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and obtuse angles; the largest are four hundredths of a millimetre, perfectly transparent, but distinctly marked edges. They are identical in the yolks in the process of formation, and in the ovarian vesicles, and whatever the size of these ovules from those which are only Om 01 in diameter, to the largest which are 0m-03. In the smaller ovules with diameter varying from 0m-001 to 0.005, the grains have the same tabular shape ("en tablettes,") but they were much smaller, and did not exceed two hundredths of a millimetre in length. Generally these grains are of one dimension in each ovule. But the differences we are about to point out, show that the grains enlarge with the development of the ovules, and that the vitellus, when they are little developed, has much smaller grains of ichthin than with those which are nearer the oviduct or more in the egg. The Ray from which these yolks were taken were hardly Om-50 long not counting the tail, with a weight of 4 or 5 kilogrammes. In our many examinations of different grains of ichthin, we met occasionally with some little tables almost square, others regular or irregular pentagons; these grains have a tendency to separate. We have not yet been able to see whether these different forms are due to some constant cause, or whether they are owing to simple accidental variations, so common in even the most elementary natural productions. We tried to crush these grains in an agate mortar, and found that generally they break according to the axes of the rectangles of these 'tablettes,' and not according to their diagonals. We studied the grains of yolk developed in the largest of our Ray such as in our markets go under the name of the soft or white Ray. This is the Raia oxyrhynchus of Linnæus. We must not forget to remark that individuals of this kind of Ray are even two metres long, not counting the tail, that they attain the weight of 100 kilogrammes, and yet the eggs of this Ray give the smallest grains of ichthin. Those of the spiked Ray (Raia fullonica,) and those of "la raie ronce" (Raia rubus,) are very much like those of "la raie bordée;" the most noticeable difference consists in their smaller dimensions. The largest are only three hundredths of a millimetre. In the eggs of these two sorts of Ray, the grains are very often regular ellipses, but the rectangular form is still more common. The vitellin grains of the marbled torpedo (Torpedo marmorata,) from La Rochelle, are very different in shape from those of the Ray, the former being elliptical or circular: there are no rectangular grains; their transparency and their other physical properties are the same. They are only two hundredths of a millimeter, but it is not to be forgotten that the Torpedos are never farge. Ichthin of sharks is in larger grains, more elongated than those of the Ray, and of a very long oval shape. We have noticed, too, some variations in the form. In a careful microscopic ex

amination, we saw one of these ovoid grains pointed at both ends. Another had the two long straight sides, terminating in two isoceles triangles; it was the figure of an elongated hexagon. The hound-fish (Squalus mustelus, Lin.), smaller than the "Melandre," has grains of ichthin almost as large as those of the latter. They are five hundredths of a millimetre; their form is different from all the others. These grains are round, but often united in the most varied forms. The Bounce (la ronsette, Squalus canicula, Lin.) has rectangular grains and obtuse angles, very like those of the Ray; their longer side is four hundredths of a millimeter. The angel-fish (l'ange, Squatina angelus, Dum.) has as large grains as the Squalus mustelus; they are elliptical like those of the shark, and as large, for they are six hundredths of a millimeter. We conclude then, from the comparison of the forms of these grains in the different species cited, that the oviparous species like the Ray and Bounce, have more or less rectangular grains, and are very much alike, while the Cartilaginous viviparous species, like Torpedos and Sharks, generally have oval grains; that if the development of the ovula influences the largeness of the grains of ichthin, the size of the fish has no effect on the size of the grains.

The grains of ichthin are insoluble in water, alcohol and ether; they are completely transparent, and do not become opaque by being kept even for a long time, in boiling water; hydrochloric acid dissolves them without producing a violet color: the two latter properties clearly establish the difference between ichthin, albumen and vitellin. All the concentrated acids dissolve ichthin; when they are dilute, they do not act upon it, excepting acetic and phosphoric acids, which immediately dissolve grains of ichthin, even when greatly diluted with water. Solutions of potash and soda are slow solvents of ichthin. It is insoluble in ammonia. Grains of ichthin when burnt, leave no ashes. When the ease with which ichthin can be obtained from the eggs of certain fish is considered, and when it is seen that grains of ichthin, by the regularity of their form, offer all the characteristics of a really pure principle, it is impossible not to consider it as one of the most interesting substances in the animal organization. Ichthin, when analyzed, gives the following composition:

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Its centesimal composition will then be :

C 510 H 6-7 Az 15.0 Ph 1.9

O 25.4

It is easy to believe, that these regularly formed tables are small crystals. To remove doubts in this respect, we have had recourse to the kindness of M. de Sénarmont, who examined our grains. with a polarizing apparatus. This proved to him and to us, that the grains of ichthin are not crystallized.

ART. VIII.-The Arabic or Indian Method of Notation; by THOMAS H. MCLEOD.

THE subject of arithmetical notation in its relation particularly to the Arabic or Indian, the Roman, and Grecian systems, has arrested the attention of mathematicians from the earliest period of modern mathematical investigation; but the mechanical structure, especially of the Indian, seems to have been overlooked, as well as the probable circumstances under which that structure originated. Barlow, in his Theory of Numbers, presents the following equation, N=ar" br"−1 + cr"−2+ &c. . . . . pr2 + qr+w,

where may be any number whatever, and a, b, c, &c. integers less than r, as expressing the scheme of the Indian method. It is undoubtedly a formula by which that method may be explained; but that it exhibits its simple primitive mechanical structure, may be justly questioned. For in the first place it does not show the first position of 0 (zero) with any reliable certainty: the only place we are left to refer it, from the explanations, is to 10, where it appears on the right of 1, which we apprehend is not its first place; secondly, r appears in the form of a power, which is unquestionably true, but accidental; thirdly, it is asserted that r may be any number whatever, yet upon examination it will be found always to assume the form 10, whatever be its significance in the deuary measure; and finally, it cannot be in any manner supposed that the scheme had its origin in philosophy, as the formula would seem to indicate, but that it arose out of the circumstances and necessities of the people from whom it sprung.

Having finished these few restrictions, of which more might be made, concerning what mathematicians have said upon this subject, we shall proceed to explain what we consider to be the simple primitive structure of the Indian method, and afterwards allude by way of comparison to that of the Grecian and Roman methods.

The first peculiarity of the Indian method is its 0 (zero), which stands as the origin of the scheme, as will be seen by writing it out thus:

0, 1, 2, 3, 4, 5, 6, 7, 8, 9,

where it evidently appears in its first place. The next characteristic worthy of note, is the 10, which does not appear among the

above figures; it is evidently made up of 1, and 0, but how from this circumstance does it get its significance? We conceive the way to be this: It is well known to land-surveyors and other lineal measurers, that the place where the measuring commences is but a point and has no lineal significance, and that the first unit of measure is at the distance of a unit from that point, and the second at two measures, and the third three, &c. ; i. e. in measuring land the first pin is stuck at the distance of one chain, or measure, from the place of beginning, the second at the distance of two chains, the third at the distance of three chains, and so on. If then the place of beginning be represented by 0, and the several distances respectively by 1, 2, 3, 4, &c., the whole will be properly expressed. But when the pins are all exhausted and a tally is to be made, how is it to be done? Very naturally by placing down a 1, and a 0 (zero) (which has no lineal significance) to the right; the fact is thus recorded, and will read, one tally and no more; at the distance of one small measure (or one chain) from this point, a 1 tally and 1 chain will be marked down (11) and the expression will read one tally, and one chain more; and so on to the second tally, which will be made and recorded by a 2 and a 0, to the right (20), which will read two tallys and no more. At the distance of one small measure or unit from this place there will be recorded 2 tallys and 1 chain more, or 21, which will read two tallys and one chain; at the distance of two small measures, the record will be 22, which will read two tallys and two chains, and so on.

This we conceive to be the simple structure and probable origin of the Indian method. In its structure it is strictly geometrical, using that term in its primitive signification.

It will at once be perceived that the tally is not necessarily made at one small measure beyond 9, but that it may be made at any measure before or beyond that place; but in each case the point of repetition will always be expressed by 10, and this from the fact that it necessarily reads one principal or large measure and no more. This verifies the statement before made that r, in Barlow's formula, will always be expressed by 10, and that its appearing as a power is accidental. The whole subject will be clearly exhibited by the following Table:

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It will here be seen that the repetitions do not necessarily begin at one measure beyond 9 but may begin at any measure before or after that point. It will also be seen how 10 obtains its significance.

The Grecian Method, was also a method of large and small measures, but it had no 0 (zero), 1 (a unit) being the first figure in the scheme, whence, it evidently had its origin in the consideration of individual objects and not in the measure of distances, i. e. it is arithmetical, using that term also in its primitive sense. It employs the letters of the Greek alphabet for its characters, the first letter representing a unit, the second letter two units, &c., to ten, which is expressed by (iota), the tenth letter of the alphabet; with this character the repetitions begin. A new character is introduced at each repetition, as at twenty, thirty, and so on to one hundred, to represent which a new character is added, when the whole is repeated; new characters are added for each hundred afterwards, to one thousand, which is also represented by a new character, as well as ten thousand, and one hundred thousand, &c. The scheme seems then to be simply this: one, ten and one, twenty, twenty and one,-one hundred, one hundred and one,-one thousand, one thousand and one, &c. It is seen, contrary to what has been stated, that any number could be expressed by this method, all that was necessary being to introduce a new character at the end of the proper repetition. If the Greeks did not express any number beyond 100,000,000, it was because they either did not understand the scheme of their notation, or because at this point it became unwieldy; the latter was probably the case. It will be also seen that the repetition can commence and be carried on with any number whatever.

The Roman Method, is rather a method of fives than of tens. It begins its repetitions with five, introducing a new character to express that number. It introduces new characters to express ten, fifty, one hundred, five hundred, one thousand, in each case making use of the previous figures in connection with the new ones to express the numbers beyond them. Thus :

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which evidently reads, five, five and one, ten, or two fives-for so do some at least explain the figures-two fives and one, three fives, three fives and one, four fives, four fives and one, &c., fifty, fifty and one, one hundred, one hundred and one, five hundred, five hundred and one, one thousand, one thousand and one, &c. It is also evident that it had its origin in contemplating and recording individual objects like the Grecian method, as like it, it

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