I think, what you are asking is how, Howitzers replaced 'flatter' trajectory cannons.

Well, Howitzers became more accurate, and technology, ranging from optics, sighting tactics, the advent of 'signaling' during the American Civil War, the rifling of barrels, the advent of radio communications, etc.

These are all factors that allowed for safer, more accurate use of indirect fire rather than direct-trajectory fire. If I had to place a 'firm' date, I'd say the Italian wars of Independence (most notably Napoleon III's campaign in N. Italy) was where the lines first began to be blurred between the 'classical' view of artillery and the more 'modern' view of artillery as indirect firing howitzers. The campaigns of the 1850s still saw the heavy use of smoothbores, but of a finer make, and many cannons were being upgraded to rifled barrels. The civil war again saw further refinement of this, but in truth 1915-1916 was when the 'modern' idea of Artillery was truly born.

Many nations, including Canada, Great Britain and Germany, began to formulate complex mathematical equations to conduct indirect counter-battery fires , a mission that before would've required quite literally a duel between two batteries that had a relatively unbroken line of sight to one another. The use of Howitzers and Artillery we can see at battles like Cambrai, in 1917, would differ heavily from lets say, a battle on the Marne, in 1914. The latter would be more recognizable to a Napoleonic soldier than to a contemporary one, whereas the former would be quite the opposite.

Communications technology had to catch up to the new abilities of cannons, once communications could match the range of modern cannon, we begin to see cannons used in a fashion much more akin to how they are used today.

Hope that cleared some things up.

Sources:

"Warfare in the Age of Napoleon, vol. 3" Dodge, Theodore. Leonaur publishing, 2011.

"Solferino, 1859 - The Battle for Italy's Freedom", Brooks, Richard. Osprey Publishing, 2009

"Vimy Ridge: A Canadian Reassessment" Geoffrey Hayes, Andrew Iarocci. Wilfrid Laurier University Press, 2007.

All these books touch upon the decisiveness of Artillery. An astute reader will no doubt be able to see the obvious differences in employment as the march of technology crawls on.