The Nature of Martial Arts in Chinese / Japanese...

自然武道 is a means the Nature of Martial Arts.

The first two characters create a word that means nature, natural, or spontaneous.

The last two characters are often translated as martial arts.

龍虎精神 means the spirit of the dragon and tiger.

It speaks to the vitality and vigor that is the nature of these two creatures.

Beyond “spirit,” the last two characters can also mean mind, soul, or heart. Therefore, you can also say this means “Heart of the Dragon and Tiger,” etc.

龍虎精神 is often titled “Ryukoseishin” in many Japanese martial arts.

拳禪一如 is a Japanese phrase that is often translated as “train both body and spirit.”

Here's the breakdown of the words in this phrase:
拳 means fist.
禅 is zen, which means meditation.
一如 is a word that means “to be just like,” “oneness,” “true nature,” or “true character.”

So to get to the translation of “train both body and spirit,” you must understand that “fist” is representing “body” and the idea of meditation is representing “mind.”

I have to say, this is not how I would translate this. To me, it's really about training with your mind and remembering that meditation is a huge part of training, not just your fist. As the Shaolin Buddhist monks show us, meditation is just as important as physical training in martial arts.