333 3333 1000, 10000, etc., and these values multiplied by 60 give the series 18, 19.8, 19.98, 19.9998, etc., which evidently approaches 20 as a limit; but the product of 60 into (the limit of the repetend 0.333, etc.) is also 20. Again, if we multiply 60 into the different values of the decreasing series, 30, 3000, 30000, etc., which approaches zero as a limit, we shall get the decreasing series 2, 3, 3, 50, etc.; and this series evidently approaches zero as a limit. In this way the pupil may easily be led to a complete comprehension of the subject of limits. The Teacher is likewise advised to give frequent written examinations. These should not be too difficult, and sufficient time should be allowed for accurately constructing the figures, for choosing the best language, and for determining the best arrangement. The time necessary for the reading of examination-books will be diminished by more than one-half, if the use of the symbols employed in this book be allowed. PHILLIPS EXETER ACADEMY, 1879. G. A. W. NOTE TO REVISED EDITION. THE first edition of this Geometry was issued about nine years ago. The book was received with such general favor that it has been necessary to print very large editions every year since, so that the plates are practically worn out. Taking advantage of the necessity for new plates, the author has re-written the whole work; but has retained all the distinguishing characteristics of the former edition. A few changes in the order of the subject-matter have been made, some of the demonstrations have been given in a more concise and simple form than before, and the treatment of Limits and of Loci has been made as easy of comprehension as possible. More than seven hundred exercises have been introduced into this edition. These exercises consist of theorems, loci, problems of construction, and problems of computation, carefully graded and specially adapted to beginners. No geometry can now receive favor unless it provides exercises for independent investigation, which must be of such a kind as to interest the student as soon as he becomes acquainted with the methods and the spirit of geometrical reasoning. The author has observed with the greatest satisfaction the rapid growth of the demand for original exercises, and he invites particular attention to the systematic and progressive series of exercises in this edition. The part on Solid Geometry has been treated with much greater freedom than before, and the formal statement of the reasons for the separate steps has been in general omitted, with a view to give a more elegant form to the demonstrations. A brief treatise on Conic Sections (Book IX) has been prepared, and is issued in pamphlet form, at a very low price. It will also be bound with the Geometry if that arrangement is found to be generally desired. The author takes this opportunity to express his grateful appreciation of the generous reception given to the Geometry heretofore by the great body of teachers throughout the country, and he confidently anticipates the same generous judgment of his efforts to bring the work up to the standard required by the great advance of late in the science and method of teaching. The author is indebted to many correspondents for valuable suggestions; and a special acknowledgment is due, for criticisms and careful reading of proofs, to Messrs. C. H. Judson, of Greenville, S.C.; Samuel Hart, of Hartford, Conn.; J. M. Taylor, of Hamilton, N.Y.; W. Le Conte Stevens, of Brooklyn, N.Y.; E. R. Offutt, of St. Louis, Mo.; J. L. Patterson, of Lawrenceville, N.J.; G. A. Hill, of Cambridge, Mass.; and G. W. Sawin, of Cambridge, Mass. Corrections or suggestions will be thankfully received. |