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"If I can make you see the world the way I see it, then you will automatically think the way I think."
- Philip K. Dick

Gravity Assist  
  Using the gravity and orbital speed of a celestial body to change speed and course of a spacecraft.  

These "manoeuvres" were an essential part of the earliest visits to the outer solar system.

In the book, an Earth spaceship by Martian brigands who are determined to get their hands on the Moon's one key resource, radiactum. Radiactum was "the catalyst mineral which was revolutionising industry."

We were at this time no more than some sixty-five thousand miles from the moon's surface. The Planetara presently would swing upon her direct course for Mars. There was nothing that would cause passenger comment in this close passing of the moon; normally we used the satellite's attraction to give us additional starting speed.
Technovelgy from Brigands of the Moon, by Ray Cummings.
Published by Astounding Stories of Super Science in 1930
Additional resources -

The mathematics of the manner in which the orbits of comets were altered by close approaches to planets was known in the late eighteenth century. However, this idea was not applied to space craft until much later.

In the early nineteen-twenties, Walter Hohmann demonstrated that the lowest energy path between any two planets is an ellipse that is tangential to the orbits of both the planets. However, no rockets existed until the 1960's that were powerful enough to generate the initial thrust, and a minimum energy orbit would take thirty years from Earth to Neptune - and forty years to Pluto.

Also in the 1920's, Hermann Oberth described a powered flyby, or Oberth maneuver, that is a maneuver in which a spacecraft falls into a gravitational well and then uses its engines to further accelerate as it is falling, thereby achieving additional speed. The resulting maneuver is a more efficient way to gain kinetic energy than applying the same impulse outside of a gravitational well.

The Oberth effect is distinct from a gravity-assist “slingshot” maneuver, which describes an elastic momentum and energy exchange between an unpowered spacecraft and a planet when both are under the gravitational influence of a third body, typically the Sun

In the volume published in 1932, Nikolai Alexyevich Rynin introduced the reader to the mysterious Yuri Vasilievich Kondratyuk. In a volume of Interplanetary Flight published in Russian in 1932, this description of gravity assisted flight was included:

Kondratyuk laid in two paragraphs a vague concept of gravity-assist. In the first paragraph, he suggested that a close approach between the spacecraft and the satellite of a planet may be used to “gather velocity” for a solar system journey and to “absorb velocity” when it is returning. In the second paragraph, a figure similar to Fig. 2 was used to explain that two approaching bodies may be used to continuously increase the velocity of a vehicle describing a trajectory around them. He correctly remembered that the vehicle will lose velocity if the bodies are moving apart.

A historical review of the theory of gravity‑assists in the pre‑spacefight era

It's not clear whether or not this published work was related to earlier, unpublished work by the author dating back as far as 1920.

To the best of my knowledge, the concept of "gravity assist" as a means of altering the speed or course of a spacecraft was apparently not discovered by the scientific world (in English, anyway) until some 20 years after the publication of Cummings' novel. In a footnote A C Clarke indicates the first ref he's aware of was the paper "Perturbation Manoeuvres" by Derek Lawden (Journal of the British Interplanetary Society, vol 13, no 5, Sept 1954). Lawden is a mathematician, and in the early fifties worked on the optimization of rocket trajectories. It is also reported that a summer intern at JPL in 1961, Michael A. Minovitch, showed that the gravity field of a planet could provide thrust to a spacecraft. He demonstrated that careful design of the trajectory to a target planet could provide a gravity assist to move from that planet to a second planet.

Historians of science still differ on this issue.

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Additional resources:
  More Ideas and Technology from Brigands of the Moon
  More Ideas and Technology by Ray Cummings
  Tech news articles related to Brigands of the Moon
  Tech news articles related to works by Ray Cummings

Gravity Assist-related news articles:
  - Gravity Assist Will Help Pluto-Bound Craft
  - The Amazing Gravity-Assisted Journey Of Rosetta

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