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ON THE ANTIQUE HOUR-LINES. | 13 
temorial lines, the radii from the centre P’ are = tan. poi. dist. 
and the co-ordinates are affected by that quantity. Ifthe gene- 
rating diameter GKR move in the direction of the dart, and 
set out from I; it describes the curvilinear branch A, and 
tan. pol. 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 ITI, on the other side ofthe 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, ne 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 ep a 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 Cravius and Mowrucia. Cxavius demonstrates 
that the antique hour-lines do not coincide with great circles; 
Vor, VIII. P. 1, kK and: 
