willingly. 21. Willing or not willing, he shall go away. 22. Will you keep my secret ? 23. I will keep it. 24. He changes his opinion with every event.

EXERCISE 170 (Vol. III., page 221).

1. Se mariera-t-il contre le gré de son père? 2. Il ne se mariera pas contre le gré de ses parents. 3. Pourquoi me savez-vous mauvais gré ? 4. Je ne vous sais point mauvais gré. 5. La chevelure de votre petite fille est-elle nouée ? 6. Elle n'est pas nouée, elle flotte au gré du vent. 7. Que pensez-vous de mon livre ? 8. C'est à mon gré le meilleur livre que j'aie lu. 9. Ne me saurez-vous pas mauvais gré, si je ne viens pas aujourd'hui ? 10. Je ne vous en saurai pas mauvais gré. 11. Ne lirez-vous pas cette lettre ? 12. Quelque bien écrite qu'elle soit, je ne la lirai pas. 13. Ces dames sont-elles belles? 14. Quelque belles et bonnes qu'elles soient, elles ne sont pas à mon gré. 15. Etes-vous faché contre mon frère ? 16. Non, Monsieur; je lui sais bon gré de ses intentions, quelles que soient les suites de sa conduite. 17. Me garderez-vous le secret ? 18. Je vous le garderai de bon gré. 19. Votre sœur garde-t-elle le lit de bon gré ? 20. Elle ne garde pas la chambre de bon gré. 21. Bon gré, mal gré, il faut qu'elle garde la chambre quand elle est malade. 22. Garderez-vous le secret sur ce point? 23. Je le garderai de bon gré. 24. Je vous sais bon gré de vos bonnes intentions. 25. Lui savez-vous bon gré de cela ? 26. Je lui en sais bon gré. 27. Le juge gardera-t-il son domestique ? 28. Il le gardera. 29. Fait-il son travail à son gré? 30. Il le fait à son gré. 31. M. votre frère est-il obligé de garder la maison? 32. Il est obligé de garder le lit. 33. N'a-t-il pas quitté sa chambre? 34. Il n'a pas encore quitté sa chambre; il est trop malade pour la quitter. 35. Je vous saurais le meilleur gré du monde, si vous

vouliez le faire.

EXERCISE 171 (Vol. III., page 221).

1. Sir, to what shall I have the pleasure of helping you? 2. I wiH ask you for a slice of that ham. 3. Have the goodness to help those gentlemen. 4. I will thank you for a slice of that boiled meat. 5. Shall I offer you a slice of that roast meat? 6. I am much obliged to you, Sir; I have enough. 7. Miss, shall I have the honour of would helping you to a wing of this partridge? 8. I thank you, Sir; prefer one of those ortolans. 9. Sir, shall I send you soup? Madam, have the goodness to help this young lady. 11. I will ask you for a little afterwards. 12. John, present this cutlet to that gentleman. 13. These vegetables are delicious. 14. Sir, I am very glad that you find them good. 15. Sir, will you not sit down? 16. I am exceedingly obliged to you, Sir, my father is waiting for me at home. 17. Have you not wished them good morning? 18. I have wished them good evening. 19. Have you bid them adieu ? 20. I have bid my brother adieu. 21. I have taken leave of them. Have you desired them to walk in ? 23. I have. 24. Gentlemen, the dinner is on the table. 25. Have the goodness to sit here,

EXERCISE 172 (Vol. III., page 221).

of a father towards you? 21. He has been to me a father and mother. 22. Will you look more closely into this affair? 23. No, Sir; I will be satisfied with what I know of it. 24. Is not that physician tenacious of his opinion? 25. He cares more about it than about the life of his patients.

PLANE TRIGONOMETRY.-III.

SOLUTION OF RIGHT-ANGLED TRIANGLES-FUNDAMENTAL

PRINCIPLES, ETC.

X. Solution of Right-angled Triangles.-Every triangle con sists of six "elements," three sides and three angles. Any three of these being given, including at least one side (this is necessary, because triangles merely equiangular can be con structed in infinite number), Trigonometry enables us to calcu late the remaining elements. The formulæ evolved as yet only enable us to do this for right-angled triangles, and as these involve one known quantity (the right angle), it is sufficient if any two of the other elements (including one side) be given. We may have (referring to Fig. 3), besides the right angle(1.) Given two sides.

(2.) Given one side and one angle.

Either of these cases may be solved by the ratios given in Section II., and by a table of natural sines and cosines, tangents and cotangents, such as that given at the end of Galbraith and Haughton's "Trigonometry." The following examples may all be solved by the annexed table of ratios for a few angles only, purposely restricted to three places of decimals:

10.

0.485

0.554

61°

22.

1. Madame, que vous servirai-je ? 2. Je vous demanderai une tranche de ce jambon. 3. Vous enverrai-je une aile de cette volaille ? 4. Non, je vous remercie, Monsieur. 5. S'il vous plait, Monsieur. 6. Monsieur, aurai-je le plaisir de vous servir une tranche de ce jambon ? 7. Je vous remercie, Monsieur, je prendrai de préférence un morceau de la perdrix. 8. Vous offrirai-je un petit morceau de ce bouilli ? Je vous remercie, Monsieur; j'en ai suffisamment. 10. Madame, vous enverrai-je un peu de potage ? 11. Mille remerciments, Monsieur. 12. Monsieur, oserai-je vous prier de servir mademoiselle ? 13. Avec beaucoup de plaisir, Monsieur. 14. Jean, présentez cette soupe à

9.

monsieur. 15. Ces ortolans sont délicieux. 16. Je suis bien aise

que vous les trouviez bons. 17. A-t-on servi? 18. Non, Monsieur; on n'a pas encore servi. 19. Il est trop tôt. 20. Vous plait-il d'y aller? 21. Il ne me plait pas d'aller chez lui, mais j'irai si vous le désirez. 22. Irai-je avec vous? 23. Comme il vous plaira. 24. Votre ami ne veut-il pas s'asseoir ? 25. Il vous est fort obligé; il n'a pas le temps aujourd'hui. 26. Avez-vous souhaité le bonjour à votre ami ? 27. Je lui ai souhaité le bonsoir. 28. Ne lui avez-vous pas dit adieu ? 29. Je lui ai dit adieu. 30. Ayez la complaisance de vous mettre ici. 31. J'ai pris congé d'eux. 32. J'ai pris congé de tous mes amis.

EXERCISE 173 (Vol. III., page 270).

......

of Angles in Column IL

a tan. B=100 x 727=72·7;

A 90° B = 54°.

b

Since tan. B:

10. A house 50 feet high abuts upon a street found to measure 33.7 feet in width. Find the length of ladder required to reach the top from the opposite side of the street, and the angle the ladder will make with the wall of the house.

11. Two trains travelling, one at 20 miles an hour, the other faster, come into collision at a level crossing, where the two lines (both being free from curves) cross each other at an angle of 36°. Some time before the collision, a passenger in the slower train observes the other exactly abreast of him on the other line of railway, and judges the trains to be a quarter of a mile apart. How far from the crossing were both trains at that moment, and what was the speed of the faster train?

12. One of two boys, flying a kite in a level field, observes that he has let out the whole of his string-60 yards-just as his companion, looking up, cries out that the kite flies perpendicularly over his head. The boys found afterwards they were standing 30 yards apart. How high was the kite, and what angle did the string make with the ground? 13. The rope holding the "captive balloon "at Chelsea against a strong wind, when 400 yards were paid out, was found to incline 15° from the perpendicular. How high was the balloon, and how far from the foot of the rope would a piece of iron ballast have fallen, if dropped from the balloon ?

XI. The Fundamental Formula.-We have hitherto examined only the relations between ratios of the same angle; we proceed now to trace the relations between ratios of two or more different angles. The number of formulæ expressing these relations may be extended almost at will, but they are all derived from the following formulæ for the sines and cosines of the sum and difference of two angles, known, therefore, as the four fundamental formulæ :—

Sin. (A + B) = sin. A cos. B + cos. A sin. B...... (33) Sin. (AB): sin. A cos. B cos. A sin. B. (34) Cos. (A+B) = cos. A cos. B sin. A sin. B...... (35) Cos. (AB) = cos. A cos. B + sin. A sin. B...... (36)

where A and B are any angles whatever.

These formulæ may be thus expressed in words :

(33) The sine of the sum of two angles is equal to the sine of the first into the cosine of the second plus the cosine of the first into the sine of the second.

(34) The sine of the difference of two angles is equal to the sine of the first into the cosine of the second minus the cosine of the first into the sine of the second.

(35) The cosine of the sum of two angles is equal to the product of their cosines minus the product of their sines.

(36) The cosine of the difference of two angles is equal to the product of their cosines plus the product of their sines.

To prove (33).-In Fig. 8 let A O B = A, and BOC B; then

To prove (34).-Let A O B (Fig. 9)= A, and BOC B; then AOC A B.

In o c take any point P, and draw the perpendiculars PQ, P R, and R 8, RT, as before (RT to P Q produced).

от PT RS PT ОР OR ОР

=

=

ОР ОР ОР

B

R

P

Fig. 9.

Again, prove (34), where A is a trigonometrical angle in the fourth quadrant, B an angle in the second quadrant, and their difference an angle in the third quadrant.

=

Let A O B in Fig. 11 = A, and BOC B; .'. A o c (A - B). Construct as before

=

S

Fig. 11.

of the stars. Delicate observations show us that they have proper motions, but it is very ifficult to determine these. We can, however, ascertain the motion of the Sun by observing the relative distances of the stars. We find that in one part of the sky the stars seem to be very gradually opening out, and getting further apart, while in the opposite quarter they are as gradually closing up, evidently showing that we are moving towards the former part, just as when we are travelling in a wood the trees in front seem opening out, while those we have passed appear to be getting closer together.

Astronomers have naturally been anxious to ascertain something of the shape of the whole cluster of stars which constitutes our

system, and have employed the telescope as a sounding-line to learn the depth in different directions. If the stars are scattered at all uniformly in space, they will, of course, appear more sparse in those parts where we look through the thinnest layer of them. Now when we observe the sky on a clear night, we at once notice a pale belt passing round it, commonly known as the Milky Way. In one part of its course it divides into two branches, which, after separating a little way, and passing about a third round the sky, again unite into Powerful telescopes show us that this consists of a dense mass of minute stars. The greater portion, indeed, of those visible are clustered along this line, while in those parts of the sky removed from it the number of telescopic stars is comparatively small; hence we may reasonably assume that this belt indicates to us the direction in which the greatest number of the stars lie, and in which our cluster extends furthest.

one.

From this we may form an idea of our system, and it seems that the best representation of it may be obtained by taking a flat circular body-as, for example, a cheese and splitting it by passing a knife about one-third of the way through, the two parts being made to diverge a little, as shown at a, b (Fig. 44). The Sun, (s) is situated somewhere near the centre, and the split side causes the divided appearance of the Milky Way. One of the nebulæ, when seen through a powerful telescope, is found to present a somewhat similar appearance, and is considered to be a cluster closely resembling our own.

they happened to lie in a straight line directed almost towards the Earth; that they were, in fact, merely optical couples, one being an immense distance behind the other. After many

observations, Herschel found that their distances and relative positions did vary, but instead of it being, as he expected, an annual fluctuation caused by the Earth's motion, it was a progressive change. He thus found that in some cases the two stars were revolving round one another in elliptical orbits, and that they were physical couples, the two forming one system. These he called binary stars or couples, to distinguish them from optical pairs. Other observers have followed up these investiga tions, and there are now upwards of 600 binary stars known and noted, and in many cases their times of revolution have been calculated. One of the best examples of this class is afforded by e Lyra, which is sometimes called the Double-double Star. To the naked

eye it appears a somewhat faint star, but a telescope of very little power will show it to be double. When, however, a more powerful instrument is employed, each of tl.ese components is in turn found to consist of two smaller ones, as shown in Fig. 45. The lower pair revolves in about 2,000 years, and the upper in about half that time, while the two couples take a very long period to revolve around their common centre of gravity.

One remarkable feature in connection with the double stars is the fact that in some instances the component stars are of different colours. In R Leporis, for instance, one is white, while the other is a deep red; in 8 Cygni again, the colours are yellow and blue; and in y Andromeda, they are orange and green.

When we come to note the colour of different stars, and compare it with former records, we find that in a few instances a change has taken place. Thus Sirius, which now shines with a pure white light, is spoken of by old observers as a ruddy star. There are also many others which exhibit changes in brilliancy, and these changes seem, in most cases, to be periodical. The star on which this discovery was made is o Ceti, called also Mira, or the Wonderful Star, a name it well deserves. At the time of its greatest brightness it is usually of the first or second magnitude, it then decreases for two or three months,

When we look at the heavens on a clear night, we observe here and there two stars in very close proximity: the telescope further reveals to us that very many of these which appear to the naked eye as single stars, consist in reality of two or more so close together that they ap. pear as one. Sir W. Herschel was the first to direct specialattention to these objects, of which he compiled a list. He hoped that by very accurate measurements of the apparent distances between them, he might be able in some instances to detect a variation, and thus ascertain their parallax, and by that their distances. The idea then was that these stars merely appeared close together because

till it becomes invisible, and remains so for about five months, its minimum brightness being about equal to that of a twelfth magnitude star. It then again appears, and the whole period occupied by these changes is about 331 days.

Algol, or B Persei, is another variable star, remarkable for its short period and rapid changes. It ordinarily appears as a star of the second magnitude, but in a period of three and a half hours it diminishes in brightness to the fourth magnitude, and after a few minutes begins again to increase, attaining its former brilliancy in another period of three and a half hours. At this it remains two days thirteen hours, and then the same series of changes recurs.

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