not generally into those of equilibrium. The speed or velocity of a cannon-ball must be considered at every varying moment of its flight; but the strains and pressures among and on the beams of the roof of a railway station are the same at all moments. Time does not affect the latter unless by wear and tear. With statics, therefore, we commence, and, of course, with the simplest class of questions, those which relate to a force or forces acting on a single point. But here I must turn back to the notion of force, and endeavour to fix it with greater accuracy in your minds. I must show you how it is said to be applied and measured to the body it moves or strains; and this will best be done under the three following heads :— 1. The Direction of a Force. 2. The Point of Application of a Force. 3. The Magnitude of a Force. 1. The Direction of a Force.-In Mechanics, forces are assumed to act in right lines. The assumption is made for the best of reasons namely, that of experience. All the simpler cases of motion confirm it, and all the more complicated can be accounted for by it. A ball falls to the ground in a right linethat which points to the centre of the earth, whence the force of attraction which moves it acts. The billiard-ball moves in a right line; and the calculations of the skilful player, which are based on the supposition that it so moves, are never found to be wrong. A ship, with her sails square set and wind aft, moves in a right line; and to make it leave that line the steersman must put the helm to port or starboard, and by turning the face of the rudder against the water, cause another force to be applied to the ship across the line of its course, and at her stern, turning her round. It is true that the stone thrown obliquely into the air moves in a curved path; but in this case we know that there are two forces-not one only-acting on it, namely, the original impulse, which makes it move in a right line, and the earth's attraction, which pulls it from that line into a curved course. Moreover, all the calculations on which are based the predictions of astronomers as to the places in which the sun, moon, and planets will be on a certain day, hour, and minute, are based on this assumption, that forces act in right lines; and the predictions invariably prove true. Our first mechanical axiom may, therefore, on the ground of experience be assumed to be true-namely, that the direction in which a force acts is that of a right line. Indeed, it is not easy to conceive how it could act otherwise. 2. The Point of Application of a Force.-The direction of a force being disposed of, we must fix our ideas as to its point of application. The rule is, that any point on the line of its direction may be considered such; but this you must understand with a limitation, or exception, which should not be forgotten. The point of application can only be on so much of the line of direction as lies within the body. For instance, suppose a person to push with an iron rod, which he holds in his hand, at the point A (as in the diagram), against a block of iron which lies on a table. Then, clearly A is the point of application of the force with which he pushes. Let now a hole be drilled through the block in the direction of the push from A to E, into which the rod may fit closely but freely; and also other holes, downwards, b B, c c, d D, to meet the passage, A E, into which thumbscrews, b, c, d, are fitted. Let the rod now be passed through the block so as to emerge at the other side, and clamp it down firmly by the thumb-screw, b. If it be now pushed against the block with the same force as before, it is clear that the force will be arrested by the thumb-scrow, b, at B, and that в will become its point of application to the body. So, in like manner, may it be applied to c and D, by tightening in succession each screw, while the others are left loose. In all these cases the force is the same, and the direction the same; but the points of application are different. But will the effects in the several cases be different? No; for the portion of the rod within the block, and extending from A to any of the points of application, performs the same part in transmitting the force from A to the point within, as the iron which was removed did when the force was first applied directly at A. The removed iron has its place filled by an equivalent of that metal in rod, and the body is virtually in its original condition. The force of the hand may still be considered applied at A, thence to be transmitted to B, or C, or D, as we please, by the portion of rod within. The second case becomes identical with the first, and the effects, therefore, must be identical in every respect; and, nothing being changed, intensity, direction, nor effect of the force, it is clearly indifferent which point we make the point of application. Another instance is the raising of a weight by a rope. Weight and rope together make one body; and whether the lifting power be applied by engine, by horse, or by man, whether it acts over a pulley or not, every point of the strained rope may be considered a point of application. Or let the case be that of three strings attached to a ring, and pulled in different directions by three persons. It makes no difference, in this compound body of ring and strings, whether the hold taken of the latter be long or short-all their points are points of application of their respective forces. We thus see that, in all cases, we may assume that the point of application of a force is any point on so much of its line of direction as lies within the body. To suppose it applied to a point outside would be absurd; for, as we have shown, though a force may act or push through a point of empty space, it can make no impression on that point, either in the way of strain or motion, and therefore cannot come under the consideration of Mechanics. 3. The Magnitude of a Force.-To find a suitable measure of the intensity or magnitude of a force, we must also look to experience. It would be very convenient to measure forces by comparing them with weights; but this is not always practicable, and, even if it were, it would not answer all the purposes of Mechanics. I may as well, therefore, explain to you the perfect method, as that is as simple as any other. Experience teaches that a double force produces a double velocity, a treble force a treble velocity, and so on, in any body to which it is applied. But then a difficulty occurs: the same force will produce different velocities in bodies of different sizes. If it make a ball of one pound weight move at a certain rate, it will give double that speed to a half-pound ball, and half to one of two pounds. As a general rule, the greater the mass of the body, the less the speed produced. Everybody is familiar with this fact. We see, then, that if we desire to measure forces by the velocities they produce, we must try them on bodies of some fixed weight or mass. Tried on this particular mass, experience teaches that that which produces the greater velocity is the greater force. Now, the mass of matter which mechanicians choose for this purpose is that of any substance which is equal in weight to a cubic inch of distilled water. That much matter is designated the Unit of Mass, and for a reason I shall hereafter more fully explain. Imagine, then, a round ball, say of ivory, whose weight is that of a cubic inch of pure water, and suppose that several forces are in succession applied to it; the velocities they produce will be accurate measures of their intensities, or of their magnitudes. But, then, how are the velocities to be ascertained? Clearly by the spaces the ball would move over in any given time, say the unit of time-a second-on the force being applied to it. Suppose, then, the unit ivory ball, put on a perfectly smooth floor, and then suddenly struck by a blow equal to the force you want to measure, By some means and there are many which may be devised-manage to ascertain the distance the ball moves over in one second. That space, or length of line, will be the measure of the force; and if any number of such forces be tried in the same way and on the same ball, that which causes it to move over the greater space is the greater force, over a double space a double force, and so on. The final result, then, is that, in considering a force in Mechanics, we must first suppose drawn within the body a line representing its direction. Then, on that line, let any point be taken for its point of application. Thirdly, on the line of direction so fixed, let as many inches be measured from the point of application as, on any scale you agree to use, represents the space the force would cause the unit ivory ball to move over in one second. Then you have a line which also in magnitude represents the force. Or in fewer words— A FORCE is represented, both in magnitude and in direction, by a finite right line passing through its point of application. If in the above explanations I have succeeded in giving you clear notions of the aim of Mechanics, and of the nature and effects of force, you are prepared for the consideration of a force, or forces, applied to a single point, which will be the subject of our next Lesson. LESSONS IN FRENCH.-II. SECTION I.-FRENCH PRONUNCIATION (continued). II. FRENCH ACCENTS. 17. THE Constant use of certain marks called accents in the French language constitutes a marked peculiarity which cannot escape the attention of the student. Rarely, except in elementary works of the English language, is the syllable of any given word which requires an emphasis marked. grave accents, and is placed over each of the vowels except y. It indicates that the letter over which it is placed has a sound twice as long as it has without it, viz. : Âge, Bête, Bûche, Côte, Gite, Mê-lée, Tête. This accent also indicates the suppression of the letter s, after the vowel over which it is placed; thus Bête, Beste, Fête, Feste, Tête, Teste. The s was not sounded, but gave to the preceding vowel that prolonged sound now represented by the circumflex accent. The circumflex accent also serves to distinguish parts of 18. But it is not so in the French language: here, accents of were formerly written various kinds are constantly meeting the eye on every page. One thing, however, must be observed, namely:-the position of the accent does not always and infallibly mark the syllable of a word which must receive the stress of voice in common pro19. Modern French grammarians have established the follow-speech from each other; thus, ing rule, namely:-to place the stress of voice on the last pronounced Crú is a participle from the verb syllable of every word. nunciation. 20. A slight inspection only of the following examples will illustrate the above remarks. 1. Dé-vo-rer (pronounced Day-vo-ray). The first syllable of this word is marked with an accent; must the stress of voice, therefore, be placed upon the syllable de? No: if the rule be applied to this word, the stress of voice falls on the last syllable, rer. It will then be asked, What is the use of this accent? We answer, It modifies the sound of the vowel over which it is placed. 2. Lé-gère-ment (pronounced Lay-zhair-mon, with the sound of the final n suspended). Again, the word used now as an example has the same kind of an accent as the word used in the previous example had; and also, it is placed over the same vowel. But it has another dif ferent accent over the first vowel of the second syllable; and, according to the rule, the stress of voice is not placed either upon the first or second syllable, but upon the last. This second accent (observe its form and position) only serves to modify the sound of the vowel over which it is placed. Sometimes, however, an accent is placed over a vowel of the syllable which, according to the rule, receives the stress of voice, viz. :— Cé-lé-bri-té. 3. Ba-ti-ment (pronounced Bat-tee-mon, with the sound of the final n* stopped). Again, in the word used here as an example, a third and still different accent is placed over the vowel a. Its presence affects the sound of that vowel only. It has nothing whatever to do with the proper accent of that word, as the term accent is understood when applied to words in the English language. As a general rule, the stress of voice is not so strong in the French as in the English language. 21. Accents, therefore, as used in the French language, are certain marks differing from each other, and placed over certain vowels only, for specific purposes. 22. There are three accents, viz.: First, to modify its sound. English language; that is, when a word is divided, so that a Its use in connecting syllables is precisely the same as in the part of it is at the extreme right hand of a line, and the rest at the extreme left of the line following. 29. The APOSTROPHE (') is like a comma placed at the upper end of letters instead of at the lower end, or at the bottom on a line with the lower end. Its use is to show the elision, or cutting off, of a vowel before words commencing with a vowel or h mute, and is much used in the French language, as: L'ami, instead of le ami. Secondly, to mark the existence of a distinct and final position between two parts of speech, viz., the verb and the syllable, as: This letter cannot be translated, because it has no meaning. It is thus used merely for the sake of euphony, or agreeable sound. 31. PARENTHESIS AND PUNCTUATION.-In the French language, the marks used in punctuation, etc., are the same, and used for the same purposes, as in the English language. (See READING AND ELOCUTION.) SECTION IV.-THE ARTICLE USED PARTITIVELY. 1. The article, preceded by or contracted with the preposition de [Sect. III. 1, 2], is placed in French before words used in a partitive sense. Such words may generally be known in English when some or any is or may be prefixed to them [§ 13 (10), Has the bookseller's son a gold pencil-case? § 78 (1)]. Du pain De la viande De l'argent Bread, or some bread. Meat, or some meat. Money, or some money. 1. Avez-vous de la viande ? 2. Oui, Monsieur, j'ai une livre de viande. 3. Votre fils a-t-il un morceau de pain? 4. Oui, Madame, il a un morceau de pain. 5. Le libraire a-t-il un livre? 6. Il a de l'encre et du papier. 7. Votre sœur a-t-elle une montre d'or? 8. Elle a une mentre d'or et un dé d'argent. 9. Le boulanger a-t-il du vin ou de la bière? 10. Le boulanger a du thé et du café. 11. Votre frère a-t-il du fromage? 12. Il a du fromage et du beurre. 13. La dame a-t-elle une cuiller d'argent? 14. La dame a une cuiller et une fourchette d'argent. 15. Le boucher a-t-il de la viande aujourd'hui ? Oui, Monsieur, il a un morceau de bœuf. 17. Le charpentier a-t-il une table? 18. Oui, Monsieur, il a une table d'acajou. 19. Avez-vous le livre du médecin ? 20. Non, Madame, mais j'ai le livre de votre sœur. 21. Qui a du café et du sucre ? 22. L'épicier a du café et du sucre. 23. La sœur du libraire a-t-elle un gant? 24. Non, Monsieur, mais elle a un livre. 25. A-t-elle une plume d'acier? 26. Non, Monsieur, elle a une plume d'or. 27. Vous avez le porte-crayon du médecin. EXERCISE 6. 1. Have you any tea? 2. Yes, Madam, I have a pound of tea. 3. Who has bread? 4. The baker has bread, butter, and cheese. 5. Has the tailor cloth? 6. The tailor has a piece of cloth. 7. Has the physician gold? 8. Yes, Sir, the physician has gold and silver. 9. Has the lady a silver watch? 10. Yes, Miss, the lady has a silver watch and a gold pen. 11. Has 12. Yes, Sir, she has silk and cotton. 13. your sister silk? Have you a knife? 14. Yes, Sir, I have a steel knife and a silver fork. 15. Have you meat to-day, Sir? 16. Yes, Sir, I have a piece of beef. 17. Has your carpenter a mahogany table? 18. Yes, Sir, he has a mahogany table. 19. Has your 22. Yes, Sir, he has a gold pencil-case and a steel pen. 23. Who has your sister's watch? 24. Your brother has the gold watch and the silk hat. 25. We have gold, silver, and steel. (See Rule 5.) SECTION V.-THE NEGATIVES, ETC. 1. To render a sentence negative, ne is placed before the verb, and pas after it. Je n'ai pas, Tu n'as pas, Il n'a pas, Elle n'a pas, Negatively. Nous n'avons pas, Vous n'avez pas, Ils n'ont pas, Elles n'ont pas, RÉSUMÉ OF Le tailleur a-t-il le bouton ? Il n'a eu ni le drap ni le cuir. Vous n'avez pas de viande. (R. 5.) N'avons-nous pas? Have we not? Has the tailor the button? The tailor has not the button. He has not had the cloth. He has had neither the cloth nor the You have no meat. Have we anything? We have nothing, or not anything. Nous n'avons jamais de café. (R.5.) We never have coffee. 1. Le chapelier a-t-il de la soie? 2. Le chapelier n'a pas sister a glove? 20. Yes, Sir, my sister has a silk glove. 21. de soie, mais il a du velours. 3. A-t-il du velours de coton ? 4. Non, Monsieur, il n'a pas de velours de coton, il a du velours de soie. 5. Avez-vous de la viande ? 6. Oui, Monsieur, j'ai de la viande. 7. Le médecin n'a pas d'argent. 8. Qui a de l'argent? 9. Le marchand n'a pas d'argent, mais il a du drap, du velours, et de la soie. 10. Avez-vous quelque chose? 11. Non, Monsieur, je n'ai rien du tout. 12. Le tailleur a-t-il deux boutons d'argent? 13. Non, Monsieur, il a deux boutons de soie. 14. Qui a votre chien? 15. Le voisin a le chien de mon cousin. 16. N'a-t-il pas votre cheval aussi? 17. Non, Monsieur, il a le cheval de votre ami. 18. Avez-vous l'histoire de France? 19. Non, Madame, je n'ai ni l'histoire de France ni l'histoire d'Angleterre. 20. N'avez-vous ni le livre ni le papier? 21. Non, Mademoiselle, je n'ai ni l'un ni l'autre. 22. Qui a du papier? 23. Le libraire n'a pas de papier. 24. Quelqu'un a-t-il un livre ? 25. Personne n'a de livre. EXERCISE 8. 1. Has the baker velvet? 2. No, Sir, the baker has no velvet. 3. Who has silk velvet? 4. The hatter has silk velvet and a silk hat. 5. Have you two silver buttons ? 6. No, Sir, I have a cloth coat, a silk hat, and a velvet shoe. 7. Has your neighbour a wooden table? 8. Yes, Sir, he has a mahogany table. 9. Has your cousin a history of England? 10. No, Sir, he has a history of France. 11. I have neither the cloth nor the velvet. 12. We have neither the meat nor the coffee. 13. Has any one a book? 14. Your cousin has a book, a velvet coat, and a silk hat. 15. Have you the physician's book? 16. Yes, Madam, I have the physician's book and the lady's gold pen. 17. Has the merchant cloth ? 18. The merchant has no cloth, but he has money. 19. Who has your neighbour's dog? 20. Nobody has my neighbour's dog. 21. Has any one my book ? 22. No one has your book. 23. Has your cousin's brother anything? 24. No, Sir, he has nothing. 25. Who has your friend's book? 26. Your brother has my cousin's book. 27. Has he the tailor's coat ? 28. He has not the tailor's coat. 29. We have neither the cloth nor the silk. LESSONS IN PENMANSHIP.-II. In our last lesson we gave the student an example of the first stroke that should engage his attention in beginning to acquire the art of writing, and explained to him that it was a downstroke square at the top and brought downwards with an equal pressure of the pen until it narrows at the bottom into a fine hair-line, which is turned upwards towards the right. This down-stroke with a fine up-turn, or "pot-hook," as it is familiarly called, but which we shall term a bottom-turn for the sake of brevity, enters into the composition of no less than ine letters of the alphabet in writing, of which four-namely, i, u, t, 1-consist of this stroke only, with certain slight modiEcations. We mention this to the self-teacher to encourage him to perseverance in the task he has undertaken, for he will see plainly enough, after a little consideration, that when he is able to imitate this bottom-turn correctly, he has not only learnt to make this simple stroke itself, but has actually advanced more than half-way towards writing the four letters we have just ramed, besides five others that will be pointed out in the course of future lessons. A brief examination of the copy-slips given in this page will be sufficient to prove the truth of our statement. The letter i, the simplest letter in the alphabet, is merely the elementary bottom-turn shown in Copy-slip No. 1, with a dot or point a little above it in the direction of the slope of the letter, or, in other words, immediately above the letter in a straight line which passes through the centre of the thick down-stroke from top to bottom. The letter u, again, is merely the bottom-turn twice repeated, the fine hair-stroke of the first bottom-turn being joined to the thick down-stroke of the second in a line passing midway between the two horizontal lines within which the letter is written; while the letter t is formed by the bottomturn, commenced at the same distance above the upper of these horizontal lines as that at which the dot is placed above the letter i, and crossed a little above that line by a short horizontal hair-stroke. It may be as well to say something about the form in which our Copy-slips are placed before our readers. The lines a a, b b, as in Copy-slip No. 4, are the lines between or within which what we may call the body of each letter is written. These lines and the space between them resemble in some measure the staff in music, portions of certain letters being carried above the upper line a a in some cases, or below the lower one b b in others, as ledger notes are carried above or below the staff in musical notation. The line cc, midway between the lines a a, b b, is that in which the letters, or component parts of letters, should be joined together, while the line d d shows the distance above a a at which the letter t should be commenced, or the dot placed above the letter i. The diagonal lines sloping from right to left show the proper inclination of the thick down-strokes of the letters, and act as guide lines to enable beginners to make all their letters of the same slope, and keep the down-strokes parallel to one another. A little trouble taken at starting to keep on the same level the heads, loops, and tails of all letters that extend above or below the lines within which the body of each is written, will go far to ensure neatness and regularity when the learner can write with ease and rapidity, and his handwriting begins to assume a character peculiar to itself. By combining these symbols according to the following rules all numbers can be represented: When two symbols-are placed together, if the one denoting the less value is on the left of the other, then the less number is to be subtracted from the greater; if on the right hand, it is to be added to it. Thus IX denotes ten with one subtracted, or nine; XI denotes eleven; XL denotes forty; LX, sixty. If the symbols are of equal value, then they are simply to be added. Thus XX denotes twenty; CC, two hundred, etc. The value represented by Io is increased tenfold by every additional placed on the right. Thus 5,000 is denoted by 100, and 50,000 by 1000. The value of the symbol CIO becomes increased tenfold by the addition of C and Ɔ, one on each side of the line I. Thus 100,000 is denoted by CCCIODO, 1,000,000 by CCCC, and so on. A straight line placed over any one of these symbols increases its value a thousand-fold. Thus I denotes 1,000; V, 5,000; L, 50,000; C, 100,000. 2,000 was usually denoted by CIOCIO, but sometimes by IICIO, or IIM, or MM. Similarly, 4,000 was denoted by IVCIƆ, etc. The above remarks will sufficiently explain the following Table of Roman Numerals : 8. Nine thousand nine hundred and ninety-nine millions, nine hundred and ninetynine thousand nine hundred and ninety-nine. 9. Write the number which follows this last one in order. 10. One trillion and three. 11. Eighty millions two hundred and three thousand and two. 12. Two hundred and two millions twenty thousand two hundred and two. 13. Twenty thousand millions. 14. Two hundred thousand and twenty millions two thousand. 15. The next number to thirty thousand billions nine hundred and ninety-nine thousand. ADDITION. 1. The process of uniting two or more numbers together, so The number as to form a single number, is called Addition. thus formed is called the sum of the separate numbers. 2. The sign + placed between two numbers indicates that they are to be added together. This symbol is called plus. The sign placed between two numbers denotes that they are equal. Thus 2+ 35, expresses that 2 and 3 added together are equal to 5. 3. Suppose that it be required to add the two numbers 3452 and 4327 together. These are respectively 3 thousands, 4 hundreds, 5 tens, and 2 units, which, added together, are equal to— 7 thousands, 7 hundreds, 7 tens, and 9 units. The sum, therefore, of 3452 and 4327 is 7 thousands, 7 hundreds, 7 tens, and 9 units, which, according to our system of notation, will be written 7779. This is got by putting down the two numbers one under the other, the units under the units, the tens under the tens, and so on; and then adding up the lower to the upper figure in each place, thus: 3452 4327 7779 CX CC CCC CCCC DC 1. Write out the names of all the numbers from one to a hundred, and express them in figures. 2. Write out the names of the numbers which immediately follow : 4. In the example we have taken, the sum of the numbers of the thousands amounts only to a number expressed by one figure, namely, 7; and similarly for the hundreds, the tens, and units. Suppose, however, that we have a case in which this is not so; for instance, to add 8976 and 4368. These are respectively equal to 8 thousands, 9 hundreds, 7 tens, and 6 units, or, added together, to 1 12 thousands, 12 hundreds, 13 tens, and 14 units. This, however, is not at present in a form which can be at once written down according to our system of notation. We must, therefore, alter its form. 13 tens and 14 units are the same as 14 tens and 4 units. Now 14 units are the same as 1 ten and 4 units; therefore But 14 tens are the same as 1 hundred and 4 tens; therefore 12 hundreds and 14 tens are the same as 13 hundreds and 4 tens. But 13 hundreds are the same as 1 thousand and 3 hundreds; therefore 12 thousands and 13 hundreds are the same as 13 thousands and 3 hundreds. Hence we see that 12 thousands, 12 hundreds, 13 tens, and 14 units, are the same as 13 thousands, 3 hundreds, 4 tens, and 4 units, which, by our notation, is written 13344. 5. The preceding process will sufficiently explain the following Rule for Addition : Write down the numbers under each other, so that units may stand under units, tens under tens, etc., and draw a line beneath them. Then, beginning with the units, add the columns separately. Whenever the sum of the figures in a column is a number expressed by more than one figure, write down the right-hand figure of such number under the column, and add the other figure or figures into the next column. Proceed in this way |