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The number of times the root must be taken as a factor to produce the given quantity, is denoted by the name of the

root.

Thus 2 is the fourth root of 16; because 2 x 2 x 2 x 2 16, where two is taken four times as a factor to produce 16. So a is the square root of as; for a3 X a3 ɑ®. Powers and roots are correlative terms. If one quantity is a power of another, the latter is a root of the former. As 63

is the cube of b, so b is the cube root of b3,

There are two methods in use, for expressing the roots of quantities; one by means of the radical sign √, and the other by a fractional index. The latter is generally to be preferred; but the former has its uses on particular occasions. When a root is expressed by the radical sign, the sign is placed before the given quantity, in this manner, √a, Thusa is the 2nd, or square root of a.

3a is the 3rd, or cube root.

The figure placed over the radical sign, denotes the number of factors, into which the given quantity is resolved; i. e. the number of times the root must be taken as a factor to produce the given quantity.

Thus ya shows that a2 is to be resolved into two factors, and a3, into three factors; and "a", into n factors.

a

The figure for the square root is commonly omitted, and the radical sign is simply written before the quantity, thus =2√ a2.

When a figure or letter is prefixed to the radical sign without any character between them, the two quantities are to be considered as multiplied together.

Thus 2a, is 2 x Va, that is, 2 multiplied into the root of a, or which is the same thing, twice the root of a. And yb, is x x vb, or x times the root of b. When no co-efficient is prefixed to the radical sign, 1 is always understood; √a being the same as 1 √ a, that is, once the root of a.

The cube root of a is a2. For a2 × a2 × a2 = a3, Here the index is divided into three equal parts, and the quantity itself resolved into three equal factors. The square root of a2 is al or a. For a Xa a2. By extending the same plan of notation, fractional indices

are obtained.

Thus, in taking the square root of al or a, the index 1 is divided into two equal parts, and; and the root is a at. On the same principle, the cube root of a, is a = 3√ a The nth root, is anya, etc.

Every root, as well as every power of 1, is 1. For a root is a factor, which multiplied into itself wil produce the given quantity. But no factor except 1 can produce 1, by being multiplied into itself.

So that 1, 1, 1, 1, etc., are all equal.

Negative indices are used in the notation of roots, as well as of powers.

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FRENCH READINGS.-No. XXV.

LA MARGUERITE ET L'E'PI DE BLE'.
SECTION I.

J'AIME bien les fleurs, ces sourires de la nature; mais je ne leur livre pas mon jardin tout entier.

Outre les gazons qui, après avoir offerta leurs tapis à mes promenades, vont garnir de foin les rateliers de mon écurie; outre les arbres qui donnent tour à tour l'ombre de leur feuillage à ma tête3 et le suc de leurs fruits à mon palais, je réserve, tous les ans, un petit coin de mon enclos' pour en faire un champ de blé.

Quelle barbarie m'allez-vous dire, vous défigurez votre jardin !

E'coutez-moi, s'il vous plaît, avant de me juger.

D'abord, je pourrais vous répondre qu'un massif de blé n'a rien de laid. Au printemps, c'est de l'émeraude, en étés c'est de l'or. Un de mes voisins, qui s'y connaît, a dans son beau paic un champ d'avoine encadré de géraniums, et qui forme un tableau splendide et varié. Mais, chez moi, le froment est séparé des fleurs et dérobé à l'oeil par une haie vive. Il ne saurait donc défigurer mon jardin.11

Je prétends au contraire, qu'il lui fait honneur,12 et je fonde ma prétention sur un souvenir d'enfance, qui m'est sacré.

J'étais enfant, et je me promenais dans le jardin de mon père.13 le même que je cultive après lui. L'hérédité est bonne aux jardins comme à tant d'autres choses.-A la même place qu'aujourd'hui, il y avait, non pas un champ, mais un simple épi de blé,15 pauvre enfant du hasard, qui avait jeté là un grain, à côté d'une plate-bande de marguerites.

Je trouvai que le voisinage de l'épi déshonorait l'éclat du parterre et j'allais l'arracher avec la tige, lorsque mon père m'arrêta la main.

-Il faut y regarder à deux fois," me dit-il avant de détruire une œuvre de Dieu, toute petite et toute modeste qu'elle soit. Qui sait ce que deviendra celle-ci ? Laissonsla vivre auprès des marguerites. Nous verrons et nous comparerons leurs destins.

Comme mon père achevait ces mots, deux enfants passèrent, derrière la haie.18 C'étaient les deux filles d'un fermier voisin; l'une vive, alerte, brune, aux yeux noirs et pétillants;19 l'autre blonde, pâle, aux yeux bleus, à l'air doux et réfléchi.20 J'ai retenu leurs noms. La première s'appelait Marie, et la seconde Louise.-Marie s'écria: Les belles marguerites! Monsieur ?

voulez-vous m'en donner une,

Mon père me fit un signe. Je choisis la fleur la plus grosse, 22 la plus variée de couleurs, la mieux disposée en couronne, et j'en fis cadeau à la petite fille, qui la mit aussitôt sur ses cheveux. Mon père alors observant sa sœur, lui demanda si elle ne voulait pas une fleur aussi ?23-Pour toute réponse, Louise regarda, en rougissant, l'épi de blé, qui se dressait avec ses pointes et était déjà gonflé de quatre rangs de grains jaunes.-Je vous remercie, Monsieur, dit-elle enfin, je vous demanderai ce bel épi," quand il sera tout à fait mûr. Ce sera ma première moisson. - Frappé de ces mots, mon père répliqua: Très bien, ma petite, tu peux compter sur ton épi.

25

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TRAVAILLEZ BIEN: A full explanation of the use of the French tenses may be found in vol. iii. pp. 383-5. If more is wanted, it may be obtained from "Cassell's French Manual," pp. 49 and 5.

CHARLES SUTTON (Old Compton-street) has solved all the first portion of the Second Centenary of Algebrical Problems except the 28 h.

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eus and our are mono

in which the or is denoted by the single characters.
syllables, eos and ous dissyllables. whas always a long sound, Twv is pro-
nounced like the English ord tone. Tov like on.

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ON PHYSICS, OR NATURAL PHILOSOPHY.

No. LII.

(Continued from page 379.)

THE EYE CONSIDERED AS AN OPTICAL

INSTRUMENT.

Structure of the Human Eye.-The eye is the organ of vision, that is to say, of the phenomenon by virtue of which the light emitted or reflected by bodies, produces in us the sensation that reveals their presence to us. It is placed in a bony cavity or socket, and supported in its position by means of muscles, tendons, and nerves, which render it capable of a great variety and delicacy of movements, while the eyelids and eye brows protect it from external injury. Its size is nearly the same in all individuals, the apparent differences of size being caused by the smaller or larger opening of the Fig. 338 represents a transverse section of the eye from the Fig. 338.

lid.

d

front to the back. As appears from that figure, the general
form of the eye is that of a spheroid, the curvature of which
is greater in front than behind. The eye is composed of the
following membranes and media-the cornea, the iris, the
pupil, the aqueous humour, the crystalline, the hyaloid and
choroid membranes, the retina and the optic nerve.
The cornea, a, is a transparent membrane situated in
front of the ball of the eye.
It has the appearance of a small
spherical cup, with a base of nearly half an inch in diameter.
Its circumference is so closely attached to the sclerotic or
outermost tunic, that some anatomists consider them as one
and the same membrane.

The Sclerotic, i, is a membrane which, with the cornea, envelops all the constituent parts of the eye. In front it has an opening, nearly circular, in which the cornea is encased; the posterior and internal portion is perforated to afford passage for the optic nerve.

a

the size of the pupil varying inversely as the intensity of the light. The iris also serves to correct the spherical aberration by preventing the marginal rays from crossing the edge of the crystalline.

The Aqueous Humour.-Between the posterior portion of the cornea and the anterior of the crystalline, is a transparent liquid called the aqueous humour. The space e, occupied by this humour, is divided into two compartments by the iris; the part 6, between the cornea and the iris, is called the anterior chamber; the part e, between the iris and the crystalline, is called the posterior chamber.

The Crystalline is a transparent body, with the form and properties of a double convex lens, placed behind the iris and very near that membrane. It is enveloped in a transparent membrane called the capsule, which adheres at its edge to the annular band formed by the ciliary processes (or hair-like projections), g.

The anterior surface of the crystalline is less convex than the posterior surface. Its tissue is composed of a series of layers almost concentric, which are harder in the centre than at the circumference. The outermost coats have almost a liquid softness. The refracting power of these coats decreases from the centre to the circumference.

The Vitreous Humour and Hyaloid Membrane.-The transparent mass, resembling the white of an egg, which occupies the whole of that portion h of the eyeball which is behind the crystalline, bears the name of the vitreous humour or vitreous body. It is enveloped in the hyaloid membrane, which covers the posterior surface of the crystalline capsule, and all the internal surface of another membrane called the retina.

The Retina and Optic Nerve.-The retina m is a membrane intended to receive the impression of the light and transmit it to the brain, by means of a nerve n called the optic nerve, which, proceeding from the brain, enters the eye and spreads over the retina in the shape of a nervous net-work.

The retina and the optic nerve have no other special function than that of receiving and transmitting to the brain impressions of images. They are altogether insensible to injury from external objects. They have been cut and pricked without appearing to cause any pain to the animals upon which these experiments have been performed.

The Choroid Membrane k, is a membrane interposed between the retina and the sclerotic. It is essentially vascular, and covered on its internal surface with a dark substance like the pigment of the negro's skin. Its object is to absorb all the rays that are not required to produce distinct vision. The choroid membrane is fringed with a series of hair-like projections g, called, as we have stated above, ciliary processes, between the iris and the crystalline capsule, to which they adhere, forming round it a disk like that of a radiate flower. By its vascular tissue the choroid membrane serves to convey the blood to the interior of the eye, and especially to the ciliary processes.

Course of Rays in the Eye.-Considering the various parts of which the eye is composed, it may be compared to a camera obscura, of which the pupil is the aperture, the crystalline the converging lens, and the retina the screen on which the The Iris, d, is an annular opaque diaphragm or partition, image is portrayed. The effect is therefore the same as that fastened at its outer circumference, and free at its central edge. produced at one conjugate focus of a bi-convex lens by an This membrane is placed between the cornea and the crystal-object placed at the other focus. Let AB, fig. 339, be an line. It is this that forms the coloured part of the eye. object placed in front of the eye, and let us consider the rays It is pierced by an opening called the pupil, which in man is emitted from any point A in the ojbect. Of all the rays procircular. In the lower animals it assumes various forms. In ceeding from it, only those which are directed towards the those of the feline tribe, it is narrow and elongated vertically, pupil penetrate the eye and contribute to vision. These rays, while in ruminating animals it is elongated in a horizontal on their entrance into the aqueous humour, undergo refracdirection. It is by the pupil that the rays of light penetrate tion, which makes them approach the axis A a, drawn through into the eye. Its diameter, which varies in the same indivi- the optical centre of the crystalline. When they meet the dual, is between one-tenth and a quarter of an inch, but these crystalline they are again refracted as by a bi-convex lens. At limits may occasionally be surpassed. The alternate enlarge- last, after having undergone further refraction in the vitreous ment and contraction of the pupil are very rapidly performed, humour, they meet in the point a, and there form the image they are constantly going on, and play an important part in of the point a. The rays which proceed from the point в go the phenomenon of vision. The pupil contracts under the to form the image of this point at b. Similar remarks apply influence of strong light, and dilates in a feeble one. The to all the intermediate points between A and B, whence we movements of the iris appear to be involuntary: see that a very small real but reversed image ab of the object AB is formed upon the retina when the eye is properly constituted.

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From what has been stated, it will be seen that the iris is a screen with an opening of variable magnitude, and its function is that of regulating the quantity of light which enters the eye,

VOL. V.

Reversing of Images.-To prove that the images formed on 130

the retina are really reversed, the eye of an Albino or white negro is employed, because in this singular race the choroid membrane is without pigment, and consequently the light can pass completely through it. The eye is stripped of the cellular tissue which envelops it at the back, and placed at an aperture in the window-shutter of a dark chamber, when by the aid of a magnifying lens it may be clearly seen that the images on the retina are reversed.

The reversing of images in the eye has much occupied the attention of philosophers and physiologists, and numerous theories have been put forward to explain how it is that we do not see objects reversed. Some consider that it is by custom

this angle increases or decreases in proportion to the size of the object, and for the same object it decreases in proportion as the distance increases, which is seen in the accompanying figure, 341. The consequence is, that the more distant objects are, the smaller they appear, for the secondary axes AO BO crossing each other in the centre of th crystalline, the size of the image projected on the retina depends on that of the visual angle A O B.

Estimation of Distance and the Size of Objects.-The estimation of distance and size depends on the concurrence of several circumstances; the visual angle, the optical angle, comparison with objects whose size is familiar to us, and the diminution

Fig. 339.

B

and a sort of education of the eye that we see objects rectified, that is to say, in their true position with relation to us. Others think that we perceive the real position of objects in the direction of the luminous rays which they send forth, and that as these rays cross each other in the crystalline, the eye sees the points A and B in the directions a A and B respectively, whence the objects appear upright. Such was the opinion of d'Alembert. M. Muller, Volkmann, and others, maintain that as we see every thing upside down, and not one object to the exclusion of others, nothing appears reversed, since we have no means of comparison. None of these theories can be considered altogether satisfactory.

Optical Axis, Optical Angle, Visual Angle.-The principal optical axis of the eye is the axis of the whole ball, that is to say, the straight line with regard to which the figure is symmetrical. In a well-formed eye, it is the straight line passing through the centre of the pupil and the crystalline, as oo, fig. 339. The lines Aa, Bb which are apparently but not really straight, are secondary axes. It is in the direction of the principal optical axis that objects are most clearly seen. The optical angle is the angle BAC, fig. 340, formed by the

of the clearness of the image by the interposition of an atmosphere more or less charged with vapour.

When the size of an object is known, as the height of a man, a tree, or a house, the distance is estimated by the magnitude of the visual angle under which it is seen. If the size of the object is unknown, it is estimated by comparison with surrounding objects.

A colonnade, or an avenue of trees appears to diminish in proportion as the distance increases, because the visual angle decreases; but the habit of seeing columns and trees, and our knowing their usual height, correct the judgment. In the same manner, although very distant mountains may be seen under a very small angle, and occupy only an insignificant space in the field of vision, yet, as we are accustomed to aerial perspective, we do not overlook or mistake their real greatness. The optical angle is also an essential element in estimating distance. This angle, increasing or diminishing as objects approach or recede, the motion that we give our eyes, in order that their optical axes may converge towards the object which we are looking at, suggests an idea of its distance. However, it is only by long custom that we are enabled thus

Fig. 340.

B

Ө

principal optical axes of two eyes directed towards the same object. This angle is smaller in proportion to the distance of the object.

The visual angle is the angle A о B, fig. 341, under which an object is seen, that is to say, the angle formed by the secondary axes drawn from the optical centre of the crystalline to the opposite extremities of the object. For the same distance

corresponding motion of the eyes. It is well known that those to determine a relation between the distance of objects and the who have been born blind and afterwards obtained sight by the operation of cataract, at first think all objects equally distant.

must be placed in order to be seen with the greatest clearness, Distance of distinct vision.—The distance at which objects Fig. 311.

is called the distance of distinct vision. This distance varies for different individuals, and even for different eyes of the same person. For small objects, like printing type, it ranges from ten to twelve inches.

Adaptation of the eye to all distances.-The eye exhibits a remarkable property, not found in an equal degree in any optical instrument, which consists in this, that though the images have a tendency to be formed at a distance in front of the retina in proportion as the objects are more distant, they are always formed exactly on this membrane, for the eye enables us to see clearly at very varied distances. However, if we can see clearly at different distances, we cannot do so at the same time, which proves that some modification takes place in the system of the eye, or at least shows the necessity of fixing our attention on the object we wish to see. And indeed if we look at two objects in a line, situated at distances of one and two yards, on directing the eye to the first, the second appears indistinct, while if we fix our eye upon the second, the first becomes indistinct. Whence we conclude that when the eye has been disposed in a way suitable for seeing at a certain distance, it is not suitably arranged for seeing at another distance, but it has the capability of adapting itself to one distance after the other.

Several hypotheses have been proposed to explain how it is that the eye has the power of seeing clearly at various distances. Some philosophers attribute the phenomenon to the dilation and contraction of the pupil. It is unquestionably true that the dilation and contraction of the pupil are in some way connected with the adaptation of the eye to various distances, but it is important to observe that they are also connected with the intensity of the light, and that for the same distance the opening of the pupil may vary very much. Others are of opinion that the diameter of the eye, from the front to the back, varies under the influence of the action of the muscles, which move the organ in such a manner as to make the retina approach or recede from the crystalline, while the image itself approaches or recedes, for we know that in convergent lenses the image approaches in proportion as the

Fig. 312.

Gassendi maintained, that at any one instant perception takes place by only one image or the other; which, however, is disproved by the experiments of Wheatstone.

Taylor and Woollaston contend that two points symmetrically situated on the right or left of each retina correspond to the same cerebral nerve on the right or left. This opinion is supported by the fact that some persons are affected with a temporary paralysis of half the retina, on the same side of each eye, in such a way that they only see the right or the left half of objects. Woollaston and Arago both experienced this in their own persons. Brewster attributes the unity of sight to our acquired habit of referring the impressions simultaneously produced upon each retina to one single object.

The following are the principal facts connected with vision by two eyes. We see more clearly with two eyes than with one. On looking at an object, first with one and then with both eyes, the difference is very perceptible.

When the two eyes are each fixed upon a different object in such a manner that the two optical axes meet on this side or beyond the two objects, remarkable optical illusions may be produced. For instance, if we look at two small objects a and b, by means of two tubes, which give the optical axes of the two eyes directions a o and вo (fig. 342), we see only one object in o, the intersection of these two lines. If this point be on the other side of the objects (as in fig. 342), the object seen is also beyond them, and vice versa when the point is between the eyes and the objects (as in fig. 343).

If the objects a and b are two small discs, the one red and the other green, we see only a white disc, green and red being complementary colours, or colours which when combined produce white. These various experiment prove that the impressions on the two eyes are simultaneous, and combine to produce a single sensation.

We are indebted to Mr. Wheatstone for many experiments showing the essential difference between sight with two eyes and with one. The result of these experiments is, that it is only with two eyes that we can obtain a clear perception of the relief of bodies, that is to say, of their solidity as well as

Fig. 343.

object recedes. Hunter and Young attributed a contracting property to the crystalline, by virtue of which it assumes a more or less convex form, in such a manner as to make the rays always converge on the retina.

Kepler, Camper, and many others, considered that by the action of the ciliary processes, the crystalline is capable of removing and approaching the retina more or less. Others again have attributed the power of seeing clearly at very different distances, not to any movement of the retina or crystalline so as to make them approach or recede from each other, but to the fact that the variations in the focal distance of the crystalline humour caused by the increasing distance of the object, are so small as not to interfere with the clearness of the image. This last theory is confirmed by the experiments of Magendie and De Haldat. Sturm has also propounded a theory not very dissimilar, according to which the light may proceed from any point in a line between two foci, and it will be sufficiently condensed to produce the sensation of clear vision. Consequently, when external objects approach or recede, it is only necessary that the retina be always between these two foci, or nearly coincide with one of them, in order that the image may be distinct.

Single sight with two eyes. When the two eyes are fixed upon the same object, an image is formed upon each retina, and yet we see only one object. To explain this phenomenon,

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superficial extent. It is probable that the solidity of objects could not be perceived with one eye, were it not otherwise known to us. In fact, when the object is at a small distance, and we look at it with both eyes, as the two axes must converge towards it, the perspective is altered for the two eyes, and the images are perceptibly unequal. This is easy to prove by looking alternately at the same object with each eye. Now it is from the simultaneous perception of these two images that the perception of relief appears to result, as is proved by the following experiment.

The Stereoscope.-Mr. Wheatstone has invented an ingenious apparatus, called the stereoscope, which serves to show the influence of vision with two eyes upon the perception of solidity. This apparatus, as modified by Sir David Brewster, consists of a small wooden box, the upper compartment of which contains two tubes for directing the optical axes. At the bottom of the box are two drawings, which each eye sees separately through a convergent lens placed in each tube. These drawings represent the same object but with different perspectives which are precisely those that would correspond to the optical axis of each eye, if it looked at the object at a short distance. Hence in looking down the tubes each eye receives the same impression as if it looked at the object itself, and the result is so distinct and vivid a perception of relief that the illusion is complete and truly surprising.

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