Mathematics like calculus can be mechanised with analogue (continuous output) mechanisms, but programmability is limited. being a continuous analogue output you can say that they are able to instantaneously (ignoring gear backlash and other mechanical tolerances) solve complex equations. The inputs can be changed freely and continuously and multiple machines could be interconnected to form complex analogue logic, but it is difficult to imagine a reasonable way that it could perform general programmable computing tasks rather than being purpose built for a specific equation or family of equations. See the analogue clockwork logic used in early autopilots and space flight computers, bombing calculators, early maritime navigation computers etc..
A general purpose mechanical computer is of course possible, but doesn't scale well to what we would consider a modern computer equivalent, as another commenter mentioned babbages analytical engine roughly scaled and parallelised to a modern computer would weigh as much as a decently sized asteroid. even if the consecutive generations of smaller scale machines allowed for micro engineering of mechanical logic gates via micro scale CNC it would still be limited to a certain physical minimum size much larger than our current transistors and could only function at a limited speed and with a limited lifetime.
Other methods that could be used are chemical logic, which can be scaled pretty small, but requires constant fuelling and would be prone to "noise" from contamination. Optical logic is another option, but requires advanced optics, filters and polarisers which would require advanced manufacturing in the first place.
The most likely, if still far fetched option for pre-industrial NON-MECHANICAL computing to me is biological, through selective breeding and intense training of humans or animals to perform computing tasks, even subconsciously. We know this is possible as we see the occasional human computer arise naturally, sometimes with extreme and unbelievable capabilities, but we don't know how to control it to "manufacture" a biological computer in this way.
Mathematics like calculus can be mechanised with analogue (continuous output) mechanisms, but programmability is limited. being a continuous analogue output you can say that they are able to instantaneously (ignoring gear backlash and other mechanical tolerances) solve complex equations. The inputs can be changed freely and continuously and multiple machines could be interconnected to form complex analogue logic, but it is difficult to imagine a reasonable way that it could perform general programmable computing tasks rather than being purpose built for a specific equation or family of equations. See the analogue clockwork logic used in early autopilots and space flight computers, bombing calculators, early maritime navigation computers etc..
A general purpose mechanical computer is of course possible, but doesn't scale well to what we would consider a modern computer equivalent, as another commenter mentioned babbages analytical engine roughly scaled and parallelised to a modern computer would weigh as much as a decently sized asteroid. even if the consecutive generations of smaller scale machines allowed for micro engineering of mechanical logic gates via micro scale CNC it would still be limited to a certain physical minimum size much larger than our current transistors and could only function at a limited speed and with a limited lifetime.
Other methods that could be used are chemical logic, which can be scaled pretty small, but requires constant fuelling and would be prone to "noise" from contamination. Optical logic is another option, but requires advanced optics, filters and polarisers which would require advanced manufacturing in the first place.
The most likely, if still far fetched option for pre-industrial NON-MECHANICAL computing to me is biological, through selective breeding and intense training of humans or animals to perform computing tasks, even subconsciously. We know this is possible as we see the occasional human computer arise naturally, sometimes with extreme and unbelievable capabilities, but we don't know how to control it to "manufacture" a biological computer in this way.