In the years 1657 to 1663 Mazarin sent him on diplomatic missions in Italy, Egypt, Greece, Turkey, Germany, Poland, Moscow (where he regretted not having seen the fortifications at Kazan against the Tatars, and found that the maritime defenses were in the Dutch manner), Prussia, Livonia (with the Swedish fortifications of Riga), and Lithuania. In the course of his travels he encountered Paul Wurz, occasioning the correspondence that resulted in Blondel's first publication, a mathematical pamphlet entitled "Epistola ad P. W. Wurzium", which discussed the breaking resistance of beams. Blondel demonstrates that a mathematical proof of Galileo, allowing the cross-section of a beam to be parabolically shaped such that its weight was reduced by one third, only applied to cantilevered beams and did not apply to the specified aim, a beam supported at both ends, for which a semicircular or elliptical shape would apply. Some of these questions were taken up again in 1673, when he published his "Résolution des quatre principaux problèmes d'Architecture".