How To Solve It

A New Aspect ofMathematical Method

G. POLYAStanford Univer,sity

SECOND EDITION

Princeton University Press

Princeton, New Jersey

VI From the Preface to the First Printing

work may be an exercise as desirable as a fast game oftennis. Having tasted the pleasure in mathematics he willnot forget it easily and then there is a good chance thatmathematics will become something for him: a hobby, ora tool of his profession, or his profession, or a greatambition.

The author remembers the time when he was a studenthimself, a somewhat ambitious student, eager to under.stand a little mathematics and physics. He listened tolectures, read books, tried to take in the solutions andfacts presented, but there was a question that disturbedhim again and again: "Yes, the solution seems to work,it appears to be correct; but how is it possible to inventsuch a solution? Yes, this experiment seems to work, thisappears to be a fact; but how can people discover suchfacts? And how could I invent or discover such things bymyself?" Today the author is teaching mathematics in auniversity; he thinks or hopes that some of his more eagerstudents ask similar questions and he tries to satisfy theircuriosity. Trying to understand not only the solution ofthis or that problem but also the motives and proceduresof the solution, and trying to explain these motives andprocedures to others, he was finaUy led to write thepresent book. He hopes that it will be useful to teacherswho wish to develop their students' ability to solve prob­lems, and to students who are keen on developing theirown abilities.

Although the present book pays special attention to therequirements of students and teachers of mathematics, itshould interest anybody concerned with the ways andmeans of invention" and discovery. Such interest may bemore widespread than one would assume without reflec­tion. The space devoted by popular newspapers andmagazines to crossword puzzles and other riddles seemsto show that people spend some time in solving unprac·

From the Preface to the First Printing vii

tical problems. Behind the desire to solve this or thatproblem that confers no material advantage, there maybe a deeper curiosity, a desire to understan~ the ways andmeans, the motives and procedures, of solutlon. .

The following pages are written somewhat conCISely,but as simply as possible, and are based on a long andserious study of methods of solution. This sort .of study,called heuristic by some writers, is not in fashlOn now·adays but has a long past and, perhaps, some future. .

Studying the methods of solving problems, ~e perceIveanother face of mathematics. Yes, mathematlcs. h~s twofaces; it is the rigorous science of Euclid but It 1~ alsosomething else. Mathematics presented in the Euchdeanway appears as a systematic, deductive science: but mat~e.matics in the making appears as an experimental, Ill­

ductive science. Both aspects are as old as the science ofmathematics itself. But the second aspect is new in onerespect; mathematics "in statu nascendi," in the proce~

of being invented, has never before been presented. III

quite this manner to the student, or to the teacher hIm-self, or to the general public. .

The subject of heuristic has manifold con~ectI~ns;

mathematicians, logicians, psychologists, educatlOnahsts,even philosophers may claim various parts of it as belong­ing to their special domains. The auth~r, well aware ofthe possibility of criticism from opposite quarter: andkeenly conscious of his limitations, has one claIm tomake: he has some experience in solving problems andin teaching mathematics on various levels.

The subject is more fully dealt with in a more exten­sive book by the author which is on the way to com·pletion.

Stanford University, August I, I944

Contents

From the Preface to the First Printing

From the Preface to the Seventh Printing

Preface to the Second Edition

"How To Solve It" list

Introduction

PART 1. IN THE CLASSROOM

Purpose

v

viii

IX

xvi

xix

1. Helping the student 1

2. Questions. recommendations,mental operations 1

3· Generality 2

4· Common sense 3

5. Teacher and student. Imitation and practice 3

Main divisions~ main questions

6. Four phases

7· Understanding the problem

8. Example

9· Devising a plan

10. Example

1]. Carrying out the plan

Xl

56

78

10

Xll Contents Contents xiii

12. Example

J 3. Looking back

14. Example

15. Various approaches

16. The teacher's method of questioning

17· Good questions and bad questions

More examples

18. A problem of construction

19· A problem to prove

20. A rate problem

PART II. HOW TO SOLVE IT

A dialogue

PART III. SHORT DICTIONARYOF HEURISTIC

Analogy

Auxiliary elements

Auxiliary problem

Bolzano

Bright idea

Can you check the result?

Can you derive the result differently?

Can you use the result?

Carrying out

'3J4

,6

'920

22

33

37

46

5°

5758

5961

6468

Condition

Contradictoryt

Corollary

Could you derive something useful from the data?

Could you restate the problem?t

Decomposing and recombining

Definition

Descartes

Determination, hope, success

Diagnosis

Did you use all the data?

Do you know a related problem?

Draw a figuret

Examine your guess

.Figures

Generalization

Have you seen it before?

:Here is a problem related to yoursand solved before

,Heuristic

,.Ileuristic reasoning

<If you cannot solve the proposed problem

;~l4nduction and mathematical induction

Iinventor's paradox

it possible to satisfy the condition?

'buitz

t Contains only cross-references.

7'73

73

73

757585

9'93

94

95

98

99

99103108

"0

"0

,""3"4114

121

'""3123

XIV Contents Contents XV

Look at the unknown

Modern heuristic

Notation

Pappus

Pedantry and mastery

Practical problems

Problems to find, problems to prove

Progress and achievement

Puzzles

Reductio ad absurdum and indirect proof

Rednndantt

Routine problem

Rules of discovery

Rules of style

Rules of teaching

Separate the various parts of the condition

Setting up equations

Signs of progress

Specialization

Subconscious work

Symmetry

Tenns, old and new

Test by dimension

The future mathematician

The intelligent problem-solver

The intelligent reader

The traditional mathematics professor

t Contains only cross-references.

123129

134

141

'48

149

154

157160

16.

171

17 1

'7'17'173173

'74178

19°197199200

.0.'°5.06

.07

.08

Variation of the problem

What is the unknown?

Why proofs?

Wisdom of proverbs

Working backwards

PART IV. PROBLEMS, HINTS,SOLUTIONS

Problems

Hints

Solutions

22 1

Introduction

The following considerations are grouped around thepreceding list of questions and suggestions entitled "Howto Solve It." Any question or suggestion quoted from itwill be printed in italics, and the whole list will bereferred to simply as "the list" or as "our list."

The following pages will discuss the purpose of thelist, illustrate its practical use by examples, and explainthe underlying notions and mental operations. By way ofpreliminary explanation, this much may be said: 1£,using them properly, you address these questions andsuggestions to yourself, they may help you to solve yourproblem. If, using them properly, you address the samequestions and suggestions to one of your students, youmay help him to solve his problem.

The book is divided into four parts.The title of the first part is "In the Classroom:' It

(:ontains twenty sections. Each section will be quoted byib number in heavy type as, for instance, "section 7:'Sections 1 to 5 discuss the "Purpose" of our list in gen.eral terms. Sections 6 to 17 explain what are the "MainDivisions, Main Questions" of the list. and discuss a firstpractical example. Sections 18, 19. 20 add "More Ex·amples."; The title of the very short second part is "How to_~lve It." It is written in dialogue; a somewhat idealized

':'f;eacher answers short questions of a somewhat idealized;fftudent.

f"."' The third and most extensive part is a "Short Diction­" of Heuristic"; we shall refer to it as the "Dictionary."

: XIX

xx Introduction Introduction xxiIt contains sixty-seven articles arranged alphabetically.For example, the meaning of the term HEURiSTIC (setin small capitals) is explained in an article with this t.itleon page 112. When the title of such an article is referredto within the text it will be set in small capitals. Certainparagraphs of a few articles are more technical; they areenclosed in square brackets. Some articles are fairlyclosely connected with the first part to which they addfurther illustrations and more specific comments. Otherarticles go somewhat beyond the aim of the first part ofwhich they explain the background. There is a key­article on MODERN HEURISTIC. It explains the connectionof the main articles and the plan underlying the Diction­ary; it contains also directions how to find informationabout particular items of the list. It must be emphasizedthat there is a common plan and a certain unity, becausethe articles of the Dictionary show the greatest outwardvariety. There are a few longer articles devoted to thesystematic though condensed discussion of some generaltheme; others contain more specific comments, still otherscross-references, or historical data, or quotations, oraphorisms, or even jokes.

The Dictionary should not be read too quickly; its textis often condensed, and now and then somewhat subtle.The reader may refer to the Dictionary for informationabout particular points. If these points corne from hisexperience with his own problems or his own students,the reading has a much better chance to be profitable.

The title of the fourth part is "Problems, Hints, Solu·tions." It proposes a few problems to the more ambitiousreader. Each problem is followed (in proper distanc~) b'ya "hint" that may reveal a way to the result whlch IS

explained in the "solution."We have mentioned repeatedly the "student" and the

"teacher" and we shall refer to them again and again. It

:Dlay be good to observe that the "student" may be a highschool student, or a college student, or anyone else whois studying mathematics. Also the "teacher" may be ahigh school teacher, or a college instructor, or anyoneinterested in the technique of teaching mathematics. Theauthor looks at the situation sometimes from the pointof view of the student and sometimes from that of theteacher (the latter case is preponderant in the first part).Yet most of the time (especially in the third part) thepoint of view is that of a person who is neither teachernor student but anxious to solve the problem before him.

PART 1. IN THE CLASSROOM

PURPOSE

I. Helping the student. One of the most importanttas~s of the teacher is to help his students. This task isnot quite easy; it demands time. practice. devotion, andsound principles~

The student should acquire as much experience ofindependent work as possible. But if he is left alone withhis problem without any help or with insufficient help,he may make no progress at all. If the teacher helps toomuch, nothing is left to the student. The teacher shouldhelp. but not too much and not too little, so that thestudent shall have a reasonable share of the work.

If the student is not able to do much. the teachershould leave him at least some illusion of independentwork. In order to do so, the teacher should help thestudent discreetly, unobtrusively.

The best is, however, to help the student naturally.The teacher should put himself in the student's place. heshould see the student's case, he should try to understandwhat is going on in the student's mind, and ask a ques­tion or indicate a step that could have occurred to thestudent himself.

2. Questions, recommendations, mental operations.Trying to help the student effectively but unobtrusivelyand naturally. the teacher is led to ask the same questionsand to indicate the same steps again and again. Thus. incountless problems, we have to ask. the question: What

1

required diagonal is the hypotenuse. Yet the teachershould be prepared for the case that even this fairly ex~

pHcit hint is insufficient to shake the torpor of the stu­dents; and so he should be prepared to use a wholegamut of more and more explicit hints.

"Would you like to have a triangle in the figure?""What sort of triangle would you like to have in the

figure?""You cannot find yet the diagonal; but you said that

you could find the side of a triangle. Now, what will youdo?"

"Could you find the diagonal, if it were a side of atriangle?"

When, eventually, with more or less help, the studentssucceed in introducing the decisive auxiliary element, theright triangle emphasized in Fig. 1, the teacher shouldconvince himself that the students see sufficiently farahead before encouraging them to go into actual .:alc4la.tions.

"I think that it was a good idea to draw that triangle.You have now a triangle; but have you the unknown?"

"The unknown is the hypotenuse of the triangle: wecan calculate it by the theorem of Pythagoras."

"You can, if both legs are known; but are they?""One leg is given, it is c. And the other, I think, is not

difficult to find. Yes, the other leg is the hypotenuse ofanother right triangle."

"Very good! Now I see that you have a plan."11. Carrying out the plan. To devise a plan, to can·

ceive the idea of the solution is not easy. It takes so muchto succeed: formerly acquired knowledge, good mentalhabits, concentration upon the purpose, and one morething: good luck. To carry out the plan is much easier;what we need is mainly patience.

The plan gives a general outline; we have to convince

12 In the Classroom IZ. Example 13

ourselves that the details fit into the outline, and so wehave to examine the details one after the other, patiently,till everything is perfectly clear, and no obscure cornerremains in which an error could be hidden.

If the student has really conceived a plan, the teacherhas now a relatively peaceful time. The main danger isthat the student forgets his plan. This may easily happenif the student received his plan from outside, and ac­cepted it on the authority of the teacher; but if he workedfor it himself, even with some help, and conceived thefinal idea with satisfaction, he will not lose this ideaeasily. Yet the teacher must insist that the student shouldcheck each step.

We may convince ourselves of the correctness of a stepin our reasoning either "intuitively" or "formally." Wemay concentrate upon the point in question till we seeit so clearly and distinctly that we have no doubt that~e step is correct; or we may derive the point in ques­bon according to fonnal nJes. (The difference between"insight" and "f?rm~<; proof" is clear enough in manyimportant cases; we may leave further discussion tophilosophers.)

The main point is that the student shOUld be honestlyconvinced of the correctness of each step. In certain cases,the teacher may emphasize the difference between "see­ing" and "proving": Can you see clearly that the step isCorrect' But can you also prove that the step is correct?

12. Example. Let us resume our work at the pointwhere we left it at the end of section 10. The student, atlast, has got the idea of the solution. He sees the righttriangle of which the unknown x is the hypotenuse andthe given height c is one of the legs; the other leg is thediagonal of a face. The student must, possibly, be urgedto introduce suitable notation. He should choose y to de­note that other leg, the diagonal of the face whose sides

In the Classroom

could utilize again the procedure used, or apply the reosuit obtained. Can you u.se the result, or the method, forsome other problem?

14. Example. In section 12. the students finally ob­tained the solution: If the three edges of a rectangularparallelogram, issued from the same corner, are a, b, c,the diagonal is

va' + b2 + c".Can you check the result? The teacher cannot expect a

good answer to this question from inexperienced stu·dents. The students, however, should acquire fairly earlythe experience that problems "in letters" have a greatadvantage over purely numerical problems; if the prob­lem is given "in letters" its result is accessible to severaltests to which a problem "in numbers" is not susceptibleat all. Our example, although fairly simple. is sufficientto show this. The teacher can ask several questions aboutthe result which the students may readily answer with"Yes"; but an answer "No" would show a serious flaw inthe result.

"Did you use all the data? Do all the data a, ,b, cappear in your formula for the diagonal?" /

"Length, width, and height play the same role in ourquestion; our problem is symmetric with respect to a, b, c.Is the expression you obtained for the diagonal sym­metric in a, b, c? Does it remain unchanged when a, b, care interchanged?"

"Our problem is a problem of solid geometry: to findthe diagonal of a parallelepiped wi th given dimensionsa, b, c. Our problem is analogous to a problem of planegeometry: to find the diagonal of a rectangle with givendimensions a, b. Is the result of our 'solid' problem anal­ogous to the result of the 'plane' problem?"

"If the height c decreases, and finally vanishes, the

J4· Example 17

parallelepiped becomes a parallelogram. If you put c = 0

in your formula, do you obtain the correct formula forthe diagonal of the rectangular parallelogram?"

"If the height c increases, the diagonal increases. Doesyour formula show this?"

"If ~ll three measures a, b, c of the parallelepiped in.crease m the same proportion, the diagonal also increasesin the same proportion. If, in your formUla, you substi­tute 12a, 12b. 12C for a, b, c respectively, the expression ofth~ diagonal. owing to this substitution. should also bemultiplied by 12. Is that so?"

"If a, b, c are measured in feet, your formula gives thediagonal measured in feet too; but if you change all meas­ures into inches, the fonnula should remain correct. Isthat so?"

(The two last questions are essentially equivalent; seeTEST BY DIMENSION.)

These questions have several good effects. First, an in­telligent student cannot help being impressed by the factthat the fonnuhl passes so many tests. He was convincedbefore that the fonnula is correct because he derived itcarefully. But now he is mote convinced, and his gain inconfidence comes from a different source; it is due to asort of "experimental evidence." Then. thanks to theforegoing questions. the details of the formula acquirenew significance, and are linked up with various facts.,!he formula has therefore a better chance of being reomembered. the knowledge of the student is consolidated.Finally, these questions can be easily transferred to simi­lar problems. After Some experience with similar prob­lems, an intelligent student may perceive the underlyinggeneral ideas: use of all relevant data, variation of thedata, symmetry, analogy. If he gets into the habit ofdirecting his attention to sueh points, his ability to solveproblems may definitely profit.

c= 6.b-8/

By 'straightforward application of the formula, x = 14·5.For more examples, see CAN YOU USE THE RESULT?15. Various approaches. Let us still !~tain, for a while.

the problem we considered in the foregoing sections 8,10, 12, 14. The main work, the discovery of the plan, wasdescribed in section 10. Let us observe that the teachercould have proceeded differently. Starting from the samepoint as in section 10. he could have followed a somewhatdifferent line. asking the following questions:

"Do you know any related problem?'"Do you know an analogous problem?""You see, the proposed problem is a problem of solid

geometry. Could you think of a simpler analogous prob-­lem of plane geometry?"

"You see, the proposed problem is about a figure inspace, it is concerned with the diagonal of a rectangular

I5. Various Approaches 19

back to his question: Can you use the result, or themethod, for some other problem? Now there is a betterchance that the students may find some more interestingconcrete interpretation. for instance. the following:

"In the center of the flat rectangular top of a buildingwhich is 21 yards long and 16 yards wide, a flagpole is tobe erected, 8 yards high. To support the pole. we needfour equal cables. The cables should start from the samepoint. 2 yards under the top of the pole. and end at thefour oorners of the top of the building. How long is eachcable?"

The students may use the method of the problem theysolved in detail introducing a right triangle in a verticalplane. and another one in a. horizontal pla.ne. Or theymay use the result, imagining a rectangular parallele­piped of which the diagonal, x, is one of the four cablesand the edges are

In the Classroom

Can you check the argument? To recheck the argumentstep by step may be necessary in difficult and importantcases. Usually, it is enough to pick out "touchy" points[or rechecking. In our case, it may be advisable to discussretrospectively the question which was less advisable todiscuss as the solution was not yet attained: Can youprove that the triangle with sides x, y, c is a right tri­angle? (See the end of section 12.)

Can you use the result or the method for some otherproblem? With a little encouragement, and after one ortwo examples, the students easily find applications whichconsist essentially in giving some concrete interpretationto the abstract mathematical elements of the problem.The teacher himself used such a concrete interpretationas he took the room in which the discussion takes placefor the paralIeJepipedof the problem. A.dull student maypropose, as application, to calculate the diagonal of thecafeteria instead of the diagonal of the classroom. If thestudents do not volunteer more imaginative remarks, theteacher himself may put a slightly different problem, forinstance: "Being given the length, the width, and theheight of a rectangular parallelepiped, find the distanceof the center from one of the corners."

The students may use the result of the problem theyjust solved, observing that the distance required is ODehalf of the diagonal they just calculated. Or they may usethe method, introducing suitable right triangles (thelatter alternative is less obvious and somewhat moreclumsyJn the present case) .

After this application, the teacher may discuss the con~

figuration of the four diagonals of the parallelepiped,and the six pyramids of which the six faces are the bases.the center the common vertex, and the semidiagonals thelateral edges. When the geometric imagination of thestudents is sufficiently enlivened, the teacher should come

parallelepiped. "\That might be an analogous problemabout a figure in the plane? It should be concerned with-the diagonal-of-a :rectangular-"

"Parallelogram."The students, even if they are very slow and indiffer­

ent, and were not able to guess anything before, areobliged finally to contribute at least a mi_TIute part of theidea. Besides, if the students are so slow, the teachershould not take up the present problem about the paral­lelepiped without having discussed before, in order toprepare the students, the analogous problem about theparallelogram. Then, he can go on now as follows:

"Here is a problem related to yours end wIved before.Gan you use it?"

"Should you introduce some auxiliary element in orderto make its use possible?"

Eventually, the teacher may succeed in suggesting tothe students the desirable idea. It consists in conceivingthe diagonal of the given parallelepiped as the diagonalof a suitable parallelogram which must be introducedinto the figure (as intersection of the parallelepiped witha plane passing through two opposite edges) . The idea isessentially the same as before (section 10) but the ap­proach is different. In section 10, the contact with theavailable knowledge of the students was establishedthrough the unknown; a formerly solved problem wasrecollected because its unknown was the same as that ofthe proposed problem. In the present section analogyprovides the contact with the idea of the solution.

16. The teacher's method of questioning shown in theforegoing sections 8, 10, 12, 14, 15 is essentially this:Begin with a general question or suggestion of our list,and, if necessary. come down gradually -10 more specificand concrete questions or suggestions till you reach onewhich elicit.-s a response in the student's mind. If you

20 In the Classroom I6. The Teachers Method of Questioning 21

have to help the student exploit his idea, start again, ifpossible, from a general question or suggestion containedin the list. and return again to some more special one ifnecessary; and so on.

Of course, our list is just a first list of this kind; itseems to be sufficient for the majority of simple cases, butthere is no doubt that it could be perfected It is impor~

tant, however. that the suggestions""from which we startshould be simple, natural, and general, and that their listshould be short.

The suggestions must be simple and natural becauseotherwise they cannot be unobtrusive.

The suggestions must be general, applicable not onlyto the present problem but to problems of all sorts, ifthey are to help develop the ability of the student and notjuSt a special technique.

The list must be short in order tbat the questions maybe often repeated. unartificially, and under varying cir­cumstances; thus. there is a chance that they will beeventually__assimilated by the student and win contributeto the development of a mental habit.

It is necessary to come down gradually to specific sug~

gestions, in order that the student may have as great ashare of the work as possible.

This method of questioning is not a rigid one; for~

tunately so, because, in these matters. any rigid, mechani·cal. pedantical procedure is necessarily bad. Our methodadmits a certain elasticity and variation, it admits variousapproaches (section 15). it can be and should be soapplied that questions asked by the teacher could haveoccurred to the student himself.

If a reader wishes to try the method here proposed inhis class he should, of course, proceed with caution. Heshould study carefully the example introduced in section8, and the following examples in sections 18, 19. 20. He

should prepare carefully the examples which he intendsto discuss, considering also various approaches. He shouldstart with a few trials and find out gradually how he canmanage the method, how the students. take it, and howmuch time it takes,

17. Good questions and bad questions. If the methodof questioning formulated in the foregoing section is wellunderstood it helps to judge, by comparison, the qualityof certain suggestions which may be offered with the in­tention of helping the students.

Let us go back to the situation as it presented itself atthe beginning of section 10 when the question was asked:Do you know a related problem1 Instead of this. with thebest intention to help the students, the question may beoffered: Could you apPly the theorem of Pythagoras?

The intention may be the best, but the question is aboutthe worst. We must realize in what situation it was of­fered; then we shall see that there is a long sequence ofobjections against that sort of "help."

(I) If the student is near to the solution, he may un­derstand the suggestion implied by the question; but ifhe is not, he quite possibly will not see at all the point atwhich the question is driving. Thus the question fails tohelp \vhere help is most needed

(2) If the suggestion is understood, it gives the wholesecret away, very little remains for the student to do.

(3) The suggestion is of too special a nature. Even ifthe student can make use of it in solving the presentproblem, nothing is learned for future problems. Thequestion is not instructive.

(4) Even if he understands the suggestion, the studentcan scarcely understand how the teacher came to the ideaof putting such a question. And how could he, the stu­dent, find such a question by himself? It appears as anunnatural surprise, as a rabbit pulled out of a hat; it isreally not instructive.

22 In the Classroom I8. A Problem of Construction 23

None of ~ese o~jectio~scan be raised against the pro­cedure descrIbed m sectlon 10, or against that in sec­tion 15.

MORE EXAMPLES

.18. A .problem of construction. Inscribe a square in agIVen trtangle. Two vertices of the square should be onthe base of the triangle, the two other vertices of thesquare on the two other sides of the triangle, one on each.

"What is the unknown?""A square:'UWhat are the data?""A triangle is given, nothing else.""What is the condition.'"

. "The four corners of the square should be on the per­Imeter of the triangle, two corners on the base, one cor­ner on each of the other two sides."

"Is it possible to satisfy the condition?""I think s~. I ani not so sure."

. ''You do not Seem to find the problem too easy. If youcannot solve the proposed problem, try to solve first somerelated problem. Could you satisfy a part Of the con.dilion'!"

"What do you mean by a part of the condition?""You see, the condition is concerned with all the ver­

tices of the square. How many vertices are there?""Four."

"A part of the condition would be concerned with lessthan four vertices. Keep only a part of the condition,drop the other part. What part of the condition is easyto satisfy?"

"It is easy to draw a square with two vertices on theperimeter of the triangle-or even one with three verticesOn the perimeterl"

"Draw a figure!"

d . d'f't has only three"The square is not eternllw:- I J"

vertices on the perimeter of the tnangle.

Ii ""Goodt Draw a gure.The student draws Fig. 3·

FIG. 3

"The square, as you said, is not determined by the part. 'I"

of the condition you kept. How can It vary.

. . . . . . f"Three corners of your square are on the penmeter a

the triangle but the fourth corner is not yet there ",:hereit should be. Your square, as you said, is undetermmed.

In the Classroom

"Try it experimentally, if you wish. Draw more squareswith three comers on the perimeter in the same way asthe two squares already in the figure. Draw small squaresand large squares. What seems to be the locus of thefourth corner? How can it vary?"

The teacher brought the student very near to theidea of the solution. If the student is able to guess thatthe locus of the fourth comer is a straight line, he hasgot it.

19. A problem to prove. Two angles are in differentplanes but each side oj one is parallel to the correspond­ing side ot the other, and has also the same direction.Prove that such angles are equal.

What we have to prove is a fundamental theorem ofsolid geometry. The problem may be proposed to stu­dents who are .familiar with plane geometry and ac­quainted with those few facts of solid geometry whichprepare the present theorem in Euclid's Elements. (Thetheorem that we have stated and are going to prove is thep,oposition 10 of Book XI of Euclid.) Not only ques­tions and suggestions quoted from our list are printedin italics but also others which correspond to them as"problems to prove" correspond to "problems to find."(The correspondence is worked out systematically in

PROBLEMS TO FIND, PROBLEMS TO PROVE 5. 6.)"What is the hypothesisr'"Two angles are in different planes. Each side of one

is parallel to the corresponding side of the other. and hasalso the same direction.

UWhat is the conclusion'!""The angles are equal.""Draw a figure. Introduce suitable notation."

Eg. A Problem to Prove

it can vary; the same is true of its fourth corner. Howcan it vary?"you

now

FIG. 2

"4The student draws Fig. 2.

"You kept only a part at the condition, anddropped the other part. How tar is the unknown

determined?"

In the Classroom

The student draws the lines of Fig. 4 and chooses,helped more or less by the teacher, the letters as in Fig. 4.

"What is the hypothesis? Say it, please, using your nota~

tion.""A., B, C are not in the same plane as A', B', C'. And

AB II A'B', AC II A'C'. Also AB has the same direction asA'B', and AC the same as A'C':'

FIG. 4

"What is the conclusion?""LBAC = LB'A'C',""Look at the conclusion! And try to think of a familiar

theorem having the same or a similar conclusion.""If two triangles are congruent, the cOlTesponding

angles are equal.""Very good! Now here is a theorem related to yours

and P'roved before. Could you use it?""I think so but I do not see yet quite how.""Should you introduce some auxiliary element in order

to make its use possible?"

"Well, the theorem which you quoted so well is about

Ig. A Problem to Prove

triangles, about a pair of congruent triangles. Have youany triangles in your figure?"

"No. But I could introduce some. Let me join B to C,and B' to C'. Then there are two triangles. /J., ABC,/::" A'B'C'."

"Well done. But what are these triangles good for?""To prove the conclusion, LBAC = LB'A'e':'"Goocll If you wish to prove this, what kind of tri~

angles do you need?"

A'

s'

FIG. 5

"Congruent triangles. Yes, of course, I may choose B J

C. B' , C' so that

AB .... A'B', AC = A'G':'

"Very goodl Now, what do you wish to prove?""I wish to prove that the triangles are congruent,

~ ABC = ~ A'B'C'.

If I could prove this, the conclusion LBAC == LB'A'C'would follow immediately."

"Finel You have a new aim, you aim at a new conclu­sion. Look at the conclusion! And try to think of a

:20, A Rate ProblemIn the Classroom

familiar theorem having the same or a similar conclu­sion."

"Two triangles are congruent if-if the three sides ofthe one are equal respectively to the three sides of theother."

"Well done. You could have chosen a worse one, Nowhere is a theorem related to yours and proved before.Could you use it?"

"1 could use it if r knew that Be = B'G'.""That is right! Thus, what is your aim?""To prove that Be = B'C'.""Try to think of a familiar theorem having the same or

a similar conclusion:'"Yes. I know a theorem finishing: '..• then the two

lines are equaL' But it does not fit in.""Should you introduce some aUXiliary element in order

to make its use possible1"

"You see, how could you prove BC = Brc, when thereis no connection in the figure between BC and B'G',"

"Did you use the hypothesis? What is the hypothesis'!""We suppose that AB 11 A'B', AC II AtC'. Yes, of course,

r must use that:'"Did you use the whole hypothesis.? You say that AB II

A'B'. Is that all that you know about these lines?""NOi AB is also equal to A'B', by construction, They

are parallel and equal to each other. And so are A C andA'C'."

"Two parallel lines of equal length-it is an interestingconfiguration. Have you seen it before?"

"Of course! Yesl ParallelogramI Let me join A to A',B to B'. and C to C'."

"The idea is not 50 bad. How many parallelogramshave you now in your figure?"

"Two. No, three. No, two. 1 mean, there are two of

29which you can prove immediately that they are paral­lelograms, There is a third which seems to be a parallelo­gram; I hope I can prove that it is one. And then theproof will be finishedl"

We could have gathered from his foregoing answersthat the student is intelligent. But after this last remarkof his, there is no doubt.

This student is able to guess a mathematical result andto distinguish dearly between proof and guess. He knowsalso that guesses can be more or less plausible, Really, hedid pTOfit something from his mathematics classes; hehas some real experience in solving problems, he canconceive and exploit a good idea.

20. A rate problem. Water is {lowing into a conicalv~ssel at the rate r. The vessel has the shape of a rightclrcular cone, with horizontal base, the vertex pointingdownwardsj the radius of the base is a, the altitude 0/ the

b

FIG. 6

cone b. Find the rate at which the surface is rising whenthe depth of the water is y. Finally, obtain the numericalvalue of the unknown supposing that a = 4 ft" b = 3 ft"r = 2 cu, ft. per minute, and y - 1 ft.

The students aTe supposed to know the simplest rulesof differentiation and the notion of "rate of change:'

"what are the data?"

'The radius of the base of the cone a = 4 ft., the alti­tude of the cone b = 3 ft., the rate at which the water isflowing into the vessel r 0= 2 CU. ft. per minute. and thedepth of the water at a certain moment, y = 1ft,"

"Correct. The statement of the problem seems to sug­gest that you should disregard, provisionally, the numeri­cal values, work with the letters. express the unknown interms of a, b, f J Y and only finally, after having obtainedthe expression of the unknown in letters, substitute thenumerical values. I would follow this suggestion. Now,what is the unknownr'

"The rate at which the surface is rising when the depthof the water is y."

"What is that? Could you say it in other terms?""The rate at which the depth of the water is in-

creasing.""What is that? Could you restate it still differently?""The rate of change of the depth of the water.""That is right, the rate of change of y. But what is the

rate of change? Go back to the definition.n

"The derivative is the rate of change of a function:'"CorrecL Now, is y a function? As we said before, we

disregard the numerical value of y. Can you imagine thaty changes?"

"Yes, y, the depth of the water, increases as the timegoes by:'

"Thus, y is a function of what?""Of the time t.""Good. Introduce suitable notation. How would you

write the 'rate of change of y' in mathematical symbols?"

"dy "dt

"Good Thus, this is your unknown. You have to ex­press it in terms of aJ b, T, y. By the way, one of these datais a 'rate.' Which one?"

30 In the Classroom 20. A Rate Problem 31

::r is th~ rate at which water is flowing into the vessel."What 15 that? Could you say it in other terms?"

"r is the rate of change of the volume of the water inthe vessel."

"What is that? Could you restate it still differently1How would you write it in suitable notation?"

" _ dV ..,--dl •

"What is n""The volume of the water in the vessel at the time t."

·'Good. Thus, you have to express dy in telll1s of a bdt ' 1

dV H .(it, y. ow WIll you do it?"

"If you cannot solve the proposed problem try to solvefirst some related problem. If you do not see yet the con-

nection between ~ and the data, try to bring in some

Si~~.l~~ connection that could serve as a stepping stone,"

"n• 0 you not see that there are other connections? ForlOstance, are y and V independent of each other?"

"No. When y increases, V must increase too.""Thus, there is a connection. What is the oonnection?"

. "Well, V is the volume of a cone of which the altitudeIS y. But I do not know yet the radius of the base:'

"You may consider it, nevertheless. Call it something.say x."

2"V=1rxy"

3 .

::Correct. Now, what about x? Is it independent of y?"~o. When the depth of the water, y, increases the

radIUS of the free surface, x, increases too.""Thus, there is a connection. What is the connection?"

In the Classroom

"One more connection, you see. I would not missprofiting from it. Do not {org~t, you wished to know the

connection between V and y."I have

ayx=7]

1ta2y3 "

V= 3b2 .

'"Very good. This looks like a stepping stone, does itnot? But you should not forget your goal. What is the

unknown?"

"Well, dy ."dt . d., dV

b f.A.J and"You have to find a connecuon etween dt' dt'

other quantities. And here you have one between y, V,and other quantities. What to do?"

"Differentiate! Of coursel

dV 7ra2y'l. dyTt=b2d"

Here it is." ""Finel And what about the numerical values?

"If a = 4, b = 3, ~ """ r = 2, Y = }:, then

'll'" X 16 X I dy )l

2= 9 d"

PART II, HOW TO,SOLVE IT .~•.A DIALOGUE o-3'ir7-~$-A

Getting Acquainted

Where should I start? Start from the statement of theproblem.

What can I do? Visualize the problem as a whole asclearly and as vividly as you can. Do not concern your­self with details for the moment.

What can I gain by doing so? You should understandthe problem, familiarize yourself with it, impress its pur­pose on your mind. The attention bestowed on the prob.lem may also stimu,late your memory and prepare for therecollection of relevant points.

Working for Better Understanding

Jf'here should I start? Start again from the statementof the problem. Start when this statement is so clear toyou and so well impressed on your mind that you maylose sight of it for a while without fear of losing it all-i)­gether.

What can I dot Isolate the principal parts of yourproblem. The hypothesis and the conclusion are theprincipal parts of a "problem to prove"; the unknown,the data, and the conditions are the principal parts of a"problem to find," Go through the principal parts ofy~ur problem, consider them one by one, consider themin turn, consider them in various combinations, relatingeach detail to other details and each to the whole of theproblem.

33

Carrying Out the Plan

Where should I start? Start from the lucky idea thatled you to the solution. Start when yOll feel sure of yourgrasp of the main connection and you feel confident thatyou can supply the minor details that may be wanting.

What can I do? Make yOUT grasp quite secure. Carrythrough in detail all the algebraic or geometric opera­tions which you have recognized previously as feasible.Convince yourself of the correctness of each step by for­mal reasoning, or by intuitive insight, or both ways if youcan. If your problem is very complex you may distin­guish "great" steps and "small" steps, each great stepbeing composed of several small ones. Check first thegreat steps, and get down to the smaller ones afterwards.

What can I gain by doing so'! A presentation of thesolution each step of which is correct beyond doubt.

far it leads you, and reconsider the situation. The situa­tion has changed, thanks to your helpful idea. Considerthe new situation from various sides and seek contactswith your Cannedy acquired knowledge.

What can I gain by doing so again'! You may be luckyand have another idea. Perhaps your next idea will leadyou to the solution right away. Perhaps you need a fewmore helpful ideas after the next. Perhaps you will beled as-tray by some of your ideas. Nevertheless you shouldbe grateful for all new ideas, also for the lesser ones, ahofor the hazy ones, also for the supplementary ideas add­ing some precision to a hazy one, or attempting the cor­rection of a less fortunate one. Even if you do not haveany appreciable new ideas for a while you should begrateful if your conception of the problem becomes morecomplete or more coherent, more homogeneous or betterbalanced.

A Dialogue34

What can I gain by doing s07 You should prepare andclarify details which are likely to playa role afterwarW.

Hunting for the Helpful Idea°d ° IWhere should I start? Start from the conSi eratlon 0

the rinei al parts of your problem. Start when theseprin~ipal ~arts are distinctly arranged and c1e~rlY con­ceived. thanks to your previous work. and w en your

memory seems responsive. .What can I do? Consider your problem from var~ou~

sides and seek contacts with your formerly acqUire

k.nowle~dgeo problem from various sides. EmphasizeConst er your . ine the

different parts. examine different detaIls. exam b'same details repeatedly but in different ways, ~~:r::the details differently, approac~ t~:r:a~o:tai;, somesides. Try to see some new meamng

interpretat'on of the whole. dnew - .th your formerly acquired knowle ge.

Seek contacts WI • • . .h' k f what helped you in simtlar sItuatIOns m

Thry to: ;ry ~o recognize something familiar in what yout e pas. . ething useful in what youexamine. try to perceIve som

recognize. I 1 .d erhaps a de·Wh t could I iherceive? A help u 1 ea, p

a r I the way to the verycisive idea that shows you at a g ance

en~ can an idea be helPful? It shows you the whole ofow f the wa . it suggests to you ID'Jre or

the way or a part 0 y, ceed Ideas are moreles-s distinctly how you c;nkPr~ yo; have any idea ator less complete. You are uc Y

aU. . h . m-hlete idea? You shouldWh t can I do wzt an mcO r 'd

• a. . oks advantageous you should const erconsider it. If. It 10 I" bl you should as-certain hoWit longer. If It looks. re la e

A Dialogue 35

A Dialogue

Looking Back

Where should I start? From the solution, complete and

correct in each detail. . .What can I do? Consider the solution from var~o:

sides and seek contacts with your formerly acqmr

knowledge. kConsider the details of the solution and try .to rna e

h'Ie as you can' survey more extensIve parts

t em as SImp ,of the solution and try to make them shorte~; try tOhs~e

, 1 T to modIfy to t elrthe whole solutlOn at a g ance. ry . to

d Iler Or larger parts of the solutlOO, trya vantage sma ., .' fi .tiIll rove the whole solution, to make It mtmt!ve, to t IPl' d knowledge as naturally asinto your fanner y acqulIe th

'ble Scrutinize the method that led you to. epOSSI. ". ke use of It forsolution, try to see Its pomt, and try to rna kother problems. Scrutinize the result and try to rna e use

of it for other problems. d d, "y mayfin anewanWhat can 1 gam by domg so. au . "

d" . new and lllteresttngbetter solution, you may lSCover " .if you get into the habit o£ surveymg

facts. In any case, .' au willand scrutinizing your solutions m thiS way, y

_ e knowledge well ordered and ready to use.acqmre som I" roblemsand you will develop your ability of so vmg P .

PART III. SHORT DICTIONARYOF HEURISTIC

Analogy is a sort of similarity. Similar objects agreewith each other in some respect, analogous objects agreein certain relations of their respective parts.

1. A rectangular parallelogram is analogous to a rec­tangular parallelepiped. In fact. the relations betweenthe sides of the parallelogram are similar to those be­tween the faces of the parallelepiped:

Each side of the parallelogram is parallel to just oneother side, and is perpendicular to the remaining sides.

Each face of the pat'allelepiped is parallel to just oneother face, and is p~rpendicular to the remaining faces.

Let us agree to Call a side a "bounding element" of theparallelogram and a face a "bounding element" of theparallelepiped. Then, we may contract the two fore­going statements into one that applies equally to bothfigures:

Each bounding element is parallel to just one otherbounding element and is perpendicular to the remainingbounding elements.

Thus, we have expressed certain relations which arecommOn to the two systems of objects we compared, sidesof the rectangle and faces of the rectangular parallele­piped. The analogy of these systems consists in this com·munity of relations.

2. Analogy pervades all our thinking, our everydayspeech and our trivial conclusions as well as artisticways of expression and the highest scientific achieve.ments. Analogy is used on very different levels. People

37

Any plane passing through the median eM of the tn­an~le contains the centers of gravity of all parallel fibersWhICh constitute the triangle. Thus, we are led to theconclusion that the center of gravity of the whole tri­angle lies on the Same median. Yet it must lie on theother two medians just as well, it must be the commonpoint of intersection of all three medians.

It is desirable to verify now by pure geometry, inde­pendently of any mechanical assumption, that the threemedians meet in the same point.

Analogy 39

has its. center of graviry in the same plane, then this planeconta~ns a,l:io. the c~nter of gravity of the whole system S.

ThIS pnnClple YIelds all that we need in the case of thetr~angle. ~irs~, it implies that the center of gravity of thetna~gle hes In. the plane of the triangle. Then. we may~?nsl~er the tnan~le as consisting of fibers (thin strips,mfinItely narrow parallelograms) parallel to a certain

side of the triangle (the side AB in Fig. 7). The centerof gravity of each fiber (of any parallelogram) is. obvi­ously, its midpoint, and all these midpoints lie on theline joining the vertex C opposite to the side AB to themidpoint M of AB (,ee Fig. 7).

38 Analogy

often use vague, ambiguous, incomplete. or incompletelyclarified analogies, but analogy may reach the level ofmathematical precision. All sorts of analogy may playarole in the discovery of the solution and so we shouldnot neglect any sort. .

3. 'l\7e may consider ourselves lucky when. trymg tosolve a problem, we succeed in discovering a simpleranalogous problem. In section 15, our original problemwas concerned with the diagonal of a rectangular paral­lelepiped; the consideration of a simpler analogous prob­lem, concerned with the diagonal of a rectangle, led us tothe solution of the original problem. We are going todiscuss one more case of the same sort. We have to solvethe following problem:

Find the center of gravity of a homogeneous tetra­

hedron.'Vithout knowledge of the integral calculus, and with

little knowledge of physics, this problem is not easy atall' it was a serious scientific problem in the days ofAr~himedes or Galileo. Thus, if we wish to solve it withas little preliminary knowledge as possible, we shouldlook around for a simpler analogous problem. The corre·sponding problem in the plane occurs here natur~l1y:

Find the center of gravity of a homogeneous tnangle.Now. we have two questions instead of one. But :wo

questions may be easier to answer than ~ust o~e questIon-provided that the two questions are lntelltgently con­

nected.4. Laying aside. for the moment, our original problem

concerning the tetrahedron, we concentrate upon thesimpler analogous problem concerning the triangle. Tosolve this problem. we have to know something ab.outcenters of gravity. The following principle is plaUSIbleand presents itself naturally. ,

It a system of masses S consists of parts, each of whtch

A M

FIG. 7

c

B

In the case of the triangle. we had three medians likeMe each of which has to contain the center of gravity. .of the triangle. Therefore, these three medians mus~ meetin one point which if> precisely the center of gravity. In

40 Analogy

5. After the case of the triangle, the case of the tetra·hedron is fairly easy. We have now f>olved a problemanalogous to our proposed problem and, having solvedit, we have a model to tallow.

In solving the analogous problem which we use now asa model we conceived the triangle ABC as consiflting offibers p~ranel to ODe of its sides, AB, Now. we conceivethe tetrahedron ABCD as consisting of fibers parallel to

one of its edges, AB.The midpoints of the fibers which constitute the tri­

angle lie all on the same sttaight line, a ~edian of thetriangle, joining the midpoint M of the side AB. to theopposite vertex C. The midpoints of the fibers whl~~ c?n.stitute the tetrahedron lie all in the same plane. ]OlDmgthe midpoint M of the edge AB to the opposite .edge CD(see Fig. 8); we may call this plane MCD a medtan plane

of the tetrahedron.

the case of the tetrahedron we have six median planeslike MCD, joining the midpoint of SOIne edge to the op­posite edge, each of which has to contain the center ofgravity of the tetrahedron. Therefore, these six medianplanes must meet in one point which is precisely thecenter of gravity.

6. Thus, we have solved the problem of the center ofgravity of the homogeneous tetrahedron. To completeoui solution, it is desirable to verify now by pure geome­try, independently of mechanical considerations, that thesix median planes mentioned pass through the samepoint.

When we had solved the problem of the center of grav­ity of the homogeneous triangle, we found it desirableto verify. in order to complete our solution. that the threemedians of the trian,gle pass through the same point.This problem is analogous to the foregoing but visiblysimpler.

Again we may use, in solving the problem concerningthe tetrahedron, the simpler analogous problem concern.ing the triangle (which we may suppose here as solved) .In fact, consider the three median planes. passing_through the three edges DA, DB, DC issued from thevertex D; each passes also through the midpoint of theopposite edge (the median plane through DC passesthrough M, see Fig. 8). Now. these three median planesintersect the plane of D. ABC in the three medians of thistriangle. These three medians pass through the samepoint (this is the result of the simpler analogous prob­lem) and this point, just as D. is a common point of thethree median planes. The straight line, joining the twocommon points, is common to all three median planes.

We proved that those 3 among the 6 median planes~hich pass through the vertex D have a common straightline. The same must be true of those 3 median planes

41Analogy

D

MFIG. 8

A

Analogy

which pass through A; and <:l1so of the 3 median planesthrollgh B; and also of the ~ through C. Connectingthese facts suitably, we Ulay prove that the 6 medianplanes have a common point. (The 3 median planespassing through the sides of 6. ABC determine a com­mon point, and 3 lines of intersection which meet in thecommon point. Now, by what we have JUSt proved,through each line of intersection one more median planemust pass.)

7. Both under 5 and under 6 we used a simpler analo­gous problem, concerning the triangle, to solve a prob­lem about the tetrahedron. Yet the two cases are differentin an important respect. Under 5, we used the method ofthe simpler analogous problem whose solution we imi·tated point by point. Under 6, we used the result of thesimpler analogous problem, and we did not care howthis result had been obtained. Sometimes, we may beable to me both the method and the result of the simpleranalogous problem. Even our foregoing example showsthis if we regard the considerations under 5 and 6 asdifferent parts of the solution of the same problem.

Our example is typicaL In solving a proposed problem,we can often use the solution of a simpler analogousproblem; we may be able to use its method, or its result,or both. Of course, in more difficult cases, complicationsmay arise which are not yet shown by our example.Especially, it can happen that the solution of the analo­gous problem cannot be immediately used for our orig.inal problem. Then, it may be worth while to reconsiderthe solution, to vary and to modify it till, after havingtried various forms of the solution, we find eventuallyone that can be extended to our original problem.

8. It is desirable to foresee the result, or, at least, somefeatures of the result, with some degree of plausibility.Such plausible forecasts arc often based on analogy.

Analogy 43

Thus, we m~y know that the center of graVity of aho~ogeneous t~1angle coi~cides with the center of gravityof Its three VertIces (that IS, of three material points withequal. masse~, placed in the vertices of the triangle).Kno~:ng thIS, we may conjecture that the center ofgraVIty of a homogeneous tetrahedron coincides with thecenter of gravity of its four vertices.

This conjecture is an "inference by analogy." Knowingthat the triangle and the tetrahedron are alike in manyrespects. we conjecture that they are alike in one morerespect. I~ would be foolish to regard the plausibility ofsu~ conjectures as certainty, but it would be JUSt asfool.Ish, or even more foolish, to disregard such plausibleconjectures.

Inference by analogy ~ppears to be the mOst common~nd of c~nclusion, a~d it is possibly the most essentialkmd. It YIelds more or less plausible conjectures whichmay o~ may not be~_confirmedby experience and stricterreasonmg. The chemist, experimenting on animals inorder to foresee the influence of his drugs on humansdraws conclusions by analogy. But so did a small boy iknew. His pet dog had to be taken to the veterinary andhe inquired: '

"Who is the veterinary?""The animal doctor:'"Which animal is the animal doctor?"

. g. An analogical conclusion from many parallel casesIS stro~ger than one from fewer cases. Yet quality is still~re Important .here than quantity. Clear-cut analogiesweIgh mo~ heaVIly than vague similarities, systematicallya:ranged mstances count for more than random collec­bons of Cases.

In the foregoing (under 8) we put forward a conjec­tur~ about the center of gravity of the tetrahedron. ThisCOnJecture was supported by analogy; the case of the

2 1

3 3 1

4 6 4 1

Very little familiarity with the powers of a binomial isneeded to recognize in these numbers a section of Pascal'striangle. We found a remarkable regularity in segment.triangle, and tetrahedron.

10. If we have experienced that the objects we com~

pare are closely connected, "it~.fere~ces b~ analogy," asthe following, may have a certam weight With us.

tetrahedron is analogous to that of the triangle. We maystrengthen the conjecture by examining one more a~al()­

gous case, the case of a homogeneous rod (that IS, astraight line-segment of uniform density) .

The analogy between

segment triangle tetrahedron

has many aspects. A segment is contained in a straightline, a triangle in a plane. a tetrahedron in space. Straightline-segments are the simplest one-dimensional boundedfigures. triangles the simplei'ot polygons, tetrahedrons thesimplest polyhedrons.

The segment has .2 zero-dimensional bounding ele­ments (2 end.points) and its interior is one-dimensional.

The triangle has 3 zero-dimensional and 3 one-dimen­sional bounding elements (3 vertices, 3 sides) and itsinterior is two-dimensional.

The tetrahedron has 4 zero-dimensional, 6 one-dimen­sional, and 4 two-dimensional bounding elements (4vertices, 6 edges, 4 faces) , and its interior is three-cHmen­

sional.These numbers can be assembled into a table. The suc­

cessive columns contain the numbers for the zero-, one-,two-, and three-dimensional elements, the successive rowsthe numbers for the segment, triangle, and tetrahedron:

44 ./fnawgy Analogy 45

The center of gravity of a homogeneous rod coincideswith th~ center of gravity of its .2 end-points. The centerof graVIty of a homogeneous triangle coincides with thecenter of wavity of its 3 vertices. Should we not suspectth~t ~he ce~ter of gravity of a homogeneous tetrahedronCOlllC1~eS WIth the center of gravity of its 4 vertices?

. '-:gam, th~ center of gravity of a homogeneous rod~vldes the dIstance between its end-points in the propor­U:Jll 1 : I. The center of graVity of a triangle divides thedistan~e ~tw~en any vertex and the midpoint of the0PPOSI te Side m the proportion .2 : I. Should we not sus.pect that. t~e cen"ter of gravity of a homogeneous tetra.hedron diVides th~ distance between any vertex and thecenter of wavity of the opposite [ace in the proportion3 : 11

It appears extremely unlikely that the conjectures sug.geste~ by these q.uestions should be wrong. that such abeautlf~l reg~lanty should be spoiled. The feeling thath~nnomous slmp,le order cannot be deceitful guides thediscoverer both III the mathematical and in the other~i~nces. an~ is expressed by the Latin saying: simplexslg!llum VeT! (simplicity is the seal of truth) .

[The preceding suggests an extension to n dimensions.It appears unlikely that what is true in the first threedimensions, for n = 1, 2, 3. should cease to be true for?igher, va~~~s ~f n. This conjecture is an "inference byInduction; It Illustrates that induction is naturally basedon analogy. See INDUCTION AND MATHEMATICAL INDUC­

TION.]

[11. We. finish the present section by considering brieflythe ~,ost Important cases in which analogy attains the

. precISIon of mathematical ideas., (I) Two systems of mathematical objects, say Sand S'.:ue so connected that certain relations between the ob.Jects of S are governed by the same laws as those betweenthe objects of S'.

2. There are various reasons for introducing auxiliaryelements. We are glad when we have succeeded in recol·lecting a problem related to ours and solved before. It isprobable that we can use such a problem but we do notknow yet how to use it. For instance, the problem whichwe are trying to solve is a geometric problem, and therelated problem which we have solved before and havenow succeeded in recollecting is a problem about tri·angles. Yet there is no triangle in our figure; in order tomake any use of the problem recollected we must have atriangle; therefore, we p.ave to introduce one, by addingsuitable auxiliary lines "to our figure. In general, havingrecollected a formerly solved related problem and wish­ing to use it for our present one, we must often ask:Should we introduce some auxiliary element in order tomake its use possible? (The example in section 10 istypical.)

Going back to definitions; we have another opportu­nity to introduce auxiliary elements. For instance, expli­cating the definition of a circle we should not onlymention its center and its radius. but we should alsointroduce these geometric elements into our figure. With­out introducing them, we could not make any concreteuse of the definition; stating the definition withoutdrawing something is mere lip-service.

Trying to use known results and going back to defini­tions are among the best reasons for introducing auxil­iary elements; but they are not the only ones. We mayadd auxiliary elements to the conception of our problemin order to make it fuller, more suggestive, more familiaralthough we scarcely know yet explicitly how we shallbe able to use the elements added. We may just feel thatit is a "bright idea" to conceive the problem that waywith such and such elements added.

We may have this or that reason for introducing an

46 Auxiliary Elements

This kind of analogy between Sand S' is exemplifiedby what we have discussed under 1; take as S the sides ofa rectangle. as S' the faces of a rectangular parallelepiped.

(II) There is a one-one correspondence between theobjects of the two systems Sand S', preserving certainrelations. That is, if such a relation holds between theobjects of one system, the same relation holds betweenthe corresponding objects of the other system. Such aconnection between twO systems is a very precise sort ofanalogy; it is called isomorphism (or holohedral iso­

morphism) .(III) There is a one-many correspondence between

the objects of the two systems Sand S' preserving ~ertai~

relations. Such a connection (which is important In varI­ous' branches of advanced mathematical study, especiallyin the Theory of Groups, and need not be discussed herein detail) is called merohedral isomorphism (or homo­morphism; homoiomorphism would be, perhaps, a bettertenn). Merohedral isomorphism may be considered asanother very precise sort of analogy.]

Auxiliary elements. There is much more in our con­ception of the problem at the end of our work than wasin it as we started working (PROGRESS AND ACHIEVEMENT,

1) . As our work progresses, we add new elements to thoseoriginally considered. An element that we introduce inthe hope that it will further the solution is called an

auxiliary element.1. There are various kinds of auxiliary elements. Solv­

ing a geometric problem, we may. introduce ne;v linesinto our figure, auxiliary lines. SolVIng an algebraIC prob­lem. we may introduce an auxiliary unknown (AUXILIARY

PROBLEMS, 1). An auxiliary theorem is a .theorem wh~seproof we undertake in the hope of promoting the soluuonof our original problem.

Auxiliary Elements 47

Auxiliary Elements4B A uxiliary Elements

auxiliary element, but we should have some reason. Weshould not introduce auxiliary elements wantonly.

3. Example. Construct a triangle. being given oneangle, the altitude drawn from the vertex of the givenangle, and the perimeter of the triangle.

A

p

49

the perimeter of the triangle. Therefore we must intro­duce p. But how?

\Ve may attempt to introduce p in various ways. Theattempts exhibited in Figs. 9, 10 appear clumsy. If we tryto make clear to ourselves why they appear so unsatis­factory, "ve may perceive that it is fo~ L~ck of symmetry.

In fact, the triangle has three unknown sides a, bJ c.We call a, as usual, the side opposite to A; we know that

a+ b + c ~ p.Now, the sides band c play the same role; they are inter­changeable; our problem is symmetric with respect to band c. But band c do not play the same role in ourfigures 9, 10; placing the length p we treated band cdifferently; the figures 9 and 10 spoil the natural sym­metry of the problem whh respect to band c, \Ve shouldplace p so that it has the same relation to b as to c.

This consideration may be helpful in suggesting toplace the length p as in Fig. 1 L \Ve add to the side a of

FIG. 9 A

FIG. 11

DBa

b

cb

.'.'...'.'.­.'.'~,,~

.'.'.'E

p

A

FIG. 10

We introduce suitable notation. Let a denote the givenangle, h the given altitude drawn from the vert~x A 0.£ aand p the given perimeter. We draw a figure III whIchwe easily place a and h. Have we used all the data? No.our figure does not contain the given length p, equal to

the triangle the segment CE of length b on one side andthe segment BD of the length c on the other side so thatp appears in Fig. II as the line ED of length

b+ a + c = p.If we have some little experience in solving problems ofconstruction, we shaH not fail to introduce into the

y = x 2•

We obtain now a new problem: Find y. satisfying theequatlOn

y2 ~ '3Y + 36 ~ o.

The new problem is an auxiliary problem; we intend touse it as a means of solving our original problem. Theunknown of our auxiliary problem, y. is appropriatelycalled auxiliary unknown.

2. Example. Find the diagonal of a rectangular paral­lelepiped being given the lengths of three edges drawnfrom the same corner.

sideration may help us to solve another problem, ouroriginal problem. The original problem is the end wewish to attain. the auxiliary probfem a means by whichwe try to attain our end.

An insect tries to escape through the windowpane,tries the same again and again, and does not try the nextwindow which is open and through which it came intothe room. A man is able, or at least should be able, to actmore intelligently. Human superiority consists in goingaround an obstacle that cannot be overcome directly, indevising a suitable auxiliary problem when the originalproblem appears insoluble. To devise an auxiliary prob­lem is an important operation of the mind. To raise aclear-cut new problem subservient to another problem,to conceive distinctly as an end what is means to anotherend, is a refined achievement of the intelligence. It is animportant task to learn (or to teach) how to handleauxiliary problems intelligently.

1. Example. Find x, satisfying the equation

x4 - 13X2 + 36 = o.

If we observe that x 4 = (X2)2 we may see the advan­tage of introducing

50 Auxiliary Problem

figure. along with ED, the auxiliary lines AD and AE,each of which is the base of an isosceles triangle. In fact,it is not unreasonable to introduce elements i~~o theproblem which are particularly simple and famll1ar, as

isosceles triangle. . .We have been quite lucky in introducing our auxIlIary

lines. Examining the new figure w~ may discover thatLEAD has a simple relation to the given angle a· In fact,we find using the isosceles triangles b:. ABD and D. ACE

that LDAE = ~+ 90°, After this remark. it is natural to2 .

try the construction of /':" DAE. Trying. thi~ construcu~n.

we introduce an auxiliary problem whIch IS much easIer

than the original problem.4. Teachers and authors of textbooks should not forget

that the intelligent student and THE INTELLIGENT RE~ER

are not satisfied by verifying that the steps of a reasonmgare correct but also want to know the motive and thepurpose of the various steps..The introduction of. anauxiliary element is a conspICUOUS step. If a trIck.yauxiliary line appears abruptly in the figur~,. witho~t an!motivation, and solves the problem surpnsmgly. mtelh­gent students and readers are disappointed; th.ey feel thatthey are cheated. Mathematic~ is in~resting III so far asit occupies our reasoning and mventlve powers. But thereis nothing to learn about reasoning and invention if t~emotive and purpose of the most conspicuous step rem.am

incomprehensible. To make such steps comprehensibleby suitable remarks (as in the foregoing. under 3) or bycarefully chosen questions and suggestions (as in s~ctions10. 18, 19. 20) takes a lot of time and effort; but it may

be worth while.

Auxiliary problem is a problem which we c~nsider.not for its own sake, but because we hope that its con·

Auxiliary Problem 5'

52 Auxiliary Problem

Trying to solve this problem (section 8) we may beled, by analogy (section 15), to another problem:Find the diagonal of a rectangular parallelogram beinggiven the lengths of two !lides drawn from the !lamevertex.

The new problem is an auxiliary problem; we considerit became we hope to derive some profit for the originalproblem from its consideration.

3. Profit. The profit that we derive from the consider­ation of an auxiliary problem may be of various kinds.We may use the result of the auxiliary problem. Thus, inexample 1, having found by solving the quadratic equa­tion for y that y is equal to 4 or to g, we infer thatx2 = 4 or x 2 ==- 9 and derive hence all possible values ofx. In other cases, we may use the method of the auxiliaryproblem. Thus, in example 2, the auxiliary problem is aproblem of plane geometry; it is analogous to, but sim·pIer than, the original problem which is a problem ofsolid geometry. It is reasonable to introduce an auxiliaryproblem of this kind in the hope that it will be instruc·tive, that it will give us opportunity to familiarize our­selves with certain methods, operations, or tools, whichwe may use afterwards for our original problem. In ex­ample 2, the choice of the auxiliary problem is ratherlucky; examining it closely we find that we can use bothits method and its result. (See section 15, and DID YOUUSE ALL THE DATA?)

4. Risk. We take away from the original problem thetime and the effort that we devote to the auxiliary prob.lem. If our investigation of the auxiliary problem fails,the time and effort we devoted to it may be lost. There·fore, we should exercise our judgment in choosing anauxiliary problem. We may have various good reasonsfor our choice. The auxiliary problem may appear moreaccessible than the original problem; or it may appear

Auxt"liary Problem 53

instru~tive; or it may have some Sort of aesthetic appeal.SometImes the only advantage of the auxiliary problemis that it is new and offers unexplored possibilities; wechoose it because we are tired of the original problemall approaches to which seem to be exhausted.

5· How to find one. The discovery of the solution ofthe proposed problem often depends on the discovery ofa suitable auxiliary problem. Unhappily, there is no in­fallible method of discovering suitable auxiliary prob­lems as there is no infallible method of discovering thesolution. There are, however, questions and suggestionswhich are frequently helpful, as LOOK AT THE UNKNOWN.

We are often led to useful auxiliary problems by VARI.

ATION OF THE PROBLEM.

6. Equivalent problems. Two problems are equivalentif the solution of each involves the solution of the other.Thus, in our example I, the original problem and theauxiliary problem are equivalent.

Consider the following theorems:A. In any equilateral triangle, each angle is equal

to 60°.

B. In any equiangular triangle, each angle is equalto 60°.

These two theorems are not identical. They containdifferent notions; one is concerned with equality of thesides, the other with equality of the angles of a triangle.But each theorem follows from the other. Therefore, theproblem to prove A is equivalent to the problem toproveB.

If we are required to prove A, there is a certain advan­tage in introducing, as an auxiliary problem, the prob­lem to prove B. The theorem B is a little easier to provethan A and, what is more important, we may foreseethat B is easier than A, we may judge so, we may findplausible from the outset that B is easier than A. In fact,

the theorem B, concerned only with angles, is more"homogeneous" than the theorem A which is concernedwith both angles and sides.

The passage from the original problem to the aux­iliary problem is called convertible reduction, or bi­lateral reduction, or equivalent reduction if thes.e twoproblems, the original and the auxiliary. are equivalent.Thus, the reduction of A to B (see above) is convertibleand so is the reduction in example 1. Convertible reduc­tions are, in a certain respect. more important and moredesirable than other ways to introduce auxiliary prob­lems, but auxiliary problems which are not equivalentto the original problem may also be very useful; takeexample 2.

7. Chains of equivalent auxiliary problems are fre.quent in mathematical reasoning. We are required tosolve a problem A; we cannot see the solution. but wemay find that A is equivalent to another problem B.Considering B we may run into a third problem C equiv~

alent to B. Proceeding in the same way, we reduce C toD, and so on, until we come upon a last problem L whosesolution is known or immediate. Each problem beingequivalent to the preceding. the last problem L must beequivalent to our original problem A. Thus we are ableto infer the solution of the original problem A from theproblem L which we attained as the last link in a chainof auxiliary problems.

Chains of problems of this kind were noticed by theGreek mathematicians as we may see from an importantpassage of PAPPUS. For an illustration, let us reconsiderour example I. Let us call (A) the condition imposedupon the unknown x:

(A) .4 - '3. 2 + 36 - o.

One way of ~lving the problem is to transform the pro--

Proceeding in the same way, we obtain

(D) (,.2 - 13)2 ~ 25

(E) 2X 2 - 13 = ±5

(F) x 2 = 13 ± 52

(G) X_±~132±5

(H) x = 3, or -3, or 2, or ~2.

Each reduction that we made was convertible. Thus, thelast condition (H) is equivalent to the first condition(A) so that 3, -3,2, -2 are all possible solutions of our

original equation.. In the foregoing-, we derived from an original condi­

hon (A) a sequence of conditions (B), (C), (D), ...each of which was equivalent to the foregoing. Thispoint deserves the greatest care. Equivalent conditionsare satisfied by the same objects. Therefore, if we passfrom a proposed condition to a new condition equivalent

•(..2) 2 - 2 (,.2) '3 + ,69 ~ 25.(C)

Auxiliary PrOblem 55

posed condition into another condition which we shallcall (B):

(B) (,.2) 2 - 2 (..2) '3 + '44 ~ o.

Observe that the conditions (A) and (B) are different.They are only slightly different if you wish to say so,they are certainly equivalent as you may easily convinceyourself, but they are definitely not identical. The pas.sage from (A) to (B) is not only correct but has a clear­cut purpose, obvious to anybody who is familiar with thesolucion of quadratic equations. ""Varking further in thesame direction we transform the condition (B) into stillanother condition (C) :

Auxiliary Problem54

Bolzano, Bernard (1781,1848), logician and mathema·tician. devoted an extensive part of his comprehensivepresentation of logic, Wissenschaftslehre, to the subjectof heuristic (vol. 3. pp. 293'575). He writes about thispart of his work: "I do not think at all that I am ableto present here any procedure of investigation that wasnot perceived long ago by all men of talent; and I do notpromise at all that you can find here anything quite newof this kind. But I shall take pains to state in dear wordsthe rules and ways of investigation which are followedby all able men, who in most cases are not even consciousof following them. Although I am free from the illusionthat I shall fully succeed even in doing this, I still hope

other, more risky than a bilateral or convertible re­duction.

Our example .2 shows a unilateral reduction to a lessambitious problem. In fact, if we could solve the originalproblem, concerned with a parallelepiped whose length,width, and height are a, b, c respectively, we could moveon to the auxiliary problem putting c = 0 and obtaininga parallelogram with length a and width b. For anotherexample of a unilateral reduction to a less ambitiousproblem see SPECIALIZATION, 3, 4, 5. These examples showthat, with some luck, we may be able to use a less am­bitious auxiliary problem as a stepping stone, combiningthe solution of the auxiliary problem with some appro-­priate supplementary remark to obtain the solution ofthe original problem.

Unilateral reduction to a more ambitious problem mayalso be successful. (See GENERAliZATION. 2, and the reduc­tion of the first to the second problem considered inINDUCTION AND MATHEMATICAL INDUCTION, 1, 2.) In fact,the more ambitious problem may be more accessible; thisis the INVENTOR'S PARADOX.

56 Auxiliary Problem

to it. we have the same solutions. But if we pass from aproposed condition to a narrower one, we lose solutions.and if we pass to a wider one we admit improper, adven­titious solutions which have nothing to do with the prD­posed problem. If, in a series of successive :eduction~, .wepass to a narrower and then again to a wIder candIllonwe may lose track of the original problem completely. Inorder to avoid this danger, we must check carefully thenature of each newly introduced condition: Is it equiv.alent to the original condition? This question is stillmore important when we do not deal with a single equa­tion as here but with a system of equations, or when thecondition is not expressed by equations as, for instance,in problems of geometric construction.

(Compare PAPPUS) especially comments 2, 3, 4, 8. Thedescription on p. 143, lines 4-21, is unnecessarily re­stricted; it describes a chain of "problems to find," eachof which has a different unknown. The example can·sidered here has just the opposite speciality: all problemsof the chain have the same unknown and differ only inthe form of the condition. Of course, no such restriction

is necessary.)8. Unilateral reduction. We have twO problems, A and

B, both unsolved. If we could solve A we could hencederive the full solution of B. But not conversely; if wecould solve B. we would obtain, possibly, some informa­tion about A, but we would not know how to derive thefull solution of A from that of B. In such a case, more isachieved by the solution of A than by the solution of ~.

Let us call A the more ambitious, and B the less ambt­tious of the two problems.

If, from a proposed problem, we pass either to a moreambitious or to a less ambitious auxiliary problem wecall the step a unilateral reduction. There are two kindsof unilateral reduction, and both are, in some way or

Bernard Bolzano 57