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[PDF page 347] INDEX. 331 Purbach on π, 216 Puzzles, Arithmetical, 4–29 — Geometrical, 48–65 — Mechanical, 67–75 Pythagorean Symbol, 148 Quadrature of Circle, 211–222 Queens Problem, Eight , 97–103 Queens, Problems with, 97–103 Racquet-ball, Cut on, 83–85 Railway Puzzles (shunting), 53–54 Ramification, 154, 155 Raphael on Astrology, chap.x Ravenna, Labyrinth at, 152 Rayleigh, 84, 86, 307 Record, R., 171 Regiomontanus on π, 216 Reimer on Delian Problem, 204 Reiss, 26, 63 Relative Motion, 69 Reneu, W., 174 Renton, 42 Reynolds, O., 312 Rhind Papyrus, 214 Riccioli on the Calendar, 292 Rich, J., 270 Richard, J., 69, 284 Richards on use of compass, 302 Richter on π, 221 Riemann, G.F.B., 276, 284, 286 Rigaud, S.P., 314 Ring-Dial, 298–300 Rockliff Marshes, Labyrinth at, 152 Rodet on Arya-Bhata, 215 Rodwell on Hyper-space, 276 Roget, P.M., 162, 163, 164–167 Romanus on π, 217 Rome, Labyrinth at, 152 Röntgen Rays, 311 Rooke, L., 171 Rosamund’s Bower, 150 Rosen on Arab values ofπ, 216 Rothschild, F., 244 Routes on a Chess-board, 25, 158 Row, Counters in a, 48–50, 58–61 Rudio on π, 211 Russell, B.A.W., 68 Rutherford on π, 220 Saccheri, J., 276, 284 Saffron Walden, Labyrinth at, 152 Sailing, Theory of, 79–82 Sand-clocks, 300 Sarrau on Ether, 307 Saunderson, N., 171 Sauveur, Magic Squares, 122, 123, 136 Scale of Notation, Denary, 10 — Puzzles dependent on, 10–12 School-girls, Fifteen, 103–109 Schubert on π, 211 Schumacher, 220 Scott, Sir Walter, 244 Scytale, The, 257 Seconds, def. of, 289, 290 Secret Communications, chap. xi Seelhoff, 224, 226, 229, 231 Selander on π, 213 Senate-House Examination, chap. vii Seneca on Astrology, 243 Senior Optimes, 173, 176, 177 Seventy-seven Puzzle, 44 Shanks on π, 220 Sharp on π, 219 Shelton, T., 270, 271 Sherwin’s Tables, 219 Shuffling Cards, 109–111 Shunting Problems, 53–54 Sidereal Time, 289 Simpson’s Euclid, 188

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[PDF page 348] 332 INDEX. Sixty-five Puzzle, 42 Skeleton Cubes, 26 Smith, A., onπ, 222 — Hen., on Numbers, 33 — R., 171, 177 — R.C.,see Raphael Snell on π, 218, 219 Solar Time, 289 Sosigenes on Calendar, 291, 292 Sound, Problem in, 87 — Velocity of, 305 Southey on Astrology, 248 Southwark, Labyrinth at, 153 Space, Properties of, chap.xii Spectrum Analysis, 305 Spectrum Top, 87 Spin on Cricket-ball, 85 Spirits, Raising, 248 Sporus on Delian Problem, 207 Sprague on Eleven Queens, 101 Squaring the Circle , 211–222 Stability of Equilibrium, 72–74 Statical Games, 48–52 Steen on the Mousetrap, 119 St Laurent on cards, 110 Stöffler on the Calendar, 292 Stokes on Ether, 307 St Omer, Labyrinth at, 152 Story on the Fifteen Puzzle, 88 Strabo on Lake Moeris, 150 Stringham on Hyper-space, 283 Sturm, A., 204 St Vincent, Gregory of, 208 Styles, 296 Suetonius, 268 Sun-cylinders, 297 Sun-dials, 296–298 Sun-rings, 298–300 Sun, the Mean, 290 Svastika, 152 Swift, 67 Sylvester, 49, 51, 108, 109 Tacitus on Astrology, 243 Tait, 46, 59, 60, 145, 148, 304, 308, 315, 319 Tanner on Shuffling Cards, 110 Tarry, 57, 140, 150 Tartaglia, 2, 16, 20, 27, 55 Tate, 269 Tavel, G.F., 190 Taylor, B., 171 Taylor, Ch., on Trisection, 210 Tennis-ball, Cut on, 83–85 Tesselation, 50–51 Theory of Numbers , chap. ix Thibaut on Baudhayana, 215 Thompson on Magic Squares, 123 Thomson, J.J., 304, 308, 311, 312, 319 Thomson, Sir Wm.,see Kelvin Thrasyllus on Astrology, 244 Three-in-a-row, 48–50 Three-pile Problem, 115–119 Three-Things Problem, 18–19 Tiberius on Astrology, 244 Time, chap. xiii — Equation of, 290 — Measurement of, 287–290 — Units of, 287–291 Tissandier, 70 Todhunter, J., 198 Tonstall, C., 171 Tower of Hanoï , 91–93 Trastevere, Labyrinth at, 152 Trees, Geometrical, 154–155 Treize, Game of, 120 Tricks with Numbers , 4–25 Tripos, Mathematical, chap. vii Tripos, Origin of term, 200–202 Trisection of Angle , 209–211 Tritheim, J., 274

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[PDF page 349] CH. XIV] INDEX. 333 Trollope on Mazes, 151 Troy-towns, 152 Turton, W.H., 39, 44, 102 Uhlemann on Astrology, 236 Unicursal Problems, chap. vi Van Ceulen onπ, 217, 218 Vandermonde, 63, 158, 163 Van Etten, 2, 10 Vase Problem, 16 Vega onπ, 220 Vick on Clocks, 301 Vieta, 208, 217 Vince, S., 188, 189, 190 Violle, Magic Squares, 122, 136 Virgil, 150 Voigt on Ether, 307 Volpicelli on Knight’s Path, 158 Von Helmholz, 78, 276, 307, 308 Vortex rings, 308 — Spheres, 308 — Sponges, 309 Voting, Question on, 26 Walecki, 107 Walker, G.T., 21 Wallis, J., 93, 95, 171, 219, 269 Ward, S., 171 Waring, E., 176, 184, 186, 189, 190 Warnsdorff, Knight’s Path, 164 Watch Problem, 13 Watches, 77, 290, 300, 301 — as Compasses, 302–303 Water-clocks, 300, 302 Waterloo, Battle of, 296 Watersheds and Watercourses, 48 Watson, R., 177, 185 Waves, Superposition of, 87 Weber-Wellstein, 229 Week, Days of, from date, 296 Week, Names of Days, 290–291 Weights Problem, the , 27–29 Western on Binary Powers, 32 Wheatstone on Ciphers, 268, 270 Whewell, W., 170, 189, 190, 191, 192, 236, 244 Whist, Number of Hands at, 26 Whiston, W., 171 Wiedemann on Lake Moeris, 150 Wilkins on Ciphers, 252, 253, 257, 265, 274 William III of England, 153 Willis on Hauksbee’s law, 82 Wilson on Ptolemy, 237 Wilson’s Theorem, 190 Wing, Labyrinth at, 152 Wood, J., 184, 188, 189 Woodhouse, R., 190, 191, 192 Woolhouse, Kirkman’s Problem, 103 Worcester, Marquis of, 271 Wordsworth, C., 170, 176, 183, 185, 202 Work, 72–75 Wranglers, chap. vii, 173 Wright, E., 171 Wright, J.M.F., 193 Year, Civil, 291–293 Year, Mohammedan, 293 Zach on π, 220 Zech, R., 301 Zeller, 296 Zeno on Motion, 67–68 Zodiac, Signs in Astrology, 238, 241 CAMBRIDGE: PRINTED BY JOHN CLAY, M.A. AT THE UNIVERSITY PRESS.

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[PDF page 350] 334 A SHORT ACCOUNT OF THE HISTORY OF MATHEMATICS By W.W. ROUSE BALL. [Third Edition. Pp. xxiv + 527. Price 10s. net. ] MACMILLAN AND CO. Ltd., LONDON AND NEW YORK. This book gives an account of the lives and discoveries of those mathe- maticians to whom the development of the subject is mainly due. The use of technicalities has been avoided and the work is intelligible to any one acquainted with the elements of mathematics. The author commences with an account of the origin and progress of Greek mathematics, from which the Alexandrian, the Indian, and the Arab schools may be said to have arisen. Next the mathematics of medieval Europe and the renaissance are described. The latter part of the book is devoted to the history of modern mathematics (begin- ning with the invention of analytical geometry and the infinitesimal calculus), the account of which is brought down to the present time. This excellent summary of the history of mathematics supplies a want which has long been felt in this country. The extremely difficult question, how far such a work should be technical, has been solved with great tact.... The work contains many valuable hints, and is thoroughly readable. The biographies, which include those of most of the men who played important parts in the development of culture, are full and general enough to interest the ordinary reader as well as the specialist. Its value to the latter is much increased by the numerous references to authorities, a good table of contents, and a full and accurate index.—The Saturday Review. Mr. Ball’s book should meet with a hearty welcome, for though we possess other histories of special branches of mathematics, this is the first serious attempt that has been made in the English language to give a systematic account of the origin and development of the science as a whole. It is written too in an attractive

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[PDF page 351] 335 style. Technicalities are not too numerous or obtrusive, and the work is interspersed with biographical sketches and anecdotes likely to interest the general reader. Thus the tyro and the advanced mathematician alike may read it with pleasure and profit.—The Athenæum. A wealth of authorities, often far from accordant with each other, renders a work such as this extremely formidable; and students of mathematics have reason to be grateful for the vast amount of information which has been condensed into this short account.... In a survey of so wide extent it is of course impossible to give anything but a bare sketch of the various lines of research, and this circumstance tends to render a narrative scrappy. It says much for Mr. Ball’s descriptive skill that his history reads more like a continuous story than a series of merely consecutive summaries.—The Academy. We can heartily recommend to our mathematical readers, and to others also, Mr. Ball’sHistory of Mathematics. The history of what might be supposed a dry subject is told in the pleasantest and most readable style, and at the same time there is evidence of the most careful research.—The Observatory. All the salient points of mathematical history are given, and many of the results of recent antiquarian research; but it must not be imagined that the book is at all dry. On the contrary the biographical sketches frequently contain amusing anecdotes, and many of the theorems mentioned are very clearly explained so as to bring them within the grasp of those who are only acquainted with elementary mathematics.—Nature. Le style de M. Ball est clair et élégant, de nombreux aperçus rendent facile de suivre le fil de son exposition et de fréquentes citations permettent à celui qui le désire d’approfondir les recherches que l’auteur n’a pu qu’effleurer.... Cet ouvrage pourra devenir très utile comme manuel d’histoire des mathématiques pour les étu- diants, et il ne sera pas déplacé dans les bibliotheques des savants.—Bibliotheca Mathematica. The author modestly describes his work as a compilation, but it is thoroughly well digested, a due proportion is observed between the various parts, and when occasion demands he does not hesitate to give an independent judgment on a dis- puted point. His verdicts in such instances appear to us to be generally sound and reasonable.... To many readers who have not the courage or the opportunity to tackle the ponderous volumes of Montucla or the (mostly) ponderous treatises of German writers on special periods, it may be somewhat of a surprise to find what a wealth of human interest attaches to the history of so “dry” a subject as math- ematics. We are brought into contact with many remarkable men, some of whom have played a great part in other fields, as the names of Gerbert, Wren, Leibnitz, Descartes, Pascal, D’Alembert, Carnot, among others may testify, and with at least one thorough blackguard (Cardan); and Mr. Ball’s pages abound with quaint and amusing touches characteristic of the authors under consideration, or of the times in which they lived.—Manchester Guardian. There can be no doubt that the author has done his work in a very excellent way.... There is no one interested in almost any part of mathematical science who

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[PDF page 352] 336 will not welcome such an exposition as the present, at once popularly written and exact, embracing the entire subject.... Mr. Ball’s work is destined to become a standard one on the subject.—The Glasgow Herald. A most interesting book, not only for those who are mathematicians, but for the much larger circle of those who care to trace the course of general scientific progress. It is written in such a way that those who have only an elementary acquaintance with the subject can find on almost every page something of general interest.—The Oxford Magazine.

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[PDF page 353] 337 A PRIMER OF THE HISTORY OF MATHEMATICS By W.W. ROUSE BALL. [Second Edition. Pp. iv + 148. Price 2s. net. ] MACMILLAN AND CO. Ltd., LONDON AND NEW YORK. Thisbook contains a sketch in popular language of the history of math- ematics; it includes some notice of the lives and surroundings of those to whom the development of the subject is mainly due as well as of their discoveries. This Primer is written in the agreeable style with which the author has made us acquainted in his previous essays; and we are sure that all readers of it will be ready to say that Mr. Ball has succeeded in the hope he has formed, that “it may not be uninteresting” even to those who are unacquainted with the leading facts. It is just the book to give an intelligent young student, and should allure him on to the perusal of Mr. Ball’s “Short Account.” The present work is not a mereréchauffé of that, though naturally most of what is here given will be found in equivalent form in the larger work.... The choice of material appears to us to be such as should lend interest to the study of mathematics and increase its educational value, which has been the author’s aim. The book goes well into the pocket, and is excellently printed.—The Academy. We have here a new instance of Mr. Rouse Ball’s skill in giving in a small space an intelligible account of a large subject. In 137 pages we have a sketch of the progress of mathematics from the earliest records up to the middle of this century, and yet it is interesting to read and by no means a mere catalogue.—The Manchester Guardian. It is not often that a reviewer of mathematical works can confess that he has read one of them through from cover to cover without abatement of interest or fa- tigue. But that is true of Mr. Rouse Ball’s wonderfully entertaining little “History of Mathematics,” which we heartily recommend to even the quite rudimentary mathe- matician. The capable mathematical master will not fail to find a dozen interesting facts therein to season his teaching.—The Saturday Review.

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[PDF page 354] 338 A fascinating little volume, which should be in the hands of all who do not pos- sess the more elaborateHistory of Mathematicsby the same author.—The Math- ematical Gazette. This excellent sketch should be in the hands of every student, whether he is studying mathematics or no. In most cases there is an unfortunate lack of knowl- edge upon this subject, and we welcome anything that will help to supply the deficiency. The primer is written in a concise, lucid and easy manner, and gives the reader a general idea of the progress of mathematics that is both interesting and instructive.—The Cambridge Review. Mr. Ball has not been deterred by the existence and success of his larger “His- tory of Mathematics” from publishing a simple compendium in about a quarter of the space.... Of course, what he now gives is a bare outline of the subject, but it is ample for all except the most advanced proficients. There is no question that, as the author says, a knowledge of the history of a science lends interest to its study, and often increases its educational value. We can imagine no better cathartic for any mathematical student who has made some way with the calculus than a careful perusal of this little book.—The Educational Times. The author has done good service to mathematicians by engaging in work in this special field.... The Primer gives, in a brief compass, the history of the advance of this branch of science when under Greek influence, during the Middle Ages, and at the Renaissance, and then goes on to deal with the introduction of modern analysis and its recent developments. It refers to the life and work of the leaders of mathematical thought, adds a new and enlarged value to well-known problems by treating of their inception and history, and lights up with a warm and personal interest a science which some of its detractors have dared to call dull and cold.—The Educational Review. Itisnottoomuchtosaythatthislittleworkshouldbeinthepossessionofevery mathematical teacher.... The Primer gives in a small compass the leading events in the development of mathematics.... At the same time, it is no dry chronicle of facts and theorems. The biographical sketches of the great workers, if short, are pithy, and often amusing. Well-known propositions will attain a new interest for the pupil as he traces their history long before the time of Euclid.—The Journal of Education. This is a work which all who apprehend the value of “mathematics” should read and study..., and those who wish to learn how to think will find advantage in reading it.—The English Mechanic. The subject, so far as our own language is concerned, is almost Mr. Ball’s own, and those who have no leisure to read his former work will find in this Primer a highly-readable and instructive chapter in the history of education. The condensa- tionhasbeenskilfullydone, thereader’sinterestbeingsustainedbytheintroduction of a good deal of far from tedious detail.—The Glasgow Herald. Mr. W.W. Rouse Ball is well known as the author of a very clever history of mathematics, besides useful works on kindred subjects. His latest production isA Primer of the History of Mathematics, abookofonehundredandfortypages, giving in non-technical language a full, concise, and readable narrative of the development

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[PDF page 355] 339 of the science from the days of the Ionian Greeks until the present time. Anyone withaleaningtowardsalgebraicorgeometricalstudieswillbeintenselyinterestedin this account of progress from primitive usages, step by step, to our present elaborate systems. The lives of the men who by their research and discovery helped along the good work are described briefly, but graphically.... The Primer should become a standard text-book.—The Literary World. This is a capital little sketch of a subject on which Mr. Ball is an acknowledged authority, and of which too little is generally known. Mr. Ball, moreover, writes easily and well, and has the art of saying what he has to say in an interesting style.—The School Guardian.

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[PDF page 356] 340 MATHEMATICAL RECREATIONS AND ESSAYS By W.W. ROUSE BALL. [Fourth Edition. Pp. xvi + 402. Price 7s. net. ] MACMILLAN AND CO. Ltd., LONDON AND NEW YORK. This work is divided into two parts; the first is on mathematical recre- ations and puzzles, the second includes some miscellaneous essays and an account of some problems of historical interest. In both parts ques- tions which involve advanced mathematics are excluded. The mathematical recreations include numerous elementary ques- tions and paradoxes, as well as problems such as the proposition that to colour a map not more than four colours are necessary, the explanation of the effect of a cut on a tennis ball, the fifteen puzzle, the eight queens problem, the fifteen school-girls, the construction of magic squares, the theory and history of mazes, and the knight’s path on a chess-board. The second part commences with sketches of the history of the Mathematical Tripos at Cambridge, of the three famous classical prob- lems in geometry (namely, the duplication of the cube, the trisection of an angle, and the quadrature of the circle) and of Mersenne’s Numbers. These are followed by essays on Astrology and Ciphers. The last three chapters are devoted to an account of the hypotheses as to the nature of Space and Mass, and the means of measuring Time. Mr. Ball has already attained a position in the front rank of writers on subjects connected with the history of mathematics, and this brochure will add another to his successes in this field. In it he has collected a mass of information bearing upon matters of more general interest, written in a style which is eminently readable, and at the same time exact. He has done his work so thoroughly that he has left few ears for other gleaners. The nature of the work is completely indicated to the mathematical student by its title. Does he want to revive his acquaintance with the Problèmes Plaisans et Délectablesof Bachet, or theRécréations Mathématiques et Physiques of Ozanam? Let him take Mr. Ball for his companion, and he will have

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[PDF page 357] 341 thecreamoftheseworksputbeforehimwithawealthofillustrationquitedelightful. Or, coming to more recent times, he will have full and accurate discussion of ‘the fifteen puzzle,’ ‘Chinese rings,’ ‘the fifteen school-girls problem’et id genus omne. Sufficient space is devoted to accounts of magic squares and unicursal problems (such as mazes, the knight’s path, and geometrical trees). These, and many other problems of equal interest, come under the head of ‘Recreations.’ The problems and speculations include an account of the Three Classical Problems; there is also a brief sketch of Astrology; and interesting outlines of the present state of our knowledge of hyper-space and of the constitution of matter. This enumeration badly indicates the matter handled, but it sufficiently states what the reader may expect to find. Moreover for the use of readers who may wish to pursue the several heads further, Mr. Ball gives detailed references to the sources from whence he has derived his information. These Mathematical Recreations we can commend as suited for mathematicians and equally for others who wish to while away an occasional hour.—The Academy. The idea of writing some such account as that before us must have been present to Mr. Ball’s mind when he was collecting the material which he has so skilfully worked up into hisHistory of Mathematics. We think this because ... many bits of ore which would not suit the earlier work find a fitting niche in this. Howsoever the case may be, we are sure that non-mathematical, as well as mathematical, readers will derive amusement, and, we venture to think, profit withal, from a perusal of it. The author has gone very exhaustively over the ground, and has left us little opportunity of adding to or correcting what he has thus reproduced from his note- books. The work before us is divided into two parts: mathematical recreations and mathematical problems and speculations. All these matters are treated lucidly, and with sufficient detail for the ordinary reader, and for others there is ample store of references.... Our analysis shows how great an extent of ground is covered, and the account is fully pervaded by the attractive charm Mr. Ball knows so well how to infuse into what many persons would look upon as a dry subject.—Nature. A fit sequel to its author’s valuable and interesting works on the history of mathematics. There is a fascination about this volume which results from a happy combination of puzzle and paradox. There is both milk for babes and strong meat for grown men.... A great deal of the information is hardly accessible in any English books; and Mr. Ball would deserve the gratitude of mathematicians for having merely collected the facts. But he has presented them with such lucidity and vivacity of style that there is not a dull page in the book; and he has added minute and full bibliographical references which greatly enhance the value of his work.—The Cambridge Review. Mathematicianswithaturnfortheparadoxesandpuzzlesconnectedwithnum- ber, space, and time, in which their science abounds, will delight inMathematical Recreations and Problems of Past and Present Times.—The Times. Mathematicians have their recreations; and Mr. Ball sets forth the humours of mathematics in a book of deepest interest to the clerical reader, and of no little attractiveness to the layman. The notes attest an enormous amount of research.— The National Observer.

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[PDF page 358] 342 Mr. Ball, to whom we are already indebted for two excellent Histories of Math- ematics, has just produced a book which will be thoroughly appreciated by those who enjoy the setting of the wits to work.... He has collected a vast amount of information about mathematical quips, tricks, cranks, and puzzles—old and new; and it will be strange if even the most learned do not find something fresh in the assortment.—The Observatory. Mr. Rouse Ball has the true gift of story-telling, and he writes so pleasantly that though we enjoy the fulness of his knowledge we are tempted to forget the considerable amount of labour involved in the preparation of his book. He gives us the history and the mathematics of many problems ... and where the limits of his work prevent him from dealing fully with the points raised, like a true worker he gives us ample references to original memoirs.... The book is warmly to be recommended, and should find a place on the shelves of every one interested in mathematics and on those of every public library.—The Manchester Guardian. A work which will interest all who delight in mathematics and mental exercises generally. The student will often take it up, as it contains many problems which puzzle even clever people.—The English Mechanic and World of Science. This is a book which the general reader should find as interesting as the mathe- matician. At all events, an intelligent enjoyment of its contents presupposes no more knowledge of mathematics than is now-a-days possessed by almost everybody.—The Athenæum. An exceedingly interesting work which, while appealing more directly to those who are somewhat mathematically inclined, it is at the same time calculated to interest the general reader.... Mr. Ball writes in a highly interesting manner on a fascinating subject, the result being a work which is in every respect excellent.— The Mechanical World. É um livro muito interessante, consagrado a recreios mathematicos, alguns dos quaes são muito bellos, e a problemas interessantes da mesma sciencia, que não exige para ser lido grandes conhecimentos mathematicos e que tem em gráo elevado a qualidade de instruir, deleitando ao mesmo tempo.—Journal de sciencias mathematicas, Coimbra. The work is a very judicious and suggestive compilation, not meant mainly for mathematicians, yet made doubly valuable to them by copious references. The style in the main is so compact and clear that what is central in a long argument or process is admirably presented in a few words. One great merit of this, or any other really good book on such a subject, is its suggestiveness; and in running through its pages, one is pretty sure to think of additional problems on the same general lines.—Bulletin of the New York Mathematical Society. A book which deserves to be widely known by those who are fond of solving puzzles ... and will be found to contain an admirable classified collection of inge- nious questions capable of mathematical analysis. As the author is himself a skilful mathematician, and is careful to add an analysis of most of the propositions, it may easily be believed that there is food for study as well as amusement in his pages.... Is in every way worthy of praise.—The School Guardian.
