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The July Revolution, though not quite the comedy of errors of Louis XIV's slow lose of all legitimacy and authority that lead to the chaos of a Revolutionary France more or less created and upheld (for the first few years at least) the sentiments of Paris, was still in many ways an unlikely and almost unexpected coup created on the fly by a assembly of liberal elites. Looking from the outside, it can be hard to beleive how easily a few National Guardsmen, a mob of townsfolk, and 400 peers managed to get the Monarchy to abdicate in the span of three days and quickly establish order under a man who dident even plan to be King of was even in on the effort. Of course, much of this can be chalked up to just how little resistance Charles actually put up before surrendering: even when he had regular army troops on the way to help relieve the Paris garrison.
So, given it was clearly in his capacity (and in alignment with his Absolutist leanings), what if Charles X, instead of abdicating once the rebels had gained the initial edge in the capital in the Three Glorious Days, followed Polignac's advice to resist and either staged a fighting escape from the city, or sent out offical orders to the Guard to fight and organize a defense of his positions while condeming the rebelling deputies as traitors who needed to be disposed and killed? A few starting questions...
Would the Duc'de Orlean even accept any offer by the Revolutionaries to be their head in such a circumstance? I doubt it, but if they can't find a suitable dynasty member what do the rebels do in terms of their demands? Like the post Proclaimation of Rebellion Americans, they're kind of stuck with something that disposes of Charles and his Ministers.
Can they get other parts of the country to raise a similar street rebellion? From what I've read, there were many sympathizers (of at least firm opposition to the Monarchal overreach of the Ordinances) in the other French cities who resisted attempts to spread the Royal orders. Would it be enough to trigger them into perceiving their own liberties at stake and Charles' tyranny too unacceptable to produce a broader front for regeime change?
Can the Parisians even hold out against the Royalist forces? If so, for how long? Certainly, urban warfare in 1830 is a nightmare, especially with the narrow streets of Paris, but there's only so much food and ammunition.
How does this affect the French position in Algiers? The invasion has just barely taken control of the capital and it's future status is still in limbo. If the Revolution bleeds on longer and spreads further from Paris, those troops occupying Algiers and Oran are going to start looking aweful tempting to pull back. Or do they even declare loyalty to the Crown?