This topic I want to relate to the concerns of Phorometry. That is to say, etymologically porism and phorea are related. The notion of a Lemma or corollary may then be properl adduced.
My earlier research into phorometry took me to the Greek notion of the limit of what the eye can see., but I wish to look at it again in the light of Pappus.
Certainly, this line of thought has the also attendant effect, visible to the rye, of determining or confirming a growing conviction, that in Isaac Barrows time, the term Geometry related not to Euclid, but to Pappus. That since that time later scholars of the European school have consistently pushed this term geometry backwards in history untill it landed on a more or less mythical figure of repute named Euclid. The motive for doing this was not to laud the poor chap, but rather to make him the fall guy. And to big up Archimedes, Apollonius and possibly Pappus, for correcting several mistakes in prior knowledge. In Particulate Apollonius is reputed to have made some pragmatic kathalou, that is practical fixed or laid down principles, a list of improvements to any future editions of the Stoikeioon.
While this is all well and good it does imply a settled view of geometry being overturned, whereas it is clearly a revision of the introduction to philosophical principles. This type of things happens all the time twist scholars. For example Newton proposed to lay down principles of philosophy, so that the praxis of discourse and rhetoric could lead to some consensual conclusions..
For Apollonius, his innovative way of thinking, notably exhibited in his books on Conics, widely accepted as the de facto presentation of that subject, so that it overturned Euckid's Conics, meant he had to set up his style of thinking and philosophising from the " start". The Start is another way of translating Stoikeioon. Thus we find him making critical suggestion for appending notions to the Stoikeioon..
The evidence is overwhelming. Geometry in Prussia was defined by the writings of Pappus, not Euclid. Thus a problem automatically exists: what are the foundations of geometry? Gauss set this task for Riemann to address in his Habilitation speech. The academic hope being that his work, would be judged in context, and promulgated by his student, Riemann. To this end he preempted Bolyai, attempted to get a jump on Lobachewski and completely ignored the work of Steiner and the Grassmanns. He also underestimated the importance of ring and group theory.
Gauss took no hostages when it came to self promotion, but in this instance he seems to have tried to establish Prussia as a centre of mathematical excellence in the world, as part of the big thing he wanted to achieve for the state and his nation.,it was his sideways contribution to the debate and reorganisation underway due to the Humboldt reforms.
When the Ausdehnungslehre is read in this light, the Formenlehre is immediately recognise able as the complete analytical method of Apollonius! The projective viewpoint that underpins the notion of the inner product leaps out and grabs you by the throat as Apollonian, and the harmonic analysis of Cotes and Apollonius are seen to be the former inspired by the latter. The immense works on Appolonius by Halley a contemporary of Cotes and Newton are a clear motivator, and a source of ideas that Newton well knew in principle, but had lost insight into in his later life due to mercurial poisoning. Votes not only proved a theorem similar to Pappus' but went on to see the harmony of measurement possible.
The work of Halley, a labour of love clearly was so exhausting he could not see the wood from the trees, but Cotes could. He died before he could elucidate, but essentially he would have revealed insights into the works of Apollonius on Conics! Newton by his great perspicacity and reading came to know through Barrow some of the theories of Apollonius as his own, but there is a great system and practice that Cotes had twigged onto which Newton could no longer remember if he ever knew, and this is the topic of Normans Universal hyperbolic geometry. But before Normann, the Grassmanns were onto it in Prussia in the early 1800's, even Gauss had not spotted it.
The formal structure of a pedagogical treatise requires problem, porism and theorem. This structure formally represents the human difficulty( problem) , the human investigation of the problem and solutions( porism) and the divine insight into a godlike generality( theorem),
A porism therefore is one of many investigations into a problem , the intention being to extend the problem out as far as the eye can see. This is analogous to analysing the problem down to the n-th partition to find common ground or indeed synthesising a similar but larger problem and finding principles in this circumstance of vision lead thinking( porism)
From these explorations, principles for Divine inspiration and expression of veracity are chosen as theorems..
Compare this with the Indian aphoristic style. This Koran style is meant to stretch the student to breaking point to allow the insights from the gods to flow into a receptive mind. Certainly, Pythagoras used this method to bring on his students to the level of mastery called Mathematikos, their own personal connection with the Musai. Porisms do the same, bringing the student through a journey to a divine solution.
phoria and poros are from the same root. one is bearing the whole the other is only what the eyes can bear.. /both involve a distant journey.
πόρ-ισμα , ατος, τό, Geom.,
A.deduction from a previous demonstration, corollary, as it were a windfall or bonus (cf. “πορίζω” 11.2), Euc.3.1, etc.: metaph., Procl. in Alc.p.139C., Hierocl.in CA23p.469M., Dam.Pr.251.
II. (“πορίζω” 111) a kind of proposition intermediate between a theorem and a problem, defined by Papp.648.18 sqq., Procl.in Euc.p.301F.
Henry George Liddell. Robert Scott. A Greek-English Lexicon. revised and augmented throughout by. Sir Henry Stuart Jones. with the assistance of. Roderick McKenzie. Oxford. Clarendon Press. 1940.
The National Endowment for the Humanities provided support for entering this text.
πορίζω 1 πόρος
1. [select] Properly, like πορεύω, to carry: to bring about, to furnish, provide, supply, procure, cause, Ar., Plat.; absol., θεοῦ πορίζοντος καλῶς Eur.:—often with a notion of contriving or inventing, id=Eur., etc.:—Mid. to furnish oneself with, to provide, procure, Lat. sibi comparare, Ar., Thuc.:— Pass. to be provided, Thuc., etc.
2. [select] πορίζεταί τινι, impers., it is in one's power to do, c. inf., Xen.
πόρος 1 περάω
I.a means of passing a river, a ford, ferry, Lat. vadum, Il., Hdt., etc.; Πλούτωνος π. the Stygian ferry, Aesch.
2.a narrow sea, strait, firth, Lat. fretum, Hes., Aesch.; Ἰόνιος π. the Ionian sea which is the passage-way from Greece to Italy, Pind.:— ἐν πόρῳ in the passage-way (of ships), in the "fair-way" Hdt.
3.periphr., πόροι ἁλός the paths of the sea, i. e. the sea, Od.; ἐνάλιοι π. Aesch., etc.; so, of rivers, πόρος Ἀλφεοῦ, Σκαμάνδρου, i. e. the Alpheus, etc., Pind.
4.a way over a river, a bridge, Hdt.
5.generally a pathway, way, Aesch., Soph.; πόρος οἰωνῶν their pathway, Aesch.
6.a passage through the skin, οἱ πόροι the pores, Plat.
II.c. gen. rei, a way or means of achieving, accomplishing, οὐκ ἐδύνατο π. οὐδένα ἀνευρεῖν Hdt.; π. ὁδοῦ a means of performing the journey, Ar.; π. κακῶν a means of averting evils, Eur.:—c. inf., πόρος τις τίσασθαι id=Eur.
2.absol. a means of providing, contrivance, device, resource, Aesch., Ar.
3.at Athens, π. χρημάτων a way of getting or raising money, Xen., Dem.: in pl., "ways and means, " resources, revenue, Dem.
III.a going, journey, voyage, Aesch., Eur.
1 πόρος, ὁ,
I.to drive right through, λευκοὺς ἐπέρησεν ὀδόντας Il.
2.commonly, to pass across or through a space, to pass over, pass, cross, traverse, περᾶν θάλασσαν, πόντον Od.; πύλας πέρησεν passed through the gates, Il.; τάφρος ἀργαλέη περάαν hard to pass, id=Il.; τὰς φυλακὰς π. to pass the guards, Hdt.:—metaph., κίνδυνον π. to pass through, i. e. overcome, a danger, Aesch.; π. ὅρκον, prob. to go through the words of the oath, Lat. jusjurandum peragere, id=Aesch.
3.rarely of Time, to pass through, complete, τοῦ βίου τέρμα Soph.; τὴν τελευταίαν ἡμέραν Eur.
II.intr. to penetrate or pierce right through, of a weapon, Il.; of rain, Od.: to extend to a place, Xen.
2.to pass across, to pass, δι᾽ Ὠκεανοῖο Od.; ἐπὶ πόντον Il.; διὰ Κυανέας ἀκτάς through the Symplegades, Eur.
3.to pass to or from a place, εἰς Ἀΐδαο Theogn.; ἔξω δωμάτων Soph.:—c. acc. loci, π. Δελφούς Eur.
4.rarely of Time, διὰ γήρως π. Xen.; εὐδαίμων π. to live happy, Orac. ap. Xen.
5.to pass all bounds, to go too far, Soph.; so, π. ὀργῆς to pass all bounds in anger (or to cease from anger), id=Soph.
I.beyond, across or over, further, Lat. ultra, Plat.
2.c. gen., Ἀτλαντικῶν πέρα ὅρων Eur.
II.of Time, beyond, longer, Xen.
2.c. gen., π. μεσούσης ἡμέρας id=Xen.
III.beyond measure, excessively, extravagantly, πέρα λέγειν, φράζειν Soph., etc.
2.c. gen. more than, beyond, exceeding, π. δίκης, καιροῦ Aesch.; π. τῶν νῦν εἰρημένων Soph.; θαυμάτων π. more than marvels, Eur.:—sometimes the gen. is omitted, ἄπιστα καὶ πέρα things incredible, and more than that, Ar.
3.also as comp., foll. by ἤ, Soph.
IV.above, higher than, τῶν ἐχθρῶν πέρα id=Soph.
φόρος 1 φέρω
1.that which is brought in, tribute, such as is paid by subjects to a ruling state, as by the Asiatic Greeks to Athens, Thuc.; φόρου ὑποτελεῖς subject to pay tribute, id=Thuc.; φόρον ὑποτελέειν to pay tribute, Hdt.; ἀπάγειν, φέρειν Ar.; φ. τάξασθαι toagree to pay it, Hdt.; τάξαι to impose it, Dem.
2.any payment, Xen., Plut.
1 φόρος, ὁ,
φέρω from Root ἘΝΕΚ or ἘΝΕΓΚ come ἤνεγκον and ἤνεικα from Root ΟΙ comes οἴσω in pl. always
Radic. sense, to bear, Lat. fero:
A.to bear or carry a load, Hom., attic; of a woman with child, Aesch., Soph.
II.to bear, bear along, implying motion, πόδες φέρον Il.; horses are said ἅρμα φέρειν id=Il.; of a wind, Hom.; ὁ βορέας εἰς τὴν Ἑλλάδα φέρει is fair for Greece, Xen.
III.to bear, endure, suffer, Od., etc.; of wine, τὰ τρία φέρων bearing three parts of water, instead of ἴσον ἴσωι, Ar.:—often with Advs., βαρέως, δεινῶς, χαλεπῶς φέρειν τι, like Lat. aegre, graviter ferre, to bear impatiently, take ill or amiss, opp. to κούφως, ῥαιδίως φέρειν, Lat. leviter ferre, to bear patiently, take easily, Hdt., attic:—such phrases are constructed mostly c. acc. rei; sometimes, c. dat. only, βαρέως φέρειν τοῖς παροῦσι Xen.
IV.to bring, fetch, Hom., attic:—Mid. to bring with one, or for one's own use, Od., etc.
2.to bring, offer, present, δῶρα id=Od.; χάριν τινὶ φ. to grant any one a favour, do him a kindness, Hom., attic
3.to bring, produce, work, cause, Hom.; φ. κακόν, πῆμα, ἄλγεα to work one woe, id=Hom.:— to produce, bring forward, cite, Dem.
4.to bring one word, to tell, announce, Aesch., etc.:—so in Mid., λόγους φ. Eur.; but also, ἔπος φέρεσθαι to have word brought one, receive, id=Eur.
5.to pay something due or owing, φόρον φέρειν to pay as tax or tribute, Thuc.; μισθὸν φ. Xen. (but also to receive pay, Ar., Thuc.):—of property, to bring in, yield as rent, Isae.
6.ψῆφον φ. to give one's vote, Lat. ferre suffragium, Aesch.; ψῆφος καθ᾽ ἡμῶν οἴσεται (as Pass.) Eur.:—hence φέρειν τινά, to appoint to an office, Dem.
V.to bear, bring forth, produce, of the earth or of trees, Od., Hdt., etc.:—absol. to bear, bear fruit, be fruitful, Hdt.
VI.to carry off or away, Il.: of stormy winds, Od.; of a river, Hdt.:—Mid. to carry off with one, Od., Xen., etc.
2.to carry off as booty or plunder, Il., etc.:—often in the phrase φέρειν καὶ ἄγειν, v. ἄγω I. 3:— φέρειν alone, to rob, plunder, θεῶν ἱερά Eur.; ἀλλήλους Thuc.:—Mid. in same sense, Hom.
3.to carry off, gain, win, achieve, Il., Soph., etc.; μισθὸν φέρειν (v. supr. IV. 5):—so in Mid. to win for oneself, Il., attic:—metaph., τὰ πρῶτα, τὰ δεύτερα φέρεσθαι to win and hold the first, the second rank, Hdt.; πλέον or πλεῖον φέρεσθαι to gain the advantage over any one, τινος id=Hdt., etc.;—the Mid. being used of that which one gets for one's own use, esp. to take home, id=Hdt.
VII.absol., of roads, to lead to a place, ἡ ὁδὸς φέρει εἰς . . , like Lat. via fert or ducit ad . . , id=Hdt., Thuc., etc.
2.of a tract of country, to stretch, extend to or towards, like Lat. vergere or spectare ad . . , φέρειν ἐπί or ἐς θάλασσαν Hdt., etc.
3.metaph. to lead to, be conducive to, ἐς αἰσχύνην φέρει id=Hdt.; ἐς βλάβην φέρον Soph.
b.to aim at a thing, hint or point at, refer to it, εἰς or πρός τι Hdt., Plat.; so, τοῦ δήμου φέρει γνώμη, ὡς . . , the people's opinion inclines to this, that . . , Hdt.; τῶν ἡ γνώμη ἔφερε συμβάλλειν their opinion inclined to giving battle, id=Hdt.
c.impers. much like συμφέρει, it tends (to one's interest), is conducive, φέρει σοι ταῦτα ποιεῖν; id=Hdt. d. intr., v. B. I. 2.
VIII.to carry in the mouth, i. e. to speak much of, Aeschin.: Pass., εὖ, πονηρῶς φέρεσθαι to be well or ill spoken of, Xen.: also absol. φέρεται, like Lat. fertur, [the report] is carried about, i. e. it is said, τοιόνδε φέρεται πρῆγμα γίγνεσθαι Hdt.
IX.imperat. φέρε, like ἄγε, used as adv. come, now, well, φέρ᾽ εἰπὲ δή μοι Soph.; so, before 1 pers. sg. or pl. subj. used imperatively, φέρε ἀκούσω Hdt.; φ. δὴ ἴδωμεν, φ. δὴ σκεψώμεθα Plat.
2.before a question, φέρε τροπαῖα πῶς ἄρα στήσεις; well then, how wilt thou erect trophies? Eur.
X.part. neut. τὸ φέρον, as Subst. fortune, fate, τὸ φέρον ἐκ θεοῦ φέρειν χρή ye must bear what heaven bears to you, awards you, Soph.
B.Pass. is used in most of the above senses, esp.,
I.to be borne along by waves or winds, to be swept away, Od.; ἧκε φέρεσθαι he sent him flying, Il.; ἧκα πόδας καὶ χεῖρε φέρεσθαι I let go my hands and feet, let them swing free [in the leap], Od.
2.often in part. with another Verb of motion, φερόμενοι ἐσέπιπτον they fell on them with a rush, Hdt.; ὠιχόμην φερόμενος Plat.;—so, in part. act. used intr., φέρουσα ἐνέβαλε νηί she bore down upon the ship and struck it, Hdt.; φέρων hurriedly, in haste, Aeschin.
II.of voluntary motion, ἰθὺς φέρεται Il.; ὁμόσε τινὶ φέρεσθαι to come to blows with him, Xen., etc.
III.metaph., εὖ, κακῶς φέρεσθαι to turn out well or ill, succeed or fail, νόμοι οὐ καλῶς φέρονται Soph.; τὰ πράγματα κακῶς φέρεται Xen.; ἐᾶν ταῦτα φέρεσθαι to let these things take their course, Dem.:—of persons, εὖ φερόμενος ἐν στρατηγίαις being
The quadrivium consisted of arithmetic, geometry, music, and astronomy. These followed the preparatory work of the trivium made up of grammar, logic, and rhetoric. In turn, the quadrivium was considered preparatory work for the serious study of philosophy (sometimes called the "liberal art par excellence") and theology. The word "trivia" has been rarely used to refer to the trivium.
These four studies compose the secondary part of the curriculum outlined by Plato in The Republic, and are described in the seventh book of that work. The quadrivium is implicit in early Pythagorean writings and in the De nuptiis of Martianus Capella, although the term "quadrivium" was not used until Boethius early in the sixth century. As Proclus wrote:
The Pythagoreans considered all mathematical science to be divided into four parts: one half they marked off as concerned with quantity, the other half with magnitude; and each of these they posited as twofold. A quantity can be considered in regard to its character by itself or in its relation to another quantity, magnitudes as either stationary or in motion. Arithmetic, then, studies quantities as such, music the relations between quantities, geometry magnitude at rest, spherics [astronomy] magnitude inherently moving.
Between 1760 and 1801 a crisis arose among the pedagogues of Geometry. Geometry, which had thought to have been around for thousands of years, and unchangeable, was suddenly being set free from its moorings!. It became clear that the pedagogues had made some fundamental teaching error. What it was and where it was not clear to those involved in the melee. However, the pedagogues were attacked ad hominem, as was the style of argument, and they responded vitriolically.
Bertrand Russels extraordinary vitriole against Euclid has to be understood as a thinly veiled attack on the established pedagogues and not on Euclid, whose work actually remains unscathed!
This turning against the pedagogues seems to have been catalysed by the French revolution and the establishment of a French Ecole for innovation and learning under a royal patronage just prior to that revolution. After the revolution, napoleons expeditions revealed the wealth of wisdom lost to Europe during the dark ages, which he was determined to restore to Europes glory, under his leadership of course.
Keeping his eye on the whole situation from a Prussian empirical perspective, Gauss tried to position Prussian Academia in a pole position. Since the defeat by the French, Prussia was seriously changing its structure to take a stronger stance in Europe vis a bis the French. Any change in Geometical thinking gass wanted to be at the forefront of. He was desperate to do something for his emperor!
This book seems to have become the European textbook on Geometry, and one which would have come into the purview of the Grassmanns , beginning with Justus.
The turmoil among the pedagogues provided young revolutionary thinkers a chance to publish their works. In that atmosphere many publications got ignored. Boiyai,Lobachevsky and the Grassmanns. The few great names were hard pressed to maintain some academic discipline, and to justify the maintenance of Acadmia. Revolution is a 2 edged sword.
My conclusion on the whole matter is that from the outset Euclid's Stoikeioon was a critical success, but it soon got taken out of context. It's very celebrity meant it was going to be fair game. But in this way, it's legendary status was assured even as the facts of its purpose we're obscured.
The political and religious use of this textbook is staggering! That such a elementary, introductory enterprise should shape humanity and define human thought, reason, rationality, science and religious certainty is astounding!
Next to the Tanakh and the Bible and now the Qu'Ran, it is a defining text on human consciousness. Alongside the Vedas, and the Upanidhads, world consciousness is defined in and by the cultures derived and associated with all these texts.