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LENGTH of the DIVISION LINE parallel to the two bases or ends.
Algebraically, Va'+b”, in which x represents the division line
2 and a and b the parallel bases (or ends) of the trapezoid.
Thus I=I; 1.5=2.25; i+2.25=3.25=sum of the squares ; 3.25 - =1.625; V 1.625=1.27475=length of the division line.
To find the length of one of the halves, or the perpendicular from the shorter base to the division line. Rule.- Divide the difference of the division line and the shorter base (or end) by the difference of the two bases (or ends), and multiply the quotient by the length of the plank, or the perpendicular from one base to the other.
x-a Algebraically, y= — X length.
Thus: 1.27475 length of division line
1. shorter base, or end
1.5 longer base, or end, 1. shorter base, or end.
.27475 divided by
:55:54950 -54950 X 12 (length) =6.594=perpendicular from shorter base (or end) to division line.
Proof: By the rule for finding the area of a trapezoid,
1.387875 Multiplied by 6.594
Multiplied by 5.406 Equals 7.503147750
7-50285 Area of one-half.
Area of the other half. This is correct to within less than .0003 of a square foot, and might have approximated more closely absolute correctness if the root of 1.625 had been more exactly taken.
The above problem was proposed in the Journal heretofore, but
the solutions sent misunderstood the intention of the problem, which is exactly stated now. The arithmetical rule is given in this number, but this rule is based on a geometrical demonstration, which it is hoped some of the readers of the JOURNAL will send to the editor. The problem seems to be of some practical interest to carpenters and surveyors.
S. T. PENDLETON. --- -- -----
Science in the Common School,
On the walls of a certain picture gallery in the Old World, hangs a painting, which, no less by its powerful limning, than by the subject it represents, arrests instantly the attention of every passer-by. It portrays a young man engaged in a game of chess—his antagonist, a horrid and sulphurous devil—the stake for which they play, a human soul. Substitute, in place of this grinning fiend, a calm, impassive angel, devoid of sympathy, but at the same time devoid of prejudice or passion—who makes no allowance for ignorance, feels no compassion for mistakes-who plays the game through till the final checkmate is given, with the same imperturbable stolidity, the same automatic precision—and we have, I think, no very unfair picture of human life, considered in its physical aspects alone, and in regard to the never-ending conflict with humanity on the one hand, and the laws of Matter and Force upon the other. If I have borrowed this illustration from one of our greatest living scientists, it is because it illustrates so much more forcibly than any conception of my own could do, the fact which meets us at every turn, from which there is no escape, of the utter unswervingness, the rigid inflexibility, of the laws which dominate the physical world. Each of us, man or woman, individual or nation, living or dead, has been, or is now, engaged in this conflict. Though the player on the other side be hidden from us, though we grope in darkness, if so be we may but touch the hem of His garment, though we dispute the laws laid down, and struggle frantically and blindly to escape the inevitable result, yet the game goes on. Ingorance does not extenuate, mistakes may not be corrected. Nature's laws, by which we mean the operations at work in the material cosmos, are as noiseless, yet as merciless, in their action, as a steel machine. Education is nothing more nor less than the right learning of these laws, and by just as much as an individual comprehends, and, comprehending, obeys them, by just so much are we justified in calling that individual educated.
In the words of Huxley, “That man, I think, has had a liberal education, who has been so trained in youth, that his body is the ready servant of his will, and does with ease and pleasure, all the work that, as a mechanism, it is capable of; whose intellect is a clear, cold, logic engine, with all its parts of equal strength, and in smooth, working order; ready, like a steam engine, to be turned to any kind of work, and spin the gossamers, as well as forge the anchors of the mind; whose mind is stored with a knowledge of the great and fundamental truths of nature, and of the laws of her operations ; one who, no stunted ascetic, is full of life and fire, but whose passions are trained to come to heel by a vigorous will, the servant of a tender conscience; who has learned to love all beauty, whether of Nature or of Art, to hate all vileness, and to respect others as himself."
But, in the present state of our pedagogical system, where is such an education to be had ? Not in our preparatory and high schools; not even-with a few shining exceptions-in our colleges and universities.
The average American pupil, who leaves the common school at the age of eighteen or twenty, to face the battle of life, has as little idea of the armor he must don, or the weapons he must wield, as Ajax or Achilles had of the mysteries of the needle gun. He may be able to extract the cube root of a surd to within one ten-billionth, and yet not know the essential difference between tartaric and sulphuric acid. He may be able to analyze the most complex and involved sentence that Robert Browning ever wrote, and yet be profoundly ignorant whether silicate or sandstone best favors the growth of wheat.
Nor does the evil stop here, or the ignorance cease with added years. I take it for granted that I am addressing an audience of more than average cultivation, and yet I dare assert, that with the exception of the few who may have received a special medical education, there is not one in ten who knows the absolute necessity for the aeration of the blood, or is aware of the intimate correlation between the muscular power and nervous action. And if the shepherds know not this, what shall the sheep do, which follow them ? And yet these are facts which meet us at the very threshold of being, and on our knowledge or ignorance of which depends in many cases our life or death. Does it not behoove us then to understand these laws, to know these facts, and that, not when the knowledge comes too late, not when the Nemesis of a violated law is upon our track, but in such ample time, that knowledge shall save, and obedience prolong our life? And where, if not in the primary school, is the proper place for this instruction?
Let me briefly answer here some objections which may arise to the views which I advance; the first one being, that it is impossible to teach a young child anything about Natural Science. Is it? You teach him the alphabet, you force him to learn the (to him) unmeaning jargon of long lists of polysyllabic words with their analyses and definitions, or the still more unmeaning croon of column after column of the multiplication table, and can you not teach him the simpler facts that fire will burn, and water drown, with their accompanying reasons and modifications? In fact, the very first questions which mark the budding of the child's intellect, are just such as comprehend natural phenomena. “What makes the sun rise and set ?” “Where do the winds and clouds come from?” “What makes the earth crack open when it is dry?" These and a thousand similar questions, are asked by children every day, and if the child finally grows too phlegmatic to feel further interest in such subjects, the blame lies with those who have repressed his childish questioning, and stifled the growing thirst for knowledge. “But,” says another objector, “common sense and experience will teach him these things.” I have but little fear of being misunderstood by any thinker, when I assert that common sense is but another name for common ignorance. (I think some other writer has made that assertion before me, but if so, it only adds to it confirmatory weight.) Common sense taught the priests of the Italian Inquisition that the earth was flat and stationary, and Galileo languished in prison. Common sense taught the solons of our own Congress that electricity could never “put a girdle round the earth in forty minutes," and Morse was derided as a madman. Scientific sense must supplement common sense, or rather, common sense must be educated till it becomes scientific sense. As to waiting till experience teaches him, as well might you expect or allow him to test the active qualities of arsenic or strychnine by actual experiment upon himself. Experience is too bitter a teacher, for any one to wish to go to school to her, unless through sheer necessity.
Experience taught the people of Europe in the sixteenth century, that narrow streets, crowded with wooden tenement houses, and reeking with filth and garbage, were contrary to Nature's laws, but they did not learn the lesson, till, as the price of their tuition, they paid to her avenging tutors, called by men fire and plague, millions of dollars worth of property, and hundreds of thousands of human lives. “But," again it is objected, " we have no time for these things. Reading, writing, arithmetic, grammar, geography,—these he must learn ; when, then, are we to find time to teach him, even if he could comprehend them, the more abstruse sciences of chemistry, natural philos
ophy, botany, &c. ?” Our answer is, teach them the same way, and to the same amount as, or we would even be content with less than, the others. Because you cannot carry a child through trigonometry, conic sections and the calculus, will you therefore teach him no mathematics at all? Because he has not time to master rhetoric, with its tropes, hyperboles and apostrophes, is it therefore not worth while that he should learn how to write an ordinary business letter correctly? Teach him one fact a day, which can be done in fifteen minutes, and by the time he leaves school, he will have such a basis of scientific knowledge and wealth as no money can supply.
But here again may arise a question, which, to us as a nation whose god is said to be the almighty dollar, is perhaps the most potential of all. “Of what practical benefit will this scientific education be? Will it beat ten per cent. ?” We have no hesitation in saying, it will. It is true now, and will be still truer in the future, that he who, in the struggle for existence, can bring to the aid of his natural shrewdness, an accurate knowledge of physical phenomena, who can bind Nature to his side, as an unwearying, unfaltering ally, who can supplement his own weak strength by the gigantic powers of Matter and of Force, will, of necessity, surpass his more feeble competitor. Gunpowder, steam, electricity, are but modifications of Force derived from Nature by man, and wisely used to his own advantage: and within her invisible storehouse she holds millions more, waiting for him who shall have the courage to seek and the knowledge to apply them.
Such are some of the reasons, imperfectly stated, why we advocate the study of science in our common schools. As to the how and when, that can perhaps be best governed by circumstances, but I would suggest, that if, instead of the vapid “rhetorical exercises ” (so called) which cling as a fungus-like excrescence to so many of our schools, an hour's talk on some familiar scientific subject were substituted, the gain would be invaluable. The question may be asked, “What branches of natural science would you teach, and in what order?” As to the order, that perhaps is immaterial, as it is the facts, and the deductions from these facts alone, which we propose to teach. As to what branches, I should answer, physiology and anatomy by all means, the prominent laws of hygiene, the fundamental facts of physical geography, or knowledge of the earth, not simply as confined to its surface, but as to the winds, tides, rain-falls, changes of seasons, diversities of climate, and kindred phenomena ; some of the elementary principles of the structure of plants, their germination, growth and decay; somewhat of natural philosophy, as embracing the me