74 ON THE ANTIQUE HOUR-LINES. 



and Montucla merely states, but without discussion, that they 

 are curves of a peculiar nature *. 



It has been shewn above, chiefly by means of a projection 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 



whose 



* The passages from Clavius and Montucla are as follows: 



Clavii Aslrolabium lib. i. lemma 39. " Circull maximi transeuntes per horas 

 insequales iEquatoris, et duorum parallelorum oppositorum, non necessario per 

 horas insequales parallelorum intermediorum transeunt in sphsera 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, ami the corresponding points of 

 division joined. 



MoyrvcLA, Hint, des Math, torn i. edition de 175S : " Les lignes de ces sortes 

 d'heures £les heures antiques^ ne sont point droites comme les precedentes, mai* 

 courbes, et meme d'une forme tres bizarre ; de sorte qifon ne petit les decrirc 

 qifen determinant plusieurs points de chacune ; la maniere de les trouver se pre- 

 sentera facilement a tout geometre ; cVst 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 Deumbre to con- 

 trovert the opinion of Montucla in the following words : " Montucla dit, en 

 parlant des heures temporaires antiques, qifelles sont courbes, et meme d'une 

 forme tres bizarre, &c. Hist, des Mathem. torn i. On ne concoit pas com- 

 ment une pareille inadvertance a pu echapper a 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." Delambre sur un cadran^anti- 

 que trouvt: dans risk de Delos, et par occasion de la gnomonique des anciens ; no? 

 lice lue a la classe des Sciences Physiques et Mathematiques de rinslitut Royal 

 ck France, le 10 Octobre 1814. 



