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THE THEORY OF PROPORTION

THE THEORY

OF PROPORTION

BY

M. J. M. HILL, M.A., LL.D., ScD., F.R.S.

ASTOR PROFESSOR OF MATHEMATICS IN

THE UNIVERSITY OF LONDON

LONDON

CONSTABLE AND COMPANY, LTD.

1914

PREFACE

This little book is the outcome of the effort annually renewed

over a long period to make clear to my students the principles

on which the Theory of Proportion is based, with a view to its

application to the study of the Properties of Similar Figures.

Its content formed recently the subject matter of a course

of lectures to Teachers, delivered at University College, under

an arrangement with the London County Council, and it is

now being published in the hope of interesting a wider circle.

At the commencement of my career as a teacher I was accus-

tomed, in accordance with the then estabHshed practice, to take

for granted the definition of proportion as given by Euclid in

the Fifth Definition of the Fifth Book of his Elements* and to

supply proofs of the other properties of proportion required

in the Sixth Book which were valid only when the magnitudes

considered were commensurable. Dissatisfied with the results

of a method which could have no claim to be considered

logical, after trying some other modes of exposition, I turned

to the syllabus of the Fifth Book drawn up by the Association

for the Improvement of Geometrical Teaching. But again I

found this hard to explain, and it was evident that my students

could not grasp the method as a whole, even when they were

able to understand its steps singly.

After prolonged study I found that, in addition to the

difficulty arising out of Euclid's notation, which is a matter

of form and not of substance, and the difficulty that Euclid

does not sufficiently define ratio, two reasons could be assigned

for the great difficulty of his argument.

(1) Of the long array of definitions prefixed to the Fifth

Book there are only two which effectively count. One of these,

the Fifth, is the test for deciding when two ratios are equal ;

and the other, the Seventh, is the test for distinguishing

* The substance of the Fifth Book is usually attributed to Eudoxus.

O/^O^ K^f^^

viii PREFACE

between unequal ratios. They are intimately related, but

when once stated they can be treated as independent.

Now it can he seen at once that if the test for deciding when two

ratios are equal is a good and sound one^ it should he possihle to

deduce from it all the properties of equal ratios^ and in order to

obtain these properties it should not he necessary to employ the

test for distinguishing hetween unequal ratios.

But Euclid frequently employs this last-mentioned test, or

propositions depending on it, to prove properties of equal

ratios. In fact, it is not at all easy for any one trying to follow

the course of his argument to see whether it leads naturally

to the employment of the Fifth or of the Seventh Definition,

or a proposition depending on the Seventh Definition. Euclid's

proofs do not run on the same lines, and are so difficult and

intricate that they have almost entirely fallen out of use. It

will be shown in this book that all the properties of equal

ratios can he proved hy the aid of the Fifth Definition, and that

the Seventh Definition is not required.

This is effected, without departing from the spirit or the

rigour of Euclid's argument, by assimilating Euclid's proofs

of those propositions in which the use of the Seventh Defini-

tion is directly or indirectly involved to his proofs of those

propositions in which he employs the Fifth Definition only.

(2)1 think it will appear to any one who reads this book that

it is in a high degree probable that the two assumptions

(i) liA=B,then(A\C) = (B\C),

and (u) If A >B, then {A:C) >{B:C)

form the real bed-rock of Euclid's ideas, and that he deduced

his Fifth and Seventh Definitions from these two fundamental

assumptions as his starting-point, but that he finally re-

arranged his argument so as to take the Fifth and Seventh

Definitions as his starting-point and then deduced the above-

mentioned assumptions as propositions.

An argument which does not follow the course of discovery

is frequently very difficult to follow. De Morgan, in his

Theory of the Connexion of Numher and Magnitude, gives

reasons for thinking that Euclid arrived at the conditions in

the Fifth and Seventh Definitions from the consideration of a

model representing a set of equidistant columns with a set of

PREFACE ix

equidistant railings in front of them, and the relation between

the model and the object it represented. However that may

be it cannot, I think, be denied that these definitions appearing

at the commencement of Euclid's argument without explan-

ation present grave difficulties to the student. I hope to

show that these difficulties can be removed and the whole

argument presented in a simple form.

I have given a few geometrical illustrations in this book,

some of which are not included in either of the two editions of

my book entitled The Contents of the Fifth and Sixth Books of

Euclid's Elements, published by the Cambridge University

Press. I desire, however, to draw special attention to the very

beautiful applications of Stolz's Theorem (Art. 40) to the proof

of the proposition that the areas of circles are proportional to

the squares on their radii (Euc. XII. 2), see Art. 61 ; and

also to the proof of the same proposition on strictly Euclidean

lines, for both of which I am indebted to my friend Mr. Rose-

Innes (see Art. 61a). These proofs differ from Euclid's in a

most important particular, viz. they do not assume the exist-

ence of the fourth proportional to three magnitudes of which

the first and second are of the same kind. I think that any one

who has tried to understand Euclid's argument will find the

proofs here given much simpler and more direct. Euclid uses

a reductio ad ahsurdum. As against methods other than

Euclid's the infinitesimals are, by the aid of Euclid X. 1,

handled in a manner which is far more convincing, at any rate

to those who are commencing the study of infinitesimals.

I am aware that in bringing this subject forward, and in

suggesting that a treatment of the Theory of Proportion,

which is valid when the magnitudes concerned are incom-

mensurable, should be included in the mathematical curricu-

lum, I have immense prejudices to overcome.

On the one hand it is the outcome of all experience in teach-

ing that Euclid's presentation of the subject is beyond the

comprehension of most people whether old or young, a view

with which I am in complete agreement. The matter is

regarded as res judicata, and most teachers refuse to look

at Euclid's work, or anything claiming kinship with it.

On the other hand, in suggesting any modification of

X PREFACE

Euclid's argument, I have before me the dictum of that great

Master of Logic, Augustus de Morgan, who said, " This same

book (the Fifth Book of EucHd's Elements) and the logic of

Aristotle are the two most unobjectionable and unassailable

treatises which ever were written," and if that be so the use-

fulness of my work would be in dispute. What is presented

here is a modification of Euclid's method, which requires for

its understanding a knowledge of Elementary Algebra. I

find no difficulty in explaining the first nine chapters, which

form Part I., to students who are commencing the study of

the properties of similar figures ; and whose intellectual

equipment in Geometry includes a knowledge of the subject

matter of the first four books of Euclid's Elements. As I have

ventured to make several criticisms on Euclid's argument, I

hope it will not be supposed that I do not appreciate either

the magnitude or the ingenuity of the work. Its ingenuity is

in fact one of the obstacles, if not the greatest obstacle to its

finding a place in the mathematical curriculum. What is

claimed for the argument set out here is that an easier road

to the same results has been found which is not deficient in

rigour to that contained in the Euclidean text. Dedekind says

in his Essays on Number''^ that it was especially from the

Fifth Definition of the Fifth Book that he drew the inspira-

tion which led him to the theory of the " cut " or " section "f

in the system of rational numbers, a theory which is funda-

mental in the Calculus. The propositions in this book furnish

a number of easily understood examples of the " cut " and

thus prepare the student for the study of irrational numbers

in the Calculus. Its subject matter is thus very closely linked

with modern ideas and well worthy of study.

The book is arranged in three parts. The first part. Chap-

ters I. -IX., contains an elementary course, which can be ex-

plained to any one with average mathematical ability. The

fourth, fifth, and sixth chapters should be carefully studied.

Any difficulty that there may be in the first part will be found

in these chapters. The table of contents gives a clear idea of

their subject matter, and the main points that have to be borne

in mind in the subsequent argument are summed up in Article

* Translated by Beman,'p. 40. f Schnitt.

PREFACE xi

41. The frequent use of Archimedes' Axiom in this work is

of great assistance to students when they enter upon the study

of the Calculus.

The second part, Chapters X. and XI., is suitable for stu-

dents preparing for an Honours Course and for Teachers. It

is too difficult for an elementary course, and is not intended

for those who are not really interested in mathematical study.

The third part, Chapter XII., is a commentary on the

Fifth Book of Euclid's Elements, and contains remarks on

matters which are of interest to those who are concerned with

the history of the ideas involved.

This commentary is not intended to be a complete one, but

deals only with some matters which have not been noticed in

the earlier chapters. The reader who is interested in this

part of the subject should consult Sir T. L. Heath's Edition

of Euclid's Elements.

My acknowledgments are due to the Syndics of the Cam-

bridge University Press for their courtesy in permitting me

to use the methods employed in the two editions of my Con-

tents of the Fifth and Sixth Books of Euclid's Elements ; and to

the Editor of the Mathematical Gazette for permission to use

a portion of the material of my Presidential Address to the

London Branch of the Mathematical Association, published

in the July and October numbers of the Gazette for 1912.

I am also under great obligation to De Morgan's Treatise on

the Connexion of Number and Magnitude, and especially in

connection with the matter of Chapter XII. to Sir T. L.

Heath's great editipn of Euclid's Elements.

Some further information will be found in my two papers

on the Fifth Book of Euclid's Elements in the Cambridge

Philosophical Transactions, Vol. XVI., Part IV., and Vol.

XIX., Part II.

M. J. M. HILL.

University of London,

University College, 1913.

CONTENTS

PART I

CHAPTER I

Abticles 1-3

Magnitudes of the same kind.

PAGE

Arts. 1, 2. Examples of Magnitudes o/ <^ same A;tmi . . . 1

Art. 3. Characteristics of Magnitudes of the same kind . . 1

CHAPTER II

Abticles 4-12

Propositions relating to Magnittides and their Multiples.

Art. 4. Statement of the Propositions ..... 4

Art. 5. Prop. I. (Euc. V. 1) 4

n{A +B + C + . . .) =nA -i-nB -hnC + . . .

Art. 6. Prop. II. (Euc. V. 2) 6

{a+h +C+ . . .)N =aN +bN -\-cN -\- . . .

Art. 7^ Prop. Ill 6

{r{s))A =r{sA) =s{rA)^{s{r))A.

Corollary.

s[{n(r)}A]=r[{n(s)}Al

Art. 8. Prop. IV. (Euc. V. 5) 7

IiA>B, then r{A -B)=rA -rB.

Art. 9. Prop. V. (Euc. V. 6) 7

If a> b, then {a-b)R =aR -bR.

Art. 10. Prop. VI 7

li A> B, then rA> rB.

li A =By then rA=rB.

li A <B, then r A <rB.

If rA > rB, then A> B.

If rA=rB, then A ^B.

If rA <rB, then A <B.

xiii

xiv CONTENTS

PAGE

Art. 11. Prop. VII. 8

If a> 6, then aR> hR.

If a =6, then aR =^hR.

If a <6, then aR <bR.

If aR> hR, then a> b.

If aR =bRf then a=b.

If aR <bR, then a <b.

Art. 12. Prop. VIII. If X, Y, Z are magnitudes oj the same kind,

and if X>Y -\-Z, then an integer t exists such that

X>tZ>Y. . 9

Corollary, li A, B, C are magnitudes of the same kind,

and if A> B, then integers n, t exist such that

nA>tC>nB ....... 10

CHAPTER III

Articles 13-18

The relations between Multiples of the same Magnitude,

Commensurable Magnitudes.

Art. 13. The ratio of one multiple of a magnitude to another

multiple of the same magnitude . . . .11

The ratio of nA to rA is defined to be -â€¢

r

Arts. 14-17. Geometrical Illustrations ..... 13

Art. 18. If^=aG',J5=6(?, C=c(? 16

and iiA>B, then {A:C)> {B-.C) ;

if A =B, then {A:C)={BiC) ;

if A <B, then {A:C) <{B:C).

CHAPTER IV

Articles 19-21

Magnitudes of the same kind which are not Multiples of the same

magnitude. Incommensurable Magnitudes.

Art. 19. Magnitudes of the same kind exist which have no common

measure ........ 18

Art. 20. The diagonal and side of a square have no common

measure ........ 18

Art. 21. Consideration of the question " Whether two magnitudes

of the same kind which have no common measure can

have a ratio to one another ? " If so, it cannot be a

rational number . . . . . . .19

CONTENTS XV

CHAPTER V

Articles 22-28

Extension of the Idea of Number.

PAGE

Art. 22. The widening of the idea of number to include negative

numbers, vulgar fractions positive and negative.

The system of rational numbers . . . .22

Art. 23. Every nimiber in the system of rational numbers has a

definite place . . . . . . .23

Art. 24. Do numbers exist which are not rational numbers ? . 24

Art. 25. Study of the square root of 2 . . . . .25

Art. 26. An irrational niunber has a definite place with regard

to the system of rational numbers, and is a magnitude

which in the technical sense of the words is of the same

kind as the rational numbers ..... 26

Art. 27. Mode of distinguishing between unequal irrational

numbers ........ 26

Art. 28. Conditions for equahty of irrational numbers . . 27

CHAPTER VI

Articles 29-41

On the Ratios of Magnitudes which have no Common Measure.

Art. 29. Principles on which the Theory of the Ratio of Magni-

tudes which have no common measure is based . . 28

{l)IiA>B {2)UA:=B {Z)liA<B

then {A:C)> {B:C) then {A:C)={B:C) then {A:C) <{B:C)

Art. 30. Prop. IX. Assimiing the above principles ... 28

then (1) if {A:0)> {B:C) (2) if {A:C)={B:C) (3) if {A:C) <(B:C)

then A>B then A=B then A<B

Art. 31. Prop. X 30

(i) JirA>sB (ii) lirA^sB (iii) Ifr^<5i5

then(^:5)>- then(^:B)=- then(^:B)<-*

(iv) If {A:B)> ~ (v) If {A:B) =- j[vi) If {A:B) <~

r r r

then rA>sB then rA =sB then rA <sB.

Art. 32. The ratio of two magnitudes of the same kind is a num-

ber rational or irrational . . . . .32

Art. 33. Definition of Equal Ratios 33

Art. 34. liA=B, then no rational number can he between {A:C)

and(B:C) 34

Art. 35. The Test for Equal Ratios 34

Art. 36. Derivation of the conditions of Euc. V. def. 5 â€¢ , â€¢ 35

xvi CONTENTS

PAGE

Art. 37. Definition of Unequal Ratios ..... 35

Art. 38. The test for distinguishing between Unequal Ratios . 36

Art. 39. li A>B then a rational number can be found which lies

between {A:C) and {B:C) 37

Art. 40. Simplification of the Test for Equal Ratios . . .37

(Stolz's Theorem)

Prop. XI. If all values of r, s which make sA>rB

also make sC> rD, and if all values of r, s which make

sA <rB also make sC <rD, then if any values of r, s,

say r =^1, 5 =5i, exist which make s^A =riB, then must

also SjC=riÂ£>.

Magnitudes in Proportion.

Art. 41. Recapitulation of the chief points of the preceding

theory 39

CHAPTER VII

Articles 42-49

Properties of Equal Ratios. First Group of Propositions.

Art. 42. Statement of the Propositions ..... 40

Art. 43. Prop. XII .40

li{A:B)={C:D),

then {rA:sB) ={rC:sD}, Euc. V. 4.

Art. 44. Prop. XIII .41

Ii{A:B)={C:D),

then {B:A) = {D:C). Euc. V. Cor. to 4.

Art. 45. Prop. XIV. If {A:B) = {C:D)=^{E:F), and if all the 42

magnitudes are of the same kind,

then iA:B)={A-i-C-\-E:B-{-D+F). Euc. V. 12.

Art. 46. Prop. XV 43

{A:B) =^{nA:nB). Euc. V. 15.

Art. 47. Prop. XVI. . . . . . . . .43

IiiA:B)={X:Y),

then {A-\-B:B) = {X+Y:Y). Euc. V. 18.

Art. 48. Prop. XVII 44

li{A+B:B)={X+Y:Y),

then {A:B)^{X:Y). Euc. V. 17.

Art. 49. Geometrical illustration (Euc. VI. 1) . . . .45

The ratio of the areas of two triangles of equal altitudes

is equal to the ratio of the lengths of their bases.

CHAPTER VIII

Articles 50-53

Properties of Equal Ratios. Second Group of Propositions.

Art. 50. Statement of the Propositions . . . . . 48

CONTENTS xvii

PAOK

Art. 51. Prop. XVIII. li A, B, C, D be four magnitudes of the

same kind ........ 49

andif (^:B)=((7:D),

then {A:C)={B'.D). Euc. V. 16.

Corollary. If, with the data of the proposition,

A>C,t\ienB>D;

but if ^ =C, then B =D ;

and iiA<C, then B <D. Euc. V. 14.

Art. 52. Prop. XIX. . 60

Ii{A:B)={T:U),

andif (B:C)=(C7:F),

then {A:C)={T:V). Euc. V. 22.

Corollary. If, with the data of the proposition,

^>0,thenT>F;

but if ^ =0, then T^V;

and if ^ < C, then T<V. Euc. V. 20.

Art. 53. Prop. XX 52

Ji{A:B)={U:V),

and a {B:C) = {T:U),

then {A:C)={T:V). Euc. V. 23.

Corollary. If, with the data of the proposition,

^>C, thenT>F.

but if ^ =C, then T = V;

and iiA<Cy then T <V. Euc. V. 21.

CHAPTER IX

Articles 54-57

Properties of Equal Ratios. Third Grov/p of Propositions.

Art. 54. Statement of the Propositions ..... 54

Art. 55. Prop. XXI. 64

li{A+CiB+D) = {C'.D),

then (^:B) = (C:Z>). Euc. V. 19.

Art. 56. Prop. XXII 65

li{A'.C)={X'.Z),

andif (B:C) = (y:Z),

then {A +B:C) =(X + FrZ). Euc. V. 24.

Art. 57. Prop. XXIII &5

li AfB,C,D are four magnitudes of the same kind, if A be

the greatest of them,

andif (^:J5)=(C:jD),

then A+D>B + C. Euc. V. 25.

xviii CONTENTS

PART II

CHAPTER X

Articles 58-67

Geometrical applications of Stolz's Theorem.

PAGE

Art. 58. Some subsidiary propositions ..... 57

If A and B be two magnitudes of the same kind, of which

A is the larger, and if from A more than its half be

taken away, and if from the remainder left more than

its half be taken away, and so on ; then if this pro-

cess be continued long enough, the remainder left

will be less than B (Euc. X. 1).

Art. 59. If a regular polygon of 2** sides be inscribed in a circle,

then the part of the circular area outside the polygon

can be made as small as we please by making n large

enough (included in Euc. XII. 2) . . . .58

Art. 60. The areas of similar polygons inscribed in two circles are

proportional to the areas of the squares described on

the radii of the circles (Euc. XII. 1) . . .60

Arts. 61, 61a, 616. The areas of circles are proportional to the

squares described on their radii (Euc. XII. 2) . . 61

Art. 62. If CijCa represent the contents of two figures, such

that it is possible to inscribe in (7 1 an infinite series of

figures Pj, and in (7 2 an infinite series of correspond-

ing figures Pg, such that {Pi:P2) has a fixed value

{Si:Sz)y and that Ciâ€”Pi and C^â€”Pz can be made

as small as we please, then will . . . .65

Art. 63. The circumferences of circles are proportional to their radii 66

Art. 64. The area of the radian sector of a circle is equal to half

the area of the square described on its radius . . 66

Art. 65. The area of a circle whose radius is r is Trr^ ... 68

Art. 66. The volumes of tetrahedra standing on the same base

are proportional to their altitudes .... 68

Art. 67. The volumes of tetrahedra are proportional to their

bases and altitudes jointly ..... 71

CHAPTER XI-

Articles 68-70

Further remarks on Irrational Numbers. The existence of the Fourth

Proportional.

Art. 68, Separation of the system of rational numbers into two

classes ........ 74

CONTENTS xix

PAGE

Art. 69. Separation of the points on a straight line into two

classes. The Cantor-Dedekind Axiom ... 75

Art. 70. The existence of the Fourth Proportional ... 76

Prop. XXIV. If A and B be magnitudes of the same

kindy and if G be any third magnitude, then there

exists a fourth magnitude Z oj the same kind as C such

that {^:B)=((7:Z).

PART III

CHAPTER XII

Akticles 71-100

Commentary on the Fifth Book of Euclid's Elements.

Art. 71. The Third and Fourth Definitions .... 81

Art. 72. The Fifth Definition 82

Art. 73. The idea of ratio need not be introduced into the Fifth

Definition. Relative Multiple Scales ... 82

Arts. 74-77. Study of the conditions appearing in the Fifth

Definition. Determination of those which are in-

dependent ........ 85

Arts. 78-79. The Seventh Definition. Reduction to its simplest

form 88

Art. 80. A point arising out of the Seventh Definition not dealt

withbyEuchd 89

Arts. 81-82. Statement of the evidence as to Euclid's view of

ratio ......... 91

Art. 83.* The First Group of Propositions. Magnitudes and their

Multiples (Euc. V. 1, 2, 3, 5, 6) . . . . 93

Art. 84. The Second Group of Propositions .... 94

Properties of Equal Ratios deduced directly from the

Fifth Definition (Euc. V. 4, 7, 11, 12, 15, 17).

Art. 85. Deduction of Euc. X. 6 from Euc. V. 17 without assum-

ing that a magnitude may be divided into any number

of equal parts ....... 94

Art. 86. The Third Group of Propositions . . . .96

Properties of Unequal Ratios depending on the Seventh

Definition (Euc. V. 8, 10, 13).

Art. 87. Euc. V. 8 97

Art. 88. Euc. V. 10 97

Art. 89. The Fourth Group of Propositions .... 98

Properties of Equal Ratios depending on both the Fifth

and Seventh Definitions (Euc. V. 9, 14, 16, and 18-25).

Art. 90. Independence of the Fifth and Seventh Definitions . 99

^x CONTENTS

PACK

Art. 91. Comparison of the proofs of Euc. V. 14 and 16 with those

given in this book ...... 99

Art. 92. Euc. V. 18. EucHd's assumption of the existence of

the Fourth Proportional . . . . .100

Art. 93. The relation between Euc. V. 20 and 22 . . . 101

Art. 94. The relation between Euc. V. 21 and 23 . . . 101

Art. 95. The Compounding or MultipHcation of Ratios. The

order of the multiphcation does not affect the result

(Euc. V. 23) 102

Art. 96. Addition of Ratios (Euc. V. 24) 103

Art. 97. The importance of Euc. V. 25 103

Art. 98-99. Deduction from Euc. V. 25 of the propositions that

as n tends to + oo , a^ tends to + oo if a > 1 ; but

to + 0, if < a < 1 104

Art. 100. The relation between the last-mentioned hmit and

Euc. X. 1 105

Index . . . . 107

THE THEORY OF PROPORTION

PART I

CHAPTER I

Articles 1-3

Magnitudes of the same kind.

Article 1

No attempt will be made to give a general definition of the

term " Magnitude." It is sufficient to give a number of

examples ; e.g. lengths, areas, volumes, hours, minutes,

seconds, weights, etc., are called magnitudes.

Article 2

It is, however, important to make precise the sense in

which the term

" magnitudes of the same kind "

will be employed.

Some examples of what is meant will first be given.

All lengths are magnitudes of the same kind.

All areas are magnitudes of the same kind.

All volumes are magnitudes of the same kind.

All intervals of time are magnitudes of the same kind.

Article 3

Characteristics of Magnitudes of the same kind.

In the next place the characteristics of magnitudes of the

same kind will be specified.*

* Stolz's account of the properties of absolute magnitudes in his Allge-

meine Arithmetik, Erster Theil, page 69, is followed in essentials.

2 THfi. lailQRY OF PROPORTION

â– Thfefej^; :s^ili, W reaidljy a'rimitted if we consider the mag-

nitudes to be segments of lines, or areas, or volumes, or

weights, etc.

A system of magnitudes is said to be of the same kind when

the magnitudes possess the following characteristics :

( 1 ) Any two magnitudes of the same kind may be regarded

as equal or unequal.

In the latter case one of them is said to be the

smaller, and the other the larger of the two.

(2) Two magnitudes of the same kind can be added

together. The resulting magnitude is a magnitude

of the same kind as the original magnitudes.

This property makes it possible to form multiples of

a magnitude.

For denoting any magnitude by A, then A-\-A is a

magnitude of the same kind as A. It will be denoted

by 2^.

Then 2A+A is a magnitude of the same kind as A.

It will be denoted by 3A . And so on, if r denote

any positive integer, rA-\-A is> a. magnitude of the

i III II III 1

$B S27 TbD

L^'

THE THEORY OF PROPORTION

THE THEORY

OF PROPORTION

BY

M. J. M. HILL, M.A., LL.D., ScD., F.R.S.

ASTOR PROFESSOR OF MATHEMATICS IN

THE UNIVERSITY OF LONDON

LONDON

CONSTABLE AND COMPANY, LTD.

1914

PREFACE

This little book is the outcome of the effort annually renewed

over a long period to make clear to my students the principles

on which the Theory of Proportion is based, with a view to its

application to the study of the Properties of Similar Figures.

Its content formed recently the subject matter of a course

of lectures to Teachers, delivered at University College, under

an arrangement with the London County Council, and it is

now being published in the hope of interesting a wider circle.

At the commencement of my career as a teacher I was accus-

tomed, in accordance with the then estabHshed practice, to take

for granted the definition of proportion as given by Euclid in

the Fifth Definition of the Fifth Book of his Elements* and to

supply proofs of the other properties of proportion required

in the Sixth Book which were valid only when the magnitudes

considered were commensurable. Dissatisfied with the results

of a method which could have no claim to be considered

logical, after trying some other modes of exposition, I turned

to the syllabus of the Fifth Book drawn up by the Association

for the Improvement of Geometrical Teaching. But again I

found this hard to explain, and it was evident that my students

could not grasp the method as a whole, even when they were

able to understand its steps singly.

After prolonged study I found that, in addition to the

difficulty arising out of Euclid's notation, which is a matter

of form and not of substance, and the difficulty that Euclid

does not sufficiently define ratio, two reasons could be assigned

for the great difficulty of his argument.

(1) Of the long array of definitions prefixed to the Fifth

Book there are only two which effectively count. One of these,

the Fifth, is the test for deciding when two ratios are equal ;

and the other, the Seventh, is the test for distinguishing

* The substance of the Fifth Book is usually attributed to Eudoxus.

O/^O^ K^f^^

viii PREFACE

between unequal ratios. They are intimately related, but

when once stated they can be treated as independent.

Now it can he seen at once that if the test for deciding when two

ratios are equal is a good and sound one^ it should he possihle to

deduce from it all the properties of equal ratios^ and in order to

obtain these properties it should not he necessary to employ the

test for distinguishing hetween unequal ratios.

But Euclid frequently employs this last-mentioned test, or

propositions depending on it, to prove properties of equal

ratios. In fact, it is not at all easy for any one trying to follow

the course of his argument to see whether it leads naturally

to the employment of the Fifth or of the Seventh Definition,

or a proposition depending on the Seventh Definition. Euclid's

proofs do not run on the same lines, and are so difficult and

intricate that they have almost entirely fallen out of use. It

will be shown in this book that all the properties of equal

ratios can he proved hy the aid of the Fifth Definition, and that

the Seventh Definition is not required.

This is effected, without departing from the spirit or the

rigour of Euclid's argument, by assimilating Euclid's proofs

of those propositions in which the use of the Seventh Defini-

tion is directly or indirectly involved to his proofs of those

propositions in which he employs the Fifth Definition only.

(2)1 think it will appear to any one who reads this book that

it is in a high degree probable that the two assumptions

(i) liA=B,then(A\C) = (B\C),

and (u) If A >B, then {A:C) >{B:C)

form the real bed-rock of Euclid's ideas, and that he deduced

his Fifth and Seventh Definitions from these two fundamental

assumptions as his starting-point, but that he finally re-

arranged his argument so as to take the Fifth and Seventh

Definitions as his starting-point and then deduced the above-

mentioned assumptions as propositions.

An argument which does not follow the course of discovery

is frequently very difficult to follow. De Morgan, in his

Theory of the Connexion of Numher and Magnitude, gives

reasons for thinking that Euclid arrived at the conditions in

the Fifth and Seventh Definitions from the consideration of a

model representing a set of equidistant columns with a set of

PREFACE ix

equidistant railings in front of them, and the relation between

the model and the object it represented. However that may

be it cannot, I think, be denied that these definitions appearing

at the commencement of Euclid's argument without explan-

ation present grave difficulties to the student. I hope to

show that these difficulties can be removed and the whole

argument presented in a simple form.

I have given a few geometrical illustrations in this book,

some of which are not included in either of the two editions of

my book entitled The Contents of the Fifth and Sixth Books of

Euclid's Elements, published by the Cambridge University

Press. I desire, however, to draw special attention to the very

beautiful applications of Stolz's Theorem (Art. 40) to the proof

of the proposition that the areas of circles are proportional to

the squares on their radii (Euc. XII. 2), see Art. 61 ; and

also to the proof of the same proposition on strictly Euclidean

lines, for both of which I am indebted to my friend Mr. Rose-

Innes (see Art. 61a). These proofs differ from Euclid's in a

most important particular, viz. they do not assume the exist-

ence of the fourth proportional to three magnitudes of which

the first and second are of the same kind. I think that any one

who has tried to understand Euclid's argument will find the

proofs here given much simpler and more direct. Euclid uses

a reductio ad ahsurdum. As against methods other than

Euclid's the infinitesimals are, by the aid of Euclid X. 1,

handled in a manner which is far more convincing, at any rate

to those who are commencing the study of infinitesimals.

I am aware that in bringing this subject forward, and in

suggesting that a treatment of the Theory of Proportion,

which is valid when the magnitudes concerned are incom-

mensurable, should be included in the mathematical curricu-

lum, I have immense prejudices to overcome.

On the one hand it is the outcome of all experience in teach-

ing that Euclid's presentation of the subject is beyond the

comprehension of most people whether old or young, a view

with which I am in complete agreement. The matter is

regarded as res judicata, and most teachers refuse to look

at Euclid's work, or anything claiming kinship with it.

On the other hand, in suggesting any modification of

X PREFACE

Euclid's argument, I have before me the dictum of that great

Master of Logic, Augustus de Morgan, who said, " This same

book (the Fifth Book of EucHd's Elements) and the logic of

Aristotle are the two most unobjectionable and unassailable

treatises which ever were written," and if that be so the use-

fulness of my work would be in dispute. What is presented

here is a modification of Euclid's method, which requires for

its understanding a knowledge of Elementary Algebra. I

find no difficulty in explaining the first nine chapters, which

form Part I., to students who are commencing the study of

the properties of similar figures ; and whose intellectual

equipment in Geometry includes a knowledge of the subject

matter of the first four books of Euclid's Elements. As I have

ventured to make several criticisms on Euclid's argument, I

hope it will not be supposed that I do not appreciate either

the magnitude or the ingenuity of the work. Its ingenuity is

in fact one of the obstacles, if not the greatest obstacle to its

finding a place in the mathematical curriculum. What is

claimed for the argument set out here is that an easier road

to the same results has been found which is not deficient in

rigour to that contained in the Euclidean text. Dedekind says

in his Essays on Number''^ that it was especially from the

Fifth Definition of the Fifth Book that he drew the inspira-

tion which led him to the theory of the " cut " or " section "f

in the system of rational numbers, a theory which is funda-

mental in the Calculus. The propositions in this book furnish

a number of easily understood examples of the " cut " and

thus prepare the student for the study of irrational numbers

in the Calculus. Its subject matter is thus very closely linked

with modern ideas and well worthy of study.

The book is arranged in three parts. The first part. Chap-

ters I. -IX., contains an elementary course, which can be ex-

plained to any one with average mathematical ability. The

fourth, fifth, and sixth chapters should be carefully studied.

Any difficulty that there may be in the first part will be found

in these chapters. The table of contents gives a clear idea of

their subject matter, and the main points that have to be borne

in mind in the subsequent argument are summed up in Article

* Translated by Beman,'p. 40. f Schnitt.

PREFACE xi

41. The frequent use of Archimedes' Axiom in this work is

of great assistance to students when they enter upon the study

of the Calculus.

The second part, Chapters X. and XI., is suitable for stu-

dents preparing for an Honours Course and for Teachers. It

is too difficult for an elementary course, and is not intended

for those who are not really interested in mathematical study.

The third part, Chapter XII., is a commentary on the

Fifth Book of Euclid's Elements, and contains remarks on

matters which are of interest to those who are concerned with

the history of the ideas involved.

This commentary is not intended to be a complete one, but

deals only with some matters which have not been noticed in

the earlier chapters. The reader who is interested in this

part of the subject should consult Sir T. L. Heath's Edition

of Euclid's Elements.

My acknowledgments are due to the Syndics of the Cam-

bridge University Press for their courtesy in permitting me

to use the methods employed in the two editions of my Con-

tents of the Fifth and Sixth Books of Euclid's Elements ; and to

the Editor of the Mathematical Gazette for permission to use

a portion of the material of my Presidential Address to the

London Branch of the Mathematical Association, published

in the July and October numbers of the Gazette for 1912.

I am also under great obligation to De Morgan's Treatise on

the Connexion of Number and Magnitude, and especially in

connection with the matter of Chapter XII. to Sir T. L.

Heath's great editipn of Euclid's Elements.

Some further information will be found in my two papers

on the Fifth Book of Euclid's Elements in the Cambridge

Philosophical Transactions, Vol. XVI., Part IV., and Vol.

XIX., Part II.

M. J. M. HILL.

University of London,

University College, 1913.

CONTENTS

PART I

CHAPTER I

Abticles 1-3

Magnitudes of the same kind.

PAGE

Arts. 1, 2. Examples of Magnitudes o/ <^ same A;tmi . . . 1

Art. 3. Characteristics of Magnitudes of the same kind . . 1

CHAPTER II

Abticles 4-12

Propositions relating to Magnittides and their Multiples.

Art. 4. Statement of the Propositions ..... 4

Art. 5. Prop. I. (Euc. V. 1) 4

n{A +B + C + . . .) =nA -i-nB -hnC + . . .

Art. 6. Prop. II. (Euc. V. 2) 6

{a+h +C+ . . .)N =aN +bN -\-cN -\- . . .

Art. 7^ Prop. Ill 6

{r{s))A =r{sA) =s{rA)^{s{r))A.

Corollary.

s[{n(r)}A]=r[{n(s)}Al

Art. 8. Prop. IV. (Euc. V. 5) 7

IiA>B, then r{A -B)=rA -rB.

Art. 9. Prop. V. (Euc. V. 6) 7

If a> b, then {a-b)R =aR -bR.

Art. 10. Prop. VI 7

li A> B, then rA> rB.

li A =By then rA=rB.

li A <B, then r A <rB.

If rA > rB, then A> B.

If rA=rB, then A ^B.

If rA <rB, then A <B.

xiii

xiv CONTENTS

PAGE

Art. 11. Prop. VII. 8

If a> 6, then aR> hR.

If a =6, then aR =^hR.

If a <6, then aR <bR.

If aR> hR, then a> b.

If aR =bRf then a=b.

If aR <bR, then a <b.

Art. 12. Prop. VIII. If X, Y, Z are magnitudes oj the same kind,

and if X>Y -\-Z, then an integer t exists such that

X>tZ>Y. . 9

Corollary, li A, B, C are magnitudes of the same kind,

and if A> B, then integers n, t exist such that

nA>tC>nB ....... 10

CHAPTER III

Articles 13-18

The relations between Multiples of the same Magnitude,

Commensurable Magnitudes.

Art. 13. The ratio of one multiple of a magnitude to another

multiple of the same magnitude . . . .11

The ratio of nA to rA is defined to be -â€¢

r

Arts. 14-17. Geometrical Illustrations ..... 13

Art. 18. If^=aG',J5=6(?, C=c(? 16

and iiA>B, then {A:C)> {B-.C) ;

if A =B, then {A:C)={BiC) ;

if A <B, then {A:C) <{B:C).

CHAPTER IV

Articles 19-21

Magnitudes of the same kind which are not Multiples of the same

magnitude. Incommensurable Magnitudes.

Art. 19. Magnitudes of the same kind exist which have no common

measure ........ 18

Art. 20. The diagonal and side of a square have no common

measure ........ 18

Art. 21. Consideration of the question " Whether two magnitudes

of the same kind which have no common measure can

have a ratio to one another ? " If so, it cannot be a

rational number . . . . . . .19

CONTENTS XV

CHAPTER V

Articles 22-28

Extension of the Idea of Number.

PAGE

Art. 22. The widening of the idea of number to include negative

numbers, vulgar fractions positive and negative.

The system of rational numbers . . . .22

Art. 23. Every nimiber in the system of rational numbers has a

definite place . . . . . . .23

Art. 24. Do numbers exist which are not rational numbers ? . 24

Art. 25. Study of the square root of 2 . . . . .25

Art. 26. An irrational niunber has a definite place with regard

to the system of rational numbers, and is a magnitude

which in the technical sense of the words is of the same

kind as the rational numbers ..... 26

Art. 27. Mode of distinguishing between unequal irrational

numbers ........ 26

Art. 28. Conditions for equahty of irrational numbers . . 27

CHAPTER VI

Articles 29-41

On the Ratios of Magnitudes which have no Common Measure.

Art. 29. Principles on which the Theory of the Ratio of Magni-

tudes which have no common measure is based . . 28

{l)IiA>B {2)UA:=B {Z)liA<B

then {A:C)> {B:C) then {A:C)={B:C) then {A:C) <{B:C)

Art. 30. Prop. IX. Assimiing the above principles ... 28

then (1) if {A:0)> {B:C) (2) if {A:C)={B:C) (3) if {A:C) <(B:C)

then A>B then A=B then A<B

Art. 31. Prop. X 30

(i) JirA>sB (ii) lirA^sB (iii) Ifr^<5i5

then(^:5)>- then(^:B)=- then(^:B)<-*

(iv) If {A:B)> ~ (v) If {A:B) =- j[vi) If {A:B) <~

r r r

then rA>sB then rA =sB then rA <sB.

Art. 32. The ratio of two magnitudes of the same kind is a num-

ber rational or irrational . . . . .32

Art. 33. Definition of Equal Ratios 33

Art. 34. liA=B, then no rational number can he between {A:C)

and(B:C) 34

Art. 35. The Test for Equal Ratios 34

Art. 36. Derivation of the conditions of Euc. V. def. 5 â€¢ , â€¢ 35

xvi CONTENTS

PAGE

Art. 37. Definition of Unequal Ratios ..... 35

Art. 38. The test for distinguishing between Unequal Ratios . 36

Art. 39. li A>B then a rational number can be found which lies

between {A:C) and {B:C) 37

Art. 40. Simplification of the Test for Equal Ratios . . .37

(Stolz's Theorem)

Prop. XI. If all values of r, s which make sA>rB

also make sC> rD, and if all values of r, s which make

sA <rB also make sC <rD, then if any values of r, s,

say r =^1, 5 =5i, exist which make s^A =riB, then must

also SjC=riÂ£>.

Magnitudes in Proportion.

Art. 41. Recapitulation of the chief points of the preceding

theory 39

CHAPTER VII

Articles 42-49

Properties of Equal Ratios. First Group of Propositions.

Art. 42. Statement of the Propositions ..... 40

Art. 43. Prop. XII .40

li{A:B)={C:D),

then {rA:sB) ={rC:sD}, Euc. V. 4.

Art. 44. Prop. XIII .41

Ii{A:B)={C:D),

then {B:A) = {D:C). Euc. V. Cor. to 4.

Art. 45. Prop. XIV. If {A:B) = {C:D)=^{E:F), and if all the 42

magnitudes are of the same kind,

then iA:B)={A-i-C-\-E:B-{-D+F). Euc. V. 12.

Art. 46. Prop. XV 43

{A:B) =^{nA:nB). Euc. V. 15.

Art. 47. Prop. XVI. . . . . . . . .43

IiiA:B)={X:Y),

then {A-\-B:B) = {X+Y:Y). Euc. V. 18.

Art. 48. Prop. XVII 44

li{A+B:B)={X+Y:Y),

then {A:B)^{X:Y). Euc. V. 17.

Art. 49. Geometrical illustration (Euc. VI. 1) . . . .45

The ratio of the areas of two triangles of equal altitudes

is equal to the ratio of the lengths of their bases.

CHAPTER VIII

Articles 50-53

Properties of Equal Ratios. Second Group of Propositions.

Art. 50. Statement of the Propositions . . . . . 48

CONTENTS xvii

PAOK

Art. 51. Prop. XVIII. li A, B, C, D be four magnitudes of the

same kind ........ 49

andif (^:B)=((7:D),

then {A:C)={B'.D). Euc. V. 16.

Corollary. If, with the data of the proposition,

A>C,t\ienB>D;

but if ^ =C, then B =D ;

and iiA<C, then B <D. Euc. V. 14.

Art. 52. Prop. XIX. . 60

Ii{A:B)={T:U),

andif (B:C)=(C7:F),

then {A:C)={T:V). Euc. V. 22.

Corollary. If, with the data of the proposition,

^>0,thenT>F;

but if ^ =0, then T^V;

and if ^ < C, then T<V. Euc. V. 20.

Art. 53. Prop. XX 52

Ji{A:B)={U:V),

and a {B:C) = {T:U),

then {A:C)={T:V). Euc. V. 23.

Corollary. If, with the data of the proposition,

^>C, thenT>F.

but if ^ =C, then T = V;

and iiA<Cy then T <V. Euc. V. 21.

CHAPTER IX

Articles 54-57

Properties of Equal Ratios. Third Grov/p of Propositions.

Art. 54. Statement of the Propositions ..... 54

Art. 55. Prop. XXI. 64

li{A+CiB+D) = {C'.D),

then (^:B) = (C:Z>). Euc. V. 19.

Art. 56. Prop. XXII 65

li{A'.C)={X'.Z),

andif (B:C) = (y:Z),

then {A +B:C) =(X + FrZ). Euc. V. 24.

Art. 57. Prop. XXIII &5

li AfB,C,D are four magnitudes of the same kind, if A be

the greatest of them,

andif (^:J5)=(C:jD),

then A+D>B + C. Euc. V. 25.

xviii CONTENTS

PART II

CHAPTER X

Articles 58-67

Geometrical applications of Stolz's Theorem.

PAGE

Art. 58. Some subsidiary propositions ..... 57

If A and B be two magnitudes of the same kind, of which

A is the larger, and if from A more than its half be

taken away, and if from the remainder left more than

its half be taken away, and so on ; then if this pro-

cess be continued long enough, the remainder left

will be less than B (Euc. X. 1).

Art. 59. If a regular polygon of 2** sides be inscribed in a circle,

then the part of the circular area outside the polygon

can be made as small as we please by making n large

enough (included in Euc. XII. 2) . . . .58

Art. 60. The areas of similar polygons inscribed in two circles are

proportional to the areas of the squares described on

the radii of the circles (Euc. XII. 1) . . .60

Arts. 61, 61a, 616. The areas of circles are proportional to the

squares described on their radii (Euc. XII. 2) . . 61

Art. 62. If CijCa represent the contents of two figures, such

that it is possible to inscribe in (7 1 an infinite series of

figures Pj, and in (7 2 an infinite series of correspond-

ing figures Pg, such that {Pi:P2) has a fixed value

{Si:Sz)y and that Ciâ€”Pi and C^â€”Pz can be made

as small as we please, then will . . . .65

Art. 63. The circumferences of circles are proportional to their radii 66

Art. 64. The area of the radian sector of a circle is equal to half

the area of the square described on its radius . . 66

Art. 65. The area of a circle whose radius is r is Trr^ ... 68

Art. 66. The volumes of tetrahedra standing on the same base

are proportional to their altitudes .... 68

Art. 67. The volumes of tetrahedra are proportional to their

bases and altitudes jointly ..... 71

CHAPTER XI-

Articles 68-70

Further remarks on Irrational Numbers. The existence of the Fourth

Proportional.

Art. 68, Separation of the system of rational numbers into two

classes ........ 74

CONTENTS xix

PAGE

Art. 69. Separation of the points on a straight line into two

classes. The Cantor-Dedekind Axiom ... 75

Art. 70. The existence of the Fourth Proportional ... 76

Prop. XXIV. If A and B be magnitudes of the same

kindy and if G be any third magnitude, then there

exists a fourth magnitude Z oj the same kind as C such

that {^:B)=((7:Z).

PART III

CHAPTER XII

Akticles 71-100

Commentary on the Fifth Book of Euclid's Elements.

Art. 71. The Third and Fourth Definitions .... 81

Art. 72. The Fifth Definition 82

Art. 73. The idea of ratio need not be introduced into the Fifth

Definition. Relative Multiple Scales ... 82

Arts. 74-77. Study of the conditions appearing in the Fifth

Definition. Determination of those which are in-

dependent ........ 85

Arts. 78-79. The Seventh Definition. Reduction to its simplest

form 88

Art. 80. A point arising out of the Seventh Definition not dealt

withbyEuchd 89

Arts. 81-82. Statement of the evidence as to Euclid's view of

ratio ......... 91

Art. 83.* The First Group of Propositions. Magnitudes and their

Multiples (Euc. V. 1, 2, 3, 5, 6) . . . . 93

Art. 84. The Second Group of Propositions .... 94

Properties of Equal Ratios deduced directly from the

Fifth Definition (Euc. V. 4, 7, 11, 12, 15, 17).

Art. 85. Deduction of Euc. X. 6 from Euc. V. 17 without assum-

ing that a magnitude may be divided into any number

of equal parts ....... 94

Art. 86. The Third Group of Propositions . . . .96

Properties of Unequal Ratios depending on the Seventh

Definition (Euc. V. 8, 10, 13).

Art. 87. Euc. V. 8 97

Art. 88. Euc. V. 10 97

Art. 89. The Fourth Group of Propositions .... 98

Properties of Equal Ratios depending on both the Fifth

and Seventh Definitions (Euc. V. 9, 14, 16, and 18-25).

Art. 90. Independence of the Fifth and Seventh Definitions . 99

^x CONTENTS

PACK

Art. 91. Comparison of the proofs of Euc. V. 14 and 16 with those

given in this book ...... 99

Art. 92. Euc. V. 18. EucHd's assumption of the existence of

the Fourth Proportional . . . . .100

Art. 93. The relation between Euc. V. 20 and 22 . . . 101

Art. 94. The relation between Euc. V. 21 and 23 . . . 101

Art. 95. The Compounding or MultipHcation of Ratios. The

order of the multiphcation does not affect the result

(Euc. V. 23) 102

Art. 96. Addition of Ratios (Euc. V. 24) 103

Art. 97. The importance of Euc. V. 25 103

Art. 98-99. Deduction from Euc. V. 25 of the propositions that

as n tends to + oo , a^ tends to + oo if a > 1 ; but

to + 0, if < a < 1 104

Art. 100. The relation between the last-mentioned hmit and

Euc. X. 1 105

Index . . . . 107

THE THEORY OF PROPORTION

PART I

CHAPTER I

Articles 1-3

Magnitudes of the same kind.

Article 1

No attempt will be made to give a general definition of the

term " Magnitude." It is sufficient to give a number of

examples ; e.g. lengths, areas, volumes, hours, minutes,

seconds, weights, etc., are called magnitudes.

Article 2

It is, however, important to make precise the sense in

which the term

" magnitudes of the same kind "

will be employed.

Some examples of what is meant will first be given.

All lengths are magnitudes of the same kind.

All areas are magnitudes of the same kind.

All volumes are magnitudes of the same kind.

All intervals of time are magnitudes of the same kind.

Article 3

Characteristics of Magnitudes of the same kind.

In the next place the characteristics of magnitudes of the

same kind will be specified.*

* Stolz's account of the properties of absolute magnitudes in his Allge-

meine Arithmetik, Erster Theil, page 69, is followed in essentials.

2 THfi. lailQRY OF PROPORTION

â– Thfefej^; :s^ili, W reaidljy a'rimitted if we consider the mag-

nitudes to be segments of lines, or areas, or volumes, or

weights, etc.

A system of magnitudes is said to be of the same kind when

the magnitudes possess the following characteristics :

( 1 ) Any two magnitudes of the same kind may be regarded

as equal or unequal.

In the latter case one of them is said to be the

smaller, and the other the larger of the two.

(2) Two magnitudes of the same kind can be added

together. The resulting magnitude is a magnitude

of the same kind as the original magnitudes.

This property makes it possible to form multiples of

a magnitude.

For denoting any magnitude by A, then A-\-A is a

magnitude of the same kind as A. It will be denoted

by 2^.

Then 2A+A is a magnitude of the same kind as A.

It will be denoted by 3A . And so on, if r denote

any positive integer, rA-\-A is> a. magnitude of the

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