cover -- copyright page -- title page -- Foreword -- Preface -- Contents -- Short Biography of H. S. Wall -- 1 Numbers -- Sample Proofs -- 2 Ordered Number Pairs -- The Simple Graph L -- Problems -- The Logarithm -- Properties of L -- The Simple Graph E -- Powers and Roots -- 3 Slope -- Straight Lines -- Computation Formulas for L -- Computation of e -- Slope of a Simple Graph -- The Simple Graph I -- 4 Combinations of Simple Graphs -- Addition -- Multiplication -- Bracket Multiplication -- The Derivative -- 5 Theorems about Simple Graphs -- The Conjectured Computation Formula -- The Addition Formulas for L and E -- The Simple Graphs S and C -- A Point Set which is not a Simple Graph -- Inequalities -- 6 The Simple Graphs of Trigonometry -- Addition Formulas for A -- Computation Problem for A -- The Simple Graphs K, C, S, and B -- Extension of S and C to the Set of all Numbers -- Connection with Triangles -- Property S -- 7 The Integral -- Property S -- Properties of Integrals -- Evaluation of Integrals -- A Theorem on Transformation of Integrals -- 8 Computation Formulas Obtained by Means of the Integral -- Computation Formulas For L, A -- Computation Formulas for E, S, C -- The Binomial Theorem -- Length of a Simple Graph -- Length of an Arc of a Circle -- Polar Equations -- Analogy Between S, C and S, C -- 9 Simple Graphs Made to Order -- The Equation f ' = g · f + h -- The Equation f" + kf = g -- The Equation f" + pf' + qf = g -- 10 More about Integrals -- g-Length of an Interval -- Integral of a Simple Graph with Respect to a Simple Graph -- Properties of Int^b_a f dg -- Bounded Variation -- Extended Integrals -- Number Sequences -- Two Questions -- 11 Simple Surfaces -- Gradient -- Combination of Simple Surfaces -- More Simple Graphs Determined by a Simple Surface -- g-Area of a Rectangular Interval -- Maxima and Minima.

Integral of a Simple Surface with Respect to a Simple Surface -- Iterated Integrals -- Case Where f is not Continuous -- Integral over a Point Set -- Area of a Simple Surface -- Path Integrals -- Vectors -- Assorted Theorems and Problems -- 12 Successive Approximations -- Existence of Certain Numbers Established by Successive Approximations -- Lemmas Concerning Sequences -- Simple Graphs Determined by Successive Approximations -- The Equation f ' − f = −H -- Ordered Pairs of Simple Graphs Determined by Successive Approximations -- Particular Examples -- (A) The Simple Graphs S and C -- (B) The Equation f ' = I^2 + f^2 -- (C) The Equation u f " + v f ' + wf = g -- 13 Linear Spaces of Simple Graphs -- Linear Transformations in the Plane -- Axioms for Linear Space -- The Inner Product Space M[a, b] -- Some Linear Transformations in M[a, b] -- Recipe for the Kernel -- Another Kind of Endpoint Problem -- Simple Kernels -- An Identity and Some Consequences -- Some Geometry in the Space M[a, b] -- Complete Orthonormal Sets -- The First Expansion Theorem -- An Important Special Case -- Second Expansion Theorem -- Third Expansion Theorem -- Every Simple Kernel has a Proper Value -- Orthogonal Polynomials -- 14 More about Linear Spaces -- Properties of the Space H -- Linear Transformations in H -- The Space C_0[a, b] -- The Space BV_0[a, b] -- The Space C_m[a, b] -- The Integral Int_a^b dXdY/dm -- 15 Mechanical Systems -- Falling Bodies -- Fluid Pressure -- Motion in a Plane -- Centroids -- Integral Tables -- Index of Simple Graphs -- Glossary of Definitions.