the ground, for a dexterous shove at the ball may sometimes be quite as effective in serving the purpose of your side at a critical moment as a swinging blow, the opportunity for which may, indeed, very rarely occur. If the ball receives a good hit, and flies forward to the goal, a general rush is made in pursuit, one side aiming to follow up the advantage, and the other to overtake the ball first and restore the balance of the game.

It will be apparent that in a rush and struggle of this description a fall or a hard knock is exceedingly likely to occur, and that Hockey is therefore not a game suited to weakly or timid players. But there are rules by which it is sought to avoid, even in the heat of the conflict, any chance of more than a comparatively slight injury to the players, and to confine that result purely to the effects of accident. It is forbidden, in the first place, to raise the head of the stick higher than the shoulder, under the penalty of a blow on the shins from the hockey-stick of one of the opposite side; and thus a check is given to the reckless and promiscuous flourishing about of the player's stick, to the imminent hazard both of his friends and opponents. Moreover, any player proved wilfully to have struck another is at once excluded from the play. Besides these rules, the following are generally accepted :

1. A player must not cross to the side of his opponents before a rush or scrimmage has commenced.

2. The ball must be fairly struck through the goal, and not thrown or kicked.

3. It is forbidden to kick or throw the ball during the general game, but the ball may be stopped by any part of the person of a player who may intervene between it and the goal.

4. If the ball be struck beyond, but not through the goal, and if it be passed through the goal otherwise than by a fair hit, the youngest player of the side owning that goal shall return it by a gentle throw towards the centre of the ground. These, with the two rules given before, comprise all that it is necessary to observe in playing the game of Hockey, except the general rules of good temper and forbearance, which are required in all games alike.

66

The Scottish form of the game, known as Shinty, calls for no special remark, more than that the goals are called "hails," and that the game itself may owe its name either to the frequent danger to the player's shins, or to the shindy which characterises the culminating struggle. 'Hurley," the Irish variation of the game, also differs but little from that here described; but in Ireland the game has been, perhaps, a more general favourite, and played occasionally on a larger scale, than in either of the sister kingdoms. We borrow from Mr. and Mrs. S. C. Hall's "Ireland" an amusing anecdote in illustration of this fact. "About half a century ago," we are told, "there was a great match played in the Phoenix Park, Dublin, between the Munster men and the men of Leinster. It was got up by the then lordlieutenant and other sporting noblemen, and was attended by all the nobility and gentry belonging to the vice-regal court, and the beauty and fashion of the Irish capital and its vicinity. The victory was contended for a long time with varied success; and at last it was decided in favour of the Munster men, by one of that party running with the ball on the point of his hurley and striking it through the open window of the viceregal carriage, and by that manoeuvre baffling the vigilance of the Leinster goalmen, and driving it in triumph through the goal."

There is no record of matches on quite so extensive a scale having been played in the sister kingdoms; but we learn on the authority just quoted that, in the last generation, several good matches at hurley were played on Kennington Common between the Irish residents of St. Giles's and those of the eastern por tions of the metropolis, the affair being got up by some of the sporting noblemen of the day. Besides Kennington Common, several of the other open spaces around London were once noted as favourite spots for the exhibition in perfection of the game of hockey, and especially, in the last century, the extensive fields which then lay at the back of the British Museum. The amusement is not so frequently seen now, having yielded somewhat before the rival attractions of football and cricket, but it is a favourite still in many parts of the country.

CQ, forming with each other a small angle PC Q. On CP set off CD equal to A, and DF equal to B, and on cq set off CE equal to B. Join D E, and through the point F draw F G parallel to DE, and cutting co in a; the straight line E G is a third proportional to A and B; that is, A is to B as B is to E G. If we know the length of A and B, we can find the third proportional to them by dividing the square of the length в by the length of A. Thus, if a be three feet, and в be six feet, the third proportional to A and B measures twelve feet, for the square of 6 divided by 3, or 36 ÷ 3 = 12.

PROBLEM XV.-To find a fourth proportional to three given straight lines.

Let A, B, and C be the three given straight lines to which it is required to find a fourth proportional. Draw two straight lines DP, DQ, forming with each other a small angle, PD Q. On DP

G

Fig. 22.

set off DE equal to A, and EF equal to c, and on DQ set off DG equal to B. Join E G, and through F draw FH parallel to EG, and cutting DQ in H. The straight line H G is a fourth proportional to A, B, and c; that is, A is to B as c is to H G. If we know the length of A, B, and c, we can find the fourth proportional to them by multiplying the length of B and c together, and dividing the product by the length of A. Thus, if A be four feet, B six feet, and c two feet, the fourth proportional to A, B, and c measures three feet; for 6 x 2 = 12, and 12÷ 4 = 3.

PROBLEM XVI.-To divide a given straight line into any number of parts which shall be to one another in a given proportion. Let A B be the given straight line, which it is required to divide into five parts, which are to one another in the following proportions-namely, 5, 2, 3, 1, 4. First draw the straight

Fig. 23.

H

AC of indefinite length, making a small angle B A C with the given straight line A B. Along A c, from a scale of equal parts, Bet off in regular succession A D equal to 5 of these equal parts, DE equal to 2, EF equal to 3, F G equal to 1, and G H equal to 4. Join H B, and through the points D, E, F, G draw the straight lines D I, E K, FL, GM, cutting the straight line A B in the points 1, K, L, M. The given straight line A B is now divided into five parts, A I, I K, KL, L M, M B, which are to one another in the required proportions-namely, 5, 2, 3, 1, and 4.

This method of dividing a straight line into any number of parts, which shall be to one another in a given proportion, is based on Problem XII. (page 192). Supposing it had been required to divide A B into 15 equal parts, it is manifestly only requisite to set off along AC 15 equal parts, denoted by the dots on the line AC, from A to H, and then draw straight lines in succession through each dot on H A, from H to A, parallel to

The process that has been described in this Problem ensures

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 arithmetical process. For example, had the line'A B in Fig. 23 measured 30 inches, we can see at once that, as the sum of the numbers which show the proportion of the lines into which it is required to divide it is equal to 15, the half of 30, we have only to multiply each number by 2, and mark off A I equal to 10 (or 5 x 2) inches, I K equal to 4 (or 2 x 2) inches, and so on. But supposing A B had measured 29 inches, instead of 30, then A I would be represented numerically by 93, I K by 31 inches, etc., and lines involving fractions of inches such as, which are not to be found on an ordinary scale, would be very difficult to mark out without making a special scale for the purpose, or resorting to the plan given above.

PROBLEM XVII.-To draw an equilateral triangle on any given straight line.

Let A B be the given straight line on which it is required to draw an equilateral triangle. From the point A as a centre, with A B as a radius, describe the arc BC; and from the point B as a centre, with BA as a radius, describe the arc A C, cutting the arc B C in the point c. Join A C, BC; the triangle A B C is equilateral or equal-sided (see Definition 19, page 53), and it is drawn on the given straight line

A B.

Fig. 24.

If the arcs C A, C B be extended to cut each other in the point D below the straight line A B, by joining D A, D B, we get another equilateral triangle A B D, which is equal to the equilateral triangle A B C, and which is also drawn on the given straight line A B. By taking any straight line as a radius, and from each of its extremities as centres striking arcs intersecting or cutting each other on opposite sides of it, we get, by drawing straight lines from the points in which the arcs cut each other to the extremities of the straight line used as a radius, a regularlyformed diamond-shaped figure, whose four sides and shortest diagonal or diameter are all of equal length, such as A C B D in the above figure. This figure with four equal sides is called a rhombus. (See Definition 30, page 53.)

The learner should construct Fig. 24 on a large scale by the aid of his compasses and ruler. On applying a parallel ruler to the opposite sides of the figure A C B D, he will find that they are parallel to each other, namely, A C to B D, and B C to AD; ACBD is therefore a parallelogram, and A B, C D are its diagonals. (See Definition 26, page 53.) From Theorem 5 (page 156) the student learnt that the greatest side of every triangle is opposite the greatest angle, and that the greater the opening of the angle the greater must be the line that subtends or is opposite to it. Now in the triangle A B C, or in any other equilateral triangle, the three stright lines or sides by which it is contained are all equal to one another, and as equal sides must necessarily subtend equal angles, the three angles of the triangle A B C—namely, A B C, BCA, CA B-are also all equal to one another. Again, from Theorem 7 (page 156) we have learnt that the three interior angles of any triangle are equal to two right angles. A right angle contains 90 degrees, and as two right angles contain just twice as many, or 180 degrees, each of the equal angles A B C, B C A, C A B, in the interior of the equilateral triangle A B C, contains 1803 or 60 degrees.

Continuing our investigations a little further, we find that each of the angles AC E, B C E is half of the angle A C B, and is therefore an angle of 30 degrees. The angles A D E, B D E are also angles of 30 degrees, because each of them is half of the angle A D B, which, like the angle A C B, is an angle of 60 degrees. The angle C A D is equal to the angles C A B, D A B, and as each of these equal angles contains 60 degrees, the angle CAD contains 120 degrees. In the same way the angle C B D also contains 120 degrees. The diagonals of the rhombus A C B D intersect each other at right angles, therefore it will be seen that each of the angles c E A, C E B, D E A, D E B is a right angle.

Fig. 24 teaches us how to draw an angle of 45 degrees without the aid of the protractor, as we will proceed to show. A CE is an angle of 30 degrees, and so is its adjacent angle

Uhr ist es? What o'clock (literally, how much upon the clock) is it ?

When a part or the whole of the last quarter of an hour is named, it is designated, as in English, by its distance from the tour following, as :—

Es fehlen fünf, acht, oder zehn Mi nuten bis (or an) zwölf.

Es fehlt ein Viertel bis zwölf.

Wir dürfen Andern nicht thun, was wir nicht wünschen von ihnen ge than zu haben.

Er hat Briese schreiben wollen.
Wird sie gehen müssen?

minutes to twelve.

It lacks a quarter to twelve.

Ich mußte den ganzen Abend lesen. Sie hatten es nicht thun sollen.

We must not do to others what we do not wish to have done by them.

He has wished to write letters.
Will she be obliged to go?

She will not be able to go.
We have not wished to do it.
You will be allowed to go.

I was obliged to read the whole evening.

They ought not to have done it.

EXERCISE 37.

1. Wollen Sie mit mir nach Mannheim gehen? 2. Ich kann nicht, ich | habe keine Zeit. 3. Wann können Sie gehen? 4. Ich werde die nächste Woche gehen, wenn Sie so lange warten können. 5. Will Ihr Lehrer mit 3hnen auf das Feld over nach der Stadt gehen? 6. Er will nicht aufs Feld, und kann nicht nach der Stadt gehen. 7. Was wollen diefe Kinder? 8. Sie wollen Aepfel und Kirschen, aber sie können keine kaufen, denn sie haben kein Geld. 9. Was wollen Sie, mein Herr? mein Fräulein? meine Dame? 10. Wollen Sie die Güte haben, mir ein Glas (Sect. LXI.) Wasser (Sect. XXV.) zu geben? 11. Können Sie mir sagen, wie viel Uhr es ist? 12. Ich kann es (Sect. XXXV. 6) Ihnen nicht sagen, ich habe keine Uhr bei mir. 13. Was wollte der Kaufmann Ihnen verfaufen? 14. Ich konnte nichts bei ihm finden, was ich kaufen wollte. Wir werden morgen schlechtes Wetter haben. 16. Es fann sein, daß es noch heute regnen wird. 17. Können Sie die deutsche Handschrift lesen? 18. Nein, ich habe genug mit der Druckschrift zu thun. 19. Der 20. Eine Neibische (Sect. XVI.) will seinen Freund nicht loben. Gelehrte ist nicht immer eine gute Hausfrau. 21. Geduld ist eine schwere Kunst; Manche (§ 53. 1) können sie lehren, aber nicht lernen. 22. Ein guter Lehrer muß Geduld haben. 23. Jeder gute Schüler wird aufmerksam sein.

EXERCISE 38.

15.

1. You can go into the garden, but you cannot remain long there. 2. These attentive scholars were allowed to go with time better. their teacher to Mannheim. 3. We can employ [anwenten] our 4. Can you speak German ? 5. We could not learn our lessons this week. 6. You must learn this week's lessons [die Aufgaben dieser Woche] attentively. 7. You may go tomorrow to your parents. 8. He may be a good man. 9. The housewife must (is obliged to) go to market to-morrow. 10. Have you written to your parents? 11. Yes, I was obliged to write. 12. It is two o'clock. 13. I shall arrive at your house at a quarter past three o'clock. 14. Will you come twenty minutes before eight o'clock? 15. I may come to your house this evening, but do not wait for me. 16. As long as [so lange als] it rains, I cannot go out. 17. Fish can only [nur] live in water, and birds in the air. 18. You should not have done that, it will not be any recommendation [keine Empfehlung] to you. 19. I wish to go to the theatre this evening. 20. We may not have the opportunity [Gelegenheit] another time.

LESSONS IN MUSIC.—IV.

THE Binary (or two-pulse) measure is the boldest of the measures, and the one most easily felt or performed. It is by fur the best for large masses of voice, and is well adapted to aid in giving majesty to a tune. Try "St. Stephen's" or "Bedford," first in the three-pulse measure (lengthening the accented notes), and then in the two-pulse measure, and you will understand the character of the Binary measure. The Trinary (or three-pulse) measure is well adapted to aid in producing a soft and soothing musical effect. When the tune is simple it is not unfit for congregational use, especially if the people have been trained to keep the accent. The adaptation of this measure to soft and soothing music is illustrated by its analogy (according to Dr. Bryce) to the breathing of health and rest. The Quaternary (or four-pulse) measure, when delicately performed, gives much eleganee to a tune. It is adapted to congregational tunes when the movement is not too slow. Try the well-known tune "Vesper Hymn," taking care to give the medium accent. The Senary (or six-pulse) measure is commonly used in connection with quick movements, and is naturally soft, light, and elegant; for this reason it is better adapted to secular compositions than to sacred music.

Take a low sound of your voice for the key-note in this exercise. If any one gives you the pattern from an instrument, tell him to play in the key of D with two sharps. You understand that the letters under the "staff" are the initials of the notes on the modulator, and direct you in tracing out the tune there. The notes are placed within the accent marks to which they belong. DоH occupies the whole of the loud "pulse" of the measure. ME fills the first soft pulse, and Soн the second. This is the Trinary measure. The second measure is easily understood. In the third measure you have the first Dон occupying two pulses (loud and soft), and the second Doн only one pulse. The horizontal stroke, as in the second pulse, always indicates that the preceding note is to be continued. Thus the last note of the exercise is continued through the whole measure. In the fourth measure the third accent-mark is followed by no note. In the time of that pulse, therefore, the voice rests. If the previous exercises have been perfectly learnt from the

modulator, you will probably be able to make this out without pattern. Be careful to give the proper accent. You are strongly recommended not to study the "staff," at present, in any of these exercises. It is printed here that you may be able to return to it when you have gained some command of voice and some knowledge of music itself, and are not likely to be perplexed by its numerous signs; but if we may suppose that you have done this, then the following remarks will be of use. [The open note is twice as long as the closed notes. The empty "pulse," during which the voice rests, is represented by a distinct character, called a "rest." It tells you to rest as long as one of the closed notes, in the same time, would be sung. A dot after a note, in the old notation, bids you sing that note half as long again. Thus you perceive that the relative length of notes is expressed by symbols, and not, as in the solfa notation, measured out pictorially by the regularly recurring accents placed along the page.]

EXERCISE 11.-DоH, ME, SOн.

if it could be so. But as it is to be found, in different tunes, on every position on the staff, it is important that we should not mislead you. We prefer, however, that this exercise should be sung in the key of D or C, not of E.]