4. (II.) _Measurement of Space._--Of quantities admitting of _continuous_ increase and decrease, (for number is discontinuous,) space is the most simple in its mode of measurement, and requires most frequently to be measured. The obvious mode of measuring space is by the repeated application of a material measure, as when we take a foot-rule and measure the length of a room. And in this case the foot-rule is the _unit_ of space, and the length of the room is expressed by the number of such units which it contains: or, as it may not contain an exact number, by a number with a _fraction_. But besides this measurement of linear space, {148} there is another kind of space which, for purposes of science, it is still more important to measure, namely, angular space. The visible heavens being considered as a sphere, the portions and paths of the heavenly bodies are determined by drawing circles on the surface of this sphere, and are expressed by means of the parts of these circles thus intercepted: by such measures the doctrines of astronomy were obtained in the very beginning of the science. The arcs of circles thus measured, are not like linear spaces, reckoned by means of an _arbitrary_ unit, for there is a _natural unit_, the total circumference, to which all arcs may be referred. For the sake of convenience, the whole circumference is divided into 360 parts or _degrees_; and by means of these degrees and their parts, all arcs are expressed. The _arcs_ are the measures of the _angles at the center_, and the degrees may be considered indifferently as measuring the one or the other of these quantities. 5. In the History of Astronomy[2\3], I have described the method of observation of celestial angles employed by the Greeks. They determined the lines in which the heavenly bodies were seen, by means either of Shadows, or of Sights; and measured the angles between such lines by arcs or rules properly applied to them. The Armill, Astrolabe, Dioptra, and Parallactic Instrument of the ancients, were some of the instruments thus constructed. Tycho Brahe greatly improved the methods of astronomical observation by giving steadiness to the frame of his instruments, (which were large _quadrants_,) and accuracy to the divisions of the _limb_[3\3]. But the application of the _telescope_ to the astronomical quadrant and the fixation of the center of the field by a _cross_ of fine wires placed in the focus, was an immense improvement of the instrument, since it substituted a precise visual ray, pointing to the star, instead of the coarse coincidence of Sights. The accuracy of observation was still further increased {149} by applying to the telescope a _micrometer_ which might subdivide the smaller divisions of the arc. [Note 2\3: _Hist. Ind. Sc._ b. iii. c. iv. sect. 3.] [Note 3\3: _Ib._ b. vii. c. vi. sect. 1.]