FOOD FOR THOUGHT.

[Send all communications to W. F. L. Sanders, Connersville, Ind. They should be received by August 18. Be prompt. Write only on one side of your paper.

PROBLEM 137.

SOLUTIONS RECEIVED.

Show that the square inscribed in a semicircle is to the

square inscribed in a quadrant of the same circle :: 8:5.

ABCD square inscribed in semicircle, and
OEFG square inscribed in quadrant.

Let OD OF R; and let DC=x; then from the triangle ODC, we have

=r2; from which x2=1 Area of square AC.

4r2
5

4

NOTE. The square BHIK is 1⁄2 of ABCD and of square OEFG, and the line MN is trisected at B and K.

[Permit us to call special attention to Prof. Gregg's note to his solution. It adds much interest to the solution.-ED.]

SOLUTION: (L. N. FOUTS, Supt. Schools, Brownstown.)

sine of 45°; while the square inscribed in a semicircle is equal to the square of the sine of an angle whose tangent is 2.

Therefore sine2 45: sine2 63° 26' 4" 5:8.

PROBLEM 134. A merchant had a number of eggs, and when he counted them by twos there was one left; counting by threes there were two left; counting by fours there were three left; counting by fives there were four left; counting in this way until he reached fourteen he had thirteen left. How many eggs did he have? (THOMAS JONES, Sulphur Springs, Ind.)

SOLUTION: If there had been one egg more he could have divided the number exactly by 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 and 14; hence, the number of eggs must have been one less than the L. C. M. of these numbers their L. C. M.=360360; 360360-1=360359, answer.

To this problem Supt. J. C. Gregg, Brazil, Ind., sends us a long solution but exceedingly ingenious.

PROBLEM 135. $7800 of 5 per cent. stock is sold at 104, and the proceeds invested in railway shares paying 6 per cent.; what is the price of the latter if the income derived is increased by $26?

SOLUTION: $7,800X.05=$390, first income; $390+$26=$416, second income; 416.06-6,933, number of dollars stock producing the second income; $7,800+1.04 $8,112, received for the first stock; 8,112+6,933}= 1.17; hence, the price per dollar of second stock is $1.17, or price per share $117. (L. E. RADY, Edwardsville, Ind.)

PROBLEM 136. Two clocks strike 9 together on Monday morning; on Tuesday morning, one is 8 min. to 11 when the other strikes 11; how much must the first be moved up so that they may both strike 9 together in the evening?

SOLUTION: From 9 o'clock, Monday, to 11 o'clock, Tuesday, is 26 hours; in 26 hours there is a loss of 8 min.; in one hour there is a loss of min.; from 11 o'clock A. M. to 9 o'clock P. M. is 10 hours; in 10 hours there would be a loss of 10 times min. 3 min., 8 min.+3 min. 11 min., the amount the first must be moved up. (W. F. ENTEMAN, Leota.)

PROBLEM 138. Given

{ x1-y1=1280
\x3y+x y3=480 )

to determine the values of x and y by quadratics.

480;

SOLUTION:-(x2—y2) (x2+y2)=1280; x y (x2+y2)=480; x2+y2= x y Substituting this value of x2+y2 in the first equation, we have—

480 x y

(x2—y2 ) ——=1280; or, 3x2-8xy-3 y2=0.

Transposing, dividing by 3, and completing the square we have8xy 16 y2 25 y2

+ 3

Extracting the square root we get

X

4 y 5y

3

V

=+g; x=3 y or- we now find x=+6 or +6 V-1, any y=+2 or

3

Let y=m x; we then get the following:

(J. F. MILLIS, Bloomington.)

(By substitution, Prof. Gregg gets the values given above.)

L. N. Fouts of Brownstown, gets the following additional values :

CREDITS.-130, Philip Abell, Velpen 135, Everett Beadles, Velpen ; L. E. Rady, Edwardsville 134, 135, Walter N. Vanscoyoc, Crawfordsville 135, 136, Alton Blunk, Crown Center; Mary Dilts, St. Paul. 134, 135, 138, Elmer E. Carter, Frankton 134, 136, 137, Otto Clayton, Fowler 134, 137, 138, J. F. Millis, Bloomington... 134, 135, 136, 137, 138, W. F. Enteman, Leota; J. A. Newlin, Plainfield; L. N. Fouts, Brownstown, J. C. Gregg, Brazil.. .134, 135, 136, H. S. Burlingame, Sparta.

NOTES AND QUERIES.

QUERY 44. We have received that old problem x2+y=7 and x+y2=11 to find the values of x and y.

QUERY 45. Another contributor sends us another old puzzle:

Divide 8 gallons of wine (in an eight-gallon vessel) into two equal parts, and use only a three-gallon measure and a five-gallon measure. Try it.

QUERY 46. Of the many different correct solutions to Ex. 36, page 254, Indiana Complete Arithmetic, which is the one intended, or the most reasonable. (WILLIAM R. EDWARDS, Nebraska.)

PROBLEMS.

139. Two cities are 200 miles apart. How high must a person ascend at one, that he may just see the other? Solve by geometry.

W. F. ENTEMAN, Leota.

140. A person reaching the railway station 30 minutes before the train was due, starts to meet it. He walked 11⁄2 miles before he met the train. If he had gone in the opposite direction, he would have been 21⁄2 miles from the station when the train overtook him. What was the train's rate per hour? ALTON BLUNK, Crown Center, Ind.

141. Which is the better investment, the 10 per cents. at 211%, or the 2 per cents. at 424? Which is the better investment, if % is paid for brokerage?

142.

Construct an equilateral triangle in which the altitude is a mean proportional between the side and the base.

143. Given x+y=10, and xly-12 to find x and y.

144. I have 3 jars, A, B and C, holding 1, 3 and 5 gallons respectively. A is empty, B full of water, C full of wine. I fill A from B; I fill up B from C, and pour the contents of A into C ; I repeat the whole process; how much wine is there in B? In C?

MISCELLANY.

THE NORTHERN INDIANA NORMAL SCHOOL.

The Northern Indiana Normal at Valparaiso has had another prosperous year. This school has had a remarkable history. It was started by its present president twenty-four years ago with thirty-five students. With but few exceptions, every succeeding year has shown an increase in attendance over the preceding year until the year just closed has shown an average attendance of over two thousand and the enrollment for the year will exceed more than four thousand different students. The new departments of biology and higher English started last year have been highly appreciated and largely attended. The coming year it is the purpose of the management to introduce a full course of electrical engineering. As electrical engineers are in increasing demand this department will doubtless prove a great success. This school employs only first-class teachers and does thoroughly whatever it undertakes. H. B. Brown is president and O. P. Kinsey vice-president.

BEGINNING with November, the state board questions in literature and science of teaching, will be based on the new reading circle books.

MONTICELLO. Six of the Monticello teachers are attending the Cook county summer normal school. The White county normal enrolls over eighty.

THE Law School, in connection with the Indianapolis University, is anticipating a large attendance the coming year. See the advertisement on an another page.

THE Worthington summer normal school, under the management of W. D. Kerlin, numbered 174. This was the largest in the state so far as heard from. At the close of the school, the students presented Mr. Kerlin a fine rocking chair.

INDIANA UNIVERSITY, at its last commencement, which was its seventysixth, sent out ninety-four graduates. President A. S. Draper, of Illinois University, made the address, his subject being "The Relations of the Individual to the Mass."

SEYMOUR.—“Par excellence" is the term E. A. Remy's paper applies to the high school graduating exercise. It further says that "Prof. Montgomery and his able corps of teachers are to be congratulated on the results of their excellent work."

FORTVILLE had a prosperous year under the direction of J. W. Jay. Its high school has received a commission, and the work generally is well done. Mr. Jay was offered more money to accept another position, but preferred to continue his work at Fortville.

WANTED TEACHERS AND STUDENTS.—Bright young men and women wishing profitable employment until the colleges and public schools open in the fall, can secure it by addressing President WM. M. CROAN, Anderson Normal University, Anderson, Ind.