298 On the Principles of Motion, tyc. 



mathematics,) depend on some species of motion, which rs 

 conceived to measure the space, or interval occupied by the 

 quantity, or numerically on the relation of the quantity to 

 some standard measure, or unit ; this mode of considering 

 quantity is virtually a motion of aggregation, and when so 

 estimated, has been denominated discrete quantity and the 

 other, continued. There are only names given to what is 

 immutable and identical, according to the two modes of ex- 

 istence in our ideas. But that which has been the subject of 

 many cavils and much ingenious discussion on the principles 

 of this science, is not any supposed want of clearness in 

 either mode of conceiving of quantity. It is the logic or the 

 legitimate deductions of the calculus in its first principles, 

 where the ratio of variables dependent on one another, is to 

 be estimated in their ultimate or vanishing state, which is 

 oppugned. That, say the objectors, must be either some- 

 thing or nothing; if it be something, it must partake of all 

 the successive variations, which arise from the variation of 

 the magnitude, and will not be that which has been assigned 

 by mathematicians as the true ratio: but if it be nothing, 

 there can be no ratio assigned. Here the mathematician 

 may come in and say that it is neither something nor nothing, 

 if by those terms are meant substance or matter, but a spe- 

 cies of quantity well known in mathematics to have a ratio, 

 though without any material existence or occupancy of space, 

 viz. the limits of quantities. Geometry, the clearest and most 

 evident of all the sciences, assumes them as the fundamentals 

 on which its towering fabric is built. Points, lines, and sur- 

 faces, are the first principles of reasoning in this accurate 

 science. It is not our object to enter on any metaphysical 

 discussion, relative to the connexion of these limits with the 

 actual quantities with which they are inherently united. It 

 will be sufficient to observe, that they have never been made 

 a ground of objection to the elegant demonstrations of Euc- 

 lid, Archimedes, Apollonius, and others; but because in the 

 higher branches of mathematics, they are more remote and 

 recondite, and less the subjects of common observation or 

 comprehension; here there has been more room for dispu- 

 tants and cavillers, such as Berkeley, to raise objections, which 

 undoubtedly must ever exist in the minds of those who have 

 never penetrated the subject in any form. 



Newton and Leibnitz had laid the foundation of this sci- 

 ence on a sufficiently tenable ground; but illustration and 



