ALBION COLLEGE ALBION COLLEGE, at Albion, Mich., was chartered as a college in 1861, by members of the Methodist Episcopal Church. The number of students is about 200, males and females. It has a preparatory, classical, and scientific course of instruction. Its endowment fund is $200,000. Its library contains about 2000 volumes. Rev. G. B. Jocelyn, D. D., is the president of the institution (1875). The tuition is free. ALCOTT, Amos Bronson, an American e lucator, was born in 1799. He first gained distinction by teaching an infant school, for which employment he evinced a singular aptitude and tact. He removed to Boston in 1828, where he manifested the same skill in teaching young children, at the Masonic Temple. His methods, however, were in advance of public opinion, and were disapproved. On the invitation of James P. Greaves, of London, the co-laborer of Pestalozzi in Switzerland, in educational reform, Mr. Alcott, in 1842, went to England; but the death of Mr. Greaves, which occurred before his arrival, interfered with his prospects. On his return to this country, he attempted with some of his English friends to establish a new community at Harvard, Mass.; but the enterprise was soon abandoned. Mr. Alcott has since written several works, one of which, Concord Days, was published in 1872.-See E. P. PEABODY, Record of School (Boston, 1834), and Conversation on the Gospels (Boston, 1836). ALCOTT, William Alexander, M. D., cousin of the preceding, noted for his zeal and success as a common-school teacher, and his lifelong efforts in behalf of popular education, was born in Wolcott, Ct., in 1798, and died at Auburndale, Mass., in 1859. He had only an elementary education; and, for several years, he taught in the district schools of his native State, distinguished for his remarkable earnestness, and the many reforms which he labored to introduce into the imperfect school management and instruction of his time. He afterwards studied medicine; but his chief labors were devoted to the cause of education, co-operating with Gallaudet, Woodbridge, and others in the endeavor to bring about much-needed reforms in the public schools of the State. Subsequently, he associated himself with William C. Woodbridge, and assisted him in the compilation of his school geographies, and also in editing the American Annals of Education. He also edited several juvenile periodicals. His newspaper contributions were very numerous, and quite effective on account of their racy and spirited style. An article which he published on the Construction of School Houses gained him a premium from the American Institute of Instruction. His labors as a lecturer on hygiene, practical teaching, and kindred subjects were severe and unintermitting. He is said to have visited more than 20,000 schools, in many of which he delivered lectures. His writings are very numerous; and some of them were widely popular. The most noted are: Confessions of a Schoolmaster, The House I Live in, The Young Man's Guide, The Young Woman's Guide, The Young Housekeeper, etc., etc. Dr. Alcott was a genuine philanthropist, though extreme and somewhat eccentric in many of his views. As one of the pioneers in the cause of common-school education and reform in practical teaching, his labors were of incalculable value. ALCUIN (Lat. Flaccus Albinus Alcuinus), a distinguished English scholar, ecclesiastic, and reviver of learning, was born in Yorkshire about 753, and died in 804. He was educated at York under the direction of Archbishop Egbert, and was subsequently director of the seminary in that city. Returning from Rome, whither he had gone by direction of the English king, he met the emperor Charlemagne at Parma, and was induced by that monarch to take up his residence at the French court, and become the royal preceptor. Accordingly, at Aix-la-Chapelle, he gave instruction, for some time, to Charlemagne and his family, in rhetoric, logic, divinity, and mathematics. It has been said with much truth, that "France is indebted to Alcuin for all the polite learning of which it could boast in that and the following ages." The universities of Paris, Tours, Soissons, and many others were either founded by him, or greatly benefited by his zeal in their behalf, and the favor which he procured for them from Charlemagne. In 796, he was appointed abbot of St. Martin's at Tours, where he opened a school which acquired great celebrity. Here he continued teaching till his death. Alcuin was probably the most learned man and the most illustrious teacher of his age; and his labors were very important in giving an impetus to the revival of learning, after the intellectual night of the Dark Ages. He left many epistles, poems, and treatises upon theological and historical subjects, all written in Latin, and noted for the elegance and purity of their style. The Life of Alcuin (Leben Alcuin's) by Prof. LORENZ, of Halle (1829) has been translated into English (1837) by SLEE.-See Allgemeine Deutsche Biographie, art. Alcuin. ALEXANDRIAN SCHOOL, a name variously applied, but chiefly designating (1) a school of philosophers at Alexandria in Egypt, which is chiefly noted for the development of Neoplatonism, and its efforts to harmonize oriental theology with Greek dialectics; (2) a school of Christian theologians in the same city, which aimed at harmonizing Pagan philosophy with Christian theology. The city of Alexandria became, soon after the death of Alexander the Great, by whom it had been founded, a chief seat of science and literature. The time during which the teachers and schools of Alexandria enjoyed a world-wide reputation, is called the Alexandrian Age, and is divided into two periods, the former embracing the time of the Ptolemies, and extending from 323 to 30 B. C., and the second embracing the time of the Romans, extending from 30 B. C. to 640 A. D. Grammar, poetry, mathematics, and the natural sciences were all taught in the Alexandrian School; and among the most illustrious teachers 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 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: ALGEBRA "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 5 and 6 make 11a, is to teach error. If this comparison teaches anything. 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. So also to read am, "x to the mth power," when m is not necessarily an integer, is to violate this fundamental characteristic of the notation. In like manner, to use the expressions greatest common divisor, and least common multiple, when literal quantities are under consideration, is an absurdity, and moreover fails to give any indication of the idea which should be conveyed. For example, we cannot affirm that 2ax-2bxy is the greatest common divisor of 2a3.c1-2a2bx1y +2ab2x2y · 2ab3x`y3· -2bxy and 4a b'x3y? 2b'xy; since a-by is a divisor of these polynomials, and whether 2ar2-2bxy is greater or less than ax— by cannot be affirmed unless the relative values of the letters are known. To illustrate, 2ar' -2b.xy=2x (ax-by). Now suppose a 500, b=10, y=2, and ; then ac-by=30, and 2ax-2bxy-6. Moreover, it is not a question 19 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 2ar-2bxy is the highest common divisor of these polynomials, with respect to a. 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 5 and 6ax is 11a. 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 Tay, 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 may be taught to find analogies in the teachings of the common arithmetic, which will at least partially remove the difficulty. When he comes to understand, that attributing to numbers the affection positive or negative gives to them a sort of concrete significance, and allies them in some sort to denominate numbers, he may at least see, that 5ay and 2ay do not necessarily make Tay; for, if one were feet and the other yards, the sum would not be Tay of either. If, then, he comes to understand that the fundamental idea of this notation is, that the terms positive and negative indicate simply such opposition 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. destroys 2ay of the 5ay, The ordinary illustra- | Other principles bearing on this important subtions in which forces acting in opposite directions, ject will be developed under the following head. 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 obtained, remove all the difficulty, and make it seem no more absurd that "minus 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 + a by+b, 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, we say 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. 66 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 at tained considerable proficiency in mathematics, have really set clearly before their own minds the fact that used as an exponent is not a fraction in the same sense as in its ordinary use, and hence that the demonstration that as given concerning common fractions, by no means proves that the exponent equals the exponent . 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 Vaxb-p, whence ab=p; and, extracting the square root of each member we have √ab = p. Hence √a× √b=ub. 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 va and a, and b into two equal factors, asb and b, ab will be resolved into four factors which can be arranged in two equal groups, thus √a、b × √a√b. Hence √√b 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 b c d 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 c (a + b) a (c + d), 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 : : : == proportion, but at every step has his attention held closely to the fundamental characteristics of a proportion. Let the ratio ab ber; hence as a proportion is an equality of ratios, the ratio ed is r; and we have a b=r, and cd =r, or a= br, and e= = dr. Substituting these values of a and e in the terms of the proportion which are changed by the transformation, we have a b = br + b, or b (r + 1), and c + 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 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 con-portion; an elementary treatment of the subject cerning the transformation of a proportion by one uniform method, which method in all cases clearly reveals the reason why the proportion is not destroyed. 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 proof 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 second class comprises what may be called high school pupils. For this grade the range of topics need not be much widened, but the study of each should be extended and deepened. This will be the case especially as regards the theory of exponents, positive and negative quantities, radicals, equations involving rad icals, and simultaneous equations, especially those of the second degree. To this should be added the arithmetical and geometrical pro This choice of a line of argument which shall be applicable to an entire class of propositions is of no slight importance in constructing a mathematical course. It enables a student to learn with greater facility and satisfaction the demonstrations, and fixes them more firmly in his memory; while it also gives broader and more scientific views of truth, by thus classifying, and bringing into one line of thought, numerous truths which would otherwise be seen only 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 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-effiof algebraic and logarithmic functions to enable cients; a sufficient knowledge of the differentiation the student to appreciate the idea of function and variable, to produce the binomial formula, the logarithmic series, and Taylor's formula, which is |