add the product of the next figure, multiply again by 10 and add the product of the next figure; and so on. (iii.) The same method may be applied in compound multiplication of either money, weights or measures, and the reasons for the steps are there made more apparent ; suppose, for instance, that 27. 11s. 94d. is to be multiplied by 425, we multiply first by 4, then by 10 and add in twice the top line, then again by 10 and add in 5 times the top line. (iv.) The process is identical with that of building up such an expression as ax+ ban-1+etc., which is the foundation of all methods of approximation to the roots of equations. (v.) In the multiplication of decimals to a given degree of approximation there is a very decided advantage in proceeding with the figures of the multiplier from left to right, and if we adopt this order from the beginning, no change is afterwards needed and much explana

tion is therefore saved.

Precisely similar reasons to those here given for the order of multiplication occur again and again throughout arithmetic. For instance, the old rules for square and cube root, which depend on the form of the expansions of (a+b) and (a+b)3, should give way to Horner's method, not only because this method is so simple and so much more easily remembered by beginners, but chiefly because the rules of square and cube root are thus presented as only two forms of the same general process which is applicable to the extraction of any root.'

As this method is not generally known, it may not be out of place here to give an example of it. Let it be required to find the cube root of 277,167,808. Divide the number into periods of three figures in the usual way.

Make three columns, and letter them from left to right-I., II., III. Place the given number under I. and 0 under II. and III., putting 1 in a column by itself in front. The column on the left (III.) is the column of three operations, the next that of two operations, that on the right of one operation.

The operations consist in multiplying the amount in the preceding column by the figure in the quotient, and adding in the cases of all columns but I.-subtracting under I. :—

III. Thirdly, both the purposes for which arithmetic is taught dictate the desirability of looking upon all sides of a question, and not restricting the number of ways in which it may be solved. For example, the use of the Rule of Three by the unitary method does not exclude the need of the Rule of Proportion, or make the Chain Rule unnecessary.

IV. The second of the purposes for which arithmetic is taught, viz., for its use in mercantile affairs, adds some lessons of its own. Speed of working must be obtained, and as the mind can reckon quicker than the hand can write, this consideration forbids the writing of unnecessary figures. Thus, in long division the method is shortened, and also the wording, by

By trial we find the first figure 6. Place 6 times 1 under III. and add, 6 times 6 under II. and add, 6 times 36 under I. and subtract. Repeat under III. and II., and again under III.

Add 0 to 18 under III., 00 to 108 under II., and bring down the next period under I. Proceed for the next figure in the same way. To find what the next figure will be, treat the number under I. as dividend, and that under II. as a trial divisor. This gives 5:—

ART EDUCATION

tion as a teacher was not that he invented any new form of discipline. His success was wholly due to his own earnest endeavour to apply the principles of Christianity to life in the school as well as out of it. The mere fact of his own genuine devotion to Christian principle had an irresistible influence with the boys under his care; the amiability of his heart, the justice of all his dealings with them, the transparent honesty of his own character, made him at once loved and feared. His method may be illustrated by the way in which he trained boys to truthfulness. In the higher forms of the school, if a boy, in replying to a question on some point of conduct, was not satisfied simply to give his reply, but attempted to support it by other statements, Arnold at once stopped him with the words, 'If you say so, that is quite enough. Of course I believe your word.' The feeling at once grew up in the school that it was disgraceful to tell the head-master a lie, and thus truthfulness became habitual. In this and other ways Arnold gained a complete mastery in directing the public opinion of the school-and there is no more powerful aid to discipline, no more effective instrument for controlling a company of boys as well as the society of men at large, than public opinion, or the general standard of moral conduct. Arnold could act with severity where he found it necessary. Once he made an example of several boys by expelling them from the school for gross breaches of truthfulness and order, and, in doing so, he said, 'It is not necessary that this should be a school of three hundred, of one hundred, or even of fifty boys. It is necessary that it should be a school of Christian gentlemen.' In June 1842 Arnold was suddenly cut short by an attack of angina pectoris at the early age of 47. Besides his labours in the school Arnold was a prolific writer. In addition to his edition of Thucydides, he wrote a History of Rome,' in three volumes, a work based on the then popular sceptical theories of Niebuhr. He also published five volumes of sermons, and contributed numerous articles to the encyclopædias, reviews, and periodicals of the day. In 1841 he was appointed by Lord Melbourne to the Professorship of Modern History in the University of Oxford. He only lived to deliver one short course of lectures, which were attended by numer

ous audiences, and were published after Arnold's death. See Stanley's Life and Letters of Thomas Arnold.

Art Education. See ESTHETIC CULTURE and SCIENCE AND ART DEPARTMENT.

Arts (Liberal).-Art is derived from.. the same root as aro, to plough, because ploughing was the first art (Max Müller); or more commonly from a root ar, meaning to fit things together. In itself it is. a wide term often used to denote everything not a direct product of nature, and in this sense we speak of nature and art. In a more restricted sense it is opposed to science on the one hand, and to manufac tures on the other. Its meaning is madefairly clear in the old definition that Science is to know that I may know; Art is to know that I may teach.' There is a more limited sense still, including a group. of arts, whose end is not use but pleasure.. These are called the fine, the liberal, or the polite arts-'liberal' here meaning: only such as the leisured classes (freemen as opposed to slaves) could follow. Theseare sometimes spoken of as art, as if they only were the arts. By common consent the five principal fine arts are-architec ture, sculpture, painting, music, poetry (See AESTHETIC CULTURE.)

Ascham, Roger, b. 1515.-One of the earliest of English educational reformers, whose claim to that distinction is established by the new method of teaching he unfolded in his celebrated Scholemasterpublished in 1570, two years after his. death. This work, in the opinion of Dr. Johnson, 'contains perhaps the best advice. that was ever given for the study of languages.' Ascham advocates the adoption of the natural in preference to all artificial methods, and maintains that the dead languages must, like mother tongue, 'begotten, and gotten only by imitation. For as ye used to hear, so ye used to speak.' He expresses his willingness to venture a good wager that an apt scholar who will translate some little book in Tully on the frequent repetition method, will in a very short time learn more Latin 'than the most part do that spend from five to six years. in tossing all the rules of grammar in common schools.' Like Locke, Ascham spoke from successful experience as a private tutor, and he tells us that his illutrious pupil Queen Elizabeth, who never took yet Greek nor Latin grammar in her

hand after the first declining of a noun and a verb, but only by this double translating of Demosthenes and Isocrates daily, without missing, every forenoon, and likewise some part of Tully every afternoon, for the space of a year or two, hath attained to such a perfect understanding in both the tongues,' as to be a more remarkable example of the acquisition of great learning and utterances than even Dion Prussæus, whom Ascham instances as having accomplished this feat with the assistance of only two books, the Phado of Plato and the de Falsa Legatione of Demosthenes. Roger Ascham was a native product of the new learning of the sixteenth century which marked the decline of monkish Latin and the rise of a more liberal scholarship with the introduction of Greek into the school curriculum. Ascham publicly read Greek at Cambridge in 1536, published Toxophilus, the Schole of Shootinge, 1545, and was Latin secretary to Edward VI., Mary, and Elizabeth. For ten years previous to the accession of Elizabeth he was her preceptor.

Assimilation. See DISCRIMINATION. Association of Ideas.-This expression refers to the well-known laws which govern the succession of our thoughts. Whenever one thing reminds us of another, this process of suggestion is due to a law of association. The first and principal one, known as Contiguity, tells us that ideas recur to the mind in the order in which the original objects and impressions presented themselves. In this way we associate events that occur together or in immediate succession, as the movement and sound of a bell, objects and events with places, one place with another, and so forth. All acquisition of knowledge, whether by direct observation or through the medium of instruction, involves the building up of a group of such associations. Thus, a child's knowledge of a particular animal includes associations between the several characteristic features, between the animal as a whole, and its proper surroundings, its habits of life, &c. In studying geography and history, complex associations of place and time have to be built up. Since, moreover, all verbal acquisition implies the working of this law, both in the coupling of names with things and in the connection of words in a given order, it is evident that the whole process of learning is concerned to a large extent with the fixing of

associations in the mind. In addition to the law of Contiguity, it is customary to specify two other principles governing the succession of our ideas, viz. Similarity and Contrast. It is a matter of common observation that natural objects, persons, words, &c., often recall similar ones to the mind. Here, however, it is evident that the connection is not due to the fact that the things were originally presented in this order, but rather to the action of the mind in bringing together what is similar. This law has an important bearing on the process of acquisition (q.v.). By discovering points of resemblance between new facts and facts already known, we are able greatly to shorten the task of learning, as is seen in the rapidity with which an accomplished linguist masters a new language. All assimilation of new knowledge evidently involves the working of this principle, since it proceeds by joining on the new acquisition to old ones which are seen to have some analogy or affinity to the first. The law of Contrast, which says that one idea tends to call up its opposite, as good, bad, seems to be by no means universal in its action, and is not a principle co-ordinate in independence and dignity with the other two. So far as it is valid, it represents a tendency of thought which springs out of the essential conditions of our knowledge of things. begin to know common objects by distinguishing one thing from another, and the broader differences or contrasts among things are among the first to impress the childish mind. In this way a child learns to think of opposites together, as sweet sour, good naughty. The well-known effect of contrast on the feelings renders it a valuable instrument for giving greater vividness to impressions, and so stamping them more deeply on the mind. The contrasts of climate, scenery, social condition, and so forth, are a great aid in the more descriptive and pictorial treatment of geography and history. (For a fuller exposition of the laws of association see Bain, Mental and Moral Science, bk. ii. chap. i.-iii.; Sully, Teacher's Handbook, chap. ix. ; and Spencer's Principles of Psycho logy, i. 228.

We

Assyria (Schools of). See SCHOOLS OF ANTIQUITY.

Astronomy (ãoтpov, a star, and vóμos, a law) is the science of the heavenly bodies. It does not form an adequate part of the