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LESSONS IN ARITHMETIC.-XIX.

SQUARE AND CUBE ROOT (continued).

9. THE square root of a fraction is obtained by taking the square root of the numerator for a numerator, and the square root of the denominator for a denominator. This follows at once from the consideration that the multiplication of fractions is effected by multiplying the numerators for a numerator, and the denominators for a denominator. When either the numerator or the denominator is not a complete square, in which case the fraction itself evidently has no exact square root, instead of finding an approximate root of both numerator and denominator in decimals, and then dividing one by the other, it will be better first to reduce the fraction to a decimal, and then to take the square root.

EXAMPLE.-To find the square root of 2.

Reducing to a decimal, we find it to be 285714 (see Lesson XVI., Art. 21).

Hence we should find by the previous method the square root of 28571428571428... to as many decimal places as we please, by continually taking in more and more figures of the recurring periods.

Similarly, in finding the square root of , we should proceed thus: 4, and then find the square root of 400000, etc., to as many places as we please.

=

Obs. It does not follow that because the numerator and denominator of a fraction are not complete squares, that the fraction has no square root; for the division of numerator and denominator by some common measure may reduce them to perfect squares. Thus, 2, when numerator and denominator are divided by 7, gives, the square root of which is . A fraction must be reduced to its lowest terms io determine whether it be a complete square or not.

10. Abbreviated Process of Extraction of Square Root. When the square root of a number is required to a considerable number of decimal places, we may shorten the process by the following

Rule for the Contraction of the Square Root Process.

Find by the ordinary method one more than half the number of figures required, and then, using the last obtained divisor as a divisor, continue the operation as in ordinary long division. EXAMPLE.-Find the square root of 2 to 12 figures.

2-0000, etc. (1·414213 | 56237

1

24) 100

96

281) 400

281

2824) 11900

11296

28282) 60400 56564

282841) 883600 282841

2828423) 10075900 8485269

15906310

14142115

17641950 16970538

6714120 5656846

10572740

8485269

20874710

19798961

1075749

Here, having obtained by the ordinary process the first seven

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To extract the cube root of a given number is the same thing as resolving it into three equal factors.

As in the case of the square root, we must content ourselves with giving, without explanation of the reason of its truth, the Rule for the Extraction of the Cube Root of a given number. Mark off the given number into periods of three figures each, by placing a point over the figure in the unit's place, and then over every third figure to the left (and to the right also, if there be any decimals). Put down for the first figure of the root the figure whose cube is the greatest cube in the first period, and subtract its cube from the first period, bringing down the next period to the right of the remainder, and thus forming a number which we shall call a dividend. Multiply the square of the part of the root already obtained by 3 to form a divisor, and then, having determined how many times this divisor is contained in the dividend without its two right-hand figures, annex this quotient to the part of the root already obtained.* Then determine three lines of figures by the following processes :

1. Cube the last figure in the root.

2. Multiply all the figures of the root except the last by 3, and the result by the square of the hust.

3. Multiply the divisor by the last figure in the root.

Set down these lines in order, under each other, advancing each successively one place to the left. Add them up, and subtract their sum from the dividend. Bring down the next period to the right of the remainder, to form a new dividend, and then proceed to form a divisor, and to find another figure of the root by exactly the same process, continuing the operation until all the periods are exhausted.

13. In decimals, the number of decimal places in the cube root will be the same as the number of points placed over the decimal part, i.e., as the number of periods in the decimal part. Obs.-If, finally, there be a remainder, then the given number has no exact cube root, but, as in the case of the square root, an approximation can be carried to any degree of nearness by adding ciphers, and finding any number of decimal places.

The rule will be best understood by following the steps of an example.

It will be found necessary sometimes, as will be seen by the example given in Art. 15, to set down as the next figure in the root, one less than this quotient.

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Placing the points as indicated in the rule, we observe that the cube of 4 is the greatest cube in the first period 78. Subtracting 43, or 64, from 78, we get a remainder 14, to the right of which we bring down the next period 314, to form a dividend. Multiplying the square of 4 by 3, we get for a divisor 48, which will go 2 times in 143 (the dividend without its two right-hand figures). We set down 2, therefore, to the right of 4 as the next figure in the root, and then proceed to form the three lines according to the rule.

1. 8 is the cube of 2. 2. 48 is 3 x 4 x 22.

3. 96 is the product of 2, the last obtained figure in the root; and 48, the divisor.

Placing these three lines under each other, but advancing each successively one place towards the left, and adding, we get 10088, which we subtract from the dividend 14314, leaving a remainder 4226. To the right of this we bring down the next period 601, thus forming another dividend.

The next divisor 5292 is 3 x 422, and is contained 7 times in 42266. Putting down, then, 7 as the next figure in the root, we form three lines as before:

1. 343 is the cube of 7, the last figure in the root.

2. 6174 is 3 x 42 x 72.

3. 37044 is 7 × 5292.

Adding these up when properly placed, we get 3766483, which we subtract from the previous dividend 4226601, leaving a remainder 460118.

There are now no more periods left. Hence 427 is the number whose cube is the nearest cube number to the given number, and less than it. If there were no remainder, the root obtained would be the exact cube root of the given number.

14. In such an example as that worked out above, we could place a decimal point and as many periods of ciphers as we may wish after the original number, and thus, by continuing the process according to the rule, get as many decimal places as may be required as an approximation to the cube root.

In finding the cube root of a decimal, the periods must be completed by adding ciphers, if necessary.

15. When the cube root of a fraction is required, the cube root of the numerator and the cube root of the denominator will be the numerator and denominator respectively of the fraction which is the cube root of the original fraction. If the numerator and the denominator are not both perfect cubes when the fraction is reduced to its lowest terms (vide 9, Obs.), the best plan generally will be to reduce the fraction to a decimal, and then to find the cube root of that decimal. In the case of mixed numbers, they must be reduced to improper fractions, in order to see whether the resulting improper fraction has its numerator and denominator both perfect cubes. Thus, 523 reduced to an improper fraction gives 3, of which the cube root is, or 1. But if, when so reduced, the numerator and denominator are not perfect cubes, then it will be better to reduce the fractional part of the mixed number to a decimal, and placing the integral part before it, find the cube root by the above rule.

15875

3675) 1725000

64

1680

14700

1486864

238136

And so on to as many more decimal places as we may desire. Obs. Exactly as in the case of the square root, when one more than half the number of figures required of the root have been found by the rule, the rest may be found by simply dividing, as in ordinary division, by the last divisor.

16. Obs. It will be observed that although 27, the first divisor, is really contained 6 times in 176, we only put down 5 in the root. The reason is that, on examination, we find that 6 would be too large, for it would make the sum of the three lines which we add up greater than the dividend 17600. This explains the note at page 318. We must, therefore, always be careful to observe whether the figure put down in that root will or will not make the sum of the three lines too large. The dividing the dividend without its two last figures by the divisor is not, therefore, an infallible guide to the next figure of the

root.

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LESSONS IN ARCHITECTURE.-I. ARCHITECTURE is the art of planning, constructing, and adorning public or private buildings according to their intended use. The word architecture is derived from the Greek apxw (ar'-ko), 1 command, and TEKTOV (teck'-tone), a workman. This etymology indicates the operatives engaged in the building on the one hand, and the leader or chief, the man of science and practical skill, putting in action all his resources in order to execute his plan on the other. Such a division as this was, no doubt, established from the beginning of the art. According, therefore, to the literal meaning of the etymology, mankind must have, at the origin of architecture, possessed a degree of civilisation sufficient for the organisation of different kinds of industrial operations, and acquired a degree of skill in the art, which enabled some men by their experience to be the leaders or directors of others. In this way, we may suppose that the art itself, or rather the symmetry, the harmony of proportions, and good taste in structures, at first began to be developed.

Before arriving at this point, mankind must have overleapt ages. One of the first wants of society was a covering or shelter from the inclemency of the weather, whether of heat or of cold. Simple was the art employed in constructions of this kind. Grottoes or caves hollowed square to make them more habitable, and cottages constructed of branches of trees and blocks of stone-such were the primitive constructions in wood and stone which formed the rudiments of architecture. From the simplicity of early structures men passed to the study of proportions;

they then dared to attempt the grand; and, at last, reached the sublime.

The origin of architecture cannot be assigned to any particular country. Every nation produced its own art, or style, by employing the various materials within its reach, and by giving to them such forms as their wants required. Proceeding at first from the high table-lands of Asia, in order to people the earth, the early fathers of our race could have but little idea of architure, or of a well-established system of construction. As wandering and pastoral tribes, like the Hottentots of the present day, they lived in tents or wretched huts, which had no pretensions to architecture. It was not until they became more settled that they sought the means of rendering their buildings more durable, by employing in their construction wood or stone, and bricks baked in the sun.

From the differences in the materials, and from the variety of tastes and feelings, arise the varied appearances which the monuments of different nations present, and which constitute their peculiar style of architecture. Thus the Egyptian, born in the hot climate of Africa, in a country destitute of wood fit for building, and near the mountains of the valley of the Nile, containing large blocks of freestone and granite, created for himself a vigorous style of buildings, which completely sheltered him from the burning rays of the sun. These buildings were formed of colossal masses, which were easily transported along the waters of that famous river. The Greek, inhabiting a milder climate, surrounded by forests and quarries, gave a lighter form to his edifices, and employed wood in their construction, which harmonised well with the marble-a material of which the fineness admitted of a greater delicacy of

study of these will be duly appreciated by the historian, the philosopher, the archæologist, and the artist, who, each with his own particular view, knows how to find a great lesson in these silent witnesses of past civilisation, as well as in those existing in full vigour around us.

Architecture is founded upon three great principles, which ought to be immutable: 1, the useful, without which states and private individuals would be led into superfluous and ruinous expenses; 2, the true, because it ought to express in all its varied forms the great principles of construction upon which it rests; 3, the beautiful, which is the end of all the arts depend. ing upon design, and no less of architecture the most useful. On these principles, every style of architecture has the same value; and an artist should not curb his genius by confining himself to the study of one particular style. It is only the man of talent, to whom the construction of an edifice is entrusted, who can combine the different arrangements and forms, harmonise the various parts, and particularly express by plans, skilfully worked out, the disposition of the whole or of every part of the building. Upon these arrangements and plans rests the reputation of an ar Ichitect, and science demands of him a well-grounded assur ance of the good construction and dura bility of his work.

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THE HUT OF THE HOTTENTOT: AN EXAMPLE OF THE PRIMITIVE ATTEMPTS OF MAN TO CONSTRUCT A DWELLING.

Architecture is not an imitative art, like her sister arts, sculpture and painting. We see nothing in nature like our buildings as a whole; or rather nothing which could serve to guide us in its applications, or in the harmony of its lines. In this art, man has done everything himself. He has employed matter; he has invented forms and proportions to produce in the minds of his fellow-creatures ideas correlative of order, harmony, grandeur, richness, and

as it were, thought to matter, without being indebted for his ideas to any of the external forms of nature. Like the poet and the musician, the architect can transport the spectator into an ideal world, by creating forms and effects formerly unknown; but, very different from them in results, he renders his creations palpable, and gives them durability. Moreover, the useful, the true, and the beautiful, must be ever present to his view; and, however fruitful his imagination may be, he cannot emancipate it from science, the eternal basis of a" the productions of his art.

structure and arrangement. The Chinese, surrounded by rivers | durability. He has been enabled, by the force of art, to give, bordered with bamboo, had only a meagre and tortuous species of architecture, as ephemeral in its duration as it was fragile in its origin and construction. The very different character exhibited in local architecture enable us to judge of a country by its monuments, inasmuch as the buildings themselves are the expression of the various wants of the people who constructed them. It is easy to understand how their different arrangements and structures are but the reflection of the religion or the manners of the people. The general style of the monuments of a country is a durable image of the different phases of its civilisation. In these, we see it in its primitive, refined, or degraded state, as civilisation arose, approached to perfection, or decayed.

Nations naturally established great divisions in their architecture. They first built their private dwellings, then their public buildings, and these, in their numerous subdivisions, constituted civil architecture. Religion caused them to build temples and other edifices, attaching to them ideas of duty and moral obligation: thus arose sacred architecture. The fortification of their frontiers, their towns, and their conquered countries, gave birth to military architecture. In this basty sketch, we see how extensive is the series of buildings which cover the face of the globe, some of which belong to the first ages of its history, and others of which are being re-discovered in our own day. The

The architect should therefore spend his youth in the study of his art, and of the splendid examples left on the face of the old world by ancient civilisation. In conjunction with these studies he should make himself master of the exact sciences, in order that he may execute his plans with precision, and study the nature of their construction. He should also become familiar with the physical sciences, in order that he may understand the nature of the materials which he must some day employ, and be able to calculate their effects. In short, he should devote himself to practical experience, and to the working part of architecture, in order to render himself capable of executing public or private buildings, and to make himself responsible for the stability of edifices entrusted to him.

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ANIMAL PHYSIOLOGY.-X.

THE ORGAN OF TASTE (concluded).

In treating of the objects which excite the sense of taste, we must draw attention to the distinction between taste proper, and the alimentary sensation of relish. That these sensations are different, will appear from the consideration that many things which are very appetising, and in the eating of which there is great pleasure, have but little distinctive taste. Butter and animal flesh are good instances of this. The tip of the tongue applied to these would give but little indication of the presence of sapid bodies; but the succeeding parts of the organ and the mouth declare them very good. On the other hand, sweet and bitter principles are detected at once by the tip of the tongue, though they be entirely indifferent to the sense of relish. Alum is thus sweet to the sense of taste, but disgusting to the sense which we have called alimentary. The sense of taste proper, or the appreciation of what is sweet, bitter, sour, etc., is more connected with the intellect than the sense of what is

Savoury; and

hence it is less

dependent on

the state of the
body, and it
leaves behind
it a multitude
of distinct
ideas which
can be held in

the memory. Thus a person when suffering from sea-sickness can well discriminate between sugar and quinine; but he would be a very indifferent judge of the flavour of a beef-steak at such a time. The multitude of flavours which can be distinguished is truly remarkable; for not only does the apricot, plum, cherry,

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and which is of a more animal character, is enjoyed in a greater degree in the brute than in man; while the true gustatory sense, being more connected with the exercise of the mental powers of comparing and distinguishing, is certainly weaker in the lower animals.

Brutes may be roughly divided into two great divisions, the carnivora, or flesh-eaters, and the herbivora, or vegetable-eaters. The type of the first class is the tiger, or, to give a more familiar example, the cat; while the other is represented by the ox. In each of these, the whole body seems to have been constructed in relation to the food. The tiger has jagged back teeth, and pointed side fangs which lock deeply into one an other, but have no grinding surface. The jaws that wield these are short, strong, and can play only to and from one another. It can therefore grip and hold, but cannot chew. The stomach is small and intestines short, because flesh is very nutritious, and needs but little digestion. The fore limbs can move freely in all directions, and are furnished with claws to strike and seize. The ox has long jaws, rough but flat hind

IV

V

III

III. TONGUE OF A FIELDFARE. V. TONGUE OF A CHAMELEON.

I. TONGUE OF A CAT. II. FILIFORM PAPILLE OF A LEOPARD.
IV. TONGUE OF AN OSTRICH.

and apple each have a characteristic taste, though they all belong to the same order of plants, but a hundred varieties of apples all challenge recognition from this sense. The grape produces a thousand wines, each with a bouquet of its own, even though alcohol and water are the main constituents of them all, and that which causes the difference is so small in quantity, that the chemist cannot separate it. Some sensations described as tastes, are but little removed from those of touch; thus, the taste of nutgalls, called an astringent taste, and the fiery taste of alcohol, are probably caused by mechanical action on the outer skin. In the first case, the forcible contraction of the parts occasions a roughness; and spirit will produce a burning sensation on any delicate part of the body.

We have now to apply our experimental knowledge of the sensation derived through the tongue and mouth to the inquiry How far do brutes participate in these sensations? In order to answer this question we must observe the gestures and exhibitions of animation of animals while feeding on those substances whose tastes we are ourselves acquainted with. Observation seems to lead to the conclusion which we should naturally have arrived at from reasoning on the question. The conclusion is this, that the sensation which we have called the alimentary feeling,

VOL. L

teeth, and a close fitting row of front ones in the

front of the lower jaw, playing on a pad in the upper, and the lower jaw can swing sideways and 80 grind the food. He can therefore clip and chew, but cannot grip.

This comparison might be

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not, then, in speaking the sense of taste in animals, speak of the class as a whole, because the objects of the sense are so different in the two divi

sions of the class. It must not be sup

posed that this division of brutes is sharply drawn; for between the two types of tiger and ox, animals of every grade of intermediate structure are found. Moreover, the division is not a good one for the purposes of zoological classification; for though both the tiger and the Tasmanian devil eat flesh, and the kangaroo eats grass like the ox, yet even the tiger is more like the ox, and the Tasmanian devil more like the kangaroo, than are those animals when crosscoupled, as in the first sentence. Further, some brutes made on the flesh-eating type, eat all kinds of vegetables, as the bear does; and others built on the plan of herb-eaters, will eat flesh, as the pig will. In fact, the division is a false one when we are treating of the classification and structure of animals, but is nevertheless a useful one when we are writing of their powers and functions. In other words, it is a good physiological but a bad anatomical division. We have entered so far into the question, not only because it bears on our special subject, but also because it explains the term "physiology," with which these lessons are headed.

Of carnivorous animals, it may be stated that the alimentary sense, which is associated not only with the tongue, but with the throat and palate, is keen and pleasurable in the extreme,

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These animals can endure privation from food for considerable periods without manifesting any signs of starvation; but the smell, sight, and, most of all, the partial taste of flesh, excite them to eager, and even ferocious craving. Hence the popular notion of the dangerous nature of wild beasts which have once tasted blood is a true one. On the other hand, when the food is once obtained, it is torn to pieces, flung to the back of the mouth, and swallowed with a rapidity which altogether forbids the idea that these animals possess to any extent the faculty of discrimination in their tastes.

This view of the question is also borne out by an inspection of the tongue. In the illustration, the reader will find a representation of a cat's tongue. This tongue is long, and has but few round papilla; but it is covered with a dense pile of long, thin, pointed, overlapping projections (filiform papilla), which are directed backwards, and towards the mid line. The circumvallate papillæ, again, are but four in number, two on each side. It is this pile of pointed papillæ which makes the cat's tongue feel rough when she licks. The covering of these papillæ is so dense, hard, and thick, when compared with that of our own, that we are justified in thinking them mechanical only in function; and yet they cover the whole tongue almost to the exclusion of the other kinds.

In the larger members of the cat family, these pointed papillæ are quite like hard thorns or spines; and with them the lion, tiger, and leopard can rasp away the last adhering fragments of flesh and ligament from the bones. A patch of these papillæ from the leopard's tongue are represented in the engraving. They are two-lobed and rounded, and have from their back part a single sharp spine running directly backward, and they are set in a very regular pattern, alternating in each row. On the summit of the leopard's tongue a number of papilla were found without spines, as though worn off, or perhaps not developed, lest the palate should be injured by them.

In illustration of these remarks we may give an incident. A gentleman had reared a tame leopard from a cub, and having always fed it on bread, etc., the animal was very docile, and showed no sign of savageness. It was often caressed by its master, and returned the blandishments after its manner. While thus engaged, it one day took its master's hand into its mouth, and began to lick it gently, but owing to the roughness of the tongue it caused some blood to flow. The gentleman, no doubt feeling some pain, tried to withdraw his hand, but, to his surprise, the beast for the first time in its life began to growl. With great presence of mind the gentleman relented from his effort to release his hand, rang the bell, asked his servant for his loaded pistol, and then shot his now dangerous favourite through the head.

In herbivorous animals, while the sense is far less keen, so far as the alimentary sensation is concerned, we have no reason to suppose that the distinguishing gustatory sense is in any degree stronger,

The main mass of the food of the ruminants is insipid. Freshness is the strongest term that can be used to express its desirability. A large bulk is required for but a little nutriment. Thus we find the ox occupies a considerable number of its wakeful hours in grazing and chewing, and it feeds along the pasture, tearing up the grass with but little discrimination. It is true that a cow will avoid noxious or disagreeable plants when they grow in clumps; for a field, otherwise closely cropped, still presents long stalks of the common buttercup. It would seem, however, that this avoidance is rather due to instinct than to disgust. Many plants have very powerful, bitter, sour, and astringent principles, and they are intimately mingled with the grass; ¡ yet, as we seldom see a cow eject the food from its mouth, we cannot suppose it to have any very delicate sense of taste. From the fact that oxen ruminate, we might suppose that they enjoy the sense of taste while chewing the cud. So doubtless they do in a minor degree; but the act by which the food is returned to the mouth is probably quite involuntary; and the lazy, dreamy way in which an ox ruminates contrasts strongly with the avidity with which a carnivorous animal feeds.

The tongue of a ruminant is very long and flexible. It is often twisted round the herbage to tear it up, or break it off; and the qualities which fit it for this use are manifested in the highest degree in the tongue of the camelopard. This animal can extend by the length of this member its already great powers of reaching high, and thus hook down the branches of the palm. Well might this animal suggest to Lamarck that its whole organism had been modified by a constant endeavour to reach higher and higher.

The position of the large walled-round papilla is very various in different animals. The reader will have observed their position in the chimpanzee, in one long line of about twelve in number down the middle of the tongue, with a few scattered ones on each side. In the pig, otter, and seal they have the V-shaped arrangement which they have in man, but are fewer in number. In the sheep they form a thick, raised ridge on each side at the back of the tongue.

One of the most singular uses to which the tongue is put in this class is manifested by the ant-eaters, whose long slimy tongues are used to thrust into ants' nests, so that when they are retracted into their long tubular mouths the ants are carried with them, adhering to the mucus.

If this article had been headed "The Tongue," instead of "The Organ of Taste," we should have a long task before us to describe the various shapes of the organ in toads and reptiles, and also in snails and insects. The organ to which the word tongue has been applied has a wonderful diversity of form, and many interesting peculiarities; but in most cases its main office is to seize or to masticate the food, and the function of taste is subordinate to this.

In birds the tongue is almost as diversified in form as the beak; but it is usually cased in horn at its fore part, and there are only a few papillæ above the air-hole. In parrots it is fleshy; and these birds seem to have more of the sense of taste than most birds, for they will turn a lump of sugar or a nat about in their beaks for some time to test its qualities before eating it. It is certainly singular that birds, whose proper food is fruit, should be so little endowed with a sense to appreciate its delightful and delicate flavour; nevertheless, it seems as though the tongue were only applied to test the softness, and therefore the ripeness of the fruit. The tongue drawn to represent that of the fieldfare, may be taken as the typical tongue of a bird. The small triangular tongue of the ostrich, supported on its slender arch of bone, is given because of its singular shape and shortness. The length of the tongue has but little relation to the length of the beak. Thus both the pelican and the toucan have enormous beaks; but the former has a tongue as short as that of the ostrich, while that of the latter is very long. The tongue of the woodpecker is a living harpoon.

In some reptiles there is evidence of a sense of taste, but it is doubtless inferior to that of higher animals. The tongue of the chameleon, given in the engraving, is of curious shape; and the mechanism by which it can be darted upon a luckless fly is elaborate and interesting; but its description would be out of place here. In the toad and frog the tongue grows as the tail drops off. It sprouts from the inside of the lower jaw, and grows backward, so that its bi-lobed end lies free in the mouth, and can be filliped forward out of that cavity. This is also rather an organ of prehension than of taste. The forked tongue of the snake is familiar to every one. Its reiterated protrusion and vibration has led the vulgar to consider this action as a threat, and to believe that it is the sting of the animal. It, however, has no such function. It may have some power of tasting, but it is more probable that it is an organ of touch; for this creature, limbless and covered with hard scales, is greatly in need of a means of feeling outward objects.

Fishes' tongues have seldom any soft parts, and cannot therefore be organs of taste. They are not unfrequently furnished with teeth. In some fish a cushion of soft substance, wel supplied with blood-vessels, is found on the roof of the mouth.

All the higher orders of mollusca have an organ to which the name of tongue has been given, and some authors have proposed to group together the head-walkers, belly-walkers, and wing. footed classes under one sub-division, calling them odontophora, or animals which have a tooth-bearing tongue. This organ in snails (gasteropods) bears transverse rows of teeth arranged in complicated and beautiful patterns, and is sometimes so long as to be called the lingual ribbon. As it is often used to file away

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