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were Ammonius, Plotinus, Hierocles, Proclus, Apollonius (poet), Galen (physician), Euclid (mathematician), Eratosthenes (astronomer), Ptolemy (geographer). When Christianity began to gain a firm footing, it was found necessary to devote to the instruction of the catechumens special care, in order to fortify them against the attacks upon Christianity by the pagan philosophers. The catechists not only gave to the candidates for admission into the Christian Church elementary instruction, but also delivered learned lectures on Christianity, and combined with it instruction in philosophy. Though, from its original character, the school continued to be called the catechetical | school of Alexandria, it was in its subsequent development something very different from a catechetical school, and may rather be regarded as the first theological faculty, or school of scientific theology, in the Christian Church. In opposition to the pagan philosophers, the teachers of the Christian schools chiefly undertook to show that Christianity is the only true philosophy, and alone can lead to the true gnosis, or knowledge. As the first teacher of the Christian theological school, Pantaenus (about 180) is mentioned, who was followed by Clement, Origen, Heraclas, Dionysius, Pierius, Theognostes, Serapion, Peter Martyr. The last famous teacher of the school was Didymus the Blind (335 to 395), who, being blind from boyhood, had learned reading, writing, geometry, etc., by means of brass letters and figures, and was equally distinguished for his piety and extent of knowledge. The method of teaching used in this, as well as in the other schools of that age, was the Pythagorean. The teacher explained, and the pupil listened in silence, though he was permitted to ask questions. Every teacher taught in his own house, there being no public school buildings. The teachers did not receive a fixed salary, but the pupils made them presents. Origen is reported to have declined all presents. He supported himself on a daily stipend of four oboli, which he received for copying the manuscripts of ancient classics.-See MATTER, Histoire de l'école d'Alexandrie (2 vols., 2d ed., Paris, 1840-1844); BARTHÉLEMY ST.HILAIRE, De l'école d' Alexandrie (Paris, 1845); SIMON, Histoire de l'école d'Alexandrie (2 vols., Paris, 1844-1845); VACHEROT, Histoire critique de l'école d'Alexandrie (3 vols., Paris, 1846 -1851); GUERIKE, De Schola quæ Alexandriae floruit catechetica (Halle, 1824); HASSELBACH, De schola quae Alexandriae floruit catechetica (Stettin, 1826); RITTER, Geschichte der christlichen Philosophie, vol. 1, p. 419-564.

ALFRED THE GREAT, king of the West Saxons and virtually ruler of all England, holds the same prominent position in the history of education in England, which Charlemagne occupies in France and Germany. He was born in 849, succeeded his brother Ethelred as king of the West Saxons in 871, and died in 901. After having thoroughly humbled the Danish invaders and secured the independence of England, he gave his whole attention to internal reforms, and specially to the promotion of education. Al

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though he is said to have been twelve years of age, before he was taught the alphabet, and although his health was always feeble, he showed a thirst for knowledge which is almost without parallel in the history of European princes. He gave eight hours every day to religious exercises and to study. He translated numerous works from Latin into Saxon, as Bede's History of England, Boethius' De Consolatione Philosophiae, and the Liber Pastoralis Curae of Gregory the Great. He invited distinguished scholars to his court from all countries, among whom Wernfried, Plegmund, and Athelstan of Mercia, Grimbald of France, the Irishman John Scotus Erigena, and the monk Asser of Wales are the most famous. A large number of schools were founded and suitably organized. The convents became, more generally than had been the case before, nurseries of science. All the public officers were required to learn to read and write; and Alfred declared that the children of every freeman without exception should be able to read and write, and should be instructed in the Latin language. A complete list of his works is given in the Encyclopædia Britannica, art. Alfred. See STOLBERG, Leben Alfred des Grossen, (Münster, 1815); WEISS, Geschichte Alfred des Grossen (Schaffhausen, 1852); Freeman, Old English History and History of the Norman Conquest.

ALFRED UNIVERSITY, at Alfred, N. Y., was founded in 1857, by the Seventh Day Baptists. The number of students in the preparatory department (in 1874) was 293, males and females, and in the collegiate department 114, of whom 42 were females. It has a classical and a collegiate course of instruction. Its endowment is $70,000; the number of volumes in its library is about 3500. Rev. J. Allen is the president. Its tuition fee is small.

ALGEBRA (Arab. al-jabr, reduction of parts to a whole). For a general consideration of the purposes for which this study should be pursued, and its proper place and relative proportion of time in the curriculum, the reader is referred to the article MATHEMATICS. It is the purpose of this article to indicate some of the principles to be kept in view, and the methods to be pursued in teaching algebra.

The Literal Notation.-While this notation is not peculiar to algebra, but is the characteristic language of mathematics, the student usually encounters it for the first time when he enters upon this study. No satisfactory progress can be made in any of the higher branches of mathematics, as General Geometry, Calculus, Mechanics, Astronomy, etc., without a good knowledge of the literal notation. By far the larger part of the difficulty which the ordinary student finds in his study of algebra proper the science of the equation and in his more advanced study of mathematics, grows out of an imperfect knowledge of the notation. These are facts well known to all experienced teachers. Nevertheless, it is no unfrequent thing to hear a teacher say of a pupil:

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"He is quite good in algebra, but cannot get along very well with literal examples!" Nothing could be more absurd. It comes from mistaking the importance and fundamental character of this notation. It is of the first importance that, at the outset, a clear conception be gained of the nature of this notation, and that, in all the course, no method nor language be used which will do violence to these principles. Thus, that the letters a, b, x, y, etc., as used in mathematics, represent pure number, or quantity, is to be amply illustrated in the first lessons, and care is to be taken that no vicious conception insinuate itself. To say that, as 5 apples and 6 apples make 11 apples, so 5a and 6a make 11a, is to teach error. If this comparison teaches any thing, it is that the letter a in 5a, 6a, and 11a, simply gives to the numbers 5, 6, and 11 a concrete significance, as does the word apples in the first instance; but this is erroneous. The true conception of the use of a, to represent a number, may be given in this way: As 5 times 7 and 6 times 7 make 11 times 7, so 5 times any number and 6 times the same number make 11 times that number. Now, let a represent any number whatever; then 5 times a and 6 times a make 11 times a. The two thoughts to be impressed are, that the letter represents some number, and that it is immaterial what number it is, so long as it represents the same number in all cases in the same problem. Again, the genius of the literal notation requires that no conception be taken of a letter as a representative of number, which is not equally applicable to fractional and integral numbers. Thus we may not say that a fraction which has a numerator a and a denominator b, represents a of the b equal parts of a quantity, or number, as we affirm that represents 3 of the 4 equal parts; for this conception of a fraction requires that the denominator be integral; otherwise, if b represent a mixed number, as 43, we have the absurdity of attempting to conceive a quantity as divided into 43 equal parts. The only conception of a fraction, sufficiently broad to comport with the nature of the literal notation, is that it is an indicated operation in division; and all operations in fractions should be demonstrated from this definition.

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as to the value of the divisor that is involved; it is a question as to the degree. Hence, what we wish to affirm is that 2ax-2bxy is the highest common divisor of these polynomials, with respect to x.

In order that the pupil may get an adequate conception of the nature of the literal notation, it is well to keep prominently before his mind the fact that the fundamental operations of addition, subtraction, multiplication, and division, whether of integers or fractions, the various transformations and reductions of fractions, as well as involution and evolution, are exactly the same as the corresponding ones with which he is already familiar in arithmetic, except as they are modified by the difference between the literal and the Arabic notations. Thus, the pupil will be led to observe that the orders of the Arabic notation are analogous to the terms of a polynomial in the literal notation, and that the process of "carrying" in the Arabic addition, etc., has no analogue in the literal, simply because there is no established relation between the terms in the latter. Again, he will see that, in both cases, addition is the process of combining several quantities, so that the result shall express the aggregate value in the fewest terms consistent with the notation. This being the conception of addition, he will see that for the same reason that we say, in the Arabic notation, that the sum of 8 and 7 is 5 and 10 (fif-teen), instead of 8 and 7, we say, in the literal notation, that the sum of 5ax and 6ax is 11ax. In fact, it is quite conceivable that the pupil, who understands the common or Arabic arithmetic, can master the literal arithmetic for himself, after he has fairly learned the laws of the new notation.

Positive and Negative.-Although the signs + and, even as indicating the affections positive and negative, are not confined to the literal notation, the pupil first comes to their regular use in this connection, and finds this new element of the notation one of his most vexatious stumbling-blocks. Thus, that the sum of 5ay and-2ay should be 3ay, and their difference 7ay, and that "minus multiplied by minus should give plus," as we are wont to say, often seems absurd to the learner. Yet even here he So also to read xm, "x to the mth power", when may be taught to find analogies in the teachm is not necessarily an integer, is to violate this ings of the common arithmetic, which will at fundamental characteristic of the notation. In like least partially remove the difficulty. When he manner, to use the expressions greatest common comes to understand, that attributing to numbers divisor, and least common multiple, when literal the affection positive or negative gives to them quantities are under consideration, is an absurd- a sort of concrete significance, and allies them ity, and moreover fails to give any indication of in some sort to denominate numbers, he may the idea which should be conveyed. For example, at least see, that 5ay and 2ay do not neceswe cannot affirm that 2ax-2bxy is the greatest sarily make Tay; for, if one were feet and the common divisor of 23-2a2bx3y+2ab2x2y other yards, the sum would not be 7ay of either. 2b3xy' and 4a2b2x2y2 — 2ab3x2y3 —2b'xy; If, then, he comes to understand that the fundasince ax-by is a divisor of these polynomials, and mental idea of this notation is, that the terms whether 2ax-2bxy is greater or less than ux-positive and negative indicate simply such opposiby cannot be affirmed unless the relative values of the letters are known. To illustrate, 2ax2 -2bxy=2x (ax-by). Now suppose a=500, b=10, y=2, and x; then ax-by-30, and 2ax2-2bxy-6. Moreover, it is not a question

tion in kind, in the numbers to which they are applied, as makes one tend to destroy or counterbalance the other, he is prepared to see that the sum of 5ay and -2ay is 3ay; since, when put together, the -2ay, by its opposition of nature,

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destroys 2ay of the 5ay, The ordinary illustrations in which forces acting in opposite directions, motion in opposite directions, amounts of property and of debts, etc., are characterized as positive and negative, are helpful, if made to set in clearer light the fact, that this distinction is simply in regard to the way in which the numbers are applied, and not really in regard to the numbers themselves.

So, also, in multiplication, the three principles, (1) that the product is like the multiplicand; (2) that a multiplier must be conceived as essentially abstract when the operation is performed; and (3) that the sign of the multiplier shows what is to be done with the product when obtaine 1, remove all the difficulty, and make it seem no more absurd that "minas multiplied by minus gives plus," than that "plus multiplied by plus gives plus": in fact. exactly the same course of argument is required to establish the one conclusion as to establish the other. When we analyze the operation which we call multiplying +abyb, we say "a taken b times gives +ab. Now the sign+before the multiplier indicates that the product is to be taken additively, that is, united to other quantities by its own sign." So when we multiply a by-b, a multiplied by b (a mere number) gives - ab (a product like the multiplicand). But the sign before the multiplier indicates that this product is to be taken subtractively, i. e. united with other quantities by a sign opposite to its own." This, however, is not the place to develop the theory of positive and negative quantities; our only purpose here is to show that the whole grows out of a kind of concrete or denominate significance which is thus put upon the numbers, and which bears some analogy to familiar principles of common arithmetic.

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Exponents. One other feature of the mathematical notation comes into prominence now for the first time, and needs to be clearly comprehended: it is the theory of exponents. Here, as well as elsewhere, it is important to guard against false impressions at the start. The idea that an exponent indicates a power is often so fixed in the pupil's mind at first, that he never afterwards rids himself of the impression. To avoid this, it is well to have the pupil learn at the outset that not all exponents indicate the same thing; thus, while some indicate powers, others indicate roots, others roots of powers, and others still the reciprocals of the latter. Too much pains can scarcely be taken to strip this matter of all obscurity, and allow no fog to gather around it. Nothing in algebra gives the young learner so much difficulty as radicals, and all because he is not thoroughly taught the notation. Perhaps, but few, even of those who have attained considerable proficiency in mathematics, have really set clearly before their own minds the fact that used as an exponent is not a fraction in 3 the same sense as in its ordinary use, and hence that the demonstration that 3 as given concerning common fractions, by no means proves that the exponent equals the exponent.

Other principles bearing on this important subject will be developed under the following head. Methods of Demonstration.-It requires no argument to convince any one that, in establishing the working features, if we may so speak, of a science, it is important that they be exhibited as direct outgrowths of fundamental notions. Thus, in giving a child his first conception of a common fraction, no intelligent teacher would use the conception of a fraction as an indicated operation in division, and attempt to build up the theory of common fractions on that notion. It may be elegant and logical, and when we come to the literal notation it is essential; but it is not sufficiently radical for the tyro. It is not natural, but scientific rather. So in the literal notation, the proposition that the product of the square roots of two numbers is equal to the square root of their product, may be demonstrated thus: Let √axvb=p, whence ab=p"; and, extracting the square root of each member we have y'ab = p. Hence a× √b=ab. Now, this is concise and mathematically elegant; but it gives the pupil no insight whatever into "the reason why." What is needed here is, that the pupil be enabled to see that this proposition grows out of the nature of a square root as one of the two equal factors of a number; i. e., he needs to see its connection with fundamental conceptions. Thus ab means that the product ab is to be resolved into two equal factors, and that one of them is to be taken. Now, if we resolve a into two equal factors, as a and a, and b into two equal factors, as ✓b and b, ab will be resolved into four factors which can be arranged in two equal groups, thus √α√b × √a√b. Hence ab is the square root of ab because it is one of the two equal factors into which ab can be conceived to be resolved. In this manner, all operations in radicals may be seen to be based upon the most elementary principles of factoring. Again, as another illustration of this vicious use of the equation in demonstrating elementary theorems. let us consider the common theorems concerning the transformations of a proportion. As usually demonstrated, by transforming the proportion into an equation, and vice versa, the real reason why the proposed transformation does not vitiate the proportion, is not brought to light at all. For example, suppose we are to prove that, If four quantities are in proportion, they are in proportion by composition, i. e., if a bed, a: a + b c c + d The common method is to pass from the given proportion to the equation be ad, then add ac to each member, obtaining ac+bc = ac+ad, or e (a + b) a (cd), and then to transform this equation into the proportion a: a+b:c:c+d. No doubt, this is concise and elegant, but the real reason why the transformation does not destroy the proportion, viz., that both ratios have been divided by the same number, is not even suggested by this demonstration. On the other hand, let the following demonstration be used, and the pupil not only sees exactly why the transformation does not destroy the

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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 cd is r; and we have a b=r, 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 cd drd, or d (r+1); whence we see that a:a+b:c: cd is deduced from a:b::c:d by multiplying both consequents by r+ 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 thre› 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 radmore scientific views of truth, by thus classi-icals, and simult ineous 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 argu. ment 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 OINEY'S University Alg bra 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 knowlledge 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 negalive 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:

74

125 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

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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 3ab 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 +Tab 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 — 2 ay, 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 pernicious 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. IIlustrations should also be required of the pupils;

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