deduced 
from experience. Thus arises the problem, to discover 
the simplest matters of fact from which the measure-relations of 
space may be determined ; a problem which from the nature 
of the case is not completely determinate, since there may be 
several systems of matters of fact which suffice to determine the 
measure-relations of space—the most important system for our 
ee purpose being that which Euclid has laid down as a 
foundation. These matters of fact are—like all matters of fact— 
not necessary, but only of empirical certainty ; they are hypo- 
theses. We may therefore investigate their probability, which 
within the limits of observation is of course very great, and 
inquire about the justice of their extension beyond the limits of 
observation, on the side both of the infinitely great and of the 
infinitely small. 
I.—WNotion of an n-ply extended magnitude 
In proceeding to attempt the solution of the first of these pro- 
blems, the development of the notion of a multiply extended 
magnitude, I think I may the more claim indulgent criticism in 
that I am not practised in such undertakings of a philosophical 
nature where the difficulty iies more in the notions themselves 
than in the construction ; and that besides some very short hints 
on the matter given by Privy Councillor Gauss in his second 
memoir on Biquadratic Residues, in the ‘*G6ttingen Gelehrte 
Anzeige,” and in his Jubilee-book, and some philosophical re- 
searches of Herbart, I could make use of no previous labours. 
§ 1.—Magnitude-notions are only possible where there is an 
antecedent general notion which admits of different specialisa- 
tions. According as there exists among these specialisations a 
continuous path from one to another or not, they form a con- 
tinuous or discrete manifoldness : the individual specialisations 
are called in the first case points, in the second case elements, of 
the manifoldness. Notions whose specialisations form a discrete 
manifoldness are so common that at least in the cultivated 
languages any things being given it is always possible to find a 
notion in which they are included. (Hence mathematicians might 
unhesitatingly found the theory of discrete magnitudes upon the 
postulate that certain given things are to be regarded as equiva- 
lent.) On the other hand, so few and far between are the occa- 
sions for forming notions whose specialisations make up a cov- 
tinuous manifoldness, that the only simple notions whose 
specialisations form a multiply extended manifoldness are the 
positions of perceived objects and colours. More frequent occa- 
sions for the creation and development of these notions occur 
first in the higher mathematic. 
Definite portions of a manifoldness, distinguished by a mark 
or by a boundary, are called Quanta. Their comparison with 
regard to quantity is accomplished in the case of discrete mag- 
nitudes by counting, in the case of continuous magnitudes by 
measuring. Measure consists in the superposition of the magni- 
tudes to be compared ; it therefore requires a means of using 
one magnitude as the standard for another. In the absence of 
this two magnitudes can only be compared when one is a part 
of the other ; in which case also we can only determine the more 
or less and not the how much. The researches which can in 
this case be instituted about them form a general division of the 
Science of magnitude in which magnitudes are regarded not as 
existing independently of position and not as expressible in 
terms of a unit, but as regions in a manifoldness. Such re- 
searches have become a necessity for many parts of mathematics, 
e.g., for the treatment of many-valued analytical functions ; and 
the want of them is no doubt a chief cause why the celebrated 
theorem of Abel and the achievements of Lagrange, Pfaff, 
Jacobi for the general theory of differential equations, have so 
long remained unfruitful. Out of this general part of the science 
of extended magnitude in which nothing is assumed but what is 
contained in the notion of it, it will suffice for the present pur- 
pose to bring into prominence two points; the first of which 
relates to the construction of the notion of a multiply extended 
manifoldness, the second relates to the reduction of determina- 
tions of place in a given manifildness to determinations of 
quantity, and will make clear the true character of an 7-fold 
extent. . 
§ 2 --Ifinthe case of a notion whose specialisations form a con- 
tinuous manifoldness, one passes from a certain specialisation in 
a definite way to another, the specialisations passed over form a 
simply extended manifoldness, whose true character is that in it 
a continuous progress from a point is possible only on two sides, 
forwards or backwards. If one now supposes that this mani- 
- foldness in its tum passes over into another entirely different, 
and again in a definite way, namely so that each point passes 
over into a definite point of the other, then all the specialisa- 
tions so obtained form a doubly extended manifoldness. In a 
similar manner one obtains a triply extended manifoldness, if 
one imagines a doubly extended one passing over in a definite 
way to another entirely different ; and it is easy to see how this 
construction may be continued. If one regards the variable 
object instead of the determinable notion of it, this construction 
may be described as a composition of a variability of +1 
dimensions out of a variability of 2 dimensions and a variability 
of one dimension. 
§ 3.—I shall now show how conversely one may resolve a 
variability whose region is given into a variability of one dimen- 
sion and a variability of fewer dimensions. ‘To this end let us 
suppose a variable piece of a manifoldness of one dimension— 
reckoned from a fixed origin, that the values of it may be com- 
parable with one another—which has for every point of the 
given manifoldness a definite valuc, varying continuously with 
the point ; or, in other words, let us take a continuous function 
of position within the given mani o!dness, which, moreover, is 
not constant throughout any parto that manifoldness, Every 
system of points where the function 42s a constant value, forms 
then a continuous manifoldness of fewer dimensions than the 
given one. These manifoldnesses pass over continuously into 
one another as the function changes ; we may therefore assume 
that out of one of them the others proceed, and speaking gene- 
rally this may occur in such a way that each point passes over 
into a definite point of the other; the cases of exception (the 
study of which is important) may here be left unconsidered, 
Hereby the determination of position in the given manifoldness 
is reduced to a determination of quantity and to a determination 
of position in a manifoldness of less dimensions. It is now easy 
to show that this manifoldness has #—1 dimensions when 
the given manifoldness is #-ply extended. By repeating then 
this operation # times, the determination of position in an 
n-ply extended manifoldness is reduced to » determinations 
of quantity, and therefore the determination of position in a 
given manifoldness is reduced to a finite number of deter- 
minations of quantity when this is possible. There are mani- 
foldnesses in which the determination of position requires 
not a finite number, but either an endless series or a continuous 
manifoldness of determinations of quantity. Such manifoldnesses 
are, for example, the possible determinations of a function for a 
given region, the possible shapes of a solid figure, &c. 
Il.—Measure-relations of which a manifoldness of n dimensions 
ts capable on the assumption that lines havea length independent 
of position, and consequently that every line may be measured by 
every other. 
Having constructed the notion of a manifoldness of 7 dimen- 
sions, and found that its true character consists in the property 
that the determination of position in it may be reduced to x 
determinations of magnitude, we come to the second of the 
problems proposed above, viz., the study of the measure-relations 
of which such a manifoldness is capable, and of the conditions 
which suffice to determine them. These measure-relations can 
only be studied in abstract notions of quantity, and their depen- 
dence on one another can only be represented by formule. On 
certain assumptions, however, they are decomposable into rela- 
tions which, taken separately, are capable of geometric re- 
presentation ; and thus it becomes possible to express geome- 
trically the calculated results. In this way, to come to solid 
ground, we cannot, it is true, avoid abstract considerations in 
our formulz, but at least the results of calculation may subse- 
quently be presented in a geometric form. The foundations of 
these two parts of the question are established in the celebrated 
memoir of Gauss—‘‘ Disquisitiones generales circa superficies 
curvas.” 
§ 1.—Measure-determinations require that quantity should be 
independent of position, which may happen .in various ways. 
The hypothesis which first presents itself, and which I shall here 
develop, is that according to which the length of lines is inte- 
pendent of their position, and consequently every line is measur- 
able by means of every other. Position-fixing being reduced to 
quantity-fixings, and the position of a point in the 7-dimensioned 
manifoldness being consequertly expressed by means of # 
variables 2, x», 7g, + + » “,) the determination of a line comes 
to the giving of these quantities as functions of one variable. 
The problem consists then in establishing a mathematical ex- 
pression for the length of a line, and to this end we must con- 
sider the quantities as expressible in terms of certain units, I 
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