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 MATHEMATICAL TEACHING 
 
 MODERN METHODS. 
 
 BY 
 
 TRUMAN HENRY SAFFORD, PH.D., 
 
 FIELD MEMORIAL PROFESSOR OF ASTRONOMY IN 
 WILLIAMS COLLEGE. 
 
 8603 
 
 BOSTON: 
 
 D. C. HEATH & CO., PUBLISHERS. 
 1896. 

 
 COPYRIGHT, 1886, 
 BY D. C. HEATH & CO. 
 
 J. S. CUSHING & Co., PRINTERS, BOSTON.
 
 A \\ 
 
 5/R 
 
 PREFATORY NOTE. 
 
 THE following pages were suggested by a long practical 
 experience in giving instruction in mathematical subjects. 
 The conclusions presented in them are believed to be in 
 agreement with the views of progressive educators, and differ 
 essentially from the ordinary traditions. I am only too sen- 
 sible how imperfectly I have been able to express in words 
 the practice of better teachers than myself, of whom two 
 especially, Professors J. M. Peirce of Harvard University, and 
 George A. Wentworth of Exeter, have given me kind assist- 
 ance, which I wish here to acknowledge. 
 
 T. H. SAFFORD, 
 WILLIAMSTOWN, Dec. 2, 1886.
 
 STATE NORMAL SCHOOL, 
 
 IiOS 
 
 INTRODUCTION. 
 
 THE history of mathematical teaching in this country is 
 yet to be written. It is necessary to pay some attention ) 
 to this history in writing upon the theory ; as the traditions of 
 the elders have a great influence, partly good and partly in- 
 jurious. If we find that a tradition in mathematical teaching 
 arose from definite reasons still in force, we must be cautious 
 about rejecting it as useless ; but such are not all the methods 
 which have been handed down. 
 
 A few salient points of history may be useful to consider. 
 About 1730 the Hollis professorship of mathematics and natural 
 philosophy was founded at Harvard College ; it goes back near 
 enough to Newton's time to enable us to see the impulse which 
 caused its establishment. Again, the successors of Winthrop, 
 the first Hollis professor, taught by English methods and essen- 
 tially English books, including Playfair's Euclid, till long after 
 the Revolution. Webber's Mathematics, compiled from Hutton 
 and the other English writers, and Hutton's own books, were the 
 standard for many years. It was the military experience of the 
 Revolution which led to the employment of French and Swiss 
 professors in West Point Academy ; and its reorganization 
 under Thayer, shortly after the War of 1812, led to the produc- 
 tion by Davies, Church, and Bartlett, of a series of text-books 
 compiled from Lacroix, Legendre, and other writers of the 
 same nation. 
 
 But about the same time similar things were done at Harvard 
 College ; and the " Cambridge Mathematics " of John Farrar,
 
 6 INTRODUCTION. 
 
 similarly compiled, took the place there and in some other 
 newer colleges of the works of Hutton and Playfair. 
 
 The marking system, so far as I can gather, seems to have 
 originated in France, I do not know whether it has yet dis- 
 appeared there, and to have been introduced into this country 
 through West Point. 
 
 The French mathematics have been the standard originals 
 of our text-books from the early part of this century till the 
 present day. In giving up Euclid, our writers have not been 
 altogether consistent ; they have modified Legendre's Geometry 
 so as to make it resemble Euclid as much as possible. 
 
 Till within a few years, there have been in this country three 
 main schools for mathematicians, West Point, Harvard Col- 
 lege, and Yale College. Of these West Point was the earliest to 
 educate men in considerable numbers, who could deal success- 
 fully with large problems of applied mathematics. The profes- 
 sors, at first partly foreigners, introduced a knowledge of the 
 modern analysis, especially in its applications to engineering 
 and military matters of other kinds ; and their pupils were so 
 apt that a single generation replaced the older men with 
 American successors : it was no longer necessary to call men 
 from Europe. 
 
 The West Point graduate who has most largely influenced 
 science is doubtless A. D. Bache, a great-grandson of Franklin, 
 The Coast Survey, first conducted by Hassler, had in his hands 
 a character of strangeness ; excellent though he was in all his 
 work, he failed to adapt his ideas to some of his subordinates 
 and to certain necessary conditions of American public service ; 
 and as he grew older, he found difficulty in carrying his plans 
 out undisturbed. After Bache succeeded to the superintendency 
 of the Coast Survey in 1844, he found available for its higher 
 geodetic works a number of West Point officers, of whom T. J. 
 Lee was one, and Humphreys, afterwards chief engineer of the 
 army, another. One of the leaders in practical astronomy of
 
 INTRODUCTION. 7 
 
 the Topographical Engineers was J. D. Graham ; and the work 
 which had been done by that corps upon the national and State 
 boundaries had familiarized a good many army officers with 
 field astronomy and geodesy. 
 
 Bache, who had been out of the army nearly twenty years, 
 employed his great organizing and scientific capacity in training 
 the Coast Survey corps (including detailed army officers) into 
 practical methods for its various problems ; and the connection 
 between the West Point officers and the able young civilians, 
 who are now the veterans of the Survey, was extremely whole- 
 some. 
 
 Lee prepared a work Tables and Formulae which has 
 served an excellent purpose in bridging the gap between 
 theory and practice ; especially for the last generation of 
 West Point officers. 
 
 The graduates of this institution, however, are now more 
 closely employed in military and other public duty; so that 
 their selection as professors of mathematics in colleges is not 
 so common as it was thirty years ago ; and they are no longer 
 employed in the Coast and Geodetic Survey. 
 
 The younger naval academy at Annapolis is beginning to 
 send out scientists of ability ; but not in the same line of 
 studies as the military engineers. 
 
 Harvard College took up the new movement in mathematics 
 quite early, as I have said. The first great mathematician of 
 Eastern Massachusetts was, however, Nathaniel Bowditch, 
 who was a practical sailor and man of business. His great 
 work the translation of Laplace's Mecanique Celeste gave 
 a stimulus to mathematics and the higher astronomy which can 
 hardly be overestimated. Among the Harvard men of Farrar's 
 time were James Hayward, Charles Henry Davis, and Sears 
 Cook Walker ; who all in after life did much to promote the 
 science. Hayward was for a while professor, and strove to 
 reform the teaching of geometry ; Davis became a naval
 
 8 INTRODUCTION. 
 
 officer, served on the Coast Survey, and. established the Amer- 
 ican Ephemeris : Walker, after a business life in which he did 
 much for astronomy as an amateur, became also an official of 
 the Coast Survey, and applied the telegraphic method of deter- 
 mining longitudes very early, if he was not its sole inventor. 
 But the great Harvard mathematician was Benjamin Peirce, 
 who made Cambridge a mathematical centre at one time quite 
 unequalled in this country. Peirce was an admirable teacher for 
 those who could receive his ideas ; his quickness and depth 
 of intellect were such, however, that he lacked the patience to 
 instruct the average student, and some even of those who wished 
 a high mathematical training were compelled to admit that they 
 could not follow him. He attracted, year by year, a few of the 
 better scholars to his lectures on the Calculus and Analytical 
 Mechanics, and drew around him many of the best mathe- 
 maticians of the whole country. So that a very large share of 
 those who attended his lectures are now in high positions either 
 as investigators or teachers. 
 
 The American Ephemeris was in fact established at Cam- 
 bridge. Peirce and C. H. Davis were nearly connected in 
 family, and worked together in its establishment ; and thirty 
 years ago the office attracted young mathematicians by the 
 intricacy of the problems then needing solution, as well as by 
 the very modest recompense which they received. At present 
 the Ephemeris is on an excellent footing, and its current work 
 can be done more mechanically : the investigators have not all 
 left it, however. 
 
 At Yale College some cause which I cannot precisely indi- 
 cate perhaps the influence of Jeremiah Day and Denison 
 Olmsted gave rise to a school of young mathematicians 
 about half a century ago. Among these, besides men now 
 living, were E. P. Mason, Stanley, both of whom died com- 
 paratively young, W. A. Norton, but above all, William 
 Chauvenet.
 
 INTRODUCTION. 9 
 
 This excellent man and lucid writer was admirably adapted 
 to promote mathematical study in this country. His father, a 
 Frenchman of much culture, trained him very thoroughly in 
 the knowledge of the French language, even in its niceties. 
 They habitually corresponded in that language ; and the son 
 was enabled to study the mathematical writings of his ancestral 
 country in a way which enabled him to reproduce in English 
 their ease and grace of style, as well as their matter. In 
 these respects his works are far more attractive than those of 
 ordinary English writers : his Trigonometry is much the best 
 work on the subject which I know in any language ; his 
 Spherical and Practical Astronomy is frequently quoted by 
 eminent continental astronomers, and his Geometry has raised 
 the standard of our ordinary text-books, of which it is by far 
 the best existing. 
 
 Of late years a new mathematical school chiefly in certain 
 branches of abstract higher algebra has been established at 
 the Johns Hopkins University. The original impulse given by 
 Sylvester will, however, be broadened as well as strengthened 
 by the somewhat altered direction lately given to it ; and I 
 think that school will produce mathematical physicists as well 
 as pure mathematicians. The great increase of polytechnic 
 schools and schools of science is also favorable to applied 
 mathematics. 
 
 /""it is an old complaint against mathematics as a mental disci- 
 pline, that it is too abstract and unpractical. When we look at 
 the ordinary courses in our colleges and schools, we shall find 
 that there is much truth in this ; but the complaints are entirely 
 groundless when mathematics takes its proper place in our 
 courses, and is taught in the proper manner. 
 
 The theory most prevalent among teachers is that mathe- 
 matics affords the best training for the reasoning powers ; and 
 this in its traditional form. [The modern, and to my mind the 
 true, theory is that mathematics is the abstract form of the
 
 IO INTRODUCTION. 
 
 natural sciences ; and that it is valuable as a training of the 
 reasoning powers, not because it is abstract, but because it is 
 a representative of actual thingsJ Thus geometry, as Euclid 
 gives it to us, is an excellent training ; not because the things 
 of which it treats are unreal, however : a Euclidian geometry 
 of four dimensions would not be a good study for an ordinary 
 pupil. Again( arithmetic, as we shall see by and by, is over- 
 done, in a certain sense, in our schools ; just so far as the 
 teaching is based upon the concrete, so far is it profitable ; but 
 when the book-makers begin to make it too abstract, as they 
 very often do, it becomes a torture to both teacher and 
 learners, or, at best, a branch of imaginary knowledge uncon- 
 nected with real life. ) 
 
 The usual requirements for the degree of bachelor of arts in 
 this country include two subjects of rather doubtful benefit to 
 the average college student, analytical geometry and spheri- 
 cal trigonometry. The justification for the first is complete 
 enough, if by the name a different thing is meant, conic 
 sections. These must, to some degree, be understood, if 
 physics or astronomy form part of the course ; but their 
 proper treatment for most students is synthetic, although alge- 
 bra may be employed. Of spherical trigonometry, a brief 
 outline may be taught to the average man ; but the real utility 
 of this subject would not compensate him for the pains he 
 would need to take to get a complete idea of it. 
 
 In Germany the introduction of both subjects, even into the 
 very thorough gymnasium course, was long debated ; in England 
 neither of these is required for the bachelor's degree at Cam- 
 bridge or Oxford ; at Harvard no one need study any mathematics 
 whatever, even solid geometry ; but the admission examination 
 covers algebra and plane geometry. So that the Harvard 
 mathematical standard for graduation is nominally lower than 
 the German, nominally higher than the English ; while in most 
 other American colleges we profess to require more than do
 
 INTRODUCTION. 1 1 
 
 the institutions of either England or Germany. My own no- 
 tion is, that the American colleges would do well to require 
 arithmetic, algebra, up to, but not including, the theory of 
 equations, plane and solid geometry, and plane trigonometry, 
 for graduation ; and to greatly increase the thoroughness of 
 teaching within this range. The Yale standard seems to me 
 about as much too high as the Harvard one is too low, so far 
 as the amount required is concerned. 
 
 A good many years' experience in teaching college students 
 astronomy, both in Western colleges and at Williams, leads me 
 to think that the average man has been " over," as the phrase 
 is, more mathematics than he has digested or can apply. It is 
 rare that I find such a pupil who is even accurate in arithmetic. 
 A short course in the art of computation which I give to an 
 elective class, numbering, on an average, six or eight (introduc- 
 tory to practical astronomy), is apt to develop great want of 
 skill in ordinary calculation among the members of the class ; 
 and similar difficulties are apt to manifest themselves in all the 
 work they have done before entering college. The men who 
 do not have to be taught very elementary matters in arithme- 
 tic, algebra, and geometry, are those who unite natural ability 
 with a thorough preparation under experienced teachers. 
 Their college course seems to me to deal largely with too 
 advanced matters for their real preparation ; but it is entirely 
 the average American course ; and I find among the pupils 
 who come from Western colleges into our upper classes just 
 the same degree of inability to calculate well and accurately. 
 In the gymnasia of Germany the teachers are warned not to 
 suffer ordinary reckoning to be forgotten. 
 
 Another quite singular experience has forced itself upon me 
 again and again, in the most diverse ways. It is that, on the 
 whole, pupils whose arithmetic was taught them in foreign 
 countries, especially Germany and the North of Europe, make 
 fewer mistakes, and are more quickly able to calculate from an
 
 12 INTRODUCTION. 
 
 algebraic formula than are those taught in this country. There 
 are German teachers in the United States whom I have known, 
 whose pupils show similar capacity. The reason I take to be 
 this : that in America, arithmetic and the other rudimentary 
 branches of mathematics are taught, not for their own sake, 
 but as a training of the reasoning faculties ; and the matter in 
 the text-books is not scientifically arranged, but traditionally ; it 
 is not usually selected by mathematicians, but by schoolmas- 
 ters. These too often require of their pupils, not an indepen- 
 dent power of accurate calculation, but the ability to follow 
 and repeat the reasonings of the books, and to solve the vari- 
 ous artificial examples with the answers usually given 
 which are utterly repugnant, in too many cases, to anything in 
 nature or practical life. 
 
 The Hon. J. W. Dickinson, Secretary of the Massachusetts 
 Board of Education, and his agent, Mr. Walton, have expressed 
 themselves very strongly on this matter in the late reports of 
 the Board ; the well-known " Norfolk County Report " of Mr. 
 Walton is extremely instructive. 
 
 What the cure in detail is for this state of things, will be 
 stated farther on, in treating of the various branches ; but in 
 general, it is as follows : 
 
 1. The employment of good, experienced teachers in all 
 schools, and their proper remuneration. 
 
 2. A rational course of study, concentrated upon a few 
 essential points, and so graded that the easier matters shall be 
 taught first, and the more difficult ones later. Years ago I 
 found a certain " Higher Arithmetic " more difficult than alge- 
 bra or trigonometry. 
 
 3. The use of small but carefully arranged text-books. 
 Mehler's Hauptsatze, the text-book used by many advanced 
 teachers in Germany, gives about 150 pages to algebra, geome- 
 try, and trigonometry. This represents an extreme view. I 
 have recommended its reading, in German, to my pupils who 
 intended to become teachers.
 
 INTRODUCTION. 13 
 
 4. Objective and oral teaching, combined with extemporalia 
 upon easy problems. 
 
 5. Laboratory work in mathematics, as well as in physics. 
 Every teacher of mathematics should have learned to practi- 
 cally apply the science to surveying and astronomy, as well 
 as to physics. Every high school where pupils are to become 
 teachers, every normal school, as well as every college, should 
 have its modest collection of astronomical instruments. Many 
 high schools in this country are already so provided, and some 
 have completely equipped observatories, the Hartford (Conn.) 
 High School, for example ; Mount Holyoke Female Seminary, 
 which is almost a college, however ; as well as, in England, 
 Rugby School ; whose astronomical teacher, Wilson, became 
 head master of Clifton College. 
 
 Elementary schools should teach carefully ordinary methods 
 of weighing and measuring, both on the common and on the 
 metric system. 
 
 The qualifications of the teacher or professor of mathematics 
 should, on the mathematical side, include two things. First, a 
 knowledge of more subjects than he expects to teach, always a 
 grade higher. That is, whoever is to teach arithmetic must 
 know algebra, including logarithms, and geometry ; the teacher 
 of plane geometry must be well versed in solid geometry and 
 plane trigonometry. Second, a knowledge of the applications 
 of the subjects to be taught ; thus the teacher of spherical 
 trigonometry must be familiar with practical astronomy and 
 geodesy; and he who only teaches plane trigonometry must 
 know ordinary surveying. The teacher of arithmetic in a com- 
 mon school must be thoroughly familiar with weighing and 
 measuring, and with the practical application of the metric 
 system.
 
 METHODS OF TEACHING THE VARIOUS 
 BRANCHES OF MATHEMATICS. 
 
 I SHALL take them up in the following order : Primary Arith- 
 metic ; Scientific Arithmetic and Algebra ; Geometry ; 
 Plane Trigonometry ; Algebraic Geometry, Analytical Geome- 
 try, and Descriptive Geometry ; Spherical Trigonometry ; Dif- 
 ferential and Integral Calculus, Higher Analysis. 
 
 This order is not that of a well-arranged programme of study, 
 but is (approximately) according to the usual practice. The 
 right position of the separate branches will be indicated later. 
 Certain general remarks can,- however, be made in this chapter 
 which refer more or less to all mathematical teaching, from that 
 of the simplest numbers up to that of the highest analysis. 
 
 The first of these is, that all teaching should have a real, 
 objective basis. 
 
 To show what this should be, I will take a moderately easy 
 example from elementary algebra; logarithms. The theorem 
 on which this depends is the following : 
 
 or or = a 
 
 This is true whether m and n be positive or negative, integral 
 or fractional. The pupil is supposed to know this theoretically 
 and practically, root and branch ; and to have applied it to a 
 sufficient number of easy cases in which, for example, the 
 denominator of m and ;/ shall not exceed 10, to be thoroughly 
 convinced of its truth and practical applicability ; and to under- 
 stand the logical limitations of the formula so far as not to
 
 l6 METHODS OF TEACHING THE VARIOUS 
 
 confuse different roots of the same number. In other words, 
 he must know that the number a is positive, and its positive 
 roots only are to be considered. 
 
 The table of logarithms is now the objective basis. The 
 teacher explains that a = 10 and that the values of m corre- 
 sponding to various t values of a m are given in it. The pupil 
 will naturally ask how this table is formed ; and the teacher, 
 if competent, can give a sufficient answer. After this is done, 
 multiplication by logarithms and division by logarithms are 
 now practised until the class is thoroughly trained in these two 
 rules as far as they can be employed without interpolation. 
 
 The omission of the characteristic in the table has to be 
 explained by the teacher during this exercise, which is not com- 
 pleted until it is thoroughly understood, and, in fact, becomes 
 mechanical. 
 
 The number of decimals employed in the pupils' tables 
 should be five ; four or six will do, however. The library of 
 the institution should have at least four tables accessible, 
 Bruhns's seven-figure tables, Albrecht's edition of the six-place 
 Bremiker, and some good five and four place tables. The two 
 just mentioned are easily obtained, and not expensive. 
 
 Interpolation and the limits of accuracy in the use of the 
 tables should now be explained and made secure by practice. 
 
 After multiplication and division by logarithms have been 
 thus taught, the subjects of involution and evolution should be 
 treated in the same manner. Those of the class who are likely 
 to become teachers must here be shown that the practical 
 method of dealing with powers and roots is the logarithmic ; 
 so that in later years they will abstain from annoying their 
 young pupils with difficult and needless problems solved in the 
 antiquated manner. And they will be shown, also, how to calcu- 
 late a compound-interest table, an excellent exercise in itself, 
 as well as a labor-saving contrivance in arithmetic. In dealing 
 with roots and compound interest, interpolation must be ration-
 
 BRANCHES OF MATHEMATICS. I/ 
 
 ally explained, and not blindly executed. If, now, we ask why 
 logarithms are so little appreciated in business and engineering, 
 we shall find the reason to be that the teachers of arithmetic 
 have not, as a rule, really learned their _use_; they have " been 
 over the subject," it is true, so that they go on wasting time in 
 arbitrary exercises in involution, evolution, and compound 
 interest done by more tedious methods, and do not really 
 appreciate how instinctively the best calculators employ 
 logarithms. 
 
 Objectivity in teaching them consists, then, in making loga- 
 rithms the object. But to give a good object-lesson of any 
 kind, the teacher must know his object not merely superficially, 
 but thoroughly, and with interest in it. The man or woman 
 who is to deal with logarithms as a teacher, must be in some 
 degree a practical computer. 
 
 Let us take another example from a subject which is often 
 badly taught, solid geometry. 
 
 The objective basis here is a perception of the relations of 
 space. It is a difficult thing to produce, and that pupil is 
 fortunate who has had really good object-lessons on form at 
 and from an early age. 
 
 Solid geometry is usually taught to Freshmen in college. If 
 the instructor has himself a vivid perception of form, he can 
 by taking pains and time, and exerting himself, do a good deal 
 with his men at the beginning of the subject, by employing 
 quite common illustrations, such as the walls of the class-room, 
 the doors, the outside walls of the building, etc. In small 
 classes the stereoscope can be used to advantage ; there are 
 stereoscopic views of geometrical solids to be had in the 
 market ; and a little pains will enable him to employ the magic 
 lantern. In all his teaching he must not forget that his end in 
 view is to produce images of the geometrical figures in the 
 minds of the pupils ; so that he and they will be looking 
 mentally at the same or similar objects ; and that neither will
 
 1 8 METHODS OF TEACHING THE VARIOUS 
 
 be lost among the abstract words of the demonstrations. Any 
 other teaching of solid geometry is a mockery, and had better 
 be replaced by something more useful. 
 
 A third example of objectivity in mathematical teaching is 
 the differential calculus in its rudiments. The object is best 
 expressed in Newton's words, the fluxion and the fluent. 
 When the great philosopher discovered the calculus, he simply 
 generalized certain principles which had preceded it ; and to 
 do this was obliged to concentrate his thoughts upon the 
 objects just named, not as functions of anything else, nor as 
 representatives of any mechanical process : but as variable 
 quantity and its variation. 
 
 The illustrations which naturally arise in a teacher's mind, 
 and the definitions and boundaries which have to be set up in 
 teaching, are not the principal matter ; the concepts themselves 
 of function and differential coefficient are the primary objects 
 of contemplation. 
 
 The pupil's mind does not grasp the elements of the 
 calculus until these primary objects are wholly present to con- 
 sciousness, and he can practically and intelligently apply these 
 conceptions. 
 
 In a far lower degree the same method is the true one for 
 the elements of number. The objects here are at first the 
 individual numbers as Grube has pointed out ; and when these 
 are mastered up to 100, the larger concept of number in general 
 takes their place. The analysis of the number 4, or the num- 
 ber n, or the number 97, is as difficult to one pupil as that 
 of the concept " function " to another. 
 
 The grand stumbling-block of all inexperienced teachers is 
 dogmatic method. Just as the old Eton Greek and Latin gram- 
 mars written both of them in Latin were incomprehensible 
 to the pupil, so are the definitions of any branch of mathematics 
 to the pupil utterly unfamiliar with the subject. It may be 
 necessary, and it certainly is unobjectionable, for the text-book
 
 BRANCHES OF MATHEMATICS. 19 
 
 to express the rudiments of knowledge on any subject in a few 
 pages. But the teacher who gives out these few pages as a 
 lesson to a pupil ignorant of the subject has not yet mastered 
 the alphabet of his profession. It is his duty to commence by 
 a careful study, with a mental sounding-line, of the depths of his 
 pupils' ignorance ; to be followed by an objective presentation 
 of the elements of the new subject. There are various ways of 
 ' exciting curiosity ; in mathematics it is unusually necessary to 
 employ them, as most scholars come to the higher branches a 
 good deal prejudiced against them. Poor teaching leads to the 
 inveterate idea that the subject is only adapted to peculiar minds, 
 when it is the one universal science, and the one whose four 
 ground-rules are taught us almost in infancy, and reappear in 
 the motions of the universe. 
 
 In fact, I cannot foresee the extent of the beneficent effect 
 of genuinely good teaching (viz., according to Grube's method) 
 of primary arithmetic. It seems to me almost as if it would 
 revolutionize the whole mathematics up to quaternions ; at 
 least, if combined with Pestalozzian teaching of geometry and 
 drawing. 
 
 PRIMARY ARITHMETIC. 
 
 The space is here lacking to set forth Grube's method in de- 
 tail.* Excellent pamphlets have been published upon it ; the 
 one written by Prof. F. Louis Soldan is the most available which 
 I have found in English. 
 
 t am inclined to think that the strict practical application 
 of the first principle analysis of every number up to 
 100 requires more arithmetical power than the ordinary 
 untrained teacher possesses. For such a one, then, to attain 
 what I consider the highest success, it is necessary to work out 
 
 * See Appendix.
 
 2O METHODS OF TEACHING THE VARIOUS 
 
 this analysis carefully in detail, and impress it upon the mem- 
 ory. Let me give an example the number 89 : 
 
 89 = a prime number 
 = ii X 8+ i 
 = 29 X 3 + 2 
 = 43 X 2 + 3 
 
 = i? X 5 + 4 
 
 = 7 X 3 X 2 2 + 5 
 
 = 83 + 6 
 
 = 41 X 2 + 7 
 
 = 3 4 + 2 3 ; and so forth. 
 
 In other words, the teaching of any one number of the hundred 
 involves on the teacher's part an immediately available percep- 
 tion of every combination of factors in every number less than 
 that one ; as well as a ready memory of all combinations of two 
 numbers which produce it by addition. Grube does well in 
 restricting to 100, the numbers to be individually analyzed ; does 
 he put too severe a strain upon the teacher in requiring this 
 method up to 100? Certainly not, if the teacher is selected 
 for his or her skill ; very possibly, if the ground of election is 
 poverty, friendship, or any other quality not available to the 
 good of the pupils. 
 
 Is Grube wrong, then, in thinking it possible to accomplish 
 the work up to 100 in two years, beginning at the age of six or 
 seven ? 
 
 Or shall we say that so thorough a knowledge of these num- 
 bers is not worth communicating ? It seems to me almost self- 
 evident that it is, if arithmetic has any usefulness. For the 
 lack of skill within these limits among business men is almost 
 inconceivable, to say nothing of those who are dependent upon 
 their daily labor. 
 
 We may, if we choose, take the ground that thorough teach- 
 ing of any kind is impracticable, theoretical, visionary ; that
 
 BRANCHES OF MATHEMATICS. 21 
 
 teachers are entitled to do bad work because their salaries are 
 so poor ; or that the instincts of a young person of either sex 
 are a better guide in this profession than the thoughts of a 
 Pestalozzi, or a Froebel, or their followers. 
 
 But the present writer can assure all who employ arithmetic 
 that the thorough analysis, strictly according to Grube, of the 
 first hundred numbers, both concrete and abstract, is a most ' 
 admirable foundation for higher arithmetic, and a most useful 
 exercise preparatory to every-day business. 
 
 Who does not sometimes wish to know what he can do with 
 a definite sum of money ? Is not, in fact, this the really fun- 
 damental problem of business arithmetic ? 
 
 In all rational methods of teaching arithmetic, the mental or 
 oral side of the subject is the most important. Here, again, 
 the inexperienced teacher is at a loss. It is comparatively 
 easy to set sums from the book, and see if the answer agrees ; 
 and but little more difficult to consult the "key" ; not so easy, 
 however, to make up, and control the solution of, oral extem- 
 poraneous examples. 
 
 It is here that the normal schools afford the oppertunity of 
 professional training, too precious to be lost. Even college 
 professors could, if they would, do some good in the same 
 direction; there are many chances for them to teach the 
 importance of mental arithmetic. 
 
 Written arithmetic should not be taught with large numbers. 
 The pupil who can add, up to five significant figures, rapidly 
 and correctly ; or multiply two numbers of four digits each by 
 each other ; or divide eight figures, or even six, by three, with- 
 out frequent failure, is well enough equipped. It is not in this 
 direction that practical weakness in arithmetic lies. 
 
 Fractions, as taught according to Grube, are first objectively 
 presented. The addition, subtraction, multiplication, and divis- 
 ion of the simpler ones should precede any dealing with more 
 complicated forms. Thoroughness in those which under
 
 22 
 
 any circumstances whatever are oftenest used is the main 
 thing to be aimed at, after all. 
 
 It will be found by any one who notices the calculations 
 made by common people that halves and quarters are very 
 much oftener used than thirds, sixths, or even tenths. The 
 carpenters' rule gives eighths and sixteenths of an inch ; the 
 mechanic is a very fine one who employs decimal fractions at 
 all. Federal money is, practically, dollars and cents, hardly 
 calculated as dollars and hundredths except in banks or other 
 large business. 
 
 Thus in teaching fractions the simpler ones should be util- 
 ized as far as they will go, and those more complicated than 
 any one uses left out of view entirely. There is a justification 
 in the employment of all denominators up to ten, but for few 
 primes beyond ten save very incidentally. 
 
 It is a common practice in modern text-books to introduce 
 decimal fractions before vulgar. I suspect it not to be the 
 natural order; those who so present the subject probably 
 expect the pupils to learn the simpler vulgar fractions earlier, 
 perhaps in an easier book. Denominate numbers are (usually) 
 overdone in our arithmetics and in those schools where the 
 book, the whole book, and nothing but the book, furnishes the 
 material of teaching. It is of some use to a surveyor to know 
 the number of feet in a mile, and perhaps the number of inches ; 
 but the dealers in square feet of land, and -those in square miles, 
 are too far apart to have any important relationship. There 
 are absurd questions of reduction ascending and descending 
 which are only a torture, and I doubt the possibility on the 
 part of ordinary people of reading an ordinary physician's pre- 
 scription so as to apply apothecaries' weight. We are more 
 liable to be poisoned by the wrong substance than the wrong 
 quantity ; if the doctor and the apothecary can learn to read 
 the dog- Latin, they can also get up the weights and measures 
 which accompany it. And I fancy the homoeopaths employ a
 
 BRANCHES OF MATHEMATICS. 23 
 
 more rational system of measurement, however great be the 
 strain they may be thought to put on the powers of numbers. 
 
 Percentage and its applications, except interest, are a pretty 
 easy subject ; and interest is actually taken up in our schools 
 and our books far beyond the necessities of life. Partial pay- 
 ments, it is true, are sometimes troublesome, but experience 
 shows that the power of doing such examples does not follow 
 the study of ordinary arithmetic unless the pupil has learned 
 (as few have) to do simpler sums correctly. The writer was 
 called upon a few years ago to do such a calculation, which had 
 been " given up " by the holder of the note, a large merchant 
 in a great city ; by the maker of the note, of course (he was 
 only an intelligent native of a foreign land) ; and by his two 
 daughters, normal school graduates, and actual high school 
 teachers in Massachusetts. It was no more difficult than many 
 given in our books of arithmetic, but none of the parties knew 
 the answer. 
 
 Compound interest should be lightly touched upon ; the 
 table of amounts for various percentages and times is easily 
 proved and understood, and is very instructive to the extrava- 
 ^nt and the improvident. 
 
 Proportion is best taught in the beginning by the " reduction 
 to unity " ; in fact, it is better called rule of three at first, and 
 when a little more abstraction begins, the use of letters for 
 numbers should begin also. 
 
 Can any one imagine a good teacher who is also a good 
 algebraist, who will not train his pupils to use letters for num- 
 bers long before arithmetic is completed? Interest and pro- 
 portion are the two most important portions of arithmetic for 
 this training ; but until the present writer actually takes a class 
 through the whole course from eight or ten years of age to 
 fourteen, he cannot say positively when he would begin the use 
 of letters. They would certainly be in place long before the 
 time indicated by the present custom.
 
 24 METHODS OF TEACHING THE VARIOUS 
 
 Mensuration is the application of arithmetic to geometry. 
 The arithmetical part is simple enough; but the geometry 
 must precede it, and be well taught, or that will happen which 
 was told me, that only one man in a village (not a college pro- 
 fessor) could measure a wall correctly. The one man became 
 extremely powerful in politics. I do not say that the "boss 
 system " in our politics is due to ignorance of mathematics ; 
 but the arithmetical power of a leading politician must evidently 
 be considerable. 
 
 In conclusion of the chapter, let me say that a good teacher 
 of arithmetic must combine the following qualities : 
 
 1. Quickness in mental operations. 
 
 2. Correctness in calculation. 
 
 3. Power rapidly to form new examples, especially in con- 
 crete numbers. 
 
 4. Knowledge of algebra and geometry. 
 
 5. Ability to teach objectively and find illustrations. 
 
 6. Patience with the slow pupils. 
 
 7. Thoroughness everywhere. 
 
 To improve in teaching arithmetic, he or she must improve 
 in all these qualities ; the best books on the advanced branches, 
 so far as I know them, will be mentioned later. 
 
 SCIENTIFIC ARITHMETIC AND ALGEBRA. 
 
 We may, if we choose, make a distinction between higher 
 arithmetic and elementary arithmetic on the one hand, and 
 algebra on the other. This course is conducive to that multi- 
 plication of books which is so dear to some authors and some 
 teachers. But a practical distinction of great importance is 
 between the arithmetic which every man even a day-laborer 
 needs, and that which is more technically business arith-
 
 BRANCHES OF MATHEMATICS. 2$ 
 
 metic. The day- laborer needs to have a very thorough acquaint- 
 ance with the four ground-rules and federal money, in order to 
 escape the snares which are set for his ignorance ; and the com- 
 mon school cannot altogether restrict its course to this amount. 
 A knowledge of the intricacies of business arithmetic is best 
 deferred till the young man needs to study them professionally. 
 
 Now my contention is, that after elementary arithmetic 
 through fractions, rule of three, and percentage, and such a 
 course of these subjects as shall deal only with moderate num- 
 bers and involve much mental arithmetic, the next subject 
 on the arithmetical side should be elementary algebra. 
 
 But this should be first taught in a propcedeutic way ; that 
 is, orally and objectively, in the proper manner to introduce a 
 new subject. The average age of the class beginning algebra 
 so will be about thirteen. The whole year's work in algebra 
 will be given to the four ground-rules, with a few simple equa- 
 tions, to excite more interest ; and negative quantities will be 
 taught after these have been dealt with, employing positive 
 quantities, added and subtracted. That is, in the early exer- 
 cises care will be taken that the difficulty of the subtraction of 
 a larger number from a smaller does not arise. 
 
 The method of treatment can be got from the first five 
 chapters of Todhunter's Algebra, with the application of a 
 little common sense in the introduction of abstract terms and 
 ideas. 
 
 But, above everything else, let the teacher practise his or her 
 class patiently, diligently, and faithfully, both orally and in 
 writing, with simple examples rather than complicated ones, 
 with lively dialogue rather than tedious recitation, on every 
 point involved. Two lessons a week for a year can be well 
 spent on these chapters. 
 
 Square and cube root applied to numbers, and compound 
 interest, should be taught in this year. 
 
 Let the pupil who is to be educated for commercial business
 
 26 METHODS OF TEACHING THE VARIOUS 
 
 learn a good deal of algebra thoroughly before he begins his 
 technical course. 
 
 After the four ground-rules of algebra are really well taught, 
 the subject is very easy until we come to radicals. The com- 
 bination of exponents is very important and necessary, easy 
 enough so long as the exponents are whole numbers, but a 
 little difficult when they are fractional or negative. Their diffi- 
 culty can be avoided, first of all, by objective, oral, practical 
 treatment ; but is greatly aggravated by weakness of the pupil 
 in the four ground-rules, the use of the bracket and negative 
 quantities included. 
 
 How many become discouraged at this point ! Our colleges 
 admit many Freshmen whose ideas of radicals are utterly con- 
 fused, who, in consequence, cannot do anything with quadratics. 
 
 Let me say here that the teacher of algebra who does not 
 keep his pupils fresh in ordinary arithmetic by continually 
 reviewing the important portions setting problems orally 
 loses a great aid in mathematical training. 
 
 Let me take a brief diversion here. The practical mathema- 
 tician, I do not care what his profession may be, spends a large 
 share of his time in the ordinary operations of arithmetic, simple 
 algebra, easy geometry; and the time given to difficult and 
 perplexing problems of the higher mathematics is small indeed. 
 If he be one of those rare men who can solve the great 
 problems of abstract mathematics, he will soon find himself 
 obliged to live alone, with hardly any sympathy ; and the few 
 disciples he can get will, like himself, be in danger of becoming 
 hopelessly unpractical. This will happen unless he takes a 
 lively interest in some branch of natural science proper. 
 
 Whoever you be, and whatever branch of mathematics you 
 wish to teach, do not forget to practise your pupils upon the 
 elements as well as upon the higher branches. Just as soon as 
 the theory of combination of exponents has been taught, both 
 for whole numbers and fractions, logarithms should be intro-
 
 BRANCHES OF MATHEMATICS. 2/ 
 
 duced and practically employed. They should certainly pre- 
 cede quadratic equations. 
 
 Algebra, "through quadratics," is in Eastern colleges the usual 
 requirement for admission. In this requirement logarithms 
 are not included. The usual result is that these are not prac- 
 tically taught until trigonometry is begun, and the average 
 pupil does not learn the difference between a logarithmic 
 sine and a natural sine. There is no cure for this confusion, 
 except to teach logarithms where they belong, as a part of 
 scientific arithmetic, and to apply them to purely arithme- 
 tical problems, and to be thorough with their practice. 
 
 The binomial theorem is often taught by the method of in- 
 determinate coefficients, very loosely applied. This is a hasty 
 and untrustworthy method ; the demonstration in this manner 
 requires several steps usually overlooked. In this, as in many 
 other cases, I think the usual fashion of teaching goes too far. 
 A boy is better prepared to take up the differential calculus, if 
 he knows the rigid demonstration of the binomial theorem for 
 positive whole exponents only, and can apply it rapidly and 
 accurately, than if he has only a superficial acquaintance with it 
 by the method of indeterminate coefficients. Of easily acces- 
 sible works upon algebra I should recommend Todhunter as 
 the best work for a teacher to study. It is not altogether 
 easy to employ it as a text-book in the ordinary schools of this 
 country, for the reason that it is pretty large, and has more 
 adaptation to English methods of instruction. It was origi- 
 nally designed for those students in Cambridge, England, who 
 were aiming at mathematical honors, generally under private 
 instruction as well as public. 
 
 The theory of equations is frequently studied in our colleges. 
 It should properly be replaced (as a required study) by more 
 thorough training in quadratics and cubic equations, using in 
 the latter Cardan's rule for reducible cases, and the trigono- 
 metrical solution for the irreducible. In point of fact, it is
 
 28 METHODS OF TEACHING THE VARIOUS 
 
 hardly worth while to take up the theory of equations at all 
 until the pupil has become thoroughly familiar with trigono- 
 metrical analysis, so as to be able to handle complex numbers 
 with facility. 
 
 The okUsystem of college tutorships, now in its decadence, is 
 responsible for a great deal of harm in mathematics as well 
 as in other subjects. The young man just out of Harvard was 
 often employed as a tutor, while pursuing law or divinity as a 
 student : the now most eminent English scholar in the country, 
 being valedictorian of his year, was selected as such a tutor in 
 mathematics a year after his graduation. Of course he went 
 through the form perfectly ; but his heart cannot but have been 
 elsewhere. Within a few years this has been changed in nearly 
 all good colleges. Men are selected to teach college students 
 with some reference to their own tastes and studies ; and the 
 corresponding requirement that those studies shall be kept up 
 is more insisted upon. 
 
 These remarks are made at this point because the attempt 
 to teach higher algebra in the Freshman year is often the 
 most conspicuous failure in our mathematical courses. The 
 young men visibly love to practise that which is not too difficult. 
 A skilful algebraist who is also an experienced teacher can make 
 the subject a delight to many, if he is not, as sometimes hap- 
 pens, simply inactive and critical. But to make it a delight to 
 any one, that pupil must be doing something well within his 
 powers, and not be attempting to aim blindly at what is far 
 beyond his capacity. 
 
 The good time, however, is coming when temporary teachers 
 of all grades will, little by little, give place to those who intend 
 to make teaching a permanent profession. 
 
 To conclude what I have said upon algebra, let me give the 
 following rules for teaching it : 
 
 i. Base it upon arithmetic, and keep up a continual set of 
 exercises in mental arithmetic and mental algebra.
 
 BRANCHES OF MATHEMATICS. 2Q 
 
 2. Use extemporalia very frequently. 
 
 3. Keep yourself fresh in the subject. 
 
 4. Learn the secret of objective teaching even in the most 
 abstract matter. 
 
 5 . Be not afraid of too much practice or too much- repetition 
 of elements ; look at the way your pupils practise ball-playing. 
 
 6. If you are compelled to use a bad text-book, supply its 
 deficiencies by much study of good ones ; and remember that 
 the text-book is, after all, not the teacher. 
 
 7. If you are an inexperienced teacher, get advice from 
 older and wiser heads ; do not think that when you have learned 
 the subject, perhaps in the antiquated manner, you can 
 teach it without experience or caution. 
 
 8. Study the writings of advanced educators. 
 
 GEOMETRY. 
 
 There is nowhere in teaching any greater gap between 
 theory and tradition than in this subject. It is quite possible 
 to obtain what are considered good results in geometry by the 
 ordinary method ; viz., that of requiring the reproduction by 
 the help of the blackboard of the demonstrations in the text- 
 book. The pupils taught in this way, however, are usually 
 unable to apply geometry, and especially solid geometry, to any 
 advantage ; and when, if it so happens, they are required to do 
 this, they are obliged to learn over again the rudiments, which 
 in all probability they have slurred over or neglected at the 
 proper time. The process is like that of building a foundation 
 for a house already completed, which is very proper in the 
 case of ready-made houses ; but the final result is not altogether 
 the ideal one, nor will the house so constructed be permanently 
 satisfactory.
 
 3O METHODS OF TEACHING THE VARIOUS 
 
 Dr. Thomas Hill, ex-president of Harvard College, insists 
 very strongly upon the early presentation of the facts of geo- 
 metry. The same idea has been carried out in foreign countries 
 quite extensively these many years, it was one of Pestalozzi's, 
 but does not yet meet with very great acceptance in America. 
 The reason I do not know, except, it may be, that the tradi- 
 tional course is quite different, and that school organization in 
 this country is often, in fact usually, in the hands of persons 
 who are not educators ; and that even the best teachers are 
 allowed but little freedom. 
 
 The kindergarten, as I have said elsewhere, introduces forms 
 in the concrete ; Pestalozzi and Pestalozzians make much of 
 forms in the primary school ; but in our practice this knowl- 
 edge is forgotten before geometry is nominally begun. The 
 contention, then, of theory, is to introduce geometry before 
 arithmetic is completed ; and I am very much inclined to 
 think that this should be done more thoroughly and skilfully 
 than is often the case, even if some arithmetical subjects need 
 to be displaced to make room for it. 
 
 The method of beginning geometry while arithmetic is going 
 on is of course the objective one. Let the teacher present for 
 analysis a cube of wood; call upon the pupils to count its 
 faces, edges, corners ; do the same with the other regular 
 solids ; afterwards, with some not regular polyedrqns, which are 
 of practical importance, and with the round bodies. Mean- 
 while, let the idea of a geometrical surface a line, a point 
 be abstracted from the concrete presentation ; and after the 
 concept of the geometrical point has been thus obtained, let the 
 teacher proceed to form the different lines straight, broken, 
 curved by the motion of a point, the different surfaces by 
 the motion of the lines. Experience must show in each case 
 how far to go. 
 
 For a class which is to study mensuration, the few theorems 
 necessary will offer no great difficulty. Strict demonstration is
 
 BRANCHES OF MATHEMATICS. $1 
 
 not needed here ; it is sufficient if the pupils are taught in a 
 semi-objective way the rational groundwork of the truths, whose 
 complete comprehension is the main thing needed. Even the 
 Pythagorean proposition can be illustrated by several methods, 
 none of which are equal to Euclid's, in my judgment, as demon- 
 strations ; for when you try to put them into words, you find 
 more words are needed ; but one or two at least are better for 
 the elementary, half- demonstrative stage preliminary to men- 
 suration. 
 
 For an older class which is beginning geometry with the 
 intention of going through an ordinary course in Euclid or 
 Legendre or the late writers, but of which none or few of the 
 members have any good preparation of geometrical ideas, the 
 teacher's difficulties are at the maximum. Even here some- 
 thing may be done by giving a number of lessons on the defi- 
 nitions, and dwelling longer than usual on the first book. The 
 definitions must be objectively illustrated ; the same course, of 
 abstracting from the concrete, and of forming straight lines 
 and curves by the motion of a point, must be fully pursued ; 
 although as the pupils are older and riper, this will not take 
 so long. In the subject of quadrilaterals a good deal of in- 
 terest can be roused by a blackboard analysis of the various 
 cases of combinations of four lines ; the pupil should be shown 
 the combinations of three lines also, but this is almost too easy. 
 
 Three lines can be all parallel. 
 
 two parallels intersected by a third, 
 no two parallel, the triangle. 
 
 The first book of geometry, either Euclid or Legendre or 
 Chauvenet, contains enough matter to occupy a year's time. 
 Not five hours a week probably ; but the mental growth of a 
 year is not greater than the step from ignorance of demonstra- 
 tion to a thorough mastery of it even in a limited matter. Let 
 me give the course laid down by Arnold (of Rugby) in his
 
 32 METHODS OF TEACHING THE VARIOUS 
 
 Miscellaneous Works (N.Y., 1845, 8vo) in the matter of 
 Euclid : 
 
 Fourth Form: Book I, Propositions 1-15. 
 
 Upper Remove : Book i , Propositions 1 5 to the end. 
 
 Lower Fifth : Book 3. 
 
 Fifth Form : Euclid to the end of Book 6. 
 
 Sixth Form : Euclid, 3-6 ; Plane Trigonometry ; Conic Sections. 
 
 These five classes represent nearly a year apiece. But after 
 the first book is accomplished, plane geometry has but few dif- 
 ficulties. The English do not learn the quadrature of the cir- 
 cle ; our boys do. It involves pretty long calculations, which 
 can be abridged, if necessary, by omitting the decimals be- 
 yond the fifth place, and made more intelligible by introducing 
 algebraic notation. This comes in just as well here as later in 
 geometry. 
 
 I strongly advise keeping a small corner of time for the 
 " modern geometry." Chauvenet's treatise has a small amount of 
 this in the text, and more in the appendix ; the last editions of 
 Loomis give a brief appendix to the subject ; Gillespie's Sur- 
 veying introduces it as a help to practical field methods ; and 
 in several ways the teacher can get possession of some of the 
 most important results. The foreign text-books already intro- 
 duce more of these theorems into their main framework. 
 Steiner, the great Swiss geometer, was Pestalozzi's pupil at 
 Yverdun ; and his work grew directly out of Pestalozzi's ideas. 
 
 It is very likely that the teachers of geometry in our high 
 schools are often so overworked that they have not much 
 time to study the many new books on this subject. Riders, as 
 they are called in England, or " original propositions," as we 
 call them here, are quite indispensable. 
 
 Solid geometry is firmly established in our higher courses ; it 
 is very singular that in England it is but slowly introduced. 
 We have excellent text-books, but I suspect the teaching is
 
 BRANCHES OF MATHEMATICS. 33 
 
 not always excellent. Objective illustration, as I have before 
 stated, is the main requirement of the teacher ; and he must 
 insist thoroughly and patiently on applications in mensuration. 
 If the little " modern geometry " hinted at above be at hand, 
 the forms in space will take upon themselves a new attraction. 
 Allusions (at least) to perspective, should be made. 
 
 Conic sections, in their synthetic form, should precede ana- 
 lytical geometry. The old " Bridge " and his imitators are 
 better than no teaching ; but it is to be hoped that the method 
 of Chasles and Steiner will soon become available. If the 
 teacher has ability, the most important properties of the conic 
 sections can be taught heuristically by this method, without 
 text-book. It is a very good experiment to try on a strong 
 class of Sophomores. 
 
 The teacher's reference-books in geometry may include 
 Chauvenet's Geometry (published by Lippincott), MacDowell's 
 Exercises in Euclid and Modern Geometry (London: Bell), 
 Cremona's Elements of Projective Geometry (Oxford : Clar- 
 endon Press; N. Y. : Macmillan), and the works of Chasles 
 and Steiner, if he understands French and German : Mulcahy's 
 Modern Geometry (Dublin) is also excellent. 
 
 PLANE TRIGONOMETRY. 
 
 If I have not sufficiently insisted upon measurement before, 
 let me now make amends. Before the pupil takes up this sub- 
 ject, he or she must have a distinct conception of the measure- 
 ment of angles. Paper protractors are good ; a pocket com- 
 pass is of great help ; but I think every high school ought to 
 have a theodolite. If nothing more can be procured, a little 
 German instrument with four-inch circles, and a telescope 
 magnifying ten times, can be bought for $60 or $70. Let the 
 teacher lay out and measure the sides and angles of some tri-
 
 34 METHODS OF TEACHING THE VARIOUS 
 
 angles, and get the pupils to repeat the measures. If a com- 
 pass only can be employed, the accuracy will not be very 
 great ; but still some approximation can be obtained by fre- 
 quent repetition of the same measures. 
 
 The subject of logarithms must be introduced in algebra 
 pretty thoroughly, as I have before stated. The first trigo- 
 nometrical calculations, however, must be made with the natu- 
 ral sines, tangents, etc. It is necessary to teach a few formulae 
 so before proceeding to the logarithmic tables. 
 
 Here, as everywhere, the objective method of teaching must 
 prevail. It is pretty easy to show by the circle what the sine, 
 cosine, tangent, cotangent, secant, and cosecant are (of course 
 to radius unity). But it is best to introduce at first the sine 
 and cosine only, and show immediately how their introduction 
 reduces the solution of right-angled triangles to order. Now 
 the table of natural sines and cosines is brought forward, and 
 its formation explained in an elementary way ; and it is then 
 used in numerical practice. It can also be shown how to plot 
 angles by the help of this table, and they can be compared 
 with the same angles as plotted by a protractor. 
 
 Then may be similarly introduced the tangent and the rest 
 of the trigonometric functions, and the solution of right-angled 
 triangles pretty thoroughly mastered. 
 
 It is now time to explain the tables of logarithmic sines, 
 etc. and the pupils are trained in their use with right-angled 
 and oblique-angled triangles, the formulae for the latter being 
 developed in their order. 
 
 The applications of plane trigonometry to surveying are 
 extremely important. Our American surveying methods, as 
 handed down by tradition, have been greatly improved within 
 a few years. The teacher of trigonometry ought to have for 
 reference the latest books ; the best are W. M. Gillespie's Sur- 
 veying, his posthumous treatise on Levelling, Topography, and 
 Higher Surveying ; and, above all, J. B. Johnson's Surveying.
 
 BRANCHES OF MATHEMATICS. 35 
 
 It seems to me proper to teach the practical side of trigo- 
 nometry first ; the theoretical part, analytical trigonometry or 
 " goniometry," should be made a strong mathematical discipline, 
 if possible ; otherwise not meddled with. The capacity to measure 
 and solve triangles is of practical value ; an imperfect knowl- 
 edge of goniometry is not useful, but rather tends to confuse 
 and dishearten the pupils. In this subject, as everywhere else, 
 teach the concrete first, thoroughly ; lead up to the abstract by 
 easy stages. , 
 
 ALGEBRAIC GEOMETRY; ANALYTICAL GEOMETRY. 
 
 By algebraic geometry, as distinguished from analytical geome- 
 try, I mean, of course, the application of algebra to geometric 
 problems. It has nothing to do with loci, which are best taught 
 in plane geometry ; nor with co-ordinates, which come later. 
 Most treatises on analytical geometry have a short introduction 
 of algebraic geometry ; this should be taken by the teacher as 
 the basis of oral lessons, accompanied with extemporalia, and 
 other enlivening exercises. I have already stated that algebra 
 ought, in my judgment, to be pretty freely used in pure geome- 
 try, after the pupil has been thoroughly trained in geometrical 
 reasoning. 
 
 When algebraic geometry has been well taught, and the con- 
 ception of a locus is well established, there is no difficulty in 
 explaining rectangular co-ordinates. In my judgment this 
 subject should be begun in a very elementary way, and applied 
 first to the ellipse. The straight line and circle are too familiar 
 for the student to see the necessity and meaning of the method ; 
 all the more as the properties of the conic sections are practi- 
 cally more important than the abstract method. 
 
 When the pupil has a pretty clear idea of rectangular co- 
 ordinates, and of the commoner properties of conies, then, I 
 think, is the time to begin the higher generalizations.
 
 36 METHODS OF TEACHING THE VARIOUS 
 
 In technical schools a course of descriptive geometry is 
 necessary ; and it is much to be wished that it might be 
 brought in as an elective in colleges, as at Harvard, and other 
 colleges which have scientific foundations annexed. The diffi- 
 culties are pretty serious arising from the general lack of train- 
 ing in drawing. 
 
 SPHERICAL TRIG ONOME TR Y. 
 
 Spherical trigonometry is a branch of the science which at 
 Harvard and in all English courses is quite optional. It is 
 rather remarkable that it is not required for all pupils at the 
 Massachusetts Institute of Technology. A certain amount of 
 this branch is needful for a complete understanding of solid 
 geometry ; but I do not think that the subject is well taught in 
 many institutions. In fact, if astronomy ever gets its rights in 
 our courses, a rational disposition can be made of spherical 
 trigonometry, which is not now possible. 
 
 Its principal uses are 
 
 i St. In crystallography. 
 
 2d. In the theory of surveying- instruments. 
 
 3d. In geodesy. 
 
 4th. In astronomy. 
 
 In each of these branches and they are all pretty advanced 
 subjects the form of treatment varies with the use. Such a 
 book as Chauvenet's, which is very complete, gives too much for 
 a required course ; if the plane trigonometry be taught from 
 this text-book, a slight taste of the spherical can be added. 
 For an elective course in spherical trigonometry or any of its 
 applications, however, the better preparation is a thorough 
 knowledge of plane trigonometry, especially ability to calculate 
 from a formula, with or without logarithms. A smattering of 
 spherical trigonometry, especially when Napier's Rules are the
 
 BRANCHES OF 'MATHEMATICS. 37 
 
 mainstay, is comparatively of no use whatever. Of course 
 spherical trigonometry must be preceded by a thorough course 
 in goniometry. 
 
 In teaching an elective class in practical astronomy I find it 
 necessary to begin with a short course in the art of computa- 
 tion. The pupils are ordinarily familiar with common six-figure 
 tables ; the first thing to be done is to teach them the use of 
 the modern standard six-place tables (Bremiker's), and to 
 practise them in the computations of spherical trigonometry, 
 employing the fundamental formulae with partial omission of 
 the logarithmic modifications ; also in computations by Gauss's 
 formulae ; in checking their work by various cross-controls ; in 
 the use of the modern methods of calculating small arcs ; in 
 the calculation of geodetic or small spherical triangles ; in the 
 determination of spherical excess and the computation of 
 geodetic triangles by plane trigonometry ; and in the use of 
 sum and difference logarithms. Parallel with this course they 
 are taught by daylight the handling of the smaller portable 
 instruments ; one of which is a theodolite (or universal instru- 
 ment) with microscopic readings ; and the delicate striding 
 level of this instrument is tested with a level-trier. Thus from 
 a good deal of practice in exact angular measurement (to 
 seconds) they are led to see the definite relationship between 
 observation and calculation, which is very important if spherical 
 trigonometry is to have any special educational value. 
 
 THE CALCULUS. 
 
 In very few large colleges at the present day the attempt is 
 made to teach the calculus as a required study. The great 
 universities of this and other countries give it a very prominent 
 place as an elective ; and in technical schools it is of course 
 required. In some " Realschulen " preparatory to polytechnic
 
 38 METHODS OF TEACHING THE VARIOUS 
 
 institutions it is required, and with success ; in others a prepa- 
 ration only is made for its future study. 
 
 In this country the subject is at a disadvantage from the 
 want of thorough preparation for it; the students have too 
 often slurred over their algebra and trigonometrical analysis. 
 Granted a thorough knowledge of these subjects, so that the 
 pupil can quickly follow a well-graded text-book or a judicious 
 lecture, and progress in the calculus will be relatively rapid. 
 But at best the instructor must go slowly over the fundamentals, 
 putting in extemporalia and oral instruction of all kinds, and 
 carefully making his ground secure as he goes along, taking 
 care also to guard his pupils from confusion as to the source of 
 their difficulties. In the use of a good modern text-book with 
 examples, the students' greatest difficulty is at first to reduce 
 the examples to the form indicated by the text-book, or to any 
 simple form whatever. This is not a difficulty in the calculus 
 itself; for the teacher's business is to explain such matters at 
 first, until the men are stronger in the subject. But the most 
 disheartening thing in mathematical teaching is to find that the 
 pupils cannot do the elementary part of their work because of 
 hurried and superficial study (of algebra or trigonometrical 
 analysis when the calculus is the advanced subject) upon the 
 lower necessary branches. 
 
 It is hardly needful to give directions for instruction in the 
 portions of mathematics which lie beyond the differential calcu- 
 lus. There are few who study them ; and the teacher will 
 almost always do better if he employs the genetic method, or 
 that which presents the matter in the natural order. For 
 example, the theory of determinants is a very important subject ; 
 while its fundamental idea is an elementary one, the pupil 
 cannot go very far without some facility in mathematical think- 
 ing. The first notions are communicated with great ease ; but 
 the study soon leads into pretty deep water. The teacher will 
 succeed best with it if he follow in some degree the historical 
 order of ideas.
 
 BRANCHES OF MATHEMATICS. 39 
 
 Higher algebra, including the theory of equations, and above 
 all, complex numbers, is a branch which should run parallel to 
 the calculus for a good while ; but not much should be done 
 with it before the rudiments of the calculus have been studied. 
 I apprehend that in the foreign universities as well as our own 
 the instinct of the younger mathematical generation is a pretty 
 safe guide as to questions of order. It is only in the schools 
 where the pupil is compelled to follow the traditional pro- 
 gramme that there is much danger of intermingling very easy 
 and very difficult matter. 
 
 The integral calculus may be divided into several portions. 
 The elementary methods of integration are easy enough, and 
 the pupil can be readily practised in them. But the instructor, 
 and especially the text-book maker, is liable to confuse easy 
 with difficult problems in a very perplexing way. I quite well 
 remember the hopeless dismay with which I read certain books 
 on the integral calculus, only to find that, as I became more 
 familiar with the differential, the difficulties seemed to lessen so 
 far as I had time or need to follow them up. This method left 
 me with vast tracts of the higher portions about which I did 
 not know or care anything until the time came up to employ 
 them, when they appeared easier. In other words, the really 
 important mathematics for practical purposes always grow out 
 of the uses which they subserve. 
 
 The university teaching of pure mathematics tends always to 
 take the form of instruction from a discoverer to his disciples. 
 This is proper, and in the right place ; for the discoverers are 
 few, and their disciples can oftener receive than originate. But 
 such instruction requires to be tempered by more methodical 
 processes in the lower institutions. Benjamin Peirce, for in- 
 stance, gave an enormous stimulus to mathematical thinking in 
 this country ; but his methods of teaching were hardly adapted 
 to the average Freshman and Sophomore.
 
 4O METHODS OF TEACHING THE VARIOUS 
 
 GENERAL REMARKS. 
 
 Mathematics, as a branch of study, is very homogeneous. 
 The difficulty, to a child of six, of understanding the processes 
 of ordinary arithmetic, is very similar to that which the riper 
 student finds in mastering the integral calculus. The reasoning 
 is abstract in either case, and the combinations rather too 
 numerous to be grasped at once. If you examine any college 
 student who claims that he cannot, and never could, understand 
 mathematics, you will find that some branches he understands 
 very well indeed ; about others, he will prove more indistinct. 
 You will finally reach the point where he became wholly dis- 
 couraged, and gave it up as hopeless. Many students cannot 
 add quickly and correctly ; the principle of carrying seems to 
 trouble them, as is painfully evident in the average computa- 
 tions of novices. Fractions are often very indistinctly taught 
 and learned ; there are many that stumble on negative quan- 
 tities, radicals, logarithms in algebra. In geometry, the meas- 
 urement of the circle is a great rock of offence ; and it is quite 
 rare to find a pupil who sees distinctly and clearly the lines in 
 space required in solid geometry. 
 
 Now, all this shows what is a great evil in our schools, 
 teaching by the inexperienced and inexpert. The pupil who 
 cannot carry correctly, has never been taught the analysis 
 (according to Grube) of the first hundred numbers. 
 
 539756 
 
 925185 
 
 Here I have set down, at random, two numbers : 56 and 29 
 are 85 ; not 9 and 6 are 15, set down 5, i to carry to 5 is 6, and 
 2 are 8. Again, 97 and 54 are 151 ; 53, 38, and i are 92. 
 
 This process, which I believe is like that of all skilled com- 
 puters (except that they usually add from left to right), saves
 
 BRANCHES OF MATHEMATICS. 4! 
 
 at once half the carrying, and the little which is left stands out 
 in most cases visibly to consciousness as having some meaning, 
 because the amount carried is far less than most of the other 
 numbers added. But can the average primary teacher perform 
 Grube's operations quickly and surely? 
 
 Now, in arithmetic, throughout, time must be taken away 
 from riddles, puzzles, operations with enormous numbers, long 
 sums on the slate or blackboard, and given to solid work in the 
 elements of numbers, especially mental work, to extemporalia, 
 to practical applications of arithmetic which really mean some- 
 thing. 
 
 This work is hard for the teacher, and requires strong, vigor- 
 ous persons, who are well educated, for the lowest classes. And 
 this again involves expense, which is often a bar to the best 
 results. Let the great city superintendents and boards of 
 education, then, fully realize their responsibilities, and keep 
 themselves fresh and free from hampering traditions. 
 
 In the middle schools, the city grammar schools, the town 
 high schools, the lower academies, etc., algebra is neces- 
 sary, and geometry indispensable. Let the use of letters to 
 express numbers and quantities, and the employment of the 
 negative sign of quantity, and the geometrical forms and ele- 
 mentary relations be made the matter of very careful and 
 thorough teaching, so that the correct conception and applica- 
 tion of these things shall be instinctive with the vast majority 
 of the pupils. 
 
 The teachers in these schools are more apt to be judged by 
 "success in teaching" than anything else. They consequently 
 are almost always anxious to improve. Superintendents and 
 principals have in those grades a heavy responsibility. The 
 danger is that they will be overworked ; their offices, like those 
 of all teachers in this country, and perhaps all others, are not 
 fully appreciated ; so that an ambitious and able superintendent 
 or principal is liable to find himself unable to keep up his
 
 42 METHODS OF TEACHING THE VARIOUS 
 
 studies, and he is often thwarted in his plans and methods. 
 The semi-political elections of teachers are a great evil. 
 
 The tradition is ingrained in us, of regarding the teacher's 
 profession as a stepping-stone to something better, and so 
 long as the teacher does not have leisure to study and power 
 to apply his ideas in his teaching, just so long will he be 
 discontented and anxious to get into a more independent 
 profession. 
 
 As America becomes richer, the great benefactions to univer- 
 sities and colleges do something to raise the teaching craft. A 
 bright young graduate who has made his mark as a teacher 
 not unnaturally looks to Alma Mater to give him a position. 
 If his work in lower schools is original and good, and he is 
 promoted to the college, he will be missed lower down, and 
 other like men be sought in his place. Thus the tendency to 
 improve, if the college has it well developed, is propagated in 
 the lower schools. But if colleges are backward, hold to anti- 
 quated and traditional methods, if their professors become 
 sleepy from too easy positions, and lose their interest in higher 
 study, their negative influence upon the lower schools will be 
 very oppressive. 
 
 Fortunately, however, the colleges are fast getting out of 
 their ruts. While the clamor for the repression of Greek is not 
 a fair one, it reveals a popular belief that the colleges have 
 been, in past times, unprogressive. But, Greek or no Greek, 
 these institutions are thoroughly awakening to new methods. 
 Just as soon as any college can get a large body of students 
 who are non-Grecians and yet thoroughly trained, will there be 
 a relaxation of the terms of admission on this point, or at least 
 some method by which these men can receive a higher educa- 
 tion. Whatever course of studies keeps unripe boys from pre- 
 mature graduation will always be unpopular with a large class 
 of well-to-do parents. What every college faculty most desires 
 is a good yearly accession of sound scholars ; and at the present
 
 BRANCHES OF MATHEMATICS. 43 
 
 da/ there is a full practical acknowledgment that these young 
 men are worth hard work on the teacher's part. 
 
 There is no department of college study in which improve- 
 ments in method of instruction are not now making. The 
 young men who return from Germany are not always well up in 
 German pedagogy, it is true ; but within a few years the older 
 ones have become far more objective in their methods. 
 
 Laboratories, observatories, scientific collections, art museums, 
 libraries, are fast advancing, and hardly any subject is now 
 taught by text-book method only. And American scholars 
 and scientists are becoming numerous enough to claim their 
 rightful place in education ; ignorance of the higher branches 
 of a subject is now more rarely a positive recommendation to a 
 teacher of the elements. 
 
 A " mathematical laboratory " is something not often men- 
 tioned ; but I think the thing, however we name it, a necessity. 
 It will contain, in part, such things as are ordinarily contained 
 'in a physical laboratory, but such as relate to ordinary, not 
 purely scientific measures. Thus a pair of scales may be con- 
 sidered to belong to mathematical apparatus rather than physi- 
 cal, so far as weighing is a branch of arithmetic ; a surveyor's 
 chain, so far as the expansion by heat or the elasticity of the 
 chain is not to be taken into account. If a good high school 
 could exist, which taught mathematics and not physics, its 
 laboratory would contain all tools relating to the weights and 
 measures needed in an ordinary community. 
 
 At any rate, let not the teacher of mathematics imagine, if 
 he does not teach physics, that he can without damage avoid 
 all opportunities for laboratory practice on his pupils' part. 
 Many young people do not know how to measure a room in 
 feet and decimals, and get its area. So, too, careful weighing 
 with good commercial scales or the steelyard is a useful arith- 
 metical lesson.
 
 44 METHODS OF TEACHING THE VARIOUS 
 
 PROGRAMME OF COURSE OF STUDY. 
 
 The ideal programme of study is about as follows. I give 
 only the order of subjects, in parallel columns : 
 
 Primary Arithmetic. Notions of Form, Drawing. 
 
 Arithmetic, through Rule of Three. Rudiments of Geometry. 
 
 Universal Arithmetic. ) r>i /- 
 
 Plane Geometry (one or two books). 
 
 Simple Equations. > 
 
 Algebra, through Quadratics. Plane Geometry. 
 
 f Solid Geometry. 
 Algebra, Series, etc. < Conic Sections. 
 
 ( Plane Trigonometry. 
 
 Theory of Equations (begun). Analytical Geometry (rudiments). 
 
 Calculus (rudiments). Spherical Trigonometry. 
 
 Commercial arithmetic, for those who are to go into business, 
 goes with algebra through quadratics, or replaces it. Of 
 course this programme is somewhat variable ; but the main 
 principle, that a course of arithmetic mst run parallel with 
 one of geometry from the beginning of a school course to the 
 end, is one which is laid down by the best educators since 
 Pestalozzi's time. 
 
 CONCLUSION. 
 
 It is likely that a large share of those who have read the 
 previous chapters will think my suggestions unlikely to be 
 realized, and that any one who has tried to get practical help 
 from them will feel somewhat discouraged. To such I would 
 say that educational theorists are often distasteful because they 
 seem to require too much at once ; but the uprooting of old 
 methods and the implantation of new ones, like the driving out 
 of the Canaanites in the Holy Land, is apt to be a very 
 gradual process.
 
 BRANCHES OF MATHEMATICS. 45 
 
 The professor, superintendent, or teacher, is but one factoi 
 in the work of education. The programme of studies is usually 
 fixed and badly arranged, and it is difficult to change it. 
 When changes are made, however, they can usually be made 
 in the right direction. 
 
 The young teacher fresh from college or the normal school 
 will usually teach as he or she has been taught ; if the college 
 teaching is good, better than the average in the locality, im- 
 provements will gradually be made round about. And college 
 teaching has, as I have said before, greatly improved of late 
 years, especially in the objective and practical direction. Such 
 a young instructor, if he is not overworked, can do much 
 towards realizing a high ideal in his profession ; and I trust the 
 modern tendency towards a higher estimation of this profession 
 will continue. 
 
 But such a young teacher, especially if but little leisure is 
 allowed, is apt to overlook the need of studying the theory of 
 teaching the science of adaptation. He is so enthusiastic 
 for the new truths he has learned in the higher school, that he 
 does not see that his average pupils cannot at once come up to 
 his standard. While admitting, at once it may be, that the true 
 order of teaching is from the known to the unknown, he does 
 not quite realize the " invincible ignorance " of the pupils ; and 
 is apt not to interest them, but to compel them to learn. Of 
 course, compulsion to industry is a good thing ; but many pupils 
 lack industry for the very reason that learning has always been 
 made hateful to them. 
 
 As a college professor I have known a great many students 
 who suffered extremely and were a grief to all their friends, 
 including their professors, simply from loose and disorderly 
 habits of work. This, of course, is partly the result of heredity, 
 partly of poor domestic training ; but partly, I am very sure, 
 from dogmatic instruction by inexperienced teachers. The 
 young teacher is apt to impart^ or try to impart, information,
 
 46 METHODS OF TEACHING THE VARIOUS 
 
 instead of producing good habits. How is it with those who 
 are more experienced? 
 
 I am inclined to think that our school systems, as a whole, 
 put too much of the work upon the pupils and too little upon 
 the teachers. Not that the latter have, on the whole, less to 
 do than those of foreign countries ; but in backward schools 
 their labor is wasted by great subdivision, to accommodate the 
 irregular progress of the pupils and the whims of parents ; 
 while in the more advanced city schools too much time is 
 often given to machinery and the marking system. In certain 
 cities the superintendent and the principals impress their 
 methods too much upon their subordinates, who, for their 
 part, have but little scholarship and individuality. These evils 
 are very closely connected with bad municipal government. 
 
 Experienced teachers, if they have leisure and disposition to 
 improve, are always the best ; but youthful enthusiasm some- 
 times attains far better results than the extreme conservatism 
 of old age. A special difficulty in America has arisen from the 
 enormous rapidity of progress ; the laudatores temporis acti still 
 hold to a state of things which is intolerably old-fashioned. In 
 trigonometry, for instance, we have text-books whose stereotype 
 plates are not yet worn out, but whose ideas are those of a past 
 century; and it is extremely difficult to modernize a pupil 
 taught with their help. The student must, to solve his triangle, 
 draw a figure and make a proportion ; he has to be trained to 
 work from formulae. 
 
 Many such teachers are skeptical about the necessity or use- 
 fulness of studying educational theory. At bottom, this refusal 
 or neglect amounts to the affirmation that tradition and the 
 general progress of the age, combined with natural ability to 
 teach, are a sufficient guide to good teaching; but it often 
 happens that such an instructor repels the effect upon him of 
 the world's progress, and becomes a conservative pure and 
 simple. Such a course is v^ry detrimental, and is a serious
 
 BRANCHES OF MATHEMATICS. 47 
 
 drawback to the benefit to be derived from his experience ; his 
 classes are apt to work mechanically and without interest. The 
 best thing for an experienced teacher to do is to progress 
 slowly but constantly in the improvement of his methods. The 
 books he must study are, in mathematics, those of a scientific 
 character in the branches he teaches, and good text-books in 
 the portions just beyond his course ; also those which treat of 
 the applications of what he is teaching, and finally works on 
 pedagogy. He will do well in these latter studies to take up 
 practical books, not theoretical ones only. 
 
 In conclusion, I may be permitted to say again, what I have 
 said before, that the mathematics which is serviceable in general 
 education is not that of tradition and the English universities of 
 half a century or more ago, but the practical form of the science 
 first developed on the continent of Europe. This is the form 
 which our colleges and technical schools need, and which can 
 be made intelligible and interesting. Much of what has been 
 taught in our schools must and will be dropped as artificial and 
 ugly, to be replaced by that which is natural, beautiful, and 
 useful. American mathematicians will do well to go on in the 
 lines traced for them by Benjamin Peirce and Chauvenet ; and 
 our teachers will need to study this science, and at the same 
 time learn the great art of Pestalozzi and his followers.
 
 APPENDIX. 
 
 THE kindly critic of this little work in the Nation of Feb- 
 ruary 24 has suggested that Grube is not well known. I find 
 his name mentioned in Professor Hall's Bibliography, and the 
 title of his book is there given. Since writing the monograph I 
 have obtained a copy of the book, " Leitfaden fur das Rechnen 
 in der Elementarschule," 6 te Auflage, Berlin, 1881 ; about 150 
 pages octavo. The price in Germany is i mark, 80 pfennigs, 
 or about 45 cents. Soldan's abstract is published in this coun- 
 try by the Interstate Publishing Company. 
 
 Grube's leading idea is that the separate numbers up to 100 
 must be studied as individual objects ; not, of course, by the 
 use of counters beyond the first few exercises, and at proper 
 intervals later ; but taking, as I have pointed out on page 20, 
 each number as a thing to be analyzed by the pupils. When 
 the class has arrived at 89 in the course of work, as the 8gth 
 step, they are already familiar with the qualities of the preced- 
 ing numbers by using proper synthetic exercises and problems. 
 
 In explaining the steps to be taken with the first hundred 
 numbers, the author occupies as much space as is contained in 
 the whole of this monograph ; all of it is very instructive. 
 
 Grube takes the ground that the systematic study of num- 
 bers, according to rules or methods of treatment, such as 
 numeration, notation, addition, etc., must be preceded by at 
 least two years' work in the numerical elements ; in order to 
 train the mind to deal with numbers systematically, it must 
 already be familiar with the numbers themselves. The usual
 
 APPENDIX. 49 
 
 practice, so far as it attempts to teach things and their syste- 
 matic arrangement at the same time, is wrong ; the true method 
 of teaching a system is to use as its material that which is 
 already well known. 
 
 I may be permitted here to say that I have, and have had 
 for many years, an almost unconscious habit of analyzing num- 
 bers which casually present themselves, such as the street 
 numbers of shops and houses. A distinguished Harvard pro- 
 fessor of science, formerly tutor in mathematics, told me the 
 same thing about himself with regard to the numbers of the 
 hymns as set up in church ; and I have no doubt my friend and 
 critic can detect the same thing in his own mental constitution. 
 
 Those who wish to improve their methods of teaching mathe- 
 matics will do well to study the excellent manuals printed, or 
 published, for the use of elementary schools ; a good one 
 is Mr. John T. Prince's "Courses and Methods" (Boston, Ginn 
 & Co.) ; as well as other pedagogical books. Much which I 
 should need to say to amplify this monograph, as the critic de- 
 sires, is merely the ordinary working-out of common school 
 methods to be found in these works. 
 
 The facts which are brought out in the Nation (as before 
 quoted) about lack of elementary training in such subjects as 
 the meaning of "seven per cent," "three-thirteenths," and 
 " two right angles " simply prove that the pupils have not had 
 these objects (using the word in its mental sense) properly pre- 
 sented and impressed upon their minds. 
 
 For example, the idea of one-half, the simplest of all frac- 
 tions, is, according to Grube, the subject of the first step of 
 the course for the fourth year ; then, when the pupil is about 
 nine years old. Grube occupies rather more than two pages in 
 explaining how to teach it, and wishes it to be followed by the 
 ." thirds," the " fourths," and so on. But in dealing with the 
 " thirds " the sixths come naturaify by subtraction from the 
 half; so that when the "sixths" are nominally reached, the
 
 5O APPENDIX. 
 
 pupil is quite ready to deal with pretty large denominators, 
 up to thirty at least. Grube recommends that for this first 
 part of the course in fractions half a year should be taken ; 
 thinking, quite justly, that after the pupil can analyze and 
 combine by addition and subtraction every fraction with de- 
 nominator six or less, he is quite ready to go on rapidly in 
 a systematic course of fractions according to the four ground 
 rules. 
 
 Grube's method for arithmetic seems to me that which best 
 prepares the mind for systematic work by varied exercises of 
 all kinds in the rudiments. It is partially employed in the late 
 edition of Warren Colburn's Arithmetic, the editors of which, 
 as I understand, were two distinguished Cambridge mathema- 
 ticians. They do not give the analysis, according to Grube, 
 beyond twenty, so that a teacher would need to study Grube's 
 book in addition to employing Colburn's. 
 
 Let me say, in concluding this short appendix, that I find 
 that on page 30 I have been misled into expressing myself 
 ambiguously about the " facts " of geometry. My opinion is 
 that in that matter I differ from Dr. Hill. The " rudimentary 
 geometry" I wish to see taught in the lower schools has been 
 systematically worked out by a great many German writers ; it 
 enables pupils to geometrize without, perhaps, the ability to put 
 their demonstrations into words as demonstrations ; just as in 
 arithmetic one multiplies both ways, and soon becomes aware 
 that 9 times 7 equals 7 times 9 without the ability to prove that 
 ab = ba. 
 
 Mere facts in geometry are, perhaps, not fruitful at all ; but 
 the theory of instruction which I have been led to accept lays 
 great stress upon the separation of difficulties for the child's 
 mind into various classes, according to the maturity which they 
 demand of the pupil, and upon the preparation of children 
 six to nine years old for later systematic instruction by a half- 
 playful, at least an objective, presentation of the rudiments.
 
 APPENDIX. 5 1 
 
 Several indications have reached me that the higher teach- 
 ers in this country are not all well aware of the vast progress 
 which the didactic art is making in the primary schools, or if 
 aware that something is going on, do not appreciate it. In 
 studying pedagogy I have continually derived great benefit from 
 familiarity with normal school work and the maxims and prac- 
 tice of the best primary teachers.
 
 SCIENCE. 
 
 For descriptions of forthcoming books in Science see announcements at end oi 
 'is catalogue. 
 
 Organic Chemistry: 
 
 An Introduction to the Study of the Compounds of Carbon. By IRA REMSEN, 
 Professor of Chemistry, Johns Hopkins University, Baltimore. 374 pages. Cloth. 
 Price by mail, 1.30. Introduction price, $1.20. 
 
 THIS book is strictly an introduction to the study of the com- 
 pounds of carbon, adapted to the use of scientific schools, medical 
 schools, schools of technology, and colleges. It takes nothing for 
 granted except an elementary knowledge of general chemistry. Spe- 
 cial care has been taken to select for treatment such compounds as 
 will best serve to make clear the fundamental principles of the sub- 
 ject. General relations as illustrated by special cases are discussed 
 rather more fully than is customary in books of the same size ; and, 
 on the other hand, the number of compounds taken up is smaller than 
 usual, though all which are of real importance to the beginner are 
 treated with some degree of fulness. There is thus less danger of 
 confusion than when a larger number is brought to the attention of 
 the student. 
 
 Eleven editions of this book have been sold for the English market. 
 It has been translated into German, Italian, and Russian. These 
 facts, together with the estimates of professors of chemistry which will 
 be found in our special circular, show that the book has already taken 
 high rank on both sides of the Atlantic. 
 
 M. M. Pattison Muir, Prof, of 
 Chemistry, Cambridge Univ., England: 
 A great many students in this university 
 are using the book; indeed, I think it 
 will soon be used in preference to any 
 other elementary work on the subject. 
 
 D. Mendeleeff, Prof, of Chemistry 
 in the Univ. of St. Petersburg, Russia : 
 The subject is treated in a manner so 
 novel and so instructive that I look for- 
 ward with confidence to seeing the book 
 translated into other languages.
 
 SCIENCE. 
 
 The Elements of Inorganic Chemistry : 
 
 Descriptive and Qualitative. A Text-Book for Beginners, based on Experi- 
 mental and Inductive Methods. By J. H. SHEPARD, Professor of Chemistry, 
 South Dakota Agricultural College, and Chemist to the U. S. Experimental Sta- 
 tion, S.D. 397 pages. Cloth. Price by mail, #1.25. Introduction price, $1.12. 
 
 IT is a practical embodiment of the modern spirit of investi- 
 gation. It places the student in the position of an investigator, 
 and calls into play mental faculties that are too often wholly neg- 
 lected. It leads him to experiment, to observe, to think, to originate. 
 Coming as it does from the working laboratory of a practical in- 
 structor, who has had the constant advice of fellow-teachers in all 
 parts of the country, this text may be fairly taken as an exponent of 
 the latest methods of teaching chemistry. 
 
 Its distinctive features are : experimental and inductive methods ; 
 the union of descriptive and qualitative chemistry, thus allowing 
 these kindred branches to supplement and illustrate each other; a 
 practical course of laboratory work illustrating the general principles 
 and their application ; a fair presentation of chemical theories, and a 
 conciseness which confines the work to the required limits. 
 
 The book covers more fully than any other the outline of work laid 
 down in the report of the Conference on Chemistry to the Committee 
 of Ten. 
 
 Our special circular on this book contains a list of over three hun- 
 dred and fifty Colleges and Schools into which it made its way by 
 merit alone, in a little Tnore than a year's time, and also gives a large 
 number of letters from teachers of the siibject showing with what ease, 
 profit, and satisfaction it is used in the class-room. 
 
 C. A. Schseffer, Prof, of Chemistry, 
 Cornell Univ. : It is excellent. The plan 
 is well conceived, and embodies the method 
 by all means the best. 
 
 C. P. Chandler, Prof, af Chemistry, 
 Columbia Univ. : An excellent book. A 
 great degree of accuracy characterizes the 
 entire work. 
 
 Otis C. Johnson, Prof, of Applied 
 Chemistry, Univ. of Michigan : I like 
 it so well I have nothing to criticise. 
 
 Ira Remsen, Prof, of Chemistry, 
 Johns Hopkins Univ.: The book teaches 
 that chemistry can only be learned in the 
 laboratory, and that the book is only a 
 guide to experimental work, a lesson 
 which certainly needs to be taught above 
 all others in chemistry. 
 
 Herbert B. Smith, Prof. f Chm 
 istry, Yale Coll.: A class working 
 through it would gain an unusually good 
 knowledge of the subject.
 
 SCIENCE. 
 
 The Laboratory Note-Book. 
 
 For Students using any Chemistry. Board covers. Cloth back. 192 page& 
 Price by mail, 40 cents. Introduction price, 35 cents. 
 
 TT contains blanks for experiments ; blank tables for the reactions of 
 1 the different metallic salts ; pages for miscellaneous matter ; and 
 an extra chart for the natural classification of the elements, similar to 
 that on page 221 of Shepard's Chemistry, which may be rolled into a 
 cylinder by the student. 
 
 The advantages of using this note-book are, briefly, these : It 
 saves time for the student and gives the teacher control of his work ; 
 its size is convenient ; it is cheaper than an ordinary blank-book ; the 
 paper is such that it readily takes ink without blotting or smearing, 
 or it may be used with a lead pencil. 
 
 The value of systematic note-taking by the the student in chemis- 
 try can hardly be over-estimated. 
 
 Our Chemistry Circular contains fac-similes of three pages, pre* 
 pared by the students in the Ypsilanti high school, showing how the 
 book is to be used. 
 
 Geo. F. Sawyer, Deft, of Chemistry, 
 Martha's Vineyard Summer School: It 
 is the best in plan I have seen. 
 
 D. W. Batson, Pres. Ky. Wesley an 
 Univ., Millersburg: It is answering our 
 purpose excellently. 
 
 Chemical Problems: 
 
 Adapted to High Schools and Colleges. By JOSEPH P. GRABFIELD and T. S. 
 BURNS, Instructors in General Chemistry in the Mass. Inst. of Technology. 
 Paper. 96 pages. Price by mail, 30 cts. Introduction price, 25 cents. 
 
 HHIS book comprises the principles of stoichiometry, with separate 
 1 chapters upon atomic and molecular weight, determinations and 
 specific gravity of gases, and upon the principles of thermo-chemistry 
 and its application to inorganic chemistry. The authors have added, 
 as a set of general questions, a series of problems and reactions and 
 the examination papers in general chemistry given at the Institute 
 of Technology during the last ten years. The book contains no 
 formulae, students being led to solve the problems from the principles 
 involved.
 
 SCIENCE. 
 
 A Short History of Chemistry. 
 
 By F. P. VENABLE, Professor of Chemistry in the University of North Carolina. 
 171 pages. Cloth. Introduction price, $1.00. By mail, $1.10. 
 
 THERE has long been a need for a short, systematically arranged 
 History of Chemistry written in the English language. This 
 book is issued with a view to filling this want. It is concise, and yet 
 full enough to give a connected view of the growth and development 
 of the science. Stress has been especially laid upon the rise of theories 
 and the discussions over them, the overthrow of the false and the sur- 
 vival of the true. 
 
 To the student, even the beginner in the science, as. Victor Meyer 
 has pointed out, the historical method is the most attractive and the 
 most promising in results. 
 
 This book is the outcome of a course of lectures, tested during 
 three years' use with students. All of the chief authorities upon the 
 subject have been consulted, and, where possible, the original writings 
 referred to. The aim has been to so digest and systematize this mass 
 of material as to render it available for those desiring a general 
 knowledge of the subject. 
 
 The book should be in every school and college library and is 
 especially adapted for use as complement to the ordinary college courses 
 in general chemistry. 
 
 A prospectus giving preface and table of contents sent free to any 
 one desiring to see it. 
 
 The following are a few of the many commendatory words that have 
 come to us concerning Dr. Venable^s book : 
 
 Ira Remsen, Professor of Chemistry, 
 Johns Hopkins University: It is short 
 and clear, and will, I think, prove of value 
 to students and general readers. There is 
 no book like it in English. 
 
 J. O. Reed, Asst. Professor of Physics, 
 University of Michigan : The appear- 
 ance of the book is timely and it contains 
 much valuable matter in small space. 
 
 The Educational Times, London, 
 Eng.: We strongly recommend this 
 charming little book to teachers of chem- 
 istry. It is attractively written, and we 
 have failed to find any inaccuracy. The 
 author has fully justified his assertion 
 that the study of the history of a science is 
 one of the best aids to its intelligent com- 
 prehension.
 
 6 SCIENCE. 
 
 A Laboratory Manual in Organic Chemistry. 
 
 Containing directions for a course of experiments, systematically arranged to ac- 
 company Remsen's Organic Chemistry. By W. R. ORNDORFF, Assistant 
 Professor of Chemistry in Cornell University. 192 pages. Boards. Intro- 
 duction price, 35 cents. By mail, 40 cents. 
 
 INTENDED to be used as a laboratory guide in the study of the 
 A Compounds of Carbon, or Organic Chemistry, and to supplement 
 the lectures and text-book work in the subject. Careful directions are 
 given for about eighty experiments on the derivatives of the most im- 
 portant series of hydrocarbons. The student is constantly questioned 
 as to the meaning of his work and the results obtained, and is led to 
 deduce his own conclusions as far as possible. 
 
 The book has been in use for the past six years in the Organic Lab- 
 oratory of Cornell University, and has given general satisfaction. It 
 has now been corrected and revised in the light of the experience 
 gained in the past, and in the belief that it will prove useful in other 
 laboratories. 
 
 A Laboratory Guide in General Chemistry. 
 
 For a twenty weeks course. By GEORGE WILLARD BENTON, Instructor in 
 Chemistry, Indianapolis High School. 165 pages. Boards. Introduction price, 
 35 cents. By mail, 40 cents. 
 
 THIS manual is intended for the use of the student in working labor- 
 atories of high school grade. It contains careful and detailed 
 instructions for the performance of 156 experiments, an abundance of 
 queries and blank pages for notes. The experiments have been selected 
 with a view to simplicity in apparatus and manipulation, and a high 
 percentage of success at the hands of the average pupil. 
 
 As a foundation for quiz and recitation, and for the development of 
 the general principles of inorganic chemistry, the course will be found 
 suggestive and pedagogical. It is adapted to any text-book. 
 
 Teachers will appreciate the attention given to details, which are so 
 essential to beginners. This feature of the book, having been 
 especially considered as the course has grown up in actual laboratory 
 practice, is most carefully and successfully carried out. 
 
 The table of References by Experiments as given in the appendix 
 will be found a new and valuable feature.
 
 SCIENCE. 
 
 The Elements of Chemical Arithmetic, 
 
 with a Short System of Elementary Qualitative Analysis. By J. MILNOR COIT, 
 Master in St. Paul's School, Concord, N. H. 95 pages. Cloth. Price by mail, 
 55 cents. Introduction price, 50 cents. 
 
 I^HIS manual is divided into two parts. Part I contains the more 
 important rules and principles of Stoichiometry, put in a very 
 simple way to suit the comprehension of the average student. These 
 rules and principles find application in a series of carefully selected 
 problems which follow. Part II contains an elementary system of 
 qualitative analysis intended for beginners. The most available and 
 satisfactory tests for the more important metals and acids are given 
 with simple tables of separation. Besides these are some carefully 
 compiled tables for reference. The book will be found valuable as a 
 companion for some regular text of descriptive chemistry or where it 
 is desired to give an elementary course in practical chemistry. It 
 has just been carefully revised and some valuable tables added. 
 
 T. H. Norton, Prof, of Chemistry, 
 Univ. of Cincinnati, O. : It is admirably 
 written, and well adapted to supplement the 
 ordinary descriptive text-book or series of 
 lectures preparatory to scientific courses. 
 
 Robert Peter, Prof, of Chemistry, 
 Kentucky State Univ., Lexington: I 
 can recommend 'it as a good, compen- 
 dious, and cheap guide for Laboratory 
 practice in Qualitative Analysis. 
 
 Stereo-Chemistry. By CHARLOTTE F. ROBERTS. Professor of Chemistry in 
 Wellesley College, Mass. 160 pages. Cloth. Introduction price, $1.00. By 
 mail, $1.10. 
 
 THE object of this little book is to present in a somewhat elementary 
 and compact form the most important results of the stereo-chem- 
 ical investigation of the past twenty years. It is hoped that it may give 
 to students of chemistry, who have not had the time or opportunity to 
 follow the original papers which have appeared from time to time, a 
 general idea of the work which has been accomplished, and of that 
 which remains to be done. Since constant references are made to the 
 original papers, the book may well be taken as the starting-point for a 
 more extended study of the subject. 
 
 The principal topics are: 1st, the general principles of stereo-chem- 
 istry ; 2d, the work done, principally by Von Baeyer, in connection 
 with closed rings; jd, the stereo-chemistry of nitrogen; 4th, light 
 thrown on theoretical problems by stereo-chemical investigation.
 
 SCIENCE. 
 
 First Book in Geology. 
 
 By N. S. SHALER, Professor of Paleontology, Harvard University. 272 pages, 
 with 130 figures in the text. Cloth: introduction price, $1.00; price by mail. 
 $1.10. Boards: introduction price, 60 cts. ; by mail, 70 cts. Teacher's Man- 
 ual. Paper, 74 pages. Retail price, 25 cts. 
 
 r^vESIGNED to give public school pupils and general readers a few 
 L^ clear, well-selected facts as a key to the knowledge of the earth. 
 The aim is to illustrate the principles of geology by reference to as many 
 facts of familiar experience as possible, such as ; Pebbles, Sand, and 
 Clay. The Making of Rocks. The Work of Water and Air. The 
 Depths of the Earth. Irregularities of the Earth. Origin of Valleys 
 and Lakes. Movements of the Earth's Surface. Place of Animated 
 Beings in the World. Sketch of the Earth's Organic Life. Nature 
 and Teaching of Fossils. Origin of Organic Life, and a Brief 
 Account of the Succession of Events on the Earth's Surface, etc. 
 
 The Teacher's Manual contains seventy-four pages of directions 
 for those who use the book in class instruction. The average reader 
 who desires to get a glance at geology and a general notion of its 
 bearings on ordinary life will find this edition of exceeding interest. 
 
 The special circular on this book shows how it has been received by 
 teachers of Geology. The book has been translated into Polish, and 
 is soon to be translated into German. It is also being used in many 
 schools as a Supplementary Reader, the edition in board covers being 
 especially prepared for this purpose. 
 
 John C. Branner, Prof, of Geology, 
 Stanford Univ. : With a view to urging 
 the use of some elementary book on geol- 
 ogy in the schools of this State, I have 
 examined Prof. Shaler's Geology. I can- 
 not do better than recommendit. 
 
 F. W. Hooper, Teacher of Nat. 
 Science, Adelpht Acad., Brooklyn : Of 
 all the attempts at making elementary 
 text-books on the natural history sciences, 
 this is the most successful. 
 
 N. A. Cobb, Teacher of Natural 
 Science, Williston Sem., Mass, : I un- 
 
 hesitatingly say that it is the best text- 
 book on geology for young people that I 
 have ever seen. 
 
 Alexander Winchell, late Prof. 
 Geology, Univ. of Mich. : Its marked de- 
 parture from the methods of the old didac- 
 tic treatises which have done so much tr> 
 put geology at a disadvantage, in compari- 
 son with botany, is in the direction of 
 common sense and in the interests o( 
 science and education. I believe the book 
 will have, as it deserves, an extensive use 
 in the school.
 
 SCIENCE. 
 
 Object Lessons and How to Give Them. 
 
 In two volumes. Manuals for Primary and Intermediate Teachers. By GEORGK 
 RICKS, Inspector of Schools, London School Board. First volume, 200 pages. 
 Second volume, 212 pages. Retail price, 90 cents each. 
 
 THESE two graded volumes are by the author of " Natural History 
 Object Lessons," which is so popular and so widely used, and 
 are designed to aid teachers in the lower grades in the same way as 
 the more advanced book gives aid for higher grades. 
 
 Two points of fundamental importance are considered : ( I ) the se- 
 lection and adaptation of lessons to the capacities of children; (2) the 
 method of giving the individual lesson. By means of the complete 
 system here given, lessons are graduated to the mental conditions and 
 previous training of pupils. Each lesson, while fulfilling its own 
 special purpose, forms a link in a chain. The method of conducting 
 the lesson is such that the child is not allowed to be a mere passive 
 recipient of information, but is led to exercise his own mental powers. 
 The lessons are suggestive rather than exhaustive ; some are little more 
 than outlines ; a few are worked out more fully as models for young 
 teachers. The experiments are numerous and interesting, yet simple 
 and inexpensive. The contents of the books are as follows : 
 
 FIRST VOLUME. Chapters I and II train to habits of observation 
 and comparison, rather than impart information. Ill shows by means 
 of experiments special properties upon which uses of familiar objects 
 depend. IV imparts useful information about common necessaries of life. 
 V completes course of ' Systematic Object Lessons " for infant schools. 
 
 SECOND VOLUME. Chapter I deals with common properties of 
 solids. II deals with similar properties of liquids. Ill, IV, and V 
 bring the reasoning faculties more fully into play. 
 
 Geo. L. Chandler, Supv. of Science, 
 Public Schools, Newton, Mass. : I think 
 Part II (which I have examined) a book 
 that will find a place on the teacher's 
 study table for help in preparing science 
 work. 
 
 F. H. Bailey, formerly Teacher of Sci- 
 ences, Mrs. Shaw's School, Boston : They 
 are invaluable. I cannot spare them a 
 day from my work. 
 
 W. D. Sheldon, Vice-Pres. of Girard 
 Coll., Phila. I have examined them with 
 a good deal of care. The lessons are ad- 
 mirably arranged and developed. I shall 
 recommend them. 
 
 A. C. Boyden, Teacher of Science, 
 Normc.l SJiool, BridgfVMtttr, Mass.: 
 They are excellent books and I strongly ad- 
 vise you to put them on this market.
 
 i6 
 
 SCIENCE. 
 
 Elementary Course in Practical Zoology. 
 
 By B. P. COLTON, A.M., Prin. of High School, Ottawa, 111. Cloth. 196 pages 
 Price by mail, 85 cents. Introduction price, 80 cents. 
 
 A BOOK to lie open before the student while he is studying the 
 animal itself. It first tells him where to find his specimens ; 
 how to observe their habits and habitats ; in a few cases their meta- 
 morphoses and modes of development ; how to collect and preserve ; 
 and, finally, how to dissect them. In short, it is a guide to the study 
 Df animals rather than a mere descriptive zoology. 
 
 A series of typical representations of all the animal sub-kingdoms 
 s represented ; after the study of each, the student is led to compare 
 :hem one with another, noting their resemblances and differences ; 
 :hus he learns how to classify animals instead of memorizing a system 
 )f classification. 
 
 This method gives the student a better view of the animal kingdom 
 han any amount of mere reading, and, at the same time, develops and 
 lisciplines his powers of observation and description. 
 
 The special circular on this book gives a large number of strong 
 'ommendations ; a list of places that at once adopted it; and an ex- 
 el lent article on the study of Zoology, written for " The Dial" by Carl 
 H. Eigenmann, Dept. of Zoology, Ind. State Univ. 
 
 David S. Jordan, Pres. of Leland 
 Stanford Jr., University, Cal.: I regard 
 t as the only text-book in general zoology 
 et published which is fit to be used in 
 igh school classes. It is on the right plan, 
 nd the method is admirably carried out. 
 
 Nature, London : In consideration of 
 he pernicious rubbish which, even yet, 
 ccasionally finds its way into our own 
 lementary schools under the guise of the 
 lementary text-book of Science, it ispleas- 
 nt to reflect upon the merits of this work. 
 
 Prof. Alpheus Hyatt, Boston So- 
 'ety of Natural History : Mr Colton's 
 esire is to cultivate the faculty of obser- 
 ition, and I think he has succeeded in 
 roducing a very valuable help to the 
 acher of manual science. 
 
 T. C. Mendenhall, Pres. Polytechnic 
 Inst., Worcester, Mass.: I am delighted 
 with the book. I think the author has 
 struck " pay-rock " for the department of 
 science. 
 
 B. Ellsworth Call, Manual Train- 
 ing School, Louisville, Ky. : We have had 
 135 copies in use. Enthusiasm has been 
 awakened by the actual study of the ob- 
 jects, an enthusiasm which has been 
 directed to definite ends by this little 
 work. The book deserves most abundant 
 success, and I believe will receive it. 
 
 Albert H. Tuttle, Prof, of Zoology, 
 Ohio State Univ. : It is the most useful 
 book of its kind ever published in this 
 country, not only for high school use, but 
 for beginners in zoology in colleges as well.
 
 SCIENCE. 23 
 
 An Astronomical Lantern. 
 
 Invented by Rev. JAMES FREEMAN CLARKE, Boston. The face (6 1-2 by 10 in- 
 ches in size) is of ground glass, behind which lights may be placed. Thirty-two 
 constellations are photographed upon seventeen slides of semi-transparent card 
 board, and stars of four magnitudes are represented by perforations of proper size 
 Price with the slides and a copy of " How to Find the Stars," (see below) $4.50. 
 The whole carefully packed in a wooden box, with sliding cover. 
 
 THIS useful piece of apparatus is designed to facilitate the study 
 of stellar astronomy. It is intended for beginners in astronomy 
 in schools and in families, and for all, both old and young, who desire 
 to become acquainted with the constellations. 
 
 The student wishing to observe any particular constellation or 
 cluster has only to light a candle within the lantern, insert the appro- 
 priate slide, and go out into the night. Holding up the lantern in one 
 hand, he can compare the constellation as it appears on the lantern with 
 that in the sky, until he becomes perfectly familiar with the latter. 
 
 C. A Young, Prof, of Astronomy, 
 Princeton Coll.: I find it to be an ad- 
 mirably contrived apparatus for its pur- 
 pose, simple, easily managed, and effect- 
 
 use, in connection with the little book that 
 accompanies it, more rapidly and easily 
 
 C. S. Lyman, Prof, of Astronomy, 
 Yale Univ. : I have never known any 
 
 contrivance that could compare with this 
 Lantern for saving alike time, trouble, and 
 eye-sight, and rendering such study attrac- 
 tive and easy. 
 
 ive. I thmk an adequate knowledge of 
 
 ,. n * ij t, v.4. A if v O. C. Wendell, Harvard Coll. Ob 
 
 the constellations could be obtained by its 
 
 servatory: It combines in a high degree 
 
 simplicity with clearness, and, for begin- 
 
 f . , v ners and amateurs, I think it has no equal, 
 
 than from the most elaborate and expen- 
 
 .,..,, I am confident that the names of the 
 
 sive celestial globe. 
 
 principal stars and constellations can be 
 learned from it in half the time required 
 to learn them from an ordinary map. 
 
 How to Find the Stars. 
 
 By Rev. JAMES FREEMAN CLARKE. Paper. 47 pages. Price, 15 cents. 
 
 THE object of this book is to help the beginner to become better 
 acquainted with the visible starry heavens ; to know the winter 
 and summer constellations and the principal fixed stars. It shows 
 the position of the constellations at different periods of the year. 
 The most interesting objects at each period of the year, especially- 
 such as can be found with a telescope of moderate power, are indi- 
 cated. A description of the Astronomical Lantern (mentioned above) 
 is appended.