RESULT.1 litre of unknown acid solution contains 2'43 grms. hydric sulphate. The students are now capable of appreciating the general laws of chemical action, and it is at this point that they are first introduced. Dalton's theory is explained, the law of Avogadro stated, and the use of formulæ adopted. In this latter connection, making use of the density of oxygen and sulphur dioxide, we deduce a formula for sulphur dioxide, and it is always gratifying to notice. the pleasure that boys find in applying their own experimental data in support of such an abstract consideration. We now return once more to sulphur, and endeavour to ascertain whether it will form any compounds with metals. The sulphides of copper and iron are prepared and their composition is determined. Their behaviour with dilute acids is observed, and the production of sulphuretted hydrogen is noted. The gas is prepared in large quantities, its properties are examined and its density determined. Adopting the same apparatus as in the case of sulphur dioxide, but using metallic In concluding this course the opportunity of exhibiting the behaviour of sulphuretted hydrogen with some metallic salts is taken, and an endeavour is made to show how useful it may prove in separating these metals from others which do not react in a similar manner. NOTES ON ARITHMETICAL CALCULATIONS.1 F By JOHN ORCHARD, M.A.(Oxon.) III. RESOLUTION INTO PRIME FACTORS. ACILITY in the resolution of numbers into prime factors is essential to the apt arithmetician. So many short processes depend on the expression of numbers as the product of primesor, at any rate, of simple factors-that to be slow in this one process is to spoil all. The fundamental method is to exhaust the possibilities of a number as a factor-container by a process of continuous division by primes. It is generally advisable to do this by commencing with the lowest and proceeding to higher primes in ascending order of magnitude, dividing the number by each prime as often as possible before passing to the next. Example: Find the prime factors of 194040. (1) If, at a glance, it could have been seen that the number 194040 was divisible by 8( 23), it would have been unnecessary to have divided by 2 three times; a single division by 8 would have been sufficient. Similarly one division by 9 could have been substituted for the double division by 3. (2) It would evidently be a great convenience to be able to tell at sight if any given prime were a factor of a number without going through the test of dividing the number by the prime. Unfortunately it is not possible to say at a glance in every case whether a prime or power of a prime is a factor of a number. But it is possible in a few cases, and in others certain criteria exist which simplify the matter somewhat. The follow 1 Concluded from p. 52. ing rules and criteria are useful in determining the divisibility of a number by the following primes and powers of primes : 2 The last digit of the number is even. 52: The last two digits are divisible by 25; i.e., Example: 1/234/324. The critical number in this case is 324+1-234-91=7×13 so that, 1234324 is divisible by both 7 and 13, but not by II. II: Another method: Find the sum of every alternate digit, beginning with the units digit, and also the sum of every alternate digit beginning with the tens digit. If the difference between these sums is o or a multiple of eleven, the number is divisible by eleven. 17: Divide the digits of the number into groups of two, starting from the units end. Then if first group 2 (second group) +22 (third group) - 2 (fourth group) - [(fifth group) 2 (sixth group) + 22 (seventh group)-2 (eighth group)]+ &c., is divisible by 17, the number is similarly divisible. Example: Test if 17 is a factor of 1,45,35,40,67,00,08,31. Critical number is 31 -2x8+0-8x67-140-2×35+4×45-8 ×1}=-663 and 63−2×6=51=17×3 .. 663 is a multiple of 17 and so is the original number. 19 Divide the digits into groups of two as for 17; the criterion is that: (first group) + 5 (second group) + 5a (third group) + &c., should be divisible by 19. 37 Divide the digits into groups of three as above, then the sum of these groups must be divisible by 37. IOI: Same as for 37 with groups of four. 73 As for 7, 11 and 13, but the digits are to be 137 divided into groups of four instead of three. Criteria can be found for any prime, but in cases other than those mentioned they are generally more interesting than useful. Example: (1) Factorise 2,308,600. (a) Last three digits are divisible by 8 and by 25 .. 23 and 52 are factors. (c) 543-11=532=7×76 .. 7 is a factor, but 11 and 13 are not. (d) Test for 17 : 7/11,543 1,649 49-2X 16=17.. 17 is a factor. 97 =a prime number Example: (2) What primes less than 20 are factors of 3,396,432? Usual tests for 2 and 3 satisfy, but 5 is not a factor. 432+3-396=39=3×13 .. 13 is a factor, but 7 and II are not. To test for 17. 32-2×64+4×39 −8 × 3=188-152=36. .. 17 is not a factor of the number, as it is not a factor of 36. To test for 19. Critical number is 32 +5 × 64+52×39+53×3 =32+320+975 +375 (use the shortened forms for multiplication by 5, 52 and 53) = 1702. Apply the process again to 1702. 2+5×17=87, which is not a multiple of 19 .. 19 is not a factor of the number. 2, 3 and 13 are the only prime factors less than 20 of the given number. Example: (3) Numbers like the following, in which the number of thousands equals the number contained in the last three digits, are always divisible by 7, 11 and 13; for 7 × 11 × 13 SQUARE AND CUBE ROOT. Instead of using the ordinary methods of extracting square and cube roots, whenever possible resolve into prime factors by the above methods. The square and cube roots are then immediately written down by dividing the index of each prime by two or three as the case may be. Example: Find the square root of 9922500. 9922500=10 × 99225 = 102 × 52 × 3969 = 2a × 5a × 32 × 441 = 2a × 51× 31× 7′′ The square root is 2 x 52 x 32 × 7=3150. COMPARISON OF VULGAR FRACTIONS. There are three ways in which the magnitude of vulgar fractions may be compared. (1) The ordinary method, viz., to reduce all the fractions to a common denominator, which is the L.C.M. of the denominators of the fractions. Example: Which is the greatest and which the least of the fractions,,? The fractions are 154, 155, 19. Therefore and are the greatest and least fractions respectively. (2) If the denominators are convenient divisors, convert the fractions into decimal fractions to as many places as is necessary. In the last example, for instance, the three fractions are ·666.... ·714 . . . ., •727 . . . ., and these decimals are easily comparable. (3) If, on the other hand, the numerators are more convenient divisors than the denominators, convert the fractions into reciprocals of decimal fractions. plier, say 76 089762. Evidently the last of the digits in the multiplicand which can affect the third place of decimals is the 2, for it represents 100000, and the maximum value of any digit which multiplies it is 7X 10, giving a product of, or 0014. It is, however, more convenient to consider the 3, which represents 10,000ths, and the 7, which represents tens, as producing a product of 1,000ths, which is the denomination of the third place of decimals. The effect of the 2 can be allowed for by an addition. Proceed as follows: Place the digit of maximum value in the multiplier under that digit in the multiplicand, multiplication by which will affect one place beyond the last decimal place required, and write the other digits of the multiplier in reverse order as in the example below. When the multiplication by 7 has been completed, strike out the 7 and 3, and proceed to multiply the 4 by 6, carrying from the product of 6 and 3. The result of multiplying thousandths by units. corresponds to the multiplication of ten-thousandths by tens, so that the 4 will be in the third decimal place. Then strike out the 6 and 4, and o and 9, and multiply the 8 by 8 and so on. 875 of 1000.. 0875 = 10 80 In the same way a decimal, part of which recurs, may be easily converted to a vulgar fraction without going through the process usually employed, provided that this recurring part, when written without non-recurring figures, represents a simple fraction. Thus : The 6 in the fifth decimal place of the multiplier only affects the result as regards the 5 in the third. decimal place, and the 2 in the sixth place does not affect it at all. Example: Find correct to the 8th decimal place the product of 002345678 and '0876532. *002345678 235678 *000187654 16420 DANS les premières opérations de l'esprit, que les sens soient toujours ses guides: point d'autre livre que le monde, point d'autre instruction que les faits. L'enfant qui lit ne pense pas, il ne fait que lire; il ne s'instruit pas, il apprend des mots. Rendez votre élève attentif aux phénomènes de la nature, bientôt vous le rendrez curieux; mais pour nourrir sa curiosité, ne vous pressez jamais de la satisfaire. Mettez les questions à sa portée, et laissez-les lui résoudre. Qu'il ne sache rien parce que vous le lui avez dit, parce qu'il l'a compris lui-même; qu'il n'apprenne pas la science, qu'il l'invente. Si jamais vous substituez dans son esprit l'autorité à la raison, il ne raisonnera plus; il ne sera plus que le jouet de l'opinion des autres.Rousseau. THE LEAVING CERTIFICATES OF THE were due to the very popularity of the examinations and to the domination they had acquired over the whole field of higher education. With teachers, school managers, and the general HE latest circular of the Scotch Education public, education was becoming more and more Department, embodying important modifica-identified with leaving certificate results. tions in the existing method of granting Leaving Certificates, affords a suitable opportunity of reviewing the history of the scheme and of estimating its influence upon the higher education of the country. The gradual evolution of an efficient system of secondary education which it illustrates, and in which it has been a most important factor, has lessons for England which she may profitably study. The Education (Scotland) Act of 1872, unlike the similar measure for England, conferred upon the Department the control of secondary as well as of primary education. But the onerous task of building up an adequate elementary system naturally tasked for many years all the energies of the Department. Secondary education was left uncontrolled and unguided, and every school was a law unto itself in organisation, curriculum, methods and standard. The best class of schools, with a strong university connection, still maintained a high level of attainments, but the general standard was very low. The institution of the Leaving Certificates in 1888, on the direct initiative of Sir Henry Craik, effectually rescued such schools from the chaos into which they were rapidly drifting. The scheme set up a standard of efficiency at which all could aim, and yet did nothing to crush the free development of individual. schools. The impetus these certificates gave to higher education cannot be over-estimated. better proof of this could be given than the fact that every recognised secondary school presents pupils for these examinations, and that the total number of candidates last year amounted to 17,000. The recognition of the certificates by the Universities of England and Scotland is a sufficient guarantee of the high standard exacted. No The success or failure of a school was determined by the number of certificates obtained, and pupils of eleven and twelve years of age have been presented at the examinations in order to swell the number of certificates. The possibility of a "Leaving" certificate of a secondary school being obtained at the age of eleven or twelve carries with it at once the condemnation of the principle on which the certificate was based. In addition, the action of some County Committees, in allocating their grants according to the number of certificates gained, threatened to introduce into higher education the vicious principle of payment by result which has justly been discarded in the elementary system. For these reasons the radical changes announced in the new circular will be generally welcomed. Its leading features are: (1) Certificates will no longer be issued for single subjects, but for groups of subjects. (2) Certificates will be of two grades, the Leaving Certificate proper, and the Intermediate Certificate. (3) Candidates for the Leaving Certificate proper must be seventeeen years of age, have attended for four years at a recognised secondaryschool, and pass in four subjects on the higher grade standard, or in three subjects on the higher standard, and two on the lower. English and Mathematics are compulsory subjects for all, and Latin for all save Science students. (4) Candidates for the Intermediate Certificate must be fifteen years of age, have attended two years, at least, at a recognised secondary-school, and pass in four subjects, of which one must be on the higher-grade standard. English and Arithmetic are compulsory subjects. received a course of instruction of adequate range and quality, and to be proficient in those elements of the curriculum which do not admit of being fully tested by written examinations. (5) The Department will only issue certificates to those who, in addition to satisfying the above These certificates were issued in three grades-conditions, are certified by the Inspector to have Lower, Higher, and Honours-for each subject in the curriculum of a secondary school. The issue of certificates for passes in isolated subjects was a radically wrong policy, and the principle of calling all such "Leaving" certificates-whether for passes in the lower grade of, say, English and Arithmetic, or in the Honours grade of Latin and Mathematics -was unjustifiable and misleading. In extenuation of the action of the Department, it should be recognised that an examination of any kind in secondary education was an innovation, and was regarded with suspicion, and the Department probably followed the line of least resistance. For many years the new certificates served admirably their intended function of stimulating the development of higher education, and of estimating to some extent its progress. But students of educational policy had noted growing defects in the system which seemed likely to impair its usefulness. Some of them, as already indicated, were inherent in the scheme, but the most serious The Group Certificate. The outstanding feature in the new conditions emphasises the essential unity of all the subjects which go to make up that general education which it is the true function of a school to give. The single-subject certificate, on the other hand, supported the fallacy that education could be divided up into water-tight compartments with no relation to one another. It is not intended, in the meantime, to insist on all the subjects in each group being passed in one year, as the Department will issue a document to each successful candidate certifying the subject or grade in which he has passed towards obtaining the group certificate. It is to be hoped that this is only a temporary expedient to bridge the old and new systems, as its retention would perpetuate the evils of the old scheme and lead to specialisation in certain subjects every year with a consequent neglect of others. Compulsory Latin. But admirable as is the principle of a group certificate, its value depends on its application. Considering the many-sidedness of secondary education, justice to all demands that the grouping of subjects should be sufficiently varied to suit every type of school. In the Intermediate Certificate this is the case, and it promises to be a highly popular and useful factor in school life. But the conditions governing the Leaving Certificate proper are so reactionary in tendency, that one is at a loss to understand how they have come to be issued by a Department which has shown itself so anxious to build up an educational system suited to modern needs. The insertion of Latin and Mathematics as compulsory subjects renders the certificate a purely academic one. Indeed, it makes it merely another name for the university preliminary examination in Arts or Science. As only a small fraction of secondary pupils look forward to a university career, the great majority of pupils are to be sacrificed to the fetish of a classical education. It is in no wise the purpose of this article to belittle the value and importance of the classics as training instruments. But it is surely too late in the day to have to enter a plea for the impartial treatment of modern languages. The teaching of modern languages in this country has now reached such a stage that a training can be given in them, which is as effective and useful as any other that can be given within the same time. Lord Balfour recently said that, while the classics at their best were the highest means of culture known, yet the effective use of even one modern language was of much more value than the fragmentary knowledge of a classical language, which was all the great majority of pupils ever attained. Yet it is this fragmentary knowledge that the new circular seeks to impose on all pupils. The insistence upon this condition will have a disastrous effect upon the commercial schools that the Department has done so much to encourage. Such schools must either make the lower certificate the goal of their efforts, or remodel their curriculum and take up Latin instead of German. The better schools will adopt the latter alternative, as no higher school can long maintain its reputation when its full curriculum can only win a lower grade of Leaving Certificate. In response to the representation of educational bodies throughout the country, the Department have agreed not to enforce the condition as to compulsory Latin till 1904. But the strong op. position with which the clause has been received should secure the entire abandonment of the regulation. So strong is the feeling in certain circles that the retention of the condition would eventually lead to the withdrawal of many schools from the whole system of Leaving Certificate Examinations. Such a result in the interests of schools and education would be very undesirable. The Age Limit. Compared with the pupils who leave the English public schools, the Scottish youth certainly leave school very early. In order to prolong the school life, and at the same time to remove the temptation to overpressure of the pupils, the Department have fixed seventeen years as the minimum age for obtaining the Leaving Certificate proper. This will mean the addition of at least one year to the present average age of leaving school. Teachers are in general sympathy with the aims of the Department in this respect, but it remains to be seen whether the public appreciation of the value of the certificate will be sufficiently high to secure this additional year of school life. If the Education Department could attach to their certificates some of the exemptions and privileges of the Abiturienten Examen of German secondary schools, the success of the attempt would be assured. Failing that, the commercial community can infinitely enhance the value of the certificate by making its possession the necessary passport into their employment. Sir Henry Craik, in last year's report on higher education, emphasises this view: "The educational machinery of the country can never have a fair chance until merchants as a body set their faces against the practice of putting boys into business at thirteen or fourteen, and until in their selection of apprentices they give preference and reasonable encouragement to those who can produce evidence of having profited by their school training." To ensure the hearty support of this class, upon whose attitude the whole success of the new departure depends, it is essential that modern. languages should have absolute equality with the classics in qualifying for the Leaving Certificate Examination. If this concession is granted, the new scheme of the department will not only deserve success, but will do much to command it. THE CRY OF PREMATURE SPECIALISATION. By H. MACAN, M.A. ORD ROSEBERY, in one of his recent Leloquent disquisitons on things in general, deplored the extent to which the people of this country are being led away by catchwords and phrases, and so refuse to think out the problems of the day for themselves without the aid of the leader writers in the newspapers. A study of the educational controversies of the last ten years on the work and limits of the spheres of technical and secondary education respectively has led me to the conclusion that the phrase "premature specialisation" is in this category of catchwords, and so has been greatly misunderstood and, I fear, often intentionally abused. The origin of the phrase lies with the firstgrade secondary school, which prepares its pupils for a university career and the learned professions. When some thirty years ago the subjects of science and modern languages first began to invade the sacred portals dedicated to classics and at a pinch to mathematics, the advocates of "modernisation" as usual pressed their claims with undue ardour. Just as under the old régime the classical boy was |