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hearing would be wanting to them. Yet, strange to say, while problems may be thus propounded :—What structures, in the the ears of the cuttle-fish and the slug have been satisfactorily fish, are the representatives of the ossicles of the tympanum detected, the seat of hearing in insects is still undetermined. called the hammer (malleus) and anvil (incus) in the mammal ? The antennæ, or jointed appendages of the head, have been To this question an answer is given by some of our best usually looked upon as the seat of the sense of hearing, but anatomists which is almost startling from its strangeness, but whether it be in the basal joint or the terminal one is a matter | which, on further examination, has much to support it. These of dispate; and in one instance it was supposed to have been anatomists affirm that the two bones, which form the joint of found in the hip joint of the front pair of legs—a singular the lower jaw in the fish, are the representatives of the hammer position, it must be confessed. To show the difficulty of deter and anvil, taken out, so to speak, of the ear-drum, much enlarged mining these matters, we have given a sketch of the external and applied to quite a different purpose. Such questions as orifices of two supposed organs of sense in the common lobster. these require much research to determine them, and are only The little conical protuberance, with a hole through the shell at mentioned here to give a slight insight into the difficulties found the summit, which is closed by a membrane, beneath which is a in unravelling the plan of Nature, though there is undoubtedly ittle bag of fluid with a nerve running to it, which is found on a plan in all her works.

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L BOSE CONTAINING THE EAR OF A RABBIT. II. EAR-BONE OF THE WHALEBONE WHALE, ONE-FOURTH NATURAL SIZE. III. INTERNAL EAR

Or a BIRD. IV. EAR OF A COD. V. EAB-STONE OF COD. VI. UNDER SIDE OF LONG ANTENNA OF A LOBSTER. VII. UPPER SIDE OF

SHORT ANTEXXA OF A LOBSTER. Rel. to Nos, in Figs.-II. 1, tympanic bone; 2, its point of attachment to the skull. III., IV. 1, cochlea; 2, vestibule: 3, oval hole; 4, 5, 6,

semi-circular canals; 7, sack of ear-stone. VI., VII. 1, organs of sense.

the under side of the first joint of the first, or long, pair of The temporal bones—which, in man, lodge the internal and aatennæ, has been long considered the organ of hearing. Now, support the external ears, and besides these functions, close in however, the opinion seems to prevail that this is an organ of the brain-case at the sides, send out strong buttresses forward to smell, while that found opening on the upper side of the first strengthen the bones of the face, and others to sling the throat. joint of the second, or short, pair of antenna, is now thought to bones upon, and also give attachment to the lower jaw-are the te the true ear. In searching for the ear, the presence of hard most difficult bones in the body to describe and remember. bories suspended by threads in a sack containing liquid, and Many vessels and nerves enter them by numerous holes, and capable of striking upon a nerve filament, is considered as these subdivide and find their way out in such strange ways, characteristic and indicative of an ear, just as the expansion of that many a poor medical student has trembled when, in an a nerve in front of black pigment and behind a transparent examination, a temporal bone has been placed in his hand. Ternbrane is thought to denote an eye. The first-named These bones are no doubt composed of many elements which are structure is found in the organ of the lobster last described, distinct in reptiles, birds, and fish: but, to make confusion bat not in the other.

worse confounded, the student of comparative anatomy finds on It will be seen that much remains to be made out abont the i the one hand that Professor Owen divides the bone into at least ear, and the subject is extremely difficult to study. Indeed, nine elements, and gives them names according to his theory ; some of the most perplexing problems of the comparative on the other Professor Huxley transposes all the relations, and sostomist seem to be associated with the ear. One of the christens them by new names.

sound.

LESSONS IN FRENCH.—XIII.

FRENCH.
PRONUNCIATION.

ENGLISE.
Bonneterie
Bo-nait-ree

Hosiery business. SECTION 1-FRENCH PRONUNCIATION (continued).

Comédie
Ko-inay-dee

Comedy. [fectionary.
V.-COMPOUND VOWELS (continued).

Confiserie
Konh-fiz-ree

Art of making con-
Democratie

Day-mo-kra-see Democracy. EU.-Name, uh; sound, like the e nute or unaccented, which Figuerie

Feeg'-ree

A fig-garden, has been already explained, except when it is a verb, or com

Pie
Pee

A magpie. mune a verb, in which latter case it has the sound of French

Vio
Vee

Life. 2, which also has been explained.

This combination is a very common ending of words in the F EXCI. PRONUN. ENGLISH FRENCH. Pronux. ENGLISH. French language. The e, however, often bears the acute accent, Dieure D'muhr Residence. Leur Luhr

Their, thus, ie. These vowels also appear very often in the body of a Eux th

Tlum. Milien Me le-ah Middle. word, with the e accented. In such cases they do not constitute Duur Fluhr Flocer. Peuple Puh: l' People.

a diphthong, and cannot be illustrated by the sound of ee in the Heureux Th-ruh Happy. Pleu-voir Pluh-v'wahr To rain.

English word bee, but each preserves its own distinct vowel Heure Thr Hour. Plusieurs Plu-ze-uhr Many. Jeane Zhuhn Youny. Veuve Vuhy Widou.

10.-Name, eo; sound, like the letters io in the last syllable Sometimes the u of this combination is under a circumflex of the Latin word cur-cu-lio. accent, thus, eû, in which case the sound of the compound vowel

FRENCH.
PRONUNCIATION.

ENGLISH. is prolonged.

Approvisionner A-pro-vee-zen-nay To victual.
The correct round of this compound vowel is no more difficult

Cautionné
Ao-seo-nay

A tarrantee. to be acquired than is the correct sound of e mute or unaccented.

Donationaliser Day-na-seo-na-le-zay To denationalise. But it often happens that the letter, or combination of letters,

Mioche

Meo-sh (long o) A lrat.
Pioche
Peo-sh (long o)

Pickars. which immediately follou's it, adds vastly to the difficulty of

Veo-lon
Violon

Violin,
pronouncing it. Bring the lips nearly together, ovally, in speak-
ing this compound vowel. Practise patiently and thoroughly

This diphthong retains the sound first illustrated in most, if upon the above and other examples, until you are satisfied you

not all, endings in sion and tion. have mastered the difficulty.

SECTION XXIII.-IRREGULAR VERBS: THEIR PRESENT OI.–Name, oah, or wah; sound, like the letters oah of the

INDICATIVE, proper name Noah. Do not give this compound vowel the

1. There are in French, as in other languages, verbs which sound of wor, or 00-ave, as is too commonly done.

| are called irregular, because they are not conjugated according FRENCH, PRONUN. ENGLISH, FRENCH. PRONUN, ENGLISH. I to the rule, or model verb of the conjugation to which they Ardoise Ar-doahz Slate. Histoire Is-t'wahr History.

belong (S 62]. Auditoire O-dit-oahr Assembly.

(trill ther)

2. Many irregular verbs have tenses which are conjugated Aroir Ay'wahr To hare. Manoir Man'wahr Manor, Bois B'wah

regularly. Tod. Noir N'wahr Black, Désespoir Day-zes-p'Desperation. | Pouvoir Poo-v'wahr To be able.

3. The singular of the present of the indicative of the irrewahr Roi R'weh (trill King.

gular verbs is almost always irregular, Devoir Der'-wahr To oue.

the r)

4. In verbs ending in yer, the y is changed into i before an e OU.--Name, 00; sound, like the letters oo in the English

muto [$ 49). word moon.

5. PRESENT OF THE INDICATIVE OF THE IRREGULAR VERBS. FRENCH. PRONUN. ENGLISH. FRENCI. PRONUN, ENGLISH. ALLER, 1, to go. I ENVOYER, 1, to send. 1 VENIR, 2, to come. Bouloversé Bool-ver-say Distracted. Pour Poor (trill For.

S. Je vais, I go, do go, J'envoie (R. 4) I send, | Je viens, I come, do Coup Koo A blone, the r)

or am going.

do send, or am send. come, or a comina. Douche Dooch Douche or Pourri Poo-ree Rottenness. Tu vas.

Tu envoies. [ing. Tu viens.
bath. Route Roote (trill A route.

Il va.
Il envoie.

n vient. Fouet Foo-ay

the r)
P. Nous allons,
Nous envoyons.

Nous venons.
Oi
Oo
Wuere, Tout and Tooto AU.

Vous allez.
Vous envoyez.

Vous venez.
Oubli Oo-blee For jetful Toute

Is vont.

Ils envoient (R. 4). Ils viennent.
Tes. Toucher Too-shay To touch.

6. All verbs ending in enir are conjugated like venir. Pondro Poodr Porder. Tour Toore Journey.

7. The student will find in § 62 the irregular verbs alphaVI.--DIPITAONGS.

betically arranged. He should always consult that table when 68. There are six diphthongs, namely :-in, ie, io, ua, ue, ui, 1 he meets with an irregular verb. whose sounds we now proceed to illustrate,

8. The expression à la maison is used for the English at But do not suppose that these combinations of vowels are home, at his or her house, etc. always diphthongs, in whatever place they are situated. If Le chirurgien est-il à la maison ? Is the surgeon at homo ! followed by two consonants, the first of which is m or n, the Mon frère est à la maison,

My brother is at home. last vowel forms with the m or n a nasal, unless the m or n be 1 9. The preposition chez, placed before a noun or pronoun,

answers to the English at the house of, with (meaning at the Sometimes, again, these vowels which now appear as diph. residence of), among, etc. ($ 142 (3)]. thonra are but parts of syllables of a word, and must be pro- Chez moi chez lui, chez elle. At my house, at his house, at ha nounced only as distinct vowels.

house. IA.-Name, in; sound, like the letters i in the English

Chez nous, chez vous, chez eux, m., At our house, at your house, at their wordd fig, and a in the word fat, pronounced as one syllable, chez elles, f.,

house, The sounds of both, however, must be distinctly heard without

That is, literally, at the house of me, at the house of him, eto. any hiatus between them.

Chez mon père, chez ma scur, At my father's, at my sister's. FRISCH. PRONUX. EXILISH. | FRENCH. PRONTN. ENGLISH.

10. The word avec answers to the English with, meaning Criard Kree-ar (amorous. 1 Pliable Plee-abi F ler ble. Coriacé Kor-eeassay Tough. Pliage Plee-azh Folding.

merely in the company of. Fiacre Fee-akr Cab. Tiare Tee-ar (trill Tiara,

Venez avec nous, ou avec lui, Come with us, or with him, Intrique latreek latrical.

the r)

11. The word y means to it, at it, at that place, there. It is Piaffe Pee-afr Ostentation. Viande Vee-anhd Mea!.

generally placed before the verb, and refers always to someIE.-Name, ee; sound, like the letters ee in the English thing mentioned [$ 39, $ 103, § 104). word bee.

Votre seur est-elle chez vous ? Is your sister at your house !
FREXCX.
PRONUNCIATIOX,

ENGLISH.
Oai, Monsieur, elle y est,

Yes, Sir, she is there.
Académie
Ak-ad-aymee

Academy.
Ao-al-ozhee
Analogie

12. In French, an answer cannot, as in English, consist merely

Analogy.
Anomalie
An-om-alee
Anomaly.

of an auxiliary or a verb preceded by a nominative pronoun; as,
Asie
Az-ee
Asia.

Do you come to my house to-day? I do. Have you books? Bonhomie Bo-20-mee Good nature.

I have. The sentence in French must be complete; as, I go

doubled.

there ; I have some. The words oui or non, without a verb, ! maker's ? 32. It (elle) is there. 33. Have you two gold Fould, however, suffice.

watches ? 34. I have only one gold watch. 35. Who intends Venez-vous chez moi aujourd'hui ? Do yo'l come to my house to-elay? to go to my father's this morning ? 36. Nobody intends to go Oni, Monsieur, j'irai, Yes, Si, I will

there.
Arez-vous des litres chez vous ? Har you books at home ?
O, Monsieur, nous en avons, Yos, Sir, we hxve.

OUR HOLIDAY.
RÉSUMÉ OF EXAMPLES.

HOCKEY.
Où est le colonel?
Where is the colonel ?

WHEN the frosts of winter have hardened the ground, and the
Il est chez son frère ainé.
He is at his eldest brother's.

air is keen and bracing, out-door amusements, to be at once V'est-il pas chez nous?

Is he not at our house? Non, Monsieur, il n'y est pas. No, Sir, he is not.

enjoyable and beneficial, must be active and exhilarating in their Madame votre more est-elle à la Is your mother at home ?

nature. Hence the popularity in the winter season of such maison ?

games as Football and Hockey, with their new competitor La Non, Madame, elle n'y est pas. No, Madam, she is not.

Crosse, which, since we wrote about it in No. 1 of the POPULAR Allez-vous chez nous, ou chez lui? Do you go to our house, or to his EDUCATOR, we are glad to find is advancing in favour, moru house?

than one club having been established for its systematic practice Nous allons chez le capitaine. We go to the captain's.

during the winter months. A new game is a new source o. Vest-il pas chez votre frère ? Is he not at your brother's ?

harmless pleasure to hundreds, and perhaps to thousands or Noa, Monsieur, il est chez nous. No, Sir, he is at our house. N'envoyez-vous pas vos habits chez Do you not send your clothes to your

tens of thousands. The great and almost sudden popularity oî vos smurs ? sister's?

Croquet shows how welcome is a suitable addition to the list of Je les envoie chez elles. I send them to their house.

popular amusements, and we therefore spare a passing word to N'allez-vous pas chez ce monsieur? Do you not go to that gentleman's comment upon the reception given to the Indian game whicla Je n'y vais pas, je n'ai pas le temps I do not (R. 12), I have not time to was the subject of our first paper. d'y aller aujourd'hui. go there to-da.

Of Football we have also treated ; and we have now to VOCABULARY.

describe the game of Hockey, which, under the names of Shinty

in Scotland and Hurling in Ireland, is popular throughout the All-er. I. ir.. to go. | Horloger, m., wateh. Relieur, m.,bookbinder. United Kingdom Ami, m., friend. maker. Rest-er, 1, to romain,

Hockey consists in driving a ball from one point to another Associé, m., partner. Hollandais, -e, Dutch. I lire.

by means of a hooked stick, and is believed to derive its namn : Capitaine, m., captain. Magasin, m., warehouse. Rasse, Russian, Demeur-er. 1. to live. ) Maison, f., house. Ven-ir, 2, ir., to come. | from the shape of the latter implement, sometimes called :) direll. Matin, m., morning. Voisin, -e, neighbour,

hookey. No precise rule is laid down as to the form this stick Gilet, m., taistcoat. Peintre, m., painter,

should take. It is simply a weapon with a bent knob or hook

at the end, large or small, thick or thin, according to the option EXERCISE 41.

of the player, and used for the purpose of striking the ball, or 1. Où allez-vous mon ami? 2. Je vais chez Monsieur votre perhaps of catching it up on the point for a throw towards the père; est-il à la maison ? 3. Il y est ce matin. 4. D'où venez- goal. Hockey-sticks, therefore, are of all shapes, sometimes vons? 5. Nous venons de chez vous et de chez votre sour. ' simply in the form of a stout walking-stick with a crook at 6. Qui est chez nous ? 7. Mon voisin y est aujourd'hui. 8. the end. Où avez-vous l'intention de porter ces livres ? 9. J'ai l'inten The Hockey ball must be one fitted to receive hard and tion de les porter chez le fils du médecin. 10. Avez-vous tort frequent blows. Anything in the nature of a cricket-ball is de rester chez vous? 11. Je n'ai pas tort de rester à la maison. found to be ill-adapted for this peculiar game, as the leather 12. L'horloger a-t-il de bonnes montres chez lui ? 13. Il n'a soon bursts, through the effects of the knocks received from all pas de montres chez lui, il en a dans son magasin. 14. Chez kinds of rugged-pointed sticks. A large bung, strongly tied qui portez-vous vos livres ? 15. Je les porte chez le relieur. and quilted over with string, is a favourite and an inexpensive 16. Allez-vous chez le capitaine hollandais? 17. Nous n'allons ball for the purpose; and the best of all is perhaps a solid indiapas chez le capitaine hollandais, nous allons chez le major russe. rubber one, or the larger part of a thick india-rubber bottle, 18. Est-il chez vous ou chez votre frère ? 19. Il demeure chez firmly closed at the end from which the neck has been cut. nous. 20. Ne demeurons-nous pas chez votre tailleur? 21. Now for the game itself, which in its principle bcars a great Vous y demeurez. 22. Votre peintre d'où vient-il? 23. Il resemblance to Football, and contains at least the germ of the Tient de chez son associé. 24. Où portez-vous mes souliers et Canadian La Crosse. The players are divided into two parties, mon gilet? 25. Je porte vos souliers chez le cordonnier et votre each of which has its goal, the goals being fixed towards either gilet chez le tailleur.

end of a tolerably spacious ground. They consist, as at FootEXERCISE 42.

ball, of two upright posts, about six feet apart, but the cross. 1. Where does your friend go? 2. He is going [Seet. XXII., pole is

pole is almost invariably employed at Hockey, and is usually 6 to your house or to your brother's. 3. Does he not intend to place au

d to placed at a height of about four feet from the ground. Through go to your partner's ? 4. He intends to go there, but he has

these goals the ball has to be driven; and the space through no time to-day. 5. What do you want to-day? 6. I want my which it has to pass at either end, before the game is won, is waistcoat, which (que) is at the tailor's. 7. Are your clothes therefore a space of about six feet by four. at the painter's ? 8. They are not there, they are at the tailor's. In commencing, the two parties meet midway between the 3. Where do yon live, my friend ? 10. I live at your sister-in.) goals, and are arranged in line, their left hands towards the ays? 11. Is your father at home? 12. No, Sir. he is not. opponents' goal, and their right directed to their own. The 13. Where does your servant carry the woods' 14' He carries ball is thrown up into the air by one of the party winning it to the Russian captain's. 15. Does the rentleman who (o) the toss, by which toss also the choice of position for the is with your father live at his house ? 16. No. Sir. he lives goal is determined. As the ball falls, it is the object of both sith me. 17. Is he wrong to live with you? 18. No. Si he sides to strike it towards the goal of the enemy, or at least to 23 right to live with me. 19. Whence (d'où) comes the carpen. prevent it from being struck in the direction of their own. Two ter? 20. He comes from his partner's house. 21. Has he two goal-keepers are stationed at each end to beat back the ball if Partners ? 22. No, Sir, he has only one who lives here in it approaches dangerously near; and, if the party playing be 23. Have you time to go to our house this morning? 24. We large enough, it is usual to place two of the opposite side near kave time to go there. 25. We intend to go there and to speak the respective goal-keepers, in order that their defensive efforts to your sister. 26. Is she at your house ? 27. She is at her may be (owti) house. 28. Have you bread, butter, and cheese at home. It may well be imagined that on the fall of the ball an 23. We have bread and butter there. 30. We have no cheese exciting scene ensues. In the attempt to striko it, the hockey. there, we do not like cheese. 31. Is your watch at the watch : $

watch. sticks are crossed in mimic warfare, and as it reaches the ground

both sides surround it in a general “scrimmage," while it is *The French, in speaking to a person whom they respect, prefis | pushed, thrust, or struck by the hockey-sticks, according to the the word Monsieur, Madame, or Mademoiselle, to the word represent

chance which the various players may get of aiming at it. The ing their interlocutor's relations or friends.

hockey-stick properly should never be raised much higher than

[graphic][subsumed][merged small]

the ground, for a dexterous shore at the ball may sometimes be The Scottish form of the game, known as Shinty, calls for no quite as effective in serving the purpose of your side at a critical special remark, more than that the goals are called “hails," and moment as a swinging blow, the opportunity for which may, that the game itself may owe its name either to the frequent indeed, very rarely occur. If the ball receives a good hit, and danger to the player's shins, or to the shindy which characterises Aies forward to the goal, a general rush is made in pursuit, one the culminating struggle. “Hurley," the Irish variation of the side aiming to follow up the advantage, and the other to over- game, also differs but little from that here described; but in take the ball first and restore the balance of the game.

Ireland the game has been, perhaps, a more general favourite, It will be apparent that in a rush and struggle of this de- and played occasionally on a larger scale, than in either of the scription a fall or a hard knock is exceedingly likely to occur, sister kingdoms. We borrow from Mr. and Mrs. S. C. Hall's and that Hockey is therefore not a game suited to weakly or “ Ireland” an amusing anecdote in illustration of this fact. timid players. But there are rules by which it is sought to “About half a century ago," we are told, "there was a great avoid, even in the heat of the conflict, any chance of more than match played in the Phænix Park, Dublin, between the Munster a comparatively slight injury to the players, and to confine that men and the men of Leinster. It was got up by the then lordresult purely to the effects of accident. It is forbidden, in the lieutenant and other sporting noblemen, and was attended by first place, to raise the head of the stick higher than the all the nobility and gentry belonging to the vice-regal court, and shoulder, under the penalty of a blow on the shins from the the beauty and fashion of the Irish capital and its vicinity. hockey-stick of one of the opposite side; and thus a check is The victory was contended for a long time with varied success; given to the reckless and promiscuous flourishing about of the and at last it was decided in favour of the Munster men, by player's stick, to the imminent hazard both of his friends and one of that party running with the ball on the point of his opponents. Moreover, any player proved wilfully to have struck hurley and striking it through the open window of the viceanother is at once excluded from the play. Besides these rules, regal carriage, and by that manœuvre baffling the vigilance of the following are generally accepted :

the Leinster goalmen, and driving it in triumph through the 1. A player must not cross to the side of his opponents before goal.” a rush or scrimmage has commenced.

There is no record of matches on quite so extensive a scale 2. The ball must be fairly struck through the goal, and not having been played in the sister kingdoms; but we learn on the thrown or kicked.

authority just quoted that, in the last generation, several good 3. It is forbidden to kick or throw the ball during the general matches at hurley were played on Kennington Common between game, but the ball may be stopped by any part of the person of the Irish residents of St. Giles's and those of the eastern pora player who may intervene between it and the go:1.

tions of the metropolis, the affair being got up by some of the 4. If the ball be struck beyond, but not through the goal, sporting noblemen of the day. Besides Kennington Commori, and if it be passed through the goal otherwise than by a fair several of the other open spaces around London were once noted hit, the youngest player of the side owning that goal shall as favourite spots for the exhibition in perfection of the game of return it by a gentle throw towards the centre of the ground. hockey, and especially, in the last century, the extensive fields

These, with the two rules given before, comprise all that it which then lay at the back of the British Museum. The amuseis necessary to observe in playing the game of Hockey, except ment is not so frequently seen now, having yielded somewhat the general rules of good temper and forbearance, which are before the rival attractions of football and cricket, but it is & required in all games alike.

| favourite still in many parts of the country.

LESSONS IN GEOMETRY.–VII. A I would be represented numerically by 99, I K by 31 inches,

etc., and lines involving fractions of inches such as is, which PROBLEM XIV.- To find a third proportional to two given | are not to be found on an ordinary scale, would be very difficult straight lines.

to mark out without making a special scale for the purpose, or Let A and B be the two given straight lines to which it is re

the two given straight lines to which it is re- resorting to the plan given above. quired to find a third proportional. Draw two straight lines CP, PROBLEM XVII.-To draw an equilateral triangle on any given

cg, forming with each other straight line.
a small angle PC Q. On Cpl Let A B be the given straight line on which it is required to
set off c d equal to A, and draw an equilateral triangle. From the point A as a centre,
D F equal to B, and on co with A B as a radius, describe the arc BC;
set off C E equal to B. Join and from the point B as a centre, with

D E, and through the point BA as a radius, describe the arc AC, cut

C F draw F G parallel to D Е, ting the arc Bc in the point c. Join A C,
Fig. 21.

and cutting coin G; the BC; the triangle ABC is equilateral or

straight line E G is a third equal-sided (see Definition 19, page 53), proportional to A and B; that is, A is to B as B is to E G.

and it is drawn on the given straight line If we know the length of A and B, we can find the third pro

AB. portional to them by dividing the square of the length B by the If the arcs CA, C B be extended to cut length of A. Thus, if a be three feet, and b be six feet, the each other in the point o below the straight third proportional to A and B measures twelve feet, for the line A B, by joining D A, D B, we get another square of 6 divided by 3, or 36 = 3 = 12.

equilateral triangle A B D, which is equal PROBLEM XV.-To find a fourth proportional to three given to the equilateral triangle A B C, and which straight lines.

is also drawn on the given straight line Let A, B, and c be the three given straight lines to which it is A B. By taking any straight line as a

required to find a fourth pro-radius, and from each of its extremities as Fig. 24. A portional. Draw two straight centres striking arcs intersecting or cutting

lines DP, DQ, forming with each other on opposite sides of it, we get, by drawing straight each other a small angle, PDQ. lines from the points in which the arcs cut each other to the

On D P set off D E equal to a, extremities of the straight line used as a radius, a regularly.

o and E F equal to c, and on formed diamond-shaped figure, whose four sides and shortest Fig. 22.

DQ set off DG equal to B. diagonal or diameter, are all of equal length, such as A CBD in

Join E G, and through F draw the above figure. This figure with four equal sides is called a FH parallel to EG, and cutting Do in n. The straight lino rhombus. (See Definition 30, page 53.) A G is a fourth proportional to A, B, and c; that is, a is to B The learner should construct Fig. 24 on a large scale by the as cis to H G.

aid of his compasses and ruler. On applying a parallel ruler to If we know the length of A, B, and c, we can find the fourth the opposite sides of the figure ACBD, he will find that they are proportional to them by multiplying the length of B and c parallel to each other, namely, A C to B D, and Bc to AD; A CBD together, and dividing the product by the length of A. Thus, if is therefore a parallelogram, and A B, C D are its diagonals. (See A be four feet, B six feet, and c two feet, the fourth propor. Definition 26, page 53.) From Theorem 5 (page 156) the tional to A, B, and c measures three feet; for 6 X 2 = 12, and student learnt that the greatest side of every triangle is opposite 12 : 4 = 3.

the greatest angle, and that the greater the opening of the angle PROBLEM XVI.—To divide a given straight line into any num. the greater must be the line that subtends or is opposite to it. ber of parts which shall be to one another in a given proportion. Now in the triangle A B C, or in any other equilateral triangle, Let A B be the given straight line, which it is required to the three straight lines or sides by which it is contained are all

H divide into five equal to one another, and as equal sides must necessarily subtend

parts, which are to equal angles, the three angles of the triangle ABC-namely, one another in the A B C, BCA, CA B—are also all equal to one another. Again,

following propor. from Theorem 7 (page 156) we have learnt that the three interior AS

tions — namely, 5, angles of any triangle are equal to two right angles. A right Fig. 23.

2, 3, 1, 4. First angle contains 90 degrees, and as two right angles contain just

draw the straight twice as many, or 180 degrees, each of the equal angles A B C, line a c of indefinite length, making a small angle BAC with the B C A, C A B, in the interior of the equilateral triangle A B C, given straight line A B. Along A C, from a scale of equal parts, contains 180 = 3, or 60 degrees. set off in regular succession A D equal to 5 of these equal parts, Continuing our investigations a little further, we find that D E equal to 2, E F equal to 3, F G equal to 1, and g 1 equal to each of the angles A CE, BCE is half of the angle A C B, and is 4. Join u B, and through the points D, E, F, G, draw the straight therefore an angle of 30 degrees. The angles A DE, B D E are lines D I, E K, FL, GM, cutting the straight line A B in the also angles of 30 degrees, because each of them is half of the points I, K, L, M. The given straight line A B is now divided angle A D B, which, like the angle A C B, is an angle of 60 into five parts, A I, I K, KL, L M, M B, which are to one another degrees. The angle Cad is equal to the angles CAB, DAB, in the required proportions, namely, 5, 2, 3, 1, and 4.

and as each of these equal angles contains 60 degrees, the angle This method of dividing a straight line into any number of CAD contains 120 degrees. In the same way the angle C BD parta, which shall be to one another in a given proportion, is also contains 120 degrees. The diagonals of the rhombus based on Problem XII. (page 192). Supposing it had been A C B D intersect each other at right angles, therefore it will required to divide A B into 15 equal parts, it is manifestly only be seen that each of the angles CEA CE B, D E A, D E B is requisite to set off along A C 15 equal parts, denoted by the a right angle. dots on the line A c, from A to H, and then draw straight lines in Fig. 24 teaches us how to draw an angle of 45 degrees with succession through each dot on H A, from 1 to A, parallel to H B. out the aid of the protractor, as we will proceed to show.

The process that has been described in this Problem, ensures AC E is an angle of 30 degrees, and so is its adjacent anglo an accurate division in cases where the different parts would be represented by fractions or mixed numbers (see Lessons on Arithmetic, page 160), if we endeavoured to arrive at them by an

ERRATA.-In a few of the earlier impressions of our last Lesson in arithmetical process. For example, had the line A B in Fig. 23

93 Geometry, some typographical errors occurred, which we must request measured 30 inches, we can see at once that, as the sum of the

the reader to correct, as follows:- Page 191, Problem VII. At the

apex of the triangle whose base is "AC" insert "B." - Page 192, numbers which show the proportion of the lines into which it is

Problem IX. At the extremity opposite to “B” of the line E F insert required to divide it is equal to 15, the half of 30, we have | "o"; at the other end of the diagonal from "E" insert "A".- Page 192, only to multiply each number by 2, and mark off A I equal to 10 Problem XI., line 3. For “ the centre d,” read "the centre a."(or 5 X 2) inches, I K equal to 4 (or 2 x 2) inches, and so on. Page 192, Problem XIII., line 9. For "De equal to 7," read "DB But supposing A B had measured 29 inches, instead of 30, then equal to B." VOL. I.

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