It has a library and apparatus valued at $8,000, amounting to more than $100,000. Students besides the buildings, which are estimated at are required to pursue a three years' elementary $30,000 ; and, in 1875, reported 4 teachers and course, after which they are permitted to choose 126 pupils. The State Normal School and Uni- one of four courses—that of scientific agriculture, versity, at Marion, and the Normal School, at of civil and mining engineering, of literature, or Huntsville, are neither of them so extensive as of science. Under agricultural chemistry, are that at Florence. They are intended for the taught the composition of soils, the relation of education of colored teachers. The former, in air and moisture to vegetable growth, the chem1875, had 3 teachers and 70 pupils; the latter, istry of farm processes, the methods of improving 2 teachers and 84 pupils. This institution is soils. etc. These are accompanied by lessons in designed to become a university for the colored practical agriculture throughout the course. Milipopulation of the state. Besides these state nor- tary training is given, but only to the extent of mal institutions, there are four schools of the improving the health and bearing of the stusame grade under the control of the American dents. Free scholarships. two in number, are proMissionary Association, and one conducted by viiled for each county in the state. The course the Methodists, having an aggregate, in the state, of study covers four years. The number of inof 659 pupils under normal instruction. structors in all the departments, in 1875, was 7; Teachers' institutes were held, during the the number of students, 50, in the regular course, year 1875, in six counties, and their organization and 5 in the special. Law is taught in departments is contemplated in four more. The interest organized for the purpose in the State University aroused, both on the part of the teachers and of and the Southern University ; theology, in the the people at the places of meeting, leads to the Southern University, in Talladega College, and, belief that their permanent establishment is only to some extent, in Howard College; medicine. a question of time. in the Southern University, and in the Medical Secon-lary Instruction. There are 218 pub- College of Alabama, at Mobile. This last inlic high schools in operation in the state, 3 of stitution provides a two years' course of study, which are for colored, the remainder, for white and, in 1875, had I instructors and 50 students. pupils. The course of study prescribed for these Special Instruction.-- The Alabama Institution institutions has been already stated. A number of for the Deaf, Dumb, and Elind was founded in high schools and academies are scattered through | 1860 at Talladega, and is maintained at an annual the state, which occupy a position intermediate expense of about $18,000. The deaf-njute departbetween the primary schools and colleges. Accu- ment is provided with a small museum of naiural rate statistics in regard to them are, however, dif- history and a library of 300 volumes. The studies ficult to procure. In Talladega College, the work pursued are n.athematics and the ordinary Enhas thus far been entirely preparatory, the colle- glish branches. Instruction is also given in agrigiate classes not having been formed. In 1875, culture and gardening. In 1875, there were it ha'l 12 instructors, and a total of 247 students in 4 instructors and 52 pupils. In the department all the departments. It is conducted by the for the blind there were, in the same year, 2 inAmerican Missionary Association for the benefit structors and 10 pupils. of the colored people. ALABAMA, University of, at Tuscaloosa, Superior Instruction. There are several in- was chartered in 1820, but not organized till stitutions of this grade in the state, the most 1831. At the commencement of the civil war important of which are enumerated in the fol- it was in a prosperous condition, but was burned lowing list : by a federal force during the war. It was rebuilt in 1868, and is now in a flourishing condition. Whn Religious 'l he value of its grounds, buildings, apparatus, etc., is estimated at $100,000; and it has an en dowment of $300,000. Its library contains 5,000 Howard College 1843 Bap. Southern University. Greensboro 1856 m. Epis.s. volumes. In 1874, the number of instructors Spring Hill College.. Near Mobile was I, and of collegiate students 76. 1820 Non-sect. demic 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 bistory; (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 profes Professional and Scientific Instruction.- sional education embraces a school of law, and The Agricultural and Mechanical College of Ala- ' a school of civil engineering. All the students, bama was established at Auburn by an act of the except those specially infirm, are subjected to legislature, its endowment being the proceeds of military drill. A special military school affords inthe land grant made by Congress for the benefit struction in military science and art, in military of agriculture and the mechanic arts. The law, and in elementary tactics. The president of amount thus derived was $218,000, to which was the institution is Carlos G. Schmidt, LL. D., added all the property of East Alabama College, i elected in 1874. NAME Location found-denomina cd tion Marion 1836 R. C. he aca Univ. of Alabama.. Tuscaloosa ALBION COLLEGE ALEXANDRIAN SCHOOL 17 ALBION COLLEGE, at Albion, Mich., was Woman's Guide, The Young Housekeeper, etc., chartered as a college in 1861, by members of etc. Dr. Alcott was a genuine philanthropist, the Methodist Episcopal Church. The number though extreme and somewhat eccentric in many of students is about 200, males and females. It of his views. As one of the pioneers in the has a preparatory, classical, and scientific course cause of common-school education and reform in of instruction. Its endowment fund is $200,000. practical teaching, his labors were of incalculable Its library contains about 2000 volumes. Rev. value. G. B. Jocelyn, D. D., is the president of the ALCUIN (Lat. Flaccus Albinus Alcuinus), institution (1875). The tuition is free. a distinguished English scholar, ecclesiastic, and ALCOTT, Amos Bronson, an American reviver of learning, was born in Yorkshire e lucator, was born in 1799. He first gained about 753, and died in 804. Ile was educated distinction by teaching an infant school, for at York under the direction of Archbishop which employment he evince la singular aptitude Egbert, and was subsequently director of the and tact. He removed to Boston in 1828, where seminary in that city. Returning from Rome, he manifested the same skill in teaching young whither he had gone by direction of the English children, at the Masonic Temple. His methods, king, he met the emperor Charlemagne at however, were in advance of public opinion, and Parma, and was induced by that monarch to were disapproved. On the invitation of James take up his residence at the French court, and P. Greaves, of London, the co-laborer of Pesta become the royal preceptor. Accordingly, at lozzi in Switzerland, in educational reform, Mr. Aix-la-Chapelle, he gave instruction, for some Alcott, in 1842, went to England; but the death time, to Charlemagne and his family, in rhetoric, of Mr. Greaves, which occurred before his arrival, logic, divinity, and mathematics. It has been interfered with his prospects. On his return to said with much truth, that “ France is indebted this country, he attempted with some of his to Alcuin for all the polite learning of which it English friends to establish a new community could boast in that and the following ages.” The at Harvard, Mass.; but the enterprise was soon universities of Paris, Tours, Soissons, and many abandoned. Mr. Alcott has since written several others were either founded by him, or greatly works, one of which, Concord Days, was pub- benefited by his zeal in their behalf, and the lished in 1872.--See E. P. PEABODY, Record of favor which he procured for them from CharleSchool (Boston, 1834), and Conversation on the magne. In 796, he was appointed abbot of St. Gospels (Boston, 1836). Martin's at Tours, where he opened a school which ALCOTT, William Alexander, M. D., acquired great celebrity. Here he continued cousin of the preceding, noted for his zeal and teaching till his death. Alcuin was probably success as a common-school teacher, and his life the most learned man and the most illustrious long efforts in behalf of popular education, was teacher of his age; and his lalors were very imborn in Wolcott, Ct., in 1798, and died at portant in giving an impetus to the revival of Auburndale, Mass., in 1859. He had only an learning, after the intellectual night of the Dark elementary education ; and, for several years, he Ages. He left many epistles, poems, and treattaught in the district schools of his native State, ises upon theological and historical subjects, all distinguished for his remarkable earnestness, and written in Latin, and noted for the elegance and the many reforms which he labored to introduce purity of their style. The Life of Alcuin (Leben into the imperfect school management and in-Alcuin's) by Prof. Lorenz, of Ilalle (1829) has struction of his time. He afterwards studied been translated into English (1837) by SLEE.-See melicine; but his chief labors were devoted to Allgemeine Deutsche Biographie, art. Alcuin. the cause of eclucation, co-operating with Gallau ALEXANDRIAN SCHOOL, a name varidet, Woodbridge, and others in the endeavor to ously applied, but chiefly designating (1) a school bring about much-needed reforms in the public of philosophers at Alexandria in Egypt, which schools of the State. Subsequently, he associated is chiefly noted for the development of Neoplatohimself with William C. Woodbridge, and as- nism, and its efforts to harmonize oriental theolsisted him in the compilation of his school geog- ogy with Greek dialectics; (2) a school of raphies, and also in editing the Americom An- Christian theologians in the same city, which niils of Education. He also edited several juve- aimed at harmonizing Pagan philosophy with nile periodicals. His newspaper contributions Christian theology. "The city of Alexandria bewere very numerous, and quite effective on ac came, soon after the death of Alexander the count of their racy and spirited style. An Great, by whom it had been founded, a chief article which he published on the Construction of seat of science and literature. The time during School-Houses gained him a premium from the which the teachers and schools of Alexandria American Institute of Instruction. His labors enjoyed a world-wide reputation, is called the as a lecturer on hygiene, practical teaching, and Ale.candrian Age, and is divided into two pekindred subjects were severe and unintermitting. riods, the former embracing the time of the Ile is said to have visited more than 20,000 Ptolemies, and extending from 323 to 30 B. C.; schools, in many of which he delivered lectures. and the second embracing the time of the RoHis writings are very numerous ; and some of mans, extending from 30 B. C. to 640 A. D. them were widely popular. The most noted are: Grammar, poetry, mathematics, and the natural Confessions of a Schoolmaster, The House I sciences were all taught in the Alexandrian Lire in, The Young Man's Guide, The Young School; and among the most illustrious teachers 18 ALFRED THE GREAT ALGEBRA were Ammonius, Plotinus, Hierocles, Proclus, though he is said to have been twelve years of Apollonius (poet), Galen (physician), Euclid age, before he was taught the alphabet, and (mathematician), Eratosthenes (astronomer), Ptol- although his health was always feeble, he showed emy (geographer). When Christianity began to a thirst for knowledge which is almost without gain a firm footing, it was found necessary to de, parallel in the history of European princes. vote to the instruction of the catechumens special He gave eight hours every day to religious care, in order to fortify them against the attacks exercises and to study. He translated nuupon Christianity by the pagan philosophers. The merous works from Latin into Saxon, as Bede's catechists not only gave to the candidates for History of England, Boethius' De Consolaadmission into the Christian Church element- tione Philosophiae, and the Liber Pastoralis ary instruction, but also delivered learned lectures Curae of Gregory the Great. He invited dison Christianity, and combined with it instruction tinguished scholars to his court from all counin philosophy. Though, from its original character, tries, among whom Wernfried, Plegmund, and the school continued to be called the catechetical Athelstan of Mercia, Grimbald of France, the school of Alexandria, it was in its subsequent Irishman John Scotus Erigena, and the monk development something very different from a Asser of Wales are the most famous. A large catechetical school, and may rather be regarded number of schools were founded and suitably as the first theological faculty, or school of scien- organized. The convents became, more generally tific theology, in the Christian Church. In op- than had been the case before, nurseries of position to the pagan philosophers, the teachers science. All the public officers were required to of the Christian schools chiefly undertook to learn to read and write; and Alfred declared show that Christianity is the only true philos- that the children of every freeman without exophy, and alone can lead to the true gnosis, or ception should be able to read and write, and knowledge. As the first teacher of the Christian should be instructed in the Latin language. A theological school, Pantaenus (about 180) is men- complete list of his works is given in the Encytioned, who was followed by Clement, Origen, clopædia Britannica, art. Alfred. See STOLHeraclas, Dionysius, Pierius, Theognostes, Sera- BERG, Leben Alfred des Grossen, (Münster, 1815); pion, Peter Martyr. The last famous teacher of Weiss, Geschichte Alfred des Grossen (Schaffthe school was Didymus the Blind (335 to 395), hausen, 1852); FREEMAN, Old English History who, being blind from boyhood, had learned read and History of the Norman Conquest. ing, writing, geometry, etc., by means of brass ALFRED UNIVERSITY, at Alfred, N. letters and figures, and was equally distinguished Y., was founded in 1857, by the Seventh Day for his piety and extent of knowledge. The method Baptists. The number of students in the preof teaching used in this, as well as in the other paratory department (in 1874) was 293, males schools of that age, was the Pythagorean. The and females, and in the collegiate department teacher explained, and the pupil listened in 114, of whom 42 were females. It has a classilence, though he was permitted to ask questions. sical and a collegiate course of instruction. Its Every teacher taught in his own house, there be- endowment is $70,000; the number of volumes ing no public school buildings. The teachers did in its library is about 3500. Rev. J. Allen is not receive a fixed salary, but the pupils made the president. Its tuition fee is small. them presents. Origen is reported to have de ALGEBRA ( (Arab. al-jabr, reduction of clined all presents. He supported himself on a parts to a whole). For a general consideration of daily stipend of four oboli, which he received for the purposes for which this study should be purcopying the manuscripts of ancient classics.—See sued, and its proper place and relative proportion MATTER, Histoire de l'école d'Alexandrie (2 vols., of time in the curriculum, the reader is referred 2d ed., Paris, 1840-1844); BARTHÉLEMY ST.- to the article MATHEMATICS. It is the purpose of Hilaire, De l'école d'Alexandrie (Paris, 1845); this article to indicate some of the principles to Simon, Histoire de l'école d'Alexandrie (2 vols., be kept in view, and the methods to be pursued Paris, 1844-1845); Vacheror, Histoire cri- in teaching algebra. tique de l'école d'Alexandrie (3 vols., Paris, 1846 The Literal Notation. – While this notation -1851); GUERIKE, De Schola quo Alexandriae is not peculiar to algebra, but is the charfloruit catechetica (Halle, 1824); HASSELBACH, acteristic language of mathematics, the student De schola quae Alexandriae floruit catechetica usually encounters it for the first time when (Stettin, 1826); RITTER, Geschichte der christ- he enters upon this study. No satisfactory lichen Philosophie, vol. 1, p. 419–564. progress can be made in any of the higher ALFRED THE GREAT, king of the West branches of mathematics, as General Geometry, Saxons and virtually ruler of all England, holds Calculus, Mechanics, Astronomy, etc., without the same prominent position in the history of a good knowledge of the literal notation. By education in England, which Charlemagne occu- i far the larger part of the difficulty which the pies in France and Germany. He was born in ordinary student finds in his study of algebra 849, succeeded his brother Ethelred as king of the proper the science of the equation and West Saxons in 871, and died in 901. After in his more advanced study of mathematics, having thoroughly humbled the Danish invaders grows out of an imperfect knowledge of the and secured the independence of England, he notation. These are facts well known to all exgave his whole attention to internal reforms, and perienced teachers. Nevertheless, it is no unfrespecially to the promotion of education. Al- quent thing to hear a teacher say of a pupil : error. " He is quite good in algebra, but cannot get, as to the value of the divisor that is involved ; along very well with literal examples !" Nothing it is a question as to the degree. Hence, what could be more absurd. It comes from mistaking we wish to affirm is that 20.2°—2bxy is the the importance and fundamental character of highest common divisor of these polynomials, this notation. It is of the first importance that, with respect to x. at the outset, a clear conception be gained of In order that the pupil may get an adequate the nature of this notation, and that, in all the conception of the nature of the literal notation, course, no method nor language be used which it is well to keep prominently before his mind will do violence to these principles. Thus, that the the fact that the fundamental operations of adletters a, b, x, y, etc., as used in mathematics, rep- dition, subtraction, multiplication, and division, resent pure number, or quantity, is to be amply whether of integers or fractions, the various transillustrated in the first lessons, and care is to be formations and reductions of fractions, as well as taken that no vicious conception insinuate itself. involution and evolution, are exactly the same as To say that, as 5 apples and 6 apples make 11 the corresponding ones with which he is already apples, so 5a and 6n make lla, is to teach familiar in arithmetic, except as they are modi If this comparison teaches anything. it fied by the difference between the literal and the is that the letter a in 5a, 6a, and lla, simply Arabic notations. Thus, the pupil will be led gives to the numbers 5, 6, and 11 a concrete to observe that the orders of the Arabic notation significance, as does the word apples in the are analogous to the terms of a polynomial in the first instance; but this is erroneous. The true literal notation, and that the process of “carrying" conception of the use of a, to represent a num in the Arabic addition, etc., has no analogue in ber, may be given in this way: As 5 times 7 the literal, simply because there is no established and 6 times 7 make 11 times 7, so 5 times any relation between the terms in the latter. Again, number and 6 times the same number make 11 he will see that, in both cases, addition is the times that number. Now, let a represent any process of combining several quantities, so that number whatever; then 5 times a and 6 times a the result shall express the aggregate value in make 11 times a. The two thoughts to be im- the fewest terms consistent with the notation. pressed are, that the letter represents some num This being the conception of addition, he will see ber, and that it is immaterial what number it is, that for the same reason that we say, in the Ara80 long as it represents the same number in all bic notation, that the sum of 8 and 7 is 5 and 10 cases in the same problem. Again, the genius (fif-teen), instead of 8 and 7, we say, in the of the literal notation requires that no concep literal notation, that the sum of 5ax and 6ax is tion be taken of a letter as a representative of llax. In fact, it is quite conceivable that the number, which is not equally applicable to frac- pupil, who understands the common or Arabic tional and integral numbers. Thus we may not arithmetic, can master the literal arithmetic for say that a fraction which has a numerator a and a himself, after he has fairly learned the laws of denominator b, represents a of the b equal parts of the new notation. a quantity, or number, as we affirm that repre Positive and Negative.- Although the signs + sents 3 of the 4 equal parts; for this conception and —, even as indicating the affections positive of a fraction requires that the denominator be and negative, are not confined to the literal notaintegral ; otherwise, if b represent a mixed num tion, the pupil first comes to their regular use ber, as 44, we have the absurdity of attempting in this connection, and finds this new element to conceive a quantity as divided into 43 equal of the notation one of his most vexatious parts. The only conception of a fraction, suf- stumbling-blocks. Thus, that the sum of 5ay ficiently broad to comport with the nature of the and— 2ay should be 3ay, and their difference literal notation, is that it is an indicated oper- 7ay, and that “minus multiplied by minus ation in division ; and all operations in fractions should give plus,” as we are wont to say, often should be demonstrated from this definition. seems absurd to the learner. Yet even here he So also to read am,“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 lay do not neceswe cannot affirm that 2ax?—2bxy is the greatest sarily make Tay; for, if one were feet and the common divisor of 2a3.ci—2a2b.c'y + 2ab?xyother yards, the sum would not be Tay of either. - 2b.cyand 40 box3y2 — 2ab3r y} — 2b\xy' ; If, then, he comes to understand that the fundasince —by is a divisor of these polynomials, and mental idea of this notation is, that the terms whether ar?——2b.cy is greater or less than ax— positive and negative indicate simply such opposiby cannot be affirmed unless the relative values tion in kind, in the numbers to which they are of the letters are known. To illustrate, 2ac? applied, as makes one tend to destroy or counter--2bxy=2.c (a.cby). Now suppose a=500, balance the other, he is prepared to see that the b=10, y=2, and to; then ac-by=30, and sum of 5ay and —2ay is 3ay; since, when put 202—2bcy=6. Moreover, it is not a question together, the —2ay, by its opposition of nature. we say destroys 2ay of the 5ay, The ordinary illustra-' Other principles bearing on this important sub tions in which forces acting in opposite directions, ject will be developed under the following head. motion in opposite directions, amounts of proper Methods of Demomstration.-It requires no ty and of debts, etc., are characterized as positive argument to convince any one that, in establishand negative. are helpful, if ma le to set in clearer ing the working features, if we may so speak, of light the fact, that this distinction is simply in a science, it is important that they be exhibited regard to the way in which the numbers are ap- as direct outgrowths of fundamental notions. plied, and not really in regard to the numbers Thus, in givin: a child his first conception of a themselves. 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 ; i 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 procluct 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- Vaxvb=p, whence ab=p; and, extracting the clusion as to establish the other. When we ana- square root of each member we have vab=p. lyze the operation which we call multiplying Hence vaxv=uly. Now, this is concise + a by + 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 - 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 v'a and Va, and b into two equal to develop the theory of positive and negative factors, as vb and vb. 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 Vvb v'avb. Hence vavo 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- i radicals may be seen to be based upon the most matical notation comes into prominence now for · elementary principles of factoring. Again, as the first time, and needs to be clearly compre- | another illustration of this vicious use of the hended: it is the theory of exponents. Here, equation in demonstrating elementary theorems, as well as elsewhere, it is important to guard let us consider the common theorems concerning against false impressions at the start. The idea ' the transformations of a proportion. As usually that an exponent indicates a power is often so demonstrated, by transforming the proportion fixed in the pupil's mind at first, that he never | into an equation, and vice versa, the real afterwards rids himself of the impression. To reason why the proposed transformation does avoid this, it is well to have the pupil learn at not vitiate the proportion, is not brought to the outset that not all exponents indicate the light at all. For example, suppose we are to same thing; thus, while some indicate powers, prove that, If four quantities are in proporothers indicate roots, ot] roots of powers, and tion, they are in proportion by composition, others still the reciprocals of the latter. Too much i. e., if a : b::c:d, a: a + b ::c:c+ d. pains can scarcely be taken to strip this matter The common method is to pass from the given of all obscurity, and allow no fog to gather proportion to the equation be ad, then add around it. Nothing in algebra gives the young ac to each member, obtaining ac + bc = ac+ad, learner so much difficulty as radicals, and all be- or c (a + b) a (c + d), and then to cause he is not thoroughly taught the notation. transform this equation into the proportion Perhaps, but few, even of those who have at- a:a+b::c:c+d. No doubt, this is concise tained considerable proficiency in mathematics, and elegant, but the real reason why the transforhave really set clearly before their own minds the mation does not destroy the proportion, viz., that fact that ž used as an exponent is not a fraction in both ratios have been divided by the same numthe same sense as j in its ordinary use, and hence ber, is not even suggested by this demonstration. that the demonstration that f = } as given con- On the other hand, let the following demonstracerning common fractions, by no means proves tion be used, and the pupil not only sees exactly. that the exponent 4 equals the exponent 3. why the transformation does not destroy the |