5.

1. Haben Sie auch gehört, ich sei vom Pferte gefallen? 2. Nein, ich bitte, Sie seien aus tem Wagen gefallen. 3. Die Geschichte meltet, ta Tills, welcher Magreburg im treißigjährigen Kriege eroberte, sehr barbarisch verfahren sei. 4. Mein Bruder sagte, Sie seien sehr gelebt worten. Die Franzosen behaupten, sie seien tie Gebildetsten in der Welt. 6. Ihre Schwester glaubte, Sie wären in ter Start gewesen. 7. Die Englänter stat ter Meinung, sie seien die Herren res Meeres. 8. Dieser Reisende erzählte, er sei zweimal in Rom gewesen. 9. Er hofft, er werre in acht Tagen in Dresten sein. 10. Sie fürchten, Sie seien zu langsam im Handeln gewesen. 11. Wir glaubten, Sie wären auf dem Lante. 12. Ich glaube, wir wären gestern zu Guch gekommen, wenn das Wetter schöner gewesen wäre. 13. Ich glaubte, er wäre ter warnenden Stimme seiner Eltern eingerent gewesen. 14. Er sagte zwar, er sei frank, aber Viele behaupten, es sei Verstellung ren ihm gewesen. 15. Seine Verwandten sagen, sein Glück habe sein Unglück berbeigeführt. 16. Ich hörte mit Verauern, Sie hätten bas Nervenfieber gehabt. 17. Da ich in tem obern Zimmer war, hörte ich Sie icht rufen. 18. Man erzählt, der Ungar habe bis in den Tod sein Vater lant treu vertheidigt. 19. Ich hörte, dieser junge Franzose werte ein großes Hermögen erben. 20. Ich glaube, daß viele Menschen hier auf Erden ihr Gutes gehabt haben werten.

EXERCISE 81.

1. People say these gentlemen have been tipsy, but they are mistaken. 2. They say that residence in Paris is more agree able than in London. 3. We could not believe that this was true. 4. It is universally believed that the enemy has crossed

the frontier. 5. He asserted that it was better to stay at home than to go out. 6. I wish that he may be treated with more kindness. 7. He tells every one that you are a very rich man; but if you were, you would not be so penurious. 8. Have you heard, too, that your friend has fallen from his horse? 9. No, but I have heard that he has fallen out of the coach. 10. I hope that you will be with your parents in a fortnight. 11. I doubt that he can be so ungrateful. 12. This stranger says that he has been twice to India, and was very sick on his last voyage.

LESSONS IN GEOMETRY.-XII. As the next lesson will put the learner in possession of the last of the problems that we intend to give on the construction of figures contained by three and four straight lines-namely, the triangle, the square, the rectangle, and the parallelogram-we would recommend him to go carefully over the whole of the present series of problems from the commencement, constructing as many figures as he possibly can, to meet the requirements of the data in each case. And in doing this we advise him to try to construct figures different in form to those which we have given in these pages, as, if he can do this, he may be sure that he has gained a thorough knowledge of the various methods of construction set forth in the different problems.

The problem in practical geometry that was brought before the notice of the student in the last lesson, showing him how to construct a square that shall be equal in superficial area to the sum of two squares described on two given straight lines, has given him the key to the construction of squares, rectangles, and parallelograms, equal in superficial area to the sum or difference of any two or more squares, rectangles, or parallelograms, as the case may be; and it has also shown him that the main principle on which their construction depends, is the relation between the triangle, the figure contained by the least possible number of straight lines (since two straight lines cannot enclose a space, although one curved line can, as in the case of the circle), and all regular figures contained by straight lines-namely, the square, the rectangle, and the parallelogram. It may be as well to repeat that this principle is, that when a square, rectangle, or parallelogram is upon the same base and between the same parallels, the area of the square, rectangle, or parallelogram (as the case may be), is double the area of the triangle.

Now supposing we have a square, rectangle, or parallelogram before us, and we wish to construct a triangle equal in area to either of these figures, what have we to do? Manifestly nothing more than to draw one of the diagonals of the figure in question, produce the base indefinitely in the necessary direction, and, after setting off on it a straight line equal in length to the side of the square, rectangle, or parallelogram, that serves as its base, to join the extremity of the line thus set off with the upper end of the diagonal. This will be evident on an inspection of Fig. 42, where, in the square (rectangle or parallelogram) A B C D, the diagonal A c is drawn; the base CD, on which the square (rectangle or parallelogram) A B C D stands, is produced indefinitely in the direction of F; a straight line, D E, set off along it from the point D, equal to DC; and the straight line EA drawn, joining the points E and A, and completing the triangle or parallelogram) A B C D.

AEC, which is equal in superficial area to the square (rectangle

And, conversely, when we wish to draw a rectangle or parallelogram equal to a given triangle, all we have to do is to bisect the base of the triangle, and on either half of the base construct the required rectangle or parallelogram, after drawing through the apex of the triangle a straight line parallel to the base. In the case of the rectangle, after bisecting the base of the triangleas, for example, in Fig. 43, where the base of the triangle ABC is bisected in D-and drawing a straight line, PQ, of indefinite length, through the apex A of the triangle A B C parallel to its basa BC, a rectangle equal in superficial area to the triangle A B C is formed by drawing the straight lines CE, D F through the extremities C and D of C D, one-half of the base B C, perpendicular to B C, and meeting P Q in E and F; or by drawing the perpendiculars D F, B G, through the extremities D and B of B D, the other half of the base meeting P Q in F and G.

In the case of the parallelogram, if it be required to make two of its opposite sides equal to a given straight line, as the straight line x in Fig. 43, or two of its opposite angles equal to a given angle, as the angle Y, we must from the extremity of one-half

make the angle DC K equal to the given angle Y, and through D draw D H parallel to C K in order to complete the parallelogram as before.

This process will only be found practicable for the construction of a square equal in area to a triangle when the triangle is a right-angled isosceles triangle; but for any other description of triangle it will be found necessary first to construct a rectangle equal in superficial area to the given triangle, and then to draw a square equal to the rectangle thus obtained. How to do this will be shown presently in Problem XXXI.

F

Fig. 43.

By Problem XXX. we are enabled to construct a square equal in area to any number of given squares. Thus, suppose we wish to construct a square equal in superficial extent to the five squares of which the length of the sides of each is represented by the straight lines A, B, C, D, E respectively (Fig. 44). Draw any straight line, F G, equal to A, and at its extremity, G, draw G H at right angles to it equal to B. Join F H: the square described on FH is by Problem XXX. equal to the squares described on FG and G H. Next draw H K equal in length to the given line c, at right angles to HF. Join K F. The square described on F K is equal to the squares described on K H, H F, or to the squares described on K H, H G, G F, since the square described on HF is equal to the squares described on H G, G F. By continuing this process we at last obtain the straight line M F. The square described on this line is equal to the sum of the squares described on the given straight lines, A, B, C, D, E. Now let us see how far this is of practical value to the artisan. Let us suppose that a cabinet-maker has a number of small squares of veneering of several kinds of choice wood, each square being of a different size, and he wishes to use up this wood in veneering a table or the panel of a cabinet without wasting a single scrap of it. By following the process just described it is manifest that he possesses the means of readily ascertaining the exact area of the square that these pieces will cover, and after finding this, he can, if it be desirable, by Problem XXXI. draw a rectangle equal in area to the square if he prefer this form for using up his squares of veneering, and then arrange his pattern in such a manner that his squares may be worked up without waste. It is also a process that is useful to the maker of floors in parquetry, or to a stonemason who wishes to know how large a square he Fcan pave with a number of smaller squares of stone or slate of different sizes. Of course in such Fig. 44. cases the operator would work to a given scale, and the process might be used as a test of the correctness of the result of the operation by which the whole content of the squares may be found arithmetically, or as one which is far more certain and involves far less trouble than the arithmetical operation, which would be a long and tedious one.

DK

PROBLEM XXXI.-To draw a square that shall be equal in superficial area to a given rectangle.

Let A B C D (Fig. 45) be the given rectangle; it is required to draw a square equal in superficial area to the rectangle A B C D. Produce C indefinitely in the direction of E,

and on the straight line D E set off D F equal to the side DA, or B C of the rectangle A B C D. Bisect C F in G, and from G as centre with the radius G C or G F describe the semicircle c H F. Produce D A until it meets the arc CHF in K. Then along the straight line DC set off D L equal to D K, and through the points K, L, draw the straight lines K M, L M parallel to C F, D K respectively, and meeting in the point м. The figure DLM K is a square, and it is equal in area to the given rectangle A B C D. If F N O L had been the given rectangle, the same process would have been followed. FL would have been produced in the direction of L, and L c set off on it equal to the side L o of the rectangle LONF; C F bisected in G; the semicircle c H F described as before, and L o produced to meet the circumference C H F in P. The square drawn on LP is equal in area to the rectangle F L O N. If it be required to draw a square equal in area to a given parallelogram, we have only to construct a rectangle equal to the given parallelogram, and proceed as above. This will be seen from Fig. 45, in which the rectangle A B C D is equal to the parallelogram D C Q R. PROBLEM XXXII.-To draw a rectangle that shall be equal to a given square, and have one of its sides equal to a given straight line.

N

H

D

Fig. 45.

M

B

Let A B C D (Fig. 46) be the given square, and x the given side of the required rectangle, and in this case let x represent the shorter of the two pairs of sides by which the rectangle is enclosed. First produce C D indefinitely both ways towards E and F, and along C E set off C G equal to x, and also along C B set off c H equal to x. Join B G, bisect it in K, and through K draw K L perpendicular to B G, meeting E F in L. Then from the point L as centre, with the radius L G, describe the semicircle G B M. Through the point м draw м N parallel to A D or C B, and through H draw H N parallel to A B or E F, and let the lines HN, M N meet in N. The rectangle C H N M is equal in area to the square A B C D.

When the longer of the two pairs of sides that enclose the rectangle is given, as Y in Fig. 46, produce C D indefinitely both ways as before, and set off c M along C F equal to Y. Join в M, bisect в м in o, and through the point o draw oL at right angles to B M, meeting E F in L. Then from L as centre, with the distance L M, describe the semicircle M B G. Set off along C B the straight line c H equal to c G, and complete the rectangle C H NM by drawing H N, M N through the points H and M parallel to c M and C H respectively.

Fig. 46.

X

The learner must remember that the side of a square is a mean proportional between the sides of any rectangle that is equal to it in superficial area; and, therefore, that to find the length of the side of a square equal to a given rectangle, we must set off on the same straight line, but in opposite directions, two lines equal in length to the sides of the given rectangle, bisect the line thus obtained, describe a semicircle on it, and find the mean proportional to the two lines of which it is composed, by drawing a perpendicular from their point of junetion to meet the semicircle, as in Problem XIII. (page 193); while, to find the lengths of the sides of a rectangle that shall be equal to a given square, we must draw a straight line at right angles to a line equal in length to the side of the square, and from a point in this line on either side of the line that repre sents the side of the given square, draw a semicircle with a radius equal to the straight line joining the point that is used as the centre of the semicircle and the more remote extremity of the line that represents the length of the side of the given square. The lines intercepted between the other extremity of this line and the extremities of the arc of the semicircle will be equal in length to the sides of a rectangle, having a superficial area equal to that of the given square.

ANIMAL PHYSIOLOGY.-XII.

THE ORGAN OF TOUCH (concluded).

Ir has been shown in the previous lesson that the sense of touch, in its wider sense, is of a highly intellectual character. As an informant of the mind it is second only to the sense of sight, and in the suggestion of abstract ideas it is, perhaps, superior even to vision itself. There is no fundamental conception in relation to matter which it cannot impart. Though devoid of every other sense, a man possessed of this can pursue the study of every science, if he will but surmount the diffi

managed that the eye is completely imposed upon, and it is almost impossible to believe that the surface is flat. A rapid sweep of the hand, however, at once dispels the illusion; and so effectually that when you move back to gaze again, it is difficult to regain the impression of an embossed surface. The unbelieving Thomas, however reprehensible his scepticism might be, expressed it both with force and delicacy; for he at once recognised that his own sense of sight might be deceived, and expressed a doubt, not of the truthfulness of the other disciples, but of their correctness of vision.

That this sense, when combined with the muscular sense, is

PSEUDOPODIA.

culties which oppose themselves to his acquisition of the results of the experience of other men. Thus, blind men have taken to the study of mathematics, and by the aid of the figures of Euclid, conic sections, etc., given in relief, have acquired a knowledge which has placed them in an honourable position in the examinations at Cambridge. The very theory of light and all its laws are quite comprehended by such blind students. The sense of touch is absolutely bounded by the surface of the body, but it makes amends for being less far-reaching than other senses by being the most real of all the senses. We make our ultimate appeal to it when the eye gives false or confusing indications. In the King's Palace at Amsterdam there is a wainscot painted to express figures as if they projected from, and were carved, upon its surface. The shading is so well

VOL. I.

of a highly intellectual character, does not at all contradict the statement that it is also the simplest and most rudimentary of the senses. That it is simple and rudimentary agrees well with the fact that satisfactory evidence may be found of its existence in most animals. The possession of this sense reaches far lower down in the animal scale than that of the other special senses. Definite organs of touch are well developed in animals in which no other organs of sense are found; and the power of extemporising feelers, or prolongations of the body into fingers or filaments, is a character of the very lowest animals with which we are acquainted. Reflection would tell us that this surface sense is almost essential to animal life. How necessary must it be for every animal that moves or feeds to know the exact limits of its body-the confines of the domain over which

25

it has control; what is part of itself, and therefore has to be nourished, cherished, and defended-what is foreign, and therefore may be used or avoided, as it is wholesome or noxious., Indeed, the sense seems indispensable to all animals that are not plunged and fixed, through every stage of their life, in the midst of a medium which is both air and food to them-to all animals, it might be said, if it were not tautological, whose life is not purely vegetable.

In the higher animals, and in all those whose means of defence lie more in their active powers than in defensive armour, the sense of touch is distributed over the surface of the skin, as in man. Every such animal may be compared to an island. The boundary of its body is the coast-line. Along the whole of this are placed, at various intervals, places of out-look, just as our own tight little island has been surrounded with Martello towers. These stations are few and far between where the coast is rocky, abrupt, and inaccessible, but nearer together at those parts where a descent could be easily made, and crowded together at the outlets of ports, creeks, and rivermouths, through which an active commerce is carried on. The comparison of the extremities of the tactile nerves to Martello towers is the more appropriate, because these have ceased to be of any use in defence, and have become stations of out-look for the coast-guard. So the tactile nerves are, in themselves, no protection, but rather, being delicate organs, they need protection; for they act as alarmi-ts, awakening and calling up the active powers to fight in defence of the common country. These two functions of the skin-namely, that of passive defence and active alarm-are complementary to one another: where one is very efficient, the other is less needed. In the scaled and mailed fishes, and in such forms as the tortoise among reptiles, and the armadillo among animals, the function of sensation is sacrificed to that of defence; but in the naked skinned animals the sense of touch had need be very acute. In comparing man with the lower animals of that class to which he belongs, we find that his sense of touch is, perhaps, better developed than that of any other animal. The lower animals have to sacrifice a certain amount of their surface sensibility to the paramount necessity of being shielded from the cold; or, to put it more truthfully, to the retention of their animal heat. Man has neither the continuous thick coating of hair of the ox, the thick skin of the rhinoceros, nor the dense accumulation of fat below it which is found in the pig and in the whale. He is only cosmopolitan because his superior intellect enables him to clothe and house himself. His nearest relatives among beasts, though much better supplied with hair than himself, are confined to the tropics. Man makes himself at home everywhere, but only by becoming a "clothes philosopher." His triple investment of ordinary, nether, and over clothing, prove him to be an exotic species. He supplements by art the line of defence at those points where nature has left him exposed. The main use of the coating of hair is, no doubt, to defend the brute from the winter's cold, but that which will keep in the heat will keep it out, so that it may also be considered as a defence against the excessive heat of the sun also. Doubtless the universal presence of hair on the heads of both sexes of the human species indicates that in his native home man had more to fear from sun-stroke than from the cold of winter. Besides this, the hair is sometimes a real defence against the rough usage of the outer world. Thus the manes of the lion and the buffalo are real shields both against trenchant blows and the worrying of the teeth of hostile animals. Even the matted hair of the negro is said to be able to resist a tolerably forcible sabre cut. The principal use, however, is, doubtless, to defend from cold; and it is remarkable how this main object is arrived at without much prejudice to the function of touch.

Few solid substances are lighter than hair, even when pressed close; and few substances are worse conductors of heat- -so that brutes retain their heat by the aid of a substance which costs them but little in the way of carriage. Beyond this, the springy, stiff, yet soft texture of hair, makes it always permeable to the air; and air, when motionless, is a bad conductor of heat, and adds, absolutely, to weight. Hence on the coldest day, when the thermometer stands below zero, the beast is still surrounded with a layer of warm air, almost equal in temperature to its body. So much to prove its efficiency for its main purpose. Now we have to show how it leaves the sense of touch, if not unimpaired, Most not obliterated. The reader must refer back to the

illustration in Lesson XI. (page 353) to understand the structur and relation of each hair to the skin in which it is developed and fixed. The hair is essentially a tubular projection of the cuticle, firmer and denser in its composition, being made up of closely. pressed, elongated, spindie-shaped cells, instead of scale-like, easily-detached ones. It is not, however, produced from the level of the surface of the body, but from a bag or follicle, which is always narrow, and more or less deep as the hair is long or short. This horny tube dilates at the bottom of its bag to enclose a vascular papilla, similar in every respect to those papilla which lie immediately under the surface of the superficial cuticle. The hair itself, like the rest of the cuticle, is without sensation, as indeed it must be for the comfort of the animal; but the papilla has not only blood-vessels but nerves, and is very sensitive, so that the hair cannot be pulled or moved in any direction without affecting the sensitive part. Though a furred animal cannot precisely tell the exact point at which it is touched, on account of the length and flexibility of its individual hairs, yet the sensation of touch is as truly conveyed to the true skin, as it is when the pressed ridges of the forefinger of man cause feeling to be excited in the papillæ beneath them. In one respect hairs are even advantageous to the sense of touch, inas much as they reach considerably beyond the surface, and thus the range of the sense is extended. This advantage is so far recognised by nature that certain hairs are specially developed which have no other use than that of touch. These may fairl be described as tactile organs. These hairs are usually, and almost exclusively, situated in the upper lip, projecting from the most prominent part of the muzzle. In quadrupeds the snout is of course the most salient part of the body, and is most used in investigation. These whiskers, as they are called (though they would be better named moustaches), are remarkable for their length and stiffness, the depth to which their large bulbs run into the skin, and even protrude in the internal surface, and also for the large nerves that enter the papillae of the bulbs. Those coming from the whiskers of a seal as they run together look like the strands of small cords as they become woven into a rope of tolerable thickness. The animals in which these whiskers are most developed are the carnivora and the rodents. This is not improbably associated with the fact that these are for the most part nocturnal animals. Moreover, many of the rodents inhabit holes in the ground, trees, etc.; and many of the smaller carnivora are always poking about in holes and crannies for prey. It certainly would be an advantage to a fox on a dark night to be able to gauge with his whiskers the size of the aperture in a hen-roost before he tried to force his way through it; and thus it has been thought that there is a relation between the width of the body and the extreme extent of the whiskers.

In birds the place of hairs is supplied by feathers. The stru ture of these is very wonderful and beautiful, but a description of it would be out of place here, because they are certainly less efficient tactile organs than hairs. Birds' feathers are coarse than hairs; they are less flexible; they are inserted only on cer tain parts of the body; and since there must be provision made for moulting, they are more definitely cut off from the sensitive skin below. For all these reasons they are not good organs for transmitting the sense of touch, although they are formed in the same manner as hairs. Probably on account of this inap titude to transmit impressions, they are sometimes replaced by hairs in certain parts of the body; but as a rule the whole of the bird's body is encircled with feathers, which lie overlapping one another, and turned in one direction towards the tail of the bird. in the same manner as tiles on a house-roof. A bird's jaws, instead of being covered with soft, flexible and sensitive iips, are covered with a hard, horny bill, and its legs, though ofter devoid of feathers, have to be defended by scales or scutes, to prevent the long tendons of their leg muscles being severed Under these circumstances, a bird enjoys little advantage froli its sense of touch. Indeed, it is only in the padded under-surface of the foot and toes, and sometimes in the beak and tongue-when the former is leathery, and the latter not capped with horn-where there can be any provision for the exposure of a sensitive surface. It has sometimes been stated that the heron, as he stands in shallow, muddy water, is guided by feeling the eels twisting in and out, or even sucking his toes. This statement seems rather suited for a fable of the biter bitten than to be regarded as a scientific fact. That the sense is pre