29. 'NIL,' Holbeck.-We think you will find Clyde's Geography far the best-the larger edition, we mean. 30. SUBSCRIBER, Nottingham.-If a+b+c=0, show that 2(ab+be+ca)2=a'+b1+cʻ. (ab+be+ca)2=a2b2 +b2c2+c2a2 +2abc(a+b+c) Also 2ab+262c2 + 2c3aa — a* — b1 — cʻ (by ordinary formula or by multiplication; see a former number of this Query Column) =(a+b+c) (a+b−c) (a−b+c) (b+c−a)=0, Tod 32. To recommend a book on algebra to you, unless you name further particulars, is hard. Hamblin Smith (Rivingtons, price 3s.) is very profuse in his explanation of difficulties. hunter's smaller work (Macmillan, 2s. 6d.) will carry you as far, and in a more scientific, though not so easy fashion. Todhunter's larger work (Macmillan, 7s. 6d.) is by all means the best, if suited to your requirements. 33. ALGEBRAIST, Montrose.-(1) Factors for x2+4x — 1, and similar sums. Solve the quadratic in x-x+4x-1=0, x=. =-2±√5. Hence the factors are x+2−√5, and x+2+√5. In general, ax2+bx+c=a(x—∞) (x — ß), where (2) There are x students in one class, y in another, but attend both. Find an expression for the whole number of students. xy-2, as is surely evident. (3) The angles of a four-sided figure, taken in order, are as the numbers 1, 2, 3, 4. Prove that two opposite sides are parallel. Let x, 2x, 3x, 4x be the angles. Then, since sum of angles of a quadrilateral is equal to 4 right angles, x+2x+3x+4x=360, x=36. Hence the angles are 36°, 72°, 108°, 144°; and as 36°+144° =180° 72°+108°, it follows by Euc. 1. 28 that two opposite sides are parallel. The sides of a triangle are 6, 7, and 8. Find the segments into which the bisector of the mean angle divides the opposite side. Let the The mean angle is the angle opposite side 7. segments be x and 7-x, x being near to the angle formed by 6 and 7. Then by Euc. VI. 2, 6: 8::x:7-x; .. 6: 14 :: x 7, componendo; .'. x=3;.'. the segments are 3 and 4. 34. PUPIL TEACHER, Stratford.-We must refer you to any work on Trigonometry or Geodesy for the method of measuring mountains. 35. The publisher of Brook-Smith's Arithmetic is Macmillan & Co., price 4s. 6d. 36. Oranges are bought for 2s. 6d. a hundred; some are sold at 3s. 6d. per hundred, and the rest at 2s. 10ld. per hundred. The same profit is made as if they had all been sold at 3s. 1d. a hundred. Of 1000 oranges sold, how many fetch 3s. 6d.? Suppose x fetch 3s. 6d., then 1000-x fetch 2s. 10d., and 7x total receipts are shillings. But we are told 2 also that these receipts are the same as if all had sold at 3s. 1d. 7x 23(1000-x) Hence = 1000. . 25, 8 +23(1000—x) + 2 8 or 28.x+23000-23x=25000; 37. J. S., Dumfries.—Solve (1) x2+y=4. Substituting from (1) in (2) for y, (4−x2)2+x— -10=0; ..x-8x+x+6=0; ..x*—8x2+7+x-1=0; •'. (x2 — 7) (x2 -1)+x-1=0; .. x=1 or (x+1) (x2 —7)+1=0. (3) The value x=1 gives y=3, and this is the only solution which is both rational and integral. The cubic equation (3), which gives us others, may be dealt with thus:-Write it x+x2-7x-6=0. Since it consists of a set of positive terms followed by a set of negatives, it has only one real root (see Theory of Equations). Call the equation f(x)=0. Then ƒ(2)= — 8, ƒ(3)=9; hence the root lies between 2 and 3. Put x=2+: I +, and multiply by y The equation becomes 83 — 9 y2 — 73 — 1=0. If this expression be (y), 0(1)= — 9, and (2)=13. Hence there is a root between 1 and 2. Hence, finally, the only other value of x that is real is about 23, whence y=-23 about. Or the equation might be dealt with in the ordinary way for cubics. 38. Write again more definitely for instructions as to Science and Art Mathematical Examination, stating what proficiency you have attained, and what books you have read. We shall then be glad to advise you. BY ELIZA A. YOUMANS. Edited, with Notes and a Supplement, BY JOSEPH PAYNE, F.C.P., Author of 'Lectures on the Science and Art of Education,' etc. "This study, according to her just notions on the subject, is to be fundamentally based on the exercise of the pupil's own powers of observation. He is to see and examine the properties of plants and flowers at first hand, not merely to be informed of what others have seen and examined.'-Pall Mall Gazette. NEW AND CHEAPER EDITION. FIRST BOOK OF BOTANY. Besigned to Cultivate the Observing Powers of Children. BY ELIZA A. YOUMANS. With 300 Engravings, crown 8vo, cloth, price 2s. 6d. 'It is but rarely that a school-book appears which is at once so novel in plan, so successful in execution, and so suited to the general want as to command universal and unqualified approbation, but such has been the case with Miss Youman's "First Book of Botany." It has been everywhere welcomed as a timely and invaluable contribution to the improvement of primary education.'-Pall Mall Gazette. LONDON: C. KEGAN PAUL & CO., I PATERNOSTER SQUARE. (ESTABLISHED 1871), To supply Teachers, School Board Members, and School Every one should read our Circular and Extracts, which are sent Post free. Apply to the MANAGER, N.W. 'We have now had plenty of time to try the piano; and I can say to you how very much we are pleased with it, and how greatly I am obliged to you for procuring me so excellent an instrument so cheap.'-MATTHEW ARNOLD, Esq., H.M.I. 'You are at liberty to make any use of my wife's and my own opinionthat the piano is excellent both for tone and touch, and that it is a very cheap piano.'-Rev. T. W. SHARPE, M.A., H.M.I. for Training College for Schoolmasters. 15,000 similar Testimonials have been received from all parts of the country, from H.M. Inspectors, Assistant Inspectors, Presidents of N. U.E.T., Principals and Masters of Training Colleges, Teachers and School Board Clerks. THE ONLY ADDRESS IS 39 OSENEY CRESCENT, CAMDEN ROAD, LONDON, N.W. Now ready, Foolscap Folio, 524 pp., elegantly bound in Leather, antique, gilt back, covers and edges (an exact imitation of the Original Edition), price 21s., THE FABLES OF ESOP, AND OTHER EMINENT MYTHOLOGISTS. Reprinted in Old-Faced Type from the Celebrated Edition of 1669. BY SIR ROGER L'ESTRANGE, KT. CONTAINING THE MORAL AND REFLECTION TO EACH FABLE; THE LIFE OF ESOP; AND THE ORIGINAL FRONTISPIECE. The Elements of Plane and Spherical Trigonometry. With an Appendix The Elements of Physiography. For the use of Science Schools. By The World We Live In; or, First Lessons in Physical Geography. For The Earth's History; or, First Lessons in Geology. For the use of LONDON: W. H. ALLEN & CO., 13 WATERLOO PLACE. Algebraic Teaching Reformed. Algebraic Factors: How to Find them, and How to Use them. By W. T. KNIGHT, F. S. Sc., London, Science Teacher, formerly Mathematical Master at New College, Southsea, Author of 'Solutions to Science Papers in Mathematics,' etc. Every Teacher and Student should at once obtain a copy of this book, since its principles and methods, when fully grasped and adopted, must inevitably produce a complete revolution in the teaching and practice of Algebra. To persons preparing for Examinations it will be invaluable. 'No one can read these pages without being convinced that a vast improvement ought to be effected in many of the methods of performing the most ordinary operations of Algebra. More than this, no beginner can work through the examples given in the book, according to the plans laid down in it, without bringing to all future work a facility of handling and transforming Algebraical expressions formerly quite wanting.'-Sheerness Guardian. 'You may freely use my name in recommending the book. Send me 50 copies.-J. B. COLGROVE, M.A., Head Master, Grammar School, Loughboro'. 'I think it a pity that all teachers of Algebra do not know there is such a capital book to be had at a very moderate price.-W. J. MANTLE, F. R. G.S., Head Master, Diocesan School, Lincoln. Other remarks by practical teachers :-'It is all it professes to be.' Valuable and not costly.' 'That rara avis in this stereotyped age--something both new and good.' 'A real boon to all teachers of Algebra.' 'Your admirable book.' To be had ONLY from the Author, SPRING HILL HOUSE, TAVISTOCK, Price 2s., post free; four Copies as three. New Edition, crown 8vo, price 3s. 6d. ; post free, 38. 10d., PARTING WORDS TO BOYS LEAVING SCHOOL. BY MAURICE CHARLES HIME, M.A., LL.D., HEAD MASTER OF FOYLE COLLEGE, LONDONDERRY. Second Edition, price is. 6d.; post free, 1s. 8d., MORALITY: An Essay addressed to Young Men. By Author of Parting Words to Boys Leaving School,' etc. etc. The subject is dealt with in an exhaustive manner, the notes are varied and extensive, and, considering the difficulty and delicacy of the task, it has been executed well.'-Vide Oxford University Herald. Price One Shilling, A CHRONOLOGICAL & GEOGRAPHICAL TREE. Showing the Development of the British Power and the Affinity of the British Nation with the Ten Lost Tribes of Israel. LONDON: W. H. GUEST, 20 WARWICK LANE. GIFT AND PRIZE BOOKS. Small 8vo, cloth extra, 2s. 6d.; gilt edges, 3s. PHŒEBE'S FORTUNES. By Mrs. R. O'REILLY. With Illustrations. THE CHILDREN'S JOURNEY, and other Stories. By the Author of 'Voyage en Zigzag. With Illustrations by the Author. THE GIRLS OF THE SQUARE. By Mrs. R. O'REILLY. MODERN WONDERS OF THE WORLD; or, The New Sinbad. By WILLIAM GILBERT. With Illustrations by ARTHUR Hughes. TALES OF MANY LANDS. By M. FRASER-TYTLER. With Illustrations. SLYBOOTS, AND OTHER FARM-YARD CHRONICLES. BY BEATA FRANCIS. With many Illustrations. ROUGHING IT IN VAN DIEMEN'S LAND, Etc. By RICHARD ROWE. MARQUISE AND ROSETTE, Etc. By the Baroness MARTINEAU DES CHESNEY. GEIKIE'S PRIZE COPY-BOOKS. 'The competition in the copy-book department of school requirements seems to be even more intense than in that of reading-books. The ingenuity displayed in giving freshness and variety to the different sets that appear is wonderful. The set before us is not new. It has established a reputation; and as its merits become more widely known, it will certainly be more widely used. As regards ruling, heading, figuring, and lettering, we know of nothing superior.'-Educational News. 'We are much pleased with these copy-books.. ... Of the penmanship and its arrangement we can speak in the highest terms. The models are beautiful specimens of caligraphy, and the exercises included in the twenty-one numbers of the series leave nothing to be desired. No. 15 of the series, consisting of beautiful specimens of "text" and "small-hand" writing, is well adapted for pupil teachers, as a preparation for the annual tests in pen Price 2d. each. OF THE PRESS. manship. Good copy-books are by no means rare nowadays, but these new candidates for popular favour can justly be classed among those in the front rank.'-The Schoolmaster. 'This long series of writing copy-books commend themselves to notice for the author's elaborate devices for guiding the hand of the pupil in the earlier stages of his work, for the good quality of the styles of writing taught, for the interest and variety of the information sought to be impressed upon the memory in the subject-matter of many of the advanced exercises, for the thought bestowed upon the commercial copies in the introduction of letterwriting, book-keeping, account-keeping, financial transactions, and generally, for the range and exhaustive completeness of the series.'-School-Board Chronicle. STRAHAN & CO., LIMITED, 34 PATERNOSTER ROW. TUITION BY CORRESPONDENCE. BRANCHES.--Matriculation, Scholarship, Certificate, Science, Drawing, Diocesan, Civil Service, Musical (including Tonic Sol-fa), Medical, Legal, Shorthand, P.T. Papers, Special Subjects, etc. DETAILS will be sent upon application. No fee is in any case required until the specific qualification sought has been obtained. Address, Mr. JAMES JENNINGS, Tuition by Correspondence Office, Deptford, London. 'Method is the Secret of Success. THE NINETEENTH SESSION of the London University Matriculation Correspondence Class is in progress. Members may be enrolled now with advantage, as all subjects begin ab initio. Private Students may receive preparation in ALL, or ANY, of the following subjects:-Latin, Greek, French, English Language, Geography, Mathe. matics, Natural Philosophy, and Chemistry. (Hebrew and German, on special terms.) Any certificated Teacher of ordinary ability may undertake the course of studies issued by the Class with the full assurance of ultimate success. The fee is moderate, and payable either quarterly or upon passing the Examination. Pamphlets containing full particulars will be sent gratis and post free on addressing TUDOK ROGERS, F.R.S.L. HARGRAVE, CHESTER. Set by the London University Examiners. MATRICULATION QUESTIONS, JUNE 1880. WITH FULL ANSWERS. BY WALTER P. WORKMAN, First in Honours at the above Examination. Price 3s. 6d. The Hints, Notes, Books of Reference, and all information necessary to intending Candidates, which have been ap pended, render this little Manual the best possible guide to the Matriculation Examination. LONDON: JOSEPH HUGHES, Just published, 12mo, price 35., FOR THE USE OF SCHOOLS AND PRIVATE STUDY. Just published, price 3d., SENTENCES FOR ANALYSIS, FOR USE IN SCHOOLS. · SELECTED BY F. HEELEY, B.A., UNIVERSITY OF LONDON. W. & R. CHAMBERS, LONDON AND EDINBURGH. TO PUBLISHERS, AUTHORS, ETC. MESSRS. AUSTING & SONS, BOOKBINDERS, JOHNSON'S COURT, FLEET STREET, LONDON, E.C., WOULD respectfully intimate that they Estimate or Contract for every description of WHOLESALE CLOTH BINDING (both Plain and Fancy), and that they are prepared to execute Large or Small Orders without delay, at the Lowest Trade Prices. Having been established nearly half a century, they can give very satisfactory references if required. A Large Assortment of Ornamental Desigr.s in Stock for Covers. H Ponder. THIS IS THE RESULT. OWARD'S MIRACULOUS FORMULA still retains its title of NEVER FAILING, and can be used with con For Senior Pupil Teachers. Price 3d. DICKINSON'S HISTORY OF ENGLISH LANGUAGE. With Saxon, Latin, and Greek Prefixes and Suffixes, and Hints on Composition and Letter-Writing. Cheap School Edition, well bound, price Is.. fidence (after everything else has failed) as a Producer of Whiskers, MRS. GOTHARD'S LESSONS ON COOKERY. Moustachios, Eye-brows, a Curer of Scanty Partings, Hair Free from observation, and post paid, 15 stamps. NOTE NAME AND ADDRESS : DANIEL HOWARD, 331 HIGH HOLBORN, GRAY'S INN, LONDON, W.C. Price 2s. 6d., extra cloth, gilt lettered. The Leeds Mercury says-'An excellent little volume.' The Schoolmaster says 'The book is well got up, strongly bound, and contains a great variety of recipes.' The Scholastic World says-'This book can be highly recommended to all who are interested in this important subject.' LONDON: JOSEPH HUGHES, Pilgrim Street, Ludgate Hill, E.C. Cox's are THE ONLY COPY BOOKS PUBLISHED which afford in Standards II. and III. (the most difficult to pass) a thorough training in working an Arithmetical Examination Paper. BOLD, ROUND WRITING. COX'S Copy Books ARE THE Children's Favourites. COX'S Copy Books SAVE THE Teacher's Time. NOW READY, IN 15 PARTS, PRICE 2d. EACH. A gross (assorted) of these popular Copy Books will be sent CARRIAGE PAID to any address in England, London: JOSEPH HUGHES, Pilgrim Street, Ludgate Hill, E.C. |