examining the treatise written by Apollonius about the comparison between the dodecahedron and the icosahedron inscribed in the same sphere, (showing) what ratio they have to one another, they thought that Apollonius had not expounded this matter properly, and accordingly they emended the exposition, as I was able to learn from my father. And I myself, later, fell in with another book published by Apollonius, containing a demonstration relating to the subject, and I was greatly interested in the investigation of the problem. The book published by Apollonius is accessible to allfor it has a large circulation, having apparently been carefully written out later-but I decided to send you the comments which seem to me to be necessary, for you will through your proficiency in mathematics in general and in geometry in particular form an expert judgment on what I am about to say, and you will lend a kindly ear to my disquisition for the sake of your friendship to my father and your goodwill to me."

The idea that Apollonius preceded Euclid must evidently have been derived from the passage just quoted. It explains other things besides. Basilides must have been confused with Baoiλeus, and we have a probable explanation of the "Alexandrian king," and of the "learned men who visited" Alexandria. It is possible also that in the "Tyrian" of Hypsicles' preface we have the origin of the notion that Euclid was born in Tyre. These inferences argue, no doubt, very defective knowledge of Greek: but we could expect no better from those who took the Organon of Aristotle to be "instrumentum musicum pneumaticum," and who explained the name of Euclid, which they variously pronounced as Üclides or Icludes, to be compounded of Ucli a key, and Dis a measure, or, as some say, geometry, so that Uclides is equivalent to the key of geometry!

Lastly the alternative version, given in brackets above, which says that Euclid made the Elements out of commentaries which he wrote on two books of Apollonius on conics and prolegomena added to the doctrine of the five solids, seems to have arisen, through a like confusion, out of a later passage' in Hypsicles' Book XIV.: "And this is expounded by Aristaeus in the book entitled 'Comparison of the five figures,' and by Apollonius in the second edition of his comparison of the dodecahedron with the icosahedron." The "doctrine of the five solids" in the Arabic must be the "Comparison of the five figures' in the passage of Hypsicles, for nowhere else have we any information about a work bearing this title, nor can the Arabians have had. The reference to the two books of Apollonius on conics will then be the result of mixing up the fact that Apollonius wrote a book on conics with the second edition of the other work mentioned by Hypsicles. We do not find elsewhere in Arabian authors any mention of a commentary by Euclid on Apollonius and Aristaeus: so that the story in the passage quoted is really no more than a variation of the fable that the Elements were the work of Apollonius.

1 Heiberg's Euclid, vol. v. p. 6.

CHAPTER II.

EUCLID'S OTHER WORKS.

IN giving a list of the Euclidean treatises other than the Elements, I shall be brief: for fuller accounts of them, or speculations with regard to them, reference should be made to the standard histories of mathematics'.

I will take first the works which are mentioned by Greek authors. 1. The Pseudaria.

I mention this first because Proclus refers to it in the general remarks in praise of the Elements which he gives immediately after the mention of Euclid in his summary. He says: "But, 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 misled. This 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."

The book is considered to be irreparably lost. We may conclude however from the connexion of it with the Elements and the reference to its usefulness for beginners that it did not go outside the domain of elementary geometry.

1 Heiberg gives very exhaustive details in his Litterargeschichtliche Studien über Euklid; the best of the shorter accounts are those of Cantor (Gesch. d. Math. 13, 1907, pp. 278-294) and Loria (Il periodo aureo della geometria greca, p. 9 and pp. 63—85).

2 Proclus, p. 70, 1—18.

* Heiberg points out that Alexander Aphrodisiensis appears to allude to the work in his commentary on Aristotle's Sophistici Elenchi (fol. 25 6): "Not only those (exo) which do not start from the principles of the science, under which the problem is classed...but also those which do start from the proper principles of the science but in some respect admit a paralogism, e.g. the Pseudographemata of Euclid." Tannery (Bull. des sciences math. et astr. 2° Série, VI., 1882, 1ère Partie, p. 147) conjectures that it may be from this treatise that the same commentator got his information about the quadratures of the circle by Antiphon and

2. The Data.

The Data (Sedoμéva) are included by Pappus in the Treasury of Analysis (Tomos ávaλvóμevos), and he describes their contents'. They are still concerned with elementary geometry, though forming part of the introduction to higher analysis. Their form is that of propositions proving that, if certain things in a figure are given (in magnitude, in species, etc.), something else is given. The subjectmatter is much the same as that of the planimetrical books of the Elements, to which the Data are often supplementary. We shall see this later when we come to compare the propositions in the Elements which give us the means of solving the general quadratic equation with the corresponding propositions of the Data which give the solution. The Data may in fact be regarded as elementary exercises in analysis.

It is not necessary to go more closely into the contents, as we have the full Greek text and the commentary by Marinus newly edited by Menge and therefore easily accessible'.

3. The book On divisions (of figures).

This work (Tepì diaιpéσewv Bißxíov) is mentioned by Proclus3. In one place he is speaking of the conception or definition (λóyos) of figure, and of the divisibility of a figure into others differing from it in kind; and he adds: "For the circle is divisible into parts unlike in definition or notion (ávópoia τ Xoy), and so is each of the rectilineal figures; this is in fact the business of the writer of the Elements in his Divisions, where he divides given figures, in one case into like figures, and in another into unlike" "Like" and "unlike" here mean, not "similar" and "dissimilar" in the technical sense, but "like" or "unlike in definition or notion" (Xóyw): thus to divide a triangle into triangles would be to divide it into "like" figures, to divide a triangle into a triangle and a quadrilateral would be to divide it into "unlike" figures.

The treatise is lost in Greek but has been discovered in the Arabic. First John Dee discovered a treatise De divisionibus by one Muḥammad Bagdadinus and handed over a copy of it (in Latin) in 1563 to Commandinus, who published it, in Dee's name and his own, in 1570°. It was formerly supposed that Dee had himself translated Bryson, to say nothing of the lunules of Hippocrates. I think however that there is an objection to this theory so far as regards Bryson; for Alexander distinctly says that Bryson's quadrature did not start from the proper principles of geometry, but from some principles more general.

1 Pappus, VII. p. 638.

Vol. VI. in the Teubner edition of Euclidis opera omnia by Heiberg and Menge. A translation of the Data is also included in Simson's Euclid (though naturally his text left much to be desired).

3 Proclus, p. 69, 4.

♦ ibid. 144, 22—26.

5 Steinschneider places him in the 10th c. H. Suter (Bibliotheca Mathematica, IV, 1903, pp. 24, 27) identifies him with Abū (Bekr) Muḥ. b. 'Abdalbāqi al-Bagdadi, Qādi (Judge) of Māristan (circa 1070-1141), to whom he also attributes the Liber judei (? judicis) super decimum Euclidis translated by Gherard of Cremona.

De superficierum divisionibus liber Machometo Bagdadino adscriptus, nunc primum Ioannis Dee Londinensis et Federici Commandini Urbinatis opera in lucem editus, Pisauri, 1570, afterwards included in Gregory's Euclid (Oxford, 1703).

the tract into Latin from the Arabic1; but it now appears certain' that he found it in Latin in a Cotton MS. now in the British Museum. Dee, in his preface addressed to Commandinus, says nothing of his having translated the book, but only remarks that the very illegible MS. had caused him much trouble and (in a later passage) speaks of "the actual, very ancient, copy from which I wrote out..." (in ipso unde descripsi vetustissimo exemplari). The Latin translation of this tract from the Arabic was probably made by Gherard of Cremona (1114-1187), among the list of whose numerous translations a "liber divisionum" occurs. The Arabic original cannot have been a direct translation from Euclid, and probably was not even a direct adaptation of it; it contains mistakes and unmathematical expressions, and moreover does not contain the propositions about the division of a circle alluded to by Proclus. Hence it can scarcely have contained more than a fragment of Euclid's work.

But Woepcke found in a MS. 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 MS. and corresponds to the description of it by Proclus. Generally speaking, the divisions are divisions into figures of the same kind as the original figures, e.g. of triangles into triangles; but there are 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" and "to draw in a given circle two parallel straight lines cutting off a certain part of the circle." Unfortunately the proofs are given of only four propositions (including the two last mentioned) out of 36, because the Arabic translator found them too easy and omitted them. To illustrate the character of the problems dealt with I need only take one more example: "To cut off a certain fraction from a (parallel-) trapezium by a straight line which passes through a given point lying inside or outside the trapezium but so that a straight line can be drawn through it cutting both the parallel sides of the trapezium." 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." Moreover the treatise is no fragment, but finishes with the words "end of the treatise," and is a well-ordered and compact whole. Hence we may safely conclude that Woepcke's is not only Euclid's own work but the whole of it. A restoration of the work, with proofs, was attempted

1 Heiberg, Euklid-Studien, p. 13.

• H. Suter in Bibliotheca Mathematica, IV3, 1905–6, pp. 321—2.

3 Journal Asiatique, 1851, p. 233 sqq.

We are told by Casiri that Thabit b. Qurra emended the translation of the liber de divisionibus; but Ofterdinger seems to be wrong in saying that according to Gartz (De interpretibus et explanatoribus Euclidis Arabicis schediasma historicum, Halae, 1823) there is a

by Ofterdinger', who however does not give Woepcke's props. 30, 31, 34, 35, 36.

4. The Porisms.

It is not possible to give in this place any account of the controversies about the contents and significance of the three lost books of Porisms, or of the important attempts by Robert Simson and Chasles to restore the work. These may be said to form a whole literature, references to which will be found most abundantly given by Heiberg and Loria, the former of whom has treated the subject from the philological point of view, most exhaustively, while the latter, founding himself generally on Heiberg, has added useful details, from the mathematical side, relating to the attempted restorations, etc. It must suffice here to give an extract from the only original source of information about the nature and contents of the Porisms, namely Pappus. In his general preface about the books composing the Treasury of Analysis (Tóπos dvaλvóμevos) he says:

"After the Tangencies (of Apollonius) come, in three books, the Porisms of Euclid, [in the view of many] a collection 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 written originally 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 complete Ms. of Thabit's translation in the Escurial. I cannot find any such statement in Gartz.

L. F. Ofterdinger, Beiträge zur Wiederherstellung der Schrift des Euklides über die Theilung der Figuren, Ulm, 1853.

Heiberg, Euklid-Studien, pp. 56—79, and Loria, Il periodo aureo della geometria greca, PP. 70-82, 221-5.

8 Pappus, ed. Hultsch, vII. pp. 648-660. I put in square brackets the words bracketed by Hultsch.

I adopt Heiberg's reading of a comma here instead of a full stop.