In models, mechanical universe refers to []

Overview
In 1902, American Edwy Hand, in his A Mechanical Universe, outlines a seemingly implicit atheism (god and atheism are not mentioned) model of the universe, based on sixteen points or “basic elements of existence”, as he refers to things, namely: [1]

1. There is only one kind of matter in the universe and its definition is set forth herein.
2. Matter only partly fills universal space.
3. Matter exists in several grades or orders, first primary matter, second atoms, third the stars, and fourth the galaxy.
4. The primary order of matter and the conditions which mark its existence in space.
5. The atom, the simplest mechanism in the universe,
6. Motion of masses of matter is the only active cause principle in the universe.
7. Collisions impacts pressure, are the only mode of active causation in existence.
8. Atoms, stars, the galaxy, are formed by a constructive process of condensation.
9. Condensation is caused by the mechanical process which follows each impact of collision and differentiates motions into higher and lower velocities.
10. The process and cause of energy radiation.
11. The nature and operation of the gravitation cause.
12. The mode of union whereby atoms combine and form molecules.
13. The mode of chemical action and change.
14. The source of chemical energy as in combustion.
15. The two basic principles of electricity.
16. The principle of affinity in chemical combination.

In 1985, David Goodstein, a Caltech physics professor, lectured to his freshman physics class in the form of a 56-part telecourse entitled “The Mechanical Universe”, which in 2016 was uploaded (Ѻ) to YouTube. The funded-show spurred the followup two-volume textbook The Mechanical Universe (1985), by Richard Olenick, Tom Apostol, and Goodstein; in volume one (Ѻ), they mention god three times: [2]

“In the center of all the celestial bodies rests the sun. For who could in this most beautiful temple place this lamp in another or better place than that from which it can illuminate everything at the same time? Indeed, it is not unsuitable that some have called it the light of the world; others, its minds, and still others, its ruler. Trismegistus calls it the visible god; Sophocles' Electra, the all-seeing. So indeed, as if sitting on a royal throne, the sun rules the family of the stars which surround it.”
Nicolaus Copernicus (1543), On the Revolution of the Heavenly Orbs

“And with respect to the general cause, it seems manifest to me that it is none other than god himself, who, in the beginning, created matter along with motion and rest, and now by his ordinary concourse alone preserves in the whole the same amount of motion and rest that he placed in it. For although motion is nothing in the matter moved but its mode, it has yet a certain and determinate quantity, which we easily see may remain always the same in the whole universe, although it changes in each of the parts of it.”
Rene Descartes (1644), Principles of Philosophy

The third mention of god (pg. 264) is in discussion of Descartes’ mechanical law following universe, wherein they state that Descartes believed, firstly, that the universe was set in motion by god, and secondly, to resolve the issue of a ‘running down universe’, since all mechanical things, e.g. clocks, machines, etc., run down over time, Descartes posited that the “quantity of motion”, aka momentum p, of all bodies in the universe, is preserved by god who created it in the beginning and constantly sustains it, is a vector quantity which is the product of the mass m of an object and its velocity v:

momentum

whereby the sum of the momenta of all the bodies in the universe is a constant:

Momentum of universe

Therefore, Descartes' god-based mechanical model of the universe is complete. Later, of course, Pierre Laplace (1802), building on Newton, produced a god-free mechanical model of the universe (see: Napoleon Laplace anecdote).

See also
Atheistic universe
● Theistic universe
● Pantheistic universe
● Physical universe
Living universe
● Spiritual universe
● Thermodynamical universe

References
1. Hand, Edwy N. (1902). A Mechanical Universe. Inland Printing Co.
2. Frautschi, Steve C., Olenick, Richard P., and Apostol, Tom S. (1985). The Mechanical Universe (god, 3+ pgs). Cambridge University Press, 2007.

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