in the central line as altered by the impact, compound them with the perpendicular velocities which remain unaltered, and the resulting velocities and directions will be those with which the balls will proceed after the impact.

To take the most simple case, suppose the ball A, moving in the direction EC, and with the velocity EC, to strike the ball B, which is at rest. Join D and c, the centres of the balls, and decompose EC into FC in the line joining the centres, and EF perpendicular to it. Then ▲ Fig. 2.

L

X K

LY

velocities CK and KD. The first is destroyed, and then partially restored by the impact; the second remains unaltered, except by the friction at the moment of impact, which we do not consider. If then we take DL equal to KD, and draw LM perpendicular to XY, and in length such a fraction of KC as e is of 1, the ball will move after impact with the velocities DL and LM, that is, with the velocity DM in the direction DM. If the system were perfectly inelastic, the ball would proceed along DL; if perfectly elastic, ML would be equal to cк, and DM and CD equally inclined to XY. If the size of the ball be taken into account, XY must be supposed to be a line parallel to the obstacle, and distant from it by the radius of the ball.

Direct impact and collision. Let the masses of the two balls or material particles be m and m', and let them move with uniform velocities v and v' in the same direction along a straight line, v being greater than v', so that m overtakes m'. Let u be the common velocity of the two balls when the compression at the moment of impact is a maximum; let P be the momentum spent in producing this compression, and ep the momentum acquired during the restitution of the form of the bodies, e being the coefficient of elasticity. Let v and v' be the velocities of the balls when collision ceases. Hence, we have the three following cases :

Oblique impact and collision. In oblique impact we assume that the mutual action of the balls during collision is along the line joining their centres at the instant when compression is a maximum, and along that line only; that is, we assume the balls to be perfectly smooth. Hence, if a smooth ball impinges obliquely on a smooth plane, the line of reaction of the plane will be perpendicular to its surface, and the momentum of the impinging ball will be affected along that line only.

In fig. 1, let the ball c impinge at D, and be carried away to M; let v and v' be the velocities before and after impact, and which are therefore proportioned to CD and DM.

The equations for the impact of two smooth balls are similarly formed, and may be easily understood from their analogy to those given above for direct impact. Thus, suppose two balls A and B to move in directions oblique to one another, and to strike each other. Decompose the velocity of each ball into two, one in the line joining the centres at the moment of impact, and the other perpendicular to it. The pair of velocities perpendicular to the central line will not be altered by the impact; and as far as the remaining velocities are concerned, the case is precisely the one already solved. Find the velocities

will only strike the ball B with the velocity Fo. Suppose that by the preceding rules it is found that A, striking B at rest with the velocity FC, will be thrown back with the velocity CG, while B is struck forward with the velocity DK. Then B will receive this velocity, and this one only; as to a it has after the impact acquired the velocity CG, which it combines with the velocity GL, equal and parallel to EF; so that CL represents the velocity and the direction of the motion of a after the impact.

In every case of impact, when the balls approach one another with uniform velocities, the centre of gravity of the balls moves uniformly, and in a straight line. After the impact, though the directions and velocities of the balls may have changed, yet their centre of gravity still continues to describe the same line, and with the same velocity as before. This proposition is proved in all works on elementary mechanics.

The following conclusions will now be readily deduced by any one who understands the preceding results :

1. If two inelastic balls move in the same direction, they do not separate after the impact, but either move on with a common velocity, or are reduced to rest. If both move in the same direction, the velocity after impact is (aa + вb)÷(A+B); but if they move in different directions, the motion after impact is in the direction of that ball of which the momentum (Aa or Bb) is the greatest, and the velocity is (Aα — Bb) ÷ (A + B) or (Bb — Aα)—(A+B). When the momenta are equal, there is no motion after the impact. If b=0, or if one of the balls be at rest before impact, the velocity after impact is aa (A+B). To deduce these results, make e=0 in the formula, and give the velocities their proper signs.

2. If two perfectly elastic balls move in the same direction they separate after the impact, the velocity of the foremost being augmented from b to b + 2(a —b)s÷(A+B). But the velocity of the hindmost is either retarded, altogether destroyed, or made to change its direction, the algebraical formula for the velocity after impact being a- -2(a - b) B (A+B). This is nothing when B exceeds A, and when b is to a as B-A is to 2B. And according as b is to a in a less or greater ratio than the preceding ratio, a's velocity is or is not altered in direction.

If two perfectly elastic balls move in opposite directions, that of a being called positive, the velocities of A and B after impact are determined in magnitude and direction by the formulæ

2B A+B

2A A+B

and-b+(a+b). a―(a+b)

4. If two perfectly elastic balls be equal in magnitude, the velocity of each after the impact is that which the other had before the impact, both in magnitude and direction.

5. In all cases perfectly elastic balls recede from each other after impact with the same velocity with which they approached before impact; since if e=1, v-u-a-b. But in every other case the rate of recess after impact is the same proportion of the rate of approach before impact which e is of 1.

6. The vis viva, or product of the mass and square of the velocity, of a couple of perfectly elastic balls is the same before and after impact; in every other case it is less after impact than before.

For further mathematical developments and deductions from these general formula, we may refer especially to Professor Price's treatise on 'Infinitesimal Calculus,' vol. iii. c. 8 and 10; to Golding Bird's Natural Philosophy,' p. 134, et seq.; and to Professor Walker's treatise on 'Mechanics,' c. vii. p. 173.

COLLISIONS AT SEA. The accidental contact of vessels which traverse the ocean and the numerous channels and roadsteads of Europe, is a painful feature of commerce; because the extension of commerce must increase in fearful ratio the chances of calamities which are known as collisions at sea.

These calamities present to us danger to human life in a new aspect. Its novelty consists not so much in the peculiarly frightful nature of accidents of this class, as in the causes which produce them. In the olden time, or even within the memory of the present generation, when ships ran foul of each other, it was seldom that loss of life attended these mishaps; and in the Channels the amount of damage would frequently extend, perhaps, to the loss of an anchor or two, the starting of a butt-end, or, perhaps, a crash not difficult to patch up until some port could be gained. The form of our shipping in those days was rather adapted to the carrying of burthens than for speed; and consequently, from the fuller lines of the bow of the ship the blow was distributed over a greater surface of timber, when ships were, moreover, built with greater strength.

Now, however, not only do ships, as a general rule, possess increase of speed, but their lines are so adjusted as to present the least possible resistance in their forward end, and their bows are absolutely wedgelike. Nor is this all; for instead of the fore-part of the stem-piece itself being 12 or 14 inches in thickness, many iron steamers of thousands of tons burthen present no more breadth of stem at the fore-part than belongs to a London wherry. Hence, in these times, circumstances seem to conspire towards the increase of danger in collisions; and from the combined influences of increased speed and the sharpness of stems, vessels at sea are really and daily in danger of being literally cut in two. In reading tales of chivalry, our horror may have been deep at the description of prowess and strength which would cleave a man to the brisket: shall we be unmoved now, when we hear of a fine ship being in like manner, and without any warning, cut through timbers and bulkhead, and windlass and deck, almost to the fore-mast itself? Yet such has occurred, and is always threatening. In the collision between the Mail and the Excelsior, off Birkenhead, in 1856, even worse than this happened, for the whole of the unfortunate passengers who were sleeping forward were in one instant crushed, and mangled, and destroyed in the general wreck, from the cutting-down blow of a sharp stem!

Thus collisions at sea must be viewed as evils of very great magnitude, requiring energetic and watchful legislation for their prevention; for, in addition to the changes in the mere forms of shipping referred to, another element of mischief is looming in the fast-diminishing distance, it is the increased size of shipping. Collisions, it is true, have only hitherto sent dozens of our fellow-creatures suddenly into eternity; but the period is at hand when these dozens may become hundreds, perhaps thousands, unless some more powerful influences are called into more active operation to prevent it. It is peculiarly the duty of a Cyclopædia to grapple impartially with any question in arts and sciences resulting from progress, and to examine the various subjects of its contents, in order to detect new features which menace, or to watch known aspects which indicate changes; and this subject will accordingly receive our proper attention.

The great difficulty of the question of collision is perhaps the prevailing erroneous idea that it is merely a nautical question; while, in reality, it affects directly the whole community, as passengers of ships are the members alike of coast-resident and inland families; and it is remarkable that ships carrying only their ordinary crews seem to have been less liable to the accidents under consideration than those carrying passengers.

An extraordinary fact has been elicited from the careful investigation of accidents which occurred in 1856. One would naturally have expected to find that collisions would especially occur during the time of fog or haze, or certainly in the hours of darkness; but the editor of the Nautical Magazine' has produced a statement or abstract in that valuable periodical, in which we find the following table of the collisions in 1856. ('Nautical Mag.,' Nov., 1857):—

From the above it actually appears that the greater number of collisions occur in clear weather, and the least number in dark or thick weather. To whom, then, can we, in a spirit of impartiality, impute culpability? When mariners themselves are so seriously implicated, we can only turn imploringly (as in the cause of humanity) to the authorities themselves. It is easy to accuse even these of neglect, but it would really be difficult to convict them even of indifference. The collection of materials for the above table is of itself a proof that the government

of the country is alive to the importance of the question, that the naval heads of the Board of Trade are anxious to remedy this crying evil, this standing blot upon our seamanship and vigilance. The only inference an impartial public can draw is, that the onus and odium of the fault must rest, in a great measure, with the officers of ships themselves. It is well to consider if the comparative exemption of ships of war from collision arises from better discipline-better look-out. From the above it is evident that, 1st, More attention is called for on board merchant-ships generally; 2nd, The rules of the road by day may be defective; 3rd, The rules of the road by night may be defective also. Probably legislation can only partially present a remedy for the defect first mentioned. Surely it is beneath the dignity of a British sailor, either to run into or be run into by any craft that floats. There was a time when vigilance formed the very key-stone in the character of the English seaman; it is hoped the present generation will maintain this character. It may be that the changes from sailing to steaming have unsettled the habits of seamen, and that, in the confusion, accidents, for a time unavoidable, may occur; but is it not reasonable to expect that every merchant vessel should have at least one able seaman in each watch whose duty and responsibility should be to prevent collision? Surely precautions and responsibilities which attach to railway officials, might without hardship be enforced on, and attached to, officers of the mercantile marine, where such very serious interests of life and property are at stake. But legislation may be greatly assisted by some conventional, better-understood rules among ship-masters themselves; as for instance, larger steamers, being often difficult to steer readily, and drawing more water, should claim a proper consideration from sinaller ones. A few examples made by the Board of Trade, in the salutary punishment of obstinate or unyielding masters of small craft, might, in river or estuary navigation, partially remove danger.

As regards the second assumption, namely, defective rules of the road by day, it is extremely difficult to know where to place the limit between stringent law and freedom of judgment. Nay, it may even be demonstrated, that compliance with any known law or rule as to the movement of the helm, irrespective of the judgment of the helmsman, would prove in many cases disastrous; and those who attempt by diagrams to illustrate any proposed rule, ought to remember that, in order to render their illustrations acceptable and useful, the vessels must not only be supposed to answer their helms with equal readiness, but their speed must be precisely similar. A discretionary power in commanders, by day, seems to be more valuable than all regulations, excepting such good old ones for sailing-vessels as starboard tack hold on, port tack give way, &c.; and for steamers, a good look-out, a good man at the helm, and passing each other on the port side, &c., ought to be enough.

In crowded rivers or channels, something might be done towards the greater safety of navigation. Something is, in fact, called for, on account of the various customs which prevail in certain districts; but a knowledge of them seems to be confined to the localities. And committees and others who have approached the question of collisions at sea, all seem to have neglected to give due prominence to the want of attention to that which is the root of evil,-namely, that when ships are approaching in opposite or oblique directions, no collision would be likely to take place if one commander knew what the other was about to do with his helm. Positive safety hinges on this.

So little however is this precious hint known or sought for in general, that, absolutely, a man-of-war steamer navigating the Thames, and meeting a river steamer, would use signs to her helmsman having opposite significations in each vessel; that is to say, for instance, the naval commander extending his hand to signify to his helmsman port, might be understood by the river steamer's rule to be about to starboard the helm. There is no rule of the road as enforced by law to prevent this: if any exists, the writer distinctly assures the public that it is not generally known, and even naval officers are left to their own judgment in this matter. And again, as if still further to complicate this serious question, a commander in each service will give his order by hand, according as he is fitted with a wheel or a tiller; for, as an example, motion of the hand generally indicates the direction in which motion is to be communicated to the part of the steering apparatus which is touched by the hand [STEERING APPARATUS], and the spokes of the wheel move in a direction contrary to that of the tiller. Without the aid, therefore, of some further legislation, can we hope to prevent collisions? It may, moreover, be remarked, that if two persons are meeting carelessly on a pavement, collision is only avoided by one of the two knowing in proper time what the other is about to do. The same remark applies to shipping: a ready means of knowing in time the intended motions of an opponent seems indispensable to safety. Some such rule, therefore, as the following is much wanted: When a commander (throughout the world, if possible), seeing another vessel approach, extends his right arm horizontally, let it signify to his own helmsman-starboard the helm; let the extension of his. left arm signify to his own helmsman-port your helm. His opponent, seeing this, could act on the moment, and would know what to do accordingly. If in a steamer and about to stop her, let both his arms be held up while he gives the word. Accidents by day could, under such simple rules, scarcely occur.

parts of Greece, who had flocked thither for security, began to send out colonies into Ionia and to many of the islands; the Peloponnesians sent theirs to Italy, Sicily, and some parts of Greece. But all these colonies were sent after the Trojan war." The Dorians from Megaris, Argos, Corinth, and other places, colonised some of the larger islands, part of Creta, Rhodes, Corcyra, as well as Ægina, Cos, &c. They founded the Hexapolis on the south-west coast of Caria, in Asia Minor, which district took from them the name of Doris. A colony of Lacedæmonians founded Cyrene. The Megarians founded Chalcedon, Byzantium, Selymbria, Heraclea, and other places on the coasts of the Euxine. Sicily also was chiefly colonised by Dorians. Syracuse was a Corinthian colony, which afterwards founded Acræ, Camarina, &c., Gela was a colony of Rhodians and Cretans, and Agrigentum was a colony from Gela. The Megarians founded Selinus. The Chalcidians built Naxus, which was the first Greek settlement in Sicily, and afterwards took Leontini and Catana from the Siculi. For a more detailed account of the numerous Dorian colonies, see K. O. Müller's History of the Doric Race.'

The Ionians from Attica, another great branch of the Hellenic stock, after the death of Codrus, the last king of Athens, emigrated to the west coast of Asia Minor, which took its name from them and established there twelve cities or communities, which quickly rose to a high degree of prosperity, and formed a kind of federal union. About eighty years before, the Eolians and Achæans, two nearly allied races, being driven away from Peloponnesus by the Dorians, had emigrated to the coast of Asia Minor, where they formed colonies from Cyzicus on the Propontis as far southwards as the Hermus. Phocæa was the most northern of the Ionian towns on the borders of Æolis. The Eolians also colonised the islands of Lesbos, Tenedos, and others in | that part of the gean. These emigrations were posterior to the time of Homer, who mentions other people as occupying that coast. The Athenians at a later date colonised Euboea, where they founded Chalcis and Eretria, and they also sent colonies to Naxos, to the islands of Ceos, Siphnos, Seriphos, and other islands of the Egean. Many of these colonies having thriven and increased, became colonisers in their turn. The enterprising mariners of Phocæa formed various colonies, the most celebrated of which is Massilia on the south coast of Gaul. The Chalcidians of Euboea founded Cumæ, on the west coast of Italy, in the country of the Opici. Pirates from Cuma founded Zancle, but a fresh colony of Samians and other Ionians escaping from the Persian invasion, in the time of the first Darius, took Zancle, and were afterwards in their turn dispossessed by Anaxilas, tyrant of Rhegium, who called the town Messene (now Messina), from the name of his original country in the Peloponnesus. The Eolians founded Dicæarchia, afterwards Puteoli, and they with the Cumaans are believed to have founded Parthenope (Naples). Ionian colonists settled on the coast of Sardinia.

The Greek colonies on the east coast of Italy, setting aside the confused traditions of Arcadian immigrations, Pelasgian, &c., supposed to have taken place before the Trojan war, consisted chiefly of Dorians and Achæans from the Peloponnesus. Croton, Sybaris, and Pandosia were colonies of the latter. Tarentum was a colony of Lacedæmonians, and Locri Epizephyrii of the Locrians. Greek colonies were settled both on the north and east sides of the Pontus (Black Sea), and also on the north coast in the modern Crimea. [BOSPORUS, in GEOG. Div.] As to the relations subsisting between the colonists and the natives or prior inhabitants of the countries which they occupied, it was undoubtedly in most cases strictly in accordance with the right of the strongest. Either the natives withdrew into the interior and left the ground to the new occupants, as the Siculi did in several instances, or they resisted, in which case, when overpowered, the men were exterminated or reduced to slavery, and the conquerors kept the women for themselves. In some instances the older inhabitants were reduced to the condition of serfs or bondmen to the new settlers. The records of authentic history do not present us with an instance of any colony being settled in a country where there were not previous inhabitants. The consequence of the immigration of a new race, who seek to possess themselves of the land, must be the extermination or gradual decay of the prior race, unless the old inhabitants are made slaves. So far as we trace the history of Greek colonies in the scattered fragments of antiquity, such were the consequences of their colonial settlements. On the coast of Italy it would appear that the Greeks pursued a more humane or more politic course. They are said to have allied themselves to and intermarried with the natives, and by their superior civilisation to have acquired great influence. It may here be remarked that the Greeks, so far from being averse to foreign intermixtures, as some have said, mingled their blood freely with that of all the nations with whom they came into contact, and thus the civilisation of the Hellenic stock was gradually introduced among nations less advanced in the useful arts.

What were the relations between these Greek colonies and the mother country, and between those colonies that were of a kindred

race?

This may be gathered pretty clearly from Thucydides. Epidamnus was a colony of Corcyra; but the leader of the colony (oikiσrǹs), the founder of the colony, or the person under whose conduct it was settled, was a Corinthian, who was called or invited, says Thucydides,

66

from the mother city (called by the Greeks the metropolis, untpótoxis, or parent state), according to an ancient usage. Thus it appears that if a colony wished to send out a new colony, this was properly done with the sanction of the metropolis. Some Corinthians and other Dorians joined in the settlement of Epidamnus, which became a thriving community, and governed itself independently of both mother countries. In the course of time, however, civil dissensions and attacks from the neighbouring barbarians induced the Epidamnians to apply to Corcyra, as their metropolis, for assistance, but their prayers were not attended to. Being hard pressed by the enemy, they turned themselves to the Corinthians, and gave up their town to them, as being the real founders of the colony, in order to save themselves from destruction. The Corinthians accepted the surrender, and sent a fresh colony to Epidamnus, giving notice that all the new settlers should be on an equal footing with the old settlers: those who did not choose to leave home were allowed to have an equal interest in the colony with those who went out, by paying down a sum of money, which appears to have been the price of allotments of land. Those who went out gave their services; those who stayed at home gave their money. "Those who went out," says Thucydides, were many, and those who paid down their money were also many." For the moneyed people it was in fact an affair of pure speculation. The Corcyræans, themselves originally a colony from Corinth, having become very powerful by sea, slighted their metropolis, and "did not pay to the Corinthians the customary honours and deference in the public solemnities and sacrifices, as the other colonies were wont to pay to the mother country." They accordingly took offence at the Corinthians accepting the surrender of Epidamnus, and the result was a war between Corcyra and Corinth. (i. 24.) Again, the Corcyræan deputies, who were sent to court the alliance of the Athenians against Corinth, stated, in answer to the objection that they were a colony of Corinth, that "a colony ought to respect the mother country as long as the latter deals justly and kindly by it; but if the colony be injured and wrongly used by the mother country, then the tie is broken, and they become alienated from each other, because, said the Corcyræans, colonists are not sent out as subjects, but as free men having equal rights with those who remain at home." (i. 34.) This shows the kind of relation as understood by the Greeks between the metropolis and its colonies. The colonies were in fact sovereign states, attached to the mother country by ties of sympathy and common descent, so long as those feelings were fostered by mutual good-will, but no further. The Athenians, it is true, in the height of their power, exacted money from their own colonies as well as from the colonies of other people, and punished severely those who swerved from their alliance, such as Naxos; but this was not in consequence of any original right of dominion as supposed to belong to the mother country over the colony. Many of the colonies, especially the earlier ones, which were the consequence of civil war or foreign invasion, were formed by large parties of men under some bold leader, without any formal consent being asked from the rest of the community: they took their families, their arms, and their moveables with them, to conquer a new country for themselves; they left their native soil for ever, and carried with them no obligations or ties. Those that went off in more peaceful times, by a common understanding of the whole Commonwealth, went also away for ever, and freely and voluntarily, though under a leader appointed by the parent state, to seek a country where they could find an easier subsistence than at home. In either case it was a complete separation of a member from the body. When the Athenians, in later times, took possession of parts of Eubœa (Thucyd. i. 114), and of Ægina (ii. 27), of Melos (v. 116), and shared the lands among their own citizens who went there, the relationship thus formed was of a different kind. In the case of Egina the whole population, which was of Hellenic stock, was turned out, and a body of Athenians occupied their place, with the express object of being as a body or community subordinate to the state of Attica, in order to prevent the annoyance to which Attica had long been subject by the proximity of an independent island so well situated both for the purpose of annoying Attica and for self-defence.

That the colonies of a kindred race should feel a common interest in opposition to those of a rival branch is natural, and is proved among other instances by the case of the deputies from Egesta in Sicily, who, while requesting the assistance of the Athenians against the Syracusans and Selinuntians, urged as an additional plea that the Leontines, who were originally Chalcidians, and therefore akin to the Athenians, had been expelled from their town by the Syracusans, and showing that it was the interest of the Athenians to assist a kindred people against the prevailing power of the Dorian colonies in Sicily. (Thucyd. vi.)

Before we pass to the Roman colonies, we must say something of the system of colonisation among the other inhabitants of the Italian peninsula in the ante-Roman times. The Etruscans extended their conquests north of the Apennines in the great plain of the Po, and founded there twelve colonies, the principal of which was Felsina (Bologna). They afterwards, having defeated the Umbrians, many years before the assumed foundation of Rome, extended themselves into East and South Italy, penetrated into Latium, and took Campania from the Oscans, where they founded likewise twelve colonies, the principal of which was Capua. The Etruscans, being skilled in architecture, surrounded their towns with solid walls built of massive