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Table of Contents 2 1. on the ratio or proportion of two magnitude 6 2. Ratio of magnitude that vanish together 8 3. Ratio of Continuous increasing or decreasing quantities 12 4. The notion of infinitely small quantities 16 5. on functions 20 6. infinite series 21 7. Convergent and divergent series 23 8. taylor's theorem derived functions 26 9. differential coefficients 29 10. the notion of differential calculus 32 11. algebraical geometry 37 12. signs and algebrical directions of geometrical magnitudes 39 13. the drawing of a tangent to a curve 45 14. Language of leibnitz 48 15. Orders of infinity 52 16. geometrical illustration 55 17. problem solved by leibnitz 60 18. An illustration from dynamics. 65 19. Simple harmonic motion 71 20. The Mrthod of fluxions 74 21. Accelerated Motion 75 22. Limiting ratios of magnitude that increase without limit. 80 23. Recapitulation of Results 90 24. Approximation 91 25. Solution of equations 94 26. Partial and total Differentiation 95 27. Practical application of preceding theorem. 102 28. Rules for Differentiation 104 29. Differential Coefficients of Differential Coefficients. 106 30. Calculus of finite differences. Successive Differentiation

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