Draw parallel circles, and the regular polygons mentioned in last by the preceding figures that they divide the curve into four equal and lesson, with circles touching all the angles, see fig. 44. Fig, 44. O Draw, also, lines composed of parts of circles. Then draw In the sweeps, and straight lines joining on to circular curves. curve lines, the component arcs of circles or other curves may be equal or unequal, as seen in fig. 45. These lines must be drawn at once, no piecing or joining being allowed; you may make them of any length you please, and you should draw them in all directions, horizontally, perpendicularly, and at all angles of inclination. Careful practice of this nature will be especially useful to those pupils who have an inclination to ornamental design. similar parts. An ellipse may be drawn by hand by a careful attention to this property. Draw the diameters in the first place, and then if you mark points, as in the case of the circle, you will not find it very difficult to produce a neat curve. In a rhombus, which is, as you know from the last lesson, a figure with four equal sides in parallel pairs, draw lines joining the angles, and bisect (that is, halve each side), joining the bisecting points. You will then get a figure in which it is easy to describe an elliptical curve. When a square is treated in this manner the curve becomes a circle, see fig. 47. Fig. 47, Curves of an elliptical shape are perhaps more prevalent in natural forms than those of any other kind. The three classes of Fig, 45. The circle is a curve of uniform character, a small circle being a | correct miniature of a large one. It is not so, however, with the ellipse; the varieties of its form are infinite, all depending on the proportions of the major and minor axes, see fig. 46. ig. 46. С form-the bell-shaped, the horn-shaped, and the egg-shaped or oval (from Lat. ovum, an egg)-take in a vast number, and the variable form of the ellipse renders the varieties of these shapes inexhaustible. Fig. 48 will show how the egg-shape or oval tapering at one end is formed from two ellipses sweeping into each other, Fig. 48. This form is seen also in the general outline of the human face. Exercises on linear drawing will be continued in the next lesson. And at this point it will be proper to say a few words on the materials used in these lessons, for it is absolutely necessary that you should practise the drawing of the same forms again and again, until you have mastered them. It is, therefore, recommended to all to use, instead of paper and pencil, the black-board and a piece of chalk. The board may be made of deal from a third of an inch to half an inch in thickness, two feet in width, and two and a half feet in length. It should be squared, that is, the sides should be The proper lengths of these axes being determined, you will see made truly at right angles to each other at all the corners. You may cut a notch at the middle point of each edge of the board, and the intersection of the lines joining these notches will give you the central point on the face of the board. The best mode of blackening the board is to get a mixture of lamp-black and size, quite hot, and lay on a thin coating with a brush, using enough of size to prevent its becoming powdery when dry. If you do not attend to this circumstance, your fingers and everything that touches the board will become soiled. When the black coating wears into light spots, rub a little ink over the places with a brush or feather. Soft white chalk may be obtained at some shops in small square or round sticks; this is well adapted from its neat form for drawing purposes; but a piece of common chalk somewhat long in shape, and cut tapering to a point, will do good service. Fasten it with twine to a thin flat stick, then roll a piece of paper round the stick fastening it with sealing wax, and in the absence of a portecrayon you will have a tolerable substitute. The form of a portecrayon is shown in fig. 49. It may be described as a metal tube widened and split at each end, and havings rings to slide upon it so as to keep the chalk tightly fixed when inserted in the tube. Fig. 49. On a board such as we have described, your practice will be on a seale sufficiently large to induce boldness and freedom of hand. Keep your wrist raised from the board in sweeping your curve lines, and give play to the shoulder and elbow; with the arm in a cramped position you will never turn them properly. We close the lesson with an observation which might be appended to each in succession, as it conveys an admonition which should never lose its force however often repeated :-" If you merely read what is here printed, and only look at the figures set before you, without performing the exercises, or (if once performing them) without persevering until you master them, it will be vain for you to expect to make yourself a good draughtsman; but if you really practise them often and well, and by this means render each successive le son a help to the next, your progress is certain, and your reward is sure.' LESSONS IN ENGLISH.-No. XXXI. By JOHN R. BEARD, D.D. We are about to lay before the student a large portion of the roots of the Latin language. In the study of them, he may become acquainted with the treasures of the Roman literature, and the tone and strength of the Roman mind. These lessons do not, indeed, lie on the surface. Nevertheless, they are to be learnt by care and diligence. For this purpose, impress on your mind the preceding remarks, and remembering that a language is the mirror of a nation's mind, accustom yourself to see and contemplate the Romans in their words-those unerring tokens of thought, those mental miniatures. LATIN STEMS. cess centum, a hundred cent cerno, I separate, sce aedes, a house edi edifice cretus, separated cret cete, a whale cet fy edify cinctus, girded cinct citus, quickly moved cit 1qui equilibrium animus, mind anim animosity clamo, I call out aer, air include, exclude coeval clausus, shut claus clus clause seclusion Of course it is only so much of the Latin vocabulary as exists in English that I shall set forth in these pages. The Latin words found in English, exist almost exclusively in combination. The combinations in which they appear vary in kind. The combination may consist of one word and a suffix, or one word and a prefix; or, again, of two words with or without terminating forms. The combinations require certain changes. The changes are effected by dropping or altering the Latin termination, or by the substitution of one vowel for another, as i for a, e for a, &c. secretion, discre(e)t cetaceous succinct, precincts cite, excitement exclamation The meaning of many of the words given as examples the student will either know already or may deduce from the etymology. In other cases some additional aid may be required. That aid I shall supply in quotations and in such remarks as the several topics may seem to require. "Those milks (in certain plants) have all an acrimony, though one would think they should be lenitive."-Bacon, "Nat. Hist." Cowper. "Most satyrists are indeed a public scourge, Their mildest physic is a farrier's purge, Their acrid temper turns, as soon as stirred, The milk of their good purpose all to curd." Acer is properly that which is sharp as the point of a spear, or the edge of a sword, that which pricks or cuts; whereas acerbus (acerbity) denotes that which is bitter to the taste. According to its derivation, edification is house-building. The spiritual house is intended, the metaphor being borrowed from the diction of the New Testament. Consult 1 Cor. iii. 9; xiv. 3; Ephes. ii. 21; iv. 12, 16. "So that it is by the equilibre of the muscles, by the aid of a considerable and equipollent (equally powerful) muscular force in constant exertion, that the head maintains its erect posture."-Paley, "Nat. Hist." "Government has coercion (restraint) and animadversion upon such as neglect their duty."- South. Articulation is properly the making of articles, that is, small limbs or joints; hence dividing a flow of sound so as to produce separate and distinct sounds, that is letters and syllables. This power belongs only to man. Accordingly, Homer, that great master of distinctive and descriptive epithets, gives as the characteristic of the human race that they articulate. "The first of these, at least, I thought denied Milton, "Paradise Lost." "The subject of the present chapter will be the offence of homicide, or destroying the life of man, in its several stages of guilt, arising from the particular circumstances of mitigation or aggravation which attend it."-Blackstone, "Commentaries." Homicide, that is, in its corresponding Saxon term, manslaughter, denotes the general act of man-killing, leaving it to be decided whether the killing was or was not murder, that is, premeditated killing. Unpremeditated man-killing is generally termed manslaughter, as contradistinguished from murder. It deserves attention, that not one of these words, homicide, manslaughter, man-killing, murder (Ger. mord, murder; Fr. mort, death; Lat. mort, and Gr. moros, death), conveys in itself the idea of "malice aforethought.' " Clinical is a scientific word, and like most of our scientific words may have come to us from the Greek; for the Greek klino is the root of the Latin clino, and is similar to it in import. A clinical lecture is a discourse on a disease, delivered by the bedside of the patient. A clinical convert is one that is converted on his deathbed. In the early history of the churches, those were called clinici or clinical, who, wishing to have all their sins washed away at once, the delay of baptism by Constantine, miscalled the great. postponed baptism till their dying hour. See Gibson's remarks on "Horror stalks around Wild staring, and his sad concomitant Philip. " 'Cremation, burning, is applied particularly to the ancient custom of The Chinois, without cremation or urnal interment of their bodies, make use of trees and much burning, while destroying corpses by fire. they plant a pine-tree by their grave."-Brown,“ Urn Burial.” Capillary signifies that which is like hair; hence it is applied to the small vessels of the body, as the ramifications (branches) of the arteries, "the capillaries;" also to tubes; and attraction in tubes as fine as hair, is called "capillary attraction." The former (fore) legs of this animal (the elephant) appear when he standeth, like pillars of flesh without any evidence of articulation."—without losse, and that losse to be manifest."-B. Jonson. Brown, "Vulgar Errors." "A strict and succinct style is that where you can take away nothing "Père Bougeant's third volume will give you the best idea of the treaty of Munster, and open to you several views of the belligerent and contracting parties"-Chesterfield. Derivatively considered, to combine is to put things together in pairs, but it is employed without this restriction to signify to put together generally. "The impediments were-first, the negligence of the pastors; secondly, combinations, that is, double benefices, when men having two cures could not sufficiently attend both."-Hales. "Few painters have obliged us with finer scenes, or have possessed the art of combining woods, lakes, and rocks into more agreeable pictures, than G. Poussin."-Hurd. It is curious to see in incubation, the act of the hen in setting on her eggs, and incumbency, the condition of a clergyman as occupant of a living, how the same stem, and very nearly the same letters, may come to signify things so very unlike. Cadaverous comes immediately from cadaver, a corpse, and denotes the colour or complexion of a corpse. Cadaver, a corpse, by its etymology points out the fact which denotes death, namely, falling, from cado, I fall. A dead body cannot stand. "To translate him line for line is impossible, because the Latin is naturally a more succinct language than either the Italian, Spanish, French, or even the English; which by reason of its monosyllables, is far the most compendious of them."-Dryden. among the ancient Greeks and Romans of gathering up and binding The idea in succinct, girded, is taken from the custom prevalent around the waist their long flowing robes, when they were about to apply to any manual occupation. Compare Is. viii. 9; John xiii, 4, 5. EXERCISES FOR PARSING. Some have termed the "Songs of Solomon, or the Canticles," Hebrew Epithalamium. The rage for autographs seems to have in a measure subsided. The autographs of the writers of the The word apology sometimes signifies defence rather than excuse. The former is the older meaning of the term. The monuments of Egypt are covered with hieroglyphics. The hieroglyphics of Egypt have for the most part been at length deciphered. Exodus is the name borne by the second book in the Bible. This name was given to the book because it recounts the departure of the children of Israel out of the land of Egypt. New Testament are not known to exist. watchmaker's practice, you must take the language to pieces, study those several pieces, and then try to put them all together bit by bit. In this operation everything depends on your acquiring a correct knowledge of the several component parts. Therefore study etymology, study the Greek, Latin, and other stems. If you fail in this you will be, and you will remain, in the condition of the watchmaker's apprentice."Surely, I may become a master by studying a good English dictionary."- "Never; the mere use of the dictionary is like looking at the watch on the outside; at the best you will thus look only a small way into it, and after all, having given much more trouble than would be necessary to acquire the language thoroughly with the aid of etymology, you will, whatever cfiorts you may make, acquire nothing more than a superficial acquaintance with English. The etymological study of a language is the only wise and proper one; it is also the shortest and the easiest in the long run."-"What do you mean by etymological study?"-" That study which is founded on etymology or a knowledge of root-meanings, a knowledge of the meanings of the component parts or the elements of a language. Etymology is the A, B, C, of a language; and as you cannot write without knowing 'your alphabet,' so you cannot read without knowing the materials you have to employ. I fancy I should ill succeed in your cabinetmaking. Why?""For one thing you don't know the tools." want to teach you what they are, and what they are for. Therefore study the Greek and Latin stems or roots."-"But you do not forbid the use of a dictionary. Some of the words given in the lessons I cannot make out-what am I to do ?"-" Consult a good English dictionary. I am not against the proper use of a dictionary; it is the abuse of a dictionary I wish to guard you against. Do not expect too much from a dictionary. Do not place your reliance on a dictionary. Do not fly to a dictionary the moment you meet with a word you do not understand. Instead of consulting the dictionary, consult your own head. Surely you will be better off if you carry a dictionary about with you."-"Yes, I will get a pocket dietionary."" "—"No! no! I don't meant that; pocket dictionaries are of little more use than pocket pistols;" it is a head-dictionary that I wish to recommend; if you have a dictionary in your own head you will never be at a loss; and the way to acquire such a treasure is by systematic study the etymological study of the English tongue."-" Still you think a dictionary may be useful; what dictionary do you recommend?""I think it indispensable that you should possess a good English dictionary; talent and industry of the first class might do without a dictionary; and you yourself will fail in your duty if you do not learn far more without than you learn by means of a dictionary; nevertheless, there are occasions when a dictionary is useful, not to say necessary, and on that account I will set before you means for determining which of the dictionaries of the English language you should purchase; but enough for the present." "I find the study of those Greek stems difficult."-"Every study is difficult at the first, and often is a study the more difficult the more valuable it is, both for the information it contains and for the mental discipline which it gives. Pursue the course which the Lessons in English take, exactly in the order in which it is pre--"No; but the tools of the English language I do know, and sented, and master each lesson in succession."-"If by mastering you mean that I should thoroughly comprehend and retain in mind every part, I must eandidly tell you that I am unable to do so.""Why? every word likely to cause difficulty is explained, and an example of its import and use is given; the etymology of the words is, too, so set forth, that I should have thought you would, from that alone, have been led to the several meanings."-"Well, I have, I believe, made out the meanings of some of the words from a knowledge of their constituent elements."-"Doubtless you have, and with practice you will succeed in thus making yourself acquainted with them all; it is by this means that I have learnt the import of thousands of the words with which I am familiar.""O you have had good dictionaries.”—“ True, I possess good dictionaries, but the best dictionaries will not suffice to give anyone even a verbal knowledge of a language; and I assure you it is very possible for a dictionary to be so used as to be a hindrance to a real, and thorough, and exact acquaintance with a language. A dictionary is a very good servant, but a very bad master. A slavish use of a dictionary retards and obstructs even a verbal knowledge of a language. You should aim to become your own dictionary; and to a great extent your own dictionary you may become, if you take the trouble to make yourself familiar with the roots of the English Do you think you would ever acquire a knowledge of the steamengine, so as to be able to make an engine yourself, if you confined your inspection to its exterior? The way to know how to put a steam-engine together is first to take it to pieces, and then carefully to examine the structure and use of every part."—"Yes, brother, there is sense in that; I had a proof last week: I took my watch to be repaired, and as I stood there at the counter chatting with the watchmaker he began to take my watch to pieces; my curiosity was excited, I watched every step, and when he had done (or rather undone the watch), he explained to me the use and function of every part. To-morrow I am to go to see him put the parts together."-"A very good illustration; now you would understand what you see to-morrow very imperfectly if you had not seen the watch taken to pieces, and if, further, you had not carefully marked and studied every piece of the machinery. After all, your knowledge of the structure and the movements of the watch will remain very much inferior to the watchmaker's knowledge; why?"-"Isuppose, because he is more exactly and more thoroughly familiar with the several parts."-"Exactly; apply this to a language; it is the parts or the elements of the English language that I want you to be master of, well knowing that when you are so, you will know and write the language well; but without that mastery you must not expect to become a proficient in our tongue. You did not, I fancy, entrust your watch to the watchmaker's apprentice ?"-"I should be very sorry to do so."-"Why?" Why? because he is an apprentice, and a young one too."-"Very well, you thought he did not understand his business; and if he did not understand his business, it was chiefly because he was unacquainted with the structure and uses of the parts of your watch."-" But why take the watch to pieces in order to acquire that knowledge ?""Simply because that knowledge cannot well be otherwise acquired. I dare say you have looked at your watch very often."-"O yes, and I have tried to look into it-but never could get to know much about its works, or its operations."-"No, and long enough might the watchmaker's apprentice look at and look into his master s watches before he would acquire the knowledge and skill requisite to make him a watchmaker. Now, in regard to the English, you wish to be a watchmaker, that is, you wish to write good English how can you succeed unless by learning the parts of the structure with which you have to deal? No, no; you must follow the LESSONS IN ARITHMETIC-No. XX. PROBLEMS IN VULGAR FRACTIONS. 13. HAVING in our former lesson on vulgar fractions explained their nature and the principles of operation employed in their management, we now proceed to show the application of these principles in the solution of those problems which are usually to be found in treatises of arithmetic, PROBLEM I. To reduce a vulgar fraction to its lowest terms. DEFINITION 5.-When a vulgar fraction is expressed in the simplest form which it will admit of, it is then said to be in its lowest terms (art. 9, p. 134). Thus, the vulgar fraction is capable of being expressed in the form; because by dividing both the numerator 12, and the denominator 16, by the same number 4, which is the greatest common measure or divisor of these terms (see p. 56, vol. II.), its value is not altered (art. 10, p. 134), and it then assumes the simplest form which it will admit of, viz., 4; this, then, is the fraction in its lowest terms. RULE 1.-Find the greatest common measure or divisor of the terms of the fraction, that is, of the numerator and the denominator P. 56, vol. 11); and divide both of them by this common measure; the quotients will be the lowest terms of the fraction, that is, they will be the numerator and the denominator respectively, of the fraction in its lowest terms. Note: If the greatest common measure should be unity or 1, the fraction is already in its lowest terms. RULE 2.-Apply the theorems relating to the factors or divisors (given in p. 56, vol. II.) of numbers, by dividing both the terms of the fraction by any divisor which may be found upon trial COMMON TO BOTH; do the same with the fraction which results from this operation, and with the next, and so on, until it be found that the fraction, ultimately resulting, has no common divisor; this will be the given fraction in its lowest terms. Note: If the greatest common divisor of the terms of a fraction involves a prime number not included in the theorems above mentioned, and not otherwise determinable from the nature of the numbers themselves, this method will fail in reducing the fraction to its lowest terms; recourse must then be had to Rule 1. RULE 3.-Resolve the terms of the fraction into their prime factors (see pages 55 and 56, vol. I.), and cancel those factors which are common to each; the products of the remaining factors of each will be the lowest terms of the given fraction respectively. EXAMPLE TO RULE 1.-Reduce the fraction to its lowest terms. 75)105(1 (d) 75 30)75(2 (e) Here, the greatest | 15)30(2 (ƒ) terms. 30 EXAMPLE TO RULE 2.-Reduce the fraction 38 to its lowest Here, by employing the first and third theorems in p. 56, and dividing both terms of the given fraction and of the successively resulting fractions, by the common divisors (16), (8), and (3), we obtain the answer required; as in the following scheme, where we have put the common divisors above the fractions to which they belong, the resulting fractions being those on the right of each :(16) (8) (3) W=1=1= lowest terms. Here, the last divisor 15 is the greatest common divisor, as before. EXPLANATION. In this example, the 1st dividend is 465, and the 1st divisor 285; these are arranged in separate columns, and the 1st quotient, 1, is put on the left of the dividend; the product of the 1st divisor and 1st quotient is put under the 1st dividend, and subtracted from it; the 1st remainder 180 now becomes the 2nd divisor, and the 1st divisor the 2nd dividend. The second quotient 1, is now put on the right of the 2nd dividend; the product of the 2nd quotient and the 2nd divisor is put under the 2nd dividend and subtracted from it; the 2nd remainder 105 now becomes the 3rd divisor, and the 2nd divisor becomes the 3rd dividend. This process is carried on in the same way, until the last divisor 15 is obtained, and there is no remainder. This divisor 15 is the greatest common divisor. In order to render this explanation clearer, we have marked the quotients in the above operation, which correspond with the quotients in the example to Rule 1, with the same letters, a, b, c, d, e, and f. 2. To abridge the process of finding the lowest terms: Arrange the first dividend and all the divisors in succession from the first to EXAMPLE TO RULE 3.-Reduce the fraction to its lowest the last; then, under each divisor put the quotient which belongs Here, cancelling the factors 2, 2, 2, 2, 2, and 3, which are common to each term, the remaining term in the numerator is 7, and to it. In a new line, put 1 under the last divisor, and put the number which is under it in the preceding line also under the second last divisor in the new line. Multiply the number under this divisor in the first line by the number under it in the new line, to the product add the number to the right of it in the new line, and place the sum I under the third last divisor in the new line. Then, multiply the number under this divisor in the first line by the number under it in the new line, to the product add the number to the right of it in the new line, and place the sum under the fourth last divisor. Proceed in the same way, until numbers be obtained in the new line for all the divisors and for the first dividend; then the last two numbers thus obtained will be the lowest terms of the fraction. Thus, taking the same example already given under Rule I, and arranging the first dividend and all the divisors in succession, with their respective quotients, and the numbers in the new line, as above directed, we have 485 285 180 105 75 30 15 31 19 12 7 5 2 1 New line. the remaining terms in the denominator are 3 and 5, whose pro- Here, the last two numbers obtained are 19 and 31; whence the duct is 15. We have the answer, therefore, as follows: Of these three rules for reducing a fraction to its lowest terms, the first is generally the most easy of application, and the most universally employed by arithmeticians. It is tedious, however, in its operation, and irksome in the case of large numbers. The following method of abridging the processes both of finding the greatest common divisor, and of finding the lowest terms, will be acceptable to our students: 1. To abridge the process of finding the greatest common divisor: Arrange the dividends and divisors in two separate columns, and fraction reduced to its lowest terms is EXPLANATION. In the preceding operation it is plain that as the number 2 in the first line under 15 the last divisor, indicates that it is contained 2 times in the preceding divisor, and the number 2 in the same line under 30 indicates that 30 is contained 2 times in the divisor preceding it, therefore 15 must be contained at least 2 times multiplied by 2 times, that is, 4 times in that divisor 75; but as 15 was the remainder, which contains itself 1 time, therefore 75 contains 15 exactly 5 times. Whence, the number 5 is placed under 75 in the new |