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 versed in ancient and modern geometry, but also had a full knowledge of the analytical methods and discoveries of the continental mathematicians. His earliest memoir, dealing with an analytical expression for the rectification of the ellipse, is published in the Transactions of the Royal Society of Edinburgh (1796); and this and his later papers on “Cubic Equations” (1799) and “Kepler’s Problem” (1802) evince great facility in the handling of algebraic formulae. In 1804 after the dissolution of the flax-spinning company of which he was manager, he obtained one of the mathematical chairs in the Royal Military College at Marlow (afterwards removed to Sandhurst); and till the year 1816, when falling health obliged him to resign, he discharged his professional duties with remarkable success. During this period he published in the Philosophical Transactions several important memoirs, which earned for him the Copley medal in 1814 and ensured his election as a Fellow of the Royal Society in 1815. Of special importance in the history of attractions is the first of these earlier memoirs (Phil. Trans., 1809), in which the problem of the attraction of a homogeneous ellipsoid upon an external point is reduced to the simpler case of the attraction of another but related ellipsoid upon a corresponding point interior to it. This theorem is known as Ivory’s theorem. His later papers in the Philosophical Transactions treat of astronomical refractions, of planetary perturbations, of equilibrium of fluid masses, &c. For his investigations in the first named of these he received a royal medal in 1826 and again in 1839. In 1831, on the recommendation of Lord Brougham, King William IV. granted him a pension of £300 per annum, and conferred on him the Hanoverian Guelphic order of knighthood. Besides being directly connected with the chief scientific societies of his own country, the Royal Society of Edinburgh, the Royal Irish Academy, &c., he was corresponding member of the Royal Academy of Sciences both of Paris and Berlin, and of the Royal Society of Göttingen. He died at London on the 21st of September 1842.

A list of his works is given in the Catalogue of Scientific Papers of the Royal Society of London.

 IVORY (Fr. ivoire, Lat. ebur), strictly speaking a term confined to the material represented by the tusk of the elephant, and for commercial purposes almost entirely to that of the male elephant. In Africa both the male and female elephant produce good-sized tusks; in the Indian variety the female is much less bountifully provided, and in Ceylon perhaps not more than 1% of either sex have any tusks at all. Ivory is in substance very dense, the pores close and compact and filled with a gelatinous solution which contributes to the beautiful polish which may be given to it and makes it easy to work. It may be placed between bone and horn; more fibrous than bone and therefore less easily torn or splintered. For a scientific definition it would be difficult to find a better one than that given by Sir Richard Owen. He says: “The name ivory is now restricted to that modification of dentine or tooth substance which in transverse sections or fractures shows lines of different colours, or striae, proceeding in the arc of a circle and forming by their decussations minute curvilinear lozenge-shaped spaces.” These spaces are formed by an immense number of exceedingly minute tubes placed very close together, radiating outwards in all directions. It is to this arrangement of structure that ivory owes its fine grain and almost perfect elasticity, and the peculiar marking resembling the engine-turning on the case of a watch, by which many people are guided in distinguishing it from celluloid or other imitations. Elephants’ tusks are the upper incisor teeth of the animal, which, starting in earliest youth from a semi-solid vascular pulp, grow during the whole of its existence, gathering phosphates and other earthy matters and becoming hardened as in the formation of teeth generally. The tusk is built up in layers, the inside layer being the last produced. A large proportion is embedded in the bone sockets of the skull, and is hollow for some distance up in a conical form, the hollow becoming less and less as it is prolonged into a narrow channel which runs along as a thread or as it is sometimes called, nerve, towards the point of the tooth. The outer layer, or bark, is enamel of similar density to the central

part. Besides the elephant’s tooth or tusk we recognize as ivory, for commercial purposes, the teeth of the hippopotamus, walrus, narwhal, cachalot or sperm-whale and of some animals of the wild boar class, such as the warthog of South Africa. Practically, however, amongst these the hippo and walrus tusks are the only ones of importance for large work, though boars’ tusks come to the sale-rooms in considerable quantities from India and Africa.

Generally speaking, the supply of ivory imported into Europe comes from Africa; some is Asiatic, but much that is shipped from India is really African, coming by way of Zanzibar and Mozambique to Bombay. A certain amount is furnished by the vast stores of remains of prehistoric animals still existing throughout Russia, principally in Siberia in the neighbourhood of the Lena and other rivers discharging into the Arctic Ocean. The mammoth and mastodon seem at one time to have been common over the whole surface of the globe. In England tusks have been recently dug up—for instance at Dungeness—as long as 12 ft. and weighing 200 ℔. The Siberian deposits have been worked for now nearly two centuries. The store appears to be as inexhaustible as a coalfield. Some think that a day may come when the spread of civilization may cause the utter disappearance of the elephant in Africa, and that it will be to these deposits that we may have to turn as the only source of animal ivory. Of late years in England the use of mammoth ivory has shown signs of decline. Practically none passed through the London sale-rooms during 1903–1906. Before that, parcels of 10 to 20 tons were not uncommon. Not all of it is good; perhaps about half of what comes to England is so, the rest rotten; specimens, however, are found as perfect and in as fine condition as if recently killed, instead of having lain hidden and preserved for thousands of years in the icy ground. There is a considerable literature (see ) on the subject of big-game hunting, which includes that of the elephant, hippopotamus and smaller tusk-bearing animals. Elephants until comparatively recent times roamed over the whole of Africa from the northern deserts to the Cape of Good Hope. They are still abundant in Central Africa and Uganda, but civilization has gradually driven them farther and farther into the wilds and impenetrable forests of the interior.

The quality of ivory varies according to the districts whence it is obtained, the soft variety of the eastern parts of the continent being the most esteemed. When in perfect condition African ivory should be if recently cut of a warm, transparent, mellow tint, with as little as possible appearance of grain or mottling. Asiatic ivory is of a denser white, more open in texture and softer to work. But it is apt to turn yellow sooner, and is not so easy to polish. Unlike bone, ivory requires no preparation, but is fit for immediate working. That from the neighbourhood of Cameroon is very good, then ranks the ivory from Loango, Congo, Gabun and Ambriz; next the Gold Coast, Sierra Leone and Cape Coast Castle. That of French Sudan is nearly always “ringy,” and some of the Ambriz variety also. We may call Zanzibar and Mozambique varieties soft; Angola and Ambriz all hard. Ambriz ivory was at one time much esteemed, but there is comparatively little now. Siam ivory is rarely if ever soft. Abyssinian has its soft side, but Egypt is practically the only place where both descriptions are largely distributed. A drawback to Abyssinian ivory is a prevalence of a rather thick bark. Egyptian is liable to be cracked, from the extreme variations of temperature; more so formerly than now, since better methods of packing and transit are used. Ivory is extremely sensitive to sudden extremes of temperature; for this reason billiard balls should be kept where the temperature is fairly equable.

The market terms by which descriptions of ivory are distinguished are liable to mislead. They refer to ports of shipment rather than to places of origin. For instance, “Malta” ivory is a well-understood term, yet there are no ivory producing animals in that island.

Tusks should be regular and tapering in shape, not very curved or twisted, for economy in cutting; the coat fine, thin, clear and transparent. The substance of ivory is so elastic 