Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/359

Rh to BO that is, AO is to OQ as QO is to OB. Therefore the triangles AOQ and QOB are similar triangles (VI. 6); and therefore AQ is to QB as QO is to BO. This shews that the ratio of AQ to BQ is constant; we have still to shew that this ratio is the same as the assigned ratio.

We have already shewn that AO is to DO as DO is to BO; therefore, the difference of AO and BO is to DO as the difference of DO and BO is to BO (V. 17); that is, AD is to DO as BD is to BO; therefore AD is to BD as DO is to BO; that is, AD is to DB as QO is to BO. This shews that the ratio of QO to BO is the same as the assigned ratio.

ON MODERN GEOMETRY.

56. We have hitherto restricted ourselves' to Euclid's Elements, and propositions which can be demonstrated by strict adherence to Euclid's methods. In modern times various other methods have been introduced, and have led to numerous and important results. These methods may be called semi-geometrical, as they are not confined within the limits of the ancient pure geometry; in fact the power of the modern methods is obtained chiefly by combining arithmetic and algebra with geometry. The student who desires to cultivate this part of mathematics may consult Townsend's Chapters on the Modern Geometry of the Pointy Line, and Circle.

We will give as specimens some important theorems, taken from what is called the theory of transversals.

Any line, straight or curved, which cuts a system of other lines is called a transversal; in the examples which we shall give, the lines will be straight lines, and the system svill consist of three straight lines forming a triangle.

We will give a brief enunciation of the theorem which we are about to prove, for the sake of assisting the memory in retaining the result; but the enunciation will not be fully comprehended until the demonstration is completed.