An introduction to the study of the elements of the differential and integral calculus. From the German of the late Axel Harnack, With the permission of the author.

CONTENTS.
First Book. (Pp. 1-109.)
Real numbers and functions of real numbers.
Page
First Chapter: Rational numbers................ 1
~ 1. Addition and Subtraction. 2. Multiplication and division.
3. Positive and negative numbers. 4. Fractional numbers.
Second Chapter: Radicals and irrational numbers in general. 6
~ 5. Powers and roots. 6. The general conception of a number.
7. Calculation with numbers in general. 8. The logarithm.
Third Chapter: Conception of a variable quantity and calculation
with variables, specially with infinite quantities... 15
~ 9. The continuous series of numbers. 10. Calculation with variables.
Fourth Chapter: Conception and notation of functions of a
variable.................... 19
~ 11. Algebraic and transcendental functions. 12, 13. Tr'igonometric
and circular functions. 14. The first problem of analysis.
Fifth Chapter: Geometric representation of a function; its
continuity and its differential quotient... 24
~ 15. Geometric representation. 16. Criterion of continuity. Uniform
continuity. 17. Points of discontinuity. Superior and inferior limits of
a continuous function. 18. Behaviour of a function for infinite values
of the variable. 19. Theorems concerning continuous functions. 20. The
differential quotients, progressive and regressive. 21. Condition for their
equality. 22. Theorem of the mean value. 23. Geometric meaning of
the differential quotient. Historical note. 24. Geometrical corollaries
and illustrations.
Sixth Chapter: Differentiation of the simplest functions. 47
~ 25. The elementary functions. 26. General rules. 27. Explicit
rational algebraic functions. 28. Exponential function. 29. Trigonometric
functions. 30. Inverse functions: logarithmic and circular. 31. Explicit
irrational function.
Seventh Chapter: Successive differentiation of explicit functions. Different orders of infinitely small quantities. 56
~ 32. The higher derivates. 33. The higher quotients of differences.
31. Infinitesimals of different orders. 35. Successive differentiation
of elementary functions. Rule of the product. 36. Note on expressing
successive derivates independently.
Eighth Chapter: Calculation of functions by infinite series.
General theorems concerning series of powers... 65
~ 37. The extended theorem of the mean value. 38. Calculation of
functions. 39. Convergence of infinite series. 40. Taylor's series.
41. Investigation of its convergence. 42. Exponential series. 43. Trigonometric series. 44. General theorems concerning series of
powers: I. Convergence. II. Sum of two series. III. Unconditional
and conditional convergence. IV. Continuity. V. Differentiation. 45.
Infinite series as a definition of trigonometric functions. 46. Binomial