= Then we have the system: y+z = 4 (1), y'+z1 square (1), y2+z2+2yz = 16 or y2+z2 16-2yz (3), = (16-2yz)2 (4), = (16-2yz)2 (5), 82 (2), (5) is a quadratic equation in yz. Its roots are 3 and 29; The system y+z or y = 1, z = = 3 gives y = 3, z = 1, hence x = = 4 and yz = -26; = 29 gives imaginary roots. The system y+z 54, Solutions were also received from NELLIE F. HENDERSON and ROXANNA E. SPILLER. Geometry. 567. Proposed by N. P. Pandya, Sojitra, India. Circumscribe a triangle about a given circle, the ratio of the angle bisectors being known. A note on this will be published in the next issue.-Editor. 568. Proposed by Murray J. Leventhal, Stuyvesant High School, New York City. The line that joins the orthocenter of a triangle to any point on the circumscribed circle bisects the line joining the feet of the perpendiculars dropped from this point upon the three sides of the triangle. Solution by Nelson L. Roray, Metuchen, N. J. Let ABC be the given triangle, O the ortho-center, P any point on the circumcircle, PD LAC, PLLCB, PK1AB, ASICB, BRIAC, CT LAB, S' point on circle where AS intersects circumference, DLK Simson's Line intersecting AS at M. Note: PO does not bisect Simson's Line nor can it, since the line is a straight. PO bisects certain parts of it, e. g., LM in the above. Also solved by Philomathe. 569. Proposed by N. P. Pandya. The vertex A of a triangle is the center of a given circle. P and Q are points of intersection of AB, AC, respectively, with the circle. The tangents at P and Q divide the base in the ratios kland m : n, respectively. Construct the triangle ABC. No solutions of this problem have been received. The editor will consider it in the next issue.-Editor. By Sturm's theorem, we see that this equation has but one real root, and it is between 3 and 4. By Horner's method, I found -3.93911, hence tg 0/2 -3.93911, and 0/2 = 104°14'40", or x 208°29'20". = Note. The general solution is n.360° +208°29′20′′. Late Solutions. 558. Angus McLeod, Aberdeen, Scotland. Answers to Queries. Concerning certain objections to proofs of problem 554, which was published in May, the editor wishes to say that although solutions II and III are not worked out in detail, to save space, it is true that when the proper algebraic reductions are performed the result may be stated in a manner similar to that used in solution VI at the close. The solver of III suggests that (a2+b2-c2)/2ab be substituted for cosC in performing the reduction. It must be remembered that a, b, and c are positive values by the nature of the problem. PROBLEMS FOR SOLUTION. 581. Proposed by A. MacLeod, Aberdeen, Scotland. The six straight lines which trisect the angles of a triangle meet in three points which form the vertices of an equilateral triangle. Could four equilateral triangles be found determined by the twelve interior intersection points of these six lines?--Editor. 582. Proposed by A. MacLeod. A and B are two points outside a circle unequally distant from the center. Find the point C on the circle such that AC+CB is a minimum. 583. Proposed by Norman Anning, France. Solve the triangle, a = 181, A = 120°, b-c = 1. 584. Proposed by Norman Anning. AD, BE and CF are the medians of the triangle ABC; show that cotADB+cotBEC+cotCFA = 0. BIRDS HELP SCHOOL GROUNDS. HOW AND WHY THEY SHOULD BE ATTRACTED TO THE YARD OR CAMPUS. Birds have such an important part in the courses of nature study given in most schools and colleges that more should be done, according to the United States Department of Agriculture, to attract them to school grounds or campuses. Unfortunately, says the department, the campus, like the park, has suffered from too formal landscape gardening. Cropped hedges are not inviting to the birds, and wooded campuses where formerly birds abounded have been so filled with buildings, so gardened and formalized that birds are now scarce. Some corner-preferably of original woodland and undergrowth if that still exists-should be set aside and allowed to run wild, as a bird sanctuary. BIRDS AN EDUCATIONAL RESOURCE. In addition to the help that birds give as protectors of trees and shrubbery against injurious insects, they furnish, on school grounds, a distinct educational resource. There is scarcely an advanced school in the country that does not offer courses in bird study; and the study of birds out of doors is a necessary supplement to that in classroom and laboratory. For this reason, says the department, the educational need should be kept in view by those in charge of college and school grounds. Plants which furnish bird foods should be added, nest boxes put up, and winter feeding carried on as an interesting and instructive part of the work in bird-study classes. It is further suggested that the teacher of ornithology should be called in to advise in relation to the planning and treatment of the campus. Those who are interested in attracting birds to public and semipublic reservations can get a publication devoted to this subject by writing to the United States Department of Agriculture at Washington for Bulletin 715. BOOKS RECEIVED. Census of the Virgin Islands of the United States, by Eugene F. Hartley. 174 pages. 15x23 cm. Paper. 1918. Government Printing Office. Airplane Characteristics, by Frederick Bell, Cornell University. Pages v+123. 16x23 cm. Cloth. 1918. $1.60 net. Taylor & Company, Ithaca, New York. Petroleum; A Resource Interpretation, by Chester G. Gilbert and Joseph E. Pogue, United States National Museum. 81 pages. 15.5x24.5 cm. Paper. 1918. Government Printing Office. Ontario High School Laboratory Manual in Chemistry, by George A. Cornish, University of Toronto, and Arthur Smith, Central Technical School, Toronto. Pages vii+135. 12.5x19 cm. Cloth. 1917. 25 cents. The Macmillan Company, Toronto. Ontario High School Chemistry, by George A. Cornish, University of Toronto, and Arthur Smith, Central Technical School, Toronto. Pages vii+297. 12x19.5 cm. Cloth. 1918. 50 cents. The Macmillan Com pany, Toronto. Essentials of Modern Chemistry, by Charles E. Dull, South Side High School, Newark, New Jersey. Pages ix +458. 13.5x19.5 cm. Cloth. 1918. Henry Holt & Company, New York City. General Science Manual, by Leland R. Thompson, High School, East Chicago, Ind. Loose-leaf. 101 experiments. 21x27 cm. Atkinson, Metzger & Company, Chicago. BOOK REVIEWS. General Science Manual, by Leland R. Thompson, High School, East Chicago, Ind. Loose-leaf. 101 experiments. 21x27 cm. Atkinson, Metzger & Company, Chicago. A splendid, practical, and up-to-date manual on general science. A book that can work into almost any laboratory and be used with any text on this subject. It deserves a wide and extensive sale. C. H. S. Economy in Food, by Mabel T. Wellman, Indiana University. 36 pages. 13x19 cm. Cloth. 1918. 30 cents net. Little, Brown & Company, Boston. A remarkable little book which ought to be in the possession of every housekeeper and person who buys food for the family or restaurant. In a small space rules and advice are given on economy in buying, economy in storing food, economy in planning meals, economy in the use of leftovers, and economy in fuel. There is a splendid table of cost of one hundred calorie portions of foods and factors for calculating them. C. H. S. Second Course in Algebra, by Herbert E. Hawkes, Colorado University, William E. Lobe, Kansas City Polytechnic High School, and Frank C. Touton, Central High School, St. Joseph, Mo. Pages vii+277. 13x19 cm. Cloth. 1918. $1.00. Ginn & Company, Boston. This is a revision of a well-known and widely circulated text. It has been considerably simplified, and some chapters have been omitted in accordance with the present tendencies to dismiss from school courses those that are nonessential. In the place of these omitted chapters the emphasis is put on the material which is of everyday and lasting importance. The exercises, of which there are a large number, are practically new, and many of them are for oral use. Instructors will find this as nearly perfect a text in algebra as it is possible to produce. C. H. S. Elementary General Science, by Daniel R. Hodgdon, Normal School, Newark, N. J. Pages xxii +553. 15x20.5 cm. Cloth. 1918. Hinds, Hayden & Eldredge, New York City. This remarkable book has been written for use in the seventh and eighth grades and the first years of the junior high school. It is one of the very latest books on general science, and without question is one of the best texts lately put on the market in this subject. The volume is written in a most fascinating style, and in such a manner as will appeal to the interests of the young mind. The author has distributed the amount of matter over the various sciences that are discussed, very uniformly, giving each science, as he understands it, its fair proportion of space. He has made good use of books and matter bearing upon the subjects treated in the text, and has given due credit. He is very wise in the selection of the 395 cuts and illustrations, many of which are new and to the point. The chapter on "Nostrums" is peculiarly apropos. Too much cannot be said in praise of the volume, and it undoubtedly will receive a very large adoption. C. H. S. The Organization of Thought, by A. N. Whitehead, Trinity College, Cambridge. Pages vii+228. 14.5x22.5 cm. Cloth. 1917. $2.00. J. P. Lippincott Company, Philadelphia. This volume is a compilation of discourses which were delivered by the author on various occasions. The book treats of the idea of thought as presented in the eight addresses included in the book. A common line of thought or reflection extends through the book. The fact that the discourses are given in an informal way makes it of a more practical value to the everyday reader than if the talks were of a more technical nature. The author is well informed on the results of the events that are now transpiring in Central Europe, and he has brought his thought into connection with the general reformation of all activities which must necessarily follow the conclusion of this great war. From cover to cover there are many suggestions that will meet the approval of the reader. The book is mechanically well made, and is printed on thick, uncalendered paper, in heavy, doubl spaced type, making it very easy to read. The volume should have a wide circulation. C. H. S. Elements of General Science, Revised Edition, by Otis W. Caldwell, Teachers College, Columbia University, and William L. Eikenberry, University of Kansas. Pages xii +404. 13.5x20 cm. Cloth. 1918. Ginn & Company, Chicago. This revised edition of a splendid book has been brought up to date, and in this treatment of the various subjects the authors have, if possible, improved upon the excellent first edition. There is nothing of a detrimental nature that can be said concerning this most excellent book. It is the result of many years' experience in the instruction of secondary pupils by these two most competent authors, and that which they present in this volume is authoritative, accurate, and to the point. The 181 cuts and illustrations have been selected with a view to illustrating directly the material treated in the text. The authors' study of the general science situation of the United States has been of such a character that they have been able to produce a book which can be used without any difficulty in any part of the country. Doubtless the new edition will have as wonderful a sale as the first edition had. School men will make no mistake in adopting this book for their classes. C. H. S. Hering, New Essentials of Physics for College Students, by Daniel W. 1914. $1.75 net. D. Van Nostrand Company, New York City. This is the second edition of this most excellent text in physics, and is the outgrowth of a series of lectures which the author delivered before the undergraduate students of New York University. There are a number of changes made in this edition over the first one, and in every case the change has been for the better. It has been the object of the author to produce a book in reasonably complete form, so that the ordinary man |