straight line PX of indefinite length towards x, and along PX from P set off PC, CD, each equal to A. Through c draw cy of unlimited length towards Y, and along it from c set off CE equal to B. Through P and D draw PF, DG parallel to CE and equal to it, and through E draw FG parallel and equal to PX. Divide PC, CD into any number of equal parts in the points a, b, c, etc., and divide PF, D G each into the same number of equal parts in the points m, n, o, etc., s, t, u, etc., as each of the straight lines PC, CD have been divided into. Draw straight lines through the points a, b, c, etc., parallel to CE, and from E draw straight lines to the points m, n, o, etc., in PF, and the points s, t, u, etc., in DG. Then through the points 1, 2, 3, etc., formed by the intersection of Er with the parallel through f, E q, with the parallel through e, E p, with the parallel through d, etc., trace the curve EP above the axis E C, and the curve ED below it. The curve PE D is the required parabola.

A

B

Fig. 91.

PROBLEM LXIV.-To describe an hyperbola by mechanical

means.

The hyperbola, instead of being considered as a single curve, is frequently represented as consisting of two equal and symmetrical curves, having their vertices opposite each other, and their branches proceeding in contrary directions. The reason of this may be understood from Fig. 92, in which two cones are represented, the one having its apex against the apex of the other, and its base turned in the contrary direction. Such a double cone as this may be generated by the revolution of two equal equiangular and similar right-angled triangles, having their vertices contiguous, and their altitudes in the same straight line as the triangles A B C, A DE, in the figure. We may also conceive the double cone to be generated by the revolution of a straight line, BA F, or D A G, round its central point, A (which is fixed), and inclined at any angle less than a right angle to a perpendicular straight line, EA C, passing through A, which perpendicular becomes the axis of the cone thus generated. Now if we suppose a plane HKML to pass through the axis EC of the double cone, and the double cone to be cut by another plane parallel to the plane HK M L, as the plane N O P Q, it is manifest that it will cut each branch of the cone in NOR, PSQ, which form two equal c symmetrical and opposite curves, and which are considered as each forming a branch of the complete hyperbola.

Fig. 92. Our readers will now more readily comprehend the method of describing an hyperbola by mechanical means, and when certain data are given; and they will also understand why an hyperbola is said to have two foci, like an ellipse.

In x Y (Fig. 93), which represents any straight line of indefinite length, let two points, A and B, be selected as the foci of the hyperbola to be described. Take a flat, narrow ruler, CD, with a hole in it near one end, through which a pin may be inserted to fasten the ruler to the paper or board on which the hyperbola is to be traced, the ruler working freely round the pin. Suppose F, in A B, be selected as the vertex of the hyperbola that is to be traced. Take another point, E in A B, so that A E is equal to FB; then E will be the vertex of the opposite branch of the hyperbola, and E F the major axis of the curve. Let a string be fastened at the end, D, of the ruler C D, and let the string be of unlimited length, or, what is as well, of the same length us the ruler. Set off along the ruler from the point A, in the direction A D, a straight line A G, equal to E F, and holding the string tightly to the edge of the ruler, mark it at the point opposite to the point G in the ruler; then thrust a pin through the string at the point thus marked, and fasten it down at the point B. Keeping the cord stretched to its utmost tension with a pencil-point, and having the edge of the ruler applied to the straight line XY, move it slowly upwards round the

pivot A. Before starting, when the edge of the ruler is contiguous to x Y, the point G will be at H, the pencil-point at F, and the string in the position B F, F K. As the ruler moves upwards, the pencil-point traces out the curve FLN P, the point G describing or moving in the path of the arc HM, and the end of the ruler D in the path of the arc Ko. The point Q, where A B is bisected at right angles by the perpendicular R s, is the centre of the hyperbola. By reversing the ruler, and repeating the operation below XY, the lower part, FT, of the curve PFT may be traced; and by fixing the ruler so that the point represented in the figure at A may be at B, and the end of the string fastened at A, the opposite branch of the hyperbola passing through E may be described. The

X

Fig. 93.

11

straight line v B U, passing through the focus B, is called the latus rectum of the hyperbola; FB the abscissa, and BV the ordinate of the point v; FZ the abscissa, and PZ, TZ the ordinates of the points P and T. The chief peculiarity of the parabola is, that the distance of every point in the curve, as the ruler passes from one position to another from the focus, is equal to its distance from the point marked G in the ruler. Thus, when the ruler is in the position A K, and G is at H, FH is equal to FB; in the position AD, LB is equal to LG; in the position A O, N M is equal to N B; while in the position AP, PW is equal to P B. The distances, AF, FB; AL, LB; AN, NB; AP, PB, are called the focal distances of the points F, L, N, P respectively, and the difference between the greater and the lesser of any of these pairs of distances from the foci of the hyperbola is equal to EF, the major axis of the hyperbola; and this is true for every point in the curve. For this reason, in the commencement of the problem, A G was made equal to E F.

PROBLEM LXV.-To describe an hyperbola by firing a number of points through which the curve may be traced, the major aris, and the abscissa and ordinate of any point in the curve being given.

In Fig. 94 let any indefinite straight line, x Y, be the axis of the required hyperbola; the portion in. P tercepted between the points A and B being set off equal to P, the given major axis ; and Q, R being the given abscissa and ordinate of

a point in the curve. From B, set off along x Y, in the direction of Y,

X

Fig. 94.

B C equal to Q, and through c draw the straight line DE of indefinite length, at right angles to x Y; and from c along D E in the directions of D and E, set off CF, CG, each equal to E. The points F and G are points in the required curve. Through F and a draw F H, G K parallel to x Y, and through в draw HK parallel to DE. Divide CF, CG each into five equal parts in

the points a, b, etc., e, f, etc., and divide HF, KG also into five equal parts in the points k, l, etc., o, p, etc. Of course, when the curve is large, the greater the number of parts into which the double ordinate, FG, and the parallels, HF, K G, are divided, the more accurately the curve can be traced, care being taken to divide the parallels into the same number of equal parts as each half of the double ordinate, FG, is divided into. From the point a draw straight lines through the points F, a, b, etc., and from в draw straight lines through the points k, l, etc., o, p, etc., and through the points of intersection of

the lines Aa, Bn, Ab, âm, etc., numbered 1, 2, etc., and the points

required.

F, B, and G, trace the curve FB G. This curve is the hyperbola Lest some of our readers may be tempted to inquire of what practical use it may be to be acquainted with the method of tracing parabolas and hyperbolas of different degrees of curvature, we may remind them that the parabola sometimes is used in forming an arch, while such articles as tazzas and wineglasses, and other pieces of useful and ornamental china-ware, may be formed by the revolution of an hyperbola about its axis, as may be seen by copying the curve in Fig. 94, so that the vertex, B, points downwards, and then adding a slender stem and foot to form a wine-glass.

PROBLEM LXVI.-To describe the curve called the cycloid. The term cycloid, derived from the Greek KUKλocions (ku-kloi'-dees), like a circle, is a name given to the curve traced by any point in the circumference of a circle during the complete revolution of the circle while rolling along a straight line. For example, as a carriage is drawn along on a road or railroad, the end of any spoke in one of its wheels, or a nail in the tire, describes a succession of curves, similar to the curve resembling half of an ellipse in Fig. 95. That the reader may understand how the curve is traced, let A B C D represent a circle, having two diameters, A C, B D, intersecting each other at right angles, and let the circle be standing on a straight line, x Y, of indefinite length, so that the diameter AC is at right angles to XY, which is a tangent to the circle A B C D, the circle touching it only in the point A. Suppose the circle to roll slowly along the straight line XY, in the direction of x, and pass into the position A' B'C' D'. It has now performed a quarter of a complete revolution, and the point ▲ in ascending into the position A' has traced a path represented by the curve A A'. In the next quarter of a revolution the point A is brought to the top in the position A", and when a complete revolution of the circle has been made it has passed from A" to A"" and """, having traced in its passage from A to A""" the curve A A' A"A" A"". Practically, the cycloid may be traced by causing a thin disc of metal, ivory, or even cardboard, having a slight nick in its circumference to receive a pencil point, to travel slowly along the edge of a ruler until a

during the revolution. It is evident that, as every point of the circumference of the circle in succession touches the straight line A c during the revolution, a c, which we may call the base of the cycloid, is equal in length to the circumference of the circle B K D. If the circle be caused to return to its position in the centre of the cycloid when B is at its highest position, as in the figure, and straight lines, such as м H, N F, be drawn through the circle parallel to the base and terminating both ways in the curve of the cycloid, these straight lines pass through opposite points in the circumference of the circle B K D, at equal distances from the diameter B D, which is perpendicular to the base, G L being equal to G o, and E K to E P. It will be found that LH is equal to the arc B L, and that the arc BH is equal to twice the chord B L, and so on for the other points, M, N, F, in the curve of the cycloid, through which straight lines have been drawn parallel to the base. The arc BC will therefore be equal to twice BD, the diameter of the generating circle, and the whole curve A B C consequently equal to four times B D. This curve is said to have been discovered and its properties first investigated by Galileo. PROBLEM LXVII.-To describe a spiral.

drawn. Draw a horizontal straight line, x Y, of indefinite length Take any point, A (Fig. 97), as the centre of the spiral to be through A, and from A as centre, with any distance, ▲ B, describe the semicircle B D C. Then from the point в as centre, with the distance B C, describe the semicircle C E F on the opposite side of a B. Next, from A as centre, with the distance ▲ F, describe . the semicircle F G H, and then from B and A, in alternation as centres, at the

R

Fig. 97.

distances BH, A L, etc. etc., describe as many semicircles in succession as may be required. A spiral of any given number of turns may be described on a given straight line by dividing the given straight line into as many equal parts as there are turns required, and bisecting the central division if the number of turns be odd, or the division on the right or left of the centre of the line if the number of turns be even. The centres to be fixed in describing the semicircles must be the point of bisection, and either of the points of division immediately contiguous to it if the number of turns be odd, or the point of bisection and the centre of the divided line if the to describe a spiral of eight turns or semicircles on the given number of turns be even. Thus, in Fig. 97, if it be required straight line R T, divide R T into eight equal parts, in the centre of the spiral. Then from the points A and B, in alterna points N, H, C, B, F, L, P, and bisect BC, or BF, in A for the tion, describe the semicircles BD C, CEF, etc. etc. PROBLEM LXVIII.-Any two straight lines being given, tɔ by which they shall determine a curve

be connected.

E

Let A B, C D (Fig. 98) be any two straight lines which it is required to connect by a curve. Produce A B, C D in the direction of B and c, until they meet in E. Bisect the angle B E c by the straight line E F. From the extremities B and c of the straight lines A B, C D, draw B F, C F perpendicular to A B and C E respectively, and intersecting each other and the straight line EF in the point F. From F as centre, with the distance F B or Fc, describe the arc BC. This arc connects the straight lines AB, CD. The same process is followed when the given straight lines are at right angles to each other, as AB, GH, which are connected by a curve, B G, struck from K as centre, the point of

Fig. 98.

indicative mood present tense? what the Latin sign? what is the Latin sign of the sub. pluperf.? what the English sign of the same? So go through all the parts.

I hope you understand what I mean by these signs. Your understanding of them is the more important, because they pertain not merely to the verb amo, or to the first conjugation, but to all the verbs; and because, when you are perfect in your knowledge of them as just given, you will easily put Latin into English and English into Latin. On account of this importance, I will subjoin a few explanations.

These signs, then, might be called a set of equivalents, and I might have indicated them after this manner :

These signs or equivalents are, you see, without any verb. They are so given because they are applicable to all verbs. Thus to -i you prefix the stem amar, and make amavi; so to have you add I and loved, and make the corresponding English, that is, the English equivalent of amavi-namely, I have loved. In some instances the English sign is arbitrary, or the best we can get; in the ind. pres. love is chosen as the E. S. (English sign) for the want of a better. Scarcely less arbitrary is the E. S. of the imp.-namely, did.

These departures from exact correspondence, precision, and uniformity are certainly drawbacks; but notwithstanding these drawbacks, great aid may be derived from a careful and systematic attention to the system here set forth.

I have said that these signs are applicable to all verbs. If so, they need not be repeated. And in general the statement is correct. You will, however, bear in mind what you have previously learnt as to the tense-endings, and the mood-endings; and then you will remember that instead of -bo, -am (es, etc.) is .the ending, and as the ending so the sign of the first future of the third and the fourth conjugations. One or two other deviations will occur to you.

Abl. Ama-ndo. EXAMPLES.-Like this model, conjugate laudo, 1, I praise; curo, 1, I take care of; voco, 1, I call.

Compare together the 2 Fut. with the Sub. Perf. You will find that the endings are the same, except in the first person, which in the former is ro, in the latter rim. In other words, the Latin language has no distinctive form beyond the first person for one or the other of these tenses. A distinction is attempted with the aid of the accent or the quantity. Thus, the first person plural of the second future is pronounced long, as amaverimus, while the first person plural of the subjunctive perfect is pronounced hort, as amaverimus; and consequently you find the sign of the long vowel over the in the former tense, and the sign of the short vowel over the i in the latter tense, showing that, although the words are spelt alike, they are not pronounced in the same way.

There is a difference between the first future, amabo, and the future formed with the aid of the future participle, thus, amaturus sum. Amabo means I will or shall love, simply indicating a future act, without determining when, or the precise point in the future when the act will take place. Amaturus sum signifies I am about to love, that is, I shall shortly love; intimating that the action signified in the verb is near at hand, and in the immediate future.

Of the first future there is properly no subjunctive tense; the import, however, is expressed by combination, thus, amaturus sim (sis, sit, etc.), I may be about to love; amaturus essem, I might be about to love. The second future also is without a subjunctive mood.

EXERCISES.-Form according to the model now given, that is, write them out in full, with all the parts in both Latin and English, these verbs-laudo, 1, I praise; vigilo, 1, I watch; comparo, 1, I procure.

COMPARATIVE ANATOMY.—IX.

ROTATORIA-MYRIAPODA.

Myriapoda.

Arachnida.

PERHAPS it is better to notice at this stage a class of animals whose relations to other classes are difficult to express. As we have before stated that it is quite impossible to place the whole array of animals in a single line according to their grades of structure, the reader will not be surprised that we have to break off in the midst of the description of a definite and well-sustained series of animals to treat of a class which cannot well be inserted into that series. The class referred to is called Rotatoria. The animals which compose it are decidedly inferior in complexity of structure to the animals we shall have to describe as coming in the next order to the Annelids, and in many respects also inferior to the Annelids themselves, and yet they lead up to a class of animals called Crustacea, which are as decidedly of a worms. In higher type than the

Annelida

Crustacea.

Helminthozoa.

Echinodermata.

Cœlenterata.

Rotatoria.

Polyson

many respects, these are also superior to the Myriapoda, which directly succeed to the worms, and of which we shall write in the subsequent part of this lesson. The difficulties under which a constructor of a system of classification labours may be best ilIlustrated by the annexed diagram, in which the lines branching upward from the single stem marked Protozoa represent the relations of the divisions of the animal kingdom to one another. These relations are so complicated, and have occasioned so much diversity of opinion among naturalists, that it would be presumptuous to assume that the diagram gives the relations exactly as they

Protozoa.

exist, and there are, moreover, cross relations which it is impossible to represent by such a diagram; but the reader may gather from it some idea of the nature of the relations, and how impossible it is to follow them in a continuous description of the animal kingdom.

Obviously, if a writer were to pursue any one of the lines indicated, describing in order the animals which successively come under his notice on that line, he would be led further and further from the other lines, and he must pursue his course until he has arrived at the highest animal of the branch which he has been ascending; and then, like an Alpine traveller who has gained the summit of a peak, he will have to look around at similar elevations, between which and his own position there is no stepping-stone. Thus he must, of necessity, retrace his steps to the lower level, from which another ascending path takes its rise.

Another course, the one we have adopted, is to break off whenever a gap in the series occurs, and look around to see that we are not leaving behind us any group of animals of a similar or lower grade of structure, and if we are in danger of doing so, to return at once to the description of the neglected group. We are the more reconciled to this method of procedure, because the relations of the classes to one another are so far from being determined, that each independent author has a different arrangement.

It will be seen by the diagram that, while the classes Coelenterata, Echinodermata, Annelida, and Myriapoda seem to follow one another in a natural succession, leading up to the Insectathat order which, of all others in the articulate sub-kingdom, is perhaps the highest and most wonderfully constituted the Rotatoria seem to start in a rather loose relationship with the Protozoa, and to lead up towards the Crustacea, a class which, as represented by its higher orders, is almost as complicated in structure as the Insecta, but whose lower orders are very much less organised. It would seem also as though the great subkingdom of the Mollusca is connected to the Articulates through their lowest class, the Polyzoa, and the class which we now have to describe.

The Rotatoria were first classed with the Infusoria by Ehrenberg. This classification was not to be wondered at, as all the rotary animals are microscopic, and they are obtained from infusions of vegetable or animal substances in water. Their outward appearance is also not unlike the higher orders of the Protozoa, and they move about by the same means as many of these do that is, by means of the vibrations of closely-set, fine, short, delicate hairs, called cilia. These cilia are so named from the Latin cilium," an eye-lash." As these are the very minute organs of animals of less thanth of an inch in length, it may be well conceived that the name ciiia has relation to the form, and not to the size of the organs. The cilia in the Rotatoria, instead of being scattered all over the surface of the animal, as in Paramecium (a Protozoon), or in the Turbellaria, are confined to flat, convex lobes, situated round or near the mouth, whose edges they fringe. When the animal fixes itself, the motion of these lashes brings food to its mouth by causing currents of water to pass towards it; and when it relaxes its hold, then the same motion causes it to progress through the water much in the same way as a screw-steamer is propelled. Some of these animals have the lobes all united into one circular disc, and as the motion of the cilia is so ordered as to cause the appearance of a number of successive waves, following one another round and round the circle, it was once thought that the disc was a kind of cogged wheel whirling rapidly about a fixed axle. Hence the name Rotifera, or wheel-bearing animals, was given to them. If this had been the right explanation of the motion, it would have furnished an instance of a locomotive apparatus met with nowhere else in the whole animal kingdom. A little reflection concerning this contrivance led some naturalists to doubt whether it really existed. Of course it is essential to the mechanical device which we call a wheel that it should be entirely disconnected with the axle upon which it plays, otherwise it could not revolve; and yet it is essential that all animal structures, especially to those employed in locomotive actions, that there should be an organic communication between them and the organs of nutrition, by means of which liquids can be sent to supply the waste caused by vital actions. This liquid must also be sent in such a way as not to be lost or wasted in the transit. It would seem, then, that the mechanism of the wheel is incon

sistent with animal organism. This consideration led to a fresh study of the so-called wheel-animalcules. It is almost needless to remark that the separate cilia were too small for their motions to be distinctly traced, otherwise the mistake could never have occurred. It is now supposed that the successive action of the cilia gives rise to an optical illusion, by which the appearance of rotation is maintained, while the organ on which the cilia is situated remains stationary. This supposition is rendered almost a certainty by observing the same motion in those nearly-allied creatures, members of the same class, whose discs are not circular, but divided into lobes. In these species it could be seen that the lobes did not participate in the revolutions. The way in which this optical illusion is effected will be best seen by reference to the illustration (Fig. VII.). From this it may be seen that if the cilia are deflected from the perpendicular only in one direction, and that a number of these act together, so as to cross one another while the down-stroke is given, it will give rise to a number of dark points where the crossing occurs, which points, by the successive action of each cilium in the series, will seem to pass rapidly round the disc, while, since each returns to its erect position separately and slowly, the eye cannot trace their motion. This method of explanation is rendered more probable by the fact that these aquatic creatures are usually examined under the microscope by means of transmitted light, and hence anything which cuts off the rays of light at a particular point will catch the eye and be followed by it.

These cilia are found so very generally throughout the range of the animal series--they are placed on such different parts of animals, and applied to such different purposes-that it is as well to give some little time to the consideration of them. We have already had occasion to mention them as covering the body of some Infusoria, and being applied to locomotion. They are also found on the inner (as well as the outer) wall of the Coelen. terata, and there cause a circulation of the fluid in the stomach. They are set on the combs of the Ctenophora, or bands on the larvæ of the Echinodermata, and in these situations are swim ming organs. We mentioned them also as set on the tufts of vessels called gills in the Annelids, and we shall find them again on the plate-like gills of Lamellibranchiata, and in these positions they cause a change in the external water, and so subserve the function of respiration. In the human subject they cover the membrane of the nasal chambers, the trachea, and the tubes leading to the lungs, and are continually employed to bring up the mucous which would else choke the passages. In all these cases, and in a thousand more which might be mentioned, their action, though applied to different purposes, is essentially the same. Their motion always creates an appearance of waves moving along in one definite direction, and never returning. It is very easy to attribute motion to ciliary action, and, of course, if the action be capable of driving liquid over the surface, it is also able to move the surface upon which the cilia are set, and the animal with it when that animal floats in liquid; but it is not an easy thing to explain the method of this action. When we say that the circulation in sponges is maintained by the ciliated chambers, the cilia of which whip the water in one direction, we are repeating what a multitude of writers have said before us, but we by no means explain the motion. If a switch be passed violently backwards and forwards through air or water, it creates a commotion, but it has no tendency to move the air or water, or the hand which holds it, in any definite direction. How, then, do these minute switches effect their purpose? Why does not the effect of the motion in one direction exactly counterbalance the effect of the motion in the other? The writer conceives the following to be the explana tion, for which the reader will be in some measure prepared by the remarks already made on the ciliary action in the Rotatoria. Suppose we conceive of a number of upright rods set on a mem. brane in a line corresponding to the line of the resulting waves, and moving in a direction at right angles to this, or in the direction of the waves caused by them. If one cilium or rod act alone, being rapidly brought down, the liquid will be thrown off from its sides to the right and left, the more obliquely in proportion to the rapidity of its motion. It will make its way by splitting the fluid, which, being thrown off laterally, will finally unite behind it. But suppose the rods on each side of this single rod are in motion in a parallel direction at the same time, then it comes in contact, not with stagnant water, but with the conjoined stream thrown off by these, which furnishes a