Various sources, such as Cicero's Republic state that Archimedes had made a machine consisting of glass spheres that represented the Eudoxian system of the world.
Considering that Callipus died over a decade before Archimedes was born, would Archimedes have been aware of Callipus' work and therefore would his mechanism use the full 34 spheres needed to account for planetary movement as Callipus suggested instead of the 27 used by Eudoxus?
There is no definite answer on this question. About the Archimedes machine we only know a very imprecise description from the authors who did not understand much about this machine, or about science in general. What Archimedes knew and what he did not we cannot tell for sure. Probably he knew about Callipus. Probably he visited Alexandria. Alexandria at that time had a library, and one may suppose it contained all important writings on mathematics and astronomy. There are no reasons to assume that Callipus writings were missing since he lived recently....
But there is no firm ground for all these speculations.
According to Sedley's Epicurus and the mathematicians of Cyzicus, "Archimedes's celebrated planetarium was said to reproduce all the planetary motions simultaneously, but whether it followed a system of concentric spheres or that of eccentrics or epicycles is unknown. Our text... suggests that comparable, though undoubtedly much simpler, mechanisms existed a century earlier". The text comes from writings of Epicureans recovered at Herculanum about a school of astronomy founded by Eudoxus at Cyzicus. Calippus "while at Cyzicus encountered the mechanical difficulties involved in constructing a unified working model of all the concentric spheres, and was familiar with whatever solutions had been attempted. The insertion of counteracting spheres was probably one of these solutions... Cyzicenes may have limited each model to depicting just one or two orbits." Aristotle already describes the model completed with counterspheres in Metaphysics as if it was reality, and even Epicureans, their philosophical opponents, were aware of what Cyzicenes were up to. Eudoxian sphere building, as it was called, and Calippus's innovations were quite well known, and Eudoxians in general had a strong influence on Alexandrian geometry, as can be seen from Euclid's Elements. It is hard to imagine that Archimedes, who corresponded with Alexandrians and wrote his own book on sphere building missed it.
However, Archimedes apparently made a big step forward, his planetarium is described as combining motions of all seven planets and powered by a single source, so if he used Calippean spheres he would have to insert counterspheres that cancel planet specific motions when the motion is transmitted from outer to inner spheres. That would increase the total to at least 47 spheres, if not all 54 Aristotle had, and make the device rather unwieldy. But Cyzicene planetaria were also quite schematic, Epicurus criticized them for failing to reduce "irregularities of sun's appearence" to regularity. If Archimedes's device, like its predecessors, was meant to demonstrate the motion of the planets only qualitatively there would be no need for extra Calippean spheres. Eudoxus's original model could already do that, including even the retrograde loops, and Calippean additions did not make it quantitatively feasible anyway. For additional discussion see Did Archimedes use epicycles in his planetarium?
