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Names of Ratios.
tangent, and secant of the arc BM, which is the complement of the
arc AM, and the measure of the angle BOM (the complement of cotangent
the angle AO M), are called respectively the cosine, cotangent, cosecant cosec.
and cosecant of the arc A M, or of the angle A o M, these terms versed sine
versin. coversed sine coversin.
being contractions for complement-sine, complement-tangent, and suversed sine suversin.
The names tangent (touching) and secant (cutting) are The latter three names are given to the following expressions : sufficiently indicative of the straight lines a T and or, inas1 - cosine, 1 - sine, and 1+cosine; that is, versin. =1-cos.; much as the one touches the circle at A, and the other cuts the circle coversin.=1-sin.; and suversin=l+cos.
at m; 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 A o n' and A on in the third and fourth limited by the curvature of the arc ma. The part A P 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
o it is (. ing the revolution of the straight line about o, from right to left, will be, according to the Elementary Triangles o n' P' and metrical lines in the other three quadrants will be as fol
According to the old system of trigonometry, the trigono
In the Second Quadrant.
of the arc A BM'
COS, - cot. os' = cosec.
sec. N'a = cos. BS
of the arc ABN'
of the salient angle A ON.
In the old system of trigonometry, a straight line urawn
from one extremity of an arc to the other extremity is called = cot.
the chord of the arc; accordingly a straight line drawn from m to A is the chord of the arc A M. This straight line is also the chord of the arc MBA'NA, which is sometimes called the explement of the arc AM. 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 According to the old system of trigonometry, which has 60° is equal to the radius. given place to the modern system to which the preceding In the construction of the Trigonometrical Lines on the definitions belong, angles were measured by circular arcs, and Plane Scałe, fig. 2, p. 13, vol. iv., the Line of Chords, marked the straight lines drawn in and about these arcs received the ca, is constructed thus: with any given distance, say the names of the Trigonometrical Ratios above explained; and distance from 0 to 60° on the scale, describe a circle, and this application of these names, which existed long before the divide the first quadrantal arc into 90 equal parts or degrees ; modern improvements, accounts for their origin, in general, in then draw chords from one of the extremities of the quadrantal a satisfactory manner.
arc to each degree: next, lay down their lengths on the scale, Thus, if in fig. 1 we suppose a circle A B A' B' to be described in a straight line from ch to 90, and you will have the Line of from the centre o and with radius o A, it will pass through the Chords as marked on the scale. points M, M, N, and n, if the parts o M, O M', on, and on be The Line of Sines, marked si, is constructed thus: describe made all equal to one another. Then, the same constructions the circle and divide it as before ; then draw perpendiculars being made, and the tangent TT and ss' being drawn through from each of the degrees, beginning at 1°, to one of the radii the points A and B, as also the chords MN, MM, M'n, and n'n, of the quadrantal arc; next lay down their lengths on the we shall then have, according to the old system, the following scale, in a straight line, from si to 90°, and you will have the definitions :
Line of Sines as marked on the scale. The perpendicular mpis called the sine of the arc A M, or of The Line of Tangents, marked ta, is constructed thus: grathe angle A 0 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 arc marked 08; next draw straight lines from the centre through y, 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 AM, or of the angle A OM; 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 AM, or of the and the successive points of the intersection of these straight angle A OM. Either of the radii 0 A, OM, OM', &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 90is | 1 care very little of what the student makes it, whether of glass, infinite, i.e., without end.
metal gas-pipe, india-rubber, gutta-percha, leather, or any other The Line of Secants marked se is constructed thus : lay down material which necessity or ingenuity may suggest'; all I do care the lengths of the straight lines extending from the centre to the about is, that the tube shall be air-tight; and this is an essential tangents of each degree in succession, on the scale in a straight point, Short lengths of metal gas-pipe are exceedingly useful in line, and you will have the line of Secants, which begins the laboratory for effecting communications like this; but one point where the Line of Sincs terminates.
in connexion with such tubes the student will soon discover, they The line marked Ru for RHUMBS, is the line of the chords do not easily admit of permanently tight adaptation to apertures, of the different points of the compass, and is constructed thus: in corks. Nevertheless this may be accomplished, and the tightness divide the first quadrantal arc into 32 equal parts; then draw retained by care. Above all things the operator should avoid chords from one of the extremities of the quadrantal arc to giving the portion of tube enveloped by the cork a short twist so each division ; next lay down their lengths on the scale in a
as to produce a distortion, of which fig. 31 is an exaggeration. 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
Fig. 31, are only s, which are marked from 1 to 8 on the scale, but every quarter point is also marked although not numbered and 5 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 tonths, 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,
It is easy to understand that such a twist being given, the beginning at 1°; then lay down the distances between the centre of the circle, and the points of the intersection of these original'air-tighting, however perfect, is henceforth destroyed.' We straight lines with the perpendicular to the radius of the have now provided all the necessary apparatus for collecting the straight line, and you will have the Line of Semitangents as blown glass, as being more accurately plain; but the latter, ren-quadrantal arc which passes through 00, on the scale, in a gas, but the storing of it requires a trifling addition in the shape
of discs or plates of glass. Plate-glass is better than common marked on the scale. Lastly, The Line of Longitudes, marked Lo, is constructed thus : divide with emery and water, answers perfectly well. The shape of
dered fat by grinding on a stone or another piece of glass along one of the radii of the quadrantal arc into 60 equal parts ; then, these discs may be either square or circular. They may be cut through each of these divisions draw perpendiculars to that either by means of a glazier's diamond, or both nearly as well, radius, intersecting the quadrantal arc in as many points, and the round ones indeed better, by a common pair of scissors used number them from 1 to 60, beginning from that formed by | under water. A plate of glass thus treated does not admit of che perpendicular nearest the centre; next, draw chords to being cut like a bit of silk or cotton. I admit, nevertheless, that these points froin that extremity of the quadrantal arc marked a careful person may, after some preliminary trials, trim plates of 60 ; lastly, lay down in a straight line, on the scale, the glass by this means to almost any shape he desires. Well, we are lengths of these chords, and you will have the Line of Longitudes now ready to commence operations. as marked on the scale. The use of these various lines on the Plane Scale, and the bottle under water; remove the bottle and disc from the water, and
(1.) Apply the glass discs successively to the mouth of each application of the Protractor in the Solution of Problems, must be deferred till we give another Lesson in Instrumental 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 Arithmetic,
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 LESSONS IN CHEMISTRY.-No. VI.
each with a little pomatum. In my preceding lesson I left off with a general description of (3.) Fill a bottle with water; invert it over the pneumatic the nature and uses of the pneumatic trough ; the student will trough ; transmit hydrogen gas into it; not beginning to collect the now proceed to use this trough according to instructions. The gas immediately it is developed, but waiting a short time until object to be aimed at is the collection of hydrogen gas; which you are certain that all the atmospheric air originally contained of course we must make before we can collect, and the process of in the generating bottle has been expelled. As soon as the bottle making it having been gone through already, requires no fresh has become full of gas, i.e., empty of water, slide under its mouth. description. All the various pieces of apparatus necessary to this one of the oiled and accurately-fitting glass plates (fig. 32). collection, save one, have been mentioned. This unmentioned one is a bent tube of some sort for transmitting the gas from the
Fig. 32, generator to the storing bottle or jar (see fig 30).
(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 aoquainted with its prominent qualities.
(5.) Attach a bit of wax taper to a stem of wire (fig. 33), by stick
ing the wire into the wax, light the taper and plunge it Does not restore reddened litmus to its original blue; therefore into an inverted bottle of hydrogen gas, as represented in fig. 34. is not alkaline.
Particularly observe two phenomena :-(a) General Remarks concerning the Nature of Flame.--Perhaps you Fig. 33.
The gas itself burns where it comes into contact observed, when the jet of hydrogen was ignited, that it burned with the atmosphere. (6) The taper when with a pale and scarcely perceptible flame. From that circumplunged up into the gas is extinguished. stance you might have inferred that very little heat is developed
Deductions. Therefore hydrogen gas is lighter by such flame. This idea is incorrect; the flame produced by the than air, otherwise it would come out of the burning of hydrogen gas is really very powerful as to heat, and inverted bottle. It is a combustible but not a generally, let it be remembered, that the heating power of a flameis supporter of combustion.
in an inverse ratio to its illuminating power. The most violent (6.) Repeat the experiment, having reversed flame, as to heating and firing effects, results from the combustion the conditions of the bottle, i.e. place it to stand of two measures of hydrogen gas and one measure of oxygen gas; mouth upward; remove the glass plate, and but the light of this flame is scarcely perceptible. plunge into the bottle the ignited taper. The The reader will here do well to again develope some hydrogen latter now continues to burn as it did in the gas in the tobacco-pipe bottle apparatus, and set the gas on fire naked atmosphere, proving again the extreme as it escapes. Whilst burning, if some powdered charcoal, or lightness of hydrogen, by showing that it has magnesia, or lime, or indeed almost any powder, be sifted into the escaped.
flame, its iliuminative property will greatly increase. The sifting (7.) Pour some lime water very rapidly into can be best effected by attaching a screw to the end of a stick;
a bottle containing hydrogen; replace the glass placing the powder to be sifted in the sieve, and striking the end plate before all the hydrogen has escaped, agitate the bottle, and of the stick with a mallet, fig. 35. remark that no change is perceptible.
N.B. Lime wator is prepared by soaking a piece of quicklime From the result of this experiment it may be deduced, firstly, in distilled water; atmospheric contact not being permitted, i.e., that red-hot hydrogen is not very luminous; secondly, that redperform the operation in a bottle filled to the stopper with water. hot solid particles are more luminous; and it may be suspected The transparent portion of the resulting liquid is called lime that red-hot solid particles exist in the flame of candle-lamps, coal water, the turbid sediment cream of lime.
and other similar illuminative sources. The suspicion is (8.) Moisten a piece of blue litmus paper with distilled water, just : every person is aware that an object immersed in a flame and hold it in an inverted bottle-full of hydrogen gas. Remark of this kind becomes sooty or black. On what then does this that no change takes place. N. B. Litmus paper and tincture of sootiness depend! On charcoal, this being the solid matter litmus are general tests of acidity. Acids turn these materials which nature designs to become red-hot in an illuminative flame. red. Deduction. Hydrogen gas is not acid.
The student will not forget, then, the fact that coal-gas, oil, tal(9.) Tinge a moistened alip of litmus paper red, by holding it low, &c., contain, as one of these elements, charcoal, or, in for"a few instants over the mouth of a bottle containing any chemical language, carbon; indeed, generally any substance that volatile acid, such as spirit of salt (hydrochloric or muriatic during combustion covers an object immersed in it with a sooty acid). Immerse this moistened slip in a bottle containing hydrogen coat contains charcoal or carbon. The student will not fail to gas as before. Remark that no change ensues, the redness of the see, moreover, that charcoal, when burned, becomes invisible; paper remaining unimpaired. N.B. Litmus paper thus reddened which invisible product must be a gas. It is called carbonic aciá is a test for alkalies generally, which class of bodies cause the gas,
It shall be the object of a future lesson to teach something original blue colour to return. Deduction. Hydrogen gas is not alkaline. Instead of litmus paper reddened, yellow turmeric paper should be made acquainted with the theory of the changes
which more about this gas; meantime it is proper that the learner
engue when sulphuric acid and water are added to zinc. He will brown. (10.) Partly fill a bottle with hydrogen gas; apply the glass result is a solution of sulphate of zinc, and the gaseous result is
say, perhaps, I already know what these changes are; the liquid plate ; agitate ; reimmerse in water ; remove the glass plate, and hydrogen gas; what more can I want to know? Yes, you remark that no fresh portion of water rushes into the bottle; thus require to know a little more than this, and the best way of proving that hydrogen gas is not perceptibly absorbable by imparting this further knowledge will be by means of a diagram water.
as follows: (11.) Finally, remember the following recapitulation of pro
escapes perties as characteristic of hydrogen gas. It is
18 Oxygen Devoid of smell (when pure.). Lighter than air.
40 Oxide of Zinc Invisible; therefore colourless. Combustible.
40 Sulphuric Acid
80 Sulphate of Non supporter of combustion.
Oxide of Zinc. Not absorbed by water.
It will appear, then, from an examination of the preceding Not affecting lime water.
diagram, that the hydrogen comes from the water used, and its Does not redden litmus; therefore is not acid.
evolution is proximately determined by the formation of oxide of
zinc, to combine in its turn with sulphuric acid. The reader I shall not say more about it at present, but shall simply conwill moreover observe that what we call, for shortness, sulphate tent myself by remarking, that chemical combinations do not of zinc is really sulphate of the oxide of zinc. Acids never com- take place in proportions a little more or a little less, but they bine with metals, but with acids.
are fixed, unvarying, definite, and therefore capable of representaWith respect to the diagram just given, I advise the student, tion by numbers, which latter are called the atomic equivalents whenever he is in doubt as to the changes which ensue during or proportional numbers of the bodies concerned. Thus 8 is chomical composition or decomposition, to have recourse to a dia- the atomic number for oxygen, and 1 for hydrogen, congram. First put down all the substances employed, then divide sequently the atomic number of water must be 9. You them into three components, then join the elements together by must learn the atomic numbers of simple badies, but do not
I attempt too much at a time. Remember on this occasion the results.
atomic numbers of hydrogen, oxygen, and zinc-1, 8, and 32 ja 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 channel tion 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, S03=S +30, S. portions.
standing for sulphur and 0 for oxygen.
LESSONS IN GERMAN.--No. LXXII.
Irregular Verbs, continued from p. 87.
(4) Mögen, to be allowed, to have liberty. (See Remark 11.)
Present. mögen, to be mögend, being
Perfecto gemodit haben, gemocht, to have been allowed. allowed.
1 lidt mag,
may du magst
, thou art allowed. tu mögeft, thou mayst 3 er mag,
he is allowed. er möge, 1 wir mögen, we are allowed.
wir mögen, we may 2 (ihr möget, you are allowed. ihr möget, you may sie mögen, they are allowed. I fie mögen, they may Imperfect Tense.
Inperfect Tense. i lic mochte, I was allowed. ich möchte, I might 2 du mochtest, thou wast allowed. du möchtest
, thou mightst er mochte, he was allowed. er möchte, he might
wirmochten, we were allowed. wir möchten, we might 2 ihr mochtet, you were allowed. ihr möchtet, you might 3 sie mochten, they were allowed. fie möchten, they might Perfect Tense.
Perfect Tense. 1 id) Habe gemacht, I have been ich habe I may have
been allowed, 2
bu hast gemocht, allowed, &c du habeft 3 er hat gemocht,
&c. 1 wir haben gemocht,
wir haben ihr Habet gemocht,
ihr habet 3 ste haben gemocht,
bu Hatteft gemocht, allowed, &c. tu hättest been allowed
&c. 3 wir hatten gemocht,
wir hätten ihr hattet gemocht,
ihr Hättet 2 fie hatten gemocht,
sie hätten 3 First Future Tense,
First Future Tense.
First Future, ich werbe mögen,
I shall be ich werde (if) I shall belich würde 2 du wirst mögen, allowed, &c bu werbest allowed, &c. du würdest 3 er wird mögen,
er würde 1 wir werden mögen,
wir würden 2 ilr werbet mögen,
ir würdet 3 sie werden mögen,
sie würden Second Future Tense,
Second Future Tense. Second Future.
lowed, &c. er würde
wir würden 2 ihr wertet
ihr würdet 3 file werten
ON THE AGENCY OF CORAL INSECTS IN THE PRODUCTION OF
(11) Remarks on miögen.
MM. Quoy and Gainard were the first to show that the Mögen marks possibility under allowance or concession from coral insects had not built up their masses of rock from great another : as, Er mag lachen, he may laugh; that is, he has per- fathoms deep, and that these incrustations rested upon some
depths, but had merely produced stony incrustations a few mission to laugh, no one hinders him. Er mag ein braver Mann sein, he may (I grant) be a brave man ; where the possibility of his underlying rocks. They also remarked that wherever land being a brave man is a tặing conceded. Kindred to this are the to the intense heat of the sun, there the polyparia abounded
was cut into bays with shallow and quiet water, and exposed other significations (chance, inclination, wish, &c.) usually attri- most, and there they incrusted the rock the most extensively. buted to this verb: thus, es möchte regren, it might rain ; that is, From circumstances of this character it was conjectured that the causes that seem to forbid, are likely not to operate; ich the coral reefs and coral islands took their form and shape möchte es bezweifeln, I am disposed or inclined to doubt it, that is, from the forms and the inequalities of the rocks on which I might doubt it altogether, but for certain circumstances they were built, and that circular or oral islands owed their seeming to forbid: mõge es der Himniel geben, may heaven grant it; form to the underlying crusts of the craters of submarine polich mag es nicht thun, I do not like to do it, that is, I am not per-canoes.
, All these hypotheses have been long ago exploded, mitted by my feelings to do it cheerfully, &c.
partly by Sir CHARLES LYELL, but chiefly by Mr. CHARLES
student of coral formations.
Coral rocks are divided into three great classes, called
respectively Atolls, Barrier Reefs, and Fringing Reefs. By Thos. W. JENKYN, D.D., F.R.G.S., F.G.S., &c.
ATOLLS used to be called Lagoon Islands, and consist of
rings of land in the midst of the ocean. The ring of land, CHAPTER IV.
sometimes oval or egg-shaped, is a few hundred yards in ON THE EFFECTS OF ORGANIC AGENTS ON
breadth. These ring islands or atolls are sometimes only a EARTH'S CRUST.
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
clear, and shallow water. It is this sheet of water that is called
a lagoon. In this water the more minute and the more found, which have all the appearance of being the work and stronger and the larger live and work on the outer margin of product of living agents-agents that knew how to secrete the ring among the waves and breakers. Every such atoil has atoms of carbonate of lime out of sea water, and had skill to
an opening at one part of it, which allows a ship of any burden unite those particles into beautiful structures, which were to
to pass from the ocean into the lagoon. form stony habitations for their own safety and comfort.
The second class of coral rocks consist of Barrier Reefs. This class of animals is constantly called coral. This is not their appropriate name, for coral is the name of the rock that These are coral rocks which either extend in straight lines in is built, and not of the animal that constructs it. They are the front of a continent or of a large island, or encircle smaller sometimes called Zoophytes, a Greek term which means islands. In both casesthey are separated from land animal plants, on account of their resemblance in form to by a broad and rather deep channel of water, they are analegrowing plants. At other times they are called Polyparia, gous to the lagoon within the atoll. and Polypifera. These and others are only names for the
The annexed illustration (fig. 99) represents a part of the barcoral insect, or the animal that constructs the coral rock. The rier that encircles an island. It is a true sketch of the Island coral insect consists of a little oblong bag of jelly, which is of Bolabola, as 'seen from one of the central peaks. You see closed at one end but open at the other. The mouth of the that the coral reef is covered with palm trees, and you must bag is surrounded by the insect's tentacles or feelers, which imagine that the reef completely encircles the island, in the are generally about six or eight in number, and dart in all centre of which you see that peaked rock. That reef was all directions like the rays of a star.
worked beneath the sea, but by a volcanic upheaval, sudden Myriads of these minute animals live close together, and or gradual, it has become dry land.
The extent and dimensions of these barrier reefs vary from unité to form a common stony skeleton called coral, in the three miles to more than forty miles in diameter. There is minute openings of which they live. When they are under water, they protrude their mouths and tentacles to seize and both ends of the island, that is 400 miles long.
near New Caledonia a reef, fronting one side and encircling receive their calcareous food; but the moment they are apprehensive of danger, they withdraw into their holes. These calcareous abodés 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 everywhere 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 Fig 99.—The Isiand of Bolabola in the Pacific, surrounded og rock over a vast area of sea bottom, and that they, in their turn,
a Coral Reef overgraun with Palms. would work up their rocky structures to the surface of the
The third class of coral rocks are Fringing Reefs. Where