Wisdom

Without this distinction we should Post Date: Sat, 2 Aug 2008 1:45:01 +0000
The difficulty is not in knowing what extension is geome trically considered, but what it is in reality. We know the geometrical essence, but what we want to ascertain is, whe ther this essence realized is something which is confounded with some other real thing, or is only a quality which we know without knowing the being to which it belongs. Without this distinction we should deny the basis of geom etry ; for, it is evident that if we should not know the essence of extension in the aforesaid manner, we could not be sure that we are not building in the air when we raise upon the idea of extension the whole science of geometry.

Autor of the post: Undefined

The dimensions in the external Post Date: Sat, 2 Aug 2008 2:02:40 +0000
29 Thus then under this aspect, we are certain that ex tension exists outside of us, and that there are true dimen sions. This idea is a necessary consequence of the idea of the external world, as we said before. The dimensions in the external world must be subject to the same principles as those which we conceive, or the very idea which we have formed of the external world is reversed.

Autor of the post: Undefined

The sharpest instrument can never Post Date: Sat, 2 Aug 2008 2:14:57 +0000
I do not mean by this that a real circle may be a geometrical circle, but only that what is true of the second must be true of the first also, in proportion as it is constructed with greater or less exactness. Beyond what can be formed by the most perfect and exact instruments, I can conceive, without pas sing from the order of reality, a circle or any other figure, as near as I please to the geometrical idea. The sharpest instrument can never mark an indivisible point, nor draw a line without breadth; but this surface, on which the pobt is marked, or the line drawn, being infinitely divisible, I can conceive a case in which the reality will come infinitely near to the geometrical idea, 30 Astronomy and all the physical sciences rest on the supposition that real extension is subject to the same prin ciples as ideal extension ; and that experience comes closer to theory in proportion as the conditions of the second are more exactly fulfilled in the first.

Autor of the post: Undefined

Therefore, extension exists not only Post Date: Sat, 2 Aug 2008 2:34:06 +0000
The art of constructing mathematical instruments, which has been brought in our day to a surprising perfection, regards the ideal as the type of the real order; and progress in the latter is the approx imation to the models of the former. Theory directs the operations of practice, and these in their turn confirm by the result the foresight of theory. Therefore, extension exists not only in the ideal order, but also in the real ; and it is something, independently of our ideas ; and geometry, that vast representation of a world of lines and figures, has a real object in nature.

Autor of the post: Undefined

This is a mistaken opinion Post Date: Sat, 2 Aug 2008 2:50:55 +0000
How far the real corresponds with the ideal, we shall ex amine in the next chapter. 31 THE disagreement which we discover between the phenomena and the geometrical theory makes us apt to think that reality is rough and course, and that purity and exactness are found only in our ideas. This is a mistaken opinion caused by want of reflection.

Autor of the post: Undefined

We shall first prove Post Date: Sat, 2 Aug 2008 3:08:40 +0000
The reality is as geo metrical as our ideas ; the phenomenon realizes the idea in all its purity and vigor. Be not startled by this seeming paradox ; for it will soon appear to you a very true, reason able, and well-grounded proposition. We shall first prove that the ideas which are the ele ments of geometry have their objects in the real world, and that these objects are subject to precisely the same condi tions as the ideas.

Autor of the post: Undefined

In the ideal order Post Date: Sat, 2 Aug 2008 3:21:28 +0000
This proved, it clearly follows that geo metry in all its strictness exists as well in the real as in the ideal order. 32 Let us begin with a point. In the ideal order, a point is an invisible thing, it is the limit of a line and its generating element, and it occupies a determinate position in space.

Autor of the post: Undefined

The point is the generating Post Date: Sat, 2 Aug 2008 3:40:06 +0000
It is the limit of a line ; for when we take away its length, we have a point remaining which we are forced to regard as the limit of the line unless we destroy it en tirely so as to have nothing left. The more the line is shortened the nearer it approaches to a point, yet can never be identified with it until its length is wholly suppressed. The point is the generating element of the line ; for we form the idea of lineal dimension by considering a point in motion.

Autor of the post: Undefined

As a general rule, geometry Post Date: Sat, 2 Aug 2008 3:56:55 +0000
The occupation of a determinate position in space is another indispensable condition of the idea of a point, if we wish to use it in geometrical figures. The centre of a circle is a point in itself indivisible, it fills no space ; but in order that it be of any use as centre, we must be able to refer all the radii to it, and this is impossible unless it oc cupy a determinate position equidistant from all points of the circumference. As a general rule, geometry acts upon dimensions, and these dimensions require points in which they commence, points through which they pass, and points in which they end, and by which distances, inclinations, and all that relates to the position of lines and planes, are measured.

Autor of the post: Undefined

sponds to the geometrical point Post Date: Sat, 2 Aug 2008 4:10:41 +0000
Nothing of all this can be conceived unless the point, although not extended, occupies a determinate posi tion in space. 33 Does there exist in nature anything which corre-. sponds to the geometrical point, and unites all its conditions} with as great exactness as science in its purest idealism can! desire ? I believe there does.

Autor of the post: Undefined

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