56. A hare is 50 leaps before a grey-hound, and takes 4 leaps to the grey-hound's 3; but two of the grey-hound's leaps are as much as three of the hare's: how many leaps must the greyhound take to catch the hare?

57. A person being asked his age, replied, that multiplied by of his age would give a product age: what was his age?

Ans. 300.

of his age equal to his Ans. 16.

58. A person has a lease for 99 years; and being asked how much of it was already expired, answered, that two thirds of the time past was equal to four fifths of the time to come: what was the time past? Ans. 54 years.

59. There is a fish whose tail weighs 9 lbs., his head weighs as much as his tail and half his body, and his body weighs as much as his head and tail: what is the whole weight of the fish? Ans. 72 lbs.

60. There is a certain number, consisting of two digits, the sum of which digits is 5; and if 9 be added to the number itself the digits will be inverted: what is the number?

Ans. 23.

61. A person found, upon beginning the study of his profession, that of his life hitherto had passed before he commenced his education, under a private teacher, at a public school, and four years at the university: what was his age?

Ans. 21 years.

62. To find a number such, that, whether it be divided into two or three equal parts, the continued product of its parts shall be equal to the same quantity. Ans. 63.

63. A person has two horses and a saddle worth 50l.: now if the saddle be put on the back of the first horse, it will make his value double that of the second; but if it be put on the back of the second, it will make his value triple that of the first: what is the value of each horse

Ans. one 30l. and the other 401.

64. To divide the number 90 into four such parts that if the first be increased by 2, the second diminished by 2, the third multiplied by 2, and the fourth divided by 2, the sum, difference, product, and quotient, shall each equal the same quantity.

Ans. The parts are 18, 22, 10, and 40.

65. By his will a father disposed of his property as follows; namely, to his oldest son he gave 100 dollars of the property, and a tenth part of the residue; to the second, 200 dollars and a tenth part of the residue; to the third, 300 dollars and a tenth part of the residue; and so on to the last, always increasing the sum first paid out by 100 dollars. It appeared that the portions of all the children were alike. Required the value of the property, the number of children, and the portion of each child.

Ans. The estate was 8100 dollars, the children 9, and the portion of each 900 dollars.

66. A and B have the same income; A is extravagant and contracts an annual debt amounting to of his income; but B lives upon of his; at the end of 10 years, B lends A money enough to pay off his debts, and has 160l. to spare: what is their income? Ans. 2801.

67. A person passed of his age in childhood, in youth, and 5 years besides in matrimony, at the end of which time he had a son, who died 4 years before his father, and reached only half his father's age: at what age did the father die?

Ans. 84.

68. A shepherd, driving a flock of sheep in time of war, meets with a company of soldiers, who plunder him of half his flock and half a sheep over; and he receives the same treatment from a second, third, and fourth company, each succeeding company plundering him of half the flock the last had left and half a sheep beside, insomuch that in the end he had only 8 sheep left: how many sheep had he in the beginning?

Ans. 143.

69. A person, fifteen years after he was married, being asked the age of himself and of his wife at the time of their marriage, replied, that he was then thrice as old as his wife, but that now he was only twice as old: what were their ages?

Ans. He was 45 and she 15.

70. It is required to find two numbers such that the first added to half the second shall make 20, and the second added to one third the first shall also make 20. Ans. 12 and 16.

71. Two travellers, distant 154 miles, set out at the same time to meet each other, the one proceeding at the rate of 3 miles in

i

2 hours, and the other at the rate of 5 miles in 4 hours: how long and how far did each travel before they met?

Ans. The time was 56 hours, the spaces travelled 84 and 70 miles.

IX. Formation of Powers and Extraction of Roots. 72. What is the square root of 9 x2? (Art. 122.)

74. What is the square root of a1 + 4 a3 x + 6 a2x2 +

76. What is the third power of 8 x3 y3? (Art. 127.)

77. What is the fifth root of 32x5 y1? (Art. 129.)
Ans.-2xy2.

78. What is the fourth power of x-a? (Art. 141.)

79. What is the square of a2+ 2 ax + x2? (Art. 145.) Ans. a4a3 x + 6 a2x2 + 4 α x3 + x^.

80. What is the cube root of x 15x26x+1? (Art. 155.)

81. What is the fifth root of 32x5

6x5+ 15x4 20 x3 +

Ans. x22x + 1.

30 x1 +80x3

3

40x2

Ans. 2x 1.

END.

ELEMENTS OF GEOMETRY,

BY

A. M. LEGENDRE,

MEMBER OF THE INSTITUTE AND THE LEGION OF HONOUR, OF THE ROYAL SOCIETY OF LONDON, &c.

TRANSLATED FROM THE FRENCH

FOR

THE USE OF THE STUDENTS OF THE UNIVERSITY

ᎪᎢ

CAMBRIDGE, NEW ENGLAND,

BY JOHN FARRAR,

PROFESSOR OF MATHEMATICS AND NATURAL PHILOSOPHY.

SECOND EDITION,

CORRECTED AND ENLARGED

CAMBRIDGE, N. E.

PRINTED BY HILLIARD AND METCALF,

At the University Press.

SOLD BY W HILLIARD, CAMBRIDGE, AND BY CUMMINGS, HILLIARD, & Co. NO. 134 WASHINGTON STREET, BOSTON.

1

DISTRICT OF MASSACHUSETTS, TO WIT.

District Clerk's Office.

BE it remembered, that on the thirtieth day of May 1825, in the fortyninth year of the independence of the United States of America, Cummings, Hilliard & Co. of the said district, have deposited in this office the title of a book, the right whereof they claim as proprietors, in the words following, viz :

"Elements of Geometry, by A. M. Legendre, member of the Institute, and the Legion of Honour, of the Royal Society of London, &c. Translated from the French, for the use of the students of the University at Cambridge, New England. By John Farrar, Professor of Mathematics and Natural Philosophy. Second edition, corrected and enlarged."

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