In this work, we present an extensive review and detailed analysis of sunspot measurements, drawings, and engravings made by John Flamsteed and, mainly, by Philippe de La Hire during the Maunder minimum. All available information and contemporary knowledge about the sunspot nature are shown. The coordinates, areas, and numbers of sunspots and sunspot groups are reconstructed. Based on these observations, La Hire, Jean-Dominique Cassini, and his son Jacques Cassini regularly published results that shed light on the purpose of sunspot measurements and the scientific paradigm of that time. In particular, astronomers believed that sunspots were recurrent over decades. We compare the reconstructed time-latitude diagram with those obtained by Spoerer (Ueber die periodicitat der sonnenflecken seit dem Jahre 1618.., 1889) and Ribes and Nesme-Ribes (Astron. Astrophys.276, 549, 1993). The sidereal differential rotation rate is estimated, and its latitudinal profile is reconstructed. We also evaluate the fraction of sunspot groups that obey or violate Joy’s law.