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The study of mathematics is apt to commence in disappointment. The important applications of the science, the theoretical interest of its ideas, and the logical rigour of its methods, all generate the expectation of a speedy introduction to processes of interest. We are told that by its aid the stars are weighed and the billions of molecules in a drop of water are counted. Yet, like the ghost of Hamlet's father, this great science eludes the efforts of our mental weapons to grasp it- 'Tis here', 'tis there', 'tis gone' and what we do see does not suggest the same excuse for illusiveness as sufficed for the ghost, that it is too noble for our gross methods. “A show of violence”, if ever excusable, may surely be “offered” to the trivial results which occupy the pages of some elementary mathematical treatises.

Contents: 1. The abstract nature of mathematics. 2. Variables. 3. Methods of application. 4. Dynamics. 5. The symbolism of mathematics. 6. Generalizations of number. 7. Imaginary numbers. 8. Imaginary numbers (continued). 9. Coordinate geometry. 10. Conic sections. 11. Functions. 12. Periodicity in nature. 13. Trigonometry. 14. Series. 15. The differential calculus. 16. Geometry.