HISTORY OF ARITHMETIC.

XIII-DECIMALS.

There are obviously two ways in which decimals may have originated. They may have come as a special form of common fractions, or they may have been the result of an extension of the decimal idea below units. These methods of origin are directly opposite. Which is the true origin will probably never be known. Some of the early writers take one view and some the other. The notion that was in the mind of the man who first gave the idea to the world can not now be determined.

The earliest indication of a decimal is found in the writings of Orontius Fineus, in a work published in 1525. In order to get the square root of 10 he extracts the square root of 10,000,000, getting 3.162. He then separates the 162 and uses it to get sexagesimal fractions. He does, however, make the statement that in the 162, 1 is a tenth, 6 is six hundredths, etc. This would indicate that he had the decimal idea. For the next fifty or sixty years various writers used Orontius's method but avoided the decimal idea.

In 1685 Simon Stevin de Bruges, better known as Stevinus, wrote a work which he called La Disme. In this book the real idea of a decimal is first made prominent. He made a very strong argument for the use of his nombres de disme. His notation was very awkward, but his thinking on the subject was very clear. By most writers Stevinus is considered the father of decimals. De Morgan, Dr. Peacock, and the Britannica all give him this distinction.

Stevinus's work was introduced into England in 1608 through a translation made by Richard Norton. Richard Witt in 1613 made free use of the decimal notation in his tables of compound interest, although he expressly states that the numbers are to be considered the numerators of common fractions whose denominators are 100. .... Napier in 1617 wrote a treatise on the subject, and by some is thought to have been the inventor of the decimal point. In 1619 Henry Lyte, Gentleman, published a work on the subject. In presenting the merits of decimals he' became so enthusiastic as to say, "If God spare my life, I will spend some time in most cities of this land for my countrie's good, to teach this,

art." William Oughtred in his Claris of 1631 so presented decimals as to greatly help in making them popular.

The exact origin of the decimal point is no better known than the origin of decimals themselves. Some very ingenious hypothetical origins have been suggested, but they lack one essential element-verification. Many of the early forms were ingenious but cumbersome. The rise of decimals to popularity would have been more rapid had a simple form of the separatrix been invented at the start. The usual law of progress was followed and the separatrix evolved from the complex to the simple.

Some of the more important forms will now be given with the date and name of the inventor. For this we will use the number 3.1416.

THE NEW INDIANA ARITHMETICS-I.

There will go into the schools of Indiana this fall a new series of arithmetics from which much is expected. The authors, Professor Cook of Illinois and Miss Cropsey of Indianapolis, are both practical and successful school people. Both have made a special study of the subject. Much of the material in each book of the series is the result of schoolroom experience. The elementary book is largely the work of Miss Cropsey, while the advanced book is due to Professor Cook.

The mistake may be made of expecting too much from the new books. There are some things which text-books can not do. No textbook in an elementary subject can take the place of the teacher. The better the book the greater aid it will be to him. The real spirit of the subject, however, must become vital to the student through the medium of the teacher. No book, however good, supplies all the needs of all pupils. It must necessarily have in view that hypothetical individual known as the average pupil. The teacher, in the use of any book, must have the courage both to supplement and to cut out. For some classes the list of problems in a particular subject may be inadequate, while other classes in the same school need only half the list. The wise teacher will always fit the conditions to his class regardless of his text-book.

A good text-book by its logical arrangement, careful development and statement of principles, and its large number of well-graded and interesting exercises, apparently diminishes the work of the teacher. This reduction is only apparent, for to realize all that the good text-book makes possible, the teacher must look deep into the subject and see it in many of its bearings which do not seem necessary in the use of a poor book.

To many teachers the elementary book will be disappointing because it does not contain any work for the first and second years. This omission should not be interpreted to mean that no number work should be done in these two years. It only means that during this period the work should be oral, and wholly controlled by the teacher. Much of it must necessarily be objective. It is the time par excellence for learning to count. The number system, which lies at the basis of all mathematics, is known through the medium of counting. Children take a keen interest in counting. Its importance, united with the child interest it arouses, makes it a thing of fundamental consideration in early school life. This counting should be by one's, two's, three's,

etc., both forward and backward. Of course no attempt should be made to hurry matters, but everything should be done carefully. The advance should be slow and every point should be clinched by frequent reviews. The counting should certainly extend to one hundred.

Although many consider it rank heresy, the use of figures in the first and second grades is a great help to the children in their counting. Figures are as necessary in number work as letters are in reading.

In the first two years it seems reasonable to expect that the children should form a good acquaintance with the number system, and through it should know all the simpler number combinations of the addition, subtraction, multiplication and division tables and that they should be able to express ordinary numbers by means of figures.

The American Book Company has just issued Tes Plane and Solid Geometry. The author

zited, in a very admirable way, the invenas and demonstrable geometry. Each propon is preceded by a number of questions deged to lead the student to a correct underanding of the truth to be demonstrated. While this is a splendid thing in the early study

the subject, it is doubtful if it is needed in the higher books. The arrangement of the book is such that this matter in whole or in part could be omitted at any stage of the pupil's progress. The number and character of the exercises is a strong feature of the work. At the end of each book there is a Summary, giving the truths established in a simple but systematic way.

There will appear in these columns a number of articles upon the use of the New Arithmetics. It is the intention to serve, in the best way we can, the interests of the common-school teachers. We invite you to send to the editor of the department whatever questions you may have upon the new books. In the articles to follow we will discuss as fully as space will allow the questions which seem to be the most important.

From the press of Harper Bros. there comes a very interesting little book under the title of Observational Geometry. In the character and arrangement of the material the author has done a fine piece of work. It is not only suggestive but also instructive. No boy or girl can open the book without desiring to know its contents. The practical construction of models, and the frequent use of level, square, straightedge and compass closely relates the book to the world of nature and art surrounding the student. The illustrations are characteristically good. The use of such a book in the seventh and eighth grades would almost revolutionize the study of geometry.

R. L. Myers & Co. of Harrisburg, Penn., are the publishers of Weidenhamer's Mental Arithmetic. The book is made up entirely of problems. It proceeds in a logical way from very simple easy problems to those that are quite difficult. The problems are of such a character as to constantly call forth the best thought of a student. The book is a valuable contribution to the neglected subject of mental arithmetic.

In announcing a department of music for the INLAND EDUCATOR it is not intended to offer a series of lessons in music, nor to outline a course of study, nor yet to advocate any particular method of teaching music. The advocates of the various plans have ably and amply set forth their respective merits, and many friends of each will be found among readers of the EDUCATOR. The purpose is rather to supplement what is already being done by presenting each month one or two songs, and a small quantity of reading matter prepared or selected with special reference to a finer appreciation of music and of its value in the public schools. It is most encouraging to feel that modern psychology recognizes the importance of the emotive power in character building, and the boundless influence that good music has to awaken this power. The poet told us long ago that "music has charms to soothe the savage breast," but it is vastly more to know and to recognize the wonderful power it has over the impressionable minds of children. The words and melody of a beautiful song fixed in the memory of a child shall do more in the shaping of his life than a score of sermons. It is possible to attune a young life to a harmonious rhythm that will influence and dominate all his years. It is this thought, then, that will guide in the selection of material for this department. We have a rich store to select from and shall demand from every candidate for admission that it possess beauty, tone, meaning and life.

A WORD ON METHOD.

It goes without saying that there must be some definite method of procedure in the music teaching of the schools. While the singing of good songs has a value that is being recognized more and more by the thoughtful educators of the country, the mere singing of songs accomplishes little or nothing by way of imparting a knowledge of the elements of music to the child and,

unless music teaching in the schools accomplishes more than mere song singing, it can hardly hold a place in the schools as an educational factor. If children are to acquire a knowledge of music, there must be some system of gradation, some method of classification, so that the material presented may be adapted to the capacity of the child and his efforts in music-study may be properly directed. And it is the principle upon which this gradation is based that is the all important one in music teaching.

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Many efforts have been made during the past few years to grade music so as to put it on a practical working basis and properly relate it to the regular branches of school work. The saying to teach music as we teach arithmetic and reading," has grown to be a very trite one. But now educators everywhere are asking themselves why these so-called educational plans of teaching music, with their rigid gradations and orderly progression of sight-reading exercises, get so little response from the child. The answer is that music cannot be treated at first as a study in the elementary facts of the science; it must be treated as a manner of expression for the child, and such phases of the art must be presented to the child at first as he will naturally respond to. The most beautiful and inspiring of all studies of the school curriculum, it loses its charm and sinks to a practical utilitarian level when robbed of its inspiration and set before the child as a study in bare technique. Most especially is this true when these studies are embodied in long successions of sight-reading exercises, the child's progress being regulated by his ability to read them.

HELPING CHILDREN TO BECOME MUSICAL.*

A beautiful song and its correct interpretation embody all the elements of music teaching, even in the most advanced stages of study. I would make the statement now that I believe that all musical difficulties may be reached and solved through the needs of interpretation, and that without a loss of interest to the child or a necessity of much musical drudgery. We are learn(Continued on page 32.)

* From a paper read before the National Educational Association.