2d. By the Mercantile Rule, principal for the whole time = payment (for 59 months) is x 2)r, x + x(n u)x + x --- putting .005r, the amount of the k(1 + nr). The amount of the first + x(n 1)r, and in like manner 3)r, etc., to the nth and last payment, The sum of these payments, which must equal the amount of the principal, is næ + rx [ (n − 1) + (n 2) + 3) +........+ 1]. The series in brackets is arithmetical and its sum is 11⁄2 (n 1)n. Hence nx + 1⁄2 rx (n − 1)n = k(1 + nr) and which with no interest is a. (n x= k(1 + nr) 1) = $18,881.63. n + nr (n 3d. Another method, which is well suited to this problem and accords with the Connecticut Rule, and also with the Vermont and New Hampshire Rules in the case of interest annually, is to find the amount of the debt and of each of the payments to the end of each year. Put 1.06 = r. Twelve payments a year are made each = 4. The first is on interest for 11 months, the second for 10, etc., to the last, which paid at the end of the year has no interest. Therefore the sum of the amounts of the 12 payments is 12 plus the interest of r dollars for 66 months or .33x. 12x + .33x = 12.33. Subtracting this from the amount of the principal for the year, leaves as a principal for the second year kr 12.33x. The second remainder is kr 12.33xr 12.33x. 12.33 and the fifth, kr 12.33xr The last remainder must be zero. Hence 12.33x = CREDIT FOR SOLUTIONS RECEIVED. J. W. Ellison, I. L. Winckler, H. E. Trefethen. Algebra 44. H. E. Trefethen. Algebra 48. R. P. Harker, H. E. Trefethen, G. J. Van Buren, Franklin T. Jones, A. J. Wile, I. L. Winckler, E. L. Brown. Geometry 45. J. R. Parker, H. E. Trefethen, E. C. Thayer. Geometry 49. H. E. Trefethen, E. L. Brown. Geometry 50. J. F. West, H. E. Trefethen, E. L. Brown, I. L. Winckler. Trigonometry 51. T. M. Blakslee, H. E. Trefethen, I. L. Winckler, E. L. Brown. Applied Mathematics 43. H. E. Trefethen. Applied Mathematics 47. H. E. Trefethen. Applied Mathematics 52. H. E. Trefethen, E. L. Brown. Total number of solutions, 28. PROBLEMS FOR SOLUTION. ALGEBRA. 58. Proposed by H. C. Whitaker, Ph.D., Philadelphia, Pa. The money that will pay the wages of Tom Jones for 614 days will pay the wages of Harry Smith for 81/, days. For how many days will the same sum pay the wages of the two men? 59. Proposed by H. C. Whitaker, Ph.D., Philadelphia, Pa. At a certain time, a train overtakes a man and ten seconds thereafter passes him. Twenty minutes after passing this man, the train meets another man and in nine seconds thereafter passes the second man. Counting from the time that the train passed the second man, how soon will the two men meet? GEOMETRY. 60. Proposed by I. L. Winckler, Cleveland, O. Find a point in a given straight line from which, if tangents are drawn to two given circles, they will make equal angles with the given line. (From Chauvenet's Geometry.) 61. Proposed by Russell P. Harker, Parker, Ind. Given the base 26 of a triangle and a, the difference of the base angles. Find the equation of the locus of the vertex. (From Bailey and Wood's Analytic Geometry.) MISCELLANEOUS. 62. Proposed by John W. Scoville, Syracuse, N. Y. A prison consists of 36 cells arranged like the squares of a chess board. There are doors between all adjoining cells. A prisoner in one of the corner cells is told that he can have his freedom, if he can get into the diagonally opposite corner cell, by passing through each of the cells once and only once. Can the prisoner win his freedom? DEPTH OF DEATH VALLEY, CAL. LOWEST POINT IN UNITED STATES. The United States Geological Survey has just completed a line of spirit levels through Death Valley, Cal., and much to the surprise of everyone familiar with the region has ascertained that the depth of that area is not so great as was supposed. The final computations of the results have not yet been made, but the preliminary figures give for the lowest point a depth of 276 feet below sea level. Bennetts Well, which is near this point, is 266 feet below sea level. These figures may be altered by two or three feet when the final computations are made, but they are probably not more than three feet in error. The Geological Survey now has elevation marks on the highest and lowest points of dry land in the United States. It is a strange coincidence that these two extremes are both in southern California and only seventy-five miles apart. Mount Whitney is a foot or two over 14,500 feet above sea level, while Death Valley, as above stated, is 276 feet below. Before the Salton Sink, also in southern California, was flooded by the Colorado River, it contained the lowest point of dry land in the United States, a spot 287 feet below sea level. Previous estimates of the depth of Death Valley based on barometer readings gave for the lowest point figures varying from 250 to 450 feet below sea level. The level line of the Geological Survey is believed to be the first accurate determination of elevations in that locality that has ever been made. DEPARTMENT OF SCIENCE QUESTIONS. FRANKLIN T. JONES, University School, Cleveland, O. This department is designed to serve as a medium for the exchange of ideas on questions and questioning in the sciences. Questions will be printed from various sources-college entrance examinations, textbooks, etc. Comment is invited. Suggestions and criticisms as to character, adaptability and usefulness are desired. Readers of this journal are invited to propose questions and problems which will be of general interest, or of a type which will be useful in the class-room. It is not expected that questions which will not be useful to pupils will be frequently printed. Since the majority of the questions will be of a comparatively simple character, solutions and answers will not be published unless specifically asked for. Teaching suggestions are wanted. Address all communications to the editor of the department. 9. In a test of the new electrical locomotive for the N. Y. C. R. R. the indicator showed a speed of thirty miles per hour in sixty-three seconds from the time the controller was put on the first notch. What was the average acceleration? As the controller was thrown over, the speed increased at the rate of five miles per hour every thirty seconds. How long did it take to reach the maximum speed of seventy miles per hour? (News Item.) 10. If a force of six hundred thousand dynes is resolved into mutually perpendicular components making equal angles with the given force, find the value of the components. (Sheffield.) 11. (a) If a rifle-bullet were shot directly upward with a velocity of one thousand feet per second, how far would it rise in a vacuum? (b) If the bullet weighs one ounce, how great will be the potential energy at the greatest height? (Name the unit of energy). (Harvard.) 12. Which requires the greater amount of heat, to raise one thousand grams of water from ninety-seven degrees to ninety-nine degrees Centigrade, or to raise the same amount of water from ninety-nine degrees to one hundred and one degrees Centigrade, under ordinary atmospheric pressure? Why? Change eighty degrees Fahrenheit to Centigrade; one hundred eighty degrees Centigrade to Fahrenheit, and to absolute scale. (Princeton.) 13. The two branches of a divided circuit have resistances of three ohms and five ohms respectively. The total current in the circuit is twenty-four amperes. What is the current in each branch? Explain. (Board.) (Sheffield.) 14. Describe the phenomena of total reflection. 15. Ten grams of crystallized sodium sulphate were found by experiment to contain 4.70 grams of water. How many molecules of water of crystallization does the crystallized salt contain? (Harvard.) 16. Give Avogadro's law. If one hundred and five million molecules of hydrogen burn in oxygen, how many molecules of water are formed? (Sheffield.) 17. What evidence have we that matter is indestructible? If we cannot create matter, how are we to account for the fact that the products of combustion of a candle weigh more than the original candle? (P1uceton.) 18. What is a flame? What are the conditions essential to (a) its luminosity, (b) its maximum temperature? Make a sketch of a Bunsen burner and its flame, illustrating in detall and explaining the parts of each. 19. (Board.) What substances are contained in ordinary gun-powder? Explain why it explodes. (Sheffield.) 20. Explain how the human body obtains energy from the sun. (Harvard.) MATHEMATICAL ANNOUNCEMENTS. The report of the committee on the Teaching of Geometry which was presented at the last meeting of the Association is now in press. Copies of this report will be mailed soon to all members of the Mathematics Section of the Association. Copies can be obtained fro those outside of the Association by enclosing a two-cent stamp to the Secre tary of the Mathematics Section, Miss Mabel Sykes, 438 57th St., Chicago. This report is important to teachers of high school mathematics in many ways. It is replete with suggestion as to ways of handling geometry. It shows that a careful study of elementary geometry by strong people is productive of great fruit, that many things may be improved in numerous ways by almost any teacher who cares to take the trouble; it gives some standards of judging of good teaching, and is valuable in so many ways that no teacher of geometry can afford to be without it. No one can read this report carefully and leave it with the impression that all has been done that can be done for the improvement of geometry as a school subject, The council of the Association of Teachers of Mathematics in the Middle States and Maryland has authorized the organization of a selfgoverning section of that association in the vicinity of Rochester, N. Y. A preliminary meeting to organize the section occurred at the University of Rochester on February 23. The friends of the teaching of mathematics will wish their New York colleagues all success in this enterprise. Incidentally, we remark another evidence of the fact that a renaissance in the teaching of mathematics is on among us. Professor H. E. Slaught's name appears on the title page of the American Mathematical Monthly of Springfield, Mo., and this well-known journal now announces its publication UNDER THE AUSPICES OF THE UNIVERSITY OF CHICAGO. METRIC VOICE FROM AUSTRALIA EXTRACTS FROM THE "MELBOURNE AGE," NOVEMBER 27, 1906. The prospects of a reform in our cumbersome British weights and measures and coinage are looking brighter. In the new British Parliament, as in the last, a majority of the members are in favor of a sweeping change. Even in the Transvaal, amid the turmoil of getting the country straight after the war, a Commission was recently appointed to take evidence and report upon improvements in weights and measures. From its report it appears that public opinion in the Transvaal is strongly in favor of a bold adoption of the metric system. But the South African Commission gives practical proof of its earnestness and sagacity by recommending that the use of the metric system be made compulsory in land surveying and the sale of drugs. The reason for selecting these two departments by way of a commencement is obvious. Almost all the important land transactions in a country are local. An inappreciable fraction of the transactions in land either in Australia or South Africa is carried on with Britain. Hence in adopting an improved method of expressing land measurements quite independently of the old country, the self-governing parts of the Empire introduce no element of complication into their trade relations with Britain. In the drug trade the measurements are in the hands of expert pharmacists, to whom a change from the absurd apothecaries' measures would present no difficulty, but a welcome relief. Moreover, the British system of measures for land is the most fantastic of all the displays of unreason afforded by our weights and measures. In this twentieth century the table for lengths sounds almost crazy in its eccentricities-12 inches 1 foot, 3 feet 1 yard, 52 yards 1 rod, 25 links 1 rod, 22 yards or 4 rods 1 chain, 10 chains 1 furlong, 8 furlongs 1 mile. In the surface table the absurdity culminates in the relations 144 square inches 1 square foot, 9 square feet 1 square yard, 304 square yards 1 perch, 40 perches 1 rood, 4 roods 1 acre, and 640 acres 1 square mile. The bare recital of these tables proves them to be freakish barbarisms and anachronisms that are quite intolerable. The Transvaal Commission's recommendation is wise and sound that these absurdities should be swept away by making the metric system compulsory in land surveying. It is estimated that it costs £1,000,000 a year to teach the British weights and measures to British children, and that it wastes a whole year out of the schooling of each child. It is simply impossible to estimate with any accuracy the loss in which adults are involved by our inconvenient weights and measures. With the metric system and decimal money the compound rules of arithmetic disappear. As a compound rule sum takes about five times as long as the corresponding simple rule sum, the total waste of time involved in the retention of our antiquated system is huge. Moreover, in all the Empire's foreign business with the fifty metric nations there is a steady loss of Britain's prestige going on. One little fact shows that the British system is doomed, and should be got rid of as soon as possible. New industries even in England itself |