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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 -•

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 • , • 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=ri£>.

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 Ci—Pi and C^—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

■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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