Mr. T. S. Davies on Pascal's Hexagramme Mystique. 333 



The presumed repugnance of patients to a heat-blister may 

 be easily obviated by judiciously representing the momentary 

 continuance of the pain, and the probable speedy alleviation 

 of a serious disease. 



Sir, your much obliged servant, 

 Langham Place, Nov. 2, 1826. ANTHONY CARLISLE. 



P.S. I am obliged to you for correcting some errors in the 

 printing of my pamphlet, which was unavoidably published 

 without mv revisal. 



L. Properties of Pascal's Hexagramme Mystique. By 

 T. S. Davies, Esq. of Bath* 

 'HE theorem discovered by Maclaurin in 1722, and pub- 

 lished at the end of his Algebraf in 1727, is one of the 

 most important properties of the conic sections with which we 

 are acquainted. It is, indeed, almost the only useful theorem 

 we possess, in which the axes, foci, centres) and diameters of 

 the sections are not introduced : and therefore it furnishes the 

 means of investigating some of the most difficult properties of 

 this class of curves. An immediate result of that principle is 

 " the beautiful property, — that if the opposite sides of a hexa- 

 gon inscribed in a line of the second order, be produced, the 

 points of concourse will range in a straight line ;" and was 

 noticed by him amongst many other equally curious results 

 in a paper in the Philosophical Transactions for 1733. This 

 particular consequence was known to the celebrated Pascal, 

 and is found in a printed fragment, entitled Essais pour les 

 Coniques, and bearing the date of 1640. The entire paper is 

 printed in the fourth volume of his Works (Paris, 1819), and 

 occupies but four very open octavo pages. In this tract the 

 property is stated in a more general form than Professor Leslie 

 has given to it<in the above quotation; but Pascal's way of 

 considering it is first as a property of the circle, and thence 

 inferring its truth as a property of the other sections, by pro- 

 jection. The whole is left without demonstrations, or even 

 reference to any other work, whence they may be deduced. 

 It appears, however, that at some period of his brief life, he 

 had pursued the subject more into detail; though the tracts 



* Communicated by the Author. 



f Page 345, ed. 4. This theorem is comprised in a more general one 

 of Brackenridge (Phil. Trans. 1735). A restricted case, viz. when all the 

 poles are in a straight line, is in reality identical with the general theorem 

 of Pappus, as restored by Simson (Phil. Trans. 1723, or Tractatus de Poris- 

 matibus, in his Opera Retiqua) : whilst the theorem of Maclaurin under a 

 similar restriction becomes virtually the same as the first of the said two 

 general propositions. 



were 



