proof that they are, at any rate, not unsuitable for such a purpose. They are, speaking generally, not too difficult for novices in the science; the demonstrations are rigorous, ingenious, and often elegant; the mixture of problems and theorems gives perhaps some variety, and makes their study less monotonous; and, if regard be had merely to the metrical properties of space as distinguished from the graphical, hardly any cardinal geometrical truths are omitted. With these excellences are combined a good many defects, some of them inevitable to a system based on a very few axioms and postulates. Thus the arrangement of his propositions seems arbitrary; associated theorems and problems are not grouped together; the classification, in short, is imperfect. That is the main objection to the retention of Euclid as a school-book. Other objections, not to mention minor blemishes, are the prolixity of his style, arising partly from a defective nomenclature, his treatment of parallels depending on an axiom which is not axiomatic, and his sparing use of superposition as a method of proof. A text-book of geometry which shall be free from Euclid's faults, and not contain others of a graver character, and which shall at the same time be better adapted to purposes of elementary instruction, is much to be desired, and remains yet to be written.
Of the thirty-three ancient books subservient to geometrical analysis, Pappus enumerates first the Data (AeSo/xeVa) of Euclid. He says it contained 90 propositions, the scope of which he describes; it now consists of 95. It is not easy to explain this discrepancy, unless we suppose that some of the propositions, as they existed in the time of Pappus, have since been split into two, or that what were once scholia have since been erected into propositions. The object of the Datais to show that when certain things lines, angles, spaces, ratios, &c. are given by hypothesis, certain other things are given, that is, are determinable. The book, as we are expressly told, and as we may gather from its contents, was intended for the investigation of problems; and it has been conjectured that Euclid must have extended the method of the Data to the investigation of theorems. What prompts this conjecture is the similarity between the analysis of a theorem and the method, common enough in the Elements, of reductio ad absurdum, the one setting out from the supposition that the theorem is true, the other from the supposition that it is false, thence in both cases deducing a chain of consequences which ends in a conclusion previously known to be true or false.
The Introduction to Harmony (Eio-aywyr) Ap/jioriK-^), and the Section of the Scale (Kararofj.^ Karovos), treat of music. There is good reason for believing that one at any rate, and probably both, of these books are not by Euclid. No mention is made of them by any writer previous to Ptolemy (140 A.D.), or by Ptolemy himself, and in no ancient codex are they ascribed to Euclid.
The Phænomena (fcaivo/j.cva) contains an exposition of the appearances produced by the motion attributed to the celestial sphere. Pappus, in the few remarks prefatory to his sixth book, complains of the faults, both of omission and commission, of writers on astronomy, and cites as an example of the former the second theorem of Euclid's Phenomena, whence, and from the interpolation of other proofs, Gregory infers that this treatise is corrupt.
The Optics and Catoptrics (QTTTLKO., KaroTrrpi/ca) are ascribed to Euclid by Proclus, and by Marinus in his preface to the Data, but no mention is made of them by Pappus. This latter circumstance, taken in connexion with the fact that two of the propositions in the sixth book of the Mathematical Collection prove the same things as three in the Optics, is one of the reasons given by Gregory for deeming that work spurious. Several other reasons will be found in Gregory's preface to his edition of Euclid's works. In some editions of Euclid's works there is given a book on the Divisions of Superficies, which consists of a few pro positions, showing how a straight line may be drawn to divide in a given ratio triangles, quadrilaterals, and pentagons. This was supposed by John Dee of London, who transcribed or translated it, and entrusted it for publication to his friend Federic Commandine of Urbino, to be the treatise of Euclid referred to by Proclus as TO Trtpi StatpeWov (3i(3Lov. Dee mentions that, in the copy from which he wrote, the book was ascribed to Machomet of Bagdad, and adduces two or three reasons for thinking it to be Euclid's. This opinion, however, he does not seem to have held very strongly, nor does it appear that it was adopted by Commandine. The book does not exist in Greek.
The fragment, in Latin, De Levi et Ponderoso, which is of no value, and was printed at the end of Gregory's edition only in order that nothing might be left out, is mentioned neither by Pappus nor Proclus, and occurs first in Zamberti's edition of 1537. There is no reason for supposing it to be genuine.
The following works attributed to Euclid are not now extant:
1. Three books on Porisms (Ilepi TWV no/Hoyzarwv) are mentioned both by Pappus and Proclus, and the former gives an abstract of them, with the lemmas assumed. A porism, according to Pappus, was neither a theorem nor a problem, but something of an intermediate form, which yet could be enunciated as a theorem or as a problem. Later geometers, he says, defined it to be a local theorem wanting part of the hypothesis, but this definition he censures as imperfect. After the publication of Commandine's translation of Pappus (1588), many attempts were made to extract from this unsatisfactory description a clear idea of what a porism was, and, with the help of the lemmas, to restore Euclid's books. The mystery, which baffled the penetration even of Edmund Halley, was not resolved till the time of Simson, who, in 1722, gained some insight into the subject, and whose posthumous treatise De Poriwnatibus appeared in 1776. Simson's views have been objected to by recent French writers, such as M. Paul Breton, and M. Michel Chasles; but for a discussion of the subject recourse must be had to the article Porism. Here it will be sufficient to state Simson's definition, which is, "A porism is a proposition in which it is proposed to demonstrate that one or more things are given, between which and every one of innumerable other things, not given but assumed according to a given law, a certain relation, described in the proposition, is to be shown to take place; and to refer to Simson's Opera Reliqua; Playfair's paper On the Origin and Investigation of Porisms; Trail's Life of Dr Simson; Breton's JRecherches Nouvelles sur les Porismes d Euclide, and his Question des Porismes; Vincent's Considerations sur les Porismes; and Chasles's Les Trois Livres de Porismes cTEuclide.
2. Two books are mentioned, named ToVcov TT/DO? eVt^aveia, which is rendered Locorum ad Superficiem by Commandine and subsequent geometers. These books were subservient to the analysis of loci, but the four lemmas which refer to them, and which occur at the end of the seventh book of the Mathematical Collection, throw very little light on their contents. Simson's opinion was that they treated of curves of double curvature, and he intended at one time to write a treatise on the subject. (See Trail's Life, pp. 60-62, 100-105).
3. Pappus says that Euclid wrote four books on the Conic Sections (/?i/3Aia reWapa Kwrt/cwv), which Apollonius amplified, and to which he added other four. It is known that, in the time of Euclid, the parabola was considered as the section of a right-angled cone, the ellipse that of an acute-angled cone, the hyperbola that of an obtuse-angled
