Newton and John Macin it turns out endeavoured to put the mathematical Principles( Principia) down in English. And one of the things about the author overseeing a translation is the nuance and responsiveness to criticism this enables in the translation. Much misunderstanding was occasioned by Newton's Principia, not the least who had the primacy in the calculus.
The first lemma in the motion of bodies is where this misunderstanding begins, and where Newton distinguishes himself from his continental rivals.
What is a lemma?
From Ancient Greek λῆμμα (lēmma, “premise, assumption”), from λαμβάνω (lambanō, “I take”).
mathematics: proposition used mainly in the proof of some other proposition
liguistics: canonical form of a term
strip off the rind
(Show lexicon entry in LSJ Middle Liddell Autenrieth) (search) λέπω verb 1st sg pres ind act
λέπω verb 1st sg pres subj act
Word frequency statistics
that which is peeled off, rind, husk
(Show lexicon entry in LSJ Middle Liddell) (search) λέμμα noun sg neut voc
λέμμα noun sg neut nom
λέμμα noun sg neut acc
Word frequency statistics
(Show lexicon entry in LSJ Middle Liddell) (search) λῆμμα noun sg neut voc
λῆμμα noun sg neut acc
λῆμμα noun sg neut nom
Word frequency statistics
λῆμμα , ατος, τό, (λαμβάνω)
A. anything received, opp. δόμα, Antig. ap. Plu.2.182e; λ. καὶ ἀνάλωμα receipt and expense, Lys.32.20, Pl.Lg. 920c, Anaxandr.26; ἀνενεγκεῖν （ἐν- Pap.) ἐν λήμματι place to credit, PEleph.15.4 (iii B.C.), cf. BGU1346.2 (i B.C.), etc.: generally, gain, profit, D.5.12, etc.; “λ. τι κέρδους” Id.45.14; esp. of unjust gain, Din. 1.45; παντὸς ἥττων λήμματος unable to resist any temptation of gain, D.19.339; “ὥσπερ ἂν τρυτάνη ἐπὶ τὸ λ. ῥέπειν” Id.18.298; “λ. λαβεῖν” Id.21.28, 27.39: freq. in pl., S.Ant.313, D.8.25, etc.; “τὰ λ. τοῦ ἀργυρίου” Id.49.57; “λημμάτων μετέχειν” Id.58.40; “τἀπὸ Θρᾴκης λ. ἕλκουσι δεῦρο” Antiph.196.
II. in Logic, statement taken as true, assumption; esp. premiss in a syllogism, “ἐπὶ λ. τῷ τοιούτῳ” A.D.Synt.245.13; “τὰ οἰκεῖα τῇ ἐπιστήμῃ λ.” Arist.Top.101a14; λήμματα τιθέναι ib. 156a21, cf. Gell.9.16, Phld.Rh.1.9 S.; prop. the major premiss (the minor being πρόσληψις), Crinis Stoic.3.269; later, ἀποδεικτικὰ λήμματα παρασχεῖν offer scientific proofs, Gal.14.627.
III. matter, substance, or argument of a sentence, etc., opp. form or style (λέξις), D.H.Dem.20, Longin.15.10, etc.: hence, title or argument of an epigram, Lat. lemma, Mart.14.2; theme or thesis, Plin.Ep.4.27.3, Mart.10.59; nutricis lemmata, 'baby songs', Aus.Ep.12.90.
IV. in LXX, burden laid on one, commission received, esp. of prophecy, Na. 1.1, Je.23.33, al.; even, “λῆμμα ἰδεῖν” Hb.1.1, cf. La.2
The root idea of "le" is to move from and /or to . Thus to move the covering from a body to some other position . This gives rise to the idea of the outer covering . But much of a fruit is outer covering and this then becomes a valued gift to be given. Thus the notion of something given, that is of value and necessary.
Quite often a lemma is the "fruit" of some other deliberation, some other conclusion ofered as fruit to sustain and support a following argument or deliberation Something to chew on while one thinks or ponders.
Does one have to accept it? No!, but what if we do? What are the consequences of that? And if we do not what are the consequences of that decision?
The ponderous and ponderful nature of a lemma are what make it intriguing to consider.
Quantities, and the ratios of quantities,
Which in any finite time converge continually to equality,
and before the end of that time
approach nearer the one to the other than by any given difference,
become ultimately equal
Savour this convoluted obviousness, because it delicately lays a heavy yoke on your conception of the world and the behaviour of quantity as perceived by human intellect.
There is a principle that says that Quantity may be divided ad infinitum. at least mentally. Now this means ceaselessly, without stopping, with no end in sight, no finish. Thus it concentrates the mind locally on the process that never stops. It does not mean infinitely, or to infinity. Thus applying this principle one naturally one deduces that quantity can be adjusted to any "size", and more particularly that quantity can be brought into equality.
Every weights and measure authority in the world operates on this principle that quantites can be brought into equality.However, the scientific method demands a tolerance, an error estimate nowadays.
Thus we accept that quantities are equal but we hedge our assertion with an error estimate. Thus we say "probably" equal.
For Newton the principle is contradicted if we accept this approach: We cannot bring quantities into equality if we do not accept that they are equal ultimately. Either they are equal or they are not.
Today we say they are not equal, but we do not know if they are or not, so they could be equal.
We cannot use the notion of equality or shall we say duality, if we do not accept this assertion. If we cannot accept equality or duality we cannot do mathematical deduction. We deduce through the notion of duality, equality.
The notion of an exact science derives from this notion of equality, and out approach to exactness depends on this notion that quantities can be exactly equal.
The rise of Quantum Mechanics, and the notion of the uncertainty principle, and the probability distribution seems to run ounter to this assertion. But we cannot comprehend these notions without equality, that out of a range of possibilities we may select one to be equal to, we just do not know precisely which one!
But now Newton is not one for seeking absolute quantities. Schooled by Eudoxus Newton avoids the controversy of abolutes by using ratios and proportions. In this case the notion of equality of a ratio is the notion of equality!, the notion of duality. The notion of duality encompasses Everything about the Euclidean and Newtonian notion of "Equality".
Now also realise, Newton does not converge to zero, he chooses to converge to equality, duality.
Now this is what Brahmagupta meant by shunya. Shunya is fullness. If you compare fullness to fullness you arrive at equality/duality. If you even do compare zero to zero you still arrive at equality. duality. In passing we can deal with the issue of dividing by "zero". Brahmagupta says it is "zero", and it should be whatever shunya is defined as. If it is defined as nothing, then when you take nothing from anything you get no pieces. Division is repeated subtraction. even to say that you can take nothing away ad infinitum still leaves you with nothing.
Suppose now Shunya means everything, subtracting everything from a distinct quantity leaves you with everything still to take away! Hardly anything of everything is in the distinct object, but it still s not zero or nothing.
The 2 different outcomes for the process of subtraction reflect the 2 different notions of shunya, but the ratios of quantities as one increases quantity to everything, or diminishes quantity to nothing tend to equality as the rate of change of the quantities allow. Those that converge either by increasing or decreasing are ultimately assigned equality.
As we have been taught that factorisation is to be called multiplication so we ave been taught that the contra of factorisation is division. But it is all the process of factorisation, the process of obtaining factors of a form. Sometimes, as children, we are taught fairy tales, but when we become adults we can identify them as such.
It is the behaviour of the ratios that Newton relies upon, not the absolute quantity, and the notion of duality/equality assigning equalness is also crucial to his development of the calculus.
It is a fair question to ask if quantities, whatever referent we may assign to that term, do indeed converge continually to equality, if this is not just a fictional or formal notion, an abstraction derived from observations but nothing more than saying:let it be so, In addition the imposition of finite time means that we are hurried to this conclusion, thus making it a pragmatic and timely one. It is a fair point , and touches on the nature of our experience of reality, and our human frailties.
Thus the lemma exposes all our weaknesses and problems and allows 2 ways out: that duality of quantity be accepted, and duality of ratios be accepted in the case of continual and timely convergence. If it is not accepted, then any alternative scenario "messes " with the supposition of continual convergence, continuous convergence and continuity of quantity.
Thus our Quantum Mechanics fails all these criteria, and Newton's Analysis cannot apply to this type of situation without major modification. The notion of probability is used, not so much as an acknowledgement that we do not know where the quantities are , but more and increasingly so, as a substitute continuous medium of quantity which enables us to apply Newton's tools with modifications.
To do so however, we have moved into la la land with the conceptions of continuous reals, infinitesimals, complex disjoint reals, quaternionic disjoint reals etc, because we wish to maintain the impact of this simply stated lemma number 1.
Newton has one more rule by which he binds us to his assertion.We are allowed by any means to measure this convergence, to ensure that the quantities defeat any difference we care to lay upon them in their converging. Thus to ultimately, in this finite time declare them not equal is to fail this test of convergence. Consequently, if the converging does not fail this arbitrary test we are begged to accept equality means the quantities are equal.
The notion of Duality/equality in a ratio was more important to Newton than we consider it. For many have gone down the road of exactness from Hippasisus onward, and it has proved divisive and meaningless, But Eudoxian Proportion Theory has proved eminently pragmatic if not mere "political correctness", and face saving. Thus for Newton it is the ratio of equality/duality that confers the notion of dual or equal on any quantities that are in comparison, and if this ratio does not exist to the sensibilities, then neither do equal quantities. Contrariwise, should such a ratio exist or appear to exist or be agreed to exist , then for god's sake let us confer equalness to the quantities, for Pity's sake!
There, at the last i have put before you the pitiful postulation of the lemma. If you accept it then you are bound in Newton's world.