HISTORIC SKETCHES. 



117 



(2.) From 

 Take 



Answer 2h 



(5.) (6.) 



3a6m xy 17 + 4aa> 



7a6m + Gary 20 ax 



til. 1 examples, it will bo neon that tho di/erenc 



between a potiitive and a negative quantity may bo greater than 



( the two quantities. In a thermometer, tho difference 



between 28 degree* above zero, and 16 degrees below, in 44 



degree*. The difference between gaining 1,000 pounds in trade, 



sing 500 pounds, is equivalent to 1,500 pounds. 



^traction may bo proved, as in arithmetic, by 

 adding the remainder to tho subtrahend. Tho sum ought to be 

 equal to the* minuend, upon tho obvious principle, that the dif- 

 ference of two quantities added to one of them, is equal to the 



MPLE8. (1.) From 2xy 1, subtract m/ + 7. 



Operation. Proof. 



Here, Minuend 2xy 1 Add xy -f- 7 Subtrahend, 



Subtrahend xy -f- 7 To 3xy 8 Remainder. 



Remainder Sxy 8 2xy 1 Minuend. 



h+ 3bx (3.) hy ah (4.) nil 76y 

 3h 96 5hy 6a7i 5nd by 



hy + 5a/i 4?ul Gby 



(7.) (8.) 



ax + 76 3a7i + axy 



4ax + 156 7a/i + axy 



10a6m Try + 3 -f 5ax -f- 5a 86 -f 10a7i 



63. When there are several terms alike in the subtrahend, 

 they may bo united and their sum bo used. Thus, 



EXAMPLES. (1.) From 06, subtract 3om + am + 7am 

 -+ 2am -f- 6am. 



Hero 06 Sam am 7am 2am 6am = ab 19am. 



Ansiver. 



(2.) From y, subtract a + a + a + a. Ans. y 4a. 

 (3.) From ax be + 3ax + Tbc, subtract 46c 2ax -f- 6c + 4ax. 



Answer. 2ax -j- be. 

 (4.) From ad -f 3dc bx, subtract 3acl -f fix dc -f ad. 



Answer. 4dc 8bx Sad. 



64. The sign , placed before the marks of parenthesis which 

 include a number of quantities, requires that, when these marks 

 are removed, the signs of all the quantities thus included should 

 be changed. Thus a (6 c + d) siprnifies that the quantities 

 5 c and + d are to be subtracted from. a. Remove the 

 pnrrvf!' expression will then become a 6 -\-c d, 

 an expression which has exactly the same meaning as the 

 former. 



EXAMPLE. From xy -f- d, take Tad xy -f- d + hm. Here, 

 ay -\- d (7aeZ xy -f- d + hm) Tad + 2xy hm. Answer. 



65. On the other hand, when a number of quantities are to be 

 introduced within the marks of parenthesis, with immediately 

 preceding it, their signs must bo changed. Thus, m + 6 

 dx + 37i= (m 6-f- dx 3/t). 





EXERCISE 5. 



1. From 6ob + 7xy + 18d/g, take 3xy + 4ob + 8df,j. 



2. From - 35o* - 21ab - 37m, take - 30m - 15ab - lOaz. 



3. From 9ay + 19bx + 22bc, take 12ay + 31bc + 50b*. 



4. From 8zy - lOab + 6d, take 12ab + lOd + 24a-y. 



5. From 7a + 6* + df + x<jz, take 3* - 4o - 3d/- 17*yz. 



6. From 18bc - xy + 22gh, take 41*y - gh + be. 



7. From 21az + y + ac ay, take 4a be + * yz dc. 



8. From 21* + 40zy - 13a, take 42 + lOab - 5bc. 



9. From 5xy, take 2ab + 30ab + ab 4ab. 



10. From Sax + 16ai/, take 4ax ay + Sax + 4ay. 



11. From o + b, take -(c + d-f+g-h- xy). 



12. From 7ob + lOry - 7ad, take - (6ob - 12iry + ad). 



13. Introduce the following quantities within a parenthesis with 

 immediately preceding, without altering their value ; viz., a + b 

 - c -d +/ + gh. 



11. Also, ab cdx + df x - y + ghf be + xyz. 



15. From 4 + Cbbb, take 3xx + 4bbb. 



16. From 20yy - 2>j + 12ooa, take 15[/y -2>j - IZaaa. 



17. From - 8 (a + b) + 10 (x + y), take 2 (a + b) - 6 (* + y). 



18. From 4 (a + b) - 1G (x - y), take 17 (a + b) + 36 (x - y). 



19. From 2a aa + bo, take o 4aa 6ba. 



20. From xx + Sx - xxx, take 2* + 3r* + lOxxx. 



21. From 18 - 25ab + 20* + 3y, take 3* + 3<j - 25ob + 1. 



22. From 6 (a - y) - 17 (a + y), take 3 (o + y) - 7 (o - y). 



23. From ax xy my 6, take 6ax 6xy ay + 46 7d/. 



24. From GCa - 4b, take 20a - b - 30o - 16a - 3b + 5a. 



88. From f&t* - a*, Uko2*V - b*. 



26. From** -f 4*y + 0*V + ** + V*. take ** - 4*>y -r * - icy* 



27. From 4as - 8a + 16*, take 8a - 4a ^ 6a. 



28. From a + b + c, tako - a -r b + e. 



29. From 4a f 3a'x + *' 2x\ take 2a> ia9x ax* x*. 



30. Take - * + 3* - 1 from * + 3* + 3 + 1. 



31. From a*' + by, t*ke rx* - dy*. 



KEY TO EXERCISES IN LESSONS IN ALGEBRA. 

 EXERCISE 4. 



1. 6ab + ed - 4m + 7. 



2. 3y - dx + hm - 1. 



3. abm + bm - 5y + x + 10. 



4. 8m + 3xy - 11. 



5. 9ah]/ + 16. 



6. Had + xy. 



7. by + 3 (b - a) x 



8. 6a* + xy. 



9. 6b + 44cd/-3*y 



10. 18a + 4ax - Sbi 



11. 8ab - 6bc + 4cd - 



+ 18/g - 2ax. 



12. 8abc + 25abd + 5aryz. 



13. 3d/+4az + 74y + 30. 



14. 55a -I- 68b. 



15. 7(o + b). 



16. 2xy (a + b). 



17. 2az + 5aa + 3e + Zxxt. 



HISTORIC SKETCHES. XXIX. 



SUMMARY OF SKETCHES FROM ENGLISH HISTORY. 



IT is proposed in the present paper not to draw a fresh sketch 

 of any historic incident, but to present a summary of what 

 sketches have been given already, to re-survey the ground over 

 which we have travelled, and to point out the chief features of 

 English history which have been deemed noteworthy. 



In our first sketch (Vol. I., page 9) we gave some account of 

 the early history of our island from the day when the Romans 

 landed in 55 B.C. until the death of Harold in 1066 ; and we 

 endeavoured to present a picture of Anglo-Saxon England 

 under the Heptarchy and under one feupremo monarch. In 

 doing this we were, of course, led to glance at the ever-recurring 

 strife between the Saxons and the Danes, and at the intro- 

 duction of Christianity into the country through the agency of 

 Augustine. 



In the three following sketches we narrated the events 

 immediately succeeding the Norman Conquest, showing how 

 for eighty years from A.D. 1066 to 1146 the country periodi- 

 cally rang with the cry of " Down with the Normans." The 

 Saxons, Danes, and Celtic Britons, the victims of cruel op- 

 pression, and frenzied with despair, repeatedly rose against 

 their Norman conquerors, but all in vain. Revolt after revolt 

 was put down by a stern hand, the last outbreak occurring in 

 1137. After this time we gradually lose all traces of dis- 

 tinctness between the inhabitants of the country. " Inter- 

 marriages, community of interest, the exigencies of domestic 

 and foreign policy, all contributed to blend the antagonistic 

 races into one. By the time that Simon de Montfort sent out 

 his writs in 1264 for tho first English Parliament to assemble, 

 we find no invidious distinction between English and Normans, 

 and we mark no longer, excepting in the names of members of 

 the Upper House, the deeply carved furrows of the Norman 

 Conquest." 



And now, arriving at the date of the first memorable event 

 in the history of free England, we showed in onr fifth sketch 

 how the chief bulwark of English freedom, the Great Charter, 

 was won. The miserable condition of the working classes, both 

 artisans and labourers, was sot forth, and it was shown how that 

 the clergy of the day, with Stephen Langton, Archbishop of Can- 

 terbury, at their head, had compassion upon them and strove to 

 help them to some measure of liberty. How the result of their 

 action fell short of what was intended was also shown, but the 

 fruits of their striving were duly noted and commended as being 

 very great. In the days of King John (1199 1215), the clergy 

 were the friends of liberty, and ever sought to make free, in every 



