cursors may be screwed points of any kind, whether steel | The divisions of the straight line AE are now marked with the tracers, pencils, or crayons, or ink points. This apparatus is represented in fig. 12. To the fixed cursor there is sometimes applied an adjusting or micrometer screw, as seen in the figure, to enable a given distance or radius to be taken with the greatest nicety. Fig. 12. In a case of mathematical instruments are also contained a Tracer and Drawing Pen, for drawing straight lines in trace, or in ink. These two are usually joined in one instrument, the tracing point being screwed into the drawing pen; this instrument is represented in fig. 13, where the ink point is Fig. 13. constructed exactly on the same principle as that of the socket-compasses. There is also an instrument called a Protractor, for measuring angles upon paper, which is represented in fig. 14, and consists of a semicircle divided into degrees, from 0° to 180° each way, the 90th degree being right above the centre o. The straight line A B in the figure, is the diameter of the semicircle, and is called the fiducial (or true) edge of the protractor to be applied to one of the legs of the angle to be measured; the arch AM B, being the fiducial edge to be applied to the other leg. Thus, in order to measure the angle xo Y, the centre of the instrument is placed on the vertex o of the angle, and the edge o A on the leg o Y, so as to coincide with it exactly; then the angle A o M, on the arch A м B, determined by the point M, through which the other leg o x passes, is the measure of the angle x o Y. In this case, the measure appears to be nearly 45 degrees, as the figure represents divisions on the arch or limb of the protractor at every five degrees. Fig. 14. numbers 1, 2, 3, &c., from c to E, to denote units. The divisions of the standard unit AB, are marked 1, 2, 3, 4, 5, 6, 7, 8, 9, from B to A, to denote tenth parts of a unit; and the divisions of the perpendicular BB' are marked 1, 2, 3, 4, 5, 6, 7, 8, 9, from в to B', to denote hundredth parts of a unit. Or, if the divisions of the straight line AB denote hundreds, those between B and A denote tens, and those between B and B denote units. The scale is rendered complete by drawing straight lines from в on BA, to 1 on B'A'; from 1 on BA, to 2 on B'A'; from 2 on BA, till one be drawn from 9 to 3 on B'A'; and so on, on BA, to A' on B'A'. By the nature of similar triangles, hereafter to be explained, the small part of the parallel to the base 1 B, within the triangle A B Fig. 15. C 987654321 D E B1 B', at the division marked 1 is one-tenth part of the base 1 B', and consequently one-hundredth part of the line AB; the smail parts of the other parallels are in succession, two-hundredths, three-hundredths, &c. Hence, if a straight line is to be measured, take its length in the compasses, and apply it to the scale from B towards E. If it measures an exact number of units, say from в to E, then the straight line may be said to measure 1'unit, 1 ten, or 1 hundred. If it does not measure from E to 3, 30, or 300 equal parts, according as AB is made to stand for B exactly, but extends from E exactly to one of the division said to measure 3.4, 34, or 340 equal parts, according to the marks between в and A, say 4, then the straight line may be standard unit, as before. If it does not extend from E to the division marked 4 between B and A exactly, but falls somewhere between 4 and 5, then move the compasses downwards, preserv ing one point always in the line E E', and both points parallel to A E, till the other point fall on the intersection of the diagonal marked 4, 4, with one of the parallel straight lines marked on BB', say 6; then the straight line may be said to measure 3-46, 34-6, or 346 equal parts, according to the standard unit, as before. For the purposes of navigation, dialling, &c., the plane scale has frequently on the side obverse to the diagonal scale just described, a set of lines, besides those of equal parts, containing divisions for the measurement of leagues, rhumbs, chords, sines, tangents, semitangents, secants, lines of longitude, &c. Such scales are considered the best, as they are generally executed with great care. The scale called Gunter's scale has the same divisions on one side of it, as are to be found on the plane scale, but of a larger size, and when well constructed, admitting of greater accuracy; but being usually made of boxwood, this is seldom the case. The obverse side of Gunter's scale, has a set of lines representing the logarithms of the num This apparatus for measuring angles is sometimes engraven on the upper side of a pair of parallel rulers, and sometimes on the obverse side of a plane scale. The protractor is more com-bers which denote these divisions; by means of the logarithmonly made so that the centre of the semicircle, and the fiducial edge containing it, shall be on the outside of the instrument rather than on the inside, as above. The Plane Scale is a flat ruler with several lines of equal parts, on one side, divided according to certain proportional parts of an inch; and having, on the other side, the diagonal scale, decimally divided so as to measure units, tens, and hundreds of equal parts, with a very considerable degree of exactness. The construction of this scale, so useful in graphical (i.e. drawing) operations, such as the construction of plans, maps, and charts, of architectural designs, of plans and sections of machinery, &c., is founded on the properties of similar triangles, as treated in the sixth book of Euclid. We shall endeavour to give our readers a practical idea of its construction. On a straight line AE, (fig. 13,) divided into any convenient number of equal parts, AB, BC, CD, DE, &c., one, AB, is assumed as the standard unit of measure. From the different points, A, B, C, D, E, &c., perpendiculars of a convenient length, as AA, BB', CC, DD, EE, &c. are drawn to the straight line AB, and terminated in the straight line A' parallel to AE. The unit AB is divided in. equal parts; then the opposite part A'B' is similarly divided; next the perpendicular BB is divided into 10 equal parts, and through each division, straight lines parallel to AE or A'E are drawn. mic lines, arithmetical calculations can be performed instrumentally, that is, without the operation of the ordinary rules. A modification of this instrument, called the sliding Gunter, is still more ingenious in its construction, and still more useful as an instrument of calculation. The explanation of these instruments, however, belongs to a more advanced state of knowledge among the generality of our readers. This we hope to reach by their perseverance. One of the most useful instruments in a mathematical case, is the sector; a mere sketch of its appearance is given in tig. 16. It is composed of two flat rulers, moveable on an axis, or jointed at one end like a pair of compasses; hence, it is called by the French, compas de proportion,-the compasses of propor tion. From the centre of the axis or joint, several scales are drawn on the faces of the rulers, so as to correspond exactly with each other. The two rulers are called the legs of the sector, and represent the radii of a circle; and the middle point of the joint, its centre. It contains a scale of inches, lines of equal parts, of chords, secants, and polygons, on one side of each leg; and on the other side of each leg, two lines of sines, tangents, &c., besides lines of the logarithms of the numbers expressing these quantities along the whole length of the sector, when stretched to an angle of 180 deg., as well as the logarithms of the natural numbers. As in the case of the plane scale, we can here only give one or two examples of the use of the sector, by way of illustration. Thus, in the figure, o is the joint of the sector, o A and o в are its legs, the marks on the legs represent the divisions of the line, of lines or equal parts. Its use is to find straight lines that shall be to one another in a given proportion. Suppose, for example, that it was required to find a straight line whose length shall be to the length of a given line as 3 to 10. Open the sector until the distance of the two points marked 10 on its legs is equal to the A writer, for James Besson, a French mechanician, who published an account of it in his "Théâtre des Machines," a work of which the plates were engraved before 1569. He says it is usually attributed to Justus Byrgius, who published his description of it in 1603. John Robertson, librarian to the Royal Society, in his "Treatise on Mathematical Instruments," London, 1775, ascribes the invention of a similar instrument to Fabricius Mordente in 1554, according to a statement made by his brother, Gaspar Mordente, in his book on the Compasses, published at Antwerp in 1584. QUESTIONS ON THE PRECEDING LESSON. Describe the common ruler or straight edge, and the various purposes to which it is applied. How is a straight line obtained in the practice of the mechanical arts? Describe the various forms of the parallel ruler, and its use; and state the defect of the common construction. How is the accuracy of a ruler to be tested, and what qualities should it possess? Describe the nature of a pair of compasses, and its different uses. What are dividers? socket-compasses? the tracing point? the ink point? the crayon point? the lengthening bar? the bow and plug compasses? the beam compasses? What is the tracer and drawing pen? the protractor and its use? the plane scale and its use? the diagonal scale? what does the distance M.N. or P.N. measure on the scale in fig. 15? take off the distances represented by the following numbers on the same scale; 376, 289, 174, 385, 3-2, 2-36 and 37-2. Give some account of the plane and sliding Gunter's scales. What is the sector, and its use? what its French name? what are the proportional compasses, and their use? what is the French name? who was its inventor? LESSONS IN MUSIC.-No. II. By the Rev. JOHN CURWEN. It is important that the learner should become thoroughly and practically familiar with the structure of that musical "scale of all nations and of all time" which was partially described in the last lesson. The following account, by Colonel Thompson, who is no less distinguished for his philosophical and learned disquisitions on the science of music than for the other great services which, by pen and speech, he has rendered to his countrymen-the following account by him, of the first attempts of philosophy to measure this scale, will interest the student: "The histories of all nations refer to very early periods the discovery that certain successions or combinations of sounds have the effect upon the ear which is implied by music; and it may be assumed that in all countries a considerable degree of practical acquaintance has been aequired with the sounds before any person has thought of investigating the cause. The story of Pythagoras listening to blacksmiths' hammers, and discovering that the different sounds had some relation to the weights, has been sufficient to secure to that philosopher the renown of being the first who sought for the explanation of musical relations in the properties of matter. The account given by Nichomachus is, that Pythagoras -DOH heard some iron hammers striking on an anvil, and giving out sounds that made most harmonious combinations with one another, all except one pair, which led him to inquire what were the peculiarities of the hammers which produced these different effects. Whether this is an exact account or not, some observation of this kind appears to have speedily led to the discovery, that of strings of the same thickness and composition, -LAH and stretched by the same weight, those gave the same musical sound (or were what is called in unison) which were of equal lengths;-that if of two strings in unison, as above, one was shortened by a half, it produced a sound which, though -SOH very far from being in unison with the sound of the other, might be heard contemporaneously with it, with a strong sensation of satisfaction and consciousness of agreement, and that the two sounds in fact bore that particular relation to each other by which two voices, of very different kinds, like those of a man and a child, can sing the same -FAH tune or air as really as if they sang in unison, being what musicians have since distinguished by the title of octaves; that if, instead of a half, the string were shortened by a third part, there was produced a note which, heard either in combination with or succession to the first, created one of those marked effects which all who had attained to any degree of musical execution by the guidance of the ear had treasured up as one of the most efficient weapons in the armoury of sweet sounds, being what modern musicians name the fifth-and that if, instead of a third part, it was -RAY shortened by a fourth, there was produced another note very distinct from the last, but which, like it, was immediately recognisable as one of the relations which experimental musicians had agreed in placing among their sources of delight, being the same which in modern times is called the fourth. "So far, Pythagoras and his followers appear to have run w 1. Instead, however, of pursuing the -DOH clue of which they already had hold, and examin. ing the effects of shortening the original string by a fifth part | following are the results of such experiments as those just and by a sixth, they strayed into shortening the results of previous referred to. Arithmeticians may notice that the proportion of experiments by a third, and lengthening them by an eighth, * the vibrations is inversely as the length of the strings given and here was the beginning of sorrows. * The attempt above. But we here print the fractions with a common (beyond these three steps) at the division of the "Canon”denominator to make the relations more obvious. in other words, at the division of a string into the lengths which produce the sounds that make music in a single key-was a failure."-See "Exercises, Political, Literary, &c., by Lieut.Colonel T. Perronet Thompson." Vol. II., Article "Enharmonic of the Ancients." See also his "Theory and Practice of Just Intonation," a shilling pamphlet, published by Effingham Wilson. The experiments of modern philosophers have been rewarded with the discovery that a musical string divided in the proportions given underneath will produce the notes of the scale as there described. Let it be noticed that the figure 1 stands for the whole length of the string, whether a foot, a yard, or any other measure, and whatever sound (in pitch) it gives -that sound being taken for the key note-Doн. It may also be mentioned that the same numbers denote the comparative lengths of organ pipes capable of sounding the corresponding notes. Perhaps these proportions will be better understood by the diagram in the preceding column. A single string thus stretched and used for these experiments is called a monochord. If the student is a mechanic, let him make one and verify the measurements here given. Let him suspend a board of four or five feet against a wall. To the upper part of this board fasten the end of a pianoforte-wire or other musical string which is of the same thickness throughout. Let the wire pass down the face of the board, over a firm wooden bridge, an inch or so high, and close to the top, and over a moveable bridge at the bottom; and let it be kept stretched by a heavy weight. Set your moveable bridge (which the weight will keep in its place) at the bottom, marking the spot, and take the sound of the whole string, by the help of a fiddle-bow, for your Doн, or key-note. Then (having properly measured and marked the board) move the bridge to the other divisions, sounding each note as before. It may be well to mention that Colonel Thompson maintains, and with good show of reason, what he calls the "duplicity" of RAY and TE. They are sometimes sounded by good singers and violin-players a very small degree lower than their usual position given above. These experiments will fix in your mind a clear notion of the scale. THE NOTES OF THE SCALE. The proportion of vibrations While the key note gives the following number Dou RAY MR FAH Son LAH TE 24 27 30 32 | 36 | 40 | 45 | 48 24 24 24 24 21 21 24 24 If our arithmetical friend will now work a few sums in proportion, he will be able to show the value of the intervals between the several notes of this scale. Thus the vibrations of Doн differ from those of RAY, in being three less, and (three being one-ninth of twenty-seven) Doн has therefore only eight-ninths of RAY's vibrations. The same proportion will be found between FAH SOн, and LAH TE. These intervals are called the “great tones." The proportion of RAY ME, and of SOH LAH is nine-tenths. These are the "small tones." proportion of ME FAH, and of TE DOн, is fifteen-sixteenths. These are called semitones, or more properly, Tonules. It you calculate from the length of the string given above you will find still the same proportions existing. The Now this evidently means that the lower note of the "great tone" has 1280 vibrations, while the higher note has 1440, and (as the lengths of string are in inverse proportion to the vibrations) that it takes 1440 degrees of the string, while the higher takes only 1280 such degrees. Therefore the propor tional difference between them, whichsoever way you look at it, is one hundred and sixty degrees. In the same way you will find that the difference between the two notes of this "small tone" is one hundred and forty-four degrees, and that the interval of the " tonule" is ninety degrees. The degrees in each case are of similar value, all measured on the same scale (common denominator) of 1440 degrees. We may therefore treat them as belonging to one scale, and adding three "great tones," two "small tones," and two "tonules" together, we shall obtain a perfectly measured scale of 948 degrees. As all these numbers, however, will divide by 2, retaining, of course, the It will be well for you to understand the connexion between same proportion to one another, it is better to regard the scale these musical notes and the vibrations of the sonorous body as composed of 474 degrees, containing three "great tones" of 80 which produces them,-whether that body be the string of a degrees, two "small tones" of 72 degrees, and two“ tonules” violin, the air in an organ pipe, a small plate of glass or metal, of 45 degrees, and this is the smallest perfect measurement or the "chordæ vocales"-the vocal chords-of that wonder- of the scale in plain figures. But if the pupil will go one step ful little box instrument, called the "larynx," which you can further, and divide each of these intervals by nine, he will see feel in your own throat. Sounds produced by irregular vibra- how we obtain the proximate scale of fifty-three degrees. tions are not musical. They form the " roar, rattle, hiss, The tonule will be exactly 5 degrees, the small tone exactly 8 buzz, crash," or some other noise. But sounds produced by degrees, and the great tone only one-ninth of a degree less equal and regular vibrations are musical. "That musical than 9 degrees. Adding these together, as before, you will notes are produced by a rapid succession of aërial impulses have the "Index Scale,' as Colonel Thompson calls it, "of at equal intervals, is very clearly illustrated by an instrument fifty-three," and you will see that it is three-ninths or one called the syren, the invention of Cagniard de la Tour. A third of a degree too large. We strongly advise the pupil to blast of air is forced through a narrow aperture in a pipe; construct a "monochord," and try for himself whether this is and a flat circular disk, perforated near its circumference with not in truth an accurate description of that scale of related a number of small holes equidistant, and in a circle concentric notes which God has made most suitable to human ears and with the disk, is so applied to the pipe, that the blast is souls. All the books of science are agreed that it is; and interrupted by it, excepting when one of the holes in the disk experience bears the same testimony. It is the more imporis opposite to that of the pipe; and when the former is made tant that you should understand these points, because the true to revolve rapidly, the resulting aërial impulses cause a series scale is dreadfully abused by the common keyed-instruments. of isochronous vibrations that produce a musical note, and the Many of these are tuned by what is called equal tempera corresponding number of its vibrations can very easily be ment;" that is, the scale is divided into twelve equal semicomputed, from knowing the number of holes and of revo-tones, and it follows that the tones are all 79 degrees (of the lutions of the plate. The results obtained by this instrument perfect scale of 474), while they ought to be sometimes 80 and agree exactly with those found by other methods." (See the sometimes 72 degrees! and the tonules (semitones) arc work on "Acoustics" in Chambers's Educational Course). both 39 instead of 45!! They might as well cut down The more rapid the vibrations of the sonorous body, the more the fingers of a statue, to "equal temperament!" Human "acute" (shriller, or higher) the note produced. The ingenuity will surely deliver us soon from this monstrous distortion. You will understand now why it is so often pleasanter to sing "without the piano." You will have noticed, in connexion with these statements, that a sound produced by twice the number of vibrations (or half the length of string), as compared with any other sound, is so much like that other sound as to be called by the same name, thus, Dон and DоH'. Notes thus related are said to be at the interval of an "octave' (eighth), the one to the NOTE. In this exercise take a middle sound of your voice (neither high nor low) for the key-note or Doн. A friend again will be needed to set you a "pattern" with voice or instrument. Tell him to play or sing G, D, B, G, B, B, G. He may understand these names better than those by which you are learning, and to which your attention must at present be confined. Take care to sing the upper SOH with a clear trumpet-like sound, and ME with a calm but firm effect. Sing the exercise slowly, but with sustained decision. It will greatly add to your pleasure if you can get a friend to sing the second line of notes while you are singing the first. This exercise, too, will give you confidence. [If you are singing from the staff above, remember that one voice will take the higher notes of each couple while the other voice is taking the lower notes. The open notes, which you have here, when they occur in the same tune with the black notes, which were used in the former exercises, are to be sung twice as long, in time, as the black notes; and the open notes without a stem, like the last note in this exercise, are to be twice as long as those with a stem. This relative length does not however hold true out of the same tune. An open note in one tune may be no longer than a black note in another, and a black note in one tune no shorter than an open note in another. Let it, however, be repeated that it will be much better for the learner not to pay any attention at present to the old "notation " (way of writing), or to the remarks thus placed between brackets. He may get his mind puzzled with the notation of music, when he ought to be giving his whole attention to music itself. Sing exclusively from the syllables, and never leave an exercise until you can sing it correctly from memory, pointing on the modulator the while.]] THE POPULAR EDUCATOR. In the [ Now, of the three propositions given above, the first is the shortest Engin firm a proposition. But of what does this proposition. It is in red a setize, of the simplist på pox, TE there is, or can be. Les loan t#) wurds, then, (8) BUS LE Cosist? It consists of the min fre, anu the Ters burns. Hence you learn that in every sentene there must be at least dyect of the proposition, for it is the agent or the caL of Ja noun or pronoun, and a verb. The noun, you see, is the we say the verb burns is the predicate. By the predicate In grammar, we have also a distanation for the ver proposition, we mean that which is asserted or declar ect. What is here asserted? this: namely, that fire fr burns, then, is the predicate. To bears the fret, and not, for the present, attempt the words. f you was to see the words, then frst warn to sing the Trust in the Lord with of the wingle note Ma. Todo 2420 trance you should divide the note (in your A st 1550 1 egin seat or them by beating on the table with your hand and then the of time (you can mark will go to the frat beat and “Lord with " to above the use show this division. The * Mize heart face early to their right notes. Inte box you eling the words “and lean not unto thine." To the is warsely beard. Linh, ng boa, like the other ** reciting note" "And" Me, into two beats (and reeling sites of a chant like this may be drted 20 thany seats as you pleasey, you wi. bave the words 86 42 201 to the first Seat, and the "ane filling the Molde word S you perceive is T. $24. Take care not to eng the of labies 'slurred” on to two stand z and harpy. Let them take as much time as the syllables "un der" quickly 1. The bus wor The sand verse of words, printed underMinato, la cual ded on the same plan as the fret. The double bar, You wi. Oberve, separates the words of the reciting note "(as it is called, of the chart. you as the quare note for Doн, because the tube bezina on Me, and Dog does not of Dow, however, is in the first space, reckoning from the bottom, tence. By adding not to are, you make it a negative sentenc sur in the air." The place to modify the sense. You will easily see how this sent:nce may receive alliti na of the staff. On the lower staff it was necessary to make an addi-You may also qualify the attribute 90s by prefixing an adIt is, as it stands, an affirmative scitional line to carry Box. This is called a "ledger line."] verb, as, very good. If you wish to make it interrogative, you have only to invert the copula and the subject, and say, are you good? the predicate consists of two words. It may even c In this case, the predicate is one word, a verb. Sometimes several words. In the instance given above, you are gind,I predicate is, are good. Hence the predicate consists of t verb are, and the adjective good. The former predicate, barna, Now, this comp and predicate has two parts; first, the verb was a simple predicate; this predicate is a compovas predicat | good, which is called the attribute, or that quality which is are, which is called the espula, or link; and the adj crive ascribed to the subject you. Taus explained, the seatsa stands as follows: LESSONS IN LATIN.-No. IIL By JOHN R. BEARD, D.D. NOUNS, SUBSTANTIVE AND ADJECTIVE. In the third of the instances given above, there is a rather different kind of sentence, boys love play. Now, according to what I have just said, boys is the subject, love the copula, or verb, and play the object. The differ ne Ore English nouns remain unchanged, whether they form the here is, that instead of an attribute in the predicate, you have subjset or the object of a proposition. Keir, do you know the exact meaning of these terms, View the matter in another way. Here is a proposition- In this proposition or sentence, dog is the subject, and man is the eject. If you will study the proposition, you will see that dog is the doer or actor, and man is that which is acted upon. Hence you may form the general rule, that the subject is the doer, the actor, or agent, and the object is the being or thing which is acted upon. You may put the same rule in these words: the subject originates the action spoken of in the verb; the object receives the action spoken of in the verb. Or, again, you may say, the subject is that from which the action comes; the object is that on which the action falls. The act of biting came from the dog, and fell upon the man. As I wish to make everything clear as we proceed, I will enter here a little more into this matter. A proposition is the enunciation or statement of a thought, or a fact. Thus, fire burns; you are good; boys love play; are each a proposition. Of course the statement must be compiete, or there is no proposition. What you say must make sense in itself, or there is no proposition, but only one word or more. Thus, if instead of saying fire burns, you say merely fire, or burns, you do not utter a proposition, for you do not make a statement. what? for you have left the sentence unfinished. So if you Ir you affirm, you are, I naturally ask, declare that boys love, the question arises, what? and only when you have added the word play, do you finish the senten by making the sens complete. an object. The proposition, viewed logically, stands thus: SUBJECT. COPULA. PREDICATE. Observe, too, here, how, having gotten the main parts, the The bad boys whom, I mentioned always love much play. But here we have a complex sentence, a complex or double simple sentence is the statement of one fact or one thought. Two facts are mentioned in the last state of the sentence. proposition. As it stood before, it was a simple sentence. Those two facts are these, I mentioned some boys, and boys lore play. And these two facts are so stated that the sense of the one is not complete without the sense of the other, for you do not say merely I mentioned some boys, and boys love play; but the boys whom I mentioned love play. You thus see that the one quently, a complex sentence, such as I have now presented to proposition is intimately connected with the other. you, is a sentence within a sentence. Consethe one is the principal, the other the subordinate one. The subordinate sentence is that which is introduced by the rela Of these two sentences, |