i^lCWRLF IL^SO SOI NUMERICAL PROBLEMS IN PLANE GEOMETRY J « VJr . t ^ I I LiL LIBRARY OF THE University of California. f GIFT OF ..>?...:.....VSr:.:. .yv:V..9vA<\^Jo^^ Class 9vt4^ .f /.£ 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; (( 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 («). 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. -^IBB^? OF THE UNfVERSITY OF «^UPORN\b< LONGMANS, GREEN, ^ CO: S PUBLICATIONS. THE HARPUR EUCLID. An Edition of Euclid's Elements revised in accordance with the Reports of the Cambridge Board of Mathematical Studies and the Ox- ford Board of the Faculty of Natural Science. By E. M. 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