Foundations of geometry and therefore are valid theorems in absolute geometry. Proposition I.31 proves the existence of parallel lines (by construction). Also, the Saccheri–Legendre theorem, which states that the sum of the angles in a triangle is at most 180°, can be proved. The theorems of absolute geometry hold in hyperbolic geometry as well as in Euclidean geometry. Absolute geometry is inconsistent with elliptic geometry: in elliptic geometry there are no parallel lines at all, but in absolute geometry parallel lines do exist. Also, in elliptic geometry, the sum of the angles in any triangle is greater than 180°. Logically, the axioms do
Leibniz harmonic triangle formula_4 Whereformula_5is a binomial coefficient Whereas each entry in Pascal's triangle is the sum of the two entries in the above row, each entry in the Leibniz triangle is the sum of the two entries in the row "below" it. For example, in the 5th row, the entry (1/30) is the sum of the two (1/60)s in the 6th row. Just as Pascal's triangle can be computed by using binomial coefficients, so can Leibniz's: formula_6. Furthermore, the entries of this triangle can be computed from Pascal's: "The terms in each row are the initial term divided by the corresponding Pascal