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Notion； inseparable and their quantitative relationship is the being…for…self of measure； is only one
measure…determination。
In regard to the absolute relations of measure； it is well to bear in mind that the mathematics of
nature； if it is to be worthy of the name of science； must be essentially the science of measures … a
science for which it is true much has been done empirically； but little as yet from a strictly scientific；
that is； philosophical point of view。 Mathematical principles of natural philosophy…as Newton
called his work…if they are to fulfil this description in a profounder sense than that accorded to
them by Newton and by the entire Baconian species of philosophy and science； must contain
things of quite a different character in order to bring light into these still obscure regions which are；
however； worthy in the highest degree of consideration。
It is a great service to ascertain the empirical numbers of nature； e。g。； the distances of the planets
from one another； but it is an infinitely greater service when the empirical quanta are made to
disappear and they are raised into a universal form of determinations of quantity so that they
become moments of a law or of measure … immortal services which Galileo for the descent of
falling bodies and Kepler for the motion of the celestial bodies； have achieved。 The laws they
discovered they have proved in this sense； that they have shown the whole compass of the
particulars of observation to correspond to them。 But yet a still higher proof is required for these
laws； nothing else； that is； than that their quantitative relations be known from the qualities or
specific Notions of time and space that are correlated。
Of this kind of proof there is still no trace in the said mathematical principles of natural philosophy；
neither is there in the subsequent works of this kind。 It has already been remarked in connection
with the show of mathematical proofs of certain relationships in nature； a show based on the
misuse of the infinitely small； that it is absurd to try todemonstrate such proofs on a strictly
mathematical basis； i。e。 neither empirically nor from the standpoint of the Notion。 These proofs
presuppose thir theorems； those very laws； from experience； what they succeed in doing is to
reduce them to abstract expressions and convenient formulae。
Undoubtedly the time will come when； with a clearer understanding of what mathematics can
accomplish and has accomplished； the entire； real merit of Newton as against Kepler — the sham
scaffolding of proofs being discarded — will clearly be seen to be restricted to the said
transformation of Kepler's formula and to the elementary analytical treatment accorded to it。
Undoubtedly the time will come when； with a clearer understanding of what mathematics can
accomplish and has accomplished； he restricted to the said transformation of Kepler's formula and
to the lem en； ta analytical treatment accorded to it。
C Being…for…self in Measure
Chapter 2 Real Measure
A The Relation of Self…Subsistent Measures
（a） Combination of Two Measures
（b） Measure of a Series of Measure Relations
（c） Elective Affinity
Remark： Berthollet on Chemical Affinity and Berzelius's Theory of it
B Nodal Line of Measure Relations
Remark： Examples of Such Nodal Lines； the Maxim； ‘Nature Does Not
Make Leaps’
The system of natural numbers already shows a nodal line of qualitative moments which emerge in
a merely external succession。 It is on the one hand a merely quantitative progress and regress； a
perpetual adding or subtracting； so that each number has the same arithmetical relation to the one
before it and after it； as these have to their predecessors and successors； and so on。 But the
numbers so formed also have a specific relation to other numbers preceding and following them；
being either an integral multiple of one of them or else a power or a root。 In the musical scale
which is built up on quantitative differences； a quantum gives rise to an harmonious relation without
its own relation to those on either side of it in the scale differing from the relation between these
again and their predecessors and successors。 While successive notes seem to be at an
ever…increasing distance from the keynote； or numbers in succeeding each other arithmetically
seem only to become other numbers； the fact is that there suddenly emerges a return； a surprising
accord； of which no hint was given by the quality of what immediately preceded it； but which
appears as an actio in distans； as a connection with something far removed。 There is a sudden
interruption of the succession of merely indifferent relations which do not alter the preceding
specific reality or do not even form any such； and although the succession is continued
quantitatively in the same manner； a specific relation breaks in per saltum。
Such qualitative nodes and leaps occur in chemical combinations when the mixture proportions are
progressively altered； at certain points in the scale of mixtures； two substances form products
exhibiting particular qualities。 These products are distinguished from one another not merely by a
more or less； and they are not already present； or only perhaps in a weaker degree； in the
proportions close to the nodal proportions； but are bound up with these nodes themselves。 For
example； different oxides of nitrogen and nitric acids having essentially different qualities are
formed only when oxygen and nitrogen are combined in certain specific proportions； and no such
specific compounds are formed by the intermediate proportions。 Metal oxides； e。g。 the lead
oxides； are formed at certain quantitative points of oxidation and are distinguished by colours and
other qualities。 They do not pass gradually into one another； the proportions lying in between these
nodes do not produce a neutral or a specific substance。 Without having passed through the
intervening stages； a specific compound appears which is based on a measure relation and
possesses characteristic qualities。 Again； water when its temperature is altered does not merely get
more or less hot but passes through from the liquid into either the solid or gaseous states； these
states do not appear gradually； on the contrary； each new state appears as a leap； suddenly
interrupting and checking the gradual succession of temperature changes at these points。 Every
birth and death； far from being a progressive gradualness； is an interruption of it and is the leap
from a quantitative into a qualitative alteration。
It is said； natura non facit saltum ＇there are no leaps in nature＇； and ordinary thinking when it has
to grasp a coming…to…be or a ceasing…to…be； fancies it has done so by representing it as a gradual
emergence or disappearance。 But we have seen that the alterations of being in general are not only
the transition of one magnitude into another； but a transition from quality into quantity and vice
versa； a becoming…other which is an interruption of gradualness and the production of something
qualitatively different from the reality which preceded it。 Water； in cooling； does not gradually
harden as if it t