NOTES ON LESSER KNOWN POINTS.

NOTE (1).-Origin of Name Plantagenet.-Derived from plantagenista-Latin term for shrub we call broom-which, as an emblem of humility, was worn by first Earl of Anjou, when a pilgrim in Holy Land. From this, his successors took their crest and

surname.

NOTE (2).—Though descended from both Saxon and Norman Kings of England, Henry was not nearest representative of Saxon line-as Edgar Atheling's sister, who married Malcolm of Scotland, had sons as well as a daughter.

NOTE (3).-Chief Provisions of Constitutions of Clarendon.(1st.) Clergy should be tried by civil courts for civil offences. (2d.) Bishops, &c., should not leave the kingdom without King's consent. (3d.) Tenants in capité should not be excommunicated, nor appeals made to Rome without King's consent. (4th.) The King should receive the revenues of vacant benefices, &c.

NOTE (4).-By the passing of Poyning's law in 1495, the English obtained control over the Irish Parliament. The Parliamentary union of England and Ireland took place in 1801.

NOTE (5). Highly beneficial alterations in the law were either made or formally sanctioned by Henry II. Thus, trial by jury acquired prevalence in opposition to the barbarous usages of the ordeal and trial by battle. For purposes of justice the country was now divided into six circuits. Ralph de Glanville, Henry's last chief justiciary, is usually considered the father of English urisprudence.

NOTE (6). To obtain money for his crusading expedition, Richard released King of Scotland from his vassalage on payment of 10,000 marks (mk. 13s. 4d.) Despite this arrangement, claim of feudal superiority renewed by Edward I.

NOTE (7).-William Longchamp, the Chancellor, was appointed viceroy of England during Richard's absence. At the instigation of the King's brother John, Longchamp was summoned to appear before a Council of nobility to answer for his tyrannical conduct, and a sentence of removal and banishment was passed 1191. "This is the earliest authority for a leading principle of our Constitution the responsibility of ministers to Parliament."

NOTE (8).-Laws of Oberon.-This code of laws has been regarded as the foundation of the maritime jurisprudence of modern Europe-a proof of Richard's attention to naval affairs. According to this code "if a vessel was wind or weather bound, the

master, when a change occurred, was to consult his crew, and be guided by the opinion of the majority, whether he should put to sea. If a storm arose, the master might, with consent of masters aboard, lighten the ship by throwing part of the cargo overboard; if they did not consent, he was not to risk the vessel, but to act as he thought proper."

Lessons in Algebra.

(For Pupil Teachers in their Fourth Year.)
BY THE EDITOR.

CHAPTER IV.-DIVISION.

18. The student will remember that if the product of any number of multipliers be divided by one or more of them, the quotient is the product of all the remaining multipliers. For example; 2 × 3 × 4=24; if we now divide this product by 2, the quotient will be 3 × 4; if we divide it by 4, the quotient will be 2 × 3; if we divide it by 12, i.e., 3 x 4, the quotient will be 2; if we divide it by 2 x 3, the quotient will be 4; &c. This being a general principle, is as applicable to algebra as to arithmetic; for example:—If a = 2, b=3, and c = 4, it will readily be seen that just as 2 × 3 × 4 divided by 4 = 2 × 3, so a × bx c (i.e., abc) divided by c=ab; and just as 2 × 3 × 4 ÷ 2 × 4 = 3, so abc÷ac-b. Just so also 2abc (i.e., 2 x ax b× c) ÷ 2a = bc; and 2abc÷ ab = 2c; and also 12abc, i.e., (3 x 4 x axbxc)÷4a= 3bc, &c., &c. These considerations are embodied in the following statement or rule:

19. To divide the product of several letters by one or more of them, it is only necessary to write the remaining letters; the coefficients being divided as in arithmetic.

Examples:-xyz÷xy=z; 3abc÷3b=ac; 6bcd2d3bc.

20. From the above considerations, and remembering that a2 × a1 = ao, it is not difficult to see that a÷a = a2; and that ao÷ a2 = a1. Thus

To divide like letters-subtract the index of the divisor from the index of the same letter in the dividend; the remainder is the index in the quotient.

Examples.-8a5 ÷ 2a3 = 4a2; 16a4b4 ÷ 8a2b2 = 2a2b2; 5x3y2z÷

5xzx2y2; 7ab2c*÷3a2b2cJack; 8a"÷4a=2a"; 10xy÷ 5xy"=2x-"y-".+

21. That the "rule of signs," viz.,-Like signs produce plus, and unlike signs produce minus,-is as applicable to division as it is to multiplication may be seen from the following:-Since - a × b= − ab, from par. 18 we see that - ab÷b= -a; and -ab÷-a=b. Here when the signs of the divisor and the dividend were alike, the quotient was a plus quantity; when the signs of the divisor and the dividend were unlike, the quotient was a minus quantity. The student may readily work out similar illustrations for himself.

22. To divide a compound expression by a simple one, we divide successively each term of the former, as in the following examples:

That we are justified in dividing the "expression" a piece or term at a time, may be shown by the following arithmetical illustration; it is, in fact, what is always done in simple arithmetical division. Suppose we have to divide 2846 by 2; this 2846 = 2000 + 800 + 40+ 6, and in the ordinary method of working we should divide these numbers 2000+800, &c., successively by 2, thus

2)2846
1423

2)2000+800+40 +6 = 2)2846
1000+400+20 +3

1423

23. To divide one compound expression by another compound expression we proceed exactly as in arithmetical "Long Division," viz.,

Divide the first term of the dividend by the first term of the divisor, and put the result as the first term of the quotient; multiply the whole divisor by this term, and subtract the product from the dividend. Bring down as many terms of the dividend as may be required, and again divide, repeating the whole process until every term in the dividend has been brought down.

*The student is reminded that where, as in this case, no index is written, the index is understood to be 1.

In this example the index of x in the first term is 1, from this we have to subtract m, the index of the same letter in the divisor, but as we do not know the value of this letter m, we can only write the result of this substraction thus 1-m.

Before beginning to divide, the divisor and the dividend should be arranged according to the descending powers of some letter common to both, as in example 3 below; or according to their ascending powers, as in example 4. In example 3, for instance, we have arranged both the divisor and dividend according to the descending powers of a, putting the highest power of a first, then the next lower, and so on; the remainders, too, must be similarly treated. If we had subtracted the second line from the first just as they stand, the remainder would have been written thus,— b4+a3b; but this was altered in the writing down that the highest power of a might be placed first in the remainder, thus,ab+b4. Again, after the second piece of subtraction, if we had not altered the order of the terms the remainder would have been written thus, ba+a2b2 - ab3; but these were altered in the writing down to place the highest powers of a first, then the next lower, and so on; thus, a2b2 — ab3 + ba.

In example 4, the divisor, dividend, and remainders are arranged in an exactly opposite manner, that is, according to the ascending powers of a.

The student will generally find that the examples as they are given are already arranged according to the former of these two plans.

In these examples the quotients are placed above the dividends merely that they might occupy less space.

The following are additional examples; but the student need not work through them until he has done to No. 16, Exercise VIII. :

(1.) a2+2ab+b2 by a+b; and a2-b2 by a-b. (2.) 6x2+x-35 by 2x+5, and by 3x-7.

(3.) 30x1 + 23x3 —– 14x2 by 5x2 – 2x, and by 6x2 +7x. (4.) 3x4-300 by

(5.) x3- a3 by x

2+10, and by 32-30.·

a; ; and x2+y3 by x+y.

(6.) a3 – b3 by a2 +ab+b2; and a3 + b3 by a2 − ab +b2.

(7.) 9a1 − 4a2b2+4ab3 – b1 by 3a2 + 2ab − b2.

1

(8.) 1 − 2x − 31x2 + 72x3 – 30x4 by 1 + 4x - 10x2.

(9.) a1+4a3x+4a2x2 − x1 by a2 +2ax+x2.

(12.) x7 – 5x5 +7x3 + 2x2 − 6x − 2 by x3 – 2x – 2.

(13.) 23-y3+23+ 3xyz by x-y+2.

(14.) 54x3y2-8x2y3- 17xy4-12y by x2-2xy-312. (15.) a6+2a3b3+b6 by a2 +2ab+b2.

(16.) x+2x3y3+y6 by x2+ 2xy + y2.

(17.) a2+6ab3 +963 - 4y by a + 2y +363.

-

(18.) 16m-n2 by 2m-n1.

(19.) a2+2a+a-4 by a+at+2.

(20.) a3+a3b§+b3 by a3+a‡b3 + b3.

*The answers are on the "Subscribers Page."