It has a library and apparatus valued at $8,000, besides the buildings, which are estimated at $30,000; and, in 1875, reported 4 teachers and 126 pupils. The State Normal School and University, at Marion, and the Normal School, at Huntsville, are neither of them so extensive as that at Florence. They are intended for the education of colored teachers. The former, in 1875, had 3 teachers and 70 pupils; the latter, 2 teachers and 84 pupils. This institution is designed to become a university for the colored population of the state. Besides these state normal institutions, there are four schools of the same grade under the control of the American Missionary Association, and one conducted by the Methodists, having an aggregate, in the state, of 659 pupils under normal instruction. Teachers' institutes were held, during the year 1875, in six counties, and their organization is contemplated in four more. The interest aroused, both on the part of the teachers and of the people at the places of meeting, leads to the belief that their permanent establishment is only a question of time. amounting to more than $100,000. Students are required to pursue a three years' elementary course, after which they are permitted to choose one of four courses that of scientific agriculture, of civil and mining engineering, of literature, or of science. Under agricultural chemistry, are taught the composition of soils, the relation of air and moisture to vegetable growth, the chemistry of farm processes, the methods of improving soils. etc. These are accompanied by lessons in practical agriculture throughout the course. Military training is given, but only to the extent of improving the health and bearing of the students. Free scholarships, two in number, are provided for each county in the state. The course of study covers four years. The number of instructors in all the departments, in 1875, was 7; the number of students, 50, in the regular course, and 5 in the special. Law is taught in departments organized for the purpose in the State University and the Southern University; theology, in the Southern University, in Talladega College, and, to some extent, in Howard College; medicine, in the Southern University, and in the Medical College of Alabama, at Mobile. This last institution provides a two years' course of study, and, in 1875, had 9 instructors and 50 students. Special Instruction.--The Alabama Institution for the Deaf, Dumb, and Elind was founded in 1860 at Talladega, and is maintained at an annual expense of about $18,000. The deaf-mute department is provided with a small museum of natural history and a library of 300 volumes. The studies pursued are mathematics and the ordinary En Secondary Instruction. There are 218 public high schools in operation in the state, 3 of which are for colored, the remainder, for white pupils. The course of study prescribed for these institutions has been already stated. A number of high schools and academies are scattered through the state, which occupy a position intermediate between the primary schools and colleges. Accurate statistics in regard to them are, however, difficult to procure. In Talladega College, the work has thus far been entirely preparatory, the colle-glish branches. Instruction is also given in agrigiate classes not having been formed. In 1875, it had 12 instructors, and a total of 247 students in all the departments. It is conducted by the American Missionary Association for the benefit of the colored people. Superior Instruction.-There are several institutions of this grade in the state, the most important of which are enumerated in the following list: NAME Howard College Location Marion Southern University. Greensboro Spring Hill College.. Near Mobile Univ. of Alabama.. Tuscaloosa When Religious 1843 Вар. culture and gardening. In 1875, there were 4 instructors and 52 pupils. In the department for the blind there were, in the same year, 2 instructors and 10 pupils. ALABAMA, University of, at Tuscaloosa, was chartered in 1820, but not organized till 1831. At the commencement of the civil war it was in a prosperous condition, but was burned by a federal force during the war. It was rebuilt in 1868, and is now in a flourishing condition. 'The value of its grounds, buildings, apparatus, etc., is estimated at $150,000; and it has an endowment of $300,000. Its library contains 5,000 volumes. In 1874, the number of instructors was 9, and of collegiate students 76. he academic department embraces eight courses of study, To the above list, must be added 9 institutions open to the selection of the students: (1) I atin which afford opportunities for the higher edu- language and literature; (2) Greek language and cation of women. In addition to the studies literature; (3) English language and literature; usually pursued in such institutions, special at- (4) Modern languages; (5) Chemistry, geology, tention is given to the ornamental branches. and natural history; (6) Natural philosophy The number of instructors in these institutions, (7) Mathematics and astronomy; (8) Mental and in 1875, was 80; the number of students, 883. moral philosophy. The department of profesProfessional and Scientific Instruction.—sional education embraces a school of law, and The Agricultural and Mechanical College of Alabama was established at Auburn by an act of the legislature, its endowment being the proceeds of the land grant made by Congress for the benefit of agriculture and the mechanic arts. The amount thus derived was $218,000, to which was added all the property of East Alabama College, ❘ elected in 1874. a school of civil engineering. All the students, except those specially infirm, are subjected to military drill. A special military school affords instruction in military science and art, in military law, and in elementary tactics. The president of the institution is Carlos G. Schmidt, LL. D., 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 Pesta lozzi 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 ALGEBRA 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. aljabr, 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 unfre quent 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 5a and 6a 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 xm, "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 2a3x-2abc1y+2ab2x2y -2bxy and 4a bˆx3y2. 2аb3x`y3 — 2b1xy; since ax-by is a divisor of these polynomials, and whether 2ac2-2bxy is greater or less than ax— by cannot be affirmed unless the relative values of the letters are known. To illustrate, 2ax -2bxy=2x (ax-by). Now suppose a 500, b=10, y=2, and ; then ax-by=30, and 2ax2-2bxy-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 2a-2b.xy 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 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, 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. 66 ject 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 So, also, in multiplication, the three principles, use the conception of a fraction as an indicated (1) that the product is like the multiplicand; operation in division, and attempt to build up (2) that a multiplier must be conceived as essen- the theory of common fractions on that notion. tially abstract when the operation is performed; It may be elegant and logical, and when we come and (3) that the sign of the multiplier shows to the literal notation it is essential; but it is not what is to be done with the product when sufficiently radical for the tyro. It is not natural, obtained, remove all the difficulty, and make it but scientific rather. So in the literal notation, seem no more absurd that "minus multiplied by the proposition that the product of the square minus gives plus," than that "plus multiplied by roots of two numbers is equal to the square root plus gives plus": in fact, exactly the same course of their product, may be demonstrated thus: Let of argument is required to establish the one con- Vaxb-p, whence ab=p; and, extracting the clusion as to establish the other. When we ana- square root of each member we have √ ab = p. lyze the operation which we call multiplying | Hence √a× √√b=yub. Now, this is concise +aby+b, we say +a taken b times gives and mathematically elegant; but it gives the +ab. Now the sign+before the multiplier pupil no insight whatever into "the reason why." indicates that the product is to be taken ad- What is needed here is, that the pupil be enditively, that is, united to other quantities by its abled to see that this proposition grows out of own sign." So when we multiply a by-b, the nature of a square root as one of the two we say -a multiplied by b (a mere number) equal factors of a number; i. e., he needs to see gives ab (a product like the multiplicand). its connection with fundamental conceptions. But the sign before the multiplier indicates Thus ab means that the product ab is to be rethat this product is to be taken subtractively, solved into two equal factors, and that one of them i. e. united with other quantities by a sign op is to be taken. Now, if we resolve a into two equal posite to its own." This, however, is not the place factors, as a and a, and b into two equal to develop the theory of positive and negative factors, as b and b, ab will be resolved into quantities; our only purpose here is to show four factors which can be arranged in two equal that the whole grows out of a kind of concrete groups, thus ab× √a、b. Hence yab is or denominate significance which is thus put the square root of ab because it is one of the two upon the numbers, and which bears some analogy equal factors into which ab can be conceived to to familiar principles of common arithmetic. be resolved. In this manner, all operations in Erponents.-One other feature of the mathe-radicals may be seen to be based upon the most matical 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 as given concerning common fractions, by no means proves that the exponent equals the exponent 3. 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 abcd, a: a b :: cc + d. The common method is to pass from the given proportion to the equation be ad, then add ac to each member, obtaining ac+ be=ac + ad, or c (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 = |