Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/281

Rh I. II. The corollary was added by Simson, It is liable to serious objection. For we do not know how the perpendicular BE is to be drawn. If we are to use I. 11 we must produce AB, and then we must assume that there is only one way of producing AB for otherwise we shall not know that there is only one perpendicular; and thus we assume what we have to demonstrate.

Simson's corollary might come after I. 13 and be demonstrated thus. If possible let the two straight lines ABC, ABD have the segment AB common to both. From the point B draw any straight line BE. Then the angles ABE and EBC are equal to two right angles, by I. 13, and the angles ABE and EBD are also equal to two right angles, by I. 13. Therefore the angles ABE and EBC are equal to the angles ABE and EBD. Therefore the angle EBC is equal to the angle EBD; which is absurd.

But if the question whether two straight lines can. have a common segment is to be considered at all in the Elements, it might occur at an earlier place than Simson has assigned to it. For example, in the figure to I. 5, if two straight lines could have a common segment AB, and then separate at B, we should obtain. two different angles formed on the other side of BC by these produced parts, and each of them would be equal to the angle BCG. The opinion has been maintained that even in I. 1, it is tacitly assumed that the straight lines AC and BC cannot have a common segment at C where they meet; see Camerer's Euclid, pages 30 and 36.

Simson never formally refers to his corollary until XI. I. The corollary should be omitted, and the tenth axiom should be extended so as to amount to the following; if two straight lines coincide in two points they must coincide both beyond and between those points.

I. 12. Here the straight line is said to be of unlimited length, in order that we may ensure that it shall meet the circle.

Euclid distinguishes between the terms at right angles and perpendicular. He uses the term at right angles when the straight line is drawn from a point in another, as in I. 11; and he uses the term perpendicular when the straight line is drawn from a point without another, as in I. 12. This distinction however is often disregarded by modern writers.

I. 14. Here Euclid first requires his eleventh axiom. For