#### Passages:

Proclus, Commentary on Euclid's Elements I (271.1-273.10)
Proclus, Commentary on Euclid's Elements I  (p. 356.6-16)

These are the same quotation from the lost work: Iamblichus, Commentary on Aristotle's Categories:
Simplicius, Commentary on Aristotle's Categories, 192.12-25
Simplicius, Commentary on Aristotle's Physics, 60.6-18

#### Curves Mentioned (P = Proclus, I = Iamblichus as quoted by Simplicius):

 Curve Mathematician Purpose Mentioned location cochloid or conchoid (described by Pappus, Collectio iv §§26-29) Nicomedes (PI) angle trisection (I) 1, 2 sibling of the cochlioid Apollonius, Nicomedes (I) circle squaring (I) 3, 4 conics Apollonius (P) 2 line from double motion Carpus (I) circle squaring (I) 3, 4 quadratrices (all sources are available) Hippias (P), Nicomedes (PI) cutting an angle in a given ratio (P), circle squaring (I) 1, 2, 3, 4 speirics (discussed by Hero, Metrica ii 13, ps.-Hero, Definitions 74-75, and Proclus) Perseus (P) none 2 Archimedean spiral (besides Archimedes, On Spirals, see Pappus, Collectio iv §§21-25) others (P) cutting an angle in a given ratio (P) 1 spiral-like (= Archimedean spiral?), the line of 'Lycomedes' certainly is certainly a mistake Archimedes (I) circle squaring (I) 3, 4 other curves not named circle squaring (I) 3, 4 the "demonstrations of Sextus the Pythagorean" Sextus the Pythagorean circle squaring (I) 3, 4

### Proclus, Commentary on Euclid's Elements I (271.1-273.10)

Prop. 9, problem 4:  To bisect the given rectilinear angle.
He mixes up theorems with problems and weaves together problems with theorems and through both accompishes the whole elements, sometimes providing the subjects and sometimes observing their symptoms.  Having then proved through the previous also with regard to one triangle that the equality of angles follows on the equality of sides, and conversely, and in the same way with regard to two triangles, except that the converse case is different for one and for two trriangles, he proceeds to the problems and requests the bisection of the given rectilinear angle.  It is also clear that the angle is here given in species.  For 'rectilinear' was mentioned and not any sort.  For it is not in the elements that one would bisect every angle, where there is also some dispute if it is even possible to bisect every angle, since you might be perplexed as to whether it is possible to divide the horn angle.
The ratio of the section is determined, and this too is appropriate.  For dividing it in any ratio departs from the present conditions, e.g., into three equal angles or four or five.  For it is possible to trisect the right angle by making use of a few of the things that follow, but it is impossible to trisect the acute angle without taking up other lines (271) which are of mixed species.  Those who produced this statement of the trisection of the given rectilinear angle make this clear.  For Nicomedes trisected every rectilinear angle from conchoid lines, whose generation, order, and symptoms he provided, inasmuch as he was the discoverer of its specific character.  And others did the same from the squaring-lines (quadratices) of Hippias and Nicomedes, and so made use of mixed lines, the squaring lines.  Others were impelled to cut the given rectilinear angle in the given ratio from the Archimedean spirals.  Since grasping them is difficult for beginners, we leave them out in the present discussion.  The right time for us to examine them might rather be, perphaps, in Book iii, where one element is bisecting the given circular-arc, since we have the same manner of investigation, not merely bisecting, but also trisecting it, and the ancients got their designs for the division of every circular-arc into three equal parts from the same lines.  Therefore, one may appropriately make mention of (construction by) straight-line and circular-arcs and pass by the species which exist by mixture from these and are contorted and hard to enumerate, without thoroughly making all such investigations, namely all those that require mixed lines.  So, in the matter of the first and simplest kinds, one proposes for investigation things that can either be provided or observed from these alone (namely straight-line and circular-arc).  And this is the sort of problem now proposed, bisecting the given rectilinear angle, since this uses for the  construction one postulate and Theorems 1 and 3, while for the demonstration it only uses Theorem 8.  For problems also always need demonstration, as we said earlier.  And what is productive of knowledge takes hold from this.
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### Proclus, Commentary on Euclid's Elements I  (p. 356.6-16)

From Prop. 26, theorem 17.
In this way, the other mathematicians, as well, were accustomed to discuss issues about lines, by providing the symptoms of each kind.  For Apollonius proves what the symptoms are for each of the conic lines, and Nicomedes in the case of the conchoids, and Hippias in the case of the squaring lines (quadratices), and Perseus in the case of the speirics.  For after the generation, the kind (or figure) constructed by us determines from all others what was taken as belonging per se and qua ipse [cf. Aristotle, Post. An. A 4].  In the same way, the author of Elements discovers first the symptoms of parallel lines.
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### Simplicius, Commentary on Aristotle's Categories, 192.12-25

He proves this in the case of the squaring of the circle.  For if it was not yet discovered at that time, he says doubtfully, if it is knowable, then knowledge of it does not yet exist, but the knowable does exist.  And squaring the circle is when we construct a square equal to a given circle.  Aristotle, as it seems, did not yet know this, but Iamblichus says that it was discovered by the Pythagoreans, "as is clear from the demonstrations of Sextus the Pythagorean, who received the method of demonstration by a succession from long ago.  And later," he says, "Archimedes through the line of Lykomedes [text should be 'spiral-like line', as is clear from the other text] and Nicomedes from the curve he called on his own the squarer (quadratrix), and Apollonius from a certain line which he named the sibling of the cochlioid, while Nicomedes had the same line, Carpus though a certain line which he calls simply 'from double motion', and many others others in various ways constructed the problem," as Iamblichus relates.
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### Simplicius, Commentary on Aristotle's Physics, 60.6-18

And so what was said is not adequate, I think, for establishing the rejection of the discovery of squaring.  For Iamblichus, in the Commentary on the Categories says that Aristotle had not yet, perhaps, found the squaring of the circle, but that it had been discovered by the Pythagoreans, "as is clear," he says, "from the demonstrations of Sextus the Pythagorean, who received the method of demonstration by a succession from long ago.  And later," he says, "Archimedes through the spiral-like line and Nicomedes from the curve he called on his own the squarer (quadratrix), and Apollonius from a certain line which he named the sibling of the cochlioid, while Nicomedes had the same line, Carpus though a certain line which he calls simply 'from double motion', and many others others in various ways constructed the problem,"  Perhaps all these used a construction by instrument of the theorem.
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