Properties of Plane Triangles. 27 



ticed by mathematicians. The most fertile source, indeed, of 

 the properties of the triangle yet explored, is when it is con- 

 sidered in relation to its circumscribing circle and its four cir- 

 cles of contact : and though this class of relations had always 

 had a place in the theoretical arrangements of geometrical 

 science, yet it owes its present extent principally to the la- 

 bours and example of Professor Thomas Simpson*. Still, 

 there has been till lately so much attention bestowed upon 

 those relations which result from taking a single side as base, 

 that the symmetrical properties arising from considering all 

 the sides alike, though affording conclusions equally interesting, 

 have not been cultivated in any degree worthy of its importance. 

 A small number of such properties are, however, already 

 known ; and the following series will add a few not inelegant 

 ones to the number. 



It may be proper here to remark, that some of the most 

 beautiful of the symmetrical properties of the triangle, admit 

 of more simple demonstrations than those commonly given. 

 Let, for example in Plate I. fig. 2. the triangle ABC have its 

 four circles of contact whose centres are e, E, F, G' ; let also 

 S signify semiperimeter ; and r, radius of inscribed circle; r„ 

 r^ 7j, those of external contact. 



* The most ample series of properties in which the triangle is viewed 

 in this connection that has yet appeared, is " The Modern Geumetri/," pub- 

 lished in '' The Student" a little work of great merit, which was issued by 

 the late Mr. Hilton, of Liverpool. This work, of which but a very few 

 copies appear to have been printed, is become extremely scarce ; and the 

 majority of those mathematicians who know anything of its contents, have 

 transcribed it from the printed copies, and in some cases from MS. copies 

 at third or fourth hand. The inquiries have in several cases been success- 

 fully followed up in those different periodicals which are principally de- 

 voted to mathematics, but in so unconnected a form (which is unavoidable 

 in those works) that they are comparatively little known, and their rela- 

 tions are as little perceived. Besides, excellent as Mr. Hilton's "Modern 

 Geometry" is, it cannot be disguised that the arrangement is extremely de- 

 fective; as classes of inquiry very different in many respects are combined 

 in the same series of propositions, and referring to the same diagrams. It 

 is likely, therefore, that a very useful work might be produced (even were 

 there but little original matter in it), by giving a systematic arrangement to 

 all the properties at present known, and classing them according to some 

 distinctly marked mode of division, and demonstrating them in the brief 

 form adopted in "The Student." Sucha work would much (iuilitalc geome- 

 trical reference, and be of great service in a course of investigation, as well 

 as very convenient in stating the authorities for propositions which wc have 

 occasion to tiuotc. It is a plan which I have some years formed, and to- 

 wards accomplishing which I have taken considerable steps : but other jiur- 

 suits have so often intervened as to much impede my progress, and espe- 

 cially in that very important part, — A minute examination of the diflcrent 

 periodical works which have appeared at various times and in several parts 

 of the country. Inilecd, it is no easy matter to procure several of them, 

 and often when procured they prove to be of little use. 



E 2 Then 



