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SUPT. R. K. BUEHRLE, LANCASTER, PA.
OW that the agitation for simplified might not be accomplisht, if this time
speling is going on, and is likely to claim devoted to the writing and pronunciation considerabl attention becaus of the great of ofen misleading letters were givn to the probability of its accomplishing material acquisition of some other language, or some results, it may not be out of place to call other practically useful study? In this reattention to the great waste of time and spect, the American pupil, as well as his material incurd by the retention and inser- English cousin, is at a very great disadvantion of silent letters. On examination of tage. While the forener who is desirous of the editorial colum of one of our local acquiring a knowledge of English at a papers, it is found by actual count that mature age when his logical faculty is supabout 91% of all the letters used ar silent, posed to be developt and should be of some or in other words, useles. This would service to him, is simply confounded when show that the quantity of paper requird in told the pronunciation of a very large the printing of books and papers, etc., is number of our words which set all logic at increast by about one-tenth, and hence the defiance, such as for instance: tough, dough, cost of the paper alone is increast by that slough, through, thorough, and bade, have, amount. Of cours, the labor requird to give and live. print is in the same proportion, and it The argument ofen herd that everybody would therfor seem that with the adoption would be expected to lern the new speling, of simplified speling the cost of all printed is entirely fallacious, becaus, while, of matter would be one-tenth les. Naturally cours, the children would be expected to ther would be a saving in its carriage by lern only the new way of speling, it is very mail.
reasonabl to suppose that those who have It will easily be seen that in a cuntry lernt the old way of speling would keep on wher ther is so much printed matter as in writing their way if they preferd that, ours, the saving would run into the mil while the proof-readers would simply do lions. In the matter of time consumed in what their office would tel them to do with the writing of silent letters in the schools, regard to the speling of words in print. we have another tremendous item of waste This, to some extent, they do now. If a and on careful examination it was found speling reformer sends his copy to the that in the third year cours of our schools printing office they politely inform him pupils can write on an average only twenty that it cannot be printed that way, and that silent letters per minute, in the fourth year he must submit to the rules of the office as thirty-one, in the fifth year forty-four, and regards the speling of his production, or els in the sixth year forty-eight. Now assum- hav it rejected. The task therfor would be ing that the proportion of silent letters is imposed only on the proof-readers. It the same in the exercises in the schools as would simply be an extension of the practis it has been found to be in the editorial of now followd in the case of the thousands of the paper referred to, the waste of time for words which are spelt in two ways by recogthe eighteen millions of pupils in the schools nized authorities. Indeed I suppose that in the cuntry is simply appalling. What in a very short time this would ceas, becaus
if it is once establisht as a practis to con- sider, fighting the sentiment of the gang. form the speling as far as possibl to the I have never known any teacher to win pronunciation, everybody wil be anxious out permanently in discipline, who set to reduce the number of letters requird to himself against the sentiment of his school. properly spel a word, to the smallest pos- On the other hand, if you are adopted by sibl number. This is the fundamental law the “crowd” as a “brother," a "good felin all language. Changes in pronuncia- low," you can, if you are capable and judition are going on constantly in the direction cious achieve anything you wish. I have of economy of effort, abbreviating words constantly to inspect the work of teachers by rejecting sounds that ar difficult or re- who, although they know much, alike of quire special effort. The aim of the advo- matter and method, are nevertheless failcates of simplifying our speling means ures because they do not understand the simply that we should make changes in psychology of the gang; they do not feel speling to correspond with the changes in easy with boys; they become self-conscious pronunciation, for many, if not all our and formal as soon as they get into a group silent letters wer once sonant. The finale of young people. So the boys draw away in have, the gh in through, the t in nestle from the teacher, more or less unconand often, wer all once sounded. Occa- sciously; antagonisms slowly spring up, sionally entire syllabls ar dropt as in and ultimately he is looked upon as an drawing-room for withdrawing-room, enemy, and a proper target for the gang's 'phone for telephone and auto for auto- aggression and ridicule. Make it your first mobile. All this is in the direction of labor- object in entering any school to show that saving; why should it not be extended to you have sympathy with the group, and speling? Why should anybody prefer to that you can, on the playground as well write programme for program?
as in the school-room, be a leader, and you
will never have any serious trouble with Controlling the Gang
discipline. The greatest word in discipline Have you in your school a boy who is is leadership in and out of the school. a member of a “gang?" If you have you Popular Educator. know that he considers whatever the gang approves is right, and may be performed;
WHERE GO THE BOATS? and if this happens to be wrong, the best
Robert Louis Stevenson you can do, by your discipline, is to hold him in check temporarily. If you cannot
Dark brown is the river,
Golden is the sand. get control of the gang and influence it as
It flows along forever, a whole, you cannot accomplish much in
With trees on either hand. the way of reforming any individual mem
Green leaves a-floating, ber thereof. Your point of attack must
Castles of the foam, always be the ringleader of the group;
B-ats of mine a-boatingyou must strive to turn his sentiment
Where will all come home? your way. In order to accomplish this,
On goes the river you must exert a strong influence outside
And out past the mill, of the school-room; you must possess
Away down the valley, ability to adapt yourself to the impulses
Away down the hill. and interests of boys in a group. You must
Away down the river, be able to become as one of them for the
A hundred miles or more, time, and work from the inside. You can
Other little children never accomplish much good as an out
Shall bring my boats ashore.
Multiplication and Division of Decimals
WILLIAM F. WHITE, PH.D., MATHEMATICS, STATE NORMAL SCHOOL, NEW PALTZ, N.Y.
FOR the multiplication of whole numbers - Professor Cajori, of Colorado College,
the Italians invented many methods and Professor David Eugene Smith, of Pacioli (1494) gives eight. Of these, only Teachers College, Columbia University. one was in common use, and it alone has Romain's method is advocated in a few survived in commerce and the schools. of the best recent advanced arithmetics, Shown in I below. It was called bericu- but Stevin's is still vastly the more comocolo (honey cake or ginger bread) by the mon; and these two are the only methods Florentines, and scacchiera (chess or checker in use. Romain's has four slight advantboard) by the Venetians. The little ages: (1) A person setting down an squares in the partial products fell into example from dictation can begin to write disuse (and with them the names which the multiplier as soon as the place of its they made appropriate) leaving the familiar decimal point is seen, while in Stevin's form II below. The Treviso arithmetic method he waits to hear the entire multi(1478) the first arithmetic printed, con- plier before he writes any of it, in order tains a long example in multiplication, to have its last (right hand) figure stand which appears about as it would appear beneath the last figure of the multiplicand on the blackboard of an American school (though this position may be regarded as today.
a nonessential feature in Stevin's arrangeIn 1585 appeared Simon Stevin's im- ment). (2) Romain's method fits more mortal La Disme, only seven pages, but naturally with the “Austrian" method of the first publication to expound decimal division (decimal point of quotient over fractions, though the same author had that of divisor). (3) After the partial used them in an interest table published products are added, it is not necessary to the year before. III (below) is from La count and point off in the product as many Disme, and shows Stevin's notation (tenths decimal places as there are in the multithe first order of decimals, etc). IV is the plicand and multiplier together, since the same example with the decimals expressed decimal point in the product is directly by the notation now prevalent in America. beneath that in the multiplicand. Let us call this arrangement of work: (4) Romain's method is more readily Stevin's method.
adapted to abridged multiplication where A quarter of a century later, an arrange
only approximate results are required. ment was used by Adrian Romain in which
On the other hand Stevin's method has all decimal points are in a vertical column.
one very decided advantage: the first (See V below.) It may have been used
figure written in each partial product is earlier. I shall be glad to have attention called to an earlier use. In the meantime
directly beneath its figure in the multiI shall, for the sake of a name, call this
plier, so that it is not necessary, as it is in arrangement Romain's method.
Romain's, to determine the place of the For the historical facts I am indebted decimal point in a partial product. So mainly to the two leading American important is this last, that Stevin's alone authorities on the history of mathematics has been generally taught to children, notwithstanding the numerous points in arrangement of work shows what figures favor of Romain's.
to omit. It occurred to the writer recently to It should be remarked that all three try to combine in one method the advant- methods of multiplication of decimals are ages of both of the Flemish methods, and alike—and like the multiplication of whole he hit upon the following simple rule: numbers—in that one may multiply first Write the units figure of the multiplier by the digit of lowest order in the multiunder the last (right hand) figure of the plier or by the digit of highest order first. multiplicand, begin each partial product The method of multiplying by the highest under the figure by which you are multi- order first was described by the Italian plying, and all decimal points in products arithmeticians as a dietro. Though it may will then be directly beneath that in the seem to be working backwards, it is not so multiplicand. Decimal points in partial in fact; for it puts the more important products may be written or not as desired. before the less, and has practical advantage The reason underlying the rule is apparent. in abridged multiplication, like that shown VI shows the arrangement of work.
in VII. But that question is distinct from In this arrangement the placing of the the one under consideration... partial products is automatic, as in Stevin's Stevin writes the last figure of the multimethod, and the pointing off in the product plier under the last figure of the multiis automatic, as in Romain's. It is avail- plicand; Romain writes units under units; able for use by the child in his first multi- the method here proposed writes units plication of decimals and by the skilled under last. In whole numbers, units figure computer in his abridged work.
is the last. To assist in keeping like decimal orders Applied to the ordinary multiplication in the same column it is recommended of decimals, as in VI or VIII, the method that a vertical line as shown in VII and here proposed seems to be well adapted to VIII be drawn before the partial products schoolroom use, possessing all the simplicity are written. This is analogous to the line of Stevin's. Methods classes in this normal separating cents from dollars in ledgers, school to whom the method was presented, billheads, etc. The first use of a vertical immediately preferred it, and a grade in line as decimal separatrix is probably in the training school used it readily. Of an example in Napier's Rabdologia (1617). course this proves nothing, for every He draws it through the partial and com- method is a success in the hands of its plete products. This is also the first advocates. example of abridged multiplication. A The analogous method for division of circumference is computed whose diameter decimals would possess analogous advanis 635.
tages. It would avoid the necessity of VII illustrates the application of the multiplying the divisor and dividend by method here advocated to multiplication such a power of 10 as will make the divisor in which only an approximation is sought. integral, as in the method now perhaps The diameter of a circle is found by most in favor, and the necessity of counting measurement to be 74.28 cm. This is to point off in the quotient a number of correct to o.o1 cm. No computation can decimal places equal to the number in the give the circumference to any higher dividend minus that in the divisor, as in degree of accuracy. Partial products are the older method still common. Like the kept to three places in order to determine latter, it begins the division at once; and the correct figure for the second decimal like the former, its pointing off is automatic. place in the complete product. The IX shows the arrangement. The figure
under the last figure of the divisor is units figure of the quotient. This determines the place of the decimal point. That part of the quotient which projects beyond the divisor, is fractional.
If the order of multiplication used has been a dietro, as in VIII, the division in IX is readily seen to be the inverse operation. The partial products appear as partial dividends in the same order.
Like each of the methods in use, it may be abbreviated by writing only the remainders below the dividend, and omitting the decimal points from them. Shown in X.
If the “little castle" method of multiplication of whole numbers, with multiplier above multiplicand, had prevailed, instead of the “chess board,” in the 15th century, the arrangement now proposed for the multiplication and division of decimals would have afforded slightly greater advantage.
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