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NUMERICAL PROBLEMS
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NUMEEIOAL PEOBLEMS
IN
PLANE GEOMETRY
NUMERICAL PROBLEMS
m
PLANE GEOMETRY
METRIC AND L0GAE1TH3IIC TABLES
BY
J. G. ESTILL
OF THE HOTCHKISS SCHOOL, T.AKEVILLE, CONN.
NEW YORK
LONGMANS, GEEEN, AND CO.
LONDON AND BOMBAY
1897
Copyright, 1896
BY
LONGMANS, GKEEN, Ax\D CO.
MANHATTAN PRESS
474 W. BROADWAY
NEW YORK
PREFATORY NOTE
When^ arithmetic was dropped from the requirements
for admission to Yale College, in 1894, the following sub-
stitute was adopted : '^ Plane Geometry (b) — Solution of
numerical problems involving the metric system and the
use of Logarithms, also as much of the theory of Loga-
rithms as is necessary to explain their use in simple arith-
metical operations. — Five-figure tables will be used in the
examination/' (1896-97 Catalogue.)
At the conference on uniform requirements for admis-
sion to college, in February, 1896, at Columbia College,
representing Harvard, Yale, Princeton, University of
Pennsylvania, Columbia, and Cornell, and nearly all the
large preparatory schools of the East, the Mathematical
Conference voted unanimously to recommend that arith-
metic be dropped from the college entrance requirements,
and that a knowledge of the metric system and the abil-
ity to solve numerical problems in Plane Geometry be
required.
These two facts account for the writing of this little
book.
The most of the problems have had class-room test.
They add interest to the study of formal geometry. They
are helpful, too, in making clear, and fastening in the
memory, the principles and propositions of formal geom-
etry. They enforce the practical application of truths
183647.
VI PREFATOBT NOTE.
which boys are apt to think have no application. They
furnish a drill that is just as valuable to those who are not
preparing for college as for those who are. These prob-
lems are not to take the place of other geometries, but are
to be used with them. And, therefore, the division into
Books is made to correspond pretty closely with that of the
geometries in most general use.
The use of the metric system is begun at the very first,
simple as that necessarily makes the problems of the first
book, for the most part. No other book contains a graded
set of problems on the first two books of geometry.
No apology is considered necessary for putting in quite
a number of problems which presuppose some knowledge
of algebra.
The order of the problems is not the same as the order
of the propositions of any geometry ; neither are all the
problems which illustrate an important principle placed
together. The reason for this is obvious. Still, the order
of the problems in the different books is approximately the
same as the order of the propositions in the most popular
text-books. On account of this difference in order it will
be best to keep the text-book work somewhat ahead, unless
one cares to select the problems beforehand to give out
with the text-book lesson. Some may prefer to use the'
problems only with the review of the geometry.
Boys preparing for college will certainly take a lively in-
terest in the questions, problems, and exercises selected
from the college entrance papers.
The entrance papers were selected with great care, with
the hope that they may prove helpfully suggestive both to
teachers and pupils.
The discussion of logarithms, the explanation of their
use, and the use of the table have been made as simple
and clear as possible.
PREFATORY NOTE. idi
Only such symbols are used as are almost universally
employed.
Some few proofs are put in because they are not found
in all the text-books.
Notice of errors, or any suggestion, will be gratefully re-
ceived.
J. G. ESTILL.
HoTCHKiss School,
Lakeyille, Conn., January 8, 1897.
CONTENTS
PAGB
Preface v
Book I . „ 1
Book II . 11
Book III 17
Book IV. 29
Book V 40
Numerical Problems, Exercises, Etc.,
Selected from Entrance Examina-
tion Papers 50
College Examination Papers in Plane
Geometry 66
Logarithms 102
Examples Ill
Tables 115
NUMERICAL PROBLEMS
IN
PLANE GEOMETRY
BOOK I.
1. What is the complement of 43° ? of 75° 15' ? of 81°
11' ir ? of 14° 18" ? of ii [^ ? of m° n' ? of 82° 40' - .4
2. What is the supplement of 28° 31' 18" ? of 115° 39" ?
of 140° 1.84" ? of 1.2 L^ ? of tV Ll ? of c°- f" ?
3. Find the supplement of the complement of 50° ; of
85° 13' 22" ; of a;° ; of ^5° - 31° 18'.
4. Find the complement of the supplement of 169° 44'
42" ; of 155° 55" ; of ^° - 15° ; of c° - 8° 5".
5. How many degrees in the difference between the
supplement and the complement of an / ?
6. How many degrees in each of the A made by two
intersecting straight lines, when one of the A lacks only
2° of being } of J of a [^ ?
7. In this figure ,_\y3, ^ = /, = ^ + 12° ; how
many degrees in each /_ ?
2 OEOMETBY— NUMERICAL PROBLEMS,
8. Of two supplementary-adjacent A , one lacks 7° of
being ten times as large as the other ; how many degrees
in each ?
9. Two complementary A are such that if 7° be added
to one and 8° to the other they will be in the ratio of
3 to 4?
10. If an / divided by its supplement gives a quotient
of 5 and a remainder of 6°, how many degrees in the / ?
11. How many degrees in each of the five A about a
point, if each, in a circuit from right to left is 5° greater
than its adjacent / ?
12. Three A make up all the angular magnitude about
a point. The difference between the first and second is
10° ; the difference between the second and third is 100° ;
how many degrees in each ?
13. When the A formed by one straight line meeting
another are in the ratio 7 : 11 how many degrees in each ?
14. Find the / whose complement and supplement are
in the ratio 4 : 13.
15. Find the / the sum of whose supplement and com-
plement is 15° less than four times its complement.
16. How many degrees in the / whose supplement
taken from three times its complement leaves 1° 18' less
than the difference between the / and 50° ?
17. If the bisector of one of two supplementary-adjacent
A makes an / equal to one-sixth of the other, how many
degrees in each of the A ?
18. How many degrees in each of the five A about a
point if they are in the ratio 1:2:3:4:5?
GEOMETRY— NUMERICAL PROBLEMS. 3
19. What answer to 18 if the ratio is 3 : 3 : 7 : 11 : 13 ?
20. If the complement of the / A is three and one-half
times as large as A, what part of 7 |_^ is the / A ?
21. Find the / whose supplement increased by 26° will
be three times its complement.
22. How many degrees in the / whose supplement and
complement added together make 144° ?
23. How many degrees in the / whose supplement, in-
creased by 9°, is to its complement, decreased by 1°, as 7
to 2?
24. Find the number of degrees in each of these A, a c/
if h is 2° less than f of « ; c is f (o^ + ^ — 1°) ; ^ is ^t^^^
13° less than the sum of a, h, and c ; and e is 2°
more than the difference between the sum of h and d, and
the sum of a and c.
25. How many degrees in the / whose complement is
one-fifth its supplement ?
26. How many degrees in the / whose supplement, in-
creased by 20°, divided by its complement, decreased by
5°, gives a quotient 4 and a remainder 25° ?
27. If a J_ is 1 foot 10 inches from one end of a line
and SS*"" from the other, at what point of the line is
this 1 ?
28. Of two lines from the same point to the same
straight line, one is 1 yard 1 foot 4 inches, the other is
130*"°, what can you say of them ?
29. Two lines from a point to the extremities of a
straight line are 15 feet 4 inches, and 11 feet 11 inches.
V
4 GEOMETRY— NUMERIGAL PROBLEMS.
respectively. Two similarly drawn are 4™ ^^"^ and 3.2™.
Which pair includes the other ? Why ?
30. Of two oblique lines from a point to a straight line
one is 3 feet 10.8 inches, the other, 1™ 1*^'° 7<^™ ; which cuts
off the greater distance from the foot of the perpendicular
from the point to the straight line.
31. What answer to 30, if the lines are 35 feet and 1^°*,
respectively ?
32. If the bisector of one of two supplementary-adjacent
A makes with their common side an / = | |_r_ lacking
6°, how many degrees in the other / ?
33. Of two lines from a point to a straight line, one is
30*"° and the other is 11 inches, which is a _L, if either is ?
Why?
34. Which is the greater of two oblique lines from a point
to a straight line, cutting off, the one 20 yards, the other
15", from the foot of the 1 from the point to the line ?
35. Answer the same when the distances cut off are 1"
^dm 5cm and 5 feet 10 inches.
36. In the A A B C and A' B' C, « = 3 feet, 5 = 7
feet, c = 8 feet, / A = /_M, V =1 feet, c' = 8 feet.
Find the length of a' in centimetres.*
37. In the A A B C, « = 4", J = 5™, c = 7°* ; find in
feet (approximately) the sides of a A eq^al to the /\
ABO.
38. One side of a A is 1" 5^™, another 7 feet 5 inches.
What is the greatest value the third side can have (1) in
metric units, (2) in English units ? What is the least ?
a, 6, c, represent the Sides of a A opposite the A A, B, C, re-
spectively.
QEOMETRT— NUMERICAL PROBLEMS. 5
39. Find the ^ of the /\ A B 0, when A is 43° more
than I of B, which is 18° less than 4 times 0.
40. In the two A A B and A' B' C, A = 37°, B =
111°, c = 2.5 feet, A' = 111°, B' = 37°, d = 7'^" S^-".
What can you say of them ? Why ?
41. In the A A B 0, « = 13 feet, ^> = 17.3 feet, and c =
22. 4 feet, find in metres (approximately) the sides of a A =
the A A B C. (Log.*)
42. One of the acute zi of a right A = 37° and the hy-
potenuse is 1.5 miles, how many kilometres in the hypo-
tenuse of an equal right A which has an acute / of 37° ?
43. In the A A B C, « = 11^^ h = 32'^'", what is the
least possible value in miles of the side c ?
44. If in two A A B C and A' B' C, a = 1^ 5"^", b =
1" 2'^'° S'^'", C = 48°, a' = 3 feet 6 inches, Z»' = 4 feet
2 inches, C = 148°, what can you say of c and (/ ? Show
by your work how you reached your conclusion.
What would your answer be if all the given values were
the same except C = 48° ? Why ?
45. If in two A A B C and A' B' C, « = 7 miles, b =
13 miles, c - 15 miles, a' = lli''^ b' = 21^'^ c' = 24^"",
what about the A B and B' ? If ^>' = 201^^ what of these
A?
46. In the A A B 0, a = 1.3 miles and b = 2^°^, what
of the A A and B ? It a were the same and b = 2.08^"*,
what could you say of the ^ A and B ?
47. The ^ A and B in the A A B C are each 49° 18'
Certain problems m each book are marked thus for those who care
for practice in the use of logarithms.
6 OEOMETRT— NUMERICAL PROBLEMS.
and a — 109 yards 1 foot 1 inch, how many metres in the
side b ? (Log.)
48. If one of the A made by a line cutting two |I lines
is 3° more than y\ |j^ , how many degrees in each of the
other A ? (Mark your answers on a figure. )
49. What answer to 48 if one of the A is eight times
its conjugate / ?
50. If the exterior / at A of the A ^ B C is 115°, and
/ C is three times / B, find B and C.
51. The exterior A at A and of the A ^ B C are 71°
and 92° respectively ; how many degrees in the / B ?
52. In the A A B C, A lacks 106° of being equal to the
sum of B and C, and C lacks 10° of being equal to the
sum of A and B ; find A, B, and C.
53. Find the ^ of a /\ which are in the ratio 3:4:5.
54. Find the A of an isosceles /\ in which the exterior
/ at the vertex is 125°.
55. Find the A of an isosceles /^ in which the exterior
/ at the base is 95°.
56. Find the perimeter of an isosceles /\, in miles, if a
base of 48'^'° is the longest side of the /\ by 12^"". (Log.)
57. In the A A B 0, a = 15 yards and h = 1^^ 2"", what
about the A A and B ?
58. The point P in the bisector of the angle ^ <^ V —
is 5 yards 2 feet from the side 1—2-, how many metres is
P from 2-3 ?
GEOMETRY— NUMERICAL PROBLEMS. 7
59. The point P within an / is 6 <^™ 6^^ from one side
of the / and 2 feet 2 inches from the other side, where
does it lie ? Show the reason for your answer by your
work.
60. The / at the vertex of an isosceles ^ is one-third
the exterior angle at the vertex, how many degrees in each
/ , exterior and interior, at the base ?
61. In the A A B C, A = 35°, B = 45°, a = J mile ;
what can you say of the length of h, in metres ?
62. Two adjacent sides of a / / are respectively 18" and
21™ ; find the lengths of the other two sides in yards. (Log. )
63. The area of one of the A made by the diagonal of a
/ / is 5.2"*. How many acres in the other ?
64. If one / of a / / = J L^ , how many degrees in
each of the other A ?
65. If two adjacent zi of a / / are in the ratio of
17 : 1, how many degrees are there in each / of the
66. How many degrees in each / of a / / where one
/ exceeds one-third of its adjacent / by two-thirds of a
degree ?
67. How many degrees in each / of an equiangular
icosagon ? in each exterior / ?
68. How many sides has the polygon each of whose ex-
terior ^ = 12** ?
69. How many sides to the polygon each of whose ex-
terior A is only one-eleventh of its adjacent interior / ?
70. One side of a rhombus is 13.6/° find its perimeter
in miles. (Log.)
8 OEOMETRT—NUMERIGAL PROBLEMS.
71. One side of a rhomboid is 4 feet longer than the
other, the perimeter is 14", what are the lengths of the
sides in feet and inches ? (Log.)
72. Find the number of acres in a rhombus in which
one of the four A made by the diagonals contains 5.11"*.
(Log.)
73. 'Find the A of an isosceles l\ when one of the A at
the base is equal to one-half the / at the vertex.
74. What answer to 73, when the / at the vertex is 9°
greater than an / at the base ?
75. What are the A of an isosceles /\ in which the /
at the vertex is 12** more than one-third the sum of the
base A't ^
76. The sides of a quadrilateral taken in order are 6
inches, 18^™, 15«", 7^ inches, respectively. What is the
nature of this quadrilateral ?
77. How many sides has the polygon each of whose in-
terior A = 171° ?
78. The line joining the middle points of two sides of a
/\ is 2.5 miles, what is the length of the third side in kilo-
metres ?
79. How many sides has the polygon the sum of whose
interior A exceeds the sum of its exterior A by 3240° ?
80. One of the diagonals of a rectangle is 40 yards 2 feet
10 inches ; find the length of the other in metres. (Log.)
81. One base of a trapezoid is 125°™, the line joining the
middle points of the non-parallel sides . 7™, find the length
of the other base.
82. How many sides has the equiangular polygon each
of whose interior A exceeds its adjacent exterior by 108° ?
QEOMETBT— NUMERICAL PROBLEMS. 9
83. How many sides has the polygon the sum of whose
interior A is double the sum of the exterior A ?
84. The line joining the middle points of the non-paral-
lel sides of a trapezoid is 13 feet 5 inches, and one of the
bases is 2J times as long as the other ; find the length of
the bases.
85. Find the length in metres of the line which bisects
one side of a /\ and is parallel to a side whose length is 9
feet 10.11 inches.
86. If you should join the extremities of two parallel
lines whose lengths are 7^" and 4.375 miles respectively,
what kind of a figure would be formed ? Why ?
87. How many sides has the polygon the sum of whose
^ is 4J times those of a hexagon ?
88. Find in inches the bases of a trapezoid in which the
line joining the middle points of the non-parallel sides =
40°™ and one base is S^'™ longer than the other.
89. How many sides has the polygon the sum of whose
interior A exceeds the sum of its exterior A by 38 |^ ?
90. One base of a trapezoid is 5.1™, the line joining the
middle points of the non-parallel sides is 2^ times the other
base ; find the other base.
91. How many sides has the polygon each of whose in-
terior A exceeds its exterior / by f | [^ ?
92. How many sides has the polygon each of whose in-
terior ^ is 6 times its exterior / ?
93. Find the difference in perimeter, in inches, between
a square whose side is 1 foot 6 inches and a rectangle whose
adjacent sides are 30'='" and 60.5'=™ respectively.
10
GEOMETRY— NUMERICAL PROBLEMS,
94. Find the number of feet of lime-line of a tennis-
court, as represented below. Keduce your answer to
metres. (Log.)
1
*s
a/.ff.
^ 2/.fi.
^
^
1
^^efi
95. Through the vertices of a /\ A B 0, lines are drawn
parallel to the opposite sides of the [\ , thus forming a
second /\ . Find the perimeter of the second l\ in kilo-
metres, if the sides of the first /\ are 5 miles, 8 miles, and
11 miles.
96. How many sides has the polygon each of whose A
= 162° ?
97. The perimeter of a rectangle is 8.04°*, and the sides
are in the ratio of 1 to 1-|, find the lengths of the sides in
inches.
98. How many sides has the polygon the sum of whose
interior A exceeds the sum of its exterior A by 1080° ?
99. A man owns a rectangular garden 55" by 34" ; he
makes a path 3.3" wide around it ; what is the perimeter
of the part that remains ?
100. Find the number of yards of lime-line for a foot-
ball field, which is 330 feet by 160 feet, including all the
five-yard lines. How long would it take a runner to cover
the total distance, if he can make 110 metres in 12 seconds ?
(Log.)
OEOMETRT— NUMERICAL PROBLEMS, 11
BOOK II.
1. If the radii of two intersecting ® are 3"" and 7" re-
spectively, what is the greatest possible distance, in feet
and inches, between their centres ? The least ?
2. Four chords are a^-" 5°"" S*^"", 0.15 miles, 0.25'^", and
330 yards, respectively. If one is a diameter, which is it ?
Which of the others is nearest to the centre ? Which
farthest from it ?
3. If a central / of 28° intercepts an arc of 3.2"", find,
in feet and inches, the arc intercepted by an equal / in an
equal O.
4. What can you say of the central zi of a O which in-
tercept, and the chords which subtend, two arcs which are
respectively 28 yards and 25" ?
5. In a given O, the chord A B is 5 yards 2 feet, the
chord C G is 4.9™. Compare the arcs A B and C D, and
the distances of the chords from the centre.
6. What can you say of two chords whose distances
from the centre are 13*'" and 5 inches respectively ?
7. One of the arcs intercepted by two chords, one of
which is a diameter, intersecting at right angles, is 41°
18' 4" ; find the other arcs.
8. A secant parallel to a tangent subtends an arc of
117° 41' ; find the arcs intercepted by the secant and the
tangent.
9. One of the arcs intercepted by a diameter and a
parallel secant is 37° 30' ; find the length, in miles, of the
12 GEOMETRY— NUMERICAL PROBLEMS.
arc subtended by this secant, if a degree of the circum-
ference is 24^°*. (Log.)
10. The line joining the centres of two 0, tangent to
each other externally, is 14" 7^"^ 3*"°, and the radius of the
less is 3™ 8*^™ 5"™, find the radius of the greater,
11. If a central / of 25° 15' intercepts an arc of 15
feet 10 inches, find the length of the semi-circumference
of the O. (Log.)
12. How many degrees in an / inscribed in J of a cir-
cumference ?
13. Find the length of the arc intercepted by an in-
scribed / of 20° 22^' in a O whose circumference is -J- of
a mile. (Log.)
14. How many degrees in an inscribed / which inter-
cepts j\ of a quadrant ?
15. An / formed by a tangent and a chord is t^ |_^ ;
how many degrees in the intercepted arc ?
16. Find the length of the arc intercepted by a central /
of 12° 15' in a O whose circumference = 1^"". (Log.)
17. If a central / of 85° 40' intercepts an arc of 32.5'°,
how many degrees and minutes in the central / which in-
tercepts an arc of 65*''° ? (Log.)
18. What part of a |_^ is an / between a tangent and
a chord intercepting an arc of JJ of a semi-circumference ?
19. The / between two cliords intersecting within the
circumference is 35°, its intercepted arc is 25° 18' ; find the
arc intercepted by its vertical /_ .
QEOMETRT— NUMERICAL PROBLEMS.
13
20. Find the / between a secant and a tangent when
their intercepted arcs are respectively \ and \ of the cir-
cumference.
21. The / between two secants, intersecting without the
circumference, is 58° 41', one of the intercepted arcs is
230° ; find the other.
22. Find the / between two tangents when the inter-
cepted arcs are in the ratio 7 : 2.
Fig. 2.
23. If, in Fig. 2, the / A B C = 67°, and the arc D
is 25°, how many degrees in the / A B D ?
24. In the same figure, B G is a diameter, B is 8°
more than G ; find the / E B 0.
25. In the same figure, the arc D B is three and one-
half times the arc D C, and the / D B G = 13^° ; find
the / D B C.
26. In the same figure, if G D and B are in the ratio
3 : 7 and the / D B = 15°, how many degrees in the /
GBC?
27. In Fig. 3, Q P is 24° less than a semi-circumference,
how many degrees in the / Q M P ?
28. The / R M K is 27°, the arc R K is 100° ; how
long is the arc T L, if a quadrant on this figure = 15"* ?
F THE \
ERSITY J
( UNIVERSITY
14 GEOMETRY— NUMERICAL PROBLEMS.
29. The / R H K is 70°, the arc R Q L is three times
as long as the arc K P ; find the number of degrees in
KP.
30. The arc R P T is 10° less than two-thirds of a cir-
cumference, the / QMTis 17°; how many degrees in
QT?
31. How many degrees in the central / which inter-
cepts an arc of 17*''", when a quadrant is 4^" 2*5" 5°*™ ?
32. The / between two tangents from the same point
is 32° 30' ; find the ratio of their intercepted arcs.
33. If a central / of 65° intercepts an arc of 10 feet
5.984 inches, how many metres will there be in an arc of
the same O intercepted by a central / of 211° 15' ? (Log.)
34. The / between two tangents from the same point,
to a O whose radius is 55^'", is 120° ; how many inches in
the chord joining the points of tangency ?
35. The centres of two © which are tangent to each
other internally are 5 feet 8 inches apart, the radius of
one is l.V ; find the radius of the other.
36. The chord joining the points of tangency of two in-
tersecting tangents forms with one of them an / of 17°
7' ; find the / between the tangents.
37. The radii of two concentric © are 8 feet 2.425
inches and 2.25™, respectively; find the radius of a tan-
gent to both. (Two solutions.) Get one answer in metric
units, the other in English units.
38. The / between two chords, one of which is a dia-
meter, is ^ L^ ; find the arc subtended by the less chord.
39. Find the circumference, in metres, of a Q in which
GEOMETRY— NUMERICAL PROBLEMS, 15
a central / of 11® 15' intercepts an arc of 3.5 inches.
(Log.)
40. The / between a tangent and a secant is 8° 11',
the smaller of the intercepted arcs is 56° 50' 40" ; find the
larger.
41. In a certain O a central / of 78° 45' intercepts an
arc of 168 miles ; how long will it take a train moving 24
miles per hour to cover the circuit ?
42. Two sides of an inscribed l\ subtend ^ and ^ of
the circumference, respectively ; find the A of the /\.
43. One / of an inscribed ^ is 35°, one of its sides
subtends an arc of 113° ; find the other A of the /\.
44. The bases of a trapezoid subtend arcs of 100° and
140°, respectively ; find its A and the / made by the non-
parallel sides produced.
45. How long would it take a train running 40 miles an
hour to go round a O in which a central / of 15° inter-
cepts an arc of 7.2^" ? (Log.)
46. The numbers of degrees in the arcs subtended by
the sides of a pentagon, in order, are consecutive ; find the
A of the pentagon.
47. The arcs subtended by three consecutive sides of a
quadrilateral are 87°, 95°, 115° ; find the A of the quadri-
lateral ; the A made by the intersection of the diagonals ;
and the A made by the opposite sides of the quadrilateral,
when produced.
48. Find the / made by the radii and the line joining
the points of contact of two tangents drawn through a
16 OEOMETRT— NUMERICAL PROBLEMS.
point 6 inches from the circumference of a O of 6-inch
radius.
49. Find the A of an isosceles /\, if the arc subtended
by one of the equal sides is 33° more than 1.6 times the
arc subtended by the base.
50. An / formed by a diagonal and a base of an in-
scribed trapezoid is 20° 30' ; find the A made by the in»
tersection of the diagonals.
51. Over how many degrees of arc of a O whose circum-
ference is 435'^"' will a train, moving 60 miles per hour, go
in 15 minutes 5 seconds ? (Log.)
52. Three consecutive A of an inscribed quadrilateral
are 140° 30', 80° 30', and 29° 30' ; find the numbers of de-
grees in the arcs subtended by the four sides.
53. If it takes light 8 minutes to come from the sun
to the earth, which distance is the same as 57.3° of the
earth^s orbit, how long would it take it to go the length
of the entire orbit, supposing the orbit a O ? (Log.)
54. Three consecutive ^ of a circumscribed quadri-
lateral are 85°, 122°, 111° ; find the number of degrees in
each / of the inscribed quadrilateral made by joining {he
points of contact of the sides of the circumscribed quadri-
lateral.
55. Find the circumference of a O in which a train go-
ing 60 miles an hour goes over an arc of 1° 35' in 17
seconds. (Log.)
56. Two arcs subtended by two adjacent sides of an in-
scribed quadrilateral are 127° and 68° 30', and the / be-
tween the diagonals, which intercepts the arc of 68° 30', is
77° 30' ; find the A of the quadrilateral.
GEOMETRY— NUMERICAL PROBLEMS.
17
57. If a star makes a complete circuit of the heavens in
23 hours 56 minutes, through what arc will it go between
9.12 P.M. and 12.13 a.m. ? (Log.)
58. If the earth in revolving about the sun moves 65,500
miles per hour in its orbit, find the entire length of this
orbit, remembering that it takes 365 days 6 hours 9
minutes 9 seconds to make a complete revolution. (Log.)
59. If Jupiter is 476,000,000 miles from the sun, and
the length of its orbit is three and one-seventh times the
diameter of its orbit, and its period of revolution is 11
years, 315 days, what is its hourly motion in its orbit ?
(Log.)
60. If the earth's radius, 3,963 miles, is equal to the
length of an arc of 57' of the moon's orbit about the earth,
what is the distance to the moon, considering the orbit a
O and the circumference three and one-seventh times the
diameter ? (Log.)
BOOK III.
1. In Fig. 4, B = 52™, A = 28'", A' B' is || to A B,
C B' =r IS'" ; find C A' and A' A.
2. If, in the same figure, A' = 10 feet. A' A = 12 feet
4 inches, and B' C = 16 feet 3 inches, what is the length
of OB?
Fig. 4.
Fig. 6.
18 GEOMETRY— NUMERICAL PROBLEMS.
3. In Fig. 5, A B = 18.7™, B C = 29.4™, A C =: 40.4™,
and B D is the bisector of the /ABC; find A D and
DC. (Log.)
4. If, in Fig. 5, A D = 3 feet 5 inches, A B = 4 feet 2
inches, and B C = 7 feet, find the length of A C.
5. In Fig. 6, C D is the bisector of the / A C F,
B E = 3.3'*'", A C = e*''", B C = 4.1'*"> ; find A B in yards.
(Log.)
G. If, in Fig. 6, A C = 65 yards, A B = 48 yards, B C
= 35 yards ; find B E in metres. (Log.)
7. If, in Fig. 6, A E = 18 feet 6 inches, B C = 14 feet,
and B E = 14 feet 2 inches ; find in metres the lengths
of A C and A B. (Log.)
8. The sides of a A are a^ = 15", h= 12", c = 10" ; find
the segments into which each side is divided by the bisec-
tor of the opposite / .
9. Find the segments into which each side is divided
by the bisector of an exterior / in the preceding problem.
10. The homologous sides of two similar A are 5
feet 3 inches and 4 feet 5 inches, respectively. If the al-
titude to the given side of the first is 3 feet 9 inches, find
the homologous altitude in the second.
11. The sides of a A are 4" 6^™, 6" 1^'", and 8" ; the
homologous sides of a similar A are a, 305'", c ; find a
and c.
12. In the A A B C and A' B' C, A = 59° = A', J = 3
feet 6 inches, c = 13 feet, J' = 5.6", c' = 20.8". Show
what relation, if any, these A bear to each other.
QEOMETRT— NUMERICAL PROBLEMS. 19
13. The perimeters of two similar polygons are 88™
and 396™, respectively. One side of the first is 15 yards
4 feet 2.4 inches; find the homologous side of the sec-
ond. (Log.)
14. The sides of two A are, respectively, 4*^, 9^™, 11^",
and 1.2 miles, 2.7 miles, 3.3 miles. Show by your work
any relation which may exist between these A.
15. One of the altitudes of a /\ = 1.5™ ; find the
homologous altitude of a similar ^, if the perimeters of
the two A are respectively 15 feet and 24 feet.
16. A series of straight lines passing through the point
intercept segments, on one of two parallel lines, of 15
feet, 18 feet, 24 feet, and 32 feet, the segment of the other
parallel, corresponding to 24 feet, is 16 feet ; find the other
segments.
17. Two homologous sides of two similar polygons are
35™ and 50™, respectively. The perimeter of the second is
8"™. What is the perimeter of the first ?
18. The legs of a right /\ are 3™ and 4™ ; find, in
inches, the difference between the hypotenuse and the
greater leg. Find also the segments of the hypotenuse
made by the perpendicular from the vertex of the right
/ ; and this perpendicular itself.
19. In a O whose diameter is 16™, find the length of the
chord which is 4™ from the centre.
20. The sides of a A are 30"=™, 40<=™, and 45*=™ ; find the
projection of the shortest side upon the longest.
21. Is the l\ of 20 acute, right, or obtuse ? Which
would it be if the sides were 30*=™, 40*=™, 55'='" ? Find the
OEOMETBT—NUMEMIGAL PROBLEMS.
projection of the shortest side upon the medium side in
the latter /\.
22. A tangent to a O whose radius is 1 foot 6 inches, from
a given point without the circumference, is 2 feet ; find the
distance from the point to the centre.
23. In the A A B C, a = 14^ h = 17™, c = 22™ ; is
the / C acute, right, or obtuse ?
24. To find the altitude oidi, /\m terms of its sides.
(1) ^2 _ ^2 _ g j)2^ (The square of either leg of a right
A is equal to the square of the hypotenuse minus the
square of the other leg.)
&2 = «2 + c2 - 2a X B D
Solving for B D, B D
The square of the side opposite
the acute / of a A is equal to
the sum of the squares of the
other two sides minus twice one
of them by the projection of
the other upon it.
a2 + g2 - h"
2a
GEOMETRY— NUMERICAL PROBLEMS. 21
Substituting in (1),
~\ 2a ) \ 2a )
Ha + cf-bH r b^-{a-cf -i
~L 2a i I 2a J
_ (a + g + Z>) ja + c—b) {b + a-c) (b-a + c)
~ 2a ^ 2a '
Let 2s = a + b-\-c,
Subtracting 2c=2cy
2s—2c=2{s—c)=a + b—c.
Similarly, 2{s—a)=b-\-c—a,
and2{s—b)=a-\-c—b.
Substituting we have
2^ X 2(s—b) 2(s-c) X 2{s—a) 4.s{s-a) (s-b) (s-c).
"'- 2a ^ 2a " a'
2 .
Extracting the square root, h=- V s{s—a) {s—b){s—c),
a
2 /
Similarly, ^'''—h ys{s—a)(s—b)(s—c),
and //'=? V s(s-a)(s-b){s-c),
c
li' and li" representing the altitude of the /\ upon b and
c, respectively.
25. To find the radius of the circumscribed O in terms
of the sides of the [\.
Fig. 8.
22 GEOMETRY— NUMEBIGAL PROBLEMS.
«c=2RxBD. (Fig. 8.)
(The product of two sides of a /\ is equal to the dia-
meter of the circumscribed O multiplied by the altitude to
the third side.)
But by 24, B D:=- V s(s-a) (s-b) (s-c)
b
b
4J{
Hence ac= — Vs(s-a) (s-b) (s-c)'
T. a b c
and K =
4 y\/ s(s—a) {s—b) (s—c)
26. To find the bisectors of the A ot a, l\m terms of the
sides.
(1) a c=x^+A D X D C. (Fig. 9.)
(The product of two sides of a /\ is equal to the square
of the bisector of the included / , plus the product of the
segments of the third side made by the bisector.)
Transposing in (1), (2) x^=a c—ABxB C.
Tj , CD a
But rTT^--
DA c
(The bisector of an / of a /\ divides the opposite side
into segments proportional to the adjacent sides.)
T5 ... D C + A D a + c
By composition — ^— — — = ,
and 5^-.=^ — = or
AD c
_L=^Lh£., and -l_=^±f.
J) C a ' AD c
Whence D C=— , and A J)=—.
a+c a+c
OEOMETRT— NUMERICAL PROBLEMS. 23
Substituting in (2) we have
_ a c{c+a-\-h) (c+a—h)
{a+cf
(Substituting as in 24.)
_«cx25x2(s— Z»).
Extracting the square root.
Similarly, o(fz=ij-— V h c s (s—a)'
_2_
a + 6
and ic"=-^ V abs(s-
Note. — In a right /\ (hypotenuse c and legs a, h) the
formula a= "s/ c^—h^ and h— ^/ c^—c?y should be written
a='\/(c + ^) {c—h), and 1= '\/ {c-{-a){c—d), when loga-
rithms are to be employed.
27. The chord A B, which is 4.2™ long, divides the chord
D into segments which are 1.4" and 2.1™, respectively.
Find the segments of A B made by C D.
28. The sides of a /\ are 25 yards, 30 yards, 35 yards.
Find the length of the median * to the side of 30 yards,
and its projection upon the same.
29. Find the diameter of the O circumscribed about
the l\ two of whose sides are 3 feet 4 inches and 4 feet 6
inches, and the perpendicular to the third side from the
opposite vertex is 2 feet 3 inches.
30. Find the length of the bisector of the opposite / to
the least side in the A whose sides are 24™, 20'™, 11'™ ; the
* A median is a line from a vertex o-f a A to the middle point of the
opposite side.
24 GEOMETRY— NUMERICAL PROBLEMS.
three altitudes of the /\ ; and the radius of the circum-
scribed O. (Log.)
31. Two secants from the same point without a are
25*='" and 35'='°. If the external segment of the less is 7'^
find the external segment of the greater.
32. A secant from a given point without a O and its
external segment are 2 feet 4 inches and 7 inches^ respec-
tively ; find the length of the tangent to the O from the
same point.
33. The greatest distance of a chord of 11 feet from its
arc is 6 inches ; find the diameter of the O.
34. Two sides of a ^, inscribed in a whose radius is
15 inches, are 9 inches and 25 inches ; find the perpen-
dicular to the third side from the opposite vertex.
35. Find the greater segments of a line of 36"" when it
is divided internally and externally in extreme and mean
ratio.
36. Find a mean proportional to two lines which are
5*^" and 2"° long, respectively.
37. Find a fourth proportional to the lines a, h, c, when
^=65'=^ J=42^^ c=26*='°.
38. Find a third proportional to m and n, when m=:17^""
and ^=51^"".
39. The chords A B and C D intersect at E ; A E =
15"^", B E=46'''", C = 115'^'" ; find C E and D E.
40. Find the distance from a given point to the circum-
ference of a whose radius is 9 inches, if the tangent to
the from the given point =1 foot.
41. If, in the preceding problem, another tangent were
GEOMETRY— NUMERIOAL PROBLEMS, 25
drawn from the same point, what would be the length of
the line joining the points of contact of these two tan-
gents ?
42. The segments of a transversal made by lines passing
through a common point are 1 foot 3 inches, 1 foot 9 inches,
and 2 feet 11 inches, respectively. If the least segment of
a parallel to this transversal, intercepted by the same lines,
is 30*="", find the other segments.
43. If a gate-post 5 feet high casts a shadow 17 feet long,
how high is a house which, at the same time, casts a shadow
221 feet long ?
44. A baseball diamond is a square with 90 feet to a
side ; find the distance across from first base to third.
45. The projections of the legs of a right l\ upon the
hypotenuse are 8'"" and 9'^'" ; find the shorter leg.
46. In a O whose radius is 41 feet are two parallel
chords, one 80 feet, the other 18 feet. Find how far apart
these two chords are. (Two solutions.)
47. If a chord of 75''" subtends an arc of m° in a O
whose radius is 415'='", how long a chord will subtend an
arc of m° in a O whose radius is 33.20"" ? (Log.)
48. The sides of a A are 1,789"^, 4,231^ and 3,438"^ ;
find the three altitudes and the diameter of the circum-
scribed O. (Log.)
49. The altitude of an equilateral /\ is 45 feet, what is
the length of a side in feet and inches ?
50. Find the radius of the O in which a chord of 40.5'"
is 14.4"" from the centre. Find also the distances from
one end of this chord to the ends of the diameter perpen-
dicular to it.
26 GEOMETRY— NUMERICAL PROBLEMS.
51. The greater segments of a line divided internally in
extreme and mean ratio is 1 foot 6 inches ; find the length
of the line.
52. The projections of the legs of a right /\ upon the
hypotenuse are 27'''" and 48'^"' ; find the lengths of the legs.
53. Find the width of a street, where a ladder 95.8 feet
long will reach from a certain point in the street to a win-
dow 67.3 feet high on one side, and to one 82.5 feet high
on the other side. (Log.)
54. Find the diameter of a O in which the chord of
half the arc subtended by a chord of dO'"^ is 17*=".
55. Find the altitude of an equilateral /\ whose side=
2.2"".
56. What is the diameter of a O when the point from
which a tangent of 6 feet is drawn is 8 inches from the
circumference ?
57. The sides of a A are 185", 227'", and 242"" ; find the
three altitudes, the bisectors of the three A, and the
radius of the circumscribed O. (Log.)
58. The sides of a trapezoid are 437.3 feet, 91 feet, 291. 7
feet, and 91 feet ; find the altitude of the trapezoid and
the diagonals.
59. The sides of a parallelogram are 24 J miles and 31 J
miles, and one of the diagonals is 28 miles ; find the num-
ber of kilometres in the other diagonal.
60. If a chord of 2 feet is 5 inches from the centre of a
O, what is the distance of a chord whose length is 10
inches ?
61. One side of a A is 136"'°, the altitude of the A to
the second side is 102'^'", the diameter of the circumscribed
O is 184'='" ; find the third side of the £\.
OEOMETRT— NUMERICAL PROBLEMS. 27
62. The common chord of two intersecting © whose
radii are 2 feet 1 inch and 1 foot 9 inches is 1 foot 2
inches ; find the distance between their centres.
63. Is the A whose sides are 38"°, 36™, 12^ acute, right,
or obtuse ?
64. In the /\ whose sides are 11"", 13"*, 14", find the
segments into which the side 14 is divided by the perpen-
dicular from the opposite vertex.
65. Find the legs of a right A when their projections
upon the hypotenuse are 11.16 feet and 19.84 feet.
QQ. The sides of a A are 23 feet, 27 feet, 38 feet ; find
the length of the median to the longest side and its projec-
tion upon the longest side.
67. What is the longest and shortest chord that can be
drawn through a point 15"° from the centre of a © whose
radius is 39'=" ?
68. How long is the shadow of a house 23»" high, when
a stake 4 feet high casts a shadow 2 feet 6 inches long ?
(Log.)
69. Find the length of the common tangent of two ©
which cuts the line joining their centres, when this line is
2 feet and the radii of the © are 5 inches and 3 inches.
70. The greater leg of a right A ^^ 1 inch, and the dif-
ference between the hypotenuse and the less leg is J
inch ; find the hypotenuse, the less leg, the perpendicular
from the vertex of the right / to the hypotenuse, and the
segments of the hypotenuse made by this perpendicular.
71. Find the product of the segments of any chord
passing through a point 8" from the centre of a whose
diameter is 20"°,
28 QEOMETRT— NUMERICAL PROBLEMS,
72. Through a point 21'='" from the circumference of a O
is drawn a secant 84'" long. The chord part of this
secant is SI""*. Find the radius of the O.
73. The diagonals A C and B D of an inscribed quadri-
lateral intersect at E, A is 59^ B E 35^", and D E IS"* ;
find A E and C E.
74. What is the length of a tangent drawn from a point
4 inches from the circumference of a O whose radius is 3
feet 9 inches ?
75. Find the diameter of a O in which two chords, 30
feet and 40 feet long, parallel and on opposite sides of the
diameter, are 35 feet apart.
76. The smaller segment of a line divided externally in
extreme and mean ratio is 12'"" ; find the length of the
greater segment.
77. Two sides of a ^ are 16^" and O^"", and the median
to the first side is 11^"" ; find the length of the third side
in miles.
78. In the preceding problem, find the lengths of the
projections of the median and the second and third side
upon the first side.
79. Find the lengths of the projections of each side
upon the other two sides in a /\ whose sides are 6"", 8"", and
12™.
80. How far apart are two parallel chords 48 feet and 14
feet long in a O whose diameter is 50 feet, if they are on
the same side of the centre ?
OEOMETBY— NUMERICAL PROBLEMS, 29
BOOK IV.
1. Find the area of a rectangle whose base and altitude
are 37 feet and 14 feet.
2. What is the area of a parallelogram whose base and
altitude are 13™ and 18™ ?
3. How many hektares in a rectangular field 53^"" by
29Dm p
4. Find the width of a rectangular field containing an
acre, if the length is 176 yards.
5. How many acres in a parallelogram whose base and
altitude are 17«" and 13»" ? (Log.)
6. How many rods in the side of a square field contain-
ing a hektare ? (Log.)
7. How many metres in the side of a square field con-
taining an acre ? (Log.)
8. A rectangle which is 7 times as long as it is wide con-
tains 32 square rods ; find its width and length.
9. Find the area of the surface of a flower-bed 4.55™
long and 2.75™ wide.
10. The perimeter of a rectangle is 24™, and the length
is 9.2™ ; find the breadth and the area of the rectangle.
11. What is the ratio of the areas of two rectangular
fields, one of which is 231™ long and 87™ wide, and the
other 58™ wide and 110™ long ?
12. Two rectangles have the same altitude, and the area
of the first is 62 acres and the area of the second 38 acres.
30 GEOMETRY— NUMERICAL PROBLEMS.
If the base of the first is 570 rods, what is the base of the
second ?
13. What part of a mile is the perimeter of a square
hektare (1^"" = !•""«) ?
14. The perimeter of a rectangle is 6 feet, and the length
is 3 times the breadth ; find the length, the breadth^ and
the area of the rectangle.
15. If the perimeter of a rectangle is 26™ and its length
is 2.5°" more than its breadth, find its length, breadth, and
area.
16. A parallelogram whose area is one acre has a base of
60 rods ; and one whose area is 1"* has the same altitude ;
find the base of the latter. (Log.)
17. Find the side of a square equivalent in area to a
rectangle whose base is 4 feet 6 inches and whose altitude
is 6 inches.
18. What is the altitude of a rectangle whose base is
23", equivalent to a square whose area is 5.06* ?
19. Find the area of a /\ whose base and altitude are
respectively 3 feet 2 inches and 5 yards 1 inch.
20. Find the base and altitude of a rectangle whose pe-
rimeter is 54™ and whose area is 182'^"'.
21. Find the side of a square whose area is 18 square
yards 7 square feet.
22. Find the area, in acres, of a rectangle whose pe-
rimeter is 156°" and whose dimensions are to each other
as 6 : 7. (Log.)
23. A l\ whose base is 35'='" contains .525'" ; how many
square inches in a /\ whose homologous base is 14'^" ?
(Log.)
GEOMETRY— NUMERICAL PROBLEMS. 31
24. Find the area of an equilateral l\ whose side is 8
feet.
25. Find the difference in area between a /\ whose base
and altitude are each 1 yard, and a /\ whose sides are
'each 1"". (Log.)
26. The bases of a trapezoid are 7.32"" and 8.45"", and
the altitude is 4.4" ; find the area in ares.
27. The altitude of an equilateral A is 6 feet 3 inches ;
find the area.
28. To find the area of a A i^ terms of its sides.
J
Let K = the area of the /\-
(1) K =\ ah.
(The area of a A is equal to one-half the product of its
base and altitude.)
By 24, Book III., h = - Vsis-a) (s-b) (s-c) '
Substituting in (1), K =^^x- Vs (s~a) (s—b) {s—c)\ or,
K= V 8(8—0) {s—b) (s—c)'
29. To find the area of a A i^ terms of its sides and
the radius of the circumscribed .
By 25, Book IIL, R = , . , ""^f ,, , «
^ Ws{s-a) (s-b) (s-c)
32 GEOMETRY— IfUMERICAL PROBLEMS.
Substituting K for its value as found in the preceding
article,
abc
Solving, ^ = VB'
30. To find the area of a /\ in terms of its sides and
the radius of the inscribed O.
Fig. 11.
By drawing lines from the centre of the O to the ver-
tices we form three A whose common vertex is 0, whose
bases are a, i, c, the sides of the given /\, and whose alti-
tudes are each r, the radius of the inscribed O.
Now, Area AOC = i br,
'' AOB = i cr,
" BOO = i ar.
Adding, '' ABO = ^{a+b + c)r.
Substituting K for area ABO, and s for ^ (« + ^ 4- c),
K= rs.
From this equation, r = — ; i.e., the radius of the in-
s
scribed O equals the area of the [\ divided by one-half the
perimeter.
* Hereafter this form of the formula for R should be held in mind.
GEOMETRY— NUMERICAL PROBLEMS.
33
31. To find the area of a /\ in terms of its sides and the
radius of an escribed ©.*
(1) Area AOB = \ cr\
(2) '' AOO = i br\
(3) " BOO = i ar\
Subtracting (3) from the sum of
(1) and (2),
Area ABO = ^ {b ■{- c — a)r'; or,
K = (s- a)r'.
Similarly, K= (5 -^>)/',
K = (s- cy,
/' and r'" representing the radii
of the escribed ® tangent to h
and c, respectively.
zr
These three formulae give r'= >
s a
Fig. 12.
K
, from which the radii of the escribed
can be found when the sides of a /\ are given.
32. The sides of a A are : a = 21", I = 17", c = 10" ;
find the area of the /\ and the radii of the circumscribed,
inscribed, and escribed ®.
33. Find the difference in area between a rectangle 4
times as long as wide, with a perimeter of 100 yards and a
square whose perimeter is 80 yards.
34. A man has a rectangular piece of ground 55" by
110". After a path 4.5" wide is made around it, is the
part left more or less than an acre ? How much ?
35. The bases of a trapezoid are 13.2'" and 15.6'", and
the altitude is 1 yard 2 inches ; find the area in centares.
An escribed O is a O tangent to one side of a A and the prolonga-
tions of the other two sides.
84 GEOMETRY— NUMERICAL PROBLEMS.
36. The side of a square is 2 feet ; find the sides of an
equivalent rectangle whose base is 4 times its altitude.
37. The area of a A is ll^*^"", its base is 14" ; find the
area of a similar /\ whose homologous base is 8°*.
38. Find the dimensions of a rectangle whose perimeter
is 8 feet 4 inches, and whose area is 4 square feet 13 square
inches.
39. Through the middle of a rectangular garden, 156"° by
140"", run two paths at right angles to each other and
parallel to the sides, the longer one 0.8" wide, the shorter
1.2" wide ; find the area not taken up by the paths.
40. The sides of a A are : « = 588 feet, h — 708 feet,
c = 294 feet ; find the area of the /\ and the radii of the
circumscribed, inscribed, and escribed 0. (Log.)
41. The area of a rhombus is 360<^% one diagonal is
7.2"^" ; find the other.
42. The area of a polygon is 5f times the area of a
similar polygon. If the longest side of the larger polygon
is 40", what is the longest side of the smaller polygon ?
43. Find the area of a square whose diagonal is 30 feet.
44. Find the number of square feet in an equilateral /\
whose side is one metre. (Log.)
45. Find the side in kilometres, of an equilateral ^
whose area is 47 acres. (Log.)
46. Find the side of a square equivalent to the differ-
ence of two squares whose sides are 115" and 69".
47. Find the area, in square feet, of an isosceles right ^
if the hypotenuse is 25". (Log.)
GEOMETRY— NUMERICAL PROBLEMS. 35
48. The sides of a ^ are 10 feet, 17 feet, and 21 feet.
Find the areas of the two parts into which the l\ is
divided by the bisector of the / formed by the first two
sides.
49. The side of a rhombus is 39"*, and its area is 540ca ;
find its diagonals.
50. The area of a trapezoid is 13 acres, and the sum of
its bases is 813 yards ; find its altitude.
51. Find the area, in acres, of a right l\ whose hy-
potenuse is 36""" and one leg 28.8"". (Log.)
52. Find the ratio of the areas of two A which have a
common /, when the sides including this / in the first
are 131"" and 147", and in the second are 211 feet and 287
feet. (Log. )
53. Two homologous sides of two similar polygons are
21"°" and 35""" ; the area of the greater polygon is 525"* ;
what is the area of the smaller polygon ?
54. Find the area of a quadrilateral whose sides are 8",
10™, 12*", 6°", and one of whose diagonals is 14™.
55. On a map whose scale is 1 inch to a mile, how many
hektares would be represented by a square centimetre ?
(Log.)
56. The homologous altitudes of two similar A are 9™
and 21™, and the area of the smaller is 405 square feet ;
find the area of the larger.
57. The area of a trapezoid is 84"*, its altitude 3.5"™,
and one base 20"™ ; find the other base.
36 GEOMETRY— NUMERICAL PROBLEMS.
58. The areas of two A are 144 square yards and 108
square yards. Two sides of the second are 12 yards and 21
yards, and one side of the first is 9 yards. Find a second
side of the first, which, with the side 9 yards includes an
/ equal to the / of the second included by the sides 12
yards and 21 yards.
59. Find the area of a square whose diagonal is 8".
60. Find the difference in perimeter between a rectangle
whose base is 16 feet and an equivalent square whose side
is 12 feet.
61. Find the diagonals of a rhombus whose side is 6 feet
1 inch and whose area is 9 square feet 24 square inches.
62. Find the area of a trapezoid whose parallel sides
are 28™ and SS"", and whose non-parallel sides are 12""
and 13".
63. Find the dimensions of a rectangle whose area is
1,452 square feet and one of whose sides is | its diagonal.
64. The sides of a A are 26"", 28", 30"" ; find its area,
the three altitudes, and the radii of the inscribed, escribed,
and circumscribed 0.
65. How many tiles, 6 inches by 4J inches, will it take
to cover a swimming pool 40 feet by 27 feet ?
66. Find the sides of an isosceles right /\ whose area is
98».
67. Find the area (in centares) and one side of a rhom-
bus, if the sum of the diagonals is 34 feet and their ratio
is 5 : 12.
OEOMETMY— NUMERICAL PROBLEMS. 37
68. The bases of a trapezoid are 197.3" and 142.7'°, and
its area 37.57'^ ; find its altitude. (Log.)
69. Find the area, in square feet, of a right /\, when
the sides are in the ratio 3:4:5, and the altitude to the
hypotenuse is 1.2^".
70. In the quadrilateral A B D, A B = 10™, B C =
17^ C D = 13°^, D A = 20"°, and A = 21'" ; find the
area in hektares, and the perpendiculars from B and D
to AC.
71. Find the area of a /\ if the perimeter is 82 feet and
the radius of the inscribed ©1.3 feet.
72. Find the ratio of the areas of two equilateral A if
the side of one is 10"" and the altitude of the other is 10'°.
73. Find the area, the altitudes, and the radii of the in-
scribed, escribed, and circumscribed O of the isosceles l\
whose leg is 5 feet 5 inches and whose base is 10 feet 6
inches.
74. The bases of a trapezoid are 13" and 61" ; the non-
parallel sides are 25" each ; find the area of the trapezoid.
75. How many yards of carpet f of a yard wide will it
take to carpet a room 15 feet by 18 feet ?
76. Find the area of a rhombus whose perimeter is 6"
and one of whose diagonals is 1.2".
77. The altitude of a given A is .32^"; find the ho-
mologous altitude, in miles, of a similar l\ 49 times as
large.
78. Find the area of a pentagon whose perimeter is
5.18", circumscribed about a O whose diameter is 1.1 ".
38 OEOMETBT— NUMERICAL PROBLEMS.
79. JFind the area in square metres of a right /\ in
which a perpendicular from the vertex of the right / to
the hypotenuse divides the hypotenuse into segments of
39H feet and ll^^V feet. (Log. )
80. Upon the diagonal of a rectangle 6" by 8" a /\ whose
area is three times the area of the rectangle is construct-
ed ; find the altitude of the /\.
81. Find the side of an equilateral /\ equivalent to the
sum of two equilateral A whose sides are respectively 5"*
and 12"".
82. Find the area of a trapezoid whose bases are 26 feet
and 40 feet, and whose other sides are 13 feet and 15 feet.
83. The three sides of a A are 417.31 feet, 589.72 feet,
and 389.6 feet ; find its area in ares. (Log.)
84. Find the radii of the inscribed, escribed, and circum-
scribed ®. (Log.)
85. Find the three altitudes. (Log.)
86. Find the median to the longest side.
87. Find the bisectors of the three A. (Log.)
88. The base of a A is 25'% its altitude 12" ; find the
area of the A ^^^ ^^ ^J a line parallel to the base and
two-thirds of the way from the vertex to the base.
89. Two homologous sides of two similar A are 12 feet
and 35 feet, respectively ; find the homologous side of a
similar A equivalent to their sum.
90. The bases of a given A and / / are equal, and the
altitude of the A is 2™ and the altitude of the / / 5" ;
find the ratio of their areas.
UNIVERSITY !
OEOMETRT— NUMERICAL PROBLEMS. 39
91. How many yards of wall paper are required to paper
a room 25 feet long, 22 feet wide, and 12 feet high, allowing
for a chimney which projects into the room 1 foot, one
door 5 feet by 7 feet, another 10 feet by 10 feet, a mantel
4 feet by 6 feet, and a window 6 feet by 11 feet ?
92. The homologous altitudes of two similar A are 5™
and 15"", respectively ; what fraction of the second is the
first?
93. Find the legs of a right /\ whose hypotenuse is
25"°" and whose area is 150"*.
94. In a /\ whose base is 22 feet, find the length of the
line parallel to the base and dividing the /\ into two equal
parts. (Log. )
95. Find the area of the /\ whose sides are to each other
as 5 : 12 : 13, and whose altitude to the greater side is
23J inches.
96. The area of the polygon Pis 735. 8"^™, and of the
similar polygon Q is 98.47'*'" ; find the side of Q homolo-
gous to a side of P equal to 81.41*". (Log.)
97. If two sides of a /\ whose area is 9 acres are 165
rods and 201 rods, what is the length of the portions of
these sides cut off by a line parallel to the base and cutting
off a /\ of 4 acres ?
98. Find the area of a right /\ whose hypotenuse is 70"*
and one of whose A is 60°. (Log.)
99. The side of a square is 12" ; find the side of a square
having the ratio 8 to 3 to this square.
100. In a trapezoid whose altitude is 10 feet and whose
bases are 21 feet and 29 feet, what is the length of a line
parallel to the bases and 2 J feet from the smaller base.
40 GEOMETRY— NUMERICAL PROBLEMS,
BOOK V.
Note I. — The answers to a large number of the problems
of this Book may be left in an expressed form, if desired.
For example : AVhat is the area of a hexagon inscribed in a
O whose radius is 15 feet ? Ans. ^ v3
4
Note II. — Quite a number of problems in this Book
which seem difficult, on a mere reading, are rendered quite
easy by drawing figures representing the given conditions
and requirements.
Note III. — In many of these problems it is well to rep-
resent the number in terms of which the answer is to be
gotten by a letter, and then replace the letter by its value
in the final form of the result, as in finding the area, etc.,
of circumscribed and inscribed polygons in terms of the
radius.
1. How many degrees in each / of a regular octagon ?
Of a regular dodecagon ? Of a regular polygon of 27
sides ?
2. How many degrees in the / at the centre of a regular
polygon of 15 sides ? Of 16 sides ?
3. Find the side of a square inscribed in a O whose ra-
dius is 91 feet.
4. Find the radius of a O circumscribed about a regular
hexagon whose perimeter is 5.1™.
5. How many degrees in each exterior / of a regular
polygon of 18 sides ? Of 25 sides ? Of 35 sides ?
6. How many sides has the regular polygon whose / at
the centre is 17° 8' 7f' ?
GEOMETRY— NUMERICAL PROBLEMS. 41
7. How many sides lias the regular polygon whose in-
terior and exterior A are in the ratio of 18 to 4 ?
8. Find the side of an equilateral /\ inscribed in a O
whose diameter is 35.8''™.
9. Find the perimeter of a regular decagon inscribed in
a whose diameter is 7 feet.
10. Find the radius of the circumscribed about a
regular hexagon whose apothem is 12-^/3"'"; also the area
of the hexagon.
11. Find the area of an equilateral /\ inscribed in a O
whose radius is 15 feet. (Log.)
12. Find the radius of a circumscribed about a square
whose area is 1 square yard 7 square feet.
13. Find the apothem of a regular hexagon whose area
is 54^3^™.
14. Find the radius of a O circumscribed about an equi-
lateral l\ whose area is 27v^3~ square feet.
15. Find the area of a regular hexagon whose perimeter
is 78^"".
16. The apothem of an inscribed square is 10 ^/^ ^^et ;
find the area of an equilateral l\ circumscribed about the
same 0.
17. Find the area of a regular polygon whose apothem is
3.75""", and whose perimeter is 15"™. Express the result
in acres. (Log.)
18. Find the side of a regular decagon inscribed in a
whose radius is 35 feet.
42 GEOMETRY— NUMERICAL PROBLEMS.
19. Find the ratio of the areas of two equilateral A, one
inscribed in, the other circumscribed about, a O whose ra-
dius is 5 inches.
20. Find a mean proportional between the areas of
problem 19. Find the area also of a regular hexagon in-
scribed in the same O (5-inch radius). Compare the two
results. (Log.)
21. Find the radius of a O circumscribed about a reg-
ular hexagon whose apothem is -— Vs ^g^^-
22. Find the area, in acres, of a regular hexagon cir-
cumscribed about a O whose radius is 7"". (Log.)
23. The area of .an equilateral /\ circumscribed about a
given O is 87"" ; find the area of a square inscribed in the
same O. (Log.)
Note. — It is customary to use the value d\ for n in
problems involving English units, and 3.1416 where met-
ric units are employed.
24. Find the circumference and area of a O whose
radius is 11 feet.
25. Find the diameter and area of a O whose circum-
ference is 53f feet.
26. Find the circumference of a O whose area is 502,-
656<=\
27. Two circumferences are in the ratio 3 : 5, and the
radius of the larger is 35°^ ; what is the radius of the
smaller ?
28. Find the radius of a O equivalent to two ® whose
radii are respectively 5.6°"* and 4.2°".
QEOMETRT— NUMERICAL PROBLEMS, 43
29. What is the length of an arc of 75° of a O whose
radius is 21 feet ?
30. The areas of two © are in the ratio of 1 : 5f . If the
radius of the larger is 4 feet 1 inch, what is the radius of
the smaller ?
31. Find the difference in area between a square and an
equilateral ^ each inscribed in a O whose radius is 15™.
(Log-)
32. Find the area of a segment of a O of 31-foot radius
cut off by the side of a regular inscribed hexagon. (Log.)
33. Find the difference in length between the circum-
ference of a O whose area is 15836.8056* and the perimeter
of the inscribed hexagon.
34. Find the circumference of a O circumscribed about
a square field containing 700 acres. (Log.)
35. Find the area of a O whose circumference is
29.53104°^
36. What is the area of a segment whose arc is 120°, in
a O whose radius is 4. 3"°" ?
37. Find the number of degrees in an arc equal in length
to the radius of its O.
38. What is the ratio of the areas of two O whose radii
are 50 feet and 65 feet ?
39. Find the apothem, the side, and the area of a regu-
lar octagon inscribed in a O whose radius is 1™. (Log.)
40. How many metres in the diameter of a O whose
area is one acre ?
41. What is the area of a sector whose arc is 175° in a
O whose radius is 24 feet ?
44 GEOMETRY— NUMERICAL PROBLEMS,
42. Find the radius of a O in which the arc subtended
by the side of a regular inscribed dodecagon is 3.1416'°™.
43. How many acres in a O, if a quadrant is one mile
in length ?
44. AVhat is the ratio of the areas of two ® whose cir-
cumferences are 35™ and 40"°, respectively ?
45. Find the side, the apothem, and the area of a regu-
lar dodecagon inscribed in a O whose diameter is 3^"".
(Log.)
46. How far apart are the circumferences of two con-
centric O which contain 5 acres and 10 acres, respective-
ly? (Log.)
47. Find the circumferences of the ® circumscribed
about and inscribed in a square whose side is 14"". (Log.)
48. Find the / at the centre subtended by an arc of
13 inches in a O whose radius is 14^\ inches.
49. What is the area between three ®, each tangent to
the other two, if each has a radius of 440 yards ?
50. Find the side of a square equivalent to a O whose
radius is 19 feet.
51. Find the length of a side and the area of a regular
octagon circumscribed about a O whose radius is a mile.
(Log.)
52. How far apart are two parallel chords in a O whose
radius is 33 feet, if these chords are the sides of regular
inscribed polygons, one a hexagon, the other a dodecagon ?
(Log.)
53. How many rotations to the mile does a wheel whose
diameter is 5 feet 6 inches make ?
OEOMETRT— NUMERICAL PMOBLEMJS. 45
54. Find the side of a regular pentagon equivalent to
the sum of three regular pentagons whose sides are 8™,
9", and 12'".
55. How much more fence would it take to enclose 500
acres in the shape of a square than it would if it were in
circular shape ?
56. Find the perimeter of a sector whose area is 77
square inches and whose arc is 45°.
57. Find the area of that part of a O whose radius is
7^°" included between two parallel chords, one of which is
the side of a regular inscribed /\ and the other the side of
an inscribed square. (Log.)
58. If a bicycle wheel makes 680 rotations to the mile,
what is its diameter ?
59. Find the side and area of a regular pentagon in-
scribed in a O whose radius is 8™.
60. Find the area of a O in which is inscribed a rectan-
gle 6 feet by 8 feet.
61. Find the area of the regular hexagon formed by
joining the alternate vertices of a regular hexagon whose
side is 20 feet.
62. Find the ratio of the areas of the two hexagons in
problem 61.
63. What is the radius of a O whose area is doubled by
increasing its radius 7 feet ?
64. Find the side and the area of a regular dodecagon
circumscribed about a O, whose circumference is 31.416"'°.
(Log.)
46 QEOMETRT— NUMERICAL PROBLEMS.
65. Find the radius of a O equivalent to three O,
whose diameters are 54 feet, 56 feet, and 72 feet.
66. What is the difference in area between an equilat-
eral /\ and a regular decagon each of which has a perim-
eter of 3 miles ? (Log.)
67. The area of a segment cut off by the side of a regu-
lar inscribed hexagon is 413"* ; what is the perimeter of
this segment ? (Log.)
68. Find the side of a square equivalent to a O, in
which a chord of 30 feet has an arc whose height is 5 feet.
69. Find the radius of a O three times as large as a O
whose radius is 3 feet.
70. What is the area of a regular octagon whose perim-
eter is 28^"' ? (Log.)
71. Find the area of the sector whose arc is 175 feet in
a O whose radius is 133 feet.
72. What must be the width of a walk which contains
1"' made around a circular plot of ground containing 5"' ?
73. Find the area of the sector whose arc is the side of
a regular inscribed dodecagon in a O in which a chord of
70 feet is 12 inches from the centre.
74. An acre of ground lies between three ®, each tan-
gent to the other two ; find the radius of one of these ® .
75. Find the radius of a O 36 times as large as a O
whose radius is 14™.
76. If a meridian circle of the earth is 25,000 miles,
what is the length of the diameter in kilometers ?
GEOMETRY— NUMERICAL PROBLEMS, 47
77. If the circumference of a Q is 34.5576^'", what is the
diameter of a concentric O which divides it into two
equivalent parts ?
78. If the side of a regular inscribed hexagon cuts off a
segment whose area is 25*, what is the apothem of this
hexagon ? (Log.)
79. A wheel whose radius is 3 feet 6 inches makes 20
rotations per second ; how many miles will a point on the
circumference go in a day ? (Log.)
80. The difference between the area of a O and its in-
scribed square is 3 acres, find the area of the square ?
81. If an 8-inch pipe will fill a certain cistern in 2
hours 40 minutes, how long will it take a 2-inch pipe ?
82. Find the radius of a O in which an arc of 18° has
the same length as an arc of 45° has in a O whose radius
is 56 feet.
83. If the radius of the earth is 3,963 miles, how many
metres is it from the pole to the equator, measured on a
meridian ? (Log.)
84. Upon each side of a 7-foot square as a diameter,
semicircumferences are described within the square, form-
ing four leaves, or lobes ; find the area of one of these
leaves.
85. Find the number of acres between two concentric
circumferences which are 2 miles and 1 mile long, respec-
tively. (Log.)
86. Find the height of an arc subtended by the side of
an inscribed dodecagon in a O whose area is 154 square
feet.
48 OEOMETBT— NUMERICAL PROBLEMS,
87. Find the area of a O inscribed in a quadrant of a
circle whose radius is 61"°.
88. Find the area of each part of the quadrant of prob-
lem 87, outside the inscribed O.
89. If the circumference of a O, whose diameter is 18™,
is divided into six equal parts, and arcs are described
within the O, with these points of division as centres,
what is the area of the six leaf -shaped figures thus formed?
90. If a bridge in the forn^ of a circular arch 18 feet
high spans a stream 150 feet wide, what is the length of
the whole circumference of which this arch is an arc ?
91. The area inclosed by two tangents and two radii is
140»\ If one of the tangents = 7"", find the distance
from the centre to the meeting of the tangents ; also the
area of the O, in acres.
92. Find the sum of the areas of the crescents formed
by describing semicircumferences on the legs and hypot-
enuse of a right /\ (all on one side), if the legs are 5 feet
and 12 feet respectively. How does this compare with the
area of the /\ ?
93. If the sides of a /^ are 40°', 50", and 60™, what is
the length of the circumference of the circumscribed O ?
94. Find the sum of the areas of two segments, cut off
by two chords, 15 feet and 20 feet respectively, drawn
from the same point to the extremities of the diameter of
their O.
95. If the radius of the earth is 3,963 miles, how high
must a light-house light be to be seen 30 miles off at sea ?
^
GEOMETRY— NUMERICAL PROBLEMS. 49
96. The areas of two concentric ® are to each other as
5 to 8. Find the radii of the two ©, if the area of that
part of the ring which is contained between two radii mak-
ing the angle 45° is 300 square feet.
97. If two tangents, including an / of 60° and drawn
from the same point without a O, with two radii drawn
to their points of contact, inclose an area of 162^3''",
find the length of these tangents and the area of the sector
formed by these two radii and their arc.
98. Find the area of the segments of the O in the pre-
ceding problem made by a chord perpendicular to its
radius at its middle point.
99. If a track, having two parallel sides and two semi-
circular ends, each equal to one of the parallel sides, meas-
ures exactly a mile at the curb, what distance does a horse
cover running ten feet from the curb ? How many acres
within the circuit he makes ?
100. Three ©, each tangent to the other two, inclose
with their convex arcs 1"* of ground. How far is it from
the centres of these ® to the middle point of this piece of
ground ?
4
NUMERICAL PROBLEMS, EXERCISES, PROPOSI-
TIONS, AND OTHER QUESTIONS
SELECTED FROM THE
ENTRANCE EXAMINATION PAPERS OF A NUMBER OF
THE LEADING COLLEGES AND SCIENTIFIC SCHOOLS.
1. From any point in the base of an isosceles triangle
perpendiculars are drawn to the sides ; prove their sum to
be equal to the perpendicular drawn from either basal ver-
tex to the opposite side. — Boston University.
2. The angle at the vertex A of an isosceles triangle
A B C is equal to twice the sum of the equal angles B and
C. If CD is drawn perpendicular to BC, meeting AB
produced at D, prove that the triangle A C D is equilat-
eral. — Wesley an University.
3. If from one of the vertices (A) of a triangle (A B C)
a distance (AD) equal to the shorter one of the two sides
(A B and AC) meeting in A be cut off on the longer one
(AB), prove that /DCB = 1 [/AC B- /AB C].— C^.
of Cal.
4. Show that the angle included between the internal
bisector of one base angle of a triangle and the external
bisector of the other base angle is equal to half the verti-
cal angle of the triangle. — Harvard.
5. If ABC be an equilateral triangle, and if B D, C D
bisect the angles B, C, the lines D E, D F parallel to A B,
A C, divide B C into three equal parts Cornell.
QEOMETRT— NUMERICAL PROBLEMS. 51
6. What is a polygon ? Prove that the sum of the in-
terior angles of an n-gon is ^ — 2 straight angles. — Dart-
mouth.
7. A D and B C are the parallel sides of a trapezoid
A B C D, whose diagonals intersect at E. If F is the
middle point of B C, prove that E F produced bisects
AD. — Mass. Inst. Tech.
8. If perpendiculars be drawn from the angles at the
base of an isosceles triangle to the opposite sides, the line
from the vertex to the intersection of the perpendiculars
bisects the angle at the vertex and the angle between the
perpendiculars. Prove Boston University.
9. Prove that a parallelogram is formed by joining the
midpoints of the (adjacent) sides of any quadrilateral.
Hint, draw the diagonals of the quadrilateral. — Bowdoiii.
10. In any triangle A B C, if A D is drawn perpendicu-
lar to B 0, and A E bisecting the angle BAG, the angle
D A E is equal to one-half the difference of the angles B
and C. — Cornell.
11. Show that in any right-angled triangle the distance
from the vertex of the right angle to the middle point of
the hypotenuse is equal to one-half the hypotenuse. —
School of Mines.
12. If D is the middle point of the side B of the
triangle ABC, and B E and C F are the perpendiculars
from B and to AD, prove that B E = F. — Wesley an
University.
13. If in a right-angled triangle one of the acute angles
is one-third of a right angle, the opposite side is one-half
the hypotenuse. — U. of Cal.
14. Prove that the diagonals and the line which joins
52 OEOMETBY— NUMERICAL PROBLEMS.
the middle points of the parallel sides of a trapezoid meet
in a point. — Harvard.
15. How many degrees in one angle of an equiangular
docedagon ? — Dartmouth.
16. If the opposite sides of a pentagon be produced to
intersect, prove that the sum of the angles at the vertices
of the triangles thus formed is equal to two right angles. —
Cornell.
17. The interior angle of a regular polygon exceeds the
exterior angle by 120°. How many sides has the polygon?
— Mass. Inst. Tech.
18. If one diagonal of a quadrilateral bisects both angles
whose vertices it connects, then the two diagonals of the
quadrilateral are mutually perpendicular. Prove. — Boston
University.
19. In a given polygon, the sum of the interior angles
is equal to four times the sum of the exterior. How many
sides has the given polygon ? — Wesley an University.
20. What is the greatest number of re-entrant angles a
polygon may have compared to the number of its sides ?
What is the value of the re-entrant angles of a pentagon
in terms of the interior angles not adjacent ? — Cornell.
21. Show what the sum of the opposite angles of a quad-
rilateral inscribed in a circle is equal to. — Columbia.
22. When and why may an arc be used as the measure
of an angle ? The vertex of an angle of 60° is outside a
circle and its sides are secants ; what is the relation be-
tween the intercepted arcs ? — Dartmouth.
23. Show that two angles at the centres of unequal
circles are to each other as their intercepted arcs divided
by the radii. — U. of Cat.
QEOMETRT— NUMERICAL PROBLEMS. 63
24. Prove that in any quadrilateral circumscribed about
a circle the sum of two opposite sides is equal to the sum
of the other two opposite sides. — Harvard,
25. Construct a common tangent to two circles. — Boston
University.
26. Three consecutive sides of a quadrilateral inscribed
in a circle subtend arcs of 82°, 99°, and 67° respectively.
Find each angle of the quadrilateral in degrees, and the
angle between its diagonals. — Yale.
27. If A C and B C are tangents to a circle whose centre
is 0, from a point C without the circle, prove that the
centre of the circle which passes through 0, A, and B, bi-
sects C. — Mass. Inst. Tech.
28. Fix the position of a given circle that touches two
intersecting lines. — Vanderhilt University.
29. Through a given point in the circumference of a
circle chords are drawn. Find the locus of their middle
points. — Cornell.
30. Give contractions for the inscribed, escribed, and
circumscribed circles of any triangle. — Sheffield 8. 8.
31. Construct a circle that shall pass through two given
points and shall cut from a given circle an arc of given
length. — Vassar.
32. Prove that the circumference of a circle may be
passed through the vertices of a quadrilateral provided two
of its opposite angles are supplementary. — Boston Univer-
sity.
33. A and B are two fixed points on the circumference
of a circle, and P Q is any diameter. What is the locus
of the intersection of P A and Q B ? — Harvard.
34. The length of the straight line joining the middle
54 GEOMETRY— NUMERICAL PROBLEMS.
points of the non-parallel sides of a circumscribed trapezoid
is equal to one-fourth the perimeter of the trapezoid. —
Mass. Inst. Tech.
35. The points of tangency of a quadrilateral, circum-
scribed about a circle, divide the circumference into arcs,
which are to each other as 4, 6, 10, and 16. Find the an-
gles of the quadrilateral. — Harvard.
36. Given three indefinite straight lines in the same
plane, no two of which are parallel, show that four circles
can be described to touch the three lines.
If two of the three lines are parallel, show that the four
circles reduce to two. — Cornell.
37. From a fixed point of a given circumference are
drawn two chords, OP, Q, so as to make equal angles
with a fixed chord, R, between them. Prove that P Q
will have the same direction whatever the magnitude of the
angles. — Harvard.
38. Draw a straight line tangent to a given circle and
parallel to a given straight line. — Yale.
39. Given two parallel lines and a secant line, also two
circles each tangent to both parallels and to the secant ;
prove that the distance between the centres equals the
segment of the secant line intercepted between the two
parallels. — Boston University.
40. The vertices of a quadrilateral inscribed in a circle
divide the circumference into arcs which are to each other
as 1, 2, 3, and 4. Find the angles between the opposite
sides of the quadrilateral. — Harvard.
41. Show how to construct an isosceles triangle with a
given base and a given vertical angle. — School of Mines.
42. Two circumferences intersect at A and B. Through
B any secant is drawn so as to cut the circumferences in
GEOMETRY— NUMERICAL PROBLEMS. 55
and D respectively. Show that the angle A D is the same
for all secants drawn through B. What value has this an-
gle when the circumferences intersect each other orthogo-
nally ? — Harvard.
43. The perimeter of the circumscribed equilateral tri-
angle is double that of the similar inscribed triangle. —
Sheffield S. S.
44. The radius of a circle is 13 inches. Through a point
6 inches from the centre a chord is drawn. What is the
product of the two segments of the chord ? What is the
length of the shortest chord that can be drawn through that
point ? — Wesley an University.
45. A B is the hypotenuse of a right triangle ABC. If
perpendiculars be drawn to A B at A and B, meeting A C
produced at D, and B produced at E, prove the triangles
A C E and B D similar.— FaZe.
46. Prove that the diagonal of a square is incommensur-
able with its side. When are two quantities said to be in-
commensurable ? — Boiudoin.
47. A B C D is an inscribed quadrilateral. The sides A B
and D C are produced to meet at E. Prove triangles ACE
and BDE similar. — Mass. Inst. Tech.
48. A chord 18 inches long is bisected by another chord
22 inches long. Find the segments of the latter. — N. J.
State College.
49. In any given triangle, if from two of the vertices
perpendiculars be drawn to the opposite sides, the triangle
cut off by the line joining the feet of the perpendiculars is
similar to the given triangle. — U. of Cat.
50. The diagonals of a certain trapezoid, which are 8 and
12 feet long respectively, divide each other into segments
66 OEOMETRY— NUMERICAL PROBLEMS.
which in the case of the shorter diagonal are 3 feet and 5
feet long. What are the segments of the other diagonal ? —
Harvard.
51. The sides of a triangle are 5, 6, and 8. Find the seg-
ments of the last side made by a perpendicular from the
opposite angle. — Rutgers 8. 8.
52. In a plane triangle what is the square on the. side
opposite to the obtuse angle equal to ? Demonstrate. —
8chool of Mines.
53. The sides of a triangle are 9, 8, 13. Is the greatest
angle acute, obtuse, or right ? — Vassar.
54. Given A B = xy, write five resulting proportions.
Need not prove. — Boston University.
55. The radii of two circles are 8 inches and 3 inches,
and the distance between their centres is 15 inches. Find
the length of their common tangents Wesleyan Univer-
56. The bases of two similar triangles are respec-
tively 12.34 and 18.14 metres. The altitude of the first is
6.12 metres ; find the altitude of the second. (Use loga-
rithms.) — Yale.
57. If A B and D are equal chords of a circle and in-
tersect at E, prove that A E = E D and B E = E Q.—Mass.
Inst. Tech.
58. One segment of a chord drawn through a point 7
units from the centre of a circle is 4 units. If the diame-
ter of the circle is 15 units, what is the other segment ?—
Brown.
59. Two parallel chords of a circle are d and h in length,
and their distance apart is/; what is the radius ? — Van-
derhilt University.
GEOMETRY— NUMERICAL PROBLEMS. 57
60. In a certain circle a chord is 10 inches long, while
another chord twice as far from the centre as the first is 5
inches long ; find the radius of the circle and the distances
of the chords from the centre. — Harvard.
61. When is a line said to be divided harmonically 9
From the point P without a circle a secant through the
centre is drawn cutting the circle in A and B. Tangents
are drawn from P and the points of contact connected by
a line cutting A B in Q. Show that P and Q divide A B
harmonically. — Sheffield 8. 8.
62. Two sides of a triangle are 17 and 10 ; the perpen-
dicular from their intersection to the third side is 8 ; what
is the length of the third side ? — Mass. Inst. Tech.
63. Prove that the sum of the squares of the sides of a
parallelogram is equal to the sum of the squares of its diag-
onals. — School of Mines.
64. In a triangle whose sides are 48, 36, and 50, where
do the bisectors of the angles intersect the sides ? What
are the lengths of the bisectors ? — Rutgers 8. 8.
65. The distance from the centre of a circle to a chord
10 inches long is 12 inches. Find the distance from the
centre to a chord 24 inches long. — Wesleyan University.
66. The diameter of a circle is 20 inches, the least dis-
tance from a certain point upon the circumference to a
diameter is 8 inches ; find the distances from this point to
the ends of the above diameter. — Boston University.
67. Let A B C be a right triangle. The two sides about
the right angle C are respectively 455 and 1,092 feet. The
hypotenuse A B is divided into two segments A E and B E
by the perpendicular upon it from 0. Compute the lengths
of A E, B E, and C l^.—Yale.
58 GEOMETRY— NUMERICAL PROBLEMS.
68. is any point on the straight portion, A B, of the
boundary of a semicircle. C T>, drawn at right angles to
A B, meets the circumference at D. D is drawn to the
centre, 0, of the circle, and the perpendicular dropped
from C upon D meets D at E. Show that D is a
mean proportional to A and D E. — Harvard.
69. The length of one side of a right triangle is 12, and
the length of the perpendicular from its extremity to the
hypotenuse is 4y\. Find the lengths of hypotenuse and
other side. — Mass. Inst. Tech.
70. The three sides of a triangle are 6, 8, 10 units long ;
compute the lengths of the three medial lines. — Cornell.
71. The area of a rectangle is 64, the difference of two
adjacent sides is 12 ; construct the rectangle. — Bowdoin.
72. Prove that if any point on one of the diagonals of a
parallelogram be joined to the vertices, of the triangles
thus formed, those having the same base are equivalent. —
U. of Gal.
73. In a triangle ABO, let be the point in which the
medians (lines drawn from the vertices to the middle points
of the opposite sides) intersect. Prove that the triangles
OAB, OAO, OBC are equivalent. — Amherst.
74. If two equivalent triangles have a common base, and
lie on opposite sides of it, the base, or the base produced,
will bisect the line joining the vertices. — Dartmouth.
75. If the perimeter of a rectangle is 72 feet, and the
length is equal to twice the width, find the area. — Johns
Hopkins University.
76. The area of a certain isosceles triangle is 50 square
feet, and each of its equal sides is 10 feet long ; find the
angles of the triangle. — Cornell,
GEOMETRY— NUMERICAL PROBLEMS, 59
77. Two mutually equiangular triangles are similar.
The base of a triangle is 32 feet, its altitude 20 feet. What
is the area of the triangle cut off by drawing a line parallel
to the base and at a distance of 15 feet from the base ? —
Wesleyan University.
78. The perimeter of a trapezoid is 56 inches. If each
of the non-parallel sides is 13 inches long, and the area is 180
square inches, what are the respective lengths of the parallel
sides? — Mass. Inst. Tech.
79. The area of a certain polygon is 5 square feet. Find
the area of a similar polygon whose perimeter is in the
ratio of M to N to that of the given polygon. — Sheffield
S. S.
80. A vertex of a parallelogram and the middle points
of the two sides adjacent to it form the vertices of a trian-
gle whose area is equal to one-eighth the area of the paral-
lelogram. — Boston University.
81. (a.) If two triangles are on equal bases and between
the same parallels, a line parallel to their bases cuts off
equal areas.
{jb.) Lines joining the non-adjacent extremities of
two parallel chords are equal,
(c.) State and prove the converse of the preceding
proposition. — Yale.
2 X
82. Given - = -• Construct x. — Cornell.
X 6
83. Find the area of a triangle in terms of its sides. —
Vanderhilt University.
84. Prove that, if in the triangle ABC the line drawn
from the vertex C to the middle point of the opposite side
is equal to half the latter, the area of the triangle is nu-
60 OEOMETRY— NUMERICAL PROBLEMS.
merically equal to half the product of A C by B C. — Har-
vard.
85. Given three rectangles, find a square whose area is
equal to the sum of the areas of the larger two minus the
area of the smallest one. — U. of Cal.
86. Prove that the square described upon the altitude of
an equilateral triangle has an area three times as great as
that of a square described upon half of one side of the tri-
angle. — Cornell.
87. A D and B C are the parallel sides of the trapezoid
A B C D, whose diagonals intersect at 0. Prove
area A D : area B C = aO^ '- 00^-
— Mass. Inst. Tech.
88. Construct a square whose area is 3 times that of a
given square. — Sheffield 8. 8.
89. Draw a hexagon having one re-entrant angle, and
construct a triangle equivalent to this polygon. — Cornell.
90. The parallel sides of a trapezoid are 12 and 18, the
non-parallel sides are each 5 ; find its area and the altitude
of the triangle formed by producing the non-parallel sid.es
until they meet. — Dartmouth. '
91. Through a point in one side of a triangle draw a
line parallel to the base which shall bisect the area of the
triangle. — Cornell.
92. The area of a polygon is 160 square feet, one side
is 6 feet long ; find the homologous side of a similar poly-
gon whose area is 800 square feet. — Boston U^iiversit^^
93. The base of a triangle is 16 feet, and the two other
sides are respectively 12 and 10 feet. Find the altitude of
the triangle, and also the area. — Yale.
GEOMETRT—NUMERIGAL PROBLEMS. 61
94. In a certain triangle ABC, AO'-BO^ = iAB^;
show that a perpendicular dropped from upon A B will
divide the latter into segments which are to each other as
3 to 1. — Harvard.
95. Construct a parallelogram equivalent to a given tri-
angle and having one of the diagonals equal to a given
line. — U. of Cal.
96. Construct a polygon similar to a given polygon and
having two and a half times its area. — Cornell.
97. How many degrees in each angle of a regular deca-
gon ? — Yale.
98. If the diagonals A C and B Gr of the regular octa-
gon ABCDEFGH intersect at 0, how many degrees
are there in the angle AOB ? — Mass. hist. Tech.
99. Show that the sum of the alternate angles of an in-
scribed hexagon (not necessarily regular) is equal to four
right angles. — School of Mines.
100. An equilateral triangle is inscribed in a circle.
Find its side, apothem, and area in terms of the radius R.
— Dartmouth.
101. Find the ratio of the area of a regular hexagon in-
scribed in a circle to that of a regular hexagon circum-
scribed about the same circle. — Johns Hopkins University.
102. What regular polygon has each angle equal to five
thirds of a right angle ? — U. of Cal.
103. A certain equilateral triangle has sides 8 V 3 inches
long ; what is the radius of the circumference circumscribed
about this triangle ? — Harvard.
104. Compute the area of a regular hexagon whose side
is 5 feet. Construct a triangle of equivalent area. — Sheffield
S. S.
62 QEOMETRT— NUMERICAL PROBLEMS.
105. The area of the regular inscribed hexagon of a cir-
cle is three-fourths of that of the regular circumscribed
hexagon . — Cor7iell.
106. Find the number of degrees in an angle of a regu-
lar pentagon and give proof of the process. — Boiodoin.
107. If the interior angles of any quadrilateral be bisected
and each bisector produced to meet two others, the quadri-
lateral formed may be inscribed in a circle. Prove. — Bos-
ton Univei'sity.
108. The diagonals of a regular pentagon divide each
other in mean and extreme ratio. — U. of Cal.
109. Show that an equiangular polygon inscribed in a cir-
cle is regular if the number of its sides is odd. — Cornell.
110. The radius of a certain circle is 9 inches ; find the
area of that one of all the regular polygons inscribed in it
which has the shortest perimeter. How long a perimeter
can a regular polygon inscribed in this circle have ? — Har-
vard.
111. A regular hexagon, ABCDEF, is inscribed in a
circle whose radius is 2 ; find the length of the diagonal
AC. — Mass. Inst. Tech.
112. To compute the area of a circle whose radius is
unity. — Dartmouth.
113. Find the area of a circle inscribed in a square con-
taining 400 square feet. — N. J. State College.
114. Find the side of a square equivalent to a circle
whose radius is 56 feet. (Use logarithms.) — Yale.
115. The area of a certain regular hexagon is 294 V 3
square inches ; find the area and the circumference of the
circumscribed circle. — Harvard.
GEOMETRY— NUMERICAL PROBLEMS. 63
116. The circumference of a circle is 78.54 inches ; find
(1) its diameter, and (2) its area. — Rutgers S. S.
117. If the areas of two regular pentagons be as 16 to 25,
and the perimeter of the first pentagon be 50 inches, what
is the perimeter of the second ? — Cornell.
118. If the radius of a circle is 5, find the area of the
sector whose central angle is 50°. — Wesleyan University.
119. The angle of a sector is 30° ; the radius is 12. Find
the area of the sector. — Amherst.
120. Prove that the area of the regular inscribed dode-
cagon is equal to three times the square of the radius. —
U. of Cat.
121. If the diameter of a circle is 3 inches, what is the
length of an arc of 80° ? — Mass. Inst. Tech.
122. In a circle whose radius is 8, what is the length of
the arc of a sector of 45** ? What is the area of this sec-
tor ? — Rutgers S. S.
123. If the radius of a circle is 5 inches, compute its cir-
cumference and its area ; also the perimeter, the area, and
the apothem of an inscribed square. — Yale.
124. The perimeter of a regular hexagon is 480 feet, and
that of a regular octagon is the same. Which is the greater
in area, and by how much ? — Cornell.
125. The area of a certain circle is 154 square inches ;
what angle at the centre is subtended by an arc of the cir-
cumference 5 J inches long ? — Harvard.
126. Find the length of the arc of 75° in the circle
whose radius is 5 feet. — iV. /. State College.
127. A M and B N are perpendiculars from points A and
B to the line M N. Find a point P on the line M N such
64 GEOMETRY— NUMERICAL PROBLEMS.
that the sum of the distances A P, B P, is the least pos-
sible. — Wellesley.
128. Two circles are tangent internally, the ratio of
their radii being 2 : 3. Compare their areas, and also the
area left in the larger circle with each. — Sheffield 8. 8.
129. A kite-shaped racing-track is formed by a circular
arc and two tangents at its extremities. The tangents
meet at an angle of 60°. The riders are to go round the
track, one on a line close to the inner edge, the other on a
line everywhere 5^ ft. outside the first line. Show that
the second rider is handicapped by about 22 feet. — Har-
vard.
130. The diameters of two water-pipes are 6 and 8 inches
respectively. What is the diameter of a pipe having a
capacity equal to their sum ? — Rutgers 8. 8.
131. (a.) There are two gardens: one is a square and
the other a circle ; and they each contain a hectare. How
much farther is it around one than the other ?
(b.) If the area of each is 2 hectares, what will be the
difference of their perimeters ? — Yale.
132. Inscribe a square in a scalene triangle. — Cornell.
133. A horse is tethered to a hook on the inner side of a
fence which bounds a circular grass-plot. His tether is so
long that he can just reach the centre of the plot. The
area of so much of the plot as he can graze over is -*/
(4 TT — 3 a/3) sq. rd. ; find the length of the tether and
the circumference of the plot. — Harvard.
134. If the apothem of a regular hexagon is -2, find the
area of its circumscribed circle. — Wesley an University.
135. Of all polygons formed of given sides the maxi-
mum may be inscribed in a circle. — Sheffield 8. 8.
OEOMETRT— NUMERICAL PROBLEMS. 65
136. If the radius of a circle is 6, what is the area of a
segment whose arc is 60° ? (Take n = 3.1416.) — Mass.
Inst. Tech.
137. A stone bridge 20 ft. wide has a circular arch of
140 ft. span at the water level. The crown of the arch is
140 {1 — i a/3) ft. above the surface of the water. How
many square feet of surface must be gone over in cleaning
so much of the under side of the arch as is above water ? —
Harvard.
138. Of all isoperimetric figures the circle has the
greatest area. — Corfiell.
139. Compute by logarithms the value of
s/
(2.3456)3 X (.301456)^
(4.02356)^ —Yale.
SELECTED EXAMINATION PAPERS IN PLANE
GEOMETRY SET FOR ADMISSION TO A NUMBER
OF THE LEADING COLLEGES AND SCIENTIFIC
SCHOOLS IN THE UNITED STATES.
Harvard, June, 1892.
[In solving probleniB use for n the approximate value Sf .]
1. Prove that if two sides of a triangle are unequal, the
angle opposite the greater side is greater than the angle op-
posite the less side.
In a certain right triangle one of the legs is half as long as
the hypotenuse ; what are the angles of the triangle ?
2. Show how to find on a given indefinitely extended straight
line in a plane, a point O which shall be equidistant from two
given points A, B in the plane. If A and B lie on a straight
line which cuts the given line at an angle of 45° at a point
7 inches distant from A and 17 inches from B, show that O A
will be 13 inches.
3. Prove that an angle formed by a tangent and a chotd
drawn through its point of contact is the supplement of any
angle inscribed in the segment cut off by the chord. What is
the locus of the centre of a circumference of given radius
which cuts at right angles a given circumference ?
4. Show that the areas of similar triangles are to each other
as the squares of the homologous sides.
5. Prove that the square described upon the altitude of an
equilateral triangle has an area three times as great as that of
a square described upon half of one side of the triangle.
6. Find the area included between a circumference of
radius 7 and the square inscribed within it.
GEOMETRY— NUMERICAL PROBLEMS. 67
Harvard, June, 1893.
[In solving problems use for v the approximate value 3f .]
1. Prove that two oblique lines drawn from a given point
to a given line are equal if they meet the latter at equal dis-
tances from the foot of the perpendicular dropped from the
point upon it.
How many lines can be drawn through a given point in a
plane so as to form in each case an isosceles triangle with two
given lines in the plane ?
3. Prove that in the same circle, or in equal circles, equal
chords are equally distant from the centre, and that of two
unequal chords the less is at the greater distance from the
centre.
Two chords of a certain circle bisect each other. One of them
is 10 inches long ; how far is it from the centre of the circle ?
A variable chord passes, when produced, through a fixed
point without a given circle. What is the locus of the mid-
dle point of the chord ?
3. A common tangent of two circumferences which touch
each other externally at A, touches the two circumferences at
B and C respectively ; show that B A is perpendicular to A C.
4. Assuming that the areas of two triangles which have an
angle of the one equal to an angle of the other are to each
other as the products of the sides including the equal angles,
prove that the bisector of an angle of a triangle divides the
opposite side into parts which are proportional to the sides
adjacent to them.
5. Prove that the circumferences of two circles have the
same ratio as their radii.
6. A quarter-mile running track consists of two parallel
straight portions joined together at the ends by semicircum-
ferences. The extreme length of the plot enclosed by the
track is 180 yards. Find the cost of sodding this plot at a
quarter of a dollar per square yard.
68 OEOMETRT— NUMERICAL PROBLEMS.
Harvard, June, 1894.
[In solving problems use for n the approximate value 3f .]
1. Prove that any quadrilateral the opposite sides of which
are equal, is a parallelogram.
A certain parallelogram inscribed in a circumference has
two sides 20 feet in length and two sides 15 feet in length ;
what are the lengths of the diagonals ?
2. Prove that if one acute angle of a triangle is double an-
other, the triangle can be divided into two isosceles triangles
by a straight line drawn through the vertex of the third
angle.
Upon a given base is constructed a triangle one of the base
angles of which is double the other. The bisector of the
larger base angle meets the opposite side at the point P. Find
the locus of P.
3. Show how to find a mean proportional between two given
straight lines, but do not prove that your construction is cor-
rect.
Prove that if from a point, O, in the base, B C, of a triangle,
ABC, straight lines be drawn parallel to the sides, A B,
A C, respectively, so as to meet A C in M and A B in N, the
area of the triangle A M N is a mean proportional between the
areas of the triangles B N O and C M O.
4. Assuming that the areas of two parallelograms which
have an angle and a side common and two other sides unequal,
but commensurable, are to each other as the unequal sides,
prove that the same proportion holds good when these sides
have no common measure.
5. Every cross-section of the train-house of a railway station
has the form of a pointed arch made of two circular arcs the
centres of which are on the ground. The radius of each arc is
equal to the width of the building (210 feet) ; find the dis-
tance across the building measured over the roof, and show
that the area of the cross-section is 3, 675 (4 n- — 3 4/3 ) square
feet.
GEOMETRY— NUMERICAL PROBLEMS. 69
Harvard, June, 1895.
One question may he omitted.
[In solving problems use for n the approximate value 3f ]
1. Prove that if two straight lines are so cut by a third that
corresponding alternate-interior angles are equal, the two
lines are parallel to each other.
2. Prove that an angle formed by two chords intersecting
within a circumference is measured by one-half the sum of
the arcs intercepted between its sides and between the sides of
its vertical angle.
Two chords which intersect within a certain circumference
divide the latter into parts the lengths of which, taken in
order, are as 1, 1, 2, and 5 ; what angles do the chords make
with each other ?
3. Through the point of contact of two circles which touch
each other externally, any straight line is drawn terminated
by the circumferences ; show that the tangents at its extrem-
ities are parallel to each other.
What is the locus of the point of contact of tangents
drawn from a fixed point to the different members of a system
of concentric circumferences ?
4. Prove that, if from a point without a circle a secant and
a tangent be drawn, the tangent is a mean proportional be-
tween the whole secant and the part without the circle.
Show (without proving that your construction is correct)
how you would draw a tangent to a circumference from a
point without it.
5. Prove that the area of any regular polygon of an even
number of sides (2 n) inscribed in a circle is a mean propor-
tional between the areas of the inscribed and the circum-
scribed polygons of half the number of sides. If n be indef-
initely increased what limit or limits do these three areas ap-
proach ?
70 GEOMETRT—NUMERIGAL PROBLEMS.
6. The perimeter of a certain church
window is made up of three equal semi-
circumferences, the centres of which form
the vertices of an equilateral triangle
which has sides 3i feet long. Find the
area of the window and the length of its
perimeter.
Harvard, June, 1896.
One question may he omitted.
[In solving problems use for n- the approximate value 3^.]
1. Prove that if two oblique lines drawn from a point to a
straight line meet this line at unequal distances from the foot
of the perpendicular dropped upon it from the given point,
the more remote is the longer.
2. Prove that the distances of the point of intersection of
any two tangents to a circle from their points of contact are
equal.
A straight line drawn through the centre of a certain circle
and through an external point, P, cuts the circumference at
points distant 8 and 18 inches respectively from P. What is
the length of a tangent drawn from P to the circumference ?.
3. Given an arc of a circle, the chord subtended by the arc
and the tangent to the arc at one extremity, show that the
perpendiculars dropped from the middle point of the arc on
the tangent and chord, respectively, are equal.
One extremity of the base of a triangle is given and the
centre of the circumscribed circle. What is the locus of the
middle point of the base ?
4. Prove that in any triangle the square of the side opposite
an acute angle is equal to the sum of the squares of the other
two sides diminished by twice the product of one of those
sides and the projection of the other upon that side.
GEOMETRY— NUMERICAL PROBLEMS. 71
Show very briefly how to construct a triangle liaving given
the base, the projections of the other sides on the base, and
the projection of the base on one of these sides.
5. Show that the areas of similar triangles are to one an-
other as the areas of their inscribed circles.
The area of a certain triangle the altitude of which is >/ 2,
is bisected by a line drawn parallel to the base. What is the
distance of this line from the vertex ?
6. Two flower-beds have equal perimeters. One of the beds
is circular and the other has the form of a regular hexagon.
The circular bed is closely surrounded by a walk 7 feet
wide bounded by a circumference concentric with the bed.
The area of the walk is to that of the bed as 7 to 9. Find
the diameter of the circular bed and the area of the hexagonal
bed.
Yale, June, 1892.
TIME ALLOWED, ONE HOUR.
1. Construct accurately, by ruler and compass, a parallelo-
gram A B C D having the angle A 45°, the side A B 6 units in
length, and the altitude 3 of the same units.
Calculate the length of A C.
2. (a) State the converse of the following proposition :
// a triangle is isosceles and if a straight line is drawn
through the vertex parallel to the base, it bisects an exterior
angle of the triangle.
(6) Prove the converse as you have stated it.
Make the demonstration as full and clear as possible.
3. Prove two of the following propositions : The work may
be limited to drawing a figure and giving a synopsis of the
demonstration.
72 QEOMETBY— NUMERICAL PROBLEMS.
(a) If the area of a regular polygon is equal to the product of
the perimeter by one-half the apothegm, it follows that the area
of a oirale = n R^.
(&) If two lines are drawn through the same point across a
circle, the products of the two distances on each line from this
point to the circumference are equal to each other.
(c) If the radius of a circle he dimded in extreme and mean
ratio, the greater segment is equal to one side of a regular in-
scribed decagon.
Yale, June, 1893.
1. Prove that if the diagonals of a quadrilateral bisect each
other the figure is a parallelogram.
3. Prove that in any right-angled triangle the square on the
side opposite to the right angle is equal to the sum of the
squares on the other two sides. •
A purely geometrical proof is preferred.
State fully each principle employed in the proof.
3. Given a straight line AB, of indefinite length, and a
point C without it. Find a point in A B equally distant from
A and C.
Make the necessary construction accurately with ruler amd
compass.
In what case is the solution impossible ?
4. Given an angle C O D at the
centre of a circle and the line C A
meeting D O produced in A so
that A B is equal to the radius of
the circle. Prove that the angle
A is equal to one-third of the
angle COD.
OEOMETBY— NUMERICAL PROBLEMS, 73
Yale, June, 1894.
GEOMETRY (A).
TIME ONE HOUR.
1. If the diagonals of a quadrilateral bisect each other, the
figure is a parallelogram.
2. To draw a tangent to a given circle, so that it shall be
parallel to a given straight line.
3. If A B is a chord of a circle, and C E is any chord drawn
through the middle point C of the arc A B cutting the chord
A B at D, prove that the chord A C is a mean proportional
between C D and C E.
4. The areas of two similar triangles are to each other as
the squares of any two homologous sides.
5. The area of a circle is equal to one-half the product of its
circumference and radius.
Yale, June, 1894.
GEOMETRY (B).
TIME FORTY-FIVE MIISTJTES.
1. What is the number of degrees in each angle of a regu-
lar decagon ?
2. Find the area in square feet of an equilateral triangle
whose side is 3 metres.
3. A B C is a right triangle. The sides A C and B C about
the right angle C are respectively 50 and 120 feet. Divide the
triangle into two parts equal in area by a line D F parallel to
B C. Compute the length of the three sides of the triangle
ADF.
74 QEOMETRT—NUMERIGAL PROBLEMS,
4. The area of a circle is a hectare. What is its diameter ?
5. Calculate in metres the length of a degree on the circum-
ference of the earth, assuming the section of the earth to be a
circle whose radius is 3, 963 miles. [Those taking the prelimi-
nary examinations must use logarithms. ]
[For preliminary candidates only.]
6. Find the value of the following expression by logarithms :
^-
(.06342)^ X 187.32
.34216 X 6.0372
Yale, June, 1895.
GEOMETRY (A).
TIME AliLOWED, SIXTY MIN^UTES.
1. (a) Define the terms " locus " and *' limit of a variable "
and give an example of each.
(&) Prove that two triangles are similar if their homologous
sides are proportional.
(c) Through a given point A within a circle draw two equal
chords.
[Both the construction (with ruler and compass), and also
the proof, are required. ]
Prove that if each of
two angles of a quadri-
lateral is a right angle,
the bisectors of the other
angles are either perpen-
dicular, or parallel, to
each other.
(6) Prove that if the radius of a circle is divided in extreme
and mean ratio, the greater part is equal to the side of a regu-
lar inscribed decagon.
[The construction is not required.}
QEOMETBT— NUMERICAL PROBLEMS. 75
Yale, June, 1896.
GEOMETRY (B).
TIME ALLOWED, FORTY-FIVE MINUTES.
One question may he omitted. Logarithmic tables should he
u^sed in calculating the answers of two questions.
1. The base of a triangle is 14 inches and its altitude is 7
inches. Find the area of the trapezoid cut off by a line 6
inches from the vertex.
Express the result in square metres.
2. Find the number of feet in an arc of 40° 12' if the radius
of the circle is 0.7539 metres.
3. The length of a chord is 10 feet, and its greatest dis-
tance from the subtending arc is 3 feet 7i inches. Find the
radius of the circle.
4. Find the area, and also the weight in grams, of the
largest square that can be cut from a circular sheet of tin 16
inches in diameter and weighing 8.2 ounces per square foot.
Yale, June, 1896.
GEOMETRY (A).
TIME, ONE HOUR.
1. The sum of the three angles of a triangle is equal to two
right angles.
2. Construct a circle having its centre in a given line and
passing through two given points.
3. The bisector of the angle of a triangle divides the op-
posite side into segments which are proportional to the two
other sides.
76 GEOMETRY— NUMERICAL PROBLEMS.
4. If two angles of a quadrilateral are bisected by one of its
diagonals, the quadrilateral is divided into two equal tri-
angles and the two diagonals of the quadrilateral are per-
pendicular to each other.
5. The circumferences of two circles are to each other as
their radii. (Use the method of limits.)
Yale, June, 1896.
GEOMETRY (B).
TIME ALLOWED, FORTY-FIVE MINUTES.
1. A tree casts a shadow 90 feet long, when a vertical rod 6
feet high casts a shadow 4 feet long. How high is the tree ?
2. The distance from the centre of a circle to a chord 10
inches long is 12 inches. Find the distance from the centre
to a chord 24 inches long.
3. The diameter of a circular grass plot is 28 feet. Find
the diameter of a grass plot just twice as large. (Use loga-
rithms. )
4. Find the area of a triangle whose sides are a = 12.342
metres h = 31.456 metres o = 24.756 metres, using the formula
^ ^ a + b + c '
Area = ^ s{s-a) («-&) >^^ where s= — ^ (Use lo-
garithms.)
Princeton, June, 1894.
What text-book have you read ?
1. Prove that the sum of the three angles of a triangle is
equal to two right angles. Define triangle, right angle, right
triangle, scalene triangle.
2. Prove that the opposite sides and angles of a parallelo-
gram are equal. Define a parallelogram, a rectangle.
OEOMETRT-—NUMERIGAL PROBLEMS. 77
3. Prove that an angle inscribed in a circle is measured by
one-half of the arc intercepted by its sides.
Consider all cases.
4. Show how to construct a triangle, having given two
sides and the angle opposite one of them.
Is the construction always possible? If not, state when
and why it fails.
5. Prove that if any chord is drawn through a fixed point
within a circle, the product of its segments is constant in
whatever direction the chord is drawn.
6. Prove the ratio between the areas of two triangles which
have an angle of the one equal to an angle of the other.
Define area.
7. Define a regular polygon and prove that two regular
polygons of the same number of sides are similar.
Define similar figures.
Princeton, June, 1895.
What text-book have you read ?
1. Prove that every point in a perpendicular erected at the
middle of a given straight line is equidistant from the extrem-
ities of the line, and every point not in the perpendicular is
unequally distant from the extremities of the line.
2. Prove that the sum of the interior angles of a polygon is
equal to two right angles taken as many times less two as the
figure has sides.
Define a polygon, also a right angle.
3. Prove that the tangents to a circle drawn from an ex-
terior point are equal, and make equal angles with the secant
drawn from this point through the centre ; also that either
tangent is a mean proportional between the secant and its
external segment.
Define circle, tangent, secant, chord, mean proportional.
V
NIVERSITY
or
r-.
78 QEOMETRT—NUMERIOAL PROBLEMS.
4. Show how to circumscribe a circle about a given triangle,
giving reasons for the process.
5. Prove what the area of a triangle is equal to ; also the
area of a trapezoid.
Define triangle, trapezoid, area.
6. Prove that the area of a circle is equal to one-half the
product of the circumference by the radius.
Express the area of a circle in terms of tt.
Define n and give its numerical value.
Princeton, June, 1896.
State what text-book you have read and how much of it.
1. Prove that the sum of the three angles of a triangle is
equal to two right angles ; and that the sum of all the in-
terior angles of a polygon of n sides is equal to {n — 2) times two
right angles.
2. Show that the portions of any straight line intercepted
between the circumferences of two concentric circles are equal.
3. Define similar polygons and show that two triangles
whose sides are respectively parallel or perpendicular are simi-
lar polygons according to the definition.
4. Prove that, if from a point without a circle a secant
and a tangent are drawn, the tangent is a mean proportional
between the whole secant and its external segment.
5. Prove what the area of a triangle is equal to ; — also of a
trapezoid ; — also of a regular polygon. Define each of the fig-
ures named.
6. Explain how to construct a triangle equivalent to a given
polygon.
7. Prove that of all isoperimetric polygons of the same
number of sides, the maximum is equilateral.
OEOMETRY—NUMERIOAL PROBLEMS. 79
Princeton, September, 1896.
State what text-book you have read and how much of it.
1. Name and define six quadrilateral figures.
Prove that in a parallelogram the opposite sides are equal,
and the diagonals bisect each other.
2. Define and show how to construct the inscribed circle and
the three escribed circles of a given triangle.
3. Prove that, if the base of a triangle is divided, either in-
ternally or externally, into segments proportional to the other
two sides, the line joining the point of section and the oppo-
site vertex of the triangle is the bisector of the angle (either
internal or external) at that vertex.
4. Prove what the area of a parallelogram is equal to, and
show how to construct a square equivalent to a given paral-
lelogram.
5. Prove that if a circle is divided into any number of equal
parts, the chords joining the successive points of division form
a regular inscribed polygon, and the tangents drawn at the
points of division form a regular circumscribed polygon.
6. Prove that the maximum of all isoperimetric polygons
of the same number of sides is a regular polygon.
Columbia, June, 1896.
TIME ALLOWED, TWO AND ONE-HALF HOURS.
Omit one question from each of the groups, A, B, C.
State what text-book you have used in preparation.
A.
1. Prove that, in a circle, a diameter is greater than any
other chord.
2. Prove that, in any triangle, a line drawn parallel to the
base divides the other sides proportionally.
80 GEOMETRY— NUMERICAL PROBLEMS.
3. Prove that an angle formed by a tangent and a chord of
a circle meeting at the point of contact of the tangent, is
measured by one-half of the included arc.
B.
4. Prove that if four quantities are in proportion, they are
in proportion by composition and by division.
5. Show how to construct a triangle equal to a given pen-
tagon.
6. Show how to inscribe a regular decagon in a circle.
C.
7. Let A, B, C, D be four points lying in the order named
upon a certain circumference. The arcs A B, B C, and C D,
are of 76°, 53°, and 118° respectively. Find the angle between
the chords A C and B D, and also the angle between A B and
C D, produced.
8. Prove that the difference of the diagonals of any quadri-
lateral is less than the sum of either pair of opposite sides.
9. Find a point in the base of a triangle such that lines
drawn from it parallel to the other side of the triangle shall
be equal to each other.
School of Mines, June, 1896.
TIME ALLOWED, TWO AND ONE-HALF HOURS.
1. Prove that if a straight line, E F, has two of its points,
E and F, each equally distant from two points, A and B, it is
perpendicular to the line A B at its middle point.
2. In equal circles incommensurable angles at the centre are
proportional to their intercepted arcs : demonstrate.
3. In the parallelogram A B C D straight lines join the
GEOMETRY— NUMERICAL PROBLEMS, 81
middle point E of side B C with the vertex A, and the middle
point F of side A D with the vertex C. Show that A E and F C
are parallel and that the diagonal B D is trisected .
4. Show that the areas of similar triangles are to each other
as the squares of their homologous sides.
5. How do you divide a line in extreme and mean ratio ?
6. What are the immediate propositions which lead up to
the determination of the area of the circle of radius unity, and
how is this area determined? No demonstrations are re-
quired.
University of Pennsylvania, June, 1893.
TWO HOURS.
1. If two straight lines intersect each other, the opposite
(or vertical) angles are equal.
The straight lines which bisect a pair of adjacent angles
formed by two intersecting straight lines are perpendicular
to each other.
2. If each side of a polygon is extended, the sum of the ex-
terior angles is four right angles.
3. In the same circle, or in equal circles, equal chords are
equally distant from the centre, and of two unequal chords,
the less is at the greater distance from the centre.
The least chord that can be drawn in a circle through a
given point is the chord perpendicular to the diameter through
the point.
4. Two triangles are similar when they are mutually equi-
angular,
5. Show how to find a mean proportional between two
given lines.
6
82 GEOMETRY— NUMERICAL PROBLEMS.
6. The square described upon the hypotenuse of a right-
angled triangle is equivalent to the sum of the squares de-
scribed upon the other two sides. {Give the pure geometric
proof.)
7. In a triangle any two sides are reciprocally proportional
to the perpendiculars let fall upon them from the opposite
vertices.
8. The area of the regular inscribed triangle is half the
area of the regular inscribed hexagon.
University of Pennsylvania, June, 1895. v
TIME : ONE HOUR AND A HALF.
Give all the work.
1. The interior and exterior bisectors of any angle of a tri-
angle divide the opposite side into segments which are pro-
portional to the adjacent sides.
2. If two of the medial lines of a triangle are equal, the tri-
angle is an isosceles.
3. The area of a rhombus is 240 and its side is 17, find its
diagonals.
4. Construct a square whose area shall be five times the area
of a given square.
5. The parallelogram formed by lines joining the middle
points of the adjacent sides of a quadrilateral is equivalent to
one-half the quadrilateral.
6. If the interior bisector of the angle C and the exterior bi-
sector of the angle B of a triangle ABC meet at D, prove
that angle B D C = i A.
7. In any triangle the product of two sides is equal to the
diameter of the circumscribed circle multiplied by the per-
pendicular to the third side from its opposite vertex.
GEOMETRY— NUMERICAL PROBLEMS. 83
8. Define tt. Give a method for computing an approximate
value of IT.
9. If the radius of a circle is r, what is the side of the in-
scribed decagon ?
TTniversity of Pennsylvania, September, 1895.
TIME : ONE HOUR AND A HALF.
Give all the work.
1. The lines joining the middle points of the adjacent sides
of any quadrilateral form a parallelogram whose perimeter is
equal to the sum of the diagonals of the quadrilateral.
2. Prove that the bisectors of the angles of a rectangle form
a square.
3. The three medial lines of a triangle intersect in one point
which divides each medial line in the ratio 1 : 3.
4. If from a point a tangent and a secant to a circle are
drawn, the tangent is a mean proportional between the whole
secant and its external segment.
5. Similar triangles are to each other as the squares of two
homologous sides.
6. Divide a given straight line in extreme and mean ratio.
7. Construct a triangle which shall be similar to, and three
times as large as, a given triangle.
8. From a given point without a circle draw a secant whose
external and internal segments shall be equal.
9. If the radius of a circle is 2, what is the area of a sector
whose central angle is 152° ?
84 OEOMETRY— NUMERICAL PROBLEMS.
University of Pennsylvania, June, 1896.
TIME : TWO HOURS.
1. Define : Altitude of a triangle, medial line, regular poly-
gon, inscribed angle, segment and sector of a circle.
2. If two parallels are cut by a straight line, the alternate
exterior angles are equal.
3. Either side of a triangle is greater than the difference of
the other two.
4. The sum of the angles of any polygon is equal to twice as
many right angles as the polygon has sides, less four right
angles.
5. The areas of similar triangles are to each other as the
squares of their homologous sides.
6. The lines joining the middle points of the sides of any
quadrilateral is a parallelogram.
7. Construct a square equivalent to a given triangle.
8. The line joining the middle points of the two non-paral-
lel sides of a trapezoid is 12^ inches, the distance between the
parallel sides is 8| inches, what is the side of a regular hexa-
gon equivalent to the trapezoid ?
9. Define n. Outline a method for computing it.
University of Pennsylvania, September, 1896.
TIME : TWO HOURS.
1. Define : An angle (right, acute, and obtuse), tangent to a
circle, regular polygon, mention all different kinds of paral-
lelograms.
2. If two straight lines are cut by a third, making the al-
ternate-interior angles equal, the two sides are parallel.
GEOMETRY— NUMEBIGAL PROBLEMS. 86
3. In any triangle the greater angle lies opposite the greater
side.
4. What is each angle in a regular pentagon, regular hexa-
gon, regular dodecagon ?
5. If in a right triangle a perpendicular be drawn from the
vertex of the right angle to the hypotenuse, the perpendicular
is a mean proportional between the segments of the hypote-
nuse.
6. The lines joining the middle points of the sides of a rhom-
bus form a rectangle.
7. Construct a square equivalent to a given pentagon.
8. The base of a triangle is 7.345 inches and the altitude
4.756 inches, what is the side of a regular triangle which has
the same area as the given triangle ?
9. Find the area of a regular hexagon inscribed in a circle
whose radius is 11.529 inches.
Cornell, 1894.
1. If two triangles have two sides of the one equal, respec-
tively, to two sides of the other, but the included angle of
the first greater than the included angle of the second, then
the third side of the first is greater than the third side of the
second. Prove this ; and state the converse.
2. Prove that lines drawn through the vertices of a triangle
to the middle points of the opposite sides meet in a point.
How do the areas of the three triangles formed by joining
this point to the vertices of the original triangle compare ?
Why?
3. If equilateral triangles be constructed upon each side of
any given triangle, prove that the lines drawn from their
outer vertices to the opposite vertices of the given triangle
are equal.
86 GEOMETRY— NUMERICAL PROBLEMS.
4. From any point P, outside of a circle whose centre is at
O, two tangents are drawn touching the circle at A and B ;
at Q, a variable point in the smaller arc AB, a tangent is
drawn cutting the other two tangents in H and K. Prove
that the perimeter of the triangle P H K is constant, and also
that the angle H O K is constant. Compare this angle with
the angle P.
5. If similar parallelograms be described upon the three
sides of a right triangle as homologous sides, prove that the
parallelogram described upon the hypotenuse is equivalent to
the sum of those described upon the other two sides.
6. Prove that the sum of the perpendiculars drawn to the
sides of a regular polygon from any point P within the figure,
is equal to the apothem of the polygon multiplied by the
number of its sides.
State this proposition, so modified, that the point P may
be without the polygon.
7. Of all isoperimetric triangles having the same base, that
which is isosceles has the maximum area.
Cornell, 1895.
One question may he omitted.
1. The sum of the lines which join a point within a triangle
to the three vertices is less than the perimeter, but greater
than half the perimeter.
2. Two triangles are equal if the three sides of one are equal
respectively to the three sides of the other.
3. Construct through a point, P, exterior to a circle, a secant
P A B so that AB^" = P A x P B.
4. The radius of a circle is 6 inches ; through a point 10
inches from the centre tangents are drawn. Find the lengths
GEOMETRY— JSrUMERIGAL PROBLEMS. 87
of the tangents, also of the chord joining the points of con-
tact.
5. Construct a polygon similar to two given similar poly-
gons, and equivalent to their sum.
6. The bisector of an angle of a triangle divides the opposite
side into segments proportional to the other two sides.
7. The perimeter of an inscribed equilateral triangle is equal
to half the perimeter of the circumscribed equilateral triangle.
8. If one of the acute angles of a right triangle is double the
other, the hypotenuse is double the shorter side.
Johns Hopkins University, October, 1896.
1. Prove that the bisectors of the two pairs of vertical an-
gles formed by two intersecting lines are perpendicular to each
other.
2. Show that through three points not lying in the same
straight line one circle, and only one, can be made to pass.
3. The bases of a trapezoid are 16 feet and 10 feet respec-
tively ; each leg is 5 feet. Find the area of the trapezoid.
Also find the area of a similar trapezoid, if each of its legs is
3 feet.
4. Define regular polygon. Prove that every equiangular
polygon circumscribed about a circle is a regular polygon.
5. Prove that the opposite angles of a quadrilateral in-
scribed in a circle are supplements of each other.
6. Construct a square, having given its diagonal.
7. Prove that the area of a triangle is equal to half the
product of its perimeter by the radius of the inscribed circle.
8. What is the area of the ring between two concentric
circumferences whose lengths are 10 feet and 20 feet respec-
tively ?
88 aEOMETRT— NUMERICAL PROBLEMS,
Sheffield Scientific School, June, 1892.
[Note.— state at the head of your paper what text-book you have studied on the sub-
ject and to what extent.]
1. Prove the two propositions relating to the sum of the in-
terior angles of a convex polygon, and the sum of the exterior
angles formed by producing each side in one direction.
2. In a circle the greater chord subtends the greater arc,
and conversely.
3. When is a line said to be divided harmonically f From
the point P without a circle a secant through the centre is
drawn cutting the circle in A and B. Tangents are drawn
from P and the points of contact connected by a line cutting
A B in Q. Show that P and Q divide A B harmonically.
4. Derive an expression for the area of a regular polygon.
5. When two sides of a triangle are given at what angle must
fchey intersect if the area shall be maximum ? Prove your
answer.
Sheffield Scientific School, June, 1896.
[NoTB.— state at the head of your paper what text-book you have studied on the sub-
ject and to what extent.]
1. Two angles whose sides are parallel each to each are either
equal or supplementary. When will they be equal, and when
supplementary ?
3. An angle formed by two chords intersecting within the
circumference of a circle is measured by one-half the sum of
the intercepted arcs.
3. A triangle having a base of 8 inches is cut by a line par-
allel to the base and 6 inches from it. If the base of the
GEOMETRY— NUMERICAL PROBLEMS. 89
smaller triangle thus formed is 5 inches, find the area of the
larger triangle.
4. Construct a parallelogram equivalent to a given square,
having given the sum of its base and altitude. Give proof.
5. What are regular polygons? A circle may be circum-
scribed about, and a circle may be inscribed in, any regular
polygon.
Wesleyan University, September, 1896.
li HOURS.
1. The exterior angles of a polygon, made by producing each
of its sides in succession, are together equal to four right
angles.
The sum of the interior angles of a polygon is ten right
angles. How many sides has the polygon ?
2. An angle inscribed in a circle is measured by one-half of
the arc intercepted between its sides.
3. Show how to bisect a given angle.
4. The radius of a circle is 6 feet. What are the radii of the
circles concentric with it whose circumferences divide its area
into three equivalent parts ?
5. Show how to inscribe in a given circle a regular polygon
similar to a given regular polygon.
6. If two polygons are composed of the same number of
triangles, similar each to each, and similarly placed, the poly-
gons are similar.
90 GEOMETRY— NUMERICAL PROBLEMS,
The University of Chicago, September, 1896.
TIME AliLOWED, ONE HOUR AND FIFTEEN MINUTES.
[When required, give all reasons in full, and work out proofs and problems in
detail.]
1. Show that if on a diagonal of a parallelogram two points
be taken equally distant from the extremities, and these points
be joined to the opposite vertices of the parallelogram, the
four-sided figure thus formed will be a parallelogram.
2. State and prove the converse of the following theorem :
In the same circle, equal chords are equally distant from
the centre.
3. Given a circle, a point, and two straight lines meeting in
the point and terminating in the circumference of the circle.
State what four lines or segments form a proportion and in
what order they must be taken :
(1) When the point is outside the circle, and
{a) both lines are secants,
(&) one line is a secant, and the other a tangent,
(c) both lines are tangents.
(3) When the point is within the circle, and the two lines
are chords.
Prove in full (1) {a). Show that (1) (c) is a limiting case of
(1) (a).
4. To a given circle draw a tangent that shall be perpendic-
ular to a given line.
5. Show how to construct a triangle, having given the base,
the angle at the opposite vertex, and the median from that
vertex to the base. Discuss the cases depending upon the
length of the given median.
GEOMETRY— NUMERICAL PROBLEMS. 91
Massachusetts Institute of Technology, June, 1896.
[Every reason must be stated in full.]
1. If straight lines are drawn to the extremities of a straight
line from any point in the perpendicular erected at its middle
point, they make equal angles with the line and with the per-
pendicular.
3. Two right triangles are equal when the hypotenuse and
a side of one are equal, respectively, to the hypotenuse and a
side of the other.
3. Prove the formula for the sum of the angles of any poly-
gon. Define a regular polygon. How many degrees in each
angle of a regular heptagon ?
4. In the same circle or in equal circles chords equally dis-
tant from the centre are equal.
5. Two triangles are similar when their homologous sides
are proportional.
6. A hexagon is formed by joining in succession the middle
points of the sides of a given regular hexagon. Find the ratio
of the areas of these two hexagons.
7. If A B and A i B i are any two chords of the outer of two
concentric circles, which intersect the circumference of the
inner circle at P, Q, and Pi, Q i, respectively, prove :
AP. PB=AiPx. PiBi.
92 OEOMETBY—NUMEBIOAL PR WLEMS
Brown University, June, 1896.
1. Have you been over all the required work ?
2. The exterior angle of a triangle is equal to the sum of
the opposite interior angles.
3. Find a point equidistant from two given points P and
Q, and at a given distance C D from a given line A B.
4. If a secant and a tangent be drawn from a point without
a circle, the tangent is a mean proportional between the secant
and its external segment.
5. Similar triangles are to each other as the squares of their
homologous sides.
6. The diagonals drawn from a vertex of a regular penta-
gon to the opposite vertices trisect that angle.
Vassar College, September, 1895.
1. Find the area of a right triangle if the perimeter is 60
feet, and its sides are as 3 : 4 : 5.
2. The sides of a triangle are 8, 9, 13 ; is the greatest an^le
acute, right, or obtuse ?
3. The perpendicular erected at the middle point of the base
of an isosceles triangle passes through the vertex and bisects
the angle at the vertex.
4. If two circles touch internally, and the diameter of the
smaller is equal to the radius of the larger, the circumference
of the smaller bisects every chord of the larger which can be
drawn through the point of contact.
5. If two similar triangles A B C, D E F, have their homo-
logous sides parallel, the lines AD, BE, C F which join their
homologous vertices meet in the same point.
GEOMETRY— NUMERICAL PROBLEMS. 93
Vassar College, June, 1896.
1. Define similar triangles.
State all the cases of similar triangles, and prove one.
2. Construct a right triangle, having given the hypotenuse
and the sum of the legs.
3. Prove that the radius of a circle inscribed in an equi-
lateral triangle is equal to one-third of the altitude of the tri-
angle.
4. Construct the fourth proportional when three are given.
5. Find the area of an isosceles triangle if the base is equal
to 36 feet and one leg is equal to 30 feet.
6. To divide a given line in extreme and mean ratio. What
regular inscribed polygons may be constructed by means of
this division ? Prove your statement.
Amherst College, June, 1895.
1. To construct a square that shall have to a given square
the ratio of 3 to 2.
2. The circumference of a circle is the limit of the perimeter
of a regular circumscribed polygon, as the number of sides of
the polygon is indefinitely increased.
3. If two polygons are composed of the same number of
similar triangles, similarly placed, the polygons are similar.
4. The sum of the squares on two sides of a triangle is equal
to twice the square on half the third side increased by twice
the square on the median to that side.
5. Find the locus of all points, the perpendicular distances
of which from two intersecting lines are to each other as 3
to 2.
94 OEOMETBY— NUMERICAL PROBLEMS.
Amherst College, June, 1896.
1. Two triangles having an angle of the one equal to an angle
of the other, and the including sides proportional are similar.
2. Inscribe a circle in a given triangle.
3. (1) When are two lines said to be ineommensurahle f
(2). Are 3f and Syy incommensurable ? Give the reason for
your answer. (3). Define a limit Mention some propositions
to which the method of limits is applied.
4. In an isosceles right triangle either leg is a mean propor-
tional between the hypotenuse and the perpendicular upon it
from the vertex of the right angle.
5. The area of an inscribed regular hexagon is equal to f of
that of the circumscribed regular hexagon.
Dartmouth College, 1894.
1. Name the different classes of triangles.
2. What are the conditions of similarity in triangles ?
3. The diameter of a circle is 25 feet. What is the perpen-
dicular distance to the circumference from a point in the dia-
meter 5 feet from either end.
4. One angle of a parallelogram is | of a right angle. What
values have the remaining angles ?
5. The segments of a given line are 4, 6, 7. Divide any
other line in the same proportion.
6. In any triangle the product of any two sides is equal to
the product of the segments of the third side formed by the
bisector of the opposite angle, plus the square of the bisector.
Demonstrate.
GEOMETRY— NUMERICAL PROBLEMS. 95
Wellesley College, June, 1896.
1. An angle formed by two tangents is how measured?
Prove.
2. The diagonals of a rhombus bisect each other at right
angles.
3. (a) If a line bisects an angle of a triangle and also bisects
the opposite side the triangle is isosceles.
(&) State and demonstrate the general case for the ratio of
the segments of the side opposite to a bisected angle.
4. With a given line as a chord, construct a circle so that
this chord shall subtend a given inscribed angle.
5. (a) On a circle of 4 feet radius, how long is an arc in-
cluded between two radii forming an angle of 30° ? Prove, de-
riving the formula employed.
(&) Find the area of the regular circumscribed hexagon of
a circle whose radius is 1.
6. Two similar triangles are to each other as the squares of
their homologous sides.
Bowdoin College, June, 1895.
1. The perpendiculars from the vertices of a triangle to the
opposite sides meet in a common point.
3. Upon a given straight line describe an arc of a circle
which shall contain a given angle.
3. In any triangle the square of a side opposite an acute
angle equals the sum of the squares on the other two sides
minus twice the product of one of these sides by the projec-
tion of the other upon it.
96 GEOMETRY— NUMERICAL PROBLEMS.
4. The length of a tangent to a circle, from a point eight
units distant from the nearest point on the circumference, is
twelve units. Find the diameter of the circle.
5. Two triangles having an angle of one equal to an angle
of the other are to each other as the product of the sides in-
cluding the equal angles.
6. Find the ratio of the radius of a circle to the side of the
inscribed square.
7. The area of a sector of sixty degrees is two hundred nine,
and forty-four hundredths square inches. Find the length
of the radius.
Bowdoin College, June, 1896.
1. The bisectors of the three angles of a triangle meet in the
centre of the inscribed circle.
2. The circumference of a circle described on one of the equal
sides of an isosceles triangle as a diameter passes through the
middle point of the base.
3. If two chords be drawn through a fixed point within a
circle, the product of the segments of one chord equals the
product of the segments of the other.
4. The radius of a circle is 10 ; inscribe within it a regular
decagon and compute the length of its side.
5. In an acute-angled triangle the side AB = 10, side
A C = 7, the projection of A C on A B is 3.4. Construct the
triangle and compute the third side, B C.
6. The area of one circle is 100 ; find the circumference of
another circle described on the radius of the first as a di-
ameter.
GEOMETRY— NUMEBIGAL PROBLEMS, 97
TTniversity of California, August, 1896.
1. Prove that if two sides of one of two triangles be equal to
two sides of the other, and the angles opposite one pair of
equal sides be equal, the angles opposite the other pair of sides
are either equal or supplementary.
2. To construct a triangle having given the base, one angle
at the base and the altitude.
3. Prove that the straight lines drawn at right angles to the
sides of a triangle at their middle points meet in a point.
4. Prove that, if an angle at the centre of a circle and an
angle at the circumference be subtended by the same arc, the
angle at the circumference is one-half of the angle at the
centre.
5. If the middle points of adjacent sides of a convex quadri-
lateral be connected by straight lines what figure is formed ?
What is the relation between the areas of this figure and the
quadrilateral ? Prove your statements.
6. To divide a given straight line internally in extreme and
mean ratio. What regular polygons may be inscribed in a
circle by means of this construction ? Show (without proof)
how one of these polygons is constructed.
7. Present, in the clearest language and most perfect form
you can command, some proposition of your own choosing.
98 OEOMETRT—NUMEBIGAL PROBLEMS.
Bryn Mawr College, September, 1896.
TWO AND ONE-HALF HOURS.
1. Show how to draw a perpendicular from a given point to
a given line, the point not lying on the line. Show that only
one such perpendicular can be drawn.
2. Prove that if two parallel lines are cut by a third straight
line, the two interior angles on one side of the transversal are
together equal to two right angles.
Prove that the lines bisecting the angles of a parallelogram
form a rectangle.
3. Define a parallelogram ; prove that the opposite sides and
angles are equal, and that the diagonals bisect one another.
Prove that any line through the intersection of the diagonals
of a parallelogram bisects the figure.
4. Prove that in any circle angles at the centre have the
same ratio as the arcs on which they stand.
Show how to divide the circumference of a circle into three
parts that shall be in the ratio 1:2:3.
5. Prove that an angle formed by two chords intersecting
within a circle is measured by one-half the sum of the inter-
cepted arcs.
A B C D is a quadrilateral in a circle ; P, Q, R, S, are the
points of bisection of the arcs A B, B C, CD, DA. Show that
P R is perpendicular to Q S.
6. Prove that the sum of the squares of two sides of a tri-
angle is equal to twice the square of half the base increased by
twice the square of the distance from the vertex to the bisec-
tion of the base. Apply this to find a line whose extremities
shall lie one on each of two given concentric circles, the line
itself being bisected at a given point.
7. Prove that three lines drawn through the vertices of a
triangle to bisect the opposite sides meet in a point, and de-
GEOMETUT— NUMERICAL PROBLEMS. 99
termine the position of tliis point on any one of the three
bisectors.
Show how to construct a triangle when the lengths of the
three medians are given.
8. Define the tangent to a circle at a point ; and prove that
the tangent at a point is perpendicular to the diameter through
the point.
Two circles whose centres are A, B, meet at a point P. Prove
that if A P touch the circle whose centre is B, then B P will
touch the circle whose centre is A.
9. State and prove the relation between the segments of in-
tersecting chords of a circle. Apply this to find a mean pro-
portional to two given lines.
Boston University, June, 1896.
TIME 1 H. 30 M.
[Candidates will quote authority for each step.]
1. The extremities of the base of an isosceles triangle are
equally distant from the opposite sides. Prove.
2. Two unequal circles have a common centre. Prove that
chords of the greater circle, which are tangent to the lesser
circle, are equal.
3. The sides of a triangle are 4, 7, 10 ; find the sides of a
similar triangle having nine times the area of the first. Prove
the principle employed.
4. Homologous altitudes of similar triangles have the same
ratio as any two homologous sides. Prove.
5. The sum of the perpendiculars from any point within an
equilateral triangle to the three sides is equal to the altitude
of the triangle. Prove.
100 GEOMETRY— NUMERICAL PROBLEMS,
Boston University, September, 1896.
TIME 1 H. 30 M.
[Candidates will quote authority for each step.]
1 . Connect the mid points of the adjacent sides of a rhom-
bus and prove character of the figure formed.
2. Chords meeting a diameter at the same point and making
the same angle with it are equal. Prove.
3. The radius of a circle is 10 feet. Find the side of an
equilateral triangle having the same area as the circle.
4. In any triangle the square of the side opposite an acute
angle is equal to the sum of the squares of the other two sides
diminished by twice the product of one of these sides and the
projection of the other side upon it. Prove.
5. Two equivalent triangles have a common base and lie on
opposite sides of it. Prove that the line joining their vertices
is bisected by the base, produced, if necessary.
Vanderbilt University, May 24, 1894.
1. The circles described on two sides of a triangle as diame-
ters intersect on the third side.
2. The diagonals of a trapezoid divide each other into seg-
ments which are proportional.
3. Similar triangles are as the squares of their homologous
sides.
4. Two quadrilaterals are equivalent when the diagonals of
one are respectively equal and parallel to the diagonals of the
other.
5. The area of a ring bounded by two concentric circumfer-
ences is equal to the area of a circle having for its diameter
OEOMETRT— NUMERICAL PROBLEMS. 101
a chord of the outer circumference tangent to the inner cir-
cumference.
6. A swimmer whose eye is at the surface of the water can
just see the top of a stake a mile distant ; the stake proves to
be eight inches out of the water ; required the radius of the
earth.
New Jersey State College for the Benefit of Agriculture
and the Mechanic Arts, New Brunswick, N. J., June,
1891.
1. Define the various kinds of triangles and quadrilaterals.
2. If two straight lines cut each other, the vertical angles
are equal.
3. An angle formed by a tangent and a chord from the point
of contact is measured by one-half the intercepted arc.
4. If a variable tangent meets two parallel tangents it sub-
tends a right angle at the centre.
5. The bisector of an angle of a triangle divides the opposite
side into segments proportional to the adjacent sides.
6. A parallelogram is divided by its diagonals into four tri-
angles of equal area.
7. The areas of two similar segments are to each other as
the squares of their radii.
8. The diameter of a circle is 5 feet ; find the side of the in-
scribed square.
9. Find a side of the circumscribed equilateral triangle, the
radius of the circle being 1/3,
10. Find the radius of the circle in which the sector of 45°
is .125 square inches.
LOGAEITHMS.
1 . The logarithm of a number is the exponent of the
power to which an assumed number must be raised to pro-
duce the first number.
2. Since logarithms are exponents, the principles estab-
lished in Theory of Exponents in Algebra, hold in loga-
rithms, and are the very principles which make logarithms
serviceable ; as follows :
I. The logarithm of a product is equal to the sum of the
logarithms of its factors.
II. The logarithm of a quotient is equal to the logarithm of
the dividend minus the logarithm of the divisor.
III. The logarithm of any power of a number is equal to
the logarithm of the number multiplied by the exponent of the
power.
IV. The logarithm of a root of a number is equal to the
logarithm of the number divided by the index of the root.
3. The only kind of logarithms with which we have to
do here are those in which the assumed number, called the
base, is 10.
Such logarithms are termed Common Logarithms.
1
10^ = 10000
i«-' = io^ =
.1
103= 1000
^'-' = w =
.01
102 ^ 100
10- = 103 =
.001
101 ^ 10
^'-'=w=
.0001
100= 1
LOGARITHMS. 103
Thus, by definition, log 10000 = 4 ; log 1000 = 3, etc.
But all numbers which are not integral powers of 10, as
the above are, must have a fractional, decimal, part to
their logarithms.
Thus, the logarithm of any number between
10 and 100
would lie between
that is, it would be
1 and 2,
1 + a decimal.
Of any number between
1 and 10,
the logarithm would be
+ a decimal ;
between .1 and 1,
— 1 4- a decimal;
between .01 and .1,
— 2 + a decimal ;
and so on.
This decimal part of a logarithm is called the mantissa ;
the integral part, the characteristic.
From the above it is seen that all mantissas are positive.
And to show that a negative sign belongs to the character-
istic only, it is placed above the characteristic, thus :
log .03152 = 2.49859.
4. Moving the decimal point to right or left in any
number multiplies or divides that number by ten or some
integral power of ten. And as the logarithm of a product
is equal to the logarithm of the multiplicand plus the
104 LOGARITHMS.
logarithm of the multiplier, and the logarithm of a quotient
is equal to the logarithm of the dividend minus the loga-
rithm of the divisor, and the logarithm of the multiplier
and divisor in such cases (moving the decimal point) is an
integer, the only part of a logarithm affected by a change
of the decimal point in a number is the integral part, the
characteristic.
Then all numbers which differ only in the position of the
decimal point have the same mantissa.
5. A careful study of Art. 3 will make plain the follow-
ing rules in regard to the characteristic :
I. If the number is greater than 1, the characteristic is
one less than the number of places to the left of the decimal
point,
II. If the number is less than 1, the characteristic is nega-
tive, and is one more than the number of zeros between the deci-
mal point and the first significant figure of the decimal.
Thus, the characteristic of log 378.37 is
^;
ii i( a c( ^0917 '' —
■3;
t( (c a a 5 391 ii
0;
(( <i a i( 8395 '' —
-1.
6. The rules for determining the position of the decimal
point in a number corresponding to any given logarithm
are just the converse of the above.
I. When the characteristic is positive, the number of places
to the left of the decimal point is one more than the number
of units in the characteristic.
II. When the characteristic is negative, the number is a
decimal, and the number of zeros between the decimal point
and the first significant figure is one less than the number of
units in the characteristic.
LOGARITHMS. 105
7. To avoid certain difficulties in the use of logarithms
every logarithm whicli has a negative characteristic should
have 10.00000 — 10 (equal to 0) added to it.
Thus, 2'.37931 should be written 8.37931 — 10 ;
4.92012 '' " '' 6.92012 — 10 ;
1.72082 '' '' " 9.72082 — 10.
8. The cologarithm of a number, or ths arithmetical
complement of the logarithm of the number, is the loga-
rithm of the reciprocal of that number.
Thus, colog 317 = log ^\^ ; but by II., Art. 2, log -^ij
= log 1 — log 317 = — log 317 = — log 317.
And so the cologarithm of any number is equal to the
negative logarithm of that number.
9. Since to subtract a quantity is to add that quantity
with its sign changed, rule II., Art. 2, may be stated :
The logarithm of a quotient is equal to the logarithm of the
dividend plus the negative logarithm, or cologarithm, of the
divisor.
This is the form of the rule that should be invariably
applied in practice.
10. Negative logarithms should always have zero in the
form 10.00000 — 10 added to them before they are em-
ployed otherwise in an example. This altered form of the
negative logarithm may well be distinguished by the name
cologarithm, and is so distinguished hereafter.
Thus,
1777 ( log 1777 + (— log 8943) = log 1777 + colog 8943
^Ogoo7o = 1 II II II II
( 3.24969 + (- 3.95148) = 3.24969 + 6.04851 - 10
9.39820 - 10.
i=
106 LOGARITHMS.
11. This method of using logarithms avoids all subtrac-
tion of logarithms, except in finding cologarithms ; and
these are very easily found by the following rule :
Begin with the characteristic of the logarithm and subtract
each figure from 9, except the last significant figure, and sub-
tract that from 10.
Thus, log 8409 = 3.92474 ; and colog 8409 =
10 — 3.92474— 10 --= 6.07526 — 10.
By subtracting from left to right in this way the colog-
arithm of any number of four figures or less can be read
right from the table almost as easily as the logarithm it-
self, after some practice.
12. The following points should be carefully noted in
using logarithms that have negative characteristics :
1. In getting the cologarithm, the 10 following the
mantissa destroys the second 10 of the 10.00000 — 10
added.
Thus,
- (9.85920 — 10)= — 9.85920 + 10 )
colog .7231 = -J 10.00000 — 10 = 10.00000 — 10 [
0.14080. ;
2. In adding or multiplying, superfluous tens should
be dropped.
Thus: adding, 9.87349 — 10
8.96454 — 10
18.83803 — 20 = 8.83803 — 10 ;
Multiplying, 9.76604 — 10
3
29.29812 — 30 = 9.29812 — 10.
3. In dividing, a sufiicient number of tens should be
added, before and after the mantissa, to make the number
of tens after the mantissa equal to the number of units in
the divisor.
LOGARITHMS. 107
Thus, 9.76155 — 10 _ 29.76155 — 30 _ „ „^„.^ ,^
^r — — y.y/cUO/c — iXj y
8,98304-W ^ 38.98304-40 ^ g_^^^^^ _ ^^_
HOW TO USE THE TABLE.
13. The first page of the table gives the characteristics
and mantissas of numbers up to 100. The remainder of the
table gives only mantissas. The characteristic is to be sup-
plied by the rules of Art. 5. The first three figures of the
number are found in the left-hand column, marked N at
the top and the bottom. The fourth figure of the number
is found in the first line at the top and the bottom. The
mantissa is then found in the same horizontal line with the
first three figures, and in the same vertical column with
the fourth figure. The first two figures of the mantissa
are printed only in the first column. In every case where
an asterisk is found the first two figures of the mantissa are
found in the first column of the next line below.
14. To find the logarithm of a number.
1. To find the logarithm of a number of four figures, as
8713.
By Art. 5, the characteristic = 3.
By the table as explained above the mantissa = .94017.
Hence log 8713 = 3.94017.
2. To find the logarithm of a number of five or more
figures, as 35647.
The characteristic = 4.
The mantissa for 3564 = .55194.
'' 3565 = .55206.
That is, an increase of one unit in the number, at this
point in the table, makes an increase of .00012 in the man-
108 LOGARITHMS.
tisga. Then an increase of . 7 of a unit (7 in the fifth place is
. 7 of 1 in the fourth place) in the number will make an in-
crease of .7 of .00012 in the mantissa = .000084.
( 4.55194
Therefore log 35647 = ] .00008
(4.55202.
Note 1. — The difference between any two consecutive
mantissas, as .00012 above, is called the tabular difference,
and is printed in the right-hand column of the table under
D.
Note 2. — When all these tabular differences are multi-
plied by the nine significant digits expressed as tenths,
they give a table of proportional parts. This table furnishes,
ready-made, the amounts to be added to obtain logarithms
of five-figure numbers. Only a portion (the most helpful,
however) of such a table of proportional parts, is given
with this table of logarithms, p. 115. It is sufficient to
make their use and meaning plain.
Note 3. — In calculating proportional parts and in all
calculations with tabular differences they are treated as
whole numbers, as they bear the same relation to their
mantissas that whole numbers do to whole numbers.
Note 4. — In calculating additions to be made to a loga-
rithm all figures that follow the fifth are rejected. When
the sixth figure is 5, or greater, the fifth figure is increased
by 1. When the last significant figure of a logarithm is 5,
it means that such an increase has been made for rejected
figures following the fifth place.
3. To find the logarithm of 18.7432.
The characteristic = 1.
[As already explained in Art. 4, the position of the deci-
mal point does not affect the mantissa in the least.]
LOGARITHMS. 109
Mantissa for 1874 = 27274.
" 1875 = 27300.
That is an increase of one in the number here makes an
increase of 26 in the mantissa. Then an increase of .32 of
one (32 following the fourth place is .32 of 1 in the fourth
place) in the number will make an increase of .32 of 26 in
the mantissa = 8.32.
( 1.27274
Hence log 18.7432 = \ 8
(1.27282.
Note 5. — The process employed in finding the logarithm
of a number of more than four figures is called interpolation.
15. How to find the number corresponding to the logarithm.
1. To find the number corresponding to the logarithm
0.56514.
The mantissa increases constantly throughout the table.
Follow the first column of mantissas till 56 is found, as
the first two figures of the mantissa. Continuing 514 is
easily found in the same horizontal line with 367 and in
the column under 4.
Hence the number (placing the decimal point by Art.
6) = 3.674.
2. To find the number corresponding to the logarithm
8.26470 — 10.
This mantissa cannot be found in the table.
The nearest mantissa less than 26470 = 26458.
'' " " larger '' 26470 = 26482.
The number corresponding to mantissa 26458 (disregard-
ing the decimal point) is 1839. For a mantissa 24 greater
(26482) the corresponding number is 1840, that is, an in-
crease of 24 in the mantissa, at this point in the table,
means an increase of 1 in the number. Then an increase
of 12, which is the amount the given mantissa, 26470, ex-
110 LOGARITHMS.
ceeds the mantissa 26458, would mean an increase of J|
of I, ^ .5.
Hence the number = .018395.
3. To find the number corresponding to the logarithm
1.71895.
The next smaller mantissa = 71892.
Then the given mantissa is 3 larger ; and as the tabular
difference is 8, | of 1 = .375 must be added to 5235 the
number corresponding to mantissa 71892.
Hence the number = 52.3538.
Note 1. — Numbers corresponding to given logarithms
should not be carried to more than five or six significant
figures, in a five-place table.
Note 2. — Art. 4 makes it clear that the mantissa for
200 is the same as the mantissa for 2000 ; for 375, the same
as for 3750, etc. So the mantissa for any number of three
figures is found in the column and in the same horizontal
line with these three figures in the N column.
Note 3. — A negative quantity cannot be a power of a
positive quantity, and hence a negative quantity, as such,
has no logarithm. Hence when negative quantities occur
in any example worked by logarithms, the negative sign is
absolutely disregarded, except so far as it affects the sign
of the result.
EXAMPLES.
16. Find by logarithms the values of the following :
394.1 X .9385 . . ^
1. Given ^ = --^^^^3—; find ^.
log 394.1 = 2.59561
log .9385 = 9.97243 - 10
colog .02003 = 1.69832
log X = 4.26636 = log 18465.4
.-. 2:=: 18465.4
^ ^. (801.012) 2 X (.0315)3.
2. Given x ^- -^ ^^j^^, ; find x.
log (801.012) 2 = 2.90364 x 2 = 5.80728
log (.0315)3 = (8.49831 - 10) x | - 7.74747 - 10
colog (1.3907)*= (9.85677 - 10) x ^ = 9.97135 - 10
loga;= 3.52610
... :z; = 3358.2
3. 95.37 X .0313.
4. (- 93985) X 1.0484.
5. .0008601 X 1.28865.
5008.4
9.394
.93284 X 91.3009
10.1029
-^4
9.8743'
112 LOGARITHMS.
9.
10.
.03494 X (- 9432)
.00411 X 3753.6
- 111.121
- 4.943
11. (3.1835) ^
12 ^
(1197)§
13. (.311)«.
14. v^. 0000009431.
15 ' "'
(-f)'
16. 34985 X (.00039)*
F9i)i
17. (51^)1.
jg (— 419)§ X (—90.071)
* (10016) « X (—.11101) X 1399*
j9 3 /(1.0642)'^x.l098
V (683.51)8
^^- v^9i7 Afnooff:
21. 7^02053 X .0010997 x .32024
^ .091352
22 (1^)'
23 / 311 X 497 X 7.3 U
I ( 1 9843000) V*
26. ^|-^^|.
LOGARITHMS, 113
27. •|(1000)?-^(80009)U!
28. (911 X 10003)1.
29 '7 .40071 X (.00 352)r
V (.09045321)* *
30. (-.l)i X (lOOO)i X ^."OT
31. (3+)2i
33. (21|)«-2-i.(80J)i-3.
34 ^-7 (444)^ X (.00041007)^-^
' V (9.8563)i
35 " / (15.434)-^ X (3897.3)1^^ x .41984
V (.000372)2-3 X (784.96)3 x 5013.4 x (.003)4*
TABLES
COMMON LOGARITHMS OP NUMBERS
Giving Characteristics and Mantissas of Logarithms of Numbers
FROM 1 to 100, AND MANTISSAS ONLY OF NUMBERS FROM 100 TO 10000.
LOGARITHMS OP NUMBERS.
N
Log.
N
Log.
N
Log.
N
Log.
1
0.00000
26
1.41497
51
1.70757
76
1.88081
2
0.30103
27
1.43136
52
1.71600
77
1.88e>49
3
0.47712
28
1.44716
53
1.72428
78
1.89209
4
0.60206
29
1.46240
54
1.73239
79
1.89763
5
0.69897
30
1.47712
55
1.74036
80
1.90309
6
0.77815
31
1.49136
56
1.74819
81
1.90849
7
0.84510
32
1.50515
57
1.75587
82
1.91381
8
0.90309
33
1.51851
58
1.76343
83
1.91908
9
0.95424
34
1.53148
59
1.77085
84
1.92428
10
1.00000
35
1.54407
60
1.77815
85
1.92942
11
1.04139
36
1.55630
61
1.78533
86
1.93450
12
1.07918
37
1.56820
62
1.79239
87
1.93952
13
1.11394
38
1.57978
63
1.79934
88
1.94448
14
1.14613
39
1.59106
64
1.80618
89
1.94939
15
1.17609
40
1.60206
65
1.81291
90
1.95424
16
1.20412
41
1.61278
66
1.81954
91
1.95904
17
1.23045
42
1.62325
67
1.82607
92
1.96379
18
1.25527
43
1.63347
68
1.83251
93
1.96848
19
1.27875
44
1.64345
69
1.83885
94
1.97313
20
1.30103
45
1.65321
70
1.84510
95
1.97772
21
1.32232
46
1.66276
71
1.8.5126
96
1.98227
22
1.34242
47
1.67210
72
1.85733
97
1.98677
23
1.36173
48
1.68124
73
1.86332
98
1.99123
24
1.38021
49
1.69020
74
1.86923
99
1.99564
25
1.39794
50
1.69897
75
1.87506
100
2.00000
116
COMMON LOGARITHMS OF NUMBERS.
N
O
1
2
3
4
5
6
7
8
9
D
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
00 000
043
087
130
173
217
260
303
346
389
43
432
860
01284
703
02119
531
938
03 342
743
475
903
326
745
160
572
979
383
782
518
945
368
787
202
612
*019
423
822
561
988
410
828
243
653
*060
463
862
604
*030
452
870
284
694
*100
503
902
647
*072
494
912
325
735
5^141
543
941
689
*115
536
953
366
776
*181
583
981
732
*157
578
995
407
816
*222
623
*021
775
*199
620
*036
449
857
*262
663
*060
817
*242
662
*078
490
898
*302
703
*iOO
43
42
42
42
41
41
40
40
40
04139
179
218
258
297
336
876
415
45i
493
39
532
922
05 308
690
06 070
446
819
07188
555
571
961
346
729
108
483
856
225
591
610
999
385
767
145
521
893
262
628
650
*038
423
805
183
558
930
298
664
689
*077
461
843
221
595
967
335
700
727
*115
500
881
258
633
3^004
372
737
766
*154
538
918
296
670
*041
408
773
805
*192
576
956
333
707
*078
445
809
844
*231
614
994
371
744
*115
482
846
883
*269
652
*032
408
781
*151
518
882
39
39
38
38
38
37
87
37
36
918
954
990 *027
*063
*099 *135
*171
*207
*243
36
08 279
636
991
09 342
691
10 037
380
721
11059
314
672
*026
377
726
072
415
755
093
350
707
*061
412
760
106
449
789
126
386
743
*096
447
795
140
483
823
160
422
778
*132
482
830
175
517
857
193
458
814
*167
517
864
209
551
890
227
493
849
j*202
i 552
899
243
585
924
261
529
884
*237
587
934
278
619
958
294
565
920
*272
621
968
312
653
992
327
600
955
*307
656
*003
346
687.
*025
361
36
35
35
35
35
34
34
34
34
N
O
1
2
3
4
5
6
7
8
9
D
PP
1
2
3
4
5
6
7
8
9
44
4.4
8.8
13.2
17.6
22.0
26.4
30.8
35.2
39.6
43
4.3
8.6
12.9
17.2
21.5
25.8
30.1
34.4
38.7
42
4.2
8.4
12.6
16.8
21.0
25.2
29.4
33.6
37.8
41
4.1
8.2
12.3
16.4
20.5
24.6
28.7
32.8
36.9
40
4.0
8.0
12.0
16.0
20.0
24.0
28.0
32.0
36.0
39
3.9
7.8
11.7
15.6
19.5
23.4
27.3
31.2
35.1
38
3.8
7.6
11.4
15.2
19.0
22.8
26.6
30.4
34.2
37
3.7
7.4
11.1
14.8
18.5
22.2
25.9
29.6
33.3
86
3.6
7.2
10.8
14.4
18.0
21.6
25.2
28.8
82.4
COMMON LOGARITHMS OF NUMBERS.
117
N
1
2
3
4
5
6
7
8
9
D
130
131
182
133
134
135
136
137
138
139
140
141
143
143
144
145
146
147
148
149
150
151
153
153
154
155
156
157
158
159
11394
438
461
494
538
561
594
638
661
694
33
737
12 057
385
710
13 033
354
672
988
14 301
760
090
418
743
066
386
704
*019
333
793
123
450
775
098
418
735
*051
364
826
156
483
808
130
450
767
*083
395
860
189
516
840
163
481
799
*114
436
893
333
548
873
194
513
830
*145
457
936
354
581
905
336
545
863
*176
489
959
287
613
937
358
577
893
*208
530
992
330
646
969
390
609
935
*339
551
*024
352
678
*001
323
640
956
*270
583
33
33
33
33
33
32
32
31
31
613
644
675
706
737
768
799
829
860
891
31
932
15 329
534
836
16 137
435
732
17 036
319
953
259
564
866
167
465
761
056
348
983
390
594
897
197
495
791
085
377
*014
320
625
927
227
534
820
114
406
*045
351
655
957
256
554
850
143
435
W6
381
685
987
386
584
879
173
464
*106
412
715
*017
316
613
909
202
493
*137
442
746
*047
346
643
938
331
523
*168
473
776
*077
376
673
967
260
551
*198
503
806
*107
406
702
997
289
580
31
31
30
30
30
30
29
29
39
609
638
667 ; 696
725
754
782
811
840
869
39
898
18184
469
752
19 033
312
590
866
20140
936
213
498
780
061
340
618
893
167
955
241
526
808
089
368
645
921
194
984
270
554
837
117
396
673
948
222
*013
298
583
865
145
424
700
976
249
*041
327
611
893
173
451
728
9^003
276
*070
355
639
931
201
479
756
*0b0
303
*099
384
667
949
339
507
783
*058
3:30
*127
412
696
977
357
535
811
*085
358
*156
441
724
*005
385
563
838
m2
385
29
29
28
38
28
28
38
87
37
N
1
2
3
4
5
6
7
8
9
D
PP
1
2
3
4
5
6
7
8
9
35
3.5
7.0
10.5
14.0
17.5
21.0
24.5
28.0
31.5
34
3.4
6.8
10.2
13.6
17.0
20.4
23.8
27.2
30.6
33
3.3
6.6
9.9
13.2
16.5
19.8
23.1
26.4
29.7
32
3.2
6.4
9.6
12.8
16.0
19.2
22.4
25.6
28.8
31
3.1
6.3
9.3
12.4
15.5
18.6
31.7
34.8
27.9
80
3.0
6.0
9.0
12.0
15.0
18.0
21.0
24.0
27.0
39
3.9
5.8
8.7
11.6
14.5
17.4
20.3
23.2
36.1
28
3.8
5.6
8.4
11.2
14.0
16.8
19.6
22.4
25.2
37
3.7
5.4
8.1
10.8
13.5
16.3
18.9
21.6
24.3
118
COMMON LOGARITHMS OF NUMBERS.
N
1
2
3
4
5
6
7
8
9
D
160
161
162
163
20 412
439
466
493
520
548
575
602
629
656
27
683
952
21219
710
978
245
737
*005
272
763
*032
299
790
*059
325
817
*085
352
844
*112
378
871
^139
405
898
*165
431
925
*192
458
27
27
27
164
165
166
484
748
22 011
511
775
037
537
801
063
564
827
089
590
854
115
617
880
141
&43
906
167
669
932
194
696
958
220
722
985
246
26
26
26
167
168
169
170
171
172
173
272
531
789
298
557
814
324
583
840
350
608
866
376
634
891
401
660
917
427
686
943
453
712
968
479
787
994
505
763
*019
26
26
26
23 045
070
096
121
147
172
198
223
249
274
25
300
553
805
325
578
830
350
603
855
376
629
880
401
654
905
426
679
930
452
704
955
477
729
980
502
754
*005
528
779
*030
25
25
25
174
175
176
24 055
304
551
080
329
576
105
353
601
130
378
625
155
403
650
180
428
674
204
452
699
229
477
724
254
502
748
279
527
773
25
25
25
177
178
179
180
181
182
183
797
25 042
285
822
066
310
846
091
334
871
115
358
895
139
382
920
164
406
944
188
431
969
212
455
993
237
479
*018
261
503
25
24
24
527
551
575
600
624
648
672 696
720
744
24
768
26 007
245
792
031
269
816
055
293
840
079
316
864
102
340
888
126
364
912
150
387
935
174
411
959
198
435
983
221
458
24
24
24
184
185
186
482
717
951
505
741
975
529
764
998
553
788
*021
576
811
*045
600
834
*C68
623
858
*091
647
881
*114
670
905
*138
694
928
*161
24
23
23
187
188
189
27184
416
646
207
439
669
231
462
692
254
485
715
277
508
738
300
531
761
323
554
784
346
577
807
370
600
830
393
623
852
23
23
23
N
1
2
3
4
5
6
7
8
9
D
PP
27
26
25
24 23
22
1
2
3
2/
5.'
8.]
2.6
5.2
7.8
2.5
5.0
7.5
2.4
4.8
7.2
2.3
4.6
6.9
2.2
4.4
6.6
4
5
6
10. {
13.{
16J
10.4
13.0
15.6
10.0
12.5
15.0
9.6
12.0
14.4
9.2
11.5
13.8
8.8
11.0
13.2
7
8
9
18.1
21.(
24.(
18.2
20.8
23.4
17.5
20.0
22.5
16.8
19.2
21.6
16.1
18.4
20.7
15.4
17.6
19.8
COMMON LOOAIilTHMS OF NUMBERS.
119
N
1
2
3
4
5
6
7
8
9
D
190
191
193
193
875
898
921
944
967
989
*013
*035
*058
*081
23
28 103
330
556
136
353
578
149
375
601
171
398
633
194
431
646
217
443
668
340
466
691
363
488
713
385
511
735
307
533
758
23
23
33
194
195
196
780
29 003
226
803
026
248
825
048
270
847
070
293
870
093
314
892
115
336
914
137
358
937
159
380
959
181
403
981
203
425
23
22
22
197
198
199
200
201
202
203
447
667
885
469
688
907
491
710
929
513
733
951
535
754
973
557
776
994
579
798
*016
601
820
*038
633
843
*060
645
863
*08l
22
22
22
30103
125
146
168
190
211
333
355
276
298
23
320
535
750
341
557
771
363
578
792
384
600
814
406
631
835
428
643
856
449
664
878
471
685
899
492
707
920
514
728
942
82
81
81
204
205
206
963
31175
387
984
197
408
*006
218
429
*027
339
450
*048
260
471
*0()9
281
492
*091
303
513
*112
333
534
*133
345
555
*154
366
576
81
21
3L
207
208
209
210
211
212
213
597
806
32 015
618
827
035
6:39
848
056
660
869
077
681
890
098
702
911
118
733
931
139
744
953
160
765
973
181
785
9i)4
201
81
81
31
222
243
263
384
305
325
346
366
387
408
81
426
634
838
449
654
858
469
675
879
490
695
899
510
715
919
531
736
940
552
756
960
578
777
980
593
797
*001
613
818
*021
30
20
20
214
215
216
33 041
344
445
063
264
465
082
284
486
103
304
506
122
335
536
143
345
546
163
365
566
183
385
586
303
405
606
234
435
636
20
20
20
217
218
219
220
221
222
223
646
846
34 044
666
866
064
686
885
084
706
905
104
726
925
124
746
945
143
766
965
163
786
985
183
806
*005
203
826
*025
223
20
20
20
243
262
282
301
331
341
361
380
400
420
20
439
635
830
459
655
850
479
674
869
498
694
889
518
713
908
537
733
928
557
753
947
577
773
967
596
792
986
616
811
*005
20
19
19
224
225
226
35 025
218
411
044
238
430
064
357
449
083
276
468
103
395
488
123
315
507
141
334
536
160
353
545
180
372
564
199
392
583
19
19
19
227
228
229
603
798
984
622
813
*003
641
832
*021
660
851
*040
679
870
*059
698
889
*078
717
908
*097
736
927
*116
755
946
*135
774
965
*154
19
19
19
N
1
2
3
4
5
6
7
8
9
D
120
COMMON LOGARITHMS OF NUMBERS.
N
1
2
3
4
5
6
7
8
9
D
230
231
232
233
36173
192
211
229
248
267 286
805
324
342
19
361
549
736
380
568
754
399
586
773
418
605
791
436
624
810
455
642
829
474
661
847
493
680
866
511
698
884
580
717
903
19
19
19
234
235
236
922
37107
291
940
125
310
959
144
328
977
162
346
996
181
365
*014
199
383
*033
218
401
*051
236
420
*070
254
438
*088
273
457
18
18
18
237
238
239
240
241
242
243
475
658
840
493
676
858
511
694
876
530
712
894
548
731
912
566
749
931
585
767
949
603
785
967
621
803
985
639
822
*008
18
18
18
38 021
039
057
075
093
112 1 130
148
166
184
18
202
382
561
220
399
578
238
417
596
256
435
614
274
453
632
292 310
471 489
650 ! 668
328
507
686
346
525
703
364
543
721
18
18
18
244
245
246
739
917
39 094
757
934
111
775
952
129
792
970
146
810
987
164
b28
*005
182
846
*023
199
863
*041
217
881
*058
235
899
*076
252
18
18
18
247
248
249
250
251
252
253
270
445
620
287
463
637
305
480
655
322
498
672
340
515
690
358
533
707
375
550
724
393
568
742
410
585
759
428
602
777
18
18
17
794
811
829
846
863
881 1 898
915
933
950
17
967
40 140
312
985
157
329
*002
175
346
*019
192
364
*037
209
381
*054 *071
226 243
398 1 415
*088
261
432
*106
278
449
*128
295
466
17
17
17
254
255
256
483
654
824
500
671
841
518
688
858
535
705
875
552
722
893
569 586
739 756
909 926
603
773
943
620
790
960
637
807
976
17
17
17
257
258
259
260
261
262
263
993
41 162
330
*010
179
347
*027
196
363
*044
212
380
*061
229
397
*078 *095
246 263
414 430
*111
280
447
^128
296
464
*145
813
481
•17
17
17
497
514'
531
547
564
581 1 597
614
631 647
17
664
880
996
681
847
*012
697
863
*029
714
880
*045
731
896
*062
747 764
913 i 929
*078 *095
780
946
*111
797
9()8
*127
814
979
*144
17
16
16
264
265
266
42160
325
488
177
341
504
193
357
521
210
874
537
226
390
553
243 i 259
406 1 423
570 ! 586
275
489
602
292
455
619
308
472
635
16
16
16
267
268
269
651
813
975
667
830
991
684
846
*008
700
862
*024
716
878
*040
732 i 749
894 911
*056 *072
i
765
927
*088
781 797
943 959
n04 *120
16
16
16
N
1
2
3
4
5
6
7
8
9
D
COMMON LOGARITHMS OF NUMBERS.
121
N
O
1
2
3
4
5
6
7
8
9
D
270
271
273
273
43136
153
169
185
201
317
233
349
265
381
16
297
457
616
313
473
633
339
489
648
345
505
664
361
531
680
377
537
696
393
553
712
409
569
737
425
584
743
441
600
759
16
16
16
274
275
276
775
933
44 091
791
949
107
807
965
123
833
981
138
838
996
154
854
»013
170
870
*038
185
886
*044
301
903
*059
217
917
*075
233
16
16
16
277
278
279
280
281
282
283
248
404
560
364
420
576
279
436
593
295
451
607
311
467
633
326
483
638
342
498
654
358
514
669
373
539
685
389
545
700
16
16
16
716
731
747
762
778
793
809
834
840
855
15
871
45 025
179
886
040
194
903
056
209
917
071
225
933
086
340
948
103
355
963
117
371
979
133
386
994
148
301
*010
163
317
15
15
15
284
285
286
333
484
637
347
500
653
363
515
667
378
530
683
393
545
697
408
561
713
423
576
728
439
591
743
454
606
758
469
631
773
15
15
15
287
288
289
290
291
293
293
788
939
46 090
803
954
105
818
969
130
834
984
135
849
*000
150
864
*015
165
879
*030
180
894
*045
195
909
*060
310
934
*075
335
15
15
15
240
355
370
285
300
315
330
845
359
374
15
389
538
687
404
553
703
419
568
716
434
583
731
449
598
746
464
613
761
479
627
776
494
643
790
509
657
805
523
673
830
15
15
15
394
295
296
835
982
47129
850
997
144
864
*013
159
879
*026
173
894
*()41
188
909
*056
203
933
*070
217
938
*085
232
953
*100
346
967
*114
361
15
15
15
297
298
299
300
301
302
303
276
493
567
290
436
583
30§
451
596
319
465
611
334
480
635
349
494
640
363
509
654
378
524
669
393
538
683
407
553
698
15
15
15
713
737
741
756
770
784
799
813
838
843
14
857
48 001
144
871
015
159
885
039
173
900
044
187
914
058
303
939
073
316
943
087
330
958
101
244
973
116
359
986
130
273
14
14
14
304
305
306
387
430
573
303
444
586
316
458
601
330
473
615
344
487
639
359
501
643
373
515
657
387
530
671
401
544
686
416
558
700
14
14
14
307
308
309
714
855
996
738
869
*010
743
883
*024
756
897
*038
770
911
*053
785
926
*066
799
940
*080
813
954
*094
837
968
*108
841
983
*133
14
14
14
N
O
1
i
2
3
4
5
6
7
8
9
D
122
COMMON LOGARITHMS OF NUMBERS.
N
1
2
3
4
1
5
6
7
8
9
D
310
311
313
313
49136
150
164
178
192
206
220
234 1 248
262
14
276
415
554
290
429
568
304
443
582
318
457
596
332
471
610
346
485
624
360
499
638
374
513
651
388 1 402
527 541
665 679
14
14
14
314
315
316
693
831
969
707
845
982
721
859
996
734
872
*010
748
886
*024
762
900
*037
776
914
*051
790
927
*065
803 ' 817
941 955
*079 *092
14
14
14
317
318
319
320
321
322
323
50 106
243
379
120
256
393
133
270
406
147
284
420
161
297
433
174
311
447
188
325
461
202
338
474
215
352
488
229
365
501
14
14
14
515
529
542
556
569
583
596
610 623
637
14
651
786
920
664
799
934
678
813
947
691
826
961
705
840
974
718
853
987
732
866
*001
745
880
*014
759
893
*028
772
907
*041
14
13
13
324
325
326
51055
188
322
068
202
335
081
215
348
095
228
362
108
242
375
121
255
388
135
268
402
148
282
415
162
295
428
175
308
441
13
13
13
327
328
329
330
331
332
333
455
587
720
468
601
733
481
614
746
495
627
759
508
640
772
521
654
786
534
667
799
548
680
812
561
693
825
574
706
838
13
13
13
851
865
878
b91
904
917
930
943 1 957
970
13
983
52 114
244
996
127
257
*009
140
270
*022
153
284
*035
166
297
*048
179
310
*061
192
323
*075
205
336
*088
218
349
*101
231
362
13
13
13
334
335
336
375
504
634
388
517
647
401
530
660
414
543
673
427
556
686
440
5(>9
699
453
582
711
466
595
724
479
608
737
492
621
750
13
13
13
337
338
339
340
341
342
343
763
892
53 020
776
905
033
789
917
046
802
930
058
815
943
071
827
956
084
840
969
097
853
982
110
866
994
122
879
*007
135
'13
13
13
148
161
173
186
199
212
224
237 1 250
263
13
275
403
529
288
415
542
301
428
555
314
441
567
326
453
580
339
466
593
352
479
605
364
491
618
377
504
631
390
517
643
13
13
13
344
345
346
656
782
908
668
794
920
681
807
933
694
820
945
706
832
958
719
845
970
732
857
983
744
870
995
757
882
*008
769
895
*020
13
13
13
347
348
349
54 033
158
283
045
170
295
058
183
307
070
195
320
083
208
332
095
220
345
108
233
357
120
245
^70
133
258
382
145
270
394
13
12
12
N
1
2
3
4
5
6 7
1
8
9
D
COMMON LOGARITHMS OF NUMBERS.
123
N
1
2
3
4
5
6
7
8
9
D
350
351
352
353
407
419
432
444
456
469
481
494
506
518
12
531
654
777
543
667
790
555
679
802
568
691
814
580
704
837
593
716
839
605
728
851
617
741
864
630
753
876
642
765
888
12
12
12
354
355
356
900
55 033
145
913
035
157
925
047
169
937
060
182
949
073
194
963
084
306
974
096
218
986
108
230
998
131
242
*011
133
255
12
12
12
357
358
359
360
361
363
363
267
388
509
279
400
532
291
413
534
303
435
546
315
437
558
338
449
570
340
461
582
352
473
594
364
485
606
376
497
618
12
12
12
630
643
654
666
678
691
703
715
727
739
12
751
871
991
763
883
*003
775
895
*015
787
907
*027
799
919
*038
811
931
5^050
823
943
*062
835
955
*074
847
967
*086
859
979
*098
12
13
12
364
365
366
56110
239
348
132
241
360
134
253
373
146
265
384
158
277
396
170
389
407
182
301
419
194
312
431
205
334
443
217
336
455
12
12
12
367
368
369
370
371
372
373
467
585
703
478
597
714
490
608
736
502
620
738
514
633
750
536
644
761
538
656
773
549
667
785
561
679
797
573
691
808
12
12
12
820
832
844
855
867
984
101
317
879
891
902
914
926
12
937
57 054
171
949
066
183
961
078
194
972
089
206
996
113
239
*008
134
341
*019
136
252
*031
148
364
*043
159
276
13
12
12
374
375
376
287
403
519
299
415
530
310
426
543
323
438
553
334
449
565
345
461
576
357
473
588
368
484
600
380
496
611
392
507
623
12
12
12
377
378
379
380
381
383
383
634
749
864
646
761
875
657
773
887
669
784
898
680
795
910
693
807
931
703
818
933
715
830
944
736
841
955
738
852
967
11
11
11
978
990
*001
*013
137
240
354
*024
*035
*047
*058
*070
*081
11
58 092
206
320
104
218
331
115
239
343
138
252
365
149
263
377
161
874
388
172
286
399
184
297
410
195
309
422
11
11
11
384
385
386
433
546
659
444
557
670
456
569
681
467
580
692
478
591
704
490
603
715
501
614
726
513
635
737
524
636
749
535
647
760
11
11
11
387
388
389
771
883
995
782
894
*006
794
906
*017
805
917
*038
816
938
*040
837
939
»051
838
950
*063
850
961
*073
861
973
*084
872
984
*095
11
11
11
N
1
2
3
4 5
6
7
8
9
D
124
COMMON LOGARITHMS OF NUMBERS,
N
1
2
3
4
5
6
7
8
9
D
390
391
392
393
59106
118
129
140
151
162
173
184
195
207
11
218
329
439
229
340
450
240
351
461
251
362
472
262
373
483
273
384
494
284
395
506
295
406
517
306
417
528
318
428
539
11
11
11
394
395
396
550
660
770
561
671
780
572
682
791
583
693
802
594
704
813
605
715
824
616
726
835
627
737
846
638
748
857
649
759
868
11
11
11
397
398
399
400
401
402
403
879
988
60 097
890
999
108
901
*010
119
912
*021
130
923
*032
141
934
*043
152
945
*054
163
956
*065
173
966
*076
184
977
*086
195
11
11
11
206
217
228
239
249
260
271
282
293
304
11
314
423
531
325
433
541
336
444
552
347
455
563
358
466
574
369
477
584
379
487
595
390
498
6U6
401
509
617
412
520
627
11
11
11
404
405
406
638
746
853
649
756
863
660
767
874
670
778
b85
681
788
895
692
799
906
703
810
917
713
821
927
724
831
938
735
842
949
11
11
11
407
408
409
410
411
412
413
959
61066
172
970
077
183
981
087
194
991
098
204
*002
109
215
»013
119
225
*023
130
236
*034
140
247
*045
151
257
*055
162
268
11
11
11
278
289
300
310
321
331
342
352
363
374
11
384
490
595
395
500
606
405
511
616
416
521
627
426
532
637
437
542
648
448
553
658
458
563
669
469
574
679
479
584
690
11
11
11
414
415
416
700
805
909
711
815
920
721
826
930
731
836
941
742
847
951
752
857
962
763
868
972
773
878
982
784
888
993
794
899
*003
10
10
10
417
418
419
420
421
422
423
62 014
118
221
024
128
232
034
138
242
045
149
252
055
159
263
066
170
273
076
180
284
086
190
294
097
201
304
107
211
315
10
10
10
325
335
346
356
366
377
387
397
408
418
10
428
531
634
439
542
644
449
552
655
459
562
665
469
572
675
480
583
685
490
593
696
500
603
706
511
613
716
521
624
726
10
10
10
424
425
426
737
839
941
747
849
951
757
859
961
767
870
972
778
880
982
788
890
992
798
900
*002
808
910
*012
818
921
*022
829
931
*033
10
10
10
427
428
429
63 043
144
246
053
155
256
063
165
266
073
175
276
083
185
286
094
195
296
104
205
306
114
215
317
124
225
327
134
236
337
10
10
10
N
1
2
3
4
5
6
7
8
9
D
COMMON LOGARITHMS OF NUMBERS.
125
N
1
2
3
4
5
6
7
8
9
D
430
431
432
433
347
357
367
377
387
897
407
417
428
438
10
448
548
649
458
558
659
468
568
669
478
579
679
488
589
689
498
599
699
508
609
709
518
619
719
528 1 538
629 1 639
729 739
10
10
10
434
435
436
749
849
949
759
859
959
769
869
969
779
879
979
789
889
988
799
899
998
809
909
*008
819
919
*018
829 839
929 1 939
*028 *038
10
10
10
437
438
439
440
441
442
443
64 048
147
246
058
157
256
068
167
266
078
177
276
088
187
286
098
197
296
108
207
306
118
217
316
128
227
326
187
237
335
10
10
10
345
355
365
375
385
395
404
414
424 ! 434
10
444
542
640
454
552
650
464
562
660
473
572
670
483
582
680
493
591
689
503
601
699
513
611
709
528
621
719
1 532
631
729
10
10
10
444
445
446
788
836
933
748
846
943
758
856
953
768
865
963
777
875
972
787
885
982
797
895
992
807
904
*002
816
914
*011
826
924
*021
10
10
10
447
448
449
450
451
452
453
65 031
128
225
040
137
234
050
147
244
060
157
254
070
167
263
079
176
273
089
186
283
099
196
292
108 j 118
205 215
302 1 312
10
10
10
321
331
341
350
360
369
379
389
398 408
10
418
514
610
427
523
619
437
533
629
447
543
639
456
552
648
466
562
658
475
571
667
485
581
677
495 i 504
591 i 600
686 1 696
10
10
10
454
455
456
706
801
896
715
811
906
725
820
916
734
830
925
744
839
9:35
758
849
944
763
858
954
772
868
963
782
877
973
792
887
982
9
9
9
457
458
459
460
461
462
463
992
66 087
181
*001
096
191
*011
106
200
*020
115
210
*030
124
219
»039
134
229.
*049
148
238
*058
153
247
*068
162
257
*077
172
266
9
9
9
276
285
295
304
314
333
332
342
351
361
9
370
464
558
380
474
567
389
483
577
398
492
586
408
502
596
417
511
605
427
521
614
436
530
624
445
539
633
455
549
642
9
9
9
464
465
466
652
745
839
661
755
848
671
764
857
680
773
367
689
783
876
699
792
885
708
801
894
717
811
904
727
820
913
736
829
922
9
9
9
467
468
469
932
67 025
117
941
034
127
950
043
136
960
052
145
969
062
154
978
071
164
987 1
080
173
997
089
182
*006
099
191
*015
108
201
9
9
9
N
1
2
3
4
5
6
7
8
9
D
126
COMMON LOGARITHMS OF NUMBERS.
N
O
1
2
3
4
5
6
7
8
9
D
470
471
472
473
210
219
228
237
247
256
265
274
284
293
9
302
394
486
311
403
495
321
413
504
330
422
514
339
431
523
348
440
532
357
449
541
367
459
550
376
468
560
385
477
569
9
9
9
474
475
476
578
669
761
587
679
770
596
688
779
605
697
788
614
706
797
624
715
806
633
724
815
642
733
825
651
742
834
660
752
843
9
9
9
477
478
479
480
481
483
483
852
943
68 034
861
952
043
870
961
052
879
970
061
888
979
070
897
988
079
906
997
088
916
*006
097
925
*015
106
934
*024
115
9
9
9
124
133
142
151
160
169
178
187
196
205
9
215
305
395
224
314
404
233
323
413
242
332
422
251
341
431
260 269
850 i 359
440 449
278
368
458
287
377
467
296
386
476
9
9
9
484
485
486
485
574
664
494
583
673
502
592
681
511
601
690
520
610
699
529
619
708
538
628
717
547
637
726
556
646
735
565
655
744
9
9
9
487
488
489
490
491
492
493
753
842
931
763
851
940
771
860
949
780
869
958
789
878
966
797
886
975
806
895
984
815
904
993
824
913
*002
833
922
*011
9
9
9
69 020
028
037
046
055
064
073
082
090
099
9
108
197
285
117
205
294
126
214
302
135
223
311
144
232
320
152
241
329
161
249
338
170
258
346
179
267
355
188
276
364
9
9
9
494
495
496
373
461
548
381
469
557
390
478
566
399
487
574
408
496
583
417
504
592
425
513
601
434
522
609
443
531
618
452
539
627
9
9
9
497
498
499
500
501
502
503
636
723
810
644
732
819
653
740
827
662
749
836
671
758
845
679
767
854
688
775
862
697
784
871
705
793
880
714
801
888
• 9
9
9
897
906
914
923
932
940
949
958
966
975
9
984
70 070
157
992
079
165
*001
088
174
*010
096
183
*018
105
191
*027
114
200
*086
122
209
*044
131
217
*053
140
226
*062
148
234
9
9
9
504
505
506
243
329
415
252
338
424
260
346
432
269
355
441
278
364
449
286
372
458
295
381
467
303
389
475
312
398
484
321
406
493
9
9
9
507
508
509
501
586
672
509
595
680
518
603
689
526
612
697
535
621
706
544
629
714
552
638
723
561
646
731
569
655
740
578
663
749
9
9
9
N
O
1
2
3
4
5
6
7
8
9
D
COMMON LOGARITHMS OF NUMBERS.
127
N
1
2
3
4
5
6
7
8
9
D
510
511
512
513
757
766
774
783
791
800
808
817
825
834
9
842
927
71012
851
935
020
859
944
029
868
952
037
876
961
046
885
969
054
893
978
063
902
986
071
910
995
079
919
*003
088
9
9
8
514
515
516
096
181
265
105
189
273
113
198
282
122
206
290
130
214
299
139
223
307
147
231
315
155
240
324
164
248
332
172
257
341
8
8
8
517
518
519
520
521
52i
523
349
433
517
357
441
525
366
4§0
533
374
458
542
383
466
550
391
475
559
399
483
567
408
492
575
416
500
584
425
508
592
8
8
8
600
609
617
625
634
642
650
659
667 675
8
684
767
850
692
775
858
700
784
867
709
792
875
717
800
883
725
809
892
734
817
900
742
825
908
750
834
917
759
842
925
8
8
8
524
525
526
933
72 016
099
941
024
107
950
032
115
958
041
123
966
049
132
975
057
140
983
066
148
991
074
156
999
082
165
^008
090
173
8
8
8
527
528
529
530
531
532
533
181
263
346
189
272
354
198
280
362
206
288
370
214
296
378
222
304
387.
230
313
395
239
321
403
247
329
411
255
337
419
8
8
8
428
486
444
452
460
469
477
485
493
501
8
509
591
673
518
599
681
526
607
689
534
616
697
542
624
705
550
632
713
558
640
722
567
648
730
575
656
738
583
665
746
8
8
8
534
535
536
754
835
916
763
843
925
770
852
933
779
860
941
787
868
949
795
876
957
803
884
965
811
892
973
819
900
981
827
908
989
8
8
8
537
538
539
540
541
542
543
997
73 078
159
*006
086
167
*014
094
175
*022
102
183
*030
111
191
*038
119
199
*046
127
207
*054
135
215
*062
143
223
*070
151
231
8
8
8
239
247
255
263
272
280
288
296
804
312
8
320
400
480
328
408
488
336
416
496
344
424
504
352
432
512
360
440
520
368
448
528
376
456
536
384
464
544
392
472
552
8
8
8
544
545
546
560
640
719
568
648
727
576
656
735
584
664
743
592
672
751
600
679
759
608
687
767
616
695
775
624
703
783
632
711
791
8
8
8
547
548
549
799
878
957
807
886
965
815
894
973
823
902
981
830
910
989
838
918
997
846
926
*005
854
933
*013
862
941
*020
870
949
*028
8
8
8
N
1
2
3
4
5
6
7
8
9
D
128
COMMON LOGARITHMS OF NUMBERS.
N
1
2
3
4
5
6
7
8
9
D
550
551
552
553
74 036
044
052
060
068
076
084
092
099
107
8
115
194
273
123
202
280
131
210
288
139
218
296
147
225
304
155
233
312
162
241
320
170
249
327
178
257
335
186
265
343
8
8
8
554
555
556
351
429
507
359
437
515
367
445
523
374
453
531
382
461
539
390
468
547
398
476
554
406
484
562
414
492
570
421
500
578
8
8
8
557
558
559
560
561
56-3
563
586
663
741
593
671
749
601
679
757
609
687
764
617
695
772
624
702
780
632
710
788
640
718
796
648
726
803
656
733
811
8
8
8
819
827
904
981
059
834
842
850
858
865
873
881
889
8
896
974
75 051
912
989
066
920
997
074
927
*005
082
935
3^012
089
943
*020
097
950
*028
105
958
*035
113
966
*043
120
8
8
8
564
565
566
128
205
282
136
213
289
143
220
297
151
228
305
159
236
312
166
243
320
174
251
328
182
259
335
189
266
343
197
274
351
8
8
8
567
568
569
570
571
572
573
358
435
511
366
442
519
374
450
526
381
458
534
389
465
542
397
473
549
404
481
557
412
488
565
420
496
572
427
504
580
8
8
8
587
664
740
815
595
671
747
823
603
610
618
626
633
641
648
656
8
679
755
831
686
762
838
694
770
846
702
778
853
709
7a5
861
717
793
868
724
800
876
732
808
884
8
8
8
574
575
576
891
967
76 042
899
974
050
906
982
057
914
989
065
921
997
072
929
*005
080
937
*012
087
944
*020
095
952
*027
103
959
*035
110
8
8
8
577
578
579
580
581
582
583
118
193
268
125
200
275
133
208
283
140
215
290
148
223
298
155
230
305
163
238
313
170
245
320
178
253
328
185
260
335
8
8
8
343
350
358
365
373
380
388
395
403
410
8
418
492
567
425
500
574
433
507
582
440
515
589
448
522
597
455
530
604
462
537
612
470
545
619
477
552
626
485
559
634
7
7
7
584
585
586
641
716
790
649
723
797
656
730
805
664
738
812
671
745
819
678
753
827
686
760
834
693
768
843
701
775
849
708
782
856
7
7
7
587
588
589
864
938
77 012
871
945
019
879
953
026
886
960
034
893
967
041
901
975
048
908
982
056
916
989
063
923
997
070
930
*004
078
7
7
7
N
1
2
3
4
5
6
7
8
9
D
COMMON LOGARITHMS OF NUMBERS.
129
N
1
2
3
4
5
6
7
8
9
D
590
591
592
593
085
093
100
107
115
122
129
137
144
151
7
159
232
305
166
240
313
173
247
320
181
254
327
188
262
335
195
269
342
203
276
349
210
283
357
217
291
364
225
298
371
7
7
7
594
595
596
379
452
525
386
459
532
393
466
539
401
474
546
408
481
554
415
488
561
422
495
568
430
503
576
437
510
583
444
517
590
7
7
7
597
598
599
600
601
602
603
597
670
743
605
677
750
612
685
757
619
692
764
627
699
772
634
706
779
641
714
786
648
721
793
656
728
801
663
735
808
7
7
7
815
822
mo
837
844
851
859
866
873
880
7
887
960
78 032
895
967
039
902
974
046
909
981
053
916
988
061
924
996
068
931
*003
075
938
*010
082
945
*017
089
952
*025
097
7
7
7
604
605
606
104
176
247
111
183
254
118
190
262
125
197
269
132
204
276
140
211
283
147
219
290
154
226
297
161
233
305
168
240
312
7
7
7
607
608
609
610
611
612
613
319
390
462
326
398
469
333
405
476
340
412
483
347
419
490
355
426
497
362
433
504
369
440
512
376
447
519
383
455
526
7
7
7
533
540
547
554
561
569
576
583
590
597
7
604
675
746
611
682
753
618
689
760
625
696
767
633
704
774
640
711
781
647
718
789
654
725
796
661
732
803
668
739
810
7
7
7
614
615
616
817
888
958
824
895
965
831
902
972
838
909
979
845
916
986
852
923
993
859
930
*000
866
937
*007
873
944
*014
880
951
*021
7
7
7
617
618
619
620
621
622
623
79 029
099
169
036
106
176
043
113
183
050
120
190
057
127
197
064
134
204
071
141
211
078
148
218
085
155
225
092
162
232
7
7
7
239
246
253
260
267
274
281
288
295
302
7
309
379
449
316
386
456
323
393
463
330
400
470
337
407
477
344
414
484
351
421
491
358
428
498
365
435
505
372
442
511
7
7
7
624
625
626
518
588
657
525
595
664
532
602
671
539
609
678
546
616
685
553
623
692
560
630
699
567
637
706
574
644
713
581
650
720
7
7
7
627
628
629
727
796
865
734
803
872
741
810
879
748
817
886
754
824
893
761
831
900
76S
837
906
775
844
913
782
851
920
789
858
927
7
7
7
N
1
2
3
4
5
6
7
8
9
D
130
COMMON LOGARITHMS OF NUMBERS.
N
1
2
3
4
5
6
7
8
9
D
630
631
633
633
934
941
948
955
962
969
975
983
989
996
7
80 003
073
140
010
079
147
017
085
154
034
093
161
080
099
168
087
106
175
044
113
182
051
120
188
058
127
195
065
134
202
7
7
634
635
636
209
377
346
216
384
353
233
391
359
339
398
366
236
305
373
243
312
380
250
318
387
257
335
393
264
332
400
271
339
407
7
7
637
638
639
640
641
642
643
414
483
550
421
489
557
438
496
564
434
502
570
441
509
577
448
516
584
455
533
591
463
530
598
468
536
604
475
543
611
7
7
7
618
635
633
638
645
653
659
665
673
679
747
814
883
7
7
7
7
686
754
831
693
760
838
699
767
835
706
774
841
713
781
848
730
787
855
736
794
862
733
801
868
740
808
875
644
645
646
889
956
81033
895
963
030
902
969
037
909
976
043
916
983
050
932
990
057
929
996
064
986
*003
070
943
*010
077
949
*017
084
7
7
7
647
648
649
650
651
653
653
090
158
334
097
164
231
104
171
238
111
178
245
117
184
251
134
191
258
131
198
265
137
204
271
144
311
378
151
318
285
7
7
7
291
298
305
311
318
325
331
338
845
351
7
358
435
491
365
431
498
371
438
505
378
445
511
385
451
518
391
458
525
398
465
531
405
471
538
411
478
544
418
485
551
7
7
7
654
655
656
558
690
564
631
697
571
637
704
578
644
710
584
651
717
591
657
723
598
664
730
604
671
737
611
677
743
617
7
7
7
657
658
659
660
661
663
663
757
833
889
763
839
895
770
836
902
776
843
908
783
849
915
790
856
931
796
862
928
803
869
935
809
875
941
816
883
948
'7
7
7
954
961
968
974
981
9S7
994
*000
*007
*014
7
83 030
086
151
027
093
158
033
099
164
040
105
171
046
113
178
053
119
184
060
125
191
066
132
197
073
138
204
079
145
210
7
7
7
664
665
666
317
382
347
823
389
854
230
295
360
236
302
367
243
308
378
249
315
880
256
331
387
263
328
393
269
334
400
276
341
406
7
7
7
667
668
669
413
478
543
419
484
549
436
491
556
432
497
562
439
504
569
445
510
575
453
517
583
458
523
588
465
530
595
471
536
601
7
7
7
N
1
2
3
4
5
6
7
«
9
D
COMMON LOGARITHMS OF NUMBERS.
131
N
1
2
3
4
5
6
7
8
9
D
670
607
614
620
627
633
640
646
653
659
666
7
671
«73
673
672
737
802
679
743
808
685
750
814
693
756
831
698
763
837
705
769
834
711
776
840
718
783
847
734
789
853
730
795
860
6
6
6
674
675
676
866
930
995
872
937
*001
879
943
*008
885
950
*014
892
956
*020
898
963
*027
905
969
*033
911
975
*040
918
982
*^046
924
988
*052
6
6
6
677
678
679
680
681
683
683
83 059
133
187
065
129
193
072
136
200
078
142
206
085
149
213
091
155
219
097
161
235
104
168
233
110
174
238
117
181
345
6
6
6
251
257
264
270
276
383
289
296
302
308
6
315
378
442
331
385
448
327
391
455
834
398
461
340
404
467
347
410
474
353
417
480
359
433
487
366
429
493
372
436
499
6
6
6
684
6&5
686
506
569
632
512
575
639
518
582
645
535
588
651
531
594
658
537
601
664
544
607
670
550
613
677
556
620
683
563
626
689
6
6
6
687
688
689
690
691
693
693
696
759
832
702
765
828
708
771
835
715
778
841
731
784
847
727
790
853
734
797
860
740
803
866
746
809
873
753
816
879
6
6
6
885
891
897
904
910
916
923
929
935
943
6
948
84 011
073
954
017
080
960
023
086
967
039
092
973
036
098
979
042
105
985
.048
111
992
055
117
998
061
123
*004
067
130
6
6
6
604
695
690
136
198
261
142
205
267
148
211
273
155
217
280
161
223
286
167
230
292
173
236
298
180
242
305
186
248
311
192
255
317
6
6
6
697
698
699
700
701
702
703
323
386
448
330
392
454
336
398
460
342
404
466
348
410
473
354
417
479
361
423
485
367
439
491
373
435
497
379
442
504
6
6
6
510
516
578^
640
702
532
528
535
541
547
553
559
566
6
572
634
696
584
646
708
590
652
714
597
658
720
603
665
726
609 ,
671 '
733 j
615
677
739
621
683
745
638
689
751
6
6
6
704
705
706
757
819
880
763
825
887
770
831
893
776
837
899
782
844
905
788
850
911
794
856
917
800
862
934
807
868
930
813
874
936
6
6
6
707
708
709
942
85 003
065
948
009
071
954
016
077
960
022
083
967
028
089
973
034
095
979
040
101 1
1
985
046
107
991
052
114
997
058
120
6
6
6
N
1
2 1 3
4
5
6
7
8
9
D
132
COMMON LOGARITHMS OF NUMBERS.
N
1
2
3
4
5
6
7
8
9
D
710
711
712
713
136
132
138
144
150
156
163
169
175
181
6
187
248
309
193
254
315
199
260
321
205
266
327
211
273
333
217
278
339
234
285
345
230
291
352
236
297
358
242
303
364
6
6
6
714
715
716
370
431
491
376
437
497
383
443
503
388
449
509
394
455
516
400
461
523
406
467
528
412
473
534
418
479
540
425
485
546
6
6
6
717
718
719
720
721
723
723
552
612
673
558
618
679
564
625
685
570
631
691
576
637
697
582
643
703
588
649
709
594
655
715
600
661
721
606
667
737
6
6
6
733
739
745
751
757
763
769
775
781
788
6
794
854
914
800
860
920
806
866
926
813
872
932
818
878
938
834
884
944
830
890
950
836
896
956
842
902
962
848
908
968
6
6
6
734
725
726
974
86 034
094
980
040
100
986
046
106
992
053
112
998
058
118
^004
064
124
*010
070
130
*016
076
136
*022
083
141
*028
088
147
6
6
6
737
738
729
730
731
733
733
153
213
273
159
219
279
165
225
285
171
231
291
177
337
297
183
243
303
189
249
308
195
355
314
201
261
320
207
367
326
6
6
6
332
338
344
350
356
362
368
374
380
386
6
392
451
510
398
457
516
404
463
532
410
469
528
415
475
534
421
481
540
427
487
546
433
493
552
439
499
558
445
504
564
6
6
6
734
735
736
570
629
688
576
635
694
581
641
700
587
646
705
593
653
711
599
658
717
605
664
723
611
670
729
617
676
735
623
682
741
6
6
6
737
738
739
740
741
742
743
747
80(i
864
753
813
870
759
817
876
764
833
882
770
829
888
776
835
894
782
841
900
788
847
906
794
853
911
800
859
917
6
6
6
6
923
929
935
941
947
953
958
964
970
976
983
87 040
099
988
046
105
994
052
111
999
058
116
*005
064
122
*011
070
128
*017
075
134
*023
081
140
*029
087
146
*035
093
151
6
6
6
744
745
746
157
216
274
163
331
280
169
237
286
175
233
291
181
339
297
186
245
303
192
251
309
198
256
315
204
262
330
210
268
326
6
6
6
747
748
749
332
390
448
338
396
454
344
402
460
349
408
466
355
413
471
361
419
477
367
435
483
373
431
489
379
437
495
384
442
500
6
6
6
N
1
2
3
4
5
6
7
8
9
D
COMMON LOGARITHMS OF NUMBERS.
133
N
1
2
3
4
5
6
7
8
9
D
750
751
753
753
506
512
518
523
529
535
541
547
552
558
6
564
622
679
570
628
685
576
638
691
581
639
697
587
645
703
593
651
708
599
656
714
604
662
720
610
668
726
616
674
731
6
6
6
754
755
756
737
795
853
743
800
858
749
806
864
754
812
869
760
818
875
766
823
881
772
829
887
777
835
892
783
841
898
789
846
904
6
6
6
757
758
759
760
761
762
763
910
967
88 024
915
973
030
921
978
036
927
984
041
933
990
047
938
996
053
944
*001
058
950
*007
064
955
*013
070
961
*018
076
6
6
6
081
087
093
098
104
110
116
121
127
183
6
138
195
252
144
201
258
150
207
264
156
213
270
161
218
275
167
224
281
173
230
287
178
235
292
184
241
298
190
247
304
6
6
6
764
765
766
309
366
423
315
372
429
321
377
434
326
383
440
332
389
446
338
395
451
843
400
457
349
406
463
355
412
468
360
417
474
6
6
6
767
768
769
770
771
772
773
480
536
593
485
542
598
491
547
604
497
553
610
502
559
615
508
564
621
513
570
627
519
576
632
525
581
638
530
587
643
6
6
6
649
655
660
666
672
677
683
689
694
700
6
705
762
818
711
767
824
717
773
829
722
779
835
728
784
840
734
790
846
739
795
852
745
801
857
750
807
863
750
812
868
6
6
6
774
775
776
874
930
986
880
936
992
885
941
997
891
947
*003
897
953
*009
902
958
*014
908
9(i4
*020
913
969
*025
919
975
*031
925
981
*037
6
6
6
777
778
779
780
781
782
783
89 042
098
154
048
104
159
053
109
165
059
115
170
064
120
176
070
136
183
076
131
187
081
137
193
087
143
198
092
148
204
6
6
6
309
215
231
236
232
337
343
248
254
260
6
265
321
376
271
326
382
276
332
387
282
337
393
287
343
398
293
848
404
298
354
409
304
360
415
310
365
421
315
371
426
6
6
6
784
785
786
432
487
542
437
492
548
448
498
553
448
504
559
454
509
564
459
515
570
465
520
575
470
526
581
476
531
586
481
537
592
6
6
6
787
788
789
597
653
708
603
658
713
609
664
719
614
669
724
620
675
730
625
680
735
631
686
741
636
691
746
642
697
752
647
702
757
6
6
6
N
1
2
3
4
5
6
7
8
9
D
134
COMMON LOGARITHMS OF NUMBERS.
N
1
2
3
4
5
6
7
8
9
D
790
791
792
793
763
768
774
779
785
790
796
801
807
812
5
818
873
927
823
878
933
829
883
938
834
889
944
840
894
949
845
900
955
851 ! 856
905 911
960 966
862
916
971
867
922
977
5
5
5
794
795
796
982
90 037
091
988
042
097
993
048
102
998
053
108
*004
059
113
»009
064
119
*015
069
124
*020
075
129
*026
080
135
*031
086
140
5
5
5
797
798
799
800
801
802
803
146
200
255
151
206
260
157
21]
266
162
217
271
168
222
276
173
227
282
179
233
287
184
238
293
189
244
298
195
249
304
5
5
5
809 1 314
320
325
331
336
342
347
352
358
5
363
417
472
369
423
477
374
428
482
380
434
488
385
439
493
390
445
499
396
450
504
401
455
509
407
461
515
412
466
520
5
5
5
804
805
806
526
580
634
531
585
639
536
590
644
542
596
650
547
601
655
553
607
660
558
612
666
563
617
671
569
623
677
574
628
682
5
5
5
807
808
8U9
810
811
812
813
687
741
795
693
747
800
698
752
806
703
757
811
709
763
816
714
768
822
720
773
827
725
779
832
730
784
838
736
789
843
5
5
5
849
854
859
865
870
875
881
886
891
897
5
902
956
91009
907
961
014
913
966
020
918
972
025
924
977
030
929
982
036
934
988
041
940
993
046
945
998
052
950
*004
057
5
5
5
814
815
816
062
116
169
068
121
174
073
126
180
078
132
185
084
137
190
089
142
196
094
148
201
100
153
206
105
158
212
110
164
217
5
5
5
817
818
819
820
821
822
823
222
275
328
228
281
334
233
2S6
339
238
291
344
243
297
350
249
302
355
254
307
360
259
312
365
265
318
371
270
323
376
•5
5
5
381
387
392
397
403
408
413
418
424
429
5
434
487
540
440
492
545
445
498
551
450
503
556
455
508
561
461
514
566
466
519
572
471
524
577
477
529
582
482
535
587
5
5
5
824
825
826
593
645
698
598
651
703
603
656
709
609
661
714
614
666
719
619
672
724
624
677
730
630
682
735
635
687
740
640
693
745
5
5
5
827
828
829
751
803
855
756
808
861
761
814
866
766
819
871
772
824
876
777
829
882
782
834
887
787
840
892
793
845 1
897
798
850
903
5
5
5
N
1
2
3
4
5
6
7
8
9
D
COMMON LOGARITHMS OF NUMBERS.
136
N
1
2
3
4
5
6
7
8
9
D
830
831
832
833
908
913
918
924
929
934
939
944
950
955
5
960
92 012
065
965
018
070
971
023
075
976
028
080
981
033
085
986
038
091
991
044
096
997
049
101
*002
054
106
*007
059
111
5
5
5
834
835
836
117
169
221
122
174
226
127
479
231
132
184
236
137
189
241
143
195
247
148
200
252
153
205
257
158
210
262
163
215
267
5
5
5
8:37
838
839
840
841
843
843
273
324
376
278
330
381
283
335
387
288
340
392
293
345
397
298
350
402
304
355
407
309
361
412
314
366
418
319
371
423
5
5
5
438
433
438
443
449
454
459
464
469
474
526
578
629
5
5
5
5
480
531
583
485
536
588
490
542
593
495
547
598
500
552
603
505
557
609
511
562
614
516
567
619
521
572
624
844
845
846
634
686
737
639
691
742
645
696
747
650
701
752
655
706
758
660
711
763
665
716
768
670
722
773
675
727
778
681
732
783
5
5
5
847
848
849
850
851
852
853
788
840
891
793
845
896
901
804
855
906
809
860
911
814
865
916
819
870
921
824
875
927
829
881
932
834
886
937
5
5
5
942
947
952
957
962
967
973
978
983
988
5
993
93 044
095
998
049
100
*003
054
105
*008
059
110
*013
064
115
*018
069
120
*024
075
125
*029
080
131
*034
085
136
*039
000
141
5
5
5
854
855
856
146
197
247
151
202
252
156
207
258
161
212
263
166
217
268
171
222
273
176
227
278
181
232
283
186
237
288
192
242
293
5
5
5
857
858
859
860
861
862
863
298
349
399
303
354
404
308
359
409
313
364
414
318
369
420
323
374
425
328
379
430
334
384
435
339
389
440
344
394
445
5
5
5
450
455
460
465
470
475
480
485
490
495
5
500
551
601
505
556
606
510
561
611
515
566
616
520
571
621
526
576
626
531
581
631
536
586
636
541
591
641
546
596
646
5
5
5
864
865
866
651
702
752
656
707
757
661
712
762
666
717
767
671
722
772
676
727
777
682
732
782
687
737
787
692
742
792
697
747
797
5
5
5
867
868
869
802
852
902
807
857
907
812
862
912
817
867
917
822
872
922
827
877
927
832
882
932
837
887
937
842
892
942
847
897
947
5
5
5
N
1
2
3
4
5
6
7
8
9
D
136
COMMON L00ARITHM8 OF NUMBERS.
N
1
2
3
4
5
6
7
8
9
D
870
952
957
962
967
972
977
982
987
993
997
5
871
872
873
94 002
052
101
007
057
106
012
062
111
017
067
116
022
072
121
027
077
126
032
082
131
037
086
136
042
091
141
047
096
146
5
5
5
874
875
876
151
201
250
156
206
255
161
211
260
166
216
265
171
221
270
176
226
275
181
231
280
186
236
285
191
840
390
196
345
395
5
5
5
877
878
879
880
881
883
883
300
349
399
305
354
404
310
359
409
315
364
414
320
369
419
325
374
434
330
379
429
335
384
433
340
389
438
345
394
443
5
5
5
448
453
458
463
468
473
478
483
488
493
5
498
547
596
503
552
601
507
557
606
512
562
611
517
567
616
522
571
621
537
576
636
533
581
630
537
586
635
543
591
640
5
5
5
884
885
886
645
694
743
650
699
748
655
704
753
660
709
758
665
714
763
670
719
768
675
734
773
680
729
778
685
734
783
689
738
787
5
5
5
887
888
889
890
891
892
893
792
841
890
797
846
895
802
851
900
807
&56
905
812
861
910
817
866
915
832
871
919
837
876
934
833
880
939
836
885
934
5
5
5
939
944
949
954
959
963
968
973
978
983
5
988
95 036
085
993
041
090
998
046
095
*002
051
100
*007
056
105
*012
061
109
*017
066
114
*033
071
119
*037
075
134
*033
080
139
5
5
5
894
895
896
134
182
231
139
187
236
143
192
240
148
197
245
153
202
250
158
207
255
163
211
260
168
316
265
173
331
370
177
336
374
5
5
5
897
898
899
900
901
902
903
279
328
376
284
332
381
289
337
386
294
342
390
299
347
395
303
352
400
308
357
405
313
361
410
318
366
415
333
371
419
• 5
5
5
424
429
434
439
444
448
453
458
463
468
5
472
521
569
477
525
574
482
530
578
487
535
583
492
540
588
497
545
593
501
550
598
506
554
603
511
559
607
516
564
613
5
5
5
904
905
906
617
665
713
622
670
718
626
674
722
631
679
727
636
684
732
641
689
737
646
694
742
650
698
746
655
703
751
660
708
756
5
5
5
907
908
909
761
809
856
766
813
861
770
818
866
775
823
871
780
828
875
785
832
880
789
837
885
794
843
890
799
847
895
804
853
899
5
5
5
N
1
2
3
4
5
6
7
8
9
D
COMMON LOGARITHMS OF NUMBERS.
137
N
1
2
3
4
5
6
7
8
9
D
910
911
913
913
904
909
914
918
923
928
933
938
942
947
5
952
999
96 047
957
*004
052
961
*009
057
966
*014
061
971
*019
066
976
*023
071
980
*028
076
985
*033
080
990
*038
085
995
*042
090
5
5
5
914
915
916
095
142
190
099
147
194
104
152
199
109
156
204
114
161
209
118
166
213
123
171
218
128
175
223
133
180
227
137
185
232
5
5
5
917
918
919
920
921
922
923
237
284
332
242
289
336
246
294
341
251
298
346
256
303
350
261
308
355
265
313
360
270
317
365
275
322
369
280
327
374
5
5
5
379
384
388
393
398
402
407
412
417
421
5
426
473
520
431
478
525
435
483
530
440
487
534
445
492
539
450
497
544
454
501
548
459
506
553
464
511
558
468
515
562
5
5
5
924
925
926
567
614
661
572
619
666
577
624
670
581
628
675
586
633
680
591
638
685
595
642
689
600
647
694
605
652
699
609
656
703
5
5
5
927
628
929
930
931
932
933
708
755
802
713
759
806
717
764
811
722
769
816
727
774
820
731
778
825
736
783
830
741
788
834
745
792
839
750
797
844
5
5
5
848
853
858
862
867
872
876
881
886
890
5
895
942
988
900
946
993
904
951
997
909
956
*002
914
960
*007
918
965
*011
923
970
*016
928
974
*021
932
979
*025
937
984
*030
5
5
5
934
935
936
97 035
081
128
039
086
132
044
090
137
049
095
142
053
100
146
058
104
151
063
109
155
067
114
160
072
118
165
077
123
169
5
5
5
937
938
939
940
941
942
943
174
220
267
179
225
271
183
230
276
188
234
280
192
239
285
197
243
290
202
248
294
206
253
299
211
257
304
216
262
308
5
5
5
313
317
322
327
331
336
340
345
350
354
5
359
405
451
364
410
456
368
414
460
373
419
465
377
424
470
382
428
474
387
433
479
391
437
483
396
442
488
400
447
493
5
5
5
944
945
946
497
543
589
502
548
594
506
553
598
511
557
603
516
562
607
520
566
612
525
571
617
529
575
621
534
580
626
539
585
630
5
5
5
947
948
949
635
681
727
640
685
731
644
690
736
649
695
740
653
699
745
658
704
749
663
708
754
667
713
759
672
717
763
676
722
768
5
5
5
N
1
2
3
4
5
6
7
8
9
D
138
COMMON LOGARITHMS OF NUMBERS.
N
1
2
3
4
6
6
7
8
9
D
950
951
952
953
772
777
782
786
791
795
800
804
809
813
5
818
864
909
823
868
914
827
873
918
832
877
923
836
882
928
841
886
932
845
891
937
850
896
941
855
900
946
859
905
950
5
5
5
954
955
956
955
98 000
046
959
005
050
964
009
056
968
014
059
973
019
064
978
023
068
982
028
073
987
032
078
991
037
082
996
041
087
5
5
5
957
958
959
960
961
962
963
091
137
182
096
141
186
100
146
191
105
150
195
109
155
200
114
159
204
118
164
209
123
168
214
127
173
218
132
177
223
5
5
5
227
232
236
241
245
250
254
259
263
268
^
273
318
363
277
322
367
281
327
372
286
331
376
290
336
381
295
340
385
299
345
390
304
349
394
308
354
399
313
358
403
5
5
5
964
965
966
408
453
498
412
457
502
417
462
507
421
466
511
426
471
516
430
475
520
435
480
525
439
484
529
444
489
534
448
493
538
5
4
4
967
968
969
970
971
972
973
. 543
588
632
547
592
637
552
597
641
556
601
646
561
605
650
565
610
655
570
614
659
574
619
664
579
623
668
583
628
673
4
4
4
677
682
686
691
695
700
704
709
713
717
4
722
767
811
726
771
816
731
776
820
735
780
825
740
784
829
744
789
834
749
793
838
753
798
843
758
802
847
762
807
851
4
4
4
974
975
976
856
900
945
860
905
949
865
909
954
869
914
958
874
918
963
878
923
967
883
927
972
887
932
976
892
936
981
896
941
985
4
4
4
977
978
979
980
981
982
983 .
989
99 034
078
994
038
083
998
043
087
*003
047
092
*007
052
096
*012
056
100
*016
061
105
*021
065
109
*025
069
114
*029
074
118
' 4
4
4
123
127
131
136
140
145
149
154
158
162
4
167
211
255
171
216
260
176
220
264
180
224
269
185
229
273
189
233
277
193
238
282
198
242
286
202
247
291
207
251
295
4
4
4
984
985
986
300
344
388
304
348
392
308
352
396
313
357
401
317
361
405
322
366
410
326
370
414
330
374
419
335
379
423
339
383
427
4
4
4
987
988
989
432
476
520
436
480
524
441
484
528
445
489
533
449
493
454
498
542
458
502
546
463
506
550
467
511
555
471
515
559
4
4
4
N
1
2
3
4
5
6
7
8
9
D
COMMON LOGARITHMS OF NUMBERS.
139
N
1
2
3
4
5
6
7
8
9
D
990
991
564
568
5';2
577
581
585
590
594
599
603
4
607
612
616
621
625
629
634
638
643
647
4
992
651
656
660
664
669
673
677
682
686
691
4
993
695
699
704
708
712
717
721
726
730
734
4
994
739
743
747
752
756
760
765
769
774
778
4
995
782
787
791
795
800
804
808
813
817
822
4
996
826
830
835
839
843
848
852
856
861
865
4
997
870
874
878
883
887
891
896
900
904
909
4
998
913
917
922
926
930
935
939
944
948
952
4
999
957
961
965
970
974
978
983
987
991
996
4
N
1
2
3
4
5
6
7
8
9
D
THE METRIC TABLES OF WEIGHTS AND
MEASURES.
The Metric System is a decimal system of weights and
measures.
The basis of the whole system is the metre.
The length of a metre is defined by a platino-iridium bar
kept in the International Metric Bureau at Paris. The
metre was meant to be one ten-millionth of the distance
from the equator to the pole, but a slight error in the
calculation has been discovered.
The Latin prefixes indicate the denominations smaller
than the unit, and the Greek prefixes the denominations
larger than the unit. Thus :
Deci designates tenth.
Centi
hundredth.
Milli
thousandth.
Deka
ten.
Hekto
hundred.
Kilo
thousand.
Myria
ten thousand.
The denominations in more frequent use are denoted by
heavier type.
142 METRIC TABLES.
LENGTH.
TABLE.
10 millimetres C^"") = 1 centimetre f"').
10 centimetres = 1 decimetre C"").
10 decimetres = 1 metre (■").
10 metres = 1 dekametre (^"').
10 dekametres = 1 hektometre ("™).
10 hektometres = 1 kilometre (^'").
10 kilometres = 1 myriametre (^'°).
SURFACE.
The units of surface are squares whose dimensions are
the corresponding linear units ; hence it takes 10 times 10,
or 100, of one denomination to make one of the next
higher. For measuring small surfaces the principal unit
is the square metre.
TABLE.
100 square millimetres («<!'"'") = 1 square centimetre («<i <='»).
100 square centimetres = 1 square decimetre (^i'^'").
100 square decimetres = 1 square metre (''J'").
100 square metres = 1 square dekametre («'''''°).
100 square dekametres = 1 square hektometre (**^"'°).
100 square hektometres = 1 square kilometre {^^^'^).
LAND.
TABLE.
100 cen tares ('^*) = 1 are (").
100 ares = 1 hektare ("*).
A centare is a square metre, an are a square deka-
metre, and a hektare a square hektometre.
METRIC TABLES, 143
VOLUME.
The units of volume are cubes whose dimensions are the
corresponding linear units ; hence it takes 10 times 10
times 10, or 1000, of one denomination to make one of the
next higher.
TABLE.
1000 cubic millimetres f^"'"'") = 1 cubic centimetre f "««>).
1000 cubic centimetres = 1 cubic decimetre ('^"'^■»).
1000 cubic decimetres = 1 cubic metre (*""").
WOOD.
TABLE.
10 decisteres (''") = 1 stere ('*).
10 steres = 1 dekastere (^*).
A stere is a cubic metre.
CAPACITY.
The unit of capacity is a litre, which equals a cubic deci-
metre.
TABLE.
10 millilitres ("') = 1 centilitre («').
10 centilitres = 1 decilitre ('^').
10 decilitres = 1 litre (').
10 litres = 1 dekalitre (^').
10 dekalitres = 1 hektolitre («').
10 hektolitres = 1 kilolitre (^').
144
METRIC TABLES.
WEIGHT.
The unit of weight is a gram, which equals the weight
of a cubic centimetre of water at its greatest density.
TABLE.
10 milligrams f"^)
10 centigrams
10 decigrams
10 grams
10 dekagrams
10 hektograms
1000 kilograms
= 1 centigram f^^).
= 1 decigram i^^),
= 1 gram (s).
= 1 dekagram (^^).
= 1 hektogram ("^).
= 1 kilogram, or kilo (^).
= 1 ton C)'
METRIC EQUIVALENTS.
1 metre=:39.37 in. =1.0936 yd.
1 kilometre = .62138 mile
1 hektare
1 litre
1 gram
1 kilogram
1 stere
= 2.471 acres
_ j .908 qt. dry
~ ( 1.0567 qt. liq.
= 15.432 grains
= 2.2046 lbs.
= .2759 cord
1 yard
1 mile
1 acre
1 qt. dry
1 qt. liq.
1 grain
1 pound
1 cord
= .9144 m.
= 1.6093 kilo-
metres.
= .4047 Ha.
= 1.1011.
= .9463 1.
= .0648 gram.
= .4536 K.
= 3.625 steres.
APPROXIMATE METRIC EQUIVALENTS.
1 cm. = I in.
1 Km. = f mile.
2| bush.
2| lbs.
2200 lbs.
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University. Crown 8vo. Over 250 pages. $1.90.* \In Press.
A TREATISE ON COMPUTATION. An Account of the Chief
Methods for Contracting and Abbreviating Arithmetical Calcu-
lations.
By Edv^ard M. Langley, M.A. i2mo. pp. viii-184. $1.00.
GRAPHICAL CALCULUS.
By Arthur H. Barker, B.A., B.Sc, Senior Whitworth Scholar 1895.
With an Introduction by John Goodman, A.M.I.C.E., Professor
of Engineering at the Yorkshire College, Victoria University.
Crown 8vo. 197 pages. $1.50.
" This is an able mathematical work, written with refreshing originality. It
forms unquestionably the best introduction to the study of the Calculus that we
have yet seen. Never once does Mr. Barker leave his reader in doubt as to the
significance of the result obtained. His appeal to the graph is conclusive, and
so unfolds the mysteries of the Calculus that the delighted student grasps the
utility of the subject from the sXaxV — Schoolmaster .
ALGEBRA FOR SCHOOLS AND COLLEGES.
By William Freeland, A.B., Head Master at the Harvard School,
New York City. i2mo. 320 pages. $1.40.
•• Excellent features of the book may be found in the adequate treatment of
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radicals. The form of the demonstration of processes is here exceptionally fine,
the whole showing a nice sense of logic which 'looks at the end from the begin-
ning.' It would be difficult to find a book cr ntaining a better selection of exer-
cises, or one in which the publishers have more carefully interpreted the author's
ideas of arrangement." — Educational Review.
Note.— The Answers to the Examples in this book are printed in a separate
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LONGMANS, GREEN, & CO., 91-93 Fifth Avenue, New York.
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