VOL. IV.] PHILOSOPHICAL TRANSACTIONS. Sig 



away the second term, that the sum of the negative roots might be equal to the 

 sum of the affirmative ones, as his constructions always require. 



We come next to speak of the last part of the book, to wit, his miscellany, 

 and because it falls in here somewhat properly, we therefore first mention his 

 4th chap. viz. De Maximis et Minimis, from which he derives this proposition : 

 If any magnitude, or number, as the whole, be divided into such parts, that are 

 to each other as a number to a number, the product of those powers of the 

 parts that are of the same degree as the parts themselves denominate, is the 

 greatest of all products of the like powers of the parts of the same magnitude 

 when otherwise divided. 



Concerning the rest of the miscellanies; our author. In chap. 1, treats of the 

 infinite spirals, and of the measure of the spaces comprehended by them and 

 the radius of the circle. Concerning which he observes, that Archimedes 

 squared that spiral which was made by an equal motion both in the radius and 

 circumference of the circle : that Stephano Angeli has done the like, when the 

 motion in the radius is equal, but in the circumference according to any degree 

 of acceleration ; which gave him occasion to render this doctrine easy and uni- 

 versal by reducing it to one analysis, when the motion is accelerated according 

 to any degree either in the radius or circumference. He applies this doctrine, 

 in chap. 3, to another sort of infinite spirals ; and in chap. 2, he treats of the 

 measure of spaces contained by the curves and right lines, also of their centres 

 of gravity, applying the former analysis or algebraic calculation thereto. Chap. 

 5 treats of the primary conchoid of Nicomedes : which point he determines 

 by the intersection of a parabola, whose axis is situated in the same line with 

 that of the conchoid ; or by a cubic parabola, whose axis is parallel to the 

 base of the conchoid, and vertex the same with the pole of the conchoid ; and 

 hence invents innumerable other conchoids of like properties, and finds the 

 curve passing through those points of flexure that are made by infinite con- 

 choids described about the same common pole and base, which in the common 

 conchoids he finds to be the perimeter of the cubic parabola here mentioned : 

 But in his own new conchoids, it is the ancient cissoid, extended beyond a 

 quadrant and running asymptotic : And he finds also the round solids made by 

 the rotation of these infinite curves, and of the cissoid line about their base 

 lines or asymptotes, equal to finite solids. 



Chap. 6. The author considering that Vincenzo Viviani, in his book De 

 Maximis et Minimis, found that if there were innumerable parabolas described, 

 having the same axis and vertex common, if from any point in that axis, the 

 shortest lines were drawn to those parabolas, all those points of incidence would 

 fall in an ellipsis; and the author's analysis taught him, that the proposition was 



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