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this enclosure containing another enclosure on its area, then graph consisting of all the rest of the contents of the enclosure may whenever it is scribed on the sheet of insertion be transformed by the insertion into it as a component of that graph which is contained in the inner enclosure
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In order to put this into a shape capable more explicit shape, I will first call your attention to a corollary from it. A corollary to a proposition of Euclid is a necessary consequence drawn from it by some editor of Euclid's Elements and inserted by him, originally, I suppose, marked with a little crown in the margin. These additions are, for the most part, trifles propositions that Euclid thought not worth too obvious for special notice. Hence, any easily drawn necessary consequence of a proposition is termed a corollary. Here I will tell you a secret about necessary consequences. It is a very useful thing to know, although most logicians are quite entirely ignorant of it. It is that not even this simplest necessary consequence can be drawn except by the aid of Observation, namely the observation of some feature of something of the
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nature of a diagram, whether on paper or in the imagination. I draw a distinction between Corollarial consequences and Theorematic consequences. A corollarial consequence is one the truth of which will become evident simply upon attentive observation of a diagram constructed so as to represent the conditions stated in the conclusion. A theorematic consequence is one which only becomes evident after some experiment has been performed upon the diagram, such as the addition to it of parts not mentioned necessarily referred to in the statement of the conclusion. In the present case, I am going to draw a conclusion about a double enclosure that is, two cuts one within the other and with the annular space between them blank like this The observation which I ask you to make is that
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in every such case there will be a graph of which one replica is in the outer close which is also while another is on the sheet of assertion outside. Namely, that graph is the blank. And since the present principle allows permits us to transform [into] whatever x may be, it allows this transformation when x is the blank; so that we can transform into We may count this as our third permission, so that we have
Permission No 3. A graph within a double enclosure on the sheet of assertion may be scribed on the sheet of assertion, unenclosed.
The consideration of what further explicit permission is involved in the predication of the definition, the definitum being the subject, had better be postponed until we have considered
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the predication of the definitum the definition being the subject. This predication is that in case the permission to scribe on the sheet of assertion a replica of a graph, x, would carry with it in every case a permission to scribe on the sheet of assertion a replica of a graph, y, then it is permissible to scribe on the sheet of assertion a scroll containing in its outer close only a replica of x and in its inner close a replica of y. Or in terms of transformation, if it would be permissible to transform a graph, x, should it occur on the sheet of assertion, into a graph, y, then it is permissible to transform a blank on the sheet of assertion into a scroll having