In a memoir titled Théorie plus complette des machines qui sont mises en mouvement par la réaction de l'eau (“A more complete theory of machines which are put in motion by the reaction of water”) [L. Euler, Mémoires de l'académie des sciences de Berlin, 10, 227–295 (1756)], Euler (1707–1783) consolidates what is considered to be the first general theory of rotating hydraulic machines and proposes a first model of a hydraulic turbine. In a time span of about nine years, Euler had addressed the core of the theory in four publications, which cover basic topics such as to find the moment applied to the rotor of the machine—from which the shaft torque could be found—to find the pressure acting on the internal channels of the machine, and to find the power that the machine is capable to deliver. Here, Euler correctly finds that the power depends only on the discharge and on the water head. From the pressure equation, Euler also anticipated the phenomenon of cavitation—later known to be a major hindrance to the performance of hydraulic machines. At Euler's time, the conservation equations of Fluid Mechanics had not been proposed yet, and, therefore, he could not count on the form of the equation that would be readily applicable to the modeling, that is, the integral form of the moment of momentum equation in relation to an axis. As we shall see, this makes his developments quite lengthy and intricate. The present work is based on primary sources and has the goal of examining the developments of the 1756 memoir, to show Euler's intuition and skills in the development of the theory which was applied to an archaic form of a hydraulic turbine (the Segner–Euler turbine), showing the pervasiveness of the theory which still undergirds the modeling of pumps and turbines today.

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15.
These are known as Eneström index, which are used to identify Euler's writings. Most historical scholars refer to the works of Euler by their Eneström index. They were introduced by Gustaf Hjalmar Eneström (1852–1923), a Swedish mathematician, statistician and historian of mathematics.
16.
Johann Segner (1704–1777) was a Hungarian scientist. In 1735 Segner became the first professor of mathematics, a position created for him, at the University of Göttingen. In 1755 he became a professor at Halle, where he established an observatory. He was the first scientist to use the reactive force of water and constructed the first water-jet, the Segner wheel, which resembles one type of modern lawn sprinkler. Segner also produced the first proof of Descartes' rule of signs. Historians of science remember him as the father of the water turbine. The lunar crater Segner is named after him, as is asteroid 28878 Segner.
17.
As shown in Fig. 3, Euler's uses the same letter q with two different meanings: to indicate the length of the segment C Q⁠, and also to indicate a vector perpendicular to the segment C Q⁠, and placed at point Q⁠.
18.
These results come from the rotation of the relative frame of reference by an angle γ whose cosine is x q and sine is y q⁠, giving the new coordinates A 1 ′ ,   A 2 ′ in terms of the old ones A 1 ,   A 2 as A 1 ′ = A 1 c os   γ + A 2 s in   γ⁠, A 2 ′ = A 2 c os   γ − A 1 s in   γ⁠.
19.
On § XX of E222, Euler explains the presence of 2 on his way of writing the second law of motion. Euler begins by writing the second law of motion as F = ( ∝ mdu ) / d t⁠, where ∝ is a constant coefficient which depends on how the mass m⁠, the velocity u⁠, and the time t are expressed. If the mass is expressed as the corresponding weight at the surface of the earth, and the velocity by the square root of the height which a free falling body would acquire the same velocity, then it would be found that the number 2 is the value of the coefficient ∝. To find this out, the general formula of free falling bodies should be applied to the mass m⁠: in a certain time t it has descended the height = x⁠, then the velocity u will be = √ x⁠, with which the height d x will be covered during the infinitesimal time d t = d x / √ x⁠, in which the motion in the infinitesimal height d x can be considered as uniform. Since u = √ x⁠, then, d u = d x 2 x = 1 / 2   d t⁠, and hence, d u / d t = 1 / 2⁠. Therefore, since F = ( ∝ mdu ) / d t⁠, then, F = 1 / 2 ∝ m⁠: or the force F is, in this case, is equal to the weight of the body, or else, it is equal to the mass m⁠; from which, it is clear that for this to be true 1 / 2 ∝ m = m⁠, then it is necessary that ∝ = 2⁠. This value of ∝ is valid as long as we maintain the same way of introducing masses, velocities, and time in the calculation.
20.
According to Euler's convention, g corresponds to the height that a free falling body descends in one second, then, the actual gravity equals 2 g . Hence, 2 g v is the velocity acquired by a body free falling from the height v⁠.
21.
Because of substantial fluid flow losses, increased mechanical friction and larger gap leakages caused by forward-facing vanes, only backwards curved blades with outlet angles of ζ = 140°–160° are used in centrifugal pumps today.
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