belongs to the special science of the concrete。 Examples of this kind concerning the law of falling
bodies and free； celestial motion will be found in the Encyclopedia。 of the Phil。 Sciences； 3rd
ed。； Sections 267 and 270； Remark。 In this connection the general observation may be made that
the different forms in which measure is realised belong also to different spheres of natural reality。
The complete； abstract indifference of developed measure； i。e。 the laws of measure； can only be
manifested in the sphere of mechanics in which the concrete bodily factor is itself only abstract
matter； the qualitative differences of such matter are essentially quantitatively determined； space
and time are the purest forms of externality； and the multitude of matters； masses； intensity of
weight； are similarly external determinations which have their characteristic determinateness in the
quantitative element。 On the other hand； such quantitative determinateness of abstract matter is
deranged simply by the plurality of conflicting qualities in the inorganic sphere a rid still more even
in the organic world。 But here there is involved not merely a conflict of qualities； for measure here
is subordinated to higher relationships and the immanent development of measure tends to be
reduced to the simple form of immediate measure。 The limbs of the animal organism have a
measure which； as a simple quantum； stands in a ratio to the other quanta of the other limbs； the
proportions of the human body are the fixed ratio of such quanta。 Natural science is stil1 far from
possessing an insight into the connection between such quantities and the organic functions on
which they wholly depend。 But the readiest example of the reduction of an immanent measure to a
merely externally determined magnitude is motion。 In the celestial bodies it is free motion； a
motion which is determined solely by the Notion and whose quantitative elements therefore equally
depend solely on the Notion （see above）； but such free motion is reduced by the living creature to
arbitrary or mechanically regular； i。e。 a wholly abstract； formal motion。
And in the realm of spirit there is still less to be found a characteristic； free development of
measure。 It is quite evident； for example； that a republican constitution like that of Athens； or an
aristocratic constitution tempered by democracy； is suitable only for States of a certain size； and
that in a developed civil society the numbers of individuals belonging to different occupations stand
in a certain relations to one another； but all this yields neither laws of measure nor characteristic
forms of it。 In the spiritual sphere as such there occur differences of intensity of character；
strength of imagination； sensations； general ideas； and so on； but the determination does not go
beyond the indefiniteness of strength or weakness。 How insipid and completely empty the
so…called laws turn out to be which have been laid down about the relation of strength and
weakness of sensations； general ideas； and so on； comes home to one on reading the psychologies
which occupy themselves with such laws。
Chapter 1： Specific Quantity
A The Specific Quantum
B Specifying Measure
（a） The Rule
（b） Specifying Measure
（c） Relation of the Two Sides as Qualities
Remark
The exposition here of the connection between the qualitative nature of something and its
quantitative determination has its application in the already indicated example of motion。 First of
all； in velocity as the direct ratio of space traversed and time elapsed； the magnitude of time is
taken as denominator while that of space is taken as numerator。 If velocity as such is only a ratio
of the space and time in a motion； it is immaterial which of the two moments is to be considered as
amount or as unit。 Space； however； like weight in specific gravity； is an external； real whole as
such … hence amount … whereas time； like volume； is the ideal； negative factor； the side of unity。
But here there essentially belongs the more important ratio； that which holds between the
magnitudes of space and time in free motion； at first； in the still conditioned motion of a falling
body where the time factor is determined as a root and the space factor as a square； or in the
absolutely free motion of the celestial bodies where the period of revolution is lower by one power
than the distance from the sun； the former being a square and the latter a cube。 Fundamental
relationships of this kind rest on the nature of the interrelated qualities of space and time and on the
kind of relation in which they stand； either as a mechanical motion； i。e。 as an unfree motion which
is not determined by the Notion of the moments of space and time； or as the descent of a falling
body； i。e。 as a conditionally free motion； or as the absolutely free celestial motion。 These kinds of
motion； no less than their laws； rest on the development of the Notion of their moments； of space
and time； since these qualities as such （space and time） prove to be in themselves； i。e。 in their
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 produ