252 Lieut. Zahrtmann on the Mathematical 



are equal to two right angles, and all the angles round a point, 

 to four right angles ; it will follow that the sum of all the angles 

 of the triangles is equal to a certain number of right angles to- 

 gether with the four angles of the quadrilateral. But, as the an- 

 gles of a triangle cannot exceed two right angles, the sum of all 

 the angles of the triangles will be equal to twice as many 

 right angles as there are triangles, wanting the sum of the de- 

 fects of the angles of every triangle from two right angles. 

 Wherefore, by comparing the two equal sums, it will appear 

 that the angles of the quadrilateral are equal to four right an- 

 gles, wanting the sum of the defects of all the included tri- 

 angles. Now suppose a triangle that has the sum of its angles 

 less than two right angles by some given angle ; then a qua- 

 drilateral may be constructed that shall have the sum of its 

 angles less than four right angles by any multiple of the 

 same angle ; which is absurd, because the multiple may be so 

 taken as to reduce the angles of the quadrilateral to zero, or 

 so as to be less than any proposed angle. We have men- 

 tioned these demonstrations because we have never met with 

 any other indirect proof of the equality of the angles of a tri- 

 angle to two right angles, which is both unexceptionable in 

 point of accuracy, and requires no peculiar postulate. 



In taking leave of parallel lines, some apology seems to be 

 due to the readers of this Journal for the great length of our 

 observations. But the subject is very curious in itself; it has 

 been keenly debated by very able men ; and the contest has 

 been lately resumed in a very high tone, and with additional 

 forces. It therefore seemed very desirable to bring the mat- 

 ter to some decision ; and we hope some light has been thrown 

 upon it. On the whole we are inclined to think that much 

 more importance has been attached to the functional investi- 

 gations than they intrinsically deserve. It seems very certain 

 that there was a time when such demonstrations would not 

 have passed current for sterling geometry in the Athens of 

 the North. 



April 5, 1824. Dis-iota. 



XLI. On the Mathematical and Astronomical Instrument 

 Makers at Paris. By Lieut. Zahrtmann.* 



"DEFORE I subject myself to the uncertain event of a sea 



voyage, 1 will endeavour to perform the task you have 



proposed to me, by communicating to you some particulars re- 



* From M. Schumacher's Astronomische Nachrichtcn, No. 42. 



specting 



