LESSONS IN GEOMETRY.-IV. INSTRUMENTS USED IN PRACTICAL GEOMETRY (continued). In addition to the mathematical instruments described in our last lesson, 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, A M B, being the fiducial edge to be applied to the other leg. Thus, in order to measure the angle x o 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 M B, determined by the point м, 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.

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 commonly 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 he, drawing) operations, such as the construction of plans, maps, and charts, architectural designs, plans and sections of machinery, etc., 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, A E (Fig. 15), divided into any convenient 4987654321B

Fig. 15. number of equal parts, A B, B C, C D, D E, etc., one, A B, is assumed as the standard unit of measure. From the different points, A, B, C, D, E, etc., perpendiculars of a convenient length, as AA, BB, CC, D D', E E, etc., are drawn to the straight line A E, and terminated in the straight line A' E' parallel to A E. The unit A B is divided into 10 equal parts; then the opposite part, AB, is similarly divided; next the perpendicular B B' is divided into 10 equal parts, and through each division straight lines parallel to AE or A' E' are drawn. The divisions of the straight line A E are now marked with the numbers 1, 2, 3, etc.,

VOL. I.

from c to E, to denote units. The divisions of the standard unit A B are marked 1, 2, 3, 4, 5, 6, 7, 8, 9, from в to A, to denote tenth parts of a unit; and the divisions of the perpendicular B B' are marked 1, 2, 3, 4, 5, 6, 7, 8, 3, from в to B', to denote hundredth parts of a unit. Or, if the divisions of the straight line A E 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 B A, to 1 on B' A'; from 1 on B A, to 2 on B'A'; from 2 on B A, to 3 on B' A'; and so on, till one be drawn from 9 on в A, 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 B 1 B', at the division marked 1, is one-tenth part of the base 1 B', and consequently one-hundredth part of the line A B; the small parts of the other parallels are in succession, twohundredths, three-hundredths, etc. 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 3, 30, or 300 equal parts, according as A B is made to stand for 1 unit, 1 ten, or 1 hundred. If it does not measure from E to B exactly, but extends from E exactly to one of the division marks between B and A, say 4, then the straight line may be said to measure 34, 34, or 340 equal parts, according to the 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, preserving 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 B B', 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, etc., 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, semi-tangents, secants, lines of longitude, etc. 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 numbers which denote these divisions; by means of the logarithmic 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 consiruction, 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 Fig. 16. It is composed of two flat rulers, movable 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 proportion. 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

8

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, etc., 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°, 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 B are its legs, the marks on the legs represent the divisions of the line into equal parts. Its use is to find straight lines that shall be to one another in a given proportion. Suppose, for example, that it is 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 length of the given line, which may be easily done by the help of a pair of common compasses or dividers; then, the distance of the two points marked 3 on its legs, will be the length of the straight line required.

Again, suppose that two straight lines are given, and it is required to find their ratio to each other in numbers. Open the sector until the distance of the two points marked 10 on its legs, is equal to the length of the greater of the two given straight lines; then, taking the length of the smaller of the two given lines in a pair of compasses, apply this distance to the two points of any number less than 10 marked on its legs, until it be found that it coincides exactly with that of two points having the same number, say 3; then the two given straight lines are to one another in the ratio of 10 to 3; or, in other words, the smaller is three-tenths of the greater.

The Proportional Compasses, called by the French compas de reduction-the compasses of reduction-are represented in Fig. 17, and consist of two legs A N, C M, intersecting (i.e., crossing) each other at any point within certain limits, according to the position of the button and screw, B, round which they are made to turn. These legs are graduated in such a manner that, by screwing the button at the proper place, the distance from A to c may be at pleasure one-half, one-third, one-fourth, etc., of the distance from м to N. By this instrument, a straight line may be easily divided into any number of equal parts, or into any other proportional parts required.

The invention of this instrument is claimed, by a recent 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.

LESSONS IN FRENCH.-VIII.
SECTION 1.-FRENCH PRONUNCIATION (continued).
III. NAME AND SOUND OF THE VOWELS

46. Y, y.-Name, EE, ee; sound, like the letters ee in the English word bee.

This letter is also a word; that is, it is one of the parts of speech in the French language. It is usually an adverb, meaning there. It is also used as a noun, and a pronoun.

When y stands alone, and thus becomes a word, its pronunciation is invariably like that of the letters ee in the Englich word bee, viz. :-Il y a, pronounced eel ee a. This last a must be sounded like a in the English word fat.

Y is also pronounced like the letters ee in the English word bee, when it begins or ends a word; and also when it occurs in the body of a word, after a consonant, namely:

FRENCH. Dey

Style

Système

Yole

PRONUNCIATION.
Day
Steel

Secss-taim
Ee-ol or E-ol

ENGLISH. Deu. Style.

System.

A yawl.

Moyen should be pronounced as if printed thus, namely, move; divided thus, namely, moi-i-en, but pronounced in two syllables, namely, moi-ien. Joyeur should be pronounced as if printed thus, namely, joiien; divided thus, namely, joi-i-eu, but pronounced in two syllables, namely, joi-ieu.

Royaume should be pronounced as if printed thus, namely, roiiaume; divided thus, namely, roi-i-aume, but pronounced in two syllables, namely, roi-iaume.

The pupil need not attempt to pronounce these three French words used as examples, because the combination of vowels and other letters occurring in them has not yet been illustrated. The pronunciation of y with these and other combinations of letters will be explained in future lessons.

In the two following words the y, though not placed between two vowels, is under the same rule, namely:Pays, meaning a country, should be pronounced as if printed paiis; divided thus, namely, pai-is, and pronounced pa-ee. Paysage, meaning a landscape, should be pronounced as if printed paisage; divided thus, namely, pai-i-sage, and pronounced pa-ee-zazh.

IV. NAME AND SOUND OF THE CONSONANTS.

As a general rule, none of the consonants, when final, have a distinct and independent sound, unless immediately followed by consonant is joined with the following word in pronunciation. a word commencing with a vowel or h mute; in which case the 47. B, b.-In any position within a word, this letter has the sound of the English letter b.

When doubled within a word, only one b is sounded, viz. :

At the end of proper names, b is always sounded. In these two words, namely, a-plomb and plomb, the b is silent, and the next two preceding letters in each word, namely, om, take the nasal sound of on.

48. C, c.-This letter has two entirely distinct sounds, namely, hard and soft. Before the vowels à, o, u, and œ, and also before the consonants c, l, n, and r, it has the hard sound of the letter k in the English word kill, namely :FRENCH. PRONUN. ENGLISH. FRENCH PRONUN. ENGLISH, Calamité Ka-la-me-tay Calamity. Succès Suk-sai Success. Ko-me-tay Committee. Classe Klauss Class, Kube Cube. Cnique Kneek Horse-thistle, Kuh-rr Heart. Crédit Kray-de Credit.

Comité

Cabe

Cœur But before e, i, and y, and also with the cedilla before a, o, and u, it has the soft sound of the letter s in the English word sea, namely:FRENCH. PRONUN

ENGLISH. Cedar.

Five.

Cycle.

FRENCH. PRONUN. ENGLISH. Façade Fas-sad Front. Fas-son Fashion. R'su

Façon

Reçu

Receipt.

In a few words, however, c final is not sounded, and these exceptions are best found out by consulting a French pronouncing dictionary. In a few words, c has the sound of the letter g in the English word go, namely, second, secondaire, secondairement, seconde, secondement, seconder, secondine.

In these words the c, which commences the second syllable, has the sound of the g, namely, second, as if printed segond; pronounced s'-gonh, etc. We do not illustrate all the sounds of these French words here, because of the nasal sounds contained

in them.

49. D, d. This letter generally has the sound of the letter d in the English word deed. It is usually silent when final, except in proper names.

The principal exception to the above rule is, when d is final just before a vowel or an h mute. In such a case, the d has the sound of the letter t in the English word top; and in pronuncia

* See foot-note, page 19.

Langue, f., tongue.

4. MALADIES, INFIRMITÉS,

Attaque, f., attack, fit.
Baume, m., balsam.

Bégaiement, m., stammering.

Blessure, f., wound.

Cécité, f., blindness.

Chanere, m., cancer.

Cicatrice, f., scar.

Colique, f., colic.

Coutusion, f., bruise.

Crampe, f., cramp.

Dislocation, f., dislocation. Emétique, m., emetic. Enflure, f., swelling.

Enrouement, m.,

Entorse, f., sprain.

hoarseness.

Epilepsie, f., epilepsy.

Eranouissement, m., fainting.

Fièvre, f., fever.

Ongle, m., nail. Orteil, m., toe. bone.

Os, m.,

Palais, m., palate. Paupière, f., eyelid. Peau, f., skin.

Pouce, m.,

thumb.

Poumon, m., lungs.

Prunelle, f., pupil of the eye.

Rate, f., spleen.

Reins, m. pl., loins.

Sang, m.,

blood.

Sein, m., bosom.

Sourcils, m. pl., eyebrows.
Squelette, m., skeleton.
Talon, m., heel.

Teint, m., complexion.
Tempes, f. pl., temples.
Trait, m., feature.
Veine, f., vein.
Visage, m., face.

ETC.—MALADIES, INFIRMITIES, ETC.
Fièvre scarlatine, f., scarlet fever.
Goutte, f., gout.
Guérison, f.,

Fièvre nerveuse, f., nervous fever.

cure.

Hydropisie, f., dropsy.
Indisposition, f., indisposition.
Louche, adj., squinting.
Malaise, m., indisposition.
Mutisme, m., dumbness.

Onguent, m., pommade, f., salve.
Ordonnance, f., prescription.
Petite-vérole, f., small-pox.
Pulmonie, f., consumption.

Remède, m., remedy.
Rhume, m., cold.
Rougeole, f., measles.

Surdité, f., deafness.

Toux, f., cough. Ulcère, m., ulcer.

Vertigo, m., dizziness.

Diamant, m., diamond.
Dentelle, f., lace.
Doublure, f., lining.
Ecrin, m., casket, jewel-box.
Épée, f., sword.
Eperons, m. pl., spurs.
Epingle, f., pin
Etui, m., needle-case.
Eventail, m., fan,
Fourrure, f., fur.

Flacon, m., smelling-bottle.

Frac, m., dress coat. Frange, f., fringe. Garniture, f., trimming. Gilet, m., vest, waistcoat. Grenat, m., garnet. Guêtres, f. pl., gaiters. Habit, m., coat.

le meilleur de tous les jardins de la ville.

6. Avez-vous encore de l'argent? 7. Je n'ai plus d'argent, mais j'ai encore du crédit. 8. Avons-nous encore de la salade? 9. Nous n'en avons plus. 10. Nous n'avons plus de viande. 11. Qui en a encore? 12. Mes frères et mes sœurs en ont encore. 13. En avez-vous encore beaucoup ? 14. Je n'en ai plus guère. 15. Votre tante a-t-elle plus de robes que votre nièce ? 16. Elle n'en a pas beaucoup. 17. Votre neveu est-il plus savant que votre nièce? 18. Il n'est pas aussi savant qu'elle. 19. Elle est plus savante que lui. 20. 21. Je n'ai plus froid, j'ai bien chaud. 22. N'avez-vous plus de nouvelles? 23. Je n'en ai plus. 24. En avez-vous beaucoup ? 25. Je n'en ai guère.

Avez-vous encore froid?

EXERCISE 28.

1. Has your brother a very good dictionary? 2. His dictionary is not very correct. 3. Has your father more courage than he ? 4. He has much more courage than your nephew. 5. Have your brothers credit? 6. They have but little credit, but they have money. 7. Is your aunt obliging? 8. My aunt is very obliging. 9. Have you still books, pens, and paper? 10. I have no more oooks, but I have still good pens and excellent English paper. 11. Who has still paper? 12. I have no more, but my brother has some more. 14. No, 13. Have you any news, Sir ? Madam, I have none to-day. 15. Have you as much wood as my brother's son ? 16. I have more than you or he. 17. Are you still wrong? 18. No, Sir, I am no longer (plus) wrong, I am right. 19. Are your sisters still hungry? 20. They are neither hungry nor thirsty, but they are still sleepy. 21. Is your niece as learned as he? 22. She is more learned than he and (que) his aunt. 23. Have you no news, Sir? 24. No, Madam, I have no more news. 25. Who has news? 26. I have

no more.

27. Have you them all? 28. Yes, Sir, I have them all. 29. Has your aunt much of it left? 30. She has but little more of it. 31. Has your brother any more English horses ? 32. He has no more. 33. He has two more. 34. Have you a handsome French shawl left? 35. I have no more French shawls, but I have an English one.

2. The adverb bien, used in the sense of beaucoup (much, many), is followed by the preposition de, joined to or blended with the article le, la, les [Sect. IV.].

Vous avez bien de la complaisance, You have much kindness.
Elle a bien des amis,

She has many friends.

How many pears have you?
We have many pears.
We have many (of them).
We have cherries enough.
We have not enough (of them).
You have but few peaches.
Your gardener has many peaches.
Have you no peaches?

I have many peaches and apricots.

Has the butcher anything good?
He has something good and bad.

He has ..ot anything (nothing) good,
What or which pears have you?
We have your sister's.
Which or what coat have you?
We have the tailor's.
What have you good?
Which (one) have you?

Which (ones) has your brother ?

VOCABULARY. Fleur, f., flower.

Légume, m., vegetable.
Magasin, m., ware-

house.
Oncle, uncle.
Poire, f., pear.
Poivre, m., pepper.

EXERCISE 29.

10. Nous en

1. Combien de pommes de terre votre frère a-t-il ? 2. Il n'en a pas beaucoup. 3. L'épicier a-t-il beaucoup de sucre dans son magasin? 4. Il n'en a guère, mais il a beaucoup de beurre et de poivre. 5. Votre jardinier a-t-il beaucoup de cerises? 6. Il a plus de cerises que de prunes. 7. Les prunes sont-elles meilleures que les cerises? 8. Les cerises sont meilleures que les prunes. 9. Avez-vous quelques poires mûres ? avons quelques unes, nous avons aussi beaucoup d'ananas et d'abricots. 11. Votre oncle a-t-il quelque chose de bon dans son jardin ? 12. Il a quelque chose de bon et de beau. 13. Il a de beaux légumes et de belles fleurs. 14. Avez-vous des fleurs étrangères ? 15. J'en ai quelques unes. 16. Lesquelles avezvous ? 17. J'ai celles de votre frère et celles de votre jardinier. 18. N'avez-vous pas aussi les miennes ? 19. Non, Monsieur, je ne les ai pas. 20. Qui en a beaucoup? 21. Personne n'en a beaucoup. 22. J'en ai quelques unes. 23. Avez-vous assez de thé ? 24. J'en ai assez. 25. J'en ai plus que lui.

EXERCISE 30.

Votre ami a quelque chose d'agréable,

Avez-vous quelque chose de bon ? n'ai rien de bon,

Your friend has something pleasant. Have you anything good?

I have nothing (not anything) good.

What or which napkin have you! What purses has your friend?

6. 1

have but few, but my friend has many. 7. What coat has your brother ? 8. He has a good cloth coat. 9. Has your uncle many peaches? 10. He has but few peaches, but he has many cherries. 11. How many plums has the tailor? 12. The tailor has no plums, he has cloth and silk. 13. What silk has your friend the merchant? 14. He has a great deal (beaucoup) of 15. Has the gardener anything good in (dans) his garden? 16. He has many pineapples. 17. Has he more vegetables than fruit? 18. He has more of these than of those. 19. Has your uncle many pears and cherries? 20. He has a few, and he has many apples and plums. 21. Have you a few ? 22. I have still many, but my brother has no more. 23. Which peaches has he? 24. He has large (grosses) peaches. 25. Which (ones) have you? 26. I have the best peaches. Has the merchant anything good in his warehouse? has nothing good in his warehouse, but he has something good in his garden. 29. How many potatoes has the foreigner? 30 He has not many. 31. Has he good vegetables? good vegetables. 33. Is he right or wrong? 34. He is right, but you are wrong. 35. He has neither this book nor that, he has the bookseller's.

27.

28. He

32. He has

LESSONS IN PENMANSHIP.-VIII. AFTER one more exercise in letters formed by combinations of the bottom-turn, top-turn, and top-and-bottom-turn, the learner, in Copy-slip No. 25, passes on to a new elementary stroke, the fourth in order of the simple forms of which the letters of the writing alphabet are compounded.

This new stroke is called the "straight stroke." It is a downstroke of uniform breadth from top to bottom, formed by bringing the pen from the top line ee to the bottom line bb, with an equal pressure throughout. The chief difficulty in forming this stroke Ees in lifting the pen smartly and quickly from the paper when it has been brought as far as the line bb, so that the termination

tu 7

h

h

aa, bb; but, as there is not a single letter into whose compo sition a straight stroke of this length enters, it is obviously absurd, as well as almost useless, to oblige the pupil to commence his lessons by copying a stroke that he is never called upon to make afterwards in any copy that he may write. In our system of teaching the art of Penmanship, we cause the pupil to write the simplest and easiest letters first, and then proceed to those that are more difficult, in all cases teaching him first to write the elementary strokes of which each set of letters in its sequence is formed, and then to combine them, so as to form the letters themselves. This, therefore, will explain why we did not commence our lessons with the straight stroke, according to the usual practice, and why we now introduce this

of the stroke on that line may be as square and clearly defined every respect as its commencement on the line ee. The learner has already had some practice in terminating a thick down-stroke on the line bb, in making the "hanger or topturn, and all letters into whose composition the top-turn enters. But these have been short strokes, and in making the letter 1, the only letter that he has yet made that is equal in length to the straight stroke, he has been accustomed to lessen the pressure on the pen before he reaches the line bb, in order to finish the letter with a fine hair-stroke turned upwards towards the right. Any trifling difficulty, however, that he may experience in making the straight stroke at first will soon vanish, if when he has brought his pen down as far as the line cc he remember that he has only to finish the stroke as if he were making the simple top-turn, which must now be easy enough to him.

In learning to write, the pupil is generally taught, first of all, to make a straight stroke, no longer than that portion of the stroke in Copy-slip No 25 which is contained between the lines

stroke as the fourth in order of the simple elementary strokes, and in the only form in which it is used in writing, instead of the short form usually given, in which shape, as we have observed, it is never afterwards used by the pupil. :

In Copy-slip No. 26 the pupil proceeds to form the straight stroke and the top-and-bottom-turn in alternation, and in Copyslip No. 27 he finds that these strokes, when joined together, form the letter h. The straight stroke enters into the composition of three letters-h, p, and k; but of these we confine ourselves to h and p for the present.

ESSAYS ON LIFE AND DUTY.-I.

INTRODUCTORY.

If we stand still for a moment in the great rush and hurry of this time, and look both around us on what is, and also backward as far as the eye can reach on what has been, we are struck at first sight with the vastness of the world's labour.