March 13, 1902] 



NA TUBE 



447 



During the present year, observational research in reference to 

 this object is likely to prove of an unusually interesting character, 

 as there are several features which deserve (and will doubtless 

 receive) special attention. During 1901, the red spot exhibited 

 a singular acceleration of motion when compared with its rate in 

 previous years, for it maintained a longitude of 45', consistently 

 with a rotation period of gh. 55m. 40s. '6, upon which system 

 II. of Mr. Crommelin's ephemerides is based. It will there- 

 fore be most important to trace the position of the spot in 

 ensuing months, as it seems probable that this curiously durable 

 marking, after a constantly increasing retardation between about 

 1878 and 1900 which augmented its rotation period from gh. 

 5;m. 34s. "5 to gh. 55m. 41s. -8, will now travel with greater 

 celerity and give a period becoming shorter with the time 

 until the minimum is reached. If these oscillations in velocity 

 are developed at regular intervals, it will soon be possible to 

 determine the length of the cycle for observations of the red 

 spot or of the hollow in the great southern belt date from 

 1831. There is some significant evidence in support of the 

 conjecture that the motion of certain markings as well as the 

 apparition of particular spots in various latitudes are recurrent 

 ■ on the planet, and some of the reappearances appear to take 

 place at periods not differing materially from the time of 

 Jupiter's revolution round the sun. The red spot may be 

 looked for near mid-transit on March 13 at i8h. 7m., on March 

 25 at iSh. 4m., on March 30 at I7h. 13m., on April 6 at 

 iSh. om., and on April 11 at I7h. gm. 



Observers should now endeavour to redetect the rapidly 

 moving dark spots which appeared in the north temperate belt 

 of Jupiter in 1S80 and 1S91. There were a number of spots 

 visible in this latitude in 1901, but the writer found their mean 

 rate of rotation gh. 55m. 50s. This is nearly 8 minutes in 

 excess of the rotation period found for certain irregular markings 

 in approximately the same latitude in the autumn of 1S80 

 which gave a rate of gh. 48m. It is a very singular circumstance 

 that in a similar latitude of Jupiter spots are developed showing 

 respectively the shortest and longest rotation periods of any 

 which have ever been observed. 



In 1901, a large dark spot was often seen in the south or 

 tropical • zone of the planet, and this may prove a repetition of 

 the object observed in the same latitude in i88g-gi. This spot 

 exhibits a rotation period of gh. 55m. iSs. '5, and its more rapid 

 movement will enable it to overtake the red spot in about June, 

 1902, should both the markings remain visible until that 

 month. The longitude of the south temperate or tropical spot 

 will be as follows during the next three months, and it will 

 follow the red spot at the time-intervals stated if the latter 

 object retains the same longitude ( = 45°) as in 1901 : — 



Follows 

 Longitude Red spot. 



'902 h. m. 



March 17 110-5 ' 48 



April 17 941 . . . t 2t 



May '7 77 7 o 54 



The writer obtained an observation of this marking on 

 February 27, when it appeared to be central at iSh. 40m., which 

 would make its longitude I23''8, but it was very imperfectly 

 seen. The instrument used was a 4-inch Cooke refractor, power 

 I7S- 



Another important feature for reobservation in igo2 will be 

 the white and dark spots plentifully grouped along the equatorial 

 region of the planet. In the three years iSg8, 1S99 and 1900, 

 the rotation period of the equatorial current differed very little, 

 the mean value from a large number of spots being gh. 50m. 24s., 

 or 5m. 17s. 7 less than the rate of the red spot. But in igoi 

 the mean rotation period of 28 equatorial spots observed at 

 Bristol was gh. 50m. 2gs., or 5m. 12s. less than that of the red 

 spot. 



When further observations of these variations have been 

 pursued during many oppositions, the outcome may be both 

 interesting and important as afiording a good clue to the physical 

 condition and phenomena of the planet. That great atmo- 

 spheric changes are in progress on the disc is evident, and it is 

 the facility with which they may be observed and compared 

 which renders this object a singularly attractive one to the 

 possessors of telescopes. W. F. Denni.ng. 



1 This interesting marking exhibited a motion coinciding with that of 

 objects placed in the planet's south temperate zone, though its position en- 

 croached on the south tropical .is well as the south temperate region. 



NO. 1689, VOL. 65] 



MAGIC SQUARES AND OTHER PROBLEMS 

 VpON a CHESS-BOARD} 

 T^HE construction of magic squares is an amusement of great 

 antiquity ; we hear of them being constructed in India and 

 in China before the Christian era, whilst they appear to have 

 been introduced into Europe by Moschopulus, who flourished at 

 Constantinople early in the fifteenth century. On the diagram 

 you see a simple example of a magic square, one celebrated as 

 being drawn by Albert Dlirer in his picture of "Melancholy," 

 painted about the year 1 500 (Fig. i). It is one of the fourth order, 

 involving l6 compartments or cells. In describing such squares, 

 the horizontal lines of cells are called " rows," the vertical lines 

 "columns," and the oblique lines going from corner to corner 



"diagonals." In the 16 compartments are placed the first l6 

 numbers, i, 2, 3, . . . 16, and the magic property consists in 

 this, that the numbers are placed in such w'ise that the sum of 

 the numljers in every row, column and diagonal is the same, 

 viz., in this case, 34. 



It is probable that magic squares were so called because the 

 properties they possessed seemed to be extraordinary and 

 wonderful ; they were, indeed, regarded with superstitious 

 reverence and employed as talismans. Cornelius Agrippa con- 

 structed magic squares of orders 3, 4, 5, 6, 7, 8, g, and 

 associated them with the seven heavenly bodies, Saturn, Jupiter, 

 Mars, the Sun, Venus, Mercury and the Moon. A magic square 

 engraved on a silver plate was regarded as a charm against the 

 plague, and to this day such charms are worn in the east. . 



However, what was at first merely a practice of magicians 

 and talistnan makers has now for a long time become a serious 



study for mathematicians. Not that they have imagined that it 

 would lead them to anything of solid advantage, but because 

 the theory of such squares was seen to be fraught with difficulty, 

 and it was considered possible that some new properties of num- 

 bers might be discovered which mathematicians could turn to 

 account. This has, in fact, proved to be the case ; for 

 from a certain point of view the subject has been found to be 

 algebraical rather than arithmetical, and to be intimately con- 

 nected with great departments of science, such as the "in- 

 finitesimal calculus," "the calculus of operations" and the 

 " theory of groups." 



In the next diagram (Fig. 2) I show you a magic square of 

 order 5, the sum of the numbers in each row, column and 



1 A discourse delivered at the Royal Institution on Friday evening, 

 February 14, by Major P. A. MacMahon, F.R.S. 



