y line 6, 1878] 



NATURE 



149 



However complex the curves described by them may be, 

 the points will be found to form a circle on the upper 

 plane ; and if we give to A different values, the corre- 

 sponding circles will be found to be all concentric. 

 Further, if we call the circle corresponding to the value 

 A = o the zero-circle, the area of the curves described by 

 the points on any other circle of the system equals N 

 times the ring inclosed between that circle and the zero- 

 circle. It is remarkable that such a singular point as the 

 centre of the circles should exist. 



In the special case in which N=o, i.e., where there 

 has been only an oscillatory movement of the upper 

 plane and no complete rotation, the system of concentric 

 circles is replaced by a system of parallel straight lines, 

 the area of the curves described by the points on any 

 straight line of the system being proportional to the 

 distance of that line from the zero-line. 



It should, perhaps, be pointed out that the area of a 

 figure 8 is zero, as the two halves are of opposite signs ; 

 also that when a point reciprocates on a curve the area 

 inclosed by it in its path is zero. For example : if we 

 take the interesting case of a circle rolling inside another 

 of twice its diameter, every point on its circumference 

 reciprocates on a straight line, and consequently the 

 circumference is the zero-circle. 



This theorem was suggested to me by reading a paper 

 by Mr. C. Leudesdorf in the Messenger of Mathematics, 

 where I have already enunciated it. It seems, however, 

 to be one which, from its somewhat startling simplicity, 

 may interest a larger class of readers than a purely 

 mathematical one. 



The proof is simple. Let P, F be two points on the 

 moving plane, and let A, A' be the areas described by 

 them. Let P P = r, and let the total movement of P' 

 perpendicular X.o P F — n. 



Then A - A'= nr + Nir r\ 

 If we take F as origin and the position of P' P in which 

 11 is a maximum and equal to n as initial line, n'=ncos 6. 

 Thus A—A'=ti'co~>&' r-\-NTTr\ the equation to a family 

 of concentric circles. Transforming to the centre, we 

 have A = NiT{r''-a% 



where a is the radius of the zero-circle. 



A. B. Kempe 



OLD MAPS OF AFRICA 



TV/TR. STANLEY, in the paper which he read at the 

 ^^^ Geographical Society on Monday, spoke of Africa 

 being brought to light after an oblivion of 6,000 years. 

 Notwithstanding the somewhat confused phraseology, Mr. 

 Stanley's meaning is clear enough : Central Africa, with 

 its great lakes and rivers, is now known, he means to say, 

 for the first time. But recent investigation seems to show 

 that the oblivion of Africa must be counted by hundreds 

 and not thousands of years ; that, in fact, it is only within 

 two or three centuries that a knowledge of Central Africa 

 has been allowed to lapse. A more rigorous search may 

 show that between the fourteenth and the seventeenth 

 centuries the great features that have been placed on 

 modern maps within the past few years were discovered 

 and recorded on the maps of the time. 



We have recently referred, on more than one occa- 

 sion, to two very curious globes that have been brought 

 to light, one in the National Library in Paris, and the 

 other in the Library of Lyons. On the Lyons globe, the 

 date of which is 1701, the Congo is made to issue from a 

 great lake, and wind its way westwards to the Adantic, 

 in a direction to some extent coincident with that recently 

 discovered by Mr. Stanley. As a sort of preparation for 

 the work of the great traveller, so soon to be issued,- some 

 account of the data on which these maps may have been 

 constructed, may not be uninteresting. Our information 

 is based on an article in La Nature, and on a report by 

 a commission of the Lyons Geographical Society, ap- 



pointed to investigate the value and origin of the Lyons 

 globe. 



The discovery made at Lyons is, in reality, no surprise 

 to those who know the history of geographical exploration. 

 Not only in the seventeenth century, does the Zaire- 

 Congo appear on most of the maps with the direction 

 definitely assigned to it by Stanley, but nearly all old 

 documents, from the fifteenth century— and the date 

 should be noted — make the great river issue from a con- 

 siderable mass of water far in the interior of the African 

 continent. 



Already, in the year 1500, the famous mappemonde of 

 Juan de la Cosa, the pilot of Christopher Columbus, gives 

 the same indications ; the picturesque mappetnonde 

 known as that of Henry II., repeats them with some 

 variations, as also the master-work of Mercator (1569), 

 the founder of modern geography. All the old geo- 

 graphers, or nearly all, repeat the same data : — Forlani 

 (1562), Castaldi (1564), Sanuto (1588), Hondius (1607), 

 Nicolas Picart (1644), Bloeu (1569), Sanson, &c. There- 

 fore there need be no surprise to find on a globe of the 

 eighteenth ceutury information which for more than 200 

 years previously had been registered on the map of 

 Africa. 



Whence, however, came this knowledge which our 

 fathers had of certain regions in Central and Equatorial 

 Africa ? The reply is simple ; from the Portuguese, who, 

 since the fifteenth century, undertook not only extensive 

 maritime voyages, but several times crossed Africa from 

 west to east and from east to west. It is even very pos- 

 sible that they discovered the sources of the Nile, the 

 great equatorial lakes ; thus, in the midst of the simplicity 

 and incoherence of their tracings we find, in their old 

 parchments, the great lines of African geography almost 

 as science now represents them. Most of these Portu- 

 guese, with the exception of some missionaries, were but 

 poorly educated ; they travelled much oftener as traders 

 than as experienced explorers ; nevertheless, we have 

 almost the certainty that before the year 1500 they had 

 furnished very precise information on the centre of Africa. 

 In nearly all these maps, and in that of Lyons, the Congo 

 flows in a nearly straight line from Lake Zaire or Zembre 

 to the Atlantic ; it bends only a very little to the north, 

 and does not pass the equator, as we now know it does. 



As a sort of exception, there has been found among the 

 riches of the National Library at Paris, a Spanish globe 

 of copper (without date, but probably between 1530 and 

 1 540), which is not content with presenting the same data, 

 but which reproduces, with wonderful closeness, the 

 course of the Congo as discovered by Stanley. The 

 river issues from a lake, flows towards the north, de- 

 scribes a large curve well to the north of the equator, 

 then turns west-south-west to the Atlantic. This is indeed 

 a summary of the last journey of the intrepid American 

 correspondent. Fig. i gives a perfectly accurate idea of 

 a portion of this valuable globe. 



From all this it must not he concluded that Stanley 

 has discovered nothing new. These discoveries of the 

 ancient travellers, if genuine discoveries they were, seem 

 to have ben forgotten as soon as they were recorded ; and 

 although the maps referred to above have been known 

 for generations, no one ever seems to have taken them as 

 trustworthy guides to the lines of African exploration. 

 Indeed, it is only now that Stanley has made a discovery 

 never to be forgotten that these old maps have come to 

 have a real interest, for we suspect that till now geo- 

 graphers regarded the tracings as having their basis 

 in the cartographers' imaginations. The glory of being 

 really the first discoverers of the two Nyanzas, Nyassa, 

 Tanganyika, Bangweolo, and the course of the Congo 

 cannot be taken away from Speke and Baker and Burton 

 and Livingstone and Stanley ; or if so it must be by some 

 ancient Arab or possibly Egyptian, many, many centuries 

 ago, for there can be no doubt that long before Europe 



