Commentary on Book V Definitions¶
Back to Book V Definitions.
DEFINITION 1.¶
Μέρος ἐστὶ μέγεθος μεγέθους τὸ ἔλασσον τοῦ μείζονος, ὅταν καταμετρῇ τὸ μεῖζον.
The word part (μέρος) is here used in the restricted sense of a submultiple or an aliquot part as distinct from the more general sense in which it is used in the Common Notion (5) which says that “the whole is greater than the part.” It is used in the same restricted sense in VII. Def. 3, which is the same definition as this with “number” (ἀριθμός) substituted for “magnitude.” VII. Def. 4, keeping up the restriction, says that, when a number does not measure another number, it is parts (in the plural), not a part of it. Thus, 1, 2, or 3, is a part of 6, but 4 is not a part of 6 but parts. The same distinction between the restricted and the more general sense of the word part appears in Aristotle, Metaph. 1023 b 12: “In one sense a part is that into which quantity (τὸ ποσόν) can anyhow be divided; for that which is taken away from quantity, quâ quantity, is always called a ‘part’ of it, as e.g. two is said to be in a sense a part of three. But in another sense a ‘part’ is only what measures (τὰ καταμετροῦντα) such quantities. Thus two is in one sense said to be a part of three, in the other not.”
DEFINITION 2.¶
Πολλαπλάσιον δὲ τὸ μεῖζον τοῦ ἐλάττονος, ὅταν καταμετρῆται ὑπὸ τοῦ ἐλάττονος.
DEFINITION 3.¶
Λόγος ἐστὶ δύο μεγεθῶν ὁμογενῶν ἡ κατὰ πηλικότητα ποιὰ σχέσις.
The best explanation of the definitions of ratio and proportion that I have seen is that of De Morgan, which will be found in the articles under those titles in the Penny Cyclopaedia, Vol. XIX. (1841); and in the following notes I shall draw largely from these articles. Very valuable also are the notes on the definitions of Book V. given by Hankel (fragment on Euclid published as an appendix to his work Zur Geschichte der Mathematik in Alterthum und Mittelalter, 1874).
There has been controversy as to what is the proper translation of the word πηλικότης in the definition. σχέσις κατὰ πηλικότητα has generally been translated “relation in respect of quantity.” Upon this De Morgan remarks that it makes nonsense of the definition; “for magnitude has hardly a different meaning from quantity, and a relation of magnitudes with respect to quantity may give a clear idea to those who want a word to convey a notion of architecture with respect to building or of battles with respect to fighting, and to no others.” The true interpretation De Morgan, following Wallis and Gregory, takes to be quantuplicity, referring to the number of times one magnitude is contained in the other. For, he says, we cannot describe magnitude in language without quantuplicitative reference to other magnitude; hence he supposes that the definition simply conveys the fact that the mode of expressing quantity in terms of quantity is entirely based upon the notion of quantuplicity or that relation of which we take cognizance when we find how many times one is contained in the other. While all the rest of De Morgan’s observations on the definition are admirable, it seems to me that on this question of the proper translation of πηλικότης he is in error. He supports his view mainly by reference (1) to the definition of a compounded ratio usually given as the 5th definition of Book VI., which speaks of the πηλικότητες of two ratios being multiplied together, and (2) to the comments of Eutocius and a scholiast on this definition. Eutocius says namely (Archimedes, ed. Heiberg, III. p. 120) that “the term πηλικότης is evidently used of the number from which the given ratio is called, as (among others) Nicomachus says in his first book on music and Heron in his commentary on the Introduction to Arithmetic.” But it now appears certain that this definition is an interpolation; it is never used, it is not found in Campanus, and Peyrard’s MS. only has it in the margin. At the same time it is clear that, if the definition is admitted at all, any commentator would be obliged to explain it in the way that Eutocius does, whether the explanation was consistent with the proper meaning of πηλικότης or not. Hence we must look elsewhere for the meaning of πηλίκος and πηλικότης. If we do this, I think we shall find no case in which the words have the sense attributed to them by De Morgan. The real meaning of πηλίκος is how great. It is so used by Aristotle, e.g. in Eth. Nic. V. 10, 1134 b 11, where he speaks of a man’s child being as it were a part of him so long as he is of a certain age (ἕως ἂν ᾖ πηλίκον). Again Nicomachus, to whom Eutocius appeals, himself (I. 2, 5, p. 5, ed. Hoche) distinguishes πηλίκος as referring to magnitude, while ποσός refers to multitude. So does Iamblichus in his commentary on Nicomachus (p. 8, 3—5); besides which Iamblichus distinguishes πηλίκον as the subject of geometry, being con- tinuous, and ποσόν as the subject of arithmetic, being discrete, and speaks of a point being the origin of πηλίκον as a unit is of ποσόν, and so on. Similarly, Ptolemy (Syntaxis, ed. Heiberg, p. 31) speaks of the size (πηλικότης) of the chords in a circle (περὶ τῆς πηλικότητος τῶν ἐν τῷ κύκλῳ εὐθειῶν). Consequently I think we can only translate πηλικότης in the definition as size. This corresponds to Hankel’s translation of it as “Grösse,” though he uses this same word for a concrete “magnitude” as well; size seems to me to give the proper distinction between πηλικότης and μέγεθος, as size is the attribute, and a magnitude (in its ordinary mathematical sense) is the thing which possesses the attribute of size.
The view that “relation in respect of size” is meant by the words in the text is also confirmed, I think, by a later remark of De Morgan himself, namely that a synonym for the word ratio may be found in the more intelligible term relative magnitude. In fact σχέσις in the definition corresponds to relative and πηλικότης to magnitude. (By magnitude De Morgan here means the attribute and not the thing possessing it.)
Of the definition as a whole Simson and Hankel express the opinion that it is an interpolation. Hankel points to the fact that it is unnecessary and moreover so vague as to be of no practical use, while the very use of the expression κατὰ πηλικότητα seems to him suspicious, since the only other place in which the word πηλικότης occurs in Euclid is the 5th definition of Book VI., which is admittedly not genuine. Yet the definition of ratio appears in all the MSS., the only variation being that some add the words πρὸς ἄλληλα, “to one another,” which are rejected by Heiberg as an interpolation of Theon; and on the whole there seems to be no sufficient ground for regarding it as other than genuine. The true explanation of its presence would appear to be substantially that given by Barrow (Lectiones Cantabrig., London, 1684, Lect. III. of 1666), namely that Euclid inserted it for completeness’ sake, more for ornament than for use, intending to give the learner a general notion of ratio by means of a metaphysical, rather than a mathematical definition; “for metaphysical it is and not, properly speaking, mathematical, since nothing depends on it or is deduced from it by mathematicians, nor, as I think, can anything be deduced.” This is confirmed by the fact that there is no definition of λόγος in Book VII., and it could equally have been dispensed with here. Similarly De Morgan observes that Euclid never attempts this vague sort of definition except when, dealing with a well-known term of common life, he wishes to bring it into geometry with something like an expressed meaning which may aid the conception of the thing, though it does not furnish a perfect criterion. Thus we may compare the definition with that of a straight line, where Euclid merely calls the reader’s attention to the well-known term εὐθεῖα γραμμή, tries how far he can present the conception which accompanies it in other words, and trusts for the correct use of the term to the axioms (or postulates) which the universal conception of a straight line makes self-evident.
We have now to trace as clearly as possible the development of the conception of λόγος, ratio, or relative magnitude. In its primitive sense λόγος was only used of a ratio between commensurables, i.e. a ratio which could be expressed, and the manner of expressing it is indicated in the proposition, Eucl. X. 5, which proves that commensurable magnitudes have to one another the ratio which a number has to a number. That this was the primitive meaning of λόγος is proved by the use of the term ἄλογος for the incommensurable, which means irrational in the sense of not having a ratio to something taken as rational (ῥητός).
Euclid himself shows us how we are to set about finding the ratio, or relative magnitude, of two commensurable magnitudes. He gives, in X. 3, practically our ordinary method of finding the greatest common measure. If $A, B$ be two magnitudes of which $B$ is the less, we cut off from $A$ a part equal to $B$, from the remainder a part equal to $B$, and so on, until we leave a remainder less than $B$, say $R_1$. We measure off $R_1$ from $B$ in the same way until a remainder $R_2$ is left which is less than $R_1$. We repeat the process with $R_1, R_2$, and so on, until we find a remainder which is contained in the preceding remainder a certain number of times exactly. If account is taken of the number of times each magnitude is contained (with something over, except at the last) in that upon which it is measured, we can calculate how many times the last remainder is contained in $A$ and how many times the last remainder is contained in $B$; and we can thus express the ratio of $A$ to $B$ as the ratio of one number to another.
But it may happen that the two magnitudes have no common measure, i.e. are incommensurable, in which case the process described would never come to an end and the means of expression would fail; the magnitudes would then have no ratio in the primitive sense. But the word λόγος, ratio, acquires in Euclid, Book V., a wider sense covering the relative magnitude of incommensurables as well as commensurables; as stated in Euclid’s 4th definition, “magnitudes are said to have a ratio to one another which can, when multiplied, exceed one another,” and finite incommensurables have this property as much as commensurables. De Morgan explains the manner of transition from the narrower to the wider signification of ratio as follows. “Since the relative magnitude of two quantities is always shown by the quantuplicitative mode of expression, when that is possible, and since proportional quantities (pairs which have the same relative magnitude) are pairs which have the same mode (if possible) of expression by means of each other; in all such cases sameness of relative magnitude leads to sameness of mode of expression; or proportion is sameness of ratios (in the primitive sense). But sameness of relative magnitude may exist where quantuplicitative expression is impossible; thus the diagonal of a larger square is the same compared with its side as the diagonal of a smaller square compared with its side. It is an easy transition to speak of sameness of ratio even in this case; that is, to use the term ratio in the sense of relative magnitude, that word having originally only a reference to the mode of expressing relative magnitude, in cases which allow of a particular mode of expression. The word irrational (ἄλογος) does not make any corresponding change but continues to have its primitive meaning, namely, incapable of quantuplicitative expression.”
It remains to consider how we are to describe the relative magnitude of two incommensurables of the same kind. That they have a definite relation is certain. Suppose, for precision, that $S$ is the side of a square, $D$ its diagonal; then, if $S$ is given, any alteration in $D$ or any error in $D$ would make the figure cease to be a square. At the same time, a person altogether ignorant of the relative magnitude of $D$ and $S$ might say that drawing two straight lines of length $S$ so as to form a right angle and joining the ends by a straight line, the length of which would accordingly be $D$, does not help him to realise the relative magnitude, but that he would like to know how many diagonals make an exact number of sides. We should have to reply that no number of diagonals whatever makes an exact number of sides; but that he may mention any fraction of the side, a hundredth, a thousandth or a millionth, and that we will then express the diagonal with an error not so great as that fraction. We then tell him that 1,000,000 diagonals exceed $1,414,213$ sides but fall short of $1,414,214$ sides; consequently the diagonal lies between $1.414213$ and $1.414214$ times the side, and these differ only by one-millionth of the side, so that the error in the diagonal is less still. To enable him to continue the process further, we show him how to perform the arithmetical operation of approximating to the value of $\sqrt{2}$. This gives the means of carrying the approximation to any degree of accuracy that may be desired. In the power, then, of carrying approximations of this kind as far as we please lies that of expressing the ratio, so far as expression is possible, and of comparing the ratio with others as accurately as if expression had been possible.
Euclid was of course aware of this, as were probably others before him; though the actual approximations to the values of ratios of incommensurables of which we find record in the works of the great Greek geometers are very few. The history of such approximations up to Archimedes is, so far as material was available, sketched in The Works of Archimedes (pp. lxxvii and following); and it is sufficient here to note the facts (1) that Plato, and even the Pythagoreans, were familiar with $\frac{7}{5}$ as an approximation to $\sqrt{2}$, (2) that the method of finding any number of successive approximations by the system of side- and diagonal-numbers described by Theon of Smyrna was also Pythagorean (cf. the note above on Euclid, II. 9, 10), (3) that Archimedes, without a word of preliminary explanation, gives out that
$$\frac{1351}{780} > \sqrt{3} > \frac{265}{153},$$
gives approximate values for the square roots of several large numbers, and proves that the ratio of the circumference of a circle to its diameter is less than $3\frac{1}{7}$ but greater than $3\frac{10}{71}$, (4) that the first approach to the rapidity with which the decimal system enables us to approximate to the value of surds was furnished by the method of sexagesimal fractions, which was almost as convenient to work with as the method of decimals, and which appears fully developed in Ptolemy’s σύνταξις. A number consisting of a whole number and any fraction was under this system represented as so many units, so many of the fractions which we should denote by $\frac{1}{60}$, so many of those which we should write $\left(\frac{1}{60}\right)^2$, $\left(\frac{1}{60}\right)^3$, and so on. Theon of Alexandria shows us how to extract the square root of $4500$ in this sexagesimal system, and, to show how effective it was, it is only necessary to mention that Ptolemy gives $\frac{103}{60} + \frac{55}{60^2} + \frac{23}{60^3}$ as an approximation to $\sqrt{3}$, which approximation is equivalent to $1.7320509$ in the ordinary decimal notation and is therefore correct to $6$ places.
Between Def. 3 and Def. 4 two manuscripts and Campanus insert “Proportion is the sameness of ratios” (ἀναλογία δὲ ἡ τῶν λόγων ταὐτότης), and even the best MS. has it in the margin. It would be altogether out of place, since it is not till Def. 5 that it is explained what sameness of ratios is. The words are an interpolation later than Theon (Heiberg, Vol. V. pp. xxxv, lxxxix), and are no doubt taken from arithmetical works (cf. Nicomachus and Theon of Smyrna). It is true that Aristotle says similarly, “Proportion is equality of ratios” (Eth. Nic. V. 6, 1131 a 31), and he appears to be quoting from the Pythagoreans; but the reference is to numbers.
Similarly two MSS. (inferior) insert after Def. 7 “Proportion is the similarity (ὁμοιότης) of ratios.” Here too we have a mere interpolation.
DEFINITION 4.¶
Λόγον ἔχειν πρὸς ἄλληλα μεγέθη λέγεται, ἃ δύναται πολλαπλασιαζόμενα ἀλλήλων ὑπερέχειν.
This definition supplements the last one. De Morgan says that it amounts to saying that the magnitudes are of the same species. But this can hardly be all; the definition seems rather to be meant, on the one hand, to exclude the relation of a finite magnitude to a magnitude of the same kind which is either infinitely great or infinitely small, and, even more, to emphasise the fact that the term ratio, as defined in the preceding definition, and about to be used throughout the book, includes the relation between any two incommensurable as well as between any two commensurable finite magnitudes of the same kind. Hence, while De Morgan seems to regard the extension of the meaning of ratio to include the relative magnitude of incommensurables as, so to speak, taking place between Def. 3 and Def. 5, the 4th definition appears to show that it is ratio in its extended sense that is being defined in Def. 3.
DEFINITION 5.¶
Ἐν τῷ αὐτῷ λόγῳ μεγέθη λέγεται εἶναι πρῶτον πρὸς δεύτερον καὶ τρίτον πρὸς τέταρτον, ὅταν τὰ τοῦ πρώτου καὶ τρίτου ἰσάκις πολλαπλάσια τῶν τοῦ δευτέρου καὶ τετάρτου ἰσάκις πολλαπλασίων καθ᾽ ὁποιονοῦν πολλαπλασιασμὸν ἑκάτερον ἑκατέρου ἢ ἅμα ὑπερέχῃ ἢ ἅμα ἴσα ἢ ἅμα ἐλλείπῃ ληφθέντα κατάλληλα.
In my translation of this definition I have compromised between an attempted literal translation and the more expanded version of Simson. The difficulty in the way of an exactly literal translation is due to the fact that the words (καθ᾽ ὁποιονοῦν πολλαπλασιασμὸν) signifying that the equimultiples in each case are any equimultiples whatever occur only once in the Greek, though they apply both to τὰ… ἰσάκις πολλαπλάσια in the nominative and τῶν… ἰσάκις πολλαπλασίων in the genitive. I have preferred “alike” to “simultaneously” as a translation of ἅμα because “simultaneously” might suggest that time was of the essence of the matter, whereas what is meant is that any particular comparison made between the equimultiples must be made between the same equimultiples of the two pairs respectively, not that they need to be compared at the same time.
Aristotle has an allusion to a definition of “the same ratio” in Topics VIII. 3, 158 b 29: “In mathematics too some things appear to be not easy to prove (γράφεσθαι) for want of a definition, e.g. that the parallel to the side which cuts a plane [a parallelogram] divides the straight line [the other side] and the area similarly. But, when the definition is expressed, the said property is immediately manifest; for the areas and the straight lines have the same ἀνταναίρεσις, and this is the definition of ‘the same ratio.’” Upon this passage Alexander says similarly, “This is the definition of proportionals which the ancients used: magnitudes are proportional to one another which have (or show) the same ἀνθυφαίρεσις, and Aristotle has called the latter ἀνταναίρεσις.” Heiberg (Mathematisches zu Aristoteles, p. 22) thinks that Aristotle is alluding to the fact that the proposition referred to could not be rigorously proved so long as the Pythagorean definition applicable to commensurable magnitudes only was adhered to, and is quoting the definition belonging to the complete theory of Eudoxus; whence, in view of the positive statement of Aristotle that the definition quoted is the definition of “the same ratio,” it would appear that the Euclidean definition (which Heiberg describes as a careful and exact paraphrase of ἀνταναίρεσις) is Euclid’s own. I do not feel able to subscribe to this view, which seems to me to involve very grave difficulties. The Euclidean definition is regularly appealed to in Book V. as the criterion of magnitudes being in proportion, and the use of it would appear to constitute the whole essence of the new general theory of proportion; if then this theory is due to Eudoxus, it seems impossible to believe that the definition was not also due to him. Certainly the definition given by Aristotle would be no substitute for it; ἀνθυφαίρεσις and ἀνταναίρεσις are words almost as vague and “metaphysical” (as Barrow would say) as the words used to define ratio, and it is difficult to see how any mathematical facts could be deduced from such a definition. Consider for a moment the etymology of the words. ὑφαίρεσις or ἀναίρεσις means “removal,” “taking away” or “destruction” of a thing; and the prefix ἀντὶ indicates that the “taking away” from one magnitude answers to, corresponds with, alternates with, the “taking away” from the other. So far therefore as the etymology goes, the word seems rather to suggest the “taking away” of corresponding fractions, and therefore to suit the old imperfect theory of proportion rather than the new one. Thus Waitz (ad loc.) paraphrases the definition as meaning that “as many parts as are taken from one magnitude, so many are at the same time taken from the other as well.” A possible explanation would seem to be that, though Eudoxus had formulated the new definition, the old one was still current in the text-books of Aristotle’s time, and was taken by him as being a good enough illustration of what he wished to bring out in the passage of the Topics referred to.
From the revival of learning in Europe onwards the Euclidean definition of proportion was the subject of much criticism. Campanus had failed to understand it, had in fact misinterpreted it altogether, and he may have misled others such as Ramus ($1515$—$72$), always a violently hostile critic of Euclid. Among the objectors to it was no less a person than Galileo. For particulars of the controversies on the subject down to Thomas Simpson (Elem. of Geometry, Lond. $1800$) the reader is referred to the Excursus at the end of the second volume of Camerer’s Euclid ($1825$). For us it is interesting to note that the unsoundness of the usual criticisms of the definition was never better exposed than by Barrow. Some of the objections, he pointed out (Lect. Cantabr. VII. of $1666$), are due to misconception on the part of their authors as to the nature of a definition. Thus Euclid is required by these objectors (e.g. Tacquet) to do the impossible and to show that what is predicated in the definition is true of the thing defined, as if any one should be required to show that the name “circle” was applicable to those figures alone which have their radii all equal! As we are entitled to assign to such figures and such figures only the name of “circle,” so Euclid is entitled (“quamvis non temere nec imprudenter at certis de causis iustis illis et idoneis”) to describe a certain property which four magnitudes may have, and to call magnitudes possessing that property magnitudes “in the same ratio.” Others had argued from the occurrence of the other definition of proportion in VII. Def. 20 that Euclid was dissatisfied with the present one; Barrow pointed out that, on the contrary, it was the fact that VII. Def. 20 was not adequate to cover the case of incommensurables which made Euclid adopt the present definition here. Lastly, he maintains, against those who descant on the “obscurity” of V. Def. 5, that the supposed obscurity is due, partly no doubt to the inherent difficulty of the subject of incommensurables, but also to faulty translators, and most of all to lack of effort in the learner to grasp thoroughly the meaning of words which, in themselves, are as clearly expressed as they could be.
To come now to the merits of the case, the best defence and explanation of the definition that I have seen is that given by De Morgan. He first translates it, observes that it applies equally to commensurable or incommensurable quantities because no attempt is made to measure one by an aliquot part of another, and then proceeds thus.
“The two questions which must be asked, and satisfactorily answered, previously to its [the definition’s] reception, are as follows:
What right had Euclid, or any one else, to expect that the preceding most prolix and unwieldy statement should be received by the beginner as the definition of a relation the perception of which is one of the most common acts of his mind, since it is performed on every occasion where similarity or dissimilarity of figure is looked for or presents itself?
If the preceding question should be clearly answered, how can the definition of proportion ever be used; or how is it possible to compare every one of the infinite number of multiples of $A$ with every one of the multiples of $B$?
To the first question we reply that not only is the test proposed by Euclid tolerably simple, when more closely examined, but that it is, or might be made to appear, an easy and natural consequence of those fundamental perceptions with which it may at first seem difficult to compare it.”
To elucidate this De Morgan gives the following illustration.
Suppose there is a straight colonnade composed of equidistant columns (which we will understand to mean the vertical lines forming the axes of the columns respectively), the first of which is at a distance from a bounding wall equal to the distance between consecutive columns. In front of the colonnade let there be a straight row of equidistant railings (regarded as meaning their axes), the first being at a distance from the bounding wall equal to the distance between consecutive railings. Let the columns be numbered from the wall, and also the railings. We suppose of course that the column distance (say, C) and the railing distance (say, R) are different and that they may bear to each other any ratio, commensurable or incommensurable; i.e. that there need not go any exact number of railings to any exact number of columns.
If the construction be supposed carried on to any extent, a spectator can, by mere inspection, and without measurement, compare C with R to any degree of accuracy. For example, since the 10th railing falls between the 4th and 5th columns, 10R is greater than 4C and less than 5C, and therefore R lies between $\frac{4}{10}\text{ths}$ of C and $\frac{5}{10}\text{ths}$ of C. To get a more accurate notion, the ten-thousandth railing may be taken; suppose it falls between the 4674th and 4675th columns. Therefore $10,000R$ lies between $4674C$ and $4675C$, or R lies between $\frac{4674}{10000}$ and $\frac{4675}{10000}$ of C. There is no limit to the degree of accuracy thus obtainable; and the ratio of R to C is determined when the order of distribution of the railings among the columns is assigned ad infinitum; or, in other words, when the position of any given railing can be found, as to the numbers of the columns between which it lies. Any alteration, however small, in the place of the first railing must at last affect the order of distribution. Suppose e.g. that the first railing is moved from the wall by one part in a thousand of the distance between the columns; then the second railing is pushed forward by $\frac{2}{1000}C$, the third by $\frac{3}{1000}C$, and so on, so that the railings after the thousandth are pushed forward by more than C; i.e. the order with respect to the columns is disarranged.
Now let it be proposed to make a model of the preceding construction in which $c$ shall be the column distance and $r$ the railing distance. It needs no definition of proportion, nor anything more than the conception which we have of that term prior to definition (and with which we must show the agreement of any definition that we may adopt), to assure us that C must be to R in the same proportion as $c$ to $r$ if the model be truly formed. Nor is it drawing too largely on that conception of proportion to assert that the distribution of the railings among the columns in the model must be everywhere the same as in the original; for example, that the model would be out of proportion if its 37th railing fell between the 18th and 19th columns, while the 37th railing of the original fell between the 17th and 18th columns. Thus the dependence of Euclid’s definition upon common notions is settled; for the obvious relation between the construction and its model which has just been described contains the collection of conditions, the fulfilment of which, according to Euclid, constitutes proportion. According to Euclid, whenever $mC$ exceeds, equals, or falls short of $nR$, then $mc$ must exceed, equal, or fall short of $nr$; and, by the most obvious property of the constructions, according as the $m$th column comes after, opposite to, or before the $n$th railing in the original, the $m$th column must come after, opposite to, or before the $n$th railing in the correct model.
Thus the test proposed by Euclid is necessary. It is also sufficient. For admitting that, to a given original with a given column-distance in the model, there is one correct model railing distance (which must therefore be that which distributes the railings among the columns as in the original), we have seen that any other railing distance, however slightly different, would at last give a different distribution; that is, the correct distance, and the correct distance only, satisfies all the conditions required by Euclid’s definition.
The use of the word distribution having been well learnt, says De Morgan, the following way of stating the definition will be found easier than that of Euclid. “Four magnitudes, $A$ and $B$ of one kind, and $C$ and $D$ of the same or another kind, are proportional when all the multiples of $A$ can be distributed among the multiples of $B$ in the same intervals as the corresponding multiples of $C$ among those of $D$.” Or, whatever numbers $m, n$ may be, if $mA$ lies between $nB$ and $(n+1)B$, $mC$ lies between $nD$ and $(n+1)D$.
It is important to note that, if the test be always satisfied from and after any given multiples of $A$ and $C$, it must be satisfied before those multiples. For instance, let the test be always satisfied from and after $100A$ and $100C$; and let $5A$ and $5C$ be instances for examination. Take any multiple of $5$ which will exceed $100$, say $50$ times five; and let it be found on examination that $250A$ lies between $678B$ and $679B$; then $250C$ lies between $678D$ and $679D$. Divide by $50$, and it follows that $5A$ lies between $13\frac{28}{50}B$ and $13\frac{29}{50}B$, and a fortiori between $13B$ and $14B$. Similarly, $5C$ lies between $13\frac{28}{50}D$ and $13\frac{29}{50}D$, and therefore between $13D$ and $14D$. Or $5A$ lies in the same interval among the multiples of $B$ in which $5C$ lies among the multiples of $D$. And so for any multiple of $A, C$ less than $100A, 100C$.
There remains the second question relating to the infinite character of the definition; four magnitudes $A, B, C, D$ are not to be called proportional until it is shown that every multiple of $A$ falls in the same intervals among the multiples of $B$ in which the same multiple of $C$ is found among the multiples of $D$. Suppose that the distribution of the railings among the columns should be found to agree in the model and the original as far as the millionth railing. This proves only that the railing distance of the model does not err by the millionth part of the corresponding column distance. We can thus fix limits to the disproportion, if any, and we may make those limits as small as we please, by carrying on the method of observation; but we cannot observe an infinite number of cases and so enable ourselves to affirm proportion absolutely. Mathematical methods however enable us to avoid the difficulty. We can take any multiples whatever and work with them as if they were particular multiples. De Morgan gives, as an instance to show that the definition of proportion can in practice be used, notwithstanding its infinite character, the following proof of a proposition to the same effect as Eucl. VI. 2.
“Let OAB be a triangle to one side AB of which ab is drawn parallel, and on OA produced set off AA2, A2A3 etc. equal to OA, and aa2, a2a3 etc. equal to Oa.
Through every one of the points so obtained draw parallels to AB, meeting OB produced in b2, B2 etc.
Then it is easily proved that bb2, b2b3 etc. are severally equal to Ob, and BB2, B2B3 etc. to OB.
Consequently a distribution of the multiples of OA among the multiples of Oa is made on one line, and of OB among those of Ob on the other.
The examination of this distribution in all its extent (which is impossible, and hence the apparent difficulty of using the definition) is rendered unnecessary by the known property of parallel lines. For, since A3 lies between a3 and a4, B3 must lie between b3 and b4; for, if not, the line A3B3 would cut either a3b3 or a4b4.
Hence, without inquiring where $A_m$ does fall, we know that, if it fall between $a_n$ and $a_{n+1}$, $B_m$ must fall between $b_n$ and $b_{n+1}$; or, if $m \cdot OA$ fall in magnitude between $n \cdot Oa$ and $(n+1)Oa$, then $m \cdot OB$ must fall between $n \cdot Ob$ and $(n+1)Ob$.”
Max Simon remarks (Euclid und die sechs planimetrischen Bücher, p. 110), after Zeuthen, that Euclid’s definition of equal ratios is word for word the same as Weierstrass’ definition of equal numbers. So far from agreeing in the usual view that the Greeks saw in the irrational no number, Simon thinks it is clear from Eucl. V. that they possessed a notion of number in all its generality as clearly defined as, nay almost identical with, Weierstrass’ conception of it.
Certain it is that there is an exact correspondence, almost coincidence, between Euclid’s definition of equal ratios and the modern theory of irrationals due to Dedekind. Premising the ordinal arrangement of natural numbers in ascending order, then enlarging the sphere of numbers by including (1) negative numbers as well as positive, (2) fractions, as $a/b$, where $a, b$ may be any natural numbers, provided that $b$ is not zero, and arranging the fractions ordinally among the other numbers according to the definition :
$$\text{let } \frac{a}{b} \text{ be } < = > \frac{c}{d} \text{ according as } ad \text{ is } < = > bc,$$
Dedekind arrives at the following definition of an irrational number.
An irrational number $\alpha$ is defined whenever a law is stated which will assign every given rational number to one and only one of two classes $A$ and $B$ such that (1) every number in $A$ precedes every number in $B$, and (2) there is no last number in $A$ and no first number in $B$; the definition of $\alpha$ being that it is the one number which lies between all numbers in $A$ and all numbers in $B$.
Now let $x/y$ and $x'/y'$ be equal ratios in Euclid’s sense.
Then $\frac{x}{y}$ will divide all rational numbers into two groups $A$ and $B$;
$\frac{x'}{y'}$ ,, ,, ,, $A'$ and $B'$.
Let $\frac{a}{b}$ be any rational number in $A$, so that
$$\frac{a}{b} < \frac{x}{y}.$$
This means that $ay < bx$.
But Euclid’s definition asserts that in that case $ay' < bx'$ also.
Hence also
$$\frac{a}{b} < \frac{x'}{y'};$$
therefore every member of group $A$ is also a member of group $A'$.
Similarly every member of group $B$ is a member of group $B'$.
For, if $\frac{a}{b}$ belong to $B$,
$$\frac{a}{b} > \frac{x}{y},$$
which means that $ay > bx$.
But in that case, by Euclid’s definition, $ay' > bx'$;
therefore also
$$\frac{a}{b} > \frac{x'}{y'}.$$
Thus, in other words, $A$ and $B$ are coextensive with $A'$ and $B'$ respectively ;
therefore $\frac{x}{y} = \frac{x'}{y'}$, according to Dedekind, as well as according to Euclid.
If $x/y$, $x'/y'$ happen to be rational, then one of the groups, say $A$, includes $x/y$, and one of the groups, say $A'$, includes $x'/y'$.
In this case $\frac{a}{b}$ might coincide with $\frac{x}{y}$;
that is
$$\frac{a}{b} = \frac{x}{y},$$
which means that
$$ay = bx.$$
Therefore, by Euclid’s definition, $ay' = bx'$ ;
so that
$$\frac{a}{b} = \frac{x'}{y'}.$$
Thus the groups are again coextensive.
In a word, Euclid’s definition divides all rational numbers into two coextensive classes, and therefore defines equal ratios in a manner exactly corresponding to Dedekind’s theory.
Alternatives for Eucl. V. Def. 5.¶
Saccheri records in his Euclides ob omni naevo vindicatus that a distinguished geometer of his acquaintance proposed to substitute for Euclid’s the following definition :
“A first magnitude has to a second the same ratio that a third has to a fourth when the first contains the aliquot parts of the second, according to any number [i.e. with any denominator] whatever, the same number of times as the number of times the third contains the same aliquot parts of the fourth” ; on which Saccheri remarks that he sees no advantage in this definition, which presupposes the notion of division, over that of Euclid which uses multiplication and the notions of greater, equal, and less.
This definition was, however, practically adopted by Faifofer (Elementi di geometria, 3 ed., 1882) in the following form :
“Four magnitudes taken in a certain order form a proportion when, by measuring the first and the third respectively by any equi-submultiples whatever of the second and of the fourth, equal quotients are obtained.”
Ingrami (Elementi di geometria, 1904) takes multiples of the first and third instead of submultiples of the second and fourth :
“Given four magnitudes in predetermined order, the first two homogeneous with one another, and likewise also the last two, the magnitudes are said to form a proportion (or to be in proportion) when any multiple of the first contains the second the same number of times that the equimultiple of the third contains the fourth.”
Veronese’s definition (Elementi di geometria, Pt. II., 1905) is like that of Faifofer ; Enriques and Amaldi (Elementi di geometria, 1905) adhere to Euclid’s.
Proportionals of VII. Def. 20 a particular case.¶
It has already been observed that Euclid has nowhere proved (though the fact cannot have escaped him) that the proportion of numbers is included in the proportion of magnitudes as a special case. This is proved by Simson as being necessary to the 5th and 6th propositions of Book X. Simson’s proof is contained in his propositions C and D inserted in the text of Book V. and in the notes thereon. Proposition C and the note on it prove that, if four magnitudes are proportionals according to VII. Def. 20, they are also proportionals according to V. Def. 5. Prop. D and the note prove the partial converse, namely that, if four magnitudes are proportionals according to the 5th definition of Book V., and if the first be any multiple, or any part, or parts, of the second, the third is the same multiple, part, or parts, of the fourth. The proofs use certain results obtained in Book V.
Prop. C is as follows :
If the first be the same multiple of the second, or the same part of it, that the third is of the fourth, the first is to the second as the third is to the fourth. Let the first A be the same multiple of B the second that C the third is of the fourth D ;
A is to B as C is to D.
Take of A, C any equimultiples whatever E, F ; and of B, D any equimultiples whatever G, H.
Then, because A is the same multiple of B that C is of D, and E is the same multiple of A that F is of C, E is the same multiple of B that F is of D. [V. 3]
Therefore E, F are the same multiples of B, D.
But G, H are equimultiples of B, D ; therefore, if E be a greater multiple of B than G is, F is a greater multiple of D than H is of D ;
that is, if E be greater than G, F is greater than H.
In like manner, if E be equal to G, or less, F is equal to H, or less than it.
But E, F are equimultiples, any whatever, of A, C ; and G, H any equimultiples whatever of B, D.
Therefore A is to B as C is to D. [V. Def. 5]
Next, let the first A be the same part of the second B that the third C is of the fourth D :
A is to B as C is to D.
For B is the same multiple of A that D is of C ; wherefore, by the preceding case,
B is to A as D is to C ;
and, inversely, A is to B as C is to D.
[For this last inference Simson refers to his Proposition B. That proposition is very simply proved by taking any equimultiples $E$, $F$ of B, D and any equimultiples $G$, $H$ of A, C and then arguing as follows :
Since A is to B as C is to D, $G$, $H$ are simultaneously greater than, equal to, or less than $E$, $F$ respectively ; so that $E$, $F$ are simultaneously less than, equal to, or greater than $G$, $H$ respectively, and therefore [Def. 5] B is to A as D is to C.]
We have now only to add to Prop. C the case where AB contains the same parts of CD that EF does of GH : in this case likewise AB is to CD as EF to GH.
Let CK be a part of CD, and GL the same part of GH ; let AB be the same multiple of CK that EF is of GL.
Therefore, by Prop. C,
AB is to CK as EF to GL.
And CD, GH are equimultiples of CK, GL, the second and fourth.
Therefore AB is to CD as EF to GH [Simson’s Cor. to V. 4, which however is the particular case of V. 4 in which the “equimultiples” of one pair are the pair itself, i.e. the pair multiplied by unity].
To prove the partial converse we begin with Prop. D.
If the first be to the second as the third to the fourth, and if the first be a multiple or part of the second, the third is the same multiple or the same part of the fourth.
Let A be to B as C is to D ; and, first, let A be a multiple of B ;
C is the same multiple of D.
Take E equal to A, and whatever multiple A or E is of B, make F the same multiple of D.
Then, because A is to B as C is to D, and of B the second and D the fourth equimultiples have been taken E and F,
A is to E as C is to F. [V. 4, Cor.]
But A is equal to E ;
therefore C is equal to F.
[In support of this inference Simson cites his Prop. A, which however we can directly deduce from V. Def. 5 by taking any, but the same, equimultiples of all four magnitudes.]
Now F is the same multiple of D that A is of B ;
therefore C is the same multiple of D that A is of B.
Next, let the first A be a part of the second B ;
C the third is the same part of the fourth D.
Because A is to B as C is to D, inversely, B is to A as D is to C. [Prop. B]
But A is a part of B ; therefore B is a multiple of A ; and, by the preceding case, D is the same multiple of C, that is, C is the same part of D that A is of B.
We have, again, only to add to Prop. D the case where AB contains any parts of CD, and AB is to CD as EF to GH ; then shall EF contain the same parts of GH that AB does of CD.
For let CK be a part of CD, and GL the same part of GH ; and let AB be a multiple of CK.
EF shall be the same multiple of GL.
Take M the same multiple of GL that AB is of CK ; therefore AB is to CK as M is to GL. [Prop. C]
And CD, GH are equimultiples of CK, GL ; therefore AB is to CD as M is to GH.
But, by hypothesis, AB is to CD as EF is to GH ; therefore M is equal to EF, [V. 9] and consequently EF is the same multiple of GL that AB is of CK.
DEFINITION 6.¶
Τὰ δὲ τὸν αὐτὸν ἔχοντα λόγον μεγέθη ἀνάλογον καλείσθω.
Ἀνάλογον, though usually written in one word, is equivalent to ἀνὰ λόγον, in proportion. It comes however in Greek mathematics to be used practically as an indeclinable adjective, as here ; cf. αἱ τέσσαρες εὐθεῖαι ἀνάλογον ἔσονται, “the four straight lines will be proportional,” τρίγωνα τὰς πλευρὰς ἀνάλογον ἔχοντα, “triangles having their sides proportional.” Sometimes it is used adverbially : ἀνάλογον ἄρα ἐστὶν ὡς ἡ ΒΑ πρὸς τὴν ΑΓ, οὕτως ἡ ΗΔ πρὸς τὴν ΔΖ, “proportionally therefore, as $BA$ is to $AC$, so is $GD$ to $DF$” ; so too, apparently, in the expression ἡ μέση ἀνάλογον (εὐθεῖα), “the mean proportional.” I do not follow the objection of Max Simon (Euclid, p. 110) to “proportional” as a translation of ἀνάλογον. “We ask,” he says, “in vain, what is proportional to what? We say e.g. that weight is proportional to price because double, treble etc. weight corresponds to double, treble etc. price. But here the meaning must be ‘standing in a relation of proportion.’” Yet he admits that the Latin word proportionalis is an adequate expression. He translates by “in proportion” in the text of this definition. But I do not see that “in proportion” is better than “proportional.” The fact is that both expressions are elliptical when used of four magnitudes “in proportion” ; but there is surely no harm in using either when the meaning is so well understood.
The use of the word καλείσθω, “let magnitudes having the same ratio be called proportional,” seems to indicate that this definition is Euclid’s own.
DEFINITION 7.¶
Ὅταν δὲ τῶν ἰσάκις πολλαπλασίων τὸ μὲν τοῦ πρώτου πολλαπλάσιον ὑπερέχῃ τοῦ τοῦ δευτέρου πολλαπλασίου, τὸ δὲ τοῦ τρίτου πολλαπλάσιον μὴ ὑπερέχῃ τοῦ τοῦ τετάρτου πολλαπλασίου, τότε τὸ πρῶτον πρὸς τὸ δεύτερον μείζονα λόγον ἔχειν λέγεται, ἤπερ τὸ τρίτον πρὸς τὸ τέταρτον.
As De Morgan observes, the practical test of disproportion is simpler than that of proportion. For, whereas no examination of individual cases, however extensive, will enable an observer of the construction and its model (the illustration by means of columns and railings described above) to affirm proportion or deny disproportion, and all it enables us to do is to fix limits (as small as we please) to the disproportion (if any), a single instance may enable us to deny proportion or affirm disproportion, and also to state which way the disproportion lies. Let the 19th railing in the original fall beyond the 11th column, while the 19th railing of the (so-called) model does not come up to the 11th column. It follows from this one instance that the railing distance of the model is too small relatively to the column distance, or that the column distance is too great relatively to the railing distance. That is, the ratio of $r$ to $c$ is less than that of R to C, or the ratio of $c$ to $r$ is greater than that of C to R.
Saccheri (op. cit.) remarks (as Commandinus had done) that the ratio of the first magnitude to the second will also be greater than that of the third to the fourth if, while the multiple of the first is equal to the multiple of the second, the multiple of the third is less than that of the fourth : a case not mentioned in Euclid’s definition. Saccheri speaks of this case being included in Clavius’ interpretation of the definition. I have, however, failed to find a reference to the case in Clavius, though he adds, as a sort of corollary, in his note on the definition, that if, on the other hand, the multiple of the first is less than the multiple of the second, while the multiple of the third is not less than that of the fourth, the ratio of the first to the second is less than that of the third to the fourth.
Euclid presumably left out the second possible criterion for a greater ratio, and the definition of a less ratio, because he was anxious to reduce the definitions to the minimum necessary for his purpose, and to leave the rest to be inferred as soon as the development of the propositions of Book V. enabled this to be done without difficulty.
Saccheri tried to reduce the second possible criterion for a greater ratio to that given by Euclid in his definition without recourse to anything coming later in the Book, but, in order to do this, he has to use “multiples” produced by multipliers which are not integral numbers, but integral numbers plus proper fractions, so that Euclid’s Def. 7 becomes inapplicable.
De Morgan notes that “proof should be given that the same pair of magnitudes can never offer both tests [i.e. the test in the definition for a greater ratio and the corresponding test for a less ratio, with “less” substituted for “greater” in the definition] to another pair ; that is, the test of greater ratio from one set of multiples, and that of less ratio from another.” In other words, if $m$, $n$, $p$, $q$ are integers and $A$, $B$, $C$, $D$ four magnitudes, none of the pairs of equations
$$\begin{aligned} (1) \quad mA > nB, \quad mC = \text{or} < nD, \\ (2) \quad mA = nB, \quad mC < nD\phantom{ = \text{or}} \end{aligned}$$
can be satisfied simultaneously with any one of the pairs of equations
$$\begin{aligned} (3) \quad pA = qB, \quad pC > qD,\phantom{ \text{or} =} \\ (4) \quad pA < qB, \quad pC > \text{or} = qD. \end{aligned}$$
There is no difficulty in proving this with the help of two simple assumptions which are indeed obvious.
We need only take in illustration one of the numerous cases. Suppose, if possible, that the following pairs of equations are simultaneously true :
$$\begin{aligned} &\text{(1)} \quad mA > nB, \quad mC < nD \\ \text{and} \qquad &\text{(2)} \quad pA < qB, \quad pC > qD. \end{aligned}$$
Multiply (1) by $q$ and (2) by $n$.
(We need here to assume that, where $rX, rY$ are any equimultiples of any magnitudes $X, Y$,
$$\text{according as } X > = < Y, \quad rX > = < rY.$$
This is contained in Simson’s Axioms 1 and 3.)
We have then the pairs of equations
$$mqA > nqB, \quad mqC < nqD,$$
$$npA < nqB, \quad npC > nqD.$$
From the second equations in each pair it follows that
$$mqC < npC.$$
(We now need to assume that, if $rX, sX$ are any multiples of $X$, and $rY, sY$ the same multiples of $Y$, then,
$$\text{according as } rX > = < sX, \quad rY > = < sY.$$
Simson uses this same assumption in his proof of V. 18.)
Therefore
$$mqA < npA.$$
But it follows from the first equations in each pair that
$$mqA > npA :$$
which is impossible.
Nor can Euclid’s criterion for a greater ratio coexist with that for equal ratios.
DEFINITION 8.¶
Ἀναλογία δὲ ἐν τρισὶν ὅροις ἐλαχίστη ἐστίν.
This is the reading of Heiberg and Camerer (who follow Peyrard’s MS.) and is that translated above. The other reading has ἐλαχίστοις, which can only be translated “consists in three terms at least.” Hankel regards the definition as a later interpolation, because it is superfluous, and because the word ὅρος for a term in a proportion is nowhere else used by Euclid, though it is common in later writers such as Nicomachus and Theon of Smyrna. The genuineness of the definition is however supported by the fact that Aristotle not only uses ὅρος in this sense (Eth. Nic. V. 6, 7, 1131 b 5, 9), but has a similar remark (ibid. 1131 a 31) that a “proportion is in four terms at least.” The difference from Euclid is only formal; for Aristotle proceeds: “The discrete (διῃρημένη) (proportion) is clearly in four (terms), but so also is the continuous (συνεχής). For it uses one as two and mentions it twice, e.g. (in stating) that, as $\alpha$ is to $\beta$, so also is $\beta$ to $\gamma$; thus $\beta$ is mentioned twice, so that, if $\beta$ be twice put down, the proportionals are four.” The distinction between discrete and continuous seems to have been Pythagorean (cf. Nicomachus, II. 21, 5; 23, 2, 3; where however συνημμένη is used instead of συνεχής); Euclid does not use the words διῃρημένη and συνεχής in this connexion.
So far as they go, the first words of the next definition (9), “When three magnitudes are proportionals,” which seemingly refer to Def. 8, also support the view that the latter is, at least in substance, genuine.
DEFINITIONS 9, 10.¶
9. Ὅταν δὲ τρία μεγέθη ἀνάλογον ᾖ, τὸ πρῶτον πρὸς τὸ τρίτον διπλασίονα λόγον ἔχειν λέγεται ἤπερ πρὸς τὸ δεύτερον.
10. Ὅταν δὲ τέσσαρα μεγέθη ἀνάλογον ᾖ, τὸ πρῶτον πρὸς τὸ τέταρτον τριπλασίονα λόγον ἔχειν λέγεται ἤπερ πρὸς τὸ δεύτερον, καὶ ἀεὶ ἑξῆς ὁμοίως, ὡς ἂν ἡ ἀναλογία ὑπάρχῃ.
Here, and in connexion with the definitions of duplicate, triplicate, etc. ratios, would be the place to expect a definition of “compound ratio.” None such is however forthcoming, and the only “definition” of it that we find is that forming VI. Def. 5, which is an interpolation made, perhaps, even before Theon’s time. According to the interpolated definition, “A ratio is said to be compounded of ratios when the sizes (πηλικότητες) of the ratios multiplied together make some (? ratio).” But the multiplication of the sizes (or magnitudes) of two ratios of incommensurable, and even of commensurable, magnitudes is an operation unknown to the classical Greek geometers. Eutocius (Archimedes, ed. Heiberg, III. p. 120) is driven to explain the definition by making πηλικότης mean the number from which the given ratio is called, or, in other words, the number which multiplied into the consequent of the ratio gives the antecedent. But he is only able to work out his idea with reference to ratios between numbers, or between commensurable magnitudes; and indeed the definition is quite out of place in Euclid’s theory of proportion.
There is then only one statement in Euclid’s text as we have it indicating what is meant by compound ratio; this is in VI. 23, where he says abruptly “But the ratio of $K$ to $M$ is compounded of the ratio of $K$ to $L$ and that of $L$ to $M$.” Simson accordingly gives a definition (A of Book V.) of compound ratio directly suggested by the statement in VI. 23 just quoted.
“When there are any number of magnitudes of the same kind, the first is said to have to the last of them the ratio compounded of the ratio which the first has to the second, and of the ratio which the second has to the third, and of the ratio which the third has to the fourth, and so on unto the last magnitude.
For example, if $A, B, C, D$ be four magnitudes of the same kind, the first $A$ is said to have to the last $D$ the ratio compounded of the ratio of $A$ to $B$, and of the ratio of $B$ to $C$, and of the ratio of $C$ to $D$; or the ratio of $A$ to $D$ is said to be compounded of the ratios of $A$ to $B$, $B$ to $C$, and $C$ to $D$.
And if $A$ has to $B$ the same ratio which $E$ has to $F$; and $B$ to $C$ the same ratio that $G$ has to $H$; and $C$ to $D$ the same that $K$ has to $L$; then, by this definition, $A$ is said to have to $D$ the ratio compounded of ratios which are the same with the ratios of $E$ to $F$, $G$ to $H$, and $K$ to $L$: and the same thing is to be understood when it is more briefly expressed, by saying, $A$ has to $D$ the ratio compounded of the ratios of $E$ to $F$, $G$ to $H$, and $K$ to $L$.
In like manner, the same things being supposed, if $M$ has to $N$ the same ratio which $A$ has to $D$; then, for shortness’ sake, $M$ is said to have to $N$ the ratio compounded of the ratios of $E$ to $F$, $G$ to $H$, and $K$ to $L$.”
De Morgan has some admirable remarks on compound ratio, which not only give a very clear view of what is meant by it but at the same time supply a plausible explanation of the origin of the term. “Treat ratio,” says De Morgan, “as an engine of operation. Let that of $A$ to $B$ suggest the power of altering any magnitude in that ratio.” (It is true that it is not yet proved that, $B$ being any magnitude, and $P$ and $Q$ two magnitudes of the same kind, there does exist a magnitude $A$ which is to $B$ in the same ratio as $P$ to $Q$. It is not yet till VI. 12 that this is proved, by construction, in the particular case where the three magnitudes are straight lines. The proof in the Greek text of V. 18 which assumes the truth of the more general proposition is, by reason of that assumption, open to objection; see the note on that proposition.) Now “every alteration of a magnitude is alteration in some ratio, two or more successive alterations are jointly equivalent to but one, and the ratio of the initial magnitude to the terminal one is as properly said to be the compound ratio of alteration as 13 to be the compound addend in lieu of 8 and 5, or 28 the compound multiple for 7 and 4. Composition is used here, as elsewhere, for the process of detecting one single alteration which produces the joint effect of two or more. The composition of the ratios of $P$ to $R$, $R$ to $S$, $T$ to $U$, is performed by assuming $A$, altering it in the first ratio into $B$, altering $B$ in the second ratio into $C$, and $C$ in the third ratio into $D$. The joint effect turns $A$ into $D$, and the ratio of $A$ to $D$ is the compounded ratio.”
Another word for compounded ratio is συνημμένος (συνάπτω) which is common in Archimedes and later writers.
It is clear that duplicate ratio, triplicate ratio etc. defined in V. Deff. 9 and 10 are merely particular cases of compound ratio, being in fact the ratios compounded of two, three etc. equal ratios. The use which the Greek geometers made of compounded, duplicate, triplicate ratios etc. is well illustrated by the discovery of Hippocrates that the problem of the duplication of the cube (or, more generally, the construction of a cube which shall be to a given cube in any given ratio) reduces to that of finding “two mean proportionals in continued proportion.” This amounted to seeing that, if $x, y$ are two mean proportionals in continued proportion between any two lines $a, b$, in other words, if $a$ is to $x$ as $x$ to $y$, and $x$ is to $y$ as $y$ to $b$, then a cube with side $a$ is to a cube with side $x$ as $a$ is to $b$; and this is equivalent to saying that $a$ has to $b$ the triplicate ratio of $a$ to $x$.
Euclid is careful to use the forms διπλασίων, τριπλασίων, etc. to express what we translate as duplicate, triplicate etc. ratios; the Greek mathematicians, however, commonly used διπλάσιος λόγος, “double ratio,” τριπλάσιος λόγος, “triple ratio” etc. in the sense of the ratios of 2 to 1, 3 to 1 etc. The effort, if such it was, to keep the one form for the one signification and the other for the other was only partially successful, as there are several instances of the contrary use, e.g. in Archimedes, Nicomachus and Pappus.
The expression for having the ratio which is “duplicate (triplicate) of that which it has to the second” is curious—διπλασίονα (τριπλασίονα) λόγον ἔχειν ἤπερ πρὸς τὸ δεύτερον—ἤπερ being used as if διπλασίονα or τριπλασίονα were a sort of comparative, in the same way as it is used after μείζονα or ἐλάσσονα. Another way of expressing the same thing is to say λόγος διπλασίων (τριπλασίων) τοῦ, ὃν ἔχει… the ratio “duplicate of that (ratio) which…” The explanation of both constructions would seem to be that διπλάσιος or διπλασίων is, as Hultsch translates it in his edition of Pappus (cf. p. 59, 17), duplo maior, where the ablative duplo implies not a difference but a proportion.
The four magnitudes in Def. 10 must of course be in continued proportion (κατὰ τὸ συνεχές). The Greek text as it stands does not state this.
DEFINITION 11.¶
Ὁμόλογα μεγέθη λέγεται τὰ μὲν ἡγούμενα τοῖς ἡγουμένοις τὰ δὲ ἑπόμενα τοῖς ἑπομένοις.
It is difficult to express the meaning of the Greek in as few words. A translation more literal, but conveying less, would be, “Antecedents are called corresponding magnitudes to antecedents, and consequents to consequents.”
I have preferred to translate ὁμόλογος by “corresponding” rather than by “homologous.” I do not agree with Max Simon when he says (Euclid, p. 111) that the technical term “homologous” is not the adjective ὁμόλογος, and does not mean “corresponding,” “agreeing,” but “like in respect of the proportion” (“ähnlich in Bezug auf das Verhältniss”). The definition seems to me to be for the purpose of appropriating to a technical use precisely the ordinary adjective ὁμόλογος, “agreeing” or “corresponding.”
Antecedents, ἡγούμενα, are literally “leading (terms),” and consequents, ἑπόμενα, “following (terms).”
DEFINITION 12.¶
Ἐναλλὰξ λόγος ἐστὶ λῆψις τοῦ ἡγουμένου πρὸς τὸ ἡγούμενον καὶ τοῦ ἑπομένου πρὸς τὸ ἑπόμενον.
We now come to a number of expressions for the transformation of ratios or proportions. The first is ἐναλλάξ, alternately, which would be better described with reference to a proportion of four terms than with reference to a ratio. But probably Euclid defined all the terms in Deff. 12—16 with reference to ratios because to define them with reference to proportions would look like assuming what ought to be proved, namely the legitimacy of the various transformations of proportions (cf. V. 16, 7 Por., 18, 17, 19 Por.). The word ἐναλλάξ is of course a common term which has no exclusive reference to mathematics. But this same use of it with reference to proportions already occurs in Aristotle: Anal. post. I. 5, 74 a 18, καὶ τὸ ἀνάλογον ὅτι ἐναλλάξ, “and that a proportion (is true) alternately, or alternando.” Used with λόγος, as here, the adverb ἐναλλάξ has the sense of an adjective, “alternate”; we have already had it similarly used of “alternate angles” (αἱ ἐναλλὰξ γωνίαι) in the theory of parallels.
DEFINITION 13.¶
Ἀνάπαλιν λόγος ἐστὶ λῆψις τοῦ ἑπομένου ὡς ἡγουμένου πρὸς τὸ ἡγούμενον ὡς ἑπόμενον.
Ἀνάπαλιν, “inversely,” “the other way about,” is also a general term with no exclusive reference to mathematics. For this use of it with reference to proportion cf. Aristotle, De Caelo I. 6, 273 b 32 τὴν ἀναλογίαν ἣν τὰ βάρη ἔχει, οἱ χρόνοι ἀνάπαλιν ἕξουσιν, “the proportion which the weights have, the times will have inversely.” As here used with λόγος, ἀνάπαλιν is, exceptionally, adjectival.
DEFINITION 14.¶
Σύνθεσις λόγου ἐστὶ λῆψις τοῦ ἡγουμένου μετὰ τοῦ ἑπομένου ὡς ἑνὸς πρὸς αὐτὸ τὸ ἑπόμενον.
The composition of a ratio is to be distinguished from the compounding of ratios and compounded ratio (συγκείμενος λόγος) as explained above in the note on Deff. 9, 10. The fact is that συντίθημι and what serves for the passive of it (σύγκειμαι) are used for adding as well as compounding in the sense of compounding ratios. In order to distinguish the two senses, I have always used the word componendo where the sense is that of this definition, though this requires a slight departure from the literal rendering of some passages. Thus the enunciation of V. 17 says, literally, “if magnitudes compounded be in proportion they will also be in proportion separated” (ἐὰν συγκείμενα μεγέθη ἀνάλογον ᾖ, καὶ διαιρεθέντα ἀνάλογον ἔσται). This practically means that, if $A + B$ is to $B$ as $C + D$ is to $D$, then $A$ is to $B$ as $C$ is to $D$. I have accordingly translated as follows: “if magnitudes be proportional componendo, they will also be proportional separando.” (It will be observed that separando, a term explained in the next note, is here used, not relatively to the proportion $A$ is to $B$ as $C$ is to $D$, but relatively to the proportion componendo, viz. $A + B$ is to $B$ as $C + D$ is to $D$.) The corresponding term for componendo in the Greek mathematicians is συνθέντι, literally “to one who has compounded,” i.e. “if we compound.” (For this absolute use of the dative of the participle cf. Nicomachus I. 8, 9 ἀπὸ μονάδος…κατὰ τὸν διπλάσιον λόγον προχωροῦντι μέχρις ἀπείρου, ὅσοι καὶ ἂν γένωνται, οὗτοι πάντες ἀρτιάκις ἄρτιοί εἰσιν. A very good instance from Aristotle is Eth. Nic. I. 5, 1097 b 12 ἐπεκτείνοντι γὰρ ἐπὶ τοὺς γονεῖς καὶ τοὺς ἀπογόνους καὶ τῶν φίλων τοὺς φίλους εἰς ἄπειρον πρόεισιν.) A variation for συνθέντι found in Archimedes is κατὰ σύνθεσιν. Perhaps the more exclusive use of the form συνθέντι by geometers later than Euclid to denote the composition of a ratio, as compared with Euclid’s more general use of σύνθεσις and other parts of the verb συντίθημι or σύγκειμαι, may point to a desire to get rid of ambiguity of terms and to make the terminology of geometry more exact.
DEFINITION 15.¶
Διαίρεσις λόγου ἐστὶ λῆψις τῆς ὑπεροχῆς, ᾗ ὑπερέχει τὸ ἡγούμενον τοῦ ἑπομένου, πρὸς αὐτὸ τὸ ἑπόμενον.
As composition of a ratio means the transformation, e.g., of the ratio of $A$ to $B$ into the ratio of $A + B$ to $B$, so the separation of a ratio indicates the transformation of it into the ratio of $A - B$ to $B$. Thus, as the new antecedent is in one case got by adding the original antecedent to the original consequent, so the antecedent in the other case is obtained by subtracting the original consequent from the original antecedent (it being assumed that the latter is greater than the former). Hence the literal translations of διαίρεσις λόγου, “division of a ratio,” and of διελόντι (the corresponding term to συνθέντι) as dividendo, scarcely give a sufficiently obvious explanation of the meaning. Heiberg accordingly translates by “subtractio rationis,” which again may be thought to depart too far from the Greek. Perhaps “separation” and separando may serve as a compromise.
DEFINITION 16.¶
Ἀναστροφὴ λόγου ἐστὶ λῆψις τοῦ ἡγουμένου πρὸς τὴν ὑπεροχήν, ᾗ ὑπερέχει τὸ ἡγούμενον τοῦ ἑπομένου.
Conversion of a ratio means taking, e.g., instead of the ratio of $A$ to $B$, the ratio of $A$ to $A - B$ ($A$ being again supposed greater than $B$). As ἀναστροφή is used for conversion, so ἀναστρέψαντι is used for convertendo (corresponding to the terms συνθέντι and διελόντι).
DEFINITION 17.¶
Δι’ ἴσου λόγος ἐστὶ πλειόνων ὄντων μεγεθῶν καὶ ἄλλων αὐτοῖς ἴσων τὸ πλῆθος σύνδυο λαμβανομένων καὶ ἐν τῷ αὐτῷ λόγῳ, ὅταν ᾖ ὡς ἐν τοῖς πρώτοις μεγέθεσι τὸ πρῶτον πρὸς τὸ ἔσχατον, οὕτως ἐν τοῖς δευτέροις μεγέθεσι τὸ πρῶτον πρὸς τὸ ἔσχατον· ἢ ἄλλως· Λῆψις τῶν ἄκρων καθ’ ὑπεξαίρεσιν τῶν μέσων.
δι’ ἴσου, ex aequali, must apparently mean ex aequali distantia, at an equal distance or interval, i.e. after an equal number of intervening terms. The wording of the definition suggests that it is rather a proportion ex aequali than a ratio ex aequali which is being defined (cf. Def. 12). The meaning is clear enough. If $a, b, c, d \dots$ be one set of magnitudes, and $A, B, C, D \dots$ another set of magnitudes, such that
$$a \text{ is to } b \text{ as } A \text{ is to } B,$$
$$b \text{ is to } c \text{ as } B \text{ is to } C,$$
and so on, the last proportion being, e.g.,
$$k \text{ is to } l, \text{ as } K \text{ is to } L,$$
then the inference ex aequali is that
$$a \text{ is to } l \text{ as } A \text{ is to } L.$$
The fact that this is so, or the truth of the inference from the hypothesis, is not proved until V. 22. The definition is therefore merely verbal; it gives a convenient name to a certain inference which is of constant application in mathematics. But ex aequali could not be intelligibly defined except with reference to two sets of ratios respectively equal.
DEFINITION 18.¶
Τεταραγμένη δὲ ἀναλογία ἐστίν, ὅταν τριῶν ὄντων μεγεθῶν καὶ ἄλλων αὐτοῖς ἴσων τὸ πλῆθος γίνηται ὡς μὲν ἐν τοῖς πρώτοις μεγέθεσιν ἡγούμενον πρὸς ἑπόμενον, οὕτως ἐν τοῖς δευτέροις μεγέθεσιν ἡγούμενον πρὸς ἑπόμενον, ὡς δὲ ἐν τοῖς πρώτοις μεγέθεσιν ἑπόμενον πρὸς ἄλλο τι, οὕτως ἐν τοῖς δευτέροις ἄλλο τι πρὸς ἡγούμενον.
Though the words δι’ ἴσου, ex aequali, are not in this definition, it gives a description of a case in which the inference ex aequali is still true, as will be hereafter proved in V. 23. A perturbed proportion is an expression for the case when, there being three magnitudes $a, b, c$ and three others $A, B, C$,
$$a \text{ is to } b \text{ as } B \text{ is to } C,$$
and
$$b \text{ is to } c \text{ as } A \text{ is to } B.$$
Another description of this case is found in Archimedes, “the ratios being dissimilarly ordered” (ἀνομοίως τεταγμένων τῶν λόγων). The full description of the inference in this case (as proved in V. 23), namely that
$$a \text{ is to } c \text{ as } A \text{ is to } C,$$
is ex aequali in perturbed proportion (δι’ ἴσου ἐν τεταραγμένῃ ἀναλογίᾳ). Archimedes sometimes omits the δι’ ἴσου, first giving the two proportions and proceeding thus: “therefore, the proportions being dissimilarly ordered, $a$ has to $c$ the same ratio as $A$ has to $C$.”
The fact that Def. 18 describes a particular case in which the inference δι’ ἴσου will be proved true seems to have suggested to some one after Theon’s time the interpolation of another definition between 17 and 18 to describe the ordinary case where the argument ex aequali holds good. The interpolated definition runs thus: “an ordered proportion (τεταγμένη ἀναλογία) arises when, as antecedent is to consequent, so is antecedent to consequent, and, as consequent is to something else, so is consequent to something else.” This case needed no description after Def. 17 itself; and the supposed definition is never used.
After the definitions of Book V. Simson supplies the following axioms.
Equimultiples of the same or of equal magnitudes are equal to one another.
Those magnitudes of which the same or equal magnitudes are equimultiples are equal to one another.
A multiple of a greater magnitude is greater than the same multiple of a less.
That magnitude of which a multiple is greater than the same multiple of another is greater than that other magnitude.