# Micaiah John Muller Hill on Euclid

M J M Hill published The Contents of the Fifth and Sixth Books of Euclid (Cambridge University Press, 1900) and The Theory of Proportion (Constable and Co., London, 1914). We give below information about these books in the form of Prefaces and reviews. One of these reviews, by Julian Coolidge in the Bulletin of the American Mathematical Society, is quite critical and we give also a reply by Hill to that review, also printed in the Bulletin of the American Mathematical Society.

Click on a link below to go to the information about that book.

The Contents of the Fifth and Sixth Books of Euclid

The Theory of Proportion

1. The Contents of the Fifth and Sixth Books of Euclid, by M J M Hill.
1.1. Preface.

The object of this work is to remove the chief difficulties felt by those who desire to understand the Sixth Book of Euclid. It contains nothing beyond the capacity of those who have mastered the first four Books, and has been prepared for their use. It is the result of an experience of teaching the subject extending over nearly twenty years. The arrangement here adopted has been used by the Author in teaching for the past three years and has been more readily understood than the methods in ordinary use, which he had previously employed.

The Sixth Book depends to a very large extent on the Fifth, but this Fifth Book is so difficult that it is usually entirely omitted with the exception of the Fifth Definition, which is retained not for the purpose of proving all the properties of ratio required in the Sixth Book, but only for demonstrating two important propositions, viz., the 1st and 33rd.

The other properties of ratio required in the Sixth Book are usually assumed, or so-called algebraic demonstrations are supplied. The employment side by side of these two methods of dealing with ratio confuses the learner, because, not being equivalent, they do not constitute, when used in this way, a firm basis for the train of reasoning which he is attempting to follow. A better method is sometimes attempted. This is to insist on the mastering of the Fifth Book, expressed in modern form as in the Syllabus of the Association for the Improvement of Geometrical Teaching, before commencing the Sixth Book.

But it is far too difficult for all but the best pupils, and even they do not grasp the train of reasoning as a whole, though they readily admit the truth of the propositions singly as consequences of the fundamental definitions, which are

1. The fifth definition, which is the test for the sameness of two ratios.

2. The seventh definition, which is the test for distinguishing the greater of two unequal ratios from the smaller.

3. The tenth definition, which defines "Duplicate Ratio."

4. The definition marked A by Simson, which defines the process for compounding ratios.
In order to make things clear, it is necessary to explain what it is that makes Euclid's Fifth Book so very difficult.

There is first the difficulty arising out of Euclid's notation for magnitudes and numbers. This has been entirely removed in most modern editions by using an algebraic notation and need not therefore be further considered.

There is next the difficulty arising out of Euclid's use of the word "ratio," and the idea represented by it.

His definition of ratio furnishes no satisfactory answer to the question, "What is a ratio?" and it is of such a nature that no indication is afforded of the answer to the still more important question, "How is a ratio to be measured?" As Euclid makes no use of the definition in his argument, it is useless to examine it further, but it is worth while to try to get at his view of ratio. He asserts indirectly that a ratio is a magnitude, because in the seventh definition he states the conditions which must be satisfied in order that one ratio may be greater than another. Now the word "greater" can only be applied to a magnitude. Hence Euclid must have considered a ratio to be a magnitude. To this conclusion it may be objected that if Euclid thought that a ratio was a magnitude he would not so constantly have spoken of the sameness of two ratios, but of their equality. One can only surmise that, whenever it was possible, he desired to leave open all questions as to the nature of ratio, and to present all his propositions as logical deductions from his fundamental definitions. Yet the question as to the nature of ratio is one which forces itself on the careful reader, and is a source of the greatest perplexity, culminating when he reaches the 11th and 13th Propositions.

The 11th Proposition may be stated thus:-

If $A : B$ is the same as $C : D$, and if $C : D$ is the same as $E : F$, then $A : B$ is the same as $E : F$.

Now if a ratio is a magnitude, this only expresses that if $X = Y$; and if $Y = Z$, then $X = Z$.

As this result follows from Euclid's First Axiom it is difficult to see the need for a proof.

This only becomes apparent when the reader realises that Euclid's procedure may be described thus:-

Let $A, B, C, D$ be four magnitudes satisfying the conditions of the Fifth Definition, and let $C, D, E, F$ be four magnitudes also satisfying the same Conditions, then it is proved that $A, B, E, F$ also satisfy the conditions of that definition.

Remarks of a somewhat similar nature apply to the 13th Proposition.

In this book it is shown that two commensurable magnitudes determine a real number; and this real number is called the measure of their ratio. The proof of the proposition that two incommensurable magnitudes of the same kind determine a real number (which is taken as the measure of their ratio) is too difficult to find a place in an elementary text-book like this.

A still greater difficulty than the preceding arises from the fact that Euclid furnishes no explanation of the steps by which he reached his fundamental definitions.

To write down a definition, and then draw conclusions from it, is a process which is useful in Advanced Mathematics; but it is wholly unsuitable for elementary teaching. It seems not unlikely that Euclid reached his fundamental definitions as conclusions to elaborate trains of reasoning, but that finding great difficulty in expressing this reasoning in words owing to the absence of an algebraic notation, he preferred to write down his definitions as the basis of his argument, and to present the propositions as logical deductions from his definitions.

Apparently he has left no trace of the steps by which he reached his fundamental definitions; and one of the chief objects of this book is to reconstruct a path which can be followed by beginners from ideas of a simpler order to those on which his work is based.

The most vital of his definitions is the Fifth, on reaching which the beginner, who has read the first four books of Euclid, experiences a sense of discontinuity. He knows nothing which can lead him directly to it, he has no ideas of a simpler order with which to connect it; and he is therefore reduced to learning it by rote. His teacher may show him that it contains the definition of Proportion given in treatises on Algebra; but even with this assistance it remains difficult for him to remember its details. He may find frequently does learn to apply it correctly in demonstrating the 1st and 33rd Propositions of the Sixth Book, but the Author's experience both of teaching and examining leads him to the belief that it is not really understood.

The explanation here given of the Fifth Definition, apart from the actual notation employed, is that given by De Morgan in his treatise on the Connexion of Number and Magnitude published in the year 1836, and is made clear by a device for exhibiting the order of succession of the multiples of two magnitudes of the same kind, when arranged together in a single series in ascending order of magnitude. This device is called the relative multiple scale of the two magnitudes. The notation employed to exhibit it is substantially due to Professor A E H Love, F.R.S. This notation attaches a graphical representation to the Fifth Definition, which appeals to the eye of the learner.

The seventh definition, as will presently be shown, is not required.

The tenth definition, which defines Duplicate Ratio, is here based on that marked A by Simson.

Definition A, which defines the process for Compounding Ratios, is fully explained in four stages, commencing with the general idea on which the process is based, and ending with the proof of the fact that the process employed will always lead to consistent results.

There remains but one great difficulty for consideration. This is the indirectness of Euclid's line of argument, arising from the fact that he uses the Seventh Definition where the Fifth alone need be employed. His Fifth Definition states the conditions which must be satisfied in order that two ratios may be the same (or if ratios are magnitudes, that they may be equal).

If this definition is a good and sound one, it is evident that it ought to be possible to deduce from it all the properties of equal ratios. This is in fact the case. It is wholly unnecessary to employ the Seventh Definition, which refers to unequal ratios, to prove any of the properties of equal ratios. Its use only renders the proofs of the propositions indirect and artificial and consequently difficult. Not only does no inconvenience result from avoiding its use, but it is possible to get rid of the latter part of the 8th Proposition, and of the whole of the 10th and 13th Propositions, which deal with unequal ratios, and of the 14th, 20th and 21st Propositions of the Fifth Book, which arc particular cases of the 16th, 22nd and 23rd Propositions respectively.

The remaining Propositions are demonstrated by means of the Fifth Definition alone; and all, with a single exception, fall under one or other of two well recognised types.

These types correspond to the two forms of the conditions for the sameness of two relative multiple scales (or two ratios).

The first form of the conditions is Euclid's Test for Equal Ratios as stated in the Fifth Definition of the Fifth Book. It is the one which springs most naturally out of the nature of the subject. It contains three classes of alternatives, one of which ,appears only when the magnitudes of the ratios are commensurable, Sometimes it is possible to examine all three classes of alternatives in the same way. On the other hand, in the extremely important Propositions Euclid V. 16, 22, 23, the examination of the cases in which the ratios are commensurable has to be conducted upon different lines to those which are applicable when they are not commensurable. This it is quite possible to do, but the line of argument is artificial and therefore difficult for a beginner, as will be seen by consulting Notes 6, 9, and 11 at the end of the book. The proofs of Props. 16, 22, 23 as completed by these Notes depend on the use of Prop. 62 (Euclid V. 4), but the way in which that proposition has to be used does not suggest itself naturally.

It is on this account that the second form of the conditions for the sameness of two relative multiple scales, has been introduced into this book. So far as the Author knows, it was first published by Stole. It contains four classes of alternatives, but it has this very great advantage over Euclid's form that the examination of all the four classes of alternatives can always be conducted upon the same lines.

Reference has been made above to one Proposition which does not fall under either of the above recognised types. This is Proposition 61 (Euclid V. 24). The proof here given is Euclid's. It is very much shorter than any direct deduction of it from either form of the conditions for the sameness of two scales. At the same time its artificial character stands out in striking contrast to the directness of the proofs of the other propositions.

The plan followed in this work is to explain at an early stage what the relative multiple scale of two magnitudes is, then to prove a few of the simpler properties of relative multiple scales, then to point out that these propositions accord with the ideas of ratio, formed by learners long before they commenced to read Geometry, in so far as those ideas have assumed definite shape. In this way the mind of the reader is led to the idea that two magnitudes of the same kind determine not only a relative multiple scale, but also a ratio; so that he sees why, whenever two relative multiple scales are the same, Euclid expressed that fact in the statement that the ratio of the magnitudes determining the first relative multiple scale is the same as the ratio of the magnitudes determining the second relative multiple scale.

In fact all that Euclid proves in regard to the sameness of two ratios may be conveniently expressed as the proof of the sameness of two relative multiple scales, and the advantage of proceeding in this way is this:-

The argument is made to relate to a thing which is completely defined, viz, the relative multiple scale of two magnitudes; whilst in Euclid's argument it is not made clear what a ratio is; and the lack of information on this point is a serious obstacle to the learner.

The determination of the stage at which the idea of ratio should be introduced into the argument is one of great difficulty. Complex as the idea is, it is formed by every one at an early age. As soon as a child can recognise an object from a drawing of it, he has formed the idea of similar figures, and therefore he is able to see that the ratio of two of the dimensions of the object is the same as that of the corresponding dimensions of the drawing. When however the use of the idea, and its introduction into Algebra and Geometry are under consideration, its complexity becomes apparent. There is no necessity, arising out of the nature of the subject, for introducing the idea of ratio into the statement of any of the Propositions in the Fifth Book, with the possible exception of the 8th, 10th and 13th, which deal with unequal ratios and are not required for the Sixth Book.

It is not until the subject of Compounding Ratios is reached that the introduction of the idea becomes desirable. I believe that it will be ultimately recognised that it is best to postpone its introduction until the stage just mentioned has been reached. At the same time I have not ventured to do this in this book, as I desire to conform to established practice, so far as is possible, consistently with clearness of treatment.

Several alterations have been made in the order of the Propositions, De Morgan pointed out that Learners found great difficulty in reading the Fifth Book on account of the abstract character of the reasoning, its application to something concrete not being easily perceived, Accordingly in this work Propositions from the Sixth Book are taken as soon as a sufficient number of Propositions from the Fifth Book have been proved to make it possible to deal logically with those in the Sixth Book.

Further alterations made in the order of the Propositions are due to the desire to indicate at an early stage of the work a line of argument which may be followed in order to reach the idea of ratio.

The principal alteration in the proofs is of course the use of the Theory of Relative Multiple Scales in the Propositions of the Fifth Book, in the 1st and 33rd Propositions of the Sixth Book, and in the proof of the first part of the 2nd Proposition of the Sixth Book, where it has the advantage not only of proving all that Euclid does, but also of giving several other propositions which (if required) must be deduced from Euclid's result by the use of other Propositions in the Fifth Book.

With regard to the enunciations no attempt has been made to adhere to Euclid's words. All those propositions which may be viewed either as expressing properties of equal ratios or of the sameness of certain relative multiple scales are enunciated from both these points of view. In the part of the book which precedes the section on ratio, these propositions are enunciated first as properties of relative multiple scales, and secondly as properties of equal ratios for the sake of reference only, inasmuch as the term "ratio" has not yet been explained. After the section on ratio the order of the two modes of enunciation is inverted, as possibly more convenient for reference.

This work contains demonstrations of all the Propositions of the Fifth Book except Nos. 8, 10, 13, which depend on the Seventh Definition, and which are not used in the Sixth Book; of all the Propositions in the Sixth Book together with those marked A, B, C in Simson's Euclid and beside these the following:-

The Proposition here numbered 7 contains the earlier part of Euclid v, 8. and is an extremely useful proposition.

Prop. 8 shows the equivalence or the two forms of the conditions to be satisfied in order that two relative multiple scales (or two ratios) ma y be the same.

Prop. 11 shows that two commensurable magnitudes have the some relative multiple scale as two whole numbers.

Prop. 12 relates to differing relative multiple scales; and with Props. 9 and 11 is very useful in developing the idea of ratio.

Props. 24, 59, 60 and 63 appear here chiefly but not entirely on account of their bearing on the theory of the point at infinity on a straight line.

Props. 36 and 37 are connected with the Theory of Duplicate Ratio.

Prop. 39 shows that the rectangle contained by the diagonals of a quadrilateral cannot exceed the sum of the rectangles contained by the opposite aides. It includes the Proposition marked D in Simson's edition of the Sixth Book.

There are given in suitable positions in the book, the definitions of Harmonic Points and Lines, of the Pole and Polar, of Inversion of the Radical Axis and the Centres of Similitude of Two Circles, and (so far as is possible without explaining the use of the Negative Sign in Geometry) of Cross or Anharmonic Ratio, with the sole object of rendering intelligible the terminology employed in a number of interesting examples in the book.

The Author believes that he has not taken without acknowledgment from other textbooks anything which is not common property.

His special thanks are due to the Cambridge University Press Syndicate, who have made the publication of the book possible; and to his friend and former pupil, Mr L N G Filon, M.A., for valuable suggestions and assistance whilst the book was passing through the press.

He will be grateful to his readers for suggestions end corrections.

1.2. Review by: Julian Lowell Coolidge.
Bull. Amer. Math. Soc. 8 (1902), 218-220.

The fact that Euclid no longer presents a living issue in American education makes it difficult to pass a fair judgment on Dr Hill's book. We are inclined to look upon attempts to "temper the wind to the shorn lamb" by elucidating particular spots in the "Elements" as a waste of time and energy. But we have no right to start from such an assumption in considering the present work. The simple fact is that Euclid has been, and will long continue to be, the foundation of geometrical knowledge in that nation which has produced Newton and Cayley and Sylvester. Parts of Euclid are undoubtedly too difficult for beginners, and the book before us attempts to remove the greatest of these difficulties, the theory of proportion. In American books we seek to reach this end by an appeal to the analogy of algebra, but herein we depart entirely from Euclid's pattern. Professor Hill adopts a different, and more euclidean device: the relative multiple scale.

Let $A$ and $B$ be two magnitudes which are supposed to be capable of comparison as well as their multiples. The order of succession of the various multiples of $A$ and $B$ is called their relative multiple scale, and may be represented symbolically by $rA ≥ ≤ sB$ where $r$ and $s$ are positive integers. The relative multiple scales of $A, B$ and $C, D$ are equal if, for every positive integral value of $r$ and $s, rA = sB$ is a sufficient condition for $rC ≥ ≤ sD$. A simple geometrical diagram is introduced to exhibit the three possible relations and this diagram is continually presented to give the learner a clear idea of what he is doing. Two sets of magnitudes having equal multiple scales are said to be proportional, and the transformations which leave the proportional relation unaltered constitute a large part of the book. With perfect propriety, the author lays great stress upon the axiom of Archimedes, and there is throughout a commendable consistency in the method and spirit of the work.

Such is the general scheme and, granted the author's aim, and point of view, it is about the best that could be adopted. In some of the details, the book is almost incredibly careless.

The first proposition (page 2) is a proof of the commutative and associative laws in the multiplication of positive integers, but the proofs are based upon the assumption that these laws hold in the case of addition. On page 6 is the proof that if $a$ and $b$ are positive integers, while $a > b$, and $R$ is a magnitude $aR > bR$. It is tacitly assumed that $bR + cR > bR$, or that $R$ obeys axiom 9, "The whole is greater than its part."

A multiple scale is defined (page 12) as a set of magnitudes, while a relative multiple scale is a portion of a diagram (page 14). On page 29 is the corollary "If $ABC$ be a triangle, and if the sides $AB, AC$ be cut by any straight line parallel to $BC$, then the sides $AB, AC$ are divided proportionally." This is given before the word "proportionally" is defined.

Section 3, pages 33-36, is entitled "A Chapter on Ratio," and from it we glean the following facts: The reader is supposed to have an innate idea of relative magnitude. "The ratio of one magnitude to another (which must be of the same kind as the first) is the relative magnitude of the first compared with the second." Having now a clear and precise idea of what a ratio is, from this soul-satisfying definition, we are prepared for the next step, which is this: If we accept the axiom that a number $p$ may always be found of such a sort that the relative multiple scale of p and 1 is equal to that of the two comparable magnitudes $A$ and $B$, then $p$ may be taken as the measure of the ratio of any pair of magnitudes having the same scale as $A$ and $B$. It is interesting to notice that the author himself seems to have a certain diffidence about these definitions, for he does not use them in any of his subsequent work, but sticks to the safe path of the relative multiple scale. Apparently their only use in the whole chapter is that it is convenient to use the words "ratio" and "proportion" while Euclid's definition of the latter is cumbersome.

As a last error in detail, we notice the following definition on page 73:

"If four straight lines (presumably in the same plane) be cut by any transverse in four harmonic points, they are called four harmonic lines, or said to form a harmonic pencil." This may be interpreted in two ways. If the author means that they either form four harmonic lines or a harmonic pencil, the former seems a perfectly useless definition of four harmonic tangents to a conic; if, on the other hand, he looks upon the two forms of words as synonymous, and this seems the more natural interpretation, he is departing from the usual English practice of defining a pencil as composed of coplanar and concurrent lines.

To sum up: the book seems to us well planned but carelessly executed. It has, of course, no message for the average American teacher, and whoever uses it must do so with circumspection. With this caution it may prove to be of some value to teachers who do not care to depart entirely from Euclid's theory of proportion, yet find the ordinary presentation beyond the grasp of their pupils.

JULIAN LOWELL COOLIDGE.

HARVARD UNIVERSITY.

1.3. M J M Hill, Reply to Mr J L Coolidge's review of Hill's Euclid.
Bull. Amer. Math. Soc. 8 (1902), 479-481.

I desire to thank the editors of the Bulletin for their courtesy in acceding to my request that they should insert a reply to the review of my edition of the fifth and sixth books of Euclid's Elements by Mr Coolidge, published in the February number of the Bulletin, as it contains statements which give an erroneous impression of the contents of the book.

The book differs from previous editions in two important particulars. These are:

1. The explanations of the fundamental definitions of the fifth book of Euclid.

2. The removal of the indirectness from Euclid's line of argument.

The second of these matters, though emphasised by italics on page viii of the preface, has been passed over without notice by the reviewer. The discovery of this indirectness and the possibility of removing it, were published by me in the Cambridge Philosophical Transactions, volume 16, part 4; and the importance of the work was recognised in the review of that paper in the Jahrbuch über die Fortschritte der Mathematik, volume 28 (1897), page 152.

The first of the above-mentioned particulars is partially noticed by the reviewer in his account of the relative multiple scale. But he has penetrated so little into the spirit of the book that he describes relative multiple scales as being "equal" when certain conditions are satisfied. As relative multiple scales are not magnitudes the term "equal" is inappropriate. Under the conditions referred to, the relative multiple scales should be described as "the same." The reviewer also says: "The order of succession of the various multiples of A and B is called their relative multiple scale." It is not the "order," but "the device for showing the order."

The objections which the reviewer appears to feel most strongly, relate to the chapter on ratio, although the object of this chapter is only to make it possible to use the words "ratio" and "proportion." It is difficult to understand why the reviewer should describe my translation of Euclid's definition of ratio as a "soul-satisfying definition," and state that "a clear and precise idea of what a ratio is" is derivable therefrom, when the definition is preceded in the text by the statement that "It is only an endeavour to express in English the idea contained in the definition of ratio as stated in Euclid's Greek," and that "no use will be made of the definition in the argument."

The next statement of the reviewer must be quoted in full. "If we accept the axiom that a number p may always be found of such a sort that the relative multiple scale of p and 1 is equal to that of the two comparable magnitudes $A$ and $B$, then $p$ may be taken as the measure of the ratio of any pair of magnitudes having the same scale as $A$ and $B$." This is a misconception. The above statement is not an axiom, but a particular case of the fundamental proposition in the theory of relative multiple scales, which is proved in the paper referred to above. Euclid's determination of the fourth proportional to three given straight lines is a particular case of the same fundamental proposition which suffices for those who do not desire to go beyond the sixth book of Euclid. The proof of the above mentioned fundamental proposition is too difficult for those for whom this book is meant, and it has therefore been omitted.

The reviewer's objections to the omission of a reference to the axiom that "the whole is greater than its part," and to the omission of the statement of an assumption in the proofs given of the commutative and associative laws in the multiplication of positive integers, are beside the point in regard to an elementary textbook like this. To call the attention of beginners to such matters generally results in preventing them from learning those parts of the subject which it is desirable that they should master at this stage.

There are two other criticisms on points of detail. The word "proportionally" is used on page 29 before the definition of proportion has been given. In this case a footnote, identical with that on each of the two following pages, was unfortunately omitted.

The second criticism relates to the accidental omission of the single but important word "concurrent" in the definition of four harmonic lines. This omission I greatly regret.

I have now dealt with the whole of the criticisms, and leave it to those who may be interested to determine whether they justify the charge that "In some of the details the book is almost incredibly careless."

There is one matter left which is of interest to teachers:

The reviewer says: "Parts of Euclid are undoubtedly too difficult for beginners, and the book before us attempts to remove the greatest of these difficulties, the theory of proportion. In American books we seek to reach this end by an appeal to the analogy of algebra, but herein we depart entirely from Euclid's pattern." I have not been able to determine the exact significance of the words "an appeal to the analogy of algebra," but I believe that those who will take up the fifth book of Euclid and examine how readily the ideas of the irrational number as developed by Dedekind can be used in connection with its results, will find that in rigour it far surpasses the modern attempts to turn the difficulties which Euclid faced and overcame.

UNIVERSITY COLLEGE, LONDON,
May 31, 1902
2. The Theory of Proportion, by M J M Hill.
2.1. 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 accustomed, in accordance with the then established 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 tong 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 between unequal ratios. They are intimately related, but when once stated they can be treated as independent.
Now it can be seen at once that if the test for deciding when two ratios are equal is a good and sound one, it should be possible to deduce from it all the properties of equal ratios, and in order to obtain these properties it should not be necessary to employ the test for distinguishing between 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 be proved by 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 Definition is directly or indirectly involved to his proofs of those propositions in which he employs the Fifth Definition only.

(2) I think it will appear to anyone who reads this book that it is in a high degree probable that the two assumptions

(i) If $A = B$, then $(A : C) = (B : C)$, and (ii) 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 rearranged 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 Number 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 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 explanation 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 to the proof of the proposition that the areas of circles are proportional to the squares on their radii (Euclid XII. 2); 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. These proofs differ from Euclid's in a most important particular, viz. they do not assume the existence of the fourth proportional to three magnitudes of which the first and second are of the same kind. I think that anyone who has tried to understand Euclid's argument will find the proofs here given much simpler and more direct. Euclid uses a reductio ad absurdum. 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 incommensurable, should be included in the mathematical curriculum, I have immense prejudices to overcome.

On the one hand it is the outcome of all experience in teaching 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 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 Euclid'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 usefulness 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 inspiration which led him to the theory of the "cut" or "section" in the system of rational numbers, a theory which is fundamental 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, Chapters I-IX, contains an elementary course, which can be explained to anyone 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 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 XL is suitable for students 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 Cambridge University Press for their courtesy in permitting me to use the methods employed in the two editions of my Contents of the Fifth and Sixth Book 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 edition 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.

2.2. Review by: J.
Isis 3 (2) (1920), 307.

Hill maintains that a treatment of the theory of proportion, which is valid when the magnitudes concerned are incommensurable, should be included in the mathematical curriculum. He has arrived at the conclusion that, in addition to the difficulties arising out of Euclid's notation and out of the fact that Euclid did not sufficiently define ratio, two reasons could be assigned for the great difficulty, of this argument: (1) Of the many definitions prefixed to the fifth Book, the only ones which effectively count are the fifth, the test for deciding when two ratios are equal, and the seventh, the test for distinguishing between unequal ratios. Euclid has the logically unnecessary practice of deducing some of the properties of equal ratios from the seventh definition. Hill merely uses the fifth for this purpose. (2) It seems probable to Hill that the two assumptions: if $A ≥ B$, then $(A : C) ≥ (B : C)$ were fundamental with Euclid in the sense that he first deduced the fifth and seventh definitions from them and only afterwards reversed the process. The appearance of the above definitions at the beginning of Euclid's argument and without explanation presents grave difficulties to the student, which are avoided in this work. This work is a modification of Euclid's method, which requires for its understanding a knowledge of elementary algebra. The first three chapters are devoted to an indication of what "magnitudes of the same kind" are (in essentials after Stolz), some propositions on their integral multiples, and the definition of the "ratio" of two such multiples as a rational number. The (Chap. IV-V-XI) concept of ratio and number are extended so that irrational numbers are introduced, and Chap. VI-IX contain the theory of ratio of commensurable and incommensurable magnitudes based on the above indications; the test for equal ratios is that no rational number lies between them. Applications of Stolz's theorem simplifying this test are given in Chap. X; and Chap. XII is a commentary on Euclid's fifth Book.

2.3. Review by: G B Mathews.
The Mathematical Gazette 8 (117) (1915), 87-89.

This treatise will be read with great interest and profit, because the author has not only taught the subject for a number of years, but has devoted a great deal of time and labour to the task of putting the theory in a form at once strict and intelligible. In this he has attained a marked success, and it is unlikely that any great improvement on it can be made, to the ordinary pupil.

Briefly, the procedure is as follows. First, we have a discussion (mainly after Stolz) of a set of magnitudes of the same kind, their multiples, and the ratio of two such quantities, when they are commensurable. This ratio is defined as a rational number; namely,

if $nA = mB$, we have $A : B = m/n$.

Next a proof is given of the fact that there are geometrical quantities which are not commensurable; and this is followed by a very clear account of Dedekind's theory of irrational numbers. Then comes the definition of equal ratios, including a proof of Stolz's theorem; then three chapters giving the main propositions of the theory; then a chapter called "Applications of Stolz's theorem," which is practically an account of the method of exhaustion, with applications to circles, tetrahedra, etc.; and finally, a commentary on Euclid's fifth book.

There are a great many interesting points on which one is inclined to make remarks; only a few of them can be considered here. First of all, there is the psychological question as to the process by which mankind, and particularly the Greeks, arrived at a complete theory of ratio and proportion. In the absence of sufficient evidence, we can only arrive at vague and imperfect conclusions, but some inferences seem fairly certain. After the notion of equality, that of one quantity being double or half of a like quantity was acquired; and apparently this was, for a long time, the only consciously realised application of ratio. Of course, in commerce and the arts, multiples of quantities were constantly used, but they were determined by counting, and such evidence as there is seems to show that acquaintance with rational fractions as ratios, or more accurately as "parts", even in the case of aliquot parts, was slowly acquired. This is illustrated by such awkward terms as sesquiunx and sesquipedalis in Latin (sesqui, alone, is only once found).

Sometime or other it must have been realised that if the price of a cubit of linen from a particular roll was (say) a drachm, then there was a certain relation between 3 cubits and 4 cubits of linen similar to that of the 3 drachms and 4 drachms which would have to be paid for them. In this way, a sort of notion of the proportion of two couples of like commensurable quantities would arise; and it is not improbable that this preceded the geometrical cases where all the four quantities are of the same kind, even when practical use was made of the properties of similar figures. Eventually, it would be seen that there are innumerable cases where we have simultaneously

$nA = mB$ and $nA' = mB'$.

Whether $A'$ is of the same kind as $A$ or not, we can express this relation by saying that $A, B, A', B'$ (in that order) "form an analogy" or "are in proportion"; and on this definition we can construct a theory of proportion without any discussion of ratio, so long as we confine ourselves to commensurable quantities. How did the term "ratio" arise? Probably by noting cases where $nA = mB$, but $nA' < nB$. Here it would be inferred that $nA > nA'$, and hence that $A > A'$. This being so, it might be said that since $A > A'$ absolutely, $A > A'$ in relation to $B$; and by a change of expression, we might say "the ratio of $A$ to $B$ is greater than the ratio of $A'$ to $B$." This practically agrees with Prof Hill's conclusion. In a similar way, from

$nA > nB$ but $nA' < mB$,

we have $A > A'$, and the same conclusion about ratios as before. To make a strict theory out of this, we must show that we cannot have simultaneously such cases as

$mA > nB, mA'< nB', pA < qB, pA' < qB'$.

It is just possible that Euclid's arithmetical theory of proportion arose from this; for the inequality $mq > np$, which follows from $mA > nB, pA < qB$, leads, if $e$ is any unit, to $me/n > pe/q$, and this can be brought into connexion with Euclid's seventh book. However that may be, as soon as it was discovered that the length of the diagonal of a square is incommensurable with its side - and this was done when the Greeks were fully conscious that a square with one diagonal drawn is in some comprehensive sense "similar" to any other square with one diagonal drawn, and when it could be verified experimentally that a right-angled triangle with sides $3e, 4e, 5e$ is similar to one with sides $3e', 4e', 5e'$ whatever the units $e, e'$ may be, and that in this case the ratios of corresponding sides are equal - it would be almost inevitable to suppose that the diagonal and side of any one square and the diagonal and side of any other square are in proportion, although the terms are incommensurable. The question now was to frame a definition of proportion which would apply both to the commensurable and to the incommensurable case; and this led, in a way which seems impossible to trace, to the invention of the relative scales of equimultiples, of which De Morgan's case of the trees and the railings is a classical illustration.

Here I venture to say something about Stolz's theorem. I must say that its importance, or at least its novelty, seems to me to be exaggerated. Euclid was quite aware that if $A, B$ are incommensurable, the case $mB = nA$ cannot arise, so that the relative scale of (A, B) cannot give any coincidences; hence, in this case, for a proportion $A : B :: A' : B'$ we must establish the correspondences $mA > < nB, mA' > < nB'$ for all values of $m, n$ and such relations as $mA = nB, mA' = nB'$ will not occur. But he meant to frame his definition of equality of ratios in such a way as to cover all cases, and therefore $mA =nB$ is included as a possible case, though it need not, and, in general, will not occur.

In this connexion I wish to say that I entirely disagree with Max Simon's assertion that "they (the Greeks) possessed a notion of number in all its generality." On the contrary, whatever the Greeks of Euclid's time meant by the ratio $A : B$, when $nA = mB$, I am very doubtful whether they ever thought of it as equivalent to the number $\large\frac{m}{n}\normalsize$. And I am convinced that they never thought of the ratio of the diagonal of a square to its sides as the number $√2$; otherwise a great deal of Euclid's Elements would be superfluous; and superfluity is a defect to (or from) which the Greek mind is constitutionally averse. It is true, of course, that, on the basis of what the Greeks did know about ratios, it is possible to construct (with the Cantor-Dedekind axiom) a strict theory of number applied to geometrical quantities; but this is quite a different thing. A man may have all the materials for building a cathedral: it does not follow that he will build it; still less that his building will be perfect. Prof Hill's reference to that most original mathematician, Michael Stifel, should be noted.

As to Euclid's definitions, Book V. The third is absolutely silly, as it stands, for the only meaning I can make out of it is "the ratio of two homogeneous magnitudes is their relation with respect to quantuplicity, whatever that relation may be." Definition 4 restricts definition 3 (such as it is) by saying that the relations of quantuplicity must be such that the relations $mA > B$ and $nB > A$ must be satisfied by some integral values of $m, n$. (One, of course, may be taken to be 1, on Euclid's assumptions; i.e. either $A > B$, or $B > A$.)

What Euclid does not see is that we can define the symbols >, =, <, with regard to ratios without defining the term "ratio" itself; just as we can define the equality of two segments $AB, CD$ without defining the term segment. In our definition we are treating "multiple" and "equimultiple" as more primitive terms than "ratio."

A former pupil of mine, not specially brilliant, but named Newton, pointed out to me that the theory of proportion might be based, not on relative scales of equimultiples, but on relative scales of submultiples; that is to say,

$A : B = C : D$ if for all integers $m, n$

$\large\frac{1}{m}\normalsize A ≤ ≥ \large\frac{1}{n}\normalsize B$ according as $\large\frac{1}{m}\normalsize C ≤ ≥ \large\frac{1}{n}\normalsize D$.

From the Greek point of view, this theory is hardly so satisfactory as the classical one: because, although everybody admitted that any quantity could be multiplied to any extent, it was a question whether every quantity could be divided indefinitely into a number of equal parts.

In conclusion it may be remarked that the successive primes $p_{1}, p_{2}, p_{3}, ...$, arranged in order of magnitude, form a system of similar magnitudes, but do not satisfy the laws of a set of "like" magnitudes because we cannot satisfy

$p_{i} + p_{j} = p_{k}$

for all values of $i, j.$

G B MATHEWS.

2.4. Review by: Philip E B Jourdain.
Science Progress in the Twentieth Century (1906-1916) 10 (39) (1916), 499-500.

Almost all teachers agree that Euclid's treatment of proportion in the fifth book of his Elements is so difficult, both in form and matter, that it is quite unsuitable for purposes of teaching. Prof Hill completely assents to this, but maintains that a treatment of the theory of proportion, which is valid when the magnitudes concerned are incommensurable, should be included in the mathematical curriculum. It must be remembered that Prof Hill has a very large experience of teaching and is also author of an admirable Contents of the Fifth and Sixth Books of Euclid's Elements and three important papers on the Fifth Book in the Cambridge Philosophical Transactions. The third of these papers is noticed in this number in a "Recent Advances in Science: Mathematics," but the other two were published some years ago. Prof Hill arrived at the conclusion that, in addition to the difficulties arising out of Euclid's notation and out of the fact that Euclid did not sufficiently define ratio, two reasons could be assigned for the great difficulty of his argument. In the first place, the only two of the many definitions in the Fifth Book which effectively count are the test for deciding when two ratios are equal (fifth definition), and the test for distinguishing between unequal ratios (seventh definition). Further, Euclid takes the unnecessary course of deducing some of the properties of equal ratios from the seventh definition. In this book Prof Hill only uses the fifth for this purpose. In the second place, Prof Hill thinks that it is very probable that the two assumptions: If $A r B$ , then $(A : C) r (B : C)$, where the relation $r$ may be either = or > , form the real bed-rock of Euclid's ideas, and that he first of all deduced his fifth and seventh definitions from these two fundamental assumptions. In any case, the appearance of the above definitions at the beginning of Euclid's argument and without explanation presents grave difficulties to the student, which are avoided in this work. This work is modification of Euclid's method, which requires for its understanding a knowledge of elementary algebra.

Prof Hill remarks that Chapters I -IX contain an elementary course and Chapters X and XI contain an advanced course. However, for those who are more interested in what the book contains than in what parts of it are suitable to be taught to whom, the following summary may be useful. The first three chapters introduce "magnitudes of the same kind" as undefined entities with which examples make us more or less familiar and of which characteristics are given in essentials after Stolz, to whose Allgemeine Arithmetik Prof Hill's work is greatly indebted. Then some propositions on their positive integral multiples are given; and the "ratio" of two such multiples of the same magnitude is defined as an integer or fraction. Chapter IV: By a proof probably due to Pythagoras himself, it is shown that there are magnitudes of the same kind (the side and diagonal of a square) which are not multiples of the same magnitude. In such cases, can the magnitudes have a ratio to one another?: "if so there must be numbers which are not rational". Prof Hill is evidently more interested in the process of discovery of mathematical objects than in their logical nature. Logically, Prof Hill's method is like first defining "foreigners" as Frenchmen and then asking if there are any non-French foreigners. But we all know how easily one can make fun of the "extension of the idea of number," which is described much as usual in Chapter V, and what important heuristic methods badly expressed have been used in this "extension." We can all, with a little sympathy, appreciate the value of the vague question: "Does anything exist which is not a rational number, which is nevertheless entitled to be ranked as a number?". And most of us will agree that "an argument which does not follow the course of discovery is frequently very difficult to follow."

Chapter VI: Construction of a new theory of the ratio of commensurable or incommensurable magnitudes which satisfies the conditions: If

$A r B$, then $(A : C) r (B : C)$,

where the relation $r$ is either >, <, or =. "Two ratios are said to be equal when no rational number lies between them; and the test for equal ratios is that, if

$(A : B) r \large\frac{p}{q}\normalsize$ then $(C : D) r \large\frac{p}{q}\normalsize$

where $r$ has the above meaning and $p$ and $q$ are unrestrictedly variable integers. "Stolz's theorem" is that the condition obtained when = is put for $r$ is superfluous; and applications of this theorem of exhaustion are dealt with in Chapter X. Chapters VII -IX give properties of equal ratios; Chapter XI contains further remarks on irrational numbers, and includes a statement of the Cantor-Dedekind axiom; and Chapter XII is a commentary on the Fifth Book of Euclid.

On p. x it is said that Dedekind acknowledged that he drew his inspiration especially from Euclid's fifth definition (of the equality of ratios) of his Fifth Book. This does not seem to be the case. In the passage referred to by Prof Hill (Beman's translation of Dedekind's Essays, Open Court Co., p. 40), Dedekind merely remarks that the conviction that an irrational number is defined by the specification of all rationals that are less and all that are greater than it was, put in another way of course, at the bottom of Euclid's definition, and was the source of Bertrand's and others' considerations and of his own theory.

This is a book which should stimulate an intelligent student to research.

Philip E B Jourdain.

Last Updated June 2021