investigations are incorporated into habitual daily practice. It is one of our immediate tasks to make sure that this knowledge is disseminated throughout our schools, and that the principles and practices which are thoroughly established are made effective in our classroom teaching, for, after all, good teaching is the basis of all true progress. If our schools are to fulfil the task given them, the citizen must be made not only economically or socially efficient-he must be trained in the proper use and enjoyment of leisure time. Art, music, and literature ennoble and enrich life, and their joys and pleasure must be constant possessions. Recreation should be a phase of definite instruction. Libraries, museums, and concert halls are filling an increasing place in civic life. More and more these activities are becoming identified with our public schools, and this work should be largely extended. To have a taste for good literature, fine art, and worthwhile music, to find pleasure in the competition of games, and to enjoy the wonders of the great outdoors insures an extended measure of happiness to the individual and increased tranquillity for the state. Some means must be found for making available the experience of the teaching body when the superintendent of schools is considering educational plans and policies. To interpret democracy to children, teachers must themselves understand and appreciate the principles of democracy. To grow and do their best work they must see the large problems and the relation which their work bears to these problems. To plan wisely, the superintendent heeds the counsel of his entire corps. The teachers' council, rightly conceived, and administered as a help to the superintendent, has come to serve the schools. Finally, the school system must be made to commend itself to the community. In a great metropolitan city the teachers, cooperating with the board of education and numerous organizations vitally interested in and friendly to public education, undertook the task of "selling the schools" to those most interested in the children. A three-year campaign resulted in 150 per cent advance in educational revenues. In a dignified way, the people have been induced to advance from a levy of fourteen million in 1919 to thirty-four million in 1922 for the education of their children. The average community is interested in its children; it is a part of the duty of a city system, as a professional organization, to lead the way, to interpret the schools to the community. GRAPHS. BY CHARLES H. SAMPSON, 320 Huntington Ave., Boston, Mass. More and more the subject of graphs demands attention as a department of the subject of algebra that needs to be stressed. Heretofore we have not given this mathematical method of indicating values the attention that it deserves. Neither have we realized the important place that it occupies in the field of those things that can be practically used. Algebra is not always considered a practical subject and properly so. We do not use it as a whole in much of the work that we are called upon to do from day to day. It is not a practical working tool in the same way that other branches of mathematics are. It has, of course, other values that are of great benefit to the student; for example, that of mathematical mental exercise. But graphs differ from most of the other members of the algebra family in that the rules and principles that are used with them can be practically applied. There are very good reasons why we, as students, should understand how and why graphs are used. We must also realize that the study of them is worthwhile because of value of their use in many fields of everyday work. One can easily understand the value of plotting methods (graphs) if he will use his powers of observation. Instances are everywhere evident of this. In every large city one may see practical applications of the use of graphs. In Boston, for example, on the "common" we have the daily temperatures plotted as the changes take place. The result is not merely temporary. The graph gives a permanent record of a large amount of information on a comparatively small sheet of paper. And best of all the record is in a form that is easily understood and in a form that is so compact that data for a long space of time can be filed away without occupying much room. There are many other examples of the value of graphs as a means of keeping in a satisfactory manner records of importance. One can think of several of the things that we like to have recorded. Birth rates, as they change from year to year; averages in the stock market; production of material from day to day in a factory or manufacturing plant, etc. All of these and similar things may be compactly and completely represented by employing graphs. The teaching of graphs is not so much a question of how as when. The writer feels that the subject should be covered as a whole rather than by the "now and then" method so generally followed. It does seem wise to discuss the subject for a little while, drop it, and then take it up again at a period when the mental attitude of the student is apt to be different than when the subject was first discussed. The whole question simmers down to "Is it not better to devote the first year of algebra study entirely to hammering on the fundamentals?" The job there ought to be that of building a firm foundation. Graphs may be given a little attention merely as a matter of interest, but to treat them too seriously at this stage in the course is somewhat of a mistake. They can be studied much more effectively later. The pad method of studying graphs is the most practical and effective. Every sheet on these pads has an explanation at one side and a place for the solution of a similar problem at the other side. One is taught how to do a graph problem in a neat and accurate manner if the pad method is followed, and there is the added advantage of tying the explanation and the problem together. When should the study of graphs be undertaken? The answer "the last subject covered should be graphs." The student has by this time learned all processes needed for their proper solution. He can then attempt them with some assurance that he will know what he is doing. THE SETTLING OF PRECIPITATES BY CENTRIFUGATION. By WALTER O. WALKER, High School, Carthage, Mo. The application of centrifugal force in settling of precipitates in liquids is by no means new. Advanced agricultural, biological, and chemical laboratories have made use of machines for centrifugation for a number of years. However, the High School and even the college chemical laboratories have failed almost unanimously in recognizing the great amount of time saved in filtering processes by centrifugation. It is astonishing the little use made of the Babcock apparatus, and machines of a like nature. Most High School chemical laboratories have in their equipment Babcock milk testing apparatus. By a simple modification, this tester can be changed to accommodate the average sized test tube. A cork is selected which will fit snugly in the top of the receptacle, which ordinarily carries the Babcock bottle. The cork is then cut in two, parallel to its base. The smaller piece, after having had a small hole scooped out in its center to accommodate the bottom of the test tube, is placed in the bottom of the Babcock receptacle. The larger upper half of the cork is fitted in the top of the Babcock receptacle and a hole (large enough to admit the test tube) is bored through it. If the receptacles are all fitted in this manner, several test tubes can be accommodated at one time. The usual precaution with regard to a balanced load must necessarily be followed. The time saved in filtering processes for a class in one day will warrant the installation of this modified apparatus in any chemical laboratory. The method of operation is as follows: The test tube containing the liquid and the precipitate is placed in the machine and properly balanced with another test tube of water or another test tube containing an equal amount of precipitate. The machine is rotated at normal speed. The length of time of rotation depends upon the nature of the precipitate. Barium sulphate, copper oxide, or any heavy precipitate requires about two minutes. A light gelatinous precipitate, such as aluminum hydroxide, requires about five minutes. A semi-colloid, such as arsenic, sulphide, requires more time and may not entirely be settled out by this process. In every case, the major portion of the precipitate settles out during the first minute of rotation. The additional time is necessary to clear up the filtrate (that which corresponds to the filtrate in the ordinary filtering process). For the average precipitate, the separation of precipitate and filtrate is as complete and satisfactory as in the old filtering process. The points of advantage in the process are: 1. Time saved. The ordinary process of filtering is long and tedious. 2. A clear separation of filtrate and precipitate. 3. The precipitate is left in a compact form in the bottom of the test tube while the filtrate is poured off. This makes the process a great time-saver in qualitative chemistry. 4. Immiscible liquids which have become emulsified may be settled rapidly without resort to waiting for the natural gravity process. The laboratory operation of this modified Babcock machine has proven its great value. Once installed it is of inestimable value. Two machines will generally accommodate the average laboratory section. SQUARE ROOT OF A LINE WITHOUT USE OF THE CIRCLE. BY JOHN B. WOOD, Los Angeles, Calif. The unit parallelogram has been neglected by mathematicians. I have not seen it in any modern textbook. Sir Oliver Lodge in his Easy Mathematics (by the way-is there such an animal?), says something almost on the lines of it. His thought is (this is not a quotation but my expression of it) x2+2ax+a2 is an area, the square on x+a; so x2-2x+1 is an area, the square on x-1; it seems as if a line was added to an area and twice a line subtracted from it. But it isn't so-2x is a parallelogram: x(2), 1 is a square 12 = 1(1). So, if he had further noticed (and said) r2 is the parallelogram x21 = xx, he would have introduced the unit parallelogram to mathematical teaching. There is another idea which in a sense is introductory to the unit parallelogram. That of the equality of what may be named cross parallelograms. It is, of course, understood that Euclid is "Victorian" and as much out of date mathematically as Wilkie Collins or Anthony Trollope in fiction. But for all that, as my eometry was learned more than fifty years ago, the credit must be Euclid's 43, 1. To Euclid is also due the geometric relations expressed in the algebraic equation due to Descartes, y = 2rx-x2. (Sir Oliver Lodge uses this numerically in his Easy Mathematics). It is evident that the square on y equals the parallelogram whose base is 2r-x, altitude x. If x here is 1, we have the unit parallelogram (2r-1)1. Now given a line its square root may be found by drawing the circle, 2r the diameter, etc. The vertical y standing on the line at 1 inside the rim is "found" by the circle arc. We have it in hand now to show how x as the square root of x21 may be had without use of the compass. Here, before proceeding, is the place to apologize for a slight inaccuracy in language. The parallelograms are in strict scentific phrase rectangles. The other word has been chosen as more significant, in fact; the parallel feature being the important thing for the mathematics. = 12+2dv+(dv)", Algebraically putting (1+dv) = x, x2 (dr, difference in the vertical between 1 and x). Geometrically this is the unit paralellogram x21 which is made up of 12+dv1+dv 1 +(dv) 21. And we see that x2 = xx is made up of them, noting that dvdv or (dv)2 = (dv)21. But, of course that exposition takes du as known, whereas outside of the 1X1 = 12, we do not know how 221 is divided. The critical or deciding point for the pur |