The Elements

¶ Another is the rhomboid original: "helmuaym", a term derived from Arabic translations of Euclid used in the Middle Ages which has opposite sides equal and opposite angles equal; however, it is contained neither by right angles nor by equal sides. Beyond all these, however, all quadrilateral figures are called trapeziums original: "helmuay/riphe". ¶ Parallel lines are those which, being located in the same plane and extended in either direction, do not meet even if they are drawn out to infinity.

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There are five Postulates Petitiones: basic geometric "requests" or assumptions required to perform constructions: ¶ To draw a straight line from any point to any point. ¶ To extend a finite straight line continuously in a straight path as far as one wishes. ¶ To describe a circle around any center using any distance as a radius. ¶ That all right angles are equal to one another. ¶ If a straight line falls upon two straight lines and the two interior angles on one side are less than two right angles, those two lines, if extended in that direction, will undoubtedly eventually meet This is the famous "Parallel Postulate," the foundation of Euclidean geometry. ¶ That two straight lines enclose no surface.

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These are the Common Notions Communes animi conceptiones: self-evident mathematical axioms that apply to all sciences, not just geometry: ¶ Things which are equal to one and the same thing are also equal to each other. ¶ And if equals are added to equals, the wholes will also be equal. ¶ And if equals are subtracted from equals, the remainders will be equal. ¶ And if you take away equals from unequals, the remainders will be unequal. ¶ And if you add equals to unequals, the totals will also be unequal. ¶ If two things are equal to one, they will be equal to each other. ¶ If there are two things, each of which is half of one and the same thing, each will be equal to the other. ¶ If any thing is placed upon another and applied to it, and one does not exceed the other, they will be equal to each other This refers to the principle of superposition. ¶ The whole is greater than its part.

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It should be known, however, that besides these common notions—or "common knowledge"—Euclid omitted many others which are incomprehensible in number, of which this is one: ¶ If two equal quantities are compared to any third of the same kind, both will be either equally greater than that third, or equally smaller, or both equal at once. ¶ Also another: whatever the ratio of one quantity is to any other of the same kind, any third quantity will have that same ratio to some fourth of the same kind in continuous quantities Continuous quantities refer to physical magnitudes like lines or planes; this is universally true whether the antecedents are larger or smaller than the consequents. For magnitude decreases toward infinity. In numbers Discrete quantities or integers, however, it is not so: but if the first is a submultiple of the second, any third will be an equal submultiple of some fourth; because number increases toward infinity, just as magnitude is diminished toward infinity.

Proposition 1.

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To construct an equilateral triangle upon a given straight line.
¶ Let the given straight line be a.b. I wish to construct an equilateral triangle upon it. At one of its ends, at point a, I shall place the fixed foot of the compass; and I shall extend the other, movable foot as far as b; and I shall describe a circle c.b.d.f. according to the length of the given line, by the second postulate.

A geometric diagram showing two intersecting circles of the same radius. The centers of the circles are marked 'a' and 'b'. The intersection points are marked 'f' (top) and 'c' (bottom). A triangle is formed by connecting points 'a', 'b', and 'f'. The outer arcs are labeled with 'f', 'b', and 'h' on the right circle, and 'f', 'a', and 'd' on the left circle.