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early inhabitants, of their language and manners, of General Wolfe, of M. Papineau, of the recent rebellion. And with what unmixed delight does the intelligent student of geography look upon the map of such a country as Spain. He forgets for the moment its size, and form, and colouring. He thinks of Hannibal and of Carthage, of Scipio and the Celtiberian Prince, of the Goths, of the victorious Moors, of their gorgeous palaces, of their dances, and their songs. He glows with the chivalry of Rodrigo the Cid, or he thinks of the petty jealousies of Gallegan, Catalan, Andalusian, and Castilian, in a country which nature has made one; he recollects Ferdinand and the inquisition, Charles V., and America ; or fancies he sees an auto da , a bull-fight, or a castanet dance. There is not a country of Europe, and there is scarcely a country of the world, that would not afford the materials for a most interesting essay, in which all these suggestions could be appropriately applied, and beautifully blended, with every important town, province, mountain, and river.

INSTRUCTIONS IN ARITHMETIC.

(Continued from page 301.)

CHAPTER III.-ON THE POUR FUNDAMENTAL OPERATIONS OF ARITH

METIC, AND THE NATURE OF NUMBERS IN RELATION TO THEM.

58. The numbers we have hitherto added or subtracted have been either abstract numbers, or, if concrete, of the same kind; and this follows from the definition of addition and subtraction. For it is evident that the sum or difference of any numbers, can only be the sum or difference of units of the same kind. Thus the sum or difference of nine shillings and four shillings, can only be thirteen shillings or five shillings; and when we treat of abstract numbers, their sum or difference must be abstract also.

59. In multiplication, the multiplier is an abstract number, and the product, for the most part, of the same species as the multiplicand.

For, if I inquire the price of eight yards when one yard costs five shillings, I must repeat the price of one yard as many times as there are yards; that is, I must multiply five shillings by eight. Now eight is an abstract number, for I say, eight times five shillings, and to say eight yards multiplied into five shillings, would be an expression without meaning

We have seen in the case of abstract numbers, that 5 X 8= 8 X 5; but in the previous example, to say that eight yards are to be repeated five times, would be incorrect, seeing that the answer to the question would then be in yards, and not in shillings.

Hence, we should always take care to assume for the multiplicand that which is so in reality, and is therefore to be repeated. The fact, however, that 5 X 8 = 8 X 5, leads to this inference;—that eight yards at five shillings the yard, cost as much as five yards at eight shillings, and in like manner, if the question be the price of 20 yards at four shillings the yard, we might say that it would be equivalent to the

B

D

A E

C

price of four yards at 20 shillings, or 4 yards at one pound, that is, four pounds; and this change in the order of the factors is sometimes useful.

60. When two factors are measures of length of the same kind, and we are treating of surfaces, the product is formed of squares, which have for their sides a unit of that kind.

And again, if we are treating of solids, and there are three factors, being measures of length of the same kind, the product is formed of cubes, each of which has for its side a unit of that kind.

61. Sometimes the two factors and the product are lines; in which case, however, one of the factors is considered as an abstract number, ex. gr.: :-Let it be asked, what is the inclination or the difference of level of two points, which are distant eight yards from each other, when for each yard the inclination is two inches ?

Let A B, being the line of inclination, be equal to eight yards; and let AC be the horizontal line. The inclination of a yard, AD, is D Е, which by supposition is two inches.

For two yards the inclination will be double; and for BC, or the inclination of the line A B, we shall have eight times DE, or 2 inches x 8 = 16 inches.

Hence we see that eight is an abstract number, and that the result, or 16,

represents inches. 62. In division, the divisor is in general an abstract number, and the product is of the same nature as the dividend. This is always the case in questions which are referable to the two first mentioned modes of considering the process of division (vide Sect. 49 and 50); for let it be proposed to divide 12 shillings between three persons, and it is evident that the number 3,

which indicates into how many parts we must divide the twelve shillings, ceases to be a concrete number, and no longer represents three persons, but solely the number 3, and is consequently abstract; and in dividing 12 shillings into three equal parts, each part consists of four shillings, or a concrete number of the same kind as the dividend.

If we reason according to the first mode (Sect. 49), calling x the share of each person, x will be so many shillings, and if we repeat them three times, we shall have 12 shillings, and thus the factor 3, which is the divisor, is an abstract number.

63. According to the third mentioned mode (Sect. 51) of considering division, the dividend and the divisor, being concrete numbers, are of the same kind, and the quotient is an abstract number, in order that it may subsequently be a concrete number of another kind; ex.gr., When the question is how many yards can be bought for 20 shillings, when one yard costs five shillings, and we ask how often five shillings is contained in 20 shillings, the quotient, or 4, is an abstract number; but as the same number, 4, indicates also how many yards we can buy with 20 shillings, it thus becomes a concrete number, though of a differ

ent kind from the divisor and dividend. When we have problems or questions, the converse of those given in Sect. 60 and 61, it will be easy to determine the nature or kind of each number, since the dividend is the product of two factors, of which one is the quotient and the other the divisor.

F. CURIE. (To be Continued.)

ON CERTAIN CURIOUS PROPERTIES OF NUMBERS, WITH

THEIR APPLICATION TO THE CONSTRUCTION OF QUES

TIONS FOR THE USE OF TEACHERS. INDEPENDENTLY of the utility of the following methods of constructing questions, the properties of numbers upon which they depend are curious and instructive, and have not, so far as I know, been noticed by any writer upon this subject. Although some of the methods here proposed may be less simple, and therefore less generally useful, than the method given in my treatise on the “First Principles of Arithmetic, ” yet they will enable the intelligent teacher to give a greater variety to the form of his questions.

Addition. Ex. 1.-142,524 Here, taking the central point of the first hori

213,120 zontal row, the figures on the left hand, added to 340,215 the corresponding figures on the right, produce 023,421 sixes; in the second they produce threes; in the

third fives ; in the fourth fours; and so on, ob719,280 serving that the sum of the numbers 6, 3, 5, and 4

is some multiple of nine ; then, in the answer, the figures to the left of the central point, added to the corresponding figures to the right, produce nines. The only precaution to be observed is simply, that the sum of the last column should not exceed nine. Ex. II.-14,52 In this example the numbers made up by the

23,43 figures in the question are sixes, and the figures 32,34 in the answer make up nines, as in the last ex

ample,

70,29 Ex. III.—23,54 In this example the numbers made up by the

14,74 figures in the question are taken at pleasure, and 28,93 in the answer the figures make up eights.

67,21 When questions are constructed as in example 3, the law of the figures in the answer will frequently appear in a disguised form; but the true law may be exhibited by adding a certa number (determined from the form of the question) to the answer, as in the following example :

Ex. IV.-2461 Here the numbers made up by the figures in

1342 the question, are, 8, 5, 3, and 6; subtracting each 3201 of these numbers from 9, we have 1+4+6+3= 165 9+5; that is, after casting out the nines, the re

mainder is 5; we therefore add 55 to the answer, 7172 and then the result gives the law of nines. The 55 number of figures to be added must always be half

the number of figures in the answer.

7227 Ex. V.-124653 Here the figures in the first row produce sevens,

53835 in the second eights, and in the third fourteens; 587967 then 2+1+4=7; and by adding 777 to the an

swer, the law of nines will be produced. 766455

777

:

767232 As the most general form has been given to the construction of questions according to this method, in the rule of subtraction, in the “ First Principles of Arithmetic," any further exposition, on this part of the subject may be deemed unnecessary.

Multiplication. Ex. I.-214,7,563 X 36=7731,2268. Here the figures in the multiplicand make up sevens, with a central seven ; the addition of the figures in the multiplier make up nine ; and the figures in the product, or answer, observe the usual law of nines. The figures in the multiplicand may make up any number whatever. The only precaution to be observed is simply, that the product of the first figures in the multiplicand and multiplier, that is, 2 x3, does not produce a number greater than nine, and for this purpose it is only necessary to keep these figures low numbers. When there are three figures in the multiplier, there must be two central figures in the multiplicand, and so on, as in the following example :

Ex. II. 132,88,756 x 243=32291,67708. Here the figures in the multiplicand make up eights.

Ex. III.-14,6,52 x 48=703,296. Here the figures in the multiplicand make up sixes; the addition of the figures in the multiplier gives twelve; and the figures in the product, or answer, follow the law of nines. Now 6 and 12 are both multiples of 3. The numbers respectively made up by the figures in the multiplicand and multiplier, may be any multiples of 3. When there are three figures in the multiplier, there must be two central figures in the multiplicand, and so on as in the last case.

Ex. IV.-121,3,212 x 57=6915,3084. In this example, the figures in the multiplicand make up threes; the addition of the figures in the multiplier gives 12, which is a multiple of 3, and the figures in the product follow the law of nines.

Ex. V.- 132,6,534 x 42=5571,4428. In this example, the figures

in the multiplicand make up sixes, and the figures in the multiplier, added together, give six ; now 6 being a multiple of 3, the figures in the product will follow the law of nines.

Ex. VI.-201,465 X3=604,395.
Ex. VII.-143,66,523 x 342=49133,50866.

Let us now consider the most general form in which these questions may be proposed. Ex. VIII.-153,7,624 x 48=7380,5852

6666

7381,2618 In this example, the figures in the multiplicand make up sevens, with a central seven ; the two figures of the multiplier are written at pleasure; then to obtain the test for the accuracy of the answer, we cast out the nines from the figures of the multiplier; thus, 4+8=9+3, which gives us 3 for the remainder; we now subtract 7 (the number made up by the figures in the multiplicand), from 9, which gives us 2, and multiplying these remainders together, we have 3 X 2=6; then, 6666 being added to the product, the resulting figures will follow the law of nines. The figures in the multiplicand may make up any number. When there are three figures in the multiplier, there must be two central figures in the multiplicand, and so on as before mentioned. Ex. IX.-243,5,312 x 37=9010,6544

4444

9911,0988 In this example, the figures in the multiplicand make up fives ; casting out the nines from 3+7, we have 1 for the remainder; then 9–5=4, and 4x1=4; we now add fours to produce the law of nines. Ex. X.-1273,9837 x 7=8917,8859

2222

8918,1081 Ex, XI.—231,44,213 x 361=83550,60893

55555

83551,16448 The method given in the “ The First Principles of Arithmetic,” is a particular case of the method here proposed.

T. TATE,
Mathematical Master of the National Society's

Training Institution, Battersea.
(To be continued.)

HINTS TO TEACHERS ON SETTING SUMS. Rev. Sir,-A correspondent in the October number has given us “ A Help to Teachers in Setting Sums.” I apprehend that but few teachers

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