Authors: Piotr Błaszczyk and Anna Petiurenko

We present a new model of a non-Euclidean plane, in which angles in a triangle sum up to \(\pi\). It is a subspace of the Cartesian plane over the field of hyperreal numbers \(\mathbb{R}^*\), adheres to Hilbert’s axioms of absolute geometry, deviates from the parallel and Archimedean axioms, and is not hyperbolic. The model enables one to represent the negation of equivalent versions of the parallel axiom, such as the existence of the circumcircle of a triangle, and Wallis’ or Lagendre’s axioms, as well as the difference between non-Euclidean and hyperbolic planes. In our model, we present counterexamples to various versions of the parallel axiom, including those found in Euclid’s Elements, such as the circumcircle proposition. Additionally, we demonstrate the failure of the Wallis axiom and Legender’s axiom, showing discrepancies between their predictions and our model’s outcomes, thus challenging the equivalence of these axioms to the parallel axiom. We emphasize that besides the well-known non-Archimedean Pythagorean fields, there exist non-Archimedean Euclidean fields as well. The model has unique educational advantages because, unlike standard models of non-Euclidean planes involving non-Euclidean representations of straight lines (Poincaré) or angles (Klein), the key idea of our model requires only the basics of Cartesian geometry and non-Archimedean fields.

Bibliography

1. Max Dehn,Legendre’schen Sätze über die Winkelsumme im Dreieck,Mathematische Annalen,vol. 53 (1900), no. 3, pp. 404–439.
2. Piotr Bł,aszczyk, Anna Petiurenko,New model of non-Euclidean plane,arXiv:2302.12768 [math.HO],2023.
3. Piotr Bł,aszczyk, Anna Petiurenko,New model of non-Euclidean plane,arXiv:2302.12768 [math.HO],2023.

Authors: Piotr Błaszczyk and Anna Petiurenko

In De Analysi, Newton derives three primary achievements of calculus: the area under the curve \(y(x)=x^{\tfrac mn}\) equals \(\frac {n}{m+n}x^{\tfrac {m+n}n}\) (Rule I), the power series of arcsine, and the power series of sine. Two further Rules introduced without proof reinforce Rule I. Rules II and III state that the area under finitely/infinitely many curves equals the sum of areas under each curve.

The standard interpretation of De Analysi runs through calculus: adopting the Riemann integral, it presents Rule I as the Fundamental Theorem of Calculus \((\int_0^x f(t)dt)'=f(x)\). Accordingly, term-by-term integration of series explains Rule III. However, this interpretation does not correspond to the argument’s structure regarding the series of arcsine and sine. In calculus, one first expands the series of sine and then gets the expansion of arcsine by the theorem on the inverse function derivative. On the contrary, Newton finds the power series of arcsine first and then the series of sine. The core of this difference is that Newton does not apply the derivative or limit concept.

We interpret De Analysi within the framework of nonstandard analysis, providing a coherent account of Newton’s technique of indivisibles, measuring the area under a curve by rectangles and ‘infinitely close’ relation as a deductive tool. We represent Newton’s arguments on a hyperfinite grid, meaning a discrete domain rather than a continuous one. We measure the area under a curve by a hyperfinite sum, reconstruct Newton’s proof of Rule I, and prove his Rule III.

Bibliography

1. Nigel J. Cutland,Loeb Measure in Practice,Springer,2000.
2. Tom Whiteside, editor,The Mathematical Papers of Isaac Newton: Vol. 2,Cambridge University Press,London,1968.