|Left: the pneumatical engine. Right: the rack-and-pinion piston part of the engine.|
In late 1658 or early 1659, the pneumatical engine, or English air pump, was built by Robert Hooke under the direction of Robert Boyle, a device which, according to Boyle, had two principal parts “a glass vessel and a pump to draw the air out of it.”
The first gas law, now known as Boyle's law, was determined using this pneumatical engine, and first enunciated by Boyle in 1662 as:
“The pressures and expansions are in reciprocal proportion.”
or in modern formulation:
which says that for a body of gas at constant number of particles n and temperature T the product of the measure of the pressure P and volume V of gas will be a constant K.
The following is an photo (Ѻ) of a reconstructed Boyle vacuum pump or pneumatical engine:
Parts and operation
The pump A, comprised a single brass barrel about 14 inches in length and 3 inches in internal diameter. It stood upon a strong wooden tripod, with its mouth turned upwards. The piston or sucker C was solid. The shank or piston-rod had teeth cut in it, so as to form a rack; and was moved by a toothed wheel or pinion E working into the rack, and turned by a handle F.
A hole was bored in the side of the upper end of the cylinder, provided with a plug G, which could be drawn out or pushed in by the hand. This was the only valve of the engine. The object of the inversion of the cylinder was to allow the globular or pear-shaped glass-receiver, H, from which it emptied the air, to be placed in a vertical position above the pump. The receiver had a large opening in the top for inserting objects into it. This opening could be narrowed by a tight-fitting brass ring I, in the center of which was an aperture provided with a brass stopper K, to close it. The receiver terminated below in a narrow neck, cemented into a brass stopcock L, which was ground to fit an opening in the upper end of the cylinder, near to the stoppered valve G.
The figure represents the arrangement of the parts, when the pump had completed on exhaustion. The valve of the cylinder G is shut, the stopcock L of the receiver open. L was then shut, G opened, and the piston driven home. The valve was then closed, the stopcock again opened, and the piston drawn down; and so on ad infinitum. By reversing the order in which the valve and stopcock were closed and opened, the pump could be made to condense instead of rarefying the air of the receiver. The valve for that purpose was opened, whilst the stopcock was ****, and the piston drawn down, so as to allow the cylinder to be filled with atmospheric air. The valve was then shut, the stopcock opened, and the piston as it ascended condensed the air into the receiver.
In circa 1649-54, German engineer Otto Guericke had built the first vacuum pump. News of this device soon reached the likes of English physicist Robert Boyle, who had been mediating on the results of Italian physicist Evangelista Torricelli’s 1643 mercury experiments on the vacuum. Guericke’s designs and results had been published in the Mechanica Hydraulo-Pneumatica of Schottus in 1657. Boyle stated that he did not set out to construct his own pneumatical engine (air-pump) until he had heard of Guericke’s “way of emptying glass vessels by sucking out the air at the mouth of the vessel.” In retrospect, Boyle states that he put both Mr G. (Greatrex) and R. Hooke to contrive an air-pump which should be more manageable than the German one, and free from defects; and adds “after an unsuccessful trail or two of ways proposed by others, Hooke fitted me with a pump.” Hooke commented on this “In 1658 or 59 I contrived and perfected the air-pump for Mr. Boyle.”
Boyle, it seems, was the supervisor of the construction of his pump. He called in the assistance of a mister Greatrex, a well-known London instrument-maker, but the two of them could not succeed in making a serviceable machine.  Boyle then turned to his assistant Robert Hooke, known to be a great mechanical genius, to make the devise. Hooke studied Greatrex’s contrivance, commenting that “it was too gross to perform any great matter.” Hooke commenced with his design in 1658 finishing early in 1659 . The results of the experimenting with the devise were published in Boyle's 1660 New Experiments on the Spring of the Air. 
1. (a) Wilson, George. (1849). “On the Early History of the Air-Pump in England”, The Edinburgh New Philosophical Journal, pgs. 330-54; esp. pg. 331.
(b) Shapin, Steven; Schaffer, Simon. (1985). Leviathan and the Air Pump: Hobbes, Boyle, and the Experimental Life (rarefying engine, pg. 258; vacuum machine, pg. 267; Magdeburg machine, pg. 280). Princeton, 2011.
2. Mr. Greatrex could be “Valentine Greatrex” or “Ralph Greatrex”; Hooke calls him Gratorix; Waller’s Life of Hooke, pg. iii, Birch names him Greatericks and Greatoricks.
3. Boyle, Robert. (1660). New Experiments: Physico-Mechanical, Touching the Spring of the Air, and its Effects: Made, for the most part, in a New Pneumatical Engine. Publisher.