the greatest of them as often as there are Terms, is illuftrated by comparing the Triangle with a Parallelogram of the fame Height and Bafe; and what he has demonftrated of the Sum of the Squares of the Terms compared with the Square of the greateft Term, may be illuftrated by the Proportion of the Pyramid to the Prifm, or of the Cone to the Cylinder, their Bases and Heights being equal; and by the Ratios of certain Fruftums or Proportions of thefe Solids deduced from the elementary Proportions. He appears folicitous, that his Demonstrations fhould be found to depend on thofe Principles only, that had been univerfally received before his Time. In his Treatife of the Quadrature of the Parabola, he mentions a Progreffion, whofe Terms decreafe conftantly in the Proportion of four to one; but he does not fuppofe this Progreffion to be continued to Infinity, or mention the Sum of an infinite Number of Terms; tho' it is manifeft, that all which can be understood by those who affign that Sum, was fully known to him. He appears to have been more fond of preferving to the Science all its Accuracy and Evidence, than of advancing Paradoxes; and contents himself with demonftrating this plain Property of fuch a Progreffion, That the Sum of the Terms continued at Pleafure, added to the third Part of the last Term, amounts always to of the firft Term: Nor does he fuppofe the Chords of the Curve to be bifected to Infinity; fo that after an infinite Bifection, the infcribed Polygon might be faid to coincide with the Parabola. Thefe Suppofitions had been new to the Geometricians in his Time, and fuch he appears to have carefully avoided. This is a fummary Account of the Progress that was made by the Ancients in measuring and comparing curvilineal Figures, and of the Method by ! which they demonftrated their Theorems of this Kind. It is often faid, that curve Lines have been confidered by them as Polygons of an infinite Number of Sides; but this Principle no where appears in their Writings: We never find them refolving any Figure or Solid into infinitely fmall Elements: On the contrary, they feem to have avoided fuch Suppofitions, as if they judged them unfit to be received into Geometry, when it was obvious, that their Demonftrations might have been sometimes abridged by admitting them. They confidered curvilineal Areas as the Limits of circumfcribed or infcribed Figures, of a more fimple Kind, which approach to these Limits (by a Bisection of Lines or Angles, continued at Pleafure) fo that the Difference between them may become less than any given Quantity. The incribed or circumfcribed Figures were always conceived to be of a Magnitude and Number that is affignable; and from what had been fhewn of thefe Figures, they demonftrated the Menfuration, or the Proportions of the curvilineal Limits themfelves, by Arguments ab Abfurdo. They had made frequent Ufe of Demonftrations of this Kind from the Beginning of the Elements; and these are, in a particular Manner, adapted for making a Tranfition from right-lined Figures, to fuch as are bounded by curve Lines. By admitting them only, they established the more difficult and fublime Part of their Geometry, on the fame Foundation as the firft Elements of the Science; nor could they have propofed to themselves a more perfect -Model. But as these Demonftrations, by determining diftinctly all the feveral Magnitudes and Proportions of these infcribed and circumfcribed Figures, did frequently extend to very great Lengths, other Methods of demonftrating have been con trived by the Moderns, whereby to avoid thefe circumftantial Deductions. The first Attempt of this Kind known to us, was made by Lucas Valerius; but afterwards Cavalerius, an Italian, about the Year one thousand fix hundred and thirty five, advanced his Method of Indivifibles, in which he propofes, not only to abbreviate the antient Demonftrations, but to remove the indirect Form of Reasoning used by them, of proving the Equality or Proportion between Lines and Spaces, from the Impoffibility of their having any different Relation; and to apply to these curved Magnitudes the fame direct Kind of Proof that was before applied to right-lined Quantities. This Method of comparing Magnitudes, invented by Cavalerius, fuppofes Lines to be compounded of Points, Surfaces of Lines, and Solids of Planes; or, to make use of his own Defcription, Surfaces are confidered as Cloth, confifting of parallel Threads; and Solids are confidered as formed of parallel Planes, as a Book is compofed of its Leaves, with this Restriction, that the Threads or Lines, of which Surfaces are compounded, are not to be of any conceivable Breadth, nor the Leaves or Planes of Solids of any Thicknefs. He then forms thefe Propofitions, that Surfaces are to each other, as all the Lines in one to all the Lines in the other; and Solids, in like Manner, in the Proportion of all the Planes. This Method exceedingly fhortened the former tedious Demonftrations, and was eafily perceived; fo that Problems, which at firft Sight appeared of an infuperable Difficulty, were afterwards refolved, and came, at length, to be defpifed, as too fimple and eafy: The Menfuration of Parabolas, Hyperbolas, Spirals of all the higher Orders, and the famous Cycloid, were among the early Productions of this Period. The Difcove ries made by Torricelli, de Fermat, de Roberval, Gregory, St. Vincent, &c. are well known. They who have not read many Authors, may find a Synopfis of this Method in Ward's Young Mathematician's Guide, where he treats of the Menfuration of Superficies and Solids. Notwithstanding, as this Method is here explained, it is manifeftly founded on inconfiftent and impoffible Suppofitions; for while the Lines, of which Surfaces are fuppofed to be made up, are real Lines of no Breadth, it is obvious, that 'no Number, whatever, of them, can form the leaft imaginable Surface: If they are fuppofed to be of some sensible Breadth, in order to be capable of filling up Spaces, i. e. in Reality to be Parallelograms, how minute foever be their Altitude, the Surfaces may not be to each other in the Proportion of all fuch Lines in one, to all the like Lines in the other; for Surfaces are not always in the fame Proportion to each other with the Parallelograms infcribing them. The fame contradictory Suppofitions do obviously attend the Compofition of Solids by parallel Planes, or of Lines by fuch imaginary Points. This heterogeneous Compofition of Quantity, and Confufion of its Species, fo different from that Distinctnefs, for which the Mathematics were ever famous, was oppofed at its firft Appearance by several eminent Geometricians, particularly by Guldinus and Tacquet, who not only excepted to the first Principles of this Method, but taxed the Conclufions formed upon it as erroneous. But as Cavalerius took Care, that the Threads or Lines of which the Surfaces to be compared together were formed, fhould have the fame Breadth in each (as he himself expreffes it) the Conclufions deduced by his Method, might generally be verified by founder Geometry; fince the Comparison of thefe Lines was, in Effect, the comparing together the infcribed Figures. As in the Application of this Method, Error, by proper Caution, might be avoided, the Af fiftance it feemed to promise in the analytical Part of Geometry, made it eagerly followed by thofe who were more defirous to difcover new Propofitions, than folicitous about the Elegance or Propriety of their Demonftrations. Yet fo ftrange did the contradictory Conception appear, of compofing Surfaces out of Lines, and Solids out of Planes, that, in a fhort Time, it was new modelled into that Form, which it ftill retains, and which now univerfally prevails among the foreign Mathematicians, under the Name of the differential Method, or the Analysis of infinitely Littles. In this reformed Notion of Indivifibles, Surfaces are now fuppofed as compofed not of Lines, but of Parallelograms, having infinitely little Breadths and Solids, in like Manner as found of Prifms, having infinitely little Altitudes. By this Alteration it was imagined, that the heterogeneous Compofition of Cavalerius was fufficiently evaded, and all the Advantages of his Method retained. But here, again, the fame Abfurdity occurs as before; for if by the infinitely little Breadth of these Parellelograms, we are to understand what thefe Words literally import, i. e. no Breadth at all; then they cannot, any more than the Lines of Cavalerius, compofe a Surface; and if they have any Breadth, the right Lines bounding them cannot coincide with a Surface bounded by a curve Line. The Followers of this new Method grew bolder than the Difciples of Cavalerius, and having tranfformed his Points, Lines, and Planes, into infinitely little Lines, Surfaces, and Solids, they pretended, they no longer compared together hete rogeneous |