Names of Ratios. versed sine suversed sine Contractions. cosec. coversin. The latter three names are given to the following expressions: 1 —cosine, 1 — sine, and 1+cosine; that is, versin. 1 —1— cos. ; coversin. 1-sin.; and suversin. 1+cos. tangent, and secant of the arc вM, which is the complement of the arc A м, and the measure of the angle в OM (the complement of the angle AOM), are called respectively the cosine, cotangent, and cosecant of the arc A M, or of the angle A o м, these terms being contractions for complement-sine, complement-tangent, and complement-secant. The names tangent (touching) and secant (cutting) are sufficiently indicative of the straight lines A T and or, inasmuch as the one touches the circle at A, and the other cuts the circle at м; the name sine is Eot so clearly indicative of its meaning, Referring to fig. 1 again, the trigonometrical ratios belonging cavity), the straight line MP being drawn within the circle, and although it be said to be derived from sinus (a curvature or to the salient angles AON' and AON in the third and fourth limited by the curvature of the arc MA. The part AP of the quadrants, that is, the angles whose initial and terminal radius o A is called the versed sine of the arc A M, or of the angle positions are respectively o A and O N', and o A and ON, reckon-AOM; it is turned (verto, I turn) from the sine MP. ing the revolution of the straight line about o, from right to left, will be, according to the Elementary Triangles O N' P' and O NP, as follows: metrical lines in the other three quadrants will be as folAccording to the old system of trigonometry, the trigono lows: Third Quadrant. NP sin. According to the old system of trigonometry, which has given place to the modern system to which the preceding definitions belong, angles were measured by circular arcs, and the straight lines drawn in and about these arcs received the names of the Trigonometrical Ratios above explained; and this application of these names, which existed long before the modern improvements, accounts for their origin, in general, in a satisfactory manner. Thus, if in fig. 1 we suppose a circle A B A B′ to be described from the centre o and with radius o A, it will pass through the points M, M', N', and N, if the parts o M, O M', ON', and on be made all equal to one another. Then, the same constructions being made, and the tangent T T and ss' being drawn through the points A and B, as also the chords MN, M M′, M′N', and N'N, we shall then have, according to the old system, the following definitions: NP = sin. AT' tan. In the old system of trigonometry, a straight line urawn from one extremity of an arc to the other extremity is called the chord of the arc; accordingly a straight line drawn from M to A is the chord of the arc ▲ M. This straight line is also the chord of the arc MBA'N A, whieh is sometimes called the explement of the arc A M. The chord of a sixth part of the circle is equal to the radius of the circle, that is, the chord of an arc of 60° is equal to the radius. In the construction of the Trigonometrical Lines on the Plane Scale, fig. 2, p. 13, vol, iv., the Line of Chords, marked CH, is constructed thus: with any given distance, say the distance from 0 to 60° on the scale, describe a circle, and divide the first quadrantal arc into 90 equal parts or degrees; then draw chords from one of the extremities of the quadrantal arc to each degree: next, lay down their lengths on the scale, in a straight line from CH to 90, and you will have the Line of Chords as marked on the scale. The Line of Sines, marked si, is constructed thus: describe the circle and divide it as before; then draw perpendiculars from each of the degrees, beginning at 1°, to one of the radii of the quadrantal arc; next lay down their lengths on the scale, in a straight line, from si to 90°, and you will have the Line of Sines as marked on the scale. The perpendicular MP is called the sine of the arc a м, or of The Line of Tangents, marked TA, is constructed thus: grathe angle A o M, of which this arc is the measure; the straight duate the circle as before; then draw an indefinite perpendiline AT intercepted between the point of contact A, one cular to the radius passing through the extremity of the quadextremity of the arc AM, and the straight line or drawn rantal are marked 0°; next draw straight lines from the centre through M, the other extremity of the arc, is called the tangent through each of the degrees in the quadrantal arc to meet that of the arc a M, or of the angle A O M; and the straight line or perpendicular; then lay down on the scale, in a straight line, intercepted between the vertex of the angle or centre of the the distances between the fixed extremity of the perpendicular circle and the tangent, is called the secant of the arc A M, or of the and the successive points of the intersection of these straight angle A O M. Either of the radii o A, O M, O M', &c., is generally lines drawn from the centre and that perpendicular, and you called unity, and the lengths of the other lines, as compared will have the Line of Tangents as marked on the scale. On with the length of this unit, are determined accordingly. the scale the Line of Tangents extends only to 70°; beyond Again, the straight lines QM, BS, and os, which are the sine, I this point the scale would require to be greatly lengthened for the tangents of degrees beyond 70°; and the tangent of 90° is infinite, i.e., without end. The Line of Secants marked SE is constructed thus: lay down the lengths of the straight lines extending from the centre to the tangents of each degree in succession, on the scale in a straight line, and you will have the Line of Secants, which begins where the Line of Sines terminates. The line marked RU for RHUMBS, is the line of the chords of the different points of the compass, and is constructed thus: divide the first quadrantal arc into 32 equal parts; then draw chords from one of the extremities of the quadrantal arc to each division; next lay down their lengths on the scale in a straight line from Ru to 8, and you will have the Line of Rhumbs, as marked on the scale; the number of points in a quadrant are only 8, which are marked from 1 to 8 on the scale, but every quarter point is also marked although not numbered; and S points make 32 quarter points. The Line of Leagues, marked LE, is only a scale of equal parts; and so is the line marked E. P.; the first division of each being divided into tenths. The line of Semitangents, marked S. T., is merely a Line of Tangents of Half the Arcs, and is constructed thus: divide the first quadrantal arc as before, and from the remote extremity of the second quadrantal arc (that is, the point of the semicircle marked 180°), draw straight lines to the successive degrees of the semicircle, beginning at 19; then lay down the distances between the centre of the circle, and the points of the intersection of these straight lines with the perpendicular to the radius of the quadrantal are which passes through 0°, on the scale, in a straight line, and you will have the Line of Semitangents as marked on the scale. Lastly, The Line of Longitudes, marked Lo, is constructed thus: divide one of the radii of the quadrantal arc into 60 equal parts; then, through each of these divisions draw perpendiculars to that radius, intersecting the quadrantal arc in as many points, and number them from 1 to 60, beginning from that formed by the perpendicular nearest the centre; next, draw chords to these points from that extremity of the quadrantal are marked 60; lastly, lay down in a straight line, on the scale, the lengths of these chords, and you will have the Line of Longitudes as marked on the scale. The use of these various lines on the Plane Scale, and the application of the Protractor in the Solution of Problems, must be deferred till we give another Lesson in Instrumental Arithmetic. LESSONS IN CHEMISTRY.-No. Vl. In my preceding lesson I left off with a general description of the nature and uses of the pneumatic trough; the student will now proceed to use this trough according to instructions. The object to be aimed at is the collection of hydrogen gas; which of course we must make before we can collect, and the process of making it having been gone through already, requires no fresh description. All the various pieces of apparatus necessary to this collection, save one, have been mentioned. This unmentioned one is a bent tube of some sort for transmitting the gas from the generator to the storing bottle or jar (see fig 30). Fig. 30. 1 care very little of what the student makes it, whether of glass, metal gas-pipe, india-rubber, gutta-percha, leather, or any other material which necessity or ingenuity may suggest; all I do care about is, that the tube shall be air-tight; and this is an essential point. Short lengths of metal gas-pipe are exceedingly useful in the laboratory for effecting communications like this; but one point in connexion with such tubes the student will soon discover, they do not easily admit of permanently tight adaptation to apertures. in corks. Nevertheless this may be accomplished, and the tightness retained by care. Above all things the operator should avoid giving the portion of tube enveloped by the cork a short twist so as to produce a distortion, of which fig. 31 is an exaggeration. Fig. 31. It is easy to understand that such a twist being given, the original air-tighting, however perfect, is henceforth destroyed. We have now provided all the necessary apparatus for collecting the gas, but the storing of it requires a trifling addition in the shape blown glass, as being more accurately plain; but the latter, ren-of discs or plates of glass. Plate-glass is better than common dered flat by grinding on a stone or another piece of glass along with emery and water, answers perfectly well. The shape of these discs may be either square or circular. They may be cut either by means of a glazier's diamond, or both nearly as well, the round ones indeed better, by a common pair of scissors used under water. A plate of glass thus treated does not admit of being cut like a bit of silk or cotton. I admit, nevertheless, that a careful person may, after some preliminary trials, trim plates of glass by this means to almost any shape he desires. Well, we are now ready to commence operations. bottle under water; remove the bottle and disc from the water, and (1.) Apply the glass discs successively to the mouth of each see, as you easily can, whether contact between the plate and mouth of the bottle takes place all round; if not, grind the mouth or the plate as necessity may require with emery or silver sand and water, until contact is perfect. (2.) Dry the face of the bottles' mouths, also the discs; smear each with a little pomatum. (3.) Fill a bottle with water; invert it over the pneumatic trough; transmit hydrogen gas into it; not beginning to collect the gas immediately it is developed, but waiting a short time until you are certain that all the atmospheric air originally contained in the generating bottle has been expelled. As soon as the bottle has become full of gas, i.e., empty of water, slide under its mouth. one of the oiled and accurately-fitting glass plates (fig. 32). This tube is indicated in the preceding diagram by the letter t. (4.) Next place the bottle to stand on a table until wanted, taking the precaution to lay some sort of weight upon the glass plate to prevent its being raised up by the probable expansion of the gas. Having thus collected a few bottles of hydrogen, you can proceed to make yourself acquainted with its prominent qualities. (5.) Attach a bit of wax taper to a stem of wire (fig. 33), by stick Fig. 33. ing the wire into the wax, light the taper and plunge it into an inverted bottle of hydrogen gas, as represented in fig. 34. Particularly observe two phenomena :—(a) The gas itself burns where it comes into contact with the atmosphere. (6) The taper when plunged up into the gas is extinguished. Deductions. Therefore hydrogen gas is lighter than air, otherwise it would come out of the inverted bottle. It is a combustible but not a supporter of combustion. (6.) Repeat the experiment, having reversed the conditions of the bottle, i.e. place it to stand mouth upward; remove the glass plate, and plunge into the bottle the ignited taper. The Îatter now continues to burn as it did in the naked atmosphere, proving again the extreme lightness of hydrogen, by showing that it has escaped. (7) Pour some lime water very rapidly into a bottle containing hydrogen; replace the glass plate before all the hydrogen has escaped, agitate the bottle, and remark that no change is perceptible. Fig. 34. Does not restore reddened litmus to its original blue; therefore is not alkaline. General Remarks concerning the Nature of Flame.-Perhaps you observed, when the jet of hydrogen was ignited, that it burned with a pale and scarcely perceptible flame. From that circumstance you might have inferred that very little heat is developed by such flame. This idea is incorrect; the flame produced by the burning of hydrogen gas is really very powerful as to heat, and generally, let it be remembered, that the heating power of a flame is in an inverse ratio to its illuminating power. The most violent flame, as to heating and firing effects, results from the combustion of two measures of hydrogen gas and one measure of oxygen gas; but the light of this flame is scarcely perceptible. The reader will here do well to again develope some hydrogen gas in the tobacco-pipe bottle apparatus, and set the gas on fire as it escapes. Whilst burning, if some powdered charcoal, or magnesia, or lime, or indeed almost any powder, be sifted into the flame, its illuminative property will greatly increase. The sifting can be best effected by attaching a screw to the end of a stick; placing the powder to be sifted in the sieve, and striking the end of the stick with a mallet, fig. 35. N.B. Lime water is prepared by soaking a piece of quicklime in distilled water; atmospheric contact not being permitted, i.e., perform the operation in a bottle filled to the stopper with water. The transparent portion of the resulting liquid is called lime water, the turbid sediment cream of lime. (8.) Moisten a piece of blue litmus paper with distilled water, and hold it in an inverted bottle-full of hydrogen gas. Remark that no change takes place. N. B. Litmus paper and tincture of litmus are general tests of acidity. Acids turn these materials red. Deduction. Hydrogen gas is not acid. (9.) Tinge a moistened slip of litmus paper red, by holding it for a few instants over the mouth of a bottle containing any volatile acid, such as spirit of salt (hydrochloric or muriatic acid). Immerse this moistened slip in a bottle containing hydrogen gas as before. Remark that no change ensues, the redness of the paper remaining unimpaired. N.B. Litmus paper thus reddened is a test for alkalies generally, which class of bodies cause the original blue colour to return. Deduction. Hydrogen gas is not alkaline. Instead of litmus paper reddened, yellow turmeric paper might have been used; alkalies change the colour of this to brown. (10.) Partly fill a bottle with hydrogen gas; apply the glass plate; agitate; reimmerse in water; remove the glass plate, and remark that no fresh portion of water rushes into the bottle; thus proving that hydrogen gas is not perceptibly absorbable by water. From the result of this experiment it may be deduced, firstly, that red-hot hydrogen is not very luminous; secondly, that redhot solid particles are more luminous; and it may be suspected that red-hot solid particles exist in the flame of candle-lamps, coal gas, and other similar illuminative sources. The suspicion is just: every person is aware that an object immersed in a flame of this kind becomes sooty or black. On what then does this sootiness depend? On charcoal, this being the solid matter which nature designs to become red-hot in an illuminative flame. The student will not forget, then, the fact that coal-gas, oil, tallow, &c., contain, as one of these elements, charcoal, or, in chemical language, carbon; indeed, generally any substance that during combustion covers an object immersed in it with a sooty coat contains charcoal or carbon. The student will not fail to see, moreover, that charcoal, when burned, becomes invisible; which invisible product must be a gas. It is called carbonic acid gas, It shall be the object of a future lesson to teach something should be made acquainted with the theory of the changes which more about this gas; meantime it is proper that the learner ensue when sulphuric acid and water are added to zinc. He will result is a solution of sulphate of zinc, and the gaseous result is say, perhaps, I already know what these changes are; the liquid hydrogen gas; what more can I want to know? Yes, you require to know a little more than this, and the best way of imparting this further knowledge will be by means of a diagram as follows: PLUR. SING. PLUR. SING. PLUR. SING PLUR. SING. PLUR. SING. PLUR. SING. zinc, to combine in its turn with sulphuric acid. The reader will moreover observe that what we call, for shortness, sulphate of zinc is really sulphate of the oxide of zinc. Acids never combine with metals, but with acids. L With respect to the diagram just given, I advise the student, whenever he is in doubt as to the changes which ensue during chemical composition or decomposition, to have recourse to a diagram. First put down all the substances employed, then divide them into three components, then join the elements together by lines or brackets in a manner that shall be accordant with actual I shall not say more about it at present, but shall simply content myself by remarking, that chemical combinations do not take place in proportions a little more or a little less, but they are fixed, unvarying, definite, and therefore capable of representation by numbers, which latter are called the atomic equivalents Thus 8 is or proportional numbers of the bodies concerned. the atomic number for oxygen, and 1 for hydrogen, conYou. sequently the atomic number of water must be 9. must learn the atomic numbers of simple badies, but do not attempt too much at a time. Remember on this occasion the atomic numbers of hydrogen, oxygen, and zinc-1, 8, and 32; One point connected with the preceding diagram requires fur- this is surely no difficult matter. If you choose to remember ther explanation; I mean the numbers there given. My first the atomic weight of sulphuric acid to be 40, well and good; hereintention was to have omitted them, because the general explana- after you will get at this information through another channeltion of what took place would have been equally comprehensible you will be told that sulphuric acid is a compound of three equiwithout them. Further reflection caused me to alter this deter-valents of oxygen and one of sulphur; now the equivalent number mination; let the reader, then, consider them as the shadow of a of sulphur being 16 and of oxygen 8, it follows that 16+(3x8). coming doctrine-the atomic theory and doctrine of definite pro- 40; or, in the symbols of chemical algebra, SO3S+30, S. portions. standing for sulphur and O for oxygen. results. A (11) Remarks on mögen. Mögen marks possibility under allowance or concession from another: as, Er mag lachen, he may laugh; that is, he has permission to laugh, no one hinders him. Er mag ein braver Mann fein, he may (I grant) be a brave man ; where the possibility of his being a brave man is a thing conceded. Kindred to this are the other significations (chance, inclination, wish, &c.) usually attributed to this verb: thus, es möchte regnen, it might rain; that is, the causes that seem to forbid, are likely not to operate; ich möchte es bezweifeln, I am disposed or inclined to doubt it, that is, I might doubt it altogether, but for certain circumstances seeming to forbid: möge es der Himmel geben, may heaven grant it; ich mag es nicht thun, I do not like to do it, that is, I am not permitted by my feelings to do it cheerfully, &c. LESSONS IN GEOLOGY.-No. XLV. CHAPTER IV. ON THE EFFECTS OF ORGANIC AGENTS ON THE SECTION III. ON THE AGENCY OF CORAL INSECTS IN THE PRODUCTION OF N almost every district on the surface of the globe, and at This class of animals is constantly called coral. This is not their appropriate name, for coral is the name of the rock that is built, and not of the animal that constructs it. They are sometimes called Zoophytes, a Greek term which means animal plants, on account of their resemblance in form to growing plants. At other times they are called Polyparia, and Polypifera. These and others are only names for the coral insect, or the animal that constructs the coral rock. The coral insect consists of a little oblong bag of jelly, which is closed at one end but open at the other. The mouth of the bag is surrounded by the insect's tentacles or feelers, which are generally about six or eight in number, and dart in all directions like the rays of a star. Myriads of these minute animals live close together, and unite to form a common stony skeleton called coral, in the minute openings of which they live. When they are under water, they protrude their mouths and tentacles to seize and receive their calcareous food; but the moment they are apprehensive of danger, they withdraw into their holes. These calcareous abodes form, over the bottom of the sea, stony cases, called coral banks or coral reefs, which they build up from a moderate depth, not much exceeding a hundred feet. It is found that at different depths, and in different areas, corals of different species develope themselves. Their range extends on each side the equator between 32° north latitude and 28° south latitude. The amount of coral rocks in different oceans is enormous; but not so enormous as was at first apprehended by the earlier navigators. The scientific men, who accompanied exploring expeditions, found the Indian and Pacific Oceans studded every where with the products of these polyparia. As the seas in the immediate neighbourhoods of coral rocks were always well-nigh unfathomable, it was conjectured that the coral insects had built up their masonry from a sea bottom at immense depths. In the coral rocks which appeared above the surface of the sea, the insects had finished their work and died; but it was conjectured that other zoophytes were, in the meantime, just commencing their architecture at the bottom of deep seas, were spreading their sheets of coral rock over a vast area of sea bottom, and that they, in their turn, would work up their rocky structures to the surface of the ocean. MM. Quoy and Gainard were the first to show that the coral insects had not built up their masses of rock from great fathoms deep, and that these incrustations rested upon some depths, but had merely produced stony incrustations a few underlying rocks. They also remarked that wherever land was cut into bays with shallow and quiet water, and exposed most, and there they incrusted the rock the most extensively. to the intense heat of the sun, there the polyparia abounded From circumstances of this character it was conjectured that the coral reefs and coral islands took their form and shape from the forms and the inequalities of the rocks on which they were built, and that circular or oval islands owed their form to the underlying crusts of the craters of submarine volcanoes. All these hypotheses have been long ago exploded, partly by Sir CHARLES LYELL, but chiefly by Mr. CHARLES DARWIN, the most distinguished and the most successful student of coral formations. Coral rocks are divided into three great classes, called respectively Atolls, Barrier Reefs, and Fringing Reefs. ATOLLS used to be called Lagoon Islands, and consist of rings of land in the midst of the ocean. The ring of land, sometimes oval or egg-shaped, is a few hundred yards in breadth. These ring islands or atolls are sometimes only a mile in diameter, but sometimes as much as thirty miles. Land of this description is generally low, rising but little above the level of high water, but covered with cocoa-nut trees and pandanus of great height (see illustration, fig. 102, at the close of this lesson). Within these rings of land is a bed of calm, clear, and shallow water. It is this sheet of water that is called a lagoon. In this water the more minute and the more delicate kinds of coral insects find a tranquil abode, while the stronger and the larger live and work on the outer margin of the ring among the waves and breakers. Every such atoll has an opening at one part of it, which allows a ship of any burden to pass from the ocean into the lagoon. The second class of coral rocks consist of Barrier Reefs. These are coral rocks which either extend in straight lines in the front of a continent or of a large island, or encircle smaller islands. In both cases, as they are separated from the land by a broad and rather deep channel of water, they are analogous to the lagoon within the atoll. The annexed illustration (fig. 99) represents a part of the barrier that encircles an island. It is a true sketch of the Island of Bolabola, as seen from one of the central peaks. You see that the coral reef is covered with palm trees, and you must imagine that the reef completely encircles the island, in the centre of which you see that peaked rock. That reef was all worked beneath the sea, but by a volcanic upheaval, sudden or gradual, it has become dry land, The extent and dimensions of these barrier reefs vary from three miles to more than forty miles in diameter. There is near New Caledonia a reef, fronting one side and encircling both ends of the island, that is 400 miles long. |