Page:Encyclopædia Britannica, Ninth Edition, v. 15.djvu/660

Rh 628 M A T M A T According to analyses made by Alonzo Robbins it also contains about 1 5 of a peculiar tannin which does not precipitate potassio- tartrate of antimony, nor tan leather. The glutinous substance resembles in consistence common birdlime, and is considered by Byasson to be a compound ether, the alcohol of which would be near cholesterin. Since the beginning of the 17th century mate has been drunk by all classes in Paraguay, and it is now used through out Brazil and the neighbouring countries. In 1855 the amount of mate annually consumed in South America was estimated by Von Bibra at 15,000,000 lb, and the consumption is now probably three or four times as great ; in Brazil it brings in a revenue of about 410,000. In the Argentine Republic alone the consumption is not less than 27,000,000 lt&amp;gt; per annum, or about 13 lb per head, while the proportion of tea and cofl ee consumed is only about 2 tb of the former and i Ib of the latter per head. The export of mate from Brazil to foreign countries has also increased from 2,720,475 kilos in 1340 to 5,206,485 kilos in 1850, 6,808,056 kilos in 1860, 9,507,086 kilos in 1870, and 14,063,731 kilos in 1879-80. See Scully, Brazil, London, 1866 ; Mansfield, Brazil, te., London, 185G ; Phar maceutical Journal (3), &quot;vol. vii. p. 4 ; (3), vol. viii. p. 615, 16-27 ; Christy, New Commercial Plants, No. 3, London, 1880 ; Mulhall, Progress of the World, 1880, p. 488 ; Zeitschrift Oesterreichischer Apothekerverein, 1882, pp. 273, 285, 310. MATERA, a city of Italy in the north-east of the province of Potenza, 48 miles from Potenza, on the high road to Bari. Part of it is built on a level plateau and part in deep valleys adjoining, the tops of the campaniles of the lower portions being on a level with the streets of the upper. The prin- cipaf building is the co-cathedral of the archbishopric of Acerenza and Matera, formed in 1203 by the union of the two bishoprics, dating respectively from 300 and 398. In 1871 the population of the commune was 14,312 (that of the city 14,262), in 1881 15,700. Under the Normans Matera, the ancient Mateola, was a count- ship for William Bras do Fer and his successors. It was the chief town of the Basilicata from 1664 till 1811, when the French trans ferred the administration to Potenza. MATHEMATICAL DRAWING AND MODELLING. The necessity for geometrical drawings and models is as old as geometry itself. The figure has formed the basis of many a geometrical truth; and demonstration by mere inspection of this has frequently to do service for more rigorous proof. So necessary is this visual representation of an idea that there is hardly a branch of mathematics which does not make use of it in the form of tables, symbols, formulae, &c. The visual method is especially important in geometry. The figure is to the geometer what the numerical example is to the algebraist on the one hand limiting the horizon, on the other imparting life to the con ception. Herein lies the didactic value of the figure, which is the more indispensable the more elementary the stage of instruction. To be able to dispense with it is a faculty acquired only after a long and special training. The power of mental picturing is a talent which can be so strengthened by use that even a slightly gifted mind may acquire the power of carrying out a series of geometrical operations without the aid of a figure, provided these do. not lead into unfamiliar regions. But each new group of ideas which the geometer would master requires a new graphic setting forth, which not even the experienced can dispense with. Drawings are sufficient in plane geometry; but solid geometry requires models, except in specially simple cases, in which delineation by means of perspective or some conventional method may suffice. Then, again, in passing from the geometry of the plane, straight line, and point in space to that of curved surfaces, tortuous curves, &c., new and distinct graphical methods are necessary. The difficul ties encountered in understanding new groups of geometrical forms are best removed by a careful study of a small number of characteristic models and drawings. As a means of education, the model is lively and suggestive, forming in this way a completing factor in the course of instruction. We remember the pleasure experienced when, after a discussion which has yielded a series of hardly reconcilable properties of one and the same geometrical figure, a model is exhibited which combined these pro perties in itself; or the striking manner in which a deformable model either of pasteboard or thread executes its transformations before the eye of the observer and scientific student. The study of the model raises new and unexpected questions, and can even do valuable service in leading to new truths. In the more elementary departments of plane and solid geometry and descriptive geometry, models are abundant and easily obtainable; but there are comparatively few collections of drawings and models for instruction in higher geometry. There are numerous drawings of algebraic and transcendental curves in the well-known treatises on analytical geometry of Cramer, Euler, Salmon, in Frost s Curve Tracing, &c. ; but there is still a deficiency in systematic enumerations of the forms of curves and surfaces of a given order or class. In this connexion we may mention Pliicker s System der analytischen Geometric (curves of the third order), and Beer s Enumeratio linearum IV. ordinis. A graphical representation of all the characteristics of the singular points of an algebraic curve of the fourth order is given in Zeuthen s Systemer af plane Kurver (1873). As regards tridimensional figuring, the oldest known models for instruction in the higher geometry are the thread models of skew surfaces constructed about the year 1800 under the direction of G. Monge for the Ecole Polytechnique in Paris. In 1830 Th. Olivier of Paris got the same executed in movable form. The great development in modern times of certain branches especially of the higher geometry has given a new importance to such methods of graphical representation. Amongst the larger collections we must mention the elegant series of complex surfaces, consisting of twenty- seven items, constructed by the celebrated J. PHicker of Bonn. After Pliicker s death copies, not very satisfactory, were made from zinc casts. The collection of plaster and thread models published by Muret of Paris (now Delagrave), and intended for instruction in descriptive geometry, con tains many architectural forms. The wire models of tortuous curves by Professor Wiener of Carlsruhe, and the thread- models of developable surfaces by Professor Bjb rling of Lund, merit notice amongst others. Perhaps the largest and most extensive of the collections is that of L. Brill, bookseller in Darmstadt. These represent every depart ment of the higher and applied mathematics. The cata logue embraces some seventy numbers, with over a hundred plaster, thread, and metal models. Several series of this collection were prepared in the mathematical department of the technical college of Munich. In the preparation of these models, involving the development of a comparatively novel art, certain practical lessons were gained, especially in the working of plaster models, to which we may direct attention. We assume that the preliminary designs are prepared with the aid of board, ruler, square, compasses, and such well-known instru ments as are used by the draughtsman. The material to be employed, whether wire or thread, interlaced pasteboard strips, or plaster, depends upon the special circumstances of each case, and upon the purpose aimed at in the construction of the model. Two bundles of parallel disks of cardboard_or metal- sheeting, inclined at an adjustable angle, may be used with advan tage in representing a series of different but mutually transformable surfaces. For ruled and developable surfaces the thread model is to be recommended. The surface is enclosed in a cube, or more generally in a region of space bounded by plane walls. _Upon these bounding walls are marked the series of points in which they are cut by the generative lines that are to be represented by threads. Through these points the threads are drawn, and parts of the sup porting walls are then cut away so as to allow a convenient glance into the interior of the region. The more densely the threads are strung, the liker is the appearance to that of a continuous surface. In the majority of cases plaster will be found the most convenient substance, being easily worked, and giving a result convenient and clear to the eye. There is the disadvantage, however, that one of the regions of space bounded by a surface is filled up. Should the