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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 ofhcers 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. 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); VACHEROT, 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 quæ Alexandriae is not peculiar to algebra, but is the char

floruit 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- 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 2a.c——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 6a make lla, is to teach familiar in arithmetic, except as they are modi

If this comparison teaches any thing, 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 Araso 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 1lax. 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 bequal 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 4}, 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 .xm, "x to the mth power", when may be taught to find analogies in the teachm is not necessariiy 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 2 axe -2bcy is the greatest sarily make Tay; for, if one were feet and the common divisor of 20.c* —2a+buy+ 2ab®c®y? other yards, the sum would not be Tay of either. - 26xy' and 4abx*y?

- 2ab%x*y;

26*xyo; If, then, he comes to understand that the fundasince ac-by is a divisor of these polynomials, and mental idea of this notation is, that the terms whether 2012 —2bxy is greater or less than 10- 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, 2ax? applied, as makes one tend to destroy or counter-2bxy=2x (arby). Now suppose a=500, balance the other, he is prepared to see that the b=10, y=2, and x=tu; then ac-by=30, and sum of 5ay and — ay is 3ay; since, when put 2ar? _3bxy=6. Moreover, it is not a question together, the 2ay, by its opposition of nature.

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destroys 2ay of the bay, 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 proper- Methods of Demonstration. It requires no iy and of debts, etc., are characterized as positive argument to convince any one that, in establishand negative, are helpful, if made 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 giving 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 ; , 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, obtaine I, 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- Vax vb=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 vaxvb=Vah. Now, this is concise + aby + b, we say, “ +a taken b times gives and mathematically elegant; but it gives the

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 6, 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 Vab 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 va 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 Vavb x Vavb. Hence vab 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

Ecponents.-One other feature of the mathe- 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, others 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+be=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 i used as an exponent is not a fraction in both ratios have been divided by the same numthe same sense as 3 in its ordinary use, and hence ber, is not even suggested by this demonstration. that the demonstration that 4 = 3 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 i equals the exponent z. why the transformation does not destroy the proportion, but at every step has his attention mar school, or, if in the country, never have other held closely to the fundamental characteristics of school advantages than those furnished by the a proportion. Let the ratio a :b be r; hence as common or rural district school. Nevertheless, a proportion is an equality of ratios, the ratio many of these will receive much greater profit cid is r; and we have a = b = r, and c = d from spending half a year, or a year, in obtaining =r, or a=bi, and c= dr. Substituting these a knowledge of the elements of algebra (and values of a and c in the terms of the proportion even of geometry) than they usually do in studywhich are changed by the transformation, we ing arithmetic. (See ARITHMETIC.) For this have a+b=br + b, or b (r + 1), and c + d class the proper range of topics is, a clear expo= dr + d, or d (+ l); whence we see that sition of the nature of the literal notation ; (1:a+b::c:c+d is deduced from a:b::c:d the fundamental rules, and fractions, involvby multiplying both consequents by r + 1 (the ing only the simpler forms of expression, and ratio +1), which does not destroy the equality excluding such abstruse subjects as the more of the ratios constituting the proportion, since it difficult theorems on factoring, the theory of divides both by the same number. Moreover, lowest common multiple and highest common this method of substituting for the antecedent of divisor ; simple equations involving one, two, each ratio the consequent multiplied by the ratio, and thre: unknown quantities; ratio and proenables us to demonstrate all propositions con- portion; an elementary treatment of the subject cerning the transformation of a proportion by one of radicals with special attention given to their uniform method, which method in all cases clearly nature as growing out of the simplest principles reveals the reason why the proportion is not of factoring; pure and affected quadratics indestroyed.

volving 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 extende:l 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 erponents, positive and negatire his memory; while it also gives broader and quantities, radicals, equations inrolring radmore scientific views of truth. by thus classi- ! icals, and simulteneous 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- Igressions, a practical knowledge of the binomial fully illustrated in the higher algebra by the use formula, and log rithms, and a somewhat exof the infinitesimal method of developing the tended treatment of the applications of algebra binomial formula, logarithmic series, etc., in con

to the business rules of arithmetic. A wide trast with the cumbrous special methods which acquaintance with the results attained in our have so long held their place in our text-books. high schools in all parts of the country, and an By the old method of indeterminate co-efficients, observation extending over more than twenty the pupil is required to pursue what is to him years satisfy the writer that time spent in these always an obscure, long, and unsatisfactory process schools in attempts to master the theory of for the development of each of these series. indeterminate co-efficients, the demonstration Nor are these processes so nearly related to each of the binomial and logarithmic formulus, or other, but that, to the mind of the learner, they upon the higher equations, series, etc., is, if would be even more perplexing than if absolutely not a total loss, at least an absorption of time independent. Moreover, they are styles of argu. which might be much more profitably employed ment which he never meets with again during on other subjects

, such as, for example, history, his subsequent course. On the other hand, after literature, or the elements of the natural sciences. having learned a few simple rules for differentiat- The course taken by such pupils gives them ing algebraic and logarithmic functions.* he is no occasion to use any of these principles of the enabled to develop these, and several other im- higher algebra : and the mastery of them which portant theorems, in one general way, which is as they can attain in any reasonable amount of time remarkable for its concise simplicity, as it is for is quite too imperfect to subserve the ends of its extensive application and habitual recurrence good mental discipline. This second course is in the subsequent course.

entirely adequate to fit a student for admission Range of Topics to be Embraced.—We may into any American college or university. The distinguish three different classes of pupils, who third course is what we may call the college require as many different courses in this study. course. The principal topics which our present First

, there is a very large number of our youth arrangements allow us to add to the second course who, if in the city, never pass beyond the gram- as above marked out, in order to constitute this

course, are the theory of indeterminate coeffielementary method of proving the rule for differentiat. I of algebraic and logarithmic functions to enable

* It may be new to some that there is a simple cients; a sufficient knowledge of the differentiation ing a logarithm without reference to series. This | the student to appreciate the idea of function and method was discovered by Dr. Watson of the University variable, to produce the binomial formular

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

i logarithinic series, and Taylor's formula, which is

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necessary in treating Sturm's thcorem, and to ap- this idea clearly before the mind, the teacher will preciate also the demonstration of that theorem; proceed to the 1st principle. If —-3ab be added indeterminate equations; a tolerably full prac- to 7ab how much of the Tab will it destroy? tical treatment of the higher numerical equa- (Here again we proceed from a fundamental contions; and the interpretation of equations; ception—the nature of quantities as positive and adding, if may be, something upon interpolation negative, thus deducing the new from the old.) and series in general.

Repeat such illustrations of this principle as may Class-Room Work. It is probably unneces- have been given in addition If several boys are sary to say, that a careful and thorough study of urging a sled forward by Tab pounds, and the text-books should be the foundation of our class strength of another boy amounting to 3ab room work on this subject; nevertheless, so much pounds is added, but exerted in an opposite is said, at the present time, in disparagement of direction, what now is the sum of their efforts ? "hearing recitations" instead of “teaching,” that it What kind of a quantity do we call the 3ab? may be well to remark that, if our schools succeed (Negative.) Why? How much of the + Tab in inspiring their pupils with a love of books, and does — 3ab destroy when we add it? If then in teaching how to use them, they accomplish in we wish to destroy + 3ab from + 7ab, how may this a greater good than even in the mere knowl- we do it? Proceeding then to the 2d principle, ledge which they may impart. Books are the it may be asked, how much is 6 ay 2 ay? If great store house of knowledge, and he who has now we add + 2 ay to 6 ay 2 ay, which is 4ay, the habit of using them intelligently has the key what does it become? What does the + 2ay to all human knowledge. But it is not to be destroy? What then is the effect of adding a denied, that there is an important service to be positive quantity ? Such introductory elucidarendered by the living teacher, albeit that service, tions should always be held closely to the plan of especially in this department, is not formal lect- development which the pupil is to study, and uring on the principles of the science. With should be made to throw light upon it. It is a younger pupils, the true teacher will often pref- common and very pernicious thing for teachers ace a subject with a familiar talk designed to to attempt to teach in one line of development, prepare them for an intelligent study of the while the text-book in the pupil's hands gives lesson to be assigned, to awaken an interest in it, quite another. In most cases of this kind, either or to enable them to surmount some particular the teacher's effort or the text-book is useless, or ilifficulty. For example, suppose a class of young probably worse—they tend to confuse each other. pupils are to hate their first lesson in subtrac- Such teaching should culminate in the very lantion in algebra ; a preliminary talk like the fol-guage of the text; and it is desirable that this lanlowing will be exceedingly helpful, perhaps guage be read from the book by the pupil, as the necessary, to an intelligent preparation of the les conclusion of the teaching. Moreover, there is

Observe that, in order to profit the class, great danger of overdoing this kind of work. the teacher must confine his illustrations rigidly Whenever it is practicable, the pupil should be to the essential points on which the lesson is required to prepare his lesson from the book. based. In this case these are il) Adding a neg- | A competent teacher will find sufficient opporalive quantity destroys an equal positive tunity for “ teaching ” after the pupils have gathquantity ; (2) Adding a positive quantity de- ered all they can from the book. Another imstroys an equal negative quantity; (3) As the portant service to be rendered by the living teacher minuend is the sum of the subtrahend and is to emphasize central truths, and hold the pupils remainder, if the subtrahend is destroyed from to a constant review of them. So also it is his duty out the minuend, the remainder is left. Now, in to keep in prominence the outlines of the subject, what order shall these three principles be pre- that the pupil may always know just where he is sented ? Doubtless the scientific order is that just at work and in what relation to other parts of the given; but in such an introduction to the subject the subject that which he is studying stands. All as we are considering, it may be best to present definitions, statements of principles, and theorems the 3d first; since this is a truth already familiar, should be thoroughly memorized by the pupil and and hence affords a connecting link with previous recited again and again. In entering upon a new knowledge. Moreover, this being already before subject, as soon as these can be intelligently learnthe mind as a statement of what is to be done.ed, they should be recited in a most careful and the 1st and 2d will follow in a natural order as formal manner; and, in connection with suban answer to the question how the purpose is ac- sequent demonstrations and solutions, they should complished. To present the 3d principle, the be called up and repeated. Thus, suppose a high teacher may place on the blackboard some sim- school class entering upon the subject of equaple example in subtraction as :

tions. Such a class may be supposed to be able He will then question the class thus: to grasp the meaning of the definitions without

What is the 125 called? What the 74? preliminary aid from the teacher, save in special What the 51 ? How much more than 74 is 125 ? cases. The first lesson will probably contain a If we add 74 and 25, what is the sum? Of what dozen or more definitions, with a proposition or then is the minuend composed ? What is 51+74? two; and the first work should be the recitation If we destroy the 74, what remains ? If in any of these by the pupils individually, without any case we can destroy the subtrahend from out the questions or suggestions from the teacher. Ilminuend, what will remain ? Having brought lustrations should also be required of the pupils;

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