1900s-1930s Cross frames were popular in Britain and also Holland, but the only other countries that liked them were the British and Dutch colonies. France has no known history of 20th century cross frames, and this reinforced frame design is the closest thing I’ve come across that could be described as a French cross frame.
The two extra tubes, which are half the diameter of the other tubing, join the headstock between the top and down tubes. They join the top and down tubes two-thirds of the way along.
The joint at the bottom bracket has a triangular metal brace; likewise the joint between the top tube and the seat tube. Small protrusions in the top and bottom triangular braces create points to mount an inflator pump. This machine was obviously designed with extra frame-strength in mind.
The only information I can find on the company itself is as follows:
Manufacture Ardennaise, Marcel Cosset Magasins , bureaux & ateliers – 4.6.10 & 12 Rue Demarquay à Paris . Usines à Mouzon – Ardenne. Production annuelle de 30 000 bicyclettes , motos , cadres ou series de piéces détachées ( 160.000 cycles vendus de 1895 à 1903 ).
It would appear that this frame was supplied through the french cycle trade for retailers to add their own head badge.
1920s ‘Manufacture Ardennaise’
Cross-frame / Reinforced Frame