indication of a division to be performed, of which the quotient will be a decimal; and, conversely, the decimal will be shown to be capable of being represented in the forin of a vulgar fraction, leaving for its denominator a power of ten. The pupil will consequently regard a vulgar fraction as only a particular case of division; and having been previously accustomed to simplify his divisions by getting rid of common factors, he will naturally simplify his fractions in the same manner. Having already learned that equality of ratio constitutes proportion, he will easily understand that, if two fractions give the same decimal for a quotient, their numerators and denominators are proportionals. The compound rules will also afford many opportunities of illustrating vulgar fractions, by treating shillings, pence, and farthings, as fractions of a pound. This forms a good exercise in mental arithmetic; for example : How many pence are there in a pound ? How many twopences ? How many threepences ? &c. Then-What part of a pound is a penny? What part are two pence? What part are three pence ? &c. Children can be taught to answer such questions in a very short time, and they derive a much more distinct idea of fractions from such exercises than from any other, as the value of money enters into their every-day thoughts and occupations. Regard for your space has deterred me from developing these Notes and Queries, but your readers are mostly well able to fill up the outline which I have sketched. The great advantage of the method which I. venture to propose over existing ones is, that the intellect of the pupil is trained from the very beginning for the comprehension of the higher rules. The present method does the very reverse. Numeration and notation are taught only upon whole numbers, and not a trace of the existence of decimals appears till the chapter specially devoted to that subject. The consequence of this is to produce such an intimate association between the ideas of numeration, and notation, and integers, that a considerable effort is required to realise the idea of the numeration and notation of decimals. The mind is trained to bend in one direction in the first part of arithmetic, and then suddenly bent in the opposite direction. This occasions a considerable resistance in all minds, and a total incapacity of change in many. This is, I am convinced, the reason why a decimal is something recondite and mysterious to many even highly-cultivated minds. If they had been from the very first accustomed to see figures on the right of the unit figure as well as on the left, and to numerate and note both indiscriminately, the special chapter on Decimals would only be the development of a familiar idea. Similarly, the chapters on Proportion and Vulgar Fractions would lose all their strangeness. They would be simply offshoots of Division. Every person who has reflected on the laws of the human understanding must be aware of the great advantage of giving one uniform direction to the scholar's mind, and not creating associations of ideas in his mind which must afterwards be destroyed. I shall only add that I have tried these methods of teaching in schools of various degrees of proficiency, and have always found that the analogy of decimals and integers, and of fractions, and proportion, and division, were quickly apprehended by the children, and appeared to strike them as a new light. Yours, .. H, G. B. TOPOGRAPHICAL NOMENCLATURE. CELTIC. I. Traces of Names :1. Celts (Celtæ, Keltai, Taláral), Galatian, Gaul, Gael, Caledonia, and Calet-es (a tribe of Belgic Gaul). 2. Cymry, Cimbri (Kéubpoi), Cumberland, Cambria, Cimmerii, (Kepuépiol) and Crimea. 3. Erin, Er-se, and Ire-land. Compare this with the word Arii (the name of a German tribe who lived near the Sudetan Mountains). “ The name of the Arii,” says Dr. W. Smith, “ appears to contain the same root which we find in the names of many nations of the IndoGermanic family; and it is not improbable that all the different branches of the Indo-Germanic race may have originally been called by this name.” According to Herodotus (vii. 61, 62), the Medes were originally called Arii, and the Persians Artæi. These names are identical with the Sanscrit word Arya (which means 'honourable,' entitled to respect), by which, in the ancient writings of the Hindus, the followers of the Brahminical law are designated. (See Rosen, in “ Quarterly Journal of Education,” vol. ix. p. 336.) India proper is called, in the most ancient Sanscrit works, Arya-varta (* Holy-land'). The same name was retained in the provinces of Aria and Ariana (called in the Zend language, Aíryáne), whence the modern Persian name, Iran, is derived. In the ancient Persian traditions, the Arii are celebrated as the most generous and heroic nation of the primitive ages, just as the Asae (a softened form of Arii ?) occupy the most distinguished place in the northern mythology. (See F. V. Schlegel's “ Philosophy of History.") 4. Alben, Albion (Alb-inn, white heights), and Albania. 7. Belgoe, Fir-bolgs (men of Belgæ) and Belgium. 8. Bibroci, Berkshire. 23. Elusates, Euse. 9. Cantii, Kent. 24. Helvetii, Helvetia. 10. Catini, Caithness. 25. Lemovices, Limoges. 11. Cassi, Caishow. 26. Lepontinii, Lepontine, Alps. 12. Damnonii, Devon. 27. Lexovii, Lizieux. 13. Durotriges, Dorset. 28. Lingones, Langres. 14. Iceni, Ixworth. 29. Namnetes, Nantes. 15. Selgovce, Solway. 30. Norici, Noric Alps. 16. Ambiani, Amiens. 31. Parisii, Paris. 17. Arverni, Auvergne. 32. Rhedones, Rennes. 18. Boii, Bohemia, Bavaria. 33. Rheini, Rheims. 19. Bituriges, Bourges. 34. Scordisci, Mount Scordus. 20. Cadurci, Cahors, Quercy. 35. Tigurines, Zurich. 21. Caturiges, Cottian, Chorges. 36. Turones, Tours, Rouen. 22. Centrones, Centron, Courtray. 37. Veneti, Vienne, Vannes. II. Traces of Celtic Names on the Continent of Europe :1. Ab, 'water ;' old Gaulish, hab (ab-us, the Humber), cognate with ab (Sanscrit), water; imber (Lat.), a rain-storm; VOCABULARY FOR THE EXPLANATION OF CELTIC NAMES. Aber, confluence of waters, a port. (Br.) Lin, linn, llyn, lyun, a deep pool, lake. (Cel.) (To be continued.) EviL OF NEGLECTING THE DULL CHILDREN IN SCHOOLS.--One practical evil of the most serious kind results from the mistake to which I am referring. You are tempted to give to the intelligent boys of your school more than their fair share of your time and attention, while the dull and slow are neglected, as not likely to contribute to the credit of the school and its master. * * * Yet the dull boy, who lags at the bottom of his class, may possess germs of the highest promise, languishing for want of patient and intelligent culture. * The pains which you honestly bestow on the dull boys of your school, though they may fail to obtain praise of men (and yet a wise inspector will not fail to notice them), will surely be appreciated by the only infallible Judge of every man's work.--Lecture on Moral Influence, by S. A. Pears, B.D. |