Were line formations used during the early stages of World War 1? If not, when was the last instance of line infantry?

You will find that in the early stages of the war, as you say, that there were cases of this happening regularly. While the scale of this column like advance was more of an ancestor of the old line formations of even the American Civil War, it was still a common occurence to see relatively deep lines advancing - mainly in 1914.

1914 was, in hindsight, the year of learning for every side. Every misconception of the pre-war were shattered in several months and at the feet of a million dead from both sides. Austrians advancing towards Belgrade, the open field maneveurs of Tannenberg and the like all were home to instances of generals believing it was 1866 instead of 1914.

The case of why this was done is well documented - much like the open tank warfare of 1940 turned the establishment on its head, so too did the destructive power of artillery and fast firing machine guns/rifles re-write the rulebook. Only the far-sighted and clear seeing, ala Liddell-Hart and de Gaulle in the second war could predict the change the tide would bring.

There are two examples we can use in the first year of the war which can be used to demonstrate the two major causes of this column thinking - this 'Napoleonic' school of thought that had driven European and American military thinking since the days of the great man himself.

The first is that of the French themselves in 1914. The idea of charging at the enemy in a scene of dashing courage, using built-up reserves of patriotic elan was one not only dominant from the officers to the NCOs, but was actively fostered in the military establishment itself. Ferdinand Foch was a man who, in the pre-war years of 1904, had actively encouraged at the French military academy the idea of adhereing to a Napoleonic standard of unstoppable aggression and offence to overwhelm the enemy. It was his thinking that no amount of bullets could eventually stop the spirit of a courageous poliu full of confidence and patriotism. As such, the ideas of a pro-Napoleonic school Foch eventually encouraged great general Joseph Joffre to devise Plan XVII in 1913.

As such, when the time came for the French to advance into Alscae-Lorraine in 1914 during the Battle of the Frontiers, the average French infantryman was - by official French doctrine - to relentlessly pursue the destruction of the enemy by advancing without halt, ignoring casualties and using elan and bravery to overwhelm either odds, bullets or the like. It meant that we have scenes in August, September and October where artillerymen watch in horror as rows and rows of infantrymen march, as you imagine, in columns and parade order, into the fiery death of German rifles.

Our second example is one of an accidental replication of Napoleonic tactics. German warefare by 1914 was highly advanced, and one of the best systems in Europe. It had innovative officers and was more merit based then aristocracy. It also had millions of soldiers ready on the Belgian border, waiting to execute the Schlieffen Plan. As we know, this plan, readied in 1904 by the German General Staff, was a series of very strict daily planned timetables which catalogued the quick fall of the Belgian forts before a snapping German entry into France - before von Kluck's descent onto Paris.

This timetable forced the hand of many German generals. While they did manage to - just - make the timetable in Belgium, by bringing up bigger armaments to bring down the Belgian forts, it meant there was no excess time when they reached the French border. So, when the British finally made their stand at Mons - the Germans were forced by the constraints of time to send human wave after wave into the jaws of British guns. Line after line of German infantrymen was gunned down by the marksman British Expedtionary Force. By no explicit order was this line attack demanded, but rather the stress of German generals meant they threw their armies - superior in numbers to the Allied defences at that point - in a display of hopeful overwhelming numbers.

So, yes, column attacks did occur in 1914. While it wasn't quite like the scale of pitched line battles of, say, Austerlitz or Gettysburg, it was a small throwback to the column advances of the old Napoleonic days; be that due to an resurgent 'Napoleonic' doctrine of offence (for the French), or the massive timetable pressures of time changing tactics (for the Germans).