These mallets are almost the same in construction as those used in the game of Croquet, while the ball also is similar, FIG. 1.-CROQUET MALLETS. and is made of lancewood; the head is slightly curved, and | half inches in diameter, perfectly round, and each painted of a measures outwardly five and a half inches, the inner curve different colour, so that each player may know his own. being four and a half inches. The diameter of the maile-ends The mallets for striking the balls should bear a proper prois two and a half inches, each shod with a thin iron hoop: the portion to the size and weight of the latter. They are usually handle, which is very elastic, is made in one or other of the forms bound with white leather to the shown in the annexed engraving, breadth of two hands, and teralthough more fanciful ones have minated with a collar of jagged been devised, and are occasionally leather. The ball is of boxwood, seen in use. The handles should two and a half inches in diabe made of ash wood, about two meter. A pair of mailes and a feet nine inches long, and thinner ball have been presented to the in the centre than at the two British Museum." ends, which allows a spring in the stroke. In the thickest part, which is grasped by the hand, they should be about an inch in diameter, and taper to five-eighths of an inch in the middle. The heads of the mallets should be of box-wood, their greatest diameter two and a half inches, and their length about four and a half inches. The pegs for a single set of implements are two in number, one being driven in at each end of the ground. They are about two feet in length, an inch and a quarter in diameter at the top, and brought to a point at the bottom, so as to be readily driven into the earth. They are usually painted with rings of colour, which show the order of players -blue first, then pink, black, yellow, brown, orange, green, and red. TURNING PEG. 70.8 although smaller in size. The English game of Pall-mall, We have gone into this matter, as questions are frequently asked concerning it, and surprise has sometimes been expressed that so attractive a pastime was not devised until winin the last few years. 3 ·-16.FF-· -16.FF-- FIG. 2-THE CROQUET Ground. --16.F‡--- ·-16.F!- 10 412 The hoops are either eight or ten in number, according to the arrangement adopted in laying out the ground. They are made of round iron-wire, arched in form, and each leg about fifteen inches long from the crown of the arch. Being driven about two inches into the ground, they should stand thirteen inches high when fixed, the span of the arch being about nine inches. Flat-topped hoops are occasionally employed, but are not recommended. If the hoops are painted white, they are more easily distinguished. The arrangement of the hoops in laying out the croquet ground is varied according to the size of the ground, or the preference of those who may use it; but the plan shown in Fig. 2 is that most commonly adopted, and it forms the basis for all other arrangements. The relative distances here given are suited to a ground where space is limited. The great charm and attraction of Croquet, and the secret of its popularity, lie in the fact that it stands almost alone as a game in which persons of both sexes can join in the open air, and find both health and recreation together. Occupying this position, it is likely that the popularity of Croquet will increase rather than diminish. We see no reason why a croquet ground should not be formed on many a village green, or why the game should not have, like cricket, its allotted space in our public parks. In this way it might soon be found taking the place of more objectionable | his ball, and to return through hoop No. 7 (which thus practically amusements, and add another source of health and happiness to the people. STARTING PEG. We pass on now to a description of the game, and the apparatus and rules for playing it. The implements required are balls, mallets, hoops, and pegs. The balls should be of box or beech-wood, about three and a The arrow lines show the direction in which the balls are played, and the figures indicate the order in which the hoops are taken in playing the game. Thus, commencing at a short distance on either side of the starting peg, each player aims to drive the ball with his mallet through hoop No. 1, and so on through each in order until he passes No. 7. Then he has to hit the turning peg with becomes No. 8 also), and along the other side of the ground taking all the other hoops in downward succession, until he repasses No. 1; he has then to strike the starting peg with his ball, and this completes the game. A more detailed description of the game, with a code of rules to be observed in playing it, we must reserve for another paper. LESSONS IN CHEMISTRY.-X. THE protoxide of nitrogen, nitrous oxide, or laughing gas (NO, The gas cannot be collected over cold water unless with much loss, for 100 cubic inches of water at 13° Cent. absorb 77 of the gas. However, if the water be hot, a very small quantity is retained, hence the gas must be collected over hot water or mercury, upon which metal it has no action. Properties.-The gas is colourless and without smell, but possesses a sweetish taste. The When subjected to a pressure of about 30 atmospheres at the temperature of 0° Cent., by a process which will be described in a succeeding lesson, the gas becomes a clear liquid. same phenomenon is exhibited if the gas be cooled down to -88° Cent., that is to say, liquid nitrous oxide boils at -880, or the vapour of that liquid (that, is nitrous oxide gas) has a tension equal to one atmosphere at the temperature of -88° Cent.; and as it requires a pressure of 30 atmospheres at 0° Cent. to liquefy it, we say that at 0° Cent. the gas has a tension of 30 atmospheres. When liquid nitrous oxide is mixed with bisulphide of carbon, and caused rapidly to evaporate under the exhausted receiver of an air-pump, the greatest known degree of cold is obtained-140° Cent. When a body in a state of ignition is plunged into this gas, the gas is decomposed into nitrogen and oxygen, this latter causing the body to burn with the same brilliance as if it were in pure oxygen. With sulphur the process is somewhat peculiar: if the sulphur be only ignited feebly, the flame will be extinguished, showing that there is some effort required to determine the decomposition of the gas; therefore the sulphur must be in a state of thorough ignition. The resulting compounds are exactly those in the cases alluded to in the lesson on oxygen. Fig. 35. Fig. 35 is filled with mersium having been introduced into the bent part by an iron wire; cury, a small piece of potasthe tube is inverted into a capsule of mercury, and nitrous oxide end of the tube beneath the mercury being covered by the finger gas passed into it. Heat is then applied to the potassium, the The potassium takes the oxygen from the gas, forming potash, to prevent the escape of the gas when combustion takes place. and the nitrogen is left. finger may be removed, and it will be found that the mercury After the combustion is perfected, the the oxygen has been abstracted, it retains its original volume. stands at the same point in the tube, thus proving that although which it takes the name of " The most remarkable property of nitrous oxide, and from laughing gas," remains to be mentioned. It may be inhaled from an india-rubber bag. Place the tube from the bag in the mouth, close the nostrils, and breathe the gas in the bag. After about 2 minutes a pleasurable sensation will be experienced, which expresses itself in uncontrolled fits of laughter, or it may be exhibited in muscular exertion, which, if the subject be strong, amounts to "dangerous." If the experiment of burning phosphorus in this gas be There is little or no danger in this experiment to the majority arranged as in Fig. 34-that is, a jar of the gas over water, of persons; but where the heart is diseased, or in persons of the stopper of the jar being rapidly removed, and in its place a few months the pure gas, unmixed with air, has displaced full habit, it had better not be attempted. Within the last cork fitted, through which is passed a deflagrating spoon," in which is a piece of ignited phosphorus-it will be observed chloroform in minor operations, especially in dentistry. The that the water in the jar will have patient breathes the gas and becomes unconscious and insensible no tendency to rise; whereas if to pain. The effect lasts a minute, or a little longer. the gas had been oxygen instead Nitric Oxide (N,O, NO; combining weight, 30; density, 15). of nitrous oxide, as the phospho-As the termination ie will indicate, this gas contains more rus consumed the oxygen, of course oxygen than nitrous oxide. It is easily obtained by acting on the water would rise to fill the copper with nitric acid diluted with 2 times its volume of place of the gas. The explanation water. The action is thus expressed :of this fact is, that in nitrous oxide 3Cu + 8HNO, 3 (Cu2NO,) + 4H,O + N,O,. there is as large a volume of nitro- The flask will be found full of red fumes, for nitric oxide comgen as of the compound gas, that bines at once with oxygen to form nitric tetroxide, which is a is, in two volumes of the gas there deep-coloured gas. This action is an infallible test for the are two volumes of nitrogen and presence of nitric oxide. The gas may be collected in the one volume of oxygen, these three usual way over water, and it will be found to be colourless. It volumes being condensed into is less ready to support combustion than nitrous oxide, seeing two. The combination of gases by volume offers no difficulty that it requires a greater heat to decompose it, so that its if the student remember that simple gases are reckoned as one and compound gases as two volumes. To illustrate this immersed in this gas, must be very thoroughly ignited, or it will oxygen may take part in the combustion. Phosphorus, when be extinguished. Pelouze suggests the following manner for procuring this gas perfectly pure :-Digest hydrochloric acid with iron filings till it will dissolve no more; decant—that is, pour off-the clear liquid, and add to it its own bulk of hydrochloric acid. Place the liquid in a retort, and add potassium nitrate (saltpetre), and the nitric oxido begins to come off in large quantities. The composition of the gas may be determined as in the case of nitric oxide. If the formula be N,O,, it will be evident that we must have as much nitrogen left after the combustion as there was nitric oxide before. Thus :=2N+ 20. statement Fig. 34. = 5N,0 + 2P P20, +10N represents the process which takes place when phosphorus is burnt in nitrous oxide. The P2O, (phosphoric acid), which is formed, being a solid, does not enter into our calculations of the volumes of the gases; but it will be observed that there are five atoms of a compound gas (NO), and on the other side of the equation ten atoms of the simple gas N: hence, according to the above statement, if we reckon the compound gas as two volumes, we shall have ten volumes of N,O, and after the combustion has taken place we shall still have an equal quantity (ten volumes) of gas left-namely, ten. volumes of the simple gas N, which is verified by the experiment. : 2 = 2 + 2. The oxygen, being in a solid state in the potash, is not taken into account: hence two volumes of N2O2 two volumes of nitrogen; but this is found not to be the case, for upon removing the finger, the mercury rises in the bent tube exactly one-half the volume occupied by the gas. Therefore the formula for nitric oxide must be NO, and not N2O2 : NON + 0. 2 = 1 + −; The gas must either be collected over mercury or by displacement, as in the case of hydrogen (Fig. 23), as it is one-half as heavy as air. Its specific gravity is 0.59, and it possesses the well-known pungent odour of "smelling salts." When breathed it has a violent irritating power on the pulmonary passages. It is a powerful base, neutralises the strongest acids, and returns the colour to litmus paper reddened that is, the volume of nitrogen is one-half that of the nitric by an acid. oxide, which agrees with the result of the experiment. This gas has not yet been liquefied. Nitric trioxide, or nitrous acid (N,O,; combining weight, 76; density, 38). This gas is noted for its deep-red colour. The most ready method of preparing it is by heating in a capacious retort 1 part of starch with 8 of nitric acid. The gas liberated is almost pure N2O,. It forms compounds called nitrites. A very minute trace of any nitrite may be detected by mixing a dilute solution of potassium iodide with starch and a little dilute hydrochloric acid. Render the liquid to be tested also acid with hydrochloric acid; then mix the two liquids; if any nitrite be present, the liquid will become blue. These salts may frequently be detected in the well-water of towns. When this gas is reduced to a temperature of -18° Cent., the red fumes become a dark-blue liquid. When added to water, it is at once decomposed into nitric oxide and nitric acid, thus:3N,O,+ H2O=2HNO, + 4NO. Nitric tetroxide, or nitric peroxide (NO; combining weight, 46; density, 23).-The reddish-brown fumes which appear when nitric oxide meets with oxygen are chiefly of this substance. It is best prepared by heating lead nitrate in a small glass retort. The fumes which are given off are a mixture of peroxide of nitrogen and free oxygen; if they are conducted through a bent tube which is surrounded by ice and salt, the peroxide becomes condensed into a liquid. The reaction is thus expressed: 2(Pb2NO,) = 2 PbO + 4NO, +0 ̧. The red fumes are very suffocating, but will support the combustion of a taper immersed in them. This compound may be distinguished from the former, nitrous acid, by its power of imparting to a neutral solution of potassium sulpho-cyanide, a red tint; an excess of the peroxide, however, renders the liquid again colourless. The compounds of nitrogen and hydrogen are: Amidogen Ammonia Ammonium. NH, NH, NH. Amidogen (NH) is not known in a separate state, but it is believed to exist as a constituent of numerous compounds which chiefly belong to organic chemistry, and are termed amides. Ammonia (NH,; combining weight, 17; density, 8.5).—This compound receives its name from the fact that it was first prepared from the dung of the camels which the Arabs collected at the temple of Jupiter Ammon, the halting-place before the journey of the desert of Libya was undertaken. Nitrogen and hydrogen do not combine directly with each other, but it seems whenever they are liberated together by the decomposition of any compound containing them, they unite to form ammonia, and it appears frequently to be formed when hydrogen, in its nascent state-that is, just liberated from its combinationmeets with nitrogen of the air. Thus, if iron filings be moistened and exposed to the air they become oxidised, partly at the expense of the oxygen of the water; and the hydrogen as it is liberated forms, with the nitrogen, ammonia, which is found in the compound. This is also exhibited when tin, zinc, iron, and some other metals are acted on by dilute nitric acid, thus:9HNO, + 4Zn = = 4(Zu2NO2) + 3H2O + NH ̧. The whole action is not expressed by this formula-for the water which dilutes the acid is decomposed-and the equation may be rectified by using the liberated constituents of the water. Any organic bodies which contain nitrogen when distilled in a closed vessel give off ammonia. Formerly this method was resorted to for its production from horn clippings, hence its name -spirits of hartshorn. It is now obtained from the refuse products of the distillation of coal in the manufacture of gas. For the laboratory it may be prepared by gently heating equal weights of quick-lime (oxide of calcium), made into a paste with water and sal-ammoniac, which is ammonium chloride. CaO + 2NH,C1 = CaCl, + 3H,O + 2NH. It is very soluble in water. The liquid at 0° Cent. and 760 mm. pressure is capable of absorbing 1149 times its volume. If a jar of the gas be held with its mouth downwards to the surface of water, the water will rush into the jar as into a vacuum, and unless the glass be strong the jar will probably break. When water containing ammonia is heated, the gas is given off, so that at 20° Cent. only half the quantity of gas is retained which the water possessed at 0° Cent. When submitted to a pressure of 7 atmospheres at the ordinary temperature of the air, the gas becomes a liquid, which boils at -38.5° Cent., and freezes into a transparent solid at -175° Cent. This fact has been advantageously applied by M. Carré to freeze water. A saturated solution of ammonia is placed in a strong iron vessel, which is connected by a pipe with a "receiver," which is a cavity in the thick wall of a cylindrical vessel. When heat is applied to the liquor ammonia, the gas is given off in large quantities; but not being able to escape, it finds itself under great pressure, and begins to condense into a liquid in the receiver. The interior of the cylindrical vessel is filled with water, the heat is now removed from the other vessel, and the temperature of the water it contains is reduced by pouring cold water over it. But this renders the water it contains capable of absorbing the gas again, and therefore the liquefied gas in the "receiver" begins to evaporate rapidly; this, however, it cannot do, without absorbing a large quantity of latent heat, and hence the water which the "receiver" surrounds freezes. Ammonium (NH ̧).—Place a globule of mercury in a cavity in a piece of sal-ammoniac, and moisten it with liquor ammonia; then if the positive wire of a battery be attached to the salt, and the mercury be touched with the negative, the globule will swell and assume all the appearance of an amalgam. When the current is suspended, the mercury returns to its ordinary state, giving off ammonia and hydrogen. There is only one way of accounting for this, namely, that sal-ammoniac is a chloride of a metal (NH,Cl), and that in the ordinary way electrolysis took place the metal combining with But this compound only the mercury formed an amalgam. having permanence under the influence of the current, decomposes when the current is interrupted. Nessler's test discovers the most minute quantity of ammonia Saturate the solution supposed to contain ammonia with potash, then add potassic iodido saturated with mercuric iodide. If any ammonia be present, a "brick-dust" precipitate will appear. The composition of ammonia is discovered by leading the gas through a red-hot porcelain tube, or by passing a series of electric sparks. Either of these methods resolves the compound into its components, which are found to occupy double the volume of the gas, as might be expected from this equation, which has been previously alluded to:- NH =N+ 3H 2 = 1 + 3; Plural. Nom. ῥίνες. δελφίνες. γιγαντες. Gen. ῥίνων. δελφίνων. γιγαντων. Dat. ῥίτσι. δελφι-σι. γιγά-σι. Acc. δίνας. δελφίνας. γιγαντας. Dual. οδοντ-ες. οδοντ-ε. 1. In us, -αινα, -αν, as μελας, μελαινα, μελαν (gen. μελανος, μελαίνης, μελανος), black; and ταλας, ταλαινα, ταλαν, unhappy. EXERCISE 32.-ENGLISH-GREEK. 1. We have ivory. 2. Ivory is produced (γιγνομαι) in districts of Africa. 3. The rays of the sun delight the shepherds. 4. The brothers and the sisters are delighted by the rays of the sun. 5. The sister is lovely. 6. We admire fine ivory. 7. Many elephants are in Africa. 8. The business of the teeth is to masticate the food. 9. It is the duty of every man to worship the divinity. 10. To the gods there once was (in idiomatic English, the gods once carried on) a war against (προς) the giants. According to οδους are formed words compounded with οδους, ας δ, ἡ μονόδους (gen. μονοδοντος), having one tooth; according 2. Πας, πασα, παν (gen. παντος, πασης, παντος), all, every to γίγας, adjectives in -as, (gen. -αντος), as ὁ, ἡ ακαμας, unsub and its compound ἅπας, ἅπασα, ἁπαν. 3. Εκων, έκουσα, έκον (gen. έκοντος, έκουσης, έκοντος), willing; and ακων, ακουσα, ακον, unwilling (& privative making έκων into ακων). 4. The adjectives in -εις, εσσα, -εν. For example, χαρίεις, χαρίεσσα, χαριεν, lovely, which have in the dative plural of the masculine and neuter gender -εσι instead of -εισι, as it is in λειφθεις, left behind; for the participles in -εις, -εισα, -εν, form the case regularly in -εισι. I pass on to this second great division of nouns (for the first great division the learner must refer to page 195), and here, first, I must take up substantives which end in -ευς, -as, and -οῦς. The stem of these ends in v. The remains at the end of the word and before consonants, but disappears in the middle between vowels. Nouns in -eus have in the accusative singular -a, and in the accusative plural fas; take in the genitive singular what is called the Attie form in -ws, instead of -os; and in the dative singular, as well as in the nominative plural, admit contraction, which, however, is commonly not found in the accusative plural. If a vowel precedes -eus, the whole singular and plural is contracted, as in χους. Nouns in -avs and -oûs take the contraction only in the accusative plural. The words about to be declined are ὁ βασιλευς, a king; ὁ χοευς, a measure of liquid (about a gallon); d, ʼn Boûs, a bull or cow, an ox (Latin, bos, bovis); and ʼn ypaûs, an old woman. Nom. Gen. δ χαρίεις, Dat. Acc. χαρίεντα, χαρίεσσαν, Voc. χάριεν, χαρίεσσα, EXERCISE 33.--GREEK-ENGLISH. 1. Οἱ βασιλεῖς επιμελειαν εχουσι των πολιτῶν. τῷ νομεῖ ἑπεται. 3. Έκτωρ ὑπ' Αχιλλέως φονεύεται. 4. Of ἱερεις τοις θεοῖς βοῦς θύουσιν. 5. Κυρος παῖς ην αγαθων γονέων. 6. Οι αχαριστοι τους γονέας ατιμάζουσιν. 7. Πείθου, ω παι, τοις γονεῦσιν. 8. Τηλεμαχος ην Οδυσσεως υἱος. 9. Βουλου τους γονέας προ παντος εν τιμαις έχειν. 10. Οἱ των γραῶν ληροί τα ώτα τειρουσιν. 11. Καλως αρχεις, ω βασιλεῦ. 12. Αἱ γράες πολυλογοι εισιν. 13. Οἱ νομεῖς την βοῶν αγέλην εις νομήν αγουσιν. 14. Όμηρος τους Ήρας οφθαλμούς τους των βοῶν εικάζει. 15. * That is, xocus is contracted into xows, xoea into xoi, xoεwv into χουν, and eas iata χρᾶς, Πατροκλος φιλος ην Αχιλλεως. 16. Kupov, TOV TWV Перσwv angle, and therefore a body will travel down it with a greater Βασιλέα, επι τη τε αρετη και τη σοφίᾳ θαυμαζόμεν. EXERCISE 34.-ENGLISH-GREEK. 1. The flocks follow the shepherd. 2. The king has care of (for) the citizen. 3. Ears are tired by the idle talk of the old woman. 4. An old woman is talkative. 5. The shepherd leads the herd of oxen to the city. 6. Oxen are sacrificed to the gods by (o with gen.) the priests. 7. O priests, sacrifice an ox to the gods. 8. Children love their (the) parents. 9. Parents are loved by their children. 10. It is the business of a good shepherd to take (have) care of his herds. KEY TO EXERCISES IN LESSONS IN GREEK.-IX. 1. The ravens croak. 2. Avoid flatterers. 3. Keep away from the deceiver. 4. Men delight in the harp, in the dance, and in song. 5. Horses are driven by whips. 6. The harps delight the minds of men. 7. A grasshopper is friendly to a grasshopper, and an ant to an ant. 8. The shepherds sing to their pipes. 9. Among the Athenians there were contests between quails and cocks. 10. The shepherds drive the flocks of goats into the meadows. 11. The life of ants and quails is very laborious. 12. Many have a good countenance, but a bad voice. EXERCISE 26.-ENGLISH-GREEK. 1. Φευγω κολακα. 2. Κορακες κρώζουσι. 3. Τέρπεσθε φόρμιγγι. 4. Ορχηθμοι τους ανθρώπους τέρπουσι, 5. Ελαύνουσιν ίππους μαστιγγι. 6. Οἱ θύμοι των ανθρώπων ελαυνονται φόρμιγγι. 7. Αἱ συριγγες τέρπουσι τους ποιμένας. Αἱ αίγες εις τον λειμώνα ελαύνονται. 9. Ὁ ποιμην φάει προς την σύριγγα. Καλην μεν ωπα έχει ή θυγατηρ, κακήν δε οπα. EXERCISE 27.-GREEK-ENGLISH. 8. 10. 1. The birds sing. 2. Favour begets favour, (and) strife (begets) strife. 3. We count youth happy. 4. Need begets strife. 5. Rich men often conceal their baseness by (means of) wealth. 6. O fair boy, love your good brother and your fair sister. 7. Avarice is the mother of every kind of baseness. 8. The poor are often happy. 9. Wisdom in the hearts of men stirs up marvellous longings for the beautiful. 10. Death sets men free from their cares. 11. Friendship springs up by means of resemblance (in disposition). 12. Wine creates laughter. 13. Deliberation comes to the wise in the night. 14. The wise punish baseness. 15. Men often delight themselves with light (or vain) hopes. EXERCISE 28.-ENGLISH-GREEK. 1. Ορνιθες άδουσι. 2. Χαρις χάριν τίκτει, έρις έριν. 3. Σοφια εγείρεται εν TOLS των ανθρώπων θύμοις θαυμαστος έρως αγαθών. 4. Τερπομαι ωδη των ορνίθων. 5. Αἱ ωδαι των ορνίθων τέρπουσι τον ποιμένα. 6. Τερπόμεθα όρνισι. 7. Οι άνθρωποι επονται τους ανάξι. 8. Οι άνθρωποι πείθονται EXERCISE 29.-GREEK-ENGLISH. την ανάκτι 1. In difficult matters few companions are faithful. 2. The suppliants touch our knees. 3. Death is a separation of the soul and body. 4. Wealth furnishes men with various aids. 5. Do not yield to the words of wicked men. 6. Do not, my son, be a slave to the service of the body. 7. The Greeks pour cups of milk as libation-offerings to the nymphs. 8. Accustom yourself to, and exercise your body with, toil and sweat. 9. Chatterers vex (or weary) the ear with repetitions (of the same story). 10. Accustom your soul, my sou, to good deeds. 11. Evil stories do not lay hold of our ears. 12. We listen with our ears. 13. Do not hate a friend for a small fault. 14. My son, taste the milk. 15. The soldiers bear lances. EXERCISE 30.-ENGLISH-GREEK. 1. Ω νεανίαι, εθίζετε τα σώματα συν πονῳ και ιδρωτι. 2. Ορεγόμεθα των αγαθών πραγμάτων. 3. Πολλοι τέρπονται χρυσῳ. 4. Εξ αγαθου πραγματος γίγνεται κλέος. 5. Τους καλούς μύθους των σοφων θαυμαζομεν, 6. Τα των αγαθών ανθρωπων αγαθα πραγματα θαυμάζεται. 7. Οἱ στρατιῶται μαχονται λόγχαις. 8. Οι διαμειβομαι τον πλούτον της αρετης τοις αναξι. πείθεσθε τοις λόγοις των φαύλων. MECHANICS.-XVIII. 9. Mn LAWS OF FALLING BODIES-PROJECTILES-COLLISION OR IMPACT. THERE are two remarkable facts that have been discovered in connection with the laws of bodies falling down an incline that we must just notice here. The first is, that if we take any number of chords, A E, B E, etc. (Fig. 100), all meeting in E, the lowest point of the circle, and make inclined planes parallel and proportional in length to them, a body will take the same time to fall down each of these inclines. B E, for instance, is much longer than DE, yet it is inclined at a much greater velocity, and it is found that this increase of speed exactly makes up for the greater distance. The The other fact is, that if a body has to fall from one point to another not in the same vertical line, as, for instance, from D to E, the line of quickest descent is not along the B straight line joining these two points, but along some curve, as D F E. reason of this is, that if the body be moving down the curve it will at any moment be at a lower level than it D would if falling down the incline DE; and since the velocity of a falling body, as we have seen, depends upon the vertical distance passed over, its velocity is all along greater. The space passed over is, however, greater too; but this is more than compensated for by the increased velocity. The curve of shortest descent of all is found to be that which has the greatest curvature without rising as it approaches E. If a pencil be fixed so as to project horizontally from the rim of a wheel, and made to trace a curve on paper while the wheel is rolling on, it will be exactly that of shortest descent. As we shall see further on, there are other remarkable and important properties possessed by this curve, which is called a Cycloid. (See Lessons in Geometry, XXIII, page 309.) PROJECTILES. Fig. 100. Having thus seen the laws which govern the motion of falling bodies, we pass on naturally to notice the movements of projectiles. Here, of course, as before, the resistance of the air impedes motion to a greater or less extent. This resistance increases as the square of the velocity, for if the speed of a body be doubled, it not only has to displace twice the bulk of air, but it must do it with twice the velocity, and for this a fourfold force is needed. As, however, our calculations would be much complicated if we took this into consideration, we will neglect it, but we must remember to make allowance for it in our results. The path of a projectile is in a curve called a parabola, that is, a curve similar to the one which we should obtain if we were to cut a cone in a direction parallel to one side. (See Les sons in Geometry, XXI., page 251.) We can, however, trace this path in a simpler way. When a body is projected with any velocity, as, for example, when a bullet is fired from a gun, it is acted upon by two forces -the original velocity with which it was started, which, as we are not considering the resistance of the air, we may consider to be a uniform force; and, secondly, the attraction of the earth, which is an accelerating force, causing it to fall 16 feet in the first second, 48 in the next, and so on. Now from a knowledge of these two motions we can easily tell at what point the body will be at any given moment; and by thus finding several different points in its course we can trace out its path. Let the bullet be projected from the point a (Fig. 101), in the direction A B, with any given velocity, and take AC of such a length as to represent the space it would pass over in one second. Draw A D vertically D downwards to represent 16 feet, and complete the parallelogram ADE C. Fig. 101. AC and A D represent, then, the two forces acting on the ballet; and, since each produces its full effect, it will at the end of one second have arrived at the point E. Since, however, the force of gravity is not uniform, the line A E, which represents its path, will not be straight, but curved upwards, for when a half of a has been passed over, gravity will only have caused it to move over a quarter of a D. If now we draw through E a straight line BF, parallel to A B and equal to a c, it will represent the motion of the bullet from its original impulse during the next second. To |