530 SCHOOL SCIENCE AND MATHEMATICS

LIMITS IN ELEMENTARY GEOMETRY.

BY ERNEST B. LYTLE,University of Illinois.

Recently in a teachers’ course in geometry in an eastern uni-versity the professor said, "In ten years I have not seen morethan five undergraduate students in this university who couldunderstand the theory of limits." In a teachers’ course givenlast summer by the writer he received numerous requests to dis-cuss the topic of limits and, while this class had many memberswho had taught geometry in some high school, not a singlemember of that class could give a satisfactory definition of alimit. College instructors testify that rarely do any freshmencome to them with a correct notion of a limit. These facts surelypoint to some weakness in teaching limits in our high schoolsand it will be useful to endeavor to determine the cause andremedy for this weakness.

Students generally enter college remembering just two wordsabout limits, namely, "never reaches." This is probably due tothe over-emphasis of these words in connection with the defini-tion o’f a limit by both teachers and text-books. To say that avariable never reaches its limit is not only incorrect but also castsabout the notion of a limit an atmosphere of mystery in the mind’sof young students. They lose some faith in the exactness ofmathematics in using a constant as if equal to a variable when.their definition says a variable never reaches its limit. It wouldgreatly clarify the notion of a limit if the words "never reaches"were crossed out of every single definition of limit in our text-books. Whether or not a variable reaches its limit has nothingwhatever to do with the notion of a limit but is entirely concernedwith the question of continuity, a notion which need not botherelementary students. "A limit of a variable is a constant whichthat variableapproaches as near as you please in value and re-’maw nearf .While it is true that the variables considered inelementary geometry are generally discontinuous and hence do;ilot reach ^their limits, yet this fact is not a defining characteristic^UheUirmt notion,, and making it such spoils the definition fortee in continuous cases met later; and besides tends to inexact-ness and mystery in the minds of our students. The writer knowsbut three American elementary geometry texts which correctlydefine a limit. Within a month he has received copies of two

LIMITS IN GEOMETRY 531

new geometries which contain the old and incorrect definitionof a limit.

Since the variables of elementary geometry always increaseor always decrease (are monotone), a limit can first be presentedas a value to which a variable comes as near as you please butcannot go beyond, which, is the common everyday notion of alimit as a boundary which stops or hems in something. In laterwork the above definition needs no modification but only a moreprecise quantitative form of statement.

Besides incorrect definitions, the attempt to prove difficultexistence theorems is another cause for the present difficultiesin the theory of limits in elementary classes. To prove that aparticular variable has a limit, leaving out all question as to thevalue of this limit, is frequently a difficult task of higher mathe-matics. It is questionable whether or not existence proofs anddifficult evaluations of limits have any place in elementary courses;certainly if retained their old treatment as found in most textsto-day should be modified and simplified. This simplificationcan well be made by assuming the following fundamental the-orem: "If a variable always increases {or always decreases) andis always less than {or greater than) some finite constant then ithas a limit." No attempt should be made to prove this theoremin an elementary course, but students with a correct limit notionwill readily accept the truth of this theorem on intuition from afew illustrations. There is no objection to a rather free use ofunproved assumptions in beginning courses so long as they areconsciously made and explicitly marked as assumptions but donot creep in unconsciously under cover of such terms as obviouslyand the like. In many cases it is quite simple to show that avariable always increases and is always less than some finiteconstant; then by the use of the above theorem the variable isknown to have a limit. Existence theorems which do not comeunder this simple treatment should be omitted entirely from ele-mentary geometry.The theorem ordinarily given as the fundamental theorem of

limits, namely, "If two variables are constantly equal and eachapproaches a limit, then their limits are equal," is almost trivial,for if two variables are constantly equal they are the same vari-able and of course have equal limits if any at all. This theoremby no means deserves the emphasis generally given to it. -

It is being suggested in some quarters that limits and all workinvolving limits should be entirely omitted from courses in ele-

^32 SCHOOL SCIENCE AND MATHEMATICS

mentary geometry because of their difficulty and the presentunsatisfactory results. The writer disagrees with this suggestionbecause of the importance of the limit notion in the treatmentof incommensurables and because the limit idea is needed in cer-tain definitions. Such notions as length of a circumference, areaof a circle, volume of a cylinder, lateral area of a cylinder, vol-ume and the lateral area of cone, and surface of a sphere requirethe idea of a limit in their definitions. Many present day textsin geometry try to prove as theorems statements which properlyshould be assumed as definitions; e. g., ^The circumference of acircle is the limit which the perimeters of regular inscribed poly-gons and of similar circumscribed polygons approach if thenumber of sides of the polygons is indefinitely increased"{Wentworth, Book V, Prop. VIII, p. 220). This is a definitionof the length of a circle and a definition needs no proof.

The notion of a limit is too important and too useful to bedropped out of elementary geometry. The writer strongly be-lieves that limits can be presented in a form sufficiently simpleto be understood by high school students and to this end sumszip his suggestions as follows: i. Teach a correct but not themost precise definition of a limit, taking care to avoid the words"’never reaches" which are incorrect and tend to mystify. 2.

Prove the existence of limits only in such simple cases of mono-tone variables which come under the assumed fundamental the-orem quoted above. 3. Eliminate many difficult "proofs" ofpresent day texts by defining all lengths of curves, areas bounded.by curves, areas of curved surfaces, and volumes bounded bycurved surfaces as particular limits.

MAKING BOARDING HOUSES SANITARY FOR STUDENTS.The following shows how carefully the authorities of the University of

Wisconsin are looking after the health of its students:A thorough inspection of all rooming and boarding houses for students

sat the University of Wisconsin has been begun under the auspices of theliygiene committee of the faculty, to determine fully the sanitary andliygienic condition of each.

This is the first time that the score-card system of regulation of living(conditions for students at the university has been applied.A full report of .the ventilation, plumbing, cleanliness, lighting, heating,

$md general surroundings is to be made out by the inspector and kept on�file in the office of the committee. Defective conditions found are to bereported to the owner and occupant, and steps will be taken to compel alloffering accommodation for students to maintain a reasonable standard ofliving conditions.