TOL. XVII.] PHILOSOPHICAL TRANSACTIONS. 467 



notwithstanding the proportions to one another, and to the to irxv, or vast abyss 

 of infinite space (wherein is the locus of all things that are, or can be ; or 

 the solid of infinite length, breadth, and thickness taken all manner of ways) 

 are easily assignable. For the space between two planes is to the whole, as the 

 angle of those planes is to the 36o degrees of the circle. As for cones and 

 pyramids, they are as the spherical surface, intercepted by them, is to the sur- 

 face of the sphere, and therefore cones are as the versed sines of half their 

 angles, to the diameter of the circle ; these three sorts of infinite quantity are 

 analogous to a line, surface and solid, and after the same manner cannot be 

 compared, or they have no proportion to one another. 



Besides these, there are several other species of infinite quantity, arising from 

 the contemplation of curves and their asymptotes, which, by reason of the dif- 

 ficulty of the subject, cannot be made so plain to most readers ; but what has 

 been already said, may be sufficient to evince what we undertook to explain. 



On the Defects and the Musical Notes of the Trumpet, and Trumpet Marine. 

 By Francis Roberts, Esq. R.S.S. N° igS, p. 559- 



The trumpet so famous in all ages for its use in war, the loudness and noble- 

 ness of its sound peculiarly suiting it to that purpose, is nevertheless to be rec- 

 koned among the imperfect musical instruments. For though it has a large 

 compass, the greater part of the intermediate notes are wanting, and some of 

 them imperfect. The extent of this instrument cannot be strictly determined ; 

 it reaches as high as the strength of the breath can force it ; but by considering 

 its notes within the ordinary compass of the scale of music, from double C 

 fa-ut to C sol-fa in alt, the nature of the higher notes will plainly appear. 

 These are all set down in the table, fig. 12, pi. 11, where the pricked notes are 

 imperfect, not being exactly in tune, but a little flatter or sharper than the 

 places where they stand, according as f or s is set over them. 



Here we may make two inquiries. 1 . Whence it is, that the trumpet will 

 perform no other notes in that compass besides those in the table, which are 

 usually called by musicians trumpet notes. 2. What is the reason that the 7th, 

 11th, 13th, and 14th notes, are out of tune, and the others exactly in tune? 

 In this matter we may receive some light from the trumpet marine, an instru- 

 ment though as unlike as possible to the trumpet in its frame, one being a 

 wind instrument the other a monochord, yet it has a wonderful agreement with 

 it in its effect. The sound is so like as not to be easily distinguished by the 

 nicest ear, and as it performs the very same notes, so it has the same defects ai 

 a trumpet ; for if the strings be stopped in any part but such as produces a 

 trumpet note, it yields a harsh and uncouth (not a musical) sound. 



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