SCIENCE. 



573 



ice, 

 Mtics 



Construc- 

 tion of the 

 circles. 



1 i 



3 



1. (A) is filled up with the majors in tiie same 

 way that the others must be done. The sum of the 

 two extremes, as also that of the two means in th- ho- 

 rizontal column, in to he made equal to the gum of the 

 first and last number of the progression. Thus, in the 

 square of 1- it will he 17, in the square of 8 it will be 

 ftV), in that of rj it will be Ik"), &c. when the series be- 

 gins wit!) unity. 



The preceding square of 8 fs constructed from four 

 vanities of the same kind of arrangement as these, al- 

 though any one of them would have done quite as well 

 for the magic circle. In the square, the sum of each 

 vertical is equal to that of each horizontal column, as 

 likewise their halves, but the diagonals do not agree 

 with them. 



By making use of two of the 

 above varieties, and of the distri- 

 bution No. I . in page 5? 1 , a square 

 of 8 may be formed, having its 

 diagonals the same sum as the 

 other columns, and which will give 

 to the circle an additional proper- 

 ty. Any two of the varieties 

 will do, provided one of them be 

 taken from the row A, and the 

 other from the row B. 



Properties 

 of the ma- 

 gic circle. 

 PLATE 

 CCCCLJCXXIV. 

 *'i- 7. 



The annexed square is composed of arrangements 

 No. 2. A, and No. 4. B, the first being used in the up- 

 per half, and the second in the lower half of the square. 

 The numbers commence at 1^, and end at 75, as in Dr. 

 Franklin's, and from the square they are transferred to 

 the circles No. 1, and 1 2, the two circles being necessa- 

 ry to avoid confusion in the figure, and to render the 

 explanation of its properties intelligible. 



Construction of the Circles. 



The diameter of the exterior circle is divided into 1.9 

 equal parts. The diameter of the interior circle em- 

 braces three of these parts, and the remaining divisions 

 mark the diameters of the other seven circles. The 

 circumference of the large circle is divided into eight 

 equal parts, and radii drawn to these divisions, as shown: 

 in the figures. There are thus O'l cells formed between 

 the interior and exterior circles for the numbers of the 

 square, which are placed so as to appear as radii at 

 equal distances between the linear ones. 



Properties of the Magic Circle. 



1. The sum of the numbers in each of the con- ) 

 centric circles added to the central number V SCO 

 12, is . J 



2. The numbers in each radius added to 12 is 360 



3. The numbers in half the circles, taken above") 



or below the horizontal line AB, added to > 180 



C= J 



4. The numbers in half Ihe radius added to C= 180 



5. Any four adjoining numbers forming a square ? fi 

 added to Cs: 



6. If from O, Plate CCCCLXXXIV. Fig. 7. 

 as a centre, circles be described through the 

 points of intersection at a, b, c, d, c, sanij', in > 3(iO 

 the radius of, the numbers in the spaces be- 

 tween each pair of these circles added to 12= 



Fl 





7. The number* in the half of etch of these ec- 1 

 centric circles abuv aud below ontalf' 180 

 line, added o (j= . . J 



If circle*, in like manner, be described from fie oen- 

 tres j>, i/, and r, the number* between each pair, as alto 

 their halves, will amount to the same tun) 



These are the principal prop- Iranklia's cir 



cles. 



Additional Property shown in Fig. 8. 



8. If from the interior circle eight upiraU be de- 

 scribed at equal distance from each other, J 

 and in the haute direction, so that etch makes f 

 one comple.te revolution in iti patsagc to the f" 

 exterior circle, as in t ; ig. 7, the cpocea be- I 

 tween each pair of these s-piruls will cui.uiii' 



8 numbers, whose amount added to I J 

 If the spirals are made to revolve in a contrary di- 

 rection, the amount of the numbers in the similar pace*> 

 will still be the same. . i 



AStBOKOMT. 



Description of an Aslromc'.cr. 



This name was given by M. Jeurat to a sunpte in- 

 strument for finding the rising and setting of the. star* oftnasuw- 

 and planets, and their position, which he has dt> . ultter- 

 scribed in the Memoirs of the Academy of Sciences. 



The following is an improved instrument of the, 

 same kind, and which is capable of solving many other 

 astronomical problems. 



This instrument represented in Plnte CCCCLXXXV._ri 4 T r 

 Fig. 1. consists of four divi Jed circumferences. Tueinr.. - ' xv. 

 mostof these is moveable round the centre A, and is divid- **' ' 

 ed into twenty-four hours, which are again subdivided 

 into quarters and minute?, when the circle is sufficient- 

 ly large. The second circumference is composed of 

 four quadrants of declination, divided by means < : 

 table of semidiurnal arcs, adapted to t;e latitude of the 

 place. In order to divide these quadrants, move 

 horary circle, so that 12 o'clock noon may be exactly 

 opposite to the index B : then since thi 

 equal >r, and its declination 0, when the :ial . 



arc is VI hours, the zero of the scales of 'declination . 

 will be opposite VI. VI. and as the declination 'of 

 star is equal to the cotatitude of the place, when its se- 

 midiurnal arc is 0, or when it just comes to t. 

 point of the horizon, without rising above it, the 

 g'ree of declination at the other < 

 drant, or opposite XII. XII. will be th.2 s.-ur. 

 colatitude of the place, which in the prese;.: 

 the latitude of the place being supposed 5; North. 

 The intermediate degrees of declination are then to be 

 laid down from a table of semidiurnal arcs by plac- 

 ing the degree of declination opposite to the arc 

 which it corresponds, thus the' 10th degree pf south 

 declination must stand opposite V h IS' in the after- 

 noon, and Vl h -17' in the morning, because a (K 

 tion of ten degrees south gives a semidiurnal arc of 

 V h 13'. When the scales of declination are thus com- 

 pleted, the instrument is ready for showing the rising 

 and setting of the stirs. For this purpose move the 

 horary circle till the index B points to the time of the 

 star's southing ; then opposite to the "star's declination 

 in the scale C, if the declination is south, or in the 



