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ARITHMETIC AND ALGEBRA.

As it is usually found that the blind have a strong inclination to cultivate their reflective powers, it is by no means surprising to observe the fondness they generally display for arithmetical calculations. It is undoubted that under ordinary circumstances the practice of mental arithmetic should be especially encouraged, and that this mode of calculation is sufficient to meet the wants of the blind as a class. Nevertheless, there is evidence to show that in the more advanced branches of the science of arithmetic, persons without sight need the assistance of a slate, adapted to their wants, almost as much as those who are enabled to see, and we accordingly find that various efforts have been made to supply this necessity. The first known attempt to construct a tangible arithmetical apparatus was made by the great Saunderson, early in the eighteenth century, who, finding, doubtless, that the abstruse calculations in which he constantly engaged, made a great strain on his memory, devised an appliance which enabled him to perform and record all the operations connected with arithmetic and algebra; and as this contrivance was the parent of all that followed it, and as it was introduced by the greatest blind mathematician who ever lived, we shall proceed to give some explanation of its details, taken from the 'Encyclopædia Britannica ' article" Blind :”

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Imagine a square divided into four equal parts by perpendicular lines at the sides, in such a manner that it may present the nine points, 1, 2, 3, 4, 5, 6, 7, 8, 9. Suppose this square pierced with nine holes capable of receiving pins of two kinds, all of equal length and thickness, but some with heads a little larger than the others. The pins with large heads are never placed anywhere else but in the centre of the square; those

with smaller heads, never but at the sides, except in one single case, which is that of making the figure 1, where none are placed at the sides. The sign of 0 is made by placing a pin with a large head in the centre of the little square, without putting any other pin at the sides. The number 1 is represented by a pin with a small head placed in the centre of the square, without putting any other pin at the sides; the number 2, by a pin with a large head placed in the centre of the square, and by a pin with a small head placed on one of the sides; the number 3, by a pin with a large head placed in the centre of the square, and by a pin with a small head placed on one of the sides; the number 4, by a pin with a large head placed in the centre of the square, and by a pin with a small head placed on one of the sides; the number 5, by a pin with a large head placed in the centre of the square, and by a pin with a small head placed on one of the sides; the number 6, by a pin with a large head placed in the centre of the square, and by a pin with a small head placed on one of the sides; the number 7, by a pin with a large head placed in the centre of the square, and by a pin with a small head placed on one of the sides; the number 8, by a pin with a large head placed in the centre of the square, and by a pin with a small head placed on one of the sides; the number 9, by a pin with a large head placed in the centre of the square, and by a pin with a small head placed on one of the sides." Besides the appliance contrived by Saunderson, the writer is acquainted with twelve other inventions for the same object; in all of these, however, the figures and other signs are represented by pegs or pins (of wood, bone, or metal), being placed into holes in a board, or cushion, as required by the operator. In some cases the pegs employed have only a character or sign at one end, and in others both ends of the pegs are so employed; in some inventions also the holes in the board are square, in others pentagonal, and in others octangular; and in

some contrivances the figures are represented simply by common pins, with heads and sharp points, which are thrust in different positions into an ordinary pincushion. In general, a number of little boxes are attached to the board to receive the different kinds of pegs, which have to be carefully sorted into the boxes every time they have been used. In connection however, with the pentagonal and octangular systems, only one kind of peg is employed, and consequently no sorting is required. The pentagonal mode was adopted at Edinburgh and Glasgow, forty years since, and the octangular was invented by the Rev. W. Taylor about the year 1852. Whenever in any of the above inventions more characters are required for algebra, additional pegs are used.

As a detailed description of each of the methods of tangible arithmetic would be altogether superfluous, we will merely give as a sample the following explanation of the pentagonal system: "The board generally used is 16 inches by 12, and contains 400 pentagonal holes with a space of a quarter of an inch between each. The pin is simply a pentagon with a projection at one end on an angle, and on the other end on the side. Being placed in the board, with a corner projection to the left upper corner of the board, it represents 1; proceeding to the right upper corner it is 3; the next corner in succession is 5; the next 7, and the last 9. In like manner the side projection, by being turned to the sides of the hole progressively gives 2, 4, 6, 8, 0."

The writer having found that none of the contrivances to which reference has been made really gave to the blind what was desirable in a system of tangible arithmetic, constructed about the year 1860 an apparatus, which will be fully described in the Appendix, and of which it need only be said here that it is sold at half the price of any similar appliance giving anything like the same advantages. Before quitting this subject,

it may be as well to remark that very good common English figures may be made in gutta percha; the pegs themselves should be of gutta percha, and on each end should be a figure in relief. An arithmetical appliance on this plan was made by the writer some years ago, which was very useful in enabling a sighted nurse to instruct her blind charge in the first principles of calculation. The figures were half an inch deep and three-eighths wide, and the pegs were an inch long.

TANGIBLE MAPS AND GLOBES, ETC.

Various attempts have been made to form suitable maps for the blind, the earliest recorded effort being that of Herr Weissemborg, of Mannheim. This German gentleman having lost his sight when seven years old, and finding on reaching maturity no means provided for the education of the blind, set about contriving a number of appliances to carry out that object. Among these inventions were a writing-desk, the fundamental principle of which is still the basis of similar contrivances, and an arithmetical apparatus resembling, it is said, that by Saunderson. Herr Weissemborg also conceived the idea of forming tangible maps, for which purpose he availed himself of the embroiderer's art; the boundaries being represented by narrow lines of work, the mountains by thicker lines, and towns and cities by glass or steel beads of various sizes. These maps were moderately effective, but their costliness prevented their being widely used. Since 1780 many and varied have been the exertions made in almost every country in Europe and in the United States of America, to give the sightless a cheap and easy means of studying Physical Geography; but we regret to say that the results of these efforts have proved far from satisfactory. This want of success, however, has not been caused by any inherent difficulty in the case.

but has proceeded simply, as it would appear, from the caterers not sufficiently understanding the wants of the blind in connection with tangible maps, and the means best adapted to meet those wants.

Efficient tangible maps, in addition to possessing the indispensable requisite of cheapness, must also contain the following qualities:-the land and water must be readily distinguished from each other, the courses of rivers and mountains easily traced, and the situation and relative importance of towns accurately indicated; they should also be capable of being consulted by the blind student without sighted aid. The various maps invented naturally divide themselves into two classes, viz. those which are made by hand and those printed from relief plates. In the former class, besides those made by embroidery and needlework, may be mentioned such as are formed by having the shapes of countries, etc. cut out in wood, and such as are made by having the land of an ordinary map used by the sighted, pasted upon the land of another similar map, thus causing the whole surface of the land to be raised above the water, and also enabling a sighted person easily to teach the blind.

In these maps mountains and towns are shown by brass-headed pins of different sizes, which are put through the map and clinched at the back. Maps printed in relief from plates are necessarily much cheaper than those made by hand. In general the boundaries in printed maps are indicated by raised lines, and the towns marked by dots, no distinction being usually made between land and water. those, however, printed by Dr. Howe, of Boston, U.S., the water is raised above the land, which injurious practice has been imitated by Mr. Moon, of Brighton, in some of his productions.

In

The idea of water being raised above land is so unnatural, that the mind never thoroughly becomes accustomed to it, and this produces constant irritation