3 Units stage. Tens stage. Hundreds stage. 87 87 4698 8 Tens. Tens of from 9. Hundreds. 2 6 2 256 (i.) It is not necessary to use tongue as well as The units with the negative sign over them are, brain. Do not, therefore, say nine and eight are of course, subtracted when finding the sum of the seventeen, and five are twenty-two, and four are column in which they are situate. twenty-six, and seven are thirty-three, but go through the same process mentally, as follows :Nine, seventeen, twenty-two, twenty-six, thirty MultiPLICATION. three. To find the product of two numbers of two or three (ii.) If any number occurs more than once in a digits in one line. Example: multiply 87 by 54. column make use of multiplication. Thus, in the second column, at a glance there are three eights and two other numbers, making together a fourth 8 7 8 7 eight. The sum of the column is therefore 32, 1 making 35 with the three carried from the first 54 5 4 column. (iii.) Group sets of numbers whose sum is ten ; these are easily recognised and the addition of ten 98 4698 [2+32+35=69) [6+40= 46) to a number is simplicity itself. Thus, in the third column, carrying three the process would run as First, to obtain the units, multiply units by follows :3, 13 (8 + 2), 23 (7 + 3). units. 4x7=28. Put down the 8 and carry the When adding two numbers one of which is a 2 to the tens stage. The other tens are obtained little less than a power of ten, the process is by multiplying the units in one number by the tens simplified as follows:-.g., 893 + 997 = 890 + in the other. Thus 2+(4x8)+( 57)=69. Put 1000 = 1890, 1.c., replace the number in question down the 9 and carry the 6 to the hundreds which by the power of ten, and make a balancing re- are obtained by multiplying the tens digits toduction in the other number. gether. 6+(5 x 8) = 46. After a little practice In the subtraction of decimals of many places the whole process can be done very rapidly. from whole numbers it is advisable to remember Example: multiply 256 by 394. that the process of carrying one and then subtracting from ten is equivalent to a subtraction Units. Thousands. thousands. 256 5 2 5 6 5 6 a ↑ 个 Thus subtract 8.4379246 from 9 KI 394 9* 3 9 6 4 100864 -5620754 (2+20+54 [7+8+45 17+18+15 [4+6= 10] = 76] +18=78] = 40) After saying six from ten leaves four, instead of carrying one to the four and subtracting from ten, Here the process is similar. It is well to bear subtract four from nine and so on. The result can in mind the diagrams formed by the arrows. They then be written from left to right. The process are symmetrical, and are easily remembered. The units and tens are obtained in the same way will be more easily seen if for 9 we write 8:9999999; as in the first example, but hundreds can be the ti over any digit indicating that I should obtained both by multiplying tens digits together be added to that digit. and by multiplying the hundreds digits in one Thus 8.9999999 number by the units digits in the other. Thousands are obtained by the product of the tens digits with 8:4 3 7 9 2 4 6 the hundreds digits, and tens of thousands by the 56 20 7 54 product of the two hundreds digits. The process can of course be continued to higher numbers, but It is well to remember this when dealing with it becomes rather complicated in such cases. the logarithms of numbers less than unity where, Certain cases of multiplication by numbers of though the logarithm is negative the mantissa is special form call for notice. to be positive, the characteristic being a negative (1) To multiply by a power of ten add as many whole number. Thus : – 4879632 = 7:5120368 noughts to the multiplicand as there are noughts in the and vice versa. Similarly negative whole numbers multiplier. may be transformed into positive with a com- (2) Since 530 25 100°, to pensating negative power of ten : e.g., -78963 = multiply by 5, 25 and 125 add one, two or three noughts, 121037. This is sometimes useful when finding as the case may be, and divide by 2, 4 or 8. the sum of a number of positive and negative (3) If the multiplier is a little less than a power numbers. All the negative numbers are trans- of ten, e.g., 995 = 1000 5, proceed as follows: formed by the above method, and the sum is 8976 x 995=8976x (1000 - 5)=8976000 - |(89760)' reduced to an addition. =8976000 c.g. : 5469-87+4278-8034+768-7 49880 = 5469+113+.4278+11966+768+13 8926120 =2387 9 4 3 4 I “A GRAMMAR OF THE GERMAN LANGUAGE,” by A MODERN LANGUAGE MASTER'S H. C. G. Brandt (Boston, Allyn & Bacon, about LIBRARY OF FRENCH AND GERMAN. 45. 6d.), and study along with it Heyse's“ Deutsche Grammatik oder Lehrbuch der deutschen Sprache,” By Orro SIEPMANN. 25th edition, by Dr. Otto Lyon (Leipzig, 1893, Head of the Modern-Language Department at Clifton College. about 4s.), or "Handbuch der deutschen Sprache, von Karl F. Becker, with ed. (Prag, 1876, about German. 55.), which should be supplemented by a little work entitled “SATZBAU UND WORTFOLGE IN DER N a previous article which appeared in this Magazine (March, 1901), I have laid stress on DEUTSCHEN SPRACHE," von Prof. Dr. Daniel the necessity that a young modern-language scientific work on the same subject is “ Der SANDERS (Berlin, 1883, about 2s. 6d.). A more master who wishes to equip himself for his difficult task must, in the first place, try to acquire a good deutsche Satzbau," dargestellt von Hermann pronunciation. It is sometimes thought that it is Wunderlich (Stuttgart, 1892, 252 pages), of which much easier for an Englishman to learn to pro a very much enlarged second edition is appearing nounce German correctly than French; and, in in two vols. ; vol. i., gs. unbound. consequence, students and teachers do not take the The most complete work on modern German same amount of trouble in trying to master the grammar I know is the “ Neuhochdeutsche Gram matik mit Berücksichtigung der historischen Entsounds of German, and are satisfied with a lower standard of correctness. This is a grave mistake. wickelung der deutschen Sprache,” von Friedrich Even if it were true that German sounds come more Blatz, 3rd edition (Karlsruhe, 1895, vol. i., naturally to the English organs of speech, there 856 pp., 1os. ; vol. ii., 1,314 pp., 155.). With can be no doubt that the difficulties of a really these works, a taste for the historical treatment good German pronunciation are so great as to of the German language is sure to develop, and BEHAGHEL, Die DEUTSCHE make the study of phonetics indispensable. It is a SPRACHE (Leipzig, delusion and a snare to believe that a short stay in 1887, IS.) may be recommended as a guide Of this there exists Germany will be sufficient to set things right in for beginners. a good matters of pronunciation. My advice to young English translation by Trechmann (London, Macmodern-language masters is to study carefully one millan). of the books on general phonetics, and of French and German in particular, which I have mentioned Wasserzieher's booklet “Aus dem Leben der deutschen in my last article, and add to this Vietor's excellent Sprache” (Leipzig, 64 pp., stitched, for 4d.) is very small, but little treatise, entitled “GERMAN PRONUNCIATION, the student who buys it certainly gets his money's worth. PRACTICE, AND THEORY," 2nd edition (Leipzig, Before mentioning any of the larger historical works, I must 1899, 2s.), and Georg Hempl's “ German Ortho. not omit two books which are excellent food for those who like graphy and Phonology" (Strassburg, Trübner). to leave the beaten track and to think for themselves ; they are After these, Bremer's “ Deutsche Phonetik Kern's “ Deutsche Satzlehre " (Berlin, 1888, 184 pp.) and Ries' (Leipzig, 1893, 5s.) will be found interesting and “Was ist Syntax”? (Marburg, 1894, 163 pp.). A study of stimulating. It goes without saying that a few these cannot fail to arouse the teacher's argumentative faculties, hours' help from a native with a keen ear for and he will be quite prepared to enjoy some other books of sound differences and a good knowledge of a critical description written by men who are dissatisfied with phonetics will be found invaluable at the initial much that is used in present-day German, and aim at reform. stages; but it may be pointed out that a German “ Allerhand Sprachdummheiten-Kleine deutsche Grammatik with an untrained ear and no acquaintance of des Zweifelhaften, des Falschen und Hässlichen-Ein Hilfsbuch phonetics will in all probability be found of little fiir alle, die sich öffentlich der deutschen Sprache bedienen" von use; in fact, it is not impossible that his observa Dr. Gustav Wustmann (Leipzig, 1891, 320 pp., 2s.)—is a tions may prove positively misleading, for as a rule worthy lead off in this group of interesting publications. Erbe's the statements of such natives about their own pro · Randbemerkungen” (Stuttgart, 1892, 50 pp., 6d.), Minor's nunciation and manner of producing speech “Allerhand Sprachgrobheiten " (10d.) and a host of other “ Streitschriften" followed in its train. Another book, sounds are most untrustworthy, and when they proceed to correct the pronunciation of a foreigner Sprachleben und Sprachschäden,” by Matthias, though not their criticism and advice are as a rule of a most directly connected with the polemic raging round Wustmann's excellent little work, should be read by every teacher of deplorable kind. If a thoroughly competent German, and in conjunction with it Lehmann's Sprachliche native cannot be found, the student should try to Sünden der Gegenwart” (Braunschweig, 1882, 175 pp., obtain the help of an Englishman who knows about 2s.), Heintze's “Gut Deutsch” (Berlin, 1898, 200 pp., German and phonetics. Along with the study of about 2s.), Brunner's “Schlecht Deutsch” (Leipzig, 1895, phonetics and its practical application, which 199 pp., about 25.), and Schroeder's “Vom papiernen Stil” may quite well be carried on in one's study, the (Berlin, 1896, 102 pp., about 2s.). “ Sprachgebrauch und mastery of grammar and wide and deep reading Sprachrichtigkeit” von Andresen (8th ed., Leipzig, 1899, are considerable aids for future proficiency as a 430 pp., 5s.) is a standard work which needs no teacher. Assuming that one of the ordinary school- mendation. A stimulating volume is Reichel's “Sprachgrammars, such as those by Eve, Aue, Meissner, psychologische Studien.” Vier Abhandlungen über WortstelWeisse, Kuno Meyer or Whitney, is in the teacher's lung und Betonung des Deutschen in der Gegenwart, Sparsampossession, he might with advantage take in hand keit, Begründung der Normalsprache (Halle, 1897, 337 pp., 1 recomor von about 4s.). And most excellent is Sütterlin's new work “Die deutsches Lesebuch ” (Halle, 1881, 2nd. ed., 3s. 6d.) are the deutsche Sprache der Gegenwart (1900). most reliable works on the older dialects. There are some works on composition and style Bachmann, Mittelhochdeutsches Lesebuch mit Grammatik of Modern German which are most suitable und Wörterbuch ” (Zürich, 1892, about 5s.) is a suitable for teachers, of which I must mention a few:- | book for teachers who have not sufficient time to study the “Der DEUTSCHE AUFSATZ," VON ERNST LAAS, older portions of the German language and literature fully. 2nd ed. of Part I. (Berlin, 1877); 3rd ed. of Part There are several other first-rate books which form good introII., by Imelmann (Berlin, 1894, price about nos.); ductions ; such are Heiderich, “ Einführung in das Studium des “ Der deutsche Stil von R. F. Becker, neubear Gotischen" (1s. 3d.), for beginners, then “ Zupitza,” “ Einfühbeiter," von Dr. Otto Lyon, 3rd ed. (Leizig, 1884, rung in das Studium des Mittelhochdeutschen zum Selbstun. about 6s.); Sanders' “ Deutsches Stil Musterbuch' terricht für jeden Gebildeten,” 5th ed., by Nobiling (Oppeln, about 3s. 6d.), also Michels, “Mittelhochdeutsches Elemen(Berlin, 1886, about 5s.) ; “Die Sprichwört. Lichen REDENSARTEN VON BORCHARDT,” in gänz tarbuch,” in “Sammlung von Elementarbüchern der altger manischen Dialekte Herausgeben von Dr. W. Streitberg" (Heidlicher Umarbeitung, herausgegeben von Gustav elberg, 5s.). For the transition from Middle High German to Wustmann, 5th ed. (Leipzig, 1895, about 6s.); and “ Der Bilderschmuck der deutschen Modern High German the following will be of good service : Arndt, “Der Übergang vom Mittelhochdeutschen zum NeuhochSprache in tausenden volkstümlicher Redensar deutschen der Breslauer Kanzlei" (55.), Rückert, “Geschichte ten," nach Ursprung und Bedeutung erklārt von Dr. Hermann Schröder, 5th ed. (Weimar, der neubochdeutschen Schristsprache,” 2 vols., 75. each, and von Bahder, “Grundlagen des neuhochdeutschen Lautsystems.” 1896, about 6s.). I would also advise every Teachers who have no wish to study the older dialects I would master mistress of German to become a strongly urge to take up at least the following iod. editions member of “ Der Allgemeine Deutsche Sprach- of the “Sammlung Göschen":-“Gotische Sprachdenkmäler verein," which publishes a Zeitschrift, with valu- übersetzt und erläutert ; Althochdeutsche Literatur mit Gramable articles for teachers, and also “ Wissenschaft- matik, Übersetzung und Erläuterung";-Hartmann von Aue, liche Beihefte," with excellent contributions by Wolfram von Eschenbach und Gotfried von Strassburg ;eminent authors. Waltharilied Walther der Vogelweide, Minnesang With “ Unsere Muttersprache, ihr Werden und ihr und Spruchdichtung - Der Nibelunge Nôt — Kudrun und Wesen," von Prof. Dr. O. Weise (Leipzig, 2nd ed., 1896, Dietrichepen-and Sebastian Brant, Hans Sachs, Luther, 270 pages, about 35. 6d.) we enter again upon the Fischart. historical platform, which may induce teachers to Others may wish to go further afield to Paul's “ Altembark on the study of some of the greatest works, deutsche Textbibliothek”: (1) “Die Gedichte Walthers von such as Von der Gabellentz' “ Die Sprachwissen der Vogelweide," hrg. von Paul, 1882, Is. rod. (2) “Gre. schaft” (14s.), Wundt's “Völkerpsychologie,” gorius von Hartmann von Aue, hrg. von Paul, 1882, 1s. I. Sprache (14s.) [which should be supplemented (3) “ Der arme Heinrich" von Hartmann, von Aue, von Paul, by Delbrück's "Grundfragen der Sprachfor 1882, 6d. (4) “ Heliand," hrg. von Behaghel, 1882, 2s. 6d. schung" (1901, 4s.)], or at any rate, of Whitney's (5) “Kudrun,” hrg. von B. Symons, 1883, 25. 10d., and “Life and Growth of Language" (6s.), and the others, or to editions of the great masterpieces of the older interest taken in these philosophical works will German literature. I would mention “ Wolfram von Eschenmost probably whet the appetite for a bach, Fünfte Ausgabe," von Karl Lachmann, Berlin, 1891, 640 pp. ; Iwein, “ Eine Erzählung von Hartmann, von Aue, mit thorough study of historical grammar, for which Cerf's “Short Historical Grammar of the German Anmerkungen,” von G. F. Benecke und K. Lachmann, zweite Ausgabe (Berlin, 1843, 565 pp.); “Der Nibelunge Nól, mit Language " (Williams & Norgate, 1894, Part I., den Abweichungen von der Nibelnnge Liet, den Lesarten Introduction and Phonology), and PROF. JOSEPH sämtlicher Handschriften und einem Wörterbuch," herausgegeben Wright's “Gothic, OLD AND Middle High von Karl Bartsch (I. teil text, Leipzig, 1870, 394 pp. ; II. teil, German PRIMERS ” will be found excellent helps Erste Hälfte, Lesarten, Leipzig, 1876, 292 pp. ; II. teil, and, at the same time, a good introduction to the Zweite Hälste Wörterbuch, Leipzig, 1880, 411 pp.); “Kudrun," study of the “ Deutsche Grammatik (Gotish, Alt-, herausgegeben und erklärt, von Ernst Martin (Halle, 1872, Mittel-, und Neuhochdeutsche ”), von W. Wil 336 pp.); “Walther von der Vogelweide, by Wilmanns (2nd. mann's Erste Abteilung-Lautlehre (Strasburg, ed., Halle, 1883) in Zacher's “Germanistische Handbibliothek." 1893, 332 pages, about 6s.), Zweite Abteilung- Wilmanns has also published“Leben und Dichtungen Walther's," Wortbildung (Strassburg, 1896, 663 pages, about von der Vogelweide (Bonn, 1882). For the sixteenth and sevenros.), well to Erdmann-Mensing's teenth centuries I would recommend the “ Liederbuch aus dem “Grundzüge der deutschen Syntax." sechsehnten Jahrhundert von Goedeke und Tittmann (2nd ed., The most excellent books on individual dialects for ad. Leipzig, 1881, 400 pp., 35. 6d.), and the same authors' “ Deutsche Dichter des sechzehuten und siebzehnten Jahrhunderts mit yanced students are found in Braune's “ Sammlung kurzer Grammatiken Germanischer Dialekte” (Halle, Max Niemeyer). Einleitung und Anmerkungen ” (each volume 3s. 6d.). Then Braune's “Gotische Grammatik” (Halle, 1882, 2 aufl., 129 for the eighteenth century Seuffert's excellent collection of “ Deutsche Literaturdenkmale des 18 Jahrhunderts." pp., about 35. 6d.) might be preceded by Sheitberg's “Gotisches Elementarbuch (35. 60.), Braune's “ Althochdeutsche Gram. The editions of the great classical writers, matik” (Halle, 2nd. ed., 5s. 6d.), Paul's “ Mittelhochdeutsche Grammatik" (Halle, 1884, about 3s. 6d.). Braune's “ Althoch. Klopstock, Wieland, Lessing, Herder, Goethe, Schiller, Grillparzer, Heine, are so numerous that 1 The annual subscription is gs. Intending members should write to the it is difficult to make a selection, and most teachers President of the London Branch, Professor Weiss, Royal Military Academy, will not be able to purchase and work through the Woolwich, or the Secretary, Dr. Hirsch, 102, Embletoa Road, Lewisham, London, S.E. complete works of even the greatest writers. How more as as von or ever, they should use their influence that the best lin, Hertz, 1896), C. F. Meyer's “ JÜRG JENATSCH editions of the most eminent representatives of (Leipzig, Haessel); Fontane, “Vor dem Sturm," German literature are procured for their school Spielhagen's “Problematische Naturen,” Paul libraries, such as the scholarly works of Old ande Heyse's “Novellen,”Otto Ludwig's "Heiterethei,” Middle High German mentioned above, and then Storm's “Immensee,” Rosegger's "Schriften some, at least, of the following : Luther's DES WALDSCHULMEISTERS,” Wildenbruch's “ Neue Werke, “ Kritische Gesamtausgaube (Wei- Novellen,” SUDERMANN'S “Frau Sorge," HAUPTmar), Lessing's “ Sämtliche Schriften herausge- MANN'S « VERSUNkene Glocke," Scheffel's geben von Karl Lachmann," 3rd edition, by Franz “TROMPETER VON SÄKKINGEN,” Weber's “DreizehnMuncker (Stuttgart, since 1886); Herder's “Sämt- linden,” Heine's “ Buch der Lieder,” Buchheim's liche Werke herausgegeben," Bernhard.“ Deutsche Lyrik ” and Buchheim's “ Balladen Suphan; “Goethe's Werke herausgegeben im und Romanzen” (Macmillan); and read a few Auftrage der Grossherzogin Sophie von Sachsen," modern plays, such as WildenBRUCH's "Harold" the so-called Weimar edition ; Schiller's “ Sämt. and “König Heinrich,” SudERMANN'S “ Johannes," liche Schriften herausgegeben von Karl Goedeke"; “Ehre” or “HEIMAT,” HAUPTMANN's “ Weber Heine's Werke von Dr. E. Elster, ; Grill- “ Einsame Menschen.” These will give the parzer's Sämtliche Werke, 20 vols., herausgegeben reader an idea of the tendencies and struggles of und mit Einleitungen versehen von August Sauer. modern German literature. The latter edition forms part of the “Cotta'sCHE BIBLIOTHEK DER WeltliTERATUR," in which Turning to the history of the German language and literature, most of the other classics have also appeared. I would call special attention to the “ Histoire de la langue They are very convenient editions for teachers, allemande” par Henri Lichtenberger (Paris, 1895, 478 pages), and can be bought separately, each volume which is likely to suit many teachers who prefer to read a bound in cloth, for is. Or cheaper still are the book written in French. Among histories of literature I have paper-cover editions of “Reclam's UNIVERSAL so many good works before me on my shelves that the embarras BIBLIOTHEK,” which cost 3d. each. Along with de richesse which has been troubling me all along is taking a any of these should be used Düntzer's “Erläute very acute form, and I hope readers of this article will forgive me if I call their attention to some works which they may rungen zu den deutschen Klassikern " (Leipzig, about is. each). Students of German who find it very likely not think of purchasing. Let me begin with a book written in English, entitled “ Outlines of German more convenient to use editions with English notes Literature,” by Gostwyck and Harrison, of which, I think, a cannot do better than take up Breul's editions of new revised edition by a competent German scholar will appear some of the German classics for the Pitt Press, before very long (London, 1873, nos.). Of Hermann Kluge's and Buchheim's editions for the Clarendon Press. “Geschichte der deutschen National-Literatur" (Altenburg, I may also mention Rippmann's edition of Grill 22nd ed., 1891, about 2s. 6d.), there is, I believe, also an parzer's “Sappho," and Cotteril's edition of English translation. Excellent little books are “Grundzüge “ Iphigenie” in my own series; and there are der deutschen Litteraturgeschichte" von Dr. Gotthold Klee many excellent edition's in Macmillan's German (Berlin, 3rd ed., 1898, about 35. ), and " Grundige der deutschen Classics, edited by Fassnacht, among them Litteraturgeschichte von Dr. Gottlob Egelhaaf” (Heilbronn, 6th Goethe's “ Faust," Part I. Teachers who wish ed., 1888, about 25. 6d.). Larger works and more suitable for to study this great work thoroughly should take serious students are :--Wilhelm Scherer's “Geschichte der the edition by Colvin Thomas (Boston, Heath & Co.), deutschen Litteratur" (Berlin, 8th ed., 125.), of which there which is most scholarly, and covers both the first exists an English translation by Mrs. Conybeare. Koberstein's and second part, as does Prof. Schröer's edition of “Grundrisz der Geschichte der deutschen Nationalliteratur," Faust,” Part I., 55. ; Part II., 6s. 6d. in 5 vols. (5th ed., 1892, by Karl Bartsch), is a most thorough work. An edition with many facsimiles and illustrations is Some of the editions mentioned contain biographies of the König's “ Deutsche Litteraturgeschichte,” which costs about £1. great writers, but they are necessarily short and sketchy. Those A similar and in its text more scholarly work is the “ Geschichte who wish to study the lives of the great German classics more der deutschen Litteratur von den ältesten Zeiten bis zur Gegendeeply I would refer to Erich Schmidt's “ Lessing, Geschichte wart” by Vogt and Koch (Leipzig, 1867, 16s.). A standard work seines Lebens und seiner Schristeu ” (Berlin, 1884, 2 vols.), for advanced students who wish to study the subject thoroughly Bielschowsky's “Goethe," 2 vols., or Eugen Wolff's “Goethe's is Karl Goedeke's “Grundrisz zur Geschichte der deutschen Leben und Werke,” or R. M. Meyer's preisgekrönte Arbeit Dichiung” (2nd edition, after the death of the author continued "Goethe” (628 pages), also Düntzer's “Goethe's Leben,” ios. by E. Goetze). All that a teacher may require by way of a should be mentioned, and perhaps Lewes' “ Life of Goethe.” history of the German literature of the older period is contained Among the biographies of Schiller, Minor's “Schiller, sein in “ Paul's Grundriss der Germanischen Philologie,” of which Leben und seine Werke " (Berlin, Weidmann) takes the first a second edition is appearing, and without which no serious place ; Weltrich, Brahm and Düntzer have likewise furnished student of German can exist. For the eighteenth and ninevaluable contributions. teenth centuries a few special works will be necessary. For the eighteenth century Hermann Hettner's “Geschichte der If I am to give some guidance to the more recent deutschen Litteratur im XVIIIten Jahrhundert” (2 vols., works of German literature I cannot do better, I 4th edition, by Harnack) can be thoroughly recommended. think, than quote a few authors and works without wishing to say that there are not many others There is as yet no standard book dealing with equally good. Take Gustav Freytag's “ AHNEN," the nineteenth century, but Rudolf von GottScheffel's“ Ekkehard" (Stuttgart, 1893), Gott- schall “ Die Deutsche Nationalliteratur des neunFRIED Keller's “ Züricher Novellen (Ber- zehnten Jahrhunderts ”(6th ed., 4 vols, 1892), and |