Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/295

Rh and the square on AC is equal to the squares on CD, DA, because the angle BDA is a right angle. [I. 47. Therefore the squares on CB, BA are equal to the square on AC, together with twice the rectangle CB, BD; that is, the square on AG alone is less than the squares on CB, BA, by twice the rectangle CB, BD.

Next, suppose AC perpendicular to BC. Then BC is the straight line intercepted between the perpendicular and the acute angle at B. And the square on AB is equal to the squares on AC, CB. [I. 47. Therefore the square on AC is less than the squares on AB, BC, by twice the square on BC.

II. 14. This is not required in any of the parts of Euclid's Elements which are usually read; it is included in VI. 22.

THE THIRD BOOK.

The third book of the Elements is devoted to properties of circles.

Different opinions have been held as to what is, or should be, included in the third definition of the third book. One opinion is that the definition only means that the circles do not cut in the neighbourhood of the point of contact, and that it must be shewn that they do not cut elsewhere. Another opinion is that the definition means that the circles do not cut at all; and this seems the correct opinion. The definition may therefore be presented more distinctly thus. Two circles are said to touch internally when their circumferences have one or more common points, and when every point in one circle is within the other circle, except the common point or points. Two circles are said to touch externally when their circumferences have one or more common points, and when every point in each circle is without the other circle, except the common point or points. It is then shewn in the third Book that the circumferences of two circles "which touch can have only one common point.

A straight line which touches 'a circle is often called a tangent to the circle, or briefly, a tangent.

It is very convenient to have a word to denote a portion of