The original manuscript of Israel Lyons' 'Treatise of Fluxions', divided into twelve sections.
1. The direct method of fluxions: i. Definitions; ii. Fluxions of the sum and differences of quantities; iii. Fluxions of rectangles and powers; iv. To find the relation of the fluxion from the relation of the fluents; v. Of 2nd, 3rd and 4th fluxions.
2. The inverse method of fluxions: i. Of the fluents of simple quantities; ii. Newton's binomial theorem demonstrated; iii. Demoivre's multinomial theorem and quantities; iv. Of the fluents of compound and surd quantity; v. Theorems for finding fluents; vi. Finding fluents by successive fluxions and fluents; vii. Of the resolution of adfected literal equations; viii. Of the resolution of fluxional equations; ix. Reversion of series.
3. Logarithms and measures of ratios: i. Composition of ratios; ii. To find the measure of any ratio; iii. Series for finding logarithms; iv. Of exponentials; v. Finding of fluents by the measures of ratios.
4. Of curves and drawing of tangents: i. Definitions; ii. Of lines of the first and second order; iii. Of the conchoid of Nicomedes; iv. Of the cissoid of Diocles, paraboloids and hyperboloides; v. To draw tangents to any curve; vi. To draw tangents to the conic sections; vii. To draw tangents to the conchoid; viii. To draw tangents to the cissoid; ix. To draw tangents to paraboloids and hyperboloides; x. Of mechanical curves; xi. Of the logarithmic curve; xii. Of cycloids; xiii. Of spirals; xiv. Of the quadratrix.
5. Of the greatest and least ordinates: i. Of Tayler's theorem; ii. To find when an ordinate becomes a maximum or minimum of the first kind; iii. Of the second kind; iv. Examples; v. Of the curve of swiftest descent; vi. To draw a perpendicular to a curve.
6. Of the curvature of curves: i. Curvature of different circle; ii. To find the radius of curvature in any curve; iii. Of the evoluta; iv. Variations of curvatures; v. To find the curvature of a conic section; vi. To find the curvature of a conchoid; vii. To find the curvature of a cissoid; viii. To find the curvature of a paraboloid; ix. To find the curvature of a logarithmic curve; x. To find the curvature of a cycloid; xi. To find the curvature of a quadratrix; xii. To find the quadrature in the curve of swiftest descent; xiii. To find the curvature of spirals; xiv. To find the curvature of the epicycloid; xv. Of the curvature in the figure mentioned by Sir Isaac Newton, 'Princip Lib 3 prop 28'; xvi. To investigate the curve in which the radius of curvature is inversely as any power of the ordinates.
7. Of the points of contrary flexion: i. Definition; ii. Examples in the conchoid, paraboloid and cycloid; iii. To find the points of contrary flexure in a spiral; iv. Of cuspids; v. Examples in the cissoid and paraboloid.
8. Of the quadrature of curves: i. Comparison of fluents; ii. Of the fluxions of curvilinear areas; iii. To find the area of the parabola, paraboloid, hyperbola and circle; iv. Of fluents that may be compared with the circle; v. To find the area of an ellipsis and hyperbola; vi. Of fluents that may be compared with the hyperbola, cissoid, conchoid, logarithmic curve, cycloid and quadratrix; vii. To find the area when the equation is an adfected equation, and the area of a spiral; viii. Compute the area of the differences of the fluxions of the ordinates; ix. To compute the area by equidistant ordinates.
9. The rectification of curve lines: i. To find the length of a parabola, paraboloid, circle by infinite series, and the circle by logarithms; ii. The demonstration of Mr Cotes' theorem; iii. To find the length of an ellipse, hyperbola, cissoid, conchoid, logarithmic curve, cycloid, quadratrix and spirals.
10. Of the contents of solids: i. Fluxions of solids; ii. The content of the solid generated by a paraboloid, ellipse and hyperbola, cissoid, conchoid, logarithmic curve and quadratrix.
11. Of the quadrature of curvilinear surfaces: i. The surface of a cone; ii. The fluxions of curved surfaces; iii. Area of the surface generated by a paraboloid; iv. Surfaces of a sphere, hyperbolic conoid, spheroid, cissoid and logarithmic curve; v. When a surface is finite and when infinite.
12. Of the centre of gravity: i. To find the centre of gravity of a paraboloid, circle and ellipse, hyperbola, circular arc, solid described by a paraboloid, segment of a sphere or spheroid and hyperbolic conoid; ii. Centre of oscillation; iii. To find the centre of oscillation of a right line, circle, sphere and paraboloid; iv. Of the casenaria; v. Solutions to some logometrical problems.