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 SYMPOSIUM ON MATHEMATICS 
 FOR ENGINEERING STUDENTS 
 
 BEING THE 
 
 PROCEEDINGS OF THE JOINT SESSIONS 
 
 OF THE 
 
 CHICAGO SECTION OF 
 /THE AMERICAN MATHEMATICAL SOCIETY 
 
 *" AND 
 
 SECTION A, MATHEMATICS, AND 
 SECTION D, MECHANICAL SCIENCE AND ENGINEERING 
 
 OF THE 
 
 AMERICAN ASSOCIATION FOR THE ADVANCEMENT OF SCIENCE 
 
 HELD AT 
 
 THE UNIVERSITY OF CHICAGO 
 DECEMBER 30 AND 31, 1907 
 
 Report of the Meeting Wm. T. MagbudeB, Secretary, Ohio State University 
 
 Present Condition of Mathematical Instruction for Engineers in American Colleges 
 
 Edgab J. Townsend, University of Illinois 
 The Teaching of Mathematics to Engineering Students in Foreign Countries 
 
 Alexander Ziwet. University of Michigan 
 
 The Teaching of Mathematics for Engineers Chas. F. Scott, Pittsburg, Pa. 
 
 The Point of View in Teaching Engineering Mathematics R. S. Woodward, Carnegie Institution 
 
 The Teaching of Mathematics to Students of Engineering: 
 From the Standpoint of the Practising Engineer 
 
 Ralph Modjeskt, Chicago, 111.; J. A. L. Waddell, Kansas City, Mo. 
 From the Standpoint of the Professor of Engineering 
 
 Gabdneb S. Williams, University of Michigan ; Arthur N. Talbot, University of Illinois ; 
 George F. Swain, Massachusetts Institute of Technology. 
 From the Standpoint of the Professor of Mathematics in the Engineering College 
 
 Chas. S Sltchteb. University of Wisconsin ; Fbedebick S. Woods, Massachusetts Institute 
 of Technology ; Feed W. McNaib, Michigan College of Mines. 
 General Discussion: Calvin M. Woodward, Washington University ; B. F. Groat, University of Min- 
 nesota ; C. S. Howe, Case School of Applied Science ; Clarence A. Waldo, Purdue University ; 
 0. P. Williams, Kalamazoo College ; J. Burkitt Webb, Stevens Institute of Technology ; 
 FT. T. Eddy, University of Minnesota ; S. M. Barton, University of the South ; Arthur E. 
 Haynes, University of Minnesota ; Arthur S. Hathaway, Rose Polytechnic Institute ; Edward 
 V. Huntington, Harvard University ; Donald F. Campbell, Armour Institute of Technology. 
 
 This symposium was arranged for by a committee of the Chicago Section of the American Mathematical Society 
 appointed at its meeting of December 28, 1906, and consisting of E. B. VanVleck, University of Wisconsin, Chair- 
 man ; H. E. Rlaught, University of Chicago, Secretary ; E. J. Townsend, University of Illinois; Alexander Ziwet, 
 University of Michigan; E. B. Skinner, University of Wisconsin ; A. G. Hall, Miami University; H. L. Rietz, 
 University of Illinois ; together with the co-operation of Wm. T. Magruder, Ohio State University, Secretary of Sec- 
 lion D, Mechanical Science and Engineering, of the American Association for the Advancement of Science. 
 
 Reprinted from Science, x. s. Vol. XXVIII., Nos. 707, 708, 709, 710, 713, 714, July 17, 24, 31, 
 
 August 7, 28, and September 4, 1908 
 
 
 OF 
 
 

 L 
 
[Reprinted from Science, N. S., Vol. XXVIIL, No. 707, Pages 65, 67-79, July 17, No. 708, 
 
 Pages 109-118, July 24, No. 709, Pages 129-188, July 81, No. 710, Pages 161-170, August 
 
 7, No. 718, Pages 257-268, August 28, No. 7U, Pages 289-299, September 4, 1908.'] 
 
 THE AMERICAN ASSOCIATION FOR THE ADVANCEMENT 
 
 OF SCIENCE 
 
 SECTION D MECHANICAL SCIENCE AND ENGINEERING 
 
 ENGINEERING-MATHEMATICS SYMPOSIUM 
 
 The meeting of the section for organiza- 
 tion was held in Cobb Hall of the Uni- 
 versity of Chicago on December 30 and 31, 
 1907, and January 1, 1908. The vice- 
 president of the section, Olin H. Landreth, 
 professor of civil engineering, Union Uni- 
 versity, acted as chairman of the section. 
 
 At the meeting of the general committee 
 on January 2, 1908, on the recommendation 
 of the sectional committee, Dr. George F. 
 Swain, professor of civil engineering, 
 Massachusetts Institute of Technology, 
 was elected vice-president and chairman of 
 the section for the ensuing year; and Mr. 
 George W. Bissell, dean of engineering and 
 professor of mechanical engineering, Mich- 
 igan Agricultural College, was elected sec- 
 retary for the next five years. 
 
 The promotion of acquaintance and 
 personal knowledge was an important 
 factor in the success of the meeting, which 
 was in large part due to the labors and 
 foresight of Professor H. E. Slaught, of 
 the department of mathematics of the Uni- 
 versity of Chicago, and Secretary of the 
 Chicago Section of the American Mathe- 
 matical Society. 
 
 A subscription dinner for engineers and 
 mathematicians and their friends brought 
 about one hundred persons together at 
 Hotel Del Prado on Monday evening, De- 
 cember 30. The speakers at the dinner 
 were introduced by E. B. Van Vleck, pro- 
 fessor of mathematics, University of Wis- 
 consin, Chairman of the Chicago Section of 
 
 the American Mathematical Society. They 
 were Calvin M. Woodward, dean of the 
 School of Engineering and Architecture, 
 Washington University, St. Louis, Mo.; 
 Charles- F. Scott, consulting engineer of 
 the Westinghouse Electric & Manufactur- 
 ing Co., Pittsburg, Pa.; George F. Swain, 
 professor of civil engineering, Massachu- 
 setts Institute of Technology, Boston, 
 Mass. ; and Edward V. Huntington, assist- 
 ant professor of mathematics, Harvard 
 University, Cambridge, Mass. 
 
 The first session of the engineering- 
 mathematics symposium was held on Mon- 
 day afternoon, December 30. Professor 
 Van Vleck acted as chairman. Four 
 papers were presented, as follows: 
 The Present Condition of Mathematical 
 Instruction for Engineers in American 
 Colleges: Edgar J. Townsend, professor 
 of mathematics, University of Illinois. 
 
 The Teaching of Mathematics to Engineer- 
 ing Students in Foreign Countries: 
 Alexander Ziwet, professor of mathe- 
 matics, University of Michigan. 
 
 The Teaching of Mathematics for Engi- 
 neers: Charles F. Scott, consulting 
 engineer, Westinghouse Electric and 
 Manufacturing Co. 
 
 The Point of View in Teaching Engineer- 
 ing-Mathematics: Robert S. Woodward, 
 president of the Carnegie Institution of 
 Washington. 
 The two sessions, held on the morning 
 
 180924 
 
SCIENCE 
 
 and afternoon of December 31, were de- 
 voted to a symposium on the question: 
 "What is needed in the Teaching of 
 Mathematics to Students of Engineering? 
 (a) Range of Subjects; (&) Extent in the 
 Various Subjects; (c) Methods of Prepara- 
 tion; (d) Chief Aims." The speakers 
 represented three phases of the subject, 
 namely: (a) Prom the standpoint of the 
 practising engineer; (&) from the stand- 
 point of the professor of engineering; (c) 
 from the standpoint of the professor of 
 mathematics in the engineering college. 
 
 Professor Landreth and Professor 
 Slaught were the chairmen of the two 
 sessions. The speakers were as follows: 
 Ralph Modjeski, consulting engineer, Chi- 
 cago, 111.; J. A. L. Waddell, consulting 
 bridge engineer, Kansas City, Mo.; Gard- 
 ner S. Williams, professor of civil, hy- 
 draulic, and sanitary engineering, Univer- 
 sity of Michigan; Arthur N. Talbot, pro- 
 fessor of municipal and sanitary engineer- 
 ing, University of Illinois; George F. 
 Swain, professor of civil engineering, 
 Massachusetts Institute of Technology; 
 Charles S. Slichter, consulting engineer, 
 U. S. Reclamation Service, and professor 
 of applied mathematics, University of Wis- 
 consin; Frederick S. Woods, professor of 
 mathematics, Massachusetts Institute of 
 Technology; and Fred W. McNair, presi- 
 dent of the Michigan College of Mines. 
 
 Following the presentation of the four 
 formal papers, and of the eight prepared 
 discussions above recorded, a general dis- 
 cussion was held on the entire subject. 
 The following persons took part in this 
 general discussion: Calvin M. Woodward, 
 professor of mathematics and applied me- 
 chanics, Washington University; Ben- 
 jamin F. Groat, professor of mechanics 
 and mathematics, School of Mines, Uni- 
 versity of Minnesota; Charles S. Howe, 
 president, Case School of Applied Science; 
 Clarence A. Waldo, professor of mathe- 
 
 matics, Purdue University; Clarke B. 
 Williams, professor of mathematics, Kala^ 
 mazoo College; J. Burkitt Webb, late 
 professor of mathematics and mechanics, 
 Stevens Institute; Henry T. Eddy, pro- 
 fessor of mathematics and mechanics, Col- 
 lege of Engineering, University of Min- 
 nesota; Arthur E. Haynes, professor of 
 engineering-mathematics, University of 
 Minnesota; Arthur S. Hathaway, professor 
 of mathematics, Rose Polytechnic Institute ; 
 Edward V. Huntington, assistant professor 
 of mathematics, Harvard University; and 
 Donald F. Campbell, professor of mathe- 
 matics, Armour Institute of Technology. 
 
 On motion of Professor Campbell, the 
 chairman was authorized to appoint a com- 
 mittee of three persons, they to increase 
 their number to fifteen, to be chosen from 
 among teachers of mathematics and engi- 
 neering and from the practising engineers 
 of the country; and this committee of 
 fifteen was authorized by the meeting to 
 take into consideration the whole subject 
 of the mathematical curriculum in the 
 engineering and technical departments of 
 colleges and universities, and to report to 
 the Chicago Section of the American 
 Mathematical Society. On motion of Wm. 
 T. Magruder, ex-secretary of the Society 
 for the Promotion of Engineering Educa- 
 tion and professor of mechanical engineer- 
 ing, Ohio State University, the motion was 
 amended that the committee of fifteen shall 
 submit its report to the Society for the 
 Promotion of Engineering Education at 
 its annual meeting in the summer of 1909. 
 The motion as amended was unanimously 
 adopted by those present. It is hoped 
 that at the meeting of the society in 1909, 
 a second engineering-mathematics sym- 
 posium may be held. 
 
 The selection of this important com- 
 mittee was entrusted to Professor Edward 
 V. Huntington, Harvard University, Pro- 
 fessor Gardner S. Williams, University 
 
SCIENCE 
 
 of Michigan, and Professor Edgar J. 
 Townsend, University of Illinois. They 
 will select the remaining members of the 
 committee, choose a chairman and secre- 
 tary, and determine the scope of the in- 
 vestigation that they will make. 
 
 The papers will be printed in Science in 
 the next few weeks. They will prove to be 
 interesting reading to those engaged in 
 either mathematical or engineering work 
 and will show the tendencies of the thought 
 of the meeting. The key-note of all the 
 discussions was that we need more sym- 
 pathy and knowledge of the ideals, aims 
 and work of the other fellow. 
 
 The meeting was without doubt the best 
 attended that the sections have held for 
 many years, the interest never seemed to 
 flag and, while no wonderful contributions 
 were made to scientific knowledge, every 
 one went away feeling either that he had 
 gained much information as to the other 
 man's point of view concerning scien- 
 tifically instructing engineering students in 
 mathematics and of the wishes and needs 
 of the engineering instructor, or that he 
 appreciated more the quality of work that 
 was now being done by teachers of mathe- 
 matics in engineering colleges. 
 
 m. T. Magruder, 
 Secretary, Section D 
 
 PRESENT CONDITION OF MATHEMATICAL 
 
 INSTRUCTION FOR ENGINEERS IN 
 
 AMERICAN COLLEGES 1 
 
 Our country has witnessed in recent 
 years a most marvelous industrial expan- 
 sion and development. Along with this 
 movement has come a rapidly increasing 
 demand for trained men, equipped with all 
 that science can contribute, to direct and 
 
 1 Opening address before the joint meeting of 
 Sections A and D of the American Association 
 for the Advancement of Science with the Chicago 
 Section of the American Mathematical Society 
 for the discussion of the topic " Mathematical 
 Training for Engineers." 
 
 carry forward this development of our 
 natural resources and our industrial power. 
 In meeting this demand our technical 
 schools have experienced a remarkable 
 growth, and not a little of the educational 
 thought and activity of the country is 
 being directed toward the problems con- 
 nected with technical instruction. Well- 
 equipped engineering schools have grown 
 up in the larger centers of population and 
 most of the larger state universities now 
 include strong engineering departments. 
 Mathematics is so fundamental to all of 
 this work, and so large a proportion of the 
 students now receiving mathematical in- 
 struction in this country anticipate making 
 use of it later in connection with engineer- 
 ing work, that it has been thought best by 
 the Chicago Section of the American 
 Mathematical Society to invite to a joint 
 discussion of the "Mathematical Training 
 of Engineering Students," representatives 
 from some of the leading engineering 
 schools and some of those consulting engi- 
 neers whose wide experience has brought 
 them into contact with demands of actual 
 practise. 
 
 That we may all know what the actual 
 conditions are with respect to this instruc- 
 tion and consequently have some common 
 basis for our discussion and our conclu- 
 sions, I have been asked to present a state- 
 ment of the work in mathematics which 
 is now being given to engineering students. 
 
 As the basis of our consideration, I have 
 selected seventeen institutions where engi- 
 neering work is an important feature. Of 
 these, eight give their attention largely or 
 exclusively to technical work, and the re- 
 maining institutions have strong engineer- 
 ing departments ; so that the mathematical 
 work given in these institutions may be 
 said to fairly represent the preparation in 
 this subject for engineering students in 
 American institutions. 
 
 The three most important factors enter- 
 
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 ing into the consideration of our topic are : 
 the entrance requirements, the require- 
 ments for graduation, and the qualifica- 
 tions of the instructional force. 
 
 As will be seen from Table I., all of 
 these seventeen institutions require for 
 entrance algebra through quadratics, to- 
 gether with plane and solid geometry. 
 Five of the institutions require plane trig- 
 onometry, while at several others it may be 
 counted for entrance if the student so 
 elects. It will be observed that four insti- 
 tutions require elementary algebra through 
 progressions, four require the subject of 
 logarithms, and two, Sheffield and Cornell, 
 require the whole of college algebra. 
 
 There is a general tendency over the 
 country to increase rather than diminish 
 the entrance requirements in mathematics. 
 Several institutions have recently done so, 
 and at a number of others there is a feel- 
 ing that both trigonometry and college 
 algebra should be required. This disposi- 
 tion to increase the entrance requirements 
 has come about not so much because of a 
 feeling that these subjects can be as well 
 or better taught in the secondary schools, 
 but because of a feeling on the part of 
 the technical schools that the entrance re- 
 quirements should be made as high as 
 possible in order to give room in the cur- 
 
 riculum for those professional and tech- 
 nical branches which are now deemed 
 essential. It may well be questioned 
 whether we are not in some danger of 
 going too far in increasing the require- 
 ments. I am sure that we should all agree 
 that the guiding principles should be the 
 limitations of the secondary school pro- 
 gram and the ability of the pupil at that 
 stage of his maturity to readily grasp in 
 a comprehensive manner the subjects pre- 
 sented. For example, the advisability of 
 adding college algebra to the entrance re- 
 quirements is certainly open to the ob- 
 jection that portions of it are clearly be- 
 yond the maturity of the average high 
 school pupil, and the introduction of plane 
 trigonometry would seem inadvisable in the 
 average high school on the accredited list 
 of the state universities of the Mississippi 
 Valley. When either of the fundamental 
 principles mentioned is violated, we shall 
 have coming to our freshman class, stu- 
 dents with a decided and a justifiable dis- 
 like for anything mathematical. Bather 
 than to encounter this danger, it would be 
 far better to extend the engineering course 
 over five years or to require a year of 
 college work in science and mathematics 
 before the student enters upon his tech- 
 nical course. In this connection, it is in- 
 
SCIENCE 
 
 teresting to note that the University of 
 Minnesota has recently extended its course 
 to five years for students in civil, me- 
 chanical and electrical engineering, dis- 
 tributing the required work in mathe- 
 matics throughout the first four years. 
 
 The writer does not share with some the 
 feeling that a greater uniformity in en- 
 trance requirements is either desirable or 
 of any particular consequence. Each in- 
 stitution, and especially the state institu- 
 tions, must take into consideration what 
 the secondary schools contributary to it 
 can do satisfactorily and then shape its 
 work accordingly. The size of the city, 
 the general interest in educational affairs, 
 the trend which local interests give to the 
 public-school curriculum, all tend to make 
 it possible to accomplish in one community, 
 or in one section of the country, what 
 would be quite impossible in another. We 
 must accept our students with such prepa- 
 ration as our normal constituency can give, 
 stimulated, to be sure, and to a certain 
 extent guided by the higher institution of 
 learning, and build our technical courses 
 upon that preparation as best we may. 
 
 More general dissatisfaction is expressed 
 with reference to the preparation of our 
 students in algebra than in any other sub- 
 ject. This comes from both eastern and 
 western institutions as well as from those 
 of the Mississippi Valley. At the Uni- 
 versity of Illinois last year forty per cent, 
 of the freshman class failed to pass a quiz 
 covering the main points of elementary 
 algebra and that after a two weeks' re- 
 view of the subject, and twenty-three per 
 cent, of the class failed on a second exami- 
 nation some weeks later. Of the one hun- 
 dred and ninety students who failed on 
 the first test, seventy- four per cent, entered 
 the university without conditions from 
 schools where the work had been examined 
 and approved by the high-school visitor. 
 The poor results which we get in algebra 
 
 are not due, in my estimation, exclusively 
 to poor instruction in the subject or to the 
 lack of attention in the high school. It 
 is the one subject in mathematics which is 
 begun in the high school and completed 
 in the college course. Often the high- 
 school algebra is completed in the sopho- 
 more year and then not taken up again 
 until the student enters upon his technical 
 course. All know how difficult it is to 
 retain the details of any course of study 
 during an interval of several years 
 in which the subject has been but little 
 used. That this lapse of time between the 
 completion of the high-school work and 
 the beginning of the college work is an 
 important element in the case is shown by 
 the fact that of the one hundred and 
 ninety failures mentioned over fifty per 
 cent, had not had algebra for at least four 
 years, and only ten per cent, had studied 
 the subject the year before. 
 
 A substantial gain would be made if we 
 should urge upon the high schools the 
 desirability of putting the last half year 
 devoted to algebra in the senior year of 
 the high-school curriculum and include 
 in that work the more difficult parts of 
 the subject as well as a general review 
 of the parts presented earlier. This 
 arrangement has become quite common in 
 Illinois, and the best argument that can be 
 presented in favor of such an arrangement 
 is that of the one hundred and ninety cases 
 of failure cited over sixty-three per cent, 
 had completed the work in the sophomore 
 year and less than eight per cent, had had 
 any work in algebra in the senior year. 
 Similar records have been kept at Illinois 
 for the past seven or eight years and the 
 data given are typical of the other years. 
 
 Unfortunately, we can have no assur- 
 ance that when a student has once mastered 
 a subject, he will forever afterwards retain 
 it. Neither can we hope that algebra will 
 ever be anything other than the weakest 
 
G 
 
 SCIENCE 
 
 place in the preparation of our students 
 so long as the present division of the sub- 
 ject so largely prevails. It is a situation 
 which we must accept, and the only thing 
 we can do is to make such recommenda- 
 tions as will tend to reduce the number of 
 fatalities as the boy passes from his sec- 
 ondary school to his technical course. The 
 technical school must expect to commence 
 its course in college algebra by a brief 
 review of the important points covered in 
 the high school, by taking a back-stitch, so 
 to speak, into the work already done. 
 Most of the western schools admit by cer- 
 tificate to the freshman class, and when a 
 pupil is once graduated from an accredited 
 school, he has earned the right to com- 
 mence upon his technical course. At the 
 University of Illinois, the problem has 
 been solved by saying to the freshmen in 
 mathematics that while there is no disposi- 
 tion to deprive them of their entrance 
 credit, the department of mathematics may 
 nevertheless determine the conditions 
 under which credit in college algebra can 
 be secured. Accordingly, those students 
 who fail to pass the review quiz are re- 
 quired to take two additional hours per 
 week in the subject for the remainder 
 of the semester in order to earn the same 
 credit that is given to others at the close 
 of the course. This has the advantage of 
 placing all of the students practically upon 
 the same basis, so far as attainments in 
 algebra are concerned, when they enter 
 upon the second semester's work. 
 
 A somewhat similar plan as that out- 
 lined here is followed also at the Univer- 
 sity of Wisconsin, and perhaps at other 
 institutions. It will be seen from Table 
 I. that a large number of technical schools 
 are now requiring work in logarithms for 
 entrance. This might very well be intro- 
 duced in connection with theory of ex- 
 ponents and used with advantage in high- 
 school physics. It is also gratifying to 
 
 observe that the more recent texts on 
 algebra provide work in the use of the 
 graph and in the plotting of curves. It is 
 very desirable that the work in elementary 
 algebra, including the work of curve-plot- 
 ting, should also include applications to 
 some of the simpler phenomena studied in 
 the high-school course in physics, and this 
 again is made a feature in some of the 
 more recent texts. Such an arrangement 
 affords an additional reason for putting 
 some of the work in algebra late in the 
 high-school course in order that it may 
 follow rather than precede the work in 
 physics, thus making it possible to intro- 
 duce a wider range of physical applica- 
 tions than could otherwise be done. 
 
 In Table II. is shown the number of 
 restrictions given in each of the various 
 mathematical subjects required of engi- 
 neering students. The average number 
 given to each subject for the seventeen 
 institutions is approximately as follows: 
 college algebra 50, plane trigonometry 46, 
 analytic geometry 80, and calculus 130. 
 In a number of the institutions named, 
 spherical trigonometry is taught by one of 
 the engineering departments, usually the 
 civil-engineering department, in connec- 
 tion with its applications to geodesy. The 
 number of recitations assigned to cal- 
 culus usually includes also a short course 
 in differential equations. In two cases 
 where a course of more than usual length 
 in the subject is given for the students of 
 a particular engineering department, the 
 subject has been listed separately. 2 One 
 institution, Rose Polytechnic Institute, is 
 unique among strictly engineering schools 
 in offering throughout the four years of 
 undergraduate work a rather large amount 
 of elective mathematics, including short 
 courses in advanced calculus, least squares, 
 
 * Table III. shows the number of recitations 
 given to differential equations in each case when 
 that subject was reported separately. 
 
SCIENCE 
 
 
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 90 
 
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 72 
 
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 60 
 
 
 35 
 
 
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 45 
 
 Spherical Trig- 
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 80 
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 3 
 
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 55 
 
 100 
 
 60 
 
 90 
 
 60 
 
 108 
 
 110 
 
 90 
 
 108 
 
 72 
 
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 54 
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 69 
 
 90 
 
 64 
 
 
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 125 
 
 120 
 
 144 
 
 108 
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 180 
 
 126 
 
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 70 
 
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 180 
 
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 96 
 
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 /45 
 \E.E. 
 
 
 
 
 
 
 
 
 
 
 
 
 Equations.... 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 C E. = civil engineers, M. E. = mechanical engineers, E. E. = electrical engineers, [ ] = elective. 
 
 * Massachusetts Institute of Technology now offers a course which combines the instruction in algebra, 
 analytic geometry and calculus rather than teaching these subjects as separate branches. In tables 
 II. and III., the distribution of time formerly given to these subjects is indicated as showing better 
 the relative emphasis placed upon each. 
 
 projective geometry and quaternions. In 
 all of the universities listed, and at the 
 Massachusetts Institute of Technology the 
 mathematical department offers a rather 
 wide range of advanced subjects, all of 
 which are open to engineering students so 
 far as the demands of their technical 
 course will permit. 
 
 By a study of Table II., it will be seen 
 that a considerable difference exists in the 
 amount of attention given to the various 
 subjects. In making comparisons in 
 algebra and trigonometry, however, the 
 difference in entrance conditions must be 
 taken into consideration. The amount of 
 work given in algebra ranges from fifteen 
 recitations at Stevens Institute, with an 
 entrance requirement of elementary alge- 
 bra through progressions, to ninety recita- 
 tions at Purdue with a requirement of ele- 
 
 mentary algebra through quadratics for 
 entrance. Likewise the work in plane tri- 
 gonometry ranges from thirty recitations 
 at the Massachusetts Institute to seventy at 
 Purdue. Analytic geometry and calculus 
 are naturally the most important subjects 
 for engineers in the mathematical curric- 
 ulum. One would naturally expect to 
 find a greater uniformity here. This, 
 however, is not the case. In analytic 
 geometry, it will be noticed that Armour 
 Institute requires but fifty-five recitations, 
 while the University of Minnesota gives 
 one hundred and ten recitations to the sub- 
 ject. Again in calculus the work varies 
 from seventy recitations at Rensselaer to a 
 maximum of one hundred and eighty at 
 Missouri, Wisconsin and Rose. A word 
 should be said, perhaps, concerning the 
 number of recitations recorded in the case 
 
8 
 
 SCIENCE 
 
 of Rensselaer. The department of mathe- 
 matics of that institution reports that the 
 recitations are from an hour and a quarter 
 to an hour and a half in length and that 
 the efficiency of the work is still further 
 increased by the fact that but two aca- 
 demic studies are carried simultaneously. 
 
 It will be of interest also to compare the 
 total amount of time spent upon mathe- 
 matics at these various institutions. As 
 will be seen from Table III., this ranges 
 from one hundred and eighty recitations 
 at Cornell to three hundred and ninety-six 
 at Purdue. In making this comparison, 
 we should again take into consideration 
 the difference in entrance requirements. 
 When this is done, the difference is more 
 apparent than real. For example, if we 
 add to the number of recitations given at 
 Cornell the number of recitations given at 
 Purdue to college algebra and trigonom- 
 etry, which are required for entrance at 
 Cornell as compared with three hundred 
 and ninety-six at Purdue. 
 
 It would seem that the technical schools 
 generally might well afford to make more 
 ample provision for elective mathematics. 
 Such courses as spherical trigonometry, 
 least squares, differential equations, might 
 well be placed in such a list. In this way 
 certain subjects which are desirable for 
 some branches of engineering, but not so 
 essential for others, could be taken by 
 those students interested. Sheffield offers 
 as an elective another course which might 
 be given with advantage at other technical 
 schools, namely, a course in scientific com- 
 putation in which the use of modern cal- 
 culating machines of various kinds is 
 explained and made use of. It would also 
 be well if the stronger institutions could 
 go still farther and introduce elective 
 courses in spherical harmonics, vector an- 
 alysis, theory of functions and the mathe- 
 matical theory of heat, electricity, etc., to 
 the end that the student with exceptional 
 
 mathematical ability might lay a broader 
 foundation for the theoretical side of engi- 
 neering. In this connection, it may well 
 be questioned whether the technical schools 
 of this country are in general offering suf- 
 ficient opportunity for that training which 
 has made it possible for such men as Stein- 
 metz, Osborne Reynolds and Stodola to 
 accomplish the work which has made them 
 famous. 
 
 Table III. shows also the sequence and 
 the distribution by years of the required 
 work in mathematics. "We are quite as 
 much interested, however, in the character 
 as in the amount and distribution of the 
 mathematical instruction given to engineer- 
 ing students. The close observer will have 
 noticed the change which has been made 
 and is now being made in this respect. In 
 recent years there has swept over the coun- 
 try a wave of enthusiastic discussion con- 
 cerning a closer and better correlation of 
 mathematics with the physical sciences. 
 This has been due for the most part to the 
 influence felt in this country of the Perry 
 movement in England. Much is to be 
 learned from this movement, and still more 
 is to be avoided. The discussions which 
 have arisen from it have on the whole had 
 a beneficial effect upon the teaching of 
 mathematics both in America and in Eng- 
 land. 
 
 It has first of all led to the introduction 
 into our text-books, and still more gener- 
 ally into our teaching, of a very much 
 better selection of problems problems 
 which widen the student's fund of infor- 
 mation of physical phenomena and apply 
 the mathematical principles which he is 
 acquiring more extensively than was for- 
 merly the case to the physical laws with 
 which he is familiar. Such problems as 
 the following, taken from a recent number 
 of an educational journal purporting to 
 serve the interests of mathematical teach- 
 ers in the secondary schools, is no longer 
 

 
 SCIENCE 
 
 
 9 
 
 
 TABLE 
 
 m 
 
 Institution 
 
 Freshman 
 
 Sophomore 
 
 Junior 
 
 Senior 
 
 Total 
 
 
 Al 65 ; An 55 ; C 50 
 
 C 85 ; Diff. eqs. 20 
 
 
 
 275 
 
 
 Al 45 ; Tr 70 ; An 55 
 
 An 45 ; C 125 
 
 Diff. eqs. 34 (E.E.) 
 
 Least squares 
 48 (C.E.) 
 
 M.E.; 340 
 
 E.E 374 
 
 
 C.E.; 388 
 
 
 An 60; C 120 
 
 Al 55 ; Tr 35 ; An 90 
 
 C144 
 
 
 
 180 
 324 
 
 
 An 80 ; C 108 
 
 Al 40 ; Tr 22 ; M 25 
 
 (C.E.) An 58; C 96 
 
 
 
 E.E.; M.E.; 188 
 C.E.; 241 
 
 Mass. Inst.. 
 
 Al 30 ; Tr 30 ; An 60 
 
 C 90; 
 SphTrlO (C.E.) 
 
 Diff. eqs. 45 (E.E.) 
 
 
 M.E.; 210 
 E.E.; 255 
 C.E.; 220 
 
 Michigan... 
 Minnesota.. 
 Missouri.... 
 
 Al 36 ; An 108 
 
 Al 70 ; Tr 70 ; An 40 
 
 Al 36 ; Tr 55 ; An 90 
 
 C126 
 
 An 70 ; C 110 
 
 C180 
 
 Diff. eqs. 18 
 
 
 288 
 360 
 360 
 
 Nebraska... 
 
 A136;Tr54; An 72; 
 C18 
 
 An 36 ; C 108 
 
 
 
 324 
 
 Purdue 
 
 Kensselaer.. 
 
 Al 90 ; Tr 90 
 
 Al 45 ; Tr 40 ; An 39 
 
 An 72 ; C 72 
 An 21 ; C 70 
 
 C72 
 
 
 396 
 225 
 
 
 A172; Tr54; C 36 
 [S Tr 18] ; [Proj. 
 Geom. 18] [Al 18] 
 
 An 54 ; C 90 
 [Quat 18] [An 18] 
 
 An Dyn 54 (calculus) 
 
 [C72] 
 
 [Least squares 18] 
 
 
 360 
 
 
 [180] 
 
 Sheffield .... 
 
 An 90 
 
 Logl5;SphTrl0; 
 An 43 ; C 30 
 
 C100 
 
 An 26 ; C 86 
 
 Diff. eqs. 28 
 
 [Probs. and 
 computing] 
 
 190 
 [Probs.] 
 
 238 
 
 "Wisconsin .. 
 
 Al and Tr 90 ; An 90 
 
 C160; 
 
 Diff. eqs. 20 (M.E.; 
 E.E.) 
 
 
 
 M.E.; E.E.; 360 
 C.E.; 340 
 
 Worcester... 
 
 Al 64 ; Tr 48 
 
 An 64 ; C 96 
 
 Vect. and 32 (E.E.) 
 
 
 272 
 E.E.; 304 
 
 Al = algebra, An = analytic geometry, C = calculus, Tr = trigonometry, M = mensuration, 
 Quat = quaternions, An Dyn = analytical dynamics, [ ] = elective. 
 
 thought to be in good form by our best 
 instructors : 
 
 "I bought 674,867 sheep at less than $10 
 per head; I paid for them in ten-dollar 
 bills and received back in change $7.39. 
 How many bills did I give ? ' ' 
 
 Need I call attention to the absurdity of 
 putting such problems into the hands of 
 pupils? How many farmers in any wool- 
 producing state of the country ever even 
 saw that many sheep in his entire life, and, 
 should he have occasion to buy them, would 
 for a moment think of paying for them by 
 counting out 663,395 ten-dollar bills. So 
 long as such problems are given out for 
 
 the consideration of pupils, just so long we 
 may expect even the best of them to ask the 
 question so often heard in our algebra 
 classes: "What is all of this 'stuff' good 
 for, anyway?" 
 
 Contrast with this problem the following, 
 taken at random from an algebra recently 
 published : 
 
 "Two boys, A and B, having a 30-lb. 
 weight and a teeter board, proceed to de- 
 termine their respective weights as follows : 
 They find that they balance when B is 6 
 feet and A 5 feet from the fulcrum. If 
 B places the 30-lb. weight on the board 
 beside him, they balance when B is 4 and 
 
10 
 
 SCIENCE 
 
 A is 5 feet from the fulcrum. How heavy 
 is each boy?" 
 
 In solving this problem the boy has 
 learned just as much mathematics as in 
 solving the first. In addition, his mathe- 
 matics has been brought into contact with 
 a fundamental physical law, and incident- 
 ally he is made to feel that, after all, his 
 mathematics is of consequence to him in 
 solving the sort of questions in which he 
 is interested or is likely to have experience 
 with in the future. 
 
 As has been pointed out, a change in the 
 character of the problems is gradually 
 taking place in our mathematical texts. 
 Perhaps a word of caution should be given 
 lest we go too far in the opposite direc- 
 tion, by introducing problems which re- 
 quire a technical knowledge and experience 
 beyond the comprehension of our students. 
 Perry's calculus is a conspicuous illustra- 
 tion of this danger. The subjects discussed 
 in that book would form a good sequel to a 
 certain work in engineering, but the book 
 seems to be hardly suited to meet the needs 
 of American schools as a preparation for 
 engineering study. We should aim to 
 make the mathematical work practical and 
 in harmony with engineering practise, but 
 without making it at the same time tech- 
 nical in its applications, or without going 
 too far afield by teaching mathematical 
 physics. 
 
 Another improvement which has recently 
 become noticeable in the teaching of mathe- 
 matics in this country is the breaking down 
 of the traditional barriers between the dif- 
 ferent branches and a corresponding closer 
 correlation of the different subjects in the 
 mathematical curriculum. In several of 
 our institutions the sharp division of 
 freshman work into algebra, trigonometry, 
 and analytic geometry is being more or 
 less disregarded and these subjects taught 
 as a single unit. It is thought that the 
 
 student is thus enabled to grasp more read- 
 ily these subjects as a whole, and that the 
 instructor can introduce much earlier the 
 principles of analytic geometry and of cal- 
 culus and postpone to the later part of the 
 course those topics which are relatively 
 difficult and not so essential to the elemen- 
 tary work of the course. 
 
 This plan is now being followed some- 
 what closely at the University of Wiscon- 
 sin. In the first semester fifteen or twenty 
 recitations are devoted to the elementary 
 portions of trigonometry. This is followed 
 by work in algebra, including the theory 
 of complex numbers, using trigonometry 
 and a large amount of graphic work, and 
 the elementary principles of analytic geom- 
 etry. In this work trigonometric compu- 
 tation and the use of the slide rule form 
 an important part. In the second semester 
 the algebra and trigonometry are continued 
 and combined with the essentials of an- 
 alytic geometry. 
 
 This correlation of the work of the fresh- 
 man year seems to have been most thor- 
 oughly worked out at the Massachusetts 
 Institute of Technology, where Professors 
 Woods and Bailey have recently prepared 
 a text covering the work given there in the 
 freshman year, excluding, however, trig- 
 onometry. The indications are that other 
 institutions are also contemplating a revi- 
 sion and better correlation of the work of 
 the first year. 
 
 In some of the recent books, the sharp 
 division of the calculus into differential 
 and integral calculus is done away with, 
 thus making it possible to introduce the 
 student to a wide range of easy applica- 
 tions at an early point in the course and 
 to relegate to its proper place some of the 
 more difficult parts of the differential cal- 
 culus. There is a tendency also to intro- 
 duce the methods of the calculus earlier and 
 make them the basis of portions of the an- 
 
SCIENCE 
 
 11 
 
 alytie geometry. For example, Rose Poly- 
 technic Institute gives a short course of 
 thirty-six recitations in the subject before 
 analytic geometry is taken, and what is 
 accomplished there in this formal way is 
 undertaken at other institutions by intro- 
 ducing into the analytics the elementary 
 notion of derivatives or by teaching the 
 two subjects simultaneously. 
 
 "While all are agreed that for engineers 
 mechanics should stand in a close and vital 
 relation to the calculus, that in fact it is 
 the principal reason for teaching calculus, 
 not all are agreed, however, as to the best 
 method of accomplishing this purpose. 
 Some would maintain that it should be 
 taught by the mathematical department 
 and in connection with calculus ; others and 
 perhaps the larger number feel that it 
 should be given by the engineering depart- 
 ments and made to follow and supplement 
 the calculus, giving the student his first 
 real introduction into the applications of 
 his mathematics to the fundamental prin- 
 ciples underlying all engineering courses. 
 However this may be, there is little 
 doubt that more applications to mechanics 
 should be introduced into the course in 
 calculus than is now usually the case, even 
 to the exclusion, if need be, of some of the 
 applications to geometry frequently given. 
 Problems in work, energy and stress form 
 just as legitimately an integral part of a 
 course in calculus as problems in order of 
 contact, asymptotes or envelopes. The 
 applications to geometry and to mechanics 
 should be given about the same relative 
 importance in a well-balanced course in 
 calculus. 
 
 Descriptive geometry is another subject 
 in the engineering course which might well 
 be revised and made more mathematical in 
 its treatment. It is to be regretted that 
 the subject has in this country degenerated 
 into little more than mechanical drawing. 
 
 It would be greatly improved for engi- 
 neers, as well as the general student, if we 
 should inject into it something of the scien- 
 tific spirit given it in European schools. 
 
 No presentation of the subject under dis- 
 cussion would be complete without some 
 consideration of the preparation which the 
 teacher of mathematics has, or should have, 
 who is to teach the subject to engineering 
 students. There is a strong feeling in some 
 quarters that such an instructor should be 
 a trained engineer in order that he may the 
 better appreciate the kind of applications 
 which are best suited to the training of an 
 engineer and to make sure that the proper 
 emphasis be placed upon those topics con- 
 sidered essential in such training. Some 
 would go still farther and insist that even 
 in the elementary courses in mathematics 
 usually given in the first two years, the 
 purpose and aim of the prospective engi- 
 neer is so radically different from that of 
 the general student that the content of the 
 course itself should be very different from 
 what is best suited to the student who 
 elects mathematics as a part in a general 
 education. 
 
 It goes without saying that we should 
 eliminate from the courses for engineering 
 students that which is non-essential, and 
 we should make them as practical as we 
 may by the generous use of those physical 
 applications which will give the students 
 both skill and facility in applying mathe- 
 matics to such concrete cases as may arise 
 later in his experience. On the other hand, 
 it would be disastrous to go to the extent 
 of teaching any of the principles of mathe- 
 matics empirically or of permitting stu- 
 dents to assume as already established for- 
 mulas which he has merely to learn how to 
 apply. We should avoid the danger of 
 going too far in allowing the student to 
 disregard the necessity of a formal demon- 
 stration and to regard lightly the logic and 
 
12 
 
 SCIENCE, 
 
 the philosophy of mathematics. What is 
 needed first of all is the ability on the part 
 of the student to think mathematically and 
 to have not only a ready but an intelligent 
 command of the fundamental principles of 
 the subject. We should introduce the ap- 
 plications of mathematics not for the sole 
 purpose of giving the student a foretaste of 
 the things which are in store for him, but 
 because such applications give him addi- 
 tional opportunity for gaining a clearer 
 comprehension of mathematical processes 
 and principles which might otherwise be 
 hazy; and I wish to add that this is more 
 essential for the sound training of the spe- 
 cial student of mathematics with his limited 
 opportunity for the application of his sub- 
 ject to physical phenomena than it is for 
 the engineering student who in the future 
 is to have opened up to him that wide 
 range of applications which his technical 
 studies provide. In other words, what is 
 essential in the way of applications for the 
 engineering student in the first two years 
 of his mathematical work gives the very 
 best training for the student who is taking 
 mathematics as an element in a liberal edu- 
 cation. The proper place for differentia- 
 tion, so far as the content of the course is 
 concerned, would seem to be after the com- 
 pletion of the course in calculus rather 
 than before. I present this as a plea for 
 the general student, that he should have 
 more of the applications of his mathematics 
 rather than that the engineering student 
 should have less. Both should have thor- 
 ough drill in the fundamental principles 
 of the subject and in addition all of the 
 applications of those principles which their 
 limited experience and knowledge of phys- 
 ical phenomena will permit. No student, 
 enginering or otherwise, should be led to 
 regard his mathematical work in the same 
 light in which a carpenter may properly 
 regard his jack-plane, a mere tool with 
 
 which to accomplish certain results ; neither 
 should the instructor teach mathematics in 
 the spirit in which a skilled operator might 
 regard a finely-equipped machine shop 
 whose sole purpose is to make more ma- 
 chines. Both extremes are to be avoided 
 in the early courses in mathematics. The 
 opportunity for specialization and differ- 
 entiation should come later; and any stu- 
 dent who is not capable of grasping the 
 fundamental principles of the mathematics 
 usually required in an engineering course 
 should not aspire to a bachelor's degree 
 from a large university or technical school. 
 What training is essential or desirable, 
 then, on the part of the mathematical in- 
 structor of engineering students to best 
 accomplish the general results here set 
 forth? There is no doubt that the ideal 
 thing would be to take men who have 
 completed an engineering course and later 
 supplemented it by special work in mathe- 
 matics. This, however, does not seem feas- 
 ible because of the few who could be in- 
 duced to take such a course of training. 
 It would be quite impossible to induce a 
 sufficient number of engineers to take up 
 the teaching of mathematics to meet the 
 demand, even if that seemed desirable. In 
 most cases the boy enters the engineering 
 course with the view of practising his pro- 
 fession when he has completed the course. 
 As a rule he has little taste or inclination 
 for teaching, and those few who can be in- 
 duced to enter the less remunerative pro- 
 fession of teaching are absorbed, as indeed 
 they should be, by the engineering depart- 
 ments of our technical schools. To put an 
 engineering graduate at teaching mathe- 
 matics without first having had special 
 training in mathematics would be wholly 
 undesirable. Such an instructor knows 
 but little about pure mathematics beyond 
 the elementary courses which he is present- 
 ing, and, what is even worse, often has but 
 little interest. If he can be induced to 
 
SCIENCE 
 
 13 
 
 take up in a serious way the study of 
 mathematics, he is in a fair way of becom- 
 ing a good teacher of the subject. I am 
 thoroughly convinced that mathematics 
 should be taught by mathematicians just 
 as engineering should be taught by trained 
 engineers; but the mathematical instructor 
 who wishes to teach engineers should be 
 familiar with the general field of applied 
 mathematics mechanics, strength of ma- 
 terials, thermodynamics, and in addition 
 so much of the broader field of mathe- 
 matical physics as possible. 
 
 While the mathematical instructor should 
 have some knowledge of its applications, it 
 is equally desirable that the teacher of 
 engineering should from time to time both 
 refresh and revise his knowledge of the 
 fundamental things in mathematics, to the 
 end that he may keep his methods up to 
 date and adapt his teaching to the kind of 
 mathematical instruction which his stu- 
 dents have had and avoid those methods 
 and those forms of expression which have 
 long been out of use. 
 
 In closing, I wish to add that the rapid 
 increase in engineering students has so 
 greatly increased the demand for mathe- 
 matical instructors having some knowledge 
 of engineering that it would be highly de- 
 sirable if more attention should be paid to 
 the preparation of men for such positions. 
 This can best be accomplished, perhaps, in 
 those universities having large engineering 
 departments by a closer correlating of the 
 work of the mathematical department with 
 theoretical work in engineering and mathe- 
 matical physics. It is to be regretted that 
 so little attention in this country is now 
 being given to these two fields of mathe- 
 matical activity. Institutions so situated 
 as to undertake it should offer to its stu- 
 dents graduate work in these lines in every 
 respect worthy of a doctor's degree, and 
 likewise to its instructors both opportunity 
 
 and encouragement to do research work in 
 this broad and fruitful field of human en- 
 deavor. Edgar J. Townsend 
 University of Illinois 
 
 THE TEACHING OF MATHEMATICS TO 
 
 ENGINEERING STUDENTS IN 
 
 FOREIGN COUNTRIES 1 
 
 Your committee has asked me to speak 
 of the teaching of mathematics in foreign 
 engineering colleges. My remarks will 
 have reference almost exclusively to the 
 German colleges and schools, partly be- 
 cause I am most familiar with the condi- 
 tions existing in Germany and partly on 
 account of the rather instructive campaign 
 for reforming the whole teaching of mathe- 
 matics, recently inaugurated in Germany. 
 
 As regards other countries I will only 
 say that the situation in England and Scot- 
 land where, during the last quarter of a 
 century, technical education has rapidly 
 developed on quite characteristic and indi- 
 vidual lines, deserves careful attention. 
 But I am not sufficiently well acquainted 
 with the facts to discuss this educational 
 movement. In France, it is well known 
 that the theoretical training given to engi- 
 neers is on a very high level, higher even 
 than in Germany, I believe. Thus, the re- 
 quirements for admission to the Ecole 
 Potytechnique, or even to the Ecole Cen- 
 trale, include in mathematics almost as 
 much as our engineering students get in 
 their college course. On the top of this 
 preparation, the student receives in the 
 Ecole Polytechnique an excellent two 
 years' course in higher analysis and theo- 
 retical mechanics, and then only is he 
 allowed to enter upon his special technical 
 work. It must also be taken into account 
 that admission to the Ecole Polytechnique 
 is by competitive examinations held 
 1 Read before Sections A and D, American Asso- 
 ciation for the Advancement of Science, and the 
 Chicago Section of the American Mathematical 
 Society, Chicago meeting, December 30, 1907. 
 
14 
 
 SCIENCE 
 
 throughout France, so that this institution, 
 receiving as it does the pick of students 
 from the whole country, can maintain a 
 high level of theoretical excellency. The 
 Ecole des Ponts et Chaussees and the Ecole 
 des Mines to which the student passes 
 from the Ecole Poly technique, are thus 
 what we might call graduate schools of 
 the highest rank. 
 
 Turning now to the German engineering 
 colleges, a comparison with our own best 
 engineering colleges shows apparently but 
 little difference, both as regards require- 
 ments for admission and as to the schedule 
 of courses offered in the schools themselves. 
 Nevertheless, I believe that the scientific 
 standard is decidedly higher in the German 
 than in the American engineering college. 
 I am not here concerned with the question 
 whether such a high standard of theoretical 
 knowledge is essential, or even desirable, 
 for the engineer; I merely state the fact. 
 Moreover, it is quite possible that ulti- 
 mately the average German engineer knows 
 no more mathematics than the average 
 American engineer. All I wish to main- 
 tain is that, in my opinion, an able German 
 student, in his Technische Hochschule, or 
 engineering university, can gain a more 
 thorough scientific equipment than an 
 equally able American student in his alma 
 mater. 
 
 The mathematical requirements for ad- 
 mission are about the same in Germany as 
 with us: algebra, geometry, trigonometry. 
 Not a few students now enter the German 
 engineering college with some knowledge 
 of analytic geometry and even of calculus, 
 but many still come without this knowl- 
 edge. The important point is that the pre- 
 paratory training in mathematics (inclu- 
 ding arithmetic) is distributed systematic- 
 ally and continuously over a period of nine 
 years. The same is true of other prepara- 
 tory studies. It is obviously quite impos- 
 
 sible to attain in a four-year high-school 
 course the results attained in the nine-year 
 course of a German Gymnasium, Realgym- 
 nasium, or Oberrealschule. This difference 
 in preparation must always be kept in 
 mind in making comparisons between Ger- 
 man and American universities. 
 
 The mathematical courses offered in the 
 German engineering colleges and required 
 for a degree cover plane and solid analytic 
 geometry, differential and integral calculus 
 and differential equations i. e., about the 
 same subjects that are required in this 
 country. The subject of theoretical me- 
 chanics, which is treated rather differently 
 in different schools, and even in the same 
 school for different degrees, I shall here 
 leave out of consideration, for the sake of 
 simplicity. The amount of time devoted 
 to the higher mathematics, not including 
 mechanics, appears roughly from the fol- 
 lowing table, in which the first figure in 
 each case gives the number of hours per 
 week devoted to lectures, the second the 
 
 Karlsruhe 
 
 Stuttgart 
 
 Munich 
 
 Hannover... 
 
 Danzig 
 
 Braunschweig ... 
 Zurich 
 
 k 
 
 u 
 
 
 
 
 T3S 
 
 B 
 
 O X) 
 
 
 H 
 
 pl, a 
 
 a> B 
 
 CI Q) 
 
 J a 
 
 V 
 
 02 
 
 ~ o 
 
 02 
 
 02 
 
 02 
 
 02 
 
 6+2 
 
 6+2 
 
 3 
 
 2 
 
 7+3 
 
 6+4 
 
 3+1 
 
 
 6+3 
 
 6+2 
 
 5+2 
 
 2+2 
 
 8+1 
 
 6+2 
 
 
 
 6 
 
 5 
 
 4+1 
 
 3+1 
 
 8+2 
 
 6+2 
 
 2 
 
 
 8+4 
 
 8+4 
 
 4+1 
 
 
 17+4 
 16+8 
 19+9 
 14+3 
 18+2 
 16+4 
 20+9 
 
 number of hours devoted to "exercises." 
 These exercises are a comparatively recent 
 innovation. In my time the student had 
 nothing but lectures; to gain a working 
 knowledge of the subject he had to take a 
 text-book and work for himself. Even 
 now, these exercises are optional; they 
 probably exist everywhere, although the 
 table may not show them. There are no 
 periodic examinations such as we have at 
 the end of each semester; but most stu- 
 dents take at the end of their course the 
 
SCIENCE 
 
 15 
 
 Staatsexamen, or if particularly ambitious, 
 the Diplomexamen. The lectures in mathe- 
 matics are rather more advanced and more 
 complete than those in our engineering col- 
 leges. But the requirements in the final 
 examinations are not very high. 
 
 In addition to the more thorough prepa- 
 ration of the German student and to the 
 somewhat higher standard of the lectures 
 on pure mathematics, and largely owing to 
 these circumstances, the treatment of ap- 
 plied mathematics is, I believe, on a higher 
 level in Germany than in this country. 
 The student is better prepared; no time is 
 lost in ' ' recitations, ' ' i. e., in trying to find 
 out whether the student has committed 
 things to memory; the professor is thus 
 enabled to treat scientific questions scien- 
 tifically. Besides, on an average, the Ger- 
 man professor of an engineering subject 
 has himself a higher degree of scientific 
 training and is more interested in the 
 mathematical, and in general the scientific, 
 aspects of his subject than his American 
 colleague. 
 
 It is of course always hazardous and, 
 moreover, of little use to make such gen- 
 eral statements and comparisons; and I 
 do not wish to attach any great importance 
 to them. Neither the German nor the 
 American engineering college is as good as 
 it might be or should be; no institution 
 ever is; an institution is good only in so 
 far as it is continually changing, develop- 
 ing, rising. The above comparisons are, 
 therefore, given merely as a basis for better 
 understanding the efforts that are now 
 made in Germany for the improvement of 
 mathematical teaching in all its phases. To 
 these efforts I wish to call your special 
 attention. 
 
 The German movement for the reform 
 of the teaching of mathematics is of a 
 somewhat complex nature; at least three 
 different movements may be distinguished. 
 
 One of these, originating with the German 
 association of engineers (Verein Deutscher 
 Ingenieure) had as its direct object the im- 
 provement of the mathematical instruction 
 in the engineering colleges, with a view to 
 making the instruction less abstract and 
 theoretical and more practically useful to 
 the engineer. To a certain extent, this 
 object has been attained. Practical exer- 
 cises for acquiring a working knowledge 
 of mathematics have been introduced every- 
 where, and the lectures on pure mathe- 
 matics have become less theoretical. Some 
 of the originators of this movement, espe- 
 cially Professor Riedler, of the Charlotten- 
 burg College, went so far as to demand that 
 in engineering colleges mathematics should 
 be taught by engineers. Whether or not 
 this was meant as more than a threat I do 
 not undertake to say; certainly, as far as 
 my knowledge goes, no attempt has ever 
 been made in a German engineering college 
 to put the teaching of mathematics in the 
 hands of any one but a trained mathe- 
 matician. But I believe that in the selec- 
 tion of men for such positions more atten- 
 tion has been paid in recent years to the 
 qualifications of the aspirants; mathe- 
 maticians with a bent towards applied sci- 
 ence being given the preference for posi- 
 tions in engineering colleges. 
 
 The second of the three movements re- 
 ferred to above has for its object the re- 
 form of the teaching of mathematics in the 
 universities. It is the oldest of these move- 
 ments, and has borne fruit in a variety of 
 ways. But I can here only advert to it 
 very briefly. The tremendous creative 
 mathematical activity that characterized 
 the last three quarters of the nineteenth 
 century in Germany led to a condition in 
 the universities that was injurious to the 
 preparation of teachers for the secondary 
 schools (Gymnasium, Realgymnasium, 
 Oberrealschule) . Too much stress was laid 
 
16 
 
 SCIENCE 
 
 on leading the student as fast as possible 
 to original research in some special line. 
 The system has been described as a system, 
 not of double entry, but of double for- 
 getting; upon entering the university the 
 students, most of whom are fitting for 
 teaching in the secondary schools, are made 
 to forget and almost despise the more ele- 
 mentary mathematics, and when beginning 
 their professional teaching career they are 
 again compelled to forget as fast as possible 
 all the higher and highest mathematics to 
 which they had devoted most of their time 
 at the university. The remarkable devel- 
 opment of mathematical activity in our 
 country during the last fifteen or twenty 
 years may bring about a similar situation. 
 Fortunately, the leaders of American math- 
 ematics are well aware of the danger of 
 losing the healthy contact with the more 
 elementary mathematics and with applied 
 science. Of course, it is, and always will 
 be, the chief object of a real university to 
 foster original research and productive 
 scholarship. But it is well even for the 
 most advanced specialist not to burn the 
 bridges behind him, but to keep in mind 
 the connection of his specialty with the 
 foundations of knowledge, on the one hand, 
 and with kindred branches of science on 
 the other. As Sir Isaac Newton expressed 
 it in his quaint way in a letter to Dr. Lord : 
 "He that in ye mine of knowledge deepest 
 diggeth, hath, like every other miner, ye 
 least breathing time, and must sometimes 
 at least come to terr, alt for air." 
 
 The desire to make the university teach- 
 ing of mathematics more practically useful 
 and bring it into live contact, as far as 
 possible, with the whole tendency of mod- 
 ern scientific thought led, on the one hand, 
 to a strengthening of all branches of ap- 
 plied mathematics, not only by courses 
 offered in the universities, but also by such 
 publications as the Encyklopadie, which 
 
 includes applied mathematics in the widest 
 application of the term ; on the other, it led 
 to reforms in the courses offered to future 
 teachers of mathematics, and ultimately to 
 a thorough investigation of the teaching of 
 elementary mathematics in the secondary 
 schools of Germany. 
 
 The improvement of the teaching of ele- 
 mentary mathematics is the aim of the 
 third and most recent mathematical reform 
 movement in Germany. The reforms 
 proposed in this connection by the com- 
 mittee of the German Association of Natur- 
 forscher und Aerzte, at the Meran meeting, 
 in 1905, appear to me to deserve very care- 
 ful consideration. They would apply, in 
 this country, to the teaching of mathe- 
 matics not only in the high schools, but 
 just as much in the engineering colleges. 
 For, with the preparation that our students 
 actually have, I am convinced that the best 
 method of imparting a good working 
 knowledge of the elements of analytic 
 geometry and calculus is not through lec- 
 tures, but through actual teaching based 
 mainly on solving problems, that is, by the 
 methods not of the German university, but 
 of the German secondary school. 
 
 The proposals of the committee 2 do not 
 change very essentially the number of 
 hours required for mathematics. These 
 are to be : in the Gymnasium as well as in 
 the Realgymnasium, four hours per week 
 in each of the nine years ; in the Oberreal- 
 schule generally four hours per week, in 
 the third and fourth years six hours. The 
 first three years are devoted to common 
 arithmetic and intuitional geometry, the 
 next three years to algebra and geometry 
 carried along together, the last three years 
 to advanced algebra, trigonometry, ad- 
 vanced geometry, conic sections (treated 
 synthetically and analytically) and, in the 
 
 2 See Zeitsohrift fur mathematischen und natur- 
 wissenschaft lichen Unterricht, Vol. 36 (1905), 
 pp. 533-580. 
 
SCIENCE 
 
 17 
 
 Oberrealschule, the elements of the cal- 
 culus. Apart from matters of detail this 
 distribution does not vary very much from 
 the practise now followed in the best Prus- 
 sian schools. 
 
 While thus the general program can not 
 be said to constitute a radical departure 
 from existing conditions, the statement of 
 what should be the principal aim of mathe- 
 matical teaching and the indications given 
 for carrying out this aim throughout the 
 whole course 3 appear to me as the most 
 important features of the report. In addi- 
 tion to the well-recognized object of mathe- 
 matical teaching to train the mind in rigor- 
 ous logical reasoning the report insists par- 
 ticularly on the training of geometrical 
 intuition and on acquiring the habit of 
 functional thinking. The carefully pre- 
 pared explanations accompanying the de- 
 tailed program for the nine-year course 
 show how these aims should guide the in- 
 struction at every step. The insistence on 
 the idea of the functional relation can not 
 be recommended too strongly to our writers 
 of college text-books, from trigonometry 
 to differential equations. But, as this re- 
 port demands, it should even enter into the 
 very elements of algebra and geometry. 
 
 It should be observed that the committee 
 that prepared this report was not composed 
 of mathematicians only; all branches of 
 science taught in the secondary schools 
 were represented in it; and all these 
 branches received equally careful attention. 
 While the portion of the report devoted 
 to mathematics covers almost the whole 
 range of the subject, from arithmetic to 
 the elements of the calculus, required of 
 our engineering students, there is nowhere 
 any reference to students of engineering 
 or to any other special class of students. 
 I might, therefore, appear out of order in 
 speaking of this report at the present occa- 
 
 "Loc. tit., pp. 543-545, 550-553. 
 
 sion. But I wish to say most emphatically 
 that, in my opinion, there is no special 
 "mathematics for engineers"; nor is there 
 any method of teaching mathematics, spe- 
 cially adapted to engineering students. 
 If it is wrong to present mathematics in a 
 form so abstract as to make it unintel- 
 ligible to the student, it is just as wrong to 
 present the results of mathematics in a 
 form so concrete as to reduce the science to 
 a mere art of performing certain mechan- 
 ical operations, to make it, as the saying 
 goes, a mere tool, and not a habit of think- 
 ing. 
 
 In conclusion allow me to say that I 
 should be the last to advocate a remodeling 
 of our institutions of learning on the Ger- 
 man plan, or the French plan, or any other 
 existing plan. But I believe that the time 
 has come in this country when one or two 
 years of general college study can be de- 
 manded as preparation for the professional 
 engineering course, at least for those more 
 able students who wish to obtain a thor- 
 oughly scientific preparation for their pro- 
 fessional career. An opportunity should 
 then be offered to students of engineering 
 of scientific ability to extend their knowl- 
 edge on the theoretical side. 
 
 Alexander Ziwet 
 University of Michigan 
 
 THE TEACHING OF MATHEMATICS FOR 
 ENGINEERS * 
 
 Mathematics, from the standpoint of 
 the engineer, is a means, and not an end. 
 It is an instrument or tool by which he may 
 determine the value and relations of forces 
 and materials. 
 
 The usefulness of tools depends upon 
 the sort of work which is to be done, upon 
 
 1 Read before Sections A and D of the American 
 Association for the Advancement of Science and 
 the Chicago Section of the American Mathemat- 
 ical Society, at the Chicago meeting, December 
 30, 1907. 
 
18 
 
 SCIENCE 
 
 the kinds of tools which are available and 
 upon the skill of the man who uses them. 
 "We may inquire, therefore, what are the 
 uses to which the engineer may apply 
 mathematics? What kind of mathematics 
 does he need? And what skill should he 
 possess in their use? 
 
 First, then, what work is to be done by 
 the young men who are now taking engi- 
 neering courses? A few and only a few 
 will be original investigators or de- 
 signers who will need mathematics as an 
 instrument of research. A considerable 
 number will regularly employ elementary 
 mathematics in more or less routine cal- 
 culations. Many will have little use for 
 mathematics, as engineering courses are 
 recognized as affording excellent training 
 for various business, executive and other 
 non-technical positions, particularly in con- 
 nection with manufacturing and operating 
 companies. It has been stated by the vice- 
 president of a large electric manufacturing 
 company that not over ten per cent, of 
 the technical graduates employed by that 
 company are fitted by temperament or by 
 education to take up with success the work 
 of pure engineering. A recent classifica- 
 tion of the graduates of Sibley College, 
 Cornell University, shows that about half 
 are in occupations which require no ad- 
 vanced mathematics and it is probable 
 that many of the SG'per cent, classed as 
 mechanical and electrical engineers seldom 
 go beyond the rules of arithmetic. Hence 
 a goodly proportion of engineering gradu- 
 ates do not need to be mathematical ex- 
 perts. Their mathematical studies need 
 not aim to produce experts, but should 
 have as a principal object the mathematical 
 training which is a most efficient kind of 
 training in an engineering course. On the 
 other hand, the engineers who will have 
 practical use for the higher mathematics 
 will find their ability as engineers is in a 
 
 large measure determined by their ability 
 as mathematicians. 
 
 Second, the question, what kinds of 
 mathematics does the engineer need? is 
 closely related to the class of work he is 
 to do. In general a great deal of engineer- 
 ing work is done with much less use of 
 higher mathematics than most professors 
 probably imagine ; and, furthermore, it 
 may be remarked, with much less than 
 could profitably be employed. Engineers 
 are apt to use ordinarily the mathematical 
 methods with which they are most familiar 
 and which will bring the result with the 
 least effort. One man employs calculus, 
 another draws a diagram, another writes 
 out formula?, while another gets his results 
 by mental arithmetic. The object is to get 
 the result. 
 
 The fundamental idea that mathematics 
 is something for the engineer to use finds 
 many illustrative analogies in ordinary 
 tools. Adaptation is the first requisite. 
 Tools should be suited to the work to be 
 done. An expensive machine tool with its 
 refined adjustments is quite unnecessary 
 for executing a piece of work which can 
 be done with sufficient accuracy by a few 
 minutes' application of a file. An ordi- 
 nary calculating slide rule is infinitely 
 better than a table of seven-piece logar- 
 ithms in every-day work. 
 
 On the other hand, it is particularly 
 wasteful to attempt to execute a difficult 
 and intricate piece of work with inade- 
 quate tools. But more important than the 
 tool is the skill of the man who uses it. 
 A skillful workman can accomplish results 
 with a few simple tools Avhich others can 
 not get with the most elaborate special 
 equipment. 
 
 Third, therefore, skill in the use of 
 mathematics is the really essential thing. 
 A judicious use of arithmetic with a little 
 algebra or a simple diagram often leads 
 to more satisfactory results than others 
 
SCIENCE 
 
 19 
 
 secure through elaborate processes involv- 
 ing lengthy equations and complicated 
 operations. In the latter, errors are liable 
 to occur, the common-sense import of the 
 problem is apt to be overlooked, assump- 
 tions may be made to facilitate calculations 
 which are physically unwarranted as one 
 loses sight of the physical problem in the 
 intricacy of the mathematical solution. 
 Abstract mathematical studies, if pursued 
 as a kind of intellectual calisthenics, may 
 produce a pure mathematician, but they 
 may unfit a man for practical engineering. 
 A mathematician is not necessarily an 
 engineer; nor is an elocutionist necessarily 
 a good lecturer, nor is a tool expert a suc- 
 cessful manufacturer. 
 
 Mathematics is used in engineering to 
 express the quantitative relations of nat- 
 ural phenomena. The mathematician de- 
 lights in the relations: he divorces them 
 from the phenomena and gives them ab- 
 stract expression, while the engineer is con- 
 cerned with the natural phenomena; he 
 demands the physical conception; the me- 
 dium of expressing these relations is of 
 secondary consequence. 
 
 The mathematician evolves the equation 
 for a parabola and finds a convenient 
 illustration in the law of projectiles. The 
 engineer finds that a physical result fol- 
 lows from the application of certain forces, 
 and uses the formula merely as a con- 
 venient method of expressing the law. The 
 analogue in the case of mechanical tools is 
 found by regarding a set of drawing in- 
 struments or a transit or a lathe, as some- 
 thing intelligently designed, properly pro- 
 portioned, accurately made and finely 
 finished, the merit of which lies in its own 
 inherent excellence; or, on the other hand, 
 by considering them as tools adapted for 
 doing a certain range and character of 
 work with a sufficient degree of accuracy 
 and at low cost. 
 
 A manual-training school gives familiar- 
 
 ity with mechanical tools and mathe- 
 matical study gives familiarity with in- 
 tellectual tools. In work with the manual 
 tool the boy uses it for making something 
 he learns the principle on which it 
 operates and the way to use it, by making 
 something; if it is something useful it 
 awakens a higher interest than does some 
 fancy device. Likewise training of engi- 
 neers in mathematics should be by doing 
 something, by the solving of problems, by 
 dealing with real rather than abstract con- 
 ditions. Let this training be secured while 
 applying mathematics to its normal and 
 legitimate purpose as an auxiliary in the 
 study of other branches. 
 
 In the teaching of mathematics for its 
 own sake stress is apt to be laid upon the 
 processes of deriving results rather than 
 the real meaning of the results themselves. 
 An engineer who uses logarithms has no 
 more concern regarding their derivation 
 than the ordinary user of the dictionary 
 for finding the pronunciation of words has 
 in their etymological derivation. The 
 ability to reproduce demonstrations in 
 higher mathematics from memory with the 
 book shut is often not as important as it 
 is to understand them with the book open. 
 In general an engineer, who has occasion to 
 use higher mathematics, will not be inter- 
 ested in evolving difficult equations, nor 
 will he appeal to his memory, but with 
 text-book or reference before him he will 
 seek the things he wants to use. He should 
 know where to find them and how to use 
 them. 
 
 In emphasizing what a skilled mechanic 
 can make with very ordinary tools, or the 
 true engineer can accomplish with the 
 parallelogram of forces and the rule of 
 three, there is no intention of discrediting 
 the value of fine equipments, either me- 
 chanical or mathematical, if there be the 
 ability to use them. 
 
 Possibly the practical utility of mathe- 
 
20 
 
 SCIENCE 
 
 matics may appear to be urged too 
 strongly, particularly as the writer really 
 believes in thorough mathematical train- 
 ing, but he has seen so many cases in which 
 mathematical instruction has never been 
 digested and assimilated, he has seen simple 
 problems confused by unnecessary mathe- 
 matical complications, he has seen men 
 satisfied with results which are absurd 
 because of some mathematical equations 
 sometimes quite unnecessary which seem 
 to obliterate common-sense perspective, 
 and he recalls the new insight into mathe- 
 matics which came through "Analytic 
 Mechanics" under Professor S. W. Eobin- 
 son at the Ohio State University, and 
 ''Problems in Mechanics," under Dr. 
 Fabian Franklin at Johns Hopkins 
 University, that he feels there is little 
 danger in over-emphasizing the importance 
 of concrete training in mathematical 
 study. 2 
 
 The practical questions which the dis- 
 cussion of this subject presents are these: 
 
 What mathematical subject-matter should 
 be covered? And, 
 
 How should it be taught? 
 
 The first difficulty is that there is not, 
 and can not be, a differentiation in tech- 
 nical education which is at all comparable 
 with the wide range of occupations into 
 which graduates will enter. We may as- 
 sume, therefore, that we are considering the 
 case of the average engineering student, 
 taking for granted that options may be 
 used by the best students for enabling 
 them to take up the more advanced and 
 difficult mathematics. Obviously the stu- 
 dent should have enough mathematics to 
 enable him to demonstrate the important 
 engineering laws and formulas and to read 
 intelligently mathematically written engi- 
 neering literature. While only the rela- 
 
 2 Both of these teachers of mathematics had 
 been trained as engineers and had practised the 
 profession. 
 
 tively simple mathematics is commonly 
 used by engineers, yet the ability to handle 
 new problems with confidence requires a 
 thorough understanding and appreciation 
 of the significance of the mathematical and 
 physical basis of the laws and phenomena 
 he is to use. A man who is a thorough 
 mathematician and knows how to apply 
 his knowledge has a great advantage over 
 the pure mathematician or the man with- 
 out mathematical equipment. The better 
 knowledge one has of the complex, the 
 more certainty he has in applying the 
 simple. A student should understand 
 something of the power of the advanced 
 mathematics and the field of its effi- 
 cient application. Although he may 
 not be expert in using it himself, he 
 will know when to call for a mathematical 
 expert. 
 
 An engineer of fairly wide experience 
 remarked a short time ago : ' ' The ordinary 
 engineer does not use higher mathematics 
 because he doesn't know how. He does 
 not have the proper conception of the 
 fundamental principles of the calculus be- 
 cause the subject has been taught by men 
 whose ideals are those of pure mathe- 
 matics. ' ' 
 
 If mathematics is something for engi- 
 neers to use, let its use be taught to engi- 
 neering students. After the fundamentals 
 are learned, the students should attack the 
 engineering problem at once and bring in 
 mathematics as a means of solving it. 
 Mathematics is often advocated for de- 
 veloping the reasoning powers and the 
 ability to reason from cause to effect. 
 There is danger, however, that mathe- 
 matical machinery may make the mere 
 process obscure the cause and the effect. 
 Let them be foremost, with the process 
 secondary or auxiliary to them. 
 
 The way mathematics is brought to bear 
 on some engineering problems reminds one 
 of the story of the old lady who greatly 
 
SCIENCE 
 
 21 
 
 admired her preacher because he could 
 take a simple text and make it so very- 
 complicated. 
 
 Old traditions have not wholly disap- 
 peared, the fear of degrading the pure 
 science of mathematics by applying it to 
 useful things still lingers in influence, if 
 not in precept. We must go further and 
 adapt mathematics to engineering, not only 
 in subject matter, but in method. A 
 mathematical teacher with no patience for 
 anything except mathematics will probably 
 teach a kind of mathematics which has no 
 connection with anything except mathe- 
 matics. Engineering mathematics may be 
 better taught as a part of engineering by 
 an engineer, than as a part of mathe- 
 matics by a pure mathematician. The 
 marker of levels and transits who is expert 
 in the construction of the instruments and 
 an enthusiast over the accuracy of the sur- 
 faces, the excellence of the bearings, the 
 near approach to perfection in the gradua- 
 tion and the general refinement and beauty 
 of workmanship, may make a good in- 
 structor on instruments, but a poor teacher 
 of civil engineering. 
 
 After all, it is not so much abstract 
 courses as it is personal men with which we 
 have to do, it is not mere knowledge of 
 facts or facility in mathematical manipula- 
 tion, but it is training. The young man 
 is to be developed, his native individuality 
 is to be the basis, he is to increase not only 
 his knowledge, but his powers and the 
 ability to use them. It is not mathematical 
 skill so much as a mathematical sense, or 
 mathematical common-sense, which is 
 wanted. With pure mathematics as a sci- 
 ence we have no quarrel and little affilia- 
 tion. 
 
 If you ask men who use engineering 
 graduates what qualities they should pos- 
 sess, you will find that special prominence 
 is given to ' ' common-sense " and "the 
 ability to do things." In mathematical 
 
 training it is quality rather than quantity 
 which is of first consequence. It should 
 develop the facility for systematic and 
 logical reasoning, thus furnishing a gen- 
 eral method as well as a specific means of 
 getting results. 
 
 We are concerned with applied mathe- 
 matics. The ability to state a problem ; to 
 recognize the elements which enter into it; 
 to see the whole problem without over- 
 looking some important factor ; to use good 
 judgment as to the reliability or accuracy 
 of the data or measurements which are in- 
 volved ; and, on the other hand, the ability 
 to interpret the result; to recognize its 
 physical significance; to get a common- 
 sense perspective view of its meaning and 
 the consequences which may follow; to 
 note the bearing of the various data upon 
 the final result ; to determine what changes 
 in original conditions may change a bad 
 result into one which is practical and effi- 
 cientsuch abilities as these are of a 
 higher order than the ability to take a 
 stated problem and work out the answer. 
 It may be urged that all this is not strictly 
 mathematics. But it is just this sort of 
 judgment and insight which makes mathe- 
 matics really useful, and without them 
 there is danger that they may be neither 
 safe nor sane. 
 
 The trend in education is to a closer re- 
 lation to the affairs of life. Science and 
 applied science, scientific and engineering 
 laboratories, are overcoming old ideas and 
 prejudices. Modern engineering develop- 
 ment brings its transforming influence to 
 bear upon education as well as the utilities 
 of modern life. The engineering school 
 has had a phenomenal growth within the 
 lifetime of the recent graduate a growth 
 in ideals and methods as well as students 
 and equipment. It has raised and agitated 
 broad questions as to what constitutes 
 efficient education for producing effective 
 men. It has aimed to combine not only 
 
22 
 
 SCIENCE 
 
 the abstract with the concrete, the lecture 
 room with the laboratory, and the scientific 
 experiment with the practical test; but it 
 has sought by various means to bring the 
 work of the school into close relation with 
 active professional and commercial prac- 
 tise. It has a definiteness of aim and pur- 
 pose which other educational courses are 
 apt to lack. It sets out to produce men 
 who can deal with forces and materials 
 according to scientific principles. It de- 
 velops men whose contact with physical 
 facts and natural laws are first hand and 
 whose ability to reason logically fit them 
 for dealing with new problems. The 
 training which fits men for handling engi- 
 neering problems is the kind that is needed 
 for dealing with the organization and di- 
 recting of men. The sphere of the engi- 
 neer is one the scope of which will con- 
 tinue to increase as engineering education 
 and training produce men whose contact 
 with natural phenomena gives them an in- 
 herent respect for facts as their premises, 
 who are able to think straight to logical 
 and common-sense conclusions, who have 
 an equipment of technical knowledge and 
 who can produce results. 
 
 In discussing the teaching of mathe- 
 matics to engineers, we should emphasize 
 not the mathematics nor the engineers, but 
 the teaching. Aside from the imparting 
 of knowledge and technical ability, the 
 teaching of mathematics gives opportunity 
 for training in the use of logical methods 
 and in the drawing of intelligent conclu- 
 sions from unorganized data which will 
 make efficient men, whether they follow 
 pure engineering, or semi-technical, or 
 business pursuits. Such teaching does not 
 come from the text-book; it must be per- 
 sonalit comes from the teacher. He 
 must be in sympathy with engineering 
 work and have a just appreciation of its 
 problems and its methods. He must be 
 
 imbued with the spirit and the ideals of 
 the engineer. 
 
 Chas. F. Scott 
 
 Pittsburgh, Pa. 
 
 THE POINT OF VIEW IN TEACHING 
 ENGINEERING MATHEMATICS 
 
 I hardly know why I should have been 
 asked to address you at this conference. 
 Possibly, however, the fact that I am a 
 civil engineer by profession, without hav- 
 ing been permitted ever to practise this 
 profession, and the additional fact that I 
 have been a professional teacher of mathe- 
 matical physics, without having been per- 
 mitted to continue in this work, have led 
 your committee to think that I might 
 furnish a conspicuous illustration of the 
 failures to which colleges and universi- 
 ties may lead in these lines of endeavor. 
 
 Having listened attentively to the three 
 formal papers just read, I find it essential 
 to revise my program and instead of fol- 
 lowing similar lines to those of the preced- 
 ing speakers, it seems essential to take 
 direct issue with them. This I am dis- 
 posed to do, not so much because I differ 
 wholly from the views they have set forth, 
 as because it seems necessary to have other 
 sides of the questions they have discussed 
 represented. The preceding speakers ap- 
 pear to me to have taken themselves some- 
 what too seriously. This is a general 
 fault of both theoretical and practical edu- 
 cationalists. My own experience leads me 
 to conclude that in educational affairs the 
 teacher, the school, the college and the 
 university play a much less important role 
 than we commonly suppose. In fact, I 
 have reached the provisional conclusion 
 that the majority of our students turn out 
 fairly well in the world not so much by 
 
 1 Extempore remarks before Sections A and D 
 of the American Association for the Advancement 
 of Science and the Chicago Section of the Amer- 
 ican Mathematical Society, at the Chicago meet- 
 ing, December 30, 1907. 
 
SCIENCE 
 
 23 
 
 reason of the academic instruction they 
 receive as in spite of it. 
 
 My impression also is that in taking our- 
 selves too seriously as teachers of one sub- 
 ject or another, we have, as a rule, quite 
 underestimated the magnitude and the dif- 
 ficulty of the psychological problems with 
 which we have to deal. We have, as a 
 rule, quite overestimated the capacity of 
 our average student, and have thus 
 usually expected too much from him. It 
 is, of course, desirable to set our ideal 
 high and try to rise to an elevated in- 
 tellectual level; but in doing so we have 
 commonly neglected the influence of 
 heredity as well as of environment. I am 
 inclined to think Dr. Holmes was right 
 when he said that it is essential in the 
 generation of a gentleman to begin four 
 hundred years before he is born. So also 
 is it necessary, if we wish to develop a 
 student into a first-class scholar, to begin 
 back some generations before we take up 
 the formal work of training in our col- 
 leges or schools of engineering. It is an 
 important fact, also too commonly over- 
 looked, that the fundamental ideas in- 
 volved in the mathematics and in the 
 mathematical physics essential to the pre- 
 liminary training of a prospective engi- 
 neer are far more difficult of compre- 
 hension than we are wont to suppose. As 
 a rule, I think we begin our elementary 
 mathematics somewhat too early for the 
 average mind. The result is that our stu- 
 dents acquire a mere literary knowledge 
 of the subject without grasping the basic 
 ideas essential to clear thought and espe- 
 cially essential to applications. I am go- 
 ing to give you some illustrations of this 
 fact. They will show how difficult it is 
 for the average mind to attain a proper 
 understanding of mathematico-physical 
 concepts. The difficulties here are much 
 the same as the difficulties of grammar. 
 As you know, children learn to speak, and 
 
 often speak very well, long before they 
 know anything of formal grammar, and 
 this is the natural mode of development, 
 for the logic and subtleties of grammar 
 can be appreciated only by rather mature 
 minds. 
 
 But if the concepts which belong to the 
 study of language and of grammar are 
 rather formidable, those which belong to 
 the higher mathematics and mathematical 
 physics are profoundly more difficult of 
 adequate comprehension. Let me illus- 
 trate this point by a citation from experi- 
 ence furnished by the case of a graduate 
 from one of our universities who pre- 
 sented himself to me a few years ago, while 
 I was dean of a graduate school of Co- 
 lumbia University, as a candidate for a 
 higher degree in mathematical physics. 
 This student had studied mechanics and 
 had attained a degree in engineering. In 
 order to learn something of the breadth 
 and depth of his knowledge, I asked him 
 what it is that makes the trolley car run 
 after the current is cut off. He answered, 
 "It is the force of the momentum of the 
 power of the energy of the car. ' ' There is 
 no reason to suppose that he had not re- 
 ceived good mathematical and physical 
 training, and yet it is plain from the 
 answer he gave me that he knew next to 
 nothing of the meaning of the terms he 
 used. I may cite another case of a suc- 
 cessful practising engineer, who was a 
 pupil of no less authorities in mechanics 
 and engineering than Lord Kelvin and 
 Rankine. This man wrote me a letter in 
 which he sought to convince me that 
 Newton and his followers are all wrong 
 with regard to the parallelogram of im- 
 pulses. "Thus," he said in his letter, "if 
 a particle starts out from a given point 
 under the simultaneous action of two im- 
 pulses, it will not move in the parallelo- 
 gram of the impulses, but it will move in a 
 
24 
 
 SCIENCE 
 
 tautochronous, brachistochronic, plane cate- 
 nary curve of a resilient character." 
 
 These illustrations show how extremely 
 difficult it is to master the fundamental 
 ideas which belong to a great science ; and 
 the difficulties are so great that I am dis- 
 posed to excuse, or at any rate palliate, the 
 blunders made by our average student. 
 He is, in fact, with all his blunders, not 
 very far behind many of his teachers, for 
 it is not uncommon for them to use in 
 their lectures and text-books words not at 
 all free from ambiguity. Witness, in fact, 
 the loose use of such words as force, power, 
 pressure, stress, and strain in some of the 
 best text-books and treatises of the nine- 
 teenth century. The word ''power," for 
 example, is often used in two radically 
 different senses in the same sentence. 
 
 These difficulties and ambiguities lead 
 me to suggest, in opposition to the precepts 
 laid down by a previous speaker, that we 
 may well consider the desirability of print- 
 ing mathematical books free from demon- 
 strations but containing plain statements 
 of facts. I have used such books myself 
 and am disposed to think they are amongst 
 the best books we may place in the hands 
 of a student. The simple fact is that we 
 do not follow a logical order of develop- 
 ment in acquiring knowledge. We pro- 
 ceed rather by the method of "trial and 
 error," and we often find out the facts 
 with regard to an item of learning long 
 before we become aware of the principle 
 involved. 
 
 Hence I think the reason why few of our 
 engineers know much about the formalities 
 of mathematics and mathematical physics 
 after they get through college is plain 
 enough. They are driven over so many 
 subjects during the four years of their 
 college life that they have little or no time 
 for reflection. This latter must come later 
 in life when the mind has developed a suf- 
 ficient degree of maturity to appreciate 
 
 the more recondite principles which lie at 
 the foundation of all the higher learning. 
 This fact is well illustrated also by the 
 case of our friends, the humanists, who 
 have, as you know, for a long time pro- 
 posed the study of geometry for "mental 
 discipline." As a matter of fact, those 
 who have acquired anything like a grasp 
 of geometrical principles known that very 
 few students of Euclidean geometry ac- 
 quire anything like an adequate appre- 
 ciation of the ideas involved, and it 
 is only in the rarest instances that these 
 students pursue the subject after leaving 
 college. 
 
 I have not much sympathy with the 
 engineers who would like to have their 
 own kind of mathematics, and I am not 
 disposed to commend very highly the 
 works on calculus and other branches of 
 pure mathematics designed especially for 
 engineers. On the other hand, our modern 
 mathematicians have generally failed to 
 understand the needs of the engineer. 
 Our more recent type of mathematician 
 has devoted himself too largely to the re- 
 fined questions of convergence and diverg- 
 ence of series and of existence theorems to 
 properly equip him for the numerous and 
 important applications which the ideal 
 engineer should be able to make of his 
 mathematical knowledge. The modern 
 mathematician seems prone to make the 
 engineer with some degree of mathematical 
 talent afraid of himself. I have met some 
 students whose early training had filled 
 them with caution to such a degree that 
 they would not use infinite series for fear 
 that a divergent one might be encountered. 
 It is known, however, as a matter of fact, 
 that most series essential in the applica- 
 tions of mathematics to mathematical 
 physics are safe in this regard, and one 
 of the best ways for the elementary student 
 to learn of the degree of convergence is 
 
SCIENCE 
 
 25 
 
 to apply numerical computation to these 
 series. 
 
 This leads me to say a few words con- 
 cerning numerical computations, in which 
 very few engineers and still fewer mathe- 
 maticians show any degree of proficiency. 
 It seems to me this is one of the most 
 lamentable defects of our elementary teach- 
 ing in mathematics, though here as else- 
 where the intrinsic difficulties are much 
 greater than we commonly suppose. This 
 fact is in evidence at almost every meeting 
 of our scientific societies, for it oftenest 
 happens that the author of a paper involv- 
 ing numerical calculation will talk of the 
 decimals involved instead of the significant 
 figures. Thus, he will say, "this result is 
 correct to five places of decimals," when 
 he should say, "this result is correct to a 
 specified number of significant figures," 
 the latter form of expression being requi- 
 site to indicate the degree of precision at- 
 tained. There is a grave defect in our 
 elementary teaching in these matters; but 
 it arises from the fact that almost none of 
 our teachers of elementary mathematics 
 are qualified to understand the refinements 
 and the difficulties of precision in compu- 
 tation. Thus, it often happens that stu- 
 dents will give results to five or seven 
 significant figures when the data do not 
 justify any such apparent precision. 
 
 To correct these evils we must have a 
 convention of mathematicians, engineers 
 and professional computers who will show 
 authors how to produce elementary text- 
 books giving adequate attention to these 
 matters. 
 
 As regards numerical computation, there 
 is in general need of more practise, since 
 it is through the concrete that we learn of 
 the abstract and the fundamental. No im- 
 portant formula in any text-book or 
 treatise should go without an appropriate 
 illustrative numerical example. 
 
 I would like to take advantage of this 
 
 occasion to express a hope with regard to 
 the future of our country and to the possi- 
 bility of development which may come 
 through suitable cooperation between 
 mathematicians and engineers. Nothing 
 delights me more than to attend a meeting 
 of this kind where mathematicians and 
 engineers have come together. It is an 
 auspicious sign of the times. It is one of 
 the results I have been looking forward to 
 for the past thirty or forty years. Some 
 of us here are old enough to have lived in 
 two epochs, namely, the pre-scientific and 
 the present epoch. We can remember a 
 time when engineers could not have got a 
 hearing such as they have to-day. The 
 history of their rise and development, at 
 least in this country, is well known to some 
 of us. It dates back to a time only about 
 forty years ago. During this time the 
 engineers have fought their way forward 
 to the position now accorded them in con- 
 temporary society. They have won a place 
 in public esteem without which it would 
 have been impossible to hold such a con- 
 ference as we are holding to-day. This 
 esteem has been won in spite of much op- 
 position, coming especially from the older 
 academic institutions; but now having at- 
 tained adequate recognition especially as 
 practising engineers, we have a much 
 higher duty to perform, and this I trust we 
 shall be able to meet adequately through 
 cooperation with our friends the pure 
 mathematicians. I know of no work more 
 important to the general advancement of 
 mathematico-physical science than that 
 which may lead to the development of 
 mathematical physicists, men who possess 
 at once good mathematical knowledge and 
 correspondingly adequate equipment in 
 physical science. Here is a field greatly 
 in need of concentrated effort and of ade- 
 quate appreciation. It is a lamentable 
 fact that while we can easily develop pure 
 mathematicians of a high order and experi- 
 
26 
 
 SCIENCE 
 
 mental physicists of an equally high order, 
 it seems very difficult for us to develop 
 minds possessing both qualities. To a 
 large extent I thing the development of 
 pure mathematics in the future will de- 
 pend, as in the past, on the stimulus fur- 
 nished by mathematico-physical ideas ; and 
 in like manner success in the development 
 of mathematical physics will depend 
 equally in the future on mathematical 
 ability of the highest order. In this line 
 of work we Americans have not done our 
 full duty, and it behooves us as mathe- 
 maticians and engineers, now that we have 
 got together on the plane of mutual in- 
 terest, to give attention to this important 
 field of work. 
 
 The French engineers led by Navier and 
 followed by Lame, Clapyron, and espe- 
 cially by the "dean of elasticians, ' ' Barre 
 de Saint- Venant, have contributed to sci- 
 ence the most important branch of mathe- 
 matical physics, namely, what is commonly 
 called the theory of elasticity. This is 
 superbly difficult in its purely mathe- 
 matical aspects and exquisitely beautiful in 
 its physical aspects, and it stands as a 
 splendid example of the possibilities which 
 may result from adequate cooperation be- 
 tween mathematicians and engineers. 
 
 The chief difficulty in the way of de- 
 veloping mathematical physicists appears 
 to lie in the inadequate appreciation of 
 this type of work by contemporary society. 
 Pure mathematics has a prestige of more 
 than twenty centuries behind it, and 
 the practical work of the engineer appeals 
 even to the dullest of intellects; but we 
 have failed thus far, in this country espe- 
 cially, to adequately esteem the worker in 
 the intermediate field. We must look to 
 it that more attention is given to this field 
 in our colleges and universities. Every 
 university should have two or three men 
 eminent in mathematical physics as well 
 as two or three men eminent in pure 
 
 mathematics. Thus, while I would not 
 advocate the pursuit of pure mathematics 
 or the pursuit of practical engineering less, 
 I would urge the pursuit of mathematical 
 physics more. It is only by the cultiva- 
 tion of this branch of study and investiga- 
 tion that we can keep alive the sources of 
 engineering knowledge. Important and 
 indispensable as the practical work of the 
 engineer is, the cultivation of investigation 
 and discovery in his science is still more 
 important and indispensable. Hence I 
 would urge that when the more pressing 
 questions of elementary instruction in 
 mathematics and engineering have been 
 adjusted, we give attention to the more 
 inspiring and more important questions of 
 the clarification and enlargement of the 
 fundamental ideas of our sciences. 
 
 R. S. Woodwakd 
 Washington, D. C. 
 
 THE TEACHING OF MATHEMATICS TO 
 STUDENTS OF ENGINEERING 1 
 
 FROM THE STANDPOINT OF THE PRACTISING 
 ENGINEER 
 
 I am honored by being asked to say a 
 few words to you about the results of my 
 experience as to the needs of the teaching 
 of mathematics to students of engineering 
 from the point of view of a practical engi- 
 neer. I have had the good fortune of re- 
 
 1 What is Needed in the Teaching of Mathemat- 
 ics to Students of Engineering? (a) Range of 
 Subjects; (b) Extent in the Various Subjects; 
 (c) Methods of Presentation; (d) Chief Aims. 
 A series of prepared discussions following the 
 formal presentation of the subject by Professor 
 Edgar J. Townsend, Professor Alexander Ziwet, 
 Mr. Charles F. Scott and President Robert S. 
 Woodward. (See Science, July 17, 1908, pp. 69- 
 79; July 24, 1908, pp. 109-113, and July 31, 1908, 
 pp. 129-138.) Presented before Sections D and A 
 of the American Association for the Advancement 
 of Science and the Chicago Section of the American 
 Mathematical Society, at the Chicago meeting, 
 December 31, 1907. 
 
SCIENCE 
 
 27 
 
 ceiving quite a thorough mathematical 
 training in the Ecole des Ponts et 
 Chaussees of France, and I have also had 
 the good fortune of developing into a fairly- 
 practical engineer ; my remarks will there- 
 fore be backed by actual experience. 
 
 Mathematics is to an engineer what 
 anatomy is to a surgeon, what chemistry is 
 to an apothecary, what the drill is to an 
 army officer. It is indispensable. I think 
 we all agree on this point. 
 
 There is a considerable agitation at this 
 time in France and Germany, especially 
 the former, favoring the limitation of the 
 present mathematical program of the engi- 
 neering schools on the ground that it is 
 unnecessarily extensive. From personal 
 observation, I can say that the program 
 there covers a considerably wider range 
 than in the average American college. In 
 the first place, a student entering an engi- 
 neering college on the European continent 
 must already know the analytical geom- 
 etry, the descriptive geometry, the rudi- 
 ments of differential and integral cal- 
 culus, none of which are taught here until 
 the student enters college. The average 
 length of a college engineering course 
 abroad is four years, one of the exceptions 
 being the Ecole Centrale, of Paris, France, 
 where the course is only three years, but 
 where the entering examinations are of a 
 comparatively high standard and the stu- 
 dents must be above the average in ability 
 and application in order to hold their own 
 during the college course. It is obvious, 
 therefore, that in American colleges, time 
 is spent on pure mathematics which could 
 be devoted to practical study. I believe 
 the time will come when only applied 
 mathematics will be taught in colleges, and 
 all necessary abstract mathematics will 
 form a part of the conditions for enter- 
 ing. 
 
 As time goes on, every profession tends 
 more and more towards specialization. 
 
 This tendency is quite marked in the engi- 
 neering profession. It would take too 
 long to enumerate all of these special 
 branches of engineering, but nearly every 
 branch demands a somewhat different 
 mathematical training. The time may 
 come when this specialization will extend 
 over the study of abstract mathematics, 
 differing with each student according to 
 the branch of engineering he intends to 
 follow. For instance, a railway engineer 
 who may aspire to become a railroad 
 official requires less knowledge of calculus 
 than an electrical or a bridge engineer; on 
 the other hand, he requires a greater 
 knowledge of geology than the electrical 
 engineer, and a greater knowledge of com- 
 mon law than the bridge engineer. As 
 my remarks are merely intended to fur- 
 nish topics for discussion, I will put the 
 following question: In view of the fact 
 of the steadily growing scope of special 
 education will it be desirable and possible 
 to specialize mathematical courses in col- 
 leges and adapt them to each branch of 
 engineering? This, as I understand, is 
 done at present only to a small extent in 
 applied mathematics. 
 
 Bridge engineering, of which I have 
 made a specialty, requires probably as high 
 a mathematical training as any other 
 branch of the profession, and yet, I find 
 that part of the higher mathematics which 
 I have studied in college, apart from the 
 drilling features of such studies, has been 
 entirely useless ; for instance, the theory of 
 differential equations. The time I spent 
 on it, though considerable, was not suffi- 
 cient to make me understand it thoroughly, 
 and would have been better employed in 
 the study of the methods of least work, for 
 instance, which no bridge engineer should 
 neglect to study. 
 
 On perusing the elementary books used 
 in high schools, I have been often struck 
 with the dry, uninteresting manner in which 
 
28 
 
 SCIENCE 
 
 the various subjects are being treated. The 
 examples are mostly abstract, very few 
 practical problems to work out. Unless 
 the student is very intelligent, his mind re- 
 tains nothing beyond a chaos of formulae 
 hard to remember and a few mechanical 
 means of solving abstract problems. He is 
 incapable of applying an equation to a 
 practical problem. The methods of pre- 
 sentation should, therefore, be such that the 
 student knows the why and wherefore of 
 each operation in other words, that he 
 learns to think mathematically. This 
 training in mathematical thinking should 
 also be the chief aim: one does not know 
 a foreign language unless one is able to 
 think in that language ; one does not know, 
 mathematics unless one is able to think 
 mathematically. It is not necessary for 
 that to go up into the highest mathematics, 
 but it is necessary to be thoroughly drilled 
 in elementary principles of each subject. 
 These elementary principles should be- 
 come a second nature to the student, just 
 as a language becomes a second nature 
 when it is thoroughly acquired. Problems 
 arise every day in the practise of an engi- 
 neer, which a mathematical mind can solve 
 without going into calculations, such prin- 
 ciples as those of maxima and minima, 
 those of least work, of cumulative effect of 
 forces and others are invaluable in assist- 
 ing to arrive at a logical solution of many 
 problems without the use of a scrap of 
 paper; but in order that they may be ap- 
 plied, one has to be able to think mathe- 
 matically. With a proper foundation, the 
 engineer's mind becomes so trained that he 
 applies those fundamental principles un- 
 consciously ; they direct his line of thought 
 automatically, so to speak. How to secure 
 such a foundation in a student must be left 
 to those who make a life-study of teach- 
 ing. 
 
 Ralph Modjeski 
 Chicago, III. 
 
 The methods of teaching mathematics to 
 engineering students in vogue twenty years 
 or more ago, while often sufficiently 
 strenuous, were invariably far from satis- 
 factory, in that they failed to show the ap- 
 plication of the subjects to engineering 
 practise and to explain that mathematical 
 quantities represent something real and 
 tangible, not merely abstractions. Possibly 
 methods have changed of late years; but 
 nothing that the writer has seen or heard 
 indicates to him that any fundamental im- 
 provement has been effected. Most people 
 continue to believe that mathematical sub- 
 jects are taught mainly for the purpose of 
 training the mind, and that the manipula- 
 tions involved in this branch of science are 
 simply mental gymnastics. Moreover, even 
 among engineers and professors, only a few 
 recognize adequately the great importance 
 of mathematics in engineering and that it 
 is something real and substantial instead of 
 fictitious and imaginary. It is true that 
 higher powers than the third are not con- 
 ceivable entities; but the mathematician 
 recognizes them as temporary multiples for 
 future reduction to entities. 
 
 The engineering student in his pure- 
 mathematical classes is not taught what 
 equations really mean, nor what are their 
 denominations or those of their component 
 parts. All that he learns is how to juggle 
 with quantities in order to produce certain 
 results. It is left to the professor of 
 rational mechanics to teach engineering 
 students the reality of mathematics; and 
 too often he fails to do so, sometimes, per- 
 haps, because his own conception thereof is 
 rather vague. 
 
 Concerning the teaching of pure mathe- 
 matics by the professor of rational me- 
 chanics the writer speaks from personal ex- 
 perience ; for more than a quarter of a cen- 
 tury ago he taught that branch of engi- 
 neering education in one of America 's lead- 
 ing technical schools. Notwithstanding 
 

 SCIENCE 
 
 29 
 
 the fact that the courses in pure mathe- 
 matics then given there were rigid and even 
 severe, the students, as a rule, had no idea 
 of how properly to apply the knowledge 
 they had accumulated; nor did they know 
 what the mathematical terms employed 
 really meant. It was necessary for the 
 writer not only to teach his own branch, 
 but also to supplement the students ' knowl- 
 edge of pure mathematics by explaining 
 such things as limits, differential coeffi- 
 cients, total and partial differentials, and 
 maxima and minima. 
 
 Throughout the entire course in rational 
 mechanics the writer either demanded from 
 the students or gave them demonstrations 
 of all difficult or important formulse; and 
 the students in explaining their blackboard 
 work were repeatedly asked to state the 
 denominations, not only of the equations as 
 a whole, but also of their factors and com- 
 ponent parts. The answers to such ques- 
 tions evidenced clearly whether the student 
 had a true conception of the mathematical 
 work he was doing, or whether he had 
 merely memorized certain manipulations of 
 quantities. 
 
 It was the writer's custom also to supple- 
 ment as much as possible all analytical 
 work by graphical demonstrations; and if 
 he were to resume the teaching of me- 
 chanics, he would adhere to this method. 
 
 In teaching technical mechanics the 
 writer followed only to a certain extent the 
 manner of instruction just described; for 
 by the time his students had reached the 
 technical studies, they were so well drilled 
 and weeded out that constant quizzing on 
 fundamentals was no longer necessary; 
 nevertheless the question, "what is the 
 denomination of that equation or of that 
 quantity," was one that was very likely to 
 be asked any student who gave his demon- 
 strations haltingly or who evidenced at all 
 a lack of conception of the principles in- 
 volved. 
 
 In the writer's opinion, the manner of 
 teaching pure mathematics to engineering 
 students should differ materially from that 
 usually employed in academic courses; for 
 while in the latter case it suffices if the 
 instructors be good mathematicians, in the 
 former they should also be engineers, and 
 should have taught, or at least should have 
 studied specially, both rational and tech- 
 nical mechanics. 
 
 Some institutions still adhere to the anti- 
 quated custom of teaching pure mathe- 
 matics by lectures. This method has 
 always appeared to the writer to be per- 
 fectly absurd; for the primary benefit to 
 be obtained from the study of mathematics 
 is mental training ; and the student can get 
 this only by severe effort, and not by hav- 
 ing another man's mind do the reasoning 
 for him. Midnight oil and the damp towel 
 are for most students necessary accessories 
 to the courses in pure mathematics. 
 
 The writer believes that the only legiti- 
 mate lectures in pure-mathematical courses 
 for engineering students are as follows : 
 
 First; A short opening lecture to outline 
 the work that is to be covered in the course 
 and to explain how best to study the sub- 
 ject. 
 
 Second: Frequent informal talks to in- 
 dicate the application of the mathematics 
 studied to engineering practise, to explain 
 clearly the meaning of all equations, 
 factors and terms, and to show the true 
 raison d'etre of all that is being done. 
 
 Third: A concluding lecture in the na- 
 ture of a resume to call attention to what 
 has been accomplished during the entire 
 course and to the importance thereof. 
 
 Fourth : Personal and forcible lectures to 
 lazy students so as to give them clearly to 
 understand that they must either study 
 harder or drop out of the class. 
 
 All mathematical work done by engineer- 
 ing students should be so thorough and 
 complete that the subject shall be almost as 
 
so 
 
 SCIENCE 
 
 much at command as the English language 
 or the four simple rules of arithmetic. 
 Only such thorough knowledge will enable 
 the engineer to use mathematics readily as 
 a tool, rather than as a final resource to be 
 employed solely in extreme need. 
 
 Analytical geometry should be taught 
 graphically as well as analytically in order 
 that the student shall comprehend it fully 
 and shall realize that the work is real and 
 tangible and that the equations represent 
 lines, surfaces, and volumes, and are not the 
 results of mere gymnastics. A knowledge 
 of the graphics of analytical geometry is 
 especially valuable in mechanical work, in 
 the investigation of earth pressures, in 
 suspension, bridge work, and in many 
 other lines of engineering. 
 
 The proper conception of the meaning of 
 the calculus is rarely carried away by the 
 student. He knows the rules and can per- 
 form the operations, but their significance 
 is beyond him; consequently he does halt- 
 ingly and bunglingly the original work 
 which facility in the use of the calculus 
 should enable him to perform easily and 
 well. This state of affairs is a crying evil 
 which should be corrected in all schools that 
 aim to give first class engineering courses. 
 
 Descriptive geometry is of very large 
 value in the preparation of drawings; but, 
 in addition, a thorough knowledge of it 
 greatly aids in the conception of an object 
 in space, and, consequently, is of large 
 assistance in the evolution of original de- 
 signs. A knowledge of it prior to the 
 study of the courses in pure mathematics 
 assists materially in the conception of what 
 the latter really mean; consequently de- 
 scriptive geometry should be one of the 
 earliest courses in an engineering curric- 
 ulum. 
 
 A sound knowledge of mechanics, the 
 foundation of engineering, is impossible 
 without a thorough understanding of 
 mathematics. It is true that mechanics 
 
 may be learned by rote or by so-called 
 common-sense methods; but the "rule of 
 thumb" or "pocket-book" engineer never 
 rises to noticeable heights. Such an engi- 
 neer almost invariably fails at the critical 
 moment, when a decision must be sup- 
 ported by fundamental principles. It is 
 true that the actual use of analytical 
 geometry, calculus, least squares, or even 
 higher algebra and spherical trigonometry, 
 is rare in the practise of most engineers; 
 but an engineer's grasp of technical work 
 depends upon his knowledge of these sub- 
 jects; and it is generally conceded that a 
 heavy structure can not be continuously 
 supported on a weak foundation. 
 
 Mathematics higher than the calculus is 
 of small value to the engineer, except pos- 
 sibly as a training for the mind; but the 
 writer is of the opinion that any such 
 further study of mathematics is a detri- 
 ment rather than a help, in that it tends to 
 a desire to reduce all work to mathematical 
 calculation and thus to weaken the judg- 
 ment. In other words, excess of mathe- 
 matical development sometimes produces 
 an unpractical engineer. 
 
 Most graduate engineers immediately 
 after leaving their alma mater drop for- 
 ever the study of mathematics, both pure 
 and applied, except in so far as they are 
 forced to use them by their professional 
 work. No greater mistake than this can be 
 made, for it takes very few years of non- 
 use of these subjects to cause one to forget 
 them utterly. Every young engineer 
 should make it a point to devote a certain 
 portion of his time to the reviewing of the 
 mathematical studies of his technical course 
 so as never to become rusty in them; and 
 the writer believes that it is the duty of 
 every professor of mathematics and me- 
 chanics to impress this fact continually 
 upon the minds of his students, even up to 
 the very day of their graduation. 
 
 Kansas City, Mo. J - A " L ' Waddell 
 
 
SCIENCE 
 
 31 
 
 FROM THE STANDPOINT OF THE PROFESSOR 
 OF ENGINEERING 
 
 When I come to think of what the Math- 
 ematical Society has brought upon itself, 
 I fear that it may feel something like the 
 football when it is kicked back and forth 
 upon the field. On the one hand we have 
 the trade-school element demanding more 
 knowledge of rules and, on the other, the 
 engineer demanding more knowledge of 
 principles. No fair discussion of this sub- 
 ject can be had without considering for a 
 moment the conditions and definition of 
 engineering itself. The most common defi- 
 nition was promulgated more than half a 
 century ago by Thomas Tredgold, to the 
 effect that civil engineering, which was the 
 only branch of engineering then known, so 
 the" definition may be considered as being 
 general, that "civil engineering is the art 
 of directing the great sources of power in 
 nature to the use and convenience of man. ' ' 
 I should say that "civil engineering to- 
 day is the art and science of directing the 
 great sources of power in nature to the use 
 and convenience of man," and from that 
 standpoint I am willing to discuss the ques- 
 tion as to how much and how far mathe- 
 matical instruction should enter. 
 
 If engineering is merely an art, then 
 mathematics as a science has no place in 
 the training of the engineer, but if engi- 
 neering is a science, then mathematics has 
 a place. Engineering stands to-day in the 
 act of rising to the status of a science, but 
 is still hampered by the tradesman. On 
 the one hand, we have the demand that the 
 student's training be such as primarily to 
 make him useful to some one to-morrow; 
 and, on the other side, that it make him 
 useful to the world perhaps ten years 
 hence. The two requirements are incon- 
 sistent and do not belong together. One is 
 that of the trade school, and many should 
 not go farther than that because they have 
 not the mental capacity, and the other is 
 
 the demand of the profession into which a 
 smaller number are qualified to enter. The 
 trade school has caused most of the trouble 
 with the teaching of mathematics because 
 those who are products of the trade school 
 have no use for mathematics as a science. 
 The complaint about the teaching of mathe- 
 matics does not come from engineers ; they 
 are ready to use mathematics as a science. 
 In civil engineering it is fortunate that the 
 profession has developed along lines laid 
 down by Rankine rather than by Traut- 
 wine. Both have had their use, but one 
 of them produced the scientist and the 
 other produced the tradesman. 
 
 It is maintained in the institution which 
 I have the honor to represent that they 
 who would teach engineering must prac- 
 tise it, and by analogy we might say that 
 those who teach mathematics to engineers 
 should themselves be engineers. It seems 
 to me that a time may come when such a 
 condition will be desirable, but let me say 
 now that there are few engineers to-day 
 who have had sufficient training in mathe- 
 matics to teach it themselves, much less to 
 tell mathematicians how it should be 
 taught. We can perhaps judge of the 
 deficiency of the student who comes to us, 
 but my feeling is that the remedy is not a 
 question of what, but of how. Men in my 
 institution are sending us students well 
 prepared in mathematics. Others do not 
 seem to be so fortunate. Both are teach- 
 ing the same subjects. We have to realize 
 that the student himself is a factor in this 
 question. Some students become mathe- 
 maticians under any one; others would not 
 under any one. To be taught mathematics 
 properly, the point at which engineering 
 minds must begin, is a long way back. I 
 am inclined to think they must begin some 
 generations before birth. The mathematics 
 of grammar schools needs overhauling more 
 than the mathematics of any other part of 
 our educational system, and probably the 
 
32 
 
 SCIENCE 
 
 mathematics of high schools stands next. 
 The essential thing that we ask of mathe- 
 matics is that it should develop the quanti- 
 tative reasoning power, and the student 
 must be able to think mathematically. If 
 he has not acquired that, then he should 
 drop out of engineering and take up a 
 trade. It was mentioned by a previous 
 speaker that a relatively small percentage 
 of the graduates from a certain engineer- 
 ing school were engaged in occupations in 
 which mathematics was of importance. 
 From a somewhat intimate acquaintance 
 with the graduates of that institution, I 
 may add that a much less proportion had 
 sufficient mathematical training to take 
 positions in which mathematics was an im- 
 portant requirement. Until recently, that 
 college has stood for hardly more than a 
 highly developed trade school, and it is not 
 fair to cite its statistics as showing condi- 
 tions of engineering schools. The director 
 of that institution stated many years ago 
 that he did not consider descriptive geom- 
 etry necessary for mechanical engineers, 
 and his students, having had their course 
 in machine design in the junior year were 
 frequently found taking their only course 
 of descriptive geometry when seniors. 
 
 The question has been raised as to the 
 increase of mathematics for entrance to 
 engineering schools. My view of that is 
 that it would not be wise to raise the 
 requirements at this time. Cornell has, it 
 is true, increased the requirements, but at 
 the sacrifice of both physics and chemistry, 
 and to my mind it is best that physics and 
 chemistry be taught at the age of high 
 school students, rather than analytics and 
 trigonometry. If you can not do both it 
 is better that the young mind have im- 
 pressed upon it some physical science 
 rather than encounter the more abstract 
 demands of mathematics. In the training 
 of students in mathematics I would wipe 
 out formulae. We want principles. There 
 
 is generally taught too much of the for- 
 mula, as that is what the trade school has 
 demanded. Some have objected to the 
 statement that mathematics should be a 
 tool. To my mind it is certainly an in- 
 strument. It is one of the things that the 
 engineer must use, and in order that he 
 may use it, he must be sufficiently familiar 
 with it, so that it will respond to his use 
 when he desires it. The question of elec- 
 tion in mathematics has been suggested. 
 I am certainly favorable to elections in that 
 subject, but I question the advisability of 
 such opportunity in any subject for the 
 ordinary student, before the fourth year. 
 My own observation leads me to conclude 
 that very few students are able to elect 
 intelligently before that time. The re- 
 marks relative to the employment of inex- 
 perienced instructors instead of competent 
 professors show a fault to lie with the 
 heads of the various departments them- 
 selves. If they are willing to accept, for 
 the purpose of instructing students, the 
 men who have been unable to find positions 
 elsewhere, and employ only such as will 
 work for seven to nine hundred dollars per 
 year, the unsatisfactory results are their 
 own fault. The responsible parties, the 
 trustees and regents of educational institu- 
 tions, will furnish what is shown to be 
 necessary. If it is necessary that you have 
 better men, then say so and get them, but 
 if you are satisfied with what you now 
 have, then you can expect to see decorative 
 cornices and stained glass windows, rather 
 than intellect and culture, the characteris- 
 tics of our universities. 
 
 Gardner S. Williams 
 Univeesity of Michigan 
 
 It may save time to state briefly at the 
 beginning my thought on what is needed 
 in the teaching of mathematics to engi- 
 neering students. It seems to me that, 
 

 SCIENCE 
 
 33 
 
 outside of the general cultural and devel- 
 opmental purpose of the study of mathe- 
 matics, the instruction of engineering stu- 
 dents may be discussed under three dif- 
 ferent phases, which for want of better 
 terma^may be named: (1) theory, (2) 
 practise, (3) philosophy; that successful 
 teaching of mathematics to engineering 
 students depends upon giving the right 
 relative proportion or emphasis to these 
 three phases of instruction; that the con- 
 tent of the instruction, within the limits 
 of present usage in engineering schools, is 
 of minor importance; that thoroughness is 
 essential, and that it is better to cut down 
 the extent of the matter gone over if there- 
 by a more thorough grasp of the subject 
 is secured; and that the instructor must 
 always keep in mind that he. is training an 
 average boy of average preparation with 
 a view to using mathematical principles 
 and methods of attack and mathematical 
 operations and conceptions in the mastery 
 of his engineering studies and in the treat- 
 ment of the varied problems which will 
 arise in his later engineering experience. 
 
 The great mass of our engineering stu- 
 dents, like the great mass of our engineers, 
 are not mathematical geniuses. In the dis- 
 cussion of the subject we must keep ever 
 in mind that the average engineering stu- 
 dent is not of strong mathematical bent. 
 Many of those with only mediocre mathe- 
 matical ability make successful engineers, 
 and the student of strong mathematical 
 turn may lack in some direction or may 
 have a disproportionate measure of the 
 importance of his analytical powers and 
 drop behind his less mathematical class- 
 mate. I want to make a plea for the aver- 
 age student; the boy whose analytical 
 powers have to be encouraged and devel- 
 oped. The methods of presentation must 
 be made elastic enough to include this great 
 class of students, or we shall fail to do our 
 duty as teachers. 
 
 I have mentioned three phases in the 
 presentation of mathematical subjects. 
 These may be considered in order. It must 
 be understood that these phases are not 
 mutually exclusive. 
 
 1. Theory. Analysis, demonstration and 
 the general derivation and presentation of 
 mathematical principles. The derivation 
 and exposition of mathematical principles 
 and operations and the appreciation of 
 mathematical concepts are universally 
 accepted as important elements in the 
 education of an engineer. The use of 
 mathematical forms of attack, the training 
 in processes of reasoning, the formation 
 of logical habits of thought, are hardly sec- 
 ondary in importance. And yet much less 
 emphasis is placed on formal demonstra- 
 tion and reasoning than formerly fre- 
 quently this element is overlooked or 
 treated in a slipshod way. The student 
 comes to feel that he is after facts and that 
 the derivation and proof of principles in- 
 volves useless effort he is willing to accept 
 their authenticity. It may be that years 
 ago our instructional methods carried for- 
 mal processes to an extreme and that as a 
 result mathematical work became meaning- 
 less lingo or memorized facts to many stu- 
 dents. This does not furnish argument 
 for the abandonment of training in formal 
 reasoning. For the young mind, practise 
 in analysis, in formal demonstration is 
 illuminating and developing. Even the 
 repetitive forms of analysis in the old-time 
 mental arithmetic had great mathematical 
 educational value. The speaker feels that 
 in the effort to avoid barren formalism the 
 pendulum has swung too far the other way, 
 and that both in high school and in tech- 
 nical school, and in the applied engineer- 
 ing subjects as well, the training in an- 
 alytical methods and formal processes is 
 weak. He believes that good results would 
 follow putting greater emphasis on this 
 
34 
 
 SCIENCE 
 
 phase of instruction than now seems to be 
 the trend. 
 
 2. Practise. The use and applicability 
 of mathematical principles and processes 
 in the solution of problems, drill on these 
 principles, and the acquisition of facility 
 in their use. To the average student the 
 working of examples is illuminating. 
 Without it the concept is but vaguely com- 
 prehended, the derivation only faintly 
 understood, the process may seem merely 
 verbal legerdemain. Properly used, this 
 phase of mathematical instruction is of 
 great advantage to the student of average 
 mathematical ability. It opens up the 
 view; it clears away uncertainties; it fixes 
 principles and concepts ; it gives life to the 
 subject. The problems used should be 
 within the field of the students' experience 
 and comprehension and may well bear some 
 relation to his future work, both in the 
 engineering class-room and beyond. And 
 the second part of this heading is not less 
 important. Mathematics is a tool for the 
 engineering student, and he must acquire 
 facility in its use. This does not mean 
 that the instructor should attempt to make 
 him a finished calculator or an expert 
 workman time is too short but mathe- 
 matical principles and processes must be 
 more to the student than a vague some- 
 thing which he recognizes when his atten- 
 tion is directed thereto. Instead, he must 
 have a mastery of at least the fundamen- 
 tals and he must be able to use such prin- 
 ciples and processes in his later studies 
 without having to divert his attention and 
 energy too much from the engineering fea- 
 tures involved. To acquire this facility 
 requires drill and repetition, and this drill 
 must constitute a part of the mathematical 
 training of the engineering student. The 
 multiplication table had to be learned, and 
 many other important things have to be 
 acquired in the same way. 
 
 But it seems that this important side of 
 
 instruction may be abused. The student 
 who thinks that to accept facts and work 
 problems is sufficient and the instructor 
 who thinks that illustrations and practise 
 work alone constitute mathematical train- 
 ing or that mere laboratory methods suffice 
 are greatly mistaken. The mere substitu- 
 tion in formulas is only rule-of-thumb 
 work, so much decried in engineering; and 
 the mechanic who knows how to use tools, 
 and no more, is not an engineer. There 
 must be a direct connection with the theory 
 and the philosophy of the subject to make 
 the practise side serve its proper purpose. 
 In teaching mathematics years ago, expres- 
 sions of approval came to me because I 
 was so "practical," but the underlying 
 purpose of the practical part was not 
 always understood, though this lack of 
 understanding did not affect the results of 
 the method. Inside the "sugar coating' r 
 there should always be a principle to fix, 
 a concept to illumine, a process to ex- 
 emplify, a derivation to expound. There 
 seems to be a tendency among some to over- 
 do this side of the work to the detriment 
 of the first side. While the practise fea- 
 ture is a valuable auxiliary in mathemat- 
 ical instruction, it should never be the lead- 
 ing motive. Student and instructor alike 
 should recognize this. 
 
 3. Philosophy of the Subject. The basis 
 on which the science rests, the underlying 
 meaning of the mathematical processes 
 used, a philosophical study of the method 
 of treatment and of the concepts used, 
 their connection with related things. This 
 is difficult to discuss in a general way, and 
 of course this phase is intimately connected 
 with the first and second. To my mind 
 this phase should not be neglected. It 
 must be apportioned according to the abil- 
 ity of the student. An understanding of 
 the philosophy of the subject will widen 
 his field of view and lessen the chances of 
 error. The better grasp of the meaning 
 
SCIENCE 
 
 35 
 
 will be advantageous. Its presentation 
 involves difficulties, and text-books gener- 
 ally disregard it. It must not be over- 
 emphasized, as is illustrated by the treat- 
 ment in a recent text-book in applied math- 
 ematics, where it is used largely to the ex- 
 clusion of analysis and demonstration. 
 
 Effective methods in mathematical sub- 
 jects involve, then, the skillful selection in 
 proper proportion from these three phases, 
 and the best teacher will make for himself 
 the best selection. The derivation and 
 elucidation of mathematical principles, 
 facility in their use and application, and 
 an understanding of the basis on which 
 principles and methods rest are all essen- 
 tial. A good text-book one properly pro- 
 portionedaids greatly in the work of in- 
 struction. However, it is the teacher on 
 whom reliance is placed in the end, and for 
 the student of average mathematical abil- 
 ity the teacher's influence constitutes a 
 large element. It is highly advantageous 
 for the teacher to have a fair knowledge 
 of the applications of mathematics which 
 the student will make in later work and 
 to have sympathy and interest in such 
 work. Let us also emphasize the impor- 
 tance of having the best of teachers for 
 mathematical instruction. 
 
 Let me add to this that it is my belief, 
 growing stronger after many years of ob- 
 servation, that the average engineering 
 student gets relatively little from lectures 
 on mathematical subjects; that many in- 
 structors talk too much themselves; that 
 the student must have the opportunity to 
 express himself and must be required to 
 use the mathematical language and to try 
 his own skill, and this in other than formal 
 quizzes ; and that recitation and drill work 
 are essential factors in giving training to 
 this average student. 
 
 Little can be said in the time at my dis- 
 posal on the ground which should be cov- 
 ered in mathematical instruction. Two 
 
 classes of matter are studied: (1) funda- 
 mental principles forming the skeleton of 
 the work, and (2) the more complicated 
 topics, involving further detail and insight. 
 There will be little difference of opinion 
 on the first class. There will be more on 
 the second. I have found in the teaching 
 of mechanics and of various engineering 
 subjects that certain topics and methods 
 not ordinarily given in mathematical in- 
 struction may advantageously be used in 
 the presentation of the work. The teacher 
 of thermo-dynamics or of electro-dynamics 
 has other topics to suggest, and still other 
 topics will come from other sources. Not 
 all of these may be allowed. In fact, it 
 makes little difference what particular 
 topics are included so long as the student 
 has thorough training in some of the more 
 complex work. The difficulty of giving 
 instruction in complex work lies not so 
 much in the time required, as in the ob- 
 stacle that the concepts lie beyond the 
 student's experience and that he is not 
 ready to comprehend their meaning. If 
 he had the opportunity to study these top- 
 ics after he has reached the subject in 
 which they are to be used, or if he could 
 go back over a part of mathematics after 
 his study has taken him into their field of 
 application, as indeed his instructor has 
 done for himself, the result would be more 
 satisfactory. All these limitations must be 
 considered in choosing the ground to be 
 covered in mathematical instruction. 
 
 Arthur N. Talbot 
 Univebsity of Illinois 
 
 THE TEACHING OF MATHEMATICS TO 
 STUDENTS OF ENGINEERING 1 
 
 FROM THE STANDPOINT OP THE PROFESSOR 
 OF ENGINEERING 
 
 I feel that in this discussion we engi- 
 neers occupy rather an unfortunate posi- 
 tion, on account of the fact that we are 
 1 Continued from the issue of August 7. 
 
36 
 
 SCIENCE 
 
 compelled to assume the position of critics. 
 The student comes to us from the teachers 
 of mathematics, presumably equipped with 
 a knowledge of that subject, and it becomes 
 our duty to teach him subjects in which he 
 makes use of this preparation, and to find 
 out whether he has learned to use mathe- 
 matics as a tool. However, I believe that 
 only by friendly criticism can progress be 
 made, and that every one ought to be will- 
 ing to accept such criticism when given in 
 the proper spirit. I had much rather be 
 criticized than criticize others, and we 
 teachers of engineering hope that we are 
 always ready to receive suggestions, not 
 only from other teachers, but from prac- 
 tising engineers. 
 
 I must first insist that for the engineer 
 mathematics is to be regarded as a tool 
 not as something which is studied simply 
 for the development of some mental powers, 
 but for the ability which it ought to give 
 a man to do something to use the results 
 and methods which he has been taught in 
 solving the problems of his profession. 
 
 There has been a good deal of discussion 
 in the past as to the value of mathematics 
 simply as a means of mental training, with- 
 out reference to its use, and perhaps most 
 of us remember the paper by Sir William 
 Hamilton written seventy-six years ago, in 
 which he maintains that there is no one 
 of the subjects in the curriculum which 
 develops a smaller number of mental facul- 
 ties or develops them in a more imperfect 
 and inadequate manner than mathematics. 
 I have never seen what has seemed to me a 
 conclusive refutation of Sir William 
 Hamilton's main arguments, and for my 
 part I am disposed to agree with him in 
 general, and to assign a comparatively low 
 value to mathematics simply as a training, 
 aside from its applications. I have not 
 observed that students trained in this sub- 
 ject are able to reason any better than stu- 
 dents who have ignored mathematics; in- 
 
 deed, I believe that many non-mathe- 
 matical subjects afford a better training in 
 reasoning than the study of mathematics. 
 This view may perhaps be justified by re- 
 membering that mathematics, aside from 
 geometry, deals with questions of quantity 
 and number, but not with questions of 
 quality. The student puts certain fixed 
 data into his mathematical machine and 
 grinds out the result. He does not learn 
 to observe and to discover the finer and 
 more elusive, but equally important, 
 sources of error likely to occur in the ordi- 
 nary questions of daily life, because he is 
 dealing with a rigid, unyielding, logical 
 machine. In this way his mind may be- 
 come hardened he deals with rigid demon- 
 strations and is unwilling or unable to ap- 
 preciate or submit to a less rigid method, 
 which is often the only possible one. The 
 best student of mathematics is frequently 
 one of the poorest of engineers. Give him 
 fixed data and he will get the proper result, 
 but he may be entirely incapable of attack- 
 ing a practical problem, or of deciding 
 what the proper data are. 
 
 I have not observed that students of 
 mathematics are, as a rule, more accurate 
 than other students, or that a training in 
 the branches of mathematics above arith- 
 metic leads to accuracy. Indeed, it more 
 often appears to pervert the sense of per- 
 spective, and to lead students to work out a 
 result to several figures in cases where a 
 smaller number only may be significant. 
 Mathematics does not train the observation, 
 neither does it train the imagination, except 
 in the geometrical branches, which are now 
 comparatively neglected since the powerful 
 modern methods in analysis have been in- 
 troduced. 
 
 Hamilton only allowed, as I remember, 
 that mathematics adequately trained one 
 faculty, namely, that of continuous atten- 
 tion: but I fail to see that this is trained 
 any better by the study of mathematics 
 
SCIENCE 
 
 37 
 
 than by that of language, chemistry or by 
 other natural sciences. Unfortunately, as 
 at present taught it does train the memory, 
 in a way that it ought not to do. The 
 ordinary student of mathematics subor- 
 dinates perception to a ynemorization of 
 formulae and rules. 
 
 I believe, therefore, that from the point 
 of view of the engineer, mathematics should 
 be taught with the object of giving the 
 student power to use it as a tool. With 
 reference to this I think it is fair to say 
 that the consensus of opinion among engi- 
 neering teachers and practitioners is that 
 the results of the present mathematical 
 training are very poor. The average stu- 
 dent who has completed his mathematical 
 course is frequently quite helpless when 
 called upon to attack a concrete engineer- 
 ing problem, and it is a common remark 
 by civil engineering students that they did 
 not really learn any mathematics until they 
 studied mechanics or the theory of struc- 
 tures. The results seem to be almost 
 equally poor no matter what institution the 
 student comes from, for in my classes there 
 have been students from most of the prin- 
 cipal universities and technical schools in 
 the country and I have failed to notice any 
 great difference in them in that respect. 
 They very generally lack the power to 
 do anything with the mathematics which 
 they have been taught. 
 
 With reference to the reasons for this 
 state' of things, I venture to state what 
 seem to me to be some of them, and the 
 suggestions which have occurred to me by 
 which possibly the results might be im- 
 proved. 
 
 1. In one of the previous papers a state- 
 ment was made that many students who 
 studied advanced algebra in the technical 
 schools had not studied algebra in the pre- 
 paratory schools for the two years previous. 
 This illustrates what I believe to be one 
 failing in our so-called system of educa- 
 
 tion, namely, the lack of continuity. The 
 remedy is to reform and simplify the cur- 
 riculum, and to unify and simplify the 
 entrance examinations to our colleges and 
 technical schools. So long as these en- 
 trance examinations are so extended and 
 cover so large a range of subjects, our pre- 
 paratory schools will be unable to carry 
 out their true purpose, which is, as it seems 
 to me, no less and no more than that of 
 all education, namely, to train a man 
 thoroughly in a few things and to give him 
 the power to do some little thinking for 
 himself and to take up new subjects with- 
 out assistance. 
 
 2. The great inherent difficulty which 
 teachers of mathematics as well as teachers 
 of every other subject meet with is the 
 attitude of the student, and his inability 
 to realize the seriousness and the im- 
 portance of his work. I am fond of ex- 
 pressing my view in regard to this by the 
 statement that the school is not a restau- 
 rant, but a gymnasium ; not a place where 
 a student comes to be filled up, but a place 
 where he finds apparatus and the instruc- 
 tion, by making use of which he may 
 strengthen his mental muscles. 
 
 The manufacturer can take his raw ma- 
 terial and shape it into the form which 
 he desires. The raw material of the 
 teacher is the student, but the teacher can 
 not take this material and shape it ; he can 
 only show it how it can shape itself. I 
 believe, however, that much may be done 
 in impressing upon students the proper 
 attitude which they should take toward 
 their work, and by a proper cooperation 
 between teachers and parents, which is un- 
 fortunately lacking as a rule in this 
 country, and the responsibility for which 
 must largely fall upon the parents. 
 
 3. I believe that one cause of the poor 
 results in mathematical teaching is that too 
 great a stress is laid upon analysis. Mathe- 
 matics is, of course, divided into geometry 
 
38 
 
 SCIENCE 
 
 and calculus, using the words in their 
 widest sense. Geometry is concrete; and 
 the mind perceives the steps in a geo- 
 metrical demonstration. This branch, the 
 oldest branch of mathematics, however, has 
 been largely supplanted by the modern 
 analytic methods which have been de- 
 veloped during the past three centuries, 
 largely to the detriment, it seems to me, of 
 the educational results obtained. Analysis 
 is abstract it is a powerful machine, an 
 invention for doing certain things. Into 
 one end of the machine we put the data; 
 we turn the crank, and the result comes 
 out with absolute correctness so far as is 
 warranted by the data. Now I believe that 
 too much stress is laid on these analytical 
 processes; that the student is not urged to 
 visualize his results, to express them 
 geometrically and to interpret his equa- 
 tions. I warmly second the remarks of 
 Professor Ziwet with reference to descrip- 
 tive geometry, which I believe should be 
 treated as a branch of mathematics and 
 taught more thoroughly, as it is taught in 
 Germany. For my part, I derived as much 
 benefit from my study of descriptive 
 geometry, and afterward from the study 
 of projective geometry, as from any other 
 mathematical studies. These studies train 
 the imagination, which analysis does not 
 do. But in the use of analysis, the first 
 step, namely, the formulation of a problem, 
 is really concrete. This, too, is neglected 
 in our usual courses. Our examination 
 papers are full of questions which involve 
 simply the analytic processes the differ- 
 entiation, the integration, the twisting and 
 turning of equations, while much less at- 
 tention is paid to the formulation in mathe- 
 matical language of practical problems. 
 Our students, therefore, when they meet a 
 practical problem, are unable to select or 
 judge of the correctness of the data, and 
 even if they can do this, are unable to 
 formulate the data as a preliminary to the 
 
 solving of the problem by the use of the 
 mathematical machine. 
 
 One of the great defects which I find in 
 students of mathematics is one already re- 
 ferred to, namely, that they do not in- 
 terpret their equations. The average stu- 
 dent who has completed his mathematical 
 course, for instance, has not the slightest 
 conception of what a parabola is. I make 
 this statement advisedly, because I have 
 tested it again and again for years. If he 
 could tell you what a parabola really is in 
 his mind, he would probably tell you that 
 it was a curve of more or less beauty 
 represented by letters. Perhaps he could 
 tell you what the letters are, but give him a 
 concrete problem and he would convince 
 you immediately that he did not know 
 what the letters mean. 
 
 4. Another defect, as it seems to me, in 
 our present methods, is the lack of training 
 in mental operations. In the good old 
 days menial arithmetic was taught, but 
 that seems to have gone out of fashion, 
 with so many of the other good old 
 methods. Ask the ordinary graduate of 
 our mathematical courses to tell you the 
 square of 20.75 without using pencil or 
 paper and he will look at you open- 
 mouthed with astonishment, but if he had 
 really grasped the meaning of the binomial 
 theorem and had learned to do a few 
 "sums" in his head, any grammar-school 
 boy would, of course, be able to give the 
 result immediately. 
 
 5. Another reason for poor results is, I 
 believe, inadequate class-room methods, and 
 especially the use of the lecture system. 
 In Germany, where the students in the uni- 
 versities have had the advantage of a 
 thorough preliminary training, they may 
 be able to appreciate lectures on mathe- 
 matical subjects, although I doubt even 
 this in the case of the average student. 
 For students in our American universities, 
 however, I believe that lectures in mathe- 
 
SCIENCE 
 
 39 
 
 matics are almost useless, except for a very 
 small number of students; and yet, I am 
 told that even in some of our high schools 
 mathematics is taught to a considerable 
 extent by lectures. The lecture system is 
 easy for the teacher. It involves no cross- 
 questioning, no endeavor to discern what 
 is going on in the student's mind, no adap- 
 tation of question with the object of put- 
 ting him on the right track. 
 
 Again, some mathematical exercises are 
 conducted by sending the students to the 
 board, each with a problem to solve, and 
 then marking that on the correctness of 
 their work. Occasionally a formal expla- 
 nation of his problem is required of the 
 student. This, again, seems to me to be a 
 mistaken method. Many a student can go 
 through a demonstration of a principle, or 
 solve a problem by substitution in a 
 formula, while knowing nothing of the real 
 meaning of the subject. In my opinion 
 class-room instruction should be conducted 
 by the Socratic method by question and 
 answer the teacher endeavoring to put 
 and keep the student upon the right track 
 by showing him what he can do for him- 
 self if he will only learn how. 
 
 6. Eeference has been made to the kind 
 of teachers of mathematics. Personally I 
 believe that in teaching the subject to engi- 
 neering students the best results would be 
 obtained if the teachers were engineers, or 
 at least if they were near enough to being 
 engineers to take an interest in the con- 
 crete problems themselves as distinct from 
 their solution. If I am correct in the be- 
 lief that mathematics should be taught as a 
 tool, then it can be taught best by those 
 who know how to use it as a tool. Un- 
 fortunately, however, it is difficult to get 
 engineers who are sufficiently interested in 
 mathematics and sufficiently masters of 
 that subject, who are willing to devote 
 themselves to teaching. The men who are 
 interested in the problems prefer to devote 
 
 themselves to those problems, and to go 
 into practical work. It is not necessary, 
 however, as suggested above, that the 
 teachers of mathematics should be engi- 
 neers if only they will take an interest in 
 the problems themselves, and in the point 
 of view which the student should take. 
 They can do this by cooperation with the 
 engineering teachers, by attending engi- 
 neering courses, and, perhaps, by a little 
 more realization than they now have that 
 their work is preliminary to other and 
 more important work, and that as a matter 
 of fact if the engineering student does not 
 learn to use his mathematics as a tool it 
 is practically of no value to him. For the 
 engineer, mathematics is the servant, and 
 the mathematical teacher should aim to 
 teach the subject in such a way as to obtain 
 as nearly as possible the results which in- 
 telligent engineering teachers and practi- 
 tioners desire to have obtained. 
 
 George F. Swain 
 Massachusetts Institute of Technology 
 
 FROM THE STANDPOINT OP THE PROFESSOR OF 
 
 MATHEMATICS IN THE ENGINEERING 
 
 COLLEGE 
 
 We must not take too seriously what 
 engineers have to say in an educational 
 discussion, nor take too much to heart their 
 views on the mathematical curriculum. 
 Practising engineers are not in the habit 
 of thinking very continuously on any edu- 
 cational question, although, of course, they 
 must not confess inability to respond when 
 they are called upon for pedagogical opin- 
 ions. Every practitioner in the law would 
 doubtless express views concerning legal 
 education if summoned to do so, but he 
 would be a rash educator who would at- 
 tempt to follow their advice without much 
 circumspection. I, myself, prefer to judge 
 of the engineer's views upon educational 
 matters by studying his actions rather than 
 his words. The things engineers "do" 
 
40 
 
 SCIENCE 
 
 may be taken as a true expression of their 
 deliberate judgment what they "say" is 
 often ill thought out and in contradiction 
 to their deeds. I therefore prefer to judge 
 of the present needs in the mathematical 
 instruction for engineers by the actual 
 tendencies that I observe in the evolution 
 of technology itself. 
 
 What are the great changes that the 
 engineering profession has made in tech- 
 nical science in this country in the last 
 quarter of a century? The changes are 
 quite obvious and not difficult to state. In 
 former days engineering technology was 
 founded chiefly upon current practise 
 rather than upon established principles; it 
 was more closely allied to the crafts than 
 to science. Not only is that day past, but 
 it is no longer the case that technical sci- 
 ence looks entirely to pure science for its 
 fundamental material. It has so grown 
 that it is investigating for itself and, in 
 greater and greater measure, developing 
 the basal principles for its own needs. 
 There are very few American treatises in 
 pure science which will compare in scien- 
 tific thoroughness with several treatises 
 which have lately issued from the engineer- 
 ing press. This is a very hopeful sign in 
 the growth of knowledge to see applied 
 science and pure science approaching each 
 other at numerous points, so that it is in- 
 creasingly difficult to distinguish any line 
 of demarcation between them. In this 
 change, science is not sacrificing any of its 
 strength nor compromising its ideals. It is 
 technology that is changing that is be- 
 coming less empirical, more systematic, 
 more quantitative, more scientific. 
 
 With these well recognized changes in 
 applied science before us, what should be 
 our attitude toward the mathematical sci- 
 ence that is necessarily associated with 
 engineering education? What is tech- 
 nology really requiring of the basal sci- 
 ences? Judging the engineers by their 
 
 acts and not by their words, what is the 
 real demand that they are making of the 
 physicist, of the chemist or of the mathe- 
 matician ? Is the demand to teach physics 
 or chemistry in this or that particular way, 
 or is the demand of a profounder and more 
 radical sort? The most superficial ob- 
 servation shows that the demand is of the 
 latter kind. The engineer in this twentieth 
 century is saying to the physicist, and 
 chemist, and mathematician : ' ' Know more 
 science. Discover more facts in electricity 
 in light in all properties of matter. 
 Give to the world more men like Kelvin, 
 Hertz, Helmholtz. Fill the shelves with 
 ten times the knowledge we now have." 
 These words more truly express the real 
 pressure that engineers are putting upon 
 workers in pure science, than do the words 
 they have uttered in this discussion. As a 
 single example, note that the great elec- 
 trical and other manufacturing companies 
 are impatient at the rate at which pure 
 science grows, and large sums are spent by 
 them each year in the search for new 
 truth and in filling up the gaps in exist- 
 ing knowledge. 
 
 The real demand of the engineer is not 
 for better instruments or tools with which 
 to do his work, nor is the demand for more 
 difficult projects to test his skill, nor even 
 for more capital with -which to construct 
 them. The real demand is for more 
 knowledge, more science, and for more of 
 the spirit of science in technology and in 
 technical education. I take as my text a 
 saying of Ostwald: "Science is the best 
 technology." If we teach a trade and not 
 a science the time is largely wasted. If 
 we teach dyeing and not chemistry, the 
 graduate is already out of date when he 
 begins his career, and he has not the funda- 
 mental principles wherewith to bring him- 
 self abreast of the times. I therefore re- 
 gard it of greatest importance that mathe- 
 matics be taught to engineering students 
 
SCIENCE 
 
 41 
 
 with real enthusiasm for the science itself. 
 It should be taught by men who themselves 
 are actively contributing to the growth of 
 mathematical science. The present spirit 
 of engineering science is such that no in- 
 structor in any of the basal sciences is 
 satisfactory who does not see that it is his 
 duty not only to teach what is old, but 
 to be interested in and to take an active 
 part in the development of what is new. 
 
 I regard of secondary importance the 
 particular things we do in the mathe- 
 matical course in the engineering school. 
 Different instructors, equally successful, 
 will have different opinions. Various 
 changes and improvements have been tried 
 at various institutions. At the University 
 of Wisconsin we have made innovations 
 whenever we thought it best, but I regard 
 them all of secondary importance to the 
 first requirement of all, namely, that we 
 demand the right sort of teachers, and that 
 the teaching be done in the right sort of 
 scientific spirit. 
 
 The only imperative requirement put 
 upon the mathematics in engineering 
 schools that does not rest as heavily upon 
 the mathematics of the ordinary college 
 course is the demand for compactness. It 
 is possible that there is some room in the 
 eourses in colleges of pure science for the 
 whims and fads of the various instructors, 
 for at some later place in the course the 
 balance may be restored. This, however, 
 is not true in a school of engineering. 
 There is very little room for the practise 
 of fads and new schemes. It is easy to 
 exaggerate the need of a special sort of 
 subject matter in mathematics and a spe- 
 cial class of problems for engineering stu- 
 dents. We are apt to make some very 
 foolish mistakes, if we undertake to change 
 too freely the scientific material that is 
 presented to engineering students. A good 
 engineer is worthy of the best science and 
 the best instruction that can be brought to 
 
 him he himself would be the first to 
 object if a different program were carried 
 out. 
 
 I have had a little experience in employ- 
 ing engineering graduates in engineering 
 work. In the past ten years I have given 
 employment, in various capacities, to about 
 one hundred and thirty engineering grad- 
 uates. This work has been scattered over 
 quite a wide territory and the men have 
 come from the institutions of the east, from 
 the Pacific Coast, from the Mississippi 
 Valley and from the south. I have been 
 able to judge within the limits of my ex- 
 perience what the young engineering grad- 
 uates know, and what they have forgotten. 
 I find it true that the boys have forgotten 
 a great deal of the material they had in 
 college, and that they have remembered 
 other things. They remember the manual 
 and the mechanical things how to swim, 
 how to ride a horse, how to fish, how to 
 play ball, how to run the level, how to work 
 the plane table, and how to do stadia work. 
 Now what have they forgotten ? The men 
 have forgotten the intellectual things- 
 hydraulics, electrical science, thermody- 
 namics, etc. The human mind possesses 
 an unlimited capacity for forgetting. But 
 my experience shows that the young men 
 forget their hydraulics just as quickly as 
 they forget their mathematics or their 
 mechanics. The enginee in the field ob- 
 serves that a boy remembers the right end 
 of an instrument and seems to be amazed 
 that the same man does not know the right 
 end of an integral sign. He therefore 
 concludes that the mathematics has not 
 been "taught right." If he will compare 
 intellectual things with intellectual things 
 he will find that a miscellaneous group of 
 engineers will pass as good an examination 
 in mathematics ten years after graduation 
 as they would pass in thermodynamics or 
 hydraulics. 
 
 It grates on me to hear mathematics 
 
42 
 
 SCIENCE 
 
 spoken of as a tool. Mathematics is to the 
 engineer a basal science and not a tool. 
 The spirit of that science is of more value 
 to the engineer than the particular things 
 that can be accomplished. The engineer 
 need not be a mathematician, but he needs 
 to think mathematically, and, to my mind, 
 he needs the power of mathematical 
 thought more than skill in manipulating 
 a few mathematical tools in mechanical 
 fashion. There are already too many fac- 
 tory-made products turned over to the col- 
 lege by the secondary schools. I make a 
 fundamental contrast between the engineer 
 with his mind endowed with the power of 
 creative and rational design, and the ar- 
 tisan with his hands equipped with tools 
 for physical construction. A great engi- 
 neer must be trained in correct seeing and 
 thinking, and must have the power of rea- 
 soning concerning some of the highest ab- 
 stractions of the human mind. In this 
 aspect mathematics is not a tool it is a 
 basal science. 
 
 Chas. S. Slichter 
 Univebsity op Wisconsin 
 
 At the close of Professor Townsend's 
 address he urged the desirability of tech- 
 nical schools offering more elective ad- 
 vanced work in mathematics. It may not 
 be out of place, therefore, for me to call 
 attention to the fact that in the Massa- 
 chusetts Institute of Technology we have 
 offered and given, among others, the fol- 
 lowing courses: advanced calculus, vector 
 analysis, fourier series, least squares, the- 
 ory of surfaces, theory of functions, el- 
 liptic functions, hydrodynamics and dif- 
 ferential equations of mechanics and phys- 
 ics. Some of these subjects are required 
 in one or more of our courses, but not in 
 any one of the larger engineering courses, 
 which are taken as the basis of Professor 
 Townsend's tables. This elective work, 
 therefore, while valuable in many respects, 
 
 is not the main work of the mathematical 
 department. 
 
 The mathematical teacher is in the engi- 
 neering school primarily to teach to stu- 
 dents of engineering the amount of mathe- 
 matics which is necessary to them for the 
 proper understanding and practise of their 
 profession. The object is to give the stu- 
 dent a grasp of mathematical concepts and 
 processes through their use, as one learns 
 grammar by speaking a language. Hence 
 there is no place in the required mathe- 
 matics of a technical school, nor indeed in 
 the first courses in a college of liberal arts, 
 for the refinements of modern "rigor." 
 At the same time there should be no pa- 
 tience with a loose or unscientific presen- 
 tation of first principles. The teacher him- 
 self must be thoroughly conversant with 
 modern thought, else he will teach false- 
 hood for truth, and must be enthusiastic in 
 his interest in his subject, else he will fail 
 to inspire his pupils. Hence the teacher 
 of mathematics should be primarily a 
 mathematician and not an engineer. It is 
 hard to find an engineer who has any 
 knowledge of mathematics other than a 
 small fragment which he habitually uses, 
 and any elementary teacher whose instruc- 
 tion goes to the very limits of his knowledge 
 is sure of failure. It may, of course, be 
 possible to superimpose a mathematical 
 training upon an engineering one, but in 
 that case the engineer becomes a mathe- 
 matician and my contention that mathe- 
 matics should be taught by a mathema- 
 tician is not invalidated. 
 
 On the other hand, the mathematician 
 should know something of the uses to which 
 an engineer wishes to put mathematics. 
 For that reason such meetings as this are 
 helpful, but I must confess to feeling a 
 little disappointment in not obtaining from 
 the engineers any new light on the concrete 
 problem which confronts the teacher of 
 mathematics in an engineering school. I 
 
SCIENCE 
 
 43 
 
 have met the same disappointment else- 
 where in similar meetings. It has hap- 
 pened, elsewhere if not here, that engineers 
 will tell the mathematicians what and how 
 they should teach, in apparently total ig- 
 norance of the fact that what the engineer 
 promulgates as a new gospel has been the 
 commonplace thought of the mathematician 
 for years. This ignorance may be due to 
 the fact that the engineer remembers his 
 own training of twenty or thirty years ago 
 and does not know that improvements 
 have taken place. That such is the 
 case may be seen by a comparison 
 of modern with older text-books. Such 
 criticism from the engineers is amusing, 
 but another kind of criticism is not. I 
 refer to the kind which seizes upon the 
 failure of a student to have learned mathe- 
 matics thoroughly as evidence of poor aims 
 and inefficient teaching of the mathemat- 
 ical instructor. "We all know that students 
 pass through our classes and graduate from 
 our schools whose attainments are not what 
 we wish, but while the mathematical 
 teacher delivers his product to the engi- 
 neering departments and hears of his com- 
 parative failures, the engineering professor 
 delivers his product to the world and rarely 
 hears of the specific blunders of his stu- 
 dents. Another unfair criticism is some- 
 times heard from the professor of engineer- 
 ing who says that students can not use their 
 mathematics, when the truth is they have 
 simply forgotten some particular fact, for- 
 mula, or process, which is a fad of that 
 professor. It is unfair to test mathemat- 
 ical training by tenacity of memory or 
 mere quickness in reasoning. 
 
 I have said that we must teach our stu- 
 dents to use their mathematics. Now in 
 the application of mathematics to a con- 
 crete problem there may be distinguished 
 three steps: 
 
 1. The interpretation of the data of the 
 problem into mathematical language. 
 
 2. The formal operations upon the ex- 
 pression or equations thus obtained. 
 
 3. The interpretation of the results back 
 into the terms of the original problem. 
 
 The first and third of these steps are 
 really the most important, but there seems 
 to be a popular impression that the second 
 comprises the whole of mathematics. This 
 impression is doubtless responsible for some 
 criticisms of the educative value of mathe- 
 matics. It is true that relatively a great 
 amount of time must be spent in the class- 
 room in teaching the mechanical processes 
 involved in the second step, and many stu- 
 dents in school and college get no farther. 
 To object to the amount of time spent in 
 this way and to demand, as some do, that 
 we confine our time to teaching general 
 principles and applications is to talk as 
 sensibly as a fond mother who objects to a 
 child beginning his musical education by 
 playing finger exercises instead of tunes. 
 The technique of mathematics must be 
 learned first, but the student who never 
 gets beyond the technique has not learned 
 mathematics. 
 
 The teacher of mathematics should, then, 
 use all possible means of teaching the first 
 and third of the above steps and should 
 bring his pupils to think of them as the 
 real thing. For that purpose he should 
 seek for applications and illustrations from 
 as wide a range of subjects as possible. 
 He will find himself handicapped, however, 
 in using many problems of real scientific 
 or engineering importance because of the 
 ignorance of his pupils, especially in the 
 first year in the technical school. To illus- 
 trate a new mathematical principle by an 
 application to a science with which a stu- 
 dent is not familiar is to befog and not 
 illumine the subject. Hence there is some- 
 thing to be said in favor of some of the 
 much-criticized problems of the older text- 
 books. To my mind a problem is success- 
 ful if it causes the student to take the three 
 
44 
 
 SCIENCE 
 
 steps just enumerated and is couched in 
 terms familiar to the student, even though 
 it may not be "practical." On the other 
 hand, a type of problem lately coming into 
 use, in which the student is given some 
 formula from a science of which he knows 
 nothing, and is asked to find, say, a maxi- 
 mum value, is as fruitless as if the prob- 
 lem were stated in terms of x, y and z, 
 unless it may serve to convince a sceptical 
 student that the matter he is studying has 
 some practical application. 
 
 And this leads me to the most important 
 thing I have to say, and that is that after 
 the mathematical professor has done his 
 utmost to teach the use of mathematics the 
 engineering professor must take up and 
 complete his work. I doubt if any one 
 really learned the use of mathematics in a 
 first course. Facility in using mathematics 
 comes from actual use and not from the 
 solution of illustrative examples. In the 
 course in mathematics the student expects 
 his problem to be solved mathematically 
 and has his mind alert to find the solution, 
 and that too with mathematical principles 
 fresh in his mind. In a course in engi- 
 neering, his point of view has widely 
 changed. The practical problem has now 
 his main interest, mathematical concepts 
 are in the background, and he often fails 
 to see the possibility of using mathematical 
 principles until he is trained to do so by 
 the professor of engineering. If the pro- 
 fessor, through lack of knowledge or lack 
 of interest, avoids the use of mathematics, 
 the student will soon lose the little he has 
 learned. 
 
 In other words, the mathematical train- 
 ing of a student is not complete when he 
 leaves the department of mathematics. It 
 is possible that better results could be ob- 
 tained if the mathematical department had 
 more time, say for a course in applications 
 of mathematics to miscellaneous problems. 
 But, as a rule, in our technical schools the 
 
 department of mathematics is allowed 
 barely time to teach the necessary tech- 
 nique with what illustrations and applica- 
 tions can be squeezed in. Hence the math- 
 ematical department delivers to the engi- 
 neering department an unfinished product 
 and it is the engineer's duty to teach the 
 student to use the mathematics he has 
 learned. Unfortunately, the professor of 
 engineering is too often a poor mathema- 
 tician and avoids this duty. 
 
 One of the hardest things a student has 
 to do is to combine two different domains 
 of knowledge, each somewhat unfamiliar, 
 so that he may work freely in both at once, 
 using each as a help in the other. It is 
 this difficulty which makes analytical geom- 
 etry traditionally hard, and which the stu- 
 dent meets again when he studies any form 
 of applied mathematics. It is partly to 
 help overcome this difficulty that we have 
 just made a rearrangement of our mathe- 
 matical instruction in the Massachusetts 
 Institute of Technology. We no longer 
 have courses in algebra, analytic geometry 
 and differential and integral calculus, but 
 have combined these into one "course in 
 mathematics" extending through two 
 years. Into this course the elements of 
 analytic geometry and of calculus are in- 
 troduced early and continued late. We 
 hope thus to give these principles more 
 time to become completely domiciled in the 
 student's mind. We have also been en- 
 abled to carry out two principles : the first 
 is to introduce no subject until some use 
 is to be made of it, and the second to 
 handle each problem by the method best 
 adapted to it, rather than by the methods 
 of the particular branch of mathematics 
 which one might at the moment be study- 
 ing under the old classification. We hope 
 in this way to increase the efficiency of our 
 mathematical teaching. 
 
 F. S. Woods 
 
 Massachusetts Institute of Technology 
 
SCIENCE 
 
 45 
 
 The program shows three standpoints 
 from which discussion is to emanate. I 
 occupy no one of them. It is true I have 
 had some engineering practise, but I can 
 not be termed a practising engineer. I 
 have had charge of mathematics for engi- 
 neering students in two engineering col- 
 leges, but for nearly a decade now I have 
 not met students in mathematics; and, 
 indeed, I have taught, all told, but an insig- 
 nificant amount. I am in somewhat close 
 touch with engineering students, but they 
 belong to a particular field, namely, 
 mining, which is possibly less dependent 
 on mathematics than are other branches 
 of engineering. My view-point is, there- 
 fore, somewhat of a compromise or average 
 of the three specified in the announcement. 
 
 The present discussion seems to me sig- 
 nificant. It may bring forth results. In 
 fact it seems to have had some immediate 
 consequences. Last evening after the din- 
 ner I heard a very clever mathematician 
 admit that he felt really humble, and I 
 heard a well-known engineer say that to 
 his great surprise some mathematicians had 
 a human side. I asked a pure mathema- 
 tician sitting near me to show me his hu- 
 man side, but he only shrugged his shoul- 
 ders. Perhaps he was not yet sufficiently 
 humbled. 
 
 This occasion appears to me to be signi- 
 ficant, but as showing conditions which 
 exist rather than as forecasting future 
 changes. It is a symptom of the approach 
 the arrival, perhaps of healthful condi- 
 tions rather than a cause. It may, of 
 course, in its turn become a cause, and 
 operate toward good results. That is not 
 so certain. At the moment it indicates 
 conditions surrounding the teaching of 
 mathematics to engineering students, in- 
 cluding the relations between the teachers 
 of mathematics and those of engineering 
 which have been the growth of many years. 
 Those young and virile gentlemen whom we 
 
 all delight to honor, the Woodwards, have 
 been striving for decades to bring about a 
 closer relation between the teaching of 
 mathematics and the subsequent study of 
 practise of engineering. Ten years ago at 
 the Toronto meeting of the Society for 
 Promotion of Engineering Education I pre- 
 sented a paper looking to this end. 2 There 
 are gentlemen here present who discussed 
 that paper and who may perhaps recall the 
 remarkable unanimity between the teachers 
 of mathematics and those of engineering as 
 to the results most to be desired in teaching 
 mathematics to engineering students, and, 
 indeed, as to the best available methods for 
 producing such results. This movement is 
 old. Most of the ideas which have been 
 brought out here were first conceived a 
 long time since. Nevertheless, it is good 
 to get together and talk them over, and 
 such discussions may result in help to the 
 individual teacher. 
 
 We have heard here much of the ideal 
 which the engineering school should set 
 before itself, but it might well be asked 
 what problem is presented first to the 
 school as a matter of fact? President 
 Woodward put it in part when he spoke of 
 the difficulty of getting the right men in 
 the schools when operators are so eager 
 for good men and are competing on the 
 basis of ' ' so much per month. ' ' And what 
 do the employers demand? They call for 
 men who can do something, men who can 
 think in a logical and common-sense way, 
 but, withal, when they leave the school can 
 be put to some immediate use. The first 
 problem confronting the engineering col- 
 lege is how to meet this demand, for the 
 demand must be met in some degree at 
 least or the college will cease to train men. 
 
 It is inevitable that the character of this 
 demand shall influence largely what the 
 school must do. The call is not for men 
 
 1 See Proceedings of Society for Promotion of 
 Engineering Education, Vol. V., 1897, p. 139. 
 
46 
 
 SCIENCE 
 
 highly trained in mathematics, however 
 much we may feel it ought to be. It is 
 for men who know well a little mathe- 
 matics, and who can do something with it, 
 who can use it "as a tool." And, however 
 obnoxious that expression may be to a 
 mathematical teacher, he who forgets or 
 disregards the fact which lies behind it 
 will surely weaken his instruction of engi- 
 neering students. 
 
 I do not defend the specification of the 
 employer, I point to the fact with which we 
 must deal. Personally I am inclined to 
 find fault with it, but the matter rests 
 largely in the hands of the practising engi- 
 neer. He, though he often objects to the 
 college product, is to a great extent re- 
 sponsible for its general make-up. In the 
 long run and within reasonable limits he 
 can have what he wants. Sometimes he is 
 inclined to require too much technical 
 knowledge on the part of the graduate. 
 His brother teaching in the college in order 
 to meet his requirement says to the teacher 
 of mathematics I must have those students 
 ready earlier with their mathematics. This 
 fact, together with the general tendency in 
 the colleges to raise the standards, causes 
 the mathematical training to be crowded 
 into the first year and a half or two years, 
 when the student is least mature. More 
 of it is being pushed back to the second- 
 ary school, and, in turn, into the grades. 
 Mathematical concepts are difficult, and 
 with President Woodward I am inclined to 
 think we are demanding too much, and 
 calling for it too soon. Covering less 
 ground and at a slower pace will help to 
 make better engineers. 
 
 The student comes to the engineering 
 school with the notion that he is to be 
 
 filled up with a lot of technical knowl- 
 edge, the items of which will be used by 
 him when he is a practising engineer. He 
 seems unable to comprehend that he is in 
 college to acquire mastery over his own 
 powers. He is eager for useful facts and 
 of course he forgets most of those he 
 learns not a great while after leaving col- 
 lege. The forgetting is to be assumed. 
 Under such conditions the task before the 
 teacher of mathematics, and quite as well 
 before the teacher of engineering, is to do 
 his utmost to train his student to think 
 logically and accurately about things. To 
 this end there seems to me nothing so effi- 
 cient as the solution of a large number of 
 carefully chosen problems. Indeed what is 
 one's life, if it be active, except meeting a 
 never ending succession of problems which 
 must be solved if success is to be gained? 
 If you can teach your student to take 
 vigorous hold of a problem, to first as- 
 semble all the facts which bear on the 
 question, then from the facts to reason 
 logically to a sound and safe conclusion, 
 you have started him well whether his aim 
 be engineering or otherwise. 
 
 Of transcendent importance is the 
 teacher, his personality, his attitude toward 
 his work, his knowledge of his students, 
 not as a class, but of each as a human 
 being. If we can procure the teacher who 
 can idealize his work, who can show sus- 
 tained enthusiasm for it and perform 
 cheerfully the drudgery we heard men- 
 tioned a few minutes ago, we can safely 
 leave detailed methods to him. Whatever 
 methods such a man adopts in the class- 
 room are likely to be effective. 
 
 Fred W. McNair 
 
 Michigan Coixege of Mines 
 
SCIENCE 
 
 47 
 
 THE TEACHING OF MATHEMATICS TO 
 STUDENTS OF ENGINEERING 1 
 
 WHAT IS NEEDED IN THE TEACHING OP 
 MATHEMATICS TO STUDENTS OP ENGINEER- 
 ING? (a) RANGE OP SUBJECTS; (6) EX- 
 TENT IN THE VARIOUS SUBJECTS; (c) 
 METHODS OP PRESENTATION; (d) CHIEF 
 AIMS. 
 
 By Calvin M. Woodward, Professor of 
 Mathematics and Applied Mechanics, 
 and Dean of the School of Engineering 
 and Architecture, Washington Univer- 
 sity. 
 
 I want to emphasize the point which Mr. 
 Scott has just touched on, and that is that 
 we often attempt too early to teach the 
 subjects that require mature and reflect- 
 ing minds. I want to tell you a story, a 
 true biography of some one you all know 
 of. He went through, in the city of New 
 York, the whole range of mathematics, in- 
 cluding analytic geometry and calculus. 
 He learned his formulae and definitions 
 and "passed" in some manner, but, he told 
 me, he did not know anything about them. 
 He believed he was a dunce, and whenever 
 he was required to make an intelligible 
 demonstration, he could not do it; his 
 teachers and his parents concluded that he 
 was a dunce in mathematics, and could 
 never do anything in it. He would have 
 gone through life with that notion, if some 
 one had not offered him an appointment 
 to West Point. He doubted his ability to 
 pass the entrance examination in arith- 
 metic; but his friends advised him to get 
 an arithmetic and study. He bought a 
 bcok and sat down and read the book 
 
 1 General discussion following the presentation 
 of four formal papers (see Science, July 17, 26, 
 31, 1908), and of the eight prepared discussions 
 (see Science, August 7 and 28, 1908). Presented 
 before Sections D and A of the American Associa- 
 tion for the Advancement of Science and the Chi- 
 cago Section of the American Mathematical So- 
 ciety, at the Chicago meeting, December 31, 1907. 
 
 through, and to his astonishment he found 
 it easy. He passed his examination with 
 flying colors. He entered West Point and 
 graduated at the head of his class in 
 mathematics, and is now at the head of a 
 high grade technical school. If it had not 
 been for the opportunity of going again 
 over his whole course of mathematics, he 
 would have gone to his grave thinking he 
 had no capacity for mathematical analysis. 
 That comes from poor or premature teach- 
 ing. 
 
 I am opposed to putting college mathe- 
 matics in high schools. Those young 
 people may get a glimmer of it, but they 
 get false impressions from it which are 
 hard to remove. I have been teaching 
 mathematics for forty years or more, and 
 have been teaching applied mechanics for 
 the same time. I taught Rankine for 
 twenty-five years. It has always been my 
 duty and my privilege to make my stu- 
 dents see what mathematics was good for. 
 And I want to defend the teachers of high 
 school and freshman mathematics from 
 what I think is unjust criticism. It is 
 charged they do not make their students 
 understand what mathematics is good for. 
 It is simply impossible for them to do so, 
 as I can do in mechanics. A man is very 
 fortunate who can teach mathematics and 
 then show what it is good for. I am old 
 enough to quote a little of my early experi- 
 ence. I am led to it by something Pro- 
 fessor Swain said in regard to mental 
 processes. There is nothing so valuable to 
 mathematical success as a clear grasp of 
 fundamental principles. When I was pre- 
 paring for college I gave all my time to 
 Latin and Greek. I had done all my 
 freshman mathematics and was reputed to 
 be strong on that branch, when a new 
 teacher came into the school who said, 
 "Here's a new book in intellectual arith- 
 metic, and I would like to have every stu- 
 dent in the school go through it." It was 
 
84 
 
 SCIENCD 
 
 fun for me, of course, but I went through 
 the book from A to Z; no other mathe- 
 matics that I ever studied did me so much 
 good. The teacher's maxim was, "Take 
 hold of the thread at the right end." 
 That was the secret of his splendid teach- 
 ing. I have applied that maxim to every 
 branch of mathematics I have ever studied 
 or taught. I have learned to take hold of 
 mathematics at the right end, and in a 
 measure I have taught my students to do 
 so. 
 
 By B. F. Groat, Professor of Mechanics 
 and Mathematics, School of Mines, Uni- 
 versity of Minnesota. 
 Most of the speakers have stated that 
 what they were about to say had already 
 been said by preceding speakers. I am 
 going to try to state a general principle I 
 have not heard clearly put since I came 
 here. During the lunch hour Professor 
 Slaught said that he had not heard a 
 single general pedagogical principle 
 brought out. I am going to take the 
 honor to myself, to give expression to what 
 seems to me to be a general educational 
 principle. 
 
 Mathematics is mathematics and engi- 
 neering is engineering. There is just as 
 much art, science or principle in the teach- 
 ing of mathematics as there is in the teach- 
 ing of engineering and these two subjects 
 should be distinguished, separated and 
 kept separate. If you are going to teach 
 engineering you must teach the pure prin- 
 ciples. If you are going to teach mathe- 
 matics you have got to teach pure mathe- 
 matics. Let it be pure or applied mathe- 
 matics, it is the principle involved which 
 must be taught. If this rule is not ad- 
 hered to we shall find ourselves teaching 
 something different from that which it was 
 intended to teach. 
 
 The principle is that the technical 
 courses in our engineering schools must be 
 
 separated from our general educational 
 courses. The technical courses are for the 
 purpose of fitting the man for a special 
 life work which is to come later on. The 
 general education which he should have, 
 by way of preparation, should precede his 
 technical course as far as possible. 
 
 The straight technical course should be 
 given as a course of two years extent, 
 while the general and preparatory sub- 
 jects should precede in a three- or four- 
 year course. 
 
 The University of Minnesota has adopted 
 a five-year engineering course. This is 
 along the lines I am recommending and I 
 prophesy that it will soon be extended to 
 other schools and separated into two parts. 
 
 Let your professor of engineering teach 
 engineering and your professor of mathe- 
 matics teach mathematics. That is the 
 general pedagogical principle I want to 
 announce. 
 
 By C. S. Howe, President, Case School of 
 
 Applied Science. 
 
 I have been very much interested in the 
 discussion of this subject because for 
 thirteen years I was a professor of mathe- 
 matics in an engineering school and dur- 
 ing the past five years I have been en- 
 deavoring to reconcile the differences be- 
 tween professors of mathematics and pro- 
 fessors of engineering. One thing in this 
 discussion which strikes me as very pecul- 
 iar is the sad lack of knowledge displayed 
 by the engineering professors as to what is 
 being done in mathematics in their own 
 schools. I believe from my experience and 
 from what I have seen in other institutions 
 that the professors of mathematics are 
 teaching mathematics most admirably as 
 mathematics, but they are not teaching 
 mathematics as a department of engineer- 
 ing. I do not believe that mathematics 
 should be taught as a department of engi- 
 neering. Mathematics is a science in itself 
 
SCIENCE 
 
 49 
 
 and should be taught by specialists in that 
 science if our students are to be trained 
 in the proper way. The professor of 
 mathematics has two duties to perform. 
 One is to teach his students the principles 
 of mathematics that is, to teach them to 
 reason and to understand why certain 
 processes are right and why others are 
 wrong. The student must also be taught 
 how to use his mathematics so that he can 
 solve any problem as soon as that problem 
 is expressed in mathematical terms. 
 Another duty of the mathematician is to 
 teach the student to be exact. Unless the 
 engineer is exact, unless he can obtain 
 definite and reliable results in his engi- 
 neering work, he can not succeed in his 
 profession. This accuracy must be very 
 largely taught in the mathematical depart- 
 ment and much of the time and care be- 
 stowed upon classes is for the purpose of 
 accomplishing this result. 
 
 I believe also that the professors of engi- 
 neering are teaching engineering thor- 
 oughly and well. The difficulty which we 
 are discussing to-day is not in the teaching 
 of mathematics alone nor in the teaching 
 of engineering alone, but in the connection 
 between the two. The technical student 
 is, I believe, taught pure mathematics well, 
 but when he enters the class in engineering 
 he finds that he has to deal with mathe- 
 matics under a new form that is, the par- 
 ticular engineering subject he is studying 
 must be translated into mathematical 
 terms and this is where he frequently 
 meets with great difficulty. The student 
 in algebra who has learned to solve equa- 
 tions of the first degree may have great 
 difficulty with problems involving equa- 
 tions of the first degree because he has not 
 learned to state the problems in mathe- 
 matical language. So the student who 
 begins electrical work finds certain prob- 
 lems containing known and unknown 
 quantities, but not yet expressed in mathe- 
 
 matical terms. Now I can not believe that 
 it is the duty of the professor of mathe- 
 matics to teach the student to express prob- 
 lems in the various branches of engineer- 
 ing in the form of equations or other 
 mathematical terms. In order to do this 
 it would be necessary for him to under- 
 stand all the various branches of engineer- 
 ing and it is manifestly impossible for 
 him to do this. The professor of civil 
 engineering understands the problems of 
 that subject and he should show the stu- 
 dent in his department how to express 
 these problems in such terms that the stu- 
 dent can deal with them mathematically. 
 The same may be said of each of the de- 
 partments of engineering. When the pro- 
 fessors of engineering have taught their 
 students to state the problems of their own 
 departments in mathematical language, 
 then the student who has had the course 
 in mathematics ought to be able to solve 
 the problems, and if he can not he has not 
 been taught his mathematics thoroughly or 
 so much time has elapsed since he studied 
 the subject that he has forgotten some 
 parts of it. 
 
 Again, I believe that the professor of 
 engineering should ascertain in a general 
 way how mathematics is being taught in 
 his institution and in just what form the 
 student is using certain terms so that he 
 may express his own problems in a way 
 familiar to the student. If, for instance, 
 in calculus the mathematical department 
 has been using derivations, the professors 
 of engineering in writing their problems 
 should use differential coefficients and not 
 attempt to express problems in terms of 
 differentials. I know from experience that 
 many professors of engineering do not do 
 this and their students are confused by a 
 difference of terms and not by a lack of 
 knowledge of the subject. It is evident 
 that the professors of engineering must 
 conform to the methods of the department 
 
 
50 
 
 SCIENCE 
 
 of mathematics because the department of 
 mathematics can use but one method while 
 the five or more departments of engineer- 
 ing might have several different methods. 
 It is obvious, then, that for the sake of 
 simplicity one method must be used and 
 that method must be the method of the 
 department of mathematics. 
 
 I also believe that the professor of 
 mathematics should occasionally confer 
 with the professors of engineering in order 
 to find out from them just what mathe- 
 matical subjects engineering students are 
 weak in and what subjects it is especially 
 desirable to have them well trained in and 
 to see that his students are taught these 
 things. Friendly conferences between the 
 departments are of great value and should 
 be encouraged by both the mathematicians 
 and the engineers. 
 
 By Clakence A. Waldo, Professor of 
 
 Mathematics, Purdue University. 
 
 In the table of hours for mathematics 
 in the various institutions cited by Pro- 
 fessor Townsend, the largest total stands 
 against Purdue. Also a whole semester is 
 assigned to trigonometry. Both of these 
 conditions are in a measure due to the fact 
 that we have recently passed through a 
 transitional period in which for engineers 
 solid geometry has been relegated to the 
 secondary schools. The first semester was 
 formerly divided between solid geometry 
 and trigonometry. Now it is wholly given 
 to the latter, while the second semester is 
 set aside to college algebra. Experience 
 shows that for the ordinary student college 
 algebra is more difficult than trigonometry 
 and this determines their order in our 
 program. 
 
 Placing trigonometry first and giving it 
 so much time has developed with us several 
 interesting facts. 
 
 1. Being easy to understand and having 
 
 interesting applications, it naturally fol- 
 lows secondary work. 
 
 2. While trigonometry is easy to under- 
 stand, yet to acquire facility in its use and 
 absolute mastery over it as a fundamental 
 science requires close and long-continued 
 study, yet the student, ambitious to become 
 an engineer, quickly sees that he must have 
 facility in this subject and mastery over it. 
 
 As a subject of study, therefore, at the 
 beginning of a young man's college career 
 it is well adapted to give power and to 
 instill habits of thoroughness, application, 
 concentration and mastery. 
 
 3. Engineers have been recommending 
 that a generous amount of time shall be 
 given to trigonometry, at the expense of 
 the calculus if necessary. 
 
 4. The subject is used to review and 
 emphasize much of the preparatory mathe- 
 matics, while it is also used to clear the 
 way for that which is to come. 
 
 Another peculiarity in which Purdue 
 stands almost alone we are quite prepared 
 to defend. We do not crowd the pure 
 mathematical work into the first two years, 
 much less into the first year, but give it 
 an hour less in the second year, than the 
 first, yet at the outset of the third year, 
 with his first course of calculus fairly 
 mastered, we have the student well pre- 
 pared to begin attack upon theoretical me- 
 chanics and kindred subjects. However, 
 with two hours a week during junior year 
 devoted to the further exploration of the 
 calculus carried on side by side with its ap- 
 plication to studies of a nature more or 
 less professional, like thermodynamics, the 
 student is likely to come finally into living 
 contact with calculus ideas. Through 
 three years, then, mathematical ideas are 
 held persistently and prominently before 
 the mind of the student, so that at the end 
 of that time the mental change which I 
 call the mathematical transformation is 
 quite complete. If you are intent upon 
 
SCIENCE 
 
 51 
 
 making a physical transformation by 
 which a weak man becomes robust and 
 powerful, you give one, two or three years 
 for the muscles to grow and the chest to 
 expand through long-continued and sys- 
 tematic exercise. Similarly the average 
 student does not become habitually mathe- 
 matical and exact in his thinking unless 
 you give him careful direction and devote 
 plenty of time to his development. The 
 man who uses his memory and copies 
 slavishly must disappear. In his place 
 must stand the man of trained intellect, 
 thoughtful, persistent, rich in expedients, 
 powerful in attack. To produce him there 
 are on the mathematical side two in- 
 dispensable requisites, thoroughness in the 
 fundamentals, and a sufficient time to 
 make the mathematical attack of a prob- 
 lem habitual and natural, and to give such 
 a control of and power in the use of the 
 tools of mathematics that the solution of a 
 problem of average difficulty shall be easy 
 and pleasurable. 
 
 In the required mathematical part of the 
 engineering courses at Purdue these are the 
 considerations that determine the distribu- 
 tion of the work in the four-year program, 
 and all of the time we are teaching not 
 alone the particular subject that happens 
 to be named in the curriculum but mathe- 
 matics. 
 
 Some years ago it was my fortune to 
 study descriptive geometry under Marx 
 and Von Derlin in Munich. They taught 
 their subject from the standpoint of the 
 mathematician rather than that of the 
 draftsman. They made their students 
 visualize geometric form in space and by 
 the use of that power discover methods of 
 solving on paper synthetic problems of 
 much difficulty. The German schools 
 teach descriptive geometry as a mathe- 
 matical subject, the American schools as a 
 body of problems to be solved by rule on 
 the drawing board. The former method 
 
 makes descriptive geometry the finest dis- 
 cipline of the four years' course; from 
 the other method little educational benefit 
 arises. Some years ago at the Rose Poly- 
 technic, where for a time we taught &e- 
 scriptive geometry in the German way, it 
 was not unusual to meet students who 
 declared enthusiastically that they got 
 more real good from this subject than from 
 anything else in their entire course. 
 
 I would ask the new committee to inquire 
 how and by whom descriptive geometry 
 should be taught? 
 
 By C. B. Williams, Professor of Mathe- 
 matics, Kalamazoo College. 
 The teachers of mathematics in the 
 small colleges of the middle west are pre- 
 paring many men for work in the better 
 technical schools. From our standpoint 
 there is substantial agreement between the 
 two representatives of the Massachusetts 
 Institute of Technology (Professors Wood 
 and Swain). They expressed themselves 
 so differently that one might easily fail to 
 see how closely they agree. Both want 
 longer and stronger courses in mathe- 
 matics in the secondary schools. I would 
 like to know the college teacher of mathe- 
 matics who does not agree with them. 
 They want more mathematics taught and 
 to have it taught better, to have longer and 
 more consecutive mathematical courses in 
 the secondary and primary schools. In 
 other words, the faculties of the technical 
 schools and colleges are working toward 
 the same end, that is, to have more effective 
 courses in primary and secondary mathe- 
 matics so that college students can do more 
 and better mathematical work. If we 
 could have properly prepared students, we 
 could turn out the kind of men the better 
 technical schools should have. 
 
 The engineers and teachers of engineer- 
 ing have insisted that the most necessary 
 qualification for a real engineer is that he 
 
52 
 
 SCIENCE 
 
 should be able to realize his mathematics, 
 to ' ' think mathematically, ' ' as they express 
 it. The mathematicians want the same 
 thing. We are trying to make use of and 
 to train the faculty of geometric intuition, 
 to emphasize the functional notion and to 
 develop functional thinking. There is 
 substantial agreement that the best way 
 to do this is through geometry, with per- 
 haps some help from elementary me- 
 chanics. It is true that sometimes we are 
 tempted to use too big and complicated 
 machines for little problems, but this is 
 only because we are attempting to develop 
 methods powerful enough to solve big 
 problems. 
 
 By J. B. Webb, Professor of Mathematics 
 
 and Mechanics, Stevens Institute. 
 
 Every practical problem requiring 
 mathematics for its solution consists of 
 three parts: 
 
 (a) An Analysis, which resolves the 
 problem into its elements, examines these 
 in the light of natural laws, rejects unim- 
 portant ones and defines the relations exist- 
 ing between those upon which the solution 
 depends. This involves the adoption or 
 discovery of methods of measuring the 
 elements, so that they may be expressed 
 quantitatively by symbols, and of the re- 
 duction of the relations between them to 
 the standard mathematical forms of ex- 
 pression. The result is a mathematical 
 statement of the problem by one or more 
 equations. 
 
 (&) A solution of the equations by which 
 the relations sought for between the quan- 
 tities are clearly expressed or the quanti- 
 ties put in proper form to have their values 
 calculated. 
 
 (c) The interpretation of the result, 
 which involves a translation of the same 
 from the mathematical language in which 
 it has been obtained into the original 
 
 language of the problem and a discussion 
 of the practical bearings of the same. 
 
 In conversation with a fellow mathe- 
 matician at this meeting he surprised me 
 by saying that he expected a problem to 
 be put into mathematical language before 
 it was submitted to him and I presume he 
 did not feel bound to interpret his results. 
 Now if "pure mathematicians" regard 
 practical problems in this way, engineers 
 and other practical men have just cause 
 for finding fault with "pure mathe- 
 matics," and to teach mathematics in this 
 way is to render it valueless to most stu- 
 dents. Personally I should refuse to 
 undertake a problem unless I made the 
 analysis and interpretation as well as the 
 solution. 
 
 In many if not in most problems the 
 analysis and interpretation are the main 
 parts. They require a broad knowledge of 
 practical conditions and of other sciences 
 and are far more interesting than the mere 
 solution, especially as they often bring into 
 play a large amount of ingenuity and in- 
 vention, as well as imagination and judg- 
 ment. A mathematician who can not 
 make the analysis and interpretation of a 
 problem is not to be trusted with the solu- 
 tion and an engineer who is fully compe- 
 tent to make them had better undertake 
 the solution himself or put the whole prob- 
 lem into the hands of a mathematician 
 fully competent to undertake it. 
 
 There is no excuse for a "pure mathe- 
 matician" remaining ignorant of the prac- 
 tical side of the problems he teaches, and 
 his mathematics will not be interesting or 
 trustworthy. Let him cultivate the ac- 
 quaintance of the truly educated engineer, 
 who will be only too glad to discuss prob- 
 lems with him and give him all the prac- 
 tical information he needs. But there are 
 too many engineers who are not truly edu- 
 cated and who know less about mathe- 
 matics than the "pure mathematician" 
 
SCIENCE 
 
 53 
 
 does about practical things, and they ought 
 to cultivate the acquaintance of the mathe- 
 matician and rub off the worst parts of 
 their ignorance before they attempt to 
 criticize the teaching of mathematics. But 
 it is much easier to find fault and say that 
 they never found any use for such and 
 such mathematical branches, when they 
 never gave them enough attention to make 
 them of any use. 
 
 Every mathematical teacher should teach 
 all three parts of a problem, but the aver- 
 age engineering student is so indifferent to 
 real progress and his limited time is so 
 taken up with other things that he may get 
 through his course knowing very little 
 about mathematics, no matter how well it 
 may be taught. 
 
 Students with fair ability that really 
 want to learn a particular subject can do 
 it even under indifferent teachers, but un- 
 less students exert themselves to learn, the 
 best teacher can not put knowledge into 
 them. Discuss the subject to the limit, 
 analyze and adjust the engineering courses 
 to a nicety, write new text-books, adopt 
 new systems and get new teachers and the 
 thing will remain about as it is; teachers 
 will teach and students will expect them 
 to, while only a few will learn, whether 
 the teacher expects them to or not. 
 
 By H. T. Eddy, Dean of the Graduate 
 School and Professor of Mathematics 
 and Mechanics, College of Engineering, 
 University of Minnesota. 
 Complaint has been made that in our 
 teaching of mathematics we do not pay due 
 attention to psychological and pedagogical 
 principles. I want to consider for a mo- 
 ment the application of two of these 
 principles. 
 
 First, it is necessary for the engineering 
 student to have an ample undergraduate 
 course in mathematics, and such an ex- 
 tended drill in and habitual acquaintance 
 
 with its processes that when he has for- 
 gotten nine tenths of it, just as he will of 
 this and all other subjects which he studies 
 in college, what remains with him will be 
 a sufficient equipment in this line for his 
 professional career. In other subjects his 
 residuum of knowledge is easily refreshed 
 and increased. Not so in mathematics. 
 The stock of mathematical knowledge of 
 which he is easily master on entering his 
 profession will practically be the end of 
 his attainments in that direction. Ke- 
 stricting the course in mathematics to bare 
 essentials is suicidal, for of it a small frac- 
 tion only will remain as a permanent pos- 
 session, and that fraction is likely to be 
 smaller, the smaller the amount originally 
 attempted. 
 
 Second, the teacher of mathematics is 
 prone to think that a clear presentation of 
 mathematical truth on his part, and a 
 logical demonstration by the student, are 
 all that is required in this subject. But 
 important as these things assuredly are, 
 they are insufficient to produce successful 
 results. The question is one in which 
 human interest is really of more im- 
 portance than logic, for mathematical 
 knowledge can not be successfully im- 
 parted unless genuine interest on the part 
 of the student can be in some way aroused. 
 It goes without saying, that the teacher 
 must first of all have that interest himself 
 or he ceases to be a fit teacher. How he 
 will awaken interest in his pupil depends 
 upon his own personality. Many do this 
 by help of problems which elucidate and 
 apply the principles. Just here lies the 
 reason for the usual inability of pro- 
 fessional engineers to teach mathematics. 
 They have no interest in mathematics 
 itself. It is the engineering problem alone 
 that interests them. To this matter of 
 interest, or the lack of it, may be traced 
 the failure which is apt to attend the sepa- 
 ration of classes into divisions according 
 
54 
 
 SCIENCE 
 
 to scholarship, for in that case the divisions 
 made up of poor students lose the impetus 
 to be derived from the interest which the 
 good students exhibit in their work in 
 which all participate to some degree. 
 
 By S. M. Barton, Professor of Mathe- 
 matics, University of the South. 
 While standing here in the heart of the 
 modern, bustling city of Chicago, and 
 listening to this discussion, my mind goes 
 back to the ancient city of Tarentum and 
 her distinguished governor, Archytas. 
 Archytas, while an able mathematician, 
 was too practical, as we learn, to suit the 
 ideas of the Platonic School, who objected 
 to his mechanical solutions of certain 
 mathematical problems as interfering with 
 pure reasoning. Now, while I take an im- 
 mense interest in applied mathematics 
 (what mathematician at this day would 
 not?) yet I confess to a feeling of sym- 
 pathy with Plato in his condemnation of 
 Archytas. At any rate I wish to enter 
 my protest against a possible tendency to 
 degrade mathematical teaching to the 
 memorizing of thumb-rules, and to urge the 
 advantage of a strong backbone of pure 
 mathematics in our engineering courses. 
 
 I read with interest a paper presented 
 at the Ithaca meeting of the Society for 
 the Promotion of Engineering Education, 
 by Professor Arthur E. Haynes of the 
 University of Minnesota, in justification of 
 the use of the expression "engineering- 
 mathematics." I must say I was at first 
 somewhat shocked by the expression, for 
 I had always believed that mathematics is 
 mathematics take it when and where you 
 will. While I would agree heartily with 
 much that Professor Haynes said, and I do 
 not doubt that his courses are interesting 
 and instructive, yet I question the wisdom 
 of drawing any sharp distinction in the 
 college curriculum between the mathe- 
 
 matics given to the engineering student 
 and to any other class of students. 
 
 I find myself differing absolutely from 
 the gentleman from the Massachusetts In- 
 stitute of Technology, who apparently sees 
 no beauty, much less utility, in the higher 
 branches of pure mathematics. How Pro- 
 fessor Woods, who has, by the way, written 
 such a sound text-book on mathematics, 
 can live amicably in the same state, much 
 less in the same college, as his engineer- 
 colleague, I am at a loss to understand- 
 perhaps they have an occasional fight. 
 But, joking aside, there is a dangerous 
 tendency to adopt rules (slide and mental) 
 and short-cut, approximate solution to the 
 utter exclusion of rigid proofs. Is it wise 
 to make a mere machine of the young engi- 
 neer, even if thereby he becomes rich 
 faster or grows poor less slowly ? I freely 
 admit, however, that too much theory 
 would be disastrous, and that there is great 
 room for improvement in the teaching of 
 mathematics. The student should be 
 taught how to use his mathematics, and the 
 existing gap between theory and practise 
 be bridged. While affording every pos- 
 sible facility to the student for making ex- 
 periments, collecting data, becoming expert 
 in handling instruments, making calcula- 
 tions, etc., I urge that we give them, one 
 and all, a good rigid course in pure mathe- 
 matics. 
 
 By Arthur E. Haynes, Professor of En- 
 gineering-Mathematics, University of 
 Minnesota. 
 
 I have been called upon, by name, to 
 defend the use of the term, "Engineering 
 Mathematics." The justification of the 
 term will be found in my paper on the 
 subject in Volume XIV. of the Proceed- 
 ings of the Society for the Promotion of 
 Engineering Education. As the paper 
 was not read before this association, many 
 
SCIENCE 
 
 55 
 
 of the members present are not acquainted 
 with its contents. 
 
 In brief, the reasons there given for the 
 use of the term are: 
 
 (a) Because of the main object of the 
 study of mathematics in engineering 
 courses, viz : its use as a tool. 
 
 (6) Because of the proper method of 
 teaching the mathematics of such courses. 
 
 (c) Because of the content of the mathe- 
 matics of such courses. 
 
 It is not a degradation of mathematics 
 to make it practical, it is rather an added 
 glory. It is as justifiable to use this term 
 as to use the corresponding terms agricul- 
 tural chemistry, agricultural botany, engi- 
 neering drawing, etc. We do not degrade 
 chemistry or botany or drawing by the 
 use of these terms: but their employment 
 is justified by the objects of the study, by 
 the methods required in teaching them and 
 by their content, as in mathematics. 
 
 It has been suggested that a less thor- 
 ough study of mathematics is advocated. 
 In reply to this, may I quote from an 
 article in Volume VIII. of the Proceed- 
 ings of the Society for the Promotion of 
 Engineering Education, on "The Teaching 
 of Mathematics to Engineering Students," 
 where in speaking of such teaching I said : 
 
 (a) It should be of such a character as 
 to produce an enduring stimulating effect 
 upon the mind of the student. 
 
 (&) It should give the student the power 
 to properly interpret mathematical lan- 
 guage, and to accurately and skillfully use 
 it. 
 
 (c) To secure these results, the teaching 
 must be based upon a proper order of 
 studies and carried forward in a rational, 
 intelligent manner. 
 
 By Arthur S. Hathaway, Professor of 
 
 Mathematics, Rose Polytechnic Institute. 
 
 In a paper on "Pure Mathematics for 
 
 Engineering Students," published in the 
 
 Bulletin for March, 1901, I expressed opin- 
 ions which coincide with those given here 
 to-day. I then said that instruction in 
 mathematics for engineering students 
 should have two objects (1) to develop an 
 engineering mind, and (2) to develop 
 mathematics as an instrument of research 
 for the engineer. I came to these conclu- 
 sions at that time as a result of inquiries 
 made of graduates of several institutions, 
 who were in engineering practise, and of 
 their employers. From the latter, I have 
 had the statement that it is inadvisable to 
 place a man in the higher positions in engi- 
 neering who has not had a good mathe- 
 matical training, especially, in the cal- 
 culus, which, they assert, develops those 
 modes of thought which are necessary to 
 the engineer. 
 
 I wish to call your attention to the fact 
 that the fifty-four hours of analytical 
 dynamics credited to Rose Polytechnic In- 
 stitute on this chart are spent on applied 
 calculus. There is a regular course of 
 one hundred and forty-four hours in 
 Rankine not mentioned here, which is 
 given by my colleague, Professor Gray. 
 In applied calculus we take up problems 
 which require the use of the calculus, such 
 as motions in constant, elastic and central 
 fields, the bending of beams, the twisting 
 of shafts, problems in electricity, in chem- 
 istry, etc. We take problems gathered 
 from all sources, text-books, magazines, 
 engineering professors, and discuss them in 
 the class-room, with special reference to 
 the analysis and its mode of application. 
 
 By Edward V. Huntington, Assistant 
 Professor of Mathematics, Harvard Uni- 
 versity. 
 
 I desire to call attention to the fact that 
 besides the analogy of mathematics as a 
 tool or instrument, there is also the perhaps 
 more significant analogy of the mathe- 
 matician as the discoverer of quantitative 
 
56 
 
 SCIENCE 
 
 relations which already exist in the prob- 
 lems themselves. Logarithmic relations 
 between varying quantities, for instance, 
 are not dragged into the problem from 
 some artificial tool-chest, but are already 
 present in the problem, and are analyzed 
 out of the problem much as the precious 
 metal is analyzed out of the ingot by the 
 metallurgist. The practical mathema- 
 tician is simply a scientist specially trained 
 to perceive the quantitative aspects of 
 physical phenomena. 
 
 By Donald F. Campbell, Professor of 
 Mathematics, Armour Institute of Tech- 
 nology. 
 
 We have had a number of good ideas 
 set before us in the last two days ideas 
 which we ought to make an effort to 
 crystallize. I think that the present time 
 is the psychological moment to have a com- 
 mittee appointed to draw up a report on 
 mathematics for colleges of engineering. 
 This report perhaps might be in the nature 
 of a symposium, but it would be especially 
 valuable if it considered in detail the sub- 
 jects which should be emphasized in a 
 course in mathematics for engineering stu- 
 dents. These, however, are merely sug- 
 gestions. I would not hamper the com- 
 mittee in their deliberations by outlining 
 any particular course which they should 
 pursue. The only condition which I would 
 impose is that the committee be representa- 
 tive enough that all of us can look towards 
 their report with the utmost confidence. 
 
 I would move that the chairman be em- 
 powered to appoint a committee of three, 
 these three to increase their number to 
 fifteen, chosen from among the teachers of 
 
 mathematics and engineering and the 
 practising engineers, these fifteen to con- 
 stitute a committee authorized by this 
 meeting to make such a report on mathe- 
 matics for colleges of engineering as in 
 their opinion will be of service to teachers 
 in such institutions, and to submit this 
 report when completed to the Chicago Sec- 
 tion of the American Mathematical So- 
 ciety. 1 
 
 1 Professor Campbell's motion, as amended by 
 Professor Magruder, requires the Committee of 
 Fifteen to report to the Society for the Promotion 
 of Engineering Education at its meeting in the 
 summer of 1909. 
 
 The chairman appointed Professors Huntington, 
 Williams and Townsend as the committee of three. 
 See pages 2 and 3 of this report. 
 
 The committee of three appointed the following 
 persons as members of the Committee of Fifteen: 
 E. V. Huntington, Harvard University, Cam- 
 bridge, Mass., Chairman; Philip R. Alger, U. S. 
 Naval Academy, Annapolis, Md.; D. F. Campbell, 
 Armour Institute of Technology, Chicago, 111.; 
 E. A. Engler, Worcester Polytechnic Institute, 
 Worcester, Mass.; C. N. Haskins, University of 
 Illinois, Urbana, 111.; C. S. Howe, Case School of 
 Applied Science, Cleveland, Ohio; Emil Kuichling, 
 New York, N. Y.; W. T. Magruder, Ohio State 
 University, Columbus, Ohio; Ralph Modjeski, 
 Chicago, 111.; W. F. Osgood, Harvard University, 
 Cambridge, Mass.; C. S. Slichter, University of 
 Wisconsin, Madison, Wis.; C. P. Steinmetz, Sche- 
 nectady, N. Y. ; G-. F. Swain, Massachusetts Insti- 
 tute of Technology, Boston, Mass.; E. J. Town- 
 send, University of Illinois, Urbana, 111.; F. E. 
 Turneaure, University of Wisconsin, Madison, 
 Wis.; C. A. Waldo, Washington University, St. 
 Louis, Mo.; G. S. Williams, University of Mich- 
 igan, Ann Arbor, Mich.; C. M. Woodward, Wash- 
 ington University, St. Louis, Mo.; R. S. Wood- 
 ward, Carnegie Institution, Washington, D. C; 
 Alexander Ziwet, University of Michigan, Ann 
 Arbor, Mich. 
 
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