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CN THE* ANTIQUE HOUR-LINES. 73 



temorial lines, the radii from the centre P' are zr tan. 'pal. dist. 

 and the co-ordinates are affected by that quantity. If the gene- 

 rating diameter GKR move in the direction of the dart, and 

 set out from I ; it describes the curvilinear branch A, and 

 tan. pot. dist. is positive till the generating diameter come to the 

 situation II, where it is infinite. If its revolution be continued, 

 it re-appears at III, on the other side of the centre with the nega- 

 tive sign, and gives the curvilinear branch B ; at IV. it is again 

 infinite, and re-appears positive at V, going on to form the curvi- 

 linear branch C ; at VI. it is infinite ; and at VII. it becomes 

 negative, and forms the branch D ; it then comes out positive 

 at I, and goes over its former path. The way that the change 

 of sign takes place is apparent, by considering that whilst the 

 diameter is moving in the direction of the dart, it has another 

 motion at right angles to the plane of projection. Let GPK 

 be a section of the generating sphere, at right angles to the 

 plane of projection ; NPR being the section of that plane ; 

 GKR is the generating diameter ; and when in the position 

 GKR, then tan.pol. dist. — PR, and is positive ; when GK is 

 parallel to PR, then tan. pol. dist. is infinite. If the motion in 

 this plane be continued, tan.pol. dist. passes to the other side 

 of P, as at PN, and is affected with the contrary sign. 



Most of the writers who have spoken of the hectemorial 

 lines, have treated them as great circles, because their intertro- 

 pical parts, at a moderate height of the pole, coincide sensibly 

 with great circles ; and it is this case with which authors had 

 to do in considering the gnomonic projection of the hectemo- 

 rial lines for the climates of Greece or of Italy. The writings 

 of a considerable number of authors on this subject have been 

 consulted, and they all take the hectemorial lines for great 

 circles, except Clavius and Montucla. Clavius demonstrates 

 that the antique hour-lines do not coincide with great circles ; 

 Vol. VIII. P.I. K andl 



