74 ON THE ANTIQUE HOUR-LINES. 
and Monrucra merely states, but without discussion, that they 
are curves of a peculiar nature *. 
It has been shewn above, chiefly by means of a paced on 
a plane touching the sphere at the pole, that the hectemorial 
hour-lines on the oblique sphere are not great circles; and be- 
cause the describing diameter, in order to form a continuous 
and uniform surface, must go on moving during its whole re- 
volution with that motion which it had in the beginning of its 
course, and must be always included between the two parallels 
that touch the horizon, it is concluded that the curved surface 
whase 
* The passages from Cravius and Montucta are as follows: 
Crarix Astrolabium lib. i. lemma 39.  “ Circuli maximi transeuntes per horas 
inequales A¢quatoris, et duorum paratlelorum oppositorum, non necessario per 
horas inzequales parallelorum intermediorum transeunt in sphera obliqua.” He 
gives a demonstration of this, and concludes, in the scholium, that in order to de- 
lineate the antique hours with strict accuracy, a considerable number of the se- 
midiurnal arcs are to be divided into six parts, and the corresponding points of 
division joined. 
Monrvcra, Hist. des Math. tom i, edition de 1758S: ‘ Les lignes de ces sortes 
Wheures {les heures antiques] ne sont point droites comme les precedentes, mais 
courbes, et meme d’une forme tres bizarre ; de sorte qu’on ne peut les decrire 
qu’en determinant plusieurs points de chaewne; la maniére de les trouver se pre- 
sentera facilement a tout geometre ; c’est pourquoi nous ne nous y arretons pas.” 
The circumstance mentioned in the beginning of the paragraph to which this 
note refers, has led the celebrated and profound astronomer Deramere to con- 
trovert the opinion of Monructa in the following words: “ Monrtucta dit, en 
parlant des heures temporaires antiques, qu’elles sont courbes, et méme d’une 
forme tres bizarre, &c. Hist. des Mathem. tom i. On ne congoit pas com- 
ment une pareille inadvertance a pu echapper 4 un homme aussi instruit ; car si 
la surface est spherique, ces lignes seront des grands cercles; et si la surface est 
plane, elles seront des lignes droites, puisqu’elles seront les intersections des plans 
de ces grands cercles avec le plan du cadran.”  Detampre sur un cadran“anti- 
que trouvé dans Visle de Delos, et par occasion de la gnomonique des anciens ; noz 
tice lue & la classe des Sctences Physiques et Mathematiques de [Institut Royal 
de France, le 10 Octobre 1814. 
