Although four States were involved in the struggle, South American historians are unanimous in giving the strife which broke out in 1864 the name of the Paraguayan War.
This is appropriate enough, for a number of reasons, one of them being that, after the first invading expedition on the part of the Paraguayan armies, the war was fought out on Paraguayan soil.
The capture by the Paraguayans of the Brazilian steamer Marques de Olinda demonstrated to South America that the moment of contest had arrived.
The position of the neighbouring States was far less satisfactory from a military point of view than that of Paraguay.
During the two years of his reign Lopez had steadily continued to prepare his forces for this event. At the time the Paraguayan army was, numerically, the most formidable in South America.
It had, moreover, been brought to an unusual degree of efficiency.
The condition of the Brazilian forces was very different.
In the first place, little heed had been taken to make ready for anything of the kind, and another factor which proved greatly to the disadvantage of the fighting material involved lay in the difficulty of communication between Rio de Janeiro and those portions of the great Empire which bordered on Paraguay.
Thus Lopez's invading army, when it swept through the Brazilian province of Matto Grosso, met with practically no resistance worthy of the name, and, in the absence of defending troops, it might, undoubtedly, have taken possession of vast tracts of country, and have continued to hold these indefinitely.
It was Lopez's bizarre and wild ambition which frustrated his own schemes. A single tide of invasion was not sufficient to satisfy a mind such as his.
Gathering together a second powerful army, he determined to strike at the south-eastern portion of Brazil in addition to its province of Matto Grosso.
In order to effect this he demanded in arrogant tones from Argentina permission for his troops to cross the Argentine province of Corrientes.
To this, as neutrals, it was impossible for the Argentines to consent.
As a result, Lopez in a fury declared war upon Argentina, and, as though even this did not suffice, he next found himself at grips with the Uruguayan forces.
Thus Brazil, Argentina, and Uruguay were now leagued together against the armies of the despot Lopez.
With a view of alienating the sympathies of the oppressed subjects of the Dictator from their tyrannical leader, the allies caused it to be widely proclaimed that the war they were waging was not directed against the Paraguayan people in general.
It was against Lopez alone that they were fighting, they asserted. The claim was true enough, since this was in reality the position of affairs.
Nevertheless, owing to the methods of Lopez, the proclamation carried far less weight than had been anticipated.
The Paraguayan forces now penetrated into the Argentine province of Corrientes, seized the capital, Corrientes itself, and took possession of a couple of steamers - the Gualeguay and the 25 de Mayo - which were anchored in the river opposite to that town.
The Paraguayan fleet now held command of the river system up-stream of Corrientes.
On June 11, 1865, the allied naval forces, steaming up the Paraná, came into contact with the hostile fleet.
A battle was fought, which ended in the defeat of the Paraguayan squadron, which was forced to retreat, crippled and damaged, to the north.
A succession of actions now took place on land, and the Paraguayans, although fighting with a desperate heroism, were gradually beaten back and driven across their own frontiers.
At the same time, the army which had invaded Brazil retired in sympathy, and the scene of the war changed to Paraguay itself, which was in its turn invaded by the forces of the triple alliance.
One of the most sanguinary battles of the war was fought on May 24, 1866 - very nearly a year after the first naval action off the river port of Corrientes.
At this Battle of Tuyuti the Paraguayans lost no fewer than 8,000 men, and the casualties of the allies amounted to an equal number.
Another important action was fought at Curupaiti two months later, when the progress of the allies was abruptly checked, and they were compelled to retire to some distance with a loss of 9,000 men.
This was only one of a fair number of Paraguayan victories, for the defenders, although in the main they preserved an attitude of strenuous resistance, were occasionally enabled to exchange this for active aggression.
The history of this war, which lasted for four years, is one of the most remarkable in the whole category of struggles of the kind.
Undoubtedly one of the most extraordinary features to be met with is the tremendous courage and grim determination with which the Paraguayans opposed the forces of the allies.
Every yard of the country was contested with a fierceness which left the entire countryside covered with dead and wounded.
When, moreover, the modern arms in the possession of which the Paraguayan armies had commenced the war had become lost and depleted in numbers, their place was taken by improvised weapons of all kinds, and it was frequently with the crudest firearms and lances that these devoted armies continued to fight.
The encouragement these troops received from their leaders - or, rather, from Lopez - was in one sense of a negative order.
Rewards for valour were unknown, but punishments for defaults, on the other hand, whether real or imaginary, were abundant and terribly severe.
Men were shot for having in the course of private conversation uttered words which the suspicious mind of Lopez classed as discouraging.
Thus a trooper was on one occasion executed for having ventured the remark that, although the Paraguayans rejoiced over the numbers of their enemies who were slain, they invariably forgot to count their own dead.
A second soldier met with a similar fate for having, on his return from a reconnaissance, stated that the enemy lay in great strength to the front.
Lopez conceived that a report such as this could serve no good end, and ordered its maker to be executed forthwith.
It is curious to remark that even with the astonishing proofs of their bravery and devotion which the army had shown, Lopez could never bring himself to repose any real confidence in his troops.
The tasks which were set them were frequently superhuman.
Indeed, as a rule they received the treatment of beasts rather than of men, and in order to insure the winning of his battles Lopez encouraged his officers to treat their men in a fiendish manner.
Thus, when a body of men had been placed face to face with an infinitely superior force of the enemy, and were being mowed down in hundreds by deadly volleys at close range, a line of Paraguayans were frequently stationed at the rear of their own fighting forces, with the strictest orders to pour a volley into their comrades should they show any signs of retreat.
In circumstances such as these it is not to be wondered at that the ranks of the sublime Lopez dwindled and became thin to the point of extermination; nevertheless, the gaps were caused by death and disease rather than by desertion.
One of the most pathetic circumstances of the campaign was the deep fidelity of the Paraguayans.
This was as a rule sufficiently ill-requited, as will be evident from the fate of a number of troops who, having been made prisoners by the allies, succeeded after a time in escaping and in rejoining their suffering and starving comrades.
In order to keep faith in this manner they had left a neighbourhood of peace and comparative plenty. But Lopez gave them no thanks.
On the contrary, he ordered them to be executed for not having returned to their regiments before!
Towards the end of the war scarcely a man of mature age and whole body was left in the ranks.
These were filled largely now by youths and, indeed, mere boys.
Many children of twelve and fourteen were to be found in the later stages of the war carrying their rifles and fighting with the rest, while the women of the country, including in their numbers all those of good estate and of gentle birth were, under the guardianship of lancers, set to march through the desolate forest tracts and over the countryside in order to establish new agricultural colonies.
Here they were made to dig the soil and to plant cereals and sweet potatoes in order that the armies might be fed; and should any one of these women on the march fall by the wayside, her body was transfixed by the spear of one of the escort as an example to the rest.
Thus the roadway was littered with the corpses of these slain women.
All this while Lopez was sufficiently busy in his own way.
His dreams of Empire appear to have died hard, and not until the very end came could he be brought to believe that his armies could effect no more.
He permitted his own comforts to be very little affected by the dire hardships which his troops - and, indeed, the entire nation - were undergoing.
Although he refrained as much as possible from entering into the neighbourhood of the battles themselves, he took an important share in the direction of the campaign, and it was undoubtedly owing largely to his crass ineptitude in all strategical matters that many of the disasters came about.
Although some of his moves were of the nature to render surrender or death inevitable to the actual combatants engaged in the grim struggle, a capitulation on the part of one of his officers was, in the eyes of Lopez, an unpardonable crime, and not only was the offending officer himself wont to be executed on account of the deed, but on several occasions his family was made to share his fate.
Seeing that the male members and connections of his own family had suffered tortures and execution at his hands, and that even his sisters had been flogged by his orders, it was not to be expected that the average Paraguayan would meet with mercy from Lopez.
Certainly it is no exaggeration to say that none was ever shown unless with some special object in view.
There is no doubt that a Paraguayan field-officer had, if anything, rather more to dread from his own Dictator than from his official enemy.
The end of the war, unduly protracted, came at last.
The capital, Asuncion, had fallen into the hands of the allies, and Lopez, failing any other refuge, had taken his place with the last remaining body of the defenders - a ragged and tragic army, many of whom were practically nude, and very few of whom could boast anything beyond the remnants of a shirt or a hide loin-cloth.
Others flaunted a crude poncho or a leather cap, while many possessed no weapons but an old flint-lock rifle or a worn lance.
Although nominally an army of a thousand and odd men composed this last hope, they were little more than fugitives.
Nevertheless, these last atoms of the once great Paraguayan host turned and resisted grimly each time the pursuing forces came within reach of them and delivered an attack.
At last the few remnants of even this remnant found themselves at a spot - Cerro Cora, in the forests of Paraguay - where they were overtaken and brought to bay.
There, in the face of an attack on the part of overwhelmingly superior Brazilian forces, the little party finally lost its grim determination and broke up, leaving Lopez, Madame Lynch, and their family to shift for themselves.
Madame Lynch escaped for the time being in a carriage. She had not, however, travelled far before her pursuers came up with her, and she was eventually brought back to Asuncion.
Lopez, attempting to follow her from the battle-field on horseback, became bogged in the midst of some treacherous country.
Here he was overtaken and, showing resistance, was slain by the pursuing Brazilians. With his death ended the first and last reason for the invasion of Paraguay.
The condition of Paraguay at the conclusion of the war was utterly deplorable.
Indeed, the state of the country was one which very few lands have experienced since the beginning of history.
The natural resources of Paraguay lay in agriculture.
Since all the men had been engaged in fighting, and merely a few itinerant bands of weak women had been employed in this occupation in the meanwhile, the cessation of hostilities disclosed the fact that agriculture was to all practical purposes no more.
One of the few really wise moves which Lopez had made during the war was the wholesale planting of orange-trees, the growth of which was wont to flourish to an extraordinary degree in Paraguayan soil.
The numerous new groves now proved, to a certain extent, the salvation of the population, and the fruit was eagerly devoured.
For the time being there was little else upon which the unfortunate people could live.
It is true that there were fewer mouths to feed, since the population of the land at the close of the war was insignificant compared to that which the country had supported at its beginning.
Thus, in 1863, the people of Paraguay had been estimated roughly as numbering 1,340,000 souls.
When peace was declared there were less than a quarter of a million Paraguayans left to enjoy its benefits, and of these only 28,000 were men!
A holocaust such as this would scarcely seem to come within the range of sane and modern history.
When it is realized that, roughly, only one Paraguayan out of five was left of the entire population at the end of the five years' war, the extent of the deep horrors of that period may begin to be understood, although its full tragedy can scarcely be imagined by the dwellers in more settled and peaceful countries.
It was the women of Paraguay who, having been driven at the point of the lance to labour in the fields in order to feed the army, now came forward of their own free-will in the time of peace and utter need, and heroically set themselves to agricultural toil.
After a while the rich soil of the Republic yielded sufficient harvest to satisfy the attenuated population of the land, but it was many years ere anything approaching a normal state of affairs was able to assert itself.
The war, indeed, had caused every nation involved a heavy amount of blood and treasure. In some respects it is said to have served a useful purpose.
The Argentines, for instance, claim that this struggle intensified the national spirit of the Republic, since it was the first modern war on a large scale in which the South American States had been concerned. It seems likely enough that there is some justification for this claim.
The result was, perhaps, evident in a rather lesser degree in the case of both Brazil and Uruguay.
The political effect of the campaign upon Paraguay was, of course, still more important. The allies had announced that they were fighting, not against the Republic, but against the personality of its despot, Lopez.
His death marked the end of the despotic era, and, although Paraguay has suffered greatly from revolutions from that day to this, there has been no attempt at a repetition of a reign of terror.