344; On Staudt's Theorems of Polijgons and Polyhedrons. 



by the Greek letters hi situ shall always preserve the same sign. 

 Hence we derive two geometrical formulai concerning the pro- 

 ducts of polyhedrons^ viz. 



(1.) ABCD X EFGH = ABCE x FGHD-ABCF x GHED 



+ ABCG X HEFD - ABCH x FGED. 



(2.) ABCD xEFGH = ABEFxGHCD + ABGII xEFCD 



+ ABEG X HFCD + ABHF x EGCD 



+ ABEHxFGCD + ABFGxEHCD. 



These formulse give rise to an exceedingly interesting observa- 

 tion. In order that they shall be numerically true, we must 

 have a rule for fixing the sign to be given to the solid content 

 represented by any reading off of the four points of a tetrahe- 

 dron, i. e. we must have a rule for determining the sign of solid 

 contents of figures situated anywhere in space analogous to that 

 which, as applied to linear distances reckoned on a given right 

 line, is the true foundation of the language of trigonometry, 

 and the condition precedent for the possibility of any system of 

 analytical geometry such as exists, and which, not altogether 

 without surprise, I have observed in the pages of this Magazine 

 one of the learned contributors has thought it necessary to vin- 

 dicate the propriety of importing into his theory of quaternions. 



Various rules may be given for fixing the sign of a tetrahedron 

 denoted by a given order of four letters. One is the following : 

 the content of ABCD is to be taken positive or negative, accord- 

 ing as to a spectator at A the rotation of BCD is positive or 

 negative. Another, again, is to consider AB and CD as repre- 

 senting, say two electrical currents, and to suppose a spectator 

 so placed that the current AB shall pass through the longitudinal 

 axis of his body from the head towards the feet, and looking 

 towards the other current CD ; the sign of the solid content of 

 the tetrahedron (and, indeed, also the effect, in a general sense, 

 of the action of the two currents upon one another) will depend 

 upon the circumstance of this latter current appearing to flow 

 from the right to the left, or contrariwise in respect of the spec- 

 tator. Last and simplest mode of all, the sign of the solid con- 

 tent of ABCD will depend upon the nature (in respect to its 

 being a right-handed or left-handed-screw) of any regular screw- 

 line (whetlier the common helix or one in which the increase or 

 decrease of the inclination is always in the same direction) ter- 

 minating at B and C, and so taken that BA shall be the direc- 

 tion of the tangent produced at B, and CD the direction of the 

 tangent ])rocluced at C. Inasmuch as of the twenty-four permu- 

 tations of a quaternary arrangement a defined twehc have one 

 sign, and the other twelve the contrary sign, these various de- 



