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. We 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 gare 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 bis noble use of powers which had survived through soul. that dreadful season when it seemed to his best But the sweet influences of the Olney Hymns, friends that all the better mental qualities of the how hallowed through many years, to all who knew man must be lost. them, whose way has been cloudy or dark or stormy His manner of life was simple and apparently or filled with obstructions, or cold or weary! Depure, in companionship with one whom he always votion has been increased, the strong emotions oi loved, who devoted herself to him as a mother t« men have been soothed and chastened; hope, at her feeble boy, who accompanied him in his daily first feeble, has gained strength, and has finally walks, nursed him in sickness, watched orer him soared above the world into a higher region of 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 repard 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. l'hough 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 of hope and the sweet blossoms of joy spring there, prayer. He was convinced that he had no right to and so the seed that was sowo in sadness has come pray; that his wickedness was so great it could forth amid peace and joyfuluess in many hearts. not be forgiven. In the latter portion of his life he trusted to the revelations of an ignorant or an For the Schoolmaster. imprudent schoolmaster, who pretended to prevail A Hymn. in prayer in favor of the poet. Cowper kept constant record of what the poor schoolmaster pre- WRITTEN ON THE THIRTEENTH OF APRIL, A. D. 1861. tended were revelations from above. This is a sad TUNE - Dundee. spectacle, but it should not hide from view the hallowed labors of his happier hours, wherein he earn I. ed a lasting and grateful remembrance. There To thee, O Lord, in time of fear, are some passages of his poetry which will not We look for strength and skill; soon be forgotten. And of all, those which betray We shall be safe when Thou art near, the kindly sympathy of his heart with the hearts And strong beneath Thy will. of true men are most precious to readers who ad. mire his poetry most. He has embalmed the me II. mory of Mrs. Unwin — his “ Mary”-in a series The past reveals Thy Providence : of beautiful and touching stanzas, that show how Our way is unrevealed. grateful were his feelings to a friend who was true Thou wast our fathers' confidence ; through her life and dearer to him than all other Be Thou our sun and shield. friends. In his Task he sometimes gives the read. er a glimpse into his character, particularly in those stanzas, trite, because so often quoted : God of our nation, hear our prayer ! Our fathers prayed to Thee. I was a stricken deer, that left the herd Thy voice can calm the troubled air, And quell the angry sea. 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. III. L. B. cone. L. B. Mathematics. “6. It has troice the volume of the inscribed sphere. COMMUNICATIONS for this Department, if relating to “7. Its circle of contact on the inscribed sphere the higher branches, should be addressed to J. M. Ross, is so situated that the surface of the segment beLonsdale ; otherwise to N, W. DeMunn, Providence. low it is twice that of the segment above it. “8. The surface of the inscribed sphere is For tbe Schoolmaster. trice the area of the base of the cone." SOLUTION OF PROBLEM IN JUNE NUMBER.-The apparent direction of the wind is a resultant of its To these we may add the following: true direction and of the resistance of the air, and 9. The base of the cone, surface of the sphere, this resistance is equal in intensity and directiou convex surface of cone and the entire surface of the to a wind from the west having a velocity of six cone are respectively as the numbers 1, 2, 3, 4. miles per hour. Let AB represent the wind, AD 10. The centre of gravity of the solid cone is the resistance of the air ; the diagonal AC is the the same as that of the sphere; and the centre of effect produced. By the question, BAC = 100, gravity of the surface of the cone is at the centre BCA = 450; by trigonometry, of the circle of contact. sin 100 ; sin 490 :: BC; BA ;; 6; Ans. V (a.) Let DHO be a 6sin 450 4.24264 great section of the = 24.4324 miles = velocity sphere, and AMV a cosin 130 .17365 incident triangular secrequired. tion of the circumscribed Put a = CO, raPROBLEM.–With what velocity must the man dius of sphere, z = AH, travel toward the east, that he may feel as little E radius of base of cone, wind as possible ? y= HV, altitude. O (1.) To prove propoFor the Schoolmaster. sition 1: put entire sur. The Cone and Spbere Problem. face of cone = A, vol ume = V; then The following interesting propositions relating to the sphere and a certain circumscribed cone, A=r&V (od + y) + ***, V=**t*y, wbich is to be originated with E. S. Snell, Professor of Mathe- A H M a maximum. Reducing 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 u = Ax?- 270*;* readers in The SCHOOLMASTER, believing that du they will be interested in the remarkable results, whence --= 2Ax - 8px = 0. though all may not understand the methods by da which they are obtained. And should any be dis А posed to charge us with plagiarism of any sort, we Reducing, 士 are sure the learned Professor will not. du There is another cone, inscribed in a given Either value of a substituted in gives the sphere, which has equally remarkable properties, dx propounded in the same number of the Monthly same result, which is as it should be, since x 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 « is found that will ren. other, and have constructed our diagram to repre- der the function a maximum. In all cases heresent this interesting trio at one view. after it may be understood that this test has been “1. The right cone, whose slant height is to applied. the radius of base as 3:1, has the greatest volume 2A 3 A within a given surface. We now find y=v(-), and slant height 2'p “ 2. It has less entire surface than any other ... slant height : radius of base :: 3 : 1. cone circumscribing the same sphere. (2.) DO represents the circle of contact, ... " 3. It has less volume than any other circum OM = HM = x. By similar triangles, scribing the same sphere. ay ay “4. It has twice the height of the inscribed #:y::a: -= OV; ... MV = x + sphere, “6. Its entire surface is twice that of the inscribed sphere. * Hereafter find p for y. . 24 mum. ag 3r - Yo 31 + 9 (8.) Surface sphere = 4a p ; base of cone = and A = px + pas rep = 2a2p; .'. Surface sphere : base of cone::2:1; to which we may add that, the base is one-third which is to be minimum. the convex surface of the cone, or one-fourth the From these two equations we find entire surface, and the sphere is two-thirds the du 2.25 — 4a2 convex surface of the cunc; hence representing = 0. the base of the cone by unity, we have = 1, Reducing, x = I ap2. Surface of sphere = 2, We now find y=fa, and slant height=v(6a?+-2a) Convex surface of cone = 3, = 3a/2; slant height : rains of base :: 3a/2; Entire surface of cone ap2 :: 3 : 1, the same result as before. (10.) Since CH has been found to be one-fourth (3) To prove proposition 3: we find volume the altitude HV, C, the centre of the sphere, is the 2ap centre of gravity of the solid cone; and since FH which is to be a maximum. is equal to one-third the altitude, F, the centre of 3 32 -a the circle of contact is the centre of gravity of the Omitting the constant factor we have us surface of the cone. x-a? (6.) 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=av2, y=4a ; •• slant a given sphere when the convex surface is a maziheight : radius of base ::3:1; therefore the cone of greatest rolume within a given surface, and the Using the same sphere as above, put x = 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 = 7 (x12 + y)2). 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 x/2 + (y! — a)' = a'. as the second, or third, otherwise the cones would be Convex surface = px! y(x12 + yl'), to be a maxi. similar, but not equivalent. mum. Reducing these equations as above, we find (4.) We have found above that y=4a, which du proves that the altitude of the cone is twice the di- u = 2ay!? – ys; whence 4ay - 3y = 0. ameter of the sphere. 4 (5.) The minimum value of the function of the Reducing, y = -a. entire surface from (2.) will now be found by sub 3 stituting ay2 for x, giving A = Ba2p; but the area In our figure, altitude HV=y= 4a, of the sphere = 4a2p; ... entire surface cone : 4 surface sphere : :2:1. ...yliyi - a : 4a :: 3:1, .'. y' = } HV=FH, (6. The minimum value of the function of the 3 8 which must therefore be the ultitude of the inscribvolume from (3) is V=- asp; but volume of ed cone, consequently DO, the circle of contact of 3 the circumscribed cone, must be the base of the 4 sphere ap; ... volume of cone : volume of inscribed cone, which is also proved by finding 3 2u sphere :: 2:1. V2 ; but by (3 )x=ay2, from which 3 (7.) To prove this proposition we have only to 2a show that, EF = }FH. AD being equal to AH, is FO = $ a v2 V2; ... x = FO, etc. Hence, equal to } AV, ... FH = | HV; but HE = EV, 3 ... FH = .2HE = 3 HE, ... HF:FE:: 2:1; if lines DH, OH be drawn, HOD will be a section hence one-third the spheric surface is cut off abore of the required cone. 8a? 16a 22 the circle of contact DO. Slant height 76;... REMARK.-Hence if the eye be placed at a dis 9 3 tance from the Earth equal to the diameter, one 2a 2a third of its surface would be visible, a fact which slant height : radius of base :: 16: -72::/3:1. dr is corroborated by the general formula which (2.) Required the shape of a cone inscribed in a dtr given sphere when the volume of the cone is a maziexpresses the ratio of the visible to the invisible mum. portion, r being the radius, and d the distance of In this case V = $ poly', to be a maximum. the eye from the centre ; for in this case d = 3r ;l Proceeding as before, we find u = layl: + 9 х ume = cones, х which is identical with the value of u in problem Wu ritten Examinations. (1); therefore the two cones are identical; hence the cone of greatest convex surface inscribed in a giren sphere is also the cone of greatest volume, and COMMUNICATIONS for this Department shouid be ada drəssed to A. J. MANCHESTER, Providence. has the slant height to the radius of the base as v3:1. Its altitude : diameter of sphere :: 2:3; and its ARITHMETIC.COMPLEX FRACTIONS. 4a 2a altitude : slant height :: 16:: 2 : 73. The 1. How many sevenths in 3 3 1 23 8aạp 5 Х ? 9 .0163 convex surface : area of base :: V3:1:: slant .00 3-14 1 .0 6-6 2. is what per cent. of ? 4a$p 1 48 volume of cone : volume of sphere::23:33. 3 3} 65 We will now develop some of the interesting relations between the circumscribed and inscribed 23 .0 5.6 3. How many eighths in Х ? 4-5 8-15 1. We have seen that the base of the inscribed cone is equal to the circle of contact of circum 23 .007 (7-11 .0063 4. Х scribed cone; and the altitude of the former equals .034 7 7-9 5-9 28-55 one-third that of the latter; so that, supposing the inscribed cone to take an inverted position the 49-61 32-45 8 2-11 .051 vertex will meet the centre of the base — point of contact — of the circumscribed cone. 5. .05 4-9 3 .0121 73 2. The centre of gravity of the volume of the 6 2-3 6 4-7. inscribed cone is at the centre of the sphere, since 4a 7} CF =y-a= -= {FH ; therefore 3 .05 7-9 6 2-5 0182 the three solid bodies have the same centre of gravi 6. 13-20 9-30 9 3-5 ty. 3205p 16 3. The volume of the inscribed cone is 81 8абр (3-11 94 .04 1-5 .0 5-7 that of the circumscribed cone is ;'. vol. Х 1 ume inscribed cone : rolume of circumscribed cone 71-7 .000 :: 22 : 33. 4. Base of inscribed cone : base of circumscribed 4 3-5 cone :: 21 : x2 :: 2 : 32. 6. What has been shown above in respect to .4073 .011 1-9 11-11 8. Х the division of the spheric surface by the circle of 69-500 9 .00% contacı, is true of the base of the inscribed cone, i. e., the base of the inscribed cone divides the sur 10 2-7 1 face of the sphere into parts as 1 : 2. Other interesting properties of the inscribed cone 17 1-5 .0 5-6) .003 are mentioned by Prof. Kirkwood in the article in the Monthly, above alluded to; and it is possible 9. 99-10 1 7 8-13 there are still others not yet developed, the dis 5 5-12 covery of which we leave for the investigation of the reader. J. M. R. (6.33} 15 PROBLEM.-Given the three lines that bisect the .009 8} 8-57 angles of a triangle, passing to the opposite sides, 10. to find the sides. 11 1 A few communications lie over to next month. 3 ; х Х 2.09 .0001 P. G. MENTAL ARITHMETIC. 7. 2-5 the sum received for goods is gain ; what 1. How long does it take the minute hand of a is the gain per cent. ? clock to gain one minute space on the hour-band? 8. Three boys, A, B and C, bought 15 oranges, EXPLANATION. The ininute-hand in one hour passes A paid for four, B for 44 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 honr-hand 55 minutě 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 9. I spent 1-6 my money, lost 2-5 the remainder minutes. and gave away $25 more than of what then re2. At what time between 3 and 4 o'clock do the mained and had $65 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 orertake 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 sprces 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 lhe 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? advance of A, and how far was A ahead of C? 4. At what time after 12 o'clock are the hands first together? Our Book Table. 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 TIM THE SCISSORS GRINDER, or Loving Christ and Serving Him. By Mrs. Madeline Leslie. first in the same line pointing in the same direc author of " Home Life," “ Juvenile Stories," tion? At what time pointing in opposite direc- • The Dermott Family, or the Catechism," etc., tions ? etc. Boston: Henry Hoyt, No. 9 Cornhill. 7. At what times between 5 and 6 o'clock are Here is a book which fully repays a careful readthe hands 8 minutes apart ? ing. The utter despair of the poor household of 8. At what times between 7 and 8 o'clock are the Gowers; the half-starred children; a drunken the hands 19 4-5 minutes apart? father crippled by inebriety; a mother dying with 9. At what times between 2 and 3 o'clock are apparent consumption ; little Tim with his grindthe hands 1-100 of the circumference of the dial stone starting out for work, to bring support to the apart ? 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 lore the hands make might angles ? and fidelity to his dear Saviour; and the ultimate MENTAL ARITHMETIC.-XUMBER TWO. result of all, make a wonderful book. 1. How long does it take the hour and minute We have a new song, entitled “ Our Good Ship hands, together, to pass over a distance equal to Sails To-Night,” illustrative of the departure to one minute space ? 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 Brignoli and Mr. Harrison Millard. Composed by minutes. Stephen C. Massett, and published by Firth, Pond 2. At what time between 2 and 3 o'clock do the & Co., 547 Broadway, New York. Upon the vig. hands make equal angles with the II, mark? nette is a beautiful representation of our glorious Note. It is evident the minute-hand must go to with old flag, faintly indicating that this song will stir in the same distance of the II. mark as the bour-hand is the blood in the heart of every true patriot. It is beyond it ; that is, both hands must pass over a distance truly the geni of the times. equal to 10 minute spaces. To pass over a distance equal to one minute space, it takes them, &c. BEAUTIFUL AND I'RUE. — Well has a forcible 3. At what time between 10 and 11 o'clock do writer said: “Flowers are not trifles, as one migbt the hands make equal angles with the XI, mark ? know from the pains God has taken with them 4. At what time between 3 and 4 o'clock do the everywhere; not one unfinished, not one bearing hands make equal angles with the VI. mark ? the marks of brush or pencil. Fringing the eter5. At what time between 9 and 10 o'clock does nal borders of mountain winters, gracing the pulsethe hour-hand lack as much of being up to the XI. less heart of the grey old granite, everywhere they mark as the minute hand is beyond the V. mark : are charming. Murderers do not ordinarily wear 6. At what time between VII. and VIII. o'clock roses in their button-holes. Villains seldom train does the hour-hand want as much of being up to vines over cottage-doors." the VIII. mark as the minute-hand is beyond the PEACE is the evening star of the soul, and vir VII. mark? tue its sun ; the two are never far apart. |