In every generation the Church is forced to address a particular evil and promote peace throughout the world. This was true in the 19th century and remains true today.
Pope Leo XIII encountered a variety of evils during the last half of the 19th century and felt compelled to combat them with spiritual weapons.
He believed the Rosary stood out as a singularly powerful weapon.
The Rosary
On September 1, 1883, Pope Leo XIII promulgated the encyclical Supremi Apostolatus Officio. In it he explained how the Rosary has always been a supreme aid of the Church and that he wanted the whole Church to turn to it again:
We desire that that same devotion should be offered by the whole Catholic world with the greatest earnestness to the Blessed Virgin, that by her intercession her Divine Son may be appeased and softened in the evils which afflict us. And therefore We determined, Venerable Brethren, to despatch to you these letters in order that, informed of Our designs, your authority and zeal might excite the piety of your people to conform themselves to them.
He explains in his encyclical that St. Dominic used the Rosary as a way to combat evil in his day:
That great saint indeed, divinely enlightened, perceived that no remedy would be more adapted to the evils of his time than that men should return to Christ, who "is the way, the truth, and the life," by frequent meditation on the salvation obtained for Us by Him, and should seek the intercession with God of that Virgin, to whom it is given to destroy all heresies. He therefore so composed the Rosary as to recall the mysteries of our salvation in succession.
Leo XIII then went on to write how the Rosary should be used again when the Church is in need:
We, who seek a remedy for similar evils, do not doubt therefore that the prayer introduced by that most blessed man with so much advantage to the Catholic world, will have the greatest effect in removing the calamities of our times also.
In our own lifetime Pope Francis has has a similar mind, frequently inviting the world to pray the Rosary for peace.
The Rosary is not itself a magic formula, but when prayed in faith, it opens the hearts of us all and can dispose the world to receiving a message of peace.