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. a * g y. 3 E h-r ex in. i 00 o bfi a 1 a a 8 H O w d s-s w ity of la, Al niv Tu w w .J-s M ^ University of Tuscon, Ariz. w o fl w w : s Leland Stanford Stanford, Cal. ".a > s ** '.S"O 4> . t rt .C y Qv W 60 rt M-l * -4 M- v 8 o o 9- - r o t/5 o8eaj.| bb^^' d bb 08 " 3 ' c > . rt - * . ^ C rt O ^^ o3 "^ C i "^J . . c^j . 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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. <*< - 44. 40 ? 'NteSfvt PO Cu < JO. . . . . . < S>A V. -Q A) 4-: /2 ^5 3S 8 4 VO ^ M ^? frs-i ^ i sl ^ ^ ^? ^ J 7^ p ^ ^ -5 ^ J 2 ^ 6 B. 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 ( ' 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. *- " 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 ployed. UC SOUTHERN J JGIONAL LIBRARY FACILITY A 000933320 4 STATE NORMAL SCHOOL