The loss of Galileo's Essay on Continuous Quantity is particularly to be regretted, as it would be highly interesting to see how far he succeeded in methodizing his thoughts on this important topic. It is to his pupil Cavalieri (who refused to publish his book so long as he hoped to see Galileo's printed) that we owe "The Method of Indivisibles," which is universally recognized as one of the first germs of the powerful methods of modern analysis. Throughout Galileo's works we find many indications of his having thought much on the subject, but his remarks are vague, and bear little, if at all, on the application of the method. To this the chief part of Cavalieri's book is devoted, though he was not so entirely regardless of the principles on which his method of measuring spaces is founded, as he is sometimes represented. This method consisted in considering lines as made up of an infinite number of points, surfaces in like manner as composed of lines, and solids of surfaces; but there is an observation at the beginning of the 7th book, which shews clearly that Cavalieri had taken a much more profound view of the subject than is implied in this superficial exposition, and had approached very closely to the apparently more exact theories of his successors. Anticipating the objections to his hypothesis, he argues, that "there is no necessity to suppose the continuous quantities made up of these indivisible parts, _but only that they will observe the same ratios as those parts do_." It ought not to be omitted, that Kepler also had given an impulse to Cavalieri in his "New method of Gauging," which is the earliest work with which we are acquainted, where principles of this sort are employed.[31] FOOTNOTES: [26] Vitæ Italorum Illustrium. [27] Almagestum Novum, vol. i. [28] Math. Coll. vol. ii. [29] Constructio Circini Proportionum. Moguntiæ, 1605. [30] Treatise of the Nature of Bodies. London, 1665. [31] Nova Stereometria Doliorum—Lincii, 1615. CHAPTER VI. _Invention of the telescope—Fracastoro—Porta—Reflecting telescope—Roger Bacon—Digges—De Dominis—Jansen—Lipperhey—Galileo constructs telescopes—Microscopes—Re-elected Professor at Padua for life._