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can increase horizontal movement?
Depends on whether you want a true ratio or not. The simple pin-jointed linkage gives a good approximation, but it gets worse as the angle of articulation increases. Good enough for a huge range of practical machinery, but not for instrumentation.
The racks and coaxial pinions method gives a consistent ratio throughout the range of travel, but has higher friction and more backlash unless you literally build it instrument grade. Some instruments do use such a system due to the elimination of the cosine error inherent in simple pin-jointed linkages.
An hydraulic system gives a true and consistent ratio, but it's very costly to adjust the ratio later, if that's a consideration. It's generally not suited to instrument grade precision due to elastic effects, although they can be almost eliminated by coupling the two cylinders very closely, either side by side or in tandem. TRP HY/Rd brakes are an example of close coupled cylinders used to translate a long input into a short output. They are simplified as they only have to operate in one direction, but double acting systems can be built.
Another question. Is there some kind of mechanism, linkage, gearing etc that can increase horizontal movement? IE my input movement is 10mm but the output is 50mm