SECOND PART

Regarding those things which must be understood in advance generally for the Analytical Doctrine of Plane Triangles. Chapter 1. 91. The term "Analytical Doctrine" refers to a method of solving geometric problems through calculation and algebraic rules rather than purely through physical drawings or geometric constructions.

On the calculation of Right-angled Triangles, regarding the first Axiom of Plane Triangles, from which flow 7 logarithmic rules for solving right-angled triangles according to various cases. Chapter 2. 99. Axiom In this context, an axiom is a fundamental mathematical principle or theorem used as the basis for further calculations.

On the calculation of Oblique-angled Triangles, and the second Axiom of Plane Triangles, from which arise two logarithmic rules. Chapter 3. 122. Oblique-angled triangles are those that do not contain a right angle.

On the Third Axiom of Plane Triangles, from which we derive one logarithmic rule. Chapter 4. 133.

On the Fourth Axiom of Plane Triangles, and the logarithmic rule emanating from it. Chapter 5. 153.

On certain extraordinary rules for Oblique-angled triangles, which we shall call their secondary rules, just as those above are called primary. Chapter 6. 157.

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THIRD PART

Regarding those things which must be understood in advance generally for the Analytical Doctrine of Spherical Triangles. Chapter 1. 177. Spherical trigonometry deals with triangles drawn on the surface of a sphere, which is essential for navigation and astronomy.

On the calculation of quadrantal triangles, regarding the first axiom of proportions within them, and the logarithmic rules flowing from it. Chapter 2. 186. Quadrantal triangles A spherical triangle that has at least one side equal to 90 degrees, which is one quarter of a circle's circumference.

On the calculation of simple quadrantal triangles, regarding the second axiom of proportions within them, and a single, most general rule for completing all their calculations. Chapter 3. 203.

On the calculation of Oblique-angled Triangles, and the Third Spherical Axiom. Chapter 4. 214.

On the universal calculation of Oblique Spherical triangles, by dropping a perpendicular, or by reduction to Right-angled triangles, in which there is no problem that cannot be resolved by only two additions of logarithms in general. Chapter 5. 242. Dropping a perpendicular is a technique where a line is drawn from a vertex to the opposite side at a 90-degree angle, splitting a complex triangle into two simpler right-angled triangles.

On the fourth Spherical Axiom, and the logarithmic rules emanating from it. Chapter 6. 280.