fimilar Triangles, the fame Propofition was extended to thefe Polygons alfo. But when they came to compare curvilineal Figures, which cannot be refolved into rectilineal Parts, this Method, failed. Circles are the only curvilineal plain Fi gures confidered in the Elements of Geometry. If they could have allowed themselves to have confidered these as fimilar Polygons of an infinite Number of Sides, (as fome have fince done, who pretend to abridge their Demonftrations) after proving that any fimilar Polygons infcribed in Circles, are in the duplicate Ratio of their Diameters, they would have immediately extended this to the Circles themselves, and would have confidered 2 Euc. 12. as an eafy Corollary from the first: But there is Reason to think they would not have admitted a Demonftration of this Kind, for the old Writers were very careful to admit no precarions Principles, or ought elfe but a few felfevident Truths, and no Demonftrations but fuch as were accurately deduced from them. It was a fundamental Principle with them, that the Difference of any two unequal Quantities, by which the greater exceeds the leffer, may be added to itfelf till it fhall exceed any propofed finite Quantity of the fame Kind: And that they founded their Propofitions concerning curvilineal Figures upon this Principle. in a particular Manner, is evident from the Demonftrations, and from the exprefs Declaration of Archimedes, who acknowledges it to be a Foundation upon which he established his own Difcourfes, and cites it as affumed by the Ancients in demonftrating all the Propofitions of this Kind: But this Principle feems to be inconfiftent with the admitting of an infinitely little Quantity or Difference, which, added to itfelf any Number of Times, is never fuppofed to become equal to any finite Quantity foever. They proceeded, therefore, in another Manner, lefs direct indeed, but perfectly evident. They found that the infcribed fimilar Polygons, by having the Number of their Sides increased, continually approached to the Areas of the Circles; fo that the decreafing Difference between each Circle and its infcribed Polygon, by ftill further and further Divifions of the circular Arches, which the Sides of the Polygon fubtend, could become lefs than any Quantity that could be affigned; and that all this while the fimilar Polygons obferved the fame conftant invariable Proportions to each other, viz. that of the Squares of the Diameters of the Circles. Upon this they founded a Demonstration, that the Proportion of the Circles themselves could be no other than that fame invariable Ratio of the fimilar ihfcribed Polygons. For they proved, by the Doctrine of Proportions only, that the Ratio of the two infcribed Polygons cannot be the fame as the Ratio of one of the Circles to a Magnitude lefs than the other, nor the fame as the Ratio of one of the Circles to a Magnitude greater than the other; therefore the Ratio of the Circles to each other, must be the fame as the invariable Ratio of the fimilar Polygons infcribed in them, which is the Duplicate of the Ratio of the Diameters. In the fame Manner the Ancients have demonftrated, that Pyramids of the fame Height are to each other as their Bafes, that Spheres are as the Cubes of their Diameters, and that a Cone is the one third Part of a Cylinder on the fame Base, and of the fame Height. In general, it appears from their Way of Demonstration, that when two variable Quantities, which always have an invariable Ratio to each other, approach at the fame Time to two determined Quantities, fo that they may differ lefs from them than by any affignable Measure; Measure; the Ratio of thefe Limits or determined Quantities must be the fame as the invariable Ratio of the two variable Quantities: And this may be confidered as the moft fimple and fundamental Propofition in this Doctrine, by which we are enabled to compare curvilineal Spaces in fome of the more fimple Cafes.. The next Improvement in the Way of demonftrating among the ancient Geometricians, feems to be that which we call the Method of Exhauftions, which, for the further Illuftration of this Subject, may be reprefented thus. Suppose there are two curvilineal Spaces, ACB and MON, wherein are drawn Parallelograms, whofe Breadth may be continually diminished; it is then obvious, that the firft circumfcribing, and laft infcribing Figures, may be made to differ from that curvilineal Space ACB, and from each other, by lefs than any Space, how minute foever, that fhall be named; i. e. the circumfcribed Figure can be made lefs than any Space that exceeds the Curve, and the infcribed Figure greater than any Space that is lefs than the Curve. If by confidering the Properties of these infcribed and circumfcribed Figures, which arife from the Nature of the Curve they are adapted to, a right-lined Space LMN may be affigned, that fhall be greater than every infcribed Figure, and lefs than every circumfcribed Figure, this rightlined Space LMN may be proved to be equal to the curvilineal Space ACB. For were it greater, a circumfcribed Figure might be made lefs; and if it were lefs, an inscribed Figure might be made greater. M N If, therefore, Parallelograms, whofe Breadth may be any how diminished, are drawn infcribing and circumfcribing thefe Curves; and if they are defcribed in fuch a Manner, that the circumfcribed Fi-. gure of one Curve to the circumfcribed Figure of the other, and the infcribed to the infcribed, has one and the fame conftant Proportion in every Defcription: I fay, that the curve Figure ACB, is to the Curve MON, in the fame Proportion which the infcribed and circumfcribed Figures conftantly bear to each other. For no Space greater than ACB can have to MON this Ratio, fince if it could, a Figure might be circumfcribed about ACB lefs than this fuppofed greater Space; and this circumfcribed Figure, to the correfponding Figure circumfcribing MON, would be in the fame Ratio as the fuppofed greater Space to the Curve MON; i. e. four Quantities being in the fame Proportion, the firft would be lefs than the third, and the fecond greater than the fourth. Nor can any Space lefs than ACB have to MON the conftant Ratio of the Figures in one Curve to the Figures in the other. 2 other. For if it could, a Figure might be infcribed within ACB, which would be greater than this supposed leffer Space; and this infcribed Figure, to its correfpondent Figure infcribing MON, would be in the fame Ratio as this imagined leffer Space to the Curve MON; i. e. four Quantities being in the fame Proportion, the firft would be greater than the third, and the fecond less than the fourth. Thus no Space but ACB can be to MON in the conftant Ratio of the circumfcribed and infcribed Figures. In the Manner here defcribed did the antient Geometricians demonftrate whatever they difcovered relating to the Dimenfions or Proportions of curve Lines, curvilineal Spaces, and Solids bounded by curve Surfaces; and of which, Sir Ifaac Newton's Doctrine of prime and ultimate Ratios, is no other than an Abbreviation or Improvement in the Form. Archimedes, indeed, takes a different Way for comparing the Spheroid with the Cone and Cylinder, that is more general, and has a nearer Analogy to the modern Methods. He fuppofes the Terms of a Progreffion to increase constantly by the fame Difference, and demonftrates feveral Properties of fuch a Progreffion relating to the Sum of the Terms, and the Sum of their Squares; by which he is able to compare the parabolic Conoid, the Spheroid, and hyperbolic Conoid, with the Cone; and the Area of his fpiral Line with the Area of the Circle. There is an Analogy betwixt what he has fhewn of thefe Progreffions, and the Proportions of Figures demonftrated in the Elementary Geometry; the Confequence of which may illuftrate his Doctrine, and ferve, perhaps, to fhew that it is more regular and compleat in its Kind than fome have imagined. The Relation of the Sum of the Terms to the Quantity that arifes by taking C 2 the |