§ 2. If in the case of a notion whose specialisations form a continuous 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 manifoldness in its turn 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 specialisations 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 n + 1 dimensions out of a variability of n dimensions and a variability of one dimension.
I shall show how conversely one may resolve a variability whose region is given into a variability of one dimension 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 comparable with one another - which has for every point of the given manifoldness a definite value, varying continuously with the point...
This is a 1-ply extended magnitude (a bendy 2D line) 'passing over' into a 3-ply extended magnitude (a 4D Volume). This 'skips' the 2-ply extended magnitude because we're rotating the curve, rather than just 'stretching it out' along a single dimension, like was done in fdrake's post. — StreetlightX
I shall show how conversely one may resolve a variability (manifoldness-me) whose region is given into a variability of one dimension and a variability of fewer dimensions. To this end let us suppose a variable piece of a manifoldness of one dimension (simply-extended, me) - reckoned from a fixed origin, that the values of it may be comparable with one another - which has for every point of the given manifoldness a definite value, varying continuously with the point; or, in other words, let us take a continuous function of position within the given manifoldness, which, moreover, is not constant throughout any part of that manifoldness.
There are manifoldnesses 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
A point of clarification for me. I'm trying to wrap my head around the idea that you only need one number to specify your location on a 2-dimensional line. — Moliere
I'm still confused about the OD point: does it count as an extended magnitude or not? — StreetlightX
Magnitude-notions are only possible where there is an antecedent general notion which admits of different specialisations. According as there exists among these specialisations a continuous path from one to another or not, they form a continuous or discrete manifoldness; the individual specialisations are called in the first case points, in the second case elements, of the manifoldness
According as there exists among these specialisations a continuous path from one to another or not, they form a continuous or discrete manifoldness; the individual specialisations are called in the first case points
According as there exists among these specialisations a continuous path from one to another or not, they form a continuous manifoldness (whose specialisations consist of) points
If in the case of a notion whose specialisations form a continuous manifoldness, one passes from a certain specialisation (point 1) in a definite way to another (point 2), the specialisations passed over form a simply extended manifoldness (a path between points - a line or curve), whose true character is that in it a continuous progress from a point is possible only on two sides, forwards or backwards (in the direction of increasing or decreasing arclength).
Measure consists in the superposition of the magnitudes to be compared; it therefore requires a means of using one magnitude as the standard for another.
Having constructed the notion of a manifoldness of n dimensions, and found that its true character consists in the property that the determination of position in it may be reduced to n determinations of magnitude (breaking a sentence in two here)
... (Now) 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 dependence on one another can only be represented by formulæ. On certain assumptions, however, they are decomposable into relations which, taken separately, are capable of geometric representation; and thus it becomes possible to express geometrically the calculated results. In this way, to come to solid ground, we cannot, it is true, avoid abstract considerations in our formulæ, but at least the results of calculation may subsequently be presented in a geometric form. The foundations of these two parts of the question are established in the celebrated memoir of Gauss, Disqusitiones generales circa superficies curvas.
These measure-relations can only be studied in abstract notions of quantity, and their dependence on one another can only be represented by formulæ.
certain assumptions (which ensure that the length ascriptions) are capable of geometric representation; and thus it becomes possible to express geometrically the calculated results.
§ 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 independent of their position, and consequently every line is measurable by means of every other.
Position-fixing being reduced to quantity-fixings, and the position of a point in the n-dimensioned manifoldness being consequently expressed by means of n variables x1, x2, x3,..., xn, 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 expression for the length of a line, and to this end we must consider the quantities x as expressible in terms of certain units. I shall treat this problem only under certain restrictions, and I shall confine myself in the first place to lines in which the ratios of the increments dx of the respective variables vary continuously. We may then conceive these lines broken up into elements, within which the ratios of the quantities dx may be regarded as constant; and the problem is then reduced to establishing for each point a general expression for the linear element ds starting from that point, an expression which will thus contain the quantities x and the quantities dx.
I shall suppose, secondly, that the length of the linear element, to the first order, is unaltered when all the points of this element undergo the same infinitesimal displacement, which implies at the same time that if all the quantities dx are increased in the same ratio, the linear element will vary also in the same ratio (1). On these suppositions, the linear element may be any homogeneous function of the first degree of the quantities dx, which is unchanged when we change the signs of all the dx (2), and in which the arbitrary constants are continuous functions of the quantities x.
To find the simplest cases, I shall seek first an expression for manifoldnesses of n - 1 dimensions which are everywhere equidistant from the origin of the linear element; that is, I shall seek a continuous function of position whose values distinguish them from one another.
In going outwards from the origin, this must either increase in all directions or decrease in all directions; I assume that it increases in all directions, and therefore has a minimum at that point. If, then, the first and second differential coefficients of this function are finite, its first differential must vanish, and the second differential cannot become negative; I assume that it is always positive.
This differential expression, of the second order remains constant when ds remains constant, and increases in the duplicate ratio when the dx, and therefore also ds, increase in the same ratio; it must therefore be ds2 multiplied by a constant, and consequently ds is the square root of an always positive integral homogeneous function of the second order of the quantities dx, in which the coefficients are continuous functions of the quantities x.
The next case in simplicity includes those manifoldnesses in which the line-element may be expressed as the fourth root of a quartic differential expression. The investigation of this more general kind would require no really different principles, but would take considerable time and throw little new light on the theory of space, especially as the results cannot be geometrically expressed; I restrict myself, therefore, to those manifoldnesses in which the line element is expressed as the square root of a quadric differential expression.
Such an expression (a quadratic in differentials) we can transform into another similar one if we substitute for the n independent variables functions of n new independent variables. In this way, however, we cannot transform any expression into any other; since the expression contains ½ n (n + 1) coefficients which are arbitrary functions of the independent variables; now by the introduction of new variables we can only satisfy n conditions, and therefore make no more than n of the coefficients equal to given quantities. The remaining ½ n (n - 1) are then entirely determined by the nature of the continuum to be represented, and consequently ½ n (n - 1) functions of positions are required for the determination of its measure-relations.
Manifoldnesses in which, as in the Plane and in Space, the line-element may be reduced to the form \sqrt{ \sum dx^2 }, are therefore only a particular case of the manifoldnesses to be here investigated; they require a special name, and therefore these manifoldnesses in which the square of the line-element may be expressed as the sum of the squares of complete differentials I will call flat. In order now to review the true varieties of all the continua which may be represented in the assumed form, it is necessary to get rid of difficulties arising from the mode of representation, which is accomplished by choosing the variables in accordance with a certain principle.
... If we have 3 variables x y z, there are 6 possible quadratic terms, x^2, y^2, z^2, xy, xz, yz, and so on. ...
This differential expression, of the second order remains constant when ds remains constant, and increases in the duplicate ratio when the dx, and therefore also ds, increase in the same ratio; it must therefore be ds2 multiplied by a constant, and consequently ds is the square root of an always positive integral homogeneous function of the second order of the quantities dx, in which the coefficients are continuous functions of the quantities x
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