Mathesis enucleata, or, The elements of the mathematicks by J. Christ. Sturmius ; made English by J.R. and R.S.S.

SHCOLIƲM II.

CLavius was also of this Opinion in the book afore menti∣oned, who thought that if the quadratrix be excluded out of the number of geometrical curves, by the same reason you may also exclude the ellipse, parabola, and hyperbola, since they as well as this are commonly described thro' innu∣merable points. But by that great Man's leave, we may de∣ny this consequence, by the same reason as Des Cartes has de∣ny'd the converse of it in his Geom. p 18. and 19. by vertue of which he suspects the ancients took the conick sections, &c. for mechanick or non-geometrick lines, because they did the spiral, quadratrix, &c. for such. But this is the difference between the description of the quadratrix and the conick secti∣ons thro' points, that all and every of the points of the conick sections, relating to any given point of the axis, may be geo∣metrically determin'd; but all the points of the quadra∣trix promiscuously related to any point of the generating qua∣drant, cannot be geometrically determin'd, but only those which respect some certain point, from which the quadrant may be divided into two arches of known proportion. For if, e g. in the quadrant BD the point X be given at pleasure, it will be impossible by Clavius's Rule to define a point of the quadra∣trix answering to it, because the proportion of the arches DX and BX is unknown, and consequently neither can a propor∣tional section of the right line AD be made: Not to mention that the last point E (which is the primary and most necessary one to the quadrature) even by Clavius's own confession cannot be geometrically defined. We may pass the like judgment on Archimedes's spiral and such like curves, which are conceiv'd to be described by two motions independent on one another; as

will be manifest to any one who compares the genesis of the spiral with that of the quadratrix and what we have hitherto said. Whence neither will Monantholius's trisection of a gi∣ven angle (which he essays) by means of a spiral be enough geometrical; which in his Book de Puncto, Cap. 7. p. 24. he attempts to perform thus: To the centre of a described spi∣ral and its first helical or spiral line BA (Fig. 151.) he applies the angle ABC equal to the given one abc; then having drawn circles thro' F and A where the legs of the angle cut the spi∣ral, he divides the intermediate space DA into three equal parts in 1, and K: And then thro' these points he draws circles cut∣ing the helix in L and M; and lastly having drawn BLN, BMO, he easily demonstrates from the genesis of the spiral that the arches AO, ON, NC are equal. And so after the same manner not only any angle or arch, but the whole peri∣phery may be geometrically divided into as many parts as you please; only supposing that this spiral line may be numbred among geometrical ones; as we have heretofore hinted that the cycloid, conchoid, cissoid, and logarithmical curve, &c. might be; and we have above sixteen Years ago declared our opinion for it in our German Edition of Archimedes; and now are therein confirm d by those celebrated Mathematicians Leib∣nitz, Craige, &c. who number lines of this kind, altho' they cannot be expressed by our common equations, among geome∣trical ones, notwithstanding the contrary opinion of Des Car∣tes, &c. because they admit of equations of an indefinite or transcendent degree, and are capable of a Calculus as well as others, tho' it be of a nature and kind different from that commonly used. See the Acta Erud. Lips. ann. 84. p. 234. and ann. 86. p. 292. and 294.