We ing disgrace through failure, or the mortification middle life were not of a class fitted to elevate or of relinquishing what he and his friends desired improve his life. But after his removal into Olney, should be retained. It was soon after he had final- friends of another class were around him. ly given up all intentions of serving in this office cannot forget the stern, yet kind influence of Mrs. that he endeavored to destroy his life by his own Newton, neither the gentle encouragement Lady act, and not long after that he passed into total Austen gave him in his sadder moods, nor the insanity and was confined in a mad-house. cheering character of the literary companionship It is well to let the cloak of a charitable silence of Mr. Hayley. These are bright spots in an atcover the features of this dreadful experience, mosphere of clouds. They give a warm coloring which the unfortunate poet, with a determination to a picture else but too sad to look upon with rather to be regretted than condemned, has expos-emotions other than of pity. It was a sad picture ed to view. Cowper rallied, partially recovered, indeed, described in a brief space. Living through retired into the seclusion of the country, and amidst long years of gloom and despondency, he came to the charms of rural life, encompassed by kind the close of life as gloomy and despondent as he friends, and surrounded by love and sympathy, had been all his days, and at last expired without passed year after year in peace with men, and in a any token that the cloud had been lifted from his noble use of powers which had survived through soul. that dreadful season when it seemed to his best friends that all the better mental qualities of the man must be lost. But the sweet influences of the Olney Hymns, how hallowed through many years, to all who knew them, whose way has been cloudy or dark or stormy or filled with obstructions, or cold or weary! Devotion has been increased, the strong emotions of men have been soothed and chastened; hope, at first feeble, has gained strength, and has finally soared above the world into a higher region of of hope and the sweet blossoms of joy spring there, and so the seed that was sown in sadness has come forth amid peace and joyfulness in many hearts. For the Schoolmaster. His manner of life was simple and apparently pure, in companionship with one whom he always loved, who devoted herself to him as a mother to her feeble boy, who accompanied him in his daily walks, nursed him in sickness, watched over him in his hours of mental wandering, and to whom, light and enlargement. We pass by the gloom during the last days of her life, he repaid all that and darkness of his weary life, and know and resuch a man could supply, in kindness, love and member Cowper as we love to know and remember sympathy. him by the precious gifts he has bequeathed to us. It is well known that during a large part of Cow-Though in all the way he walked, withered flowers per's life, he was constantly oppressed with a belief and faded leaves lay scattered, still the fresh buds that he was denied by Deity access to his favor by prayer. He was convinced that he had no right to pray; that his wickedness was so great it could not be forgiven. In the latter portion of his life he trusted to the revelations of an ignorant or an imprudent schoolmaster, who pretended to prevail in prayer in favor of the poet. Cowper kept con-) stant record of what the poor schoolmaster pretended were revelations from above. This is a sad spectacle, but it should not hide from view the hallowed labors of his happier hours, wherein he earned a lasting and grateful remembrance. There are some passages of his poetry which will not soon be forgotten. And of all, those which betray the kindly sympathy of his heart with the hearts of true men are most precious to readers who admire his poetry most. He has embalmed the memory of Mrs. Unwin - his "Mary"-in a series of beautiful and touching stanzas, that show how grateful were his feelings to a friend who was true through her life and dearer to him than all other friends. In his Task he sometimes gives the reader a glimpse into his character, particularly in those stanzas, trite, because so often quoted: "I was a stricken deer, that left the herd My panting side was charged when I withdrew To seek a tranquil death in distant shades." WRITTEN ON THE THIRTEENTH OF APRIL, A. D. 1861. I. To thee, O Lord, in time of fear, We look for strength and skill; II. The past reveals Thy Providence : III. God of our nation, hear our prayer! WE often pretend to fear what we really despise, It cannot be doubted that his friends influenced and more often to despise what we really fear.— very much the character of Cowper. Those in his PRENTICE. Mathematics. COMMUNICATIONS for this Department, if relating to the higher branches, should be addressed to J. M. Ross, Lonsdale; otherwise to N. W. DEMUNN, Providence. For the Schoolmaster. 221 "6. It has twice the volume of the inscribed sphere. "7. Its circle of contact on the inscribed sphere is so situated that the surface of the segment below it is twice that of the segment above it. 8. The surface of the inscribed sphere is twice the area of the base of the cone." To these we may add the following: 9. The base of the cone, surface of the sphere, convex surface of cone and the entire surface of the cone are respectively as the numbers 1, 2, 3, 4. SOLUTION OF PROBLEM IN JUNE NUMBER.-The apparent direction of the wind is a resultant of its true direction and of the resistance of the air, and this resistance is equal in intensity and direction to a wind from the west having a velocity of six miles per hour. Let AB represent the wind, AD the resistance of the air; the diagonal AC is the effect produced. By the question, BAC=10°, gravity of the surface of the cone is at the centre BCA = 45°; by trigonometry, sin 100 sin 45°:: BC: BA:: 6: Ans. 10. The centre of gravity of the solid cone is the same as that of the sphere; and the centre of of the circle of contact. H M (a.) Let DHO be a great section of the sphere, and AMV a coincident triangular section of the circumscribed cone. Put a = CO, radius of sphere, z = AH, radius of base of cone, y= HV, altitude. (1.) To prove proposition 1 put entire surface of cone = A, volume V; then A = xx√ (x2 + y2) +πx2, V=y, which is to be a maximum. Reducing THE following interesting propositions relating to the sphere and a certain circumscribed cone, originated with E. S. Snell, Professor of Mathe-A matics and Natural Philosophy in Amherst Col the first equation with reference to y, substituting lege, from whom we received the problem as a class its value in the second, and expanding and rejectexercise some years since, and who published the ing constant factors preparatory to differentiation, results of his investigations nearly two years ago in and calling this simplest state of the function, here the Mathematical Monthly. We have been intend- as well as in all the cases that follow, u, we have ing for some time to present the subject before our readers in THE SCHOOLMASTER, believing that they will be interested in the remarkable results, whence though all may not understand the methods by which they are obtained. And should any be disposed to charge us with plagiarism of any sort, we are sure the learned Professor will not. Reducing, du = 2Ax · dx 8px = 0. d'u Either value of a substituted in, gives the da There is another cone, inscribed in a given sphere, which has equally remarkable properties, propounded in the same number of the Monthly same result, which is as it should be, since z is by Prof. Kirkwood, of Indiana University. By a radial distance the double sign indicating opporecent investigations we have ascertained that site directions merely; and this result is negative, these two cones sustain curious relations to each showing that, a value of x is found that will renIn all cases hereother, and have constructed our diagram to repre-der the function a maximum. after it may be understood that this test has been sent this interesting trio at one view. applied. Omitting the constant factor we have u = (8.) Surface sphere 4a3p; base of cone = x2p=2a2p; ... surface sphere: base of cone::2:1; to which we may add that, the base is one-third the convex surface of the cone, or one-fourth the entire surface, and the sphere is two-thirds the convex surface of the cone; hence representing the base of the cone by unity, we have Base of cone Surface of sphere (9.) Convex surface of cone Entire surface of cone = 1, = = 2, 3, = 4. (10.) Since CH has been found to be one-fourth the altitude HV, C, the centre of the sphere, is the centre of gravity of the solid cone; and since FH is equal to one-third the altitude, F, the centre of the circle of contact is the centre of gravity of the surface of the cone. mum. -, (b.) We will now solve the following problems: an expression identical with that above in (2). (1.) Required the shape of the cone inscribed in hence the same result: x = ..slant = a√2, y = 4a; a given sphere when the convex surface is a mazi. height radius of base::3: 1; therefore the cone of greatest volume within a given surface, and the Using the same sphere as above, put a radius cone of least entire surface circumscribing a given of base of required cone, y' altitude; then slant sphere, and the cone of least volume circumscribing height = √(x2 + y2). In the right-angled trianthe same sphere, are one and the same cone, whose gle formed by the radius of sphere, the radius of slant height is to the radius of the base as 3: 1, pro- base of cone and the line joining the centres, we vided that, the first cone has the same entire surface have x2 + (y! — a)2 = a2. as the second, or third, otherwise the cones would be similar, but not equivalent. Convex surface = · px! √(x12 + y2), to be a maximum. Reducing these equations as above. we find (4.) We have found above that y=4a, which proves that the altitude of the cone is twice the di-u2ay2-y; whence ameter of the sphere. (5.) The minimum value of the function of the entire surface from (2.) will now be found by substituting a√2 for x, giving A = Sap; but the area of the sphere 4a2p; ... entire surface cone: surface sphere: 2 1. (6. The minimum value of the function of the volume from (3) is V: 4 8 -= 3 Reducing, du which must therefore be the ultitude of the inscriba3p; but volume of ed cone, consequently DO, the circle of contact of the circumscribed cone, must be the base of the inscribed cone, which is also proved by finding sphere = - a3p; ... volume of cone volume of sphere 2 1. 3 (7.) To prove this proposition we have only to show that, EFFH. AD being equal to AH, is FO= } a √2 √2; ... x' FO, etc. Hence, equal to AV, FHHV; but HE EV, ... FH.2HE = HE, ... HF: FE:2:1; hence one-third the spheric surface is cut off above the circle of contact DO. if lines DH, OH be drawn, HOD will be a section of the required cone. 8a2 16a2 2a === 9 2a √6; 3 2a which is identical with the value of u in problem Written Examinations. (1); therefore the two cones are identical; hence altitude slant height :: -: COMMUNICATIONS for this Department should be addrassed to A. J. MANCHESTER, Providence. ARITHMETIC.-COMPLEX FRACTIONS. 1. How many sevenths in Sa p√3 8a2p 1 23 31-5 5 11 3-7 1 .0163 .00 3-14 is what per cent. of 2 4. .03 .007 X7 7-9 7-11 .0063 ume inscribed cone volume of circumscribed cone :: 2:33. 4. Base of inscribed cone base of circumscribed cone x2 x2:: 22: 32. 6. What has been shown above in respect to the division of the spheric surface by the circle of contact, is true of the base of the inscribed cone, i. e., the base of the inscribed cone divides the surface of the sphere into parts as 1 : 2. Other interesting properties of the inscribed cone are mentioned by Prof. Kirkwood in the article in the Monthly, above alluded to; and it is possible there are still others not yet developed, the discovery of which we leave for the investigation of the reader. MENTAL ARITHMETIC. 1. How long does it take the minute hand of a clock to gain one minute space on the hour-hand? 7. 2-5 the sum received for goods is gain; what is the gain per cent. ? 8. Three boys, A, B and C, bought 15 oranges, EXPLANATION. The minute-hand in one hour passes A paid for four, B for 4 and C for the remainder. over sixty minute spaces and the hour-hand passes over They were joined by D and the four shared the five; therefore, the minute-hand in 60 minutes gains upon oranges equally, D paying 15 cents for his share. the hour-hand 55 minute spaces, and to gain one space How must A, B and C divide the money? it will take it one-fifty-fifth of 60 minutes, equal to 60-55 minutes. 9. I spent 1-6 my money, lost 2-5 the remainder and gave away $20 more than of what then re 2. At what time between 3 and 4 o'clock do the mained and had $60 left. How much had I at hands point in opposite directions? first? NOTE. The minute-hand must gain upon the hour- 10. A, B and C started from the same point and hand 15 minute spaces to overtake it, and to point in an ran in the same direction. A ran 77 rods, and 1-12 opposite direction it must gain 30 minute speces more, the distance B ran equals the distance he was in making 45 in all. To gain one minute space, &c. advance of A, and 2-5 the distance C ran equals 3. At what time between 10 and 11 o'clock are the distance he was behind B. How far was B in the hands together? 4. At what time after 12 o'clock are the hands first together? 5. At what time between 9 and 10 o'clock do the hands point in opposite directions? 6. At what time after 6 o'clock are the hands first in the same line pointing in the same direction? At what time pointing in opposite directions? 7. At what times between 5 and 6 o'clock are the hands 8 minutes apart? advance of A, and how far was A ahead of C ? Our Book Table. TIM THE SCISSORS GRINDER, or Loving Christ 8. At what times between 7 and 8 o'clock are the Gowers; the half-starved children; a drunken the hands 19 4-5 minutes apart? 9. At what times between 2 and 3 o'clock are the hands 1-100 of the circumference of the dial apart? father crippled by inebriety; a mother dying with apparent consumption; little Tim with his grindstone starting out for work, to bring support to the dear ones at home; his first success; the little 10. At what times between 4 and 5 o'clock do prayer, which a kind lady taught him; his love the hands make right angles? MENTAL ARITHMETIC.-NUMBER TWO. 1. How long does it take the hour and minute hands, together, to pass over a distance equal to one minute space? and fidelity to his dear Saviour; and the ultimate result of all, make a wonderful book. We have a new song, entitled "Our Good Ship Sails To-Night," illustrative of the departure to the war, and dedicated to the gallant patriots now EXPLANATION. In one hour the minute-hand passes volunteering in the service of our country, sung over sixty minute spaces and the hour-hand over five; with the most enthusiastic applause by Madame both together, therefore, pass over a distance equal to 65 | Anna Bishop, Miss Isabella Hinckley, Signor minute spaces. To pass over a distance equal to one minate space, they will require one-sixty-fifth of 60 minutes. 2. At what time between 2 and 3 o'clock do the hands make equal angles with the II. mark? NOTE. It is evident the minute-hand must go to with in the same distance of the II. mark as the hour-hand is beyond it; that is, both hands must pass over a distance equal to 10 minute spaces. To pass over a distance equal to one minute space, it takes them, &c. 3. At what time between 10 and 11 o'clock do the hands make equal angles with the XI, mark? 4. At what time between 3 and 4 o'clock do the hands make equal angles with the VI. mark? 5. At what time between 9 and 10 o'clock does the hour-hand lack as much of being up to the XI. mark as the minute hand is beyond the V. mark 6. At what time between VII. and VIII. o'clock does the hour-hand want as much of being up to the VIII. mark as the minute-hand is beyond the VII. mark? Brignoli and Mr. Harrison Millard. Composed by Stephen C. Massett, and published by Firth, Pond & Co., 547 Broadway, New York. Upon the vignette is a beautiful representation of our glorious old flag, faintly indicating that this song will stir the blood in the heart of every true patriot. It is truly the gem of the times. |