“Go Forth and Multiply”

It's been a while since i realised thar multiplication was repeated addition. It slowly took precedence and organized my understanding of mathematics. It lead to insight and apprehensions of forms biological and crystalline. So it is a big thing to realise that i possibly have made a mistake.

In Euclid the monad is followed by the multiple, the plethos. The pletheros comes from the idea of a form. The starting place is a form for the arithmoi. That form is the epihaneia. Subconsciously it has sides or pleura, but these are thought of a gramme, the symbol of pleura. Thus a length is deduced as a magnitude without form.

An epipaneia has at least 2 lengths subconsciously and these are named a magnitude and form but this time it is called platos, almost plethos but not quite, just as mekos is almost megas but not quite.

Plethos is therefore fullness of form of the epiphaneia, and this is what the monads populate and fill . Each population of monads is an arithmos, a kind of mini epiphaneia. Thus we have almost self similarity between the Arithmoi and the epiphaneia, essentially fractal.

The monas is multiplied to form the epiphaneia. In the meta physical philosophy the axiom is the monad multiplies to form the the diad. This is the first step of the tractys also. Thus multiples are introduced first in the philosophy and the multiple monads or Henads take a prominent position.

Similarly at the level of the epiphaneia the monad multiples to form the arithmoi. However this is in Reciprocal to the henads. The henads form monads within the Monad, whereas arithmoi are formed by monads filling a form, the epiphaneia.but each arithmos is formed out of the monads. Thus division and multiplication are hinted at right at the outset.

Immediately with the notion of filling the form is "laying together" sugkeimenon. This i took to mean aggregation, but it is aggregating an already multiplied group of monads.
thus in the pythagorean analysis multiplication and division precede aggregation.

In the metaphysical philosophy after the henads the suntemata and the sumbola appear, just a the sugkeimenon appears after the plethos or full form.

Thus in the euclidean development of the arithmoi, and in the metaphysical pythagorean philosophy on which it is based the initial form is given. The form: The monad, the epihaneia is subconsciously understood to have bounds. but more importantly they are plethorate, full of monads. The monads of the monads are henads, the monads of the epiphaneia are called arithmoi. The henads a formed within the Monad, tha arithmoi are formed out of the monads . Tthe henads are separated (exaston) out of the Monad, the arithmoi are aggregated together (sugkeimenon) out of the monads.

After the multiples and the aggregation parts are defined as arithmoi arithmou: forms made up of minor arithmoi of the Greater arithmos. They are a kind of fraction. Measurement and comparisons are made within the concept to decide how much is the minor arithmos a part of the greater arithmos (ελασσων του μειζονος οταν καταμετρη τον μειζονα)

Thus multiples lead to aggregation and comparison and distinction

A meros is mentioned in relation to a semeion. A semeion has no part no meros and thus cannot be an arithmoi or an arithmos arithmou, it is effectively "zero", but its supernatural links makes it a good equivlent to shunya.

We may use meros to define monad as an arithmoi if we wish, but this is a corollary of the definition of meros not a fundamental definition of the subjective experience Monas

Euclid distinguishes between a meros and Mere, but only to point out that mere are not measured. By this he means that mere is a general undefined term relating to meros which is a specifically defined measurement.

This king of definition looks at a fraction of a form and considers forms to be divisible into minor forms. These forms do not carry any specific measurement in the notion, just the notion of being fractional or lesser or minor in some way.

Euclid then defines "multiple form" or multiple (πολλαπλάσιος). This is where the greater,plural form arises out of the minor forms or rather as he precisely puts it when the greater plural form consists of the minor forms , when the measurement is based on the minor form.

Euclid defines multiple form now to include both monads part and arithmoi in the definition along with the action of measurement by a lesser. Thus he has 2 forms of multiples one which consits of meros and one which consists of elasson. Therefore ther is not much aggregation by addition or subtracion, but there is a lot of multioplying and measuring and comparing and distinguishing.

Pollapleisios is either a multiple form (arithmoi) or a multiple no form(gramme) but it is a greater form that consists of minor forms. The kata netreethai is the "measuring down" a form by placing one form down another. I would say by but the greek kata has this wondrful sense of downward motion, which i think is vectorially important to preserve. Thus the measuring down is done under the smaller or minor form. http://www.perseus.tufts.edu/hopper/morph?l=polu%2Fs&la=greek&can=polu%2Fs0&prior=on

In no sense up to nw has Euclid defined accurate measurement by a monad. He has defined arithmoi as covering a gorm that arises from placing them carefully together, but still not a notion of tessellation. I think this is what he moves onto next: accurate and approximate measurement by arithmoi.

1. Proto-Indo-European root *me-
the Proto-Indo-European root *me-
Derivations in other languages
English mine, English my, English mod, English mowing, English mower, English mow, English mown, Latin mos
2. Proto-Indo-European root *me-
derived from the Proto-Indo-European root *me-
Derivations in other languages
English month, English moon, Greek metron, Latin metiri, Latin mensis, Sanskrit mimite
3. Proto-Indo-European root *me-
derived from the Proto-Indo-European root *me-

μέτρον an apparently primary word
Transliterated Word Phonetic Spelling
metron met'-ron
Parts of Speech TDNT
Noun Neuter 4:632,590
measure, an instrument for measuring
a vessel for receiving and determining the quantity of things, whether dry or liquid
a graduated staff for measuring, a measuring rod
proverbially, the rule or standard of judgment
determined extent, portion measured off, measure or limit
the required measure, the due, fit, measure

μετρέω from (3358)
Transliterated Word Phonetic Spelling
metreō met-reh'-o
Parts of Speech TDNT
Verb 4:632,590
to measure, to measure out or off
any space or distance with a measurer's reed or rule
metaph. to judge according to any rule or standard, to estimate
to measure out, mete out to, i.e. to give by measure


(Show lexicon entry in LSJ Middle Liddell Autenrieth) (search) μετρ-έω verb 1st sg pres ind act epic doric ionic aeolic parad_form
μετρ-έω verb 1st sg pres subj act epic doric ionic aeolic

Word frequency statistics

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μετρ-έω , Heraclean I pl. impf.
A. “ἐμετρίωμες” Tab.Heracl.2.45: pres. part. Pass. μετριώμεναι ib.1.22, 28: (μέτρον):—measure:
I. of Space, measure, i. e. pass over, traverse, “πέλαγος μέγα μετρήσαντες” Od.3.179; προτέρω μετρεῖν (sc. θάλασσαν) to sail farther, A.R.2.915, cf. 4.1779:—in Med., “ἅλα μετρήσασθαι” Mosch.2.157; μετρούμενον ἴχνη τὰ κείνου measuring them with the eyes, S.Aj.5:—Pass., to be measured, A.Ch.209; to be measured round, D.P.197.
II. of Time, “μακροὶ . . ἂν μετρηθεῖεν χρόνοι” S.OT561.
III. of Number, Size, Worth, etc.,
1. count, Alc.142; “ἐπ᾽ ᾐόνι κύματα μ.” Theoc. 16.60, cf. AP4.3b.10 (s. v. l., Agath.).
2. measure, χώρην ὀργυιῇσι, σταδίοισι, etc., Hdt.2.6; “χώρας κατὰ παρασάγγας” Id.6.42; τῇ γαστρὶ μ. τὴν εὐδαιμονίαν measure happiness by sensual enjoyments, D.18.296; “μ. πορφύρᾳ τὸ εὔδα᾽ μον” Luc.Nigr.15, etc.; ὁπηνίκ᾽ ἂν εἲκοσι ποδῶν μετροῦντι τὸ στοιχεῖον ᾖ when you measure it, Eub.119.7, cf. 9; “ἀριθμεῖν τἀγαθὰ καὶ μετρεῖν” Pl.R.348a; μ. καὶ ἀριθμεῖν καὶ ἱστάναι ib.602d: —Pass., “Πόντος . . καὶ Ἑλλήσποντος οὕτω μοι μεμετρέαται” Hdt.4.86; “μετρεῖσθαι πρὸς ἄλληλα” Pl.Plt.284d, etc.
b. Math., of magnitudes or numbers, measure, Arist.Cael.273b12, Euc.7 Def.14, Eratosth. ap. Nicom.Ar.1.13 (Act. and Pass.), etc.; μετρηθῆναι κοινῷ μέτρῳ πρός . . to be commensurable with, ibid.
3. measure out, “τἄλφιτ᾽ ἐν ἀγορᾷ” Ar.Eq.1009, cf. Ach.548 (Pass.); “πώλοισι χόρτον μ.” E.Rh.772; “μέτρησον εἰρήνης τί μοι” Ar.Ach.1021; μετρεῖν τὴν ἴσην give measure for measure, Paus.2.18.2; ἢ μετάδος ἢ μέτρησον ἢ τιμὴν λαβέ lend by measure, Theopomp.Com.26:—Med., to have measured out to oneself, in buying or borrowing, εὖ μετρεῖσθαι παρὰ γείτονος get good measure from one's neighbour, Hes.Op.349; “τὰ ἄλφιτα καθ᾽ ἡμίεκτον μετρούμενοι” D.34.37, cf. Herod.6.5, SIG976.61 (Samos, ii B. C.), Plu. Caes.48.
4. deliver, pay, of corn and other measurable commodities, “σῖτόν τινι” D.46.20, PHib.1.39.3 (iii B. C.); ἔλαιον ib.131 (iii B. C.):—Med., receive in payment, ib.103 (iii B. C.), etc.
IV. moderate, of pain, Pall.in Hp.12.273 C.

μέγας [including the prolonged forms, feminine megale, plural megaloi, etc., cf also (3176), (3187)]
Transliterated Word Phonetic Spelling
megas meg'-as
Parts of Speech TDNT
Adjective 4:529,573
of the external form or sensible appearance of things (or of persons)
in particular, of space and its dimensions, as respects 1a
mass and weight: great 1a
compass and extent: large, spacious 1a
measure and height: long 1a
stature and age: great, old
of number and quantity: numerous, large, abundant
of age: the elder
used of intensity and its degrees: with great effort, of the affections and emotions of the mind, of natural events powerfully affecting the senses: violent, mighty, strong
predicated of rank, as belonging to
persons, eminent for ability, virtue, authority, power
things esteemed highly for their importance: of great moment, of great weight, importance
a thing to be highly esteemed for its excellence: excellent
splendid, prepared on a grand scale, stately
great things
of God's preeminent blessings
of things which overstep the province of a created being, proud (presumptuous) things, full of arrogance, derogatory to the majesty of God

μείζων irregular comparative of (3173)
Transliterated Word Phonetic Spelling
meizōn mide'-zone
Parts of Speech TDNT
Adjective None
greater, larger, elder, stronger

Before proceeding i thought it necessary to get an understanding of metreo. It is as fundamental as logos in the greek language and its context takes us back etymologically to the proto indian root "me"

"Me" is the fundamental notion of subjective comparison with self. The notion is man as a measure, as a comstandard for comparison. The notion extends to the whole as well as the parts of man.

Thus the notion of metron objectifies a subjective activity by using a symbol to stand for a part of man, and then to compare with that.
The notion extends to megas as a symbol for comparison.

What is compared is anything that is comparable, and that is anything that can be symbolised . Thus sumbola lies at the heart of the motion of metreo.

With that background i think it is clear the metreo and meizon are terms in the activity of comparing. That activity is vectoral as well as distinguishing, and thus it allows spatial orientations. Kate metreo is therefore to compare in the downward direction, usually by laying a metron on top of a meizon. The meizon is thus upo the metron.

The elassonos metron is thus used to katametreetai, to lay down upon to compare. and thus the comparison is done "upo" the elassonos metron.

Euclid thus defines Monas, arithmos, meros and Pollaplasios before he modifies the arithmoi with artios (accurate) and perissos (approximate) and enters the space filling realm of tessellation.

As you can see we have quite a ways to go in the concept of form before we come to simple addition and sutracttion and multiplication as repeated addition. And yet sunthemata has already been acknowledged and applied,and so has sumbola.

These notions which undelie Algebra precede simple arithmetic in the development of mathematical skills.

A note on what is used as a metronn for determining a part: the meizonos arithmos is used as the metron when spacifying a meros , and the elassonos metron is used when specifying a pollaplasios.


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