t4 PHILOSOPHICAL TBANSACTIONSw [aNNO 1703. 



Specimen of a General Method of determining the Quadrature of Figures. By 



the Rev. John Craig.* Addressed to Dr. Geo, Cheyne.-f N° 284, p. 1346. 



Translated from the Latin. ^f,,v sHi 'V 



You will easily believe, sir, that I am not a little pleased that the method 1 

 use for determining the quadrature of figures, is so well approved of by M. 

 Leibnitz and yourself: so that he jieknowledges it somewhat resembles the 

 method discovered by himself, and yoq conjecture it has some affinity with the 

 method of Mr. Newton. And by your own success in such pursuits you have 

 greatly improved the inverse method of fluxions, by your books on that sub- 

 ject, dedicated to Dr. Archibald Pitcairne, the ornament of our age and 

 country. But many things still remain to be discovered for perfecting this 

 inverse method ; and I shall now give some reasons which show, that what 

 remains, cannot be obtained by any methods yet in use. 



And first, when from the given relation between z and y, the fluent of zy is 

 required, all those methods demand that z should be expressed by y and given 

 quantities ; which however cannot be done when the equation defining that rela- 

 tion rises above a cubic or biquadratic ; for here the common algebra stops, to 

 the great reproach of that science. Secondly, although a general rule were 

 known, for finding the roots of equations of any degree, yet it would be quite 

 useless in this inverse method ; for the root z would be involved in so many 

 cx)mplicated surds, that by no art yet could we revert from the fluxion to the 

 fluent. For these reasons I have tried another course, and with some success, 

 a specimen of which is here given in what follows. 



* The Rev, Mr. John Craig was a very respectable Scotch divine and mathematician, as appears by 

 tis writings, both those printed in these Transactions, and the books published by himself. The dates 

 of his birth or death do not appear ; but it would seem he was minister to acong regation at Gilling- 

 ham, as his communications to the Royal Society are dated from that place. Besides his numerous 

 papers in the Transactions, his own publications are chiefly these: 1. Methodus Figurarum Quadratus, 

 &c. An. l6&5 ; 2 De Quadraturis et Locis, l693 j 3. De Calculo Fluentium, 1718 j 4. Theologiae 

 Christians Principia Mathematica, 1699. In this curious work, which was afterwards reprinted at 

 Leipsic, the author maintains by mathematical calculation, that Christianity will last only 1454 year* 

 from the date of his writing. 



f Dr. George Cheyne, eminent both as a physician and mathematician, was born in Scotland 1671, 

 and was educated at Edinburgh. Like his friend Mr. Craig, he applied himself to cultivate the new 

 science of fluxions, lately discovered by Newton. At the age of 30, Dr. Cheyne came to London, 

 where from his manner of living he became so corpulent, that his life was a burden to him. But by 

 means of a vegetable diet he recovered his health and activity, and did not die until he was in the 

 fid year of his age. Besides his treatise on the inverse Method of Fluxions, be published also 

 Philosophical Principles of Religion, natural and revealed. As also the English Malady, or a 

 Treatise on Nervous Diseases. 



