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* TRANSVAAL. 42S TRANSVERSAL. turers of all nations. In the Witwatersrand was founded the populous mining city of Jo- hannesburg, inhabited very largely by these Uit- landers, or outlandcrs. The Boers felt that the primitive life they had wished to preserve was invaded, circumscribed, and likely to be over- whelmed. They therefore sought to restrict the privileges of citizenship, in order to retain the political control in their own hands. This led to constant friction and to attempts to secure Brit- ish intervention. It is impossible in the limits of this article to discuss the many points of dif- ference which brouglit to a head the troubles be- tween Great Britain and the Republic. In the autumn of 1805 a plan was arranged between the leaders of the British South Africa Company — Cecil Rhodes (q.v.) .his coIleaOTe, Mr. Beit, and Dr. Jameson, administrator of Rhodesia — on the one hand, and several leaders of the Uitlanders — LionelPhillips.CharlesLeonard. and .John Hays Hammond re[iresenting them — on the other, for an armed raid into the Transvaal from Rhodesia, for the purpose of bringing about by a display of force the reforms desired. Dr. Jameson made it understood finally that he should act in be- half of the British supremacy. The .Johannes- burg committee did not api)arently intend to overthrow tlie Government, and when this was known sought to staj' action while they issued a manifesto calling for the desired reforms. Jame- son, disregarding their warning, entered the Transvaal December '29th with 600 men. He was defeated, surrounded, and obliged to surrender (January 2, 1896). The Transvaal Govern- ment turned the prisoners over to the Brit- ish Government for trial. They were con- victed in England and received light sen- tences. Four of the Johannesburg leaders were condemned to death by the Transvaal courts, but this was commuted to a heavy fine. The raid caused a great excitement and assumed inter- national importance. It brought the agitation and the bitter feeling between the two countries to an acute stage. The controver.sy, concealing under diplomatic phraseology much irritation on both sides, continued for some time longer. Mr. Chamberlain, the English Colonial Secretary, seemed determined to force the position of the Republic, and President Kruger was obstinate in his refusal to yield any important point. In the autumn of 1899, when war seemed imminent, the Orange Free State decided to make common cause with the South African Republic. On October 9th an ultimatum which made peace and continued negotiation impossible was presented to the Brit- ish agent at Pretoria and Mr. Chamberlain sum- marily closed the correspondence. For the sub- sequent history of the Transvaal, see South African War. Bibliography. Aylward, The Transvaal of To-Day (Edinburgh, 1881); Wangemann, Siid- afrika vnd seine liewohner (Berlin, 1881) ; Al- brecht. La rcpublique siidafricaine (Brussels, 1889) ; Kliissel, Die Yerfassung und Verwaltung dcr siidafrikanischen Freistaaten (heipzig, 1896) ; Abraham, Die siidafrikanisehe Republik (Berlin, 1896) ; Statham, fioiifh Africa as It Is (London, 1897) ; Seidel. Traiisrnal, die siidafrikanische Republik (Berlin, 1898); Bryce, Impressions of South Africa (London, 1899) ; Younghusband, South Africa To-Day (ib.. 1899) ; Tangye, In New South Africa (ib., 1900) ; Keane, The Doer Slates, Land, and People (ib., 1900) : Col- quhoun, Renascence of South Africa (ib., 1900) ; Belloch, The Xew South Africa (ib„ 1091) ; Le- clerq, Les Boers et les oripines dcs rcpubliques sudafricaines (Brussels, 1901); Alford, Geologi- cal Features of the Transvaal (London, 1891); Distant, A l'aturalist in the Transvaal (ib., 1892) ; Pollack, Les mines d'or du Transvaal (Paris, 1894) ; Goldman, South African Mines (London, 1895-96) ; Talbert, Kach den Trans- vaal-Goldfeldcrn (Berlin, 1896). History. Theal, History of South Africa (London. 1887- 93), the fullest and most reliable history of South Africa as a whole; id.. History of the ISoers in South Africa (ib., 1887) ; Cloe'te, History of the Great Boer Trek (ib., 1899); Stathani, Paul Kruger and His Times (ib., 1898) ; Voigt, Fifty Years of the History of the Republic of South Africa, 170o-lS',5 (ib., 1899) ; Aubert, Le Trans- vaal et I'Angletcrre en Afrique du Sud (Paris, 1899) ; Bryce, Brooks, and others. Both Sides of the South African Question (New York, 1900); Creswicke, South Africa and the Transvaal War (Edinburgh, 1900); Fisher, The Transvaal and the Boers (London, 1900) ; JMahan, The War in South Africa (ib., 1900) ; Dovle, The Great Boer War (ib., 1900) ; Ireland, The Anglo-Boer Con- flict (ib., 1900), a concise account, pro-British, but moderate and not unfair; Ogden, The War Against the Dutch Republics (Manchester, 1901). TRANSVERSAL (ML. transversalis, from Lat. tran-svcrsus, traversus, transverse, p. p. of transvertere, to cross, transverse, from trans, across, through + vertere. to turn ). In geome- try, a term commonly applied to a line cutting a pencil of parallels. In modern geometry the term is extended to mean any straight line cutting the other lines of a figure. Thus any line intersect- ing the three lines forming a triangle ABC in P, Q, R, is a transversal of the triangle. The theory of transversal is one of the most impor- tant in modern geometry. It has its origin in a theorem attributed to Ptolemy (q.v.), but which is found in the Spherics of IVIenelaus (q.v.), and which has been thought to go back to Hipparchua (q.v.). This states that a straiglit line drawn arbitrarily in the plane of a triangle determines on the lines of its sides six segments such that the product of three not having a common ex- tremity equals the product of the other three. Pappus (q.v.) in his Collections approaches the theory from another standpoint and shows that if a pencil of four lines be cut by a transversal in the points A, B, C, D, the ratio -—=• = - is AD BD constant for any position of the transversal. Pappus also showed that if a transversal cuts the sides and diagonals of a complete quadri- lateral, the si.x segments determined on this