|(39)||I.||Let it be granted that a right line may be drawn from any one point to any other point.|
|(40)||II.||Let it be granted that a finite right line may be produced to any length in a right line.|
|(41)||III.||Let it be granted that a circle may be described with any centre at any distance from that centre.|
(42) The object of the postulates is to declare, that the only instruments, the use of which is permitted in Geometry, are the rule and compass. The rule is an instrument which is use to direct the pen or pencil in drawing a right line; but it should be observed, that the geometrical rule is not supposed to be divided or graduated, and, consequently, it does not enable us to draw a right line of any proposed length. Neither is it permitted to place any permanent mark or marks on any part of the rule, or we should be able by it to solve the second proposition of the first book, which is to draw from a given point a right line equal to a another given right line. This might be done by placing the rule on the given right line, and marking its extremities on the rule, then placing the mark corresponding to one extremity at the given point, and drawing the pen along the rule to the second mark. This, however, is not intended to be granted by the postulates.
The third postulate concedes the use of the compass, which is an instrument composed of two straight and equal legs united at one extremity by a joint, so constructed that the legs can be opened or closed so as to form any proposed angle. The other extremities are points, and when the legs have been opened to any degree of divergence, the extremity of one of them being fixed at a point, and the extremity of the other being moved around it in the same plane will describe a circle, since the distance between the points is supposed to remain unchanged. The fixed point is the centre; and the distance between the points, the radius of the circle.
It is not intended to be conceded by the third postulate that a circle can be described round a given centre with a radius of a given length; in other words, it is not granted that the legs of the compass can be opened until the distance between their points shall equal a given line.
Book I: Euclid, Book I (ed. Dionysius Lardner, 11th Edition, 1855)
Next: Euclid, Book I, Axioms (ed. Dionysius Lardner, 11th Edition, 1855)
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