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but neither illustrations nor demonstrations | full consciousness of duty nobly done. The should be memorized, although great care should fact is, all that he has said is useless, nay, worse be taken to secure a good style of expression, than useless. He has simply intimated what modeled on that of the text. To this first rec- processes he has performed. That he could solve itation on a new subject all the class should give the problem was sufficiently apparent from his the strictest attention; and every point in it work. There was no need that he should tell should be brought out, at least once in the hear. us what he had done, when he had performed ing of every pupil
. In the course of subsequent the work before our eyes. What is wanted is recitations in the same general subject, individ a clear and orderly exposition of the reason why uals will be questioned on the principles thus he takes every step. This involves two points, developed. For example, what algebra is will since he is to show (1) that the step taken tends have been brought clearly to view in this first to the desired end, that is, the freeing of the unrecitation; but when a pupil has stated and known quantity from its connections with known solved some problem, and has given his expla quantities so as finally to make it stand alone as nation of the solution from the blackboard, the one member of the equation ; and (2) that the teacher may ask: Why do you say you have step does not destroy the equation.* Something solved this problem by algebra ? The answer like the following should be the style of explawill be: Because I have used the equation as an nation: "Given 7x*—28.0+14=238, to find the instrument with which to effect the solution. value of .x. In order to do this, I wish so to transCan you solve this problem without the use of form the equation that, in the end, x shall stand an equation? What do you call such a solution ? alone, constituting one member of the equation, What is algebra ? Again, suppose the solution while a known quantity constitutes the other has involved the reduction of such an equation as member. Hence 1 transpose the known quantity 2.0—1=} (3.:-1)+1(0+1). Of course, in the 14 to the second member. This I do by subtractfirst place the pupil will solve the example and ing 14 from each member, which may be done give a good logical account of the solution; but without destroying the equation (or the equality the teacher will make it the occasion for review of the members), since, if the same quantity be ing certain definitions and principles with this subtracted from equals, the remainders are equal. particular student, in such a practical connec
I thus obtain 70*_28x=224. I now observe tion. Thus he will ask: What is your first equa- that the first term of the first member contains tion? Wbat is your last ? [.r=2.) Do you look the square of x, while the second contains the first upon these as one and the same equation, or as power. I wish to obtain an equation which shall different equations? In how many different forms contain only the first power of .c. In order to do have you written your given equation? What this. I make the first term a perfect power by general term do you apply to these processes of dividing each member of the equation by 1, changing the form of an equation? What is which does not destroy the equality, since equals transformation ? Similarly, every principle and divided by equals give equal quotients, and I have definition will be reviewed again and again 2—-4x=32. Now, observing that ac?—4x conin such practical connections. But the great,
and stitutes the first two terms of the square of a almost universal, evil in our methods of conduct- binomial of which the square of half the coeffiing recitations is the habit of allowing mere cient of x, or 4, is the third term, 1 add 4 to this statements of processes to pass for expositions of unember to make it a complete square, and also add principles, as given by the pupil from the black- 4 to the second member to preserve the equality board in explanation of his work. The writers of the members, and have 2?–4x+4=36. Exobservation satisfies him that this most pernicious tracting the square root of 3—4x+4, I have practice is, as he has said, almost universal Let x—2, an expression which contains only the first us illustrate the common practice, and then point power of x; but to preserve the equality, I also out the better way. The pupil has placed the extract the square root of the second member, obfollowing work upon the board:
taining .c-2=+6. Finally, transposing —2 to 7x*—28x+14=238
the second member by adding 2 to each member, 7x*_-28x=224
which does not destroy the equation, I have x=8, x?_4.2=32
and —4.” If it is desired to abbreviate the ex32_4x+4=36
planation, it is far better to make it simply an 1-2=+6
outline of the reasons than a mere statement x=26=8, and -4. of the process. In this case, an outline of the He is then called upon to explain his work. reasons may be given thus: The object is to Something like the following is what we hear in disengage = from its connections with other the majority of our best schools:
quantities so that it shall stand alone, constitut"Given Tx?_28x+14=238, to find the value ing one member while the other member is a of .
known quantity. The first process is based upon "Transposing, I have 7x02—28x=224. the principle that equals subtracted from equals “Dividing by 7, C24.C=32.
leave equal remainders; the second, upon the “Completing the square, x?—4.c+4=36. “Extracting the square root, x_2=+6.
"Destroy the value of the equation", is an absurd “Transposing, x=2+6=8, and _4" expression which we frequently hear. An equation is
not a quantity, and hence has no value. The equality And the pupil turns to his instructor in the i of the members is meant.
principle that equals divided by equals give equal remainder to military, government. The popuquotients,” etc. Again, while it is admissible lation according to the census of 1872 was when the purpose is to fix attention upon any 2,416,225, of whom 245,117 were Europeans particular transformation, to omit the reasons for and their descendants; 34,574 native Jews; the some of those previously studied, it is far better remainder were Mohammedans. In regard to rethat these be omitted pro forma, than that ligion, 233, 733 were Catholics, 6,006 Protestants, something which is not an exposition of reasons 39,812 (including those of European descent) be given. Thus, if the present purpose is to Jews, and 140 had made no declaration. The secure drill in the theory of completing the Catholics have an Archbishop and two Bishsquare, after having enunciated the problem, the ops; the Protestants three Consistories, under pupil may say: “ Having reduced the equation to which both the Lutheran and Reformed Churches the form x?_42=32,” etc., proceeding then to are placed. In regard to public instruction, give in full the explanation of the process under Algeria constitutes a division, called the Academy consideration. But it is well to allow no recita- of Algeria and headed by a rector. The number tion on such a subject to pass without having at of free public schools in 1866 was 426, with least one full explanation. These remarks apply 45,375 pupils; for secondary instruction there to study and recitations designed to give intel- are four colleges and one Lyceum (at Algiers, ligent facility in reducing equations. In what may Bona, Constantine, Philippeville, and Oran), the be called “ Applications of equations to the solu- secondary institution at Tlemcen, and the free tion of practical problems” the purpose is quite school at Oran. A special system of instruction different, and so should be the pupil's explanation. has been arranged for the Mohammedan popuIn these, the statement is the important thing, and lation. It comprises the douar (village or camp) should be made the main thing in the explanation. schools, the law schools (zuüouas), the schools of In most such cases, it will be quite sufficient, if, law and literature (medresas ), the French Arabic after having given the reasons for each step in schools, and the French Arabic colleges. Algiers, the statement, thus fully explaining the principles the capital, has special schools of theology and of on which he has made the equation, the pupil medicine. The educational progress of this counconclude by saying simply: "Solving this equa- try derives a special interest from the fact that tion, I have," etc. Outlines of demonstrations it illustrates the influence which the government and synopses of topics are exceedingly valuable of a Christian country can exercise upon a Mohamas class exercises. l'or example, it requires a far medan dependency.— See BLOCK, Dictionnaire gebetter knowledge of the demonstration of Sturm's neral de la politique. A full account of the French theorem to be able to give the following outline laws regulating public instruction in Algeria may than to give the whole in detail: (1) No change be found in GRÉARD, La Législation de l'Instrur. in the variable which does not cause some one tion Primaire en France, tom. III., art. Algéri, of the functions to vanish, can cause any change ALLEGHENY COLLEGE, at Meadville, in the number of variations and permanences of Pa., was founded in 1817, and is under the the signs of the functions; (2) No two consec- direction of the Methodist Episcopal Church. utive functions can vanish for the same value The number of students in 1874–5 was 132, of the variable ; (3) The vanishing of an inter- more than one half of whom were pursuing the mediate function cannot cause a change in the collegiate course. It has classical, scientific, and number of variations and permanences; and biblical departments, and is open to both sexes. (4) The last function cannot vanish for any its library contains about 12,000 volumes. Rev. value of the variable; and, as the first vanishes L. H. Bugbee, D.D., is the president of the faculty. every time the value of the variable passes ALMĀ MATER (Lat., fostering mother) is. through a root of the equation, it by so doing a name affectionately given by students of colleges causes a loss of one, and only one, variation. We, and universities to the institution to which they therefore, have the theorem (giving the theorem). owe their education. Finally, no subject should be considered as mas- ALPHABET. The alphabet of any language tered by the pupil until he can place upon the is the series of letters, arranged in the customary blackboard a synoptical analysis of it, and discuss order, which form the elements of the language each point, either in detail or in outline, without when written. It derives its name from the first any questioning or prompting by the teacher. The two letters in the Greek alphabet, which are order of arrangement of topics, i. e., the sequence named alpha, beta. The letters in the English of definitions, principles, theorems, etc., is as alphabet have the same forms as those of the much a part of the subject considered scientifically Latin language, which were borrowed from the as are the detailed facts; and the former should Greek. The Latin alphabet, however, did not be as firmly fixed in the mind as the latter. contain all the Greek letters. The letters of the
ALGERIA, a division of N. Africa, which Greek alphabet were borrowed from the Phæniwas formerly a Turkish pashalic, but has since cian, which was that used by many of the old 1830 been in possession of the French. The Semitic nations, and is of unknown origin. It boundaries are not defined, and the tribes dispute consisted of 22 signs, representing consonantal the claims of the French to large tracts on the sounds. Into this alphabet the Greeks introduced border. The territory claimed by the French is many modifications, and the changes made by estimated at about 258,317 sq. m.; of which the Romans were also considerable. Its use about 150,568 are subject to the civil, and the in English presents many variations from its
final condition in the Latin language. Thus, I the name of each, so as to associate arbitrarily and J, and V and V, instead of being merely the form with the name; or, in simultaneous graphic variations, were changed so as to represent class instruction, to exhibit the letters on sepadifferent sounds, during the 16th and 17th cent-rate cards, and teach their names by simple repetiuries. W was added previously, in the middle tion. This process must, of course, be not only ages. The twenty-six letters of our alphabet have long and tedious, but exceedingly dry and uninterbeen thus classitied with regard to their history: esting to a child, since it affords no incentive to (1) B, D, H, K, L, M, N, P, Q, R, S, T, letters mental activity, — no food for intelligence. By from the Phænicians; (2) A, E, I, O, Z, origin- a careful selection and discrimination, however, ally Phænician, but changed by the Greeks; in presenting the letters to the attention of the (3) U (same as V), X, invented by the Greeks ; child, its intelligence may be addressed in teach(4) C, F, Phoenician letters with changed value; ing the alphabet by this method. The simple (5) G, of Latin invention ;: (6) Y, introduced forms, such as I, O, X, S, will be remembered into Latin from the Greek, with changed form; much more readily than the others; and these (7) J, V, graphic Latin forms raised to inde- being learned, the remainder may be taught by pendent letters; (8) W, a recent addition, formed showing the analogy or similarity of their forms by doubling U (or V), whence its name.
with the others. Thus ( becomes C when a The imperfections of the English alphabet are portion of it is erased ; one half of it with I, manifold: (1) Different consonants are used used as a bar, forms D; two smaller D's form B; to represent the same sound; as c (soft) and s, and so on. This method is very simple, and may 9 (soft) and j, c (hard) and k, 9 and k, x and ks. be made quite interesting by means of the black(2) Different sounds are expressed by the same board. letter; as c in cat and cel, g in get and gin, s in The letters which closely resemble cach other sit and as, jf in if and of, etc. (3) The vowels in form, such as A and V, M and N, E and F, are constantly interchanged, as is illustrated in and C and G, among capitals, and b and d, cand the following table of the vowel elements of the e, p and q, and n and u, among small letters, language and their literal representations, the should be presented together, so that their minute diacritical marks used being those of Webster's differences may be discerned. When the blackDictionary.
board is used (as it should always be in teaching Long.
classes), the letters may be constructed before as in ape, they
the pupils, so that they may perceive the elements of which they are composed. Thus the children will at once notice that b, d, p, q, are composed
of the same elements, differently combined, * her, sir, myrrh
straight stroke, or stem, and a small curve. By ** do, rule, too
“ wolf, put, book an appropriate drill, the peculiar forms, with the
name of each, will then be soon impressed upon
the pupils' minds; and, besides that, their sense “ oil, boy
of analogy, one of the most active principles of
a child's mind, will be addressed, and this will From this table it will be seen that the letter render the instruction lively and interesting. In a is used to represent seven different sounds; e, five carrying out this plan, the teacher may use the sounds; o, six sounds, etc. (See PHONETICS.) 'The blackboard, and as a review, or for practice, renames given to the letters are not in conformity quire the children to copy, and afterwards draw, with a uniform principle of designation. Thus, from memory, on the slate, the letters taught. the names of b, c, d, 9, p, t, v, and z are be, ce, Cards may also be used, a separate one being de, ge, etc.; while the names of f, l, m, n, s, and employed for each letter. With a suitable frame & are ef, el, em, en, etc.; and the names of j, k, in which to set them, these may be used with are ja, ka. The heterogeneity of these names good advantage, the teacher making, and the and of their construction will be obvious. It is children also being required to make, various important that the teacher should take cogni- combinations of the letters so as to form short zance of these incongruities in giving elementary and familiar words. A horizontal wooden bar instruction, as they dictate special methods of with a handle, and a groove on the upper edge presentation. (See ALPHABET METHOD.)
in which to insert the cards, forms a very useful ALPHABET METHOD, or "A-B-C piece of apparatus for this purpose.
LetterMethod. This has reference to the first steps Blocks may also be used in a similar manner by in teaching children to read. According to this both teacher and pupils. These blocks are some method, the pupil must learn the names of all the times cut into sections so as to divide the letter letters of the alphabet, either from an A-B-C book, into several parts, and the pupil is required to from crds, or from the blackboard; that is, he adjust the parts so as to form the letter. This must be taught to recognize the various forms of method affords both instruction and amusement the letters, and to associate with them their re- to young children, and at the same time, gives spective names. The method of doing this, once play to their natural impulse to activity. These very general, was to supply the pupils with books, various methods will be combined and others and then, calling up each one singly, to point to devised by every ingenious teacher. In some the letters, one after the other, and to pronounce schools a piece of apparatus, called the reading
** care, ere
oy OU OW "
" out, owl
frame, is used. This is constructed like a black- ago from funds contributed for the purpose by board with horizontal grooves, in which the let- W. F. Stearns, son of the president. This in. ters can be placed so as to slide along to any stitution occupies twelve public buildings, besides required position. By the use of assorted letters, the president's house, including an edifice for scithe teacher can construct any word or sentence, entific instruction, and the college church. There building it up letter by letter, as types are set. are also a gallery of art, a cabinet of natural Many interesting exercises in reading and spelling history, containing about 100,000 specimens, and may be given by means of such an apparatus, the an astronomical observatory. The department children being required to construct words and for physical training is very efficient. It comsentences themselves, as well as to read those prises an extensive and well appointed gymnaformed by the teacher. The A B C Method of sium ; and, at a certain hour, each class is reteaching the elements of reading has now, quite quired to attend, and engage in exercise under the generally, been superseded by the Word Method. direction of the professor, who is a thoroughly
See CURRIE, Early and Infant School Edu- qualified physician. The faculty includes twentycation, and Principles (md Practice of Common three instructors, and there are several endowed School Education; WICKERSHAM, Methods of professorships. The number of students in 1874 Instruction. (See WORD METHOD.)
was about 340. The college library contains ALUMNEUM, or Alumnat (Lat., from more than 30,000 volumes; and those of the alere, to feed, to nourish), the name given in societies, about 10,000. There is a scientific as Germany to an institution of learning which af- . well as a classical course; also a post graduate fords to its pupils board, lodging, and instruc- course, established in 1874, in history and polittion. The first institutions of this kind arose in ical science, with especial reference to a “science the middle ages in connection with the convents. ; of statesmanship;" while any graduate may Among the most celebrated are those founded by arrange to pursue a course of study in any deMaurice of Saxony, in the 16th century, at Pforta, partment additional to the college course. The Meissen, and Grimma. When the pupils were tuition fee is $90 per annum. received and instructed gratuitously, they were ANALYSIS, Grammatical, or Sentenexpected to perform various services for the tial.-By the analysis of a sentence is meant a school and church, such as singing in the choir. decomposition of it into its logical elements. The pupils of these schools were called alumni. Every sentence must either be a single proposi(See ALUMNES.)
tion, or be composed of propositions more or ALUMNUS, pl. Alumni (Lat., from alere, less intimately related ; and every proposition to feed, to nourish) originally the name of a must contain a subject and a predicate, the forstudent who was supported and educated at the mer expressing that of which we speak, and the expense of a learned institution (see ALUMNEUM), latter, what we say of it. The entire or logical now generally applied to a graduate of a college subject must contain a noun or pronoun, either or similar institution. The graduates of higher alone or with related words called modifiers or seminaries or colleges for females are sometimes adjuncts, or it may be a phrase or a clause. The called alumnæ.
entire or logical predicate, in the same manner, AMHERST COLLEGE, at Amherst, Mass., must consist of a verb with or without adjuncts. is one of the chief seats of learning in the These constitute all the parts, and all the relations, United States. It was founded in 1821 by the involved in the construction of a sentence. A few Orthodox Congregationalists, especially for the words, such as interjections, may be used indeeducation of young men for the ministry; but pendently of them. "Grammar has been defined its charter was not obtained till 1825. Its first as the “art of speaking and writing correctly," president was the Rev. Zephaniah S. Moore, who or as the “practical science which teaches the in 1823 was succeeded by the Rev. IIeman right use of language"; and for general purHumphrey, to whose strenuous and prudent poses this account is, perhaps, sufficiently exefforts the college owed much of its success. He plicit. It does not, however, truly distinguish continued in office till 1845, when he was suc- grammar from the other arts concerned in teachceeded by the Rev. Edward Hitchcock ; and, ing the “ right use of language," and hence does on the resignation of the latter, in 1854, the not correctly point out its peculiar province. present incumbent, the Rev. William A. Stearns, From a want of precision in defining the limitaD.D., was elected. This institution has been the tions of any art or science, there must necessarily recipient of very large donations from private follow a corresponding inaccuracy and looseness persons, and appropriations from the State in its treatment; since, before we can reason amounting to upward of $50,000. The college properly as to the best methods of attaining any funds amount in the aggregate to more than object, we must clearly conceive what that object $650,000. Its charity fund for the gratuitous is, and carefully distinguish it from all others. education of clergymen amounts to about $70,000; The special province of grammar does not exand its fund for free scholarships is at least tend beyond the construction of sentences ; but $100,000. The names of the principal donors it is quite obvious that to use language correctly, to the institution are Dr. William J. Walker, those principles and rules must be understood to the extent of $240,000 ; Samuel A. Hitch- which underlie the proper method of combining cock, $175,000; Samuel Williston, $150,000 ; sentences so that they may constitute elegant and and a college church was erected a short time logical discourse. A person may be sufficiently familiar with grammatical rules to construct sen- I laid for the intelligent study of all other gramtences with perfect correctness, but may so ar- matical terms and distinctions ; and this being range them as to express only nonsense ; and the foundation, should, of course, be the first such a person could scarcely be considered as un- thing done. Those who oppose the analytical derstanding the “ right use of language." The method assert that words are the real elements of sentence being the peculiar province of grammar, a sentence, and that any consideration of these it follows that the only subjects of investigation involves, therefore, an exhaustive analysis of the embraced within it are words, their orthography, sentence itself. With the same propriety might inflectional forms, and pronunciation, and their it be said that pieces of iron of various shapes arrangement in sentences. All grammatical de are the elements of the steam-engine. They infinitions and rules are founded upon the relations deed compose the machine, and it can ultimately of the parts of a sentence to each other; and, be resolved into them ; but could its structure therefore, these relations should be first taught. and workings be explained by taking these fragIt is with reference to these relations, that words ments of metal in a hap-hazard way, and noticing are classified into parts of speech, or, as they how they are related to others in immediate juxmight properly be called, parts of the sentence. taposition, without regard to the general struct
To define or explain these parts of speech before ure of the machine, and the dependence of its giving any definition of a sentence, is, therefore, operation upon a few elementary or primary parts, clearly illogical ; yet this has been the method of as the cylinder, piston, condenser, etc.? Words many grammarians, words being explained and are not necessarily the real elements of a senparsed as if they had only individual properties. tence. These are the subject and predicate and It is in this that the distinction between parsing their adjuncts; and, unless these component parts and grammatical analysis consists. Both are, in of the general structure be first observed, the fact, only different kinds of analysis, and are relations of the separate words cannot be underbased on precisely the same relations,—those in stood. Hence, we find that those writers who which the words stand to each other as parts of have ignored a definite consideration of these a sentence.
logical elements, have fallen into many errors Parsing, as uniformly employed by gram- and inconsistencies. marians, is a minute examination of the in- The various systems of analysis in use differ dividual words of a sentence, with the view to in no essential respect, the chief variation being determine whether the rules of grammar, proper in the nomenclature employed to designate the to the particular language in which the sentence elements of the sentence. The name generally is written, have been observed or violated. Anal- applied to a proposition forming a part of a senysis, on the other hand, deals with the relations tence is a clause, and any group of related words upon which those rules are based, and which not making a proposition is called a phrase. The are common to all languages. Thus, in parsing, modifying elements are by some called adjective the pupil is obliged to scrutinize all the inflec- or adverbial, according as they perform the functional forms in which the words composing the tions of adjectives or adverbs. Instead of the sentence are used ; and, in order to determine term adjective, adnominal is sometimes employed. whether they are proper or not, must not only The term adjunct is generally employed to desknow the rules of syntax, but the relations of the ignate an element subordinate to either subject words to each other, so as to be able to apply or predicate. Such adjuncts may be modifying, those rules. The relations are invariable in all descriptive, or appositional. A modifying adlanguages, but the rules which refer to the in- junct changes the meaning of the element to flections are founded on particular usage, and which it is applied, generally, by making it more hence are in no two languages exactly alike. On specific, or by restricting the class to which it bethis account, since the general logically precedes longs. Thus animal is a more general term than the special, the treatment of sentential analysis four-footed animal; hence.four-footed is a modishould precede any exercises in parsing. Other fying adjunct. But the term man is no more wise, how, for example, could a pupil be required general than man that is born of a woman, or to distinguish the cases of nouns and pronouns, mortal man ; the acdjuncts, that is born of a womand the person and number of verbs, before be- an and mortal being only descriptive, not modiing taught the relations of the words to each other? fying. Appositional adjuncts only explain;
By means of the analytical method, when rightly as : He, the chieftain of them all, in which the applied, the study of grammar is made clear, phrase, the chieftain, etc., is only explanatory, or logical, and easy from the very beginning. The appositional. Adjuncts may be single words, pupil is first taught the nature of the sentence, phrases or clauses; and one of the chief adits essential parts, and their relations to each vantages of sentential analysis is to show the other, and is shown how to analyze sentences of pupil that groups of words are often used so as a simple character. He is troubled with but to perform the same office as single words. In little phraseology ; for all the terms that are es- teaching this subject, a proper gradation of topics sential to the complete distinction and designa- should be observed; and much caution exercised tion of the parts of a sentence are subject, verb to avoid the perplexing of the young pupil by or predicate, object, attribute, and adjuncts. These presenting to his mind distinctions too nice to be being defined, and the pupil taught how to dis- discerned by his undeveloped powers of analysis. tinguish them, a complete foundation has been Various methods have been devised in order to