Military strategy
The rupture in the enemy lines allowed Napoleon's cavalry to flank both lines and roll them up leaving his opponent no choice but to surrender or flee.
The second strategy used by Napoleon I of France when confronted with two or more enemy armies was the use of the central position. This allowed Napoleon to drive a wedge to separate the enemy armies. He would then use part of his force to mask one army while the larger portion overwhelmed and defeated the second army quickly. He would then march on the second army leaving a portion to pursue the first army and repeat the operations (defeat in detail). This was designed to achieve the highest concentration of men into the primary battle while limiting the enemy's ability to reinforce the critical battle. The central position did have a weakness in that the full power of the pursuit of the enemy could not be achieved because the second army needed attention. Napoleon used the central position strategy during the Battle of Waterloo.

Waterloo: Napoleon masked Wellington and massed against the Prussian army, and then after the Battle of Ligny was won, Napoleon attempted to do the same to the Allied/British army located just to the south of Waterloo. His subordinate was unable to mask the defeated Prussian army, who reinforced the Waterloo battle in time to defeat Napoleon and end his domination of Europe.
It can be said that the Prussian Army under Blücher used the "maneuver de derrière" against Napoleon who was suddenly placed in a position of reacting to a new enemy threat.
Napoleon's practical strategic triumphs, repeatedly leading smaller forces to defeat larger ones, inspired a whole new field of study into military strategy. In particular, his opponents were keen to develop a body of knowledge in this area to allow them to counteract a masterful individual with a highly competent group of officers, a General Staff. The two most significant students of his work were Carl von Clausewitz, a Prussian with a background in philosophy, and Antoine-Henri Jomini, who had been one of Napoleon's staff officers.
One notable exception to Napoleon's strategy of annihilation and a precursor to trench warfare were the Lines of Torres Vedras during the Peninsular War. French Armies lived off the land and when they were confronted by a line of fortifications which they could not out flank, they were unable to continue the advance and were forced to retreat once they had consumed all the provisions of the region in front of the lines.
The Peninsular campaign was notable for the development of another method of warfare which went largely unnoticed at the time, but would become far more common in the 20th century. That was the aid and encouragement the British gave to the Spanish to harass the French behind their lines which led them to squander most of the assets of their Iberian army in protecting the army's line of communications. This was a very cost effective move for the British, because it cost far less to aid Spanish insurgents than it did to equip and pay regular British army units to engage the same number of French troops.
As the British army could be correspondingly smaller it was able to supply its troops by sea and land without having to live off the land as was the norm at the time. Further, because they did not have to forage they did not antagonise the locals and so did not have to garrison their lines of communications to the same extent as the French did. So the strategy of aiding their Spanish civilian allies in their guerrilla or 'small war' benefited the British in many ways, not all of which were immediately obvious.

Clausewitz and Jomini: Clausewitz's On War has become a respected reference
for strategy, dealing with political, as well as military, leadership, his most famous assertion being:

"War is not merely a political act, but also a real political instrument, a continuation of policy by other means."
Clausewitz saw war first and foremost as a political act, and thus maintained that the purpose of all strategy was to achieve the political goal that the state was seeking to accomplish. As such, Clausewitz famously argued that war was the "continuation of politics by other means",
and as such, suggested that the amount of force used by a warring state would and should be proportional to whatever political aim that the state sought to achieve via war. Clausewitz further dismissed "geometry" as an insignificant factor in strategy, believing instead that ideally all wars should follow the Napoleonic concept of victory through a decisive battle of annihilation and destruction of the opposing force, at any cost. However, he also recognized that his ideal of how war should be fought was not always practical in reality and that limited warfare could influence policy by wearing down the opposition through a "strategy of attrition".
In contrast to Clausewitz, Antoine-Henri Jomini (1779-1869) dealt mainly with operational strategy, planning and intelligence, the conduct of a campaign, and "generalship" rather than "statesmanship". He proposed that victory could be achieved by occupying the enemy's territory rather than destroying a opposing army.
As such, geometric considerations were prominent in Jomini's theory of strategy. Jomini's two basic principles of strategy were to concentrate against fractions of the enemy force at a time and to strike at the most decisive objective.
Clausewitz and Jomini remain required reading for today's military professional officer.

Industrial age: The evolution of military strategy continued in the American Civil War (1861–1865). The practice of strategy was advanced by generals such as Robert E. Lee, Ulysses S. Grant and William Tecumseh Sherman, all of whom had been influenced by the feats of Napoleon (Thomas "Stonewall" Jackson was said to have carried a book of Napoleon's maxims with him.)
However, the adherence to the Napoleonic principles in the face of technological advances such as the long-range infantry breechloader rifles and minie ball guns generally led to disastrous consequences for both the Union and Confederate  forces and populace. The time and space in which war was waged changed as well.  Railroads enabled swift movement of large forces but the manoeuvring was constrained to narrow, vulnerable corridors. Steam power and ironclads changed transport and combat at sea. Newly invented telegraph enabled more rapid communication between armies and their headquarters capitals. Combat was still usually waged by opposing divisions with skirmish lines on rural battlefields, violent naval engagements by cannon-armed sailing or steam-powered vessels, and assault on military forces defending a town.
There was still room for triumphs for the strategy of manoeuvre such as Sherman's March to the Sea in 1864, but these depended upon an enemy's unwillingness to entrench. Towards the end of the war, especially in defense of static targets as in the battles of Cold Harbor and Vicksburg, trench networks foreshadowed World War I. Many of the lessons of the American Civil War were forgotten, when in wars like the Austro-Prussian War or the Franco-Prussian War, manoeuvre won the day.
In the period preceding World War I, two of the most influential strategists were the Prussian generals, Helmuth von Moltke and Alfred von Schlieffen. Under Moltke the Prussian army achieved victory in the Austro-Prussian War (1866) and the Franco-Prussian War (1870–71), the latter campaign being widely regarded as a classic example of the conception and execution of military strategy.
In addition to exploiting railroads and highways for manoeuvre, Moltke also exploited the telegraph for control of large armies. He recognised the need to delegate control to subordinate commanders and to issue directives rather than specific orders. Moltke is most remembered as a strategist for his belief in the need for flexibility and that no plan, however well prepared, can be guaranteed to survive beyond the first encounter with the enemy.
Field Marshal Schlieffen succeeded Moltke and directed German planning in the lead up to World War I. He advocated the "strategy of annihilation" but was faced by a war on two fronts against numerically superior opposition. The strategy he formulated was the Schlieffen Plan, defending in the east while concentrating for a decisive victory in the west, after which the Germans would go on to the offensive in the east. Influenced by Hannibal's success at the Battle of Cannae, Schlieffen planned for a single great battle of encirclement, thereby annihilating his enemy.
Another German strategist of the period was Hans Delbrück who expanded on Clausewitz's concept of "limited warfare" to produce a theory on the "strategy of exhaustion". His theory defied popular military thinking of the time, which was strongly in favour of victory in battle, yet World War I would soon demonstrate the flaws of a mindless "strategy of annihilation".
At a time when industrialisation was rapidly changing naval technology, one American strategist, Alfred Thayer Mahan, almost single-handedly brought the field of naval strategy up to date. Influenced by Jomini's principles of strategy, he saw that in the coming wars, where economic strategy could be as important as military strategy, control of the sea granted the power to control the trade and resources needed to wage war. Mahan pushed the concept of the "big navy" and an expansionist view where defence was achieved by controlling the sea approaches rather than fortifying the coast. His theories contributed to the naval arms race between 1898 and 1914.