MAGIC SQXJAKE. 679 MAGISTBATE. It will be observed that the diagonals are al- ready magic, since each equals the constant sum n s {n^ + 1). It' therefore only becomes necessary to arrange the rows and the columns to meet the definition. In the rows, the sum of the numbers in n n^ the />-th from the top is^in^ + 1) — 2" (" ~ 2^' + ^^ ' 71 and the sum in the /oth from the bottom is ^{n^ + 1) -(- ^ ()! — 2i ■+ 1). Furthermore, the number in each cell of the /.th row from the top is less than the number in the corresponding cell in the A;th from the bottom by »(ji — 2/. + 1 J. Hence, if in these two rows we make s interchanges of the numbers in corresponding cells, we increase the numbers in the 7.th row from the top, and de- crease those in the kth from the bottom, by s • n {n — 2k + 1), and therefore make each of these rows magic. But in doing this it is, of course, necessary to leave the sum of the diag- onals unchanged. It ne.xt becomes necessary to consider the columns. The procedure here is much the same as in the case of the rows. The sum of the numbers in the /vth column from the left was originalh' equal to while in the Ath colunin from the right it was |(»2-|-1)+ '^(n-2k+l). The number in each cell of the fcth column from the left was originally less than the correspond- ing number of the kth column from the right by n — 2fc + 1. Hence j; interchanges are necessary as before, care being taken as to the diagonals. The difficulty conies "in making the double inter- change without disturbing the sums of the diag- onals, but it can be shown that this interchange is always possible if n > 2. Starting with the initial square, the changes for a 4= square are shown as follows: 1 14 15 4 1 15 U 4 9 6 7 12 12 6 7 9 5 10 11 8 8 10 11 5 13 2 3 16 13 3 2 16 Flo. 6. Fig. 7. Besides magic squares, magic polygons and solids of various forms have been studied. BiBLiOGR.vpiiY. On the history of the subject, consult: Giinther. ^'ermis!chte Unterstichungen zii}- Geschichte der mnthemntischcn TVisscn- schaften (Leipzig, 1876) ; Lucas, Rccrrntions mathdmatiques (Paris. 4 vols., 2d ed.. 1801-04) ; McClintock. "On the Jlost Perfect Form of Magic Squares," in the American Journal of Mathe- matics (1897); Horner, "On the Algebra of Alagic Squares," in the Quarterly Juurml of Mathematics (1870); Schefller, Die mugiselieii t'iguren (Leipzig, 1882) ; Ball, "Even ilagic Squares," in the Messenger of Mathcinuiies (London, 1894) ; Ball, Mathematical Kccreatioiis (London, 1892) ; Ahreus, Matlwmatisvhe L'nter- haltungen und Hpiele, giving a very complete bibliography (Leipzig, I'JOl); Schubert, Mathe- Diatiaehe Mussestunden (Leipzig, 3 vols., 2d ed., lOUU). HAGIE, ma-ge', William Fbancis (1858—). An American physicist, born at Elizabeth, N. J. He graduated at Princeton University in 1879, and was appointed professor of physics there in 1885. His papers on the contact angle of liquids and solids and on the specific heats of solutions are noteworthj'. His principal books are : A revision of Anthony and Brackett's Physics (180U), a translation of Christiansens Theoretical Physics (180G), and an edition of Papers on Thermodynamics (Harper's Scientific Memoirs, 1899). MAGINN, ma-gIn', William (1793-1842). An Irish author, born in Cork. He graduated at Trinity College, Dublin, in 1811. From 1813 to 1S23 he conducted a scliool at Cork, and beginning in 1819 with the doggerel-Latin ver- sion of Chevy Chase, for nine years contributed to lilacKicood's Maga::ine a series of clever articles. In 1823 he went to London, where he was employed by ilurray as European corre- spondent of the daily Pepresentatire. On his return to England lie was for a time joint editor of the Standard, and in 1830 assisted in estab- lishing Fraser's Magazine. In this periodical first appeared his inimitable (fallery of Lite- rary Characters, with excellent accompanying sketches by Maclise. In 1836 his attack on a novel by Grantlcy Berkeley occasioned an in- nocuous duel. The three essays on The Learning of Shakespeare (1837) and the Ilomrric Ballads (1838), versifications of portions of the Odyssey, display something of his real .scholarship. Brought by intemperance and debt to the Fleet, where he dwelt even as Thackeray's "Captain Shandon" shows him, the one-time wit eniergi'd thence, through the good offices of Sir Robert Peel, to die in extreme poverty. A collection of his writings was published in 1855-57 (New York, 5 vols., edited by R. S. Jlaekenzie). MAGISTER' SA'CRI PALATII, 11a Ifi'shi I Lat.. master of the sacred ])alace). An officer of the Pajjal Court. It is said that Saint Dominic while in Rome noticed that the servants of the cardinals were compelled to pass much time in idleness hanging around the doors of the Vatican while their masters were in audience. So he received permission to read the Scriptures to these people. When he went away another was appointed to succeed him in this labor. Out of this practice arose the present office, which is a very important one, as the incumbent i* the censor of all books and pamphlets published l)y Catholics in Rome, a member of the congrega- tions of the Index, the Tnqiiisition and Rites, and has the right to grant the degree of doctor in theology and philosophy. He is nUvays a mem- ber of the Dominican Order. MAGISTRATE (OF., Fr, inagistrat. from Lat. magistrnfiis. magistrate, mngistcr, master; connected with magnus, Gk. ixiyat, megas, Skt.
