Page images
[blocks in formation]

=1", or (I =

proportion, but at every step has his attention mar school, or, if in the country, never have other hell closely to the fundamental characteristics of school advantages than those furnished by the a proportion. Let the ratio a :b be r; hence as common or rural district school. Nevertheless, a proportion is an equality of ratios, the ratio many of these will receive much greater profit cid is r; and we have (1 = b=r, and c < d from spending half a year, or a year, in obtaining

bio, and < = dr. Substituting these a knowledge of the elements of algebra (and values of a and c in the terms of the proportion even of geometry) than they usually do in studywhich are changed by the transformation, we ing arithmetic. (See ARITHMETIC.) For this have 1 + b = br + b or 6 (+1), and c + d class the proper range of topics is, a clear expo= dr til, or d (r + 1); whence we see that sition of the nature of the literal notation ; 11:0+6::8:0+ dis deduced from a :b::c:d the fundamental rules, and fractions, involvby multiplying both consequents by r + 1 (the ing only the simpler forms of expression, and ratio +1), which does not destroy the equality excluding such abstruse subjects as the more of the ratios constituting the proportion, since it difficult theorems on factoring the theory of divides both by the same number. Moreover, lowest common multiple and highest common this method of substituting for the antecedent of divisor ; simple equations involving one, two, each ratio the consequent multiplied by the rati and three unknown quantities; ratio and proenables us to demonstrate all propositions con- portion; an elementary treatment of the subject cerning the transformation of a proportion by one of radiculs with special attention given to their uniform method, which method in all cases clearly nature as growing out of the simplest principles reveals the reason why the proportion is not of factoring; pure and affected quantiratics indestroyed

volving one, and two unknown quantities. The This choice of a line of argument which shall second class comprises what may be called high be applicable to an entire class of propositions school pupils. For this grade the range of is of no slight importance in constructing a topics need not be much widened, but the mathematical course. It enables a student to study of each should be extended and deepened. learn with greater facility and satisfaction the This will be the case especially as regards the demonstrations, and fixes them more firmly in theory of erponents, positive and negatire his memory; while it also gives broader and quantities, railicals, equations incclring radmore scientific views of truth, by thus classi- icals, and simult meous equations, especially fying, and bringing into one line of thought, those of the second degree. To this should numerous truths which would otherwise be seen be added the arithmeticul and geometrical proonly as so many isolated facts. This is beauti- gressions, a practical knowledge of the binomial fully illustrated in the higher algebra by the use formula, and logarithms, and a somewhat exof the infinitesimal method of developing the tended treatment of the applications of algebra binomial formula, logarithmic series, etc., in con to the business rules of arithmetic. A wide trast with the cumbrous special methods which acquaintance with the results attained in our have so long held their place in our text-books. high schools in all parts of the country, and an By the ol'l method of indeterminate co-efficients, observation extending over more than twenty the pupil is required to pursue what is to him years satisfy the writer that time spent in these always an obscure, long, and unsatisfactory process schools in attempts to master the theory of for the development of each of these series. indeterminate cortficients. the demonstration Nor are these processes so nearly related to each of the binomial and logarithmic formulus, or other, but that, to the mind of the learner, they upon the higher equations, series, etc.. is, if would be even more perplexing than if absolutely not a total loss, at least an absorption of time independent. Moreover, they are styles of arguwhich might be much more profitably employed ment which he never meets with again during on other subjects, such as, for example. history, his subsequent course.

On the other hand, after literature, or the elements of the natural sciences. having learned a few simple rules for differentiat- The course taken by such pupils gives them ing algebraic and logarithmic functions.* he is no occasion to use any of these principles of the enabled to develop these, and several other im- higher algebra : and the mastery of them which portant theorems, in one general way, which is as they can attain in any reasonable amount of time remarkable for its concise simplicity, as it is for is quite too imperfect to subserve the ends of its extensive application and habitual recurrence good mental discipline. This second course is in the subsequent course.

entirely adequate to fit a student for admission Ringe of Topics to be Embraced.We may into any American college or university. The distinguish three different classes of pupils

, who third course is what we may call the college require as many different courses in this study. course. The principal topics which our present First, there is a very large number of our youth arrangements allow us to add to the second course who, if in the city, never pass beyond the gram- as above marked out, in order to constitute this

course, are the theory of indeterminate coettiIt may be new to some that there is a simple of algebraic and logarithmic functions to enable

cients; a sufficient knowledge of the differentiation ing : logarithm without reference to series. This the student to appreciate the idea of function and method was disenvered by Dr. Watson of the University variable, to produce the binomial formula, the of Michigan, and was first presented to the public in OLNEY's University Algebra in 1873.

logarithmic series, and Taylor's formula, which is

[blocks in formation]

necessary in treating Sturm's thcorem, and to ap- this idea clearly before the mind, the teacher will preciate also the demonstration of that theorem; | proceed to the 1st principle. If 3ab be added indeterminate equations; a tolerably full prac- to 7ab how much of the Tab will it destroy? tical treatment of the higher numerical equa- (Here again we proceed from a fundamental contions; and the interpretation of equations; ception—the nature of quantities as positive and alding, if may be, something upon interpolation negative, thus deducing the rew from the old.) and series in general.

Repeat such illustrations of this principle as may Class-Room Work. It is probably unneces have been given in addition If several boys are sary to say, that a careful and thorough study of urging a sled forward by Tab pounds, and the text-books should be the foundation of our class- strength of another boy amounting to 3ab room work on this subject; nevertheless, so much pounds is added, but exerted in an opposite is said, at the present time, in disparagement of direction, what now is the sum of their efforts ? "hearing recitations" instead of teaching," that it What kind of a quantity do we call the 3ab? may be well to remark that, if our schools succeed (Negative.] Why? How much of the + Tab in inspiring their pupils with a love of books, and does - 3ab destroy when ve add it? If then in teaching how to use them, they accomplish in we wish to destroy + 3ab from + Tab, how may this a greater good than even in the mere knowl we do it? Proceeding then to the 2d principle, le lge which they may impart. Books are the it may be asked, how much is 6 ay 2 ay? If great store house of knowledge, and he who has now we add + 2 ay to 6 ay 2 ay, which is 4ay, the habit of using them intelligently has the key what does it become? What does the + 2ay to all human knowledge. But it is not to be destroy? What then is the effect of adding a denied, that there is an important service to be positive quantity ? Such introductory elucidarendered by the living teacher, albeit that service, tions should always be held closely to the plan of especially in this department, is not formal lec- development which the pupil is to study, and turing on the principles of the science. With should be made to throw light upon it. It is a younger pupils, the true teacher will often pref- common and very pernicicus thing for teachers ace a subject with a familiar talk designed to to attempt to teach in one line of development,

, lesson to be assigned, to awaken an interest in it, quite another. In most cases of this kind. either or to enable them to surmount some particular the teacher's effort or the text-book is useless, or difficulty. For example, suppose a class of young probably worse—they tend to confuse each other. pupils are to ha re their first lesson in subtrac- Such teaching should culminate in the very lantion in algebra ; a preliminary talk like the fol- guage of the text; and it is desirable that this lanlowing will be exceedingly helpful, perhaps guage be read from the book by the pupil, as the necessary, to an intelligent preparation of the les conclusion of the teaching. Moreover, there is

Observe that, in order to profit the class, great danger of overdoing this kind of work. the teacher must confine his illustrations rigidly Whenever it is practicable, the pupil should be to the essential points on which the lesson is required to prepare his lesson from the book. based. In this case these are i1) Alding a neg A competent teacher will find sufficient opporative quantity destroys an equal positive tunity for teaching " after the pupils have gathquantity ; (2) Adding a positive quantity de- ered all they can from the book. Another imstroys an equal negatire quantity; (3) As the portant service to be rendered by the living teacher minuend is the sum of the subtrahend and is to emphasize central truths, and hold the pupils remninder, if the subtrahend is destroyed from to a constant review of them. So also it is his duty out the minuend, the remainder is left. Now, in to keep in prominence the outlines of the subject, what order shall these three principles be pre- that the pupil may always know just where he is sented? Doubtless the scientific order is that just at work and in what relation to other parts of the given; but in such an introduction to the subject the subject that which he is studying stands. All as we are considering, it may be best to present definitions, statements of principles and theorems the 3d first; since this is a truth already familiar, should be thoroughly memorized by the pupil and and hence affords a connecting link with previous recited again and again. In entering upon a new knowledge. Moreover, this being already before subject, as soon as these can be intelligently learnthe mind as a statement of what is to be done. ed, they should be recited in a most careful and the 1st and 2d will follow in a natural order as formal manner; and, in connection with suban answer to the question how the purpose is ac- sequent demonstrations and solutions, they should complished. To present the 3d principle, the be called up and repeated. Thus, suppose a high teacher may place on the blackboard some sim- school class entering upon the subject of equaple example in subtraction as :

tions. Such a class may be supposed to be able He will then question the class thus: to grasp the meaning of the definitions without

What is the 125 called? What the 74 ? | preliminary aid from the teacher, save in special What the 51? How much more than 74 is 125 ? cases. The first lesson will probably contain a If we add 74 and 25, what is the sum? Of what dozen or more definitions, with a proposition or then is the minuend composed ? What is 51+74? two; and the first work should be the recitation If we destroy the 74, what remains ? If in any of these by the pupils individually, without any case we can destroy the subtrahend from out the questions or suguestions from the teacher. Il minuend, what will remain ? Having brought lustrations should also be required of the pupils ;


1 2 5

5 1.

[blocks in formation]

but neither illustrations nor demonstrations | the 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 re processes he has performed. That he could solve citation 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 ore 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 7x2_-282+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, 4shall 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. — * =} (3.r— 1) + + (x+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 Tx -28.1=224. I now observe tion. Thus he will ask, What is your first equa- that the first term of the first member contains tion? What is your last? [.r=2.) Do you look the square of .c, 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 , changing the form of an equation? What is which does not destroy the equality, since equals trmsformation ? Similarly, every principle and divided by equals give equal quotients, and I have definition will be reviewed again and again < —4.r=32. Now, observing that c?—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 il, or 4, is the third term, I add 4 to this statements of

processes to pass for expositions of member to make it a complete square, and also add principles, as given by the pupil froin the black- 4 to the second member to preserve the equality board in explanation of his work. The writer's of the members, and have 24x+4=36. Exobservation satisfies him that this most pernicious tracting the square root of .x --4x+4, I have x-2, practice is, as he has said, almost universal. Let an expression which contains only the first power us illustrate the common practice, and then point of x; but to preserve the equality, I also extract out the better way. The pupil has placed the the square root of the second member, obtaining following work upon the board :

X—256. Finally, transposing —2 to the sec7.0 —28.0+14=238

ond member by adding 2 to each member, which 7x'—28.0=224

does not destroy the equation, I have r=8, and X-4.r=32

-4.” If it is desired to abbreviate the explaX2—4x+4=36

nation, it is far better to make it simply an x^2=+ +6

outline of the reasons, than a mere statement x=2+6=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 ic from its connections with other the majority of our best schools :

quantities so that it shall stand alone, constitut“Given 7x?—28.0+14=238, to find the value ing one member while the other member is a

known quantity. The first process is based upon " Transposing, I have 7.r?—28x=224. the principle that equals subtracted from equals “ Dividing by 7, 22—4.0=32.

leave equal remainders; the second, upon the Completing the square, c'—4.0+4=36.

Extracting the square root, 2—2=+6. *) “Destroy the value of the equation," is an absurd * Transpoeing, 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

of x.


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 formu, than that ligion, 233, 733 were Catholics, 6,006 P’rotestants, 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 1860 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 'i lemcen, 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 (znüouas), the schools of In most such cases, it will be quite sufficient, if, law and literature (metresas), 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 irom 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. For example, it requires a far medan dependency.—See Block,Dictionnaire gebetter knowledge of the demonstration of Sturm's nerul 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'Instrucin the variable which does not cause some one tion Primaire en France, tom. III., art. Algérie. 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 ALMA 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. t., the sequence named alphu, betu. 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 X. 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

[blocks in formation]

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 classified 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 use I 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, q 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 cell, 9 in get and gin, s in The letters which closely resemble each other sit and as, f 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 “ love, luck 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 seren different sounds; e, fire carrying out this plan, the teacher may use the sounds; 0, sic 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, s, 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 x are ef, el, em, en, etc.; and the names of j, k, in which to set them, these may be used with are ja, kr. 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 somemethod, 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 crus, 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

16 art
" " all, orb

“ eve, p que

[ocr errors]
[ocr errors]

as in end
" hat

** what, not


[ocr errors]


[ocr errors]

" old

[ocr errors]

ou ou

[ocr errors]
[ocr errors]

€ urn " " use

" ice, my


oi oy


" out, owl

« ՆախորդըՇարունակել »