rogeneous Quantities, and infifted on their Principles, being now become genuine; but the Miftakes they frequently fell into, were a fufficient Confutation of their Boafts; for, notwithstanding this new Model, the fame Limitations and Cautions were still neceffary: For Inftance, this Agree ment between the infcribing Figures and the curved Spaces to which they are adapted, is only partial and in applying their Principles to Propofitions already determined by a jufter Method of Reasoning, they eafily perceived this Defect; both in Surfaces and Solids, it was evident, at first View, that the Perimeters difagreed. And as no one In→ stance can be given, where these indivisible or infinitely little Parts do fo compleatly coincide with the Quantities they are fuppofed to compound, as in every Circumftance to be taken for them, without producing erroneous Conclufions, fo we find, where a furer Guide was wanting, or difregarded, these Figures were often imagined to agree, where they ought to have been supposed to differ. Leibnitz, in two Differtations, one on the Refiftance of Fluids, the other on the Motion of the heavenly Bodies, has, on this Principle, reasoned falfly concerning the Lines intercepted between Curves and their Tangents. Berneulli has, likewife, made the fame Miftake in a Differtation on the Refiftance of Fluids, and in a pretended Solution of the Problem concerning ifoperimetrical Curves. Nay, Mr. Parent has had the Rafhness to oppofe erroneous Deductions from this abfurd Principle, to the most indubitable Demonstrations of the great Huygens. Thus it appears, that the Doctrine of Indivifibles contains an erroneous Method of Reasoning, and, in Confequence thereof, in every new Subject to which it shall be applied, is liable to fresh Errors. It is also manifeft, that the great Brevity it gave to Demonstrations, arofe entirely from the abfurd Attempt of comparing curvilineal Spaces in the fame direct Manner as right-line Figures can be compared; for, in order to conclude directly the Equality or Proportion of fuch Spaces, no Scruple was made of fuppofing, contrary to Truth, that rectilineal Figures, capable of fuch direct Comparifon, could adequately fill up the Spaces in Queftion; whereas, the Doctrine of Exhauftions does not attempt, from the Equality or Proportion of the infcribing or circumfcribing Figures, to conclude, directly, the like Proportions of these Spaces, because thofe Figures can never, in Reality, be made equal to the Spaces they are adapted to: But as thefe Figures may be made to differ from the Spaces to which they are adapted, by less than any Space propofed, how minute foever, it fhews, by a juft, tho' indirect Deduction from these circumfcribing and inscribing Figures, that the Spaces whofe Equality is to be proved, can have no Difference; and that the Spaces, whofe Proportion is to be shewn, cannot have a different Proportion than that affigned them. The Arithmetica Infinitorum of Dr. Wallis, was the fulleft Treatife of this Kind that appeared before the Invention of Fluxions. Archimedes had confidered the Sums of the Terms in an arithmetical Progreffion, and of their Squares only, (or rather the Limits of thefe Sums only) these being fufficient for the Menfuration of the Figures he had examined. Dr. Wallis treats this Subject in a very general Manner, and affigns like Limits for the Sums of any Powers of the Terms, whether the Exponents be Integers or Fractions, pofitive or negative. Having difcovered one general Theorem that includes all of this Kind, he then com pofed pofed new Progreffions from various Aggregates of these Terms, and enquired into the Sums of the Powers of thefe Terms, by which he was enabled to measure accurately, or by Approximation, the Areas of Figures without Number: But he compofed this Treatife (as he tells us) before he had examined the Writings of Archimedes; and he proposes his Theorems and Demonftrations in a lefs accurate Form: He fuppofes the Progreffions to be continued to Infinity, and investigates, by a Kind of Induction, the Proportion of the Sum of the Powers, to the Production that would arise by taking the greatest Power as often as there are Terms. His Demonstrations, and fome of his Expreffions, (as when he speaks of Quantities more than infinite) have been excepted against ; however, it must be owned, this valuable Treatife contributed to produce the great Improvements which foon after followed. But Sir Ifaac Newton has accomplished what Cavalerius wifhed for, by inventing the Method of Fluxions, and propofing it in a Way that admits of ftrict Demonstration, which requires the Suppofition of no Quantities, but fuch as are finite, and easily conceived; by his Doctrine of prime and ultimate Ratios, he has found out the proper Medium, whereby to avoid the impoffible Notion of Indivifibles on the one hand, and the Length of Exhauftions on the other. The Computations in this Method, are the fame as in the Method of Infinitefimals, but it is founded on accurate Principles, agreeable to the antient Geometry; in it the Premifes and Conclufions are equally accurate, no Quantities are rejected as infinitely small, and no Part of a Curve is fuppofed to coincide with a right Line: But as the Explication of their Nature and Ufe has employed fome of our greatest Mathematicians to write exprefs Treatifes thereon; and as the Invention can never be fufficiently applauded, we will conclude with Mr. Ditton, that the next Improvement must be the Science of pure Intelligences. CONIC SECTIONS. Of the PARABOL A. The GENESIS. I F from a Point V, in any indefinite right Line, there be taken VD=VK, and from the Point K, as a Center, with the Distance DG, you interfect CM drawn perpendicular to DG, thofe Points will be in the Curve of a Parabola; and proceeding in this Manner, an indefinite Number of Points may be found, thro' which, if a Line be fuppofed drawn, the Space (CVM) comprehended thereby, and any right Line drawn at right Angles, to the above indefinite Line, will be a Parabola. |