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A. It is proper to limit them. Although a person may understand a thing, exceeding slowness units him for a teacher. It is indicative of some mental or physical deficiency.
In accordance with an intimation in the last number, we wish to devote very little space hereafter to mathematical questions, and much more to general subjects. The Query Box has enabled those of our readers who take an especial interest in mathematics to get acquainted with each other, and they can keep up the acquaintance in other ways, if they desire. The great majority of our readers will be better pleased with more variety.
We are under great obligations to our mathematical contributors, whose problems and solutions indicate a very creditable degree of scholarship in that direction.
CORRECTION.–Page 101, March number, third line from the top, for first number, read first member; fifth line from top, number should read member.
ANSWERS TO QUESTIONS. 65.-Suppose an opening to be made directly through the center of the earth, and a cannon ball dropped into it, where would the ball come to a state of rest?
I cannot agree with the reasoning of W. H. on this question. The ball is no more destitute of gravity and inertia when in equilibrium at the center of the earth, than is a ball projected upward, at the moment it ceases to ascend, or the ball of a pendulum when at rest under its point of suspension. The ball on entering the abyss would fall toward the center with a motion constantly accelerated, though not in the same ratio as happens in bodies above the surface. sing the center the momentum acquired in falling would be expended in overcoming the resistance of gravitation to its motion from the center. If we suppose the opening to be a vacuum, there is an equilibrium of forces and an eternal oscillation of the ball from one side of the earth to the other. If we suppose a resisting medium admitted the oscillations are gradually contracted, and the ball finally come to rest at the center. It may be remarked that we must suppose our opening to be made from pole to pole, else the eastwardly motion of the earth's surface will give the ball a direction that carry it wide of the center. Even if we ignore the effect of the earth’s motion, its spheroidal form will oblige us to make provision for a curved line of motion in our opening, if made elsewhere than at the poles or equator.-S. LITTLEFIELD, Hingham.
68.—Besides the solutions to this problem, sent by Messrs. Anderson (B. R. A.), Minaghan and Powers, as mentioned in last month, one was furnished by M. Kirwan of Manitowoc. Wishing to close up mathematical discussions in the Journal, for the present, we quote some remarks from Mr. Anderson's answer, and omit the work of all the solutions.-EDRS.
This equation cannot be solved by any of the ordinary methods; but remembering, “Every equation having unity for the coefficient of the first term and for all the other coefficients whole numbers, can have only whole numbers for its commensurable roots,” we take advantage of this fact and obtain the roots of the
equation by separating the absolute term into its divisors and divide it by those divisors; then we will add the coefficient of y, and divide the sums thus obtained as before; again adding coefficient of the next term to the left until all the coefficients except coefficient of y–4 have been added; then, the divisor standing above and corresponding to minus 1, will be the root of the equation.
74.–What is the width of the Frigid Zone, and why are the Tropics of Cancer and Capricorn so called?
The width of the Frigid Zone is 47°, or 3,243 miles. The Tropics which show the boundaries of the sun's apparent path north and south of the equator, derive their name from a Greek word signifying to turn. On the 21st of June, when sun enters the sign of Cancer, and its rays fall vertically on its boundary-line to the north, it appears to stop, and return toward the equator. This boundary-line is, therefore, called Tropic of Cancer. The other one to the south is called Tropic of Capricorn, on account of the sun's return toward the equator, when it enters the sign of Capricorn on the 21st of December.
85.—A fountain has four receiving pipes—A, B, C and D. A, B and C can fill it in six hours; B, C and D in eight hours; C, D and A in ten hours; D, A and B in twelve hours. It has also four discharging pipes—W, X, Y and Z. W, X and Y will empty it in six hours; X, Y and Z in five hours; Y, Z and W in four hours; Z, W and X in three hours. Suppose the pipes are all open, and the fountain full, in what time will it be empty?-A. N. SEARLE, Lynxville.
Since A, B and C can fill the fountain in six hours, they can fill one-sixth of it in one hour. By the same reasoning B, C and D can fill one-eighth, C, D and A, onetenth, and D, A and B one-twelfth of it in an hour. Now, since each of the letters are used three times, the part A, B, C and D can fill in one hour is
of (++++++i)=12* By the same method we find that W, X, Y and Z can empty nineteen-sixtieths of the fountain in one hour, consequently there will be emptied at the end of an hour that part which is the difference between 18-1940=120, and the fountain will and the fountain will become empty in as many hours as 10 is contained times in 1(=148). 198+18=6%, answer.-C. A. THOMPSON, Lyndon; also by W. P. MASSUERE, Arcadia, and R. L. D., Trempealeau. 86.-Place five fives in such a manner that they will exactly equal 556.-D. M.
Ans.-5555=556.-G. H. D., Janesville; BENJAMIN SIMONDS, Baraboo, ard L. A. P., Glenbeulah.
Second Ans.-5+5+555=556.-C. A. T., Lyndon. 87.—Will
some one please inform the readers of the JOURNAL, of the names and terms of office of the several Chief Justices of the State of Wisconsin? Chief Justice-LUTHER S. Dixon. Term expires May 31, 1875. Associate Justice-ORSAMUS COLE. Term expires May 31, 1873. Associate Justice–BYRON PAINE. Term expires May 31, 1877. Each elected for six years.--R. L. DICKENS, Trempealeau.
[Associate Justice PAINE died over a year ago, and WM-P. LYON was elected to serve out the term. The question, it will be observed, has reference to the Chief Justices of the State. Who will give them all, from the beginning?—EDs.]
88.-Will some of the contributors to the JOURNAL furnish the readers of the same the names and respective terms of office of the Speakers of the House of Representatives?
1st Congress-F. A. Muhlenburg, of Penn., April 1, 1789 to March 3, 1791. 2d Congress-Jonathan Trumbull, of Conn., Oct. 24, 1791 to March 3, 1793. 3d Congress-F. A. Muhlenburg, of Penn., Dec. 2, 1793 to March 3, 1795. 4th and 5th Congresses—Jonathan Dayton, of N. J., Dec. 7, 1795 to Mar. 3, 1799.
3—[VOL. II.-.No. 4.]
6th Congress-Theodore Sedgwick, of Mass., Dec. 3, 1799 to March 3, 1801.
March 3, 1821.
June 3, 1834, when John Bell, of Tenn., was elected for the balance of the
23d Congress, which ended March 3, 1335. 24th and 25th Congresses--J. K. Polk, of Tenn., Dec. 7,1835 to March 3, 1839. 26th Congress - R. M. T. Hunter, of Va., Dec. 16, 1839 to March 3, 1811. 27th Congress. -John White, of Ky., May 31, 1841, to March 3, 1843. 28th Congress-J. W. Jones, of Vå., Dec. 4, 1843 to March 3, 1815. 29th Congress—J. W. Davis, of Ind., Dec. 1, 1845 to March 3, 1817. 30th Congress-R. C. Winthrop, of Mass., Dec. 16, 1847 to March 3, 1819. 31st Congress--Howell Cobb, of Ga., Dec. 24, 1849 to March 3, 1851. 32d and 33d Congresses-Lyman Boyd, of Ky., Dec.4, 1851 to March 3, 1855. 34th Congress—N. P. Banks, Jr., of Mass., Feb. 2, 1856 to March 3, 1857. 35th Congress-James L. Orr, of S. C., Dec. 7, 1857 to March 3, 1859. 36th Congress-Wm. Pennington, of N. J., Feb. 1, 1860 to March 3, 1861. 37th Congress-Galusha A. Grow, of Penn., July 4, 1861 to March 3, 1863. 38th, 39th and 40th Congresses—S. Colfax, of Ind., Dec. 7, 1863 to Mar. 3, 1869. 41st and 42d Congresses-J. G. Blaine, nf Maine, March 4, 1869 to the present. By W. H., Louville and R. L. D., Trempealeau.
89.-Will some one furnish the names and terms of office of the several Vice Presidents of the United States ; and mention those who have afterwards held the office of President.
John Adams, 8 years ; Thomas Jefferson, 4 years ; Aaron Burr, 4 years ; George Clinton, 7 1-12 years; Elbridge Gerry, 3 3-4 years ; Daniel D. Tompkins, 8 years ; John C. Calhoun, 8 years ; Martin Van Buren, 4 years ; Richard M. Johnson, 4 years ; John Tyler, 1 month ; George M. Dallas, 4 years ; Millard Fillmore, 1 1-3 years ; William R. King, 4 years ; John C. Breckenridge, 4 years ; Hannibal Hamlin, 4 years ; Andrew Johnson, 1.8 year ; Schuyler Colfax, now in office. Of these, John Adams, Thomas Jefferson, Martin Van Buren, John Tyler, Millard Fillmore, and Andrew Johnson afterwards became Presidents.
Answered by Mrs. S. C. SIRRINE, Plainfield, E. A. HAYES, Waterloo, R. L. DICKENS, Trempealeau, and W. H., Lowville.
90.-What are the names of the respective State Superintendents of Wisconsin? -HENRY CROUSE.
I give them in order, commencing with the first: Eleazer Root, Waukesha, 184850; Azel P. Ladd, Shullsburg, 1852–3; H. A. Wright, Prairie du Chien, 1854-5; A. C. Barry, Racine, 1856–7; Lyman C. Draper, Madison, 1858–59; Josiah L. Pickard, Platteville. 1.60 4, two terms and nine months; J. G. McMynn, Racine, 1864-7, one term and three months; A. J. Craig, Madison, 1868–70, one term and six months; Samuel Fallows, now holding first full term.-W. H., Lowville, R. L. DICKENS, Trempealeau, and A. O. W., Nero Lisbon.
94.-What is the best evidence we have that the city of Rome was founded 753 years before the Christain era?-L. C.
The best evidence that we that Rome was founded, B. C. 753, is Roman tradition, as given þy Varro and many other writers. Some writers vary a few years, but it is the universal voice of all Roman authors that Rome was founded some time within thirty years of that date. But there is not as much difference of opinion about the exact date of the founding of Rome as there is about the exact date of the bizth of Christ. The date of the founding of the city is about the most cerain thing in early Roman history, mythical as much of it is.-A.0. W., New Lisbon.
92.-Is it consistent with the health and educational progress of scholars to lunch at recess, between nine and twelve, and at recess, between one and four o'clock?-H. CROUSE, Eau Galle.
It is evident, that in general this is not consistent, as nearly all our food requires from three to four hours for digestion; and the organs, to remain healthy, must have rest. With small children, this will do; but with adults, at least six hours should intervene between regular meals.-E. A. Hayes, Waterloo.
[Mr. Hayes is doubtless of the opinion that the school-house is a place rather for the exercise of the brain than the stomach.-EDs.]
Second Ans.-As study draws to the brain blood that is needed in the work of digestion, thereby hindering such work, it cannot be otherwise than injurious to go directly from eating to study.-W. H.
93.-Given, 8412 square feet of three inch plank. Required, the length of one edge of a cubical box, which can be made from the plank, allowing no waste in sawing. Arithmetical solution desired ?–M. BRIER, Baraboo.
(1.) 847=6=1412, the number of square feet in one side of the box; (2.) N141=3.75+ft., the length of one edge.—R. W. 96.- Are diagrams of any real use in teaching grammar ? I think not, for these reasons. In order to be able to diagram properly, the student must know the correct use of the words in the sentence; hence the diagram is but a written expression of what he knows of (the analysis of) the sentence, not a means by which that knowledge was obtained. Whatever aid the diagram may afford in parsing tends to make the student dependent upon such disposition of the words, and to lessen his ability to determine the correct parsing when found in the sentence as it is.-L A. PRADT.
Another Answer.- Ideas can be answered to the mind through the eye as well as through the ear. Diagrams are of considerable value in teaching syntax and analysis, as they show to the eye a representation of the relation or words. For the other parts of grammar they are of no great value. There are many pupils who can scarcely grasp the abstract ideas of gramınatical relations, unless presented in some visible form, and for these diagrams are a great help.-A. O. W., Nero Lisbon.
We think that if the diagranıs are rightly used, they may be of more relative úse in gran.mar that geometrical figures are in geometry. They often point to an error in analysis which was not noticed before an attempt was made to diagram the sentence.-R. L. D., Trempealeau.
I think they are. But too much dependence should not be placed in them. They aid the learner in getting to understand the relation of words, etc., to each other, by making such relation visible to the eye.-C. A. T., Lyndon.
DEFINITIONS.—A definition, in order to become perfect, must include the whole thing, or class of things, which it pretends to define, and exclude everything which comes not under the name.-GOOLD BROWN.
The definition of Case given in the February number is not complete, because it says nothing of pronouns, which have cases as well as substantives. That of Prepositions is too copious, and the one for Interjections is not quite correct; for the greatest part of interjections do never express a complete thought. I prefer the definitions given by GOOLD BROWN, viz. : (1). Cases, in grammar, are modifications that distinguish the relations of nouns or pronouns to other words. (2). A Preposition is a word used to express some relation of different things or thoughts to each other and is generally placed before a noun or pronoun. (3). An Interjection is a word that is uttered merely to indicate some strong or sudden emotion of the mind P. SCHNEIDER, Gibbsville.
NEW QUESTIONS. 97.--If, at an annual school meeting a motion for a clerk is proposed and seconded by qualified voters of the district, and the chairman refuses to put the motion, and another motion is proposed that suits the chairman better than the first, is put and carried, is that clerk legally elected?
98.-If, at the same meeting it is proposed and seconded to elect a clerk by ballot and the same evening they elect a clerk by acclamation, can the clerk so elected perform the duties of a clerk?
99.-If the district clerk purchase blackboards, maps and the like for the diştrict, against the will of the director, and the director also refuse to sign the order for the payment of such purchase, what is to be done?
100.-Which of the district officers has the most authority?
[The name of the person who sent the last four questions for the Query Box has been lost. Having reached 100 questions, a new series is begun.]-EDRS.
NEW QUESTIONS-NEW SERIES. 1.-In the sentence, “It is wrong to do 80," how are the indicated words to be parsed?-L. A. PRADT.
2.-In the sentence, “ The pipers loud and louder blew; the dancers quick and quicker flew,” are loud and quick different parts of speech? The sentence is in Kerl's Grammar, page 261, 4th exercise, and is so noted as to imply that the author considers the words as different parts of speech.-16.
3.-Parse the indicated words: It is forty feet high. He is aged twenty years. It is well worth the money.-J. N.
4.-A body weighing 12 lbs., moving with a velocity of 6 inches a second has a momentum of 72. Its striking force is 432. If we say it moves 2 foot in a second, the striking force is 3 and the momentum 6. Which is correct, and why?
6.-Where and when was the first session of the territorial legislature of the State of Wisconsin held?-D. MOWRY, Windsor.
7.—What are the corporate names of the several universities and colleges in the State of Wisconsin, and where located and when founded ?-IB.
What are the names of the persons that figured conspicuously in the War of the Revolution ?-İB.
8.- What is the name and age of the youngest and oldest President of the United States, when the oath of office was administered ?-IB.
9.-“Whatever is, is right.” “ To be, or not to be, that is the question.” Will some of the contributors to the Query Box produce us an analysis of the above sentences ?-IB.
10. In Article VI of the United States Constitution we find the following words: « This Constitution and the laws of the United States, &c., &c., shall be the supreme law of the land ; and the judges in every State shall be bound thereby, anything in the constitution or laws of any State to the contrary notwithstanding.” Article V of the Amendments reads : “No person shall be held to answer for a capital or other infamous crime, unless on a presentment or indictment of a grand jury,” &c. Notice the punctuation, there being no comma after the word presentment. Now in view of these provisions, how can this State, or any other, constitutionally abolish the Grand Jury system? Will some Solon "rise to explain ?"-A, S., Broadhead.
11. Will some one please to inform us through the JOURNAL, if Noah and Daniel Webster were related ? if so, in what way ?--NORA C. WATERS, Portage City.