LESSONS IN MUSIC.-XIV.

CONSONANCES OF NOTES.

In addition to the observations on the consonances of DOH, ME, SOH in a former lesson, let the following remarks be examined and tested.

FAH forms a more "perfect" consonance with the key-note than LAH. (It is more like it, and has a greater number of coinciding vibrations.) But the consonance of LAH with the key-note is more soft and pleasing.

The best consonances with FAH are RAY and LAH. The best notes to sound with LAH are FAH and DOH1.

It may be noticed that when the notes of a consonance are in their closest position, as DOH1 with LAH, or LAH with FAH, the proper mental effect of each is sweetly blended with that of the other; but when, by raising or lowering one of them an octave, they are more distant, as DOH with LAH or LAH1 with FAH, each produces its own effect with greater distinctness, though still with good agreement.

Two persons can easily try these experiments by singing the chord DOH, ME, SOH together, and then "striking out" each into the separate note previously agreed on. It would be difficult to contrive any better preparation for the study of harmony.

vaulted roofs or domes at the extreme or lower end above the chancel, and sometimes also above the aisles. They had commonly in front the accessory buildings, including the baptistery and the rooms for the catechumens, or those who were preparing for admission into the church. These adjoining erections were then separated from the church by an atrium or hall, surrounded by columns which led from outside to the door of the church, which always had a porch, or narthex, as it was called. In the interior, two rows of columns divided the edifice into the nave and the aisles. The nave alone in the basilicas or churches of the primitive Christians, terminated in a semicircular arch or choir. At a later period the aisles had also their choirs; it was then that the first story or gallery which was erected upon the aisles disappeared, and the women, for whom it was reserved, began to occupy the left side of the building. The different parts of the Basilica of Parenzo will convey to the mind of the reader the arrangement of the principal parts of the churches now described.

The exterior of the basilicas was very simple; but the interior was richly decorated with mosaics, with paintings, and particularly with splendid marbles, which were used in great profusion; also with monoliths, or columns, made of a single stone, usually stolen from antique edifices, all of the most magnificent description; and the pedestals, capitals, and cornices were of equally splendid and rich materials. A good example of the old basilica PLAN OF THE BASILICA is the church of St. Agnes at Rome.

OF PARENZO.

a. The principal nave. b. The chancel. c. The hall, or atrium. d. The baptistry. e. The steeple, or belfry. f. The sacristy, or vestry. h. The choir. k k. Small choirs. 1. The triclinium, or supper

room.

The Romanesque style was formed from the combination of the two former, the Latin and the Byzantine; and in these, the arcade played the most prominent part. Arches, indeed, were multiplied in endless variety, from the choirs which rose on a circular plane, to domes and arched buttresses, those appendages to buildings which were first employed in this style of architecture. The Romanesque period, however, produced edifices different from the basilicas, by characters well defined. Orientation, or building churches east and west, became the decided rule; a transverse nave, or transept, changed the arrangement of the interior of churches, and gave them the form of a cross. The choir or recess, of a semicircular form, was commonly unique, and spanned the whole width of the edifice, including the nave and the aisles. Some Romanesque churches, however, have preserved the three recesses or choirs; and these are generally the oldest. The columns were replaced by square piers, ornamented on each side by a column carrying groined arches; and when stone mouldings were introduced at the edges of the domes, the number of the columns was increased to eight. The shafts of the four intermediate columns were then disproportionally lengthened, and departing from the proportions established by antiquity, lost their rational proportion to their diameter; the choir alone preserved the isolated columns. dependently of the principal altar, secondary ones were erected, of which the number was afterwards augmented by those of the chapels built round the choir. The exterior of the Romanesque churches also presented a very different aspect from that of the basilicas. The belfries, which were at first small, became then of great importance, and were raised above the porch, or above the cross aisle. These constructions were very solid, and had several storeys, partially open, except at top; and were surmounted by lofty stone spires. At a later period, the abutments, or spur-walls, became insufficient to sustain the thrust of the great domes, and were then detached from the walls and transformed into buttresses. Specimens of this kind of edifice may be seen in the church of Rosheim, in the Department of the Lower Rhine; St. Germain-des-Prés, at Paris; of L'Abbayeaux-Hommes, at Caen; of Bocherville, near Rouen; and others in France. In the most of these churches, there is a small galbery over the aisle, like that of St. Germain-des-Prés at Paris,

In

which represents the gallery of the early basilicas. The columns or pillars are united by circular arches, ornamented by archivolts in simple profile, and the whole of the edifice is covered with groined vaulting, and with stone mouldings of the same description.

SPECIMEN OF THE ARABIC OB MOORISH ARCH.

At the commencement of the Romanesque period in the West, the Eastern style of art, the parent of so many architectural novelties, was developed anew in Asia and Africa, and the Arabs, who played so prominent a part in the history of this epoch, covered Arabia and Egypt, and penetrated the interior of Persia, with their edifices. The Moorish branch extended along the northern side of Africa to the ocean, and there all that an Eastern imagination could produce that was fantastical and marvellous, was reproduced in their architectural works. During the most ancient period of this art the semicircular arch predominated; but it soon overstepped its central point, and the arch becoming more than a semicircle, took the shape nearly of a horseshoe. At a later period, the intrados, or inner ring of the arch, was carved in the manner exhibited in the celebrated Mosque of Cordova. The second period was that of the pointed arch very ancient specimens of which are found at Cairo. These monuments or erections are purely Arabic, such as the Mekias or Nilometer, and various mosques of that city. In Persia and in India the same style is exhibited, and is always found in connection with the pointed arch, and the same principles are fol lowed in the architecture of that country at the present day. We give in the annexed engraving a specimen of the Arabic arch. An important question has never yet been answered respecting the origin of the pointed arch, which was first, as we have seen, used by the Pelasgians, but which, abandoned until the Middle Ages, was again taken up by the Arabian and the Western architects, at an epoch when it is difficult to ascertain whether the East preceded the West in its application, or whether the reverse was the case. Whichever was the case was of little moment, until it was extended as a complete system of construction, and became the foundation of the Gothic or ogival style of architecture. It appeared in the West, in the twelfth century, in several rare edifices, in which it usually occupied the lower part of the building, as presenting more resistance in supporting its elevated portions. From this use, first originating in the demand for solidity, it was extended to all parts of the building. Thus applied, this new system of architecture was developed, took its flight, and the thirteenth, fourteenth, and fifteenth centuries saw its rise, its full vigour, and its decay. The Gothic churches are in general larger than those of the Romanesque period, they are disposed in the same way, but their architecture becomes softer, and the forms more graceful.

LESSONS IN ARITHMETIC.-XXXVIII.
EQUATION OF PAYMENTS.

18. THIS is a method by which to find the time at which two or more debts, due at different times, may be equitably paid by one payment, equal to the sum of the amounts of the debts. This time is called the equated time of payment. We will first take the case of two sums.

The principle upon which the time is calculated is this-that the sum of the present values of the two debts is equal to the present value of the sum of the amounts of the debts supposed due at the equated time.

Thus, if £100 were due nine months hence, and £50 twelve months hence, at 4 per cent., we must find the present value of each, and add them together. We must then find the time in which, at the given rate of interest, the sum so formed would amount to £150.

EXAMPLE.-Find the equated time of payment of £309, due

The interest of £300 for 9 months is the same as that of £2,700 for 1 month.

The interest of £400 for 12 months is the same as that of £4,800 for 1 month.

Hence the interest on £700 (the sum of the present values) for the equated time must be equal to the interest on £2,700+ £4,800, or £7,500, for one month. Now the time in which the same interest will be produced by two different sums will be inversely proportional to the sums; hence the equated time will be 1500, or 10 months; 7500 (9 x 300) + (12 × 400). 700 300 + 400

Noticing that

we see the truth of the following

Rule for finding the Equated Time of two or more Debts due at different times, at a given rate per cent., True Discount being reckoned.

Find the true present value of each debt, multiply it by its corresponding time, and add the products. Divide the sum of the products by the sum of the present values.

2. Reckoning Mercantile Discount.

Interest of £309 for 9 months is the same as that of 9 × £309, or £2,781, for 1 month.

That on £416 for 12 months is the same as that of 12 × £416, or £4,992, for 1 month.

Hence the interest on £309 + £416, or £725, for the equated time must be the same as that on £2,781 + £4,992, or £7,773,

for one month.

Hence the equated time is 1773, or 10.72 months nearly.
We get, then, the following

Rule for finding the Equated Time of two or more Debts due at different times, Mercantile Discount being reckoned: Multiply the amount of each debt by its corresponding time, and add the products. Divide the sum of the products by the sum of the debts.

N.B. When mercantile discount is reckoned (as is the case in practice), the rate per cent. does not affect the calculation. The times of both debts must of course be expressed in the same denomination, and the result will appear in that denomination.

EXERCISE 58.-EQUATION OF PAYMENTS.

1. Find the equated time of £800, payable in 3 years, and of £1,200, payable in 4 years, at 5 per cent. simple interest, by reckoning (1) true, (2) mercantile discount.

2. Find the equated time of payment of £261 5s., due 6 months hence, and of £209, due 18 months hence, at 4 per cent., reckoning (1) true, (2) mercantile discount.

3. Find the equated time of £692, payable in 60 days, and £254, payable in 96 days.

4. I owe £500, due 50 days hence, and £750, due 100 days hence; when should I liquidate the debt equitably by paying down £1,500,

interest being reckoned at 4 per cent. per annum ?

STOCKS, SHARES, BROKERAGE, INSURANCE, ETC. 19. Suppose that I lend a sum of money to the Government or to a company, on the understanding that I am to receive a certain fixed annual per-centage upon it (say 3 per cent.), and that at any time after this transaction it is found that more than 3 per cent. can be commonly got for money; it is clear that if I sell then my claim upon the Government or company to another person, he will not give me so much as I gave. The name given to money so lent is Stock, and the price given at any time for £100 of this stock is the price of stock at that time.

The Funds are properly the money raised by the Government, by taxes, etc., to pay the interest of the debt, but the term is often applied to the debt itself. Thus, when we hear that the Funds are at 90 it means that £90 5s. must be paid for £100 worth of stock, this entitling the purchaser to receive from the Government the sum of £3 (in the Three per Cents.) agreed to be paid upon the £100 originally lent.

Different names are given to different descriptions of stock, according to the original conditions of the formation of the debt. For instance, the Three per Cent. Consols-i.e., the Three per Cent. Consolidated Annuities, etc.

20. Given the price of Stock, to find the actual Rate per Cent. received.

23. When Government stock is purchased, the transaction is effected through the agency of a broker, who charges th per cent. upon the stock bought-i.e., 2s. 6d. upon every £100 of stock purchased.

Thus, if £500 worth of stock be purchased when the funds are at 92, the actual price paid will be (5 x £92) + (5 × £¦), or £460 12s. 6d. And, similarly, the seller of stock pays his broker 4th per cent. upon the amount of stock sold for him. This charge is called Brokerage, or Commission. In the examples we give, however, it need not be reckoned unless it is expressly mentioned.

24. Exactly the same principles hold with reference to Shares of any kind. Originally they are fixed at a certain price, and then, according to the success or failure of the company, and the probable amount of dividend it will pay, etc., the value of the shares fluctuates.

When a share, or £100 of stock, will sell for the original price which was paid for it, then the shares are said to be at par. When the price is less by a certain amount than the original price, they are said to be at so much discount; and when the price is more by a certain amount, they are said to be at so much premium.

25. EXAMPLE.-The income derived from investing a sum in the Three per Cents. at 90 differs by £1 from that derived from cent. upon the original shares. Find the sum. an equal sum invested in railway shares at 140, paying 5 pen

In the first investment,

26. By the yearly payment of a certain sum called a Premium to an insurance company, a person can secure at his death the payment of a certain larger sum. The document by which the company binds itself to pay over the money at the death of the insurer is called the Policy of Insurance. Thus, a man of 30, in ordinary health, by paying about £25 a year to a company, is able to "insure his life" for £1,000.

The principles which determine the amount of premium to be paid depend upon carefully prepared tables of statistics, showing the average rate of mortality at different ages, and also upon the doctrine of chances and annuities, but they are too compli cated to be introduced here.

There are various other kinds of insurance, as, for instance

insurance against loss by accidental fire, where a man pays a certain per-centage upon the value of the property insured. Farmers can insure their crops against damage by storms. Owners of vessels can insure them and their cargoes, etc.

EXERCISE 59.-EXAMPLES IN STOCKS, SHARES,
INSURANCE, etc.

1. When the Three per Cents. are at 89, find the rate per cent. received.

2. When the Three and a Half per Cents. are at 92, find the rate per cent. received.

3. When the Four per Cents. are at 96, find the rate per cent. received.

4. Three per Cents. are offered at 901, Five per Cents. at 150;; in which should one invest?

5. Shares paying an interest of £1 0s. 6d. cost 22; what is the rate per cent. received? What is the rate per cent. received after a deduction of income tax of 7d. in the pound?

6. Is it better to invest in the Three per Cents. at 89; or the Three and a Half per Cents. at 981 ?

7. When the Three per Cents. are at 86, find what the Three and a Half per Cents. should be at, so that an investment in both should produce the same income.

8. Find the income derived from investing £2,500 in the Three per Cents. at 86.

9. Find the income derived from investing £1,000 in the Three per Cents. at 87, after deducting an income tax of 7d. in the pound. 10. What must a person have invested in the Three per Cents. at 90%, if a transfer of of his capital to the Four per Cents. at 115 would increase his income by £7?

11. A person invests £5,187 10s. in the Three per Cents. at 83, and when the funds have risen to 85 he transfers two-fifths of his capital to the Four per Cents. at 96; find the alteration in his income.

12. A person sells out of the Three and a Half per Cents. at 923, and realises £18,550; if he invests two-fifths of the produce in the Four per Cents. at 96, and the remainder in the Three per Cents. at 90, find the alteration in his income. Find also the alteration in the per-centage he gets. Cents. when £10,000 will What will be the annual

13. What is the price of the Three per purchase £11,111 28. 23d. worth of stock? income produced?

14. When the Three per Cents. are at 87, what must the Three and a Quarter per Cents. be at in order that an investment of £1,580 in the latter may produce an income greater than the same sum invested in the former by £1 18s. 64d.?

15. A person buys £500 worth of stock at 982; what must he sell at to gain £21 58. by the transaction?

16. Which is the better investment, bank stock paying 10 per cent. at 319, or Three per Cent. Consols at 96?

17. A man invests £4,297 108. in the Three per Cents. at 951; he sells out one-third of his stock when the funds have fallen to 94, £1,600 worth of stock when they have risen to 96, and the remainder at par. What does he gain ?

18. The income derived from a sum invested in the Three per Cents. at 90 differs from that derived from an equal sum in the Five per Cents. at 150 by £1. Find the sum.

19. A person finds that if he invest a certain sum in railway shares paying £6 a share when the £100 share is at 132, he will obtain £10 168. a-year more for his money than if he invest in the Three per Cent. Consols at 93. What sum has he to invest ?

20. A person invests £10,000 in the Three per Cents. at 92, and sells out two-thirds when they have risen 1 per cent. above, and the remainder when they have fallen 1 per cent. below, the price at which he bought; he then invests the proceeds in the Four per Cents, at par. Find the change in his income.

Vous êtes-vous démis de votre Have you resigned your situation? place ?

Je m'en suis démis [§ 131, 11].
Nous ne pouvions nous empêcher
de sourire pendant ce récit.
Vous êtes-vous emparé de ce livre ? Have you seized that book?
Je m'en suis emparé.
I laid hold of it.

I have resigned it.

We could not help smiling during
that narration.

De quoi vous inquiétez-vous ?
Je ne m'inquiète de rien.
Comment ce jeune homme se com-
porte-t-il ?

Il se comporte comme il faut.
Je ne m'attendais pas à une telle
réponse.

Je ne m'y attendais nullement.

À l'avenir, in future.
Bras, m. arm.
Cass-er, 1. to break.
Droit, e, right.
Durant, during.
Ecritoire, f. inkstand.

Ennemi, m. enemy.

What do you trouble yourself about?
I trouble myself about nothing.
How does that young man behave!

He behaves properly.

I did not expect such an answer. I did not expect it by any means.

8. 11

2. Je ne me 3. Si vous alliez en Amérique, vous démettriez-vous de votre place ? 4. Je serais obligé de m'en démettre. 5. Y a-t-il longtemps que votre cousin s'est démis de la sienne ? 6. Il y a un mois qu'il s'en est démis. 7. L'ennemi s'est-il emparé de la ville? s'en est emparé. 9. Votre fils se comportera-t-il mieux à l'avenir? 10. Il s'est très bien comporté durant son séjour en Prusse. 11. Vous attendiez-vous à un pareil traitement de sa part? 12. Je ne m'y attendais pas. 13. A quoi vous attendiez-vous ? 14. Je m'attendais à être traité comme il faut. 15. Pourquoi vous êtes-vous moqué de lui? 16. Parce que je n'ai pu m'en empêcher. 17. Si vous laissiez votre écritoire ici, le paysan s'en emparerait-il? 18. Il s'en emparerait certainement. 19. Votre associé se comporte-t-il bien envers vous? 20. Il se comporte bien envers tout le monde. 21. Qui a remis le poignet à votre sœur ? 22. Le Dr. G. le lui a remis. 23. M. votre père ne s'est-il pas démis le bras droit ce matin? 24. Il ne se l'est pas démis; il se l'est cassé ce matin à cinq

heures.