March 2, 1891.] 



K N O WL EDGE 



45 



The nearest allies of the Moas are the small Kiwis ; but 

 whereas the latter have long pointed bills for probing in 

 soft mud after worms, the bills of the Moa were short and 

 broad like those of the Ostrich. Moreover, although the 

 Kiwis have no wings visible externally, they retain rudi- 

 mentary wing-bones, which have totally disappeared in 

 the Moas. The plumage of the Moas appears to have 

 been of the hair-like nature of that of the Kiwis. Since 

 the latter differ from the Ostriches in that the females are 

 larger than the males, we may assume that the same 

 condition obtained among the Moas. The Kiwis are 

 further remarkable for the enormous proportionate size 

 of their eggs ; and if anything like the same relative 

 proportions held good with those of the Moas, the egg 

 of the Giant Moa must have been of stupendous dimen- 

 sions. It is, however, probable that the eggs of the 

 larger Moas were relatively smaller than those of the 

 Kiwis. 



Passing to Australia, we find in the superficial deposits 

 remains of a bird as large as some of the medium-sized 

 species of Moa, but at once distinguished by the absence 

 of a bridge of bone at the lower end of the leg-bone. 

 This bird, which has been named Dnimnrnis, is, however, 

 as yet but very imperfectly known, so that we are to a 

 great extent in the dark as to its affinities, though it was 

 probably a distant giant relation of the Cassowaries. 



Before we again meet with fossil giant birds we have to 

 cross the whole extent of the Indian Ocean to Madagascar. 

 Here there occurs the enormous bird known as the 

 /Epyornu, the existence of which was first revealed by its 

 eggs, which are foimd sunk in the swamps, but of which 

 bones — mostly imperfect — were subsequently discovered. 

 One of these enormous eggs measures three feet in its 

 longer circumference, and 2i feet in girth ; its cubic 

 contents being estimated at rather more than two gallons. 

 The leg- bone of this bird has no bony bridge at its lower 

 end, and the cannon-bone (of which only a portion is 

 known) is as wide as that of the Elephant-footed Moa, 

 but is much longer and thinner. The natives search after 

 the eggs of this bird by probing for them in the soft mud 

 of the swamps with long iron rods. 



The Moas, the Dromornis, and the iEpyomis indicate, 

 then, three totally distinct groups of Giant Birds ; and 

 since their various habitats occupy islands on both sides 

 of the Indian Ocean, it is a fair presumption that their 

 common ancestors originally inhabited some part of 

 the great contmental mass of the Old World. Hupport 

 is ailbrded to this hypothesis by the occurrence of the 

 Ostriches on the west, and the Cassowaries and Emus on 

 the eastern side of the same great ocean. IMoreovcr, 

 there is historic evidence to the effect that Ostriches, 

 which are now confined to Africa and Arabia, formerly 

 existed in Baluchistan and Central Asia ; and since their 

 fossil remains occur in the Pliocene deposits of Northern 

 India, there is little doubt that at least this group of 

 Giant Birds originated in the northern part of the Old 

 World. Again, the Indian deposits already mentioned 

 have also yielded remains of a bird dift'ering from the 

 Ostrich in having three in place of two toes, and thereby 

 agreeing with the Cassowaries and Imuus, to which it 

 was doubtless allied, and thus indicating that these 

 birds likewise had their original home on the great Euro- 

 Asiatic continent, from whence they have gradually 

 migrated southwards till they reached regions free from 

 the large carnivorous mammals of the continents. 



Looking back through the Tertiary rocks of I'hirope to 

 see if we can find there traces of ancestral Giant Birds, 

 it is not till we come to the Lowest Eocene, or period 

 immediately below the Loudon clay, that our search is 



rewarded. Here, however, both in England, France, and 

 Belgium, we meet with limb-bones and other remains of 

 Giant Birds, which, from their huge size, must almost 

 certainly have belonged to the group under consideration. 

 In this bird, which is known as (Jaxtornis, the lower end 

 of the leg-bone has a bony bridge, as in the Moas ; and 

 since this is a feature common to the great majority of 

 flying birds, it suggests a community of origin between 

 them and the Giant Birds ; the loss of this bridge in the 

 living members of the latter thus being an acquired 

 character. Although we are still very much in the dark 

 as to the real affinities of the Gnstornia, yet it appears to 

 be more nearly related to the Moas and the Dromornis 

 than to any other birds, and it might, therefore, have 

 well been one of the ancestors of the group. 



This is at present the extent of our knowledge of the 

 former distribution of Giant Birds ; but it may be confi- 

 dently expected that whenever the Tertiary formations of 

 Northern Africa and Southern and Central Asia are fuUy 

 explored, we shall be rewarded by the discovery of other 

 kinds, which will tend to more or less completely connect 

 together those at present known to us, and which will 

 also show how these have gradually migrated, since the 

 Eocene Period, fi-om the great continental northern mass 

 to those southerly areas wherein some have existed up to 

 a comparatively late period, and where others still remain 

 as the sole living witnesses in the Old World of a group 

 which has all but passed away. 



THE MAGIC SQUARE OF FOUR. 



By T. Squire Baerett, F.S.S. 



TO treat fully of the square of 4 alone would take a 

 good-sized volume. The ancient Egyptians and 

 Pythagoreans, in their ignorance of mathematics, 

 thought it so wonderful that a series of numbers 

 could be arranged to add up alike, upward, across, 

 or diagonally, that they regarded such combmations with 

 superstitious veneration. We, however, know that it would 

 be much more wonderful if magic squares could not be 

 made. For example, considering the difficulty with 

 which a person, without some knowledge of the subject, 

 could make a magic square with an arithmetical series of 

 sixteen numbers, it would naturally be thought that it 

 could be done in very few ways. But it was shown by 

 Frenicle that it could be accomplished in at least 880 

 ways. Nor is this surprising when we consider that 

 there are nearly three billion (2,615,348,736,000) ways of 

 arranging 16 things in the form of a square. 



It is more than possible that Freuicle"s list does not 

 exhaust the number of such squares. I have never seen 

 his collection, and was therefore rather surprised to find 

 that I could make exactly the same number, but no more. 

 Nevertheless, I should not like to say that others could 

 not be constructed. 



These 880 squares, consisting as they do of numbers 

 in arithmetical progression, may obviously be classified 

 according to the relative position of each pair of comple- 

 mentary numbers. When the numbers ai-e the natural 

 series from 1 to 16, each complementary pair wUl sum 

 17; 1 -f 16, 2 + 15, 3 -t- 14, and so on. 



This classification shows the existence of 12 different 

 types, some of which are very curious. The most perfect 

 of them is the mixik square. [For definition of a uasik 

 magic square, see foot-notf, in KNowi.KixiE, p. 277.] Of 

 this type (which I call A) it can be mathematically proved 

 that only 48 variations are possible. I give an example 

 on the next page. 



