UC-NRLF 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 rB, then A> B. If rA=rB, then A ^B. If rA 6, then aR> hR. If a =6, then aR =^hR. If a <6, then aR hR, then a> b. If aR =bRf then a=b. If aR 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 {2)UA:=B {Z)liA {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 AsB (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 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 . 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 iiA0,thenT>F; but if ^ =0, then T^V; and if ^ < C, then TC, thenT>F. but if ^ =C, then T = V; and iiA). 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 same kind as A and will be denoted by (r + 1)^. The Commutative and Associative Laws apply to the Addition of magnitudes of the same kind. So that A-}-B=B-{-A. The Commutative Law. {A-{-B)+C=A-\-(B+C). The Associative Law. These laws can be conveniently illustrated by taking the case in which A, B and C represent lengths. (3) If A and B be two magnitudes of the same kind, and A be greater than B, then another magnitude X of the same kind as A and B exists such that B+X=A. This may also be written X==A-B. This can be illustrated by taking for A and B two lengths of which A is the longer. If, then, a length equal to B be cut off from A the remainder left is X. CHARACTERISTICS OF MAGNITUDES 3 (4) li A be any magnitude, and n any positive integer whatever, then a magnitude X of the same kind as A exists such that nX=A. This may also be written in either of the forms n or X=d. n It can be illustrated by dividing a segment of a straight line into any number of equal parts. It should be mentioned that if A represent an arc of a circle, although it is not in general possible by the aid of the ruler aijd compasses to divide A into n equal parts, yet it is assumed that an arc X=~ does exist. 71 (5) If ^ be greater than B, a multiple of B exists which is greater than A. The fifth characteristic is known as the Axiom of Archi- medes. It is not a consequence of the preceding four char- acteristics. The following deduction from the above is specially useful in the Theory of Proportion : If A and B are two magnitudes of the same kind, and any multiple whatever of A, say rA, is chosen, and any multiple whatever of B, say sB, is chosen, then one and only one of the alternatives rA >sB, rA=sB, rA B, then r{A -B) ^rA -rB. lia>b, then (a~b)R=aB-bE. If A >B, then rA >rB. If A =B, then rA =rB, li ArB, then A >B. If rA =rB, then A =B. li rA&, thena^>6i?. li a=b, then aR = bR. li abR, then a>b. If aR = bR, then a=b. If aRY-\-Z, then an integer t exists such that X >tZ > Y. Corollary. li A, B, C are magnitudes of the same kind, and A >By then integers n, t exist such that nA >tC >nB. Prop. I. Prop. II. Prop. III. Prop. IV. Prop. V. Prop. Conve VI. rsely. Prop. I. (Euc. V. 1.) n(A+B+C-[-, Article 5 )=nA-{-nB-{-nC+ MAGNITUDES AND THEIR MULTIPLES 5 The simplest case of this is n{A-\-B)=7iA+nB. For a rigid deduction of this from the Associative and Com- mutative Laws I refer to my edition of Euclid V and VI, 2nd edition, pp. 125-6. It is tedious, and the beginner should not be stopped at this stage with it. It is sufficient to say that the effect of the Associative and Commutative Laws is this, that when any number of magnitudes are to be added together, they may be arranged in any order and grouped in any way, the magnitudes in each group may be first added together, and then finally the sum of the groups can be found, and that the result so obtained will always be the same. Thus n(A-{-B) is the sum of n groups, each of which is A+B, The magnitude A occurs n times ; and therefore taking these together, their sum is nA. The magnitude B occurs n times ; and taking these to- gether, their sum is nB. The sum of the two groups is nA -{-nB. .'. n{A+B)=nA-\-nB. If on both sides B be replaced by C, and then A by A-\-By it follows that n{(A +B) +C) =n(A +B) -i-nC, .\n(A -\-B-\-C) =nA -\-nB-\-nC, Proceeding in this way, it follows that n{A +5+C+ . . . ) =nA -{-nB+nG-\- Article 6 Prop. II. (Euc. V. 2.) The simplest case of this is Now (a-\rh)N means that N is taken a-\-h times. Group the first aiV's together. Their sum is aN. Group the remaining &^'s together. Their sum is hN, .-. (a-^h)N=aN-{-hN. 6 THE THEORY OF PROPORTION If on both sides b be replaced by c, and then a by (a +6) it follows that ({a+b)+c)N=(a+b)N +cN :. (a+b+c)N=aN-^bN+cN. Proceeding in this way it follows that (a-\-b-\-c-{-...)N=aN-\-bN+cN-^ For a rigid deduction of the proposition from the Associa- tive and Commutative Laws see my Euclid V. and VI,, 2nd edition, p. 127. Article 7 Prop. III. (r{s))A=r(sA)=s(rA) = (s(r))A, In Prop. I., suppose that each of the magnitudes B,C, ... is equal to A, and that there are, including A, altogether s magnitudes. Then n(A+B+C+ . . .) isn(sA), and nA-\-nB-^nC-\- ... is s(nA). .'. n(sA)=s{nA). Or replacing n by r, r{sA)=^s(rA) (I). Next in Prop. II. suppose that each of the integers ayb,c, ... is equal to s ; and that there are r such integers. Then {a-{-b-\-c+ . . .)N becomes (r(s))N, and , aN-\-bN+cN-\- ... becomes r{sN). .'. {r{s))N=r(sN), or replacing iV by ^, (r{s)) A=r {sA) (II). Interchanging s and r, {s[r))A=s(TA) (HI)- Then from (I), (II), (III) it follows that {r(s))A=^r(sA)=s{rA) = (s{r))A. Corollary : 5[{7i(r)}^]=r[{ri(5)}^] To prove this, observe that [n{r)'\A=n{rA)=r{nA) .-. s[{n(r)}A]=s[r(nA)] =r[s(nA)] =r[{n(s)}A] MAGNITUDES AND THEIR MULTIPLES 7 This Corollary is not required until Propositions XVIII. and XX. are reached (see Arts. 51, 53). The efiFect of Prop. III. and the Corollary amount to this, that the factors of a product when they are all positive integers may be taken in any order and grouped in any way. Beginners will find the Corollary a little difficult, and too much time ought not to be spent on it. It is enough to call attention to the effect of the Proposition and Corollary as just stated. Article 8 Prop. IV. If A >B, then r(A -B)^rA -rB. (Euc. V. 5.) Since A >B, then by Art. 3 (3) a magnitude C exists such that A=B+C, .-. rA=rB+rC, ,\rC=rA-rB, hut C=A-B, .-. r(A -B) =rA -rB. Article 9 Prop. V. If a>h, then (a-b)R=aR-bR. (Euc. V. 6.) Since a, h are integers, and a>h, an integer c exists such that a=h-\-c, :. aR={h-^c)R=hR-\-cR (Prop. II.) .'. cR=aR-bR, but since a=b-i-c, :. c =a—b, :. (a-b)R=aR-bR. Article 10 Prop. VI. If A >B, then rA >rB, If A =B, then rA =rB, If Ar5, then A>B, If rA =rB, then A =B, ' If rA By then, as in Prop. IV., rA =rB-]-rC, where A =B+C, .-. rA >rB. If A =Bi then rA means (A-\-A-\- . . . . to r terms) =(jB + 5 + to r terms) =--rB, :, rA =rB. U A A, and .'. by the first case rB>rA, .'. rArB, then since A and B are supposed to be of the same kind., one of the three alternatives must hold : A >B, or A =B, or A rB. Hence A is not equal to B. If ArB. Hence A is not less than B, Consequently A is greater than B. The remaining cases can be proved in like manner. Article 11 Prop. VII. If a >b, then aR >bE, If a =b, then aR =hR, If a<6, then ai?<6i?. Conversely, If aR >bR, then a >6, If aR =bR, then a =b, li aRb, then, as in Prop. V., aR=bR+cR, where a=&-|-c, .-. aR >bR. MAGNITUDES AND THEIR MULTIPLES 9 If a =6, then aR means (R-\-R-{- to a terms) ={R+R-\- to 6 terms) =bR, :,aR=hR. If a<6, then 6>a, .'. by the first case bR >aR, /. aRY-\-Z, then an integer t exists such that X>tZ>Y, Corollary. If ^, B, C are magnitudes of the same kind, and if A >B, then integers n, t exist such that nA >tC >nB. Since X>Y+Z, . . X >Z. It may be that X is also greater than 2Z or 3Z or 4Z, and so on. Suppose that tZ is the greatest multiple of Z which is less than X. Then (t-\-l)Z must be either greater than X or equal to X, li{t + l)Z>X, then since X >Y -\-Z, .:(t + l)Z>Y+Z, .-. tZ>Y. U(t-\-l)Z=X, then since X>Y-\-Z, .-. (t'-{-l)Z>Y+Z, :, tZ>Y, Hence in both cases tZ>Y. But also tZtZ>Y. This proposition is not an easy one for beginners to grasp. It may be illustrated graphically thus : 10 THE THEORY OF PROPORTION Suppose that X, F, Z are lengths. On one side of a straight line mark off a length OB — Y, and BC=Z. On the other side mark off OA =X. Then since X>Y+Z, ' ' .'. OA>OB+BG, :. OA >0C. I If now, starting from 0, successive lengths equal to BC be marked off, one of the markings must fall between B and A, because BA >Z. Let this mark- ing be D, Let OD be equal to t{BC), i.e. tZ, Then since OA >0D >0B, :.X>tZ>Y. o Fig. 1. To prove the Corollary, observe that since A >B, .'. A —B is a magnitude of the same kind as A and B, and therefore of the same kind as C. Hence, by Archimedes' Axiom an integer 7i exists such that n{A-B)>C, :. nA—nB>C, .-. nA>nB-\-C. Putting, in Prop. VHL, X=nA, Y=nB,Z=C, it follows that an integer t exists such that nA>tC>nB. CHAPTER III Articles 13-18 The Relations between Multiples of the same Magnitude. Commensurable Magnitudes, Article 13 If two magnitudes are multiples of the same magnitude, they may be said to be measured by that magnitude. Thus lengths of 7 feet and 1 3 feet can be exactly measured by an undivided foot rule. These two lengths are said to have a common measure, viz. 1 foot, and are called commensurable. If a length of 2 feet and a length of 1 foot be taken, the first is said to be twice as great as the second, whilst the second is said to be half as great as the first. Thus if these two lengths be considered, not separately, but in relation to one another, they determine two numbers, viz. 2 and |. Note that in each case from the two lengths and the order in which they are taken, a number which is not a length has been determined, and that the unit in terms of which the lengths are measured does not appear in the result. In this case it is said that The ratio of 2 feet to 1 foot is 2. „ ,, „ 1 foot to 2 feet is i. Similarly : The ratio of 3 inches to 1 inch is 3. ,, ,, ,,3 inches to 2 inches is f . ,, ,, ,,2 inches to 3 inches is §. ,, ,, ,, 3 yards to 2 yards is f . ,, ,, „ 2 yards to 3 yards is §. 11 12 THE THEORY OF PROPORTION The ratio of 3 yards to 2 feet =the ratio of 9 feet to 2 feet=f . The ratio of 2 yards to 3 feet =the ratio of 6 feet to 3 feet=|^=2. The ratio of 5 miles to 7 miles=f . The ratio of 5 miles to 7 furlongs =the ratio of 40 furlongs to 7 furlongs ==4/^. The ratio of 7 minutes to 105 seconds =the ratio of 420 seconds to 105 seconds =f§^= 4. The ratio of 13 hours to 2 days =the ratio of 13 hours to 24 hours=|^. In all these cases the unit in terms of which the magnitudes are measured does not appear in the result. [When there are two units, those units are magnitudes of the same kind, e.g. an hour and a day, and the two magnitudes are expressible in terms of the same unit.] Similarly it may be said that The ratio of 3A to 2^=|- . „ „ „ 2^to3^=f. „ „ „ r^to^=^=r. „ „ „ nA to rA = ~, „ „ „ r A to nA = L Thus if two multiples of the same magnitude are given, and the order in which they are taken is fixed, then these par- ticulars determine a number. This number is the quotient of one positive whole number by another. It is usually called a vulgar fraction. All positive and negative whole numbers or fractions, which are quotients of one whole number by another, are called rational numbers, but we shall only have to deal with those which are positive in this book. The usual notation for the ratio of X to Y is X:Y, and consequently the ratio of Y to X is denoted by Y:X. It is advisable to write these in brackets, thus (Z: 7)and(r:Z), COMMENSURABLE MAGNITUDES 13 because beginners who have not grasped the idea that the whole symbol represents a single number not infrequently imagine that X : Y still represents the two distinct things X and Y. Geometrical Illustrations Article 14 (i.) There are two parallelograms on bases 3 inches and 2 inches respectively. The height of each parallelogram is 1 inch. Prove that the ratio of the areas of the parallelograms is equal to the ratio of the lengths of the bases. Let the parallelograms be ABCD and EFGH. D M . L C H Q G \ \ \ \/ / / A J K BE P F Fig. 2. Since they have the same height they may be placed be- tween the same parallels as in the figure. Let the base AB represent 3 inches, and the base EF 2 inches. Then (AB : EF)=%. On AB mark off AJ—JK=KB to represent 1 inch, and on EF mark off EP=PF to represent 1 inch. Draw JM, KL parallel to AD, and PQ parallel to EH. Then the five parallelograms AJMD, JKLM, KBCL, EPQH, PFGQ are equal in area, for they stand on equal bases and are between the same parallels. Consequently ABCD=3(AJMD), EFGH=2{AJMD), .'. (ABCD : EFGH)=^, but (AB:EF)=^, .-. (ABCD : EFGH)=(AB : EF). 14 THE THEORY OF PROPORTION Article 15 (ii.) There are two triangles on bases 4 inches and 3 inches respectively. The height of each triangle is 2 inches. It can be shown as in the last example that the ratio of the lengths of the bases of the triangles is J, and also that the ratio of the areas of the triangles is ^. Hence the ratio of the areas of the triangles is equal to the ratio of the lengths of their bases. Article 16 (iii.) In two equal circles there are arcs whose lengths are 5A and 7 A respectively, A representing the length of a certain arc. Suppose that the arc A subtends an angle a at the centre of either circle ; then the arc 5A, being divisible into 5 equal parts, each of which subtends an angle a at the centre of its circle, will subtend an angle 5a at that centre. Similarly the arc 7 A subtends an angle 7 a at the centre of its circle. A 7A)=^ Fig. 3. Now the ratio of the arcs =(5^ . i^;—^. The ratio of the angles subtended by the arcs = (5a : 7a)=f Hence the ratio of the arcs is equal to the ratio of the angles they subtend at the centre. It will be noticed that in each of these examples the par- ticular numbers which occur, viz. 3 and 2 in the first, 4 and COMMENSURABLE MAGNITUDES 15 3 in the second, 5 and 7 in the third, do not appear in the final result, which is a general proposition having no apparent con- nection with the numbers that occur. And in fact each of these propositions can be generalised. It will be sufficient to take the first. Article 17 Two parallelograms, situated between the same parallels, have commensurable bases, to prove that the ratio of the area of the first parallelogram to the area of the second parallelogram is equal to the ratio of the length of the base of the first parallelo- gram to the length of the base of the second parallelogram. D P Q R C H S T G AKLMB ENOF Fig. 4. Let the parallelograms ABCD, EFGH have their bases AB, EF commensurable. Let AK be a common measure of AB, EF. Suppose that AB=r{AK), EF=s(AK). Let AB, EF be divided as in the figure into parts each equal to AK, and through the points of division of AB let straight lines be drawn parallel to AD ', and through the points of division of EF let straight lines be drawn parallel to EH, so that each parallelogram is divided up into equal parallelograms. Since the bases of all these parallelograms are equal, and they are situated between the same parallels, they are equal in area. Since AB contains r lengths each equal to AK, therefore the parallelogram ABCD contains r parallelograms each equal to AKPD. :. ABCD=r{AKPD), Also EF=s(EN)=s(AK). 16 THE THEORY OF PROPORTION Thus EF contains s lengths each equal to AK, :. the parallelogram EFGH contains s parallelograms each equal to AKPD, :. EFGH=s(AKPD). Since AB=r{AK), and EF=s(AK) ; :.(AB:EF)='^^. Since ABCD=r(AKPD), and EFGH=s(AKPD) ; .-. {ABCD : EFGH)=l. :. (ABCD : EFGH) = (AB : EF). Another step may now be taken. Article 18 Suppose that there are three magnitudes A, B,C which are all multiples of the same magnitude G. Let A =aGy B =hG, C =cG, where a, h, c are some positive integers. Then the ratio of A to C, i.e. of aG to cG, a * is by definition -. Similarly the ratio of 5 to C is ^. Now if A >B, then aG >bG, :. a>b, a^h :. (A:C)>(B:C). If A=B, then aG=bG, :. a=b, c c' .'.{A:C) = {B:C). liAB, (A:C)>{B:C). If A{B :C), and it will be proved later that the statement has a meaning when A, B, G have no common measure. I think it must be evident to any one who compares these two fundamental ideas with the Fifth and Seventh Definitions of Euclid's Fifth Book that they are of a far simpler nature than those definitions, and that they must have formed the starting-point from which the book was built up. CHAPTER IV Articles 19-21 Magnitudes of the same kind which are not Multiples of the same Magnitude. Incommensurahle Magnitudes, Article 19 It will now be shown that if two magnitudes of the same kind be chosen at random they may not have a common measure. To prove this all that is necessary is to show that it is possible to choose some two magnitudes of the same kindj out of the infinite number that exist, which have no common measure. Article 20 Take the diagonal and side of a square. If possible let them have a common measure, viz. a length L. Let the side of the square be p times L, and let the diagonal of the square be q times L, where p, q are some positive whole numbers. Now the square on the diagonal has twice the area of the square on the side. .*. q^=2p^. If p, q have a common factor, let their greatest common factor be g. Let l=r, 1=.. Then r, s have no common factor and r^=2s^. As r, s have no common factor they cannot both be even, and therefore the following cases only need be considered : (i.) r odd, s odd. (ii.) r odd, s even, (iii.) r even, s odd. In the first and second cases r^ is odd, and cannot there- 18 INCOMMENSURABLE MAGNITUDES 19 fore be equal to 25 2, which is even. Hence the first and second alternatives cannot hold. In the third case put r=2m. This gives 2m^=^s^. But 2m 2 is even, and 5^ is odd. Hence the third alternative- cannot hold. Hence the equation r^—2s" cannot hold. Consequently the side and diagonal of a square have no common measure. It has thus been proved that magnitudes of the same kind exist which have no common measure. Article 21 Having reached this result the question arises If two magnitudes have no common measure, can one of them have a ratio to the other, and if so how is it to be measured ? Let us return to the study of the case of the diagonal and side of a square and see where it leads us. Fig. 5. 20 THE THEORY OF PROPORTION Suppose that OA is the diagonal and OB the side of the square. On OA measure off OP=OB, and 0Q=20B. It will be found that OPhc, o a The case in which ad=bc can only occur when either 24 THE THEORY OF PROPORTION n C •^ or ^ or both of them have not been reduced to their lowest terms. In this case they are said to be equal. Suppose that when reduced to their lowest terms the result is |. Then both are replaced by the single number |.* Akticle 24f Let us now put to ourselves the question : Does anything exist, which is not a rational number, which is nevertheless entitled to be ranked as a number ? If so we may agree that it must be in the technical sense of the words, a magnitude o/ the same kind as the rational numbers. Now the first of the characteristics of magnitudes of the same kind, enumerated in Chapter I (Article 3 (1)) is this : " 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." Suppose that | is any rational number whatever, and that i is a magnitude of the same kind in the technical sense as the rational numbers, but yet is not a rational number. Thus i and | are in the technical sense magnitudes of the same kind. Hence either i is equal to |, or 1 is not equal to |. Now 1 cannot be equal to -, for then i would be a rational number contrary to the hypothesis. r Hence i is not equal to ^, and .*. either i>^ or ^<^. S 6 * The negative numbers precede the positive numbers, and are arranged according to the following rule : If I precede ^, then —% precedes — ^« These may be written ^ 252, and .'. either (|)^<2orelse (|)^>2. Then, as in the preceding cases, we shall say if (?1V<2, then (^\ is less than the square root of 2 ; but if (p\ ^ >2, then ^ is greater than the square root of 2. On this understanding the square root of 2 has a definite place WITH REGARD TO the system of rational numbers. It is not itself a rational number, but whenever any rational number is assigned, it is possible to say whether it is greater or less than the square root of 2. The square root of 2 fills the gap between those rational numbers whose squares are greater than 2 and those whose squares are less than 2. We proceed to generalise the idea we have reached. Article 26 An irrational number i will be regarded as known, when- ever any rule has been given which will make it possible to distinguish those rational numbers which are greater than i from those which are less than i, because this knowledge makes it possible to determine all the properties of i. The effect of the adoption of such a rule is that Any irrational number has a definite place with regard to the system of rational numbers. It fills the gap between those rational numbers which are greater than it and those rational numbers which are less than it. Mode of Distinguishing between Unequal Irrational Numbers. Article 27 Suppose next that i and j are two irrational numbers. If they are not equal to one another, they occupy different EXTENSION OF THE IDEA OF NUMBER 27 places with regard to the system of rational numbers, and therefore some rational number - must fall between them. s Hence either i^ >j, and then i is said to be greater than j. Conditions for Equality of Irrational Numbers. Article 28 If, however, i and j are the same irrational number, then they have the same place with regard to the system of rational numbers, hence no rational number can lie between them. If.-. ?=/, and if - represent any rational number whatever, then if i >-, it is necessary that j >- ; but if ^<^, it is necessary that 7<|. Conversely, if - represent any rational number whatever, and if it be known that whenever i >^, then j >^ ; s s and whenever i'<-, then y J5, then (A : C) >(B : C) ; (2) if A=:B, then (A : C)=(B : C) ; (3)* if^<^, then(^ :C)<(B:C). The new theory of ratio is constructed so as to satisfy the above conditions (1), (2), and (3) in all cases, whether A, B, C have or have not a common measure. Since these conditions are exactly those which held good when A, B^ C have a common measure, there will be no con- tradiction between this and what precedes. This will have prepared the way for Article 30 Prop. IX. Let it be assumed to be true that, when A, ByC * It is to be observed that condition (3) is not distinct from (1). 28 RATIOS OF MAGNITUDES 29 are magnitudes of the same kind, whether they are multiples of the same magnitude or not* (1) if^>J5, then (A : C) >(B : C) ; (3) iiA(B:C), then A >B : (5) ii(A:C) = (B:C), then ^=5; (6) ii{A:C)<(B:C), then A (B:C). Now since A and B are magnitudes of the same kind, .-. A>BotA=Bot A(B : C). Hence A is not equal to B. And if A(B : C). Hence A is not less than B, and it was shown that A was not equal to B. .'. A must be greater than B, The other two cases can be proved in like manner. We shall refer to the assumptions (1) if^>J5, then (A : C) >(B : C) ; (3) if^<^, then(^ :C)<(B:C), (2) if^==^, then(^ :C) = (B:C); in what follows as the fundamental assumptions or principles on which the theory of ratio is constructed. * It will be proved later on that in this case the symbols (A : C) and (B : C) have a meaning. 30 THE THEORY OF PROPORTION Since the third is included in the first, they are equivalent to only two assumptions. It will be found that they make it pos- sible t9 construct a theory of the ratio of magnitudes of the same kind, whether they have or have not a common measure, which is consistent with and includes the former theory. We shall require the following Proposition : Article 31 Prop. X. (i.) If rA >sB, then is (A : 5) >^ ; (ii.) If rA=sB, then is (A : B)r-^ • r ■'1 r' Conversely (iii.) If rAf, then is rA >sB ; (v.) If (A : B)=f., then is rA=sB ; (vi.) If (A : B)sB, then rA >s{rX), .-. rA >r{sX) Prop. III. .-. A >sX Prop. VI. .*. by the fundamental assumption (Art. 29) (A:B)>(sX:B), .'. iA:B)>(sX:rX), :.{A:B)>f. To prove (ii.) : In this case r^=55, but B=rX, :. rA=s{rX), .-. rA=r(sX) Prop. III. .-. A=sX Prop. VI. RATIOS OF MAGNITUDES 31 /.by the fundamental assumption (Art. 29) (A : B) = (sX : rX) r' (It is to be specially noted that if any relation of the form rA=sB exist, then A and B have a common measure. This result will be useful afterwards.) To prove (iii.) : In this case rA|. NowjB=rX, and|=(5X:rX) = {sX:B), .-. (A : B) >{sX : B), .'. A >sX, Prop. IX. .-. rA >r(sX), .'.rA>s{rX) Prop. III. .*. rA >sB. To prove (v.) : In this case (A : B)=P New B=rX, f=(sX:rX) =(sX:B), :. (A :B)=(sX:B), .\A=sX, Prop. IX. .'. rA=r{sX) =s{rX) Prop. in. .-. rA=sB. 32 THE THEORY OF PROPORTION To prove (vi.) : In this case {A :B)tB; (2)uA=tB; (3) uAtB, then {A : B) >i. u If uA=tB, then (A : B)=^. u If uAt', (2) {A'.B)=l; (3) {A:B)<1^ (1) If (A : B) >|, then in order that | may not lie between (A : B) and (C : D) it is necessary that (C : D) >-. If .-. (A:B)>1, then (C:2))>|. (2) If (A : -S)=-, then in order that no rational number may lie between {A : B) and (C : D), it is necessary that (C : D)=l. If .-. {A :B)=l, then (C:D)=-; (3) If (A : B)<-, then in order that | may not lie between (A : B) and (C : D), it is necessary that (C : D)<'^. Hence if (A : B) <|, then (C : I>) <^. * This definition does not conflict with what has been said before about equal ratios, see Note in Art. 34. 34 THE THEORY OF PROPORTION NOTE ON ARTICLE 33 Article 34 In the preceding work some cases of equal ratios have been con- sidered. (1) Ratios which were equal to the same rational number were said to be equal. (2) When A =B it was laid down as a fundamental principle that {A'.C)^{B:C). In the first case it is obvious that no rational number can lie between the equal ratios, and it will now be proved that, when -4 =J5, no rational number can Ue between {A : C) and {B : C). For if possible let some rational number ~ lie between {A : C) and {B : C). Let {A : C) be greater than [B : C). Then {A : C)>'^> {B : C). Since(^:(7)>|, :.qA>pC. Since (J5 : C) <|, .-. qBpC>qB, :. qA>qB, :. A>B, which is contrary to the hypothesis that A—B. Hence if A=B no rational number can fall between {A : C) and {B : C). The Test for Equal Ratios Article 35 Conversely, if whatever the integers r and s may be, then r . if (A : B) >% it is also true that (C : D) >; * but if (A : B)=l, it is also true that (C : ^)=| ; and if (A : 5)<|, it is also true that (C : i>)<|, then will (^ : B) = (C : D), for these conditions simply express the fact that no rational number lies between (A : B) and (C : D), and /. (A : B) = {C : D). RATIOS OF MAGNITUDES 35 Euclid v., Definition 5 Article 36 Derivation from the preceding article of the conditions of the Fifth Definition of the Fifth Book of EucHd's Elements. This follows immediately by the aid of Prop. X. If sA >rB, then (^ : J5) >|, Prop. X. .-. (C :/))>§, Art. 35. .-. sC>rD Prop. X. If5^=r^, then (A : B)=^, Prop. X. .-. (C:D)=Z, Art. 35. .-. sC=rD. Prop.X. If sA rB, then sC >rD ; (i.) but when sA=rB, then sC=rD ; (ii.) and when sA ^>(C:D), OTehe(A:B)<^<(C:D), In the first case (A : B) is said to be greater than {C : D) ; in the second case (A : B) is said to be less than (C : D). In the first case, since(^:5)>|. and since ((7:D)<|, •. vA >uB ; .'. vC(C : D), then integers u, v exist such that vA >uB, but vCuD. * This definition does not conflict with what was said before about unequal ratios, see Note in Art. 39. RATIOS OF MAGNITUDES 37 NOTE ON ARTICLES 37-38 Article 39 In the preceding work a case of unequal ratios was considered. . It was laid down as a fundamental principle that then {A : G)> {B : C). In this case it will be proved that a rational number falls between {A : C) and {B : C). It was shown in the Corollary to Prop. VIII. that if A>B, then integers n, t exist such that nA>tC>nB. Since nA>tC, .UA:C)>i. Since nB t.>(B:C). Hence the rational number ^ falls between {A : C) and {B : C). Simplification of the Test for Equal Ratios (Stolz's Theorem) Article 40 I proceed now to show that the second of the three sets of conditions in the Test for Equal Ratios (Art. 35) is superfluous. Prop. XI. If all values of r, s which make sA >rB also make sC >rD (I.), and if all values of r, s which make sA r^D. .*. sfi—rj) is a magnitude of the same kind as D. 38 THE THEORY OF PROPORTION .*. An integer n exists such that n(s^C-rJ))>D, .'. nSiC>(nri-\-l)D, hut SiA=riB, .-. nSiA^nr^B<(nri-\-l)B. Hence nSiA<(nri-{-l)B, hut nSj^C >(nri-]-l)D, and /. putting s=nsi, r=7iri + l, it is seen that for these values of r, s the hypothesis (II.) is not satisfied. .'. SiC is not greater than r^D, Or (ii.) sfiD, .'. ns-fi<(nr^ — l)D, hut s^A=r^B, .'. nSiA=nr^B>(nri — l)B. Hence nSiA>{nr I — l)B, hut nsiC<(nrj^ — l)D, and .'. putting s=ns i, r=nr i — I , it is seen that for these values of r, s the hypothesis (I.) is not satisfied. /. s^C is not less than r^D. It has now been shown that s^C is neither greater nor less than r^D. .-. s^C=rJ). Hence the second set of conditions in Euclid's Definition is involved in the first and third sets of conditions. It is there- fore superfluous. Hence (^ : J5) = ((7:i)), if all values of r, s which make sA >rB also make sC >rD, and if all values of r, s which make sA |, then also (C : D) >|, and if whenever (A : ^)<|, then also (C : J^)<|, RATIOS OF MAGNITUDES 39 whatever integers r, s may be, then will (A : B) = (C : D), and it is not necessary to show also that a {A :B)=l, then (C:D)=l, Magnitudes in Proportion If (A : B)=(C : D), the magnitudes A, B, C, D are said to be proportionals, or in proportion. The proportion is usually written thus : A:B::C:D, and is read "the ratio of ^ to J5 is the same as the ratio of G to D." A and D are called the extremes, B and G the means of the proportion. D is called the fourth proportional to A, B and C. If C=B so that {A : B) = {B : D), then the three magnitudes A, Bj D are said to be in proportion, B is said to be a mean proportional between A and D, and D is said to be a third proportional to A and B. Recapitulation of the Ghief Points of the Preceding Theory Article 41 Before illustrating the preceding theory it is well to re- capitulate the chief points which have to be borne in mind in what follows. (1) Numbers exist which are not rational numbers. They are called irrational numbers. They are in the technical sense of the words magnitudes of the same kind as the rational numbers (Art. 24). (2) An irrational number is determined when a rule is given which makes it possible to decide whether the irrational number is greater or less than any rational number whatever. An irrational number has therefore a definite place amongst the rational numbers (Art. 26). (3) If ^ and B are any two magnitudes of the same kind, then the ratio of ^ to 5 is a number rational or irrational (Art. 32). (4) Two ratios are equal when no rational number lies between them (Art. 33). CHAPTER VII Articles 42-49 Properties of Equal Ratios. First Group of Propositions. Article 42 The proofs of the following propositions may be conducted on the same lines. They are independent of one another and may be taken in any order. Prop. XII. li(A:B) = {C'.D), then (rA : sB) = (rC : sD) Euc. V. 4. Prop. XIII. If (A : B) = {C : D), then (B:A) = (D:C). Euc. V. Corollary to 4. Prop. XIV. If {A : B) = (C : D) = (E : F), and if all the magnitudes are of the same kind, then (A :B) = {A +C-{-E : B+D+F). Euc. V. 12. Prop. XV. {A : B) = {nA : nB) Euc. V. 15. Prop. XVI. If {A : B) = {X : Y), then {A -\-B:B) = (X +¥:¥). ..Euc.V. 18. Prop. XVII. If (A +J5 : 5) = (X+ 7 : 7), then (A : B) = (X : 7) Euc. V. 17. Article 43 Prop. XII. If {A :B) = {C : D), to show that (rA : sB) = (rC : sD) Euc. V. 4. P Let ^ denote any rational number whatever, then it follows from Art. 40 that it is sufficient to consider the following alternatives : 40 FIRST GROUP OF PROPOSITIONS 41 Either (rA : sB) >|, :.q{rA)>p{sB), '.(q(r))A>(p{s))B, .:{A:B)> Pis) q(r) But (A :B)=(C:D), .', {q(r))C>{p{s))D, .:q{rC)>p{sD), .'.{rC:sD)>l Hence if {rA : sB) > V then (rC : sD) >l. ' q Or (r A :sB)<^, .'. q{rA)1 then tB>s A, :. sAsC, :,(D:C)>^. U(B:A)<1 then tBtB, :.(A:B)>1. But (A :B) = {C:D), .-. (0:i))>|, .-. sC>tD, .-. tD| then (D : C) >^. Henceif (5:^)<|, then(i):C)<|. Hence no rational number lies between (B : A) and {D : C). :.{B:A) = (D:C). Article 45 Prop. XIV. Ii{A:B) = {C:D) = (E:F), and if all the magnitudes are of the same kind, then {A : B) = {A-\-C+E : B+D+F). Euc. V. 12. It has to be shown that no rational number lies between (A : B) and {A +C+^ : 5+i)+i^). Let ^ be any rational number whatever. Then it is sufficient to consider the alternatives : (A:B)>1. Then since (A:B)={C:D)=^{E'.F), it follows that (C:D)>^, (E:F)>\ Hence qA >pB, qC>pD, qE >pF, :.q(A+C+E)>p(B-\-D+F), P (A:B)

Hence if (A : B) >|, then(^ -^C+E : B+D+F)>^, Hence no rational number falls between (A : B) and (A +C-\-E : B-}-D+F), .-. (A : B) = (A+C+E : B+D+F). In like manner it can be proved that ii(A,: B,)^(A,: B,)= , . .={A,,: BJ, and if all the magnitudes are of the same kind, then (A, : B,)={A, + A,+ ...+A,: B,+B^+ . . . +5„). FIRST GROUP OF PROPOSITIONS 43 Article 46 Prop. XV. To prove that (A : B) = {nA : nB), . .Euc.V. 15. It has to be shown that no rational number falls between (A : B) and (nA : nB). Let I be any rational number whatever. Then it is enough to consider the alternatives : {A:B)>i .: vA >tB, .'. n{vA)>n(tB), .'. v(nA)>t{nB), .'.(nA:nB)>i. Hence if {A : B) >|, then (nA : nB) >i (A:B)1, .-. s(A +B) >rB. It is necessary to take sep- arately the cases sr. (i.) s(r-s)B, (A:B)>^-^^ hut(A:B)=(X: 7), Ot(A+B:B)<1, .'. s(A+B)^, .-. sX>(r-s)Y, :.s{X+Y)>rY, .-. (Z+7:7)>^. (ii.) s=r,sY=rY, .'.s(X+Y)>rY, :.{X+Y:Y)>1. (iii.) s>rjsY>rY, .■.s{X+Y)>rY, .-. (X+7:F)>|. Hence if {A -\-B : B) >|;, then(Z+7: 7)>^:. .-. (X:7)<^^ :.sX<(r-s)Y, .:6(X+Y)'-. In this case sA >rB, .'.s(A+B)>{r+s)B, r+s s ' .-. (A +B:B)> hut (A+B:B)=^{X+Y: Y), .-. (X+7: 7)> r+s s ' .s{X+Y)>{r+s)Y, .\sX>rY, .-. (Z:7)>^. Hence if (^ :B)> then (X : 7) >^. (ii.) (A:B)<1. In this case 5^ l. (ii.) (BC : EF)=l, (iii.) (BC : EF)<1 It is known by Prop. XI. (Stolz's Theorem), Art. 40, that we need not consider the second alternative.* Take the first alternative : (BC:EF)>I, :. r(BC) >s(EF). JSiow set oflBR=r(BC), and ES=s(EF). Since the triangles have the same altitude they may be placed between the same parallels. A D B C R E F S Fig. 7. Join AR and DS. Then since BR=r(BC), :. by (3) above AABR=r(AABC), and since ES=s(EF), .-. by (3) above aDES=s( ADEF). * There is no difficulty in considering it in this case, as there is in some of the propositions which follow. FIRST GROUP OF PROPOSITIONS 47 :. by (2) above l\ABR>/\DE8, i.e. r(AABC)>s(ADEF), .-.(AABC: ADEF)>I. r Hence if {BC : EF) >f, then{AABC: aDEF)>^. (I.) Take now the alternative : (BC'.EFXf, :. r{BC) ES, i.e. r(AABC)C, then B >D ; but if A==C, then B=D; and if AC, thenT>F; but if ^=0, thenT=F. and if A 'j., .-. rA >sC. Hence rA—sC is a magni- tude of the same kind sls A, B, C,D. Compare it with either B or D, say with B. Then by Archimedes' Axiom an integer n exists such that n{rA -sC) >B, :. nrA >nsC-\-B, .'.an integer t exists such that nrA>tB>nsC (Prop. VIII.). Since nrA>tB, (2) (A:C)D, psC>prA-^D, /.an integer u exists such that psC >uD >prA (Prop. VIII.). Since psC>uD, 50 THE THEORY OF PROPORTION But (A : B)=(C : D), But (A:B)=(C:D), .'. nrC >tD. But tB >nsC, :. rtB>rnsC>stD, rtB>stD, rB>sD, .'.(B:D)>' u ■■■(A:B)>f, .'. psA >uB. But uD>prA, .'. ruB f, then (B : D) >| ; but if {A : C)

C, thenjB>i). If A=:C, then B=D. If A 1, :. (B:D)>1. Hence if A >(7, then B>D. (2) (A:C)=l, .'.(B:D) = l. Hence if J^=C, then B=D. (3) (A:C)<1. :.{B:D)<1, Hence if ^f, .-. lA >kC, :. lA—kCis3b magnitude of the same kind SiS A, B, C. Comparing it with B, it follows by Archimedes' Axiom that an integer v exists such that v(lA-kC)>B, .\vlA>vkC+B, .'. an integer w exists such that vlA >wB >vkC (Prop. VIII.). Since vlA>wB, But (A :B)=(T\ U), •'.(T:U)>f^, .-. vlT>wU, But wB>vkC, ,'.(B:C)>§. Now (B:C)=(U: F), .•.(C7:F)>f, /. wU>vkVy .'. vlT>wU>vkV, .\vlT>vkV, lT>kV, (T:F)>|. Oxf, (2) (A: :.lAB, nkC>nlA+B, .'. an integer r exists such that nlAf, then {T:V)>^; but if (A:C)<^, h t I I then (T\ F)< /• 52 THE THEORY OF PROPORTION Hence no rational number falls between (A : C) and (T : F), .'.(A:C) = (T:V). Corollary (Euc. V. 20) : To prove that with the data in the proposition If A>C, then T>V. If A=C, then T=V. If A|, .-. sA >rC, Now -B is a magnitude of the same kind as sA, rC. Hence by the Corollary to Prop. VIII. integers n, t exist such that n(sA)>tB>n(rC), Now (n(s))A=n(sA)>tB, But (U:V)=(A:B)>:^^^ :. (n(s))U>tV ....(1.)' Also tB>n(rC), .: tB>(n{r))C, (B:C)> n(r) Ov(A:C)\ ,(T) :.tT>(n{r))U. {11.) We have to eliminate U be- tween (I.) and {II. )> s{tT) >s[(n(r))U^ from (II.), s[(n(r))U]=r{(n(s))Ul by Cor. to Prop. III. A\.sor[{n{s))U'\>r{tV) from (I.), :.s(tT)>r(tV), :.t(sT)>t(rV), :.sT>rV, '•(T:V)>t If .•.M:(7)>|, then (T : V) >L Now (T:U)=(B:C)< 53 n{r) t ' :.tT<(n(r))U ... (IV.) We have to eliminate U be- tween (III.) and (IV.), s(tT)(7,then^>F. If^=0,thenT=F. If^<0,thenTB+C Euc. V. 25. As in the Fifth Book, the proofs of these propositions are made to depend on the proofs of the preceding propositions. They could be proved in a manner having some resemblance to those of the earlier propositions, but the proofs are compli- cated and difficult ; and altogether unsuited for an elementary course of instruction. It will be seen that the proofs of these propositions are not automatic like those which have gone before. They involve a considerable strain on the memory, but on the whole they are very much simpler than any other proofs known to me. Article 55 Prop. XXI. If(A+C:B-\-D)=(C:D), to prove that (A : B) = (C : D) Euc. V. 19. Since (A+C : B+D) = (C : D), .-. by Prop. XVIII. (A+C:C)=(B-{-D:D), 64 TfflRD GROUP OF PROPOSITIONS 55 .-. by Prop. XVII. {A:C) = {B:D), .-. by Prop. XVIII. {A:B) = (C:D). Article 56 Prop. XXII. (Euc. V. 24). U(A :C) = {X:Z), andif(J5:C) = (y:Z), to prove that {A-\-B : C)=(X-{-Y : Z). Since (B : C) = (Y : Z), .-. by Prop. XIII. (C:B) = (Z:Y). Now {A :(7) = (X:Z), and (C:B) = (Z: Y), .*. by Prop. XIX. (A:B) = (X:Y), :. by Prop. XVI. {A+B:B) = (X+Y: Y). But (B:C)=(Y:Z), .♦. by Prop. XIX. {A-\-B:C)=(X+Y:Z). Article 57 Prop. XXIII . liA,B,C,D are four magnitudes of the same kind, if A be the greatest of them, and if (A : B) = (C : D), then (A-\-D)>(B-\-C) Euc. V. 25. Since A >B, :.(A:B)>1, :.(C:D)>h :.C>D. Since {A : B) = (C : D), :. by Prop. XVIII. (A:C)={B:D). But A >C, :. {A:C)>\, :.(B:D)>\, :.B>D. 56 THE THEORY OF PROPORTION Hence D is the smallest of the four magnitudes. Since (A :B)^(C\D), .'. by Prop. XVII. (A-B:B) = (G-D:D), :. by Prop. XVIII. (A-B:C-D)=^(B\D). But B>D, .'.{B:D)>1, ... (A-B:C-D)>1, .'.A-B>C-D, :.A+D>B-{-C. PART II CHAPTER X Articles 58-67 Geometrical Applications of Stolz^s Theorem (Art. 40). Article 58 All the properties of equal ratios that can be put into an elementary course have now been given. I will now give a remarkable application of Stolz's Theorem, due to my friend, Mr. Rose-Innes, to prove that the areas of circles are proportional to the squares described on their radii. Some preliminary propositions from the Tenth and Twelfth Books of Euclid's Elements are required. They are set out here in order to make the argument complete in itself. Euclid X. 1 If A and B he two magnitudes of the same kind, of which A is the larger, and if from A more than its half he taken away, leaving a remainder R^; and if from R^ more than its half he taken away, leaving a remainder R^ ; and so on, then if this process he continued long enough, the remainder left will he less than B. This is deduced by repeated applications of the following : If X and Y be two magnitudes of the same kind, and if X be greater than Y, then if from X less than its half be taken away, and if from Y more than its half be taken away, then the remainder of X left is greater than the remainder of Y. If from X less than JZ is taken, more than JX is left. 67 58 THE THEORY OF PROPORTION If from Y more than J Y is taken, less than 1 7 is left. But since X>Y, iX>iY, .'. (more than JZ)>(less than J 7). To apply this to Euc. X. 1. Since A >B it follows by Archimedes' Axiom that an integer n exists such that nB>A. From the greater magnitude nB take away B, which is less than ^nB, The remainder is {n — l)B. From the smaller magnitude A take away more than its half. Let the remainder be R^, .'. {n — l)B>Ri by what has just been proved. From the greater magnitude (n — l)B, take away B, which is less than ^(n — l)B. The remainder is (n — 2)B. From the smaller magnitude R^ take away more than its half. Let the remainder he R2, .: (n-2)B>R2. Proceeding thus we get after s applications (n-s)B>R„ and .'. after (n—2) applications 2B>R,^,, Now take from R^_2 more than its half. Let the remainder be Rn-i- Then ii?„.2>i2„_i. But B>iR^_,, .'. B>R^_^. So that after (^ — 1) operations on A, the remainder of A left, viz. Rn-\> is less than B. Article 59 As a geometrical application of this result take the follow- ing from the Second Proposition of the Twelfth Book of Euclid's Elements : APPLICATIONS OF STOLZ'S THEOREM 59 // a regular polygon of 2° sides he 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. Take a segment of a circle which is a semicircle or less than a semicircle. c Let its chord be AB, and the middle point of its arc C. Through C draw a tangent to the circle. This is parallel to^-B. Let the perpendiculars to AB, through A and B^ cut the tangent at C at D, E. Then the triangle ABC is equal to the sum of the triangles ACD, BCE. Hence the triangle ABC is greater than the sum of the segments cut off by AC, BC. Hence the triangle ABC is greater than half the area of the segment ABC ; and therefore if the triangular area ABC be removed from the segment, the remainder left is less than half the original segment. Suppose now that a square ABCD is inscribed in a circle, then if the square be cut out from the circular area less than half the circular area is left. The part cut out is shaded. 60 THE THEORY OF PROPORTION Let each of the arcs AB, BC, CD, DA be bisected at E, F, G, H respectively. Then from the unshaded area remove the triangles AEB, BFC, CGD, DHA. Then by what has been proved the triangles AEB, BFC, CGD, DHA are greater than half the segments AEB, BFC, CGD, DHA respectively. And therefore, if the triangles are removed, more than half of the portion of the circle outside the square will have been removed. We have left, then, only the unshaded areas shown in the next figure. A This process of bisecting the arcs and removing the tri- angular areas can be continued indefinitely. At each step more than half the remaining area is removed, and therefore, if the process be carried on long enough, there will at length remain an area less than any area, say D, which may be fixed in advance. Euclid XII. 1 Article 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. APPLICATIONS OF STOLZ'S THEOREM 61 The areas of any two similar polygons are proportional to the squares described on corresponding sides. If the similar polygons are inscribed in two circles, then corresponding sides are proportional to the radii of the circles, and therefore the squares on corresponding sides are propor- tional to the squares described on the radii of the circles. Hence the areas of similar polygons inscribed in two circles are proportional to the areas of the squares described on the radii of the circles. Euclid XII. 2 Article 61 T}ie, areas of circles are proportional to the squafes described on their radii. Let Cj be the area of a circle whose radius is r*!. Let (7-2 be the area of a circle whose radius is rg. *If the ratio (Cj : Cg) be compared with any rational number 1, then it is sufficient to consider the alternatives Suppose (Ci : C2) >-» .*. MCi>Ct2, .'. uC I— tC 2=some area C. Inside Cj describe the polygon P^ as explained in Art. 59, so that (7i-PiP2, .'. tU 2 y>t£^2i but uPi>tC2, :. uP,>tP2, .•.(Pi:P2)>l. * From this point onwards the argument is applicable to many other propositions than the one under consideration. 62 THE THEORY OF PROPORTION Hence if (C^:C^>1, then (Pi:P2)>|. Now let A^i be the square described on t-j, and 8^ be the square described on r^. Then {P,:P^) = (S^:S,). Bnt(P,:P,)>l Hence if (C^ : C^) >^ ] then {S,:S,)>1\ Suppose next .'. tC 2— uC i=some area D. Inside Cg inscribe a polygon Q2 as explained in Art. 59, sothat C2-G2uCi. Inside Ci inscribe a polygon Q^ similar to Q^. ThenCi>Oi, .-. tQ2>uCi>uQi, .'. uQitC2, then uS^ >tS2 (I.) and if uC^KtC^, then uS^tC2, then by (I.) u8i>tS2i which is contrary to the hypothesis. And if uCitl2' Now let us go on doubling the number of sides of the polygons inscribed in the circle (7 2 until we reach a polygon P2 such that C2-P2<-|(w/l-^/2). Now (I,:l2)=^(8,:82), (P,:P2) = (8,:82h .-. (w7i : tl2) = (uPi : tP^), Now uli>tl2, .'. uPi>tP2. Also uli—tl2 : tl2=uP^—tP2 : tPz^ But tl2 :. tC2t82i then uC^ >tC2. In a similar way it can be shown that if u8^tC^, then uSy^>tS^\ (i.) if uCitS2, then uC^>tC2; (iii.) if uSj^KtS^, then uC^KtC^] (iv.) Suppose now that uC-^^^tC^, then if we compare uS-^ with tS^ the logical alternatives are u8i>tS2 or uSitS2y then uC^>tC2 by (iii.), which is contrary to the hypothesis that uCi=tC2. And if uSip,'C, •• Y>X„ :. every Y exceeds every X. It will be proved in the second place that The set of magnitudes X includes no greatest magnitude. The characteristic of the magnitudes X is that they are not greater than every magnitude of the form p^C. Suppose X' is one of the magnitudes X. Let X'^p^'C. Now there is no greatest pi. Suppose Pievery p^. Since Z is a magnitude Y, it is greater than every magni- tude of the form p^C. Write this thus : Z>every pjC, .'. (Z : C) >every p^. It will be proved in the fifth place That (Z : C)6ome p^C, say Z >P2"C. Then since the rational numbers p2 include no least rational number, take Then pi"Cevery p^, .-. p^" >every p^, .-. ?)2'"C>every2?iC, .•. p2"C \s>db magnitude F. But p^"Cevery p^, and (Z : (7)sB, it is necessary th^t rC>sD. (2) If the integers r, s are such that rA—sB, it is necessary that rC=sD. (3) If the integers r, 5 are such that rA qB, :. pC>qD. Also pA<(q + l)B, :.pC<(q + l)D. Hence pC lies between qD and (q-\-l)D. If, however, pA=tB, then pC—tD. To fix the ideas take a particular example. Let A repre- sent a straight line 3 inches long, and let B represent a straight line 4 inches long. Then if we arrange the multiples of A and B in ascending order of magnitude we get the following diagram : 9^ %A, 6B lA 5B 6A 5 A' t 4A, W SA 2B 2A IB lA which, of course, may be continued indefinitely upwards. This was what De Morgan called the relative multiple scale of A, B. Its properties may be seen more clearly by placing A and B at the bottom of the column, and arranging the digits above them in accordance with the following rules : If rA >sB, then r above A is to be placed higher than s above B, 84 THE THEORY OF PROPORTION If tA=sB, then r above A is to be placed on the same level as s above B. If rAsD, it will foUow that rA >sB ; (5) if rC=sD, it will follow that rA=sB ; (6) iirCsj), but r^A not greater than s^B ; so that either (i.) rfi>s^D and r^A—sJB, or (ii.) rfi>sj) and rTAsj} and r^A=s^B, then since r^A=SiB, it follows by (2) that riC=SiD, which is contrary to the hypothesis that rfi>sj). Hence r^A is not equal to s^B. (ii.) If next rfi>sJD and r^As-J). Hence r^A is not less than s^B. 86 THE THEORY OF PROPORTION Hence if rfi>SiD, then is r^A >SiB. Hence (4) holds. In like manner it can be shown that (5) and (6) hold. Article 75 In the second place it may be observed that it has already been shown in the discussion of Stolz's Theorem, Prop. XI., Art. 40, that if (1) and (3) hold, then (2) must hold. Similarly if (4) and (6) hold, then (5) must hold. Article 76 It will be proved in the third place that it is sufficient for (1) and (4) to hold. Suppose that all values of r, s which make rA >sB also make rC >5Z>, (1) and that all values of r, s which make rC>sD also make rA >8B,, (4) it will be shown that (3) holds. Suppose then that riAsjL) or (ii.) r^C^sJ). (i.) Consider first the alternative r^Asj). This is impossible, for by (4), if riC>5iA then r^A >St^B, which is contrary to the hypothesis that r^A A, :. (nr-i^-\-l)AnsJ), i.e. (nr^-^l)C>(ns^)D, but {nr^-\-\)A<(nsi)B, but by (4), putting r=nri + l, it is necessary, when (71^1 + 1)C >(ns-^)D, that {nr^-\-\)A>nSiB, and not (nr-^-\-l)AsJB, to prove that r2C>S2D, ': r^A=s^B, :. r^S2A=s-iS^B, -.' ^2^ >«52-S, .*. Si7*2^ >SiS2B, .'. Sir2A>riS2A, 88 THE THEORY OF PROPORTION .*. Sjr2C>riS2C, :. s^r^C>S2SyD, .'. r2C>S2D. Similarly it can be proved that if then r^CsB, then either rC=sD or rC{C : D) if a single pair of integers r, s exist such that either rA >sB, but rCsB, but rC=sD (2) With these might have been included also the possibility that rA=sB, but rCs'B, hut r'CSiB, but rfi=^sj). COMMENTARY ON EUCLID'S ELEMENTS 89 By Archimedes' Axiom an integer n exists such that n(riA—s^B)>B, Since r-fi=sJD, :. nriC=nSiD, :. nriC<(nSi-{-l)D, Hence putting nri=r', nsi-^l=s' it follows that integers r', s' exist such that r'A>s'B, hut r'CC, .'. (nr2 + l)Cns2B, Hence putting nr2+l=r', ns2=s^ integers r', s' exist such that r'A>s'B, hut r'CsB, but rCs'D (4) Now inasmuch as Euclid has not defined a ratio as a magnitude, it is essential from his point of view to show that no integers r, s, r\ s' can exist which satisfy simultaneously (1) and (4). To prove this : It follows from (1) that s'rA >s'sB, s'rCss'D. Since s'sB=ss'B, it follows that s'rA >sr'A, .'. s'r>sr' (5) Since s'sD=ss'D, it follows that s'rCl>{C:D), and therefore makes the number (A : B) greater than the number {C : D). Whilst (4) is equivalent to (A : B)<':^,<(C : D), and therefore makes the number (A : B) less than the number (C : D). But the number {A : B) cannot be at the same time both greater and less than the number (C : D). Consequently (1) and (4) can never be satisfied simultaneously. * See Heath's E.uclid, Book V. Def. 7. COMMENTARY ON EUCLID'S ELEMENTS 91 The Fifth and the Seventh Definitions constitute together the basis on which the structure of the Fifth Book of EucHd's Elements is reared. They alone of all the definitions prefixed to the Fifth Book effectively count. The Third and the Fourth Definitions are not sufficiently definite in form to enter in reality into the argument. Article 81 Euclid's Fifth Definition does not define ratio, but it makes it possible to decide the extremely important question whether two ratios are equal, whether they be ratios of commensurable or incommensurable magnitudes. Let us consider for a little the question, What did Euclid understand by a ratio ? In particular, did he or did he not regard it as a number ? To this question there is in his work no clear or unambiguous answer. I can only set forth the evidence on both sides. On the one hand (a) If he regarded a ratio as a magnitude, why does he give a demonstration of Prop. 11, viz. : li (A : B)=(C : D), and if (E:F) = (C :D), then (A : B) = (E : F). Simson says, " The words greater, the same or equal, lesser have a quite different meaning when applied to magnitudes and ratios, as is plain from the Fifth and Seventh Definitions of Book V." " That those things which are equal to the same are equal to one another is a most evident axiom when understood of magnitudes, yet Euclid does not make use of it or infer that those ratios which are the same to the same ratio, are the same to one another : but explicitly demonstrates this in Prop. 1 1 of Book V." (b) If he regarded a ratio as a magnitude, why does he give a demonstration of Prop. 13 ; viz : If {A :B) = (C:D), and if (C : D)>{E : F), then (A :B)>E :F), 92 THE THEORY OF PROPORTION which on the hypothesis that ratios are magnitudes amounts only to this : If each of the symbols X, Y, Z represents a ratio, and if X= F, and Y >Z, then is X >Z. On the other hand, if he did not regard a ratio as a magni- tude, (c) Why does he speak of one ratio being greater than another in the Seventh Definition ? The term greater, if used in the ordinary sense, can refer only to magnitudes. {d) Why does he not supply a demonstration in connection with the Seventh Definition showing that if the four magni- tudes A, B, C, D are such that integers r, s exist, such that rA>sB, but rCC : D ; then no integers r', s' can exist, such that r'A s'D, (II.) which are the conditions that A : B(C : D) and {A : B)<{C : D) are inconsistent if {A : B) and (C : D) are magnitudes. On the other hand, if it is only a question of a comparison of the distribution of the multiples of A amongst those of B with the distribution of the multiples of G amongst those of D, then a demonstration of the incompatibility of the conditions (I.) with the conditions (II.) is essential (see Art. 80). (e) Why should he say in the proof of Prop. 10, viz. : If (A:G)>{B:C), then A >B. If {C:A)>(C:B), then A{B : G) ? For if ratios are magnitudes, this needs no proof ; but if they are not so regarded, then the proof given by Euclid of this proposition does not follow from his definitions (see Art. 88). COMMENTARY ON EUCLID'S ELEMENTS 98 Article 82 As to this matter, modern writers are equally in conflict. On the one hand, Stolz, in his Vorlesungen uber allgemeine Arithmetik (Erster Theil, p. 94), says, " In den auf uns ge- kommenen geometrischen Schriften des Alterthumes findet sich keine deutHche Spur der Ansicht, dass das Verhaltniss zweier incommensurabelen Grossen eine Zahl sei." Whilst, on the other hand, Max Simon (EuJclid und die sechs planimetrischen Biicher), so far from agreeing in the usual view that the Greeks saw in the irrational no number, thinks it clear from EucHd, Book V., that they possessed a notion of number in all its generahty. The Arithmeticians and Algebraists of the Middle Ages called the ratios of incommensurable magnitudes " Numeri ficti " or " Numeri surdi," and regarded them as a necessary evil which had to be endured. Michael Stifel, in his Arithmetica Integra, published in 1544, treated them as real numbers. His words amount to an assertion that each irrational number as well as each rational number has a single definite place in the ordered number series.* II. Propositions (First Group). Nos. 1, 2, 3, 5, 6. Article 83 Denoting positive integers by small letters and magnitudes by large letters these propositions are : 1. r(A+B+C-\- . , .)=rA-{-rB+rC+ . . . , 2. (a-\-b-{-c+ . . .)E=aR-{-bR+cR+ 3. r{sA) is the same multiple of A as r(sB) is of B. 5. If A >B, then r(A ~B)=rA -rB. 6. If a>6, then {a-b)E=aR-bR. With regard to the first group of propositions I merely call attention to the fact that I have in the preceding chapters replaced No. 3 by another proposition which is more useful, viz. : (r(8))A=r(sA)=s(rA)={s(r))A. * EncyJdopddie der mcUhematiachen Wiasenschaften, Vol. I. A. 3, p. 51. 94 THE THEORY OF PROPORTION Euclid would have found very great difficulty in expressing this clearly in his notation. ni. Pkopositions (Second Group). Nos. i, 7, 11, 12, 15 and 17 Article 84 These propositions express properties of Equal Ratios, and are deduced by Euclid directly from the Fifth Definition. (The sign of equality will be used in place of the sign : : for " is the same as.") 4. If {A : B)=(C : D), then {rA : sB) = {rC : sD). 7. If A=B, then (A : C) = {B : C) and (C : A)=(C : B). 11. If {A : B) = {C : D), and if (E : F) = (C : D), then (A : B)={E : F). 12. If (A : B) = (C : D)=^(E : F), and if all the magnitudes are of the same kind, then (A : B)={A-\-C-^E : B^D-\-F). 15. {A : B)=(nA : nB). 17. If (A -\-B :B)=(C+D: D), then (A : B)=(C : D). No. 15 is a particular case of No. 12 (Art. 46). Article 85 From No. 17 a very important deduction can be made, bearing on Euclid X. 6, viz. : If (a:b)=(X: 7), thenX=aG, Y=bG. Suppose that X is divided into a equal parts and we call each part G. Then{a:b)=(aG:hG). But X=aG, :. (a:b) = {X:bG), but (a:b) = (X: 7), .:(X:bG)=(X:Y), .'. Y=bG. Hence X, Y have a common measure G. It will be noticed that the proof of this depends on the COMMENTARY ON EUCLID'S ELEMENTS 95 assumption made by Euclid in this and one other place only that the magnitude X can be divided into any number a of equal parts (see Art. 3 (4)). I am indebted to my friend Mr. Rose-Innes for the following proof of this proposition which does not make this assumption. It follows from Euc. V. 17, viz. : If {A-\-B:B) = (C+D:D), then {A : B) = (C : D), that U{A:B) = {C:D) and A>B, then (A-B : B)=iC-D : D), We start from (a : 6)=(Z : Y). We may suppose a >b, for if not we could begin with (6:a)=(7:Z). Suppose then a>&, {a:b)=(X:Y), :, (a-b:b)=(X-Y:Y) Euc. V. 17. If a—h>h, this can be repeated (a-2h:h)=(X-2Y'. Y). Suppose this can be done q^ times, :.(a-q^h:h) = (X-q,Y:Y). Let a—qib=ri, X—q^Y=R^, :.(r,:b)={R,:YY Then (& : ri)=(7 : i^j). Apply Euc. V. 17 again as many times as possible. Suppose we get (b-q^r^:r^) = (Y-q^R^:R^). Suppose b—q^r-^=r2i 7—^2^1=^2 we get (ri:r^) = (R^:R^). Going on thus, it is plain from the relations a-qj}=r^, 96 THE THEORY OF PROPORTION that we are in fact applying the process for finding the greatest common measure of the integers a,h. Now the integers a, h have certainly unity for a common measure. Hence the process will ultimately come to an end after a finite number of steps. Suppose we have & -9' 2^1 = ^2, rn-2-(lnrn-1 = ^' Then we have at the same time Y—q2Ri=B2, and so on. -Kn-2— 5'A-1 = 0>' .•. Rn_i measures Rn-2> :. i?„_i measures i^^.g, Rn-i measures Y and X. Thus X and Y have a common measure Rn-i- X contains Rn-i ^s many times as a contains r^.j. Y contains -R^_i as many times as b contains r^.j. It follows that Euclid might have avoided making the assumption that a magnitude can be divided into any number of equal parts. In the mode of treatment adopted in this book the assump- tion was required for the demonstration of Prop. X. (Art. 31). Without it ratios of magnitudes of the same kind could not have been compared with rational numbers. IV. Propositions (Third Group). Nos. 8, 10, 13. Article 86 8. (i.) If A >B, then (A : C) >{B : C), (ii.) liA(C : B), COMMENTARY ON EUCLID'S ELEMENTS 97 10. (i.) If {A : C) >iB : C), then A >B, (ii.) If (C : A) >{C : B), then A (E : i^), then (^ : B) >{E : i^). These propositions deal with properties of Unequal Ratios. The effective part of them is contained in the 8th Pro- position, which consists almost wholly in proving that if Aj B, C he three magnitudes of the same kind, and if A be greater than B, then integers n, t exist such that nA >tB >nC. (This is the Corollary to Prop. VIII., Art. 12, in this book.) In the proof of this the idea of ratio is not involved. Euclid's proofs of the propositions in this group depend on his Seventh Definition, but he employs them to prove properties of Equal Ratios and for this purpose only. Proposition 8 Article 87 The first part of this proposition, viz. : If A >B, then {A : C) >{B : C), is deduced by Euclid from his Seventh Definition. In the arrangement of the subject presented in this book the fact expressed by the proposition is regarded as one of the fundamental principles from which the conditions of both the Fifth and Seventh Definitions are deduced. Proposition 10 Article 88 Euclid assumes in his proof that the statements (A : C) = (B : C) and (A : C)<(B : C) are inconsistent with {A:C)>{B:C). Of course, if ratios are magnitudes, this needs no proof ; but EucUd's Fifth and Seventh Definitions do not entitle him to regard them as such. 98 THE THEORY OF PROPORTION As Simson pointed out, the proof on Euclid's lines should be as follows : If {A : C)>(B : C), then by Definition 7 a pair of integers r, s exist such that rA is greater than sC, but rB is not greater than sC. Consequently rA >rB, :. A >B. Hence if (A : C) >(B : C), then A >B. This proof is not open to criticism. V. Propositions (Fourth Group). Nos. 9, 14, 16, and 18-25. Article 89 9, (I) li (A: C)=^{B:C), then A=B. (ii.) If (C : A) = (C : B), then A=B. 14:. li A, B, C, D are magnitudes of the same kind, and if (A : B) = (C : 2)), then B=D according as A=C. 16. li A, B, Cy D are magnitudes of the same kind, and if (A : B) = (C : D), then (A : C) = (B : D). 18. If (A : B)=={C : D), then {A+B : B) = [C-\-D : D). 19. li{A+C :B+D)=-(C \D),th.en(A'.B)={A^C \B+D). 20. If {A : B)=(T : U), and if (B : C) = (U : F), then T= V according as ^=(7. 21. If {A : B)=(U : F), and if (B:C) = (T: U), then T= F according as ^=0. 22. If (A : B)=^(T : ?7), and if {B : C)=(?7 : F), then(^ :(7) = (T: F). 23. If (A : J5) = (r/ : F), and if (J5 : C) = (T : C7), then(^:C) = (T: F). 24. If {A : C) = (X : Z), and if (J5 : C) = {Y : Z), then (^-f 5 : C) = {X-\- Y : Z). 25. li A, B, C, D are four magnitudes of the same kind, and if {A : B)=(C : i)), and if A be the greatest of them, then^+J9>^+C. COMMENTARY ON EUCLID'S ELEMENTS 99 All these propositions deal with properties of equal ratios, but Euclid's proofs depend directly or indirectly on Props. 8, 10, 13, and therefore ultimately on the Seventh Definition, so that the proofs depend on properties of unequal ratios. Their ^proofs can, however, he obtained from the Fifth Defini- tion, without using the Seventh Definition, and then 14 can be deduced immediately from 16, 20 from 22, and 21 from 23. The proof of this last statement is given in the preceding chapters. Article 90 There is one point as to the connection between the Fifth and Seventh Definitions in regard to which a misconception may arise. In obtaining the conditions for the equality of ratios in the preceding chapters, use was made of such in- equalities as (^:5)>-and(^:5)<-, r r and also of the fundamental assumption that if A >B, then (A : C) >(B : C), but the Seventh Definition itself was not used. Thus the Fifth and Seventh Definitions are independent of one another. Hence arguing a priori it might he expected that it would prove to he possible to obtain all the properties of Equal Ratios by means of the Fifth Definition only, if that definition is a full and complete one ; as has in fact been shown. I proceed now to make remarks on some propositions in this group. Propositions 14 and 16 Article 91 In the preceding chapters Prop. 16 was proved first, and Prop. 14 was deduced as a Corollary. Euclid proves Prop. 14 first, and derives Prop. 16 from it. If Euclid's proofs given below be compared with the proof of Prop. 16 (see Chapter VIII, Prop. XVIII., Art. 51) it will be seen how many steps there are in Euclid's work which do not suggest themselves to any one trying to follow his argu- 100 THE THEORY OF PROPORTION ment ; whilst the proof given in this book is much more direct, the successive steps arising naturally from one another. Euclid's Proof of Props. V. 14, 16. Prop. 14. If (A :B) = (C\D), and if A >C, .-. (A:B)>(C:B), Euc. V. 8. .'.(C:D)>(C:B), .'. DD. If A=C, {A:B) = (C:B), Euc.V.7. .-. (C:D) = (C:B), :.D=B, Euc.V. 9. .-. B=D. liAB, Euc.V. 10. .-. BsC, then rB >sD, Euc. V. 14. rA=sC rB=sD, Euc. V. 14. rA positive integer, and a>l, then L a'*=+oo. n->oo To see this, let V, F^, Fg. . . F,, be in continued proportion ; ••• V : Fi=F, : F,= . . . = F„_, : F„_i=F„_, : F,. Suppose F2Fi ; :.V,-V,>V,-V. Similarly, F,-F„_i>F_i-F„_,>Fi-F. From these it follows that F„-Fi>(m-l)(Fi-F) ; and .-. V^-V>n{V,~V) and .-. F„>n(Fi-F); •••r>»(F-0 (1) F Since Fl ; and from the F continued proportion it folloAvs that —!!:=a'^ ; .*.(!) gives From this, by the aid of Archimedes' Axiom, it follows that, if a>l, and ?^ be a positive integer, L d^=-\-co. (Cf. De Morgan, ?.c., p. 72.) ^"^^^ Article 99 (ii.) If w be a positive integer, and an(Vi — V) it follows that F / F f;>«V-f; COMMENTARY ON EUCLID'S ELEMENTS 105 >«**; •.• aa n 9ic n(l—a) n{l—a) From this, by the aid of Archimedes' Axiom, it follows that L a''^=-\-0, where a5 Motor Cars and Engines • <7 Aeronautics 18 Marine and Naval Machinery 19 Turbines and Hydraulics 20 Railway Engineering 21 Reinforced Concrete and Cement 22 Civil Engineering, Building Construction, etc. . 23 Surveying, etc 24 Municipal Engineering 25 Irrigation and Water Supply 26 Telegraphy and Telephony 27 Electrical Engineering 28 Electro-Chemistry, etc. . -31 Lighting 32 Thermodynamics 32 Physics and Chemistry 33 Mathematics 38 Manufacture and Industries 39 Arts and Crafts 43 Useful Handbooks and Tables 43 Natural History, Botany, Nature Study, etc. 44 Agriculture and Farming 47 Law, Patents, etc. 48 Miscellaneous 49 Bedrock 51 Index 52 LONDON: 10 ORANGE ST. LEICESTER SQUARE Telephone: — 12432 Central. Telegrams: — Dhagoba, Lonbon. 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Text-Book of Motor Car Engineering— continued. Lubricants. Lubrication, Ball and Roller Bearings. Chassis Construction. General Principles of the Steam Car. Steam Engmes and Condensers. Steam Generators and Pipe Diagrams. The Electric Car. Materials used in Motor Car Construction. Syllabus of City and Guilds of London Institute in Motor Car Engineering. Examinatign Papers. Physical Properties of Petrols. Mathematical Tables and Constants. Vol. II. Design. Fully illustrated. ^ 16 net. Contents. — Introduction, Materials of Construction. General Con- siderations in Engine Design. Power Requirements. Determination of Engine Dimensions. Cylinders and Valves. Valve Gears. Pistons, Gudgeons and Connecting Rods. Crankshafts and Flywheels. The Balancing of Engines. Crankcases and Gearboxes. Engine Lubri- cating and Cooling Arrangements. Inlet, Exhaust and Fuel Piping, etc. Clutches and Brakes. Gearing. Transmission Gear, Frames, Axles and Springs. Torque Rods and Radius Rods, Steering Gears. 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INDEX OF TITLRS Adjustment of Observations, 24 Concrete (Reinforced) in Sub and Aerated Waters, 40 Superstructure (D.-S.), 4 Aerial Flight, 18 Contemporary Chemistry, 37 Aeroplane Patents, 19 Continuous Current Engineering, 31 Aeroplanes (D.-S.). 19 Corpuscular Theory of Matter, 34 Agglutinants and Adhesives, 42 Cotton, 41 Airships, Past and Present, 19 Cotton Seed Products, 41 Animals, Life Histories of Northern, Cranes, 13 45 Dairy Laboratory Guide, 37 Appliances, Mechanical, 13 Deinhardt-Schlomann Technical Dic- Arbitration Clause in Engineering and tionaries in Six Languages, 4 Building Contracts, 49 Designing, Theory and Practice of, Arc Lamps and Accessory Apparatus,6 24 Art of Illumination, 32 Dies, 14 Astronomy, Modern, 50 Diesel Engine for Land and Marine Basic Open-Hearth Steel Process, 15 Purpose, 8 Bedrock, 51 Direct and Alternating Currents, 31 Bells, Indicators, Telephones, etc., 6 Dust Destructors (Small), for Insti- Bleaching and Dyeing of Vegetable tutional and Trade Refuse, 25 Fibrous Materials, Chemistry of, 37 Dynamics of Mechanical Flight, 18 Biology, Outlines of Evolutionary, 46 Earth Slopes, Retaining Walls and Boiler Construction, 11 Dams, 23 Boiler Draught, 11 Elastic Arch, The, 23 Boiler Efficiencies, Engine Tests and. Electrical Machinery, Testing, 31 II Electric Central Station Distribution Boiler Explosions, Collapses, and Systems (American), 29 Mishaps, 10 Electric Cables and Networks Boilers, Marine Double-ended, 19 (Theory), 31 Boilers, Steam, 11 Electric Currents, Propagation of, in Book, The (History, etc.), 7 Telephone and Telegraph Conduc- Bridges, The Design of Simple Steel. tors, 27 21 Electric Furnaces, 32 Bridges, Reinforced Concrete, 22 Electric Installation Manuals, 6 Building in London, 48 Electric Lamps, 32 Building Materials, Introduction to Electric Mechanism, 28 the Chemistry and Physics of, 36 Electric Mining Installations, 6 Business, How to do, by Letter and Electric Power to Mines, etc., Appli- Advertising, 49 cation of, 28 Calculus, The, and its Applications, 38 Electric Power and Traction, 30 Cams, 14 Electric Power Transmission, 30 Cement, Concrete and, 23 Electric Railways, 25 Centrifugal Pumps, 15 Electric Railway Engineering, 29 Chemical Annual, Van Nostrand's, 36 Electrical Engineering (D.-S.), 4 Chemical Re-Agents, 36 Electrical Engineering, Heavy, 29 Chemical Theory and Calculations, 34 Electrical Engineer's Pocket Book, Chemistry and Physics of Building " Foster," 29 Materials, Introduction to the, 36 Electrical Measuring Instruments Chemistry of Bleaching and Dyeing, (Industrial), 30 etc., 37 Electrical Nature of Matter and Radio- Chemistry, Industrial, 33 Activity, 34 Chemistry of the Oil Industry, 39 Electricity (Hobart), 29 Chemistry of the Rubber Industry, Electricity and Matter, 34 38 Electricity through Gases (Discharge Coal, 13 of), 34 Coal Tar Dyes, Chemistry of, 37 Electro-Chemistry, Experimental, 31 Cold Storage, Heating and Venti- Electro-chemistry, Practical, 32 lating on Board Ship, 19 Electro-Metallurgy, 34 Colloidal and Crystalloidal State of Enamelling, 43 Matter, 38 Energy Diagram, fo!" Gas, 9 Compressed Air, 15 Engine Tests and Boiler Efficiencies, 1 1 Concrete and Cement, 23 Engineering Literature, Good, 50 52 INDEX OF riTl^ES—contiiiucd. Engineering Workshops, Machines and F'roce ses, 14 Engineers, The Law Affecting, 48 EngHsh-Spanish Dictionary, 49 Entropy, 32 Essays, Biographical & Chemical, 50 European Animals, 45 Extinct Animals, 44 Farm Animals, 47 Farming, First Book of, 47 Farm Management, 47 Field Engineer, 21 Fishesr A Guide to the Study of, 45 Flight, Dynamics of Mechanical, 18 Food Adulterants, Detection of, 37 Foundations and Fixing Machinery, 6 : Foundry Practice, General, 16 I From an Easy Chair. 45 | Frozen Meat Trade, History of the. 42 j Fuel, Gas and Water Analysis, etc., 12 Fuel. Introduction to the Study of, 12 Fuels, Liquid and Gaseous, 12 Furaaces, Electric, 32 Gas, Energy Diagram for, 9 Gas Engine, 9 Gas Engine Construction, 9 Gas Engine Design, 9 Gas Engineering Practice (American), 9 Gas, Gasoline, and Oil Engines, etc., 9 Gas (Town) and its Uses, 10 Gas Works Plant and Machinery, British Progress in, 10 Geographic Environment, Influences of, 46 Glasgow Text-Books of Civil Engi- neering, 2 Glass Manufacture, 40 Glass Processes, Decorative, 40 Glues and Gelatine. 42 Graphic Statics, Elements of, and General Graphic Method, 19 Grass Holding, To Work a, 47 Hardening,Tempering,etc., ot Steel, 17 Heat Engines, Vapours for, 35 Heat Power Plants, Economic and Commercial Theory of, 14 Hydraulics (D.-S. Series), 4 Hydraulics, Text-Book of, 20 Hydraulics and its Applications, 20 I Hydro-electric Developments and Engineering, 30 Identification of Organic Compounds, 37 Illumination, Art of, 32 India-Rubber and its Manufacture, 42 Industrial Accidents and Their Com- pensation, 48 Industrial Chemistry, 33 Industrial Chemistry, Outlines of, 3 Industrial Electrical T^Tca-nring Instruments. 30 Inori;anic Chemistry, Practical Methods of, 37 Insects, American, 45 Integration by Trigonometric and Imaginary Substitution, 38 Internal Combustion Engine (Wim peris). 8 Internal Combustion Engines (D.-S ),4 Internal Combustion Engines, Con- struction and Working of (Mathot), 8 Internal Combustion Engines, Desii.n and Construction (Giiklner). 8 Internal Combustion Engine, A Pri- mer of, 8 Internationa1Languai:eandScience.50 Ionization of Gases by Collision, The Theory of. 34 Iron and Steel (Hudson), 16 Iron and Steel (Stansbic), 17 Irrigation, 26 Irrigation Works, 26 Irrigation Works Notes on, 27 Irrigation Works Practical Design, 26 Laboratory and Factory Tests, 31 Law affecting Engineers, 48 Leather, Manufacture of, 41 Lettering, 44 Life in America, Distribution and Origin of, 43 Linseed Oil and other Seed Oils, 41 Liquid Air and the Liquefaction of Gases, 36 Liquid and Gaseous Fuels, 12 Liquid Fuel and Its Apparatus, 12 Liquid Fuel and its Combustion, 12 Locomotive, The Railway, 21 Machine Design, 13 Machinery, etc.. Elements of (D.-S ). 4 Machinery, Hoisting and Conveying (D.-S.). 4 Machine Tools (D.-S.), 4 Machine Tools, Modern American, 14 Malleable Cast Iron, 16 Marine Double-ended Boilers, 11 Marine Engine Design, 19 Materials, Handbook of Testing, 13 Materials, Mechanics of, 13 Mathematics for the Practical Man, 39 Mechanical Appliances, 13 Mechanical Movements, Powers, etc., 13 Mechanics of Materials, Elements of, 13 Mechanic's Handbook, 39 Metallurgy (D.-S.), 4 Microscopy, Principles of, 35 Mill and Factory Wiring, 6 53 INDEX OF TIT'LES— continued. Mineral and Aerated Waters, 40 Mines, Application of Electric Power to, 28 Modern Sanitary Engineering, 25 Motion Study, 24 Motor Car Engineering, Text-Book of, 17 Motor Pocket Book (O' Gorman's), 18 Motors, Secondary Batteries, Meas- uring Instruments, and Switch Gear, 6 Motors, Single-phase Commutator, see Electric Mechanism Motor Vehicles (D.-S.), 18 Municipal and Sanitary Engineering Encyclopaedia, 5 Municipal Engineering, British Progress in, 25 Natural History in Zoological Gardens, 44 Nature Student's Note-Book, 45 Naval Machinery, Care and Opera- tion of, 19 Nitro-Cellulose Industry, 38, 42 Oil Industry. Chemistry of, 39 Outlines of Industrial Chemistry, 3 Painting, Materials for Permanent, 42 Paints and Paint Vehicles, The Chemistry of, 36 Paper, Manufacture of, 40 Patent Laws, Foreign and Colonial. 48 Patents, Designs & Trade Marks, 48 Photography, 7 Photography (Telegraphic), 28 Physical Chemistry, Exercises in, 36 Physical Chemistry, Problems in, 33 Physics, A Text-Book of, 36 Plant Physiology and Ecology, 46 Potters' Craft, 43 Power, Natural Sources of, 15 Power Production, Refuse Disposal,25 Practical Mechanics Handbook, 39 Precious Metals, 17 Precious Stones, 43 Precision Grinding, 14 Pricing of Quantities, 24 Propagation of Electric Currents in Telephone and Telegraph Con- ductors, 27 Pumps and Pumping Engines, British Progress in, 15 Pumps, Centrifugal, 15 Quaternions, as the Result of Alge- braic Operations. 39 Radio-Active Transformations, 34 Radio- Activity and Geology, 34 Radio-Telegraphy, 27 Railway Construction and Operation (D.-S.), 4 Railway Locomotive, 21 Railway Rolling Stock (D.-S.), 4 Railway Shop Up-to-date, 21 Railway Signal Engineering (Mechani- cal), 21 Railway Structures, Reinforced Con- crete, 21 Railway Terms, Dictionary of, 49 Rainfall, Reservoirs and Water Supply, 26 Refuse Disposal and Power Produc- tion, 25 Reinforced Concrete (Marsh), 23 Reinforced Concrete Bridges, 22 Reinforced Concrete, Compression Member Diagram, 22 Reinforced Concrete, Concise Treatise on, 22 Reinforced Concrete, Manual of, 22 Reinforced Concrete,Properties,&c.,22 Reinforced Concrete in Sub- and Superstructure (D.-S.), 4 Reinforced Concrete Railway Struc- tures, 21 Rubber Industry, Chemistry of the, 38 Sanitary Engineering (Modern), 25 Science. Seven Follies of, 50 Scientific Management, 49 Searchlights : Their Theory, Con- struction and Application, 31 Seasonal Trades, 49 Sewage Discharge Diagrams, Water Pipe and, 20 Sewage Disposal Works, 25 Ship Wiring and Fitting, 6 Shop Kinks, 14 Simple Jewellery, 43 Sizing, Materials used in, 39 Small Dust Destructors, 25 Smoke Prevention, 12 Soils, 47 Soils and Manures, 47 Solenoids, Electromagnets, etc., 29 Stannaries, The, 17 Static Transformers, The Design of, 29 Statical Calculations, Reference Book for, 43 Steam Boilers, 11 Steam Boilers and Steam Engines (D.-S.), 4 Steam-Electric Power Plants, 30 Steam Engine, Modern, 10 Steam Pipes, 11 Steam (Flow of) through Nozzles, etc., II Steam Tables, The New, 11 Steam Turbines, 10 Steel, Basic Open-Hearth Process, 15 Steel Bridges, The Design of Simple, 21 Steel, Hardening, Tempering, Annealing and Forging, 17 54 INDEX OF riTl^KS -continued. Steel, Iron and (Hudson), i6 Steel, Iron and (Stansbie), 17 Steel, Welding and Cutting, etc., 17 Stone Age (The) in North America, 45 Structural Design, The Elements of, 23 Superheat, Superheating, etc., 11 Surveying, 24 Switches and Switchgear, 28 Tables, Quantities, Costs, Estimates, Wages, etc. (Prof. Smith), 44 Telegraphic Transmission of Photo- graphs, 28 Telegraphy (Wireless), MaxwelFs Theory and, 27 Telephones, Public Ownership of, 28 Testing, Handbook of (Materials), 13 Testing and Localising Faults, 6 Textiles and their Manufacture, 41 Theory and Practice of Designing, 24 Thermodynamics, Technical, 33 Thermodynamics, A Text-Book of, 33 Thermodynamics for Engineers, Applied, 32 Thermodynamics to Chemistry, Experimental and Theoretical Application of, 35 Timber, 24 Time and Clocks, 50 Toll Telephone Practice, 27 Trees, Indian, 46 Tunnel Practice (Modern), 23 Turbine Practice and Water Power Plants, 20 Turbines applied to Marine Propul- sion, 20 Turbines (Steam), 10 Una flow Steam Engine, 10 Vapours for Heat Engines, 35 Water Hammer in Hydraulic Pipe Lines, 20 Water Pipe and Sewage Discharge Diagrams, 20 Water Power Plants, Modem Turbine Practice, 20 Water Softening and Treatment, 13 Water Supply, Rainfall, Reservoirs and, 26 Weather, Forecasting, 49 Welding and Cutting Steel by Gases or Electricity, 17 Westminster Series, 7 Wireless Telegraphy, Maxwell's Theory and, 27 Wiring of Buildings, Internal, 31 Wood Pulp, 42 Wool Growing and the Tariff, 47 55 14 DAY USE RETURN TO DESK FROM WHICH BORROWED LOAN DEPT. This book is due on the last date stamped below, or on the date to which renewed. Renewed books are subject to immediate recall. REC'D LD APRli'64-3FJfl r LD 21A-40to-11,'63 (E1602slO)476B General Library University of California Berkeley ^B 15 1958 ^ ,v^^ .iP>Mr W STACKS APfl:>.:6^4B^ UNIVERSITY OF CALIFORNIA LIBRARY