Articles

This paper investigates the constructability of angles multiples of 3 within the framework of Euclidean geometry. It makes a significant contribution by presenting the first geometric construction for all such angles, offering a rigorous solution to a longstanding geometric problem. The paper reaffirms the efficacy of Euclidean geometry in providing precise constructions and robust proofs for these angles, demonstrating the enduring strength of Euclidean principles from classical to modern times. The presented workflow goes beyond Euclidean geometry to examine non-Euclidean methods, particularly analytical approaches, revealing misconceptions that compromise the genetic and geometric rigor of Euclidean principles. The paper exposes incongruities when algebraic proofs related to angle constructability are applied to the Euclidean system, emphasizing the misalignment of fundamental geometric concepts. A notable result in the paper is the construction of a angle, introducing the “ angle chord” as a novel geometric property. This property challenges assumptions made by non-Euclidean methods and highlights the nuanced geometric properties crucial for rigorous constructions. The paper refutes the fallacy of relying solely on algebra for solutions to angles multiples of , emphasizing the necessity of embracing Euclidean geometry for geometric discoveries. The paper underscores the merits and resilience of Euclidean geometry, showcasing its independence and depth across historical and modern perspectives. The newly presented geometric construction not only resolves a longstanding question but also emphasizes the intrinsic strength and uniqueness of Euclidean principles in contrast to alternative methodologies.

The deflection of the vertical is an important parameter that combines both physical (astronomic) and geometric (geodetic) quantities. It is critical in such areas as datum transformation, reduction of astronomic observation to the geodetic reference surface, geoid modelling and geophysical prospecting. Although the deflection of the vertical is a physical property of the gravitational field of the earth; which almost all terrestrial survey measurements, with the exception of spatial distances, made on the earth surface are with respect to the Earth’s gravity vector, because a spirit bubble is usually used to align survey instruments. It has been ignored in most geodetic computation and adjustment. This research work is therefore aimed at computing the component of the deflection of the vertical component for part of Rivers State using a geometric method. This method involves the integration of Global Positioning System (GPS) to obtain the geodetic coordinate of points, precisely levelling to obtain the orthometric height of this point located within the study area. By least square using MATLAB program, the estimated deflections of vertical component parameters for the test station SVG/GPS-002 were; -0.0473” and 0.0393” arc seconds for the north-south and east-west components respectively. The associated standard errors of the North-south and East-west components were ±0.0093” and ±0.0060” arc seconds, respectively. The deflection of the vertical was also computed independently from gravimetric models of the earth as: ξ = 0.0204” ±0.0008814”, η = -0.0345” ±0.0014”; ξ =0.0157” ±0.000755”, η = -0.0246” ±0.0012”; ξ = -0.0546±0.0006014, η = -0.0208±0.0006014 for EGM 2008, EGM 1996 and EGM 1984 respectively. The two-tailed hypothesis test reveals that the estimated deflection component is statistically correct at 95% confidence interval. It was observed that the effect of the deflection of the vertical is directly proportional to the distance of the geodetic baseline. Therefore, including the derived component of deflection of the vertical to the ellipsoidal model will yield high observational accuracy since an ellipsoidal model is not tenable due to its far observational error in the determination of high-quality job. It is important to include the determined deflection of the vertical component for Rivers State, Nigeria.