making equal angles with the two straight lines A B and CD, given 360 18h 4. in position. Q. E. F.* 5. Divide by Ang, 5 10y Fig. 3 abt-1 ab-1 abat 6. Divide by 3aby-By 3 ahmamyth-AY 7. Divide Ans. 12 Divide the numerator by the given integer, when it can be done without a remainder ; but when this cannot be done, multiply the denominator by the integer. hmy by 148. To divide an integer by a fraction. Scholium. If the lines are parallel, this construction fails. For Reduce the integer to the form of a fraction, and proceed as besuch a case, it is only necessary to apply Prop. XII. to the con- fore. Or, multiply the integer by the denominator, and divide the struction, and Prop. XXVIII. to the demonstration. product by the numerator. ; and is X C d ad Ans. DIVISION OF FRACTIONS. 146. To divide a fraction by a fraction. axd aa Or, a Ans, as before. Invert the divisor, and then proceed as in multiplication of d fractions. To invert a fraction, is to turn it upside down, or to make 2.xy 12. Divide xy by Ans. the numerator the denominator, and the denominator the 2 atnumerator. EXAMPLES. 3 am 4abt4cx 13. Divide ab-tex by Ans. 120 1. Divide b 9ac-33 14. Divide 3ac-by Ans. d 3 ac Here, we have х Ans. bic 149. By a former definition - the reciprocal of a quantity is To understand he reason of the rule, let it be premised, the quotient arising from dividing a unit by that quantity.' that the productif any fraction by the same fraction inverted, b 5 is always a unit. Thus the reciprocal of IX b ab d Thus 3 1. And x Sty=1. Hence, the reciprocal of a fraction is the fraction inverted. For ab hty d 6 is mothy; the reciprocal of But a quantity is not altered by multiplying it by a unit, instance, the reciprocal of mty 5 Therefore, if the product of the dividend by the divisor inverted be multiplied by the divisor itself, the last product i will be equal to the dividend. Now, by the definition, is 31 or 3y; the reciprocal of 1 is 4. Hence the reciprocal 3y “ division is finding a quotient, which, multiplied into the divisor, will produce the dividend." And as the dividend of a fraction whose numerator is 1, is the denominator of the multiplied by the divisor inverted is such a quantity, the 1 quotient is truly found by the rule. fraction. Thus, the reciprocal of is a; of is antb, &c. ato 3h 2. Divide by 2d EXAMPLES FOR PRACTICE, 4 26 by a by 6 c 2a(x,y). Ans. zy? a? 'y and 12 crowns a-head for the cows. How many did he bay 4x+2mx y 6. Divide by Ans. of each? 3aby * = the number bought of each, a262 a the cost of the sheep. 7. Divide by Ans. and 12.30 the cost of the cows. 12 Hence, 24-123 -840 by the conditions of the question. ity xty Therefore, 143 = 840 by addition ; 8. Divide Ans. and * = 60, the number bought of each. Here, the last expression is obtained from the preceding one 3ab-6xy ab---2.cy 9. Divide by dividing each member by 14, the co-efficient of 14x. Ans, 1, 2 It will be perceived, in this example, that the unknown quantity or number sought, is represented by the letter x; and 7ab from the conditions of the problem, we obtain the quantity 10, Divide 2labo by Ans. 3cx. 14x, which is equal to the given quantity 840 crowns. This whole algebraic expression, 14x =840 crowns, is called an Zab 4cxy equation. 11. Divide 8zy by Ans, 151. An EQUATION, therefore, is a proposition expressing in algeab braic characters the equality between one quantity or set of quantities 27 cm and another, or between different expressions for the same 12. Divide 18a:c by Ans. 3m quantity. This equality is denoted by the sign=, which is read “is 18(at) 2a(0-y) бат-бт. equal to.” Thus, stabtc; and 5-7817–4, are equations, 13, Divide by in one of which the sum of cand a is equal to the sum of 6 and c; and in the other, the sum of 5 and 8 is equal to the 3ctd eto difference of 17 and 4. 14. Divide 2a + Ans, The quantities on the two sides of the sign=are called mem bers of the equation; the several terms on the left constituting ab2c3 a3b2c cay the first member, and those on the right the second member. 15. Divide by When the unknown quantity is of the first power, the hy proposition is called a simple equation; or an equation of the 2.06 622-11 first degree. 16. Divide by Ans. 152. The reduction of an equation consists in bringing the unknown 2+2 3x+1 X-6 quantity by itself to one side of the sign of equality, and all the y +-24 known quantities to the other side, without destroying the equality of 17. Divide by Ans. the members. 32+2 y2-3y+2 To effect this, it is evident that one of the members must ato (2_-72 be as much increased or diminished as the other. If a quan18. Divide by Ans. tity be added to one, and not to the other, the equality will ato be destroyed. But the members will remain equal, Sama 23-tate 1. If the same or equal quantities be added to each. Ax, 1. 19. Divide by Ans. 2. If the same or equal quantities be subtracted from each. a Ax. 2. 3. If each be multiplied by the same or equal quantities. 20. Divide by Ans. (y) #-- ΑΣ, 3. yn 4. If each be divided by the same or equal quantities. AX. 4. 1 1 21. Divide 1 + by Ans. The principal reductions in simple equations are those which are effected by transposition, multiplication, and division. xto 22. Divide by Ans. REDUCTION OF EQUATIONS BY TRANSPOSITION. sy In the equation 4-7=9, the number 7 being connected with astab *+-8+ the unknown quantity x by the sign -, the one is subtracted 23. Divide by Ans. axas from the other." To reduce the equation, let 7 be added to both sides. It then becomes x–7+7=9+7. a mat The equality of the members here is preserved, because one 24. Divide 1 by 1+ Ans. 22 +23 ***a 22" is increased as much as the other. But on one side, we have -7 and +7. As these are equal, and have contrary signs, 15a 34% they balance each other, and may be cancelled. The equation 25. Divide +-4a2 + by 4-42a + 42% will then be 9+7. m2a Here the value of x is found. It is shown to be equal to 9+7, that is, to 16. The equation is therefore reduced. The Ans 2+2a unknown quantity is on one side by itself, and all the known quantities on the other side. 3x +41-2 -b3a; 26. Divide 942-28 + by 3-4 Ans. Adding b to both sides, we have t-tbatb; And cancelling as before, we have kzatb. Ans. SIMPLE EQUATIONS. 153. When known quantities, therefore, are connected with the un the equation is reduced by 150. Most of the investigations in algebra are carried on by known quantity by the sign + or TRANSPOSING the known quantities to the other side, and prefixiny means of equations. In the solution of problems, for example, we the contrary sign. represent the unknown quantity, or number sought, by a certain This is called reducing an equation by addition or subtracletter ; and then, in order to ascertain the value of this unknown quantity or letter, we form an algebraic expression from the con- tion, because it is, in effect, adding or subtracting certain quanditions of the question, which is equal to some given quantity tities, to or from each of the members. or nomber. EXAMPLE 1.-Reduce the equation 2+36-m=1-d. EXAMPLE.-A drover bought an equal number of sheep and Here, transposing +36, we have 26-mal-13; cows for 840 crowis. He paid 2 crowns a-head for the sheep, And transposing had36-4-9. Ans. ܪ 164. When several terms on the same side of an equation are A WELL-WISHER (Arselt); “ Although offensive, sulphuretted hydrogen alike, they must be united in one, by the rules for reduction in may be inhaled, when largely mixed with air, without apparent injury; and I have known it to be inhaled in large quantity, when fresh, without oroaddition, ducing further harm than faintress. I have breathed a strong admixture of EXAMPLE 2.--Reduce the equation *+55–476. it with air repeatedly, and never experienced the slightest evil result. Here, transposing 5b-4h, we have x=76-56-4-4h; Thenard says that birds are killed by a mixture of 16oo of it in air; dogs by do of it; and a horse by to of it.” J. C. B.-A. SOHOLAR: Nitric And uniting 76–50 in one term, we have x=26+46. Ans. acid is composed or 14 parts of nitrogen and 40 parts of oxygen, or 1 propor155. The unknown quantity must also be transposed, whenever tiopal of nitrogen and 6 proportionals of oxygen. " It is used extensively it is on both sides of the equation. It is not material on which in chemistry and the arts; for etching on copper, and as a solvent of tin to side it is finally placed, though it is generally brought to the the metals to their maximuin oxidation; in medicine as a tonic. The nitric left-hand side. acid of commerce is half water, and is called double aquatørtis ; another EXAMPLE 3.-Reduce the equation 23+2h=htet 3x. kind, containing three-fourths water, is called simply aquafortis." -Gray. All of our Correspondents up to this date 16th Jan. 1854, are in error here, by transposition, we have 21-h-d=32-2.x; about the Four-ball Question. In the solutions we have received, the four And by incorporation, 1-X, Ans. balls do not touch each other.-B, CRIMARAW (Lambeth), should take the 156. When the same tern, with the same sign, is on opposite cheese " is an expression that cannot be admitted into the P. E.-R. G. advice of his friends.--T. CAVE (Gee Cross): The“ versed sine of a piece of sides of the equation, instead of transposing, we may expu.ige it BRAY (Bodmin): The rule about committirig all exercises to memory, must from each. For this is only subtracting the same quantity be taken cun grano salis; an author sometimes jocosely, like a doctor, prescribes more than he knows will be taken, in order that some inay be from equal quantities. taken.-T. H. METHVEN (Hoxton): Many thanks for his kind hints.-J. FXAMPLE 4.--Reduce the equation 2+31+d=6431+7d. YOUNG (Huddersfield): The French Lessons are finished. “Le Civilisatear" may be had of D. Nuti, Foreign Bookseller to the Queen, 270 Strand. Here, expunging 3h, we have x+d=+7l; H. HARTBEG (Dover): Study Latin, Greek and Hebrew in order, and in And by transposition and incorporation =htod. Ans. the P. E. until you can read the Scriptures in their original tongues. J. 157. As all the terms of an equation may be transposed, or sup- useful sciences; all arts are founded on these.-R. Nurse (Machen), The BENTON; Nathematics, Natural Philosophy, and Chemistry are the most posed to be transposed, and it is immaterial which member is Lessons in English are finished.-WARIN : Received. --L. MURPHY (Queenwritten first, it is evident that the signs of all the terms may be borough): He has hinted what we intend. A ray of white light is divided charged, on both sides, without affecting the equality. in the following proportions, according 10 Newton and respectively. Thus, if we have ----d-a Blue. Violet Then by transposition, we have -d-to-** Newton -6 Qr, by changing the places of the members, —+ bda. J. W.W. (Portsea) : The honours attainable at Matriculation and Gra158, If all the terms on one side of an equation be transposed, nuation in the University of London are substantial, see the P. E. vol. ii. p. 138, col. 2; and p. 215, col. 2.--A, P. T. (Cranbrook) : Your trisection of each member will be equal to 0. an angle won't do.-CONSTANT READER (Wishiel) : "The population of Thus, if atbd; then it is evident that x+b-d=0. China has been rariously computed at froin 150 millions to so high a number as 360 inillions--the latter of which is the native statement, issued under the EXAMPLES imperial authority. This is considerably more than one-third of the estimated population of the globe, and would amount to an average of upwards of 280 inhabitants to the square inile." Hughes Manual of Geography.-E. MAN: 5. Reduce a+22—8=-4+ata. Ans. x=54-4. See vol. ii. p. 137.--PASSAIG (Glasgow): Cassell's Arithmetic, Algebra, and 6. Reduce y + ab — hm=a+2y — ab to hm. Ans. y=2ab Geometry; ARE NOT REPRINTS from the P. E. but works which were dcinanded by the readers of that work, before the lessons on these subjects —2mm could be completed; they may be read before the P. E., after the P, E., or 7. Reduce h+30+7=8-674-62--04b. Ans. =b-7h along with the P. I, the last method being the best. Some i'rench books are on the tapis. The Latin Lessons in the P. E, are the best; they may be -d-22. followed by Zumpt's Latin Grammar by the higher students. 8. Reduce bh +-21 — 4x+d=12 - 3x+d-7bh. Ans. 2 = the French Dictionary it will soon be completed ; doui't crumble; editors are as liable to be taken ill as other people. As to the use of the word Firons, you 8bh+9. are riglit, my boy; but it is never used in books now; we only follow the 9. Reduce 5x+-10-40-25+4x+a. Ans. 2:15. multitude; but if a fellow does not know when a story is done, he has not surely attended to it while in progress; and in that case, he would be none 10. Reduce 5c+2x4-12-32+20+-50. Ans. k11. the wiser, il we said our story is ended. 11. Reduce at 1-3.0---2014-42-7b. Ans. 220. 12, Reduce zt-3--22:4=34-4-3.244-5.x. Ans. X=31. Green. 45 56 27 27 40 27 108 89 109 Fraunhofer 46 As to 1 LITERARY NOTICES. JOHN CASSELL'S FRENCH WORKS. Now ready, price 4s. in stiff Wrapper, or 55. strongly bound in cloth, don.-T. H. E.: Beza's Latin Testament.-SL(Sunderland): We don't the First Part complete, consisting of the French and English, of CASSELL'S know.-VOLUNTARY EDUCATION (Sheffield): “Cassell's Lessons in Ger Cassell's Lessons in Ger- FRENCH DICTIONARY: the entire work will be completed in about Twentyman" are expressly stated to be a reprint from the P. E. The “ Historical eight Threepenny Numbers, and will form one handsome 8vo Volume. Educator" is intended as a substitute for the promised Lessons in History Price 9s. 6d. bound in cloth, or the Two Divisions may be had separate. in the P. E. You will not get on well with the German, or, indeed, any language, without a dictionary ; "Cassell's German Dictionary” is now A COMPLETE MANUAL OF THE FRENCH LANGUAGE, by Professor De publishing in numbers at 2d. each.--Juvenis (Springfield) would have Lolme, juot published, price 3s. neatly bound. This forms one of the written better had he heard a good sermon; the grammar appears to be well enough.-Deutche: Yes. PLODDING GENIUS (Louth); His paper most simple, practical, and complete Guides to a thorough knowledge of the on" Perseverance in lieu of Genius,” is very good; but it is capable of in French Language which has hitherto bcen published. The plan upon which provement. Let him try again, but take this new text: “Perseverance is it is conducted is admirably calculated to accomplish the proposed object. Genius. In the first place, the Grainmatical Principles of the Language are clearly C. S. (Kinross): The magic lantern will be explained by-and-by under Optics. Lenses of every kind may be had in town.--SACRA RUTT (Atholl): laid down, and, secondly, these Principles are copiously illustrated by suitable The most common unit for measuring the earth-work of ditches is the Exerrises of English to be turned into Freich. cubic yard. The best rule is to consider the ditch, if uniform, as a prism, CASSELL'S LESSONS IN FRENCH, Parte I. and II., in a neat volume, price lying on one side, and cast up the earth-work accordingly. G. HARPER (W. Hall): We have not seen a Geographical Pronouncing each 2s. in stiff covers, or 23. 6ido neatly bound in cloth ; or bound together, Dictionary.-A. K. R. (Edinburgh): Yes.-J. MATHESON (Glasgow): You | 48. 60. are doing very well ; go on as you have begun; finish the lessons in the P. E. A KEY TO CASSELL'S LESSONS IN FRENCII, containing Translations of all first.-TYRO (Manchester): Not decided. JOHANNES (Bradford): The rule the Exercises, with numerous relerences to the Grammatical Rules, price is inaccurat-; it should be, every syllable in Greek is pronounced as in Is. paper covers, or 1s.6d. cloth. English: example, tipyyns, ei-ee-nees.-A. ALISON (Liverpool) had better A Series Or Lessons in IRENCI, on an entirely Novel and Simple Plan, learn good manners before music. R. (Edinburgh): Rome was not by means of which a krowledge of the French Language may be acquired built in a day. We can't put everything into the P. E, at once. without the Aid of a Teicher. These lessous first appeared in successive recommend you books on Logic and Moral Philosophy unless we were intimate with your previous studies. Numbers of the Working Man's friend and family Instructor." They are now reprinted in it Yilisert form.--Dy special permission of H Majesty's Pustudster:General, thoác lof's: uns may be transmitted thro the Postaotfice, and will be sent to any Addits, on the receipt of S This tern is used to include both additions and subtractions. Pustage stamps.-Price 6d, in a peat. Wrapper. We can't + ܙܫܢ LESSONS IN MUSIC.-No. XXII. in former lessons on the “ mental effect” of the note Lah (the sixth above the key-note or the minor third below), or, better (Continued from p. 226, Vol. IV.) still, let them recall all they have themselves observed and felt In explaining, in our two preceding lessons, the nature of the old in connexion with it. Was it not always, when sung slowly, and established notation, we have slightly anticipated the subject the sorrowful note? Then let us suppose ourselves trying to of the present lesson. We shall now conclude our lessons on compose a very sorrowful tune,--should we not naturally Music for the present by elucidating the subject of “ Minor employ this note in the most effective positions ? Without Tunes." Why they are so denominated we shall explain presently. composing, however, let us just recall one of the oldest tunes of But, first, let us ask our readers to recall all that we have said I this kind in existence. You notice what a sorrowful effect is produced by simply sadness. Take the example with which Mr. Hickson illusclosing on Lah instead of the key-note. Yet more strik- trates this subject :ing is this effect, if the tune also opens with this note of Two other examples, in the well-known tunes St. Bride's mind the effect of LAH when thus placed in effective and Wirksworth, will bring more clearly before the positions : Our pupils will now be prepared for the following exposition FAdo when they become key-notes by " transition") its own of the subject before us. nzusical effect. It still leaves on the mind the impression ! In some tunes-chiefly those which are intended to ex. “ sorrowful suspense.' press a mournful sentiment,the note LAH is found to predo- 8. Modern musicians, in order to give to LAH a closer resenminate. It is necessarily heard both at the beginning and at blance to the ordinary key-note, and to direct the ear to it the end of such tunes; and assumes almost the importance of more decisively as the note on which the tune closes, as well a governing or key-note, but without changing (as son and las to increase the general effect of such tunes, occasionally 97 VOT. IY. LAH introduce a new note, which we shall call NE, a tonule below bears to TE. BAH, NE, LAX, heard in succession; resemble, in This note bears-the same relation to LAH which te bears mental effect, LAH, TE, DOH. The leaner mayormetimes strike Musicians also think it necessary sometimes to intro- BAĦ more easily by thinking of it as TV. The note ne is in duce another new nole, which they then use instead of Fah.. frequent use, but BAH is very seldom used in ordinary music. It is a tone below ve, or a chromatic part-tone above FAH. Ti the following Scottish tune :--We call it BAK. It bears the same relation to XE which Lay Tunes of this kind are commonly called minor tunes, lively careless abandonment. Of this we have several examples from their having the interval called a minor (smaller) third in the old English music. It will be a good vocal exercise for immediately above their predominating note LAH_(LAH, DOH), our pupils to learn to solfa them. What a pity that such fine and in distinction from other tunes which have a major (larger) | music should have been set to the words of a foolish sentiment third above their predominating note Don. They may be said or a savage drinking-song! For the first, better words are to be in the Lah node. It is advisable to take their pitch by given in Chappell's collection of “ National English Airs." It means of DUH, as in other tunes. The signature may be written is entitled the Widow's Song. To such words the tune must in this form, “ LAI MUDE, KEY A." be sung more slowly. There is nothing comic in this sad wail. Those who studied with us the modifications in the mental To the second we have adapted words from a poem of tragic effect of the note LAH, will be quite prepared to understand truth, by James Russell Lowell. The third we have left as how this kind of tune may be used in the serio-comic style, it is. and how by quickening the speed they may even express a |