This paper covers the solutions of the Euler equations in 3-D and 4-D for incompressible fluid flow. The solutions are the spin-offs of the author's previous analytic solutions of the Navier-Stokes equations (vixra:1405.0251 of 2014). However, some of the solutions contained implicit terms. In this paper, the implicit terms have been expressed explicitly in terms of x, y, z and t. The author applied a new law, the law of definite ratio for fluid flow. This law states that in incompressible fluid flow, the other terms of the fluid flow equation divide the gravity term in a definite ratio, and each term utilizes gravity to function. The sum of the terms of the ratio is always unity. This law evolved from the author's earlier solutions of the Navier-Stokes equations. In addition to the usual approach of solving these equations, the Euler equations have also been solved by a second method in which the three equations in the system are added to produce a single equation which is then integrated. The solutions by the two approaches are identical, except for the constants involved. From the experience gained in solving the linearized Navier-Stokes equations, only the equation with the gravity term as the subject of the equation was integrated. The experience was that when each of the terms of the Navier-Stokes equation was used as the subject of the equation, only the equation with the gravity term as the subject of the equation produced a solution. Ratios were used to split-up the x-direction Euler equation with the gravity term as the subject of the equation. The resulting five sub-equations were readily integrable, and even, the non-linear sub-equations were readily integrated. The integration results were combined. The combined results satisfied the corresponding equation. This equation which satisfied its corresponding equation would be defined as the driver equation; and each of the other equations which would not satisfy its corresponding equation would be called a supporter equation. A supporter equation does not satisfy its corresponding equation completely, but provides useful information which is not apparent in the solution of the driver equation. The solutions and relations revealed the role of each term of the Euler equations in fluid flow. The gravity term is the indispensable term in fluid flow, and it is involved in the forward motion of fluids. The pressure gradient term is also involved in the forward motion. The variable acceleration term is also involved in the forward motion. The fluid flow behavior in the Euler solution may be characterized as follows. The x-direction solution consists of linear, parabolic, and hyperbolic terms. If one assumes that in laminar flow, the axis of symmetry of the parabola for horizontal velocity flow profile is in the direction of fluid flow, then in turbulent flow, the axis of symmetry of the parabola would have been rotated 90 degrees from that for laminar flow. The characteristic curve for the x-nonlinear term is such a parabola whose axis of symmetry has been rotated 90 degrees from that of laminar flow. The y-nonlinear term is similar parabolically to the x-nonlinear term. The characteristic curve for the z-nonlinear term is a combination of two similar parabolas and a hyperbola. If the above x-direction flow is repeated simultaneously in the y-and z-directions, the flow is chaotic and consequently turbulent.