The mathematical subject in which the student usually has to contend most frequently with unknowns at the beginning of his studies is the history of mathematics. The ancient Greeks had already attempted to trace the development of every known concept, but the work along this line appears still in its infancy. Even the development of our common numerals is surrounded with many perplexing questions, as may be seen by consulting the little volume entitled "The Hindu-Arabic Numerals," by D. E. Smith and L. C. Karpinski.
The few mathematical unknowns explicitly noted above may suffice to illustrate the fact that the path of the mathematical student often leads around difficulties which are left behind. Sometimes the later developments have enabled the mathematicians to overcome some of these difficulties which had stood in the way for more than a thousand years. This was done, for instance, by Gauss when he found a necessary and sufficient condition that a regular polygon of a prime number of sides can be constructed by elementary methods. It was also done by Hermite, Lindemann and others by proving that epsilon and rho are transcendental numbers. While such obstructions are thus being gradually removed some of the most ancient ones still remain, and new ones are rising rapidly in view of modern developments along the lines of least resistance.
These obstructions have different effects on different people. Some fix their attention almost wholly on them and are thus impressed by the lack of progress in mathematics, while others overlook them almost entirely and fix their attention on the routes into new fields which avoid these difficulties. A correct view of mathematics seems to be the one which looks at both, receiving inspiration from the real advances but not forgetting the desirability of making the developments as continuous as possible. At any rate the average educated man ought to know that there is no mathematician who is able to solve all the mathematical questions which could be proposed even by those having only slight attainments along this line.
THE ABORIGINAL ROCK-STENCILLINGS OF NEW SOUTH WALES
BY DR. CHAS. B. DAVENPORT
COLD SPRING HARBOR, N. Y.
IN a number of places in eastern Australia curious aboriginal markings are found on the faces of the sandstone cliffs. A good idea of them is given by the photographs. These came from Wolgan Gap near Wallerang in the Blue Mountain region of New South Wales. They are found on overhanging rocks that have served as shelters or camping places for the aborigines and which doubtless have protected their works of art.
These stencillings are made by a sort of spatter work, something like that in vogue a generation ago in this country, using leaves, etc., as forms. The rocks at Wolgan Gap are a coarse sandstone stained almost black by an iron oxide derived from included bands of ironstone. These black surfaces were selected by the artists. Nearby in the rock is a band of shale which had disintegrated at its exposed edge to a white powder. The native artist put some of this white powder in his mouth, placed his hand or foot upon the rock, and blew the moistened powder upon and around his outstretched fingers or toes. When he removed them they were outlined on the rock. Since the sandstone is coarse and deeply pitted, the moist powder was blown into minute cavities where it has remained despite the erosive activities of some generations. The presence of the powder is shown on the photographs as a sort of halo around the object. The hands are either right or left, and, in some cases, both hands seem to have been stencilled at once. Sometimes the whole arm and hand are stencilled together, and in one of the photographs a boomerang is shown. The age of these stencils is not known. They were first discovered at Wolgan Gap about sixty years ago, but others have been known for a longer time, for instance, those at Greenwich, Parametta River, near Sydney.
The significance of these stencillings has been the subject of some controversy. The natives may have been induced to make them as boys carve their names on benches or even rocks. The materials for making the stencillings were present and, the example once having been set, others would emulate it. It is interesting that similar stencillings of the hands were made by cave men on the walls of some of the European caves, as, for instance, those of Aurignac in southern France. Evidently spatter work is no modern pastime.