied by 3 = 1281.772+ cubic 
feet = 1030 bushels, the contents of the 
granary (in standard bushels). 

129. 160 acres=160 x 160=26,600 square 
rods = area. 25,600 - ^ 3.141 6 = 8148.7 = 

square of radius. V8148.7 = 90.2+ = radius of field. If } is plowed, 
80 acres remam in circular form. 80 acres = 12,800 square rods. 
12,800 -f- 3.1416 = 4074.35 = r^ of remainder. V4074.35 = 63.8 rods = 
radius of remainder. 90.2 - 63.8 = 26.4 rods = width of ring plowed. 
26.4 rods = 6227.2 inches, which divided by 18, the plow's width, gives 
290.4 = number of rounds. 

bi.lwood's tbst prob. — 12. 




178 TABLE BOOK AND TEST PROBLEMS. 

UO. The wfaeelB describe two circles, one wtthln the other. CiTcam- 
lerenoes yaiy as their radii : hence, since the oater drcomference is twice 
the inner, its ladins most be twice the radins of the inner. Bot the dif- 
ference between their radii is 6 feet, wliich must be the less radins, and \ 
of the greater.' Then the diameter of the inner circle is 6 x 2 = 12 feet» 
and its circumference is 12 x 3.1416 = 37.6902 feet. 

181. Let h = the man^s height, and r = radius of the earth. Then 
2rr = circumference of the earth, and r + A = radius of circle described 
by the man^s head. Then 2T(r + A)= distance traveled by the top of 
his head ; and (2 xr -i- 2 xA) — 2 xr = distance his head trayels farther 
than his feet = 2 xA. That is, in any such problem, to find how much 
farther the head (or top) travels than the feet, multiply the height of the 
man (or object) by 2t, or 6.2832. In this problem we have 6 x 6.2832 = 
87.6002. 

183. Let CG^CFz=:CE:=r = the length of the rope, and Cthe pomt 

where it is fastened. Since the angle BCD 
is a right angle, it cuts off } of the circle : 
hence, CFOE is } of the circle. If 10 acres 
is }, the circle contains 10 x | = 13} acres 
= 2133J square rods = xi^. Dividing by x, 
and extracting square root, we have r=26+ 
rods, the length required. 

183. First Solution. The thickness makes 
no difference so long as it is uniform. The 
area of one side is 7.0686 square feet = 
1017.8784 square inches, of which A grinds 
off f , B 1^, and C \, After A and B have ground, C has his ^ left in 
circular form. Then 254.4606 -4- x = 81 = the square of the radius of C's 
circle. Hence, Vsl = = C's number of inches. B*s share = ^"^ = 
366.2574 square inches ; and B^s share + C^s share = 610.72704 square 
inches, which is also in circular form. The radius of this = the square 
root of (610.72704^-3.1416)= 13.04 inches. This is C's radius + B's 
radius : hence B's share = 13.04 — = 4.04 inches, and A's must be 
18 - 13.04 = 4.06 inches. 

Second Solution. After A grinds, f of the stone remains for B and C. 
Since the radius is 18 inches, and similar surfaces are to each other as 
the squares of like dimensions, the radius of this remainder is V| x 18 = 
13.0428 inches. Hence A grinds off 18 - 13.0428 = 4.0672 inches. Since 
C owns \ of the stone, the radius of his share is V^ x 18 = inches. 
Then B must grind off 18 - (0 + 4.0572) = 4.0428 inches. 





ARITHMETICAL SOLUTIONS. 179 

184. Let AB = 20 = 2 m, the length of the giyen line, JR = radius of 
larger circle, and r = radius of smaller circle. 
In the right-angled triangle ADC we have 
AC = Rf DC = r, and AD = m; whence 
B^ — r^^m!^. Multiplying this equation by t, 
we have rlfi — xr^ = xw^. Since xJR* = the 
area of the larger, and xr^ = the area of the 
smaller circle, the left member expresses the 
difference between the areas of the two circles, 
or the area of the ring or track. Then xm' = 
3.1416 X 10^ = 314.16 square rods, the area of 
the ring. The second requirement is indeter- 
minate. If r = 0, ^ = 10, which is the least value R can have. For any 
value of R less than 10, the value of r is imaginary. Suppose /2 = 6 : 

then y/R^ - w* = V36 - 100 = V~ 64 = SV^. R may have any value 
from 10 to 00 : then the width of the track would vary from to 10 rods. 

Pyramids and Cones, 

186. Slant height = V122 + ^ = 13. i slant height = 6.5 feet. Qr- 
cumference of base = 10 x 3.1416 = 31.416 feet. Then 31.416 x 6.6 = 
204.204 square feet = 22.680 square yards. 

186. The top section was a cone whose volume was } of the loaf. The 
top section and the next one, taken together, formed a cone whose vol- 
ume was ) of the loaf. Hence, observing that similar solids are to each 
other as the cubes of their like dimensions, we have the following pro- 
portions : 3 : 1 : : 20« : A», whence A = 20 v^ =; ^j4 \^ = 13. 86722 + inches, 
the height of the top section ; and 3 : 2 : : 20* : A', whence h = ^ -Ms = 
17.4716+ inches, height of both upper sections. Then 17.4716 - 13.86722 
= 3.60438 inches, height of middle section ; and 20 - 17.4716 = 2.52839+ 
inches, height of bottom section. 

187. The bucket is a frustum of a cone, and its volume = x {R^ + r^ + Rr) % 

in which a = altitude, R r= radius of larger base, and r = radius of smaller 
base. The areas of the bases vary as the squares of the radlL Hence, by 
condition, R^:i^::6:S, whence 3^ = 5»«, or /P = Jr^^ and JB = rVf; 
also Rr = r^ Vf . Substituting values in the formula given above, we have 
volume = 3.1416 (r^Vf + r^ + Jr^) x J of 12, or 3.1416 (Jr^ + »^ + r^Vf) 
X 4 = 13 gallons = 3003 cubic inches. Multiplying, etc., 40.637 r^ = 3003, 
whence r = 7.77 inches, 2r = diameter = 15.54+ inches, R = rVg = 
7.77 Vj = 10.03+, and 2JB = /) = 20.06+. 



180 



TABLE BOOK AND TB8T PB0BLEM8. 



1S8. Altitude of cone completed = 100 feet. Area of larger base = xr^ 
= 1256.64 square feet. 1266.64 x 33^ Q of altitude) = 41,888, contents 
of cone. 41,888 — 12,000 = 29,888, contents of cone after frustum is cut 
off. Since similar cones are to each other as the cubes of their altitudes, 
we have 41,888 : 29,888 : : 100^ : h^ ; whence h = 89.9+ feet. Then 100 - 
89.9 = 10.1 feet, the length taken off. 

189. Let r = radius of upper base. 

Then ^ 2r = radius of lower base, 

Tf^ = area of upper base, 

4x1^ = area of lower base, 

and V4xi^ x xr^ = 2xi^ = area of mean base. 

7xr^ = sum of the three bases, which multiplied by | of the altitude 
= the volume of frustum. Hence Txr^ x (J of 12) = 28xra = 7060 cubic 
inches, and xr^ = 251.7857 + . Dividing by x = 8.1416, i^ = 80.1467+, 
whence r = 8.96+, and 2r = 17.9 inches, the diameter of upper base. 
Then 17.9 x 2 = 36.8, the diameter of lower base. 

140. The bucket, represented by ABCD, is a frustum of a cone whose 
volume = contents of bucket = 266.98 cubic inches. | of 266.98 = 88.66 

= volume Of water = EFCD. Comj^eting the 
cone, its altitude is 66 inches, and its solidity 
= x/P X V = 718.38. Then 718.88 - 266.98 
= 452.4 = volume of CDK, and 462.4 + 88.66 
= 641.06 = volume of EFK, Since similar 
cones are to each other as the cubes of their 
altitudes, we have 718.38 : 641.06 : : 66* : S*, 
whence KI = 60.9+ inches. Then 60.9 + 3 
(the rise) = 63.9 = JTP, and we have the pro- 
portion 63.9+* : 56* : ; QHK : 718.38, whence 
OHK = 642.3 + cubic inches. Then the volume 
of GHK — the volume of EFK = the volume 
displaced by the ball. Hence 642.3 -641.06 = 
101 .24 = volume of the portion of ball submerged. 

141. The diagonal of the largest square inscribed in the base of the 
cone = diameter of the cone, or 10 feet. Hence } of 10^ = 60, the square 
of one side, or the area of base of pyramid. Then 60 x } of 30 = 600 
cubic feet, the volume required. 

Similar Solids, 

142. The dimensions of similar solids are proportional to the cube roots 
of their volumes. Hence \/8 : vT : : 18 J : x : : 8 : y ; whence x = 37, the 
width ; and y = 16, the depth. 




ARITHMETICAL SOLUTIONS. 181 

143. The water forms a cone similar to the glass. Hence, by similar 
solids, J : 1 : : A« : 3* : whence h = \^6.76 = 1.8+ inches, depth of water. 

144. By similar soUds, 180 : 1015 : : (6J)» : A» ; whence h = 10|J = 10 
feet 4.6 inches. 

145. Since spheres vary as the cubes of their diameters, the diameter 
of a ball containing as much as the three is the cube root of (3' + 4* + 6*) 
= v/2i6 = 6. 

146. Consider the 10-inch shell solid. Then its weight is found by 
similar solids, thus: 8': 10*:: 8 pounds : (x = 15} pounds). Since the 
shell weighs 7} poimds, we have 15f — 7f = 8 pounds = the weight of a 
ball just filling the hollow part ; but this is the weight of the. given ball, 
whose diameter is 8 inches. Hence (10 — 8) -^ 2 = 1 inch = thickness 
of shell. 

Cube8 and Spheres, 

147. The diameter of the water = 12 — 2 = 10 inches. The contents 
of the water = 10» x .5236 = 523.6 cubic inches. Then 523.6 -*- 231 = 
2.26} gallons. 

148. 36 X 16 X 3 = 1728 cubic inches = volume of water raised, which 
is also the solidity of the cube. Hence v^l728 = 12 inches = the edge of 
the cube. 

149. The solidity of a 6-inch cylinder 4 inches long = 32 x 3.1416 x 4 = 
113.0976 cubic inches, which must be the volume of the sphere. Volume 
of a sphere = d* x J x. Hence 113.0976 = .5236 (P, whence d» = 216, and 
d = 6 inches = diameter required. 

160. The volume of any sphere equals two thirds of its circumscribing 
cylinder. In this problem the submerged ball occupies two thirds of the 
bucket : hence the bucket is just as deep as wide ; that is, the bucket {s a 
cylinder whose length and diameter of base are each equal to the diameter 
of the ball, or 10 inches. 

161. The diagonal of a cube = its side x Vs. The side of this cube, 
then, is 10 -f- VS = 5.773, and its contents = 6.773* = 192.4 cubic mches. 
Since the cube's diagonal is the sphere's diameter, we have the volume 
of sphere = 10* x } x = 523.6 cubic inches. Hence the contents required 
= 523.6 - 192.4 = 331.2 cubic inches. 

162. SoUdity of ball = 3* x ^ ir = 14.1372 cubic inches, and } of this = 
3.6343 = the share of each. After the first three have wound their shares 
off, the last one has a ball containing 3.5343 cubic inches, whose diameter 
is the cube root of (3.5343 -^ .5236) = 1.89, the last one's share of the 



182 TABLE BOOK AND TEST PROBLEMS. 

diameter. After the first and second have done, the last two have a ball 
containing 7.0080 cubic inches, and its diameter is the cube root of 
(7.0686 -^ .6236) = 2.38+. Then 2.38 - 1.89 = .49 = third lady's share. 
After the first lady has wound, the last three will have a ball whose con- 
tents is 10.6029 cubic inches, and whose diameter is the cube root of 
(10.6029 -I- .6236) = 2.72. Then 2.72 - 2.38 = .34 = second lady's share, 
and 3 — 2.72 = .28 = first. Hence the first winds off .28, the second .34, 
the third .49, and the fourth 1.89. 

15S. Will the sphere be totally or only partially immersed ? In order 

to determine this, we construct the diagram, in 
which ABC represents a vertical section of the 
glass, and FGH a section of the sphere. By the 
con ditions, DB = 3, EF= 2, and DC = 8. BC 
= \/8M^=V73. Now, by similar triangles, 
DB:BC::EF:EC, or 3:V^::2:x = |V73 = 
6.696+ inches. DE= DC -EC =S ^ 6.696 = 
2.304 inches. But, E being the center of the 
sphere, EH is a radius = 2 inches. Therefore 
ED is greater than EH, and the sphere is totally 
immersed. Hence the volume of water displaced 
is equal to the volume of the sphere, which is 
represented by f xr' = 33.51 cubic inches. 



MISCELLANEOUS PROBLEMS. 

164. We invert to find how often the divisor is contained in 1. We 
then multiply by the number of times 1 is contained in the dividend, 
which gives tlie required quotient. 

165. In such expressions the operations indicated by the signs 4- and 
X must be performed first. Hence the given expression is the same as if 
written thus, 16 + (9 -*- 3)-(2 x 3), and its value is 16 + 3 - 6 = 12. 

156. The log's solidity = 20 x 3 x 2} = 160 cubic feet. A cubic foot of 
water weighs 62^ pounds. Hence a cubic foot of oak weighs 62} x .925 
= 67.8126 pounds, and 160 cubic feet weigh 150 x 67.8125 = 8671.876 
pounds. 

157. U.'s chance of succeeding is |, and his chance of failing \, B.'s 
chance of success is ^, and his chance of failure ^. Hence probability 

(1) is f X A'^ A; (2) is i X A = A; (3) la » x a = A; W ^ 
lxA = A. 




ARITHMETICAL SOLUTIONS. 188 

168. Let r = rate of increase, p = population, < = the time. We find 
r= ^ — ^ = yiiy. At the end of first year the population is p+ty =j)(l + r) , 
at the end of second year it is j?(l + r)^ ; and so on till the end of the 
t years, when it is p(l + r)*. Substituting values, we find the population 
to be 5,000,000 (1 + jhy^ = 10,206,357. The increase = 10,206,357 
- 5,000,000 = 5,206,357. 

159. The first buyer they met was paying 1 cent for 7 eggs. At that 
price A sold 7 cents* worth, retaining 1 egg; B sold 4 cents' worth, 
keeping 2 eggs; C sold 7 eggs for 1 cent, and kept 8 eggs. The next 
purchaser was a sort of ** bull,** and offered 3 cents apiece, at which price 
they all sold out, A realizing 8 cents, B 6 cents, and C 9 cents. Thus 
each got 10 cents for his eggs. 

160. Let A denote the faster train, B the other. Then to pass B when 
going the same way, A must gain 230 + 210 = 440 feet, and 440 4- 15 = 29^ 
feet, which is A's excess of rate per second. Li opposite directions they 
together move 440 feet in 3| seconds, or 117^ feet per second, of which 
A runs 29} feet more than B. Then i of (117| - 29}) = 44 =: B*s rate 
per second, and 44 + 29J = 73 J, A's jate. (44 x 3600) -4- 5280 = 30 (miles 
per hour), and (73} x 3600) -^ 5280 = 50 (miles per hour), the required 
rates. 

161. 11 + 9 = 20, which satisfies the first condition. The least multiple 
of 11 that will contain 9 with a remainder of 4 is 22. Hence 20 + 22 = 42, 
which satisfies the first and second conditions. The next addition must 
be a multiple of 99, and contain 7 with a remainder of 5. This number 
is 495. 42 4- 495 = 537, which satisfies the first three conditions. Similarly, 
637 + 2772 = 3309, the required number. 

163. After ^ has been drawn, ^ remain; ^ from this leaves 

JW = §; AfromthisleavesTWTr = §; 3^^ from this leaves VWW = ^• 

It is thus seen that after any given drawing the part of wine remain- 
ing is expressed by the first remainder raised to the power denoted 
by the number of drawings. Hence, in this case, the part remaining 

is ^0 = AVoWWft = .3486784401. Then the number of gallons is 
100 X .3486784401 = 34.867844+. 

168. This is a problem in arithmetical progression, in which a = the 
first term, d = 4, and n = 42. Then Z = a + (n - l)(i = a + 164 ; and 
< = }(a+Z)n = 21(2a + 164) = 42 a + 3444 = the whole number delivered. 
By condition, half of these, or 21 a + 1722, were delivered in the last 18 
^yS} (A)' Taking the first 24 days, { = a + 92, and < = 24a + 1104, 



184 TABLE BOOK AND TEST PROBLEMS. 

the nmnber deliyered in the first 24 days, (B). Equating (A) and (B), 
as the condition of the problem allows us to do, we have 24 a + 1104 
= 21 a + 1722, whence 3 a = 618, or a = 206. Then 42 a = 8652, and 
8652 + 3444 = 12096, the number delivered altogether. 

164. Bepresent the names by their initials. Then, by the conditions of 
the problem, S = P + 23, (1) ; and B = A + H) (2). Since the pieces are 
all squares, we have D^ + 63, P^ + 63, and A^ + 63, each a perfect square. 
From a table of squares we find that the only squares whose difference 
let 63 are 1 and 64, 81 and 144, 961 and 1024. Consequently the boys, 
A, F, and D, must be 1, 9, and 31, and the men, S, B, and H, must be 
8, 12, and 32. To satisfy (1), S must be 32, and P 9. To satisfy (2), 
B = 12, and A = 1. Observing the last condition of the problem, it is 
readily ascertained that H and A are father and son : so S and D, and 
B and P. Hence the names of the boys are Adam Hugus, Dick Smith, 
and Paul Brown. 

166. For convenience we call the grass standing on 1 acre a crop, and 

the weekly growth on 1 acre a growth. Then, by the conditions of the 

problem, 

48 oxen in 1 week eat 3 crops and 12 growths (1) 

90 oxen in 1 week eat 5 crops* and 30 growths (2) 

Comparing 5 times (1) and 3 times (2), we see that 

30 oxen in 1 week eat 30 growths (3) 

Comparing (2) and (3), we see that 

60 oxen eat in 1 week 5 crops (4) 

From (3) and (4) we find that 1 ox in 1 week will eat 1 growth, or ^ of 
a crop. Now, since the quantity to be consumed is 6 crops and 54 growths, 
it will require 1 ox (6 -f- ^) + (54 -h 1) = 126 weeks to consume it. Hence 
the number of oxen required to consume it in 9 weeks is 126 -s- 9, or 14. 

166. 1 square acre = 43,560 square feet, or 208.7 feet square. If 
planted 3} feet apart, 60 hills can be planted in a row, and a space of 2.2 
feet remains. If we divide this space equally among the 59 spaces between 
the 60 hills, they will then be 208.7 -r- 59 = 3.537 + feet apart. Now, plant- 
ing in the quincunx order will bring the rows clo ser than 3^ feet, but every 

other row will contain only 59 hills. Then ^^{^y - ^5i5|i±V= 3.02 

feet, the distance the rows are apart. And 208.7 h- 3.02 = 69 spaces, which 
allow 70 rows to be planted, of which 35 rows contain 60 hills each, and 
85 contain 59 hills each. Therefore (35 x 60) + (35 x 59) = 4165, the 
required number. (From Dr. i. j. Wiiubbacx.) 



ABITHMSTICAL SOLUTIONS. 186 

167. TaMng the quincunx order, we have 12 trees in the first row, 11 
in the second, fourth, and so on. The distance the rows are apart is 
Vl* — .52 = .866, which is the altitude of each triangle. Hence in each 
strip between two rows there is a gain of 1 — .866 = .134 of a rod. To 
gain a row will require .866 -4- .134 = 7 strips. Then 7 -f 1 = 8 strips 
contain 9 rows of trees. The odd rows contain 12 x 6 = 60 trees. The 
others contain 11 x 4 = 44 trees. We therefore have 60 + 44 = 104 trees 
in quincunx order. The strips planted cover .866 x 8 = 6.928 rods : 
hence there remain 11 — 6.928 = 4.072 rods, which is not sufficient ground 
to allow the gain of a row by the quincunx order. Hence by that order 
there would be a small loss, and we use the square order, planting 4 rows, 
12 in a row, or 48 trees. Therefore the greatest number is 104 + 48 = 
162 trees. 

168. 90 + 3 + l + 6 + f + A» » + 233^ + 67t, or 15 + 36 + 47 = 98, 
and 98 + 02 = 100. 



186 TABLE BOOK AND TEST PROBLEMS. 



ALGEBRAIC SOLUTIONS. 

FACTORING. 
169. This may be written -(26» ± 40 >/rH- 16), or -(6 Vx ± 4)«. 



170. The factors of — 20 are 4 and — 6, and their sum is — 1. Vs? 
multiplied by — 1 = — as ; hence x^ — x — 20 = (x + 4)(a5 — 6). 

171. aj»-2x«-3aj + « = a5»-3x-.2xa + 6 = x«-3x-(2ai«-6) = 

x(xa - 3)2- 2 (x2 - 3) = (x - 2)(x8 - 3). 

178. X = x^j< X* X X* , or x' x x* x x". The latter may be written 
v^ X \^ X y/a^. 

173. x«- aV- «^*+ ajy"= «'* («- «^) + IT (a - a') = (ar" + JT) («-a^)- 

174. The form of this polynomial suggests a binomial factor, (x — 3)* 
= x^ — 6 X H- 9, which subtracted from x^ + 10 x — 39 leaves 16 x — 48 = 
16 (x - 3). Hence x* + 10 x - 39 = (x - 3)2 + 16 (x - 3), which may 
be written (x — 3 + 16) (x — 3), or (x + 13) (x — 3), the required factors. 

175. The sum of the factors of — 14 is — 5. Hence the factors are + 2 
and — 7. Therefore the required factors are (x + 2) and (x — 7). 

177. x2-y2=(x-y)(x + y). 

x2 + y2 = (x - v^^+ y)(x + \/2x^ + y). 



FRACTIONS. 
178. 

xH x + 



- . X a + x , a' ax + x2 4.(12 
1 + - x + 



a a a-{-x a-^x a' — x'x' — «• 



a a a^ax — x^ — a^ — a'— x^ x* + a* 

X X X 

X a — x a — x a — x 

~a a 

179. The reciprocal of a quantity is 1 divided by that quantity. Any 
factor may be transferred from dividend to divisor (or from numerator to 



ALGEBBAIC SOLUTION 8. 187 

denominator), and vice versa, by changing the sign of its exponent. 
Hence the required reciprocal is 

1 _ (a-\-h)-* ^ (a-by 
(a-&)-" (a - b) -•» (a + 6)»' 
(a + 6)-* 

180. The numerator may be written . — , and the denominator 

— H — or ^y ' Hence the complex fraction 



(x — 3/)* x*y* 






SIMPLE EQUATIONS. 

181. Adding, 2(x -\- y + z) = a -\- b -{- c, 

or a; 4- y + « = J (« + ^ + c) (4) 

x + y = a (1) 

Subtracting, « = J (6 + c — a). 

(4) - (2) gives y = i (o - & + c). 
(4) - (3) gives x = i(a + b-c). 

188. Let - = one part : then = ^~ = the other part. By con- 

X b X ox -, 1 

ditioUy (8x — 6)+l = 5x + a5, or 8a — 4 = 6 x, whence x = 2. /. - = i. 

Thenȣ^ = ll. * ^ 

5x 10 

18S. Let X = the number of days he was idle. Then a — x = the 
number of days he worked, ex = amount forfeited, and 6(a — x) = 
amount earned. Hence 6(a — x) — ex = d, or o6 — 6x — ex = d; whence 

X =z ?i^ — ^, the required number of days. 
b-\-c 

184. Let X = velocity. Then 14 — x = rate of rowing. (14 — x) — | x 
= rate on the return. Hence (14 — } x)10} = 42, whence x = 6. 

185. Let X = the number of first kind, and y = the number of second 
kind. 

Then x + y = c (1) 



188 TABLE BOOK AND TEST PROBLEMS. 

By the first condition, - = part of a crown made by the x pieces, and 

y ^ 

f = part of a crown made by the y pieces. But both make a whole crown. 

Hence - + ? = 1 (2) 

a 

Clearing (2), bx + ay = ah (3) 

Subtracting a x (1) from (3), we have bz — ax sz ab — be, whence x = 

«(^ - ^). Theny = c-x = MLll«I 
6-a ^ b-a 

186. Let X = the number. Then x — 100,000 = the remainder after 
removing the 1, and 10 (« — 100,000) + 1 = the new number. Then, by 
condition, 10(x- 100,000)+ 1 =3x, or 7x=999,999; whence x= 142,867, 
the number required. 

187. Let X = time denoted by minute hand. Then x — 35 = number 

of minute spaces it has passed the figure 7. — = the number of minute 

spaces hour hand has moved since 6 o^clock. Then 6 — — = the number 

of spaces it lacks of figure 7. Hence, by condition, 5 — — = x — 35, 

12 

whence x = 36}^| minutes past 6 o'clock. 

188. Let X = number of ounces of alloy. 
jg 

Then — H P = number of ounces of silver. CI) 

X 

and — g = number of ounces of copper. (2) 

Then — hpH Q = Xi whence x = — ^ • 

Substituting in (1) , — hp= ~' P^ — ?_^ ^q silver. 

tn mn — tn — ti 

Substituting in (2), g = — — — , the copper. 

189. Let X = A's rate per second. 

2^ = B's rate per second. 
Then 3600 x = A's rate per hour. 

1700 



X 



= time in which A runs a mile. 



1760-20 „^ 1760 ,„ ^ ,_ 

30 = , by first conditions. 

y X '' 



ALGEBRAIC SOLUTIONS. 189 

The second heat giyes the equation Z—Lt — 32 = • Equat- 

, 1760-20 „. 1760 -9A- «« v ^ « t, 

ing, we have 30 = ^ ^^ — 32, whence y = Sj'^. By 

Bubstituting this value of y^ we find x = 5}f , A^s rate per second. 
5]tl X 3600 = 21,120 yards = 12 miles, A's rate per hour. 

190. Let X = the duty after reduction, and 2y = any number of pounds 
consumed at 6 cents per pound. Then 12^^= the original revenue. 
After the reduction, the number of pounds consumed in the same time is 
Sy, and the revenue is Sxy. By the problem, 3a^ + j (12^) = 12y, or 
Bxy •\-4y = 12y, whence as = 2} cents per pound. 

191. At 3 o'clock the hour hand was at 12, the other at 3. Let x = the 
number of minute spaces traveled by the hour hand. Then 122 = num- 
ber traveled by minute hand. If they are opposite each other, the 
minute hand is just as far past 9 as the hour hand is past 3, or x spaces. 
Hence the minute hand has passed over (45 + x) minute spaces, and we 
have 45 + x=12a;, whence lias = 46, x = 4jJj-, and 12x = 493^=the 
time past 3. 

KoTB. — Problems of thia kind are by no means rare In the old " mental *' arithme- 
tics, and may be readily solved by analysis. The reasoning employed In such analyses, 
however, is the same as that in algebra, to which such problems properly belong. 

192. Bepresent the four numbers hy w,x, y, and z^ and let their sum 
= m. Then, by the conditions, we have 

w + tH^ _ ^^ or ^^, __ 2a - TO (1) 

x+nLzJl = a,otx=^±=^ (2) 

y +^^^ = o,ory =^li^l^ (8) 

4 



2 




4a- 


TO 


3 




5a- 


TO 


4 




73a- 


-25to 



z + — = — = a, or « = — - — Q4) 

5 

By (1) + (2) + (3) + (4), TO= ^2 , 

or 37w = 73a, and TO = Jfa. 

Substituting this value of to in (1), (2), (3), and (4), we have 

to = 2 a — J? a =37^' 
aj=j(3a-J?a)=J?a. 



190 TABLE BOOK AND TEST PROBLEMS. 



198. Let X = the distance. 

Then — = the tune going, 

and 2 — — = the time returning. 

1a 



4f 2 — — 1 = the distance = x. 



(^-s)= 

Clearing, 9a-4a; = 12«, 

whence x = 6, the required distance. 

194. Let X miles per hour = strength of wind, 
and y miles per hour = strength of tide. 

Since the rates are as 6 to 8, the times are as 3 to 6. Hence { of 
12 = 4} hours = time out, and f of 12 = 7^ = time in. 

By first statement, 60 -i- 4i = 13J = x + y (1) 

By second statement, 60-^7J = 8 = a;--y (2) 

Adding (1) and (2), we find x = 10|. 

Then 2^ = lOj - 8 = 2J. 

195. For convenience let x = John^s age 3 years ago. Then x-\- Q 
= John^s age 3 years hence, and 15 a; = the father^ s age 3 years hence. 
By the first condition, 6 x + 26 = Smithes age 2 years ago. Then 5 x + 24 
= Smith's age 3 years ago. 

15 X— («+ 6) = 14 X— 6=the difference between their ages 3 years hence (1) 
(5 X + 24) — X = 4 X + 24 = the difference 3 years ago (2) 

But since the difference always remains the same, we may put the 
expressions (1) and (2) equal to each other, thus: 14x — 6 = 4x + 24, 
whence x = 3. Therefore John is now 3 + 3 = 6 years, and his father 
42 years, old. 

196. Let X = time the second hand is equally distant from the other 

two. X — 60 = distance second hand is past 12. — = distance minute 

X ^^ 

hand is past 12. -^ = distance hour hand is 

^ 720 

past 12. Now, the distance the second hand 

is past 12 = ^ of the sum of the distances the 

hour and minute hands are past 12. Hence 

2(x - 60) = ;^ + -^, or 1440X - 86400 = 12 x + x, 
^ ^ 60 720 

whence x = QO^^^ seconds. Similarly, we obtain 
|^ = x-60 + -^, whence x = 61fff, the time 

the mimite hand is midway between the others. Finally, in like manner 




ALGESBAIC SOLUTION 8. 191 

we get (60 — a:) + — = — , whence x = SOU seconds, the time till the 
^ 720 60 

hour hand is equidistant from the other two. (From Dr. I. J. Wirbback.) 

197. When C reaches W, A will be at 2>, 5 miles from B, Let E be 
the point where C picks up A. Then, since C travels 8 times as fast as A, 
DE =\oiDW=Z%. Let EQ = the distance C carries A, and a; = G^TT. 
Then ^^ = 36 - 3| - x = 31J - a;. A walks 6 + 3 j + a. Therefore the 
time of the trip in hours is 

5 + 3| + a; + i(31J-a;) (1) 

Let RF be the distance traversed by B until C leaves A at Q, As B*s 
rate is 2 miles an hour, 

BF=2U + 8| + ^^'"~^ ^ = 26| - Jx. 

Let H be the point where C picks up B. C*s rate : B's : : 4 : 1. Hence 
FH=\0■^'-\^) = ^- ^^^ ^^ich B walks. HQ=FG-FH=l\%-ix. 
HO + (?>r= 11| + |x, the distance C carries B. 

Nora. — We know C carries B td Wy because, If he put him down at any point between 
H and Wt C would reach fT first, which is not allowed by the condition of the problem. 

While A walks from G' to TT, B walks from Flo H^ and is carried from 
^ to TT. Hence, as \ the distance he walks and J the distance C carries 
him = the time, we have J(2| — /7yx)4- J(11S + |x)= the time from F 
to W, which must be x hours, as in that time A walks from G^ to TT. 
Letting this expression for the time = x, and solving, we find x = 2| J}. 
Substituting this value of x in (1), we find the time to be 15^^ hours. 



RADICALS. 

198. (a-x)'\^±^ = A/^^^'^(a-^ = V^^^. 
Then (a - x) Va^ - x^ - Va^ - x^ =(a - x - 1) Va'^ - x^. 

199. (i-«2)-i + i=l±VTE?. 

VI --x« 

The numerator may be written 



192 TABLE BOOK AND TEST PB0BLEM8. 

Henoe the fraction becomes 

Taking the square root, we obtain 

the root required. 

200. Squaring, we have 

«-« Vx+Va 

Multiplying both terms of the right member by Vx + Va, we have 

(a; + d) — 2Vax _ g — q ^2^ 

«-« (a; + a)+2Vaic 

Now, X — a being a mean proportional between (a; + a) — 2 Vox and 
(x + o)+2\/ax, the equation is true in the forms (1) and (2), and also 
in the original form, as the ratio was preserved all through. 

201. This may be written — + - Vm« - n\ 

4 2 

I Vm2 - n2 = 2/^^ Vw2 - nA 

and ^Vii^rrT^^!^/ ^^'-^' ); 

4 2^ 2 J 

but (fT+(^^V^y=r 

Hence (|)% | VSiTT^ + / x^E^y 

= the quantity whose root is to be found, and - H — ^ ~ ^ is the root. 

2 2 

QUADRATIC EQUATIONS. 

S02. Let X = the height of stump. Then a — x = part crossing stream. 

(a - x)2 = 62 4. 3.2^ or a2 - 2ax + x2 = 62 + x\ whence x = ^^ ~ ^' ' This 

2a 
formula furnishes the following rule for finding the height of any stump 

when the conditions are similar : From the square of the height of the 

tree subtract the square of the stream^s width, and divide the remainder 

by twice the height of the tree. 



ALGEBRAIC SOLUTIONS. 198 

208. Let d = the diameter. Then cP x i ir = -^ = contents of sphere, 
and /-] X 4 IT = ircP = surface. By the condition of the problem, 

y(f3 = ^LifL, or 6 irc^ = xd^. Dividing by ircP, we have 6 = (7, the diameter. 
6 



804. First SoliUion. 






From (2), 


"i 




Substituting in (1), 


X 




Clearing, we have 


X* — ax = — &. 

n 1 




Completing square, etc., 


x = |±lv^ 


-46. 



Second Solution. 

From (1), X = a - y. 

Then (2) may be written 

or y* - ay = - 6, 

whence y = - ± - Va2-4fe. 

805. (2)-(l)^ = 2V^ = 20 - 4xy. 
Transposing, reducing, etc., 

xy + i Vxy = 6. 
Completing square, etc., v^ = 2 and — }. 
Whence try = 4 and ^. 

But X = 20 — y. 

Hence (20 - y)y = 4 or ^, 

Solving this, we find y = 10 ± 4 \/6 and 10 ± f \/l6. 

Also X = 10 ± 4 V6 and 10 ± f Vlb. 

806. Let X = the number. 
Then, by conditions, 

or Vx + 6 = 6 + Vx-6. 



Squaring, x + 6 = 36 + 12 Vx - 6 + a; - 6, 

or 12 Vx-6 = - 24, 



and Vx^:^ = -2. (1) 

BLLWOOD*S TEST PROB. — 13. 



194 TABLE BOOK AND TEST PROBLEMS. 

Squaring again, we have a; — 6 = 4, 

or as = 10. 

To verify, put 10 for x in the first equation, thus : 

\/ie-Vi = 6 (2) 

Extracting the indicated roots, 

4 -(±2) = 6. 

Hence (2) is true when Vi = — 2, as it does in this problem. [See 
Equation (1).] 

a07. Adding 2, a;« + 2 + l = 2. 

a* 

Factoring, (* + ~) ^^' 

Extracting square root, a; 4. - = ± V2. 

Clearing, etc., x* ± x V2 = — 1. 

Completing square, x^ ± xy/% + J = — J. 
Whence x = db Vj ± """^^^^ 

which may be written ^^^^^^ > 

V2 

908. Let X = number of gallons first drawn. 

Then 81 — x = remainder. 

il= the part diawn, 

^^ 2d time. 

81-x-.;^(81-x)=36. 
81^ ^ 

Clearing, 81(81 -x)-x(81 -x)=81 x 36, 
or (81 - x)(81 - x)= 81 X 36. 

Extracting square root, 81 — x = 9x6 = 54, 

whence x = 81 — 64 = 27. 

Then f{(81 - 27) = 18, number of gallons drawn 2d 

time. 

309. Let X = the width of frame. 

Then 10 + 2x = the width of picture and frame, 

and 16 + 2x =: the length of picture and frame. 



.1 




ALGEBRAIC aOLUTIONS. 



195 



(10 + 2 a;) (16 + 2 a;) = area of picture and hume. 
blem, (10 + 2a;) (16 + 2a;) = 320, or 4a;» + 52a; = 160, 
\ a;a + 13 X + ip = 40 + i}A = Ap. 

square root, a; + ^a* = ± i "^/^ = 0.069+. 

.'. X = 0.069+ - 6.5 = 2.569+ inches. 
X = length of a side in rods. 
x^ = the area, 
4x = its perimeter. 

4a; = 



m. 



x^ 



Clearing, 
and 

The area is 



160 
640ar = a;', 

a; = 640. 

:^ = 2560 acres. 
160 



«11. The first member may be written thus: vqg-v<*+2v» ^ 

1 H ^^^ — ; and the second member, 1 + — . 

Vox — V« Vn 



or 



Hence 
Clearing, 



2y/n 



yfi 



Vox— Vn Vn 

2n = aVx — y/an^ or aVic=Van — 2n- 
^ — 2n 



Vi = 



a 



s„^„^„ _,,i^i.^=ii».,5i,^i^±M 



_ n(Va-2Vn)« 
a 



218. Completing square, 



^« ^ (« « 5 + c)x + (^-^^)'- (6 - a)c + (^^^)'. 

Extracting square root, a;- ^""^"^^ ^ ^(b - a)c + ( ^ " ^ "^ ^ (1) 
Reducing the quantity under the radicsd sign, it becomes 



^ fa:^-2ab - 2ac + y+ 2&c + c^ 

and factoring, it becomes 

I Ca-hy-2c(a-b) + c^ 



196 TABLE BOOK AND TEST PROBLEMS. 

Sabstitating in (1), we have 

2 > 4 

Extracting square root of right member, 
^ a — h-\-c , a—b—c 

X : — = + . 

2 2 

Whence as = a — &, or + c* • 

218. Let X = distance army traveled before the officer turned. Then 
25 + ^ = distance the officer travels in same time. 25 — a; = distance yet 
to be traveled by the army. 25 — (25 — x) = z = distance officer travels 
in returning. Then, since the rates of travel remain the same, the dis- 
tances are proportional, as follows : z : 25 + x : : 25 — a; : x, whence x = 
17.68 + miles, and 25 + 2x = 60.36 + miles, the distance traveled by the 
officer. 

214. Let X = the distance from the louder bell, and a — x = the distance 

1 x^ 
from the other. Then by condition, 1 : 3 : : x* : (a — x)*, or - = — =i . 

3 (o — x)* 

Extracting square root, — z = — — » whence x = • a — x = a — 

V3 «-« 1±V3 

a ±av8 

1±V3 i±Vs 

216. Let X = distance from B, the stronger light. Then 132 — x = 

distance from A. By condition, we have 7 : x* : : 17 : (132 — x)*, or 

7 17 a/? VT? 

--- = — — — - — —- Extracting square root, -^-^ = ± -^ Clearing, etc., 

x" (132 -x)a ^ ^ ^ X 132-x ^ ' 

xV7±xVl7 = 132V7. Whence x= ^^^^ » 

Vl ± vT7 

12 144 

216. Let X = number of eggs for 12 cents. Then — x 12 = =^^ = price 

X X 

per dozen ; also x 12 = = decreased price per dozen. Then, 

x + 2 x + 2 

144 144 A 

by problem, i^ - =1, or x* + 2x = 288 ; whence x = 16 eggs, the 

X X "T" ^ 

144 144 
number given for 12 cents. . — = — = 9 cents per dozen. 

X 16 

217. Squaring, 1 +? = 1 -.^ + 2a/i -^+ 1. 

a X ' X 

Transposing, 2-\/l ---=- + --1. 

^ X a X 

Squaring, 4-- = (- + -) _2(- +-)+ 1 (1) 



ALGEBRAIC SOLUTIONS. 197 



4a 

X 



- + - ) , we have ( - — - ) J and adding 

to _2f- + -V we have -2f--?V 
\a xj \a xj 

Hence (1) may be written 

\a xj \a xl 

X (I 

Extracting square root, 1 = 0, 

Ct X 

or x^ — ax — a\ 

Whence x = -±-V5 = "(l± Vs). 

218. Let X = cost of husking a standing row, and y = that of a ^' down ** 
row. Then ^ x i x -\- y : : 2x + y : Zx^ whence 7 a^ = 3 a-y + y^^ (1). 
•j\ being ** down " rows, 2 y + Ox = 20, whence y = 10 — 4} x. Putting 
this value in (1), we find x = $1,655+; and 9x, the standing rows, 
= $14.90. Then $20 - J§ 14.90 = $6.10, cost of "down" rows. 6x + y 
= A's money = $10.83, and 4x + y = B's money = $9.17. 

219. This equation may be written 

x*« - 2 x8« + x«« - x*« + x« = 6. 

Factoring, (x** - x«)2 - (x*« - x«) = 6. 

From this quadratic we find 

x^-x^=\± Voj = 3 or - 2. 

Taking x*» - x« = 3, 

we find x« = } ± ^.\/l3 = }(1 ±>/i3). 

Hence x = 'Vni±\/\Z). 

Taking x«« - x« = - 2, 

we find x'» = J ± JV- 7. 



Hence x=Vj(l±V-7). 



^. Let X = the digit in tens^ place, y = that in units* place. Then 
lOx + y = the number. 

By the first condition, 

10xa + X2^ = 46 (1) 

By the second condition, 

x(x + y) or x2 + xy = 10 (2) 



198 TABLE BOQK AND TEST PROBLEMS. 

(1)- (2) gives 9a;2 = 36, 

whence x = 2. 

Then from (2) we find y = 3, 

and we have 10 x + y = 20 + 3 = 23, the required number. 

821. Let X = one, and y = the other. 

Then jc + y = xy = «' — y", 

from which we have 

x^ — y'^ =^(x ■{■ y)(x — y) — X + y. 
Dividing by (x + y), we get 

a;-y = l, 
whence x = y + 1, and y = x — 1 (1) 

Then xy = y(y + 1) = y^ + y = x + y, 

whence y^ = x, then y = Vx (2) 

Equating values of y as found in (1) and (2), we have x «^ 1 = Vx. 

Squaring, etc., x* — 3 x = — 1. 

Adding}, x2-3x+ | = |, 

whence x = f + J V6. 

Then y = x-l = J + J\/6 = }(l + v^)« 

222. Clearing, 

Va — V a — Va^* — ax = nVa + nva — Va^ — ax. 

Collecting, Va(l — n) = (n + 1) v a — Va*^ — ax. 

Squaring, a(l - 2 n + »2) = («2 + 2 n + 1) (a - Va'-* - ax), 

or a — 2an-\- av^ — an^ H- 2 aw -f a — (n + l)2\/a^ — ox. 
Collecting, — 4 an = — (n + 1)^ Va^ — ax, 

or _i«^=Vsr=Tx. 

(n + 1)3 
Squaring, ^r+^ " "*' ~ ''"'' 

(n + 1)* 
Transposing, ax = a'^-a'^ I^ilS , 



ALOEBBAIC SOLUTIONS. 199 

I. Let o, 6, and c = the numbers. Then = 1 — i , whence 6 = 

2«c 6 a c 6 



a + c 



By first condition, a + -^^ + c = 191 (1) 

a-\- c 

By second condition, ac = 4032 (2) 

From (1), a2 4. 4 flfc + c2 = 191 (a + c) (3) 

Subtracting 2 oc = 8064 from (3), we have 

(a + c)2 = 191 (a + c) - 8064. 

Transposing, and completing the squares, 

(a 4. c)2 - 191 (^a + c)-\- 96^6^ = 9120.26 - 8064 = 1066.26, 
whence a-\- c= 128, and c = 128 — a. 

Then (2) becomes ac = a (128 - a) = 4032, 

or " a^ - 128 a = - 4032. 

Adding 4096, a^ - 128 a + 4096 = 64, 

whence a = 72 or 66. 

Then c = 66 or 72, and 6 = -^^ = 63. 

a-\- c 

224. Let 2 x = length of log, 

and X = width and depth. 

Then 2 x — 6 = length of excavation, 

05 — 6= width of excavation, 

and X — 3 = depth of excavatioi^, 

2 ic* = solidity of stick, 

(2 05 — 6) (jc — 6) (a; — 3) = solidity of excavation. 

The difference between the solidity of the stick and that of the excava- 
tion must be, by the last condition, 11,772. 

Hence 2x8- {(2x-6)(x- 6)(x- 3)}= 11,772, 
or 2x» -(2x8 - 24x3 + 90x - 108)= 11,772, 
or 4x2 -16x = 1944, 

whence x = 24, the width and depth of stick, 

and 2 X = 48, the length of stick. 

Then the trough is 48 — 6 = 42 inches long, 

24 - 6 = 18 inches wide, 
and 24 — 3 = 21 inches deep. 

(From Dr. I. J. Wibbback.) 



200 TABLE BOOK AND TEST PROBLEMS. 



886. Let X = the first, 
conditions, we have 


y = the second, and 
x(y + z)=2e 


2 = the third. Then, by 

(1) 






y(x + «)=60 


(2) 






« (y + a;) = 56 


(3) 


By(2)-(i), 




«(y-a;)=24 


(4) 


By (3) + (4), 




yz = 40 


(5) 


By (3) -(4), 




a;ar = 16 


(6) 


By substituting the value otxzin (1), 








xy = 10 


(7) 


From (5), 




40 
z = — 


(8) 



From 


(5), 


From 


W, 


Hence 


J 


whence 




or 




From 


(7), 


Therefore 


From 


(10). 



1«_ 


_40 


X 


y' 


16y = 


= 40a;, 


X = 


= «y 




10 


aj = 


 y 



(10) 



2y 10 ^ « ^^ ^ 

~ = — , or 2 y2 = 50, and y = 6. 



X = — =2, and from (6), z = — = 8. 
y ^ ''^ X 

886. This solution assumes that B receives 3s. 9d. less than he would 
have received had C not been called in, in which case they would have 
worked longer than 6 days, and A would have received more than 50 
shillings. For another interpretation of the italicized clause, see ** Arith- 
metical Solutions." Let x = the time required by B, and y = C's time. 

90 
A should have 10 shillings per day, and B — shillings per day. In 1 

X 

day A does -, and B 1, of the entire work. Then 6 f 1 + i^ = ^(^ + ^^ = 
9 X \9 x) 9x 

part of the whole work A and B can do in 5 days. = time in 

X "T" v 

which A and B together can reap the field. Then.— x ^* ^^^ 



X a; + 9 a + O 
B's pay if C had not been called in. — x 6 = — = what B did receive. 



X X 



ALGEBBAIC SOLUTIONS. 201 

Therefore -^^ _ 1^ = Sh or -^ - ?5 = !, whence x^ - 87a; = 
a; + 9 » * a: + 9 x 4 

— 1080, and » = 15 days, B's time. As the work was completed in 5 

days, we have - + - + - = 1, or -H h- = l, whence y = 18 days, 

9 X y 9 15 y 

C*S time. (From Mathematical Magazine.) 

227. Let X = the length, and 5y = the original rate. Then 5y x 1^ 
by = distance traveled before accident, and x — 6y = distance yet to run. 

^-=lil^ = regular time of running this distance. Hence 5/^i^ly\ = 
by S\ by J 

increased time of running this distance = ^-^^^ — ^ hours. Then ^-^^ — ^ 

Sy 3y 

+ 2 = the time on whole line, which is 3 hours more than regular 

time, ^-~. Hence the equation 
by 

^~^y +2=-^+3 (1) 

Sy by 

Under the second condition, x—(by + 50) =distance to run after acci- 
dent, and the time to run it is a; — o y — hours. ^^-^ — = time on 

Sy by 

road before accident. Hence -^-^ h ^~ ^ ~ — + 1 = whole time 

by Sy 

on trip, which is IJ hours less than under first condition. Hence the 

equation 

(2) 



5y-i,50 x-6y~50 ^^/ a; . 3\ ^. 
by ^ Sy ^ \by J ^ 



From (1) we find a; - 20 y = (3) 

From (2) we find a; - 10 y = 60 (4) 

Subtracting, lOy = 50 ; y = b; 5y = 25, the original rate. From (3), 
a; — 20 y = a; — 100 = 0; and x = 100, the length of road. 

228. Let X = the first number, and y = the ratio. Then x, xy, xy^f and 
xy* will be the numbers. 

X + xy H- xy2 + xy^ = 15 (1) 

and x2 + x2y2 + a;^ + «^2^ = 85 (2) 

Factoring (1), we have x(l +y)(l + y^)= 15, 

whence x^ = • (3) 

(1 + 2^)^(1 + y^y 



202 TABLE BOOK AND TEST PBOBLEM8. 

Factoring (2), we have 

a;2(l + y»)(l + y*)=85, 

whence x^ = ~— 7- (4) 

Cl + y«)(l + y*) 

Equating (3) and (4), and dividing by -, we have 

46 17 



(1 + y)*(i + V) 1 + y* 

Clearing, etc., 46 + 46y* = 17 + 34y + 34y2 + 34y« + 17y*, 
or 28 y« + 28 - 34 y' - 34y = 34y«. 

Dividing by y', and factoring, we obtain 
28(y.+ i)-34(i, + 2) = 34.orw(y. + ^,)-17(y + l)=17 (5) 

Put y + - = m. Then y« + ~ = «»' - 2. Then (6) becomes 14 m^ — 

17 m = 45, whence t» = JJ = 2^. Therefore y + - = 21. Clearing, etc., 

y2 — § y = — 1. Completing the square, y' — |y + f | = A, whence y — f 
= ±}t or y = 2. Therefore x = 1 ; and the numbers are 1, 2, 4, and 8. 

229. Let X = one number, 

and y = the other. 

Then xy=p (1) 

and flc^ - y« = r»(x - y)» (2) 

Dividing (2) by a; — y, we have 

x^-{'Xy + t/» = m(x - y)« (3) 

Treating (3) as a proportion, we get 

X^ + ary 4- y* : (« - y)* : : «» : 1. 
By division, 3 xy : (x - y)^ : : w - 1 : 1. 

Multiplying the first term of each ratio by |, and clearing, 

4 ary : (» — y)^ : : 4 m - 4 : 3. 

By composition, 

(X + y)* : (X - y)* : : 4 w» - 1 : 3. 

Extracting square root, 

X + y : X — y : : V4f» — 1 : V3. 



ALOFBBAIC 80L UTI0N8. 203 

By, composition and division, 

X : y : : Vim— 1 + VS : V4m — 1 — V3. 
Multiplying t^e first ratio by y^ 

zy : y^ : I V4 TO — 1 + V3 : V4 tn — 1 — VS. 
But xy=:p, 

hence y. ^ l>(V4i;r:^- >^^ p(V4^ii^n ^ 

V4W-1+V3 4TO-4 

Then y^l/V X4TO-l)>v^ \ 

and from (1), x=- — ^^^^^^ 

Vp(4 TO — 1) — V3p 



Second Solution, 






Let 


X = one, y the other ; 




then 


xy==p 


(1) 


and 


3p3 _ y8 _ ^(jp __ y)8 


(2) 



Dividing (2) hj x — y, and subtracting 3|), 

(x - y)2 = m(a; - y)2 - Sp, 
or (TO — 1) (a; — yy = 3|) ; 

whence x = y ±\l — ^• 

\to — 1 

Then (1) becomes 

\to — 1 

whence y = 1 / ± yp(4^r3T) ± Va^ \ 

2\ VSTTT / 



204 TABLE BOOK AND TEST PROBLEMS. 



SOLUTIONS TO SPECIAL BXPBDIBNTS. 

SIMULTANEOUS EQUATIONS. 

280. First Solution, 

Adding (1) and (2), »« + a; + y^ 4- y = 18. 

Adding } to both members, 

a;« + « + l + j^ + y + l = 18}='^^ = V + V- 
By inspection we know sc > y. 

Hence x^-^x + i = ^, 

and y* + y + i = ^^. 

Whence x = S and — 4, and y = 2 and — 3. (This solution is faulty.) 

Second Solution, 

From (1), x2^y-9^2, 

or a;2 - 9 = 2 - y (3) 

From (2), x -\- y^ = ^ -\- Z, 

or a; - 3 = 4 - 2/2 =(2 _ y)(2 + y), 

Equating (3) and (4), a;^ - 9 = * ~ ^ 



or 



2 + y 
jc2-9= * 



2 + 2/ 2 + y 



Transposing, x^ — — - — = 9 — 



2 + 2^ 24-2/ 



Completing square, 



x^ — g-+ ...^ , =&--iL+ 1 



2 + 2/ 4(2 + j/)» 2 + y 4(2 + y)» j 

Extracting square root, a - _-L_ = 3 - ^^^ 

whence k = 8. 

Substituting in (1), we find y = 2. 



SOLUTIONS TO SPECIAL EXPEDIENTS. 206 

Third Solution, Make 2-\-y = 8, 

Then from (2) we have x — 3 = «(2 — y), 

or . ?-? = 2-y (3) 

8 8 '^ 

From (1), jca - 9 = 2 - y (4) 

... x2-9 = ?-?, 

8 8 

or x2 - - = 9 - ? (6) 

8 8 ^ ^ 

Adding -— - to each member, we have 
4ir 

a;-2_? + _i. = 9_5:|. J_. 
« 4»2 « 4«2 



Extractmg square root, x = ±(3 V 

28 \ 28/ 



whence one value of x is found to be 3. 

Fourth Solution. a;' + y = 11 (1) 

x + y» = 7 (2) 

From(l), y = ll-a;2. 

Then y^ = 121 - 22 x^ + x* (3) 

From (2) y2 = 7 - a: (4) 

Equating (3) and (4), we have 

121-22x2 + x* = 7-x, 
or X* - 22x2 + x+ 114 = (6) 

Factoring (5), we have 

(x - 3)(x« + 3x« - 13x - 38)= 0. 
Placing each factor equal to 0, the values of x are readily found. 



lyth Solution, x2 + y = 7 


(1) 


X + y2 = 11 


(2) 


From (1) and (2), by transposition, we get 




y3-9 = 2-x 


(3) 


and y - 3 = 4 - x« 


(4) 


Put 2 H- X = ^f', and 2 - x = m. 




Then (3) and (4) become y^-9 = m 


(5) 


y^S=mW 


(6) 



206 



TABLE BOOK AND TEST PBOSLEMS. 



Equating values of m as deriyed from (6) and (6), we have 



ya-9 = 



W 



Therefore 



or 



^ W W 



Completing square, j^ - X + _i_ = 9 - A + ^ 



Extracting square root, 

Therefore 
and 



-l. = 3- ^ 



a) 



2W 2W 

x = 2, 

(From Profe«M>r B. F. Bublbson.) 



8ixth Solution. 

a;2 + y = 11 

X 4- y2 = 7 

(l)xy = »2y + y2 = lly 

(3)-(2) = x^^a;=lly-7 

(4)+ twice (1)= a;2y + 2a;2 - X = 9y + 16, 

or a;2(y^2)-x = 9y + 15. 

Dlyiding by y + 2, and completing the square, 

^2 a; . 1 ^ 36y8+132y + 121 



(1) 

(2) 
(3) 



Extracting square root, we have 



or 



whence 



x — 



^^ 6y-f 11 



2(y -f- 2) 2(y + 2) 
^^6^±i2_, 
2(y + 2) 



a; = 3. 



(From Dr. I. J. Wibkback.) 



Seventh Solution, 








^ + a; = ll 


(1) 




aJ» + y = 7 


(2) 


From (1), 


ya = 11 -. X 


(3) 


From (2), 


ya = 49-14a;a + x4 


(4) 



SOLUTIONS TO SPECIAL EXPSDISNTS. 207 

Equating and transposing, 

a:*-14a;2 + a;=-38. 
Adding to each member 2 a;' + 4 x* — 20 a, we get 

X* -f 2x8 - lOx^ - 19ic = 2Qfi + 4x2 - 20x - 38. 
Factoring, x(x« + 2x2 - lOx - 19) = 2(x8 + 2 x^ - 10 x - 19), 
whence x = 2. 

Therefore y = 3. 

281. By(l) + (2) + (3), 

2(ra ■\-tt^ + t^)-\-r8 + 8t-\-rt-m^-{- n* + p* (4) 

By (4)2, 

By 2(1)2 ^ 2(2)2 ^. 2(3)2, ^q iiave, by factoring, 

4(r2+«2^^)2^4(ia4.32^.jjj(y,^.3^^^)_2(r»+8«+r«)2=2(mHnHlJ*)(6) 

By VC6)-(6), 
r« + «« +r« = ± V{1[(2 w2n2 ^ 2n5!p2 + 2 w2p2)-(w»* + n* +1?*)]} (7) 

Put right member = c. 

Then, by (6(7) + 2(4)}*, ^ 

2(r + « + = ± ^^{2(»»'* + n2 +jp2)+ 6c} (8) 

By (4) + (7)- 2(2), and factoring, 

2r(r + « + 0= w»* - n2 +i?* + c (9) 

By (9)^(8), 



r = (m2 - n2 + j)* + c)-*-± v{2(m2 + n2 +l)2) + 6c}. 
Similarly we find 

« =z(n2 -p2 + m2 + c)-f- ± V{2(w2 + n2 H-^)+ 6c}, 

and { = (p2 _ ^2 ^ ^2 + c)-t- ± \/{2(w2 + n2 +i)2)+ 6c}. 

(From Professor B. F. Bublsbon.) 

888. From (1), x = 10 - y. 

Then (2) becomes (10 - y) \/y = 12, 

or 10Vy-y\/y=12 (3) 

Put v^ = m. 



208 TABLE BOOK AND TEST PB0BLEM8. 

Then (3) becomes lOwi - f»« = 12, 

or fii«-10»i+12 = (4) 

(4) X TO = TO* - IOto^ + 12m = 0, 

or TO* - 6to2 + 9 = 4to« - 12to + 9. 

to2-3 = ±(2to-3). 
to«=2to. 
TO = 2 = Vy, 
Then y = 4, 

and X = 6. 

888. Adding twice (2) to (I), 

x'2 + 2xy + y2 = 64. 
Extracting square root, X'\-y = ±S (3) 

Subtracting twice (2) from (I), 

x^-2xy + y^ = i. 

x-y = ±2 (4) 

Comparing (3) and (4), we find x = 5, and y = 3. 

284. (2)-(l)=2(y-x) = (y-x)(y + x), 

or 2 = y + X. 

Then (1) becomes x^ = 2, 

x=±V2, 
y = 2±V2. 

235. First Solution. (l)2 = x^ + y2 = 16 - 2xy (3) 

(3)2 = X* + 2x2y2 H- 2/* = 256 - 64 xy + 4xV, 
or x*-2xV + 64xy + y* = 256 (4) 

X* + y* = 82 (2) 

Subtractmg, 64 xy - 2 x^y^ = 174, 

32 xy - xV = 87, 
or xV - 82xy + 256 = 266 - 87 = 169. 

xy - 16 = db 13. 

xy = 3 or 29 (5) 

From (5) and (1), x is found to be 3 or 1 ; and y, 1 or 3. 

Second Solution. 

(1)* = x* 4- 4x8y + 6xV + 4xy8 + y* = 256 (a) 

(0)4.(2)= 2x* + 4x5y + 6xV + 4xy8 + 2y* = 338, 
or X* + 2x8y + 3xV + 2xy* + y* = 169. 



SOLUTIONS TO SPECIAL EXPEDIENTS. 209 

ExtTacting square root, z^-^xy + y^— ±V^ (fi) 

(\y = x'^ + 2xy + y^z^lQ (c) 

(c)-(6)=a;y = 3 or 29. 
Hence x = 3 or 1, 

y = 1 or 3. 

836. (2) - (1)8 = xy{x - y) = 6. 

But « — y = 1. 

Hence jcy = 6 (3) 

From (1), a; = y + 1. 

Then (3) becomes (y + V)y = 6. 

Completing square, etc., y = 2 and — 3. 

Then from (1), x = S or — 2. 

887. Putting a; + y = 8, and xy = p, the equations become 

«+i)«+i)2 = 86 (1) 

and p + si^-\-ps = 97 (2) 

Adding (1) and (2), (pa + 2p» 4- ««) + (p + «) = 182. 

Completing square, (j? + »)a + (l> + «) + } = ^P ; 
whence i> + « + J = V» *^^ p + « = 13, 

or J) = 13 — 8, 

Putting this value of p in (2), we find « = 7. 

Hence p = 6. 

Then x + y = 7, 

and jcy = 6 ; 

whence a; = 6 or 1, 

and y = 1 or 6. 

888. Squaring (1), -i^ + ?J:J? = 2. 

05 + y Sx 

Putting 05 + y = «, and clearing, we have 

95c2-6a» = -s2. 

Completing square, 9x^ — Qx8-{-^ = 0; 

whence 3 ac— « = 0, and 3 x = ». 

Therefore « + y = 3 x, 

or y = 2 X. 

Substituting in (2), 2 x^ - 3 x = 54 ; 

whence x = 6 or — 4}, 

and y = 12 or — 9. 

ellwood's test prob. — 14. 



i 



210 



TABLE BOOK AND TEST PROBLEMS. 



889. lirtt Solution. 
a)-(2) = 

whence 

and 
From (2), 



a;y — y« = 16 ; 

y 

V* 
ai« = 89 - y» 



(3) 
(4) 



Equating (3) and (4), and clearing, 

225 + 30y2 + y* = 89i/a-y*, 

or y4_^^y«= -.AJi 

Completing square, y* - J>^ yS + (5^)2 = (iyi)2 - 1}a = JLjji. 



whence 
Therefore 

Second Solution. 

Let 

Then (1) and (2) become 

whence 



y = 6 or 3 Vj. 

a; = 8. 
(From Bnpt. 8. Tbaitssau, in Educational Newt.) 

(1) 
(2) 

(3) 



a;3 + y« = 89 
x^-\-xy = 104 

« = vy. 

V V + y^ = 89 

89 



y» = 



whence 



i;3-|-l 
t^a -\-vi/^= 104 
104 



(4) 



y^ = 



Equating, 
Clearing, 



89 



104 



v^+1 v^ + v 
104v2+ 104 = 89^2 + 89v, 
or 15t72-89« = -104, 

or t?a-H« = -W- 

Completing square, etc. , we find v = ^/ or |. 
Substituting in (3), we find y = ± 3Vj or ±6. 

Then x = ±^^V2 or ± 8. 

(From J. M. Fboplxb, In ^ducatumal News.) 



SOLUTIONS TO SPECIAL EXPEDIENTS. 211 

840. Let X = m + n, 

y = m — n. 

Then (1) and (2) become 2 wi» - 2 mn^ = 70 (3) 

and 2m» + 6mn« = 133 (4) 

Eluninating mn^, we find «i' = ^f *, 

or w = J. 

Substituting in (3), we get n = f . 

Hence a; = m + n = J + } = 6, 

and y = m-n = J-} = 2. 

941. ^'rse Solution. 

(1)8 = a* + 3a;2y + 3«y2 + yS = 125 (3) 

Subtracting (2) from (3) , 3 a;2y + 3 xy^ = 60, 
or xyC« + y)=20. 

But X'hy = ^. 

Hence 5 xy = 20, xy = 4 (4) 

From (1) and (4), y is found to be 4 or 1, and 

x = l or 4. 

Second Solution. Dividing (2) by (1), we obtain 

«^ - «y + y2 = 13 (3) 

Squaring (1), x^ ^. 2 xy + y^ = 25 (4) 

(4)-(3)= 3xy = 12, 

«y = 4. 
X and y may now be founa as above. 

242. Let X = vy. 

Then (1) becomes t?V + y^ = 34, and y^ = ^^ (3) 

t?* + 1 

(2) becomes v^ - t?y2 = 10, and y^ = -^^ (4) 

Equating (3) and (4), -^ = -15_. 

Clearing, completing, etc., we find « = IJ or —J. 
Substituting in (3), y yS + y2 = §il!!! = 34, 

whence y = 3. 

Then a; = 6. 



212 TABLE BOOK AND TEST PBOBLSMS. 



948. Let 




z = one number. 




and 






y = the other. 




Then 






ay = X* — y* 


(1) 


and 






a:' + y« = x»-y« 


(2) 


Put 






x=zvy. 




Then 


(1) 


becomes 


vy^-vY-y^; 




whence 






t? = r2 - 1, 




or 






i^-t? = l. 




Then 






« = i + }V6 = J(l+V6). 





From (2), putting the value of v equal to a, we have 

oV + y^ = a«y8 - y'» 
or a^ 4- 1 = a*y — y, 

whence w = ^ "^ » 

*^ o»-l 

Substituting the value of a, } (1 + V5), we have 











y — 


1+V6 ' 




Then 








• 


:K6+V^)- 




.344. Let 






05 = 


one number, 




and 








y = 


: the other. 




Then 




(X 


-y)(x« 


-y^)= 


:32 


(1) 


and 




(^ 


+ y)(x3 


+ y^)= 


:272 


(2) 


Put 








a; + y = 


«» 




and 








a^ = 


p.* 




Then 


(1) becomes 




«8. 


-4p« = 


32, 




and (2) 


becomes • 




«». 


-2i)« = 


272, 




whence 








p« = 


:120. 




Then 






«8 


-2jo« = 


: «» - 240 = 272, 




or 








«» = 


512, - 




and 








« = 


:8 = « + y 


(3) 


Also 








1> = 


iJA = 15 = xy. 




From 


(3), 






« = 


8-y. 




Then 








0^ = 


:K8-y)=16, 




whence 








y = 


6. 




Then 


• 






« = 


3. 





SOLUTIONS TO SPECIAL EXPEDIENTS. 218 

S46. Let xy—p. 

Substituting the value of as + y from (2), andp for x/yt (1) becomes 

65=(9-2|))(27-8p). 
Developing, completing square, etc., we have 

p = f f or 2. 
Hence f f or 2 = isj^ (8) 

From (2) and (3), we find x = 2, and y = l, 

846. Let 05^ = r, 

y' =«. 

Then r-\'$ = r^ (3) 

and r« + «a = 3ra (4) 

From (8), « = r»-r, and 38 = ^4 _ 2r» + f« (5) 

From (4), «a = 3ra-r» (6) 

Equating (6) and (6), and dividing by i4, 

f«-.2r+l=8-.r. 

Collecting, etc., r^ — r + J = }, 

whence r = 2 or — 1. 

Then from (8), « = 2. 

Therefore a; = 4 and 1, 

and y = 8. 

947. Squaring (1), we obtain 

aj*-8aJ^9 + y* = (8) 

«* + 2 ajay* + y* = («'' + y)« (4) 

Subtracting (3) from (4), 6 a^V = («* + y*)^. 

Extracting square root, ± xy Vb = a^ + y* (6) 

Adding (1) and (6), we get 

av(l±V6) = 2a^, 
whence a; = } y(l ± VS) (6) 

Putting this value of x in (2), we have 

i^(Q ± 2V5)+ y« = 4y»(16 ± 8V6)- y». 
Dividing by y^, and multiplying by 4, 

6±2\/5+4 = iy(ld±8V5)-4y, 
or 6±\/5 = 2y(li:V^) (7) 



214 TABLE BOOK AND TEST PB0BLEM8. 

Whence y=± JV^. 

Diyiding (7) by 4, and comparing with (6), we find 

948. (l)-*-ay= 35 + 1+ y + 1=18 (3) 

(2)+a;V= «« + ^ + |^ + -i = 2a8 (4) 

Adding 4 to both sides, 

Put f X + - J = m, 

and (^ "^ y) ~ **' 

Then from (3) and (6) m-\-n = 18, 

and m^ + n^=i 212. 

From these equations we find * m = 14. 

But »» = as 4- - = 14, 

a: 

whence » = 7 db 4 V3. 

Then y is easily found to be 2 d: >/3* 

NoTX. — Neat aoIaUonfl may b« made by patting x + y^^i, and xy^sp. To make 
theee diiferent solnUona is excellent exerciee. ^ 

M9. Dividing (2) by (1), 

a:* + xy{x^ + y2)+ xV + V* = 121 (3) 

Squaring (1), a;« + y* = 4 + 2fly (4) 

Squaring (4), ic* + y* =(4 + 2a5y)« - 2ieV, 

or 16 + 16xy + 2a;V = «* + y* (5) 

(3) may now be written 
16 + 16a^ + 2xV + xK*" + y*) + a^y* = 121 (6) 

But from (4), x« + y'^ = 4 + 2xy. 

Substituting in (6), we have 

16 + 16xy + 2xy + xy(4 + 2a^) + xV = 121 (7) 

or 6xV + 20xy = 106, 

or xV + 4xy = 21; 

whence xy = 3 or — 7 (8) 



SOLUTIONS TO SPECIAL EXPEDIENTS. 216 

From (1), x = 2 + y. 

Substituting in (8), we have y (2 + y) = 3 or - 7, 
or y2 + 2y = 3 or -7; 

whence y = ±2 — 1 = 1 or— 3, ordb V— 6 — 1. 

Hence x = 3, — 1, or 1 db V— 6. 

250. Dividing (1) by 2, and adding \ x, we get 

- 4 Vi + y/x'Vy^ - ^y/x + Jx = a. 



Extracting square root, 



Vya-4Vi + \y/x = ± Vx, 

or Vy* — 4 Vx = jVi or — jVx. 

Squaring, y" — 4>/ic=:Jx or \x. 

Clearing, etc., 4y2 = IQVx + x or 9x (3) 

Factoring and transposing (2), we get 

2V2y-2Vx-l = y-Vx+l (4) 

(4)x 2= 4V2y~2Vx-l=2y-2V^ + 2. 

Subtracthig 3, 4V2y - 2Vx - 1 -3 = 2y-2Vx- 1 (5) 

Put V2y-2Vx-l = w. 

Then (6) becomes 4 m — 3 = wi^, 

whence w = 3 or 1. 



Hence V2y-2Vx-l = 3 or 1. 

Squaring, transposing, etc., y = 5 + Vx or 1 + >/x (6) 

Squaring (6), y* = (5 + Vx)* or (1 + Viy (7) 

(7)x4= 4y2 = 4(5 + Vx)aor4(l + Vx)« (8) 

Equating (3) and (8), 4 (1 + v^)« = 16 Vx + x, 
or 4 + 8\/x + 4x = l«Vx + x. 

Transposing, 3x — SVx = — 4 (9) 

(9) X 12 = 36 X - 96 Vx = - 48. 

Adding 64, 86x- 96Vx + 64 = 16. 

Extracting square root, 6 Vx = 12 or 4, 

whence x = 4 or |. 

But y = 5 H- Vx or 1 + y/x. 

Therefore y = 7, Sj, 3, or 1}. 

(From LISTIB B. FlLUUX.) 



216 TABLE BOOK AND TEST PROBLEMS. 

251. Let 2e = to + 2, 

and y = j? + 1. 

Then (1) and (2) become *» + y = 11 (3) 

and ya + a; = 7 (4) 

Put 2 + y = n. 

Then from (4) we have ac — 3 = n (2 — y), 

or ?-? = 2-y (5) 

From (8), «« - 9 = 2 - y (6) 

Equating (5) and (6), a;* - 9 = - - ?, 

or a;2--=9-- (7) 

Adding to each memher -=—^ we have 

4n^ 

a^-? + -L = 9-? + -l-. 
n 4n2 n 4n* 

Therefore x- — = ±^3-— V 

2n V 2n/ 

Taking the + value, we have x = 3. But as = to + 2, hence to = 1. 



RECIPROCAL OR RECURRING EQUATIONS. 

252. Multiplying, we obtain 

a^ + »* + x* + 2a5' + a;a + aj + l = 80»'. 

Dividing bya^, a!^ + a^ + ajH-2H--4-i + i = 30. 

Collecting, a:^ + -^ + «« + - + x + - + 2 = 30 (1) 

Let y = X + — 

*^ X 

Then a' + i = y*-3y, 

at** 

and a;^ + i = ^ - 2. 

Substituting in (1), and collecting, y> + y^ - 2 y = 30 (2) 

(2)xy= y* + y»-2y« = 30y (3) 

(8)-(2)= y* - 3ya = 28y - 30. 



SOLUTIONS TO SPECIAL EXPEDIENTS, 217 

Adding 16 ^^ + ^$^ to both members, 

2^ + 13y3 + iJA = 16y2 + 28y + ^. 

Extracting square root, y^ + -y = 4y + l, 

or y2-4y = -3, 

whence y = 3 or 1. 

But « = a;4-- = 8orl. 

X 

Therefore ac = } (8 ± Vs) or i (1 dh V^). 

258. Dividing by 8 ofi^ and arranging, we get 

^ + ^-2(- + i) = ¥ (1) 

Let x-\'- = m, 

X 

Then w^ = ^8 + ^ + 3 ^oj + iV 

and a^ -f ~ = m* — 8 TO, 

Substituting in (1), we have 

TO«-3TO-2m = ¥, 

or TO* — 6 TO = Y (2) 

(2) X TO = TO* - 5 TO^ = y TO. 

Adding to both sides ^ to^ + |{, we obtain 

TO* + }to2 + II = jyLma + y to + H. 
Extracting square root, to^ +f = f to + f , 

to' = f TO, 

TO = } = a; + J; 

or ic«-Jaj = -l, 

whence a; = 2 or }. 



(From L. B. FiujuN.) 



864. Dividing by Jl - -, VSTTl - 1 = ^^ "" ^ « 
Squaring, « + 1 -2\^T1 + 1 =^-^^ 



218 TABLE BOOK AND TEST PBOBLEMS. 

Transposing, changing signs, etc, 

2V5rn[ =^^t*^^= 1 + a; + l. 

X X 

Squaring again, 4« + 4 = i + 3 + ? + x* + 2x, 

X* X 

which may be written 

x2-2 + i-2x + -+l=0. 
x^ X 

Factoring, Ix- -V- 2(x--\'\-l=0. 
Extracting square root, x 1=0. 

X 

Clearing and transposing, x^ — x = 1, 

whence x = i(l ± VE). 

255. Dividing by a; + !> 

ic* - a^ + x2 - « + 1 = a(l + x)* = a(xl^ + 4a5'» + 6a;« + 4« + 1), 
or 05* + ! — ax* — a — as* — 4aa5* — X — 4ax + «2__5fl{jpa==0. 
Factoring, we obtain 

X* + 1 - a(x* + 1)- a^(l + 4a)- x(l + 4a)+ **(1 - 6a) = 0, 
or (x* + 1)(1 - a)-(x8 + xyi + 4a)+ x2(l - 6a) = 0. 
Dividing by (1 — a), 

(a4 + 1) _ Lti«(x8 + x) + i-=-^«^ = 0. 
1 — a 1 — a 

Dividing by x*. 



x2 



X* 1 — a\ x/ 1 — a 



Put X + i = y. 

X 

Then aj* + i = y« - 2. 



or 



'•-(\^>-(t^)=»- 



SOLUTIONS TO SPECIAL EXPEDIENTS. 219 

^ Vl-ar 4(l-a)a 4(l-a)a 1-a' 

whence y = 1±A± ^JS±Ml±Il±MSZM 
^ 2(1 - a) Af 4(1 - 0)2 

^l + 4qzt:V6(l + 4a) 
2(1 - a) 

Put this = n. 

aj2 — na; = — l, 

whence x = J(n ± V»'^ — 4). 

l + 4a±V5(l + 4a) 
' 4(1 - a) 

j,.^ aH-4(i)a 6a+4a) 2(l + 4a)V6(l + 4a) 
4(1 - a)2 4(1 - a)2 "^ 4(1 - o)^ 

v/Sa34 = ^ /lOC6"a-l)J:2(l + 4a)V6(r+f^ 
Af 4(1 - a)a 

Hence i(n± y/n^ - 4) 

_ l4 4a-t,V6(^l + 4a)d:2Vl0(6q-l)j:2(lH-4q)V6(l+4a) 

4(1 - a) 

866. First Solution. Squaring and arranging, 

a%B^ + 4 SB? + 2 a^x* - 4 «« + a« = 0. 
Dividing by x*, 

a2x* + 4x« + 202-4 + 3 = 0, 

X* X* 

or o*(x* + iW4fx2-i\ + 2aa = (1) 

Put ^"""h"^' 

Then x*-~24--^ = 



Therefore x* + 4 = w* + 2. 



X* 



220 TABLE BOOK AND TEST PROBLEMS. 



Hence (1) becomes 

or aV + 4y= — *^» 

2 



Therefore y = - 4 (^ =t "^^ - ^) J 

ft* 



a»^ " I 






I 



2 



Therefore v^ + -^zO- ±y/l - a*)a^ = 1. 



Solving, 



>_ -li:Vl-0*i:V2a±Viri^"q^ 



or 



x=±i^-i±vn=^±V2(i±\/r=^). 



a 
Second Solutioti. Squaring, dividing by a^, and factoring, we have 

which may be written f a^ + ^ I = — ^l ** -^ A h 

\ xV a*\ a^/ 

Hence, subtracting 4 from each member, we have 
Completing square, 

{''-h)H{''-iH'i-^ 

Extracting square root^ we have 





1 o 


or 


»*'-;^=-^(l±^l-«*)- 


Put second member = 2 m*. 




Then 




and X* 


-2m%»=l. 



SOLUTIONS TO SPECIAL EXPEDIENTS. 221 

Completing square, 

x* - 2m«a;2 + w* = 1 + w* ; 
whence »* = m'* ± VT+w*, 



and « = ± Vm2±vT+^ (1) 

But wa = - i(l ± Vr=^), 



a^ 



and m* = 3(l±2Vnr^ + l-a*). 

Substituting in (1), we have 

or x = ±iV-l±Vn^±V2(l±vT^^. 

3%t>d 5oZtt«ion. Dividing by 1 + ie*, 
2a?Vl -a^ _q 

1+x* r 

By composition and division, 

1 + a;* + 2gVl -a^ _ l-{-a 
l + x*-2xvT^^ 1~« 
which may be written 

x^(l4-g^)+2gVl -a:* + l-a?^ _. l-i-ff 
«2(l + xa)- 2xVir^ + 1 - x^ 1 - o 

Extracting square root, 

X VI 4- x'* 4- VI - a^ _. Vl + « 
xVl + x^-Vl-xa VI -a 
By comi)osition and division, 

xVl -f oe^ __ VI -f g + VI - g _ Vl - q' + 1 . 

Vl^=^ VrTa~vT^=^ « 

Squaring, 

g'4-g* _ (Vl-q-^4-l)* 
1 - x^ a^ 

Clearing, 

a^x' 4- a^x* = ( VT^^ + l)^ - (VH^ 4- l)«a?. 

Transposing, etc., 

a«x* 4- 2(Vr^^4- l)xa =(\/n:^ H- 1)^. 



222 TABLE BOOK AND TEST PB0BLEM8. 

Completing square, 

a* Or 

Extracting square root, 

a a 

or ox^ = ^ 1 

a a 

whence « = ± - {( VH^ + 1) ( VTTo'^ - 1)}*. 

(From Profesaor C. Hobnung, professor of mathematics, Heidelberg College, Tiffin, O.) 

367. First Solution. Expanding, 

a;io + a;» + a;8 + 2x"' + 2x''' + 2x5 + 2a;* + 2x' + aj2 + X 4- 1 
= 8x? + 10x7 - 8x5 + lox' + 8x^ 
Dividing by x^, 

a*4.a;* + x» + 2xa + 2x4-2 + ? + 4 + :i + A + -^ 

•(/ iX/ <v tX/ 

= 8x» + 10x2 -8 + ^ + 4- 

X^ X* 

Arranging and factoring, we have 

x6 + i + x* + i + x»4--,+ 2x2H-4 + 2x4-- + 2 

X» X* x' X^ X 

= 8(x.+ i) + 10(x« + l)-8 
Transposing, 

Adding to both sides 8^ x + - W 5^ x« + i) + 4^x2 + i W 4, we have 
= 8(x + l)+12(x. + l)+12(.» + i)-4 (1) 



SOLUTIONS TO SPECIAL EXPEDIENTS. 223 

Pat X + i = y. 

X 

Then a;2+l-y2_2, anda;8-.l = y8-3y. 

Futimg these values in (1), we have 

y* + y* = 8y + 12 (y« - 3y)+ 12 (y2 - 2)- 4, 
or y* + y* = 8y + 12y8 ^ 36y + 12y2 - 24 - 4 ; 

whence, by factoring, 

y*(y + 1) = 12 y2(y 4. i) _ 28(y + 1), 
and y* = 12 y2 - 28, or y* - 12 y* = - 28 ; 

whence y = ± Vg ± 2 v^ = x + - (2) 

X 



Qearmg, x^ - xV6±2V2 = - 1, 

whence x ="± }(^^ ± 2v^ ± V2 ± 2 V2). 

NoTB.— A better solution is given below. 

Second Solution, Expanding and refactoring, without transposing 
terms, 

(xw + l) + (a;* + «) + (x« + x2)+ 2(x7 + x»)+ 2(x9 + x*)+ 2x5 

= 8(x8 + x2)+ 10(x7 + x«)~ 8x6 (1) 
(1)^x5 = 

= 8(^ + i) + 10(.« + i)-8 (2) 



Put 


x + l = y. 


Then 


^ + i-2'^ 2, 




x» + i = y8-3y, 
x* 




x* + i = y*-4y2 + 2, 

X* 



and x5 + — = y* — SyS + Sy. 

X* 

Substituting these several values in (2), we obtain by transposition, etc., 
y» + y* - 12y« - 12y» + 28y -|- 28 = (3) 



224 



TABLE BOOK AND TEST PB0BLEM8. 



Since by changing the signs of the second and every alternate term 
in (8) the algebraic sum of the coefficients would be zero, by the theory 
of equations — 1 must be a root of equation (8). Therefore it is divisi- 
ble by y + 1. 

Performing the division, 



virhence y = ± Ve ± 2 v^. 



Therefore xH-- = -l 

X 



and 



X 



-f-l = ±V6±2V2 

X 



(6) 



From (4) we find 



^^±V33-i 



From (5) we find 

x = ± jV6±2V2 + }V2db2V2. 

Therefore x has 6 imaginary values, and 4 real ones. The imaginary 
values are, — 



x = 



_V-3-l 



}V(6 + 2v^)-|- jV(2^2v^). 
- }V(6-f 2\/2)+ J V(2 - 2V2). 
}V(6-2V2)+ J V(2 - 2 V2). 
^ - }V(6-2v^)+ }V(2-2v^) 



The real values are, — 



x = 



jVe + 2 V2 + }V2 + 2V2 = 2.5843176. 

}V6-2>/2 + iV2 + 2 V2 = 1.9891296. 

_ J Ve - 2 V2 + }V2 + 2\/2 = ,2082386. 

- jV6 + 2\/2 + jV2 + 2v^ = - .3869493. 

(From Profoaior B. F. Bitblbsok, Oneida CaiUe, N.Y.) 



SOLUTIONS TO SPECIAL EXPEDIENTS. 225 

HIGHER EQUATIONS. 

868. Squaring the given equation, we obtain 

2V[(aJ^ - <i^){!x? - 62)]- 2 V[(x^ - c^){x^ - d^)'\^a^ -f 6^ - c^ - (P (1) 
Squaring (1), etc., we have 

8v/[(«* - a2)(x2 - 62)(a.2 _ c2)(aj2 - dO]= 8x* - 4(a2 + fta + c^ + d2)x2 
+ 4a2&2 + 4c2di -(a2 + 52 _ c2 - <«2)2 (2) 

Squaring (2), canceling equal terms in opposite members, refactoring, 
etc., we obtain a pure quadratic from which we find that 

a; = ± V{(«H62- c2_d2)4__8(a2&24.c2<p) (a2_|. ja _ ^2 - (i2)2 
+ 16(a262 _ c8d^)2} -«- 2 V{8(a-62 _ c^i){a2 + &2 - c2 - d^) 

- 2(a2 + 62 + c2 + (|2)(a2 + 52 _ c2 - (?2)2J 

= (when a = 36, 6 = 40, c = 13, and d - 61)db 85. 

The formula found for the value of x is evidently also that which will 
be found for any equation obtained by permutating the signs of the given 
equation. To illustrate : if a = 375, h = 408, c = 180, and d = 297, we 
find by the formula that x = ± 425, which are the roots only of the 

equation Vx-^ - a2 - Vx2 _ b2=Vx^ - c2 - Vx2 - dK 

Again, if a = 94, 6 = 95, c = 96, and d = 20, we find by formula that 

= ± 100.549351213668963, 
which are the roots only of the equation 

Vx2 -a^-\- Vx-* - 62 ^. Vx2 - c2 = Vx^ - d^ 
(correct to 15 decimal places). (From B. F. Bublksok.) 

259. Clearing from the radical, we obtain, by factoring, 

16m*(l - x2)2 - 8x2(2 - w*)(l - x2) = - m*x8 (1) 

Cdnsidering (1) as a quadratic equation whose unknown quantity is 
(1 — x2), we find by its resolution that 



\ 4 m* / 

ellwood's test prob. — 15. 



226 TABLE BOOK AND TEST PB0BLEM8. 

which is also a quadratio equation whose unknown quantity is a^. Hence 



j.^ _ i: 2toM V2 ± 2(1 - m*)l - m'^\ 
2±2V1 -i»*-m* 

Bedudng this value of x^ to its lowest terms, we have 

2m> 



a;a = 



t»^ 



i3±V(2^2Vf^m*) 

whence a; = ^ *"^^ 

iV^w^ ± V[2 ± 2 V(l - m*)] 1 

(From Professor B. F. Bublbsoh.) 

Note.— In 1870, F. P. Matz, A.M., Ph.D., published this as a prize problem in 
Barnes's " Educational Monthly,*' and awarded the prise to Mr. Bnrleson of Oneida 
Castle, N.Y., for the eolation given above. 

S80. Raising to the fourth power, we have 

16x*-16x«=(2-ai2)4. 
Now, (2 + a;9)4 = (2 - x^Y + 16x^ + 64«2. 

Hence, adding to each member 16 x^ + 64 x^, we have 
16iK* + 64x8 =(2 4- »»)* =(4 + 4x2 -|- x*)«. 
Put 4 + 4x« + x4 = y. 

Then the equation becomes 

16y-64 = y«, 
or ya-16y = -64, 

and y = 8. 

Then x* + 4x» + 4 = 8, 

xa + 2=V8, 

x=V2(V2-l). 

961. Put Vx = y». 

Then the equation becomes y» — j^ = 100 (1) 

(l)xy= y*-y» = 100y (2) 

(2) + (l)= y* - y* = lOOy + 100. 



SOLUTIONS TO SPECIAL EXPEDIENTS. 227 

Adding to each member 16^^ + ^f^, 

whence y = 6) 

y« = 126 = Vx. 
Then x = 125« = 16,626. 

. Factoring, 8^x- 1^x2 + ^ + 1] = 0. 

Therefore as — i = 0, and x = }, the real root 

x« + |4-| = 0, orx» + | = -?. 

Completing square, a;2 + | + l = i_| = -g. 



Extracting square root, a; + i = ± i V— 36, 

and « = -J ± JV— 36 = i(— 1 ± V— 36), the imaginary roots* 

263. x» + 2x2 + x = 18 (1) 

(1) X X = X* + 2x» + x« = 18x (2) 

(2)-2x(l)= x*-3x2 = 20x-36. 

Adding 16x2 + J^p, 

X* H- 13 x« + ^F = 16x2 + 20x + V- 

Extracting square root, x^ + ^ = 4 x + {> 

whence x = 2. (From Dr. I. J. Wibiback.) 

364. Let _l_=r»2. 

1 H-x 

Then the equation becomes 

12 



144 1 + X 
Clearing, etc., x^ + x = 72, 

whence x i^ 8 or — 9. 



228 TABLE BOOK AND TEST PB0BLEM8. 

865. This may be written 

a!^ + 8ai»-6x-8 = (1) 

Put 05 = y — 1. 

Substituting in (1), we have 

y« - 9y = 0, or yCy« - 9)= 0. 
Hence y = 0, and y» - 9 = 0, y« = 9, y = ± 3. 
But 05 = y — 1 : consequently x = — 1, 2, and — 4. 

966. Transposing, x» - 6x* + llac - 6 = 0. 
Factoring, (x - 2) (x^ - 4 x + 3) = 0. 
Therefore x — 2 = 0, 

or X = 2, 

and x' - 4 X 4- 3 = 0. 

Adding 1 to both members, x^ — 4 x + 4 = 1. 

x~2 = ± 1. 
x = 2±l = 3orl. 
Hence x has three real roots : 1, 2, and 3. 

967. Adding 4 to each member, and factoring, 

(x2 - 3 X - 2)2 =64. 
x2 - 3x = 2 ± 8 = 10 or - 6. 
Completing square, x^ — 3 x -|- 1 = ^ or — i^. 

X - f = ± J and ± } V- 15. 

X = 6, -2, and J(3 ± V- 16). 

968. Multiplying by X, x* + 16x2 =128 x. 
Adding 16x2 + 256, we have 

x* + 32x2 + 256 = 16x2 + 128x + 266. 

x2 + 16 = 4x+16. 

x2 = 4x. 

x = 4. 

(From Dr. I. J. Wibkbick.) 

269. Put \/x = y. 

Then y* - y = 14. 

Adding 2 to each side, and factoring, 

(y2 + 4)(y + 2)(y-2)=y-2, 
or (y« + 4)(y + 2)=l (1) 



SOLUTIONS TO SPECIAL EXPEDIENTS. 229 

But (ya + 4)(y + 2)>l. 

Therefore (1) is an absurdity, which could only be brought about by 
using a zero factor. 

Hence y — 2 i^ 0, y = 2, and x = 4. 

(From Dr. I. J. Wibeback.) 

270. Adding x^ + 2x^ to each side, we have 

X* + 2aj8 + aj^ = X* + 2a;2 + 1. 

x^ -f » = a;2 + 1, 
whence x = 1. 

871. Multiplyfaig by X, x*-6x2 4-4x = 0. 

Adding 4 x^ + 1 to each member, and transposing, we have 

x*-2x2 + l =4x2-4x4-1; 
whence x* — 1 = 2 x — 1, 

x« = 2x, x = 2. 

(From Lebteb 6. Filuian.) 

272. x»- 3x2 + 4 = (1) 

(l)xx = x*-3x8 + 4x = (2:) 

J2)- 3 x (1)= X* - 9x2 =- 4x - 12. 
Adding 16x2 + ^, x* + 7x2 4-^ = 16x2 - 4x + J. 

Extracting square root, x2 + J = 4 x — }. 

Transposing, x* — 4 x + 4 = 0, 

whence x = 2. 



278. 


Vi-Vx = 100 


(1) 




X* - x^ = 100 


(2) 


Let 


x* = y. 




Then (2) becomes 


y« - y2 = 100 


(3) 


(3)xy = 


2/*-y8 = 100y 


W 


(4)-H(3) = 


y4-y2 = i00y + 100. 




Adding 16 y2 + i ja, 


y* + 15 y2 + ip = 16 y2 + 100 2^ + i^A. 




Extracting square root. 


y» + -»sf- = ±(4y + ¥). 




Taking + value, 


y2-4y + 4 = 9, 

y = 6 or — 1, 




Therefore 


x* = 5 or — 1. 
X = 6^ and - le = 15,625 and 1. 





280 TABLE BOOK AND TE8T PB0BLEM8. 

274. This equation may be written thus : — 

9a;« - «♦ - 18a;« - 2x» + 45 x^ - 25 «« - 30x - 24a; = 144 - 38. 
Transposing, 

905* - 18x» + 45x2 -36x + 36 = x*-f2x« + 25x2 + 24x + 144. 
Extracting square root, 

3x2 - 3x 4- 6 = x2 + X + 12. 
Transposing, 2x2 - 4x = 6, or x* - 2x = 3. 

Adding 1, x2- 2x4- 1=4, 

whence x = 3 or — 1. 

(From Dr. I. J. Wibsback.) 

276. x« + x2 = 80 (1) 

Multiplying by x, x» + «■ = 80 x (2) 

(2)-(l)= X* - x2 = 80x - 80. 

Transposing, x* + 80 = x2 + 80x. 

Adding to each member 24 x2 + 64, we have 

X* + 24x2 + 144 = 25x2 -f 80x + 64 ; 
whence x2 + 12:=5x + 8, 

or x2 — 5 X = — 4, 

and X = 4 or 1. 



276. Transposing and squaring, 



V___W_2a_/ _a\ 



Arrangiiig, ^ „ -^ = (l-pl (1) 

Let y = l-i- 



Then (1) becomes yi-lL = -tl, or 3?y^ -xu=-2a. 



2a 
X x^ 
Putting xy = m, we have 

m^ — rn=—2a, 

whence fn = i(l ± Vl — 8a). 

But y =z 1 — — . .-. xy = X — - = t», 

x^* X 



m 



whence x = - ± - v4a-fm2. 



SOLUTIONS TO SPECIAL EXPEDIENTS. 231 

Substitutmg value of m, we have 

a; = t j]±Vl-8a±\2±2(l-8a)^+8a}. 

877. Squaring, ^ "^ ^"^ ~ ^ = x^ - 4x4- 4. 

x-\/x2-9 

By compoBitioQ and division, 

2x x2__4a; + 6 

— ■-■■■  ^^ I • 

2Vx3_9 a;2-4x + 3 

gg -. a^* -- 8x* + 26xg - 40g + 25 
X" - 9 X* - 8x8 + 22x2 - 24x + 9 ' 



Squaring, 



By division, x«^x* - 8x3 + 20x2 - 40x + 26^ 

^ ' 9 4(x2-4xH-4) 

Extracting square root, - = ^-^^ — x-\- b ^ 

^ ^ 3 2(x-2) 

Clearing and collecting, 

x2-8x=-15,- 

whence X = 3 or 6, (From LxanB B. Fillxan.) 

278. Reducing fractions, 

1+a; 1-x ^^ 

1-X4-X2 l + « + a;' 
Clearing, etc., 2H-4x2 = a4-ax + aic*» 

or ax^ -\-{a — 4)x2 = 2 — a. 

Completing square, etc., we find that 

. q~4 .^ /16^3a^ 



x*-* 



, whence x =-v/— (4 - a ± \y/lQ - 3 a^). 

'2a 

(From Lbsteb B. Fillman.) 

279. It is easily seen that the first member is equal to (x + Vx)^. 
Hence the equation may be written 

(x H- Vx)« = 32(x + Vx) - 240. 
Completing square, 

(x + Vx)« - 32(x + Vx) + 266 = 16. 

Extracting square root, x + Vx — 16 = ±4, 
or X -f Vx = 20 or 12. 

Hence Vx = 4, or — 6, or 3 ; 

and X = 16, 25, and 9. 

Note. — To latisfy the equation with a; = 25, we must nee v'x » — 6 inatead of V^ » 5. 



232 TABLE BOOK AND TEST PROBLEMS. 

880. First Solution. Clearing, 

x2-2aj>/x-7« = 8VS- 16. 
Adding 8a;, x^ -2xy/x + x = 8x + %Vx - 16, 

which may be written x* — 2x Vx + x = 8 x — 8 v^ — 16 + 16 v^. 
Factoring, then transposing, 

(x - ^/xy - 8(x - Vx) + 16 = 16 Vx. 
Extracting square root, x — Vx — 4 = 4v^, 
or x = Vx + 4Vx + 4, 

and Vx = y/x + 2, 

or Vx — v^ + J = |. 

v^ - J = ± }. 

y/x = 2 and — 1. 
Therefore x = 16 and 1. 

Second Solution, Let Vx = y. 

Then y *—- = ^. 

Clearing and transposing, 

y4_2y8-7y2_8y+l6 = 0. 
Adding 16 j/ to both members, 

y4 _ 2y8 _ 72^2 ^ 8y + 16 = 16y. 



Let 


y = Vy-\-2. 
Vy = m. 


Then 


w2 - m = 2. 




w* - wi + J = {. 

wi = 2 and — 1. 


Therefore 


y = 4 or 1, 


and 


X = 16 or 1. 




(From Dr. I. J. Wirsback.) 



Third Solution, Clearing, and freeing from radicals, we obtain 

a4 - I8x» + 49x2 - 288x + 266 = (2) 

Factoring (2), 

(xa - 17x + 16)(x2 _ X + 16) = (3) 



SOLUTIONS TO SPECIAL EXPEDIENTS. 233 



Therefore x^ - 


17 


« + 16 = (4) 


and ^' 


i __ 


« -1- 16 = (5) 


From (4) we find 




X = 1 or 16, 


and from (6) we find that 




a = Kl±3V-7). 

(From Professor B. F. BuBLKflON.) 


Fourth Solution, Put 




Vx = y. 


Then 


y- 


7 8 
y-2 ^ 


DiYiding by 8, ^ - 


o 


7 1 

r-. c»\ —■>* 



y^-2y-7 _l 
8(y-"2) y^ 

Adding — — to both members, 
y — 2 

8(y - 2^ 2/2 y - 2 
Clearing, etc., y^Cy^ - 2 y + 1) = 8(2/2 + y - 2). 

Factoring, 2^(2/ - 1) (2/ - 1) = 8(2/ - 1) (2^ -h 2). 

2/8 _ 2,2 = 82/ + 16 (1) 

Multiplying by y, 2^ = 2^ 4- 8 2/2 + 16 2/ (2) 

Adding (1) and (2), y* = 9 2/2 + 24 2/ + 16. 

2^2 = T(32/ + 4)i 
whence 2^ = 4, - 1, and - » ± i V"^. 

Therefore S/2 or a; = 16, 1, and }(1 ± 3\/^7). 

JFY/it/^ Solution. Let 2^ = Vi. 

Then y-_L^=:i. 

2/ -2 2/2 
Clearing and transposing, 

y4_22/8-7y2_82/+ 16 = 0. 

Factoring, 

(2/-4)(2/-l)(2^ + 32/ + 4) = 0. 

Hence 2^ — 4 = 0, y = 4, and a; = 16. 

2/ — 1 = 0, 2/ = 1, and a; = 1. 

2/2 + 32/ + 4 = 0, 2/ = - K3 ± V^), and x = Kl ± SV^::?). 



234 TABLE BOOK AND TEST PROBLEMS. 



281. This may be written 

Voic^ — a , /x* -— a 



or i Vax=' — a 4- - Vx* — a = x". 

X X 

Multiplying by x, 

Vox"^ — a + Vx* — a = x*. 
Transposing and squaring, ox^ — a = x' — 2x'Vx* — a + x* — a, 
whence, dividing by x^, we have a = x* — 2 xVx* — a -f x"^. 
Transposing and squaring, 

4x2(x* - a) = x* + 2x8 + (1 _ 2a)x* ~ 2ax2 -|- a^. 
Collecting terms, and dividing by x*, we get 

x»-2x2+l~2a + ^ + - = 0. 

X"^ X* 



Combining, 

Extracting square root, x^ — -^ — 1 = 0; 

X 



hence 






x«. 


- x2 - a, and x = ± Vj ± J\/4a 


+ 1. 




282. 


0)-^ 


x'y: 


= 


x«4-- = y-Hi 
x^ y 




(3) 


(2)^ 


3xy8 


= 




i(»"*^)-K-a 




(4) 


Put 








X -f i = m. 

X 






Then 








x8 + i = »»8_3^. 

x' 






Let 








y+J=«. 






Then 








y« + i = n» - 3 tt. 







Adding (3) and (4), and substituting, we get 

w8 = 3m8, or n = wv^ = ^x + -\ v^. 
Substituting in (3), we have 



X8 + 

X« 



SOLUTIONS TO SPECIAL EXPEDIENTS. 235 

Dividing by fx + i V a;2 - 1 + i = ^ (5) 

\ Xj X2 ' 

1 8/- 

Completing square, x^ — 2 + -i- = v3— 1, 

whence a;-i = (v^-l)* (6) 

Completing square again by adding 8 to (5), 

x2 + 2+i- = v^ + 3, 



whence a; + ^ = (v^ + 3) (7) 

From (G) and (7), we easily find x = i {(3 + \/3)i+ (\/3 - 1)*}. 
Then y = i{</3(y/S + 3)*±(3v/9 - 1)*}. 

283. Let 2 X + 3 = wi', and 2 x - 3 = ««. 

Then !» + iL=lf?«! + 2^\ (i) 

Clearing, etc., 13 mV(m'^ + n^) = 4 w^ + 4 n^ (2) 

or 13w2n2 = 4m*-4w2n2 + 4n* (3) 

Taking 4 w^n^ from both members of (3), 

9w2n2 = 4to*- 8 w^n^ + 4nS 
whence 3 win = 2 w^ - 2 w^ = 2 (7712 - n^) (4) 

Adding 12 m^n^ to both sides of (3), 

25m2n2 = 4m* + 8 w^n^ + 4n*, 
whence 5 «m = 2 (m^ + ««) (5j 

From (4) and (6), w = 2 w, and w»^ = 8n' (6) 

Substituting values of m and n in (6), we get 

2x + 3 = 8(2x-3), 

whence * = f } 

284. Factoring, 

Dividing by x*, 



236 TABLE BOOK AND TEST PROBLEMS. 

Fatting 1 1 + -— j = j^, the equation becomes 

x-* X* 
completing square, ^-ly + ^ = m + ^ = ^ 

Extracting square root, 

3 , 17 ,, 10 ^, 7 
« — — = 4- — , w = — or • 

But y = 1 + _L. 

3x 

Therefore 1 + JL = 15 or - -, 

3x x2 x2' 



whence a; = i (_ 1 j- V- 251) and 3. 

285. Developing, l-|-x' = a-|-3ax + 3ax2 + ax«. 

Dividing by x2, 1 + a; = ^ + ?-^ -f- 3a + ax. 

X2 x2 X 

Factoring, 1 + x = a^l + aj^+ 3a/^-+ iV 

Dividing by - + 1, we have 

X 

x + ^-1 =a(x + l-l)+3a (1) 

X X 

(1) may be written x + 1— 1 = a( x H-i — 2 ) + 4a. 

X \ X J 

Subtracting 1 from both sides, and factoring, 

Fut (Vx-—y=y\ 

\ Vx/ 

Then (2) becomes ay* - ya = i - 4 a. 



y2 = ^ = w2, suppose. 

a — 1 



Therefore I Vx ) = m*. and Vx = m. 

\ Vi) Vx 



SOLUTIONS TO SPECIAL EXPEDIENTS. 237 

Clearing and transposing, x — mVx = 1. 
Completing square, 

4 4 4 

Extracting square root, Vx-^ — =±- ^4 + m-* ; 

whence Vx = J(m ± V4 -f- m-'), 

and X = J(m« ± 2 mV4 + «»•-« + 4 + m'), 



or x = l(ni« ± m>/4 + m=' + 2). 

Substituting value of m, and reducing, 

^ H-2a±V12a-3 

X = — ! = • 

2(1 - a) 

286. Multiplying by x, 

«♦ - 6x« 4- 11 fl;2 - 6x = (1) 

The square root of the left member is x^ — 3 x, with a remainder of 
2 x^ — 6 X. Hence (1) may be written 

(x2 - 3x)2 + 2(x2 - 3x) = 0. 

Completing square, 

(x2 - 3x)2 + 2(x2 - 3x)+ 1 = 1. 

Extracting square root, x^ — 3x + l = ±l; 
whence x^ — 3 x = or — 2, 

and X = 3, taking value. 

Taking the other value, x = f±J = 2 or 1. 

287. First Solution. 

Clearing, x^ - 6 x = 12 -f- 8 Vx. 

« 

Adding x + 4 to both sides, 

x2-4x + 4 = 16 + 8>/x + X. 
X - 2 = 4 H- Vx. 
X — Vx = 6. 

x-Vx-^i=^-. 

\^ - J = ± l. 

Vx = 3 or - 2. 
Therefore x = 9 or 4. 



288 TABLE BOOK AND TEST PB0BLEM8. 

Second Solution, Put Vx = y. 

Then, clearing, etc., 

y4_6y3-8y-12 = 0. 
Factoring, 

(y + 2)(y-3)(y» + y + 2)=0. 

Then y + 2 = 0, y = -2. 

y - 3 = 0, y = 3. 

y« + y + 2 = 0, y = }(l±VZl). 

Substituting value of Vx, we find that 

a; = 9, 4, and i(- 3 ± V^). 



Let X = m -f-n, and y = m — n. 

Substituting in (1) and (2), the equations become 
2m« + 6mn2 = 72, and 2w»-2mn2 = 48. 
From these we find m = 3, and n = 1. 
Therefore « = 4, and y = 2. 

Second Solution. To (1) add 3 x (2). The sum is 

«» + 3x2y + 3a^a + y» = 216 (3) 

Extracting cube root, 

X -f y = 6, or a; = 6 — y (4) 

(1)^(4)= x2«xy + ya = 12 (5) 

(4)2 - (5) = Sxy = 24, or xy = 8. 

Substituting value of x from (4), we get 

(6-y)y = 8; 
whence y = 2 or 4, 

and X = 4 or 2. 

289. First Solution. Let x + y = «> and xy = p. 

From (2), « = VlS+Tp. 

36 



From (1), s = 

Equating, Vl3 + 2p = 



13 -p 
35 



13-1)' 
whence 2p» - 39p2 + 972 = 0, 

or 2p« - 36p« - 3pa 4- 972 = 0. 



SOLUTIONS TO SPECIAL EXPEDIENTS. 239 

Factoring, 

2i)2 (p - 18)- 30 - 18)0 + 18)= 0. 

Reducing, 2p^ = 3(p + 18), 

whence p = 6 or — J. 

Adding 12 (or 2p) to both members of (2), 

a;2 + 2p + y2 = 25, 
whence sc + y = ± 5. 

Subtracting the same from both members, we find 

X - y = 1. 

Therefore aj = 3, 

and 2^ = 2. 

Second Solution. 

Factoring (1), (ac + y) (x^ -xy-\- y^) = 36, 

or z^-xy-\-^ = -^ (3) 

(3)-(2)= _a:j^ = _13+ 3^ 



Multiplying by - 2, 2 xy = 26 - 



x-hy 
70 



aj + y 
70 



Addmg (2), x^ -\- 2 xy -\- y^ = S9 , 

x + y 

or (X + yy = 39 - "^^ 



x-\-y 
Multiplying by x + y, (x + yy = 39(x + y) - 70. 

Put x + y = m. 

Then w«-39w=.-70. 

Multiplying by m, w* — 39 m^ = — 70 m, 

which may be written w* — 14 m* = 25 w^ — 70 m. 

Adding 49 to both members, 

m* - 14m2 + 49 = 25«i3 - 70m + 49. 

w2 — 7 = 5 m — 7. 
TO = 6 = X + y (4) 



240 TABLE BOOK AND TEST PROBLEMS, 

(4)2= a;2 + 2xy-f-y2:=26 {A) 

{A) - (2) = 2xy = 12, ajy = 6 (5) 

(^)- 4 X (fi)= x2 - 2a^ + y2 = i. 

«-y = i. 

Therefore x = 3 or 2, y = 2 or 3. 

(From Lesteb B. Fillman.) 

Third SoluUon. «« + y» = 13 (1) 

a^ + y« = 36 (2) 
Let x = 7n+Pf y = m—p. 

Then 2 m^H- 2^)3 = 13 (3) 

and 2 m^ + 6i)2 = — (4) 

(4) -(3)= 4j,« = ?i^ (5) 

Substituting value of 2 p^ in (3), we have 

2m2_t, 35-13m ^.^3 
2m 

Clearing, 4 ni» + 35 - 13 w = 26 m, 

or 4m«-39m=-36. 

Multiplying by m, 4 m* — 39 m^ = — 35 m. 

Adding to each side 4 m^ + (^)2, we have 

4 m* - 35 m2 + (^^)2= 4 w»2 - 36 m+ (5^)^. 

Extracting square root, we have 

2w|2-^=db(2ni-i!^), 

whence w = 1 or 2i. 

Substituting m = 2} in (5), we find 

P = h 
But aj = w+i> = 2J + 1 = 3, 

and y = TO— j) = 2J — J = 2. 

(From Dr. I. J. Wibebaok.) 

Fourth Solution. 

Let 8=:X'\-y,p = xy, 

Then (1) and (2) become «« - 3 ap = 36 (3) 

«2 _ 2^) = 13 (4) 

Eliminating sp, we obtain «* — 39 « = — 70. 



SOLUTIONS TO SPECIAL EXPEDIENTS, 241 

* Multiplying by «, «* - 39 «« = - 70 «. 

Adding to both sides 25 s- + 49, we get 

s* -- 14s2 + 49 = 26«2 - 70s + 49. 

s2-7=±(5s-7). 
Taking the + value, we have 

s2 = 6 s, or 5 = 5 = a; + y, 

whence » and y are readily found. (From Educational News,) 

290. jFYr»« Solution. Developing and subtracting (1) from (2), 

2a^(aj + y)=420 (3) 

Dividing 2 x (3) by (1), and reducing, 

4xy _84 





(X - yy^ 1« 


By composition, 


ix + yy _ 100 
(X - yy 16 


Evolving, 


« + y _ 10 

x-y 4 


This may be written 


a; + y = 10, 




x-y = ^\ 


lence 


x = lj and y = 3. 



(From Dr. I. J. Wibsback.) 

Second Solution. Developing (1) and (2), we obtain 

Qfi-x^y-xy^ + y^ = 160 (3) 

ofi + x^y-h xy2 -f. y8 = 580 (4) 

Adding (3) and (4), and dividing by 2, 

«« + y8 = 370 (5) 

Subtracting (3) from (4), we get 

2a;2y + 2xy2 = 420 (6) 

Multiplymg (6) by f , Sx^ + S xy^ = 630 (7) 

Adding (5) and (7), 

a^ + Sxh/ + Sxy^-\-y^ = 1000 (8) 

Extracting cube root, x + y = 10 (9) 

Factoring (6), and substituting value of x-\-y, 

2xy = i2 (10) 

Subtiactmg 2 x (10) from (9)2 gives 

a;2 - 2xy + y2 -- 16, a; - y = 4 (11) 

From (9) and (11), x = 7, and y = S. 

ellwood's test prob. — 16. 



242 TABLE BOOK AND TEST PROBLEMS. 

Third Solution. Subtracting (1) from (2) gives 

2a5y(a5 + y)=420, 

or 2xy = ^^^ (3) 

x-l-y 

From (2), x^ + y'^ = ^^ (4) 

x-\-y 

Adding (3) and (4), x^ + 2xy + y^ = ^^^ 



x-^y 

Clearing, (x + y)« = 1000. 

Extracting cube root, ir -|- y = 10 (5) 

Substituting in (2), x^ + y^ = 58 (6) 

Subtracting (6) from (6)2, xy = 21 (7) 

From (5), a; = 10 — y. 

Substituting in (7), y(10 - y) = 21, or ya - 10 y = - 21. 

Completing square, y^ __ iq y + 26 = 4, 
whence y = 7 or 3. 

Hence a; = 3 or 7. 

891. Let X + y + « = », xy -^ xz -\- yz = m, and xyz = n ; also, for con- 
venience, put a + b + c:= d=^ 216, ab -^ ac + be = g = 13248, and 
a6c=|) = 207360. 

(1) + (2) + (3) = [by involving terms and factoring] n3 = d (4) 

Similarly, (1) x (2) + (1) x (3) + (2) x (3) = 4 mn'^ - nV = g (5) 

and (1) X (2) x (3) = 4 smn^ - n»«« - 8 n* = p (6) 

(6) X n« — (6) = 8 n* = nag —p^dg —p. 

Therefore n = J ^[2 (d^ - p) ] (7) 

Now, the given equations are equivalent to 

n8 — 2nz = a (8) 

n8-2ny=b (9) 

?w - 2 nx = c (10) 

Substituting in (8), (9), and (10) the value of n« as given in (4), and 
the value of n as found in (7), and resolving severally as simple equations, 
we find that 

« = - ^L^« = ^±1 = 4. 

^[2(dg -1))] </[2(a + 6 + c)(ab -hac-^ 6c)- 2a6c] 

y_ d-b a-^ c __g 

\/[2(dflr-i))] ^[2(a + 6 + c)(a6 + ac + 6c)-2a6c] ^ 

X = <^-g q + 6 2 

\/l2(dg -p)] </[2(a + 6 -|- c) (aft + ac -|- be) - 2 aftc] 

(From B. F. Bublkson, in Notes and Queries.) 



SOLUTIONS TO SPECIAL EXPEDIENTS, 248 



292. Simplifying, we have 



- Vx^' - fti + Wa=* - a2 = ^ Va - d^ + - Vaj-* - c^ 

X X X X 



Clearing, aVx^ - 6^ + bVx;^ - a^ = cVx^ - d^ + dVx^ - tf*. 
Transposing and squaring, 

= (l-'x'2 - (Pea _ 2 6d V{x* - (a*^ + c-i)x-» + a^6^} + 6V - a^fr'. 
Collecting and canceling terms, 

= (,d^ + 6^x2 - 2 fed V{x* - (a^ + c^)x^ + a^^c^}. 
Transposing, 



= - (a-2 + c2)x2 - 2 6d V{x*^ (a^ + ^ '>'^ + aV-^}. 

We can form two perfect squares by adding to each side of the equation 
the fourth power of x and the square of the half of the coefficient of each 
radical ; that is, x* + (^ac)^ + (bd)\ Performing this addition, we have 

{x* - (62 + (r^)x2 + 62(P} - 2 acV{x* - (62 + d^)x^ + b-^d^+ a'^c^ 

={x* - (a2 + c2)x2 + a2c2} - 2 6d[ V{x* - (a^ + c2)x2 + a2c2}-f- 62(P. 
Extracting square root, 



V{x* - (62 + d')x^ + b-d^} -ac = d: [ V{x* - (a^ + c2)x2 + a2c2}-6d]. 
Using the minus sign, and transposing ac, we obtain 

V{x*-(62 + d'«)x2+62d^ = - V{x* - (a2 + c2)x2 + a2c2} + (ac + 6d). 
Squaring, canceling, and collecting, 
(a2 - 62 + c2 - d2)x2 - 2 ac(ac + 6d) 



= - 2(ac + 6<f) V{x* - (a2 + c2)x2 + a2c2}. 
Squaring again, dividing by x2, canceling and collecting, 
4(ac + 6d)2x2 - (a2 + c2 - 6* - d0^a;2 

= 4(a2 + c2)(ac -f- 6d)2 - 4ac(ac + 6d)(a2 + c^ - 62 - (P); 
whence 

a; _ ^ l^(a^ 4- c2) (oc + 6d)2 ~ 4 ac(qc + 6d)(a2 -\- c^ - 62 -^^ 
^ 4(ac -I- 6(«)2 - (a2 + c2 ~ 62 - d^y ' 

which may be reduced to the form 



4 



4(a6 + cd)(ac + bd)iad + 6c) 



(a 4- 6 + c - d)(a + 6 + d-c)(a + c + d- 6)(6 + c + <i - a) 



244 TABLE BOOK AND TEST PROBLEMS. 

8^8. First Solution, 

Multiplying by 4 «, 4 a^ - 12 «' + 16« = 0. 

Extracting square root, (2 x"* - 3 x)2 - 9 x^ + 16 a; = 0. 
Adding to each member x^^Ax, 

(2x2 - 3x)2 - 4(2x2 - 3x)= x2 - 4x. 
Completing square, and extracting square root, 

(2x2-3x)-2 = x-2, 
whence x = 2. 

Second Solution, 

Multiplying by x, x* ~ 3 x« + 4 x = 0. 

Subtracting from this 3 x the given equation, 

x*-9x2 = -4x- 12. 
Adding x^ + 16 to each member, x* — 8 x^ + 16 = x^ — 4 x + 4, 
Extracting square root, x^ — 4 = x -- 2, 

whence x = 2. 

294. Multiplying by 4 x, 4 x* - 24 x^ + 16 x = 0. 

Extracting square root, (2 x'^ - 6)2 + 16 x - 36 = 0, 
or (2x2-6)2-36=-16x. 

Adding to each member 16 x^ — 12, 

(2x2 - 6)2 + 8(2x2 - 6)= 16x2 - 16x - 12. 
Completing square, and extracting square root^ 

(2x2-6) + 4 = 4x-2, 
whence x = 2. 

296. Multiplying by 4 x, 

4x* - 32x5 + 76x2 - 48x = 0. 
Extracting square root, 

(2x2 -8x)2+ 12x2 -48x = 0. 
Factoring, etc., (2 x2 - 8 x)2 -|- 6(2 x2 - 8 x) + 9 = 9, 
whence 2 x2 — 8 x = or — 6. 

x2 — 4x = or— 3; whence x = 4, 3, or 1. 



MISCELLANEOUS SOLUTIONS. 



245 



MISCELLANEOUS SOLUTIONS. 



APPLICATIONS OF ALGEBRA. 



296. First Solution. Let ABCD be the rectangle, and E its center. 
Let X = its length, AB ; and y = its width, AC. Then x -f y = 70, (1). 
In the triangle AIE, Aff = 26, AT = j x, 
and IE= ipj I being a right angle. 
Hence 



^J'^{f)=^^<orf^t^m,i2y 




Clearing, «« + y« = 2600, (3). From 

(1), y = 70 — as. Putting this Talue of 

y in (3), we obtain, by collecting terms, 

etc., a;-^ - 70a; = - 1200, (4). Completing square, x'^-lOx-h 1226 = 26. 

Extracting square root, as- — 36 = ± 6, and a; = 35 ± 6 = 40 or 30. Hence 

y = 30 or 40. The area = 40 x 30 = 1200 square feet. 

Second Solution. AB + BD = 70. Let x = AB. Then BD = 70- x. 
AE= 26 = ED. Hence AD = 50. In triangle ABD we have xH (70 -x)* 
= 60^, whence x- — 70x = — 1200, the same equation as (4) above. 

897. Let X = the base, and y = the perpendicular. Then VxM-^ = the 
hypothenuse. By the problem, x — y = a, (1); and Vx-* -\- y^ — x = b, (2). 
Adding (1) and (2), VxM^ -y = a+b, (3). Putting a-\-b = c, trans- 
posing, and squaring (3), x^ + y® = c* -|- 2 cy + y^^ (4^. From (1)^ 
x^ + y^ = a2.+ 2 a?y, (6). Subtracting (6) from (4), c^ - a^ + 2 y (c - x) 
'{-y^ = Of (6). From (1), x = a + y. Putting this value in (6), we have 
<^-a* + 2y{c~(a-|-y)}-j-y2-o. Transposing, etc., y^-^2y(^a-c) 
= c^ — aK Completing square, ^^ + 2 y (a — c) + ( a~c)'^=(« — c) ^ + c^ — a-. 
Whence y = c — a ± v^=* — 2 ac = (c — a) db V2 c (c — a). Then x = a 
+ y = c ± V2 c?* - 2 ac = c ± \/2 c (c — a). 

298. Let ^5C2> represent the floor. Put DC = a = 40, 5C = 6 = 13, 
and EF= GH = 3. Let x = ^^, 
and y ^ AG. In the triangle IT/K?, 
X* + y* = 9, whence x = V9 — y^ (1). 
The triangles ^HG^ and GDE being 
similar, x:y: :6 — y:a — x, whence 
X* — ax = y' — 6y. Substituting value 




Ah 



of X from (1), transposing, squaring, etc., we get, after dividing by i. 



246 



TABLE BOOK AND TEST PBOBLEMS. 



y* - 13 y» + 433.252(3 + 68.5 y - 3579.75 = 0. By do uble position, or Hor- 
ner's method, we tind y = 2.88. Then x = V9 — y' = .84. Hence 

EG = V{(40 - .84)-^ + (13 - 2.88)='} = 40.4+ feet. 

899. Let X = number of acres = number of rails. Then 160 x = number 

of square rods, -%/ = radius = 4-v/ — -, and 8a/ — - = diameter ; also 

8»-v/ — -= 8 VlO Ttx = circumference. Since the boards are 1 rod long, 

and the fence is 5 boards high, - = number of rods in circumference. 

Hence - = 8 VloTx/whence x = 16,000 w = 60,265.5. 
5 

800. Let X = length (in feet) of the inside. Then a; 4- J = outside 
length, and 6(x + J)'^ = entire outer surface, (1). Since the outside sur- 
face includes several edges, it is greater than 49|. The gain is 4 edges of 
the inside length, and 8 edges of the outside lengUi ; that is, 4x-\-S(x + \) 
feet in length; and this multiphed by i (or 1} inches) gives the area 
of the said edges = f x -f J." Then 6(« -I- i)^ - (f jc -h J)= 49f, whence 
as = J^ = 2| feet, and x + J = 3 feet = 36 inches, the required length. 



301. Let AB = the shorter, and CD = the longer, of the poles ; CA = 
the plane; E^ the point of intersection. We then have JBC= 40, DA 

= 60, and EF = 16. By similar triangles, 
we have AE: AF: : AD. AC, BE: AF : : BC 
: AC ; whence BE: AE: : BC: AD, and like- 
wise DE: CE: : AD:BC. But AD: BC::S 
:2: hence DE:GE::S:2, Let2x=2?(7, 
Sx = DE, BE = 40 - 2x, and AE = 60 - 
8x. By similar triangles, we have AE: EF 
: : AD : DC, whence, substituting values, DC 
900 300 




60 -3x 20 -x' 



also CE:EF: :BC: 



BA, whence BA = — • AD^- CD» ^AC\ 

X 

(1) ; and BC^ - AB^ = AC^, (2). Hence, equating (1) and (2), AL^- 
CD» = BC^ - ABK Substituting values, 602 _ /_300_y^ ^ __ /300y^ 

Reducmg this, we get x* - 40x' + 400 x- - 1800 x + 18,000 = 0, whence, 
by Homer's method, x = 13.954+. Hence 300 -4- x = 21.5 feet, the shorter 
pole. Then 300 -*- (20 - x) = 49.62+ feet, the longer pole. 



MISCELLANEOUS SOLUTIONS. 



247 



802. Let ABC= the triangle, ED = the parallel ; and make BC 
AC=h, AB-c, CD = x, and BE 
= y. Then AD =zb — x, AE —c — y, 
and DE = x — y. By similar trian- 
gles, b — x:c — y::b:c, wlienco ex 
= &J/» (^) » *^o a:x — y::b:b — x, 
whence bx—by-hax = ab, (2). From 
(1) and (2), 

ab 



ct» 




a-\- b — c 



X = 



a -^-b — c> 



and :g^y= ^(V^) = Z>^. 

a + 6-c 



J? 



808. Let AB = pole, ^2> = the hillside, C = the point where the break 

occurs. BC = DC = a. AC=EG = x. AD = 
b, ED = c. CF being a perpendicular, and 

b-c 



ACE isosceles, AF=FE = 



2 



, and FD = 
Substi- 




^Jt_?. CE^ = c/>2 _ FD^ + £2^2. 
2 

tuting values, we have x'^ = a^ — I ] H- 

/b-cv^ ^ 2 ; 

r —7-^ ) • Clearing, transposing, and reduc- 
ing, we have x^ = «2_ (>c, and x = "n/^^^^~6c. 
Then a -|- Va^~^^bc = the original height of 
pole. (From Dr. I. J. Wirbback.) 

804. First Solution, Let x ■\- y = AB, 
X — y = BC, Now, the product of the 
sides divided by their sum is equal to the 
side of the inscribed square. Therefore 

(x-\-y){X'-y) _ j2, or x'^-y^ = 24 x, (1) ; also 

AB^ + BC^ = (X + yy +(x- yY = 362, or 2 x^ ■\-2y^ 
= 1225, (2). Adding 2 x (1) to (2), we have 4a;2 
— 48 a = 1225. Completing square, 4 a;^ _ 48 a; -f- 
144 = 1369, whence x = 24}. Putting this value of 
x in (2), we find y = 3}. Then x + y = 24} + 3] = 

28, and x-y = 21. 

/■' 

Note. — The truth of i^e statement made in the beginning 
of the foregoing solution m^y be seen by examining Equation 
(2) in the following soIutioQxof the same problem. 

Second Solution. (See figure in above solution.) Let AB = x, BG = y, 
Then ofi-^^ = 1225, (1). By similar triangles, AB:BC::AF: ED, or 




248 



TABLE BOOK AND TEST PROBLEMS. 



a? : y : : X — 12 : 12 ; whence xy = 12(a; + y), or ?^= x + y, (2). Squaring 

(2)» ^ = a:« + 2xy4-y^. Transposing, ??i(^ - 2 xy = x^ + y^, (3). 

Taking the value of x^ + y^ from (1), we have =^ — 2xy = 1225, or 

144 

x2jf2 - 288 xy = 176,400. Completing the square, xV - 288xy + 20,736 
= 197,136, whence xy = 688, (4). Putting this value of xy in (2), we 

have ^ = X + y = 49. Then x = 49 - y, and (4) becomes (49 - y)y 

= 588, or y« - 49y = -588, whence y = 21 or 28. Hence x = 28 or 21. 

305. Let x = number of acres = number of boards. Then 160 x = 
number of square rods. - = number of boards on one side. There are 

14 boards in 1 rod. Hence — = number of rods square, and ( — j = 

66 V56y 

160 X. x' = 601,760 X, and x = 501,760. (Prom Dr. I. J. Wikbback.) 

806. Volume of globe is 12^ x .5236 = 12« x Jir. Put Z) = mside 
diameter of hollow sphere. Then 2) + J = the outside diameter. (2> + J)' 
X J IT = space occupied by hollow sphere. D^ x J ir = space or volume 
inside the shell. Then Jir(Z> + J)8 - JxZ>5 = 12^ x Jir. Dividing by 
Jit, we have (2? + J)8 -D^= 12^, or 2?> +IJ)'^ + ^%D + ^^ -I>^= 1728. 
Clearing, etc., 482>2 + 122? = 110,591, or 162?-' + 42> = 36,863.?. Complet- 
ing square, 16 2X- + 4 2) + J = 36,8635 + J, whence, extracting square 
root, etc., 2> = 47.87 -f-, and 2> + J = 48.12+ inches. 

807. Let X = altitude. Then 2 x = base, each of the *equal sides = 
x\/2, and the area = x*. The area -7- perimeter = J radius of inscribed 
chrcle. Hence radius = 2x2 -?- (2x + 2xV2) = — - — Then — - — ^ x 



r _ 

X 

808 



I+V2 
Rationalizing the denominator, we have 



I + V2 



v^-1 



= V2-1. 



I + V2 ^ 2-1 

Let ^BC=the triangle, ^B = 2x, ^0=3x, BC=b, On BC 

let fall the perpendicular AD, and 
let y = BD. Then CD^b-y. 
In right triangle ADC, AD^ = 9a^ 
- 62 + 2 6y - y2, (1). In right 
triangle ADB, AD'- =? 4 x' - y«, 
(2). Equating (1) and (2), we 
have 9x2- 62+ 2 hy-y^=^x^- y\ 
or 5 x2 4- 2 6y = 62, (3). The area 
of ABC = J {BC X AD) = a. 




Hence ^2) = ? ^, and AD^ = ^. 

6 62 



Substituting in (2), we have 4x2 — 



MISCELLANEOUS SOLUTIONS. 



249 



y2=_ii-^ (4). Comparing (3) and (4), and eliminating «, we get 6y^ 
+ 86y = 46«-?^, whence y = i&V(:366* - lOOa^-— • PuttingtMs 
value of y in (3), and solving, we have 

16^56 6/^25 25 y 

and AB = 2 a; = twice this ; and AC = Sx = three times this. 



i 




809. JYrs< Solution. Denote the angle A by 9. Then the angle ECD 
= 90-6, the angle EDB = 2 0. The angle DEA = ^, the angle BEC = 0, 
and the angle BE A = 90 — ^. 

Hence the triangle ADE is ^ 

isosceles, and AD = DE = 80. 

Now, by similar triangles, 

AB : BE : : BE : BC. Let 

X = BD. Then B^ = (80 -f- x) 

(20 + x) = x2 + loox + 1600, 

(1). In right triangle EBD, 

BE^= DE^- BZ)2= 6400 - x\ 

(2). Equating (1) and (2), 

a;2 -I- 100 a; + 1600 = 6400 - x\ whence x = 30. Then EB^ = 802 - 30"^ 

= 5500, and BE = lOVsS, 

Second Solution. The triangle ADE is isosceles, hence 2)iF = 80. 
Put DB = x, EB = y. «2 ^ y^ = DJF* - 0400, (1). The angle AEB 
= 90 — ^ = the angle ECD, and the angle BEC = ^ = the angle A : 
hence the triangles AEB and EOB are similar, and we have suiAiy 
: : sin AEB : x + 80, and sin ^(or BEC): x -f 20 : : sin AEB(ot ECD) : y ; 
whence y : x 4- 80 ; ; x -|- 20 : y, or y2 - a;2 _^ 100 » + 1600, (2). Substitut- 
ing in (1), we have x^ + x^ + 100 x + 1600 = 6400 ; whence x = 30, and x^ 
= 900. Using this value of x^ in (1), we have j^ = 5500, y = 74.16+ = EB. 

810. First Solution. (See first figure on next page.) Let AB = 74 
feet, height of tree ; BD = 34, the hillside ; BE =18, the horizontal dis- 
tance ; BC = x; DE = y. Then 

y = 74-x-Vx^ + 324 (1) 

The triangle BED = triangle BCD - triangle BCE. Applying the 
rule for finding the area when the three sides are given, we have 



/ (52^ - fHf Zm = V io80(64 - ^Kx - 20) - 9; 
1 16 

From (1) and (2), we find x = 24, and y = 20. 



(2) 



260 



TABLE BOOK AND TEST PROBLEMS. 



Second Solution. Let BC = i, BF = y, FD = z, BFD being a right 
angle. Then we have (x + y)^ + 34^ = (74 — jc)* + y\ or li.x + xy 

= 2160,(1). y2 + ^-^ = 34-2, (2). «: 18: :a: + y:«, 
or 18x + 18y = xjj, (3). From (1), y = 

Substituting in (3), 18x2 _ 1332 x 



2100- 74 X 

 ■■— I.  — -  .M .  

X 

+ 38,880 = xH, (4). From (4), 



z = 



18(x2-74x + 2160) 
x^ 




Substituting values of y and z in (2), and 
reducing, we have 1161 x» - 91,908 x' + 1,959,- 
876x'- 25,894,080x +377,913,600=0. Solving, 
X = 24 = height of stump. 

811. Let ABCD represent the square, Pthe 
point within, PA = a, PB = 6, and PD = c. 
Draw EF perpendicular to CD. Put AB = BD 
= X, AE=y, EP=z. Then BE=x - y, and 
PF= X — 2f. By the Pythagorean theorem, we 
have AE^ + EP^ = AP^, or 2/2 ^. ^.a = ^2^ (i). 
^^2 + EP^ = 5P2, or (x - y)2 + 5?2 = 52^ (2). 
FD^ (or ^^2) + FP2 - p2>2^ or (x - yy + 
(x-i?)2 = c2, (3). Subtracting (2) from (3) 
gives x"^ — 2 X2? = c'^ — b\ (4). Subtracting (1) 
from (2) gives x2 - 2 xy = 62 _ a\ (5). Trans- 
posing, etc., 2xy = x2— (62 — a2)^ (6). From 
(1), z = Va2 — y2. Putting this in (4), we 

^^® x2-2xVa^-y-' = c2-62, 

or 2 X Va^ - y^ = x2 - (c2 - 62) (7) 

Adding (6)2 and (7)2, we have 
4a2x2={x2-(62-a2)}2-h{x2-(c2~62)} (8) 

For convenience put 62 — a- = m\ and c^ — 62 = n\ Then (8) 
becomes 4rt x^ =(x2 — w2)2 +(x2 — n2)2, (9). Expanding (9) and col- 
lecting terms, x* - x2(wi2 + ^2 + 2 a2) = _ ^* + ^\ (10). Putting m^ + 
n2 + 2 a2 = 2/), and ^* "*" ^* =q, (10) becomes x* - 2px'^ = - g. Com- 
pleting square, x* — 2 j9x2 + i>2 = |)2 _ g^ whence x^=p ± Vp^ — g. Then 

X = V p ± yp2 _ g. 




MISCELLANEOUS SOLUTIONS. 



251 



812. Let ABC (Fig. 1) be the triangle; AD, CO, and BE, the poles ; 
and H, K, and /, the points on the sides which are equally distant from 
the tops of the poles between which they lie. Then AD = 50, BE = 40, 





-* Jf 



CG^=30, -4B = BC=^C = 200. Let-4H=a, B ff = b, a.nd DH = EH 

= X. Then a;2 _ 50a 4. «j = 492 + j2^ whence 6^ _ a* = 900, (1). BH-\- 
AH = b-\-a = 200, (2). Dividing (1) by (2), 6 ~ a = 4.6, (3). Adding 
(2) and (3), 2 6 = 204.5, and 6 = 102.25 = BH, Then a = 200 - & = 
97.75 = AFI, In a similar manner we find AK= 96, CK= 104, BT = 
08.25, and CI= 101.75. Perpendiculars erected at the points H, /, and 
K, will meet in a point, which is the required point, being equally distant 
from />, Oy and E, the tops of the poles. In Fig. 2 erect perpendiculars 
at H, /, and K. Let be the point at which they intersect. Produce 
10 to M in AB, Bisect BC at F, and join AF, AO, BO, and CO. The 
triangle ^B(7 being equilateral, AF and 3f/ are parallel, and therefore 
AM = 2 IF. BF-Br = IF=\00^ 98.25 = 1.75. Hence AM= 1.75 x 2 
= 3.5. 3fff =:AH- AM = 97.75 - 3.5 = 94.25. The triangles ABF 
and MOH are similar. Therefore MO = 2 OZf. Let OH = x. Then 3/0 
= 2 «. J«r-2= JfO^ - Ofl'a = 4 x'i - x2 = 3 a;2. But MH = 94.25. Hence 
3a ;^ = 94.25g, an d x = 54.415-f = OH. In the triangle AOH, AO = 
y/iAH^ + OH^) = V (97.76-^ + 54.415-) = 111.876 feet, the distance the 
fourth j)ole must be placed from the foot of the 50- foot pole. In the tri- 
angle BOH, B0 = V{BH'^ -f 0H^=» 115.82 7 feet, the dista nce from the 
40-foot pole. In the triangle AOK, OK = y/ {AO^ - AK'^)- 57.428 feet. 
In the triangle COK, CO = y/{C K'^ -f Q/T-^) = 11 8.811 feet, the distance 
from the 30-foot pole. Then Vll8.8il-* + 30=* = 122.53-t- = length of 
fourth pole. 



• I 



252 



TABLE BOOK AND TEST PROBLEMS. 




818. Let MBN be any quadrant, A the centre of its inscribed circle, 

and F the centre of the required circle. 
Through A and F draw the radii of the 
quadrant, BAG and BFH. Let fall -4 D 
and FK each at right angles to BN, and 
draw FE at right angles to AD. Put 
B = BG or BN, r = AD =^ B(\/2 - l) = 
.4142136624 R ; and let z = FK, the radius 
of the required circle. Then BF = /2 — a;, 
AB = R- r, AF = r + x, AE = r - x, 
and FE = BK - BD =y/{{B - a;)^ - x*} 
- \/{(^ - ry - H}, (1). Also FE^ = FA^ 
- AE^ = (r + xy - (r - «)«, (2). Equating values of FE^, we have 

{V[(ig ^ xy ~ a ;-*] . V[(^ - ry - r^jP = (r + a;)^ - (r - x)« (3) 
ShnpUfying (3), {V(/t« - 2 i?x) - V(/P - 2ifr)}2 = 4ra; (4) 

Expanding (4), and dividing by 2, 

- V{R* - 2(r + aj)A8 + 4 /f^rx} = 2ra; - 2?2 +(r + 35)7? 
Squaring (5), expanding, uniting like terms, etc., we have 
(4 2?r + 4 r2 + B^)x^ - (6 TT^r - 4 Br'^)x = - 722^ 

From (6), a. = 3i?^r - 2Jgr^ ^ 2 i?rV (2 i?^ - 4 /fr) 

But r = R(^/2 — 1); and substituting this in (7), we have 

^_ SRHy/2-l)-2 BK^ - W2) - 2 ffl(3 v^ - 4) 
4i?^(>/2 - 1)+ 4 R\S - 2V^)+ fi^ 
^ R(V2-1) ^ ig(5 V2 ~ 1) ^ .1238993431 iJ. 
9 - iy/2 49 

Hence the radius of the required circle in any quadrant is found by mul- 
tiplying the radius of the quadrant by the constant decimal .1238993431. 



(5) 

(6) 
(7) 



GEOMETRICAL DEMONSTRATIONS, ETC. 

814. Separate JVinto any two factors, m and n. Then mn = N. With 

center O, and radius equal to ^ (w -h n), describe 
a circle ABD. On the diameter CD take CI 
equal to the smaller factor, and through /, at 
right angles to CD, draw the chord AB. Then 
AIoT IB is the square root required. Since the 
chords AB and CD intersect each other, we 
have, by geometry, AI y. IB = CIx ID. But 
by construction AI = IB. Therefore AT^ or 
IB^ = CI X ID = mn. Extracting the square 
root, A I or IB =. y/mn = VN. q.e.d. 




MISCELLANEOUS SOLUTIONS, 



253 



816. (See cut in preceding solution.) Take m = 3. Then n = 16. 
Describe a circle whose radius is ^(3 + 15), or 9 units, and take CI = 3. 
ThenAI=IB = V^. 



316. As 19 is a prime number, we must take m = 1, and n = 19. 
Describe a circle with a radius = j(l + 19) = 10 units, and take C7= 1 
(see cut in Solution 314). Then AI=IB=i vl9. 

817. First diameter = Z> = first term of series. 

TU^ seventh diameter = — • Hence the ratio is expressed thus : 2> : — • 
This we see is 8. 

818. The first term is 2>, the ratio — , and the last term 0. In geo- 

V2 



metrical progression, s = 



Ir — a 



r-1 

- D D 



Substituting, we have 



J— 1 1— L 

>/2 V2 



810. Let abed and NAPO be the two 
inscribed squares, and C the center of 
the circle. By geometry, 

he^ = Ne x eP= CP^ - Ce^. 

But Ce2 = J fte^. 

Therefore 

6e2 = CP2 - J he^, or CP^ = } be^, 
OP^ =2 CP2. 

Hence OP^ = f fte^^ or 2 OP^ = 5 b^. 
Therefore bef^: 0P^::2:b. q. e. d. 

820. Let AEBD and ^Mc be the in- 
scribed squares, and c the center of the 
circle. By Pythagorean theorem we have 
hc^ = 2 ck-, and cD^ = DE^ -\- cE\ But 
he = Dc. Therefore 2 ck^ = 2>j;* + cJer^ 
= DE^-^\DE^ = iDE\ or 8 ci^ _- 
5 I>i?2 ; that is, DE^ : c*;^ : ; 8 : 5. q.e.d. 




B 


^ 




D 


y/^ 





/ 


\ 


^ 


f 


^ 


/ 


^ 


N A 


L < 


> J 


? i 


fc 



254 



TABLE BOOK AND TEST PROBLEMS. 







821. Let ABC be the inscribed equilateral triangle, and O the center 

of the circle. Draw CD at right angles to 
AB, and join AG, We are to prove that 
CA : CO{otAO)::VS:1. Since by similar 
triangles AC: AD:: AO: CD, and AC =2 AD, 
therefor e AO = 2 OD. Then DC = SOD. 
AD = ViAW' - DIP) = ODVI Then AC 
= 2 ODVS, and 0C=0A = 2 OD. There- 
fore AC : C0i0TA0)::2ODy/'i:2 0D:: 
VS:l. 

822. The area of the triangle is 61.48+ 
square feet. But the area of a triangle is 
equal to its perimeter multiplied by half the radius of its inscribed circle. 
Hence 61.48 = 36 x i r = 18 r. Whence r = 3.415 feet. 

828. Let ABCD be the board. Then AC =10 feet, and AB = 2 feet. 
Take BI=l foot, and cut off BAL Take AO and IK each = 5 feet, 
and cut off the two pieces AIG and 
lOK. We now have the four pieces. 
Since AI = V6, we have BJ: A I: : AT: 
AG. Hence, the angles AIB and GAI 
being equal, the triangles are similar, 
and AIG is a right angle = IGK. To 
form a square, let AIG and GKI occupy V^^'' 

the positions KHF and ECG, and ABI ^ 

the comer CDF. EFGH is the square . GK= KH= AI = VE, and 
GH = 2 V5. EG = GI= y/{AG^ - JL/^) = V20 = 2 V5. 

824. Let AB be a diameter, and CD any chord of the circle. OP = the 

distance from the center to any point, P, of the 
chord. 

BP = 0P-h0B, 

and AP = AO or OB - OP. 

Hence, multiplying, 

BP X AP = OB^ - 0P\ 

But, by geometry, 

BPxAP = CPxPD. 

Therefore OB^ - OF^ = CPx PD. Transposing, OB^ = OP^ + CP 

X PD. Q.E.D. 





MISCELLANEOUS SOLUTIONS. 



265 




835. Let MNA be a vertical section of vessel, EFC the water not 
frozen, and OH the thickness of the ice. Volume of hemisphere = ^^ tD^ 
= 7.0686 cubic feet. The water not 
frozen foims a segment of a sphere 
whose height Ib HC = AO -(OH -^ AC) 
= 18 — 10 = 8 inches = h; and the radius 
of the sphere is OC = AO - AC = 18 
— 5 = 13 inches = B. The volume of a 
spherical segment of one base, as EFC, 
is ir^^(jK — ih). Substituting in this 
formula the values of h and liy we find 
the volume of the segment to be 1.20234 
cubic feet. Then 7.0686 - 1.20234 = 
5.86626 cubic feet, volume of ice. 

326. Let FBG be a circle tangent 
to major axis A A' at focus F, and 
passing through B, the extremity of 
the minor axis. Since the circle is 
tangent at focus F, its center will lie 
in the latus rectum produced. Join 
FB and GB. The triangles FBG 
and FCB are similar, and therefore 
BCiFB:: FB: FG. But FB = AC. 
Hence BC : AC : : AC : FG. q.e.d. 

827. First Solution. Let FEIH be the circle; ABCD the circum- 
scribed, and lEFH the inscribed, square ; and EH the diameter of the 
circle. The difference between the squares 
is the four equal triangles, EAI, IBH, 
HDF, and FCE. Hence IBH = J of 72 
= 18 square feet. Let x^ IB: then area 

IBH= X X - = — = 18, whence x^ = 36, 
2 2 

and X = 6. AB = /B -f ^7 = 2 x = 12 = 
EH = diameter required. 

Second Solution. The equal triangles 
EAI, ECF, HDF, and IBH are each } 
of a square, and combined are equivalent 
to two equal squares, the side being 
AJ=^ AE= EC, etc. The area of the four triangles = 72 feet. Hence 
the area of each of the two squares is 36 feet. Therefore the side ia 
6 feet = AI; and A3 or EH= 12 feet. 





256 



TABLE BOOK AND TEST PROBLEMS. 



Third Solution, By condition, AE^ - El^ = 72, (1). AB- EH, and 
AB^ = EH^. But EH^=:EP-\'IU^ = 2ErK Hence EI^ = iAB^. 
Substituting in (1), we have A B^ - ^ AB^ = 72, whence -45^ = 144, 
AB= 12 = EH. 

Fourth Solution. Assume an isosceles right triangle whose sides are 
1, 1, and V2. Its area is J. The assumed triangle is similar to each of 
the four equal triangles, IBH, etc. Hence, by similar triangles, | : 18 : : 
1 : 1B\ whence IB^ = 36, IB = 6. Then AB = 12 = EH 

Fifth Solvtion. The side of an inscribed square is to the side of a cir- 
cumscribed square as 1 is to V^. Therefore AB^-.EP : : 2 : 1, or EP = 
J ABK But AB^ - EI^ = 72. Hence AB^ - J AB^ = 72, or AB^ = 144, 
AB= 12 = EH. 



328. Let ABCD be the parallelogram, AC its diagonal, to be trisected. 

Bisect BC and AD in E and F, 
and join ED and FB. Then will 
BF2i\i&ED trisect AC, and ^fi^ 
= HG= GC. Since BiST and FD 
are equal and parallel, BF and 
^2> are also parallel. The trian- 
gles AFH and ADO are similar, 
and therefore AF: FD : :AH: HG. 
But by construction AF= FD. Therefore AH = HG. Similarly CE: 
EB.iCG: GH. But CE = EB. Therefore GC = GH. Therefore, by 
SAiom, AH = HG =0C. q.e.d. 




829. Let ABCD be the lot, ^its center, EG = 60 feet, EK =iO feet. 

Then GK = VSO^ - 40'2 = 30 feet 
The horse can graze over the rect- 
angle FGHI, and over the two 
equal segments GI and FH. FK 
= KG = SO feet, and FG = 60 
feet. Hence the rectangle is 60 
X 80 = 4800 square feet. FH = 
01= 80 feet = base of segment, 
'^ and the height is EM - OK = 60 
— 30 = 20 feet. The area of one segment is therefore J(80x20)-h 
(20» -s- 160) = 11 16 J square feet. Then both segments = 2233} feet, which, 
added to the rectangle (4800 square feet), gives 7033 J square feet, the 
area grazed over. 




MISCELLANEOUS SOLUTIONS. 



257 



330. Let A, B, and C be the centers of the circular fields. Put the 

area of each field = 80 x 160 = a square rods. Then radius = a/- = r, 

and 2r = AB = AG=BC, the sides of 

the equilateral triangle ABC. The area 

of the triangle ABC=t^VS. But this 

includes three sectors, each of which = J 

of a circle, since its angle is 60°. The 

three sectors = J of a circle = ia. Hence 

the area inclosed by the circles must be 

r^V3 — J a; or, substituting the value of 

r = A/— » we have Vs — ia. Substitut- 

ing values as given in the special prob- 
lem, we find the inclosed area = 656.78 
square rods. 




331. Let a = area of each field. The area of a circle = irr^ = a. Then 
ia = Substituting this in the formula for inclosed area as given 

above, we have r^VS - } irr* = 1 acre = 160 rods, or ( V3 - } T)r^ = 160, 
whence r = 31.6-i- rods, and diameter = 2 r = 63-f rods. 



332. Let ABCD be the rhombus, and 
AD = 24^ inches. Draw BOC, the other di- 
agonal. BO is perpendicular to AD, because 
the diagonals of a rhombus intersect at right 
angles. The area of the triangle ABB zn 
iABCD = 108 inches. 108 -4- 12 (half the 
hase) = 9= OB. AO^ -f OB^ = AB^; or 12^ 
-f 92 = AB^, and AB = 15. Hence the perim- 
eter (four sides) = 60 inches. 




333. From the center of larger ball to either wall is 12 = JB inches. 
Hence from its center to the corner is \/3i?^"= BVS. Subtracting from 
this the radius of the ball, we have bVS - B = jB( V3 - 1), the distance 
from the corner to the surface of the ball (in the direction of its center). 
Let r = radius of smaller ball. Then from its center to corner is r V3, 
and rV3 -fr = r(\/3-fl)= distance from comer to surface of larger 
ball. But this was found above to be i?(\/3 — 1). Theref ore r( V3 + 1) 
= ^( V3 - 1). But B = 12, and VS = 1.732. Hence 2.732 r = .732i? 
= 8.784, whence r = 3.215 +, and the diameter is 2 r = 6.43 inches. 

bllwood's test prob. — 17. 



258 



TABLE BOOK AND TEST PROBLEMS. 




834. Let AHFE represent a vertical section 
through the earth^s center, P the position of 
the person, and AB the base of the zone to be 
seen. By geometry, the surface of hemisphere 
is to the surface of zone as HC is to DH, or 
^ : } : : r : DH, whence, assuming radius of earth 
= 4000, 2>jEr=2666J miles. By similar tri- 
angles, DC: BG: : BG: CP. But DG= HG 
-2>ir=4000-2666§ = 1333J. Hence the 
preceding proportion becomes 1333^:4000:: 
4000 : CP, or 12,000. CP-GH==i 12,000 - 
4000 = 8000 miles = HP, the required dis- 
tance, which is the same as the earth^s 
diameter. 

835. Let ABC be the given triangle, P the 



point witnin, CP = a, AP = 6, and PB = c. On PB 
construct the equilateral triangle PBD, and join 
CD, In the triangles CBD and APB, DB = BP, 
AB = BG, and the angle O^JD = the angle PBA, 
each being equal to 60^ — the angle PBC. Hence 
GD = AP=h. We now have PC = a, CD = 6, and 
PD = c, to find the angle CPD = 0. Then the angle 
GPB = 60° 4- B, and we have two sides (CP and PB) 
and included angle to find BG, the required side. 




836. Let be the center of the circle, OC = OE=Sj the radius; 

AB = 4; CD = 6; ^F=2; C/ = 2.5. 

Then 0/= iVU = 1.658+, 

PO = Vo = 2.236, 

PJgr = 3 - V6 = .764, 

-&/=3~ J\/n = 1.342. 

Area of segment AEB, whose height is 
EF = .764, and base is AB = 4, is found by 
rule to be 2.093 +. Area of segment CED, 
whose height is EI=l .342, and base is CD 
= 6, we find by rule to be 4.715. Then 4.715 - 2.093 = 2.622, the re- 
quired area. 




MISCELLANEOUS SOLUTIONS. 



259 



887. Let be the center of the sphere, OB = 2, AB = 2, the diameter 
of the auger. The volume removed is a cylinder, AJBCD, and two seg- 
ments, AEB and CDL FE = height, and 
AB = base of segment. OB^ - BF^ = F0\ 
or 4-1 = 3. Hence OF = VS = 1.732. 
EF= OE -- 0F = 2 -- 1.732 = .268. The 
volume of a segment may be found from the 
formula J ir^(A'^ + 3 r^). Using this, we find 
the segment AEB = .431 of a cubic inch, and 
both segments = .802 of a cubic inch. The 
length of the cylinder = diameter of sphere 
less the height of the two segments, or 4 — 
.636 = 3.464 inches. Area of its base = xr^ 
= 3.1416. Volume of cylinder = irr'^h = 3.1416 x 3.464 = 10.8825. Then 
10.8825 H- .862 = 11.7445 cubic inches, the amount bored away. 

838. Let ABCD be the semicircle, E its center, and F and I the centers 
of the two given circles. Then 

FH = a,IK=hy 
and ED = EC = r, EF= r-a, 
EI=r-b, FI=za + 6, 




= Vr^ — 2 ar. 




EK = V(r ~ by - 62 = Vr* - 2 6r. GI z= HE -h EK, 
Hence {EH-\- EKy + FO^ = FI^; and, since FG = FH- IK, we have 
{ Vr* - 2 ar + Vr^ - 2 ftrf + (a - 6)^ = (a + 6)*, 
or 2r2 - 2r(a + 6)+ 2\/{(H - 2ar)(r^ - 26r)} = 4<i6, 
or r3 _ ^(e, ^ 5)^. rV{ra-2(a+6)r + 4a6} = 2a6. 
Transposing, squaring, and reducing, we get 



l + /i + 4\1^6__l_l. 
f^ \a bjr ab a« 62' 



whence 



r \V«6/ 2U 6y' J2__l/l.l\ 

'^ab 2\a b) 



889. Let r= the number of cubic inches of copper, t = the thickness 
of the shell in inches, B = the outer radius, and r = the inner radius. 
Then, by the problem, B-r=zt, (1). |ir(/2» - r^) = F, (2). By an easy 
process of elimination, we have 

' = '{>^-}-«'«=t{Af=!-'}- 



260 



TABLE BOOK AND TEST PROBLEMS. 



Hence the capacity of the kettle in cubic inches is expressed by 

i-=s{>^-'}" 

840. First Solution* On the hypothenuse of the right triangle BGE 
(Fig. 1) draw the half square BEA. From A let fall AD, a perpendicular 
on CE produced. The triangles BCE and ADE are equal, BC= DE, 
and EC = AD. The area of the quadrilateral ABCD is measured by CD 
multiplied by half the sum of the parallel sides AD and BC. Hence 
KreA ABCD = CDxi(BC + AD) = i (EC -\- BC)^. But area ABCD = 
area BEA + BCE-hADE=i(BEy-\- (ECx BC). Therefore {EC+ BC)^ 
=BE'^-^2(ECxBC), or EC^+BC^+2{ECxBC)=BE^-\-2(ECxBC). 
Therefore EC^ + BC^ = BE^. q.b.d. (From Mathematicdt Magazine.) 

NoTK. — Tbia is known as ibe Pythagorean Theorem, and the demonstration taken 
from the " Mathematical Magazine " is tho one given by President Garfield. 





Fio. 2. 

Second Solution. Let ABC (Fig. 2) be a right-angled triangle. We are 
to prove AC'^ = AB'^ + BC^. Let fall on the hypothenuse the i)erpen- 
dicular BD. Then AB^ =AC x AD, and BC'^ =AC x CD. Adding, we 
have AB^ -\-BC^ = AC x (AD + CD). But AD -\- CD = AC. There- 
fore AB^ + BC^ =ACxAC = AC^ q.e.d. 

Hiird Solution. Let ECF be any right-angled triangle. Produce CF 
to 2), making FD = CE. On CD describe a square ; also on EF. CD =z 

CF+FD, and CD^^ CF^ ■\- FD^-\-2CF x 
FD. But FD = CE (by construction). Hence, 
substituting, CD^=CF^-\- CE^-\-2CFxECy (1). 
Since DF := EC, FK= FE, and /) is a right 
angle, therefore DK= FC, and the triangle 
FDK = the triangle ECF. Similarly the tri- 
angles HBK and HAE are equal to the triangle 
ECF. The area of the square CDBA is equal 
to the area of the square EFKII + the area of 
the four equal triangles. Now, the area of the 
triangle ECF = J CF x CE, and the area of the four triangles = 2 CF x 




MISCELLANEOUS SOLUTIONS. 



261 




CE. Hence <72>* = EF^ -^2CFx CE, (2). Equating values of CD^ in 
(1) and (2), and subtracting equals from each side, we have £F'^ = CF^ 

+ CE^. Q.K.D. 

Fourth Solution, Let ABC be a triangle, right-angled at B. On AC 
construct the square ACEB, Draw BG perpendicular to AC, and pro- 
duce BQ to K, making BK= AD. Join DK 
and EK, and produce AB to IT. We are to 
prove AC^ = AB^ + BC^. ABKD and CBKE 
are parallelograms, each by construction hav- 
ing two opposite sides equal and parallel ; but 
ABKD = the rectangle AGFD, both having the 
same base and altitude. For similar reasons 
CBKE = the rectangle OCEF, Hence the two 
parallelograms are together equivalent to the 
two rectangles which form the square ACED 
= AC\ The triangles ABC and BHK are 
equal, being mutually equi-angular, and having 
their homologous sides perpendicular each to each, AC being by construc- 
tion equal to BK. Hence BC = BH, and AB = HK. Area CEKB = 
BC X BH= BC\ and the area ABKD =BAxUK = BA^. Since the 
two parallelograms = the two rectangles = AC^, it follows that AC'^ = 
BC^ H- BA\ Q.E.D. 

Fifth Solution. Let ABC be a triangle, right-angled at B. We are to 
prove AC'^ = AB'^ -{- BC^. Construct squares on each side. Draw BI 
parallel to CK, and HB and 
CD. ABC and ABE both 
being right angles, CBE is a 
straight line. Likewise ABF 
is a straight line. DAC ^ 
BAH, each being a right 
angle increased by the angle 
BAC. Therefore the triangle 
i>^C7 = the triangle BAH. 
Now, the parallelogram AI 
is double the triangle BAH, 
and the square BD is double 
the triangle DAC. Therefore 
the parallelogram AI = the 
square BD. Similarly, by 
joining BK and AG, it may 
be shown that the parallelo- 
gram CI = the square BG. 




262 



TABLE BOOK AND TEST PROBLEMS. 



But the square on -4C = the two parallelograms =-4/+ C/= the square 
BD + the square BQ, Therefore AC^ = AB'^ + BCK q.b.d. 

841. Let ABD represent a vertical section of the wine-glass, passing 
through the axes NFCKD and FCK of the glass and immersed ball, 

whose center is at C. Draw the radius 
CGt €r being the point of tangency of 
the ball and glass. Let the water line 
HE be tangent to the ball at the point 
F. Put ^iV= NB = r = S inches, ND 

= d = 9 inches, and - = — We shall 

n 3 

have, by geometry, F= J rr^ = vol- 

ume of wine-glass, and v = ^^—^ = vol- 

3n 

ume of water in the glass ; also BD 
= \/(rM^. Let x= CG- CF-CK, 
the required radius of the immersed 
ball. Comparing the similar right tri- 
angles BNB and CGB^ we have NBi 

BD'.iCGiCD; that is, r:V(r^ + (f2) 
'.:x:CD. Therefore CD = ^CSM. 




whence FD = CD-\-CF=: ^^(^ + ^ ) ^ x. By comparing similar solids 

r 
(geometry), we have (ArZ))8:(FZ>)': : V: volume of cone HED ; that is, 

^ . I gV(H -i- d^) -j. a; I ' : : - r2<2 : volume of cone HED. Therefore vol- 



ume of cone HED = — / ^"^^^'^ + ^'1 ^ x V- Now if, from the volume 

of the cone HED, we subtract the volume of the immersed ball, the 

remainder must equal v. Whence — [ ^-^(^ + ^\ +«]-'- ^^ = t? = 

3d2 I r i 3 

!r^,(l). Multlplying(l)by§,^:{5^5E±l!l+a'_4^=rf^,(2). 
on »a-^r J n 

Or -i-{«v^(^^ + d-; + ra;}8- 4 a;3 = — , or n {a;V(H + d-)+ r5c}8- 4 ndWoi? 
d-r n 



= r'd«, or nx^ {y/ {r^ ■\- d^) ■{■ rf - 4t nd'-roc^ = r^d\ (3). Whence a^ = 

^'' ^' ^ , and x= ^ _, ^^ 

n { V(r2 + d') + r }» - 4 nd^ V[n {^(7-2 + ^i) + r}8 - 4 nd^ ] 

8 V (^^^^^^^^) = 1.88796741 - mches. (From B. P. Bublbboh.) 



MISCELLANEOUS SOLUTIONS. 



268 



TRIGONOMETRICAL SOLUTIONS. 



842. Let be the center of the earth ; OP, OA, 
and OB, radii. By the problem, a& = } AB, But 
ab : AB ::oa: OA, since similar arcs are as their 
radii. Therefore oa = J OA. Let fall the perpen- 
dicular ac. Then Oc = oa = i OA = J r. By 
trigonometry, Oa : sin 90° :: Oc: sin OaCy or r : 1 
: : Jr : sin Oac; whence sin Oac = .6, which is the 
sme of 30°. Hence the angle AOa = 00°- 30° = 
60°, the required latitude. 

848. Let a = the angle ACB, r = AC = 6, AD = rBma; CD = 

r cos a. Area of circle = tj'2, area of sector ACB = -^irr^, area of tri- 

360 

angle ACD = J r^ sin a cos a ; semi-segment 
Now, the sector — the triangle = 








a 



the semi-segment. Hence -—  irr'^ — J i^ sin a 



360 



a 



cos a = T^^irr^. Dividing by irr^, we get 

1 180 
sin a cos a = tw- Clearing, a sin a 

COS a = 36. But sin a cos a = } sin 2 a. Sub- 
stituting this value in the equation, we have 
a - 28.648 sin 2 a = 36. Solving by method of 
approximation, we find a = 59° 55' 25''. Sina = .86536. AD = r sin a 
= 4.3268 feet = J of the required chord. Hence chord = 8.6536 feet. 

844. First Solutmi. Let C be the center of pond, P the point with- 
out, and A and B the points of tangency. Since the angle APB is 60°, 
the angle ^PCis 30°. PAC being a right angle, the 
angle ACP = 60°. Then, by trigonometry, sin 60° : P 

18 : : sin 30° : AC, whence AC= 10.39 = radius, and 
20.78 = diameter. 

Second Solution, The angle P being 60°, and AP 
and BP being equal, the triangle ABP is equilateral, 
and AB = 18. PC bisects AB in E, and EA = 9. 
The triangles ^^Pand ACP, being right-angled, and 
having the common angle A PC, are similar. Hence 
EP:AP::AE:AC. Now, EP = VW^ -9^ = 9y/S, 
Then 9V^ : 18 : : 9 : AC, whence AC= 10.39 ; and AD, the diameter = 
20.78 rods. 




264 



TABLE BOOK AND TEST PROBLEMS. 



845. Area of wheel = irr^ = 4 tt. Let ABD = segment in mud. Then 
^/> = lfoot, CE=: CD- ED=l foot, CB=r=^2 feet, EB = V(22 - l^) 

= a/3, ^J5 = 2\/3. Areaof triangle ^(75 = 2 \/3xi 
= V8 = 1.732 4- feet, (1). By trigonometry, BC : 
sinOO^: : CEiainEBC, or 2 : 1 : : 1 : .6. But .5 is 
the Bine of 30<^. Hence the angle BCE = 60°, and 
AGB = 120°, or J of the circle = |ir = area of the 
sector ACBD. Then }T-V3[see (1)] = 2.4567 
= area of segment ABDy which is nearly ^ of the 
wheel. 




346. Let PAB represent a portion of the surface of the earth, P being 
the north pole, and AB the equator. Let AF = the distance sailed by 

the ship = 1000 miles, and AE = the difference 
iVss:?-^ ^^ latitude. By "plain sailing,'' we may con- 

sider AFE as a plane triangle, and we have 
B : distance : : cosine course : difference of 
latitude, or 1 : 1000 : : cos 45° : difference of 
latitude = 600V2 = 707.106 + nautical miles 
= 11° 47' 6", the ship's north latitude. 

847. The equation of the path of a projectile 

B isy = x tan a ^ , (1). Transposing, 

2 v'^ cos'-^ a 

clearing, etc., 2 v^ cos"^ a(^z' tan a — y') = (gx')^, 
(2). Time = t = 




and «2 _ 



V cos a v^ cos=2 a 

(see p. 113, Olmsted). Substituting this value 

of ^ in formula (20» p. 112, ( tan a = ^, ) we have tan a = ^ , 

'^ ^^^^ 'V 2ry 2t^«co8-^a 

Substituting in (2), we have 



whence 2 r^ cos^ a = 



gr 



tan a 



-Hi- (x' tan a - y') = (^•«0*- 
tan a 



Reducing, ^ = tan a — ~ tan a. Substituting values, we have 
tan a — 



10 



X 

1700 



3500 



r 1700 

tana, whence tana = .011437+, which corresponds to 



39' 19'' = angle. Velocity may easily be found to be 2218.3 feet. 



848. Let ABCD be the bam ; and ^(? = 100 feet, length of rope. The 
horse is tied at A, and grazes over f of a circle whose radius is 100, or 



MISCELLANEOUS SOLUTIONS. 



265 



23,562 square feet, (1). By trigonometry, DF : sin DEF : : DE : sin 

DFE^ whence the angle DFE is found to 

be 13° 37' 55". Now, 75 cos DFE = FE = 

72.88 feet, and DE x 72.88 = the area of 

BDF. The triangle BDF- J of the barn = 

975.85 feet = area of DFBC, (2). The angle 

CDF = 45° - 13° 37' 55" = 31° 22' 5", and 

90° - 31° 22' 5" = 58° 37' 55" = angle FDH. 

Then the sector FDHj whose radius is DH 

= 75 feet, has an area of 2878 square feet. 

The area of the sector FBG is the same : 

hence the area of both is 5756, (3). Adding the three areas, (1), (2), 

and (3), we have 30,293.85 square feet, the area grazed. 




349. Let FBA be a vertical section through center of glass, C the cen- 
ter of sphere, and CG a radius drawn to BA ^ 
at point of tangency. We have AH= 6, HB 
= 2\,EC= CD = 2. ^J5=v6^ + 2.5'^ = 6.5. F^ 
By trigonometry, AB : sin H:: HB: sin HAB, 
or 6.5 : 1 : : 2.5 : .384615. Hence the angle 
MAB = 22° 37' 11". In right triangle AGO, 
sin 22° 37' 11" : 2 : : sin 90° : ^ C, whence A G 
= 5.2. AH-AC=0-&.2=.S = HC. Then 
HC 4- (72) =.8 + 2 = 2.8 = height of segment 
immersed. In right triangle CHI, IH = 
V2^ - .8-^ = 1.833. 1.8332 x 3.1416 = 10.557 + 
= area of base of segment. 10.557 x J of 2.8 
= 14.779, (1). (2.8)8 X .5236 = 11.493, (2). 
Adding (1) and (2), we have 26.272 cubic 
inches. 

NoTB. — j^C may be found by geometry as follows: The triangles BHA and AOC 
are right-angled, and have the angle in common. Hence they are similar. Then 
HB : CO :: AB : AC, or 2.b ; 2 :: 6.6 : AC, or 5.2. 




860. Let Cbe the center of the 10-acre field, A the stake, and AEHD the 
segment grazed. Put CA=R, AD=r, and the angle BDA=a. Then the 
angle DBA = a, and the angle DCA = 2 a. The triangle BDA is right-an- 
gled, hence sin 90 : BA : : sin a : AD, whence r =2 ^ sin a, (1). The area of 
the triangle DCE=i B^ sin 4 a, (2). 4 a : 360 : : sector CDAE: irB^, whence 

irR^a _ ir^a^ ^3^ Therefore the segment DAEB = 



sector CDAE = 



360 



90 



B^ 



— *i?*8in4a = -^ (wa — 45 sin 4 a), (4). In like manner, 180 — 2 a 
90 90 y^ \ y 



266 



TABLE BOOK AND TEST PROBLEMS. 



: 360 : : sector DAEH : trr^, whence sector DAEH = ^^^^^ZT ^ ^^ = 
rrg(90 - g) ^ ^gj rj^^ ^^^ ^^ triangle EDA = i r^ sin (180 - 2 a), (6). 

Therefore segment JDHJ^ = "^^^^ " ^^ - J r^ sin (180 - 2 a), which 

(remembering that the sine of an 
angle is equal to the sign of its sup- 
plement) 

= -^ (90 ir - ra - 90 sin 2 a), (7). 
180 

By condition, these two segments 

(4) and (7) are together equal to 1 

acre. Therefore — (xa — 45 sin 4 a) 
r2 ^ 

180^ 




90sm2a)= 160 



rods. Clearing, we have 2 R\ira - 45 sin 4 a) -f 90 xr2 - rr^a - 90 r^ 

sin 2 a = 28,800, (8). From (1), r = 2 i? sin a. Then r^ = 4 ^ sin2 a. 

Substituting this value of r^ in (8), we have 2 ^^^a — 90 /^^ sin 4 a + 

360 R^ sin^a - 4 wR^a sin^a -- 360 R^sin^a sin 2 a = 28,800. Dividing by 

2 R^, we have wa — 45 sin 4 a + 180 sin^ a — 2 ira sin^ a — 180 sin^ a sin 2 a 

14 400 
= * = 9 T, (9). From trigonometry we have sin 2 a = 2 sin a cos a. 

Hence 45 sin 4 a = 90 sin 2 a cos 2 a. Using this, (9) becomes ira — 90 
shi2acos2a + 180irsin2a - 2irasin2a - 180 sin^ a sin 2 a = 9t, (10). 
From trigonometry we have sin^ a = J(l — cos 2 a). Substituting in (10), 
we have ira — 90 sin 2 a cos 2a + 90T — 90ir cos 2 a — ira + xa cos 2 a 
— 90 sin 2 a + 90 sin 2 a cos 2 a = 9 r. Collecting or canceling terms, we 
have — 90 sin 2 a — 90 IT cos 2a + ira cos 2 a = — 81 ir. Changing signs, 
90 sin 2 a + 90 ir cos 2 a — ira cos 2 a = 81 ir. Dividing by ir, 28.65 sin 
2 a + 90 cos 2a — a cos 2 a = 81. Factoring, 28.65 sin 2 a -f cos 2 a(90 — a) 
= 81. Solving this equation (see Ray's **New Higher Algebra," Art. 
436), we find 2 a = 27^^ 18'. Then a = 13039'. Then in triangle BDA, 

sma:r::l:2R, or sin 13° 39' : r :: 1 : 2^^522 = J5., 
rods, the required length of line. 



whence r = 10.6+ 



851. Let A and D be centers of the circles, BCb, perpendicular to AE, 
Since 2>^is about if of a rod, BC more than 7 rods, and AE nearly 32 
rods, it is evident that the angle BDE is obtuse. Put AB = 31.91638 
= By DB = r, the angle BAD = 0, DE = .60606 of a rod = a. Now, 
AC = R cos e, and BC=R sin $. Hence CD=AE- AC - ED =R -R 
cos 6 — a. Again, should we join the middle point of the arc BHE 
with the center A, we should have half the chord BE as ^sin}^. 



MISCELLANEOUS SOLUTIONS. 



267 



Hence BE = 2 RsiniO. By Proposition 13, Book IV., Loomis^s 
♦» Geometry," BE-^ = BD^ + DE^ -\- 2 DE 
XCD; that is, ^ R^ sin^ i 6 = r^ + a'^ + 
2a(^R — Rcos0 — a), (1). From trigo- 
nometry we have 2 sin^ ^ 6 = 1 — cos 0. 
Putting this equivalent in (1), we have 
4B^Bm^i$ = r2+ a2+ 2 a (2 /Zsin^ J ^ - a), 
(2). Developing (2), etc., we get 

r2 = 4 /22 8in2 i^ - 4 aft sin2 J <? + a^ 
or r^ = 4 /2(ft - a) sin2 J ^ + a\ Therefore 

R^ 




r = V{4 R{R - a)sin2 J ^ + a^}, (3). Area of sector ^^JIS; = ^, (4). 
Area of triangle ABE = iAEx BC = :?ilHL?, (6). Then area of seg- 



-awrfiW ^(^-«^^^) 



2 
, (6). Area of triangle BDE — 



ment JJlTff = (4) - (6) = 

DExBC ^ aR sin 6;^ ^^^ ^^ trigonometry the sine of the angle BDA 



= BC-irBD = 



2 
R sin ^ 



, (8). Taking the inverse functions of each mem- 

r 

ber of (8), or, in other words, measuring the angle BDA by its subtend- 
ing arc, we have, angle BDA = sin-^( — ^^^ j. But this is the measur- 
ing arc of the angle BDA when radius is unity. Hence, when radius is 

DO=r, the measuring arc of the angle BDA is arc OKB=rshr^ I j . 

Now, by the same principle with which equation (4) was obtained, we find 

the area of the sector D0KB=rxir8ur^l — -^ — J =J?'2sin""M j, 

(9). Evidently, OKBHE = half the area grazed, which is correctly ex- 
pressed by the sum of (6), (7), and (9). Therefore J^HLlL^^FL^ 
^ aRBine ^^^ sin-i^ ^ ^^" ^ \ = 80, (10); and iP(^ - sin^)+ aJJsin^ 

-I- ffl sin-if ^ ^^^^ \ = 160, (11). Substituting in (11) the value of r 
from (3), we have R^(0 - sin 0) + aR sin + {iR{R - a) sin* J ^ + a^} 

X sin-i I ^^^^ — \ = 160, (12). Solving (12) by 

( V[4 R(R - a) sin2 J ^ + a^] J 

** Position" (which may require several hours), we find = 18° 16' 9.4". 
Substituting in (3), we find r = 10.04588 rods. (From B. F. Bublbson.) 



268 TABLE BOOK AND TEST PROBLEMS. 



SOLUTIONS INVOLVING CALCULUS. 

1 

868. Let X = the number. Then a?» — x = a maximum = y. Differen- 

/l L-i \ , ,.^ . dy 1 ~i ^ ^ ^ --1 

tiating, f -x" - 1 jdx = dy. Therefore ■^ = -^ — 1=0, and a?» = », 

whence x = n*~», Ans. When n = 2, we have x = 2*-* = 2-« = — = ±. 

8 2« 4 

When n = 3, we have a; = 3^' = 3"' = — = — = -i-. When n = 4, 

3i V38 v^ 

— -ill I 

we have a; = 4^-^ = 4 ' = — = — ^ = — ^^:z.* When n = 6, we have 

— -* 1 1 

X = 51-5 = 6 ^ = - = ^ etc. 

6* V3125 

368. When the hypothenuse is a minimum, the base = altitude. Hence, 
in the figure, AC= CB, and the angle B = angle A = 46°. Bisect the 
^ right angle by CD. Then the angle ACD = the an- 

gle A, and CD = D^, and ^£: = EC. But ^C = 12. 
Hence AC =2i, and (75 = 24. This result is veri- 
fied by the differential caZci^us, as follows : Put AG 
= x, BC==y, EC =12 = a. Then AE=x-a. 
By similar triangles, we have the proportion AE: ED 
::AC:BC, or x— a:a::x:y. Hence xy — ay = 
ax, (1). By the problem, x"^ -f y^ = AB^ = a mini- 
mum, (2)., Differentiating (1) and (2), we have xdy -\- ydx — ady = 

adx, (3), and 2 a:6te + 2 ydy = 0, (4). From (4), dy = - — Substitut- 

y 

ing this value of dy in (3), we obtain 1- ydx -\ = adx, whence 

y y 

y^ — ay = x^ — ax. Completing squares, y'^ — «y + J a"^ = a;^ — ax -I- } «*. 
Extracting square root, y — ia = X'-ia, whence y = x. That is, when 
the hypothenuse is a minimum, the base = the altitude, or perpendicular. 
Hence the construction and solution are same as above. 

364. Let X = the height, r = 2^2 = radius. Then x^ + r^= the square 

of the distance. Therefore the light varies as —^ By trigonometry, 

a;2 + r* 

we have y/x^ -f- r^ : 1 : : x : sin d, being the angle of inclination to the 
plane, whence sin ^ = — ^ which also expresses the variation of 

the light. Therefore the light varies as 

1 _ a; X 




x^-^1^ Va;2 H- r2 (aj'2 _,. ^)i 



= y. 



MISCELLANEOUS SOLUTIONS. 26 Q 

Differentiating, ^i^i^' '^^)^ " ^\K^+ ^)^^^) = o. Clearing, 

dividing by (x^ + r^)*, we have 2 a'^ = r^, whence x = i-^* By testin^, 

r ^ 

we find that x = ± — corresponds to a maximnm. In the specs^^j^i 

2v^ 
problem a; = ± = ± 2 rods, the required distance. 

V2 



PROMISCUOUS SOLUTIONS. 

855. Let X = .46, the repetend. Then 9 -\- x = the given number. 
100x = 46.45, and lOOx - x = 99x = 46.45 - .45 = 45. Then x = ^^ 

= ^r ; and 9 + X = 9 + A: = ^A- 

856. 1100 X 8 = 8800. 8800 -r- 5280 = 1.6+ miles. 
NoTK.— Souod travel! through air about 1100 feet per second. 

857. Every time the wheel revolves, the nail-head describes a cycloid, 
whose length is found by dividing four times the distance by ir. Hence 
3.1416 X 4 -f- 3.1416 = 4 rods. 

858. First Solution, Suppose A works x times as fast as B. Then 
digging must be x times as difficult as weeding. By second condition, 
the faster man has the easier work, and finishes four times as many 
rows. Hence x x x = x^ = 4, and x = 2. If A works twice as fast as 
B, he should receive $4, and B $2, per day. 

Second Solution. Let A's rate be x times B's. Then A's rate : B's 
rate : : x : 1. By second condition, A's rate : B's rate : : 4 : x. Hence 
X : 1 : : 4 : X, whence x^ = 4, x = 2, and A should receive i of $6, or $4. 

859. Let X = number of pounds of gold. Then a — x = number of 

pounds of silver. 1 pound of gold loses - pounds in water. Hence x 

nx t) 

pounds lose — pounds. 1 pound of silver loses , and (a — x) pounds 

lose ^ — pounds. Since the whole loss in water is m pounds, 

nx . (a — x)p , a(m —p) , a(n — m) 

— -f ^^ ^ = m, whence x = -^^ ^ and a — x = ^ -' 

a a n —p n — p 

860. First Solution. In 1 day A and B dig ^^, B and C ^\j, and A and 
C i^. Adding, 2 (A and B and C) dig i in a day, or all dig } of J = ^^^j in 
1 day. Hence ^^ -s- jij^ = 10 days, the time in which all can dig it. Then 
1 -i- (tV - i*i) = ^ days = C's time, 1 -*- (i^y - ^) = 20 days = A's time, 
and 1 -5- (^J^ - ^3) =r 30 days = B's time. 




270 TABLE BOOK AND TEST PROBLEMS. 

Second Solution. A and B dig i'^ in 1 day, and A and C j^. Hence 
B digs iV ~ i^ = A T^oie than C. But B and C dig ^. Hence C digs 
}(A - A)= A "i 1 day* or ^^ ditch in 60 days ; B digs ^i^ - ^ = ^\j 
in 1 day, or the ditch in 30 days ; A digs ^^^ — ^ = ^ in 1 day, or the 
ditch in 20 days. 

861. Let A, B, and C be the lower ends, and O the point beneath the 

tops. 0, being the center of the triangle, is ^ of AD, a perpendicular. 

A But AD = VAB^ - BD^ = V2700 = 61.9+ 

feet. Hence ^0 = § of 61.9 = 34.6+ feet. 
This is the base, and the pole erected at A 
is the hypothenuse of a right-angled tri- 
angle, of which the required plumb-line is 
the perpendicular. Hence the required line 
= V(60^ - 34.6-') = 36.1 feet, nearly. 

862. The semi-annual interest on the 
^ bonds is 3} per cent of $7000 = |246, to 
secure which at 8 per cent would require $246 -s- .04 = $6126. This 
would allow me the interest on the bonds and $6126 of their face value 
at the end of the 20 years. To be entitled to the remainder, I must 
invest an amount equal to the present value of $7000 — $6126 = $876 
for 40 intervals at 4 per cent compound interest, or $876-5-4.8010206+ 
= $ 182.26. I therefore paid $6126 + $ 182.26 = $6307.26. 
NoTX. — The 4.8010206+ is the 40th power of 1.04, or 1.04«>. 

863. Let a = y + 3. Then x^ = y^ + 6y + 9, and 6a; = 6y + 18. Sub- 
stituting, the equation becomes y^ + 6y + 9 — 6 y — 18 = 7, or y^ = 16, 
y = 4, and x = 4 + 3 = 7. 

864. Let a; = y - 1. Then x^ = y^ _ 2y + 1, and 2x = 2y - 2. Sub- 
stituting, we find y2 = 9, y = 3, x = 2. 

865. A square field containing 10 acres is V160 x 10 = 40 rods square, 
and the fence is 160 rods in length. In the rectangular field the length is 
4 times the width. Hence the width multiplied by 4 times the width = 4 
times the square of the width = the area = 1600 square rods. Then the 
square of the width = 400 square rods, and V400 = 20 rods = the width ; 
and the length = 20 x 4 = 80. The fence is 2(80 + 20)= 200 rods 
long ; and 200 — 160 = 40 rods, the difference. 

866. First Solution. Assume x2+x = a'x*. Then x = — • Assum- 

ed- 1 

ing values for a, we easily find x ; and substituting these values of x in 
x^ + X, we obtain square numbers. 



MISCELLANEOUS SOLUTIONS. 271 

Second Solution, aj^ + a; = a square number. Assume x^ + a; = m^. 
Then x^ = m^ — x. Put m^ — a; = any square number, less than »»*2, m 
terms of m and a;. For instance, (m — x)^ < m^. Then x^^m^ — 2mx 
+ a;2, a; = m — a, or w = 2 «, m^ = 4 a2. Or aj^ + x = 4a;2, aj + 1 = 4 a, 
or 3 X = 1, X = J. Values of x may be found at pleasure. 

(From Dr. I. J. Wibbback.) 

367. Let m and n represent the stumps and stones that give an earning 
of $8 ; and M and N, those that give $12. Then 26m + Jn = 800, or 
60mH-n = 1600,(1); and 25 if + J iV = 1200, or 60 Af + iV= 2400, (^). 
Since 1 stump requires the time of 40 stones, the relative times may be 
represented by 40 m + n, and 40 M + N, so that 40 w + « : 40 3/ + jV 
::3;3i::4:6; whence 200m + 6n = 160 Jif + 4iV; (2). Subtracting (2) 
from 6 times (1), we get 160 itf + 4 iV + 60 m = 8000, (3). Subtracting 
(3) from 4 times (^), we get 4ilf=5m + 160, (4). We have four 
unknown quantities, and only three equations: hence the problem is 
indeterminate. From (4), JIf = }(6m + 160), (6), an integer; and Jm 
must be an integer, say p. Then m = 4p. The lowest value assignable 
to |) is 0, in which case m = 0. Substituting in (6), 3/ = 40, and in (1), 
gives n = 1600 ; also M in (^), gives JV = 400. The next higher value 
of p is 1, making m = 4, n= 1400, itf"=45, and JV= 160. The next 
value of m is 8, which makes iV^ = — 100, which is not admissible. As 
all succeeding values of N are negative, there can be but two combina- 
tions, making stumps and 1600 stones to earn $8, and 40 stumps and 
400 stones to earn $ 12 ; or 4 stumps and 1400 stones to earn ^8, and 45 
stumps and 160 stones to earn $ 12. Consequently 40 stumps and 2000 
stones, or 49 stumps and 1660 stones, will satisfy all conditions. 

(From WiLUAM Wiley, Detroit, Mich., in the School VitUor.) 



272 TABLE BOOK AND TEST PROBLEMS. 



CURIOUS RESULTS. 

DIGITS. 

868. To get any required digit in the product, multiply the given num- 
ber by 9 taken a number of times denoted by the digit required. Thus, 
to get all 4'8, multiply by 9 x 4 = 36. 12,345,679 x 36 = 444,444,444. 

86». TTTi^ifirij = G W ; i^.'^v = (7««; TirTA77nr = (iW; and 

"ONE CENT." 

870. We will make the calculation by means of logarithms, and honoe 
will give results in round numbers. The amount of $1 for 1 year at 
6 per cent is $1.06; for 2 years it is ($1.06)2; for 3 years, (^1.06)«; 
and so on. Hence the compound amount of 1 cent for 1881 years is 
-4 = .01 X (1.06)1881. Applying logarithms, we have log A = log .01 + 
1881 log 1.06 = - 2 + 1881 ?< .02530 - 686626 = 45.600333. Therefore 
A = $3,984,130,000,000,000,000,000,000,000,000,000,000,000,000,000, which 
may truly be called a ** round '* sum. 

(From the late E. B. Bbitz, in MathemaiiccU Magazine.) 

INVOLUTION OF IMAGINARY QUANTITIES. 

871. Squaring, —a^ = x^. Squaring again, a* = x*. Extracting fourth 
root, a = X. But Vo^ = a, and V— a-* = x. Therefore Vcfi = V— a\ 
or a^ = — a^^ or 1 = — 1. The beginner will understand this better by 
studying the involution of imaginary quantities. By definition, the 
square root of any quantity multiplied by itself should be the quantity 
itself. Hence V— a x V— a = — a. But if we multiply the quantities 
under the radicals, we have V— a x V— a = Vo^ = ±a; that is, there 
appear to be two products of V— a x V— a. The true value of Va^ 
in this case, however, is — a. We know this, because the factors that 
produced a^ are known. Were the factors not known, the value of Vo^ 
would be in general ± a. 

878. Since squaring a radical removes the symbol V, we have 
(\/^)2 = - 1, (1). (vZ~i)8^ yiTi X V^n: X >/^ =(V31)2x 
>/- 1 = - 1 V^Il, (2). c>An)* = (V- l)2(\/^)2 = (- 1) X (- 1) 
= 4 1, (3). 



CURIOUS RESULTS. 273' 

878. It is because the fewer times a negative quantity is taken, the 
greater will be the product. This will be clearly seen from the following : 

-7x 3=-21, 
T 7 X 2 = - 14, 
-7x 1=- 7, 

- 7 X = 0, 

- 7 X - 1 = 7, 

- 7 X - 2 = 14, 

- 7 X - 3 = 21, etc. 

We notice that each product is 7 greater than the preceding, and that as 
the multiplier decreases, the product increases. Or take the expression 

— 6 X (rt — c). This may be written as follows : (— 6 x a) — (— 6 x c), 
which is equivalent to — ah —{— be). But minus a minus quantity = 
plus the quantity. Hence — a6 — (— 6c) = — a6 + 6c, from which we 
see that — 6 x — c = 4- 6c. 

THE ZERO FACTOR. 

874. First Solution. If a = x, then a^ = x^, and ax = x^. Subtract- 
ing 05^ from both sides, we have ax — x^ = x^ — x^ = a^ — x^ (since a^ = x^). 
Factoring, x(a — a;) = (a + 2c)(a — sc), (1). Dividing by a — x, we get 
x = a-^x = 2x. Whence 1=2. 

NoTX. — While all the operations appear to be legitimate, the result is evidently 
absurd. The error must be the using of the zero factor (a — x) in dividing (1). 
Instead of striking out the common factor in (1), we may indicate the division thus: 

_i s= A Ll 1. Now, since a = a;, a — a? = 0, and the fractions become * = 8, 

a—x a—x 

each member consisting of the symbol of indetermination. 

Second Solution. If a = a, then a^ = o^. Transposing, a^ — x^= 0, (1). 
Factoring (1), (a + x)(a — a) = 0. Hence a + a; = 0, (2); and a — x 
= 0, (3). From (2) we see that 2 a, or 2 a;, = 0, (4). If a = x = 1, 
then (4) becomes 2 = 0. Ifa = aj = 2, then (4) becomes 4 = 0, etc. 

876. This proportion must be true, since the product of the extremes 
is equal to the product of the means. Now, as a is greater than 0, it is 
therefore greater than — a. But if in the first ratio the second term is 
the greater, it must also be the greater in the second ratio. Hence we 
have these two inequalities: «> — a, and —a^a. If a = 1, then 
1 > - 1, and - 1 > 1. If a = 2, then 2 > - 2, and - 2 > 2, etc. Again : 
solving the proportion, we have (— a)"^ = a^. Dividing by — a, we get 

— a = , which may be written . Now, the quotient of 

— a a — 2a 

a2-^(a-2a) is a + 2a + 4a4-8a + 16a + 32a + 64a, etc.; that is, 

— a is infinitely greater than + a. 

ellwood's test pros. — 18. 



274 



TABLE BOOK AND TEST PROBLEMS, 



SOMETHING TO INVESTIGATE. 

876. 127,364 x 3 = 382,092. 3 + 8 + 2 + + 9 + 2 = 24. 2 + 4 = 6. 
Starting with this, we may proceed as follows, grouping the nine digits 
thus: 1, 2, 3; 4, 5, 6; 7, 8, 9. Take any number and multiply by the 
first digit of each group, proceeding as above, then by the second and 
third, as follows : — 



8425 
1 


8425 
4 


8425 

7 


8425 = 19 = 10 = 1 


33700 = 13 = 4 


58975 = 34 = 7 


8425 
2 


8425 
5 


8425 

8 


16850 = 20 = 2 


42125 = 14 = 5 


67400 = 17 = 8 


8425 
3 


8425 
6 


8425 
9 



25275 = 21 = 3 50550 = 15 = 6 75825 = 27 = 9 

In this case the resultant digits are the same as the multipliers, and 
the nine digits occur in regular order when taken vertically. Likewise 
when the multiplicand is 28,747, etc. Taking 25,432, we have the 



following : — 






25432 
1 


25432 
4 


25432 

7 


25432 = 16 = 7 


101728 = 19 = 10 = 1 


178024 = 22 = 4 


25432 
2 


25432 
5 


25432 

8 


50864 = 23 = 5 


127160 = 17 = 8 


203456 = 20 = 2 


25432 
3 


25432 
6 


25432 
9 



76296 = 30 = 3 152592 = 24 = 6 228888 = 36 = 9 

The resultant figures are the same as in the previous examples, and in 
the same horizontal line, though not in the same order. 

THE PROPOSITION OF ARCHIMEDES. 

377. ** This he might easily have done could he have brought his lever 
to bear upon it ; for it rests upon nothing, impinges against nothing, and 
floats in space, a body perfectly free to move in any direction. His lever, 
therefore, would have been a useless thing, as the slightest force brought 
to bear upon it would have caused it to move. He need only have 
stamped his foot, and the ponderous globe would have moved obedient to 
the impulse. His idea of the subject must have been that the world 
rested in all its mass like a rock upon some other ponderous body, and 
that he could apply a sufficient force by his leverage to lift it up and over- 



CURIOUS BESULTS. 



276 



turn it. His calculations and conclusions were undoubtedly correct, but 
the element of time he overlooked in his computations. Calling the 
diameter of the earth 7920 miles, and each cubic foot of its volume 
to weigh, as has been estimated, 800 pounds, we find that the earth would 
weigh 5,765,171,439,674,306,792,000 tons. Supposing Archimedes could 
exert a continual force of 30 pounds at the end of his lever, we find that 
one arm of the lever must be 384,344,762,638,287,062,800,000 times longer 
than the other in order that he might move it. Hence in order that he 
might move the earth to the height of one inch, it would have been neces- 
sary for him to have moved with the long arm of his lever 384,344,762,- 
638,287,052,800,000 inches. Now, constantly pulling with a force of 30 
pounds, he could not, with his single-man power, have traveled more 
than 10,000 feet per hour ; and at that rate, too, not more than 10 hours 
per day. He could therefore, at his utmost, have moved his end of the 
lever but 100,000 feet per day. Hence it may be readily calculated that 
to have raised the earth only one inch, it would have required his con- 
tinual labor for 8,774,994,680,737 centuries." Admitting this, we must 
deny his ability to ** move the world," not on account of a lack of power 
or principle, but on account of the shortness of human life. 

1888. 
878. 



487 


459 


461 


481 


465 


477 


475 


471 


473 


469 


467 


479 


463 


483 


485 


467 



SUMMATION BY SUBTRACTION. 

879. Assume any number greater than the sum, and from this subtract 
one of the given numbers; from the remainder take another of the 
numbers ; and so on till all have been subtracted. Then take the last 
remainder from the assumed number. 

Assume 



10000 
379 

9621 
8452 

1169 



1169 
31 

1138 
60 

1078 



10000 
1078 

8922 Ans, 



276 TABLE BOOK AND TEST PROBLEMS. 



SOLUTIONS TO SERIES. 

880. First, sum the series 1 1 = \- — 

1.23 2.3.4 3. 4. 6 

Let — = the nth term of the series = , and /Si, = the 

. Mn n(n + l)(n + 2)' 

sum of n terms of the same. 
We shall have /S; = --?_+ c, 

and Sn+i = -^ -«- + c = -i |±^ + c 

_ n+ 1 



2/An+l 



Therefore c = 



2 Ml' 



and Sn = A-T^= ^ " 



2/Ai 2/i„ 2(1.2.3) 2n(n + l)(n + 2) 

= [when n = Qo]-. 



4 2(n + l)(n + 2) " "'4 

Second, sum the series 

1 + i^+,_i^+ 



1.28 3.4.6 6.6.7 
In the series for log« (1 + «), put ac = 1, and we have 

loge 2 = l-i + i-l + i-i + •••= — + — + — + 
2 3 4 6 6 1.2 3.4 6.6 

=1-1 1 1 



23 4.6 6.7 

Therefore 2 log« 2 = 1 + — \- — \- — - — 4- . ... 

^ 1.2.33.4.66.6.7 

Whence — h — - — H = 1- ... = loge2 - i. 

1.2.3 3.4.5 6.6.7 2 

Finally, by subtracting this second series and its summation from that 
of the first, we obtain 

+ 7-4-^ + ;;-4^ + - = 7 - loge2 



2.3.4 4.6.6 6.7.8 4 

= .066862819440064690682767878764034- 



SOLUTIONS TO SERIES. 277 

(correct to 82 decimal places, as we have computed the Napierian loga- 
rithm of 2 to that number of decimal places). 

NoTX. — Logf » Kapierian logarithm. 



881. Put 



n+(n + l) + (n + 2) + (n + 3) _^ B C ^ D 

n(n + l)(«H-2)(n + 3) n n+1 n + 2 « + 3 



(1) 



Clearing (1) of fractions, 
4n + 6=(-4 + B4- (7+ 2>)n8 +(6^ + 5B + 4 (7+ 3i))n2 

+ (11^ + 65 + 3 C+ 2 Z))n + 6^ (2) 

Equating coefficients of like powers of n in (2), and solving the equa- 
tions, we find that 

^=1, 5 = -l, C = -l, and 2>=1. 
Therefore 

n+(n + l)H-(H + 2) + (» + 3) ^l 1 1. 1 . 

w(n4- l)(n + 2)(n + 3) n n+1 n + 2 » + 3' 

that is, the sum of the given series is equal to the sum of the two series 

whose general terms are - and , minus the sum of the two 

^ n n + 3 

series whose general terms are and 



n + 1 n + 2 

Therefore 



V3 4 n M + 1 n + 2y 

+ n + ...i+_i_+_i_+^L_\ 

\4 n»+ln + 2n + 8; 

= 1_1 1 I 1 ^2 2 ^^ 

3 n+ln + 3 3 (n+l)(n + 3)' 

When n = 1, we have 8i = ^^j. 
When n = 2, we have 8^ = A* 
When n = 10, we have Sia = }f J. 
When n = 1000, we have Smi = ISMXS?- 
When n = «, we have 8^ = J. 



278 TABLE BOOK AND TEST PROBLEMS. 

8SS. Let X and y = the extremes of the proportion, and z and to = the 
means, or vice versa. We shall then have 

xy = zw (1) 

a;4-y + 2f + to = a (2) 

aj2 + y* + «* + w^ = 6 (3) 

a5» + y» + «« + t«8 = c (4) 

From (2), « + 10-= a — (a + y) (6) 

(6)3 = «2 -f. 2zw 4- K^ = o* - 2a(a + y)+ a;^ + 2a^ + j^ (6) 

Suppressing 2 zu' in the first member of (6) and its eqoal 2 x^ in the 
second, and then adding x^ + y^ to both members, we have 

a2 + j^ + «^ + «^ = & = a^-2a(x + y)4-2x2 + 2y2 (7) 

From (7), 2 a(x + y) = a2 - 6 + 2(x2 + y2) (g) 

(6)« = «« + 32rw(« + to)+ «7^ = a'- 3o2(x + y)^. 8a(x + yy 

-x8-3xy(x4-y)-y' (9) 

Transposing terms in (9), and observing that Zzw = ^ xy, we have 

x8+y'+««+tr«+3xy(x+y+«4-io) = a*-3a2(x+y)-}-3rt(x+y)*; 

that is, c 4- 3axy = a^-Sa%x+y)+Sa{x'^-\-y^)+Qaxy, 

or c-8axs( = a«-3o2(x + y)4-8a(x« + y2) (lo) 

(8)x5i! = 3a«(x + y) = ?^--^^ + 3a(x« + y«) (H) 

ai)-(10) = ^'~^^^ = 3flxy-c (12) 

From (12), *y or 2w = <f*-\«b + 2e ^^g^ 

4(13) + (8) = 2 a(x + y) + <(«' - « «» + 2 '') 

oa 

= a2 - 6 + 2(x + yy (14) 

Resolving (14) as a quadratic, we obtain 

x+y and z-\-w=ia ± l^^?^^i:M±«!.j (15) and (16) 

From (13) and (16), and from (13) and (16) we easily find x and y, and 
also z and w 

, .1 //8c-6a6+o«\^l /r 9q&-2a^-4c . ^^ // 8c-6a6+fl» M 

=*^=^4Vl — 3^ — J^WL — Q-a — ^*Nl — 3;^i IT 

Ans, 

The sign before the first and third radical must be taken as positive for 
determining x and y, and as negative for determining z and to, or vice 



SOLUTIONS TO 8EBIES. 279 

versa. If we take a = 21, b = 126, and c = 819, we shall find by the 
formula that x = 8, ^ = 3, z = 6^ and w = 4; and the proportion is 
8 : 6 : : 4 : 3. 

888. We have 

a+(a+d) + (a+2d)+ — H-[a4-(n-l)<?]=}n[2a+(»-l)<2]=« (1) 
a^-^-Ca-h d)2 4- (a 4- 2d[)2 4- — + [a + (n - l)d]^ 

= [by expanding the terms, and summing those that are similar] 
na'2 + n(n - l)ad + i n(n - 1)(2 n - l)d^ = b (2) 

a' + (o + <0*+(a + 2d)8 4- — + [a+(»-l)^* 

= [by expanding the terms, and summing those that are similar] 
?io84.1«(»-l)a-d+}n(n-l)(2n-l)adHi»2(n_l)2d3=c (3) 

Combining (1) and (2), we find that 

^ = ±2 j/3n6-_3^\ (4) 

and a^i. I ir^Znb-Ss^(n^l)-i ^^^ 

n « \L »+ 1 J 

Substituting in (3) the values found for d and a, we obtain by develop- 
ing the quadratic equation, 

cii^-3s6»=-2s» (6) 

Therefore, by resolution, n = — [3 6 — y/i^b'^ — 8«c)] = [by substitut- 

ing the numerical values of the letters] 12. By substituting the numer- 
ical value found for n in (4) and (6), we find that d= ±11, and a = 35 
and 166, the first and last terms of the series. By taking a = 36, and 
d= 11, the series will be ascending ; but if we take a = 156, and d = — 11, 
the series will be descending. 

884. We have 

a + ar-\-ar'^ + — ar^-^ = ^^^ " ^^ = g (1) 

r — 1 

a« + aV +a-rt + ... aV^*-^ = ^'^^ "" ^) = 6 (2) 

fi _ 1 

a8 + o3,^ 4- aSr'J 4- - ah^"^ = ^lO^Liiil = c (3) 

r* — 1 

(2)^(1)= a(!!L±l} = 6 (4) 

r+l« 



280 TABLE BOOK AND TEST PROBLEMS, 

Combining (1) and (4), we find that 

28 

and r" = ^^-Cf-J^D (6) 

(3) -^ (1) = a^[t^(r^+ 1)4-1] ^ c ^ J 

Substituting in (7) the values found for a and r^, we obtain, by develop- 
ing the quadratic equation, 

(a* _ 4«; -f 362)r2 ^ 2(«» + 2«c ^ 362)r = -(«* - 4«c + 3&2) (8) 

Therefore, by resolution, 

r = {«♦ H- 2«c - 363 ± ^[I2a(«c ^ 62)(sS _ c)]}-^(«* - 4»c + 3 62) 

= [by substituting the numerical values of the letters] 1} or }. 

The former value gives an ascending series, and the latter a descend- 
ing one. By substituting the values found for r in (5), we find that 
a = 64 and 729, the first and last terms of the series. We finally have 

_ log [q -f (r - l)g] -- log a _ y . 
logr 

and the series is 64, 96, 144, 216, 324, 486, 729. 

386. The series is 55, 1079, 4599, 12151, etc. 

Fttst order of differences = 1024, 3520, 7562, etc. 

Second order of differences = 2496, 4032, etc. 

Third order of differences = 1536, etc. 

The first terms of the several orders of differences are D^ = 1024, 
D^ = 2496, 2>3 = 1536, and the first term of the series = a = 55. Sub- 
stituting these values in the formula 

1-2 ^ 1.2.3 ^ 1.2.3.4 ' ' 

we obtain, by reduction, « = 64 n* + 32 n' — 32 n^ — 9 n. 

Factoring, « = 16(2 n* + } n)^ - 18(2 n^-\-in) (1) 

Solving for s = 22395834549559, 

« , o *• . 1 18929704 
we find 2 n^ H- * n = — » 

^ 16 

whence n = 769. 



SOLUTIONS TO SERIES. 



To make the solution general, consider n in (1) as the unknown qixs^,: 
' tity, and resolve as a double quadratic. This gives 

« = }(Vl9 + 2Vl6s + 81 - 1) 
= [when 8 = 65, 1134, and 22396834549569 respectively] 1, 2, and 

886. First Solution, (a) Let /a„ = the nth term of the series 

Therefore ^in - /*n-i = - KA*n-i - Mn-2)i 

whence /«, — ^tj, /Uj — /«,, fi^ — fi^^ etc., is a geometrical series virith tJi^ 
common ratio — J. The first term of the series, or /i, — /*,,= 6 — cr^ 
where a = 1 (the first term of the series), and 6 = 19 (the second). Tlje 
last term of the series, or 

Mn -/*»-! = (ft - a)(- l)«-2. 
Whence, by addition, 
A^ - /ii = (6 - a)[l + (- i) + (- \Y +... + (- i)-2] 

= [by summing the series in brackets, etc.] ^ ~^^ [1 — (—})••"*]. 

Therefore 

a) 



2(& 



[l-(-i)"-^] + « 



= [when a = 1, and 6 = 19] 13 - 12(- J)"-^ Ans, 



(1) 



(6) Giving to n in the value found for the general term all possible 
Integral values from 1 to n, we obtain two distinct geometrical progres- 
sions : one for the odd powers of — J, and the other for the even powers 
of — j^. Summing these two series and adding them, then substituting 
the result in formula (1), after multiplying the last term in it by n, we 
obtain for the required sum Sn of the given series, by simplification, 

Bn =Jn(26 + a)- K& - «)[! "(- J)"] 

= [when a = 1, and 6 = 19] 13 n - 8 + 8(- J)» (2) 

We find by formula (2), when 



n = 1, «n = 1 ] 
n = 2, Sj = 20 ' 



^ n = 3, «3 = 30 J 



n = 4, 8^ =44J 
w = 7, 8, = 82} J 
n = 12, 8,, = 148^3r J 

» = 500, «5oo = ^^2 + 1 H- 26187124863169134960105517674620798217733 
1363683446183158663309447690703712373964390661607386072332572070934 
78020480568073738052367083144426628220715008, exactly. 



282 TABLE BOOK AND TEST PROBLEMS. 

Second Solution. This is a recurring series of the second order, whose 
scale of relation is w, n(= J), and is equal to two geometrical recurring 
series of the second order, in each of which the scale of relation is m, n. 
Let X and y be the first terms of these series. We then have • 

x-^xB-h xB^ -f xB^ 4- xB^ — xB''-\ 

and y 4. yJBj + y^i 4. y^a ^ y^^* ... yBj^-K 

Adding, we have 

(« + y) + (xB + yB^) + ... (a;22"-i + y2?i"-i), 
a recurring series of the second order, whose scale of relation is m, n. 
We now have a; 4- y = 1» and xB + yBi = 19, 

whence x = l?^l^, and 2/ = ^-^=^' 

B — jffj B — Bi 

In a geometrical series that is also a recurring series of the second 
order, as x-{- xr-\- xr^ + xr^ + etc. , we have 



xr'^ = mx + nxr, whence r = i(n± V-^m + w-*). 

In our problem, w = n = J. Hence r = 1 = 5, and ~ J = i?i. 
Using these values in expressions for x and y, we have 

a; = linizii} = 13, and y = -l=ili- = _ 12. 

« 
The nth or general term is 

xB^-^ + yB^^-^ = 13 ^-1 - 12 Bj^-^ = 13 - 12(- i)*»-i. 

387. (a) Put a = 4, and 6 = 64, the first two terms of the series 
respectively. Let fj^ = the nth term of the series ; then will 

/*« = V/i„-i/i„_2, and log fin = }(log/A«_i + log/Xn-2). 

Hence, with logarithms, the problem becomes exactly similar to the 
preceding one, and we have 

log /*„ = log a + I (log 6 - log a) [1 - ( - J)--i]. 

Whence, by reverting again from logarithmic functions to natural ones, 
we have 

^ = \a) 

= [when a = 4, and & = 64] 4(16)*t^-(- i>" ^\ Ans. (1) 

(5) Giving to n in the value found for the general term all possible 
integral values from 1 to n, and taking their continued product by 
adding the resulting exponents, etc. , we obtain 



SOLUTIONS TO SESLES. 283 



= [when a = 4, and 6 = 64] 4»«(16)H3 « - 2 - ( - i)» -i] ^^^ 

We find, by formula (2), when 



n = 1, Pi = 4 
w = 2, Pj = 256 
n = 3, P, = 4096 . 



n = 4, P^ = 131072 
n = 7, Pt =47(16)*A 
» = 12, Pi, = 412(16) ^^"Sc 



388. Expanding the first member of the equation into a series by 
division, we have. 

4x - 9x2 + 30a;8 - 96 x* '+ 298 oS — 941 «' + etc. = J = - (1) 

a 

The law of the coefficients is, that they are alternately plus and minus ; 
and that any one after the second is equal to the arithmetical sum of 
twice the preceding coefficient, plus three times the next preceding, 
plus twice the next. Eeverting the series, we have 

^,19 42 235 6058 247506 ^^^ 

4 a 64a2 1024 «» 16384 a* 262144 a^ 4194304 a^ 

H o = 10, then x = .026363628+ = ^ 



1100 

389. We have 

/S'„ = 1 + 16aj + 63aj2 + I60x8 + 325 x* + 576x6 

+ ... (n - 2)2(3 n - 8)x«-3 + (n - 1)2(3 a - 5)x«-2 + n2(Sn - 2)x"-i (1) 

As there' are three orders of differences in the coefficients of this series, 
we shall obtain, if we multiply (1) by (1 ~ x)^, 

/^A(l-«)'=l + 13x+18ar2(l+x+x2+xH — «"-*)- (3n8+7n2-f5n-17)x« 

+ (6 w2 + 5 n2 - 13 n + 5)x«+i - (3 w^- 2 n2)x«+2 (2) 

Summing the geometrical series in the second member of (2), and 
resolving, we obtain 

/S'„ = [l + 12x+6x2-(3n8+7»2-|.5n+l)x«+(9n8+12n2-8n-12)x»+i 
- (9 »8 + 3 n2 - 13 » + 5)x»+2 + (3 »» - 2 n2)x«+8] -^ (1 - x)* 
= [when X = 6, and n = 100] ^«53907 fflioo + 93 

= 58334974030254270215112384630875216003412751797352342464364483 
02119970321656. 

When X = .999, and n = oo, our general formula becomes 

g ^ 5x2 + 12x4-1 ^ 17998Q05000. 
(1 - x)* 




284 TABLE BOOK AND TEST PROBLEMS. 

890. Annual interest is simple interest on the principal, and on each 
year's interest after it is due. Put P=$400, /= $441.60 - f 400 = 
$41.50, and n = 4 years. Let r = the required rate. The simple interest 
on the principal for n years is Pm. The simple interest on one year's 
interest of the principal for n — 1 years is Pr^{n — 1); for n — 2 years, 
Pr^(n — 2), etc. ; the interest on the last year's interest being simply JV*. 
Hence we have 

7= Pm + Pr^l(n - l) + (n - 2)+ ... 1] 
= [by summing the series within brackets, etc.] Pr[» + J nV — J nr^ (1) 
Resolving (1) for r, we get 

r = [ V(^^»^ + 2 Pln^ - 2P/n) - Pn] -r-P*i(n - 1) = .025. 
Therefore the required rate is 2} per cent. 

891. Put a = $ 1000, r = .06, n = 10 years, and t = 20 years. Let P = 
the present value of the annuity a as a perpetuity ; Pj = the present value 
of the same for n years ; P^ = the present value of the same for t years, 
in reversion n years; and P, = the present value of the perpetuity 
deferred (n + t) years. 

We evidently have P = — The amount of an annuity of fa for n 

years and rate r is a -f a(l + r) + a(l + r)^ H a(l + r)"-i. The 

amount of its present value Pj for the same time is P|(l + r)». 

Therefore, a + a(l + r) + a(l + r)^ + ... a(l + r)«-i = Pi(l + r)* (1) 

Summing the geometrical series in the first member of (1), and resolv- 
ing, we get 

Pi=-fl ?— -^ = $7360.085, the son's bequest. 

Similarly and evidently, 

P^ = '(l —^ — ] - P, = (l \ = $6404.746, 

r\ (l + r)«+V * r(l + r)"V (1 + OV 

the daughter's bequest. 
Finally, 

Pj =P - Pa - Pi = = $2901.835, the institution's bequest. 

r(l + r)*+« 

892. Let m = the number of balls in length of the rth course, and n = 
the number in width of same. We shall have 

mn = b, (1); and (m + r — g)(n + r — 5)= c, (2). 

From (1) and (2) we obtain, by resolution, 

w and n = {c - 6 -(r - g)2 ± ^[(c - 6)a - 2(c + 6)(r - g)2-f (r - g)*]} 

-^ 2(r-g)=8and4. 



SOLUTIONS TO SERIES. 285 

Therefore the namber of balls in the length and breadth of the bottom 
coarse in the pile is m + r — 1 and n + r — 1 respectively ; whence, by 
adding the number of bails in the several courses, we have the following 
series for the number in the pile : — 

(m + r - 1) (n + r - 1) + (w + r - 2) (n + r - 2) 

+ (w + r-3)(n-f r~3)+ ... (m - n + 2; x 2H-(w-n + l)x 1, 

in which n + r — 1 is the number of terms. Developing the terms in 
this series and summing those that are serial, we obtain for «, the num- 
ber of balls in the pile, 

s=z J(n+r)(n + r- l)(3m-n + 2r- 1) 
= [when n = 4, w = 8, and r = 13] 2040 balls, Ans, 

398. Keeping trace of the pairs of doves produced at the end of the 
first, second, third, fourth, etc., years, we easily perceive that the num- 
bers would have been 0, 2, 3, 4, 14, 20, 44, 96, etc., respectively. We 
therefore have to sum this series to n = 100 terms, and add to the result 
the parent pair, to find the number of pairs /^n + 1 required in the 
problem. 

Making x + y + z the scale of relation in the series, we have 

3a; + 2y = 4 (1) 

ix-{-Sy-^2z = U (2) 

14a; + 42^ + 3;? = 20 (3) 

Resolving (1), (2), and (3), we find that 

a; = 0, y = 2, and « = 4. 

Assume that the series is equal to the sum of the following three geo- 
metrical series : 

a-^ ar + ai^-{- etc. (4) 

b-\-bp + bp^+ etc. (6) 

c + eg + cg2 4- etc. (6) 

that is, to 

(a+6+c) + (ar+6p+cg) + (ar2-|-6p2^cg2) + (ar«+&pHc^)+etc. (7) 
The scale of relation in (7) , being 

(r + p + q)- (rp -h rq -^ pq)-^ rpq, 
must be the same as that of the original series. Hence we have 

r-fi) + 5 = (8) 

rp + rg+p5 = -2 (9) 

and rpg = 4 (10) 



286 



TABLE BOOK AJUD TEST PROBLEMS. 



Therefore the yalues of r, p, and q are the roots of the cubic equation 

X«-2a; = 4 (11) 

thatis, r = 2, p = -(H-v^^), and 5=-(l-V^). 
Hence, to make (7) identical with the original series, we must have 

a + ft + c = (12) 

(13) 

(14) 



2a-(l-\/^)6-(l+V3T)c = 2 
4 o + (1 -. >/31)25 ^. (1 4. v:ri)2c = 3 
Resolving (12), (13), and (14), we find that 

a = -ft, 6=-(7+\/^^)-20, and c = - (7 - V^=T) - 20. 

Substituting the numerical values found for a, 6, c, r, p, and q in (4) 
(6), and (6), we obtain 

i5„ +1 = 1+ (0 + 2 + 3 + 4+ 14 + 20 + 44 + 96 + etc.) 

^^ [1 + 2 + 4 + 8 + 16 + 32 + 64 + 128 + 266 + etc.] 

- [(7 + v^Ti)^ 20][i -(1 + v^)+(i +>/:n)2 

= 1 + ^ - (1 + V^^)« + etc.] 

[(7 _>/i:i)^20][l -(1 - V^) + (l - V^)a 

_ (1 _ y/~iy + etc.] 

= [by summing the three geometrical series within brackets to n 
terms, and multiplying each by their respective coefficients, etc.] 

7+v:ri ^^ (-i)"(i+-v/:n)'*-i 



1 + M^n _ 1) 



20 



2-V-l 



7 -\/^ ^ (- 1)''(1 - V^)«« - 1 

20 _2 + v:n 

= [by simplification when n = 100, observing that 

(- 1)100(1 +V=n[)100=(- 1)100(1 - V^T)100 = _ (2)50] 

7 (2)100 - 3 (2)60 
10 
= 887356420169760243277720190976 pairs of doves, Ans. 

12 92 32 

894. The series — I 1 1- ••• is summed as follow^: — 

[2 (3 [4 

The general term of the series is 

n^ ^ n(n + l)-Cn + l)+l ^ 1 "^ | ^ 

|n — 1 \n 



n+ 1 



n+ 1 



n+ 1 



SOLUTIONS TO SERIES. 287 

Hence, decomposing each term in the series, beginning with the second, 
into three parts by the aid of the formula obtained from the general term, 
we have 

= [by the exponential theorem] 
.i + c-l-(6-2)+c-2- J = c-1 (1) 

QS 4,2 Rl 

The series — H 1 — is summed as follows : — 

Li ^ li 

Let 8 = the sum of the series ; then, subtracting series (1) from this 
series, term by term, we obtain 

. r^ ^\ 8, 12, 16, _4.4,4,4, 

l^ ii ii [2 [3 [4 

= [by the exponential theorem] 4 e — 4. 

Therefore « = 5? + ,^- + — + ... = 56 - 6' (2) 

[2 [3 [4 ^ ^ 

Adding (1) and (2), we have 

12 ■!. 32 2^ + 48 . 3^ + 52 . ft. ,. .^ ,QN 

— — - + — x_ + __r — +... = 6(«-l), Ans. (3) 

[2 [3 |4 

In formula (3), e is the base of the Napierian system of logarithms, and 
has been computed accurately to 200 decimal places. Hence we are able 
to give the summation of the infinite series in this problem correct to that 
number of places, which will be 6 (e - 1)= 10.309690970754271412161724 
8281169749865434826621997574498018057663444697821212865674282930711 
6099866456479836169201836963090448167977426143740200577166367378442 
879397176666094467940297928448924246038296 + .