Did Lahiri Mahasaya defeat Swami Dayanand Saraswati in a debate in Kashi?
Arya Samaj accounts don't mention such a debate, but this excerpt from the Yogiraj Charit, a traditional biography of Lahiri Mahasaya, describes him defeating Dayananda Saraswati in a debate on idol worship, in the course of which he magically silences Dayananda Saraswati's supporters.
I would to narrate an incident from the life of Yogiraj Rev. Lahiri Babaji. Once Shri Dayanand, the founder of Arya Samaj visited Varanasi. He was totally against idol-worship and considered himself to be brahmagyani i.e the self realized. He announced in Kashi that idol-worship was useless and should be stopped immediately and that God could only be realized through Veda-recital. He also challenged the scholars of Kashi to participate in a religious debate on this. He proposed that if he won, all the idols from the different temples of Kashi would be consigned to the Ganges. Everybody knew Shri Dayanand and his logical viewpoints. So no body came forward. The news spread like a wild fire. When it reached the king of Kashi, Yogiraj Lahiri Babaji was teaching the prince (as a tutor) in the palace. On seeing the worried face of the king, the legendary Yogiraj said “Do not worry, everything will be all right”. The next day Sri Dayanand and Rev. Yogiraj were face to face at Dashashvamegha Ghat of Kashi in order to discuss the religious issues raised by Sri Dayanand. Sri Dayanand asked his question “Is God formless or has God a form?“. Rev. Yogiraj in his usual composure said “God has many forms but at the same time God has no form”. Hearing this, the disciples of Sri Dayanand started shouting and used objectionable words aimed at Yogiraj. Yogiraj requested Sri Dayanand to exercise control over his disciples but Dayanandji could not discipline the hooligans. Indeed they wanted Yogiraj to concede defeat and make a retreat. At last Rev. Yogiraj told Dayanandji “Since you could not make them quiet so now I am trying to make them quiet”. Having uttered these words Rev. Yogiraj closed his eyes. All of a sudden there was complete silence. All the hooligans who were shouting found themselves helpless, their mouth remained open and the jaw-movements stopped. Sri Dayanand was taken aback. At this moment Rev. Yogiraj said “Water does no have a form of its own but when it is transformed into ice, it has a form”. In the same way so long as we try to perceive God through our sense organs, God has various forms but when we transcend our sense-organs we realize the truth (in that light of soul and self-awareness) God is formless. This God has various forms but at the same time God does not have a form i.e. God is formless”. Everybody was satisfied with this irrefutable explanation emanating from a realized master. Sri Dayanand had nothing to ask after listening to such a speech loaded with nothing but truth. At last Sri Dayanand requested Rev. Yogiraj to bring his disciples back to normalcy. Then Yogiraj ordered one of Dayanand´s disciple to bring water (the Gangajal) offered on Kashi Viswanath Shivalinga. When the water was sprinkled on the faces of those hooligans by Rev. Yogiraj they came back to normalcy. Sri Dayanand left Kashi immediately.
By the way, this isn't the only time Dayananda Saraswati lost a debate; he also lost one to a Sri Vaishnava at some point, as described here.
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