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ALGEBRA

proportion, but at every step has his attention held closely to the fundamental characteristics of a proportion. Let the ratio a: b ber; hence as a proportion is an equality of ratios, the ratio ed is r; and we have a br, and cd =r, or a = br, and c = dr. Substituting these values of a and c in the terms of the proportion which are changed by the transformation, we have a + b = br+ b, or b (r + 1), and e+d = dr + d, or d (r + 1); whence we see that a: a+b::c:c+d is deduced from a:b::c:d by multiplying both consequents by + 1 (the ratio +1), which does not destroy the equality of the ratios constituting the proportion, since it divides both by the same number. Moreover, this method of substituting for the antecedent of each ratio the consequent multiplied by the ratio, enables us to demonstrate all propositions concerning the transformation of a proportion by one uniform method, which method in all cases clearly reveals the reason why the proportion is not destroyed.

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mar school, or, if in the country, never have other school advantages than those furnished by the common or rural district school. Nevertheless, many of these will receive much greater profit from spending half a year, or a year, in obtaining a knowledge of the elements of algebra (and even of geometry) than they usually do in studying arithmetic. (See ARITHMETIC.) For this class the proper range of topics is, a clear exposition of the nature of the literal notation; the fundamental rules, and fractions, involving only the simpler forms of expression, and excluding such abstruse subjects as the more difficult theorems on factoring, the theory of lowest common multiple and highest common divisor; simple equations involving one, two, and three unknown quantities; ratio and proportion; an elementary treatment of the subject of radicals with special attention given to their nature as growing out of the simplest principles of factoring; pure and affected quadratics involving 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 exponents, positive and negative his memory; while it also gives broader and quantities, radicals, equations involving rad more 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 arithmetical 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 of the infinitesimal method of developing the binomial formula, logarithmic series, etc.. in contrast with the cumbrous special methods which have so long held their place in our text-books. By the old method of indeterminate co-efficients, the pupil is required to pursue what is to him always an obscure, long, and unsatisfactory process for the development of each of these series. Nor are these processes so nearly related to each other, but that, to the mind of the learner, they would be even more perplexing than if absolutely independent. Moreover, they are styles of argument which he never meets with again during his subsequent course. On the other hand, after having learned a few simple rules for differentiating algebraic and logarithmic functions.* he is enabled to develop these, and several other important theorems, in one general way, which is as remarkable for its concise simplicity, as it is for its extensive application and habitual recurrence in the subsequent course.

Range of Topics to be Embraced. We may distinguish three different classes of pupils, who require as many different courses in this study. First, there is a very large number of our youth who, if in the city, never pass beyond the gram

*It may be new to some that there is a simple elementary, method of proving the rule for differentiating a logarithm without reference to series. This method was discovered by Dr. Watson of the University

of Michigan, and was first presented to the public in OLNEY'S University Algebra in 1873.

formula, and logarithms, and a somewhat extended treatment of the applications of algebra to the business rules of arithmetic. A wide acquaintance with the results attained in our high schools in all parts of the country, and an observation extending over more than twenty years satisfy the writer that time spent in these schools in attempts to master the theory of indeterminate co-efficients, the demonstration of the binomial and logarithmic formulas, or upon the higher equations, series, etc.. is, if not a total loss, at least an absorption of time which might be much more profitably employed on other subjects, such as, for example. history, literature, or the elements of the natural sciences. The course taken by such pupils gives them no occasion to use any of these principles of the higher algebra: and the mastery of them which they can attain in any reasonable amount of time is quite too imperfect to subserve the ends of good mental discipline. This second course is entirely adequate to fit a student for admission into any American college or university. The third course is what we may call the college course. The principal topics which our present arrangements allow us to add to the second course as above marked out, in order to constitute this course, are the theory of indeterminate co-efficients; a sufficient knowledge of the differentiation of algebraic and logarithmic functions to enable the student to appreciate the idea of function and variable, to produce the binomial formula, the logarithmic series, and Taylor's formula, which is

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necessary in treating Sturm's theorem, and to appreciate also the demonstration of that theorem; indeterminate equations; a tolerably full practical treatment of the higher numerical equations; and the interpretation of equations; adding, if may be, something upon interpolation and series in general.

Class-Room Work. It is probably unnecessary to say, that a careful and thorough study of text-books should be the foundation of our classroom work on this subject; nevertheless, so much is said, at the present time, in disparagement of "hearing recitations" instead of "teaching," that it may be well to remark that, if our schools succeed in inspiring their pupils with a love of books, and in teaching how to use them, they accomplish in this a greater good than even in the mere knowlle lge which they may impart. Books are the great store house of knowledge, and he who has the habit of using them intelligently has the key to all human knowledge. But it is not to be denied, that there is an important service to be rendered by the living teacher, albeit that service, especially in this department, is not formal lecturing on the principles of the science. With younger pupils, the true teacher will often preface a subject with a familiar talk designed to prepare them for an intelligent study of the lesson to be assigned, to awaken an interest in it, or to enable them to surmount some particular difficulty. For example, suppose a class of young pupils are to have their first lesson in subtraction in algebra; a preliminary talk like the following will be exceedingly helpful, perhaps necessary, to an intelligent preparation of the lesson. Observe that, in order to profit the class, the teacher must confine his illustrations rigidly to the essential points on which the lesson is based. In this case these are (1) Adding a negative quantity destroys an equal positive quantity; (2) Adding a positive quantity destroys an equal negative quantity; (3) As the minuend is the sum of the subtrahend and remainder, if the subtrahend is destroyed from out the minuend, the remainder is left. Now, in what order shall these three principles be presented? Doubtless the scientific order is that just given; but in such an introduction to the subject as we are considering, it may be best to present the 3d first; since this is a truth already familiar, and hence affords a connecting link with previous knowledge. Moreover, this being already before the mind as a statement of what is to be done, the 1st and 2d will follow in a natural order as an answer to the question how the purpose is accomplished. To present the 3d principle, the teacher may place on the blackboard some simple example in subtraction as:

125

74

He will then question the class thus: 51. What is the 125 called? What the 74? What the 51? How much more than 74 is 125? If we add 74 and 25, what is the sum? Of what then is the minuend composed? What is 51+74? If we destroy the 74, what remains? If in any case we can destroy the subtrahend from out the minuend, what will remain? Having brought

this idea clearly before the mind, the teacher will proceed to the 1st principle. If -3ab be added to 7ab how much of the Tab will it destroy? (Here again we proceed from a fundamental conception--the nature of quantities as positive and negative, thus deducing the new from the old.) Repeat such illustrations of this principle as may have been given in addition If several boys are urging a sled forward by 7ab pounds, and the strength of another boy amounting to 3h pounds is added, but exerted in an opposite direction, what now is the sum of their efforts? What kind of a quantity do we call the 3ab? [Negative.] Why? How much of the +7ab does -3ab destroy when we add it? If then we wish to destroy +3ab from +7ab, how may we do it? Proceeding then to the 2d principle, it may be asked, how much is 6 ay — 2 ay? ˆ If now we add +2 ay to 6 ay -2ay, which is 4ay, what does it become? What does the + 2ay destroy? What then is the effect of adding a positive quantity? Such introductory elucidations should always be held closely to the plan of development which the pupil is to study, and should be made to throw light upon it. It is a common and very pernicicus thing for teachers to attempt to teach in one line of development, while the text-book in the pupil's hands gives quite another. In most cases of this kind. either the teacher's effort or the text-book is useless, or probably worse-they tend to confuse each other. Such teaching should culminate in the very language of the text; and it is desirable that this language be read from the book by the pupil, as the conclusion of the teaching. Moreover, there is great danger of overdoing this kind of work. Whenever it is practicable, the pupil should be required to prepare his lesson from the book. A competent teacher will find sufficient opportunity for "teaching" after the pupils have gathered all they can from the book. Another important service to be rendered by the living teacher is to emphasize central truths, and hold the pupils to a constant review of them. So also it is his duty to keep in prominence the outlines of the subject, that the pupil may always know just where he is at work and in what relation to other parts of the the subject that which he is studying stands. All definitions, statements of principles. and theorems should be thoroughly memorized by the pupil and recited again and again. In entering upon a new subject, as soon as these can be intelligently learned, they should be recited in a most careful and formal manner; and, in connection with subsequent demonstrations and solutions, they should be called up and repeated. Thus, suppose a high school class entering upon the subject of equa tions. Such a class may be supposed to be able to grasp the meaning of the definitions without preliminary aid from the teacher, save in special cases. The first lesson will probably contain a dozen or more definitions, with a proposition or two; and the first work should be the recitation of these by the pupils individually, without any questions or suggestions from the teacher. II. lustrations should also be required of the pupils;

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ALGEBRA

but neither illustrations nor demonstrations should be memorized, although great care should be taken to secure a good style of expression, modeled on that of the text. To this first recitation on a new subject all the class should give the strictest attention; and every point in it should be brought out, at least once in the hearing of every pupil. In the course of subsequent recitations in the same general subject, individuals will be questioned on the principles thus developed. For example, what algebra is will have been brought clearly to view in this first recitation; but when a pupil has stated and solved some problem, and has given his explanation of the solution from the blackboard, the teacher may ask, Why do you say you have solved this problem by algebra? The answer will be, Because I have used the equation as an instrument with which to effect the solution. Can you solve this problem without the use of an equation? What do you call such a solution? What is algebra? Again, suppose the solution has involved the reduction of such an equation as 2x-4=1 (3.x — 1) + § (.x +1). Of course. in the first place the pupil will solve the example and give a good logical account of the solution; but the teacher will make it the occasion for reviewing certain definitions and principles with this particular student, in such a practical connection. Thus he will ask, What is your first equation? What is your last? [=2.] Do you look upon these as one and the same equation, or as different equations? In how many different forms have you written your given equation? What general term do you apply to these processes of changing the form of an equation? What is transformation? Similarly, every principle and definition will be reviewed again and again in such practical connections. But the great, and almost universal, evil in our methods of conducting recitations is the habit of allowing mere statements of processes to pass for expositions of principles, as given by the pupil from the blackboard in explanation of his work. The writer's observation satisfies him that this most pernicious practice is, as he has said, almost universal. Let us illustrate the common practice, and then point out the better way. The pupil has placed the following work upon the board:

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the full consciousness of duty nobly done. The fact is, all that he has said is useless, nay, worse than useless. He has simply intimated what processes he has performed. That he could solve the problem was sufficiently apparent from his work. There was no need that he should tell us what he had done, when he had performed the work before our eyes. What is wanted is a clear and orderly exposition of the reason why he takes every step. This involves two points, since he is to show (1) that the step taken tends to the desired end, that is, the freeing of the unknown quantity from its connections with known quantities so as finally to make it stand alone as ore member of the equation; and (2) that the step does not destroy the equation.* Something like the following should be the style of explanation: "Given 7a2-28x+14=238, to find the value of x. In order to do this, I wish so to transform the equation that, in the end, shall stand alone, constituting one member of the equation, while a known quantity constitutes the other member. Hence I transpose the known quantity 14 to the second member. This I do by subtracting 14 from each member, which may be done without destroying the equation (or the equality of the members), since, if the same quantity be subtracted from equals, the remainders are equal. I thus obtain 7.x-28-224. I now observe that the first term of the first member contains the square of .r, while the second contains the first power. I wish to obtain an equation which shall contain only the first power of .c. In order to do this, I make the first term a perfect power by dividing each member of the equation by 7, which does not destroy the equality, since equals divided by equals give equal quotients. and I have x-4.r-32. Now, observing that 2-4 constitutes the first two terms of the square of a binomial of which the square of half the co-efficient of x, or 4, is the third term, I add 4 to this member to make it a complete square, and also add 4 to the second member to preserve the equality of the members, and have 4.r+4=36. Extracting the square root of 4+4, I have x-2, an expression which contains only the first power of ; but to preserve the equality, I also extract the square root of the second member, obtaining x-2=±6. Finally, transposing -2 to the second member by adding 2 to each member, which does not destroy the equation, I have .r=8, and -4." If it is desired to abbreviate the explanation, it is far better to make it simply an outline of the reasons, than a mere statement of the process. In this case, an outline of the reasons may be given thus: The object is to disengage from its connections with other quantities so that it shall stand alone, constituting one member while the other member is a known quantity. The first process is based upon the principle that equals subtracted from equals leave equal remainders; the second, upon the

"Destroy the value of the equation," is an absurd expression which we frequently hear. An equation is not a quantity, and hence has no value. The equality of the members is meant.

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principle that equals divided by equals give equal quotients," etc. Again, while it is admissible when the purpose is to fix attention upon any particular transformation, to omit the reasons for some of those previously studied, it is far better that these be omitted pro forma, than that something which is not an exposition of reasons be given. Thus, if the present purpose is to secure drill in the theory of completing the square, after having enunciated the problem, the pupil may say: "Having reduced the equation to the form -4x=32," etc., proceeding then to give in full the explanation of the process under consideration. But it is well to allow no recitation on such a subject to pass without having at least one full explanation. These remarks apply to study and recitations designed to give intelligent facility in reducing equations. In what may be called "Applications of equations to the solution of practical problems" the purpose is quite different, and so should be the pupil's explanation. In these, the statement is the important thing, and should be made the main thing in the explanation. In most such cases, it will be quite sufficient, if, after having given the reasons for each step in the statement, thus fully explaining the principles on which he has made the equation, the pupil conclude by saying simply: "Solving this equation, I have," etc. Outlines of demonstrations and synopses of topics are exceedingly valuable as class exercises. For example, it requires a far better knowledge of the demonstration of Sturm's theorem to be able to give the following outline than to give the whole in detail: (1) No change in the variable which does not cause some one of the functions to vanish, can cause any change in the number of variations and permanences of the signs of the functions; (2) No two consecutive functions can vanish for the same value of the variable; (3) The vanishing of an intermediate function cannot cause a change in the number of variations and permanences; and (4) The last function cannot vanish for any value of the variable; and, as the first vanishes every time the value of the variable passes through a root of the equation. it by so doing causes a loss of one, and only one, variation. We, therefore, have the theorem giving the theorem]. Finally, no subject should be considered as mastered by the pupil until he can place upon the blackboard a synoptical analysis of it, and discuss each point, either in detail or in outline, without any questioning or prompting by the teacher. The order of arrangement of topics, i. e., the sequence of definitions, principles, theorems. etc., is as much a part of the subject considered scientifically as are the detailed facts; and the former should be as firmly fixed in the mind as the latter.

ALGERIA, a division of N. Africa, which was formerly a Turkish pashalic, but has since 1830 been in possession of the French. The boundaries are not defined, and the tribes dispute the claims of the French to large tracts on the border. The territory claimed by the French is estimated at about 258,317 sq. m.; of which about 150,568 are subject to the civil, and the

ALPHABET

remainder to military, government. The population according to the census of 1872 was 2,416,225, of whom 245,117 were Europeans and their descendants; 34,574 native Jews; the remainder were Mohammedans. In regard to religion, 233, 733 were Catholics, 6,006 Protestants, 39,812 (including those of European descent) Jews, and 140 had made no declaration. The Catholics have an Archbishop and two Bishops; the Protestants three Consistories, under which both the Lutheran and Reformed Churches are placed. In regard to public instruction, Algeria constitutes a division, called the Academy of Algeria and headed by a rector. The number of free public schools in 1866 was 426, with 45,375 pupils; for secondary instruction there are four colleges and one Lyceum (at Algiers, Bona, Constantine, Philippeville, and Oran), the secondary institution at Tlemcen, and the free school at Oran. A special system of instruction has been arranged for the Mohammedan population. It comprises the douar (village or camp) schools, the law schools (zaïouas), the schools of law and literature (medresas), the French Arabic schools, and the French Arabic colleges. Algiers, the capital, has special schools of theology and of medicine. The educational progress of this country derives a special interest from the fact that it illustrates the influence which the government of a Christian country can exercise upon a Mohammedan dependency. See BLOCK, Dictionnaire général de la politique. A full account of the French laws regulating public instruction in Algeria may be found in GREARD, La Législation de l'Instruction Primaire en France, tom. III., art. Algérie.

ALLEGHENY COLLEGE, at Meadville, Pa., was founded in 1817, and is under the direction of the Methodist Episcopal Church. The number of students in 1874-5 was 132, more than one half of whom were pursuing the collegiate course. It has classical, scientific, and biblical departments, and is open to both sexes. Its library contains about 12.000 volumes. Rev. L. H. Bugbee, D.D., is the president of the faculty.

ALMA MATER (Lat., fostering mother) is a name affectionately given by students of colleges and universities to the institution to which they owe their education.

ALPHABET. The alphabet of any language is the series of letters, arranged in the customary order, which form the elements of the language when written. It derives its name from the first two letters in the Greek alphabet, which are named alpha, beta. The letters in the English alphabet have the same forms as those of the Latin language, which were borrowed from the Greek. The Latin alphabet, however, did not contain all the Greek letters. The letters of the Greek alphabet were borrowed from the Phoenician, which was that used by many of the old Semitic nations, and is of unknown origin. It consisted of 22 signs, representing consonantal sounds. Into this alphabet the Greeks introduced many modifications, and the changes made by the Romans were also considerable. Its use in English presents many variations from its

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final condition in the Latin language. Thus, I the name of each, so as to associate arbitrarily and J, and U 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 ages. The twenty-six letters of our alphabet have been thus classified with regard to their history: (1) B, D, H, K, L, M, N, P, Q, R, S, T, letters from the Phoenicians; (2) A, E, I, O, Z, originally Phoenician, but changed by the Greeks; (3) U (same as V), X, invented by the Greeks; (4) C, F, Phoenician letters with changed value; (5) G, of Latin invention; (6) Y, introduced into Latin from the Greek, with changed form (7) J, V, graphic Latin forms raised to independent letters; (8) W, a recent addition, formed by doubling U (or V), whence its name.

The imperfections of the English alphabet are manifold: (1) Different consonants are usel to represent the same sound; as c (soft) and s, g (soft) and j, c (hard) and k, q and k, x and ks. (2) Different sounds are expressed by the same letter; as c in cat and cell, g in get and gin, s in sit and as, ƒ in if and of, etc. (3) The vowels are constantly interchanged, as is illustrated in the following table of the vowel elements of the language and their literal representations, the diacritical marks used being those of Webster's Dictionary.

Long.
as in ape, they

"art

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Short.

as in end

46

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"care, ere

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"hat

66 ask

44 what, not
66 sit

tion. This process must, of course, be not only long and tedious, but exceedingly dry and uninteresting to a child, since it affords no incentive to mental activity, - no food for intelligence. By a careful selection and discrimination, however, in presenting the letters to the attention of the child, its intelligence may be addressed in teaching the alphabet by this method. The simple forms, such as I, O, X, S, will be remembered much more readily than the others; and these being learned, the remainder may be taught by showing the analogy or similarity of their forms with the others. Thus O becomes C when a portion of it is erased; one half of it with I, used as a bar, forms D; two smaller D's form B; and so on. This method is very simple, and may be made quite interesting by means of the blackboard.

a

By

The letters which closely resemble each other in form, such as A and V, M and N, E and F, and C and G, among capitals, and b and d, c and e, p and q, and n and u, among small letters, should be presented together, so that their minute differences may be discerned. When the blackboard is used (as it should always be in teaching classes), the letters may be constructed before 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, straight stroke, or stem, and a small curve. Qu oo" "wolf, put, book an appropriate drill, the peculiar forms, with the 46 "love, luck name of each, will then be soon impressed upon the pupils' minds; and, besides that, their sense of analogy, one of the most active principles of a child's mind, will be addressed, and this will render the instruction lively and interesting. In carrying out this plan, the teacher may use the blackboard. and as a review, or for practice, require the children to copy, and afterwards draw, from memory, on the slate, the letters taught. Cards may also be used, a separate one being employed for each letter. With a suitable frame in which to set them, these may be used with good advantage, the teacher making, and the children also being required to make, various combinations of the letters so as to form short and familiar words. A horizontal wooden bar with a handle, and a groove on the upper edge in which to insert the cards, forms a very useful piece of apparatus for this purpose. LetterBlocks may also be used in a similar manner by both teacher and pupils. These blocks are sometimes cut into sections so as to divide the letter into several parts, and the pupil is required to adjust the parts so as to form the letter. This method affords both instruction and amusement to young children, and at the same time, gives play to their natural impulse to activity. These various methods will be combined and others devised by every ingenious teacher. In some schools a piece of apparatus, called the reading

From this table it will be seen that the letter a is used to represent seven different sounds; e, five sounds; o, six sounds, etc. (See PHONETICS.) The names given to the letters are not in conformity with a uniform principle of designation. Thus. the names of b, c, d, g, p, t, v, and z are be, ce, de, ge, etc.; while the names of f, l, m, n, s, and x are ef, el, em, en, etc.; and the names of j, k, are ja, ka. The heterogeneity of these names and of their construction will be obvious. It is important that the teacher should take cognizance of these incongruities in giving elementary instruction, as they dictate special methods of presentation. (See ALPHABET METHOD.)

ALPHABET METHOD, or A-B-C Method. This has reference to the first steps in teaching children to read. According to this method, the pupil must learn the names of all the letters of the alphabet, either from an A-B-C book, from cards, or from the blackboard; that is, he must be taught to recognize the various forms of the letters, and to associate with them their respective names. The method of doing this, once very general, was to supply the pupils with books, and then, calling up each one singly, to point to the letters, one after the other, and to pronounce

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