SOURCE: A History of Greek Mathematics, Clarendon Press, 1921, 446 p.

[In the following excerpt, Heath discusses the significance and content of several of Euclid's lesser-known works.]

The Data

…Most closely connected with the Elements as dealing with plane geometry, the subject-matter of Books I-VI, is the Data, which is accessible in the Heiberg-Menge edition of the Greek text, and also in the translation annexed by Simson to his edition of the Elements (although this translation is based on an inferior text). The book was regarded as important enough to be included in the Treasury of Analysis … as known to Pappus, and Pappus gives a description of it; the description shows that there were differences between Pappus's text and ours, for, though Propositions 1-62 correspond to the description, as also do Propositions 87-94 relating to circles at the end of the book, the intervening propositions do not exactly agree, the differences, however, affecting the distribution and numbering of the propositions rather than their substance. The book begins with definitions of the senses in which things are said to be given. Things such as areas, straight lines, angles and ratios are said to be 'given in magnitude when we can make others equal to them' (Defs. 1-2). Rectilineal figures are 'given in species' when their angles are severally given as well as the ratios of the sides to one another (Def. 3). Points, lines and angles are 'given in position' 'when they always occupy the same place' : a not very illuminating definition (4). A circle is given in position and in magnitude when the centre is given in position and the radius in magnitude (6); and so on. The object of the proposition called a Datum is to prove that, if in a given figure certain parts or relations are given, other parts or relations are also given, in one or other of these senses.

It is clear that a systematic collection of Data such as Euclid's would very much facilitate and shorten the procedure in analysis; this no doubt accounts for its inclusion in the Treasury of Analysis. It is to be observed that this form of proposition does not actually determine the thing or relation which is shown to be given, but merely proves that it can be determined when once the facts stated in the hypothesis are known; if the proposition stated that a certain thing is so and so, e.g. that a certain straight line in the figure is of a certain length, it would be a theorem; if it directed us to find the thing instead of proving that it is 'given', it would be a problem; hence many propositions of the form of the Data could alternatively be stated in the form of theorems or problems….

On divisions (of figures)

The only other work [in addition to Elements and Data] of Euclid in pure geometry which has survived (but not in Greek) is the book On divisions (of figures)…. It is mentioned by Proclus, who gives some hints as to its content¹; he speaks of the business of the author being divisions of figures, circles or rectilineal figures, and remarks that the parts may be like in definition or notion, or unlike; thus to divide a triangle into triangles is to divide it into like figures, whereas to divide it into a triangle and a quadrilateral is to divide it into unlike figures. These hints enable us to check to some extent the genuineness of the books dealing with divisions of figures which have come down through the Arabic. It was John Dee who first brought to light a treatise De divisionibus by one Muhammad Bagdadinus (died 1141) and handed over a copy of it (in Latin) to Commandinus in 1563; it was published by the latter in Dee's name and his own in 1570. Dee appears not to have translated the book from the Arabic himself, but to have made a copy for Commandinus from a manuscript of a Latin translation which he himself possessed at one time but which was apparently stolen and probably destroyed some twenty years after the copy was made. The copy does not seem to have been made from the Cotton MS. which passed to the British Museum after it had been almost destroyed by a fire in 1731.² The Latin translation may have been that made by Gherard of Cremona (1114-87), since in the list of his numerous translations a 'liber divisionum' occurs. But the Arabic original cannot have been a direct translation from Euclid, and probably was not even a direct adaptation of it, since it contains mistakes and unmathematical expressions; moreover, as it does not contain the propositions about the division of a circle alluded to by Proclus, it can scarcely have contained more than a fragment of Euclid's original work. But Woepcke found in a manuscript at Paris a treatise in Arabic on the division of figures, which he translated and published in 1851. It is expressly attributed to Euclid in the manuscript and corresponds to the indications of the content given by Proclus. Here we find divisions of different rectilinear figures into figures of the same kind, e.g. of triangles into triangles or trapezia into trapezia, and also divisions into 'unlike' figures, e.g. that of a triangle by a straight line parallel to the base. The missing propositions about the division of a circle are also here: 'to divide into two equal parts a given figure bounded by an arc of a circle and two straight lines including a given angle' (28), and 'to draw in a given circle two parallel straight lines cutting off a certain fraction from the circle' (29). Unfortunately the proofs are given of only four propositions out of 36, namely Propositions 19, 20, 28, 29, the Arabic translator having found the rest too easy and omitted them. But the genuineness of the treatise edited by Woepcke is attested by the facts that the four proofs which remain are elegant and depend on propositions in the Elements, and that there is a lemma with a true Greek ring, 'to apply to a straight line a rectangle equal to the rectangle contained by AB, AC and deficient by a square' (18). Moreover, the treatise is no fragment, but ends with the words, 'end of the treatise', and is (but for the missing proofs) a well-ordered and compact whole. Hence we may safely conclude that Woepcke's tract represents not only Euclid's work but the whole of it. The portion of the Practica geometriae of Leonardo of Pisa which deals with the division of figures seems to be a restoration and extension of Euclid's work; Leonardo must presumably have come across a version of it from the Arabic.

The type of problem which Euclid's treatise was designed to solve may be stated in general terms as that of dividing a given figure by one or more straight lines into parts having prescribed ratios to one another or to other given areas. The figures divided are the triangle, the parallelogram, the trapezium, the quadrilateral, a figure bounded by an arc of a circle and two straight lines, and the circle. The figures are divided into two equal parts, or two parts in a given ratio; or again, a given fraction of the figure is to be cut off, or the figure is to be divided into several parts in given ratios. The dividing straight lines may be transversals drawn through a point situated at a vertex of the figure, or a point on any side, on one of two parallel sides, in the interior of the figure, outside the figure, and so on; or again, they may be merely parallel lines, or lines parallel to a base. The treatise also includes auxiliary propositions, (1) 'to apply to a given straight line a rectangle equal to a given area and deficient by a square', the proposition already mentioned, which is equivalent to the algebraical solution of the equation ax − x² = b² and depends on Eucl. II. 5 …; (2) propositions in proportion involving unequal instead of equal ratios:

If a.d > or < b.c, then a:b > or < c:d respectively.

If a:b > c:d, then (a∓b): b > (c∓d):d.

If a:b < c:d, then (a-b): b < (c−d):d….

Lost geometrical works

The Pseudaria

The other purely geometrical works of Euclid are lost so far as is known at present. One of these again belongs to the domain of elementary geometry. This is the Pseudaria, or 'Book of Fallacies', as it is called by Proclus, which is clearly the same work as the 'Pseudographemata' of Euclid mentioned by a commentator on Aristotle in terms which agree with Proclus's description.³ Proclus says of Euclid that,

Inasmuch as many things, while appearing to rest on truth and to follow from scientific principles, really tend to lead one astray from the principles and deceive the more superficial minds, he has handed down methods for the discriminative understanding of these things as well, by the use of which methods we shall be able to give beginners in this study practice in the discovery of paralogisms, and to avoid being ourselves misled. The treatise by which he puts this machinery in our hands he entitled (the book) of Pseudaria, enumerating in order their various kinds, exercising our intelligence in each case by theorems of all sorts, setting the true side by side with the false, and combining the refutation of error with practical illustration. This book then is by way of cathartic and exercise, while the Elements contain the irrefragable and complete guide to the actual scientific investigation of the subjects of geometry.⁴

The connexion of the book with the Elements and the reference to its usefulness for beginners show that it did not go beyond the limits of elementary geometry. We now come to the lost works belonging to higher geometry. The most important was evidently

The Porisms

Our only source of information about the nature and contents of the Porisms is Pappus. In his general preface about the books composing the Treasury of Analysis Pappus writes as follows⁵ (I put in square brackets the words bracketed by Hultsch).

After the Tangencies (of Apollonius) come, in three Books, the Porisms of Euclid, a collection [in the view of many] most ingeniously devised for the analysis of the more weighty problems, [and] although nature presents an unlimited number of such porisms, [they have added nothing to what was originally written by Euclid, except that some before my time have shown their want of taste by adding to a few (of the propositions) second proofs, each (proposition) admitting of a definite number of demonstrations, as we have shown, and Euclid having given one for each, namely that which is the most lucid. These porisms embody a theory subtle, natural, necessary, and of considerable generality, which is fascinating to those who can see and produce results].

Now all the varieties of porisms belong, neither to theorems nor problems, but to a species occupying a sort of intermediate position [so that their enunciations can be formed like those of either theorems or problems], the result being that, of the great number of geometers, some regarded them as of the class of theorems, and others of problems, looking only to the form of the proposition. But that the ancients knew better the difference between these three things is clear from the definitions. For they said that a theorem is that which is proposed with a view to the demonstration of the very thing proposed, a problem that which is thrown out with a view to the construction of the very thing proposed, and a porism that which is proposed with a view to the producing of the very thing proposed. [But this definition of the porism was changed by the more recent writers who could not produce everything, but used these elements and proved only the fact that that which is sought really exists, but did not produce it, and were accordingly confuted by the definition and the whole doctrine. They based their definition on an incidental characteristic, thus: A porism is that which falls short of a locus-theorem in respect of its hypothesis. Of this kind of porisms loci are a species, and they abound in the Treasury of Analysis; but this species has been collected, named, and handed down separately from the porisms, because it is more widely diffused than the other species] … But it has further become characteristic of porisms that, owing to their complication, the enunciations are put in a contracted form, much being by usage left to be understood; so that many geometers understand them only in a partial way and are ignorant of the more essential features of their content.

[Now to comprehend a number of propositions in one enunciation is by no means easy in these porisms, because Euclid himself has not in fact given many of each species, but chosen, for examples, one or a few out of a great multitude. But at the beginning of the first book he has given some propositions, to the number of ten, of one species, namely that more fruitful species consisting of loci.] Consequently, finding that these admitted of being comprehended in our enunciation, we have set it out thus:

If, in a system of four straight lines which cut one another two and two, three points on one straight line be given, while the rest except one lie on different straight lines given in position, the remaining point also will lie on a straight line given in position.

This has only been enunciated of four straight lines, of which not more than two pass through the same point, but it is not known (to most people) that it is true of any assigned number of straight lines if enunciated thus:

If any number of straight lines cut one another, not more than two (passing) through the same point, and all the points (of intersection situated) on one of them be given, and if each of those which are on another (of them) lie on a straight line given in position—

or still more generally thus:

if any number of straight lines cut one another, not more than two (passing) through the same point, and all the points (of intersection situated) on one of them be given, while of the other points of intersection in multitude equal to a triangular number a number corresponding to the side of this triangular number lie respectively on straight lines given in position, provided that of these latter points no three are at the angular points of a triangle (sc. having for sides three of the given straight lines)—each of the remaining points will lie on a straight line given in position.⁶

It is probable that the writer of the Elements was not unaware of this, but that he only set out the principle; and he seems, in the case of all the porisms, to have laid down the principles and the seed only [of many important things], the kinds of which should be distinguished according to the differences, not of their hypotheses, but of the results and the things sought. [All the hypotheses are different from one another because they are entirely special, but each of the results and things sought, being one and the same, follow from many different hypotheses.]

We must then in the first book distinguish the following kinds of things sought:

At the beginning of the book is this proposition:

I. If from two given points straight lines be drawn meeting on a straight line given in position, and one cut off from a straight line given in position (a segment measured) to a given point on it, the other will also cut off from another (straight line a segment) having to the first a given ratio.

Following on this (we have to prove)

II. that such and such a point lies on a straight line given in position;

III. that the ratio of such and such a pair of straight lines is given;

&c. &c. (up to XXIX).

The three books of the porisms contain 38 lemmas; of the theorems themselves there are 171.

Pappus further gives lemmas to the Porisms.⁷

With Pappus's account of Porisms must be compared the passages of Proclus on the same subject. Proclus distinguishes the two senses of the word [porisma]. The first is that of a corollary, where something appears as an incidental result of a proposition, obtained without trouble or special seeking, a sort of bonus which the investigation has presented us with.⁸ The other sense is that of Euclid's Porisms. In this sense

porism is the name given to things which are sought, but need some finding and are neither pure bringing into existence nor simple theoretic argument. For (to prove) that the angles at the base of isosceles reference are equal is matter of theoretic argument, and it is with reference to things existing that such knowledge is (obtained). But to bisect an angle, to construct a triangle, to cut off, or to place—all these things demand the making of something; and to find the centre of a given circle, or to find the greatest common measure of two given commensurable magnitudes, or the like, is in some sort intermediate between theorems and problems. For in these cases there is no bringing into existence of the things sought, but finding of them; nor is the procedure purely theoretic. For it is necessary to bring what is sought into view and exhibit it to the eye. Such are the porisms which Euclid wrote and arranged in three books of Porisms.⁹

Proclus's definition thus agrees well enough with the first, the 'older', definition of Pappus. A porism occupies a place between a theorem and a problem; it deals with something already existing, as a theorem does, but has to find it (e.g. the centre of a circle), and, as a certain operation is therefore necessary, it partakes to that extent of the nature of a problem, which requires us to construct or produce something not previously existing. Thus, besides III. 1 and X. 3, 4 of the Elements mentioned by Proclus, the following proposition are real porisms: III. 25, VI. 11-13, VII. 33, 34, 36, 39, VIII. 2, 4, X. 10, XIII. 18. Similarly, in Archimedes's On the Sphere and Cylinder, I. 2-6 might be called porisms.

The enunciation given by Pappus as comprehending ten of Euclid's propositions may not reproduce the form of Euclid's enunciations; but, comparing the result to be proved, that certain points lie on straight lines given in position, with the class indicated by II above, where the question is of such and such a point lying on a straight line given in position, and with other classes, e.g. (V) that such and such a line is given in position, (VI) that such and such a line verges to a given point, (XXVII) that there exists a given point such that straight lines drawn from it to such and such (circles) will contain a triangle given in species, we may conclude that a usual form of a porism was 'to prove that it is possible to find a point with such and such a property' or 'a straight line on which lie all the points satisfying given conditions', and so on.

The above exhausts all the positive information which we have about the nature of a porism and the contents of Euclid's Porisms. It is obscure and leaves great scope for speculation and controversy; naturally, therefore, the problem of restoring the Porisms has had a great fascination for distinguished mathematicians ever since the revival of learning. But it has proved beyond them all. Some contributions to a solution have, it is true, been made, mainly by Simson and Chasles. The first claim to have restored the Porisms seems to be that of Albert Girard (about 1590-1633), who spoke (1626) of an early publication of his results, which, however, never saw the light. The great Fermat (1601-65) gave his idea of a 'porism', illustrating it by five examples which are very interesting in themselves¹⁰; but he did not succeed in connecting them with the description of Euclid's Porisms by Pappus, and, though he expressed a hope of being able to produce a complete restoration of the latter, his hope was not realized. It was left for Robert Simson (1687-1768) to make the first decisive step towards the solution of the problem.¹¹ He succeeded in explaining the meaning of the actual porisms enunciated in such general terms by Pappus. In his tract on Porisms he proves the first porism given by Pappus in its ten different cases, which, according to Pappus, Euclid distinguished (these propositions are of the class connected with loci); after this he gives a number of other propositions from Pappus, some auxiliary propositions, and some 29 'porisms', some of which are meant to illustrate the classes I, VI, XV, XXVII-XXIX distinguished by Pappus. Simson was able to evolve a definition of a porism which is perhaps more easily understood in Chasles's translation: 'Le porisme est une proposition dans laquelle on demande de démontrer qu'une chose ou plusieurs choses sont données, qui, ainsi que l'une quelconque d'une infinité d'autres choses non données, mais dont chacune est avec des choses données dans une même relation, ont une propriété commune, décrite dans la proposition.' We need not follow Simson's English or Scottish successors, Lawson (1777), Playfair (1794), W. Wallace (1798), Lord Brougham (1798), in their further speculations, nor the controversies between the Frenchmen, A. J. H. Vincent and P. Breton (de Champ), nor the latter's claim to priority as against Chasles; the work of Chasles himself (Les trois livres des Porismes d'Euclide rétablis… Paris, 1860) alone needs to be mentioned. Chasles adopted the definition of a porism given by Simson, but showed how it could be expressed in a different form. 'Porisms are incomplete theorems which express certain relations existing between things variable in accordance with a common law, relations which are indicated in the enunciation of the porism, but which need to be completed by determining the magnitude or position of certain things which are the consequences of the hypotheses and which would be determined in the enunciation of a theorem properly so called or a complete theorem.' Chasles succeeded in elucidating the connexion between a porism and a locus as described by Pappus, though he gave an inexact translation of the actual words of Pappus: 'Ce qui constitue le porisme est ce qui manque à l'hypothèse d'un théoréme local (en d'autres termes, le porisme est inférieur, par l'hypothèse, au théorème local; c'est à dire que quand quelques parties d'une proposition locale n'ont pas dans l'énoncé la détermination qui leur est propre, cette proposition cesse d'être regardée comme un théorème et devient un porisme)'; here the words italicized are not quite what Pappus said, viz. that 'a porism is that which falls short of a locustheorem in respect of its hypothesis', but the explanation in brackets is correct enough if we substitute 'in respect of for 'par' ('by'). The work of Chasles is historically important because it was in the course of his researches on this subject that he was led to the idea of anharmonic ratios; and he was probably right in thinking that the Porisms were propositions belonging to the modern theory of transversals and to projective geometry. But, as a restoration of Euclid's work, Chasles's Porisms cannot be regarded as satisfactory. One consideration alone is, to my mind, conclusive on this point. Chasles made 'porisms' out of Pappus's various lemmas to Euclid's porisms and comparatively easy deductions from those lemmas. Now we have experience of Pappus's lemmas to books which still survive, e.g. the Conics of Apollonius; and, to judge by these instances, his lemmas stood in a most ancillary relation to the propositions to which they relate, and do not in the least compare with them in difficulty and importance. Hence it is all but impossible to believe that the lemmas to the porisms were themselves porisms such as were Euclid's own porisms; on the contrary, the analogy of Pappus's other sets of lemmas makes it all but necessary to regard the lemmas in question as merely supplying proofs of simple propositions assumed by Euclid without proof in the course of the demonstration of the actual porisms. This being so, it appears that the problem of the complete restoration of Euclid's three Books still awaits a solution, or rather that it will never be solved unless in the event of discovery of fresh documents.

At the same time the lemmas of Pappus to the Porisms are by no means insignificant propositions in themselves, and, if the usual relation of lemmas to substantive propositions holds, it follows that the Porisms was a distinctly advanced work, perhaps the most important that Euelid ever wrote; its loss is therefore much to be deplored. Zeuthen has an interesting remark à propos of the proposition which Pappus quotes as the first proposition of Book I, 'If from two given points straight lines be drawn meeting on a straight line given in position, and one of them cut off from a straight line given in position (a segment measured) towards a given point on it, the other will also cut off from another (straight line a segment) bearing to the first a given ratio.' This proposition is also true if there be substituted for the first given straight line a conic regarded as the 'locus with respect to four lines', and the proposition so extended can be used for completing Apollonius's exposition of that locus. Zeuthen suggests, on this ground, that the Porisms were in part by-products of the theory of conics and in part auxiliary means for the study of conics, and that Euclid called them by the same name as that applied to corollaries because they were corollaries with respect to conics.¹² This, however, is a pure conjecture.

The Conics

Pappus says of this lost work: 'The four books of Euclid's Conics were completed by Apollonius, who added four more and gave us eight books of Conics.'¹³ It is probable that Euclid's work was already lost by Pappus's time, for he goes on to speak of 'Aristaeus who wrote the still extant five books of Solid Loci … connected with, or supplementary to, the conics'.¹⁴ This latter work seems to have been a treatise on conics regarded as loci; for 'solid loci' was a term appropriated to conics, as distinct from 'plane loci', which were straight lines and circles. In another passage Pappus (or an interpolator) speaks of the 'conics' of Aristaeus the 'elder',¹⁵ evidently referring to the same book. Euclid no doubt wrote on the general theory of conics, as Apollonius did, but only covered the ground of Apollonius's first three books, since Apollonius says that no one before him had touched the subject of Book IV (which, however, is not important). As in the case of the Elements, Euclid would naturally collect and rearrange, in a systematic exposition, all that had been discovered up to date in the theory of conics. That Euclid's treatise covered most of the essentials up to the last part of Apollonius's Book III seems clear from the fact that Apollonius only claims originality for some propositions connected with the 'three- and four-line locus', observing that Euclid had not completely worked out the synthesis of the said locus, which, indeed, was not possible without the propositions referred to. Pappus (or an interpolator)¹⁶ excuses Euclid on the ground that he made no claim to go beyond the discoveries of Aristaeus, but only wrote so much about the locus as was possible with the aid of Aristaeus's conics. We may conclude that Aristaeus's book preceded Euclid's, and that it was, at least in point of originality, more important. When Archimedes refers to propositions in conics as having been proved in the 'elements of conics', he clearly refers to these two treatises, and the other propositions to which he refers as well known and not needing proof were doubtless taken from the same sources. Euclid still used the old names for the conic sections (sections of a right-angled, acute-angled, and obtuse-angled cone respectively), but he was aware that an ellipse could be obtained by cutting (through) a cone in any manner by a plane not parallel to the base, and also by cutting a cylinder; this is clear from a sentence in his Phaenomena to the effect that, 'If a cone or a cylinder be cut by a plane not parallel to the base, this section is a section of an acute-angled cone, which is like a shield….'

The Surface-Loci.…

Like the Data and the Porisms, this treatise in two Books is mentioned by Pappus as belonging to the Treasury of Analysis. What is meant by surface-loci, literally 'loci on a surface' is not entirely clear, but we are able to form a conjecture on the subject by means of remarks in Proclus and Pappus. The former says (1) that a locus is 'a position of a line or of a surface which has (throughout it) one and the same property',¹⁷ and (2) that 'of locus-theorems some are constructed on lines and others on surfaces';¹⁸ the effect of these statements together seems to be that 'loci on lines' are loci which are lines, and 'loci on surfaces' loci which are surfaces. On the other hand, the possibility does not seem to be excluded that loci on surfaces may be loci traced on surfaces; for Pappus says in one place that the equivalent of the quadratrix can be got geometrically 'by means of loci on surfaces as follows'¹⁹ and then proceeds to use a spiral described on a cylinder (the cylindrical helix), and it is consistent with this that in another passage²⁰ (bracketed, however, by Hultsch) 'linear' loci are said to be exhibited … or realized from loci on surfaces, for the quadratrix is a 'linear' locus, i.e. a locus of an order higher than a plane locus (a straight line or circle) and a 'solid' locus (a conic). However this may be, Euclid's Surface-Loci probably included such loci as were cones, cylinders and spheres. The two lemmas given by Pappus lend some colour to this view. The first of these²¹ and the figure attached to it are unsatisfactory as they stand, but Tannery indicated a possible restoration.²² If this is right, it suggests that one of the loci contained all the points on the elliptical parallel sections of a cylinder, and was therefore an oblique circular cylinder. Other assumptions with regard to the conditions to which the lines in the figure may be subject would suggest that other loci dealt with were cones regarded as containing all points on particular parallel elliptical sections of the cones. In the second lemma Pappus states and gives a complete proof of the focus-and-directrix property of a conic, viz. that the locus of a point the distance of which from a given point is in a given ratio to its distance from a fixed straight line is a conic section, which is an ellipse, a parabola or a hyperbola according as the given ratio is less than, equal to, or greater than unity.²³ Two conjectures are possible as to the application of this theorem in Euclid's Surface-Loci, (a) It may have been used to prove that the locus of a point the distance of which from a given straight line is in a given ratio to its distance from a given plane is a certain cone. Or (b) it may have been used to prove that the locus of a point the distance of which from a given point is in a given ratio to its distance from a given plane is the surface formed by the revolution of a conic about its major or conjugate axis.²⁴

We come now to Euclid's works under the head of

Applied mathematics

The Phaenomena

The book on sphaeric intended for use in astronomy and entitled Phaenomena has already been noticed … It is extant in Greek and was included in Gregory's edition of Euclid. The text of Gregory, however, represents the later of two recensions which differ considerably (especially in Propositions 9 to 16). The best manuscript of this later recension (b) is the famous Vat. gr. 204 of the tenth century, while the best manuscript of the older and better version (a) is the Viennese MS. Vind. gr. XXXI. 13 of the twelfth century. A new text edited by Menge and taking account of both recensions is now available in the last volume of the Heiberg-Menge edition of Euclid.²⁵

Optics and Catoptrics

The Optics, a treatise included by Pappus in the collection of works known as the Little Astronomy, survives in two forms. One is the recension of Theon translated by Zambertus in 1505; the Greek text was first edited by Johannes Pena (de la Pène) in 1557, and this form of the treatise was alone included in the editions up to Gregory's. But Heiberg discovered the earlier form in two manuscripts, one at Vienna (Vind. gr. XXXI. 13) and one at Florence (Laurent. XXVIII. 3), and both recensions are contained in vol. vii of the Heiberg-Menge text of Euclid (Teubner, 1895). There is no reason to doubt that the earlier recension is Euclid's own work; the style is much more like that of the Elements, and the proofs of the propositions are more complete and clear. The later recension is further differentiated by a preface of some length, which is said by a scholiast to be taken from the commentary or elucidation by Theon. It would appear that the text of this recension is Theon's, and that the preface was a reproduction by a pupil of what was explained by Theon in lectures. It cannot have been written much, if anything, later than Theon's time, for it is quoted by Nemesius about A.D. 400. Only the earlier and genuine version need concern us here. It is a kind of elementary treatise on perspective, and it may have been intended to forearm students of astronomy against paradoxical theories such as those of the Epicureans, who maintained that the heavenly bodies are of the size that they look. It begins in the orthodox fashion with Definitions, the first of which embodies the same idea of the process of vision as we find in Plato, namely that it is due to rays proceeding from our eyes and impinging upon the object, instead of the other way about: 'the straight lines (rays) which issue from the eye traverse the distances (or dimensions) of great magnitudes'; Def. 2: 'The figure contained by the visual rays is a cone which has its vertex in the eye, and its base at the extremities of the objects seen'; Def. 3: 'And those things are seen on which the visual rays impinge, while those are not seen on which they do not'; Def. 4: 'Things seen under a greater angle appear greater, and those under a lesser angle less, while things seen under equal angles appear equal'; Def. 7: 'Things seen under more angles appear more distinctly.' Euclid assumed that the visual rays are not 'continuous', i.e. not absolutely close together, but are separated by a certain distance, and hence he concluded, in Proposition 1, that we can never really see the whole of any object, though we seem to do so. Apart, however, from such inferences as these from false hypotheses, there is much in the treatise that is sound. Euclid has the essential truth that the rays are straight; and it makes no difference geometrically whether they proceed from the eye or the object. Then, after propositions explaining the differences in the apparent size of an object according to its position relatively to the eye, he proves that the apparent sizes of two equal and parallel objects are not proportional to their distances from the eye (Prop. 8)….

From Proposition 6 can easily be deduced the fundamental proposition in perspective that parallel lines (regarded as equidistant throughout) appear to meet. There are four simple propositions in heights and distances, e.g. to find the height of an object (1) when the sun is shining (Prop. 18), (2) when it is not (Prop. 19): similar triangles are, of course, used and the horizontal mirror appears in the second case in the orthodox manner, with the assumption that the angles of incidence and reflection of a ray are equal, 'as is explained in the Catoptrica (or theory of mirrors)'. Propositions 23-7 prove that, if an eye sees a sphere, it sees less than half of the sphere, and the contour of what is seen appears to be a circle; if the eye approaches nearer to the sphere the portion seen becomes less, though it appears greater; if we see the sphere with two eyes, we see a hemisphere, or more than a hemisphere, or less than a hemisphere according as the distance between the eyes is equal to, greater than, or less than the diameter of the sphere; these propositions are comparable with Aristarchus's Proposition 2 stating that, if a sphere be illuminated by a larger sphere, the illuminated portion of the former will be greater than a hemisphere. Similar propositions with regard to the cylinder and cone follow (Props. 28-33). Next Euclid considers the conditions for the apparent equality of different diameters of a circle as seen from an eye occupying various positions outside the plane of the circle (Props. 34-7); he shows that all diameters will appear equal, or the circle will really look like a circle, if the line joining the eye to the centre is perpendicular to the plane of the circle, or, not being perpendicular to that plane, is equal to the length of the radius, but this will not otherwise be the case (35), so that (36) a chariot wheel will sometimes appear circular, sometimes awry, according to the position of the eye. Propositions 37 and 38 prove, the one that there is a locus such that, if the eye remains at one point of it, while a straight line moves so that its extremities always lie on it, the line will always appear of the same length in whatever position it is placed (not being one in which either of the extremities coincides with, or the extremities are on opposite sides of, the point at which the eye is placed), the locus being, of course, a circle in which the straight line is placed as a chord, when it necessarily subtends the same angle at the circumference or at the centre, and therefore at the eye, if placed at a point of the circumference or at the centre; the other proves the same thing for the case where the line is fixed with its extremities on the locus, while the eye moves upon it. The same idea underlies several other propositions, e.g. Proposition 45, which proves that a common point can be found from which unequal magnitudes will appear equal. The unequal magnitudes are straight lines BC, CD so placed that BCD is a straight line. A segment greater than a semicircle is described on BC, and a similar segment on CD. The segments will then intersect at F, and the angles subtended by BC and CD at F are equal. The rest of the treatise is of the same character, and it need not be further described.

The Catoptrica published by Heiberg in the same volume is not by Euclid, but is a compilation made at a much later date, possibly by Theon of Alexandria, from ancient works on the subject and mainly no doubt from those of Archimedes and Heron. Theon²⁶ himself quotes a Catoptrica by Archimedes, and Olympiodorus²⁷ quotes Archimedes as having proved the fact which appears as an axiom in the Catoptrica now in question, namely that, if an object be placed just out of sight at the bottom of a vessel, it will become visible over the edge when water is poured in. It is not even certain that Euclid wrote Catoptrica at all, since, if the treatise was Theon's, Proclus may have assigned it to Euclid through inadvertence.

Music

Proclus attributes to Euclid a work on the Elements of Music…;²⁸ so does Marinus.²⁹ As a matter of fact, two musical treatises attributed to Euclid are still extant, the Sectio Canonis… and the Introductio harmonica…. The latter, however, is certainly not by Euclid, but by Cleonides, a pupil of Aristoxenus. The question remains, in what relation does the Sectio Canonis stand to the 'Elements' mentioned by Proclus and Marinus? The Sectio gives the Pythagorean theory of music, but is altogether too partial and slight to deserve the title 'Elements of Music'. Jan, the editor of the Musici Graeci, thought that the Sectio was a sort of summary account extracted from the 'Elements' by Euclid himself, which hardly seems likely; he maintained that it is the genuine work of Euclid on the grounds (1) that the style and diction and the form of the propositions agree well with what we find in Euclid's Elements, and (2) that Porphyry in his commentary on Ptolemy's Harmonica thrice quotes Euclid as the author of a Sectio Canonis.³⁰ The latest editor, Menge, points out that the extract given by Porphyry shows some differences from our text and contains some things quite unworthy of Euclid; hence he is inclined to think that the work as we have it is not actually by Euclid, but was extracted by some other author of less ability from the genuine Elements of Music by Euclid.

Works on mechanics attributed to Euclid

The Arabian list of Euclid's works further includes among those held to be genuine 'the book of the Heavy and Light'. This is apparently the tract De levi et ponderoso included by Hervagius in the Basel Latin translation of 1537 and by Gregory in his edition. That it comes from the Greek is made clear by the lettering of the figures; and this is confirmed by the fact that another, very slightly different, version exists at Dresden (Cod. Dresdensis Db. 86), which is evidently a version of an Arabic translation from the Greek, since the lettering of the figures follows the order characteristic of such Arabic translations, a, b, g, d, e, z, h, t. The tract consists of nine definitions or axioms and five propositions. Among the definitions are these: Bodies are equal, different, or greater in size according as they occupy equal, different, or greater spaces (1-3). Bodies are equal in power or in virtue which move over equal distances in the same medium of air or water in equal times (4), while the power or virtue is greater if the motion takes less time, and less if it takes more (6). Bodies are of the same kind if, being equal in size, they are also equal in power when the medium is the same; they are different in kind when, being equal in size, they are not equal in power or virtue (7, 8). Of bodies different in kind, that has more power which is more dense (solidius) (9). With these hypotheses, the author attempts to prove (Props. 1, 3, 5) that, of bodies which traverse unequal spaces in equal times, that which traverses the greater space has the greater power and that, of bodies of the same kind, the power is proportional to the size, and conversely, if the power is proportional to the size, the bodies are of the same kind. We recognize in the potentia or virtus the same thing as the [dynamis] and [ischys] of Aristotle.³¹ The property assigned by the author to bodies of the same kind is quite different from what we attribute to bodies of the same specific gravity; he purports to prove that bodies of the same kind have power proportional to their size, and the effect of this, combined with the definitions, is that they move at speeds proportional to their volumes. Thus the tract is the most precise statement that we possess of the principle of Aristotle's dynamics, a principle which persisted until Benedetti (1530-90) and Galilei (1564-1642) proved its falsity.

There are yet other fragments on mechanics associated with the name of Euclid. One is a tract translated by Woepcke from the Arabic in 1851 under the title 'Le livre d'Euclide sur la balance', a work which, although spoiled by some commentator, seems to go back to a Greek original and to have been an attempt to establish a theory of the lever, not from a general principle of dynamics like that of Aristotle, but from a few simple axioms such as the experience of daily life might suggest. The original work may have been earlier than Archimedes and may have been written by a contemporary of Euclid. A third fragment, unearthed by Duhem from manuscripts in the Bibliothèque Nationale in Paris, contains four propositions purporting to be 'liber Euclidis de ponderibus secundum terminorum circumferentiam'. The first of the propositions, connecting the law of the lever with the size of the circles described by its ends, recalls the similar demonstration in the Aristotelian Mechanica; the others attempt to give a theory of the balance, taking account of the weight of the lever itself, and assuming that a portion of it (regarded as cylindrical) may be supposed to be detached and replaced by an equal weight suspended from its middle point. The three fragments supplement each other in a curious way, and it is a question whether they belonged to one treatise or were due to different authors. In any case there seems to be no independent evidence that Euclid was the author of any of the fragments, or that he wrote on mechanics at all.³²

Notes

¹ Proclus on Eucl. I, p. 144. 22-6.

² The question is fully discussed by R. C. Archibald, Euclid's Book on Divisions of Figures with a restoration based on Woepcke's text and on the Practica Geometriae of Leonardo Pisano (Cambridge 1915).

³ Michael Ephesius, Comm. on Arist. Soph. EL, fol. 25^V, p. 76. 23 Wallies.

⁴ Proclus on Eucl. I, p. 70. 1-18. Cf. a scholium to Plato's Theaetetus 191 B, which says that the fallacies did not arise through any importation of sense-perception into the domain of non-sensibles.

⁵ Pappus, vii, pp. 648-60.

⁶ Loria (Le scienze esatte nell'antica Grecia, pp. 256-7) gives the meaning of this as follows, pointing out that Simson first discovered it: 'If a complete n-lateral be deformed so that its sides respectively turn about n points on a straight line, and (n−1) of its ½ n (n−1) vertices move on as many straight lines, the other ½ (n−1) (n−2) of its vertices likewise move on as many straight lines: but it is necessary that it should be impossible to form with the (n−1) vertices any triangle having for sides the sides of the polygon.'

⁷ Pappus, vii, pp. 866-918; Euclid, ed. Heiberg-Menge, vol. viii, pp. 243-74.

⁸ Proclus on Eucl. I, pp. 212. 14; 301. 22.

⁹Ib., p. 301. 25 sq.

¹⁰Œurres de Fermat, ed. Tannery and Henry, I, p. 76-84.

¹¹ Roberti Simson Opera quaedam reliqua, 1776, pp. 315-594.

¹² Zeuthen, Die Lehre von den Kegelschnitten in Altertum, 1886, pp. 168, 173-4.

¹³ Pappus, vii, p. 672. 18.

¹⁴ Cf. Pappus, vii, p. 636. 23.

¹⁵Ib. vii, p. 672. 12.

¹⁶Ib. vii, pp. 676. 25-678. 6.

¹⁷ Proclus on Eucl. I, p. 394. 17.

¹⁸Ib., p. 394. 19.

¹⁹ Pappus, iv, p. 258. 20-25.

²⁰Ib. vii. 662. 9.

²¹ Pappus, vii, p. 1004. 17; Euclid, ed. Heiberg-Menge, vol. viii, p. 274.

²² Tannery in Bulletin des sciences mathématiques, 2^e série, VI, p. 149.

²³ Pappus, vii, pp. 1004. 23-1014; Euclid, vol. viii, pp. 275-81.

²⁴ For further details, see The Works of Archimedes, pp. lxii-lxv.

²⁵Euclidis Phaenomena et scripta Musica edidit Henricus Menge. Fragmenta collegit et disposuit J. L. Heiberg, Teubner, 1916.

²⁶ Theon, Comm. on Ptolemy's Syntaxis, i, p. 10.

²⁷Comment. On Arist. Meteorolog. ii, p. 94, Ideler, p. 211. 18 Busse.

²⁸ Proclus on Eucl. I, p. 69. 3.

²⁹ Marinus, Comm. on the Data (Euclid, vol. vi, p. 254. 19).

³⁰ See Wallis, Opera mathematica, vol. iii, 1699, pp. 267, 269, 272.

³¹ Aristotle, Physics, Z. 5.

³² For further details about these mechanical fragments see P. Duhem, Les origines de la statique, 1905, esp. vol. i, pp. 61-97.