THE 1 negative; because the latter must be subtracted from the OUR HOLIDAY. former, to determine the clear profit. If the sums of a book account are brought into an algebraic process, the debit and GYMNASTICS.--X. the credit are distinguished by opposite signs. PANGYMNASTIKON. 38. The terms positive and negative, as used in the mathe- This is the name given to a most useful gymnastic apparatus maties, are merely relative. They imply that there is, either in invented by Dr. Schreber, Director of the Medical Gymnastic the nature of the quantities, or in their circumstances, or in the Institution at Leipsic. It is from the Greek, and signifies purposes which they are to answer in calculation, some such something belonging to all gymnastic exercises. It is so called opposition as requires that one should be subtracted from the because its inventor claims for it that it affords a combination other. But this opposition is not that of existence and non- of the advantages of all other apparatus ; and since its invenexistence, nor of one thing greater than nothing, and another tion it has come into extensive use both in Germany and America, less than nothing. For in many cases either of the signs may meeting very high approval. It is less known in this country be, indifferently and at pleasure, applied to the very same than it deserves to be, but we hope to make its merits familiar quantity; that is, the two characters may change places. In to our readers, determining the progress of a ship, for instance, her easting The ring exercises and the stirrup exercises which we demay be marked +, and her westing —; or the westing may scribed in our last paper are adapted from the Pangymnastikon, be +, and the easting — All that is necessary is, that the two but this apparatus in its complete form is a combination of both signs be prefixed to the quantities, in such a manner as to show rings and stirrups, each capable of being raised or lowered to which are to be added, and which subtracted. In different any height that may be desired. It is designed for use in an processes they may be differently applied. On one occasion, ordinary apartment, say of eight feet high, although a height of a downward motion may be called positive, and on another ten or twelve feet is preferable ; but it may be put up in an occasion negative. open yard, by the erection of a suitable framework, to which 39. In every algebraic calculation, some one of the quantities the ropes, etc., may be attached. It is not, it must be observed, must be fixed upon to be considered positive. All other quantities intended for use in the public gymnasium, but is, in fact, a which will increase this must be positive also. But those which simple contrivance for practising at home the most beneficial will tend to diminish it, must be negative. In a mercantile of the exercises for which the elaborate apparatus of such an concern, if the stock be supposed to be positive, the profits will institution is intended. be positive; for they increase the stock ; they are to be added. The apparatus may either be made at home or purchased to it. But the losses will be negative; for they diminish the complete at the price of about £2 10s. to £3. For the benefit stock; they are to be subtracted from it. of those who may wish to make it for themselves, we give the · 40. A negative quantity is frequently greater than the positive following description and instructions, written by the inventor, one with which it is connected. But how, it may be asked, can and translated by Dr. Dio Lewis, the great teacher of gymnastic the former be subtracted from the latter? The greater is training in America :certainly not contained in the less : how then can it be taken * Two large hand-rings are suspended from the ceiling by out of it? The answer to this is, that the greater may be ropes, which, running through padded hooks, are carried to the supposed first to exhaust the less, and then to leave a remainder walls. Two other ropes extend from the walls directly to the equal to the difference between the two. If a man has in his hand-rings. A strap with a stirrup is placed in either handpossession 1,000 pounds and has contracted a debt of 1,500; ring. By a simple arrangement on the wall, the hand-rings are the latter subtracted from the former, not only exhausts the drawn as high as the performer can reach, or let down within a whole of it, but leaves a balance of 500 against him. In com- foot of the floor; or at any altitude they can be drawn apart mon language, he is 500 pounds worse than nothing. to any distance. The distance between the stirrups and rings 41. In this way, it frequently happens, in the coarse of an can be likewise varied. The usefulness of the Pangymnastikon algebraic process, that a negative quantity is brought to stand depends upon the facility with which these changes can be alone. It has the sign of subtraction, without being connected made. The rings must be raised, let down, drawn apart, the with any other quantity, from which it is to be subtracted. stirrup-straps changed or removed altogether from the rings, This denotes that a previons subtraction has left a remainder, each and all with a single motion of the hand, and in a moment. which is a part of the quantity subtracted. If the latitude of a There are various simple mechanical contrivances by which these ship which is 20 degrees north of the equator is considered multifarious changes can be made. An ingenious mechanic can positive, and if she sails south 25 degrees : her motion first scarcely be at fault. I will suggest that in splicing the ropes diminishes her latitude, then reduces it to nothing, and finally with the rings, the splice should be long and drawn close ; else, gives her 5 degrees of south latitude. The sign — prefixed to giving way, an unpleasant surprise may occur. The ropes should the 25 degrees, is retained before the 5, to show that this is run through strong padded hooks at the ceiling, which are what remains of the southward motion, after balancing the 20 fastened on the upper side of the timber with thick nuts. The degrees of north latitude. fastenings on the walls must be made secure. The ropes with 42. A quantity is sometimes said to be subtracted from 0. By which the rings are separated should be armed with wroughtthis is meant that it belongs to the negative side of 0. But à iron snap-hooks, which should be caught into wrought-iron quantity is said to be added to 0, when it belongs to the rings which have been firmly lashed into the suspension rope, positive side. Thus, in speaking of the degrees of a thermometer, at the point where it connects with the hand-rings. The 0 + 6 means 6 degrees above 0; and 0 – 6, 6 degrees below 0.' stirrup-straps must be of very strong white leather, with edges AXIOMS. 50 rounded that the parts will not be worn. In shortening 43. An Axiom is a self-evident proposition. the straps, a buckle should not be used, for, in removing the straps from the hand-rings, much time would thereby be lost; 1. If the same quantity or equal quantities be added to equal nor should a simple hook be employed, as the leather is liable quantities, their sums will be equal. to give way, and the hook to slip out. A brass H, with one 2. If the same quantity or equal quantities be subtractel! side sewed into the end of the strap doubled, and the other from equal quantities, the remainders will be equal. slipped through slits in the body of the strap, is a perfect thing. 3. If equal quantities be multiplied into the same, or equal With this simple contrivance, the strap can be altered or taken quantities, the products will be equal. out altogether in a second, and can never give way. The 4. If equal quantities be divided by the same or equal quan- stirrups should be very strong, with serrated bottoms, and tities, the quotients will be equal. fastened into the ends of the straps with strong sewing and 5. If the same quantity be both added to and subtracted from copper rivets." another, the value of the latter will not be altered. When once this apparatus is fixed in a house, all its occupants, 6. If a quantity be both multiplied and divided by another, from the young even to the old, may use it with advantage. the value of the former will not be altered. Many of the exercises to which it is adapted are so simple that 7. Quantities which are respectively equal to any other quan- a child may practise them, and the steady motion of the muscles tity, are equal to each other. involved in others is so free from violent or undue exertion, that 8. The whole of a quantity is greater than a part. even the aged may derive pleasure and benefit from them. The 9. The whole of a quantity is equal to all its parts. inventor himself gives a list of more than one hundred exer cises which may be performed with the pangymnastikon, gra- the body backward again by the exertion of the arms and a duating in difficulty from the simplest imaginable, until they simultaneous movement of the legs forward, until you have become arduous enough to test a man's strength and skill. reversed the position, and you hang with the face upward, the Some of these were put before the readers in our last paper, to body stretched out to its full extent, with the back hollowed which we must refer our readers for many hints on the use of and the chest well arched. the pangymnastikon, the only difference being that in the latter 9. The rings should be at the height of the shoulders, and, apparatus the rings and stirrups are used in combination in being grasped firmly from the inside, should be stretched as stead of separately. Without going again over the same far apart as possible. Then cross the legs one after the other ground, we shall give a description of some of the as far as possible in front, the toes, as they rest in chief pangymnastic exercises, from the easiest to the stirrups being turned outward. The mutual the most difficult, referring our readers who may resistance created between the arms and the legs desire further details on the subject to Dr. Lewis's in this way is considerable, and forms another capi. translation of the inventor's elaborate treatise. tal muscular exercise. 1. The plain swing is shown in our first illustra 10. The rings hang rather higher than the head, tion (Fig. 31). The rings may be as high as either and wide enough apart for the arms just to reach the waist or the chest, the toes only should be in. them when extended. (Observe that the degree serted in the stirrups, the legs should be kept of distance separating the rings in this and other straight and close together, and the learner simply exercises is adjusted by the side ropes attached to swings backward and forward, with greater or less the wall.) The stirrups hang so that the feet can velocity, according to inclination. just rest in them when the legs are extended. 2. Let the rings be placed as high as the shoul Thus the whole body hangs in something like this ders, then pass the fore-arm through each ring, so that you hang by the elbow joints. Now swing to heels touch, to do which you must raise the and fro with vigour as you stand in the stirrups, body by the exertion of the muscles of the arms; and arch the chest well forward as you swing: and then return again to the extended position with This will develop the muscles of the chest more the legs stretched out. Avoid clumsy or inelegant effectually than the first exercise. movements in accomplishing this and other feats; 3. The sitting exercise is performed in the fol. for, although such motions may facilitate the perlowing manner. Stand in the stirrups with the formance of an exercise, they deprive it of half its rings grasped at the height of the waist; then value. bend the knees forward (keeping them close to 11. Practise diagonal movements in the followgether) and sit down so as to touch the heels. Now ing way. Stand in the stirrups (clear of the floor) rise again to the upright position by the use of the with the rings at the height of the chest. Hang legs alone, employing the arms merely to steady Fig. 31.-THE SWING. by the left hand from its ring, the right hand the body. being placed upon the hip, and let the right 4. Another good exercise for the muscles of the chest is the foot only rest in its stirrup, the other being placed behind following. Let the rings be as high as the chest, and the stir- it. Then move freely forward and backward, the body being rups so low that they will just rest on the floor when the rings kept quite straight; and afterwards reverse the position by are held out at arm's length from the body. The feet are put hanging by the right-hand ring with the foot in the left through the stirrups as far as to the heels. Now grasp the stirrup. rings as they hang before you, and stretch out the arms to These exercises will be sufficient to show the general scope the full reach in front of the body; next, keeping the arms and design of the pangymnastikon movements. But the appaquite straight, carry them backward as far as possible, the feet ratus may also be turned to good account in leaping exercises. all the while remaining firmly fixed upon the ground, and The addition of a cord suspended horizontally between the the legs close together. The feet being fixed in the stirrups, rings and the stirrups at any height that may be desired, is all the ropes become tightened as the arms are thrust out, and that is necessary; then you have a leaping apparatus which is the tension thus arising will give excellent play to the muscles. superior to the ordinary bar on a wooden framework. The 5. Let the rings and stirrups be as in the leaping cord may be attached by wooden pegs last exercise, with the exception that the legs or small weights slipped through the holes in are stretched apart as wide as possible, in the straps. stead of being kept close together. Now take The instructions given for leaping exercises the rings, stretching the arms out wide from in a previous paper (Vol. I., page 143) will the shoulders, and gradually bring them to apply equally to practise in this way with the gether in front of you. Let the legs remain pangymnastikon, and to these we must here stretched out during the exercise, and if the refer the learner. But in addition to these, feet slip, recover the position and begin again. the gymnast may practise vaulting, by taking 6. The twisting swing is practised as follows. a ring in one hand, and leaping with a swing Stand in the stirrups with the rings as high as over the cord which hangs below. The body, the waist; hold the rings from the inside, and in passing over, assumes almost the horizontal let the body rotate from side to side until it position, like that in other vaulting exercises. describes a semicircle. As the ropes cross It must be kept straight, the weight resting from the ceiling, the stirrup straps are made upon the ring as you pass over, and the disto cross each other likewise by the action of engaged hand being placed upon the hip. the legs. The description of a larger figure Fig. 32.-THE Bow. This is a very useful exercise, the ability to than the semicircle in this way is not recom perform which may often be turned to account mended, as it may produce too great a strain upon the appa- in passing a fence or a barrier. ratus. The seizing leap is another which may sometimes prove 7. Stand erect in the stirrups, with the rings at the height of service. At the moment when you are leaping over the either of the chest or the waist, and grasped as seen in the cord, seize cne of the rings in each hand, and hold them illustration (Fig. 32). Then from the perpendicular position tightly until you reach the ground. Or you may vary this let the body fall gradually backward until it assumes the po- exercise by placing the rings higher, so that the hold sition shown in Fig. 32; and from this return to the upright cannot be retained to the end, but is simply a catch in posture by the use of the arms alone. passing, which is relinquished before you come upon your 8. Take the rings at the height of the chest, and let the feet. Quickness both of eye and of hand will be required stirrups hang so that they will swing clear of the floor. Hold and exercised here. the rings with a firm grasp, and throw the body forward between We shall return in our next paper on gymnastics to some them, and the legs backward, so that the whole figure describes of the various kinds of gymnastic apparatus used in our public a curve, with the face directed towards the floor. Now draw gymnasia. LESSONS IN ARCHITECTURE.—XI. Romans, was very common at the period above mentioned. The pendentives, or portions of the vaults suspended out of the perARCADES—CUPOLAS-DOMES-CHURCHES-BASILICAS 1 pendicular of the walls—that is, the portions between the arches ROMANESQUE STYLE-ARABIC ARCH, ETC. 1 and the dome-were of Byzantine invention, and formed a new The Byzantines, the successors to the arts of the Romans, in and bold application of the arch in building, of which they soon consequence of the transference of the seat of the empire from began to make an improper use in the erection of towers, bel, Rome to Byzantium (afterwards called Constantinople), fol. fries, spires, and steeples of every description. An example of lowed their arched system of architecture, and even extended this application of the arch and vault will be seen in the anit to such a degree in their edifices that the architrave, which nexed illustration of the Catholicon at Athens. The first Christheir predecessors had hitherto preserved in the construction of tians of the West, in their adoption of the style of architecture their temples, was at last almost entirely abandoned. The which we have called the Latin style, substituted the arcade Byzantine architects not only used the arcade as the connecting for the architrave; but possessing less skill as builders than link from column to column in their erections, but they sur those of the East, their innovations terminated at this point. mounted their churches with cupolas or domes of an immense The great basilicas—that is, the palatial or royal churches of the size. This kind of vault, which had been seldom used by the l West-were edifices covered with plain woodwork, and had only VOL. III. 57 MOORISH ABSA. vaulted roofs or domes at the extreme or lower end above the which represents the gallery of the early basilicas. The columns and the aisles. The nave alone in the ba- covered Arabia and Egypt, and its central point, and the arch be- rials. A good example of the old basilica purely Arabic, such as the Mekias PLAN OF THE BASILICA is the church of St. Agnes at Rome. or Nilometer, and various mosques of that city. In Persia and OF PARENZO. The Romanesque style was formed from in India the same style is exhibited, and is always found in cona. The principal nave. b. the combination of the two former, the nection with the pointed arch, and the same principles are fol The chancel. c. The Latin and the Byzantine ; and in these, lowed in the architecture of that country at the present day. hall, or atrium. d. the arcade played the most prominent part. We give in the annexed engraving a specimen of the Arabic The baptistry. e. Arches, indeed, were multiplied in endless arch. The steeple, or bel An important question has never yet been answered fry. f. The sacristy, variety, from the choirs which rose on a respecting the origin of the pointed arch, which was first, as or vestry. h. The circular plane, to domes and arched but we have seen, used by the Pelasgians, but which, abandoned choir. Kk. Small tresses, those appendages to buildings until the Middle Ages, was again taken up by the Arabian and choirs. 1. The tri- which were first employed in this style of the Western architects, at an epoch when it is difficult to ascerclinium, or supper. architecture. The Romanesque period, tain whether the East preceded the West in its application, or room. however, produced edifices different from whether the reverse was the case. Whichever was the case was the basilicas, by characters well defined. of little moment, until it was extended as a complete system of Orientation, or building churches east and west, became the construction, and became the foundation of the Gothic or ogival decided rule; a transverse nave, or transept, changed the ar- style of architecture. It appeared in the West, in the twelfth rangement of the interior of churches, and gave them the form century, in several rare edifices, in which it usually occupied the of 1 cross. The choir or recess, of a semicircular form, was lower part of the building, as presenting more resistance in commonly unique, and spanned the whole width of the edifice, supporting its elevated portions. From this use, first origiincluding the nave and the aisles. Some Romanesque churches, nating in the demand for solidity, it was extended to all parts however, have preserved the three recesses or choirs; and these of the building. Thus applied, this new system of architecture are generally the oldest. The columns were replaced by square was developed, took its flight, and the thirteenth, fourteenth, and piers, ornamented on each side by a column carrying groined fifteenth centuries saw its rise, its full vigour, and its decay. arches; and when stone mouldings were introduced at the edges The Gothic churches are in general larger than those of the of the domes, the number of the columns was increased to eight. Romanesque period, they are disposed in the same way, but their The shafts of the four intermediate columns were then dispro- architecture becomes softer, and the forms more graceful. portionally lengthened, and departing from the proportions established by antiquity, lost their rational proportion to their diameter; the choir alone preserved the isolated columns. Independently of the principal altar, secondary ones were erected, LESSONS IN ARITHMETIC.-XXXVIII. of which the number was afterwards augmented by those of the EQUATION OF PAYMENTS. chapels built round the choir. The exterior of the Romanesque 18. This is a method by which to find the time at which two churches also presented a very different aspect from that of the or more debts, due at different times, may be equitably paid by basilicas. The belfries, which were at first small, became then one payment, equal to the sum of the amounts of the debts. of great importance, and were raised above the porch, or above This time is called the equated time of payment. We will firs! the cross aisle. These constructions were very solid, and had take the case of two sums. several storeys, partially open, except at top; and were sur- The principle upon which the time is calculated is this-that mounted by lofty stone spires. At a later period, the abut- the sum of the present values of the two debts is equal to the ments, or spur-walls, became insufficient to sustain the thrust of present value of the sum of the amounts of the debts supposed the great domes, and were then detached from the walls and due at the equated time. transformed into battresses. Specimens of this kind of edifice Thus, if £100 were due nine months hence, and £50 twel may be seen in the church of Rosheim, in the Department of months hence, at 4 per cent., we must find the present valne o the Lower Rhine ; St. Germain-des-Prés, at Paris; of L'Abbaye- each, and add them together. We must then find the tim aux-lIommes, at Caen ; of Bocherville, near Rouen ; and others in which, at the given rate of interest, the sum so formed in France. In the most of these churches, there is a small gal- would amount to £150. lery over the aisle, like that of St. Germain-des-Prés at Paris, EXAMPLE. -- Find the equated timo of payment of £309, du per cent. 800 3600 47 nine months hence, and of £416, due twelve months hence, at 4 EXAMPLE.--If the Three per Cents. are at 937, find the rate per cent. of interest received. 1. Reckoning True Discount. For £93 5s, paid, £3 is received yearly, Present value of £309, due 9 months hence, is , £300 Hence, as 931 : 3 :: 100 : rate per cent. received. Present value of £416, due 12 months hence, is £400 The interest of £300 for 9 months is the same as that of 9,700 for Therefore 3.217, the rate per cent. 93.23 I month. 21. Given the sum invested, and the price of the Stock, to find The interest of £100 for 12 months is the same as that of £4,800 for 1 month. the Income. Hence the interest on £700 (the sum of the present values) when they are at 88, find the income. EXAMPLE.-If £1,200 be invested in the Three per Cents. for the equated time must be equal to the interest on £2,700 + £4,800, or £7,500, for one month. Now the timo Since for every £88) paid, £100 worth of stock is received, every in which the same interest will be produced by two different 488 paid will produce an annual income of £3. Hence, as £884 : £3 : : £1,200 : required income. sums will be inversely proportional to the sums; hence the 1920 Equated time will be "!, or 105 months; Therefore the required income = £10 178. 01 . - 83% 1920 (9 x 300) - (12 x 400). 22. Given the income received from an investment of a given we see the truth of the following Sum of Money, to find the price of the Stock. Rule for finding the Equated Time of two or more Debts due at duces a yearly income of £40 17s. 043d., find the price of the EXAMPLE,---If £1,200 invested in the Three per Cents. prodiferent times, at a given rate per cent., True Discount being stock, recloned. £40 17s. 0:7d, is 40- of a pound. Find the true present value of each debt, multiply it by its corresponding time, and add the products. Divide the sum of Hence f x 497 is the number of times £3 is contained in the the prodncts by the sum of the present values. income; that is, it is the number of times £100 is contained in 2. Reckoning Mercantile Discount. the amount of stock bought. Interest of £309 for 9 months is the same as that of 9 * £309, or Hence, £1,200 divided by this will give the actual monoy paid €2,781, for 1 inonth. for each £100 of stock. 3 x 47 £89). $4,992, for 1 inonth. Hence the interest on £309 + £416, or £725, for the equated 23. When Government stock is purchased, the transaction is time must be the same as that on £2,781 + £4,992, or £7,773, effected through the agency of a broker, who charges th per for one month. cent. upon the stock bouglt-i.e., 25. 6d. upon every £100 of Hence the equated time is us, or 10.72 months ncarly. stock purchased. We get, then, the following Thus, if £500 worth of stock be purchased when the fanda Rule for finding the Equated Time of two or more Debts due at are at 92, the actual price paid will be (5 * £92) + (5 * .£), diferent times, Mercantile Discount being reckoned: or £460 12s. 6d. And, similarly, the seller of stock pays his Multiply the amount of each debt by its corresponding time, broker Ath per cent. upon the amount of stock sold for him. and add the products. Divide the sum of the products by the This charge is called Brokerage, or Commission. In the examples sum of the debts. we give, however, it need not be reckoned unless it is expressly N.B.-When mercantile discount is reckoned (as is the case mentioned. in practice), the rate per cent. does not affect the calculation, 24. Exactly the same principles hold with reference to Shares The times of both debts must of course be expressed in of any kind. Originally they are fixed at a certain price, and the same denomination, and the result will appear in that then, according to the success or failure of the company, and denomination. the probable amount of dividend it will pay, etc., the value of EXERCISE 58.--EQUATION OF PAYMENTS. the shares fluctuates. 1. Find the equated time of £800, payable in 3 years, and of £1,200, When a share, or £100 of stock, will sell for the original payable in 4 years, at 5 per cent. simple interest, by reckoning (1) price which was paid for it, then the shares are said to be at tzue, (2) mercantile discount. par. When the price is less by a certain amount than the .. Find the equated time of payment of £261 58., due 6 months original price, they are said to be at so much discount ; and hance, and of £209, due 18 months hence, at 4 per cent., reckoning (1) when the price is more by a certain amount, they are said to be true, (2) mercantile discount. at so much premium. 3. Find the equated time of £692, payable in 60 days, and £254, 25. EXAMPLE.—The income derived from investing a sum in payable in 96 days. I owe £500, dne 50 days hence, and £750, due 100 days hence; the Three per Cents. at 90 differs by £1 from that derived from when should I liquidate the debt equitably by paying down £1,500, an equal sum invested in railway shares at 140, paying 5 per interest being reckoned at 4 per cent. per annum ? cent. upon the original shares. Find the sum. In the first investment, STOCKS, SHARES, BROKERAGE, INSURANCE, ETC. £90 produces annually £3; 19. Suppose that I lend a sum of money to the Government Therefore £1 £s, or 23. 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 In the second investinent, that at any time after this transaction it is found that more £140 produces annually £5; than 3 per cent. can be commonly got for money ; it is clear Therefore £1 £rs, or Li that if I sell then my claim upon the Government or company Hence the difference of income produced by £l is - nu, or Lil to another person, he will not give me so much as I gave. The Hence the difference of income produced by £420 is £120, or £1. The answer is therefore £120. Dumne 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. INSURANCE The Funds are properly the money raised by the Government, 26. By the yearly payment of a certain sum called a Premiu.n 15 taxes, etc., to pay the interest of the debt, but the term to an insurance company, a person can secure at his death the is often applied to the debt itself. Thus, when we hear that payment of a certain larger sum. The document by which the the Funds are at 904 it means that £90 5s, must be paid for company binds itself to pay over the money at the death of the $100 worth of stock, this entitling the purchaser to receive insurer is called the Policy of Insurance. Thus, a man of 30, in from the Government the sum of £3 (in the Three per Cents.) | ordinary health, by paying about £25 a year to a company, is agreed to be paid upon the £100 originally lent. able to "insure his life" for £1,000. Different names are given to different descriptions of stock, The principles which determine the amount of premium to be according to the original conditions of the formation of the paid depend upon carefully prepared tables of statistics, showdeht. For instance, the Three per Cent. Consols-i.e., the Three ing the average rate of mortality at different ages, and also upon pe: Cent. Consolidated Annuities, etc. the doctrine of chances and annuities, but they are too compli20. Given the price of Stock, to find the actual Rate per Cent. cated to be introduced here. received. There are various cther kinds of insurance, as, for instance, |