enunciates thus, To inscribe an equilateral triangle in a circle, he states a problem; for it is also possible to inscribe in it a triangle which is not equilateral. Again, if we take the enunciation On a given limited straight line to construct an equilateral triangle, this is a problem; for it is possible also to construct one which is not equilateral. But, when any one enunciates that In isosceles triangles the angles at the base are equal, we must say that he enunciates a theorem; for it is not also possible that the angles at the base of isosceles triangles should be unequal. It follows that, if any one were to use the form of a problem and say In a semicircle to describe a right angle, he would be set down as no geometer. For every angle in a semicircle is right'." "Zenodotus, who belonged to the succession of Oenopides, but was a disciple of Andron, distinguished the theorem from the problem by the fact that the theorem inquires what is the property predicated of the subject-matter in it, but the problem what is the cause of what effect (τίνος ὄντος τί ἐστιν). Hence too Posidonius defined the one (the problem) as a proposition in which it is inquired whether a thing exists or not (ei ěσtiv ǹ μý), the other (the theorem3) as a proposition in which it is inquired what (a thing) is or of what nature (Tí ¿OTIV ǹ TOîóv Ti); and he said that the theoretic proposition must be put in a declaratory form, e.g., Any triangle has two sides (together) greater than the remaining side and In any isosceles triangle the angles at the base are equal, but that we should state the problematic proposition as if inquiring whether it is possible to construct an equilateral triangle upon such and such a straight line. For there is a difference between inquiring absolutely and indeterminately (ἁπλῶς τε καὶ ἀορίστως) whether there exists a straight line from such and such a point at right angles to such and such a straight line and investigating which is the straight line at right angles." "That there is a certain difference between the problem and the theorem is clear from what has been said; and that the Elements of Euclid contain partly problems and partly theorems will be made manifest by the individual propositions, where Euclid himself adds at the end of what is proved in them, in some cases, 'that which it was required to do,' and in others, 'that which it was required to prove,' the latter expression being regarded as characteristic of theorems, in spite of the fact that, as we have said, demonstration is found in problems also. In problems, however, even the demonstration is for the purpose of (confirming) the construction: for we bring in the demonstration in order to show that what was enjoined has been done; whereas in theorems the demonstration is worthy of study for its own sake as being capable of putting before us the nature of the thing sought. And you will find that Euclid sometimes interweaves theorems with problems and employs them in turn, as in the first 1 Proclus, pp. 79, 11-80, 5. 3 In the text we have τὸ δὲ πρόβλημα answering to τὸ μὲν without substantive: πρόβλημα was obviously inserted in error. • Proclus, pp. 80, 15-81, 4. book, while at other times he makes one or other preponderate. For the fourth book consists wholly of problems, and the fifth of theorems1." Again, in his note on Eucl. I. 4, Proclus says that Carpus, the writer on mechanics, raised the question of theorems and problems in his treatise on astronomy. Carpus, we are told, "says that the class of problems is in order prior to theorems. For the subjects, the properties of which are sought, are discovered by means of problems. Moreover in a problem the enunciation is simple and requires no skilled intelligence; it orders you plainly to do such and such a thing, to construct an equilateral triangle, or, given two straight lines, to cut off from the greater (a straight line) equal to the lesser, and what is there obscure or elaborate in these things? But the enunciation of a theorem is a matter of labour and requires much exactness and scientific judgment in order that it may not turn out to exceed or fall short of the truth; an example is found even in this proposition (I. 4), the first of the theorems. Again, in the case of problems, one general way has been discovered, that of analysis, by following which we can always hope to succeed; it is this method by which the more obscure problems are investigated. But, in the case of theorems, the method of setting about them is hard to get hold of since 'up to our time,' says Carpus, 'no one has been able to hand down a general method for their discovery. Hence, by reason of their easiness, the class of problems would naturally be more simple.' After these distinctions, he proceeds: 'Hence it is that in the Elements too problems precede theorems, and the Elements begin from them; the first theorem is fourth in order, not because the fifth is proved from the problems, but because, even if it needs for its demonstration none of the propositions which precede it, it was necessary that they should be first because they are problems, while it is a theorem. In fact, in this theorem he uses the common notions exclusively, and in some sort takes the same triangle placed in different positions; the coincidence and the equality proved thereby depend entirely upon sensible and distinct apprehension. Nevertheless, though the demonstration of the first theorem is of this character, the problems properly preceded it, because in general problems are allotted the order of precedence"" Proclus himself explains the position of Prop. 4 after Props. 1-3 as due to the fact that a theorem about the essential properties of triangles ought not to be introduced before we know that such a thing as a triangle can be constructed, nor a theorem about the equality of sides or straight lines until we have shown, by constructing them, that there can be two straight lines which are equal to one another. It is plausible enough to argue in this way that Props. 2 and 3 at all events should precede Prop. 4. And Prop. I is used in 1 Proclus, p. 81, 5-22. 2 TO TEμTTOV. This should apparently be the fourth because in the next words it is implied that none of the first three propositions are required in proving it. Proclus, pp. 241, 19—243, 11. ibid. pp. 233, 21—234, 6. Prop. 2, and must therefore precede it. But Prop. I showing how to construct an equilateral triangle on a given base is not important, in relation to Prop. 4, as dealing with the "production of triangles" in general for it is of no use to say, as Proclus does, that the construction of the equilateral triangle is "common to the three species (of triangles)," as we are not in a position to know this at such an early stage. The existence of triangles in general was doubtless assumed as following from the existence of straight lines and points in one plane and from the possibility of drawing a straight line from one point to another. Proclus does not however seem to reject definitely the view of Carpus, for he goes on2: "And perhaps problems are in order before theorems, and especially for those who need to ascend from the arts which are concerned with things of sense to theoretical investigation. But in dignity theorems are prior to problems....It is then foolish to blame Geminus for saying that the theorem is more perfect than the problem. For Carpus himself gave the priority to problems in respect of order, and Geminus to theorems in point of more perfect dignity," so that there was no real inconsistency between the two. Problems were classified according to the number of their possible solutions. Amphinomus said that those which had a unique solution (μovaxos) were called "ordered" (the word has dropped out in Proclus, but it must be TeTayμéva, in contrast to the third kind, ǎтаκтα); those which had a definite number of solutions "intermediate" (uéoa); and those with an infinite variety of solutions "unordered" (ǎTakтa). Proclus gives as an example of the last the problem To divide a given straight line into three parts in continued proportion. This is the same thing as solving the equations x+y+3=a, xz=y. Proclus' remarks upon the problem show that it was solved, like all quadratic equations, by the method of "application of areas." The straight line a was first divided into any two parts, (x+8) and y, subject to the sole limitation that (x+2) must not be less than 27, which limitation is the duopiouós, or condition of possibility. Then an area was applied to (x+2), or' (a − y), “falling short by a square figure" (èλλeîπov eidei teтpaywvw) and equal to the square on y. This determines and a separately in terms of a and y. For, if & be the side of the square by which the area (i.e. rectangle) "falls short," we have {(a-y)-z} z=y, whence 2z=(a−y) ± √{(a − y)2 - 4y2}. And y may be chosen arbitrarily, provided that it is not greater than a/3. Hence there are an infinite number of solutions. If y = a/3, then, as Proclus remarks, the three parts are equal. Other distinctions between different kinds of problems are added by Proclus. The word "problem," he says, is used in several senses. In its widest sense it may mean anything "propounded" (πpотELÓμevov), whether for the purpose of instruction (ualnσews) or construction (Tonσews). (In this sense, therefore, it would include a theorem.) 1 Proclus, p. 234, 21. 2 ibid. p. 243, 12-25. But its special sense in mathematics is that of something "propounded with a view to a theoretic construction'." Again you may apply the term (in this restricted sense) even to something which is impossible, although it is more appropriately used of what is possible and neither asks too much nor contains too little in the shape of data. According as a problem has one or other of these defects respectively, it is called (1) a problem in excess (πλeováčov) or (2) a deficient problem (Très poßλnua). The problem in excess (1) is of two kinds, (a) a problem in which the properties of the figure to be found are either inconsistent (åσúμßara) or non-existent (ávúпаρкта), in which case the problem is called impossible, or (b) a problem in which the enunciation is merely redundant: an example of this would be a problem requiring us to construct an equilateral triangle with its vertical angle equal to two-thirds of a right angle; such a problem is possible and is called "more than a problem” (μeîšov πрóẞinμа). The deficient problem (2) is similarly called "less than a problem" (eλaσσov ǹ πρóßìnua), its characteristic being that something has to be added to the enunciation in order to convert it from indeterminateness (dopioría) to order (Táğıs) and scientific determinateness (ὅρος ἐπιστημονικός): such would be a problem bidding you "to construct an isosceles triangle," for the varieties of isosceles triangles are unlimited. Such "problems" are not problems in the proper sense (kupiws Xeyóμeva πрoßλýμaтa), but only equivocally3. § 5. THE FORMAL DIVISIONS OF A PROPOSITION. "Every problem," says Proclus3, "and every theorem which is complete with all its parts perfect purports to contain in itself all of the following elements: enunciation (πpóraσis), setting-out (exdeσis), definition or specification (Siopioμós), construction or machinery (κατασκευή), proof (ἀπόδειξις), conclusion (συμπέρασμα). Now of these the enunciation states what is given and what is that which is sought, the perfect enunciation consisting of both these parts. The setting-out marks off what is given, by itself, and adapts it beforehand for use in the investigation. The definition or specification states separately and makes clear what the particular thing is which is sought. The construction or machinery adds what is wanting to the datum for the purpose of finding what is sought. The proof draws the required inference by reasoning scientifically from acknowledged facts. The conclusion reverts again to the enunciation, confirming what has been demonstrated. These are all the parts of problems and theorems, but the most essential and those which are found in all are enunciation, proof, conclusion. For it is equally necessary to know beforehand what is sought, to prove this by means of the intermediate steps, and to state the proved fact as a conclusion; it is impossible to dispense with any of these three things. The remaining parts are often brought in, but are often left out as serving no purpose. 2 ibid. pp. 221, 13—222, 14. 1 Proclus, p. 221, 7—11. 3 ibid. pp. 203, 1-204, 13; 204, 23-205, 8. H. E. 9 Thus there is neither setting-out nor definition in the problem of constructing an isosceles triangle having each of the angles at the base double of the remaining angle, and in most theorems there is no construction because the setting-out suffices without any addition for proving the required property from the data. When then do we say that the setting-out is wanting? The answer is, when there is nothing given in the enunciation; for, though the enunciation is in general divided into what is given and what is sought, this is not always the case, but sometimes it states only what is sought, i.e. what must be known or found, as in the case of the problem just mentioned. That problem does not, in fact, state beforehand with what datum we are to construct the isosceles triangle having each of the equal angles double of the remaining angle, but (simply) that we are to find such a triangle.... When, then, the enunciation contains both (what is given and what is sought), in that case we find both definition and setting-out, but, whenever the datum is wanting, they too are wanting. For not only is the setting-out concerned with the datum, but so is the definition also, as, in the absence of the datum, the definition will be identical with the enunciation. In fact, what could you say in defining the object of the aforesaid problem except that it is required to find an isosceles triangle of the kind referred to? But that is what the enunciation stated. If then the enunciation does not include, on the one hand, what is given and, on the other, what is sought, there is no setting-out in virtue of there being no datum, and the definition is left out in order to avoid a mere repetition of the enunciation." The constituent parts of an Euclidean proposition will be readily identified by means of the above description. As regards the definition or specification (diopioμós) it is to be observed that we have here only one of its uses. Here it means a closer definition or description of the object aimed at, by means of the concrete lines or figures set out in the exeo is instead of the general terms used in the enunciation; and its purpose is to rivet the attention better, as Proclus indicates in a later passage (τρόπον τινὰ προσεχείας ἐστὶν αἴτιος ὁ διορισμός). The other technical use of the word to signify the limitations to which the possible solutions of a problem are subject is also described by Proclus, who speaks of duopio poi determining "whether what is sought is impossible or possible, and how far it is practicable and in how many ways?"; and the Siopioμos in this sense appears in Euclid as well as in Archimedes and Apollonius. Thus we have in Eucl. I. 22 the enunciation "From three straight lines which are equal to three given straight lines to construct a triangle," followed imme-) diately by the limiting condition (Siopioμós). "Thus two of the straight lines taken together in any manner must be greater than the remaining one." Similarly in vI. 28 the enunciation "To a given straight line to apply a parallelogram equal to a given rectilineal |