The vase looks to be porcelain roughly 18 inches tall with a base of 4 inches the wider point of the vase is roughly 6 inches. It is believed that the vase base color was painted by one artist and the flowers by another the last information we received was from 1998 so hopefully there is new information on the signatures on the bottom .
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This is an antique, late 19th century, Victorian era vase that appears to be done in glass (a view of the top of the rim at the opening can help confirm the material). This is a white body material with a pink over glaze. The decoration is hand painted enamel with raised areas. This is likely Continental, meaning a European origin, possibly French or Bohemian. The style of the decoration is influenced by Japanese moriage ware. The monogram on the underside cannot be traced to the person who decorated the surface. This was a popular style for decorative items during this era. The estimate is based on actual recent past recorded auction sales of comparables. Retail 'asking prices' can be higher and vary.
That would have been helpful pertinent information from the beginning, however it is rather unusual for museums to issue appraisals and if they identified this as porcelain as you had, instead of glass, that too would be incorrect. Please add the images of the copies of the pages of the marks that you cite from the Toledo Museum of Art.. The 19th century is the 1800s. A 'P. Paillet' is not a listed studio artist with a collector following. Nevertheless, the estimate remains the same.
Not at all impressed with the appraisal as we have one from the Toledo museum of art along with copies of the pages of the marks that they found in there research placing the vase in the 1800’s by P. Paillet. He has over 150 enamel works on display at Metropolitan museum of New York .