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metallic wires. In 1834, Professor Wheatstone made use of a revolving mirror like that already described in connexion with the velocity of light, fig. 248. From the delay, in a given time, of the image of a spark produced by the Leyden jar when electricity was passing through a long wire, the professor found that, along a brass wire about one-twelfth of an inch in diameter, electricity is transmitted with a velocity of more than two hundred and eighty-five thousand miles in a second, which is more than half as great again as that of light. Mr. Walker, in America, having made investigations upon the same subject in 1849, by means of signals transmitted along electric telegraph wires, found that the velocity of electricity was only about eighteen thousand miles in a second, which is about one-fifteenth of the above.

In 1850 Messrs. Fizeau and Gounelle, by experiments on the telegraphic wires from Paris to Amiens and to Rouen, arrived at the following results:

1. Along a wire about one-sixth of an inch in diameter electricity is transmitted with a velocity of more than sixtythree thousand miles in a second.

2. Along a wire about one-tenth of an inch in diameter, the velocity was found to be upwards of one hundred and ten thousand miles in a second.

3. Both kinds of electricity, the positive and negative, are transmitted with the same velocity.

4. The number and nature of the elements of which the battery is formed-and consequently the tension of the electricity and the intensity of the current-have no influence upon the velocity of transmission.

5. In conductors of different substances, the velocities are not proportional to their electrical conductibility.

ANIMAL ELECTRICITY.

Electrical fishes are those which possess the remarkable property of communicating to one who touches them a shock like that produced by a Leyden jar. There are several species of such fish: the torpedo, the gymnotus or electrical eel, and the silurus. The torpedo, which is very common in the Mediterranean, has been studied with great care by Messrs. Becquerel and Breschet in France, and M. Matteucci in Italy. The electrical eel has been studied by Messrs. Humboldt and Bompland in South America, and Professor Faraday in this country, who made his observations upon living specimens. The shock communicated by electrical fishes serves them as an offensive and defensive weapon. It is entirely voluntary on their part, and becomes gradually weaker the oftener it is repeated and in proportion as the fish lose their vital power, for the electrical action soon produces great exhaustion in

them.

The shock is often very violent. According to Mr. Faraday, the shock communicated by the electrical eel is equivalent to that of an electric battery of 15 jars, the total surface of the armatures being three square feet, which explains how it is that horses are sometimes overpowered by repeated shocks from electrical eels.

Several experiments prove that the shocks do really result from ordinary electricity. Indeed, if, while touching the back of the fish with one hand, you touch the belly with the other, or with a metallic rod, you will feel a violent shock in the hands and arms, while if you touch it with an isolating body there is no shock at all. Further, when you connect one end of the wire of the galvanometer with the back of the fish and the other with the belly, you will see that at each discharge the needle is deflected and immediately returns to zero, which shows that there is an instantaneous current; the direction of the deflection also indicates that the current goes from the back to the belly of the fish. Lastly, if you pass the current of a torpedo along a helix with a small steel bar at the centre, fig. 452, the bar will be magnetised by the passage of the discharge.

By means of the galvanometer M. Matteucci has established the following facts :

1. When a torpedo is lively and vigorous, it is capable of giving a shock at any part of its body; but in proportion as its vitality is exhausted, the points at which it is capable of giving a shock approach nearer and nearer the organ which serves as the seat of the development of electricity.

2. Any point whatever of the back is always positive in relation to the corresponding point in the belly. 3. Of two points situated at unequal distances from the electric organ, the nearer always performs the part of the positive pole and the more distant that of the negative pole. The reverse of this is the case with the belly.

With regard to the organ in which the electricity originates in the torpedo, it is double, and formed of two parts symmetrically situated on the two sides of the head, and attached to the bones of the skull on their internal surface. These two parts unite in front of the nose, but are separated from the skin. According to M. Matteucci, each of these organs is formed of a large number of prismatic masses placed side oy side, and extending from the external to the internal surface in such a manner that their section perpendicular to the edges of the prisms presents the appearance of honey-comb cells. These prisms are divided perpendicularly to their edges by a series of diaphragms forming a row of small vesicles of the same kind, and filled with nine parts of water for one of albumen, and a little sea-salt.

M. Matteucci, relying upon the following experiment, regards each of these vesicles as the elementary organ of the electrical apparatus. He took from the organs of a living frog, a mass of these vesicles about the size of a pin's head, and put it in contact with the nerves of a dead frog prepared after the manner of Galvani, fig. 417. He then observed that when he excited this vesicular mass by pricking it with any pointed substance, contractions took place in the dead frog.

M. Matteucci also endeavoured to trace out the influence of the brain upon the discharge. For this purpose, he stripped bare the brain of a living torpedo, and observed that the first three lobes might be irritated without any discharge, and that if they were removed, the fish still retained the power of giving a shock. The fourth lobe, on the contrary, cannot be irritated without a discharge being instantly produced; and if it is taken away, all indication of electricity disappears, even though the other lobes remain intact. We are therefore led to believe that the original source of the electricity is the fourth lobe, whence it is transmitted by means of nerves to the two organs above described, which act as multipliers. In other electrical fishes, also, it is the brain which appears to be the original source of electricity.

Considering the large quantity of electricity discharged by some fish, certain philosophers have inquired whether there may not be a similar elaboration of electricity in other fish and animals, not perhaps sufficient to give shocks like those of the Leyden jar, but still enough to produce gentle action and perform the functions essential to life, such as digestion, secretion, etc. There seems much to favour the supposition, but at present it is not sufficiently confirmed by experiment to be any thing more than a probable supposition.

BIOGRAPHY.-No. XXIV.

SARAH MARGARET FULLER.

THIS remarkable woman was the daughter of Timothy Fuller, a member of the Boston bar, but a resident of Cambridge, Massachusetts, where Margaret was born. From 1817 until 1825, Mr. Fuller was sent to Congress, representative of the district of Middlesex. At the close of these political duties, he retired from his profession, and settled in the country as an agriculturist; soon afterwards he died.

Margaret was the oldest child of the family, and at an early age evinced remarkable aptitude for study. It became her father's pride and pleasure to cultivate her intellect to the utmost degree. We are told that his tasks were often oppressive, and thus her juvenile brain was taxed to the disadvantage of her physical healthy development. Most particularly did the father instruct his daughter in the learning he considered of the first importance-the classic tongues. An acquaintance with these subsequently led her to study the modern languages; and Miss Fuller was, from her youth, distinguished for her extraordinary philological acquirements. Of course, the German literature exerted a potent sway over her taste and genius

such influence being now-a-days too common with both adepts | Italy to Rome, they spent the ensuing winter in the "Eternal and dabblers in learning to excite wonder.

Miss Fuller was, however, besides her classical studies, most thoroughly exercised in every solid and elegant department of literature, and probably no American woman was ever before so fully educated, as that term is usually applied. After her father's decease, she devoted her talents and acquirements to the assistance of her mother and sisters, by opening classes for the instruction of ladies, both single and married, first in Boston, then in Providence, Rhode Island, and afterwards in Boston again. During this period, her womanly characteristics self-sacrificing generosity, industry, untiring kindness in the domestic circle-were beautifully displayed.

Her memory is more sanctified by the love of her exemplary qualities called forth in the privacy of home, than by all the literary laurels her admirers wish to offer her.

City," where she continued, while her friends returned to America. In the following year, Miss Fuller was married, in Rome, to Giovanni, Marquis d'Ossoli, an Italian. She remained in Rome till the summer of 1849, when, after the surrender of that city to the French, the Marquis d'Ossoli and his wife, having taken an active part in the republican movement, considered it necessary to emigrate. They went to Florence, and remained there till June, 1850, when they determined to go to the United States, and accordingly embarked at Leghorn, on board the brig "Elizabeth," bound for New York. The deplorable and melancholy catastrophe is well known: the ship, as she neared the American shore, encountered a fearful storm, and on the morning of the 8th of August was wrecked on Fire Island, south of Long Island, and the d'Ossoli family-husband, wife, infant son, and nurseall perished!

In 1839, she made a translation of Goethe's "Conversa- Margaret Fuller, or the Marchioness d'Ossoli, possessed, tions." This was her first work. She was, in the following among a host of professed admirers, many grateful, loving year, concerned with Ralph Waldo Emerson in editing the friends, to whom her sad untimely death was a bitter grief. "Dial," a periodical of some note in its day, to which both These mourn also that she left her mission unfinished, because these writers contributed essays, highly applauded by their they believe a work she had prepared, "On the Revotranscendental readers. To those who require perspicuity as lution in Italy" (the MS. was lost with her), would give her a condition of excellence in literature, such "wanderings round enduring fame. One indication of considerable mental change about a meaning," however fine may be the diction, are never she exhibited-her enthusiasm for Goëthe had abated, and appreciated; yet it is but fair to say, that the meaning of a friend of hers, a distinguished scholar, asserts, that "with Miss Fuller was always honest and generous. She was so the reformers of the Transcendental School she had no comfar from being in adoration before herself, that she seemed munion, nor scarcely a point in common." Whatever she ever aiming to enlarge the moral good of her "brother man might have done, we are constrained to add, that, of the books and sister woman.' "" she has left, we do not believe that they are destined to hold a high place in female literature. There is no true moral life in them. The simple "Prose Hymns for Children" of Mrs. Barbauld, or the "Poems" of Jane Taylor, will have a place in the hearts and homes of the Anglo-Saxon race as long as our language endures, but the genius of Margaret Fuller will live only while the tender remembrance of personal friendship shall hold it dear. Her fame, like that of a great actor or singer, was dependent on her living presence gained more by her conversational powers than by her writings. Those who enjoyed her society, declare that her mind shone most brightly in collision with other minds, and that no adequate idea of her talent can be formed by those who never heard her talk.. This was also true of Coleridge; and Dr. Johnson is certainly a greater man in "Boswell's Life" than in the 'Rambler.” Margaret Fuller had no Boswell.

In 1843, she published a volume, "Summer on the Lakes," being an account of a tour to Illinois. This book contains, with irrelevant matter, some sensible remarks, but there is little in it, as far as regards style or story, beyond what might be found in the letters of any well-educated gentlewoman of moderate abilities, who thought it worth while to journalise on a summer's ramble. About this period, Miss Fuller resided for a time in New York, where she edited the literary department of the "Tribune," contributing papers various subjects, but chiefly critical notices of the works of distinguished authors, for which task both education and genius seemed peculiarly to fit her.

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In 1845, her most important work, "Woman in the Nineteenth Century," was published in New York. It is evident that a strong wish to benefit her own sex moved her heart and guided her pen. One male critic, whose title of reverend should have inspired more charity, has flippantly remarked, that Miss Fuller wrote because she was vexed at not being a man. Not so. Though discontented with her woman's lot, she does not seek to put aside any duty, or lower the standard of virtue in order to escape the real or imagined evils in her position. Nor was it for herself she sought freedom; she wanted a wider field of usefulness for her sex, and unfortunately for her own happiness, which would have been secured by advancing that of others, she mistook the right path of progress. With her views we are far from coinciding: she abandoned the only safe guide in her search for truth. Whatever be the genius or intellectual vigour possessed by a woman, these avail nothing without that moral strength which is nowhere to be obtained save from the aid GOD has given us in his revealed Word. Experience and observation prove, that the greater the intellectual force, the greater and more fatal the errors into which women fall who wander from the Rock of Salvation-Christ the Saviour, who, "made of woman, ," is peculiarly the stay and support of the sex.

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But, though Miss Fuller's theories led to mazes and wanderings, her mind was honest in its search for truth; and with mach that is visionary and impracticable, "Woman in the Nineteenth Century" contains many useful hints and noble sentiments.

In 1844, a selection from her contributions to various periodicals was issued under the title of "Papers on Literature and Art," a work much admired by those who profess to understand the new thoughts, or new modes of expressing old apothegms, which the transcendental philosophy has introduced. It was her last published work. In the summer of 1845, Miss Fuller accompanied some dear friends to Europe; after visiting England, Scotland, France, and passing through

THEORY AND PRACTICE OF TEACHING.
No. III.

LITERARY QUALIFICATIONS OF THE TEACHER. We are now about to enter an extensive field. Since the teacher is to be the life of the school, it is of great consequence that he have within him the means of sustaining life.

The profession of teaching is advancing. The present standard of acquirement demanded of the teacher excludes many who were considered quite respectable in their vocation ten years ago. This may well be so; for within that time, quite an advance has been made in the compensation offered to teachers. It is but reasonable that acquirement should keep pace with the reward of it. Indeed, the talent and attainment brought into the field must always be in advance of the rate of compensation. The people must be first convinced that teachers are better than they were years ago, and then they will be ready to reward them. Their system of supervision has increased in strictness, during this time, in an equal ratio; and many teachers who were entirely incompetent for their places, have thus been driven to other employments. The cause is still onward; and the time is not far distant when the people will demand still more thorough teachers for the common schools, and they will find it for their interest to pay for them.

Under these circumstances, it will not be our design to give the very lowest qualifications for a teacher at present. We shall aim to describe those which a teacher ought to possess, in order to command for some time to come the respect of the

enlightened part of the community. We will not say that a man, with less attainment, than we shall describe, may not keep a good school; we have no doubt that many do. Yet if the profession is to be really respectable, and truly deserving of the regard of an enlightened people, we must have a still higher standard of qualification than we shall now insist on. The following is a list of the studies of which every teacher should have a competent knowledge. We add also to each such word of comment as appears to be necessary.

1. ORTHOGRAPHY.-This implies something more than mere spelling. Spelling is certainly indispensable. No person should ever think of teaching who is not an accurate speller. But the nature and powers of letters should also be mastered. We have in our language about forty elementary sounds; yet we have but twenty-six characters to represent them. Our alphabet is therefore imperfect. This imperfection is augmented by the fact that several of the letters are employed each to represent several different sounds. In other cases, two letters combined represent the element. There are also letters, as c, q, and x, which have no sound that is not fully represented by other letters. Then a very large number of our letters are silent in certain positions, while they are fully sounded in others. It were much to be desired that we might have a perfect alphabet, that is, as many characters as we have elementary sounds, and that each letter should have but one sound. For the present this cannot be; and the present generation of teachers, at least, will have to teach our present orthography. Those systems of orthography are much to be preferred which begin with the elementary sounds, and then present the letters as their representatives, together with the practice of analysing words into their elements, thus showing at once the silent letters and the equivalents. These systems may be taught in half the time that the old systems can be; and when acquired, they are of much greater practical utility to the learner.

2. READING.-Every teacher should be a good reader. Not more than one in every hundred among teachers can now be called a good reader. To be able to read well, implies a quick perception of the meaning as well as a proper enunciation of the words. It is a branch but poorly taught in most of our schools. Many of the older pupils get above reading before they have learned to read well; and, unfortunately, many of our teachers cannot awaken an interest in the subject, because very likely they cannot read any better than their scholars.

It would be interesting to ascertain how large a proportion of our youth leave the schools without acquiring the power readily to take the sense of any common paragraph which they may attempt to read. In this way we account for the fact that so many cease to read as soon as they leave school. It costs them so much effort to decipher the meaning of a book, that it counteracts the desire for the gratification and improvement it might otherwise afford. It should not be so. The teacher should be a model of good reading; he should be enthusiastic in this branch, and never rest till he has excited the proper interest in it among the pupils, from the oldest to the youngest in the school.

It would be well if our teachers could be somewhat acquainted with the Latin and Greek languages, as this would afford them great facilities in comprehending and defining many of our own words. As this cannot be expected for the present, a substitute may be sought in some analysis of our derivative words. Several works have somewhat recently been prepared, to supply, as far as may be, the wants of those who have not studied the classics. We should advise every teacher, for his own benefit, to master some one of these.

3. WRITING. It is not respectable for the teacher of the young to be a bad writer; nor can it ever become so, even should the majority of bad writers continue to increase. The teacher should take great pains to write a plain, legible hand. This is an essential qualification.

line map were not at hand, he ought to be able to draw one from memory,-at least, of each of the grand divisions of the earth. 5. HISTORY.-The teacher should be acquainted with history, at least, the history of his own country. He can hardly teach geography successfully without a competent knowledge of both ancient and modern history. It should, in the main, be taught in our common schools in connection with geography.

6. MENTAL ARITHMETIC.-Let every teacher be thoroughly versed in some good work on this subject. Cassell's is probably the best that has been prepared. It is not enough that the teacher is able in some way to obtain the answers to the questions proposed. He should be able to give the reason for every step in the process he takes to obtain them, and to do it in a clear and concise manner. It is this which constitutes the value of this branch as a discipline for the mind. 7. WRITTEN ARITHMETIC. This everybody demands of the teacher; and he is scarcely in danger of being without fair pretensions in this branch. He should, however, know it by its principles, rather than by its rules and facts. He should so understand it, that if every arithmetic in the world should be burned, he could still make another, constructing its rules and explaining their principles. He should understand arithmetic so well, that he could teach it thoroughly though all text-books should be excluded from his school-room. This is not demanding too much. Arithmetic is a certain science, and used every day of one's life, the teacher should be an entire master of it.

8. ENGLISH GRAMMAR.-It is rare that a teacher is found without some pretensions to English grammer; yet it is deplorable to observe how very few have any liberal or philo sophical aquaintance with it. In many cases it is little else than a system of barren technicalities. The teacher studies one book, and too often takes that as his creed. In no science is it more necessary to be acquainted with several authors. The person who has studied but one text-box on grammer, even if that be the best one extant, is but poorly qualified to teach this branch. There is a philosophy of language which the teacher should carefully study; and if within his power, he should have some acquaintance with the peculiar structure of other languages besides his own. It can hardly be expected that the common teacher should acquire an accurate knowledge of other languages by actually studying them. As a substitute for this, we would recommend that the teacher should very carefully read the little work of Dr. Beard, "Cassell's Lessons in English," also the article "Grammer in the Edinburgh and other encyclopedias. In this science the mind naturally runs to bigotry; and there is no science where the learner is apt to be so conceited upon small aquirements as in grammer. Let the teacher spare no pains to master this subject.

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9. ALGEBRA.-This branch is not yet required to be taught in all our schools; yet the teacher should have a thorough acquaintance with it. Even if he is never called upon to teach it (and it never should be introduced into our common schools till very thorough attainments are more common in the other branches), still it so much improves the mind of the teacher, that he should not be without a knowledge of it. He will teach simple arithmetic much better for knowing algebra. We consider an acquaintance with it indispensable to the thorough teacher, even of the common school.

10. GEOMETRY.-The same may be said of this branch that has been said of algebra. Probably nothing disciplines the mind more effectually than the study of geometry. The teacher should pursue it for this reason. He will teach other things the better for having had this discipline, to say nothing of the advantage which a knowledge of the principles of geometry will give him in understanding and explaining the branches of mathematics.

11. PLANE TRIGONOMETRY AND SURVEYING.-In many of our schools these branches are required to be taught. They are important branches in themselves, and they also afford good exercise for the mind in their acquisition. The young teacher, especially the male teacher, should make the acquirement.

4. GEOGRAPHY.-A knowledge of the principles of geography is essential. This implies an acquaintance with the use of globes, and the art of map-drawing. The teacher should be so well versed in geography, that with an outline map of any country before him, he could give an intelligent account 12. NATURAL PHILOSOPHY.-This branch is not taught in of its surface, people, resources, history, etc.; and if the out-most of our district schools. The teacher, however, should

understand it better than it is presented in many of the simple text-books on this subject. He should have studied the philosophy of its principles, and be fully acquainted with their demonstration. If possible, he should have had an opportunity also of seeing the principles illustrated by experiment. This is a great field; let not the teacher be satisfied with eropping a little of the herbage about its borders.

13. CHEMISTRY.-As a matter of intelligence, the teacher should have acquaintance with this branch. It is comparatively a new science, but it is almost a science of miracles. It is beginning to be taught in our common schools; and that department of it which relates to agriculture is destined to be of vast importance to the agricultural interests of our country Instead of conjecture, and hazard, and doubt, and experiment, as heretofore, a knowledge of the composition of soils, the food of plants, and the processes of nature in the culture and growth of crops, would elevate agriculture to a conspicuous rank among the exact sciences. The teacher should not be behind the age in this department.

14. HUMAN PHYSIOLOGY.-The teacher should well understand this subject. There is an unpardonable ignorance in the community as to the structure of the human body, and the laws of health, the observance of which is, in general, a condition of longevity, not to say of exemption from disease. By reference to statistics, it has been ascertained that almost a fourth part of all the children that are born die before they are one year old. More than one-third die before they are five years of age; and before the age of eight, more than one-half of all that are born return again to the earth! Of those who survive, how many suffer the miseries of lingering disease, almost sighing for death to deliver them from the pangs of life! There is something deplorably wrong in our philosophy of living, else the condition of man would not so commonly apt pear an exception to the truth that God does all things well. Dr. Woodward, late of the Massachusetts State Lunatic Hospital, says: "From the cradle to the grave, we suffer punishment for the violation of the laws of health and life. I have no doubt that half the evils of life, and half the deaths that occur among mankind, arise from ignorance of these natural laws; and that a thorough knowledge of them would diminish the sufferings incident to our present state of being in very nearly the same proportion." We know not how an acquaintance with these laws can be in any way so readily extended as through the agency of our teachers of the young. At any rate the teacher himself should understand them, both for his own profit and the means thus afforded him of being directly usefuljin the discharge of his duties to others. We have already shown that he is responsible to a great extent for the bodily health of his pupils. A thorough knowledge of physiology will enable him to meet this responsibility.

15. INTELLECTUAL PHILOSOPHY.-This is necessary for the teacher. His business is with the mind. He, of all men, should know something of its laws and its nature. He can know something, indeed, by observation and introspection; but he should also learn by careful study. His own improvement demands it, and his usefulness depends upon it.

16. MORAL PHILOSOPHY.-A knowledge of this may be insisted on for the same reasons which apply to intellectual philosophy. It is so important that the moral nature of the child be rightly dealt with, that he is a presumptuous man who attempts the work without the most careful attention to this subject.

17. RHETORIC AND LOGIC.-These are of great service to the teacher personally as means of mental discipline and the cultivation of his own taste. Even if he is never to teach them, they will afford him much assistance in other departments of instruction. He certainly should have the advantage of them.

18. BOOK-KEEPING.-Every teacher should know something of book-keeping, at least by single entry; and also be conversant with the ordinary forms of business. The profound ignorance on this subject among teachers is truly astonishing. Book-keeping should be a common-school study. But this study, which in practical life comes home to the interest not only of every merchant, but of every farmer, every mechanic, in short every business man, is almost entirely neglected in the schools. Some persons still keep their accounts on bits of

paper; others use books, but without any system, order, or intelligibility; and others still mark their scores in chalk or charcoal upon the panel of the cellar-door!

The teacher should qualify himself not only to understand this subject, but to teach it in such a way that it can be easily comprehended by the classes in our common schools.

19. DRAWING. The good teacher should understand the principles of drawing. He should also be able to practise his art. It is of great consequence to him. Without neglect of other things, children can be very profitably taught this art in the common schools. In the absence of apparatus, it is the teacher's only way of addressing the eye of his pupils in illustrating his teaching. Every teacher should take pains not only to draw, but to draw well.

20. VOCAL MUSIC.-It is not absolutely essential, though very desirable, to the good teacher, that he should understand music theoretically and practically. Music is becoming an exercise in our best schools; and wherever introduced and results. It promotes good reading and speaking, by disciplin judiciously conducted, it has been attended with pleasing ing the ear to distinguish sounds; and it also facilitates the cultivation of the finer feelings of our nature. It aids very much in the government of the school, as its exercise gives vent to that restlessness which otherwise would find an escapement in boisterous noise and whispering,-and thus it often proves a safety valve, through which a love of vociferation and activity may pass off in a more harmless and a more pleasing way. "The schoolmaster that cannot sing," says Martin Luther, "I would not look upon." Perhaps this language is too strong; but it is usually more pleasant to look upon a school where the schoolmaster can sing.

We have thus gone through with a list of studies which, it seems to us, every one who means to be a good teacher, even of a common school, should make himself acquainted with. We would not condemn a teacher who, having other good qualities, and a thorough scholarship as far as he has gone, might lack several of the branches above named. There have been many good teachers without all this attainment; but how much better they might have been with it!

We have made this course of study as limited as we possibly could, taking into view the present condition and wants of our schools. No doubt even more will be demanded in a few years. We would have the present race of teachers so good, that they shall be looked upon by those who succeed them as their "worthy and efficient predecessors."

influence, and consequently his usefulness, in proportion as We ought in this place to add, that the teacher increases his he makes himself conversant with general knowledge. This is too much neglected. The teacher, by the fatigue of his employment and the circumstances of his life, is strongly tempted to content himself with what he already knows, or at best to confine himself to the study of those branches which he is called upon to teach. He should stoutly resist this temptation. He should always have some course of study marked out, which he will systematically pursue. He should, as soon as possible, make himself acquainted generally with the subject of astronomy, the principles of geology, in short, the various branches of natural history. He will find one field after another open before him, and if he will but have the perseverance to press forward, even in the laborious occupation of teaching, he may make himself a well-informed man.

We will venture one other suggestion. We have found it a most profitable thing in the promotion of our own improvement, to take up annually, or oftener, some particular subject to be pursued with reference to writing an extended lecture upon it. This gives point to the course of reading, and keeps the interest fixed. When the thorough investigation has been made, let the lecture be written from memory, embodying all the prominent points, and presenting them in the most striking and systematic manner. It should be done, too, with reference to accuracy and even elegance of style, so that the composition may be yearly improved. In this way certain subjects are for ever fixed in the mind. One who carefully reads for a definite object, and afterwards writes the results from memory, never loses his hold upon the facts thus appropriated.

184

LESSONS IN TRIGONOMETRY.-No. I.

PLANE TRIGONOMETRY. TRIGONOMETRY is the science which teaches how to determine the several parts of a triangle from having certain parts given.

Plane Trigonometry treats of plane triangles; Spherical Trigonometry treats of spherical triangles.

The circumference of every circle is supposed to be divided into 360 equal parts, called degrees; each degree into 60 minutes, and each minute into 60 seconds. Degrees, minutes, and seconds are designated by the characters,',". Thus 23° 14′ 35′′ is read 23 degrees, 14 minutes, and 35 seconds.

Since an angle at the centre of a circle is measured by the arc intercepted by its sides, a right angle is measured by 90°, two right angles by 180°, and four right angles are measured by 360°.

The complement of an arc is what remains after subtracting the arc from 90°. Thus the arc DF is the complement of AF. The complement of 25° 15′ is 64° 45'.

In general, if we represent any are by A, its complement is 90°-A. Hence, if an are is greater than 90°, its complement must be negative. Thus, the complement of 100o 15' is -10° 16'. Since the two acute angles of a right-angled triangle are together equal to a right angle, each of them must be the complement of the other.

M

D Cotan. LI
K Cosine

sec

Tangent

Cos. Versi
G

H

E

The supplement of an arc is what remains after subtracting the arc from 180°. Thus the arc BDF is the supplement of the arc AF. The suplement of 25° 15′ is 154° 45'.

In general, if we represent an arc by a, its supplement is 180° Hence, if an arc is greater than 180° its supplement must be negative. Thus the supplement of 200° is -20°. Since in every triangle the sum of the three angles is 180°, either angle is the supplement of the sum of the other two.

The sine of an arc is the perpendicular let fall from one extremity of the arc on the radius passing through the other extremity. Thus FG is the sine of the arc a F, or of the angle ACF.

Every sine is half the chord of double the arc. Thus the sine FG is the half of FH, which is the chord of the arc FAH, double of FA. The chord which subtends the sixth part of the circumference, or the chord of 60°, is equal to the radius; hence the sine of 30° is equal to half of the radius.

The versed sine of an arc is that part of the diameter interThus GA is the versed cepted between the sine and the arc. sine of the are a F.

The tangent of an arc is the line which touches it at one extremity, and is terminated by a line drawn from the centre through the other extremity. Thus AI is the tangent of the arc AF or of the angle A C.E.

The secant of an arc is the line drawn from the centre of the circle through one extremity of the arc, and is limited by the tangent drawn through the other extremity.

Thus CI is the secant of the arc AF, or of the angle A CF. The cosine of an arc is the sine of the complement of that arc. Thus the arc DE, being the complement of a F, F K is the sine of the arc DF, or the cosine of the arc a F.

The cotangent of an arc is the tangent of the complement of that arc. Thus DL is the tangent of the arc DF, or the cotangent of the arc AF.

The cosecant of an arc is the secant of the complement of that arc. Thus CL is the secant of the arc D F, Cr the cosecant of the arc & E.

Since, in a right-angled triangle, either of the acute angles is the complement of the other, the sine, tangent, and secant of one of these angles is the cosine, cotangent, and cosecant of the other.

The sine, tangent, and secant of an arc are equal to the sine, tangent, and secant of its supplement. Thus, G is the sine of the arc AF, or of its supplement, DF. Also, ▲ 1, the tangent of the arc▲ F, is equal to BM, the tangent of the arc BDF. And c1, the secant of the arc AF, is equal to CM, the secant of the arc BDF.

The versed sine of an acute angle, ACE, is equal to the radius minus the cosine o G. The versed sine of an obtuse angle BCF, is equal to radius plus the cosine ca, that is,

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66

46

cosec. A

sec. (90° A).

66 66

90°

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In general, if we represent any angle by ▲,

80°

700

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50%

4C

30%

20

Sine

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