11 
 
 Southern Branch 
 of the 
 
 (niversity of California 
 
 Los Angeles 
 
 Form L 
 
 CIA
 
 This book is DUE on the last date stamped below 
 
 OCT 2 
 MAR 2 2 1931 
 
 Form L-9-15m-8,'26
 
 MATHEMATICS FOR 
 
 FRESHMEN STUDENTS OF 
 
 ENGINEERING 
 
 THEODORE LINDQUIST
 
 o-hc llmurraitij nf (thtragn 
 
 FOUNDED BY JOHN D. ROCKEFELLER 
 
 MATHEMATICS 
 
 FOR FRESHMEN STUDENTS 
 
 OF ENGINEERING 
 
 A DISSERTATION 
 
 SUBMITTED TO THE FACULTY 
 
 or THE 
 OGDEN GRADUATE SCHOOL OF SCIENCE 
 
 IN CANDIDACY FOR THE DEGREE OF DOCTOR OF PHILOSOPHY 
 
 DEPARTMENT OF MATHEMATICS 
 7 
 
 BY 
 
 THEODORE UNDQUIST 
 
 1911
 
 QA\\ 
 
 CONTENTS. 
 
 Chapter I. Historical Summary 1-16 
 
 I. Periods of Engineering History I 
 
 II. First Period to 1824 I 
 
 1 ) The Apprenticeship System 1-2 
 
 2) The United States Military Academy 2-3 
 
 III. Conditions in Engineering at the Transition from 
 
 the First to the Second Period 3 
 
 1) Development of Natural Resources 3 
 
 2) Transportation 
 
 i) Canals 3 
 
 ii) Railroads 4 
 
 iii) Steamboats and Bridges 4 
 
 IV. Second Period 1824 to 1862 4-5 
 
 1 ) Purely Technical Institutions 5 
 
 i) Rensselaer Polytechnic Institute 5-7 
 
 2) Literary-Technical Institutions 7 
 
 i) Union College 7-8 
 
 ii) Lawrence Scientific School of 'Harvard 
 
 University 8 
 
 iii) Brown University 8-9 
 
 iv) Cumberland University 9 
 
 v) Sheffield Scientific School of Yale Uni- 
 versity 9 
 
 vi) University of Michigan 9-10 
 
 3) Military and Naval Institutions 10 
 
 i) Virginia Military Institute IO-H 
 
 ii) United States Naval Academy ii 
 
 V. Conditions in Engineering at the Transition from 
 
 the Second to the Third Period 1 1 
 
 1 ) Transportation 12 
 
 2) Invention 12
 
 iv MATHEMATICS FOR ENGINEERS. 
 
 VI. Third Period 1862 to the Present 12 
 
 1) Morrill Land Grant Act and Subsequent Acts 12-13 
 
 2) Colleges of the Third Period 13 
 
 3) Growth of the Engineering Colleges 13 
 
 VII. Sketch of Changes in Mathematics in Engineering 
 
 Colleges 14 
 
 1) Changes in Entrance Requirements 14-15 
 
 2) Collegiate Curriculum 15 
 
 3) General Trend 15 
 
 VIII. Progress of Instruction in Mathematics 16 
 
 IX. Summary 16 
 
 Chapter II. Engineering Colleges in the United States in 
 
 1908 17-49 
 
 I. Contents 17 
 
 II. Sources of Data 17 
 
 III. Explanation of Tables 18-20 
 
 IV. Tables of Data on Engineering Colleges 21-41 
 
 V. Collected Results of the Tables 42 
 
 1 ) Growth of Engineering Activity 42 
 
 2) Entrance Requirements 
 
 i) Time for Various Subjects 43 
 
 ii) Requirements of Majority of Institutions 44 
 
 3) Curriculum 
 
 i) Position of Subjects 44~45 
 
 ii) General Trend of Position of Subjects in 
 
 Curriculum 45 
 
 iii) Comparison of Time for Various Sub- 
 jects 46 
 
 4) Percentage of Time Given to Mathematics. . 46 
 
 5) Discrepancies and Their Causes 46-49 
 
 VI. Percentage of Time Given Various Groups of 
 
 Work 49 
 
 VII. Summary 49 
 
 Chapter III. Recent Modifications in the Work in Mathe- 
 matics for Students of Engineering 50-66 
 
 I. Contents of the Chapter 50 
 
 II. General Nature of the Modified Work in Mathe- 
 matics 50-51
 
 CONTENTS. v 
 
 III. Sources of Data 51 
 
 i ) Questionaire Letter 52 
 
 IV. Synopsis of Principal Features of Modified Work 52 
 Institution No. i Coherent Course of Essential 
 
 Elements 5 2 ~53 
 
 i) Study and Use of Curves 53 
 
 ii) Early Introduction of the Calculus 53 
 
 Institution No. 2 Maximum Usable Mathemat- 
 ics in First Year 53~54 
 
 i) Origin of Course 54 
 
 Institution No. 3 Mathematical Schedule a Co- 
 herent Whole 54 
 
 i) College Algebra 54-55 
 
 ii) Analytic Geometry and the Calculus. ... 55 
 
 iii) Material Used 55~56 
 
 Institution No. 4 A Modified Coherent Sched- 
 ule of Subjects 56 
 
 Institution No. 5 Arrangement of Mathematics 
 
 Courses for Future Work . . 57 
 Institution No. 6 Combined Course in Algebra 
 
 and Trigonometry 57 
 
 i) Mathematics as a Tool 57 
 
 Institution No. 7 Algebra throout the Course in 
 
 Mathematics 57 
 
 Institution No. 8 Early Calculus . 58 
 
 Institution No. 9 Mathematics as Basis of Tech- 
 nical Training 58 
 
 Institution No. 10 Applied Contrasted with Pure 
 
 Mathematics 58-59 
 
 i) Problems from "Real Material" 59 
 
 ii) Laboratory Work 59 
 
 Institution No. n Nature of Applied Mathe- 
 matics 59-6o 
 
 i) Graphic Algebra and Trigonometry 60 
 
 ii) Analytic Geometry and the Calculus. ... 60 
 
 iii) Problems from Engineering 60 
 
 Institution No. 12 Mathematical Power and For- 
 mal Course 61 
 
 i) Problems from Engineering 61 
 
 Institutions 13 and 14 Engineering Problems. ... 61
 
 vi MATHEMATICS FOR ENGINEERS. 
 
 Institution No. 15 Extra Problems forEngineers 61 
 Institution No. 16 Division into two Classes: 
 
 Analysis and Computat'on. 61-62 
 
 i) Contents of Course in Computation. ... 62 
 
 ii) Freshman Calculus 62 
 
 Institution No. 17 Problems and Computation 
 
 Devices I 62 
 
 Institution No. 18 Computations 62 
 
 Institution No. 19 Computation Purpose 6$ 
 
 Institution No. 20 Laboratory Work 63 
 
 i) Instruments and Material Used 63 
 
 ii) Problems Considered 63-64 
 
 iii) Working's of the Course 64 
 
 iv) Object of the Course 64 
 
 v) Results 64-65 
 
 V. Summary 65-66 
 
 Chapter IV. Current Thoughts on Vital Questions 67-96 
 
 I. Mode of Obtaining the Data 67 
 
 1 ) Questionaire Letter 67-70 
 
 2) Replies 70 
 
 II. Tabulation of Data 70 
 
 III. Entrance Examinations 71 
 
 i) A Substitute for the Entrance Examination. . 71-72 
 
 IV. Specific Needs and Deficiencies 72 
 
 1 ) Trigonometry 
 
 i) Needs 73~74 
 
 ii) Deficiencies 74~75 
 
 2) Algebra 
 
 i) Fundamentals 75~76 
 
 ii) Deficiencies 7^-77 
 
 3) Analytic Geometry 
 
 i) Object 77-78 
 
 ii) Importance of Topics 78-79 
 
 iii) Deficiencies 79 
 
 iv) Derivatives and Integrals in Analytic 
 
 Geometry 80 
 
 4) Aids and Remedies for Above 80-82-
 
 CONTENTS. vii 
 
 V Modification in Mathematics for Engineers ....... 
 
 i ) Engineering Problems .................... 
 
 i) Advisability ......................... 83 
 
 ii) Nature of Such Problems .............. 84-85 
 
 2) Advisability of a Course in Computation ..... 85-86 
 
 3) Segregation of Engineers in Mathematics. . . . 
 
 i) Advisability ...................... ... 86 
 
 ii) Reasons ............................ 86-87 
 
 4) Special Texts ............................ 87 
 
 VI. Pedagogic Questions .......................... 
 
 1) Quizzes ................................. 88 
 
 2) Modes of Conducting Classes .............. 88-90 
 
 VII. General Needs and Deficiencies ................. 90 
 
 i ) General Mathematical Ability ............... 
 
 i) Need .............................. 90-91 
 
 ii) Deficiencies ......................... 92 
 
 iii) Remedies ........................... 92-94 
 
 2) Relative Importance of Topics .............. 94 
 
 VIII. Engineering vs. a General Education ............ 94~95 
 
 IX. Summary .................................... 
 
 Chapter V. Present Needs and Tendencies ............. 97-121 
 
 I. Contents of Chapter .......................... 97 
 
 II. Preparatory Mathematics ...................... 97 
 
 i ) Nature of Work Needed ................... 97-9$ 
 
 i) Algebra to be Treated as a Study of 
 
 Equations ........................ 9&~99 
 
 2) Deficiencies Found ....................... 99 
 
 3) Cause of Deficiencies ..................... 99 
 
 i) School Management ................. 100 
 
 ii) The Teacher ........................ 100 
 
 4) Suggested Remedies ...................... 100-101 
 
 i) Requirement of Entrance Examination 
 
 in Mathematics .................... 101 
 
 ii) Substitute for Entrance Examinations. . 101 
 
 iii) Demands upon the High Schools ....... 101-102
 
 viii MATHEMATICS FOR ENGINEERS. 
 
 III. Collegiate Mathematics 102 
 
 1) What Should be Taught 102 
 
 2) How It Should be Taught 102 
 
 3) Specific and General Needs and Difficulties. . 102-103 
 
 4) Specific Needs and Difficulties 103 
 
 i) Trigonometry 
 
 a) Reason for Place 'in Curriculum... 103 
 
 b) Needs 103-104 
 
 c) Deficiencies 104 
 
 d) Suggestions for Improvement 104 
 
 e) Trigonometry to be Taught from 
 
 the Utilitarian Standpoint 104-105 
 
 f ) Suggested Course 105 
 
 ii) Algebra 
 
 a) Status in Engineering Colleges. . . . 105 
 
 b) Needs and Difficulties 106 
 
 c) Suggested Improvements 
 
 I) Teach Elements 106 
 
 II) Principles of Algebra to be 
 
 Taken up When Needed. . 107 
 
 III) Algebra for Future Use.... 107-108 
 
 iii) Analytic Geometry 
 
 a) As a Mathematical Language 108-109 
 
 b) Illustrative Problems 109 
 
 c) Other Devices 1 10-1 1 1 
 
 d) Needs and Difficulties in 
 
 iv) The Calculus 
 
 a) Principles Taught 111-112 
 
 b) Advantages 112 
 
 v) Combined Course in Algebra, Coordinate 
 
 Geometry and Elements of the Calcu- 
 lus i 12-1 13 
 
 vi) Work in Computation 
 
 a) Mode of Presentation 113-114 
 
 b) Degrees of Accuracy 114-115 
 
 5) General Needs and Difficulties 115 
 
 i) Cooperation with the Professional De- 
 partments 115-116
 
 CONTENTS. ix 
 
 ii) Problems 
 
 a) Their Use 1 16 
 
 b) Kinds of Problems 116-117 
 
 c) Problems to Develop Mathematical 
 
 Thought 117 
 
 6) Pedagogic Considerations 117 
 
 i) Modes of Instruction 117-118 
 
 ii) Suggested Combinations of Modes 118-119 
 
 iii) Special Texts in Mathematics 119-120 
 
 iv) The Profession of Teaching 120 
 
 V. Summary 121 
 
 Chapter VI. Conclusions 122-130 
 
 I. Contents of Chapter 122 
 
 II. Origin and Growth of the Engineering Colleges. . 122 
 III. Progress of Work in Mathematics and Present 
 
 Needs 123 
 
 1 ) Entrance Requirements 
 
 i) Their Scope 123 
 
 ii) Present Needs 123-124 
 
 iii) Remedy 124 
 
 iv) Usefulness of Variation in Entrance Re- 
 quirements 124-125 
 
 2) Collegiate Curriculum 
 
 i) Progress 125 
 
 ii) Suggested Improvements 126 
 
 3) Instruction 
 
 i) Progress and Present Needs 126-127 
 
 ii) Special Phases to be Noted 127 
 
 a) General Aims 127 
 
 b) Fundamental Principles 127 
 
 c) Problems 127-128 
 
 d) Computation 128-129 
 
 4) Too Much Required of the Mathematical De- 
 
 partment 129 
 
 III. Mathematics for Freshmen Students of Engineer- 
 ing: 129-130 
 
 Bibliography 131-135
 
 MATHEMATICS FOR FRESHMEN 
 STUDENTS OF ENGINEERING 
 
 -2, 4^< 7 
 CHAPTER I. 
 
 HISTORICAL SUMMARY. 
 
 I. PERIODS OF ENGINEERING HISTORY. 
 
 The history of engineering education in the United States falls 
 into three periods separated by two notable events. The first of 
 these is the establishment, in the first part of the nineteenth century, 
 of courses in civil as distinct from military engineering ; the second 
 the passage of the Land Grant Act in 1862. While in this case, as 
 in many others, succeeding periods blend into each other without 
 any sharp line of demarkation, yet these three periods are very real 
 and distinct. 
 
 II. FIRST PERIOD TO 1824. 
 
 f 
 
 During the first period there were two sources of engineering 
 education ; the apprenticeship system and the United States Military 
 Academy. 
 
 i) THE APPRENTICESHIP SYSTEM. 
 
 The apprenticeship system was confined chiefly to New Eng- 
 land; it continued in existence until the Civil War and furnished 
 most of the engineering education up to about 1835.
 
 2 MATHEMATICS FOR ENGINEERS. 
 
 Instruction consisted of such information as the apprentice 
 could "pick up" through questions in connection with field and office 
 work, for which privilege he paid one hundred dollars per year for 
 a term of three years. For his work in the field he received twelve 
 and one-half cents per hour as compensation, but before the begin- 
 ning of the Civil War nothing was paid, as a rule, for office work. 
 After that date all work was paid for at the rate mentioned. The 
 above conditions were quite uniformly prevalent. This system 
 presented the extreme of "practical" instruction to the exclusion 
 of all theory. 
 
 2) THE UNITED STATES MILITARY ACADEMY. 
 
 The United States Military Academy was one of the main 
 sources of engineering education during this period since many of 
 its graduates took up work along civil lines, and remained a factor in 
 engineering education until 1850. 
 
 As early as September, 1776, a committee of the Continental 
 Congress recommended a bill for a "Continental Laboratory and 
 Military Academy." Later Washington, as president, several times 
 urged Congress to establish such an institution. Nothing was done, 
 however, until 1802 when the bill authorizing its establishment was 
 passed, March 16. July 4, of the same year, the Academy was 
 opened at West Point with ten cadets. 
 
 At first the instruction in mathematics was very limited extend- 
 ing only through Hutton's first volume. By 1810 instruction included 
 arithmetic, logarithms, elementary algebra, geometry, trigonometry, 
 mensuration of heights and distances, planimetry, stereotomy, sur- 
 veying and conic sections. In 1816 a four years' schedule of studies 
 was adopted in which the mathematical work "embraced the follow- 
 ing branches, namely: the nature and construction of logarithms, 
 and the use of the tables; algebra, to include the solution of the 
 cubic equation, with all preceding rules ; geometry, to include plane 
 and solid geometry, also ratio and proportion, construction of 
 geometrical problems, application of algebra to geometry, practical 
 geometry on the ground, mensuration of the planes and solids ; plane 
 trigonometry, with the application to surveying, and the mensuration 
 of heights and distances ; spherical trigonometry, with its application 
 to spherical problems : the doctrine of infinite series ; conic sections
 
 HISTORICAL SUMMARY. 3 
 
 with their application to projectiles; fluxions to be taught at the 
 option of the professor and students." "Fluxions," however, were 
 seldom given. This curriculum also included "A complete course 
 in engineering." 1 From 1816 there has been a gradual growth in the 
 amount of mathematics offered, up to the present schedule, which 
 was put into operation with the class entering March, icjoS. 2 
 
 III. CONDITIONS IN ENGINEERING AT THE TRANSI- 
 TION FROM THE FIRST TO THE 
 SECOND PERIOD. 
 
 A brief glance at conditions in the engineering world about the 
 time of transition from the first to the second period may be of 
 interest. 
 
 1) DEVELOPMENT OF NATURAL RESOURCES. 
 
 The output of iron had risen from 54,000 tons in 1810 to 
 165,000 tons in 1830 and to 347,000 tons in 1840. Coal mining was 
 begun about 1820 with an output of 65,000 tons for that year. Pro- 
 gress in agricultural developments gave rise to the manufacture of 
 farm implements; plows and mowers, of quite good pattern, were 
 produced as early as 1820. 
 
 2) TRANSPORTATION (i) Canals. 
 
 The first attempt at a solution of the transportation problem in 
 this country was the building of canals. As early as 1793 a short 
 canal was constructed in order to make a contour around the falls 
 in the Connecticut river at South Hadley, Massachusetts. The first 
 freight and passenger canal, which was three miles in length, was 
 begun the same year and finished in 1804, but has not been in use 
 since 1850. The principal canal building era lasted from 1810 to 
 about 1840. Only one canal of importance, the Illinois and Mich- 
 igan, in course of construction from 1836 to 1848 was finished after 
 this time. 
 
 1 Bvt. Maj.-Gen. G. W. Cullum, Biographical Register of the Officers and Gradu- 
 ates of the U. S. Military Academy with the Early History of the United States Mili- 
 tary Academy. Boston and New York, 1891. 
 
 * Chap. II, p. 41.
 
 4 MATHEMATICS FOR ENGINEERS. 
 
 () Railroads. 
 
 Canal building decreased as a result of the coming of the rail- 
 road. The first railroad, three miles in length, was constructed in 
 1826 at a cost of $34,000 and used horses as motive power. In 1829 
 the first English made locomotive was imported and used on a six- 
 teen mile track. A year later the first American locomotive was 
 constructed. The second American locomotive was -used by the 
 South Carolina Railroad which ran between Charleston and Ham- 
 burg. On its first passenger trip, made January 15, 1831, it attained 
 a speed of 15 to 20 miles per hour. The following table gives the 
 mileage of railroads up to 1900: 
 
 1830 23 miles 
 
 1831 95 miles 
 
 1832 229 miles 
 
 1835 1,098 miles 
 
 1840 2,818 miles 
 
 1845 4,633 miles 
 
 1850 9,021 miles 
 
 1860 30,626 miles 
 
 1870 52,922 miles 
 
 1900 190,082 miles 
 
 (iii) Steamboats and Bridges. 
 
 The application of steam to water traffic was made by Robert 
 Fulton with his Clermont, as early as 1807. It was not used for 
 transatlantic ships, however, before 1839. The building of tunnels 
 and bridges which had just begun at this time also greatly faciliated 
 traffic. 
 
 IV. SECOND PERIOD 1824 TO 1862. 
 
 During the first years of the nineteenth century a great step 
 forward was made in education in the United States by the 
 introduction of laboratory instruction. The principle involved was 
 further developed in the organization of courses giving work in 
 civil engineering. The Rensselaer School, founded by Stephen Van 
 Rensselaer in 1824, was the pioneer in this movement. The date of 
 its founding, therefore, serves to indicate the beginning of the second
 
 HISTORICAL SUMMARY. 5 
 
 period, although this period was not fully established until some 
 years later. Following the establishment of the Rensselaer School 
 several of the literary institutions then existing organized courses 
 in engineering. As a consequence there are to be found two distinct 
 classes of technical institutions which will here be considered briefly ; 
 purely technical institutions and literary-technical institutions. 
 
 i) PURELY TECHNICAL, INSTITUTIONS. 
 
 (t) Rensselaer Polytechnic Institute. 
 
 The Rensselaer School was the only purely technical institution 
 founded during this period. Concerning its object, Stephen Van 
 Rensselaer wrote Rev. Samuel Blatchford of Lansingburg, October 
 5, 1824, that he had established a school for those "who may choose 
 to apply themselves in application of science to the common purposes 
 of life. My object is to qualify teachers for instructing the sons 
 and daughters of farmers and mechanics, by lectures and otherwise 
 in the application of experimental chemistry, philosophy and natural 
 history to agriculture, domestic economy, the arts and manufacture." 
 The institution has, however, devoted itself to the direct preparation 
 of its students for technical occupations. The original name, Rens- 
 selaer School, was changed in 1833 to Rensselaer Institute. Instruc- 
 tion in civil engineering was first offered in the annual announce- 
 ment for 1828 but no special course was established before 1835. 
 The institution was again reorganized in 1851, when the present 
 name, Rensselaer Polytechnic Institute, was adopted. At this time 
 the course of study was also extended from one to three years. In 
 1852 there was added a preparatory course of one year, which was 
 abolished in 1862 when the course was lengthened to four years. 
 
 In the mode of instruction the Rensselaer School made a very 
 decided departure from that of the other institutions existing at the 
 time of its organization. The mode was in substance as follows: 
 each new topic was taken up by the professor in charge in a lecture 
 upon which the class was quizzed the following day, the class was 
 then divided into sections which met in various smaller rooms. At 
 the meetings of each of these sections a student would be selected to 
 repeat the kcture of the day previous. Although an increased 
 number of students has compelled a change in the form of carrying 
 on the work to that of the lecture and quiz, still the spirit of the orig-
 
 6 MATHEMATICS FOR ENGINEERS. 
 
 inal mode has been retained. This spirit that of "Learning by 
 doing" is the feature to be emphasized in connection with this insti- 
 tution. 8 
 
 The following is an outline of the course of study offered in 
 1854 after the course in civil engineering had fully crystalized. It 
 is to be regretted that the records do not show the number of exer- 
 cises devoted to each subject. 
 
 First Year. 
 
 First Term Algebra, geometry, general physics, graphics, geo- 
 desy, English composition, French. 
 
 Second Term Trigonometry, higher algebra, general chemis- 
 try, graphics, geodesy, botany, English composition, 
 French. 
 
 Second Year. 
 
 First Term Analytics, differential calculus, practical trigonom- 
 etry, general physics, minerology, chemistry, descriptive 
 geometry, English composition, French. 
 
 Second Term Integral calculus, general physics, geology, zool- 
 ogy, graphics, geodesy, English composition, German. 
 
 Third Year. 
 
 First Term Mechanics, astronomy, physical geography, geol- 
 ogy, industrial physics, English composition, philosophy. 
 Second Term Construction, mechanics, mining, geodesy, prac- 
 tical astronomy, graphics, metallurgy, industrial physics, 
 philosophy. 
 
 At first those "who have a good knowledge of arithmetic and 
 can understand good authors readily" were received into the institu~ 
 tion. The requirements were raised, however, until at the time the 
 above schedule of studies was put into operation they were, in math- 
 ematics, arithmetic, including the metric system, plane geometry and 
 algebra to equations of the second degree. In view of the fact that 
 a preparatory course of one year was offered at this time and that 
 the college course was only three years in length these were really 
 higher than the actual requirements for entrance to the institution. 
 Davies' Legendre's Geometry, Davies' Bourdon's Algebra, Chauv- 
 
 Palmer C. Rickets, History of the Rensselaer Polytechnic Institute 1824-94. New 
 York, 1895.
 
 HISTORICAL SUMMARY. 7 
 
 enet's Trigonometry, Church's Analytical Geometry and Church's 
 Calculus were the texts used. By 1893 the work in mathematics 
 had been changed to solid geometry, trigonometry and algebra for 
 the first year, analytical geometry for the second, and the calculus 
 for the third. Two years later the differential calculus was put into 
 the second year. 4 
 
 2) LITERARY-TECHNICAL INSTITUTIONS. 
 
 About the middle of the nineteenth century several literary 
 institutions recognized the advisability of adding courses in engi- 
 neering. As a rule these were at first a modification of the final 
 two years of their existing courses, but they gradually developed 
 into the highly specialized ones of the present. Because of the sim- 
 ilarity of development, changes in the schedule will be given fully 
 for two institutions and only special characteristics of the others. 
 
 (t) Union College. 
 
 Union College, at Schenectady, N. Y., was established as early 
 as 1795, but the Department of Civil Engineering was not organized 
 before 1845. Until 1852 the work required in mathematics was 
 algebra (Bourdon), one year; plane geometry (Legendre), one- 
 third year ; solid geometry, one-third year ; plane and spherical trig- 
 onometry, one-third year. Beginning with 1852 one-third year was 
 added in each of the following: algebra, conies (Jackson), analytic 
 geometry of three dimensions (Davies) and the calculus (Davies). 
 A little later the course was changed so as to call for only two years, 
 but the entrance requirements were raised so as to include a large 
 amount of work previously given in the four years' course. The 
 entrance requirements in mathematics were then two terms each of 
 algebra and geometry, and one term each of plane and spheri- 
 cal trigonometry and geometrical drawing. During the first 
 year of the two year course one term was given to each of the fol- 
 lowing: analytic geometry, accurate and approximate calculations 
 and the calculus. In 1875 the course was again lengthened to four 
 years and the entrance requirements lessened, so that in mathematics 
 algebra, to equations of the second degree, was the only requirement. 
 In 1890 this was increased to algebra thru equations of the second 
 
 4 The data for this institution and for the others to be considered are found in 
 past catalogs and announcements.
 
 S MATHEMATICS FOR ENGINEERS. 
 
 degree and plane geometry. Since that time various changes in the 
 entrance requirements in mathematics have been made until now 
 they stand as given in Chapter II where will also be found the work 
 in mathematics as scheduled for the college courses. 5 
 
 (it) Lawrence Scientific School of Harvard University. 
 
 In 1847 Abbott Lawrence gave $50,000 for the founding of a 
 scientific school in connection with Harvard University. He desired 
 "a school for the purpose of teaching the practical sciences I have 
 thought that these great branches to which a scientific education is 
 to be applied amongst us should be first, engineering ; second, mining 
 in its extended sense, including metallurgy ; third, the invention and 
 manufacture of machinery." In honor of the donor this newly 
 organized school was called the Lawrence Scientific School. The 
 course in civil engineering was not begun until two years after the 
 founding of the school. The course in mining was first offered in 
 1868, electrical engineering in 1888, mechanical engineering in 1894, 
 and architecture in 1895. In 1907 the courses in the Lawrence Sci- 
 entific School were made a part of the courses offered by Harvard 
 College and Graduate School, thus making the course in engineer- 
 ing largely elective. 
 
 (m) Brown University. 
 
 Brown University offered its first work in engineering in 1849 in 
 the form of a partial course of two years called the "Engineering 
 and Scientific Course." The work in mathematics included algebra, 
 plane and solid geometry, mensuration, trigonometry, surveying and 
 mechanics. The following year a subscription of $25,000 was raised 
 and the University reorganized by Pres. Wayland, at which time the 
 "New System" was adopted, which was in substance our full elective 
 plan. One and one-half years' professional work in civil engineering 
 was then offered which required as preparation about two full years 
 of mathematics. At the same time a course in practical "chemistry 
 applied to the arts" was also organized. In 1863 Brown University 
 was made one of the beneficiaries of the government in accordance 
 with the land grant act of 1862. Singularly enough no reference 
 to work in engineering is found in the catalogs from 1863 to 1867. 
 The catalog for 1867 offered a mechanical and agricultural course 
 
 Chap. II, pp. 21-41.
 
 HISTORICAL SUMMARY. 9 
 
 which gave in the mechanical division geometry and algebra, the 
 first year; trigonometry and engineering, the second year; besides 
 several culture studies thruout the course. The final change in the 
 mathematical schedule was made in 1901. 
 
 (iv} Cumberland University. 
 
 In 1852 Cumberland University added a School of Engineering 
 to its other departments. 
 
 (v} Sheffield Scientific School of Yale University. 
 
 The Scientific School of Yale University dates from the estab- 
 lishment of its School of Applied Chemistry in 1847. The "Course 
 in Engineering" of two years was first offered in 1853, and in 1864 
 was lengthened to three years with a variation along the lines of civil 
 and mechanical engineering during the last two years. Mining engi- 
 neering was added in 1866, electrical engineering in 1894 and sani- 
 tary engineering in 1900. The usefulness of the school was greatly 
 enhanced by a liberal endowment in 1860 from Joseph E. Sheffield, 
 in whose honor it took, two years later, the name of the Sheffield 
 Scientific School. In 1864 it received congressional recognition and 
 was made a participant in the aid given the "Land Grant Colleges." 
 This revenue was taken from it by the Connecticut Legislature in 
 1892. 
 
 The chief feature in the mathematical schedule was the intro- 
 duction in 1864 of the calculus of variations and in 1874 of "Elemen- 
 tary Theory of Numerical Approximations, Solutions of Higher 
 Numerical Equations, Methods of Interpretations" for the second 
 term of the first year. In 1880 the schedule in mathematics stood 
 as follows : first year, analytic geometry and spherical trigonometry, 
 one semester each; second year, elementary theory of functions, 
 numerical equations and differential and integral calculus, one semes- 
 ter each. The following year geometry of three dimensions was 
 put into the second year. In 1902 the first year was changed further 
 to plane and solid analytic geometry, or an introduction to the cal- 
 culus. 
 
 (rt) University of Michigan. 
 
 The University of Michigan holds the honor of being the first 
 institution supported by a state to give an engineering course. It 
 was also the only one of its kind before the passage of the Land
 
 I0 MATHEMATICS FOR ENGINEERS. 
 
 Grant Act in 1862. In 1853 its first course in civil engineering 
 was offered in connection with the general science course. The 
 work in mathematics was algebra, two terms, geometry, two 
 terms, during the first year; trigonometry, one term, conies, 
 two terms the second year; the calculus, one term the third 
 year. During 1856-57 the two semester plan was put into effect 
 and the mathematics schedule contained algebra and geometry the 
 first half and geometry, trigonometry and mensuration the second 
 half of the first year; descriptive and analytic geometry the first 
 half and the calculus the second half of the second year. In the 
 catalog for the year 1859 this statement is found: "The School of 
 Engineering commences with the second year of the Scientific Course 
 and is identical with it during the second and third years of that 
 course." The additional years of technical work were then added 
 which made algebra and geometry required subjects and placed trig- 
 onometry and analytic geometry in the first year of the engineering 
 course, with the calculus in the second year, each of which was pur- 
 sued for one-half year. A year later trigonometry was placed in the 
 first year of the scientific course and hence required for entrance to 
 the engineering course. The School of Mines was established in 
 1865 an d made a separate institution in 1885. In 1865 the first three 
 years of the Engineering and Scientific Courses were also made iden- 
 tical. This brought geometry (5-books), trigonometry and algebra 
 to the first, analytic geometry and the calculus to the second year of 
 the engineering course. Euclidean geometry was required for 
 entrance after 1866, and plane trigonometry after 1890. In 1896 the 
 School of Engineering was made a wholly separate department. 
 
 3) MILITARY AND NAVAL INSTITUTIONS. 
 
 For the sake of completeness two institutions established during 
 this period, the Virginia Military Institute and the United States 
 Naval Academy, although not organized to give work in civil engi- 
 neering will be considered briefly. 
 
 (i) Virginia Military Institute. 
 
 Virginia Military Institute, which was established as early as 
 1839, was patterned largely after the United States Military Acad- 
 emy. In 1860, thru bequests, it expanded along lines of general 
 industrial education, only to have its plant destroyed by the northern
 
 HISTORICAL SUMMARY. u 
 
 army. The cadets were transferred to Richmond, where the work 
 was continued until the evacuation of that city in 1865. Since the 
 reopening of the Institute in October, 1865, at Lexington, it has from 
 time to time been enlarging its powers. The present courses for 
 the first two and one-half years are the same, after which the student 
 elects one of the following courses ; chemical, electrical or civil engi- 
 neering. 
 
 (') United States Naval Academy. 
 
 The establishment of a Naval Academy was first suggested by 
 Hon. William Jones, Secretary of the Navy as early as 1814, but 
 it was not until 1845 that the present school was created at Annapo- 
 lis. Preceding this time the Navy had conducted a sort of midship- 
 man apprentice course. As in all such courses, the instruction was 
 largely practical, all of the theoretical matter being taken during the 
 last six months of the course. The work prescribed in mathematics 
 for this brief period was books I, 2, 3, 4, and 6 of Ray fair's Euclid, 
 McClure's Spherics and Bourdon's Algebra. The Department of 
 Mathematics in the new school was fully organized by 1850 and gave 
 work in arithmetic and algebra the first year; geometry, trigonom- 
 etry and descriptive geometry the second year; analytic geometry, 
 the calculus and astronomy the third year; navigation and survey- 
 ing the fourth year. From 1866 to 1870 little was done with ana- 
 lytic geometry. Arithmetic was omitted in 1871, at which time the 
 calculus and mechanics were organized into one course. 6 
 
 V. CONDITIONS IN ENGINEERING AT THE TRANSI- 
 TION FROM THE SECOND TO THE 
 THIRD PERIOD. 
 
 Toward the close of the second period conditions in engineering 
 were changing at a rapid rate corresponding to external influences. 
 By 1860 the consumption of iron had arisen to 919,370 tons. The 
 growth and general development of the country was marvelous 
 which resulted in many timesaving inventions. 
 
 Park Benjamin, United States Naval Academy. New York, 1900. J. R. Soley, Rear- 
 Admiral U. S. N., Historical Sketch of the United States Naval Academy. Government 
 Printing Office, 1876.
 
 12 MATHEMATICS FOR ENGINEERS. 
 
 1) TRANSPORTATION. 
 
 The first iron truss bridge is thought to have been built by 
 Trumbull in 1840 over the Erie Canal. During the ten years from 
 1850 to 1860 the railroad mileage had increased about 250%. 7 Two 
 inventions closely related to the railroad industry were made at this 
 time ; the telegraph in 1844 and the air-brake in 1865. 
 
 2) INVENTIONS. 
 
 The threshing machine was perfected about 1850 and the reaper 
 ten years later. A sucessful fire engine was made in 1853. Great 
 advances were made in dynamo machinery from 1860 to 1870. The 
 two most notable events were the invention of cassions by M. Triger 
 just before the middle of the century and the establishment of a plant 
 for the manufacture of steel by the Bessemer process in 1859. 
 
 VI. THIRD PERIOD 1862 TO THE PRESENT. 
 
 The third period is that of the land grant colleges. They are 
 the result of the "Morrill Land Grant Act" passed by Congress in 
 1862 in recognition of the need of technical education thruout the 
 United States. 
 
 i) MORRILL LAND GRANT ACT AND SUBSEQUENT ACTS. 
 
 In substance the Morrill Land Grant Act is as follows : A grant 
 of land was to be made to each state in the Union in the amount of 
 30,000 acres, or its equivalent, for each senator and representative 
 in Congress to which the state was entitled by the apportionment of 
 the census of 1860. The proceeds from the sale of these lands for 
 each state were to form an endowment for the institutions estab- 
 lished under the provisions of the act, and only the interest derived 
 from the same to be available for the support of these institutions. 
 It further required of such colleges that their "leading objects shall 
 be, without excluding other scientific and classical studies, and 
 including military tactics, to teach such branches of learning as are 
 related to agriculture and the mechanical arts, in such a manner as 
 
 1 See Table p. 4.
 
 HISTORICAL SUMMARY. ! 3 
 
 the legislatures of the states may respectively prescribe, in order to 
 promote the liberal and practical education of the industrial classes 
 in the several pursuits and professions in life." In 1892 Congress 
 passed another bill known as the "Morrill Fund Act:" "to apply a 
 portion of the proceeds of the public lands to the more complete 
 endowment and support of the colleges for the benefit of agriculture 
 and the mechanics arts established under the provisions of the act 
 of 1862." According to this act the sum of $15,000 was to be paid 
 each of the land grant colleges in 1890, and further, this was to be 
 increased by $1,000 each year until 1900, after which the sum of 
 $25,000 should be appropriated annually. March 4, 1907 Congress 
 again increased this amount by what is known as the "Nelson 
 Amendment." It provided that beginning July i, 1908, the sum of 
 $5,000 was to be added to each appropriation yearly until the amount 
 was $5o,ooo. 8 
 
 2) COLLEGES OF THE THIRD PERioo. 9 
 
 No review will be made of the colleges established during the 
 third period, for their progress presents no radical changes, except 
 at a very late date to which Chapter III will be devoted. A sum- 
 mary of the work offered by these various institutions will be found 
 in the next chapter. 10 
 
 3) GROWTH OF THE ENGINEERING COLLEGES. 
 
 The graph giving the increase in the number of engineering 
 colleges serves best to show the marvelous growth in engineering 
 education since the recognition of its need by the United States 
 Government. 11 The following figures regarding the increase in 
 students taking professional courses from 1878 to 1900 may be of 
 interest : 
 
 theology ' 8,079 or %7% 
 
 medicine 26,088 or 142% 
 
 law 1 1,835 or 2 94% 
 
 engineering 9,659 or 
 
 I. E. Clark, A.M., Education in the Industrial and Fine Arts in the United States, 
 Vol. IV, p. 838. U. S. Bureau of Education, 1898. 
 
 For list of Land Grant Colleges see foot-note p. 21. 
 10 Chap. II, pp. 21-41. 
 
 Chap. II, p. 42. 
 
 12 1. O. Baker, Presidential Address at the Annual Meeting of the Society for the 
 Promotion of Engineering Education, 1900. Proceedings vol. VIII, p. n.
 
 1 4 MATHEMATICS FOR ENGINEERS. 
 
 VII. SKETCH OF CHANGES IN MATHEMATICS IN 
 ENGINEERING COLLEGES. 
 
 The engineering courses offered by all of the institutions exist- 
 ing during the second period, with the exception of Rensselaer Poly- 
 technic Institute, were only continuations of their previously existing 
 literary or general science courses in which engineering subjects had 
 been introduced into the work of the last two, or at most, three 
 years. For this reason the work in mathematics for engineers really 
 began with that given in the general science course in those institu- 
 tions where the engineering course was a continuation of the former. 
 
 The second period may be regarded as the formative and expe- 
 rimental stage in the development of the engineering colleges; 
 changes in the mathematical work were numerous, ' sometimes 
 increasing often diminishing the amount required for entrance with 
 corresponding shifting of the collegiate curriculum. During the 
 third period changes in entrance requirements have always been in 
 the nature of increases. During this third period there have arisen 
 institutions with widely differing purposes, presenting a wide range 
 of variation both in the work required for entrance as well as that 
 offered in the curriculum, and showing corresponding irregularity 
 in the development of the mathematical courses. An exact resume 
 of the changes is, of course, impossible. The following sketch is 
 based upon the changes in twenty of the leading institutions and is 
 quite characteristic of the general development. 
 
 i) CHANGES IN ENTRANCE REQUIREMENTS. 
 
 About 1850 some algebra began to be required for entrance, by 
 1870 several of the leading institutions required one year and by 
 1908 &$% have made a requirement of one and one-half years of 
 algebra. 13 As early as 1855 a very few institutions required plane 
 geometry for entrance, and by 1875 there were still very few requir- 
 ing it, while it was quite generally required in 1880. The University 
 of Michigan and Stevens Institute were the first to require trigo- 
 nometry, which they did in 1890. Since then more and more have 
 made it a required subject until now iS% do so. 13 A few institutions 
 
 13 Table of entrance requirements Chap. II, p. 43.
 
 HISTORICAL SUMMARY. I5 
 
 required solid geometry in the early eighties and between 1885 and 
 1900 a majority of the institutions made this a requirement, since 
 which time there has been little change up to the present. 13 
 
 2) COLLEGIATE CURRICULUM. 
 
 As more work was required for entrance a proportionate amount 
 of a more advanced nature was added to the curriculum. The total 
 time given to mathematics was at first about one year but has been 
 increased by easy stages until now about two full years are given to 
 it. Trigonometry was either required for entrance or formed a part 
 of the college work in every course organized. Analytic geometry 
 has, to an extent, also been a part of nearly every course. The time 
 devoted to it has been more than doubled and its position in the cur- 
 riculum changed from the last part of the second year or first part 
 of the third year to the first year. 14 The calculus was made a part 
 of some engineering schedules as early as 1855 but it came late in the 
 course, was taken only for one-third or one-half of a year and did 
 not enter into the work of the course to any extent. The increase 
 in the attention devoted to the calculus has been more gradual than in 
 any other subject, save that of algebra. The time has been greatly 
 increased, a whole year now being given to it by most institutions, 
 and it has been placed earlier in the curriculum ; namely, in the sec- 
 ond year. 
 
 3) GENERAL TREND. 
 
 The general trend of late years has been to require one and one- 
 half years of algebra, plane and solid geometry for entrance : 15 and 
 to complete algebra, trigonometry and analytic geometry in the first 
 year, with the calculus the second year. 16 The ratio of the time given 
 to each subject \vill be found in Chapter II. 
 
 u Table of entrance requirements. Chap. II, p. 43. 
 14 Table of collegiate subjects. Chap. II, p. 44. 
 " Table of entrance requirements. Chap. II, p. 43. 
 M Table of collegiate mathematics. Chap. II, p. 44.
 
 !6 MATHEMATICS FOR ENGINEERS. 
 
 VIII. PROGRESS OF INSTRUCTION IN MATHEMATICS. 
 
 It is not our purpose here to review the progress of mathemat- 
 ical instruction in the United States." During the early years of the 
 engineering colleges there was a great deal of poor teaching, owing 
 to the fact that instructors were not especially prepared in mathe- 
 matics. The spirit of scientific instruction, the learning to do by 
 doing, which has been adopted extensively by the professional de- 
 partments has not received nearly so much recognition by the instruc- 
 tors of mathematics. But little has been done along the line of 
 formulating and presenting the work in courses especially adapted 
 and correlated to the professional subjects, and that little only recent- 
 ly. 18 So far slight attention has been paid to the training of instruct- 
 ors as teachers. 
 
 IX. SUMMARY. 
 
 The apprenticeship system and the Military Academy furnished 
 all of the engineering education until about 1835. With the found- 
 ing of the Rensselaer School in 1824 began the formative period of 
 the engineering colleges which lasted until about the time of the 
 Civil War. The courses were not highly specialized and were most- 
 ly variations for the last two years of the existing courses in the 
 literary institutions. There was no stability in the work in mathe- 
 matics, either as to time allotted, courses offered or their position in 
 the schedule. Although the National Government had quite early 
 made provision for military and naval instruction it was not until 
 1862 that it recognized the need of technical instruction along civil 
 lines, by the passage of the Land Grant Act. Since that time insti- 
 tutions of various grades have been established. Contrasted with the 
 changes of the formative period those of the present have been 
 continuously progressive ; increasing the quantity as well as the qual- 
 ity of the work in mathematics, placing the mathematical courses ear- 
 lier in the curriculum so as to make them precede most of the techni- 
 cal work, with the gradual evolution of the present schedule, while 
 close correlation of the work in mathematics with that in the techni- 
 cal subjects has but lately been taken up. 
 
 1T For a comprehensire discussion of this phase see Florian Cajori, The Teaching and 
 History of Mathematics in the United States, Government Printing Office, 1890, pp. 98- 
 CO& 
 
 "Chap. III.
 
 CHAPTER II. 
 
 ENGINEERING COLLEGES IN THE UNITED STATES 
 
 IN 1902. 
 
 I. CONTENTS. 
 
 The present chapter presents in condensed form data bearing 
 on the mathematical instruction of freshmen engineers in the various 
 institutions in the United States offering one or more courses in engi- 
 neering. It includes all institutions granting a degree for work in 
 engineering and graduating a class of at least six from such courses, 
 as compiled from the list contained in the Report of the Commission- 
 er of Education for 1908. The names of the institutions are arrang- 
 ed by states in alphabetical order. 
 
 II. SOURCES OF DATA. 
 
 The information for the following table has been obtained 
 largely from the current catalogs of the various institutions. When 
 this information was lacking or given in such a manner as to require 
 considerable interpretation, a letter was sent to the institution in 
 question, asking for the missing data or for corrections of the inter- 
 pretations made. If no corrections were received the conclusions 
 were assumed to be correct and recorded as such in the table. In a 
 few instances it has been impossible to obtain the necessary data 
 thru either catalogs or letters, and hence parts of the table have 
 necessarily been left blank. While every effort has been made to 
 guard against errors in making this table, it is too much to expect 
 that some have not crept in. It is only hoped that the errors are 
 not grave ones and that no institution has been seriously misrep- 
 resented.
 
 1 8 MATHEMATICS FOR ENGINEERS. 
 
 III. EXPLANATIONS OF TABLES. 
 
 The following explanations are necessary to an understanding 
 of the tables on pages 21 to 41. 
 
 a) Column i is self explanatory. 
 
 b) Column 2 gives the date at which the first engineering courses 
 of the various institutions were established. 
 
 c) Column 3 gives a list of the courses, as, Civil Engineering, 
 Electrical Engineering, etc., offered by these institutions at the pres- 
 ent time. The following abbreviations are used : 
 
 C. E., Civil Engineering. 
 Mi., Mining Engineering. 
 Ch., Chemical Engineering. 
 Sani., Sanitary Engineering. 
 A. E., Architectural Engineering. 
 A., Architecture. 
 E. E., Electrical Engineering. 
 M. E., Mechanical Engineering. 
 Irrig., Irrigation Engineering. 
 Mun., Municipal Engineering. 
 N. A., Naval Architecture. 
 Met., Metallurgy. 
 
 d) Column 4 contains the entrance requirements in mathematics. 
 The statements found in the various catalogs have occasionally had 
 to be modified before being recorded here, since in some instances 
 neither the time devoted to the subject nor the ground covered was 
 definitely stated and because some fixed system of units is essential 
 for comparison. The usual high school unit, which requires at least 
 four recitations per week of forty-five minutes each for thirty-six 
 weeks, was the one selected with the following time limit : algebra to 
 quadratics, one year; elementary algebra completed, one and one- 
 half year; plane geometry, one year; solid geometry, one half year; 
 and trigonometry, one-half year. This system of counting conforms 
 quite closely to the recommendation of the Committee on Entrance 
 Requirements given before the Society for the Promotion of Engi-
 
 DATA ON ENGINEERING COLLEGES. ig 
 
 neering Education in lo/n, 1 and 1902.* If no definite entrance re- 
 quirements were stated either as to time or as to work, deductions 
 had to be drawn from the statement of the courses offered in mathe- 
 matics for the freshman year. Wherever elective courses in entrance 
 mathematics are found the minimum requirement is given and the 
 elective courses placed in the freshman year of the college work. 
 
 e) Column 5 states for which course (Civil Engineering, Me- 
 chanical Engineering, or the like) the data of the succeeding seven 
 columns are tabulated. This was found necessary as no one course 
 was given by all of the institutions. To preserve as much uniformity 
 as possible, the Mechanical Engineering course, whenever given, has 
 been scheduled. 
 
 f) Column 6 contains, in condensed form, the mathematical 
 work of the curriculum. This is all required except that (as men- 
 tioned above) a few of the institutions will accept for entrance 
 some of the courses scheduled under the freshman year. For the 
 sake of uniformity the time devoted to each subject has been reduced 
 from term, semester or year hours to that of the actual number of 
 recitation hours ; again for the sake of uniformity and also to sim- 
 plify the work very considerably, thirty-six weeks have been selected 
 as the length of the school year. 
 
 g) Columns 7 to 12 inclusive contain the number of hours given 
 
 1 "a" Elementary Algebra. 
 
 1) To quadratics The four fundamental operations for rational algebraic expres- 
 sions, factoring, highest common factor, lowest common multiple, complex fractions, 
 equations of the first degree of one or more unknowns, radicals, fractional exponents. 
 
 2) Quadratics and beyond Quadratics in one or more unknowns, ratio and pro- 
 portion, progressions, elements of permutations and combinations, Binomial Theorem 
 with integral exponents and the use of logarithms. 
 
 "b" Advanced Algebra. 
 
 To include the elementary treatment of infinite series, undetermined coefficients, Bi- 
 nomial Theorem for fractional exponents, theory of logarithms, determinants and the 
 elements of the theory of equations. 
 
 Plane Geometry. 
 
 To include original exercises and numerical problems. 
 
 Solid Geometry. 
 
 To include properties of straight lines, planes, dyhedral and polyhedral angles, poly- 
 hedrons, inclined prisms, pyramids, regular solids, cylinders, cones and spheres, spherical 
 triangles and measurement of surfaces and solids. 
 
 Comittee on Entrance Requirements, Proceedings of the Society for the Promotion 
 of Engineering Education, vol. IX, p. 267. 
 
 J The next year it reported the following change desirable : that permutations and 
 logarithms be omitted ; that imaginaries should be emphasized. Ibid., vol. X, p. 200. 
 
 The part marked "Elementary" under algebra is that for the one and one-half 
 years' high school work, with the exception of the use of logarithms which are usually 
 taught with trigonometry. The "Advanced" is the usual first year's college work.
 
 20 MATHEMATICS FOR ENGINEERS. 
 
 to mathematics (M), professional subjects (P), semi-professional 
 subjects (Sp), language (L), miscellaneous subjects (Mi), and the 
 total (T). The hours are computed as for column six. The number 
 in column seven is merely the sum' of the number of hours record- 
 ed for the various subjects in the one immediatelv preceding it. 
 In some catalogs surveying and mechanics are placed under the De- 
 partment of Mathematics. Whenever it has been possible to differ- 
 eniate these from the purely mathematical subjects they are counted 
 as professional. This has been done to preserve uniformity, as in 
 most of the institutions these subjects are taught by the professional 
 departments. Semi-professional subjects include physics, geology, 
 chemistry, astronomy and mineralogy. Some of these become pro- 
 fessional work when a course other than Mechanical Engineering has 
 been scheduled. The hours given under language include English. 
 
 The general rule is to give few electives except where the work 
 in engineering is a modification of a literary course. Such electives 
 are scheduled under miscellaneous subjects. Unless otherwise men- 
 tioned the time required to complete these courses is four academic 
 years. 
 
 /i) Column 13 gives the percentages of the total time which 
 each institution devotes to mathematics.
 
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 42 MATHEMATICS FOR ENGINEERS. 
 
 V. COLLECTED RESULTS OF THE TABLES. 
 
 The facts given in detail in the tables are collected below in 
 summary form for certain important phases of the work. 
 
 i) GROWTH OF ENGINEERING ACTIVITY. 
 
 The increase in the number of institutions offering courses in 
 engineering from the founding of the United States Military Acad- 
 
 
 
 emy in 1802 until 1908 is shown in the accompanying graph. Dates 
 are plotted as abscissae and the total number of institutions giving 
 courses in engineering as ordinates. The population curve for the 
 United States covering this period is also plotted to the same axes.
 
 DATA ON ENGINEERING COLLEGES. 
 
 43 
 
 2) ENTRANCE REQUIREMENTS. (') Time for Various Subjects. 
 
 The following- table gives the numerical data of the entrance 
 requirements of the one hundred and thirty institutions recorded on 
 pages 21 to 41 as presented in collected form: 
 
 No. of Institutions. 
 
 Arithmetic 1 1 
 
 Algebra. 
 
 None I 
 
 One-half year i 
 
 One year 14 
 
 One and one-half years 95 
 
 Two years 14 
 
 Three years I 
 
 Plane Geometry. 
 
 None 9 
 
 One-half year 3 
 
 One year 114 
 
 Solid Geometry. 
 
 None 39 
 
 One-half year 87 
 
 Trigonometry. 
 
 One-half year 24 
 
 All work in mathematics (thru the calculus) i 
 
 Whether plane trigonometry is required for admission or accept- 
 ed as one of the entrance electives, the time asked for it is in all 
 cases one-half year. Nine of the institutions requiring trigonometry 
 for entrance give a review of it during the college course. The time 
 required for the completion of plane geometry is uniformly one year 
 and for the completion of solid geometry one-half year. The re- 
 quirements in algebra vary greatly both as to time and as to work, 
 a fact which is greatly to be deplored as it is the lack of preparation 
 in algebra that gives the student of engineering his greatest difficul- 
 ties, as will be fully discussed a little later. It will be seen that the 
 time requirements vary from nothing at all to three full years ; the 
 great majority or 73%, however, require one and one-half years. 
 Of the fourteen which require two years seven include college alge- 
 bra and the remaining seven a half year's review during the last 
 year of the preparatory course.
 
 44 
 
 MATHEMATICS FOR ENGINEERS. 
 
 (') Requirements of a Majority of the Institutions. 
 
 The preceding table shows that the great majority of the insti- 
 tutions require for entrance one and one-half years of algebra, one 
 year of plane geometry and one-half year of solid geometry. 
 
 % 
 3) CURRICULUM (i) Position of Subject. 
 
 The following table shows the number of institutions in which 
 the various subjects are given at the stage in the curriculum that 
 is named : 
 
 No. of Institutions. 
 
 Trigonometry. 
 
 For entrance 13 
 
 Completed first year 94 
 
 Completed second year 1 1 
 
 Completed third year I 
 
 Algebra. 
 
 All for entrance 13 
 
 Completed first year 95 
 
 Completed second year 12 
 
 Completed third year 2 
 
 Analytic Geometry. 
 
 All for entrance I 
 
 Completed first year 59 
 
 Completed second year 59 
 
 Completed after second year 4 
 
 Calculus. 
 
 All for entrance I 
 
 Completed first year 2 
 
 Begun first year, completed second year 14 
 
 All in second year 75 
 
 Completed after second year 31 
 
 Differential equations are mentioned as separate courses by 20 
 of the institutions, and theory of equations by 7. As a rule the 
 latter is included in the work scheduled under algebra and does not 
 receive special mention. The theory of least squares is included by 
 4. Only 39 mention offering spherical trigonometry, the others 
 either specify plane trigonometry only or give no hint as to the
 
 DATA ON ENGINEERING COLLEGES. 
 
 45 
 
 contents of the course. In the same way only 40 specify giving solid 
 analytic geometry. 
 
 The discrepancies in the above table are due to two things: 
 first, that no data could be obtained from 7 of the institutions, and 
 secondly, that n of the institutions give special courses 3 in mathe- 
 matics from which the subjects mentioned could not very readily be 
 differentiated. 
 
 (M) General Trend of Position of Subjects in Curriculum. 
 
 From the above table and that regarding entrance requirements 
 on page 43, we see that there is a general trend towards giving trig- 
 onometry, algebra and analytic geometry in the first year of the col- 
 lege course, and all of the calculus in the second year. 
 
 <*< 
 
 - 
 
 
 
 
 
 
 
 
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 40 
 
 
 
 
 
 
 
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 Percent-aye of To tat Time of I 
 fo 
 
 3 Chap. III.
 
 46 MATHEMATICS FOR ENGINEERS. 
 
 (t/i) Comparison of Time for Various Subjects. 
 
 The accompanying cuts, which are self explanatory, give a com- 
 parison of the time devoted to the various subjects, trigonometry, 
 algebra, analytic geometry and the calculus by the various institu- 
 tions recorded on pages 21 to 41. 
 
 4) PERCENTAGE OF TIME GIVEN TO MATHEMATICS. 
 
 The percentages of the total time of instruction in the four col- 
 legiate years given to mathematics are exhibited in the graph on 
 page 45- 
 
 5) DISCREPANCIES AND THEIR CAUSES. 
 
 The great variety in the number of hours recorded in the col- 
 umns shows the lack of uniformity among the engineering colleges 
 of the United States. Some of the more noticeable ones will here 
 be briefly considered. Where entrance requirements in mathematics 
 are low the time devoted to the collegiate work is correspondingly 
 large. The wide range in the amount of time given miscellaneous 
 subjects is due to the fact that a varying degree of importance is 
 attached to culture as compared with professional subjects, or more 
 often that the engineering courses are in varying degrees modifica- 
 tions of the general science courses. The great variation in the num- 
 ber of hours recorded under "P," "Sp" and especially under "T," is 
 accounted for in two ways. The first and real factor is, that different 
 relative values are given to recitations, lectures, laboratory and 
 industrial periods. By "industrial periods" is meant such work as 
 drawing, shop-work, etc., which requires no outside preparation. 
 The following four ratios of equivalence of the different kinds of 
 work are found in use: 
 
 No. of hours in 
 i 
 
 Mathematics 
 i 
 
 Laboratory 
 
 2 
 
 Industrial 
 . . "? 
 
 ii 
 
 i 
 
 2 
 
 .2V 2 
 
 iii 
 
 i 
 
 ..2V 2 .. 
 
 ..2V 2 
 
 iv . 
 
 . i. . 
 
 -7 
 
 . .2 
 
 In the majority of cases the institution which used the first 
 ratio recorded the lowest number of hours under "professional"
 
 DATA ON ENGINEERING COLLEGES. 
 
 47 
 
 Percentage of Total Ma.-/ie/n<zr<ca.t 
 Instruction given tro Trigonometry. 
 
 x& 
 
 
 
 i 
 
 
 
 fr 
 
 Percentage ofTotral 
 Instruction, given to College fit $e bra.
 
 4 8 
 
 MATHEMATICS FOR ENGINEERS. 
 
 3t 
 
 ;i . 
 
 ^29 
 
 12 
 
 Cv? 
 
 
 # 
 
 
 of Totnl Mat/t em at 
 Instruction $<vH fro flnafyt-t' 
 
 c ^: 
 
 
 
 
 
 
 
 
 - 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 to 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 - j 
 
 i 
 
 & 
 
 
 
 
 
 
 
 
 
 
 
 N 
 
 ^ 
 
 ,6s^ 
 
 
 
 
 
 
 s 
 
 ^ 
 
 "^ 
 
 ^ 
 
 ^ 
 
 K 
 
 
 cff- 
 
 
 ^ 
 
 ^ 
 
 ^ 
 
 -^ 
 
 f 
 
 ^ 
 
 ^ 5? 
 
 * 
 
 $ 
 
 N 
 
 *^ 
 
 VNA 
 
 
 
 4 
 
 ^^ 
 
 ^ 
 
 & 
 
 i 
 
 '(T) 
 
 Qr\ 
 
 > 
 
 V 
 
 2 
 
 S 
 
 of Total 
 Instruction give* to e/te Co.tculti$.
 
 DATA ON ENGINEERING COLLEGES. 
 
 49 
 
 subjects, which increased for the second, third and fourth ratios in 
 that order of sequence. The second cause for variation is merely 
 a clerical one, arising from insufficient data in the various catalogs. 
 
 VI. PERCENTAGE OF TIME GIVEN VARIOUS GROUPS 
 
 OF WORK. 
 
 One further computation was made: the percentages were 
 computed for each of the divisions "mathematics," "professional," 
 "semi-professional," "language" and "miscellaneous" by adding the 
 total time recorded for each. The results were as follows : 
 
 Professional 52.3 % Semi-professional 15.0% 
 
 Mathematics , . 13.7% Language 11.0% 
 
 Miscellaneous 8.0% 
 
 VIL SUMMARY. 
 
 There are one hundred and thirty instituions giving degrees for 
 work in engineering, one hundred and twenty-one of which have 
 been organized since 1860; the general entrance requirements are 
 one and one-half years algebra, one year plane geometry and one- 
 half year of solid geometry with trigonometry required by iS% of 
 the institutions; the general trend of collegiate mathematics is to 
 complete trigonometry, algebra and analytic geometry the first year 
 and to devote the second year to calculus, except that a few institu- 
 tions are beginning to introduce the calculus into the first year; 
 there is great variation in the allotment of time to the various de- 
 partments and especially to the department of mathematics where the 
 allotment varies from $% to 23% of the total time; mathematics 
 receives only one-fourth as much time as the professional studies 
 and only slightly more than those in language.
 
 CHAPTER III. 
 
 RECENT MODIFICATIONS IN THE WORK IN, MATH- 
 EMATICS FOR STUDENTS OF 
 ENGINEERING. 
 
 I. CONTENTS OF THE CHAPTER. 
 
 As already noted 1 some of the engineering colleges of the United 
 States have discarded the formal division of the work in mathematics 
 into subjects separated by sharply defined lines, and some institu- 
 tions have added new features, such as laboratory work in curve 
 tracing, special courses in computation, etc. A treatment of this 
 modified work will be the object of the present chapter. 
 
 II. GENERAL NATURE OF THE MODIFIED WORK IN 
 MATHEMATICS. 
 
 This departure from the old lines aims to arrange the work 
 in mathematics for the student of engineering to fit his special needs, 
 and is, as will be seen, in full conformity with the so-called "Perry 
 Movement." 2 In 1902 Prof. Moore advocated an immediate reform 
 in the teaching of elementary collegiate mathematics so as to bring it 
 
 1 Chap. II, pp. 21 to 41. 
 
 * It may be best to give Prof. Perry's idea of the necessary reform in the teaching 
 of mathematics in his own words : "Newton employed geometrical conies in his astron- 
 omical studies and so mechanics was developed, and therefore it is that every young en- 
 gineer must study mechanics thru astronomy, and he dares not think of differential 
 calculus till he has finished geometrical conies. The young applier of physics, the 
 engineer, needs a teaching of mathematics which will make his mathematical knowl- 
 edge part of his mental machinery, which he will use as readily and as certainly as a 
 bird uses its wings ." John Perry, On the Teaching of Mathematics, Nature, vol. 26. p, 
 319-
 
 RECENT MODI PICA T1ONS IN MA THEM A TICS. 5 1 
 
 within the realm of reality. 3 Such modifications have been worked 
 out in a number of engineering institutions where the object has 
 been to build up a logical course in mathematics based upon the 
 future needs of the student. 4 These modifications are as yet in the 
 evolutionary stage but have as their object the treatment of mathe- 
 ematics as a scientific "weapon." 5 
 
 III. SOURCES OF DATA. 
 
 Twenty-two of the institutions considered in Chapter II are 
 found to have established, to a greater or less extent, such modified 
 work in mathematics. In order to secure more complete informa- 
 tion regarding this work than is found in the catalogs a questionaire 
 letter was sent to each institution, uniformly containing the follow- 
 ing questions together with such others as were suggested by the 
 separate catalogs. 
 
 8 "Just as the secondary schools should begin to reform without waiting for the 
 improvement of the primary schools, so the elementary collegiate courses should be mod- 
 ified at once without waiting for the reform of the secondary schools." " a feeling 
 that mathematics is indeed itself a fundamental reality of the domain of thought and 
 not merely a matter of symbols and arbitrary rules and conventions." Presidential 
 Address at the Annual Meeting of the American Mathematical Society. Bull. Am. 
 Math. Soc., 1903, pp. 1-24. 
 
 * "Some technical schools have made radical changes in the mathematical work 
 of the college course. Those which have worked out the matter independently have arrived 
 at very similar results. The changes consist for the most part of modifications that nat- 
 urally follow from the movement in the secondary schools. The tendency is to blot 
 out entirely the present lines of distinction between algebra, trigonometry and analytic 
 geometry. These changes have involved, not only the early and constant use of graph- 
 ical methods and abundant use of numerical data, but have also included the early con- 
 sideration and application of vector analysis. Our most treasonable act is the deposing 
 of the conic sections, which have been the reigning family for so many years, from 
 their exalted place in our course of study. This gives an opportunity for the more 
 abundant enrichment of the course and a better comprehension of those things which 
 the student of applied science needs. Such a course not only conforms more nearly to the 
 actual needs of the students, but it also has the advantage of being more logical and 
 scientific than our old course. As a matter of fact all that the engineering student 
 learns in his usual course in mathematics is a simple comprehension of the properties of 
 the algebraic functions of a real variable. Indeed we might define the work of the 
 engineering student, during his first two years of mathematics, as the study of the expo- 
 nential function of the real variable." Chas S. Slichter, The Improvement of the Fresh- 
 man Year of Mathematical Instruction in Technical Schools, Proceedings of the Society 
 for the Promotion of Engineering Education, vol. 14, p. 146. 
 
 * " a most powerful weapon with which to unlock the mysteries of Nature." John 
 Perry, Address before the Section of Educational Science of the British Association, at 
 Glasgow, 1902.
 
 52 MATHEMATICS FOR ENGINEERS. 
 
 i ) QUESTIONAIRE LETTER. 
 
 (i) In what way does the work differ from that usually given 
 in a general science or arts course ? 
 
 (11) What is the material used and from what source is it 
 obtained ? 
 
 (in) What is the particular object sought? 
 
 (iv) When was this instruction with special reference to engi- 
 neering begun, and what suggested the same? 
 
 While some replies were quite brief, others were very compre- 
 hensive and described fully the work given. No replies were receiv- 
 ed from seven of the institutions written to, in which cases reference 
 will be made to the statements found in the catalogs wherever usable. 
 
 IV. SYNOPSIS OF PRINCIPAL FEATURES OF 
 MODIFIED WORK. 
 
 The following synopsis shows for twenty of the above institu- 
 tions separately the principal features of the instruction ascertain- 
 ed as just specified. The statements are arranged according to sim- 
 ilarity of their principal features, and are followed by a short sum- 
 mary. 
 
 INSTITUTION No. i. COHERENT COURSE OF ESSENTIAL ELEMENTS. 
 
 The reply to the letter of inquiry suggested the consultation of 
 "A Course in Mathematics," by Woods and Bailey, the first volume 
 of which is used as a text during the freshman year. 
 
 It may be well to quote from the preface in regard to the aims 
 and objects of the authors : "This book is the first volume of a course 
 in mathematics designed to present in a consecutive manner an 
 amount of material generally given in distinct courses under the var- 
 ious names algebra, analytical geometry, differential and integral cal- 
 culus, and differential equations In arranging the material, 
 
 however, the traditional division of mathematics into distinct subjects 
 is disregarded, and the principles of each subject are introduced as 
 needed and the subjects developed together. The objects are to give 
 the student a better grasp of mathematics as a whole, and of inter- 
 dependence of its various parts, and to accustom him to use, in
 
 RECENT MODIFICATIONS IN MATHEMATICS. 
 
 53 
 
 later applications, the methods best adapted to the problem in hand. 
 At the same time a decided advantage is gained in the introduction of 
 the principles of analytic geometry and the calculus earlier than is 
 usual. In this way these subjects are studied longer than is other- 
 wise possible, thus leading to greater familiarity with their methods 
 and greater freedom and skill in their application." 
 
 (*) Study and Use of Curves. 
 
 Accordingly the work in mathematics in this institution appears 
 to be as follows : A few preliminary lessons are devoted to determ- 
 inants, considering only evaluation of determinants, their use in 
 the solution of linear simultaneous equations, eliminants and the 
 testing of equations for common roots. 
 
 Some work on the different kinds of numbers and functions, 
 introduces a graphic study of polynomials. After a short considera- 
 tion of derivatives of polynomials there follows a discussion of 
 curves, considerable time being devoted to conies but only as a species 
 of curves in general, where such questions as tangents, normals, 
 diameters, cusps, asymptotes, branches, intersections of curves and 
 systems of curves are taken up. The general equation of the second 
 degree is not taken up at all. Polar and parametric representation of 
 curves are taken up as separate topics and applied to problems for 
 which they are most servicable. Solid analytic geometry is deferred 
 until the second year. 
 
 () Early Introduction of the Calculus. 
 
 Derivatives of polynomials are introduced quite early in the 
 course thru the idea of the slope, and are immediately applied to 
 problems on tangents, and maxima and minima. A more complete 
 treatment follows the study of curves as mentioned above, where 
 simple problems of physics are also considered. Quite a number of 
 transcendental functions are studied towards the close of the course, 
 which ends with a brief treatment of curvature, evolutes and invo- 
 lutes of curves. 
 
 INSTITUTION No. 2. MAXIMUM USABLE MATHEMATICS IN FIRST 
 YEAR. 
 
 In this institution a course called "An Introduction to Modern 
 Mathematics" is given the first year the aim of which is a "desire 
 to start the student with as much mathematics as possible in the
 
 54 
 
 MATHEMATICS FOR ENGINEERS. 
 
 freshman year so that he can grasp work required in the profession- 
 al schools as early as possible." It includes "much that was former- 
 ly taught under the head of Theory of Equations, combined with an 
 early introduction to the elements of analytic geometry and calculus, 
 while not anticipating the calculus to any extent." "A Course in 
 Mathematics" is also used as a text here and supplemented by Went- 
 worth's Analytic Geometry", especially for problems in solid analytic 
 geometry. 
 
 (*') Origin of Course. 
 
 This course was introduced in 1907 as many of the students 
 came well prepared in trigonometry and higher algebra so that such 
 a course could be taken advantageously, both as to time and prepa- 
 ration. A rigorous examination in trigonometry must be passed by 
 all students admitted to it. The class is divided into sections based 
 upon the ability of the students. 
 
 INSTITUTION No. 3. MATHEMATICAL SCHEDULE A COHERENT 
 WHOLE. 
 
 The catalog states, "The aim of the instruction in mathematics 
 is to present the subject so that the student may obtain a thoro 
 working knowledge of these principles which he needs to know 
 when he becomes an engineer. It is recognized that such knowledge 
 can best be obtained by an exercise of the observational faculties of 
 the student, by treating the subject as one coherent whole rather than 
 a series of more or less disconnected subjects, and by frequent appli- 
 cation of the principles taught to problems in engineering. For 
 this reason laboratory methods are sometimes employed and the 
 subjects taught are so arranged that each is a help in the development 
 of the others. We give the calculus in the freshman year in order 
 that the men may early become acquainted with the calculus method 
 so as to use the subject more intelligently in the many cases which 
 arise during the sophomore year, in physics, kinematics, etc." 
 
 (*') College Algebra. 
 
 The freshmen take five hours of algebra during the first half 
 of the year, comprising "a review of equations; plotting of curves 
 from equations ; a comprehensive treatment of surds, imaginaries,. 
 ratio, proportion, variation; the progressions; permutations; com-
 
 RECENT MODIFICATIONS IN MATHEMATICS. 55 
 
 binations ; determinants ; binomial theorem for positive integral 
 powers of the binomial ; logarithms ; partial fractions ; methods of 
 approximations to the roots of equations ; building up of equations 
 from the properties of the curves. A special feature of the course is 
 the introduction of a large number of problems and curves similar 
 to those met by the engineers in actual practice in order to drill the 
 student on many algebraic operations peculiar to engineering, espe- 
 cially those for logarithmic computations which are not found in the 
 ordinary algebra. By stating the actual engineering problem, with 
 the physical law, we try and do, to a great extent, arouse the student's 
 interest in the subject of college algebra, a study that is too often 
 found dry and uninteresting." 
 
 () Analytic Geometry and Calculus. 
 
 Five hours per week are given during the second semester to 
 analytic geometry and the elements of the calculus, when are consid- 
 ered: "Transformation of coordinates; a systematic treatment of 
 the circle, parabola, ellipse and hyperbola ; limits ; the ordinary rules 
 for differentiation with application to curve plotting, rates, maxima 
 and minima; the fundamental forms of integration with easy appli- 
 cations to problems in plane areas; those subjects in college algebra 
 and analytical geometry which lend themselves best to the calculus 
 treatment." In analytical geometry the straight line and loci prob- 
 lems receive the principal consideration; for conies only questions 
 of vertices, foci and directrices are taken up. Problems on tangents 
 are treated by the calculus and are not restricted to conies. The 
 various forms of differentiation are taken up together with their 
 applications. 
 
 (in) Material Used. 
 
 The texts on algebra and trigonometry are supplemented by 
 numerous mimeographed problems as mentioned above, of which the 
 following are types : 
 
 \ r. 
 
 ) Snout* 
 
 - 2} EVALUATE (3.i6)' 142 and 'f* x 3-*5 X .3* 
 
 A/.004 X .00032 
 
 3) SOLVE THE EQUATION Js*~+
 
 MATHEMATICS FOR ENGINEERS. 
 
 Given that (an expression in X) /* means the value of the ex- 
 pression when X = a minus its value when X = b, and given e = 
 2.71828, evaluate 
 
 4) LOG (sec X -f tan X)l ~f 
 
 fo 
 
 5) SOLVE FOR X AND Y J4 3y ~* 3 
 
 /3.4y+x= 16* 
 
 6) FIND LOG 
 
 V, 
 i8l/ 2 
 
 7) EXPAND THE DETERMINANT 
 
 iii 
 
 357 
 984 
 
 8) SOLVE ^l_^4 = x^_^Z 
 
 X-io X-6 X-7 X-9 
 
 Use logarithms in the following problems involving computa- 
 tions. 
 
 9) Within the elastic limit, the expression of a spiral due to a 
 weight suspended to one end is proportional to the weight. A spring 
 originally 18.23" l n g measures 20.35" when a weight of 3.2 oz. is 
 attached ; what will be its length when the weight is 4.5 oz. ? 
 
 10) Within the elastic limit, the extension of a rod under tension 
 varies directly as the product of the length and the tension, and in- 
 versely as the area of the cross-section. If a round wrought iron 
 rod, 3' long, .625" in diameter, is stretched .0057" by a tension of 
 6000 Ibs., how much will a bar of wrought iron io / long and of rec- 
 tangular cross-section .25" by .375" be stretched by a tension of 
 560 Ibs.? 
 
 Several problems are given under each of the conditions as 
 stated above. 
 
 INSTITUTION No. 4. A MODIFIED COHERENT SCHEDULE OF SUB- 
 JECTS. 
 
 Beginning with the work in algebra and continuing thru the 
 calculus the work of this institution is arranged as a systematic 
 whole, while not wholly disregarding the division into the subjects 
 algebra, analytic geometry, differential and integral calculus.
 
 RECENT MODIFICATIONS IN MATHEMATICS. 
 
 57 
 
 INSTITUTION No. 5. ARRANGEMENT OF MATHEMATICS COURSES FOR 
 FUTURE WORK. 
 
 The freshman work begins with a review of surds, imaginaries 
 and quadratic equations, followed by those principles needed in 
 analytic geometry. The whole mathematical schedule is arranged 
 with view to the work taken by the students in their third and fourth 
 years covering such subjects as mechanics, hydraulics, machine de- 
 sign, bridges, etc. This arrangement was reported as meeting with 
 the approval of the professional departments as well as with that 
 of the instructors of mathematics, the only difficulty being that of 
 finding suitable text-books. Trigonometry is given as a formal 
 course in the first year and all that is given of the calculus constitutes 
 the second year's work. 
 
 INSTITUTION No. 6. COMBINED COURSE IN ALGEBRA AND TRIGONO- 
 METRY. 
 
 A course in algebra and trigonometry is given the first half of 
 the freshman year in which, "the work in algebra deals with topics 
 supplementing the work in trigonometry ;" later in the course trigo- 
 nometry in turn is used in the solution of equations. 
 
 (') Mathematics as a Tool. 
 
 As to the objects and the handling of the course in mathematics 
 an instructor reports : "We lay stress upon the theory but insist 
 upon familiarity with processes and manipulations of results. For 
 instance, in the study of Taylor's Theorem, we do not go into the 
 remainder form of the theorem, but lay stress upon the actual use of 
 the theorem in the expansion of functions. In the integral calculus 
 we lay less stress upon the derivation of the reduction formulas than 
 upon the use of the tables of integration. We develop the calculus 
 as a tool rather than as a science for its own sake." 
 
 INSTITUTION No. 7. ALGEBRA THRUOUT THE COURSE IN MATHE- 
 MATICS. 
 
 No formal course is given in algebra but the subject is distrib- 
 uted thruout the whole course in mathematics. When a topic is 
 reached that calls for any principles in algebra which are new to the 
 students or which present difficulties, such principles are taken up 
 and elucidated.
 
 5 8 MATHEMATICS FOR ENGINEERS. 
 
 INSTITUTION No. 8. EARLY CALCULUS. 
 
 A course in analysis is given the second half of the first year 
 in which are treated some of the advanced topics of algebra and an 
 introduction to the calculus. This is followed by a formal course 
 in analytic geometry, where the principles of the former courses 
 are applied. 
 
 INSTITUTION No. 9. MATHEMATICS AS BASIS OF TECHNICAL TRAIN- 
 ING. 
 
 The mathematical courses are conducted by means of well 
 chosen text-books, very little supplementary material being used. 
 Special attention is given to practical problems which are interesting 
 to the students. "The ability to understand and apply mathematical 
 processes readily is the aim, and to this end special emphasis is laid 
 upon two things: elucidation of the principles and drill upon their 
 application, as furnishing the only sure basis for a thorough technical 
 and professional training." 
 
 INSTITUTION No. 10. APPLIED CONTRASTED WITH PURE MATHE- 
 MATICS. 
 
 The reply from this institution was especially clear in stating 
 the object of their special work in mathematics and its variation 
 from that held in view for the art students. To quote: "The object 
 we have in view is to give the students facility in using mathematics 
 intelligently in solving engineering problems. In the A.B. course, 
 on the other hand, we aim to give the student a broad view of the 
 field of mathematics as a branch of human knowledge and some of its 
 big results. In brief, these aims animate our methods." 
 
 For example the engineer "has to study the elements of accu- 
 racy in testing ; this demands partial derivatives, etc." For the A.B. 
 student, "Partial derivatives come under the general head of space 
 differentiation and involve a grip of the real meaning of calculus in 
 its widest sense." 
 
 "I know that there are some that contend that the engineer 
 should have this broad insight into mathematical ideas, but the 
 thing is not feasible. He is primarily interested in material construc- 
 tion, and he cannot focus his attention on two things. He is, and 
 ought to be, concerned with effective construction. In handling this,
 
 RECENT MODIFICATIONS IN MATHEMATICS. 59 
 
 mathematics is to him a different thing from what it is to one who 
 is interested in the harmonies of numbers and space. I do not believe 
 there is an engineering mathematics, but I do believe there is an engi- 
 neer's use of mathematics. 
 
 "Therefore our object primarily is to teach the engineer to use 
 mathematics. To do this he must recognize the type of his problem, 
 must understand the method of solving that type, and must under- 
 stand the construction of the method ; just as he must know in any 
 case what tool to use, how to use it, how to modify it, to adapt it or 
 even to make it." The results are reported as "on the whole very 
 good." 
 
 (i) Problems from "Real Material." 
 
 Whenever possible real material is used for problems which is 
 obtained from "laboratories, kinematics, graphic statics, etc. we 
 aim to supplement rather than duplicate the problems in these cours- 
 es." The following are two illustrations of the problems used : 
 
 "An elliptical ceiling in a church has a 20 foot span and a 
 6 foot rise at the center. Rafters run tangent to it at angles 
 of 30 and 60 to the horizontal, with a horizontal on top. 
 Find where their intersections are, their lengths, and where the 
 two at 60 strike the wall. Draw same to scale." 
 
 For algebra: "The diameter of a solid wooden column is 
 given by the formula d 2 isd (F/P i) 700 (F/P i) 
 = o; P -= ultimate strength (lb./in. 2 ), F = crushing strength 
 (lb./in. 2 ) . For white oak F = 5000, P = 4586. Find d." 
 
 (') Laboratory Work. 
 
 The students make models of the solids mostly used and do a 
 great deal of drawing to scale with pencil, using "cross-section paper, 
 polar coordinate paper ruled both for degrees and radians, loga- 
 rithmic, sine ruled, etc." They complete about one hundred illustra- 
 tions of various curves and graphic solutions in this way. 
 
 INSTITUTION No. n. NATURE OF APPLIED MATHEMATICS. 
 
 "The study of Mathematics, as pursued at this school, is chiefly 
 for the purpose of acquiring a working knowledge of its use in the 
 subsequent studies of engineering, physics, and chemistry, and not 
 merely as a component part of a general education." The chief dif-
 
 60 MATHEMATICS FOR ENGINEERS. 
 
 ference from the usual work given the arts students lies, says one 
 instructor, in "the omission of topics which, we have found, do not 
 make any impression on the student's mind, or add to his skill in 
 analysis and in manipulation, and are of no particular use in his sub- 
 sequent study. I find it useless to teach such things as undetermined 
 coefficients, partial fractions, permutations and combinations in the 
 Freshman year. I give plenty of time to the calculus and teach the 
 things when they are needed, so that the student may see what they 
 are good for, and have some chance of retaining them." 
 
 (i) Graphic Algebra and Trigonometry. 
 
 The freshmen take "Graphic Algebra, Curve Tracing and Alge- 
 braic Analysis" for the first three months. Besides the work on 
 graphics the most striking features are "convergency of infinite 
 series; use of infinite series in approximation calculations; errors 
 of observations ; methods of least squares." The next three months 
 are devoted to acquiring "a knowledge of the trigonometric functions 
 in analysis and in shortening computations." The remaining three 
 months are devoted to the solution of the oblique triangle, "limits, 
 expansion of functions in series, application of De Moivre's Theo- 
 rem," and spherical triangles. 
 
 () Analytic Geometry and Calculus. 
 
 Analytic Geometry is taken the first three months of the second 
 year. "The matter and methods are intended to aid the student in 
 his subsequent reading of technical literature, and in solving the 
 problems which arise in his work in Mechanics and Physics." The 
 work in the calculus considered from the engineer's standpoint occu- 
 pies the remainder of the year and considers differentiation, proper- 
 ties of tangents and normals and problems in maxima and minima, 
 
 (Hi) Problems from Engineering. 
 
 "No special reference is made to engineering problems in the 
 first year, and it has been considered unprofitable to use material 
 from engineering practice or engineering literature without first 
 carefully preparing it for our students. It is necessary to avoid the 
 appearance of teaching engineering or physics but to make it clear to 
 the student that we are merely teaching the kind of mathematics 
 useful in engineering. I never got satisfactory results until I cut 
 loose from all text-books, that is, class-room use of text-books."
 
 RECENT MODIFICATIONS IN MATHEMATICS. 61 
 
 INSTITUTION No, 12. MATHEMATICAL POWER AND FORMAL COURSE. 
 
 The aims and results of this institution are very well described 
 by an instructor: "The objects I have in view with students are two- 
 fold ; first, to develop in them mathematical power, including insight 
 and initiative, variety and strength of attack, all dependent of course 
 not upon guess work but upon clear, solid analysis; second, to give 
 them control over the formal part of mathematics as a tool and its 
 methods. The results have been the deepest possible interest in the 
 subject by multitudes of students and a disposition to place mathe- 
 matics study foremost in the favor and devotion of students." 
 
 (i) Problems from Engineering. 
 
 In analytic geometry and the calculus special emphasis is given 
 problems in engineering in order to bring before the student an 
 early realization of the great importance mathematics plays in engi- 
 neering practice. Cross-section paper is widely used in the fresh- 
 man work and the students taught to deduce the results directly from 
 the graphs. Texts are made the basis of the work but problems are 
 drawn from a variety of sources. 
 
 INSTITUTIONS Nos. 13 AND 14. ENGINEERING PROBLEMS. 
 
 In two of the institutions the mathematical studies are taken up 
 with special reference to their connection with mechanics. Problems 
 are selected from engineering data, the students being taught how to 
 obtain and make use of the same successfully together with the use 
 of tables and computing devices. 
 
 INSTITUTION No. 15. EXTRA PROBLEMS FOR ENGINEERS. 
 
 This institution adds a sufficient number of problems to the 
 course given to the students in engineering to double the time for 
 the same subject given to the arts students. The bulk of these prac- 
 tical problems relate to surveying while considerable attention is 
 given to the selection of trigonometric equations as a grounding for 
 future analytic work. 
 
 INSTITUTION No. 16. DIVISION INTO Two CLASSES: ANALYSIS AND 
 COMPUTATION. 
 
 Two courses are given simultaneously the first half of the 
 freshman year. The analytic side of algebra, trigonometry and ana-
 
 62 MATHEMATICS FOR ENGINEERS. 
 
 lytic geometry is considered thoroly in what is designated as "A 
 Course in Analysis." The other is "A Course in Computations;" 
 the various computations arising in connection with work not involv- 
 ing the calculus are taken up and computating instruments such as 
 planimeters, slide rules, etc., are studied and used. 
 
 () Contents of Course in Computation. 
 
 This work is given in two-hour periods; the first hour being 
 used by the instructor in a lecture, after which the students work 
 at computations under the instructor's direction. The reason for 
 this second course is that it is believed " that in the elementary 
 branches of the subject the student should be taught 'systematic com- 
 putation,' which can best be done under personal supervision of the 
 instructor and his assistants." 
 
 (n) Freshman Calculus. 
 
 The second half of the year is devoted to an elementary treatise 
 on differentiation and integration, Ransom's "Freshman Calculus" 
 being used. The purpose of the course is " to provide the student 
 of science or engineering very early in his course with a familiarity 
 with the fundamental conceptions and methods of the calculus in as 
 far as they are of use in the elementary study of the physical 
 sciences." 
 
 INSTITUTION No. 17. PROBLEMS AND COMPUTATION DEVICES. 
 
 The problems thruout the mathematical courses are selected 
 from actual questions arising in connection with "mines, stamp 
 mills, power plants, etc.," in the vicinity of the institution. Permu- 
 tations and combinations or other subjects of only a disciplinary 
 value are excluded. Tables, slide rules, planimeters, etc., are used 
 in computations. 
 
 INSTITUTION No. 18. COMPUTATIONS. 
 
 A short course for attaining accuracy is given in the freshman 
 year. Degrees of accuracy, short methods, use of tables and calcu- 
 lating instruments are the chief features.
 
 RECENT MODIFICATIONS IN MATHEMATICS. 63 
 
 INSTITUTION No. 19. COMPUTATION PURPOSE. 
 
 The general idea permeating the work is to insure a working 
 knowledge of mathematics to the students. The particular feature 
 is a course of two hours for three months of the freshman year in 
 mensuration and logarithms, which is a computation course in prob- 
 lems of physics, mechanics and engineering. 
 
 INSTITUTION No. 20. LABORATORY WORK. 
 
 The special course in mathematics is confined to one two hour 
 laboratory period per week taken in connection with one-half year 
 each of algebra and analytical geometry during the freshman year. 
 A laboratory period is also given to juniors taking analytical mechan- 
 ics. It has been necessary to discontinue laboratory work in cal- 
 culus on account of a lack of time. 
 
 (i) Instruments and Material Used. 
 
 All work is done in pencil upon "ten by ten to the inch cross- 
 section paper" which is cut to "six and three-fourths by ten, and ten 
 by thirteen and one-half inches." Drawing tables accommodating 
 four students are used. 
 
 () Problems Considered. 
 
 "The problems cover the plotting of 
 Y = sinX, Y = tanX, Y = secX, 
 
 or the co-functions; which, together with plotting in polar coordi- 
 nates, gives some review and exercise in trigonometry. In connec- 
 tion with algebra, problems in graphical solution of single equations 
 and pairs of simultaneous equations are solved. Along with ana- 
 lytic geometry are given further exercises in curve plotting, by 
 points, illustrating among other things, inflections (not the actual 
 location), asymptotes, nodes, and cusps, and to some extent the 
 relation between form of equation and form of curve (this being a 
 difficult matter with the student). We also plot certain systems of 
 conies. During the differential calculus I have had series plotted, as r 
 
 Y = X, Y = X *L, Y = X *.+ ., Y = sinX, 
 |_3_ |_3_ [5_ 
 
 illustrating convergency. Another style of problem has to do with 
 curvature ; as to draw a parabola and the circle of curvature for the 
 vertex, and then draw the circle of curvature for some other point.
 
 64 MATHEMATICS FOR ENGINEERS. 
 
 Examination of such drawings, carefully made by the student, is 
 quite instructive. Another exercise is to plot Y = f (X) and sev- 
 eral of its derivatives." 
 
 (HI) Workings of the Course. 
 
 The instructor in charge of each laboratory section gives it his 
 personal attention. He also " makes up the problems, all members 
 of the class working on the same problem at about the same time. 
 This of course results in some little copying, but on the other hand, 
 the advantages of comparison, one with the other and the suggestions 
 one student gets from another outweigh that." For calculus they use 
 the "laboratory period as a practice period in the integral calculus, 
 beginning the subject before finishing the differential, the students 
 working at the board, with the instructor passing around answering 
 questions, offering suggestions and correcting the work. In the lab- 
 oratory period the fundamental formulae were gotten thru with by 
 the time of finishing differential calculus, but the laboratory period 
 was retained thru the integral calculus." 
 
 (iv) Object of Course. 
 
 The object is " to supplement and broaden the analytical part 
 of trigonometry, advanced algebra and analytics. It has been my ex- 
 perience in assigning curves to be plotted at home to have all degrees 
 of accuracy and slovenly work presented and very little good work. 
 The student also appears to regard the graph as an unnecessary pic- 
 ture or carricature of the equation, a thing to be drawn in tree 
 hand by an artist of the impressionist school." The poor preparation 
 of the student is assigned as another reason for giving this work, 
 because the personal supervision of the instructor for a long period 
 is thought to be beneficial. 
 
 (v) Results. 
 
 As to results and working of the course we read : "Our classes 
 are small, about twenty for freshmen and ten for sophomore mathe- 
 matics. This makes it possible to provide tables and blackboard 
 room for working a whole class at once. Of course the laboratory 
 work takes more time of the instructor and we have also found 
 another difficulty ; that of finding sufficient time during the day for 
 students to take all their necessary laboratory work." 
 
 The only objections raised were those of extra demands upon
 
 RECENT MODIFICATIONS IN MATHEMATICS. 65 
 
 the teaching force and some difficulty in arranging the schedule of 
 the students. After a four years' trial the plan was reported a suc- 
 cess and well liked by the students. It was further said to aid " in 
 giving not a little practice in numerical calculations and for this 
 we encourage the use of the tables, as of squares and cubes. This is 
 made the occasion of mentioning the slide rule." 
 
 V. SUMMARY. 
 
 I ) Very few institutions mentioned the date at which they had first 
 taken up this modified work in mathematics. No date earlier 
 than 1900 was given. 
 
 2) Number of institutions that give trigonometry 16 
 
 3) Number of institutions that give algebra all 
 
 4) Number of institutions that give plane analytical geometry . . .all 
 
 5) Number of institutions that give solid analytic geometry .... 
 
 first year 7 
 
 second year 10 
 
 6) Number of institutions that give the calculus first year 7 
 
 second year 15 
 
 7) Number of institutions that give the calculus as a separate 
 
 course 14 
 
 8) Number of institutions that give the calculus in connection 
 
 with other work 8 
 
 9) Number of institutions that give courses more or less of a 
 
 laboratory nature 9 
 
 10) Material Used: 
 
 i) General text-books used only 4 
 
 ii) Text-books arranged for engineering students 3 
 
 iii) Specially prepared pamphlets, with or without texts. . 2 
 iv) Text-books supplemented by problems and exercises 
 taken from laboratories and professional depart- 
 ments 4 
 
 n) Main Features. 
 
 Problems given much more prominence than in the mathemati- 
 cal courses for arts students. 
 
 Omission of useless and uninteresting topics in algebra the use 
 of -especially prepared lists of problems in algebra. 
 Computation courses under supervision. 
 Laboratory periods for work in graphs.
 
 66 MATHEMATICS FOR ENGINEERS. 
 
 Topics arranged so as to make one coherent course without 
 division into subjects. 
 
 Aims to give an early working knowledge of mathematics, espe- 
 cially of the calculus. 
 
 Selection of topics which will form a dieect part of the future 
 work in engineering subjects. 
 
 12) Causes for the Establishment of These Modified Courses. 
 Good preparation of entering students, a cause for giving 
 
 more advanced work. 
 
 Poor preparation of entering students, a cause for giving lab- 
 oratory work. 
 
 Agitation in the journals and at the meetings of the various soci- 
 eties. 
 
 To attain the requirements of the professional departments. 
 
 13) Objects of the Course. 
 
 To create interest in algebra. 
 
 To ground students in the solution of problems. 
 
 To teach the value of the graph and its use in the interpretation 
 of results. 
 
 To introduce the calculus early so as to equip the students as 
 soon as possible with this weapon, and at the same time giving them 
 more practice in its use and for a longer time. 
 
 To give a working knowledge of mathematics and ability to use 
 it. 
 
 To familiarize the students with the mathematics they will need 
 for their future technical work. 
 
 14) Results. 
 
 Only a few of the letters received gave any expression as to the 
 results attained in these modified courses. All that did so, however, 
 were highly in favor of the new mode of procedure. Only one in- 
 stance was found of a return to the more formal treatment. That 
 was the omission of laboratory work in the calculus, necessitated by 
 a lack of time.
 
 CHAPTER IV. 
 
 CURRENT THOUGHTS ON VITAL QUESTIONS. 
 
 I. MODE OF OBTAINING THE DATA. 
 
 To obtain a survey of current thought on certain important 
 questions 651 copies of the following questionaire letter were sent 
 out. 
 
 For convenience the questions were, as far as possible, made 
 answerable by yes or no, or by numerical annotation, even at the 
 cost of making them more narrow and formal than would other- 
 wise have been desirable. In order not to make (2), (4) and (7) 
 complete tables of contents for these various subjects and thus ren- 
 der them too large to be readily answered, topics a under each had to 
 include a larger number of operations than would otherwise have 
 been desired. 
 
 i) QUESTIONAIRE LETTER. 
 
 UNLESS OTHERWISE STATED THE FOLLOWING WILL 
 
 RELATE TO THE COURSES IN MATHEMATICS 
 
 FOR FRESHMEN STUDENTS IN 
 
 ENGINEERING. 
 
 1. Is it advisable to require entrance examinations in mathe- 
 matics of all students ? 
 
 2. Number i, 2, 3, etc., in order of importance the following 
 topics which are fundamental to a course in trigonometry: 
 
 a. Derivation and manipulation of formulae. 
 
 b. Solution of triangles by use of natural functions. 
 
 c. Use of logarithms in solution of triangles and other 
 computations. 
 
 d. Problems applying the solution of triangles. 
 
 e. Higher trigonometry hyperbolic functions, DeMoiv- 
 re's Theorem, etc.
 
 68 MATHEMATICS FOR ENGINEERS. 
 
 j. In what particular topics of trigonometry do students show 
 themselves most deficient after the freshman year? 
 
 4. Number i, 2, 3, etc,, in order of importance the following 
 topics which are fundamental to a course in college algebra: 
 
 a. Review of surds, exponents, quadratics, etc. 
 
 b. Series. 
 
 c. Binomial Theorem. 
 
 d. Permutations and combinations. 
 
 e. Graphs. 
 
 f. Determinants. 
 
 g. Theory of Equations. 
 
 5. In what particular topics of algebra do students show them- 
 selves most deficient after the freshman year? 
 
 6. Should the object of analytic geometry be (a) to learn the 
 characteristics and theory of some class of curves, as conies, or (b) 
 to learn a new mathematical language, using any suitable problems 
 in curves or mechanics as the medium for the study of this language f 
 
 7. Number i, 2, 3, etc., in order of importance the following 
 topics which are fundamental to a course in analytic geometry: 
 
 a. Algebraic relations for intersection of loci, tangency, 
 etc. 
 
 b. Loci problems. 
 
 c. Right line and circle. 
 
 d. Conies. 
 
 e. Higher curves, as cycloids, etc. 
 
 f. Coordinates of points in space with application to sim- 
 ple loci. 
 
 8. In what particular topic of analytic geometry do stdents 
 ivho have completed a course in that subject show themselves most 
 deficient f 
 
 p. Would it be advisable to introduce a working knowledge of 
 elementary differentiation and integration preceding or simultane- 
 ously with the course in analytic geometry f 
 
 10. What is the remedy for (3), (5), and (8)? 
 
 11. Is it advisable to treat problems met with in actual engi- 
 neering work? 
 
 12. If so, to what extent and under which topics of (2), (4), 
 and (7)?
 
 CURRENT THOUGHTS ON VITAL QUESTIONS. 69 
 
 /j. Is a short course in computation advisable? 
 
 14. Should freshmen engineers be taught their mathematics in 
 .separate classes or together with those taking other courses? 
 
 15. Why? 
 
 16. Are special texts in mathematics for engineering students 
 advisable? 
 
 if. How often should written quizzes be given? 
 
 18. Number i, 2, j, etc., the order of usefulness of the follow- 
 ing modes of conducting classes suggesting any further modes or 
 combinations of these: 
 
 a. Assignment of work from text without any previous 
 explanation. 
 
 b. Assignment of work from text with previous explana- 
 tion. 
 
 c. New work explained, the student taking notes with or 
 without the use of a text. Written or oral quizzes at 
 regular intervals. 
 
 d. Students led to discover new principles by suggestions 
 and quizzes on the old. 
 
 e. Each student working independently, going as fast as 
 he is able. This may apply to the entire course, to each 
 topic, or to each day's work. 
 
 f. First part of hoivr devoted to the class as a whole; sec- 
 ond part given to individual work. 
 
 g. Subject treated from the laboratory standpoint; most 
 of the work being done in the class room. 
 
 /p. Number i, 2, 3, etc., the order of importance of the fol- 
 lowing pertaining to the qualifications of the engineering student 
 who has completed the courses in freshman mathematics: 
 
 a. Skill and accuracy in computations. 
 
 b. Analysis of problems. 
 
 c. Interpretations of results in solution of problems. 
 
 d. Knowledge and use of equations. 
 
 e. Representation of physical laws by means of graphs. 
 
 . 20. In which of (19) do students who have completed their 
 freshman year show the greatest deficiency? 
 21. What is the remedy?
 
 70 MATHEMATICS FOR ENGINEERS. 
 
 22. Designate by i, 2, 3, etc., the order of importance for any 
 mathematical topic of: 
 
 a. Mastery of theoretical matter involved. 
 
 b. System and neatness. 
 
 c. Accuracy in computed results. 
 
 23. Is it true, as has been said, that engineering students apply 
 themselves better than those taking a literary or scientific -course? 
 
 2) REPLIES. 
 
 The six hundred and fifty-one letters sent and the replies receiv- 
 ed are classified in the following table : 
 
 No. No. of % of 
 Sent Replies Replies 
 To instructors of mathematics and 
 professional courses, represent- 
 ing 92 institutions 544 204 37.5 
 
 To teachers of mathematics especi- 
 ally interested in the question 20 13 65.0 
 
 To professional engineers 87 20 23.0 
 
 Total 651 237 36.4 
 
 The practical side is represented more strongly in the replies 
 than the number would indicate, since several of the instructors of 
 professional subjects are also practicing engineers. 
 
 II. TABULATION OF DATA. 
 
 A summary of the replies to these questions is given in what 
 follows. The questions are taken up in order and separately. Fol- 
 lowing a short review of the important points considered in the ques- 
 tion are appended some of the most suggestive and helpful com- 
 ments in the form of quotations. The discrepancy which will often 
 be found between the total number of replies received and that re- 
 corded for any particular question is accounted for by the fact that 
 several writers omitted replies to various questions.
 
 CURRENT THOUGHTS ON VITAL QUESTIONS. 71 
 
 III. ENTRANCE EXAMINATIONS. 
 
 i. Is it advisable to require entrance examinations in mathe- 
 matics of all students? 
 
 Replies : 
 
 Yes 130 
 
 No 72 
 
 Several of those who opposed this requirement did so on the 
 ground of inexpediency but would otherwise have favored it, while 
 many who favored it voiced the same objections; namely, (a) that 
 it would cause serious disturbance in the accredited secondary schools 
 and (b) that if the rule of an institution is to receive its students 
 from the secondary schools without examination it would not be fair 
 to the other departments to make an exception in favor of mathemat- 
 ics. 
 
 Few complaints were found of poor preparation in geometry 
 but a very large number in algebra. The general opinion seemed to 
 be that something stronger than a mere request should be presented 
 to the secondary schools for better preparation in algebra. The fol- 
 lowing quotation expresses the reasons in favor of entrance exam- 
 inations: "I think that all students entering engineering courses 
 should be required to take entrance examinations for the following 
 reasons: ist, to find their deficiencies in order that these may be 
 remedied before it is too late; 2nd, to insure that the high school 
 preparation is what it should be ; 3rd, to insure that a good prepara- 
 tion has not been forgotten ; 4th, to discourage men deficient in math- 
 ematical ability from undertaking the engineering courses without 
 extra preparation, and to prevent those lacking in such ability from 
 entering such a course at all." 
 
 i) A SUBSTITUTE FOR THE ENTRANCE EXAMINATION. 
 
 One method which is, in a way, a compromise between these 
 two extremes is used by some of our foremost and rising institutions. 
 This plan, as outlined in a letter from one of the institutions using 
 it, is as follows : "Regarding Entrance Examinations in Mathematics,
 
 7 2 MATHEMATICS FOR ENGINEERS. 
 
 I would say that it seems to us not desirable to require Entrance 
 Examinations in the case of a State University, even in such a sub- 
 ject as Mathematics, on account of the fact that the University 
 should take its place as part of the general school system of the state. 
 This it cannot do if it stands aloof from these schools, and accepts 
 their students only after an examination, which explicitly refuses 
 recognition of the quality of the work done in the secondary schools. 
 
 "We recognize, however, that the students entering on certifica- 
 tion are often very deficient in mathematical knowledge. In order 
 to avoid the setting of examinations, and to provide some basis for 
 weeding out the poorest of these students, we have struck upon the 
 following scheme: Some two weeks or more after the opening of 
 school, after a rapid review of those topics in elementary algebra 
 which precede quadratic equations, we give a test covering this 
 ground. The students who fail in this test are not allowed to con- 
 tinue this course. We do not remove their credits for entrance, we 
 do not even require them to take up the study of elementary algebra, 
 but we make it rather obvious that the only way they will ever suc- 
 ceed in this course is by a thoro review of that work, and we point 
 out to them that an opportunity to do this is afforded in several of 
 the Secondary Schools in operation in this city. To the next grade of 
 students, those who do not utterly fail, but are weak on this test, we 
 give the alternative namely, that they either drop the course at once, 
 or that they continue it, together with a review of elementary alge- 
 bra, in the training school of the Teachers' College of his University. 
 We believe that this scheme is superior to the scheme of entrance 
 examinations, and highly superior to the scheme of direct admission 
 to classes on certification without restriction." 
 
 No unfavorable reply was received from any one who had ever 
 tried this plan. 
 
 IV. SPECIFIC NEEDS AND DEFICIENCIES. 
 
 The following seven questions are concerned with the needs, 
 deficiencies and remedies pertaining to topics in the subjects general- 
 ly taught the freshman year; logarithms, trigonometry, algebra, de- 
 terminants, etc.
 
 CURRENT THOUGHTS ON VITAL QUESTIONS. 73 
 
 i TRIGONOMETRY t) Needs. 
 
 2. Number i, 2, 3, etc., in order of importance the following 
 topics which are fundamental to a course in trigonometry: 
 
 a. Derivation and manipulation of formulae. 
 
 b. Solution of triangles by use of natural functions. 
 
 c. Use of logarithms in the solution of triangles and in 
 other computations. 
 
 d. Problems applying the solution of triangles. 
 
 e. Higher Trigonometry hyperbolic functions, DeMoiv- 
 
 re's Theorem, etc. 
 
 Replies : 
 
 I. 
 
 II. 
 
 III. 
 
 IV. 
 
 V. 
 
 F. 
 
 a 
 
 147 
 
 17 
 
 21 
 
 34 
 
 i 
 
 935 
 
 b 
 
 40 
 
 63 
 
 37 
 
 63 
 
 6 
 
 695 
 
 c 
 
 33 
 
 92 
 
 54 
 
 29 
 
 i 
 
 754 
 
 d 
 
 33 
 
 40 
 
 79 
 
 55 
 
 4 
 
 676 
 
 e o o 9 ii 158 207 
 
 The columns of the above table, indicated by the Roman numer- 
 als, refer to the order of importance assigned to the various topics 
 mentioned in the question. The number in any column opposite any 
 letter tells how many of those answering the question gave it this 
 particular order of importance. Thus there were 37 who placed b 
 as third in order, 11 who placed e as fourth, etc. The numbers in 
 the final column, marked F, are obtained by adding 5 times the num- 
 ber in the first column, 4 times that in the second, etc., as for a 
 147x5 -f- 17x4 -f 21x3 -f- 34x2 + ixi = 935, which shows the rel- 
 ative importance assigned to a, b, c, d, e. It should be mentioned, 
 however that a large number thought e should be wholly omitted. 
 
 Comments: 
 
 "In trigonometry, more time slhould be devoted to practical problems in 
 plane trigonometry; much less to spherical trigonometry, which is rarely 
 used even by civil engineers. It is singular that perhaps more students fail 
 in trigonometry than in other branches; as it is comparatively easy, the 
 system of instruction must be at fault." 
 
 "Few students are drilled to use logarithms intelligently, rapidly or cor- 
 rectly. Most tables of logarithms and logarithmic functions in American 
 text-books are very inconvenient and far inferior to the German tables in
 
 74 MATHEMATICS FOR ENGINEERS. 
 
 arrangement, and instructors seem scarcely to consider this in selecting a 
 text-book. Hence the student never likes to use logaritJims. Then he is 
 often required to use 6 or 7 place tables for computation in which 4 place 
 tables would be amply sufficient." 
 
 "Regarding the importance of the topics which you mentioned in trig- 
 onometry I would say that the order appears to me to be as follows : 
 b, c, d, a. I have omitted e entirely, for its importance is highly problemati- 
 cal in a course on trigonometry for freshmen, since I do not believe that 
 freshmen have any use for this matter, and I do not believe that, they can 
 get a proper grasp of it. I have placed b first simply because it seems to 
 me to involve the true spirit of trigonometry as a whole. If the student 
 grasp the essential notion of the solution of right triangles, he has a founda- 
 tion for the whole of trigonometry. I place c next since the question of 
 computation by logarithms and otherwise is the main use to which the 
 freshman will put his trigonometrical knowledge. I place d next since the 
 practical problems of trigonometry are the type to which he will apply this 
 knowledge. The derivative and the manipulation of formulae are important 
 enough, but they are certainly secondary to the general grasp of the purpose 
 of elementary trigonometry, which is essentially contained in the first three. 
 
 "The spirit of the remarks I have just made will follow through all I 
 have to say in what follows, namely, in each subject. I shall regard as of 
 first importance the general conception of that subject, next the particular 
 means of putting these into operation, and next the things to which they are 
 applied." 
 
 () DEFICIENCIES. 
 
 3. In what topics of the above in trigonometry do students 
 show themselves most deficient after the freshman year? 
 
 Replies : a b c d e 
 
 126 22 33 34 12 
 
 f~ Very few replies reported difficulties under more than one topic 
 given in question (2). It is interesting to note that 55% of the dif- 
 ficulties reported fall under a, the derivation and manipulation of 
 formulas. A few of the letters specified particular topics under a 
 as giving the greatest amount of difficulty as follows : 
 
 *) 50% in the manipulation of formulae, especially in identities, 
 functions of 2x and ^x; 
 
 2) 25% in inverse functions; 
 
 3) I2 l /i% in circular functions; 
 
 4) i2 l / 2 % in application of definitions.
 
 CURRENT THOUGHTS ON VITAL QUESTIONS. 75 
 
 Comments: 
 
 "In trigonometric equations." 
 
 "They can not apply definitions to rapid solution of triangles." 
 
 "The greatest weakness I find in the student having finished trigonom- 
 etry is an inability to interpret a given problem in the trigonometrical terms; 
 that is, he is unable to see the triangles which must be solved." 
 
 "Technical engineering courses (undergraduate) require, as a rule, little 
 mathematics beyond the elements. Hence in these courses it is only in the 
 elements of mathematics that students show either weakness or strength. 
 When the attempt is made in the freshman mathematics courses to cover 
 much ground including 'advanced' work, the inevitable result with a large 
 proportion of students is weakness in the elements, which reveals itself in 
 all subsequent work involving mathematics. The remedy, as far as a rem- 
 edy is possible, is to confine the work in the freshman and sophomore math- 
 ematical subjects mainly to the elements and to teach these as efficiently as 
 possible." 
 
 .2) ALGEBRA (i) Fundamentals. 
 
 4. Number i, 2, 3, etc., in order of importance the following 
 which are fundamental to a course in college algebra : 
 
 a. Review of surds, exponents, quadratics, etc. 
 
 b. Series. 
 
 c. Binomial Theorem. 
 
 d. Permutations and combinations. 
 
 e. Graphs. 
 
 f. Determinants. 
 
 g. Theory of Equations. 
 
 Replies : 
 
 a 
 
 b 
 
 c 
 
 d 
 
 e 
 
 / 
 
 g 9 25 24 27 26 34 28 615 
 
 The key for (2) also applies to the above table. A few replies 
 gave an equal rank to two or more topics, others omitted one or more 
 as non-essential to engineering students. Of the latter /, c, and d 
 were the principal ones mentioned and in that order. 
 
 I. 
 
 II. 
 
 III. 
 
 IV. 
 
 V. 
 
 VI. 
 
 VII 
 
 F. 
 
 177 
 
 15 
 
 ii 
 
 3 
 
 i 
 
 o 
 
 
 
 1399 
 
 2 
 
 34 
 
 42 
 
 51 
 
 26 
 
 15 
 
 10 
 
 639 
 
 19 
 
 83 
 
 56 
 
 19 
 
 15 
 
 5 
 
 o 
 
 1032 
 
 I 
 
 4 
 
 14 
 
 34 
 
 36 
 
 36 
 
 48 
 
 465 
 
 23 
 
 52 
 
 48 
 
 33 
 
 24 
 
 8 
 
 10 
 
 895 
 
 9 
 
 4 
 
 14 
 
 14 
 
 38 
 
 60 
 
 38 
 
 426
 
 7 6 MATHEMATICS FOR ENGINEERS. 
 
 Comments: 
 
 "I should omit the subject of determinants and give very little time to 
 permutations and combinations on a course in engineering algebra. Graphic 
 analysis should be introduced in the high school and should be taken up in 
 connection with all of the topics studied. 
 
 "In addition to the subjects mentioned under (4) a thorough course in 
 Limits and Logarithms is essential to the work in algebra. I believe I would 
 class these two subjects as of greater importance than the Binomial Theorem." 
 
 "A good deal of the work which we often attempt to do in advanced 
 algebra is better done after a working knowledge of analytics and calculus 
 has been obtained, and work taught without such interrelation is both more 
 difficult to master, and without its full value. For the purpose of engineer- 
 ing education, a thorough grounding in the simpler elements of higher math- 
 ematics is immensely more desirable than a perfunctory acquaintance with 
 rhe frills of mathematical theory, such as some of the higher curves having 
 icniarkable mathematical properties. Curves having remarkable properties 
 of engineering value, as for instance the involute, cycloid, logarithmic spiral, 
 etc., should be considered, as should also the properties of exponential curves, 
 and the use of logarithmic cross-section paper." 
 
 "During the college course in algebra an attempt is generally made to 
 cover too much ground rather than to do the work thoroughly. On account 
 of algebra being often poorly taught in high schools by the weakest teacher, 
 like English grammar, many students fail in this course." 
 
 "Few students are able to transform a given algebraic formula, insert 
 numerical constants, placing it in the form most convenient for practical use, 
 for example, in case of formulas relating to the safe strength of materials." 
 
 In brief, upon a defective foundation in arithmetic and algebra, the pro- 
 fessional mathematician endeavors to erect a superstructure of the higher 
 mathematics, which in most cases never becomes a permanent part of the 
 student's mental equipment and is therefore rarely utilized and is forgotten 
 quickly after entering on the practice of his profession." 
 
 (n) Deficiencies. 
 
 5. In what particular topics of algebra do students show them- 
 selves most deficient after the freshman year? 
 
 Replies: abode f 9 
 
 114 24 17 8 14 9 41 
 
 A few replies mentioned difficulties in more than one topic of 
 algebra. It appears at once that topic a, review of surds, exponents, 
 quadratics, etc., covers 50% of all the trouble recorded. In the 
 replies which gave specific details regarding a, surds were mentioned 
 the most frequently ; and then exponents, quadratics, imaginaries and 
 manipulation of algebraic symbols in about the order given.
 
 CURRENT THOUGHTS ON VITAL QUESTIONS. 77 
 
 Comments: 
 
 "In simple algebraic reductions, especially where radicals are involved." 
 
 "Fundamental operations ; such blunders as i/a -I- v/& = v/a&." 
 
 "In everything beyond quardatics." 
 
 "Solution of n linear equations with n unknowns and of equations where 
 exponential functions are involved." 
 
 "They show a general want of freedom owing to undigested loads of 
 information." 
 
 "Inability to set up equations applying to known or observed facts; 
 more practice should be given along that line." 
 
 "My students in electrical engineering are invariably deficient in their 
 algebraic training regarding the uses and powers of imaginary quantities. 
 These factors and the physical meaning attached to them are of great im- 
 portance in the treatment of alternating currents. I find, as everywhere, 
 the students are deficient in algebraic power through lack of experience 
 rather than lack of extent of algebraic training." 
 
 3) ANALYTIC GEOMETRY (t) Object. 
 
 6. Should the object of analytic geometry be (a) to learn the 
 characteristics and theory of some class of curves, as conies, or, (b) 
 to learn a new mathematical language, using any suitable problems in 
 curves or mechanics as the medium for the study of this new lan- 
 guage? 
 
 Replies : a b 
 
 56 172 
 
 It appears then that a trifle over 75% of the replies to this ques- 
 tion favored b, the learning of a new language as the object of ana- 
 lytic geometry. 
 
 Comments : 
 
 "Very decidely (b). Students generally fail to get a definite grasp of 
 the relation between the number of conditions a curve can satisfy and the 
 corresponding number of constants in the equation of the curve. Some 
 exponential and logarithmic curves should be used." 
 
 "Most decidedly (&) is the important element here, that is, to learn a 
 new mathematical language, using any suitable problems in curves or mechan- 
 ics as a medium for the study of this new language; and to become thor- 
 oughly familiar with it and to acquire power in its application." 
 
 "Regarding the alternative which you offered between (a) and (fc), 
 I should be obliged to say (&), but I would rather rephrase the matter my- 
 self thus : I do not believe that the course in analytic geometry should be
 
 78 MATHEMATICS FOR ENGINEERS. 
 
 in any sense a treatment of Conic Sections, nor that it should attempt to 
 bring out merely the geometrical properties of any certain class of curves ; 
 rather the whole object should be to familiarize the student with the notion 
 of representations of equations by graphical figures, and a representation by 
 figures of functions divorced entirely from the equation idea, together with 
 as great a familiarity with the simpler forms of curves as a thorough 
 treatment can give, so that he retains at least a rough knowledge of the 
 appearance of the curves in question; I mean forms of the class: 
 
 ax-}-b, ax* -)- bx -f- c, kx*, (ax -\- b)/(cx -f- d), sinx, cosx^e*, log x, 
 
 and other simpler forms, with perhaps some knowledge of the general 
 equation of the second degree, certainly with a knowledge of such forms as 
 x*/a* + y*/b* = i, etc.; the whole spirit of a class in analytic geometry 
 should be to familiarize the student with this possibility of geometric helps 
 in algebraic problems and the representation of functions." 
 
 (M) Importance of Topics. 
 
 7. Number i, 2, 3, etc., in order of importance the following 
 topics which are fundamental to a course in analytic geometry : 
 
 a. Algebraic relations for intersection of loci, tangency, 
 
 etc. 
 
 b. Loci problems. 
 
 c. Right line, and circle. 
 
 d. Conies. 
 
 e. Higher curves, as cycloids, etc. 
 
 f. Coordinates of points in space with application to sim- 
 
 ple loci. 
 
 Replies : 
 
 I. 
 
 II. 
 
 III. 
 
 IV. 
 
 V. 
 
 VI. 
 
 F. 
 
 a 
 
 115 
 
 31 
 
 21 
 
 10 
 
 5 
 
 4 
 
 973 
 
 b 
 
 40 
 
 77 
 
 34 
 
 32 
 
 7 
 
 o 
 
 880 
 
 c 
 
 51 
 
 54 
 
 74 
 
 9 
 
 i 
 
 4 
 
 90S 
 
 d 
 
 ii 
 
 20 
 
 39 
 
 88 
 
 22 
 
 2 
 
 622 
 
 e 
 
 ii 
 
 5 
 
 16 
 
 16 
 
 7 6 
 
 52 
 
 407 
 
 f 
 
 10 
 
 13 
 
 8 
 
 21 
 
 49 
 
 81 
 
 399 
 
 The key for the above table is the same as for question (2). 
 This question is very closely related to question (6) which should be 
 thought of in connection with it. Nearly all the answers were in 
 terms of a, b, c, etc., or such as to be directly translatable into these.
 
 CURRENT THOUGHTS ON VITAL QUESTIONS. 79. 
 
 Comments : 
 
 "Lacks ability to set up problems in algebraic symbols." 
 "No special emphasis should be given conies. They should be regarded 
 merely as one class in the discussion of curves in general together with tri- 
 gonometric, logarithmic, algebraic and transcendental curves." 
 
 (m) Deficiencies. 
 
 8. In what particular topic in analytic geometry do students 
 who have completed the course in that subject show themselves 
 most deficient? 
 
 Replies: a b c d e f 
 
 47 61 8 25 13 20 
 
 In a few instances this was not answered in a, b, c, etc., but in 
 statements which had to be interpreted into that of topics in question 
 (7). Two chief difficulties were mentioned ; the inability of students 
 to correlate algebraic and geometrical ideas and to sketch in curves 
 from the inspection of their equation without going into the detail 
 of plotting them. 
 
 Comments : 
 
 "The ability to sketch the locus from an inspection of the equations. 
 Determining the nature of a locus from its equation and interpretation of 
 results." 
 
 "In expressing laws in algebraic language and deducing the general form 
 of the locus from the equation." 
 
 "A general lack of understanding the usefulness of the subject and" 
 hence superficial knowledge of the fundamentals." 
 
 "In my work they usually show little knowledge of such curves as 
 
 y = ax* -f b ; 
 y ae* ; 
 y = asin 
 
 "After having had analytic geometry the students show themselves de- 
 ficient in two ways in their calculus. Their deficiency is largely failure to 
 grasp the general notion of plotting a function, and also failure to appreciate 
 the tangent problem. In their later engineering work their deficiency in 
 analytic geometry is largely on the side of empirical curves, and obtaining 
 of equations from empirical data. In this they are scarcely to be blamed, 
 for this is a new notation which can at the best be illustrated by lame 
 examples in their first course in analytic geometry."
 
 So MATHEMATICS FOR ENGINEERS. 
 
 (iv). Derivatives and Integrals in Analytic Geometry. 
 
 p. Would it be advisable to introduce a working knowledge of 
 elementary differentiation and integration preceding or simultan- 
 eously with the course in analytic geometry? 
 
 Replies : Yes. (together) 115 
 
 No 71 
 
 In addition to the considerable majority of opinion in favor 
 of the early introduction of the elements of the calculus (62% of 
 the total number of the replies) the striking fact was also brought 
 out, that all who reported having tried this method were, without 
 exception, in favor of it and had found it a success. On the other 
 hand several who favored the general proposition limited it to the 
 introduction of differentiation only. 
 
 Comments : 
 
 "No, haven't time." 
 
 "Should begin this early in algebra." 
 
 "I 'have combined both for the last ten years." 
 
 "Yes. I carry engineers thru analytics and calculus simultaneously." 
 
 "I think it highly advisable to treat the tangent problems by this method 
 and to give the student a working knowledge of the simpler processes of 
 differentiation along with their analytic geometry." 
 
 4) AIDS AND REMEDIES FOR ABOVE. 
 
 10. What is the remedy for (j), (5), and (8)? 
 
 The replies to this question cannot be exhibited in the tabular 
 form previously used. The remedy given most frequently was to 
 apply the theory to practical problems. Two reasons were found 
 for this; first that the greatest difficulty met by students later on 
 was in the application of principles which had been learned, and 
 second, that this would create interest and realization of the im- 
 portance of the topics studied. Another remedy frequently proposed 
 was "more drill, more drill" and in direct connection with this the 
 plea for frequent reviews.
 
 CURRENT THOUGHTS ON VITAL QUESTIONS. 81 
 
 Comments : 
 
 "Review constantly." 
 
 "Special short course sophomore year." 
 
 "Less extension and more intension in study and drill." 
 
 "Make necessary topics more important, other less so." 
 
 "Give simple problems illustrating laws of mechanics and physics." 
 
 "Smaller number of students; copious problems with engineering appli- 
 cation ; instructors who know needs of engineer." 
 
 "No one tilling. Students would do well to think more and hurry less. 
 Perhaps it would be well if writers spent more time in pointing out errors 
 that have been made and are to be guarded against." 
 
 "To extend the time devoted to the subject so as to admit of a more 
 extensive drill, and to draw upon engineering practice for concrete problems 
 which will arouse the student's interest and hold his attention." 
 
 "Greater accuracy should characterize the high school and grades. The 
 habit of accuracy can not be developed in a day or in a year." 
 
 "Emphasize mathematics as a tool and have it thoroughly mastered 
 from that standpoint." 
 
 "More drill ; selection of problems they will meet in second year and lay 
 stress upon that fact." 
 
 "Cut out at least one-third of the theoretical stuff and use the time 
 given to it in practical arithmetic and graphical problems in engineering." 
 
 "Working many problems taken as far as possible from practical appli- 
 cations either real or possible. The practical application awakens interest, 
 and interest overcomes many difficulties. We should in all cases show the 
 relation as noted in your section (6b). The problems should be carefully 
 corrected, not merely marked or other methods of solution sihown, but the 
 student's fallacies or errors pointed out. There should be frequent quizzes 
 or drills. I prefer to devote the first half of every period to drill and the 
 last half to explanations of next lesson and clearing up any difficulties in 
 quiz just given. The quizzes should overlap somewhat; i.e. should include 
 work done in the last quiz or two, or there should be a general quiz every 
 5th period." 
 
 "Too much stuffing and learning by rote." 
 
 "Early training to be analogous, to that in English and German." 
 
 "Redistribution of emphasis in teaching." 
 
 "A logical course in mathematics extending three years, not algebra, 
 trigonometry, etc., as usually taught." 
 
 "Vitalizing mathematical work by giving more time to its concrete 
 application in physics, chemistry and engineering and less to mathematical 
 training seldom of use in practice." 
 
 "I have noticed no predominating deficiency. >The general deficiency 
 is failure to recall and apply the knowledge attained earlier. To forestall
 
 82 M A THEM A TICS FOR ENGINEERS. 
 
 this as far as posible it would seem advisable that the Freshman instructor 
 make clear the purpose of the subject. Make the few fundamental principles 
 and methods stand out boldly. Aim to cultivate the mathematical type of 
 thought. Show how the few basic methods of each subject act in slightly 
 different varying ways to solve many important problems and give consider- 
 able practice in actually doing this." 
 
 "The subjects of algebra, trigonometry and analytics should be less, 
 rigorously separated into water-tight compartments. The solution of simple 
 plane triangles should be brought in with plane geometry, using only 
 natural functions of, say, acute angles. Considerable trigonometry can be 
 brought out in analytic geometry and still more in calculus. Algebra and 
 trigonometry can be united in work in series to get roots of imaginary 
 complex quantities by DeMoivre's Theorem. Whenever a new method has 
 been developed, the teachers should point out and emphasise the practical 
 problems and examinations which do not presume that a grade in a subject 
 makes continued knowledge in that subject necessary." 
 
 "The remedy for questions (j), (5), and (5) in my judgment is 
 presistent practice required of the students in application of principle to 
 practical problems. The use in explanation by the instructor of the simplest 
 non-technical language and abundant reference to the homeliest illustrations 
 at his command. I believe tihat the crucial difficulty in the case of the 
 average student of mathematics is his fear of the subject born largely of a 
 too generous use by instructors of the technical and unfamiliar language, and 
 the failure to use wherever possible, illustrations drawn from things with 
 which the student is commonly familiar. We also correlate as thoroughly 
 as may be, the subjects which are allied to mathematics with tihe mathematical 
 instruction itself; as for instance, applied mechanics, physics and applied 
 electricity." 
 
 "For question (8) the tracing of all curves met with further on." 
 
 "Some algebra should be taught every year in the high school." 
 
 "Make trigonometry more simple. Use more coordinate and graphic 
 methods." 
 
 "Connect analytic geometry with calculus." 
 
 "Drill on (a) in each witih special attention to clear statements and full 
 complete reasoning." 
 
 "For questions (j) and (8) give greater emphasis on algebraic nota- 
 tion as shorthand translation for English into algebra and vice versa." 
 
 "At the beginning of analytic geometry a short and thoro drill on 
 the kind of algebraic process and trigonometric relations to be used is 
 advised." 
 
 "A similar drill just before starting calculus on some of the most 
 needed algebraic, trigonometric and analytic relations would largely remedy 
 the difficulty. For example, a sharp drill on a few of the most fundamental 
 relations of trigonometry would help in differentiation and integration of 
 trigonometric forms."
 
 CURRENT THOUGHTS ON VITAL QUESTIONS. 83 
 
 V. MODIFICATIONS IN MATHEMATICS FOR 
 ENGINEERS. 
 
 i) ENGINEERING PROBLEMS (t) Advisability. 
 
 ii. Is it advisable to treat problems met in actual engineering 
 work? 
 
 Replies : Yes 190 
 
 No 20 
 
 The answers to this question while overwhelmingly in the 
 affirmative a little over 90% contained many warnings against 
 overdoing and the use of too technical material. They also strongly 
 condemned the use of such problems by instructors ignorant of the 
 physical meaning of the quantities involved. The principal objec- 
 tions were the lack of time and the lack of such problems. 
 
 Comments : 
 
 "Very seldom. We want principles and not applications." 
 "No, because such problems have too many unknown quantities in them." 
 "No, k is better to use ideal problems of physics, chemistry, etc.'' 
 "Yes, but there are objections : Not sufficient time. No sucih problems 
 perhaps at hand. Difficult to get problems of requisite simplicity and near 
 enough to actual engineering problems to be of any value." 
 "Yes, without duplicating teaching." 
 
 "Yes, adds interest but they must necessarily be elementary." 
 "Simple problems. Do not make the common mistake of using appli- 
 cations which are Greek to the students." 
 
 "Yes, thorough drill in problems met with in land surveying." 
 "Regarding the problems from actual engineering work, I would say 
 that I should favor very strongly th introduction of such problems, pro- 
 vided they are so selected as to be within the reach of the student's present 
 knowledge. They should not be absurd, as are some problem's; i.e., they 
 should correspond to some real facts, and they should not be fake practical 
 problems. They should not be a statement of such a somebody's formula, 
 where the various letters mean certain unheard-of things. These restric- 
 tions limit the possible material enormously, but I am convinced that they 
 should be insisted upon." 
 
 "Yes, to bring teachers of mathematics into close touch with teachers 
 of engineering."
 
 84 MATHEMATICS FOR ENGINEERS. 
 
 ('). Nature of Such Problems. 
 
 12. If so, to what extent cuid under which topics of (2), (4) 
 and (7)? 
 
 Replies : 
 For (2} 
 
 a. Derivation and manipulation of formulae 1 1 
 
 b. Solution of triangles by use of natural function 19 
 
 c. Use of logarithms in solution of triangles and other com- 
 putations 35 
 
 d. Problems applying the solution of triangles 39 
 
 For (4) 
 
 a. Review of surds, exponents, quadratics, etc 20 
 
 b. Series 19 
 
 c. Binomial Theorem 13 
 
 d. Permutations and combinations 4 
 
 e. Graphs 26 
 
 /. Determinants I 
 
 g. Theory of equations 12 
 
 For (7) 
 
 a. Algebraic relations for intersection of loci, tongency, 
 etc 18 
 
 b. Loci problems 28 
 
 c. Right line, and circle 23 
 
 d. Conies 17 
 
 e. Higher curves, as cycloids etc 7 
 
 /. Coordinates of points in space with application to simple 
 
 loci 7 
 
 The answers which referred to explicit topics in questions (2), 
 (4), and (/) are embodied in the above table. This means that 
 thirty-nine replies suggested d for question (2). d was not the 
 only suggestion in these thirty-nine replies, however, as several 
 contained two or three. While the number of answers received is 
 not so very large the opinions expressed are quite uniform, as will 
 be seen by referring to the above table.
 
 CURRENT THOUGHTS ON VITAL QUESTIONS. 85 
 
 Comments : 
 
 "To limited extent only under any topic needed in solution." 
 
 "Only a few problems. The tendency in modern text-books, is to treat 
 -too many problems of the kind." 
 
 "In all, mainly in (2) and (7)." 
 
 "Graphic solutions wherever possible." 
 
 "Follow a principle with an illustration." 
 
 "Whenever possible. They should require little technical knowledge, 
 It is easy to over do this thing." 
 
 "Problems applying solution of triangles, series, graphs, conies and higher 
 .curves." 
 
 "Paths for (7). Polygon of forces for (4) and (<?), (2)0 resolution of 
 forces, bridge structures, etc., (7)0 gear teeth, (/)/ intersections of solids, 
 with models." 
 
 "(2)b, c, d surveying and railroad curves." 
 
 "To awaken interest and show relation between abstract and applied 
 mathematics: (2) b, d; (4) c, f, g, d; (7). Call attention to their appre- 
 ciation rather than problems." 
 
 "The engineering problems should come as exercises of the application of 
 mathematical principles, rather than as a means of teaching those principles. 
 These illustrative problems should be introduced in connection with every 
 topic possible. I believe that the reason so much of the mathematics studied 
 by the engineering students falls into disuse and becomes forgotten, is that 
 the average student has to be shown how to use practically what has been 
 studied theoretically, and if not shown, will never get the practical use out 
 of what he has learned. The mind seems to be so occupied in grasping the 
 mathematics involved, that the application is lost unless specifically illustrated." 
 
 2) ADVISABILITY OF A COURSE IN COMPUTATION. 
 /j. Is a short course in computation advisable f 
 
 Replies : Yes 124 
 
 No 49 
 
 Several who favored such a course added either qualifying 
 statement, "if there is time for it," or "if there is need for it." 
 Many who opposed such a course were really in favor of the plan 
 except for time it would require. In one quotation under question 
 20 an important point is brought out; namely, that while students 
 are continually admonished to be accurate, they are, as a rule, told 
 very little about the degree of accuracy required by the various 
 problems encountered and given next to no practice in the same.
 
 86 MATHEMATICS FOR ENGINEERS. 
 
 Comments: 
 
 "Good, but no time for it." 
 
 "Should be emphasized in all courses but not given as a separate course." 
 
 "A large number of computation problems should be given in connection 
 
 with the different courses. I would not have separate computation courses." 
 "Review." 
 
 "By all means, a short course, or a course to be made as long as possible." 
 "Yes, particularly if supplemented by graphical methods; i.e., use of 
 
 slide rules." 
 
 3) SEGREGATION OF ENGINEERS IN MATHEMATICS (i) Advisability* 
 
 14. Should freshmen engineers be taught their mathematics 
 in separate classes or together u*ith those taking other courses? 
 
 Replies : Separate ............................. 136 
 
 Together ............................. 55 
 
 of the opinions expressed were thus in favor of the special 
 sections for freshmen students in engineering. A few who voted 
 "together" did so because the practical training needed by engineers 
 would also be the best for the literary and general science students. 
 The reasons given are taken up under the next question. 
 
 (ii) Reasons. 
 15. Why? 
 
 In the following quotations those under (a) state reasons why 
 separate classes are unnecessary, those under (b) why they are 
 necessary, and those under (c) why the literary and general science 
 students should take their mathematics in classes organized for 
 the engineering students. 
 
 Comments : 
 
 (a) "There is no engineering mathematics." 
 
 "Good mathematics for one is good mathematics for the other." 
 "Aids in cultivating wider social sympathy." 
 "In order to not lose its culture value apart from application." 
 "Because mathematics should be studied as a science rather than as 
 a mere working tool for engineers to be applied mechanically." 
 
 (b) "So as to emphasise engineering application." 
 
 "Mathematics for engineers has two objects: tftie development of the 
 student and the practical value."
 
 CURRENT THOUGHTS ON VITAL QUESTIONS. 87 
 
 "The engineers aim at facility in computation and the use of formulas, 
 while the art students go more into foundation and theory." 
 
 "Because the men in other courses will not probably appreciate the 
 spirit in which the subject should be attacked. Not because the method or 
 course should be different." 
 
 "Mathematical rigor need not be insisted upon. Either to select illus- 
 trative material which will apply to whole class." 
 
 "Realization of the greater certainty of the practical use made of their 
 mathematics will probably result in the average engineering student ac- 
 complishing more than will be accomplished by the non-engineering student, 
 if the former is unhampered by the company of the latter." 
 
 "Because the instruction for the freshman is the most important in- 
 struction, especially in mathematics during the four years' course. The 
 freshmen require a very specific and individual treatment as far as possible. 
 It seems so fundamental that they should be taught right ways of thinking 
 as well as right mathematical facts, that they ought to be a special and entirely 
 separate problem." 
 
 (c) "What is good for the engineer will be good for others, even the 
 practical problems." 
 
 ''The emphasis of the practical does not detract from the educational 
 nature of the subject." 
 
 4) SPECIAL TEXTS. 
 
 1 6. Are special texts in mathematics for engineering students 
 advisable? 
 
 Replies : Yes 93 
 
 No 89 
 
 Nothing of importance was brought out by this question other 
 than the results given above and the quotations which are appended. 
 
 Comments : 
 
 "They are not as a rule satisfactory." 
 
 "Rather special teachers." 
 
 "Only if it is written more clearly." 
 
 "Not necessary. Supplementary material should be chosen with re- 
 ference to students." 
 
 "Such texts may save the students valuable time." 
 
 "No, all texts should be given a turn toward engineering." 
 
 "Perhaps for analytics and calculus but not for algebra." 
 
 "Only for analytic geometry where there is room for a real working 
 text." 
 
 "Yes, if the teacher can not supply what is lacking in the theoretical 
 text-book." 
 
 "Yes and no. Theoretically it would, but I know of no such texts."
 
 88 MATHEMATICS FOR ENGINEERS. 
 
 VI. PEDAGOGIC QUESTIONS. 
 
 1) QUIZZES. 
 
 77. How often should written quizzes be given? 
 
 Of the 181 formal answers received 32 thought quizzes should 
 be given monthly, TO every there weeks, 55 every two weeks, 22 
 weekly, 25 frequently, I daily, I seldom, and 35 at the option of the 
 instructor. The general expression was, of course, that no fixed 
 rule could be given. 
 
 Comments : 
 
 "As often as instructor can stand them. Should be left entirely to- 
 him." 
 
 "Frequently, with enough time given to allow the students to think out 
 the problems." 
 
 "In my work I find it so full of interest to the pupils and the time at 
 my disposal so limited th&t I dread to lose good time for the purpose of 
 a quiz. Every quiz means a lot of good information lost to the class." 
 
 2) MODES OF CONDUCTING CLASSES. 
 
 18. Number i, 2, 3, etc., the order of usefulness of the follow- 
 ing modes of conducting classes, suggesting any further modes or 
 combinations of these: 
 
 a. Assignment of work from text without any previous ex- 
 
 planation. 
 
 b. Assignment of zvork from text ivith previous explanation. 
 
 c. New work explained, the student taking notes rvith or with- 
 
 out the use of a text. Written or oral quizzes at intervals. 
 
 d. Students led to discover new principles by suggestions and 
 
 quizzes on the old. 
 
 e. Each student working independently, going as fast as he 
 
 is able. This may apply to the entire course, to each topic 
 or to each day's work. 
 
 f. First part of the hour de^foted to the class as a whole; second 
 
 Part given to individual work. 
 
 g. Subject treated from the laboratory standpoint; most of the 
 
 work being done in the class room.
 
 CURRENT THOUGHTS ON VITAL QUESTIONS. 89 
 
 Replies: I. II. III. IV. V. VI. VII F. 
 
 a 62 18 20 17 ii 12 7 774 
 
 b 
 
 c 
 
 d 
 
 78 
 
 40 
 
 12 
 
 9 
 
 8 
 
 4 
 
 5 
 
 916 
 
 30 
 
 24 
 
 29 
 
 18 
 
 12 
 
 8 
 
 9 
 
 632 
 
 i6 
 
 23 
 
 29 
 
 17 
 
 2O 
 
 7 
 
 6 
 
 543 
 
 7 
 
 7 
 
 II 
 
 ii 
 
 13 
 
 22 
 
 28 
 
 271 
 
 40 
 
 37 
 
 31 
 
 30 
 
 II 
 
 4 
 
 2 
 
 820 
 
 ii 
 
 19 
 
 8 
 
 16 
 
 17 
 
 21 
 
 2O 
 
 39i 
 
 Several considered any definite answer to this question impos- 
 sible. The objections were that (i) these modes are so intertwined 
 with each other as to be nearly inseparable, thus making it often nec- 
 essary to use several in one recitation, (2) that different treatment 
 was necesary for different classes and (3) for different kinds of 
 work. The result of the letters was that certain definite combina- 
 tions are quite feasible, and often adhered to quite rigorously by 
 some instructors for not only a single recitation but are used as a 
 general plan of conducting classes. Some of those mentioned were 
 "a, b and d," "a and /." In a few cases one or two double periods 
 per week were given to work conducted along the line explained in g. 
 In a few of the letters e and g were considered faddish and worth- 
 less. On the other hand quite a number of instructors who could 
 not use these modes because of too large classes nevertheless ranked 
 them first and considered them to be "ideal for small classes." No 
 particular objections and no special variations in modes depending 
 upon subject matter were expressed. 
 
 Comments : 
 
 "I depend on circumstances." 
 
 "g and e desirable but generally not practicable." 
 
 "c and d can easily be overdone, also g." 
 
 "I generally combine b and /. ' 
 
 "o and /. Engineering students seem to do best if they put problems 
 and proofs on the board and explain daily." 
 
 "e is ideal if one instructor can be assigned to each three or five 
 students." 
 
 "I would favor a and /. It is a waste of time to try to explain or lecture 
 to the class until they have spent considerable time in the preparation." 
 
 "We are trying some experiments which might be classed under b, g, and / 
 but have not yet gone far enough to draw any inference. To gain good 
 results I believe it absolutely necessary that the instructor should be able
 
 90 
 
 MATHEMATICS FOR ENGINEERS. 
 
 to give individual attention to the students in his charge, and this not only 
 once a month but every hour they come to his class room or laboratory." 
 
 "The laboratory, wherever possible, is a most effective way of clinching 
 points made in the recitation room and of clearing up points that are 
 vague. A few hours' work in a laboratory having proper facilities will cause a 
 student to better grasp the real meaning of 'moment of inertia* than many 
 hours of class-room work. Laboratory and class-room work should always 
 be worked conjointly." 
 
 "My own mode is a combination of g and d. I am one of probably 
 very few teachers who can say, I have taught school twelve years and have 
 yet to conduct my first recitation. I am not sure my method could be 
 applied to the teaching of mathematics, but I surely hope such is the case. 
 I would rate e first if it were practically possible. We have faithfully tried 
 it failure." 
 
 VII. GENERAL NEEDS AND DEFICIENCIES. 
 
 Besides the needs and difficulties pertaining to any particular 
 topic as considered in question 2 to 10 there are those of a more 
 general nature as the power to think mathematically, or ability to 
 analyze problems. These are considered in the next four questions. 
 
 i) GENERAL MATHEMATICAL ABILITY (t) Need. 
 
 ip. Number i, 2, 3, etc., in order of importance the following 
 pertaining to the qualifications of the engineering student who has 
 completed the course in freshman mathematics: 
 
 a. Skill and accuracy in computations. 
 
 b. Analysis of problems. 
 
 c. Interpretations of results in solution of problems. 
 
 d. Knowledge and use of equations. 
 
 e. Representation of physical laws by means of graphs. 
 
 Replies: L II. III. IV. V. F 
 
 a 93 41 37 26 12 804 
 
 b 122 52 17 6 9 880 
 
 c 41 66 63 25 7 715 
 
 d 23 35 42 56 32 525 
 
 e 13 19 30 50 75 406
 
 CURRENT THOUGHTS ON VITAL QUESTIONS. 91 
 
 Of the general mathematical qualifications enumerated in the 
 question, the tabulated results show conclusively that a, b and c 
 are considered the chief ones, with special emphasis on b, the 
 analysis of problems. It is in this latter that the most difficulties 
 are also found, as will be seen under 20. The replies indicate that 
 the trouble students have with the application of mathematical theory 
 in problems is a case of knowing "how to do" while not knowing 
 "what to do." 
 
 Comments : 
 
 "a and b are useless without the others. They are of prime import- 
 ance." 
 
 "It is my opinion that students invariably show themselves most defi- 
 cient in analysis of problems, that is, in transferring the plain English 
 statements of a problem into mathematical form, in shape to be handled by 
 the mathematical processes with which they are often familiar." 
 
 "A practicing engineer seldom goes back to fundamentals, but works 
 largely with standard tables and formulas, and his mathematical training is 
 only of advantage to him to enable him to use these readily and intelligently, 
 and to understand the history and origin of these so that he can apply them 
 without fear of misusing or abusing them as non-tedhnical engineers are 
 likely tto do. If an engineer has the time and opportunity to go into research 
 work, he may then call into play his theoretical mathematics. It is well to 
 train young engineers for that possibility, but the great majority have little 
 opportunity for it. The mechanical branches algebra, analytic geometry, cal- 
 culus and descriptive geometry and graphics, are, however, of the utmost 
 value in the way I have indicated, and a great deal of drill work should be 
 given in these branches and plenty of problems of as original a character 
 as possible. It is of more importance to cultivate in the student the power 
 and ability to attack a problem directly and intelligently and to solve it by 
 short and sensible means than to learn many theories." 
 
 "I find difficulty in arranging the order, c is the whole thing from my 
 standpoint, if one means by that- the understanding of the theory in its 
 application to examples under that theory. I mean the grasp by the student 
 of the fundamental meaning of the subject under consideration. That is 
 the large thing. If the student does that he can be pretty well trusted to 
 get those minor details, the important formulas, the methods of solutions of 
 problems and all that, a enters very largely in trigonometry otherwise 
 -would not be of vital importance, though accuracy should always be given a 
 prominent place. In comparison with my remark under c the topics under 
 a, b, d and e are of absolutely minor importance ; still all four of these are 
 quite important and ought to be insisted upon rather largely."
 
 9 2 MATHEMATICS FOR ENGINEERS. 
 
 (it) Deficiencies. 
 
 20. In which of (/p) do students who hare completed their 
 freshman year show the greatest deficiency? 
 
 Replies: a b c d e 
 
 68 93 70 19 ii 
 
 Comments: 
 
 "They can not apply principles to practical work." 
 
 "Usually students show a greater weakness in analysis than in many other 
 topics and as a result, the accuracy, the interpretation, etc., are likewise 
 deficient." 
 
 "As an example of trouble in mathematics the following may be of 
 interest. I gave this problem to 40 senior engineers : A steam boiler when 
 tested shows an efficiency of 65%. After providing it with an economizer 
 it shows 85%. What is the percentage of saving of fuel? Half of the class 
 
 Of far 
 
 gave the answer in this fashion > ' 5 .= 30.75% and the other half 
 
 05 
 
 Q- /: f 
 
 -2- 2. 23.53%. Only one of the 40 gave a logical statement to show 
 5 
 
 why the formula he used was right." 
 
 "As regards a, one great deficiency in all students is a lack of knowledge 
 in regard to what degree of accuracy is called for in any calculation. They 
 have heard a great deal too much about accuracy, and not enough advice has 
 been given as to what is good enough for the given problem, or how to 
 determine the required degree of accuracy." 
 
 ''The weak point in the work of engineering students in the upper 
 class courses is usually their inability to apply mathematical analysis to 
 practical problems, and to formulate mathematical expressions for physical 
 facts. To my mind, the remedy for this difficulty lies in emphasizing the 
 physical rather than the philosophical aspect of mathematics, and of attach- 
 ing to every mathematical expression and operation in a problem, a definite, 
 clear-cut physical meaning. The teaching of physics and mechanics as 
 parallel courses with the calculus is a very useful method of accomplishing 
 this result. The constant use of easy practical problems, in which the student 
 can thoroly grasp the physical theory of the problem, is very useful in every 
 kind of mathematical teaching." 
 
 (Hi) Remedy. 
 
 <?/. W hat is the remedy f 
 
 The remedies proposed for difficulties met under a, may be 
 briefly summarized as follows: alwavs to demand accurate results
 
 CURRENT THOUGHTS ON VITAL QUESTIONS. 9$ 
 
 in any computation, to regard checks upon all results as a part 
 of the solution of the problem, and to urge upon all teachers of 
 secondary mathematics the necessity of insisting upon checking. 
 
 The suggestions for b and c can not be presented in any such 
 definite form as for a. Insistance upon analysis of problems from 
 the grades up, the following of every theory learned with illustra- 
 tive applications^ and the elimination of work by formulas and 
 rote as far as possible, are strong points. Practically no suggestion 
 for improvement was offered regarding d, and nothing was said 
 regarding e. 
 
 Comments : 
 
 "More concrete problems to awaken interest." 
 "Better and more attractive texts and more efficient instructors." 
 "Make students do the work rather than instructors." 
 "The remedy is to be found in further training not in the freshman 
 year." 
 
 "Application and interpretation of theory; many problems." 
 "More drill and better training in English." 
 
 "Begin with public school work and drill, drill, drill. Pretty hard in 
 one freshman year to undo years of slovenly habits and lack of thought." 
 "Before working a problem before the class give a thorough analysis 
 of the principles involved." 
 
 "Frequent exercises in analysis by use of problems slightly different. 
 Avoid monotony and sameness. Make the men defend each position and 
 make good every assertion." 
 
 "Learn to read equations in analytic geometry." 
 
 "Most decidedly more training in the use of what they are getting in 
 mathematics." 
 
 "Constant comparison and reference to actual practical work. Let stu- 
 dent remain out of school for a year and devote himself to some real 
 practical work under a boss." 
 
 "Require a rigid checking system." 
 
 "Special work in computation in connection with course in trigonometry." 
 "More drill in fundamentals and add course (13)." 
 "Require accuracy from beginning of the mathematical course." 
 "Insist on accuracy as of equal importance with other items." 
 "Have students work many problems and never use a key or answers." 
 "Give problems for home solution along the line needed." 
 "I realize that it is practically impossible in a four year course to 
 teach everything, but for the engineering graduate who expects to make 
 a financial success in his after life and not simply remain an entirely 
 theoretical man and therefore always on a salary, an employee of someone
 
 I. 
 
 II. 
 
 III. 
 
 F. 
 
 159 
 
 20 
 
 25 
 
 642 
 
 34 
 
 67 
 
 "3 
 
 349 
 
 59 
 
 108 
 
 37 
 
 420 
 
 .94 MATHEMATICS FOR ENGINEERS. 
 
 else, it has always seemed to me that if the minds of 'the students could 
 be turned a little bit more toward the practical end of the work, in addition 
 to their theoretical ideas, it would in the end be greatly to their advantage." 
 
 2) RELATIVE IMPORTANCE OF TOPICS. 
 
 22. Designate by i, 2, j, etc., the order of importance for any 
 mathematical topic of: 
 
 a. Mastery of theoretical matter involved. 
 
 b. System and neatness. 
 
 c. Accuracy in computed results. 
 
 Replies : 
 a 
 b 
 c 
 
 It is clear that the majority have placed the above in the order 
 a, c, b. Where these were said to be of equal importance all were 
 given the rank I. 
 
 Comments : 
 
 "Without accuracy all are worthless." 
 "By no means should there be any differentiation here." 
 "They are so interdependent it is difficult to give any answer. System 
 and neatness are necessary to insure accuracy." 
 
 VIII. ENGINEERING vs. A GENERAL EDUCATION. 
 
 23. Is it true, as has been said, that engineering students apply 
 themselves better than those taking a literary or scientific course? 
 
 Replies : Yes 125 
 
 No 45 
 
 Many said that as they had not had experience with both 
 classes of students they considered themselves as incompetent of 
 expressing any opinion on the question. This has reduced the numr 
 ber of replies somewhat.
 
 CURRENT THOUGHTS ON VITAL QUESTIONS. 95 
 
 Comments : 
 
 "At some institutions it is, at others not; in general it is true in the 
 east but not in the Mississippi valley and the west." 
 
 "Yes, in mathematics ; much better than the average student in literary 
 or scientific courses, but not nearly as well as the best." 
 
 "I think it is true that the engineering students apply themselves better 
 than those taking literary or scientific courses but am not sure that their 
 course is the more thorough. There are scientific students who do their 
 work much more thorough than the average engineering students, but the 
 average engineering student has a definite idea of what Hie wants and he 
 seeks those practical results." 
 
 "I believe that the training derived from an appropriately taught thorough 
 engineering course is fully the equivalent of that obtained from a general 
 science or literary course. In general, the grade of teaching done in engi- 
 neering schools is inferior to that done in classical schools, and the student 
 is not carried to that point in his work where the training is of its highest 
 developmental value. For instance, emperical machine design, taught without 
 the foundation of applied mechanics, is utterly without developmental value, 
 rational machine design has great value along such lines. When taught 
 for the purpose of developing the power of analysis in a certani branch of 
 useful knowledge, engineering courses are good. When taught tor the 
 sole purpose of turning out men capable of accomplishing certain useful 
 tasks in an orthodox and correct manner, engineering courses are of ab- 
 solutely no developmental value. The first method of engineering education 
 tends to turn out keen, clear, logical minds, capable of applying their power 
 to any task, whether it is a problem in thermodynamics, in machine shop 
 practice, or in public finance. The second method tends to turn out men of 
 immediately available but limited 1 utility, able to fill but a narrow place in the 
 life of their community, and having an education but very little better than 
 that acquired by the majority of skilled laborers." 
 
 IX. SUMMARY. 
 
 Replies to the questionarire letter were received from instructors 
 of mathematics and of professional subjects, and from practicing 
 engineers. A majority of the replies favor entrance examinations 
 in mathematics for all students while some very grave objections 
 are raised to such a practice, but there is a general demand for better 
 preparation, especially in algebra. A short review, together with 
 tests, has been found a good substitute in several institutions. Funda- 
 mental principles and the elements of the subjects taught are con- 
 sidered of primary importance, and in these are also found the
 
 9 6 MATHEMATICS FOR ENGINEERS. 
 
 greatest deficiencies, as remedies for which greater intensity of 
 work and less extension in subject matter, together with a vitaliza- 
 tion of the work are advised. The object of analytic geometry is 
 to furnish the student with a new mathematical language rather 
 than information concerning any particular class of curves, and 
 the introduction of the elements of the calculus in connection with 
 analytic geometry is favored in a majority of the replies. Inability 
 to analyze problems and to use the mathematical language learned 
 are the chief general difficulties reported, and frequent application 
 of the principles learned and teaching for future use may be helpful 
 as remedies. It is deemed advisable to teach engineering students 
 in separate classes, making some use of problems met in actual 
 engineering work for the purpose of creating interest, but special 
 texts are not strongly favored. There is a demand for work in 
 computation, with especial attention to degrees of accuracy. Little 
 attention is paid to questions concerned with modes of instruction, 
 showing the need of professionally trained teachers. Engineering 
 students are thought to be more thoro and painstaking than literary 
 students in general.
 
 CHAPTER V. 
 
 PRESENT NEEDS AND TENDENCIES. 
 
 I. CONTENTS OF CHAPTER. 
 
 Various questions regarding the needs of the work in mathe- 
 matics for freshmen engineers and the trend of thought of the 
 present day concerning it have arisen in different connections in the 
 preceding chapters. The discussion of such questions has been held 
 in abeyance until the present chapter when each can be taken up in 
 its entirety in the light of the accumulated information. Besides 
 the statements found in the previous chapters, use will be made of 
 the few works published on the subject, articles in the various 
 journals, proceedings of meetings of societies, personal interviews 
 and personal observations. As far as possible the various sug- 
 gestions for improvement have been tested by the writer's own 
 experience. 
 
 II. PREPARATORY MATHEMATICS. 
 
 If, as has been said, a man's education should begin with his 
 great grandparents, then surely the mathematical education of the 
 student of engineering should begin, at the very least in the high 
 school. The general demand for better mathematical preparation of 
 the students entering the freshman year makes the question of pre- 
 paration an essential part of our discussion. 
 
 i) NATURE OF WORK NEEDED. 
 
 The preparatory mathematics of prospective engineering stu- 
 dents should be taught with a view to its utilization. Especially 
 is this true of algebra, where each principle should be studied for 
 its future use, all the work being grouped around the equation as the 

 
 98 MATHEMATICS FOR ENGINEERS. 
 
 predominating feature. The work in geometry may profitably be 
 made more observational, and a greater number of practical prob- 
 lems introduced, especially in solid geometry. 1 Special classes 
 in high school mathematics for prospective students of engineering 
 are impracticable but all the work should be given with a view to 
 its possible application. The suggestion that the collegiate mathe- 
 matics be given from the standpoint of utility applies even more 
 forcibly to preparatory work. 2 
 
 (i) Algebra to be Treated as a Study of Equations. 
 
 ^ The writer had the opportunity, a few years ago, of teaching 
 beginning high school algebra simultaneously to two sections of 
 about equal industry and ability. With one the text was followed 
 quite closely and the results showed the usual average of interest 
 and achievement. The other section, as an experiment, was given 
 an extended course in equations, all other work being taken up as 
 necessary adjuncts. The pupils were introduced to the equation 
 through problems of arithmetic and were shown (not told) that 
 algebra is only an extension of arithmetic. This was done somewhat 
 in the following manner: two or three problems were given on 
 the relation between the sides of the right triangle which had been 
 learned in arithmetic, for example, finding the height of a tree which 
 had been blown down in such a maner that it is still in contact with 
 the stump, which is six feet high, and the end reaches the ground 
 eight feet from the foot of the stump. Then a problem was intro- 
 duced regarding a tree broken off in a similar manner but where 
 the unknown number had to be used implicitly, as where the length 
 of the tree and the distance of its top from the base of the stump 
 were given, the length of the part broken off being asked for. 
 Where the results were whole numbers some solutions were always 
 obtained, but for other cases none could be found. This brought 
 out the necessity of more general tools for the solution of problems 
 and the need of the "unknown quantity." At first the equation was 
 used only to solve for the unknown quantities. These were never 
 represented by X, Y, or Z but by L for length, W for weight, etc. 
 After some time the second great use of the equation was taken 
 
 1 Chap. IV, p. 81. 
 Chap. IV, p. 81.
 
 PRESENT NEEDS AND TENDENCIES. 99 
 
 up; namely, to show the relation between various quantities. The 
 graph in two variables was now introduced to give a geometric pic- 
 ture of this relation, and the connection between the graph and 
 the equation established. A number of simple graphs were made 
 either for quantities which could be read directly from the instru- 
 ments presented before the class, as from levers, the apparatus ex- 
 hibiting Boyl's Law, etc., or for those which could be easily computed, 
 as for interest, change of United States to English money, etc. A 
 little later this was extended so as to include any available data, 
 taken mostly from physics. Two graphs drawn with the same 
 coordinate axes furnished an introduction to simultaneous equa- 
 tions, where, of course, only linear equations were at first con- 
 sidered. Only one or two illustrations as to mode of introducing 
 the various topics ordinarily taught in algebra can be given. Frac- 
 tions were taken up when literal quantities were met with in the 
 denominators of fractions, radicals when such equations as X 2 = 8 
 
 had to be solved, and later on VX 4 3 yX -}- 2= ~\/X i, 
 etc. The modes used to introduce the various topics may have been 
 very artificial but they secured the desired results ; they created 
 a great interest in the work and produced at the end of the year a 
 class capable of using algebraic equations and of interpreting 
 results obtained from its solution in a very satisfactory manner. 
 
 2) DEFICIENCIES FOUND. 
 
 The teaching of geometry has elicted relatively light criticism. 3 
 How far this is due to good teaching and how far to the fact 
 that the future work is not largely dependant upon geometry can 
 not readily be ascertained. But in algebra deficiencies abound ; the 
 principal ones relating to surds, exponents, quadratics, imaginaries 
 and the manipulation of algebraic symbols. The one chief objection, 
 however, to the high school algebra as now taught is that the students 
 can not apply it* 
 
 3) CAUSE OF DEFICIENCIES. 
 
 There are really two causes of deficiencies ; those due to the 
 curriculum and the school management, and those due to the teacher. 
 
 8 Chap. IV, p. 71. 
 Chap. IV, p. 91.
 
 woi 
 
 sub 
 of, 
 
 ioo MATHEMATICS FOR ENGINEERS. 
 
 (*) School Management. 
 
 The school management is responsible for four great causes of 
 poor preparation: 
 
 Fully 75% of the difficulties are due to the long interval between 
 the work in algebra in the high school and that of the freshman 
 year. 
 
 Students are not always required to attend classes to obtain 
 the credits presented for entrance. This arises in two ways ; first 
 when students take all of the work in algebra at an unaccredited 
 school, and then go to one which is accredited without having to 
 "prove" their grades at the latter, and secondly, where a student 
 is permitted to do part of the work outside of the regular class 
 and "pass an examination" on the work so as "to graduate with his 
 class," or for some similarly trivial reason. 
 
 In some cities work in mathematics, especially algebra, is al- 
 loted to various teachers, who have, as a rule, been selected for 
 their ability in other subjects rather than in mathematics. 
 
 The last hindrance to be mentioned is that if the high school 
 is too small to have a physical director, the mathematics teacher is 
 often the foot-ball and base-ball coach. One never hears of a 
 teacher of Latin selected because he can coach the athletic teams. 
 
 (u) The Teacher. 
 
 Teachers who devote all their time to teaching mathematics 
 may still have poor preparation or may feel little enthusiasm in 
 their work, and will consequently produce poor results. Most of 
 the work is taught as a set of heterogeneous topics bearing no re- 
 lation to each other, and with the expectation that each will be "placed 
 upon the shelf" as soon as "completed." In no way is the work given 
 with a view to future use. No wonder that the students can not 
 apply radicals or find factors when solving quadratic equations. On 
 the other hand, an enthusiastic mathematician is liable to make the 
 work too theoretical and to over emphasize the analytic side of the 
 subject. In either case the pupils' interest, the one chief element 
 of success in secondary work, is not secured. 
 
 WMMtoi 
 
 4) SUGGESTED REMEDIES. 
 
 The engineering colleges can aid in the crusade for better teach- 
 ing of preparatory mathematics by demanding utilitarian mathemat-
 
 PRESENT NEEDS AND TENDENCIES. IQI 
 
 ics from the high schools and by showing its educational value. To 
 produce results these demands must be more than mere suggestions 
 or requests. 
 
 (i) Requirement of Entrance Examinations in Mathematics. 
 
 One way of making this demand is to require entrance exami- 
 nations in mathematics of all students. 5 Besides necessitating better 
 work on the part of the high schools, such examinations would result 
 in reviews of the high school subjects before taking up the collegiate 
 work and would also acquaint the instructors with the students' 
 weaknesses. 6 To the objection that this requirement would not be 
 fair to the other departments, it may be said that this system is in 
 force with the Department of English in well recognized institutions. 
 To the added objection that a great deal of disturbance among the 
 accredited schools would result, the reply may be that this disturb- 
 ance might have a salutary effect. 7 
 
 (it) Substitute for Entrance Examinations. 
 
 To obtain the beneficial results of entrance examinations with- 
 out their objectionable features some institutions have hit upon the 
 plan of devoting the first two or three weeks to a review in algebra 
 interspersed with tests. 8 For those who are seen to be notably 
 deficient a course in the high school is recommended, and to those 
 only slightly so, further work in review is given. This plan gives 
 the students the necessary review and saves them from the loss of 
 time and the discouragement of failure. All reports showed that 
 this plan has proved a decided success wherever tried. 
 
 (Hi) Demands upon the High Schools. 
 
 Engineering colleges that admit graduates of high schools 
 without examinations should insist that the instruction in mathe- 
 matics in these schools be given by special teachers of that subject, 
 and in no case by teachers selected because of their ability as 
 athletic coaches or on other irrelevant grounds. All work for which 
 
 s 64 per cent of the answers to the questionaire letter favored entrance examinations 
 in mathematics. Chap. IV, p. 71. 
 Chap. IV, p. 71. 
 T Chap. IV, p. 71. 
 Chap. IV, p. 71-
 
 102 .MATHEMATICS FOR ENGINEERS. 
 
 credit is given should be taken in class. A review of algebra dur- 
 ing the fourth year should also be one of the requirements for ex- 
 emption from entrance examinations. 
 
 III. COLLEGIATE MATHEMATICS. 
 
 With respect to the collegiate mathematical instruction of stu- 
 dents of engineering we consider i) what should be taught, 2) how 
 it should be taught. 
 
 1) WHAT SHOULD BE TAUGHT. 
 
 Future use should be the basis of the mathematical curriculum. 
 Practicing engineers and instructors of engineering subjects are 
 best qualified to speak of the future mathematical needs of the en- 
 gineering student. Consequently, they may justly claim the privilege 
 of designating the contents of the mathematical curriculum. Such 
 suggestions from the professional men should allow for time to 
 give all prerequisite topics, and should be really as well as nomi- 
 nally mathematical, calling for not the mere teaching of "rules of 
 thumb" but for the use and application of mathematical principles. 
 
 2) How IT SHOULD BE TAUGHT. 
 
 When the matter to be taught has been determined it is the 
 province of the mathematical department to consider the designated 
 material and all prerequisite topics without extraneous matter, and 
 to organize it into a teachable course. The duty of teaching this 
 course to the very best advantage then also falls upon the mathe- 
 matical instructors. 
 
 3) SPECIFIC AND GENERAL NEEDS AND DIFFICULTIES. 
 
 At the completion of the first year's work in mathematics the 
 students are expected to show mathematical ability of two distinct 
 kinds, specific and general ; while the difficulties encountered also 
 fall under these two heads. An example of the first would be the 
 handling of radicals ; one of the second, the ability to think mathe- 
 matically. This is a helpful distinction, especially in bringing out
 
 PRESENT NEEDS AND TENDENCIES. 
 
 103 
 
 the opinions of educators and professional men interested in these 
 questions, and accordingly was adhered to in making out the ques- 
 tionaire letter of Chapter IV. 9 
 
 The specific requirements and difficulties are somewhat the 
 easier to discuss and will be taken up first. It will also be found 
 that the general ones can often be reached thru the specific and 
 disposed of while taking care of the latter. 
 
 4) SPECIFIC NEEDS AND DIFFICULTIES. 
 
 The specific needs and difficulties will be considered under the 
 subjects of trigonometry, algebra, analytic geometry and the calculus 
 both to expedite the discussion and also because most of the institu- 
 tions observe these distinctions. Pedagogic questions relative to the 
 improvement of the work will be considered in appropriate con- 
 nections. 
 
 (t) Trigonometry. a) Reason for Place in Curriculum. 
 
 Since only 18% of the institutions require trigonometry as a 
 preparatory subject it can not be omitted in considering the work 
 of the freshman year. 10 Further, the subject is of the utmost im- 
 portance in engineering work and while comparatively easy, it pre- 
 sents an astonishing number of failures, and that in the elements. 11 
 For these reasons it requires the very best of instruction and, as a 
 consequence, some institutions which admit without examination in 
 other mathematical branches require one in trigonomery. 12 
 
 b) Needs. 
 
 A thoro grasp of the meaning of the trigonometric functions 
 
 and their use in the solution of the right triangle is really the one 
 
 thing needed. 13 Besides this the student needs to know some few 
 
 relations existing between the functions, to be able to solve a right 
 
 triangle placed in any position, to pick out the triangles in a con- 
 
 Chap. IV, pp. 67-70. 
 
 "Chap. II, p. 43. 
 
 u An instructor in railroad engineering said he had "some young men who have 
 been considered as very bright in the calculus but who can make no headway because 
 they have not the foundations of trigonometry. They have no grasp of the trigonometric 
 functions and their relations." 
 
 "Chap. Ill, p. 54- 
 
 13 Chap. IV, p. 74-
 
 104 
 
 MATHEMATICS FOR ENGINEERS. 
 
 figuration essential to his problem, and to solve oblique triangles by 
 use of formulas. Advanced topics as hyperbolic functions, De 
 Moivre's Theorem, etc., are of little value to the engineering stu- 
 dent. 14 
 
 c) Deficiencies. 
 
 The most marked deficiencies are found in the topics just men- 
 tioned as of most importance. This is probably due to the fact 
 that the elements are called into play the most frequently and points 
 out the need of special care in teaching the fundamental definitions 
 and properties of the trigonometric functions. The use of identities, 
 especially those relating to functions of 2 X and l / 2 X, inverse func- 
 tions, circular functions and application of the definitions of the 
 trigonometric functions are also found to give serious difficulty 
 while still another grave deficiency is the inability to see what 
 triangles are to be solved and to solve them in positions other than 
 the standard one with one side horizontal." 
 
 d) Suggestions for Improvement. 
 
 In consequence of the deficiencies found, the work is to be 
 simplified and confined mainly to the elements. The functions should 
 be defined from the ratio standpoint and the definitions constantly 
 reviewed. To a great number of students who have learned the 
 functions with reference to the unit circle the sine means the vertical 
 line opposite an acute angle in a triangle, etc. The meaning of the 
 trigonometric functions can be made clear by the solution of numer- 
 ous right triangles, which should be placed in various positions and 
 not in any standard one. Constant application of theoretical re- 
 sults to a variety of real problems, of which a rich store lies at hand, 
 is also indispensable. 
 
 e) Trigonometry to be Taught from the Utilitarian Standpoint. 
 
 The utility of no study is more obvious to the engineer than 
 that of trigonometry. Use can be made of this fact to arouse in- 
 terest and action on the part of the student. The very first lesson 
 is the best place to bring out this utilitarian idea in order to make 
 a lasting impression and this lesson can be made to contain a large 
 
 M Chap. IV, p. 73. 
 u Chap. IV, p. 74.
 
 PRESENT NEEDS AND TENDENCIES. 
 
 105 
 
 part of the elements of the subject. This should not be done by the 
 formal statement that trigonometry will be a most necessary part 
 of the future work in mathematics and physics but by well selected 
 illustrative problems, such as finding the height of buildings, flag 
 poles and trees by using lines of known length drawn from their 
 bases along the ground, width of rivers using base lines along their 
 banks, etc. If a surveyor's transit is not available for these first 
 lessons a model should be improvised. As another means to keep 
 up interest a formal problem where solution will be within reach of 
 the students at the end of two or three weeks' work can be proposed 
 on one of the first days to be kept constantly in view for solution. 
 The finding of the resultant of several forces passing thru a point 
 by first resolving them along two directions at right angles may 
 be cited as an illustration. For the solution of problems on the 
 right triangle similar to those mentioned above only the sine and 
 tangent functions are necessary. The functions of any angle having 
 been defined the students may be required to determine the values 
 approximately from measurements, or they may be provided with 
 tables of natural functions at once. The instruction should be con- 
 tinued in the same spirit with the solution of the oblique triangle as 
 the ultimate goal. 
 
 f) Suggested Course. 
 
 In accordance with the above it is advisable to devote two 
 hours per week during the first half of the year to trigonometry. 
 The work is to be introduced by the solution of a great number of 
 practical problems solvable by means of the trigonometric func- 
 tions applied to the right triangle. This is to be followed by 
 exercises on the relation between the functions, and also the func- 
 tions of two or more angles, leading up to the solution of the 
 oblique triangle. Logarithms are taken up just before the oblique 
 triangle. A short discussion of circular functions should also be 
 included. 
 
 (tt) Algebra. a) Status in Engineering Colleges. 
 
 Of all the mathematical subjects in the engineering curriculum, 
 algebra is in the most chaotic condition. The entrance requirements 
 vary from nothing at all to three full years, with a corresponding 
 variation in the collegiate work. The difference which is most to
 
 I0 6 MATHEMATICS FOR ENGINEERS. 
 
 the point is found in the topics given under the head of College 
 Algebra. No schematic exhibition of the subject matter can be 
 made as many of the catalogs do not state what topics are included 
 under College Algebra and those which do, present too great a 
 variety for comparison. The courses seem to have in common chiefly 
 the two topics of the Binomial Theorem and the theory of equa- 
 tions, (the latter a very indefinite term, since its scope is rarely 
 stated fully), and the failure to mention any review of previous 
 work even in quadratics. As to the rest there is a wide range of 
 variation for which no definite cause can be assigned. While some 
 topics are more especially useful to students taking a particular 
 course (such as imaginaries for the electrical engineer), this special- 
 ization is not recognized by institutions offering several professional 
 courses. Neither can the reason be that algebra is given as a cul- 
 ture study. The many serious difficulties encountered in algebraic 
 reductions by students in advanced mathematical or technical work 
 make it impossible to regard algebra otherwise than as a most vital 
 foundation for subsequent work. 
 
 b) Needs and Difficulties. 
 
 The greatest difficulty is experienced in the handling of the 
 fundamental algebraic operations, with which, first of all, the stu- 
 dent must be familiar. Determinants, permutations and combina- 
 tions, and series are of minor importance. On the other hand, 
 graphical solutions of equations, graphical representations of cer- 
 tain series, the treatment of imaginaries and the solution of equa- 
 tions of the third degree call for special attention. 16 
 
 c) vSuggested Improvements. /) Teach Elements. 
 
 The first aim is to give to the student a working knowledge of 
 fundamental principles. Every effort should be made to improve the 
 preparation given by the high school 17 but where students are still 
 deficient remedies must be applied at the earliest possible moment 
 so that they may not feel a handicap later. The review of the main 
 principles of the high school algebra as previously outlined will be 
 found helpful. 18 
 
 M Chap. IV, p. 76. 
 11 Chap. V, p. 101. 
 u Chap. V, p. 102.
 
 PRESENT NEEDS AND TENDENCIES. 107 
 
 77) Principles of Algebra to be Taken Up When Needed. 
 
 It is a mistake to conduct the instruction on the assumption 
 that students know and remember every principle previously studied. 
 Perhaps the professional men ask too much of the instructors of 
 freshmen mathematics by taking too much for granted, while the 
 latter in turn require too much of the high school teachers. Time 
 can often be saved by giving preliminary reviews of principles in- 
 volved in new work to be taken up. The plan has been carried so 
 far as to give no formal course in college algebra but to take up 
 each algebraic topic as it arises in connection with other branches 
 of mathematics. This somewhat novel idea offers some promise of 
 help in alleviating difficulties which confront the teacher of fresh- 
 man mathematics. 20 The objections to such a method are first, that 
 it may have a tendency to foster "a rule of thumb" way of treating 
 each mathematical process as applicable to but a single case; and 
 second that the student does not learn the mathematical principles 
 but only a way of connecting some mechanical manipulation with 
 the particular process in question. To avoid so great an evil the 
 method must not be carried to extremes, and the principles should 
 be applied in several different phases of use. In small sections 
 taking very specialized courses this plan commends itself most fav- 
 orably. Other plans of a more radical nature are those in which 
 college algebra is taken in connection with either trigonometry or 
 analytic geometry in such a way as to bring out its points of con- 
 tact with these subjects, and its graphic aspect. 21 As the portions 
 of college algebra that are really advanced are little used before tak- 
 ing up mechanics, real college algebra can well be deferred until 
 after analytic geometry and the calculus have been taken, the alge- 
 braic work given earlier being essentially a review and an extension 
 of the elementary work. 
 
 (wi) Analytic Geometry. a) As a Mathematical Language. 
 
 Since engineering students must use algebra, and as long as 
 there exists such extreme difficulty in applying it, collegiate work 
 in algebra should not be given them as a culture study. Let the 
 
 Chap. Ill, p. 57. 
 
 a Frederick L. Emory, Advanced Algebra in Engineering and other Colleges, Proc. 
 Soc.. for Prom. Eng. Ed., vol. Ill, p. 104.
 
 io8 MATHEMATICS FOR ENGINEERS. 
 
 instructor teach that which will be met with in the future work ; for 
 example, in radicals, let him drill on a systematic use of the three 
 fundamental rules to which all can be reduced : 
 
 a) the removal of radical denominators, 
 
 b) simplification of expressions where the number under the 
 radical is a perfect power indicated by a factor of the index, as 
 
 c) simplification of expressions containing factors of which 
 the radical root can be extracted, as V 12 - 
 
 If the students aftenvards offer a result such as V8 in the 
 solution of a problem he deserves a reduction in grade. 
 
 The following few rules are quite effective in a course in col- 
 lege algebra: 
 
 a) give a short review of exponents, radicals, quadratics, etc., 
 
 b) introduce new topics by oral quizzes on the earlier princi- 
 ples that will be involved, 
 
 c) insist that principles previously learned be employed when- 
 ever possible, 
 
 d) demand an analysis of each problem and an understanding 
 of each formula used, without, however, requiring the development 
 off-hand. 
 
 Chapter III contains several suggestions for making both the 
 position and the object of college algebra definite in the curriculum. 
 
 *) Analytic Geometry. a) As a Mathematical Language. 
 
 When the student takes up the study of analytic geometry, 
 whether designated by that formal title or not, he meets with a new 
 language just as much as when he begins the study of French or 
 German. There must, of course, be some medium to which to apply 
 this new language while learning it. Until very recently the conic 
 sections have been universally selected as this medium and often the 
 learning of the properties of the conies has been the main business 
 of the course, thereby turning it into a mere memorization of a col- 
 lection of isolated properties and formulas of the curves and losing 
 track of the analytic mode of attack, which should be its chief aim. 22 
 
 22 "Too much space should not be given to the investigation of the properties of the 
 conic sections, -- etc." Mansfield Merriman, Teachers and Text-books in Mathematics in 
 Technical Schools, Proc. Soc. for the Prom, of Eng. Ed. vol. II, p. 95.
 
 PRESENT NEEDS AND TENDENCIES, 
 
 109 
 
 It is wholly immaterial what medium is used so long as it supplies 
 the requirement of teaching the language, and interests the students 
 in its use. If in addition to this it can acquaint the student with some 
 curves met with in his future work, that is an additional advantage. 
 Any problems in mechanics or geometry possessing these qualifica- 
 tions are suitable, and conies are to be treated only as varieties of 
 curves in general. 
 
 b) Illustrative Problems. 
 
 The following are a few illustrations of problems of the nature 
 just mentioned which have been found useful by the writer: 
 
 1) Find the equation of a projectile given an initial horizontal 
 motion. 
 
 2) Ditto if thrown at an angle to the horizontal. 
 
 3) Prove that parallel rays can be brought to a focus by a para- 
 bolic reflector. 
 
 4) Proof of the method of inscribing a parabola in a rectangle 
 as used in mechanical drawing constructions. 
 
 5) Ditto for hyperbola. 
 
 The following are examples of curves for study: 
 
 i e
 
 no MATHEMATICS FOR ENGINEERS. 
 
 c) Other Devices. 
 
 It is in the application and in the appreciation of the analytic 
 language that the most frequent difficulties arise. 23 Besides the prob- 
 lems mentioned above several devices have been found useful, one 
 of which is the analytic proof of theorems found in Euclilean geom- 
 etry. Such problems as the following may be assigned as a part 
 of the following day's work : 
 
 1) "From the middle point D of the base of a right angled 
 triangle ABC, DE is drawn perpendicular to the hypotenuse BC. 
 Prove that BE 2 EC 2 = ABV 
 
 2) "CO is the line drawn from the center of a circle to any 
 point of the chord AB. Prove OC 2 = OA 2 AC X CB." 2 * 
 
 For off-hand solutions in the class room the most efficient are 
 the simpler ones involving the principles at hand, such as, to prove 
 that the perpendiculars from the vertices of a triangle upon the op- 
 posite sides meet in a point. 
 
 For bringing out the true analytic spirit the "hammer and tongs 
 process" may be used in which the unknown relations are given 
 analytic expressions however long these may be, and then applied 
 as tests to the theorem to be proved. In illustration the problems 
 above stated may be cited. The expressions for the quantities in the 
 formulas are worked out and with these the truth of the statement 
 verified or disproved. Such work is often very long but is of great 
 assistance in bringing the student to see the power of the analytic 
 expressions of geometric facts. 
 
 A full understanding of the idea of parameter is a decided aid 
 to a clear grasp of problems in which loci are found by eliminating 
 a parameter from two or more equations. The parameter should 
 be taken up early and its meaning for any particular equation dis- 
 cussed some time before the loci problems depending upon its elim- 
 ination are attempted, as otherwise the idea of the parameter is lost 
 in following the mechanical process of elimination. The parameter 
 is also useful in illustrating families of curves and in showing their 
 variations. In aiding a class to see the deductions of the equation 
 of a locus when the parameter was a point (X', Y') the writer hit 
 
 Chap. IV, p. 77- 
 
 84 Sanders, Elements of Plane Geometry. New York, 1901. p. 216.
 
 PRESENT NEEDS AND TENDENCIES. m 
 
 upon the mechanical device of using red chalk for marking the locus 
 in the graph and also for the corresponding X' and Y' coordinates 
 in the equation of the required locus. At least 90% saw at a glance 
 the relation between the graph of the locus and its equation. In 
 fact, before the solution was fully written out, the young man who 
 had asked for it exclaimed, "I could have done it if I had had 
 colored pencils." The above incident gives an excellent illustration 
 of the value of the maxim "Teach through the eye." 25 
 
 d) Needs and Difficulties. 
 
 The principal difficulties naturally arise in the same topics as 
 do the greatest needs. Besides the one great need of learning the 
 analytic language a very few special ones may be found, of which 
 loci problems, intersection of curves, tangency and the sketching 
 of curves from the equation ought to be mentioned. 26 Regular prac- 
 tice in curve sketching and the use of equations in their simplest 
 form with slightly varied terms like the following has been found 
 helpful to remedy defects in the latter case: 
 
 I) 
 
 y 2 = I2x 
 
 2) 
 
 x 2 = I2y 
 
 3) 
 
 y 2 = I2x 
 
 4) 
 
 x 2 = I2y 
 
 5) 
 
 y 2 = 12 (x 3) 
 
 6) 
 
 x 2 =i2(y-U 3 ) 
 
 (iv) The Calculus. a) Principles Taught. 
 
 Of late years the elements of the calculus have been introduced 
 more or less into the freshman work by various institutions. 27 The 
 general idea is to acquaint the student with the methods of the cal- 
 culus rather than to enter into the theory to any extent; the formal 
 course being delayed until the second year. Principles are developed 
 by problems and graphs, and the methods of the calculus are used 
 before being rigorously proved. 28 The work is, as a rule, limited 
 
 M J. W. A. Young, The Teaching of Mathematics in the Elementary and the Second- 
 ary Schools. New York, 1907, p. no. 
 *" Chap. IV, p. 78. 
 Chap. Ill, p. 65. 
 "Chap. Ill, p. S3-
 
 H2 MA THEM A TICS FOR ENGINEERS. 
 
 to problems on tangency and on maxima and minima, the former of 
 which affords a natural means for its introduction. 
 
 b) Advantages. 
 
 This early introduction to the use and methods of the calculus 
 and an acquaintance with its powerful machinery, is expected to 
 meet the demand for the calculus in practical problems. It gives 
 the most powerful and the most suitable instrument for solving a 
 great number of problems that arise in algebra, analytic geometry 
 and physics, in the first year or early in the second year. Further, 
 the calculus will not only be thus applied more extensively but will 
 also be studied during a materially longer period of time, which is 
 a decided advantage. The interest of the student is also materially 
 enhanced when the work is directed along the lines of what can be 
 done with the calculus rather than with the development of formulas 
 and their rigorous proofs. Such an introduction of the calculus 
 should make the student feel that it is not a wholly new and distinct 
 branch of mathematics but only a continuation and natural out- 
 growth of what has preceded. While the data previously recorded 
 do not show any marked tendency actually to make this early intro- 
 duction to the calculus, yet the trend of opinion is really in its favor 
 since all who reported having tried the plan had found it a success. 29 
 
 (v) Combined Courses in Algebra, Coordinate Geometry and Ele- 
 ments of the Calculus. 
 
 Coordinate geometry may be made the basis of a combined 
 course in algebra, coordinate geometry and the elements of calculus 
 to occupy two or three hours the first half and four or five hours 
 the second half year. After a two or three weeks' review of the 
 most needed preparatory algebra, the graphic representation of the 
 algebraic function is taken up, where such problems as finding points 
 of intersection are solved and such elementary notions as determin- 
 ing the distance between two points are developed. Algebraic prin- 
 ciples are thereafter taken up as occasion demands. A full treat- 
 ment of the right line and circle follows. A study of any suitable 
 interesting curves, including conies, completes the work in plane 
 analytic geometry, where the chief aim is to familiarize the student 
 
 * Chap. IV, p. 80.
 
 PRESENT NEEDS AND TENDENCIES. n 3 
 
 with this new mathematical language and to train him in its use. 
 Special stress is placed upon loci problems throughout. The meth- 
 ods of the calculus are to be used in the tangent problems, the find- 
 ing of maxima and minima of functions, and all other problems 
 which are best treated by such methods. No introduction to the 
 calculus other than the application of its elementary principles to 
 problems is to be attempted. Differentiation and integration are to 
 be carried on simultaneously. If time permits geometry of three 
 dimensions may complete the course, but this can be delayed with 
 advantage until the study of partial derivatives. 
 
 (vi) Work in Computation. a) Mode of Presentation. 
 
 The need of drill in computation is evident, 30 the only question 
 being as to the mode of presentation. Two modes suggest them- 
 selves : one, the formulating of such work into a special course as 
 has been done by some institutions, 31 and the other its distribution 
 throughout the mathematical course. While the former plan would 
 emphasize the necessity of such drill the latter is rather to be pre- 
 ferred as it will show the necessity of accuracy at all times and en- 
 hance the value of the drill by extending it over a much longer 
 period. Besides the assignment of special exercises in computation 
 as occasion arises it has been found to be quite helpful to require all 
 final expressions to be reduced to their simplest form for computa- 
 tion ; that is so as to involve the minimum amount of arithmetical 
 operations. For example, let radicals always be reduced to the form 
 
 -r-\j c which is the simplest for computation by the use of tables. 
 
 Let the student employ tables, even though unacquainted with the 
 theory involved in their construction. 
 
 By far the most potent aid in achieving accuracy is the applica- 
 tion of checks upon the work, which is always possible except in rare 
 instances. To impress the need of such checks make them a part of 
 the problems assigned and to emphasize it still further change the 
 data of the problems in the text or assign outside ones in which 
 accuracy is the main requirement. The importance of this phase of 
 the problem work for students of engineering can not be overesti- 
 
 Chap. IV, p. 85. 
 
 31 Chap. Ill, pp. 62-63.
 
 MATHEMATICS FOR ENGINEERS. 
 
 mated. An added advantage of no mean importance gained from 
 such a system of checks is the drill in insight and ingenuity which 
 it gives. 
 
 b) Degrees of Accuracy. 
 
 Of nearly equal importance with accuracy itself is the ability 
 to understand the degree of accuracy of which a problem is capable. 
 It is worse than useless to permit a student to carry out computa- 
 tions to decimal places beyond those for which the data are known 
 to be correct. The three following problems, selected from differ- 
 ent books on trigonometry, may serve as illustrations showing how 
 a much greater degree of precision is demanded in the result than 
 is found in the data if the latter are to be considered as having the 
 degree of precision ordinarily met with : 
 
 1 ) The height of a house subtends a right angle at a win- 
 dow on the other side of the street ; and the elevation of the 
 top of the house from the same place is 60. The street is 30 ft. 
 wide. How high is the house? 
 
 The result is given as 69.282 ft. This result, professing to ex- 
 press the height of the house within a thousandth of a foot, implies 
 that the width of the street has also been measured with this degree 
 of accuracy. This is not customary and not even readily possible. 
 To do so it would be necessary that the sides of the street them- 
 selves be fixed within a possible error decidedly under a thousandth 
 of a foot, and that measuring instruments of corresponding delicacy 
 be employed in measuring the width of the street. In practice these 
 conditions are never fulfilled. A good question would have been 
 to ask for the height in feet to within an error of I ft. or of I in. 
 
 2) From the top of a hill I observe two mile-stones on 
 the level ground in a straight line before me and I find their 
 angles of depression to be respectively 5 and 15 ; find the 
 height of the hill. 
 
 The result, given as 228.6307 yds., professes to give the height 
 of the hill within one ten-thousandth of a yard, or within about 
 1/250 in., which is much less than the presumable error in the plac- 
 ing of the mile-stones, and which also implies that the height of the
 
 PRESENT NEEDS AND TENDENCIES. 115 
 
 hill is a quantity that can be determined to within the diameter of 
 a grain of sand. 
 
 3) A ladder 40 ft. long reaches a window 33 ft. high on 
 one side of the street. Its foot being at the same point, it will 
 reach a window 21 ft. high on the opposite side of the street. 
 Find the width of the street. 
 
 The result is given as 56.649 ft., to which the same criticism is 
 applicable as to problem i). 
 
 When accuracy to two decimal places is meant such expressions 
 as 46 in. ought never to be allowed but 46.00 in. insisted upon. Both 
 from the engineering standpoint and from that of general mathe- 
 matical accuracy these conditions should be considered in all compu- 
 tations. 
 
 5) GENERAL NEEDS AND DIFFICULTIES. 
 
 Several needs and difficulties of a general nature have already 
 been considered in connection with the various branches of study. 
 The following are further general phases which have not been 
 touched upon: 
 
 (*') Cooperation with the Professional Departments. 
 
 The success of the instructor of mathematics can be greatly 
 enhanced if the instructors of the professional branches will, from 
 time to time, supply him with lists of mathematical difficulties en- 
 countered by their students and with the suggestions as to the prep- 
 aration needed for their respective courses. It is not meant that 
 the department of mathematics should duplicate the work of the 
 professional departments in any sense, but only supplement it so as 
 to increase the efficiency of the work of both departments. It is 
 not meant to teach a special "rule of thumb" for each particular 
 thing to be taken up in the future but that the principles and results 
 that will be necessary later are to be emphasized, in precisely the 
 same way as the instructor of trigonometry emphasizes certain prin- 
 ciples and results to be used later in the calculus, mechanics, etc. In 
 this day of correlation it seems surprising that definite steps have 
 not been taken to any great extent in the direction just mentioned. 
 The special courses in mathematics outlined in Chapter III have
 
 1 1 6 M A THEM A TICS FOR ENGINEERS. 
 
 been arranged more or less according to this plan, it is true, and 
 one is based altogether upon the work to be taken in the professional 
 departments during the last two years but the plan has received no 
 wide recognition. Surely such cooperation of the departments would 
 prove greatly beneficial by increasing the interest of the students in 
 their mathematical work thus having its service and utility pointed 
 out, by remedying much of the weakness of the students in applying 
 their mathematics to the work of the professional courses^ and finally 
 by fostering more congenial relations between the mathematical and 
 professional departments. 
 
 (it) Problems z) Their Use. 
 
 Every principle studied must be illustrated by a sufficient num- 
 ber of problems, for the purpose of fixing it in the minds of the 
 students and of clearing up any difficulties which may have arisen 
 as to its application. To emphasize and fix the mathematical lan- 
 guage many short problems should be given for immediate solution 
 during the class period : the shorter the problems and the greater the 
 number, the better. 
 
 b) Kinds of Problems. 
 
 Problems are like puzzles: if too simple they do not hold the 
 attention and if too difficult they extinguish all ambition to solve 
 them. Long and complicated problems, such as those of the old 
 "fox and hound" found in elementary algebras only a few years ago, 
 ought to be discouraged. Simplified explanations with a large num- 
 ber of simple problems will greatly aid in preventing students from 
 doing things by rote and from imitating imperfectly some sample 
 problem which has been worked out in the text-book. In order to 
 interest the students the problems should not be fantastic but related 
 to their every day life as far as possible. For the purpose of creat- 
 ing further interest the use of problems connected with engineering 
 work have been suggested. 82 There is, of course, danger of over- 
 doing and great care must be exercised in the selection and in the 
 preparation of such problems. They must not be "fake" engineer- 
 ing problems, nor too technical nor contain too many elements. To 
 make such a problem usable the useful parts may have to be selected 
 
 ' Chap. Ill, pp. S9-6i. Chap. IV, p. 83.
 
 PRESENT NEEDS AND TENDENCIES. 117 
 
 and put into such form that the student can comprehend it. On 
 the other hand the data should not be given in such a manner as to 
 make the solution a mere matter of substitution in a formula. 
 
 c) Problems to Develop Mathematical Thought. 
 
 The student who can not think mathematically has at best a 
 superficial knowledge of the mathematical language and consequent- 
 ly is unable to apply it. "Eternal vigilance" on the part of the in- 
 structor is necessary to see that his students are getting a grasp of 
 this one essential and are really thinking the problems out and not 
 using a "sleight-of-hand" trick for doing each set of problems. A 
 good illustration along this line is furnished in the study of asymp- 
 totes. The student should not be permitted to write down the equa- 
 tion of the asymptotes of 4y 2 Qx 2 = 36 as 2y 3x = o, off-hand 
 unless he can find the values of M andK which willmakey=Mx K 
 an asymptote. To insure that he really does this, he can be 
 required to find the asymptotes of 4x 2 6xy -f- 4x + 6y y 2 = 10, 
 of an ellipse or of some curve as y (2a x) =4ax. Avoid such 
 text-books as differentiate the classes of problems too highly, and 
 such as introduce each set by some one worked out in detail, leaving 
 nothing for the student but the substitution of numerical values. 
 When a problem has been solved vary the conditions just a trifle 
 and test the class on the new aspect. 
 
 6) PEDAGOGIC CONSIDERATIONS. 
 
 The following questions on the pedagogy of the subject are of 
 a general nature, not connected with any particular topics or princi- 
 ples and hence will here be discussed separately. 
 
 (*) Modes of Instruction. 
 
 It is not supposed that any instructor will hold to any one 
 mode to the exclusion of all others, nor to any fixed combination 
 at all times. Some combination of the previously mentioned modes 
 is, however, generally followed by each instructor. 33 But even 
 though an instructor may have been very successful with the use of 
 one mode still that might not be the one with which he could be the 
 
 w Chap. IV, pp. 88-90.
 
 n8 MATHEMATICS FOR ENGINEERS. 
 
 most successful. Nor does it seem likely that the instructor will in- 
 tuitively select the mode best suited to each case. The selection 
 consequently deserves careful study. 
 
 () Suggested Combinations on Modes. 
 
 The following combinations of modes are recommended as the 
 most useful : 34 
 
 A) Assignment of work with or without explanations and 
 the laboratory mode. 
 
 B) Heuristic and laboratory modes. 
 
 C) A modification of the heuristic mode applied to the 
 class as a whole during the first half of the period, the second 
 half being devoted to the individual students while working 
 problems. This is in the main as follows: the work for the 
 following day is assigned first, avoiding hurry at the end of the 
 period and incidentally giving an incentive for prompt attend- 
 ance. The theoretical matter for the day is then taken up in 
 the form of an oral quiz, in connection with which short illus- 
 trative problems are solved. This is intended to clear up points 
 of difficulty, to fix important matters and to insure preparation 
 from day to day. Whenever possible the fundamental prin- 
 ciples of the next assignment are taken up in connection with 
 the quiz on the lesson of the day and are worked out by the 
 class with slight suggestions from the instructor. Should it be 
 impossible to connect the future work with that of the day, the 
 class is reviewed on past work with which it can be connected. 
 About half of the hour is consumed in this manner. The re- 
 mainder of the time is given up to individual work by the stu- 
 dents on problems either at the boards or at their seats, pre- 
 ceded when necessary by a short general discussion on particu- 
 lar difficulties in the day's problems. During this work on prob- 
 lems the instructor "visits" each student, quizzing him and giv- 
 ing him suggestions. Individual teaching is approached as 
 much as the size of our present day classes will permit. As the 
 instructor is dealing with the individual he has even more free- 
 dom to select the mode "best suited to the case" than if he had 
 
 M Chap. IV, p. 88.
 
 PRESENT NEEDS AND TENDENCIES. 119 
 
 to consider the class as a whole, but is, of course, more limited 
 as to time. One of the real dangers is the temptation to slight 
 the good; students in favor of the mediocre, or vice versa. A 
 college class of from twenty to twenty-five and a high school 
 class of from twenty-five to thirty can be handled successfully 
 in this way. 
 
 (MI) Special Texts in Mathematics. 
 
 The texts used in mathematics are of various forms and de- 
 grees of specialization. A first class consists of those which treat 
 each topic separately and with little connection with the succeeding 
 material or future application, but lay special stress upon complete- 
 ness of the proofs and logical sequence. As a rule the problems are 
 abstract and play a minor part. A second class is not nearly so pro- 
 fuse in proofs but still preserves the sharply separated topics. The 
 problems are not so abstract but often impossible or too hypothetical. 
 This is what may be termed the usual old standard form of texts. 
 The third class confines itself to definite topics, but not so fully as 
 the two previous classes, and these topics are selected with a view 
 to their future use as the primary object. Points of special interest 
 to engineers are brought out, as use of the slide rule, degree of 
 accuracy in computations, etc. For the most part the problems are 
 simple, real and sensibly shorn of the fantastic. Those of the fourth 
 and last class are constructed along quite radical lines. They dis- 
 regard the divisions of algebra, analytic geometry and the calculus 
 and form a homogeneous whole of the three for use over a period 
 of from one to two years. Often these are written for the needs of 
 the students at some particular institution or of a particular man, 
 and as such may not be generally useful. 
 
 There is quite a division 7 of opinion as to the advisability of 
 using texts of the fourth class. In several instances where highly 
 modified courses are given, texts of the second or third class, sup- 
 plemented by problems and additional material of a more decided 
 engineering nature, are used. 
 
 The early introduction of the ideas of the calculus already dis- 
 cussed (p. in) is quite feasible under such a plan. The methods 
 of the calculus that are used would be developed in the class by the
 
 120 MATHEMATICS FOR ENGINEERS. 
 
 teacher as needed, and it is quite possible that the student would 
 thus appreciate these new appliances better than if he were intro- 
 duced to them in a more formal way. 
 
 (iV) The Profession of Teaching. 
 
 The seemingly prevalent idea that no attention need be given 
 the question of instruction must be abandoned before results of the 
 desired excellence are obtained. 33 Surely the history of mathemati- 
 cal teaching in this country, and especially in technical work, does 
 not show it to be so simple that the unprepared beginner can do the 
 work properly. It is gratifying to note that attention is being di- 
 rected to this most vital question and it is hoped it will soon receive 
 attention commensurate with its importance. 86 The reason for the 
 present excellent instruction in Latin is the fact that the teachers of 
 Latin are trained to teach; that the pedagogy of the subject is one 
 of the main things dwelt upon; that the Latin departments of our 
 advanced institutions give "teachers' courses" not only for pros- 
 pective instructors of secondary schools but also for those of colle- 
 giate grade. 87 One of the absolute requisites for good* teaching of 
 mathematics at engineering colleges is the proper preparation of 
 instructors as teachers and the placing of mathematical instruction 
 on the plane of a profession. This applies especially to teachers of 
 freshmen who are often without wide experience, while the math- 
 ematics given in the freshman year is largely the foundation of the 
 whole professional course. 
 
 Chap. IV, p. 89. 
 
 In a recent address Prof. Alexander Smith, of Columbia University, said that 
 life is a series of lessons in "problem solving" and that no subject was fit for the college 
 curriculum which could not be reduced to that basis. Continuing he said: "If the 
 American College is to be rehabilitated along these lines, namely, those of teaching 
 problem solving and giving the work of professional standards, or along similar lines, 
 two important improvements in the present situation are required. To give all subjects 
 the kind of instruction indicated will requre skillful teachers, and it is generally admit- 
 ted that the teaching of undergraduates is at present less satisfactory than is that of 
 the pupils in the grades and high school on the one hand, or of the student of the grad- 
 uate school on the other. The second need is that of more efficient methods of teaching, 
 particularly in non-linguistic subjects." Alexander Smith, Rehabilitation of the Ameri- 
 can College, Address before the Educational Sec. Am. Chem. Soc., July 1909. Science 
 vol. 30, p. 457. See also E. H. Moore, Presidential Address on "The Foundation of 
 Mathematics." Bull. Am. Math. Soc. 1903. P- 4<>*- 
 
 " E. H. Moore, Presidential Address, 1. c. 
 
 John Perry, Preliminary Education of Engineers, Address before Engineering Section 
 Brit. Soc. for Prom, of Sci., School Science and Mathematics, vol. II, p. 264. 1902.
 
 PRESENT NEEDS AND TENDENCIES. ISI 
 
 V. SUMMARY. 
 
 The present chapter has been reserved for the discussion of the 
 various questions which have arisen in connection with the data 
 previously recorded. For good work during the freshman year it 
 is necessary that the students come well prepared in the secondary 
 work; especially in algebra. Two plans have been proposed to 
 insure such a preparation. The first is to require entrance examin- 
 ations in mathematics of all students. The second is to admit with- 
 out examination in mathematics only students coming from high 
 schools which have special teachers of mathematics and which give 
 a half year's work in algebra the fourth year. The following is 
 a suggested curriculum: trigonometry for two or three hours per 
 week during the first half of the year, and a course in algebra and 
 analytic geometry which begins with a review of preparatory alge- 
 bra for the first two or three weeks, and is taken two or three hours 
 per week the first half year and for five hours per week the second 
 half year. Analytic geometry is to be the basis of this course; 
 topics of algebra being taken up as needed or as they arise in con- 
 nection with the other work of the course. The object of analytic 
 geometry is the acquisition of a new mathematical language and not 
 information regarding any particular set of curves. The elements of 
 the calculus are brought into use in the solution of the tangent prob- 
 lems and in the finding of maxima and minima of functions. As 
 much mathematics is to be taught the students in the first year as can 
 be included in the curriculum without overcrowding, and they are 
 thus to be given, at the earliest opportunity, the most powerful 
 weapons for attacking the problems they may encounter later. The 
 mathematical curriculum is to embody such principles as the instruc- 
 tors of the technical subjects consider necessary for the future work, 
 .and is to be taught by men qualified as instructors of mathematics.
 
 CHAPTER VI. 
 
 % 
 
 CONCLUSION. 
 
 I. CONTENTS OF CHAPTER. 
 
 The present chapter restates briefly the most salient results of 
 the investigation recorded in what precedes, and presents the con- 
 clusions drawn therefrom. 
 
 II. ORIGIN AND GROWTH OF THE ENGINEERING 
 
 COLLEGE. 
 
 All engineering education in the United States has been the 
 result of a demand created by economic conditions. When the ap- 
 prenticeship system proved inadequate to produce sufficiently trained 
 engineers, graduates from the Military Academy were drawn into 
 professional work along civil lines. Stephen Van Rensselaer, one 
 of the early promoters of American industrial enterprise, gave sub- 
 stantial and timely aid to the cause of industrial education by the 
 founding, in 1824, of the Rensselaer School. This school was the 
 pioneer institution of its kind and in 1835 offered the first course in 
 civil engineering in -the United States. Soon afterwards several 
 literary institutions organized courses in civil engineering and in- 
 dustrial chemistry, which at first were mostly modifications of the 
 last year or two of the regular literary courses. By the middle of 
 the ninetneenth century industries had arisen that required men 
 with technical education in mechanical as well as in civil engineer- 
 ing. This condition prompted Congress to pass the Land Grant 
 Act of 1862 as an aid to mechanical colleges. Since that time vari- 
 ous courses have been added as occasion demanded, such as Electri- 
 cal Engineering, Sanitary Engineering, etc. As a member of our 
 educational system, the engineering college is only fifty years old..
 
 CONCLUSIONS. 123 
 
 III. PROGRESS OF WORK IN MATHEMATICS AND 
 PRESENT NEEDS. 
 
 In our survey of the progress made in the mathematical work 
 of the engineering colleges up to the present time and in a discussion 
 of future improvements we have considered three phases: The 
 preparation, the collegiate curriculum, and the problem of teaching 
 the courses demanded. 
 
 i) ENTRANCE REQUIREMENTS. (i) Their Scope. 
 
 During the formative period, or until the coming of the Land 
 Grant Colleges in the early sixties, all the engineering courses of- 
 fered, with the exception of those at the Rensselaer School, were 
 modifications of the last year or two of existing literary courses. 
 Accordingly two or three years of literary collegiate work were re- 
 quired for admission to these technical courses. As more and more 
 work was added to the technical courses the amount of the literary 
 collegiate work required for admission to the technical work had 
 to be reduced until when the complete four years' technical course 
 was offered the only work common to the two courses was that for 
 entrance. The door was then opened for differentiating the entrance 
 requirements, and this has also been done to some extent. In math- 
 ematics they have been raised for the technical courses from arith- 
 metic (prior to 1850) to the present average of one and one-half 
 years of algebra, one year of plane geometry and one- half year of 
 solid geometry. Trigonometry is required by i&% of the institu- 
 tions. 1 
 
 (n) Present Needs. 
 
 The above entrance requirements are at the present adequate 
 as to extent. The need is rather for more concrete presentations of 
 the subjects and with a view to their applications, and for more 
 emphasis upon the principles that will be met in the collegiate work, 
 especially the principles concerning the handling of algebraic ex- 
 pressions and their application to problems, in both of which many 
 deficiencies are found. This demand that preparatory algebra be 
 
 1 Chap. II, p. 43.
 
 124 MATHEMATICS FOR ENGINEERS. 
 
 taught concretely and for its applied value, that the various topics 
 be treated as parts of a homogeneous whole centered about the 
 equation, that fundamental principles be made to stand out boldly 
 and clear from the working machinery involved, and that the work 
 throughout be emphasized by numerous fairly simple problems. 
 Such teaching of mathematics for future utility is conceded also 
 to be the most generally valuable from the educational point of view. 
 
 (m) Remedy. 
 
 How to secure the good preparation in entrance mathematics 
 that is so vital to successful work in mathematics during the fresh- 
 man year, is one of the most serious questions confronting the en- 
 gineering colleges of today. Unless the high school teachers have 
 been especially trained for the teaching of mathematics, satisfactory 
 preparation of students is impossible. It is the duty of the engineer- 
 ing colleges vigorously to advocate such training and to require 
 en<trance examinations in mathematics of the graduates of all high 
 schools not having a special teacher of mathematics. In addition 
 to this, only such high schools should be accredited as give a half 
 year's review of algebra in the fourth year. As a final means of 
 securing good preparation for the work of the freshman year two 
 or three weeks' review at the beginning of the year is recommended, 
 and students then found especially deficient should be required to 
 take additional preparatory work. 
 
 (iv) Usefulness of Variation in Entrance Requirements. 
 
 The great difference in entrance requirements 2 found 
 among the various engineering colleges seems, at first glance, a 
 condition to be deplored. Yet if the high schools take advantage 
 of this divergency they will find it of inestimable help in aiding 
 those of their students who intend taking an engineering course, to 
 select an institution commensurate with their capacity. Strangely 
 enough, the great number of failures in mathematics on the part of 
 freshmen students makes it appear as though little cognizance were 
 taken of this divergency. 2 The lack of uniformity in entrance 
 requirements may also serve as the entering wedge for the organi- 
 zation of some system of grading our engineering colleges. When 
 
 * Chap. II, p 43.
 
 CONCLUSIONS. 125 
 
 this is done technical education will have been placed upon a firmer 
 foundation and much of the present waste of time and energy will 
 be saved, especially in the work in mathematics. 
 
 2) COLLEGIATE CURRICULUM. (i) Progress. 
 
 Until 1850 the collegiate courses included all of algebra, to- 
 gether with plane and solid geometry, trigonometry and a little anal- 
 ytic geometry given in the latter part of the second year. After that 
 time some algebra was required for entrance and the collegiate work 
 was increased correspondingly ; calculus was first added in 1855, 
 but came late in the course and was little used in the technical work. 
 Since then the time devoted to analytic geometry and the calculus 
 has been materially increased, and these subjects have been placed 
 early in the curriculum so as to make them available for the technical 
 work. At present from seventy to one hundred and twenty class hours 
 are devoted to analytic geometry, which is completed during the 
 first year or in the early part of the second year, and from one hun- 
 dred to two hundred class hours are given to the calculus in the 
 second year. It is only within the last fifteen years, however, that 
 attention has been given to the introduction of work especially 
 planned for engineering students, and these modified curricula are 
 as yet only in the formative state. 3 Some of the more noteworthy 
 features are: i) the treatment of the subjects from the standpoint 
 of application rather than that of philosophy, that is, less attention is 
 paid to the rigorous proofs of theorems than to their application ; 
 2) less insistance upon sharp lines of demarkation between the sub- 
 jects ; 3) the introduction of laboratory courses; 4) the correlation 
 of the work in algebra and analytic geometry, without special atten- 
 tion to the conies; 5) the intrbduction of the calculus into the work 
 of the freshman year, especially in connection with the solution of 
 the tangent problem. 4 The causes for the adoption of this modified 
 work may be summarized as a desire to lessen "the gap between 
 theory and practice." 5 
 
 Chap. III. 
 
 4 Chap. Ill, pp. 52-62. 
 
 Chap. Ill, p. 66.
 
 126 MATHEMATICS FOR ENGINEERS. 
 
 (it) Suggested Improvements. 
 
 Ever since the founding of the Rensselaer School the object 
 of the engineering colleges has been the production of technically 
 trained men. The principles necessary to such a training must, 
 therefore, control the formation of the curriculum. To no subject 
 is this more applicable than to mathematics, so fundamental for the 
 more technical work. Future use, not culture value, must be the 
 criterion according to which mathematical topics are admitted or 
 rejected. It hence becomes necessary for the instructors of the 
 professional studies to cooperate with those of the mathematical 
 department by designating what mathematical principles are needed 
 in the professional courses. All mathematical topics should be 
 treated in as close proximity as possible to their application. When 
 the usual division into subjects is not adhered to principles which 
 are allied to each other will naturally be connected, and thus the 
 value of each will be enhanced. It is a decided advantage to equip 
 the student with the most powerful instrument applicable to a prob- 
 lem or set of problems the first time such problems arise, for ex- 
 ample, the use of derivatives in the solution of tangent problems. 
 The most efficient means will thus become the natural one for the 
 student to use and he will become more proficient in handling them 
 thru prolonged application. The mathematical curriculum for the 
 freshman year should contain as much usable mathematics as pos- 
 sible without overcrowding. 
 
 3) INSTRUCTION (i) Progress and Present Needs. 
 
 The men who furnished the information under the apprentice- 
 ship system could in no wise be called mathematicians or teachers. 
 From the coming of the engineering colleges there has been con- 
 tinual advancement in mathematical knowledge among the instruc- 
 tors until few deficiencies are at present found on that side. Some 
 attention is also being given to the question of preparing men as 
 teachers, 6 but on the whole little has been done in this direction as is 
 shown by the lack of interest in the technical questions of instruc- 
 
 * Chap. V, p. 1 20.
 
 CONCLUSIONS. 
 
 127 
 
 tion. 7 Until due attention is given to this most vital phase of the 
 problem and the work of mathematical instruction is raised to the 
 rank of a profession, the desired results can not be hoped for. 
 
 (M) Special Phases to be Noted. 
 
 The following are some of the salient points to be considered 
 relative to the teaching of mathematics to freshmen students of 
 engineering. 
 
 a) General Aims. 
 
 The general aim of the instruction in mathematics given to 
 freshman students should be to bring out its utilitarian value. To 
 accomplish this end it is necessary to make the relations between 
 the principles and their application the primary object, relegating 
 exhaustive proofs to the background. In thus learning to apply 
 the principles the student will also secure a clearer understanding of 
 the principles themselves than he could possibly attain thru any 
 other means. Such a utilitarian mathematics fully understood by 
 the student and applied by him in a logical and intelligent manner 
 is also of the highest educational value. 8 No study should be in- 
 troduced for its value as a mental gymnastic. 
 
 b) Fundamental Principles. 
 
 For the engineering student even more than for the literary 
 student attention to fundamental principles is of the utmost im- 
 portance. These principles should therefore first be studied thoroly 
 and illustrated by a great number of slightly varied problems ; and 
 secondly, the reduction of each problem to these fundamentals 
 should thereafter be insisted upon at all times. Such a thoro 
 grounding in fundamentals may necessitate the elimination of some 
 of the "advanced work," but without the ground work this would 
 be largely superficial, while the student thoroly equipped with the 
 basic principles will be able to do the more involved work as it 
 arises. 
 
 c) Problems. 
 
 The value of any educational work is directly proportional to 
 the interest taken by the students. In mathematics this interest is 
 
 T Chap. IV, p. 89. 
 
 8 Such a rational study has been ably described on page 95.
 
 128 MATHEMATICS FOR ENGINEERS. 
 
 largely dependant upon the problems proposed for solution. As 
 the work of the freshmen engineers is to be given mainly from 
 the standpoint of application it is readily seen what an important 
 part problems play and what great care is needed in their selection. 
 The data selected for the problems should be as real as possible and 
 entirely shorn of the fantastic. The use of problems from engineer- 
 ing data is of value not only in giving the students some little 
 insight into the nature of their future work but especially in 
 holding their interest. Still, before being used in the mathematical 
 work such data must be carefully edited so as to exemplify the 
 principles which it is desired to illustrate, without requiring the 
 student to become acquainted with the technical part of the engi- 
 neering questions involved. Too long and too complicated prob- 
 lems, on the one hand, and those requiring mere substitution in a 
 formula on the other, must be guarded against, and "fake" engineer- 
 ing problems must never be used. The instructor must also be 
 careful not to overdo matters and must take especial pains to avoid 
 the duplication of the work of the professional departments. Prob- 
 lems should never be selected for their symetry of form ; such 
 problems are rarely met with afterwards and should rather be 
 avoided, perfectly general but simple data being used instead. 
 
 d) Computation. 
 
 Accuracy in computation is a vital question with the profes- 
 sional engineer, 9 and the instructors of technical subjects can rightly 
 require some help from the instructors of mathematics along this 
 line. This can be given either by special courses in computation, 
 such as have been inaugurated by some institutions, 10 or what ap- 
 pears on the whole more satisfactory, by carrying along some work 
 in computation thruout the whole mathematical curriculum. To 
 carry out this plan the following few points should be observed by 
 the instructor: 
 
 /) Never let the value of numerical accuracy be slighted. 
 2} Drill the students to present final results in the form best 
 suited for computation; as tf V6 in the place of 
 
 Chap. IV, p. 85. 
 
 " Chap. Ill, pp. 62-63.
 
 CONCLUSIONS. 129 
 
 5) Introduce the use of tables whenever they materially shorten 
 the work and their use can be thoroly understood by the 
 students. 
 
 4) Permit the use of formulas only after the student is thoroly 
 acquainted with the quantities involved and the relations 
 expressed between them. No recognition is to be given 
 solutions based upon formulas except where perfect ac- 
 curacy is obtained. Whenever a common sense application 
 of arithmetic or the elements of algebra and geometry will 
 suffice for the solution of a problem, the use of a formula 
 should be discouraged. 11 
 
 5) Teach what is meant by the degree of accuracy needed and 
 what degree is possible in any given problem. 12 
 
 4) Too MUCH REQUIRED OP THE MATHEMATICAL DEPARTMENT. 
 
 According to the statistics previously recorded 13 the language 
 departments recieve only slightly less time than the mathematical 
 departments while the semi-professional departments receive some- 
 what more than it does. As mathematics is considered to be the 
 foundation of engineering science, is it not fair to inquire in the 
 light of these figures if too much is not asked of the instructors of 
 mathematics? As none of the other departments seem able to re- 
 linquish any of their portion of time, then' from all experiences 
 noted, two things are necessary: i) the mathematical courses 
 must be made "more intensive and less extensive" and 2) the 
 mathematical departments must have the cooperation of the pro- 
 fessional departments in order to make the courses in mathematics 
 fit the demands made upon them. 
 
 III. MATHEMATICS FOR FRESHMEN STUDENTS OF 
 
 ENGINEERING. 
 
 The mathematical courses for freshmen students of engineer- 
 ing should be composed of the topics and principles which are con- 
 sidered by the instructors of technical subjects to be prerequisite 
 
 n For confirmation and illustrations see p. 92. 
 u Chap. V, p. 114. 
 13 Chap. II, p. 49.
 
 MATHEMATICS FOR ENGINEERS. 
 
 for such subjects and this material together with what it presupposes 
 should be formulated into a compact teachable whole by the teachers 
 of mathematics. The curriculum is to contain as much usable 
 mathematics as possible; only the most powerful methods of at- 
 tack, those which will be required for use in the future, being 
 taken up. Basic principles involving the fundamental conception of 
 the subjects are to receive chief attention, but little time being 
 given to so called "advanced work." Each principle studied, should 
 be amply illustrated by simple problems based, as far as possible, 
 upon real data. Intelligent and accurate application of the principles 
 rather than their rigorous proofs should characterize the work thru- 
 out. The need of accuracy in computed results and the degree of 
 accuracy attainable should be emphasized, and the placing of re- 
 sults in the best form for computation should always be insisted 
 upon. Instructors trained as teachers as well as thoroly versed 
 in the subject matter to be taught are requisite for successful work. 
 The mathematics for the freshmen students of engineering should 
 be a "vitalized mathematics" dealing with as real and interesting 
 data as possible; it should be a utilitarian mathematics taught 
 rationally and intelligently so as to give the students power in using 
 the mathematical type of thot as well as training in its technical 
 application.
 
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 VITA. 
 
 Theodore Lindquist was born in Andover, 111., where he received most of 
 his elementary education. He was graduated from the Galesburg, 111., High 
 School in 1894, received the degree of A. B. from Lombard College (Gales- 
 burg, 111.) in 1897 and the degree of M. S. from Northwestern University 
 (Evanston, 111.) in 1899 after one year of residente study during which he 
 attended lectures by Professors Crew, Holgate and White. Subsequently 
 he has spent six quarters (the equivalent of two academic years) at The 
 University of Chicago working in the Departments of Mathematics and 
 Physics and attending lectures by Professors Bliss, Bolza, Dickson, Maschke, 
 Michelson, Millikan, Moulton and Young. He feels grateful to all of these 
 instructors for the help and encouragement received thruout his work, and 
 especially to Professor Young under whose direction the dissertation was 
 prepared; he also thanks those who so kindly contributed to the data en> 
 ployed.
 
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