In my setting, military innovation spurred by the American Civil War (1861-1865) prompts the design and construction of counterweight-driven skyhooks for the delivery of suborbital long-range munitions.
The devices, nicknamed "Tennessee Slingshots," are used to bombard Charleston and Vicksburg with barrages of small bombs with parabolic transatmospheric trajectories. The bombs are poorly aimed, and don't ultimately contribute in any significant military way to the more focused destruction wrought by heavy mortars and traditional artillery. But the relatively modestly-sized Tennessee Slingshots excite scientific and engineering interest in the possibilities of space exploration, given that the objects launched by these skyhooks are the first man-made artifacts to exit the lower reaches of the Earth's atmosphere.
After the war, continued engineering improvements lead to a decision to invest in a massive public works project - a huge space city that takes advantage of the rugged terrain of Mexico's Barrancas del Cobre. The fact that this canyon system in the western Sierra Madres is relatively close to the equator means that it's possible to take advantage of the Earth's rotational momentum, and the depth of the canyon means that the height differential allows for the construction of a number of experimental counterweight-driven skyhooks of larger and larger size.
Ultimately, a great chunk of mountain is excavated and fashioned into a massive half-million-ton counterweight on angled tracks, tethered by massive iron cables to elaborate transmission-works that translate the mechanical movement of the counterweights into the the supersonic movement of a tracked launch assembly that runs up the side of a mountain and continues on a slender rail supported by towers high into the air.
The gear ratio is such that as the counterweight slides three or four miles down a steep slide into a water pool, a looped cable is accelerated along a twenty-mile track until an object carried in a sled on the track is traveling fast enough to achieve escape velocity.
Upon reelection (following the intervening term of his rival Benjamin Harrison), Grover Cleveland announces at his inaugural address that he will authorize funds and support the goal of the Mexican-American Aeronautic Corporation to safely launch a man into orbit and retrieve that aeronaut by the end of the decade.
Is this effort doomed to fail? Could late-19th century engineers have successfully built such a skyhook, or are the available materials and technology not up to the task? Would massive iron chains snap like taffy under the strain, or would friction cause the whole building-sized gear assembly to melt and disintegrate? I'm trying to conceive of a mechanical orbital delivery system that would not rely on concussive chemical acceleration - the lead engineer would at least be aware that you can't shoot people out of a cannon - hence the long acceleration track and geared transmission. And I want to limit myself more-or-less scrupulously to the industrial and scientific capacity of the 1880s, without resort to hand-waving, anachronistic developments, or pure balonium.