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Topic: Keely Geometry Stack Section: Geometry Proposition 15 Table of Contents to this Topic |
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Proposition XV The value of the infinity which is the difference between the inscribed and circumscribed lines (axiom 4th), and which is omitted by geometers, is increased in the process of bisection of a circumference, so that at some great number of sides of a polygon it will always equal one or more in the sixth decimal place, and may be increased, until it shall equal circumference itself. Axioms as proven herein and as self-evident: First: Space is infinitely divisible. (Proposition XIV) Second: Any imaginary line (not a material line), which shall have breadth, is equal to the same portion of space. Third: Any such imaginary line is, therefore, infinitely divisible. Fourth: Any such imaginary line may, therefore, be divided, until each part or division is less than any magnitude which is, or can be, developed to our senses. Fifth: At whatever point the division of such a line may be arrested, because the sum of all parts is equal to the whole; therefore, each aprt must have breadth, though the breadth of each part may be such, that no conceivable number of them form a developed magnitude. Sixth: One line cannot occupy two places at the same time; neither can two lines be in one and the same place, at the same time. Seventh: Two lines without breadth, cannot exist with no breadth between them. Eighth: The existence of shape signifies limit; hence, no shape can exist without a boundary line definitely located, which forms no part of the shape itself, which boundary is its circumference. |
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