The 19th century was certainly the tipping point for artillery casualties in warfare. Numbers for this era are particularly rubbery, so take exact numbers with a grain (or spoon) of salt.

What kept artillery casualties down before the 19th century was essentially a technological problem. Cannons of the 18th century and earlier were generally bespoke pieces. They were individually cast from a mould or separate pieces bound together with leather or metal. Guns made in this way were extremely heavy and difficult to move around a battlefield. Their barrels were uneven inside, meaning windage (which is the gap between the cannon ball or shot and the inside of the barrel) was greater. This meant that much of the explosive power of burning powder was lost through these gaps and thus more powder was needed.

The second half of the 18th century saw improvements in cannon making which lead to lighter and more accurate guns which could be produced in higher numbers. French artillerist Jean Baptiste Vacqueville de Gribeauval designed and standardised French artillery in the 1760s with these more advanced guns in four different sizes 6lb, 8lb and 12lb cannons and a 6in howitzer. He created specialist artillery units within the army to operate the new guns and even created pre-packaged cartridges for the guns containing powder and shot, ready for use. The professionalisation and technical advances created the first truly mobile field artillery, which Napoleon would famously employ. Napoleon, being a gunner himself, doubled the number of guns in his armies, demanding a ratio of one gun for every 200 men. The British laid the final piece for a revolution in artillery in 1784 when Lieutenant Henry Shrapnel created the shell which still bares his name. However, the shrapnel shell wouldn't truly come into its own until the 1850s when a good fuse was invented.

So by the start of the 19th century we have much better guns that are lighter, more accurate and quicker to fire. There is a new professional, technical branch of the army entrusted with them and new kinds of ammunition are being invented to exploit the newfound abilities. And while they may not be able to be mass produced yet, they are certainly being made in much higher numbers.

Napoleon kicked things off with his Grand Batterie, which was basically massing all his guns in one place and concentrating fire on a single point of his enemy's line. This tactic worked very well, causing enormous numbers of casualties and breaking the morale of troops under its fire at battles like Wagram and Borodino. Like I said earlier, numbers are rubbery but Napoleon's surgeon, D. Larrey, estimated that his staff treated 100 musket or artillery wounds for every bayonet wound. Chris Duffy uses French admission records from Les Invalides which gives the following breakdown:

68.8% wounded by small arms

13.4% wounded by artillery

14.7% wounded by swords

2.4% wounded by bayonets

These are all people who survived the initial wound however, not all battlefield casualties so the numbers are a little skewed towards blades - 17% and small arms - 69% with only 15% coming from artillery. Rory Muir has done work in this area and extrapolating from the Les Invalides data reckons that the artillery casualty rate would have been around 20%. William Hodge, looking at the Crimean War in the 1850s has similar numbers with the breakdown being 65% small arms, 29% artillery and 6% blades. Take an even cut between the two and you get around 65% small arms, 25% artillery and 10% blades.

HOWEVER

The single biggest killer during the Napoleonic, or any war prior to the 20th century was disease. Hodge, writing just decades after the Napoleonic Wars, found that only 18% of deaths in the British Army came from combat, with the rest resulting from disease or accidents.

The US Civil War was in a strange period in which artillery technology had briefly stalled but also saw the introduction of the Minie ball and mass produced rifles. Numbers for this war are better known and according to Muir small arms account for 90% of all deaths with artillery causing only 10% of deaths.

As the 19th century wore on artillery technology leapt forward. First with breach loading guns and then with quick firing guns and all sorts of new ammunition. The last quarter of the 19th century saw small colonial armies of French and British troops defeating much larger forces of poorly armed locals. Battles like Ahmed Khel and Tamai show just how devastating the new artillery was.

By the end of the century France had adopted the first quick firing field gun with the Canon de 75 modèle 1897. It had a highy regarded, and tightly controlled, hydrolic-pneumatic recoil system which allowed it to fire up to 15 rounds a minute out to ranges of 6-8km. Quick firing guns like this saw their first widespread use during the Russo-Japanese War which set a lot of the foundations for what would be seen a decade later in WW1.

Edit: Formatting