8. The following application of Dedekind’s principle is the most important one in practice.

If s1, s2, s3, sn … be a sequence of positive rational nucmbers which steadily increase and never become greater than a given positive number K, then the given sequence approximates to a definite limit, K being rational.

15. Let X1, Y1, Z1 be the second points in which AH, BH, CH meet the circle GH

Then GX1 ┴ HX, and hence ║ BC,

X1X =⅓AX;

tan PX1X= 3 tan PAX = 3 tan θ,

i.e. -PX1 X= ø.

But OX1H = OGH= ø; 0X1P is a straight line.

AT the beginning of this century we find that there was a lively movement on foot to insist on the application of mathematics to actual problems occurring in daily life—a reaction from that purely formal mathematical teaching which for half a century had treated the subject-matter as of secondary importance so long as the logical training was achieved. This reaction became apparent amongst schoolmasters, especially in the meetings of the Forderungsverein, and it was even asserted by the extremists that Mathematics was nothing more than the handmaid of Science.