INSTRUMENTAL ARITHMETIC.-No. III. THE PLANE SCALE AND PROTRACTOR. In order to give our students some idea of the other lines drawn on the Plane Scale, we must explain some of the terms employed in Trigonometry. The definition of an angle has been given in the Lessons on Geometry; but in Trigonometry this definition is greatly altered and extended. Angular magnitude in general is the space generated by the revolution of a straight line about one of its extremities which remains fixed; and an angle is the space between the initial and terminal positions of the straight line, whatever be the quantity of revolution. Thus, in fig. 1, let oa be a straight line which revolves about the fixed extremity o, and let o A be its initial position in general; then, if o м be its first terminal position, A O M is an angle in what is called the first quadrant, and is less than a right angle. Now, if the terminal position of o A be in the first quadrant, as oм, the angle A o M is said to be less than a right angle; if the terminal position of o a lie in the second quadrant, as o м', the angle A o M' is said to be less than two right angles, and Fig. 2. In order to explain the different quadrants, it will be suffi so on. over o A in its revolution, and reached it a second time, then | the angular point o cut off a certain part oм of the revolving it lies in the fifth quadrant, and the angle is said to be greater straight line which generates the angle Ao M, and from the than four right angles, and less than five right angles; and point м draw MP perpendicular to the straight line oa, its initial position, we shall then form what is called the Elementary Triangle o MP. In this triangle, which by construction is right-angled, the ratio of the perpendicular MP to the hypotenuse oм, is called the SINE of the angle AOM; the ratio of the perpendicular MP to the base or is called the TANGENT of the angle AOM; and the ratio of the hypotenuse oм to the base o P is called the SECANT of the angle AO M. Again, the ratio of the base oP to the hypotenuse oм, is called the COSINE of the angle AOM; the ratio of the base o P to the perpendicular P is called the COTANGENT of the angle A O M; and the ratio of the of the angle AO M. The latter three ratios might have been defined as the reciprocals of the former three ratios; thus, the cosine is the reciprocal of the secant; the cotangent is the reciprocal of the tangent; and the cosecant is the reciprocal of the sine. Referring to fig. 1 again, if we take the angle AOM' in the second quadrant, and from the angular point o cut off a certain part o M' of the revolving straight line which generates the angle A o M', and from the point m' draw м'P' perpendicular to the initial straight line o A, produced to A', we shall then form the Elementary Triangle o M P' for the ratios belonging to the angle A O M'. Here, as in the preceding case, the ratio of the pendicular M' P' to the hypotenuse o M' is the SINE of the angle AO M'; the ratio of the perpendicular м' r' to the base o r' is the TANGENT of the angle A o M'; and the ratio of the hypote nuse o m' to the base o P' is the SECANT of the angle AON. Also, the ratio of the base op' to the hypotenuse oм' is the COSINE of the angle AO M'; the ratio of the base or to the perpendicular M' P' is the COTANGENT of the angle AOM'; and the ratio of the hypotenuse o M' to the perpendicular 'r' is the COSECANT of the angle A o M'. Angles are in practice measured by the arcs of circles intercepted between the initial and terminal positions of the revolving straight line. The circumference of every circle is by convention divided into 360 equal parts called degrees, and marked; for minuter parts, the degree is divided into 60 equal parts, called minutes; for second minuter parts, the minute is divided into 60 equal parts called seconds; this is called the sexagesimal (from Lat. sexagesimus, the sixtieth) division of the circle. In France they adopt the centesimal division of the circle, but it has its dis-hypotenuse oм to the perpendicular M P is called the COSECANT advantages. In the sexagesimal division, it is plain that a right angle is measured by a quadrant of the circle, hence it is said to contain 90°; two right angles are measured by two quadrants of a circle, or by a semicircle, and are said to contain 180°; three right angles are measured by three quadrants of a circle, and are said to contain 270°; and four right angles are measured by a complete circle, and are said to contain 360°. Moreover, five right angles are measured by five quadrants, and are said to contain 450°; and so on. The instrument used for measuring angular space, or angles in general, is called the Protractor, as shewn in fig. 2, and consists of a brass semicircle graduated (that is, marked with degrees) from 0 to 180° either way, so that every arc has its supplement marked along side of it, there being two rows of numbers, one from right to left, and one from left to right; that is, as in the figure, 0° begins at A, and 10°, 20°, 30°, &c., are marked on the outer edge of the instrument, terminating at B, which is marked 180°; and again 0° begins at B, and 10, 20°, 30°, &c., are marked on the inner edge of the instrument, terminating at A, which is marked 180o. The use of this instrument is to protract (Lat. protraho, to draw out), that is, to lay down an angle of any given number of degrees; it is also employed to measure any given angle, that is, to ascertain the number of degrees which a given angle contains. The trigonometrical ratios belonging to the angles AO M and AO M' above explained, are usually exhibited in the following abridged forms: First Quadrant. MP sine ом MP = tangent OM secant OP of the angle Ao M. OP = cosine ом OP =cotangent MP ом cosecant Besides the semicircular form of the. Protractor, there is another form which is sometimes put on an ivory Plane Scale, as that within the semicircle in fig. 2; and sometimes on an ivory Parallel Ruler, as in fig. 3. This form of the parallel ruler, although old, is in our opinion one of the most convenient; it appeared in the mathematical instrument makers' shops in London about 1760. The graduations of the Semicircular Protractor are transferred to the Scale Protractor, which is marked and numbered in the same manner as the former, by placing a ruler or straight edge on the centre c, and on the several divisions of the semicircumference A D B in succession; then marking on E F, the edge of the scale, the intersections of the straight edge with that edge by portions of the straight lines that would be drawn from c to each division of the semicircumference. The diameter A B of the semicircle is called the blank edge of the Semicircular Protractor, and part of it, as in the figure, is the blank edge of the Scale Protractor. Thus you see that in the latter, three of its edges are occupied with the transferred graduations of the semicircle, and the fourth edge is blank, but contains the centre of the semicircle c, which is at an equal distance from the extremities of the scale. Although we have combined these instruments in the figure, to show their construction, the student is not therefore to suppose that there is any such combined instrument in use; either will serve the same purpose, but one of them is enough in a case of instruments; the Semicircular Protractor is perhaps more handy in practice, but the Scale Protractor is more portable, and in some cases is more useful than the former. The graduated edges of both these instruments should be bevelled almost to sharpness, in order to admit of the easier pointing off of the divisions or degrees of the limb, the name applied to the graduated edge. In some cases, Protractors are made The usual contractions for these and some other ratios are completely circular; and for many purposes they are highly useful, especially for laying down plans in surveying. In Trigonometry, which originally meant the measuring of triangles, there are certain straight lines drawn in and about an angle, whose ratios to one another are called Trigonometrical Functions, or simply Trigonometrical Ratios. Referring to fig. 1, if we take the angle a o M in the first quadrant, and from M'P' OM OP Oм' tangent, and secant of the arc BM, which is the complement of the arc A M, and the measure of the angle B OM (the complement of the angle Ao M), are called respectively the cosine, cotangent, and cosecant of the arc A M, or of the angle A o M, 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 not so clearly indicative of its meaning, although it be said to be derived from sinus (a curvature or cavity), the straight line MP being drawn within the circle, and limited by the curvature of the arc M A. The part AP of the radius o A is called the versed sine of the arc a M, or of the angle AOM; it is turned (verto, I turn) from the sine MP. metrical lines in the other three quadrants will be as folAccording to the old system of trigonometry, the trigono lows:- of the salient angle a o N. A 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, м', N', and N, if the parts o м, o м', ON', and oN be made all equal to one another. Then, the same constructions being made, and the tangent TT' 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. A T' tan. O T' sec. NQ COS. BS cot. o s' cosec. of the arc AA'N In the old system of trigonometry, a straight line drawn 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 A M. This straight line is also the chord of the arc M BA'NA, which 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 сH to 90, and you will have the Line of Chords as marked on the scale. The Line of Sines, marked s1, 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 Line of Tangents, marked TA, is constructed thus: gra The perpendicular M P is called the sine of the arc A M, or of the 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 extremity of the arc AM, and the straight line or drawn through M, the other extremity of the arc, is called the tangent of the arc A M, or of the angle AO M; and the straight line o T intercepted between the vertex of the angle or centre of the circle and the tangent, is called the secant of the arc A M, or of the angle A O M. Either of the radii o A, O м, o м', &c., is generally called unity, and the lengths of the other lines, as compared with the length of this unit, are determined accordingly. Again, the straight lines QM, BS, and os, which are the sine, cular to the radius passing through the extremity of the quadrantal are marked 0°; next draw straight lines from the centre through each of the degrees in the quadrantal arc to meet that perpendicular; then lay down on the scale, in a straight line, the distances between the fixed extremity of the perpendicular and the successive points of the intersection of these straight lines drawn from the centre and that perpendicular, and you will have the Line of Tangents as marked on the scale. On the scale the Line of Tangents extends only to 70°; beyond this point the scale would require to be greatly lengthened for the tangents of degrees beyond 70°; and the tangent of 90° is | I care very little of what the student makes it, whether of glass, 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 are 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 8 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 1°; 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 arc 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. 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 of discs or plates of glass. Plate-glass is better than common blown glass, as being more accurately plain; but the latter, rendered 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 of the bottle takes place all round; if not, grind the mouth or the see, as you easily can, whether contact between the plate and mouth 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). 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 ing the wire into the wax, light the taper and plunge it (6.) Repeat the experiment, having reversed (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. 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. 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 during combustion covers an object immersed in it with a sooty chemical language, carbon; indeed, generally any substance that 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 say, perhaps, I already know what these changes are; the liquid result is a solution of sulphate of zinc, and the gaseous result is 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: 9 water 32 Zinc 51 Hydrogen- 40 Sulphuric Acid 40 Oxide of Zinc -escapes 80 Sulphate of It will appear, then, from an examination of the preceding diagram, that the hydrogen comes from the water used, and its evolution is proximately determined by the formation of oxide of |