upon the Antique Dials. 117 



This is the equation of the harmonic curve, whose modulus is n i, and 

 the radius of its generating circle = a'. The hectemoria, therefore* when 

 projected upon the equatorial cylinder, and that cylinder developed, is 

 the HARMONIC CURVE, the modulus of -which is n i, and radius of its ge- 

 nerating circle = a'. 



Again, the harmonic curve is the development of an elliptical section of 

 a right cylinder, and hence we are put into possession of a mechanical me- 

 thod of tracing the curves. Thus : 



A right cylinder to radius - or a' being cut by a plane which makes with 



the axis an angle = tan ~ 1 n tan I, being developed indefinitely, will give the 

 hectemoria in question, in piano. Rolling this plane hectemorial series up- 

 on a cylinder to radius a, we obtain the cylindrical hectemoria. By cutting 

 away the upper or the lower part of this cylinder by lines from the centre of 

 the equatorial section, would give a ruler upon which a line moving about that 

 centre would trace the plane hectemoria upon a plane anyhow situated with 

 respect to the ruler. The mechanical contrivances requisite to effect this 

 are not numerous nor complicated ; but great care is essential in the actual 

 structure of such an apparatus to ensure a desirable degree of accuracy. 

 These subjects, however, do not properly belong to the present place. 



XXXI. 



The decision of this long disputed question in the history of Astronomy 

 is, I trust, now effected. We see what the lines really are, and are well as- 

 sured that their deviation from these chords within the range required by the 

 ancient astronomer could not have been discovered by any methods within his 

 reach ; and that, as he had no means of tracing their prolongations, he could 

 not arrive at any true notion of their character. Even had he been in pos- 

 session of tolerably accurate methods of measuring time, he would rather 

 have been disposed to attribute any slight discrepancies between his clock 

 and his dial to errors of observation, than to any deviation of the true hec- 

 temoria from a rectilinear course. It was from considerations altogether 



VOL. XII. PART I. Q 



