We all know France had vested interest in Louisiana and Quebec, but why those two, relatively distanced areas specifically? E.g. were they trying to outflank other countries, occupying the likes of New York, Boston etc. to command a greater share of North America?
I will write in greater detail below, but, to answer your question in short, France had three main objectives that defined its geo-political strategy in North America.
1.) Curb English expansion
2.) Expand trade with friendly tribes, particularly in things like fur and natural resources
3.) Create a launching point to conquer the lucrative Spanish mines in South America.
These three objectives may seem entirely different from one another, but they were actually naturally conducive to one another. To explain, I will start a little earlier on, and show how and where France formulated its stance on North America:
Gallia Nova appeared on European maps for the first time in 1529 following Giovanni da Verrazzano's commissioned exploration of North America. Almost a century later, Champlain founded Québec city in 1608. Québec formed the nucleus of French America, and it was the Canadian experience that defined the French approach to the rest of the continent. Essentially, French North America should be thought of as an extension of French Canada. The unique society and economy that developed in French Canada formed the baseline for the rest of French North America. The economy was decentralized, and the society was highly peripatetic. There was little urbanization, and voyageurs along with coureurs des bois made up the majority of the male population of New France. These were freelance hunters and traders, the former being legally permitted and the latter being considered a nuisance to French authority.
Whereas the English are remembered for their brutal treatment of the native peoples, France was, by and large, a friendly and often welcome entity to native tribes. A major reason for this, which is also a major defining element of New France's development, was the gift economy system practiced in France. Gift economies, to oversimplify grossly, worked on an assumption of reciprocity between two people or groups which gave things as gifts to another person or group. When the French encountered native tribes, their gift economy background perfectly synced with the Native cultures' system for alliance building and trade. They too made use of gifts as a means to build trust, cultivate alliances, and create reciprocal trade routes. Scholarship on the topic of inner-continental Natives was lacking for a long time as it was prone to oversimplification or exaggerated the power of Spanish and French authority over the societies it interacted with. The Native Ground by Kathleen Du Val is an excellent work that goes into this topic in-depth. Anyway, we now know that France more or less relied on its alliances and connections with native tribes in the interior of America and, as a result, saw remarkable mingling between its own people and the natives. I am getting ahead of myself here, but wanted to mention it.
In 1673, two unassuming Frenchmen departed from La Baye Verte, Ouisconsin (Wisconsin now) on a quest for the elusive Mississippi river, hoping to discover an outlet to what they called "The Southern Sea."^(1) These two men were, of course, the famous Father Marquette and his companion Joliet. They did not find the mouth of the Mississippi because, upon arriving at an Akamsea village and partaking of the calumet pipe and *sagamité* (a food item with symbolism for friendship), the two explorers were warned by the village of warlike clans to the south who had possession of firearms.^(2) They turned back to French canada, and it would be nine years before the French attempted to find the mouth of the Mississippi again.
In the meanwhile, France was expanding downward from Québec, primarily by expanding its influence through brokering far-reaching and manifold trade alliances with native people. While it is also true that some minor settlements and military forts were constructed, such as the Arkansas Post (which was sadly destroyed by Union gunboats later on in the civil war because Confederates were using the historical site as an outpost), in 1686, it is important to note two things: first of all, French people do not like to leave France. There was not the same kind of robust settler culture or a great desire to resettle in new lands as seen by the radical protestants of England seeking to go to the New World and build a society in their image. Second of all, the French interests were largely concerned with the fur trade. To this end, Louis XIV, and through him Colbert (French economic administrator also in charge of colonial policy), actively sought to keep the economies of New France firmly dependent on the mother country. This stymied New France demographically and hindered the development of an urbanized economy, but it also allowed the frontier culture and the gift economy structure to thrive. In studying the Seven Years War, it is often noted that while Britain had an immense demographic advantage, France enjoyed a tactical advantage due to its rough-and-tumble frontiersmen and carefully cultivated alliances with dozens of tribes.
So, with the economic and social picture in mind, we can now talk about La Salle. La Salle famously found the Mississippi's mouth and declared all the tributaries of the river as French soil. La Salle is central to the development of France's strategy and intent for North America. He did two very clever things: first, he stylized the lower Mississippi as La Louisiane (Louis-Land) to stroke Louis XIV's ego. Thus, when La Salle sailed back to France and pitched the idea of colonizing and settling the Mississippi directly to Louis XIV, who was otherwise utterly disinterested in colonialism, he was successful.^(3) The second thing he did is more abstract but his suggestion of this is what basically defined French strategy. He saw the opportunity for what some historians have dubbed the Great French Arc. La Salle reasoned that, should France settle and fortify the length of the Mississippi, they would be able to essentially wall off English expansion and connect the two major ocean outlets by land. In all reality, this was an excellent idea. Naval trade was, and remains, the fastest and most reliable and profitable means of trade. Having control of the Saint Laurence, the entirety of the river system across them, alliances with the natives, and being able to militarily reinforce key areas would mean a geo-political and economic victory in North America. Look at this map to see what this would have looked like: https://www.themaparchive.com/new-france-c-1750.html
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edit: edited punctuation and italics just a bit
edit 2: fixed an ambiguous pronoun that lacked an antecedent
edit3: I typed this up too casually, and thus find myself sighing at some eyesores. fixed a redundant use of the word "really" and other small details