Preface; 1. Introduction; 2. Historical note on Landen's transformation and the various scales of moduli and amplitudes; 3. On the scale of arithmetico-geometrical means; 4. Landen's scale of increasing amplitudes; 5. The hyperbolic scale of increasing amplitudes; 6. Gauss' scale of increasing amplitudes; 7. Landen's scale of decreasing amplitudes; 8. The hyperbolic scale of decreasing amplitudes; 9. On the numerical computation of the third elliptic integral; 10. Note on the calculation of the third elliptic integral in terms of the complementary A. G. M. scale; Summary of formulae; Appendix. Examples 1-35.