By Frank Morley, F. V. Morley

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**Sample text**

As .. -~) THE EUCLIDEAN GROUP 31 is real. The first arc is then any arc joining al to as. That is, the problem of drawing 2n arcs with continuity of direction is poristic; when it is possible, it is possible in an infinity of ways. (H. ) § 18. The Product of Homologies - Geometrically, a homo- logy with centre C sends a circle into a circle, and a circle C on f into another circle CH on f. C and CH will meet again at d. Any point x of C is sent into a point xH of CH. The join of FIG. 15 these points will be on d, since the angles f, x, d and f, xH, d are given.

We have A= 0 cot a. It is manifestly proper to take the particular spiral with the angle 'TT/4. This we call the logarithmic spiral. Then That is, the circle with centre 0 and radius p cuts this spiral at one point x, and the angle actually turned through in passing from 1 to x along the spiral is A. If the angle is 1, then p is the number e. There is then associated with a number ·x=pt the number y=A+ to. We call y the logarithm of x, y =log x; when 0 is a principal angle, y is the principal logarithm.

As - aa)(a. ) ••. (as .. -~) THE EUCLIDEAN GROUP 31 is real. The first arc is then any arc joining al to as. That is, the problem of drawing 2n arcs with continuity of direction is poristic; when it is possible, it is possible in an infinity of ways. (H. ) § 18. The Product of Homologies - Geometrically, a homo- logy with centre C sends a circle into a circle, and a circle C on f into another circle CH on f. C and CH will meet again at d. Any point x of C is sent into a point xH of CH. The join of FIG.