on the hypothenuse, while the two are those on the sides. To verify this proposition in numbers, when he has learnt how to estimate the area of a square, it will be desirable to have a method of finding right-angled triangles whose sides are whole numbers. The following is the most simple: take any two numbers (4 and 7), multiply each by itself (16 and 49); take the sum and difference of the last (65 and 33): these are the hypothenuse and one of the sides of a right-angled triangle: the other side is twice the product of the numbers first chosen (56.) Book II. 12, 13. The simplest method of verifying these propositions is by means of numbers, and the following rule will find a triangle whose three sides, together with the perpendicular let fall on one side from the opposite angle, and the segments made by that perpendicular, are all whole numbers. Let ABC be the triangle, CD the perpendicular from C on AB, making the segments AD, DB. Choose any even number for the perpendicular CD, which has two even divisors; 8 for example, which is divisible by 2 and 4. Multiply and divide it by one of the even divisors, 2 for example, giving 16 and 4. Take the half sum and half difference of these, giving 10 and 6. If the first be CB, the second is DB. Proceed in the same way with the second divisor 4, which gives 17 and 15. If the first be AC, the second is AD. The side AB is the sum or difference of AD and DB, that is, 21 or 9, according as we construct CDA and CDB on opposite or the same sides of CD; that is, according as we choose both the angles A and B acute, or one of them obtuse. When propositions have been verified in the case of whole numbers, verification in some other case should be attempted by measurement and calculation. This will not be so satisfactory, on account of the errors which must arise in estimating fractions of the parts of the scale. Nevertheless, it should be attempted, and the source of error pointed out. It will be found that such numerical application will fix the proposition in the mind. We shall in the next Number proceed to show how the reasoning of geometry may be taught, on the supposition that the student is perfectly master of the language and facts of at least a large portion of the first four books of Euclid. Every beginner will not be competent to do as much as we have pointed out, but many will; and we are convinced that, by so doing, they will be prepared to look upon the reasoning of Euclid, as what it really is, one of the most admirable results of human thought, and not, as is often the case, to regard it as most unprofitable drudgery. If Euclid is usually ill understood at first, it is because, with the usual quantity of preparation, one reading of the first six books is no more than is necessary to learn the terms and modes of speech, and to acquire an indefinite notion that there is something to be learnt, and a sort of aperçu of the leading facts. These we submit ought to be known beforehand, if the reasoning part is to be made a source of enjoyment. We receive with inore satisfaction the information that charcoal and the diamond, two well-known substances imagined to be distinct, are the same, than we should have done, had we only to regard the latter as a new product, arising from the crystallization of the former. PUBLIC INSTRUCTION. THE Condition of the human race may be considered with reference to three great divisions, which comprehend its whole existence: these are its physical, moral, and intellectual state. At no period in the known history of the world have we any records of the mass of a people possessing, in any degree adequate to procure happiness, a supply of their physical, moral, and intellectual wants. At present, the most numerous class of beings which compose the human race are a prey to many physical sufferings; and all classes of society in all countries, both high and low, are generally void, we will not say of moral notions, but of moral habits. The mode of instruction followed in schools, which generally has for its sole object the cultivation of the intellectual powers, is essentially defective and incomplete. And yet we see in all countries honourable and generous men, uniting to extend knowledge, instruction, and useful information throughout society; and in England we see enormous sums annually expended with the professed, and, we may fairly admit, the real object of diminishing human suffering, and improving human character. The intentions of these true friends of humanity are certainly beyond all praise, and words are wanting to express the thanks they deserve; but the best intentions may err, if not in the design, at least in the means of accomplishing it. Now we think that the way for the public teacher to fully accomplish his noble endeavours, and one day reap the delightful fruit of his labours and his zeal, and indeed the way to make all instruction, both domestic and public, efficacious, is to let intellectual be preceded by moral education, or at least to combine them. We are, indeed, firmly persuaded APRIL-JULY, 1833, E that moral education is the basis, the foundation, and the test, not only of every system of instruction, but of the whole. social edifice. It is then to moral education, so much neglected in these times, that we must direct the skill, the attention, and the capacity of every one who devotes himself to instruction. Instruction by itself is an instrument of which either a good or bad use may be made. That which is learned in elementary schools, and which consists in knowing how to read, write, and cipher, cannot exercise much influence on morals. In fact, we should be puzzled to understand how it would be possible to give a man regular habits and just moral sentiments, by merely teaching him to perform certain operations almost mechanical, such as reading and writing are. We can much easier imagine that even a superior kind of instruction, when purely intellectual, is likely to cause a multitude of social wants to spring up, which, if they are not satisfied, often incite to crime: for instruction multiplies the social relations; it is the soul of commerce and of industry; it also creates among individuals a thousand opportunities of fraud or bad faith, which do not often exist among a rude or ignorant population. We will admit that the cultivation of the intellect alone has some effect, in so far as it tends to make immediate impulse yield to reason, and tends also to form some habits of order and industry. But it is at least insufficient. In England*, in Francet, in Germany‡, in the United States§, where a few years ago the happy influence of knowledge upon moral improvement was so much extolled, it is already acknowledged that it is not enough to establish schools in order to stop the progress of demoralization, and that it is necessary to seek out speedily some remedy for the evils which disorder society. * Minutes of Evidence taken by the Committee appointed by the House of Commons to inquire into the State of Mendicity and Vagrancy in the Metropolis and its Neighbourhood. Ordered to be printed 11th July, 1815-1816. Mr. Hume's Speech in the House, 1st July, 1812. Mr. Brougham's Speech, 28th July, 1820. George Harrison on Education.-Edinburgh Review, Nov. 1810. Rapports de la Société Philanthropique de Paris. Plan de l'Education pour les Enfans, par M. de Laborde. Rapport de M. Barbé Marbois à la Société Royale des Grisons, 1815. Etablissement et Direction des Ecoles primaires gratuites d'Adultes, etc. Journal de la Société de la Morale Chrétienne, No. 73. Letters on the United States, by Cooper. Two magistrates of the Cour Royale of Paris, who have recently made a tour through the United States of North America, where, in the course of two years, they have collected a considerable number of documents of the greatest It is not our intention here to attempt a complete essay on education; all we propose is, to fix the basis of a good system of instruction, by putting moral education before all other kinds of education, as indeed it seems to us to claim the pre-eminence. It appears that three conditions are necessary to enable a man to decide as to any course of action, and at the same time to decide well. He requires something to excite him, such as wants, instincts, feelings or imagination; something to enlighten and direct him, as intelligence and reason; and lastly, he requires an inward strength which renders him capable of submitting instinct to intelligence, the exciting to the directing faculties, and hasty determination to mature reflection. Without imagination and instincts, a man would not act at all. Without reason and intelligence, he would be in danger of acting ill, and if he had not the power of subjugating passion to reason, of what use would reason be to him? It would be better for him to have been confined to the simple impulses of nature like the brutes, than to have received reason without the power of profiting by it. Reason, if he had not the power to follow its dictates, would only serve to embitter his life, and fill it with useless remorse. These three orders of faculties, feeling, understanding, and virtue, are not developed at the same time. At first, man only follows his imagination and his passions, and his early determinations are all instinctive. Afterwards his understanding is formed; but habit and inclination continue to make him follow the track of his first impressions, and he persists in doing evil a long time after he is convinced that he could do better. At length, but slowly, he learns to make his actions coincide in some measure with his understanding, and the impulses of passion with the dictates of knowledge. The development of his activity follows precisely the same track whatever direction it may take; that is to say, in his actions as in his conduct, in his relation to things as in his interest relative to the application of the penitentiary system and to criminal statistics, report that, in the state of New York, five hundred thousand children are educated in the public schools out of a population of two millions, and more than two hundred and forty thousand pounds are annually expended for this purpose. It would appear, that an enlightened population, which is not in want of any of the capital which agriculture, commerce, and manufacturing industry offer, ought to commit fewer crimes than a population which possesses the latter advantages, without having the same knowledge to turn them to a good account; yet we do not think that the diminution of crime in the north should be attributed solely to instruction. In Connecticut, where education is still more extended than in the state of New York, crimes are seen to multiply to a frightful extent, and if we cannot blame knowledge for this prodigious increase of crimes, we must at least confess that it has not yet the power to prevent them. relation to himself or his fellow creatures, he always begins by acting instinctively; then experience begins to warn, and observation to instruct him, and at last he learns to act consistently with his judgment; he does not suffer himself to be led so blindly by his feelings, and his actions are distinguished by less impulse and more reflection. Now, in what way can his activity become moral, and when may we say that it is so ? The adjective moral is evidently derived from the Latin mos, moris, moralis. Taking this word then according to its etymology, it would appear that we ought to apply it to every mode of action which has become a habit, custom, or practice, and that we should call an action moral when it is habitual, and the constant practice of the people or the individual who performs it. This, however, is not the case; for, in the first place, we do not call those actions which relate to labour moral, however regular and habitual they may be; we reserve this qualification for those which relate to conduct. We make a distinction between the morals of a people and their skill; and whatever name we may give to those habits which direct us in the conduct of life, we do not always say that these habits are moral. We only give the name of moral to those habits which govern us when they are worthy to govern us, to serve us for rules, and are proper to form our character or manners. It is universally acknowledged that there are moral habits or manners, and immoral habits or manners. Thus as long as our activity is confined to mere labour, as long as it is confined to industry, skill, dexterity, and intelligence, it can neither be moral nor immoral; whether our labours are governed by one unchanging routine, or are in a state of progressive amelioration, they are equally destitute of morality. We may say of an ingenious workman that he is clever, but that does not mean that he is moral; of an orator that he is eloquent; of a professor, that he is a man of talent, but we do not say that they are moral. Once more, we repeat, this qualification is only applicable to those of our actions which relate to the conduct of life. Again, with respect to conduct, we must observe, that it is not moral while a man's resolves are purely instinctive, and so long as he only follows the impulses of desire, passion, and feeling. Indeed, it is well known, that the best feelings may lead a man to do wrong. It is possible that love, friendship, and paternal tenderness may induce a man to commit bad actions: much more then would those feelings which are usually connected with terms of dispraise, such as self-love, hatred, anger, pride or avarice, if he gave himself up to their im |