IQ2 PHILOSOPHICAL TRANSACTIONS. [aNNO 1715. 



Having premised thus much in general, concerning the two methods here 

 treated of, to come to a particular description of this book; in the preface I 

 give a short description of the method of increments, and an account of Sir 

 Isaac Newton's notion of the fluxions which I have already spoken of. The 

 book consists of two parts, and contains 1 1 8 pages in 4to, the propositions 

 being numbered throughout from the beginning. In the first part I explain the 

 principles of both methods: and in the second part I show the usefulness of 

 them in some particular examples. 



After having explained the notation I make use of in the introduction, in 

 the first proposition I explain the direct method, both of increments and of 

 fluxions. The second proposition shows how to transform an equation wherein 

 integrals and their increments, or wherein fluents and their fluxions are con- 

 cerned; so as instead of the integrals or fluents, to substitute their comple- 

 ments to a given quantity, with their increments or their fluxions, these 

 increasing in a contrary sense to the quantities in the first supposition. In the 

 third proposition I show how to transform a fluxional equation, so as to change 

 the characters of the fluents, making that quantity to flow uniformly, which 

 in the first supposition flowed unequally, having second, third, and other 

 fluxions, and making that quantity which in the first supposition flowed uni- 

 formly, now to flow unequally, so as to have second and third fluxions, &c. 

 This proposition is of great use in the inverse method, when we would invert 

 the expression of the relation of the flowing quantities; for example, if in the 

 supposition z flows uniformly, and x variably, by the inverse ujethod of 

 fluxions we find x expressed by the powers of z; but if we would find z expressed 

 by the powers of x, we must then transform the equation by this proposition. 

 Sir Isaac Newton and Mr. de Moivre do this by the reversion of serieses; but 

 I take this to be the more proper and more genuine method of doing it directly. 

 In the 4th and 5th propositions are explained the method of judging of the 

 nature and number of the conditions that may accompany an incremental or a 

 fluxional equation. This is a circumstance that I do not find to have been 

 explained by any one before, and the propositions are somewhat intricate. The 

 conditions that attend incremental or fluxional equations I do not know to have 

 been sufficiently taken notice of by any one; but they ought well to be attended 

 to in the inverse methods; the solutions of particular problems being never 

 perfect, unless there be provision made for the satisfying of them, by the inde- 

 termined coefficients in the equation that contains the solution of the problem. 

 Examples of this may be seen in prop. 17 and 18, where I give the solution of 

 the problems concerning the Isoperi meter and the Catenaria. 



The sixth proposition contains the general explanation of the inverse method, 



