in the morning, the unwillingness of the TEACHING ETHICS. spirit and weakness of the flesh seldom If one is inclined to preach and moral- overtaking them till an hour or so later." ize at all in school he is likely to do too A three minutes' talk of this kind with much of it. Too little is altogether better the teacher assuming an intimate, kindly, than too much, and the "unconscious tu- sympathetic attitude toward the pupils, ition" of the teacher's own conduct and reviewing the mistakes of the preceding personality is of more value than many day, finding out the causes and the best sermons. The argumentum ad hominem way to correct them; three minutes thus may, however, occasionally be used with spent are worth hours of keeping in at excellent effect. The interest of children night when both teacher and pupils are like that of grown persons is easily en- tired, fretted perhaps, and irritable with listed in the discussion of conditions a day's work in the vitiated air of the which they can themselves illustrate from schoolroom. experience and observation. Keeping in is perhaps a necessary evil, Kate Douglas Wiggin says in Patsy, as a temporary expedient it must occasion“We had our morning hymn and familiar ally be resorted to. It is productive of no talk in which we always outlined the pol- permanent good whatever, and the best icy of the day, for the children were apt schools reduce it to the least possible minto be angelic and receptive at nine o'clock imum. BRUDDER BROWN'S BLESSING. O Mahsr! let dis gath'rin’ fin' a blessin' in yo' sight! ekase, nex' week, we'll start in fresh, an’ labor twict as well. -From Irwin Russell's “Christmas Night in the Quarters.” MATHEMATICS. BIOGRAPHY IV. genius embraced a general and systematic SOPHUS LIE. treatment of the theory of differential Mathematical scholars are numerous the equations. For this purpose he devised the theory of transformation groups, the world over. All civilized nations have scope of which embraces not only the thepossessed them, and possess them now. ory of differential equations, but manifold England, for example, has among her edu other applications, including invariants, cated men a vast number of mathematical covariants, differential invariants, as well scholars, men whose knowledge of, and as problems in mechanics, physics and astraining in higher mathematics is unsur tronomy. passed. Such men have added greatly to the science; they have tinted the frescoes Lie's career as a professor of mathemat ics extended over a period of twenty-one and completed the niches; they have finished the structures only indistinctly pro years. A special chair of mathematics was created for him in the university of Chrisjected. Occasionally, however, in a cen tiania in 1877. He held this position tury there appears quite a different kind until his call to the university of Leipzig of mathematician, one whose intuition in 1886. While connected with the uniseems to comprehend the known scope of mathematics, or at least that of certain versity of Leipzig he held the chair of fields, one whose brain is afire to see be geometry. His fame spread far and wide; students from Germany, Russia, Greece, yond the known limits of mathematical knowledge, one who penetrates the un Italy, France, England, America, attended his lectures. These lectures were, as a known. Such a man may be called a mathematical genius, and such was Sophus questions, informal remarks and illustra rule, very informal, being interspersed by Lie. tions. This method of instruction was, as This remarkable man was a Norwegian, educated at the University of Christiania, a source of inspiration, far superior to the formal lecture of the ordinary German receiving his doctorate in 1865 at the age of twenty-three. During his connection professor. Lie was a giant physically and mentally, with the University as an undergraduate odd in habits, fearless in thought and acstudent Lie attracted no especial attention tion, kind to his friends, intolerant of his as an original worker in mathematics; in real or fancied enemies. During the last fact, he anticipated his career to be either few years of his life the fancied enemies philology or engineering, with a possibility of mathematics. A course of lectures gave him a source of annoyance that was entirely avoidable. He gradually grew on algebraic substitutions by Professor discontented with his surroundings at Sylow at the University of Christiania, and a careful study of Plücker's Line Geometry Leipzig, and resigned in the summer of 1898 to accept his old position at the uniinspired Lie to the highest mathematical versity of Christiania, where he died in activity. From 1868 to 1872 his genius February, 1899, at the age of fifty-six. had full sway, unhampered by the details of everyday life. It was during this brief period that the foundations for his far GRAPHICAL REPRESENTATION. reaching and epoch-making discoveries were conceived. The remainder of his life By a graph is meant a drawing used to was devoted to the amplification of this represent or illustrate some numerical or program. statistical fact. It is used very freely in The fundamental product of Lie's the public press. It has a wide applica tion, not only in the diffusion of general knowledge, but in the schoolroom as a help in all lines of mathematical work. The first essentials in a graphical representation are the lines of reference, technically known as the axes. These are two arbitrary lines at right angles to each other, the horizontal one being known as the X-axis, or the axis of abscissas, and the other the Y-axis, or the axis of ordinates. The intersection of the axes is called the origin. Anything that can be located with respect to these axes is capable of graphic representation. An example will make the matter clear. Suppose that in crossing a field 20 rods long, a man walks in a path described as follows: He crosses in 80 sec., starting at southwest corner. At the end of 4 sec. he is 1 rd. from south fence, 8 sec. 2.rds., 12 sec. 3 rds., 16 sec. 2 rds., 20 sec. 14 rds., 24 sec. 1 rd., 28 sec. frd., 32 sec. 1 rd., 36 sec. 2 rds., 40 sec. 3 rds., 44 sec. 5 rds., 48 sec. 2 rds., 52 sec. 2 rds., 56 sec. 1 rd., 60 sec. 4 rds., 64 sec. 3 rds., 68 sec. 5 rds., 72 sec. 4 rds., 76 sec. 3 rds., 80 sec. 1 rd. On the OX line let the spaces each represent 4 seconds. Then at each of these points erect perpendiculars which shall be proportional to the distance he is at that time from the south fence. A line through the tops of these perpendiculars represents his path. Such a diagram shows more at a single glance than can be gained by several minutes' study of the mere figures. (See fig. 1.) The temperature from a. m. to 5 p. m. on a certain day was as follows: my a. m. 70, 8 a. m. 75, 9 a. m. 78, 10 a. m. 80, 11 a. m. 82, 12 m. 85, 1 p. m. 88,2 p. m. 90, 3 p. m. 85, 4 p. m. 81, 5 p. m. 78. To show this in a graph we will represent the hours on the horizontal line and the temperature by the perpendiculars. A similar diagram may be used to show the price of wheat for the months of a year. Represent the months on the horizontal line and price for each month by a perpendicular proportional to it. The growth of a town or state may be shown graphically in the same way. Represent the time on th horizontal line and the population by the perpendiculars. SOLUTIONS. 35. The volume of a cylinder is 196.35 cu. ft. Its height is 10 ft. What is the area of the base of a similar cylinder whose volume is 27 times as great ? Similar solids are to each other as the cubes of their like parts. Volume of second cylinder is 27 X 196.35=5301.45. 196.35: 5301.45 = (10) 8 : x3. Lizzie Lilly, Pierceton. 36. (1) x+2y + 3z=4 (2) 2x + 3y + 4z=6 (3) 3x + 4y + 5z = 8 Eliminating x from (1) and (2), the result is y +22=2. Eliminating x from (1) and (3), the result is 2y + 4z=4. The second of these results is twice the first, hence a definite solution is not possible ; but if y + 2z=2, y can be 0 and z=1 or y= 2 and 2= -0. These are the only positive integral values possible. These give for x the values 1 and 0. R. E. Lind, Sandborn, 37. If the side of one equilateral triangle is equal to the altitude of another equilateral triangle, what is the ratio of their areas ? Suppose a a side of the first, and the altitude of the second. The altitude of the first is a2 V 3. 4 2 Similar triangles are to each other as the squares of their like parts. The triangles, therefore, are to each other as За 2 3:4. A. F. Wood, Mitchell. 38. Find the equated time for the pay. ment of the balance of a debt of $1.250, which was to run one year, without interest, one-half of it having been paid at the end of 3 months; one-fourth of the remainder at the end of 6 montbs; and one-fourth of that remainder at the end of 9 months. The various payments are $650, $156.25 and $117.1875. The balance unpaid is $351.5625. Interest on $625 for 9 mo. at 6 % $28 125. 1 Vaz . 2 2 :a : a 2 CREDITS. Lizzie Lilly, Pierceton, 35; R. E. Lind, Sandborn, 35, 36, 37, 38, 39; A. F. Wood, Mitchell, 37, 39; Lefevre Du Bois, Rochester, 37, 38, 39; L. M. Neher, N. Manchester, 35, 36, 39; W. W. Wells, Tipton, 39; 0. V. Wolfe, Walkerton, 35, 37, 38; Howard Kesler, Clark's Hill, 38; Myrtle Skinner, Gaston, 39; Clinton Moornow, South Bend, 35, 36, 37; Colonel Sentman, Stone Bluff, 35, 36, 37, 39; Ethel Bennett, Peru, 35, 37, 39; Lillian Carter, Acton, 38; 0. F. Krieger, Carthage, 37, 38; E. J. Metz, Columbia City, 35, 37, 39; R. H. Baumunk, Saline City, 37; G. E. Combs, Linton, 35; Clara Coombes, Borden, 35, 36, 37, 38, 39; P. G. Huston, Weirtown, 35, 37, 38, 39; Jos. R. Westhafer, Washington, 35, 37, 38, 39; Anna B. Partridge, 35, 37, 39; Ernest Applegate, Thorntown, 35, 36, 37, 39; F. J. Conboy, Wanatah, 35, 39; Edward Morgan, Aroma, 35, 36; R. C. Whitted, Pinhook, 35; R. L. Modesitt, Atherton, 35, 39; Olin Norman, Heltonville, 35, 37; C. O. Mitchell, Eaton, 35, 38; L. W. Clements, Elnora, 38; Flora J. Scott, Fairmount, 35; H. H. Pleasant, Tower, 35, 38; V. W. Martin, Carbon, 37, 38, 39; F. W. Donahue, Boonville, 37; John T. Campbell, Rockville, 39; Charles B. Austin, Winchester, 35; J. J. Gorhinger, Anderson, 35, 36, 37, 38, 39; A. R. Melick, Etna Green, 35; E. L. Penn, Remington, 37; Emma B. Miller, Don Juan, 35; D. W. Werremeyer, Jasper, 35; J. B. Murphy, Georgetown, 35, 36, 37, 38, 39; Ethelbert Woodburn, Ambia, 35, 36, 37, 39; Perry Snethen, Walkerton, 35, 39; Michael Kappes, Kelso, 35, 38, 39. 39. The base of a right triangle is 120 ris. and the altitude 40 rds.; how far from the vertex must lines be drawn perpendicular to the base to divide triangle into three equal parts? (See fig. 3.) Triangles ABC, MBX, NBY, are similar, since their angles are equal. Triangle ABC 40 X 120 = 2,400 sq. Triangle MBX = of triangle ABC 1.600 sa rds 1202 + 402 = CB2. 126 5 approx. ABC: MBX = CB2 : XB2. rds. area area. |